vvEPA
Unified Guidance
        STATISTICAL ANALYSIS OF
  GROUNDWATER MONITORING DATA AT
            RCRA FACILITIES

            UNIFIED  GUIDANCE
OFFICE OF RESOURCE CONSERVATION AND RECOVERY

PROGRAM IMPLEMENTATION AND INFORMATION DIVISION

U.S. ENVIRONMENTAL PROTECTION AGENCY
MARCH 2009
         EPA 530-R-09-007                    March 2009

-------
vvEPA
Unified Guidance
                 This page intentionally left blank
              EPA 530-R-09-007                               March 2009

-------
                                                                            Unified Guidance
                                     DISCLAIMER

       This Unified Guidance has been prepared to assist EPA's Regions, the States and the regulated
community in testing and evaluating groundwater monitoring data under 40 CFR Parts 264 and 265 and
40 CFR Part 258. This guidance is not a rule, is not legally enforceable, and does not confer legal rights
or impose legal obligations on any member of the public,  EPA, the States or any other agency. While
EPA has made every effort to ensure the accuracy of the discussion in this guidance, the obligations of
the regulated community are determined by the relevant statutes, regulations, or other legally binding
requirements. The use of the term "should" when used in this guidance does not connote a requirement.
This guidance may not apply in a particular situation based on the circumstances. Regional and State
personnel retain the discretion to adopt approaches on a case-by-case basis that differ from this guidance
where appropriate.

       It should be stressed that this guidance is a work in progress. Given the complicated nature of
groundwater and geochemical behavior, statistical applications describing and evaluating data patterns
have evolved over time. While many new  approaches and a conceptual framework have been provided
here based on our understanding  at the time of publication, outstanding issues remain. The Unified
Guidance sets out mostly classical statistical  methods using  reasonable interpretations of existing
regulatory objectives and  constraints. But even  these highly  developed mathematical models deal
primarily with sorting out chance effects from potentially real differences or trends. They do not exhaust
the possibilities of  groundwater  definition  using other technical or  scientific  techniques  (e.g.,
contaminant modeling or geostatistical evaluations). While providing a workable decision framework,
the models and  approaches offered within  the Unified Guidance are only approximations of a complex
underlying reality.

       While providing a basic understanding of underlying statistical principles, the guidance  doesn't
attempt to provide the reader with more thorough explanations and derivations found in standard texts
and papers. It also doesn't comprehensively cover all potential statistical approaches, and confines itself
to reasonable and current methods, which will work in the present RCRA groundwater context. While it
is highly likely that methods promoted in this guidance will be applied using commercial or proprietary
statistical software, a detailed discussion of software applications is beyond the scope of this document.

       This document has  been reviewed by the Office of Resource Conservation and Recovery (former
Office  of Solid Waste), U.S.  Environmental Protection Agency, Washington, D.C., and approved for
publication.  Mention of  trade names, commercial  products, or  publications  does not constitute
endorsement or recommendation for use.
          "It is far better to have an approximate answer to the right
          question than a precise answer to the wrong question..." — John
          Mauser
                                                                                   March 2009

-------
                                                                         Unified Guidance
                             ACKNOWLEDGMENTS

      EPA's Office of Solid Waste developed initial versions of this document under the direction of
James R. Brown and Hugh Davis, Ph.D. of the Permits and State Programs Division. The final draft
was developed and edited under the direction of Mike Gansecki, EPA Region 8. The guidance was
prepared by Kirk M. Cameron, Ph.D., Statistical Scientist and President of MacStat Consulting, Ltd. in
Colorado Springs, Colorado.  It also  incorporates the substantial efforts on the  1989 Interim  Final
Guidance of Jerry Flora, Ph.D. and Ms. Karen Bauer, both — at the time — of Midwest Research
Institute in Kansas City, Missouri. Science Applications  International Corporation (SAIC) provided
technical support in developing this document under EPA Contract No. EP-WO-5060.

      EPA also  wishes to acknowledge the  time and effort spent in reviewing and improving the
document  by a workgroup composed of statisticians, Regional and State personnel,  and industry
representatives — Dr. Robert  Gibbons, Dr. Charles Davis, Sarah Hession, Dale Bridgford, Mike  Beal,
Katie Crowell, Bob Stewart, Charlotte Campbell, Evan Englund, Jeff Johnson, Mary Beck, John Baker
and Dave  Burt.  We also wish to acknowledge the excellent comments  by a number of state,  EPA
Regional and industry parties  on the September 2004 draft. Finally, we gratefully acknowledge the
detailed reviews, critiques and comments of Dr. Dennis Helsel of the US Geologic Survey, Dr. James
Loftis of Colorado State University, and Dr.  William Huber of Quantitative Decisions Inc., who
provided formal peer reviews of the September 2004 draft.

     A  special note of thanks is due to Dave Bartenfelder and Ken Lovelace of the EPA CERCLA
program, without whose assistance this document would not have been completed.
                                                                                March 2009

-------
                                                                            Unified Guidance
                              EXECUTIVE SUMMARY

     The Unified Guidance provides a suggested framework and recommendations for the statistical
analysis of groundwater monitoring data at RCRA facility units subject to 40 CFR Parts 264 and 265
and 40 CFR Part 258, to determine whether groundwater has been impacted by a hazardous constituent
release. Specific statistical methods are identified in the RCRA regulations, but their application is not
described in any detail. The Unified Guidance provides examples and background information that will
aid in successfully conducting the required statistical analyses. The Unified Guidance draws upon the
experience  gained in the last decade in implementing  the RCRA Subtitle C and D groundwater
monitoring programs and new research that has emerged since earlier Agency guidance.

      The  guidance is  primarily oriented  towards  the  groundwater monitoring  statistical analysis
provisions of 40 CFR Parts 264.90 to .100.  Similar requirements for groundwater monitoring at solid
waste landfill facilities  under 40 CFR Part 258 are also  addressed.  These regulations  govern the
detection, characterization and response to releases from regulated units into the uppermost  aquifer.
Some of the methods and strategies set out in this guidance may also be appropriate for analysis of
groundwater monitoring data from solid waste management units subject to 40 CFR 264.101. Although
the focus of this guidance is to address the RCRA regulations, it can be used by the CERCLA program
and for improving remedial actions at other groundwater monitoring programs.

      Part I of the Unified Guidance introduces the context for statistical testing at RCRA facilities. It
provides an overview of the regulatory requirements, summarizing the current RCRA Subtitle C and D
regulations and outlining the statistical methods in the final rules, as well as key regulatory sections
affecting statistical decisions. It explains the basic groundwater monitoring framework, philosophy and
intent of each stage of monitoring — detection, compliance (or assessment), and corrective action —
and certain features common to the groundwater monitoring environment. Underlying statistical ideas
common to all statistical test procedures are identified, particularly issues involving false positives
arising from multiple statistical comparisons and statistical power to detect contamination.

     A new component of the Unified Guidance addresses issues of statistical design: what factors are
important in constructing a reasonable and effective statistical monitoring program. These include the
establishment and updating of background data, designing an acceptable detection monitoring plan, and
statistical strategies for compliance/assessment monitoring and corrective action. This part also includes
a short summary of statistical methods recommended in the Unified Guidance, detailing conditions for
their appropriate use.

      Part II of the Unified Guidance covers diagnostic evaluations of historical facility data for the
purpose of checking key assumptions implicit in the recommended statistical  tests and for making
appropriate adjustments to the data (e.g., consideration of  outliers, seasonal autocorrelation, or non-
detects). Also included is a discussion of groundwater sampling and how hydrologic factors  such as
flow and gradient can impact the sampling program. Concepts of statistical and physical independence
are compared, with  caveats provided regarding the impact of dependent data on statistical test results.
Statistical methods  are suggested for  identifying special kinds of dependence  known  as  spatial and
temporal variation, including reasonable approaches when these dependencies are observed. Tests for
trends are also included in this part.

                                              iii                                     March 2009

-------
                                                                             Unified Guidance
      Part III of the Unified Guidance presents a range of detection monitoring statistical procedures.
First, there is a discussion of the Student's Mest and its non-parametric counterpart, the Wilcoxon rank-
sum test, when comparing two groups of data (e.g., background versus one downgradient well). This
part defines both parametric and non-parametric prediction limits, and their application to groundwater
analysis when  multiple  comparisons are involved. A variety  of prediction limit possibilities are
presented  to  cover likely  interpretations of  sampling and  testing requirements  under the  RCRA
regulations.

      Substantial detailed guidance is offered for using prediction limits with retesting procedures, and
how various retesting algorithms might be constructed.  The final chapter of this Part considers another
statistical method especially useful for intrawell comparisons, namely the Shewhart-CUSUM  control
chart. A brief discussion of analysis of variance [ANOVA] and tolerance limit tests identified in the
RCRA regulations is also provided.

      Part IV of the Unified Guidance is devoted to statistical methods recommended for compliance
or  assessment monitoring  and corrective  action.  Compliance  monitoring typically involves  a
comparison of downgradient well data to a groundwater protection standard [GWPS], which may be a
limit derived from background or a fixed concentration limit (such as in 40 CFR 264.94 Table 1, an
MCL,  a  risk-based limit,  an alternate  concentration  limit,  or a  defined clean-up  standard under
corrective action).  The  key statistical procedure is the  confidence interval, and several confidence
interval tests (mean,  median, or upper  percentile)  may be appropriate  for  compliance  evaluation
depending on the circumstances.  The choice depends on the distribution of the data,  frequency of non-
detects, the type of standard being compared,  and whether or not the data exhibit a significant trend.
Discussions in this part consider  fixed compliance standards  used in a variety of EPA programs and
what  they might represent in statistical terms.  Strategies for corrective action  differ from those
appropriate for compliance monitoring primarily because statistical hypotheses are changed, although
the same basic statistical methods  may be  employed.

     Since some programs  will also utilize background as standards for compliance and corrective
action monitoring, those tests and discussions under Part III detection monitoring (including statistical
design in Part I) may pertain in identifying the appropriate standards and tests.

       A glossary of important statistical terms, references and a subject index are provided at the end
of the  main  text.  The Appendices contain additional notes on a number of topics including previous
guidance, a special study for the guidance, more detailed statistical power discussions, and an extensive
set of statistical tables for implementing the methods outlined in the Unified Guidance. Some tables,
especially those for prediction limit retesting procedures, have been extended  within the  Unified
Guidance beyond published sources in order to  cover a wider variety of plausible scenarios.
                                              iv                                     March 2009

-------
                                                                  Unified Guidance
                           TABLE OF CONTENTS

DISCLAIMER	i
ACKNOWLEDGMENTS	ii
EXECUTIVE SUMMARY	iii
TABLE OF CONTENTS	v
          PART I.  STATISTICAL DESIGN AND PHILOSOPHY


CHAPTER 1.   OBJECTIVES AND POTENTIAL USE OF THIS GUIDANCE

   1.1 OBJECTIVES	1-1
   1.2 APPLICABILITY TO OTHER ENVIRONMENTAL PROGRAMS	1-3

CHAPTER 2.   REGULATORY OVERVIEW

   2.1 REGULATORY SUMMARY	2-1
   2.2 SPECIFIC REGULATORY FEATURES AND STATISTICAL ISSUES	 2-6
      2.2.1 Statistical Methods Identified under §264.97(h) and §258.53(g)	2-6
      2.2.2 Performance Standards under §264.97(1) and §258.53(h)	 2-7
      2.2.3 Hypothesis Tests in Detection, Compliance and Corrective Action Monitoring	2-10
      2.2.4 Sampling Frequency Requirements	2-10
      2.2.5 Groundwater Protection Standards	2-12
   2.3 UNIFIED GUIDANCE RECOMMENDATIONS	         2-13
      2.3.1 Interim Status Monitoring	2-13
      2.3.2 Parts 264 and 258 Detection Monitoring Methods	 2-14
      2.3.3 Parts 264 and 258 Compliance/assessment Monitoring	2-15

CHAPTER 3.    KEY STATISTICAL CONCEPTS

   3.1 INTRODUCTION TO GROUNDWATER STATISTICS	3-2
   3.2 COMMON STATISTICAL ASSUMPTIONS	3-4
      3.2.1 Statistical Independence	 3-4
      3.2.2 Stationarity.	3-5
      3.2.3 Lack of Statistical Outliers	3-7
      3.2.4 Normality	 3-7
   3.3 COMMON STATISTICAL MEASURES	3-9
   3.4 HYPOTHESIS TESTING FRAMEWORK	3-12
   3.5 ERRORS IN HYPOTHESIS TESTING	 3-14
      3.5.1 False Positives and Type I Errors	3-15
      3.5.2 Sampling Distributions, Central Limit Theorem	3-16
      3.5.3 False Negatives, Type II Errors, and Statistical Power.	 3-18
      3.5.4 Balancing Type I and Type II Errors	3-22

CHAPTER 4.    GROUNDWATER MONITORING PROGRAMS AND STATISTICAL
ANALYSIS

   4.1 THE GROUNDWATER MONITORING CONTEXT	 4-1
   4.2 RCRA GROUNDWATER MONITORING PROGRAMS	4-3
                                        v                               March 2009

-------
                                                                        Unified Guidance
   4.3 STATISTICAL SIGNIFICANCE IN GROUNDWATER TESTING	 4-6
      4.3.1 Statistical Factors	 4-8
      4.3.2 Well System Design and Sampling Factors	 4-8
      4.3.3 Hydrological Factors	 4-9
      4.3.4 Geochemical Factors	4-10
      4.3.5 Analytical Factors	 4-10
      4.3.6 Data or Analytic Errors	 4-11

CHAPTER 5.    ESTABLISHING  AND UPDATING BACKGROUND

   5.1 IMPORTANCE OF BACKGROUND	5-1
      5.1.1 Tracking Natural Groundwater Conditions	5-2
   5.2 ESTABLISHING AND REVIEWING BACKGROUND	5-2
      5.2.1 Selecting Monitoring Constituents and Adequate Sample Sizes	5-2
      5.2.2 Basic Assumptions About Background.	 5-4
      5.2.3 Outliers in Background.	 5-5
      5.2.4 Impact of Spatial Variability	 5-6
      5.2.5 Trends in Background.	5-7
      5.2.6 Summary: Expanding Background Sample Sizes	5-8
      5.2.7 Review of Background.	5-10
   5.3 UPDATING BACKGROUND	5-12
      5.3.1 When  to Update	5-12
      5.3.2 How to Update	 5-12
      5.3.3 Impact of Retesting	 5-14
      5.3.4 Updating When Trends are Apparent.	5-14

CHAPTER 6.    DETECTION MONITORING PROGRAM  DESIGN

   6.1 INTRODUCTION	6-1
   6.2 Elements of the Statistical Program Design	 6-2
      6.2.1 The Multiple Comparisons Problem	6-2
      6.2.2 Site-Wide False Positive  Rates [SWFPR]	6-7
      6.2.3 Recommendations for Statistical Power.	6-13
      6.2.4 Effect Sizes and Data-Based Power Curves	 6-18
      6.2.5 Sites Using More Than One Statistical Method.	6-21
   6.3 How KEY ASSUMPTIONS IMPACT STATISTICAL DESIGN	6-25
      6.3.1 Statistical Independence	6-25
      6.3.2 Spatial Variation: Interwell vs. Intrawell Testing	6-29
      6.3.3 Outliers	 6-34
      6.3.4 Non-Detects	 6-36
   6.4 DESIGNING DETECTION MONITORING TESTS	6-38
      6.4.1 T-Tests	6-38
      6.4.2 Analysis Of Variance [ANOVA]	6-38
      6.4.3 Trend  Tests	 6-41
      6.4.4 Statistical Intervals	6-42
      6.4.5 Control Charts	 6-46
   6.5 SITE DESIGN EXAMPLES	6-46
                                            vi                                  March 2009

-------
                                                                    Unified Guidance
CHAPTER 7.   STRATEGIES FOR COMPLIANCE/ASSESSMENT MONITORING AND
CORRECTIVE ACTION

   7.1 INTRODUCTION	7-1
   7.2 HYPOTHESIS TESTING STRUCTURES	7-3
   7.3 GROUNDWATER PROTECTION STANDARDS	7-6
   7.4 DESIGNING A STATISTICAL PROGRAM	7-9
      7.4.1 False Positives and Statistical Power in Compliance/Assessment.	7-9
      7.4.2 False Positives and Statistical Power In Corrective Action	7-12
      7.4.3 Recommended Strategies	7-13
      7.4.4 Accounting for Shifts and Trends	7-14
      7.4.5 Impact of Sample Variability, Non-Detects, And Non-Normal Data	7-17
   7.5 COMPARISONS TO BACKGROUND DATA	7-20

CHAPTER 8.   SUMMARY OF RECOMMENDED METHODS

   8.1 SELECTING THE RIGHT STATISTICAL METHOD	8-1
   8.2 TABLE s.i INVENTORY OF RECOMMENDED METHODS	8-4
   8.3 METHOD SUMMARIES	8-9
               PART II.  DIAGNOSTIC METHODS AND TESTING

CHAPTER 9.   COMMON EXPLORATORY TOOLS

   9.1 TIMES SERIES PLOTS	9-1
   9.2 Box PLOTS	9-5
   9.3 HISTOGRAMS	9-8
   9.4 SCATTER PLOTS	9-13
   9.5 PROBABILITY PLOTS	9-16

CHAPTER 10.    FITTING DISTRIBUTIONS

   10.1 IMPORTANCE OF DISTRIBUTIONAL MODELS	10-1
   10.2 TRANSFORMATIONS TO NORMALITY	10-3
   10.3 USING THE NORMAL DISTRIBUTION AS A DEFAULT	10-5
   10.4 COEFFICIENT OF VARIATION AND COEFFICIENT OF SKEWNESS	10-9
   10.5 SHAPIRO-WILK AND SHAPIRO-FRANCIA NORMALITY TESTS	10-13
      10.5.1 Shapiro-Wilk Test (n < 50)	 10-13
      10.5.2 Shapiro-Francia Test (n > 50)	 10-15
   10.6 PROBABILITY PLOT CORRELATION COEFFICIENT	10-16
   10.7 SHAPIRO-WILK MULTIPLE GROUP TEST OF NORMALITY	10-19

CHAPTER 11.    TESTING EQUALITY OF VARIANCE

   11.1 Box PLOTS	11-2
   11.2 LEVENE'STEST	11-4
   11.3 MEAN-STANDARD DEVIATION SCATTER PLOT	 11-8
                                         vii                                 March 2009

-------
                                                                     Unified Guidance
CHAPTER 12.    IDENTIFYING OUTLIERS

   12.1 SCREENING WITH PROBABILITY PLOTS	12-1
   12.2 SCREENING WITH Box PLOTS	 12-5
   12.3 DIXON'STEST	12-8
   12.4 ROSNER'STEST	12-10

CHAPTER 13.    SPATIAL VARIABILITY

   13.1 INTRODUCTION TO SPATIAL VARIATION	13-1
   13.2 IDENTIFYING SPATIAL VARIABILITY	13-2
      13.2.1 Side-by-Side Box Plots	 13-2
      13.2.2 One-Way Analysis of Variance for Spatial Variability.	13-5
   13.3 USING ANOVA TO IMPROVE PARAMETRIC INTRAWELL TESTS	13-8

CHAPTER 14.    TEMPORAL VARIABILITY

   14.1 TEMPORAL DEPENDENCE	14-1
   14.2 IDENTIFYING TEMPORAL EFFECTS AND CORRELATION	14-3
      14.2.1 Parallel Time Series Plots	 14-3
      14.2.2 One-Way Analysis of Variance for Temporal Effects	14-6
      14.2.3 Sample Autocorrelation Function	  14-12
      14.2.4 Rank von Neumann Ratio Test.	  14-16
   14.3 CORRECTING FOR TEMPORAL EFFECTS AND CORRELATION	  14-18
      14.3.1 Adjusting the Sampling Frequency and/or Test Method.	  14-18
      14.3.2 Choosing a Sampling Interval Via Darcy's Equation	  14-19
      14.3.3 Creating Adjusted, Stationary Measurements	  14-28
      14.3.4 Identifying Linear Trends Amidst  Seasonality: Seasonal Mann-Kendall Test..... 14-37

CHAPTER 15.    MANAGING NON-DETECT DATA

   15.1 GENERAL CONSIDERATIONS FOR NON-DETECT DATA	15-1
   15.2 IMPUTING NON-DETECT VALUES BY SIMPLE SUBSTITUTION	 15-3
   15.3 ESTIMATION BY KAPLAN-MEIER	 15-7
   15.4 ROBUST REGRESSION ON ORDER STATISTICS	15-13
   15.5 OTHER METHODS FOR A SINGLE CENSORING  LIMIT	  15-21
       15.5.1  Cohen's Method.	  15-21
       15.5.2  Parametric Regression on Order Statistics	15-23
   15.6 USE OF THE 15% AND 50% NON-DETECT RULE	15-24

              PART III.  DETECTION MONITORING TESTS


CHAPTER 16.    TWO-SAMPLE TESTS

   16.1 PARAMETRIC T-TESTS	16-1
      16.1.1 Pooled Variance T-Test.	 16-4
      16.1.2 Welch's T-Test.	  16-7
      16.1.3 Welch's T-Test and Lognormal Data	16-10
   16.2 WILCOXON RANK-SUM TEST	16-14
   16.3 TARONE-WARE TWO-SAMPLE TEST FOR CENSORED DATA	16-20
                                         viii                                 March 2009

-------
                                                                      Unified Guidance
CHAPTER 17.    ANOVA, TOLERANCE LIMITS, AND TREND TESTS

   17.1 ANALYSIS OF VARIANCE [ANOVA]	17-1
      17.1.1 One-Way Parametric F-Test	 17-1
      17.1.2 Kruskal-Wallis Test	 17-9
   17.2 TOLERANCE LIMITS	17-14
      17.2.1 Parametric Tolerance Limits	  17-15
      17.2.2 Non-Parametric Tolerance Intervals	  17-18
   17.3 TREND TESTS	  17-21
      17.3.1 Linear Regression	  17-23
      17.3.2 Mann-Kendall Trend Test.	  17-30
      17.3.3 Theil-Sen Trend Line	  17-34

CHAPTER 18.    PREDICTION LIMIT PRIMER

   18.1 INTRODUCTION TO PREDICTION LIMITS	18-1
      18.1.1 Basic Requirements for Prediction Limits	 18-4
      18.1.2 Prediction Limits With Censored Data	  18-6
   18.2 PARAMETRIC PREDICTION LIMITS	18-7
      18.2.1 Prediction Limit for m Future Values	18-7
      18.2.2 Prediction Limit for a Future Mean	18-11
   18.3 NON-PARAMETRIC PREDICTION LIMITS	18-16
      18.3.1 Prediction Limit for m Future Values	18-17
      18.3.2 Prediction Limit for a Future Median	18-20

CHAPTER 19.    PREDICTION LIMIT STRATEGIES WITH RETESTING

   19.1 RETESTING STRATEGIES	19-1
   19.2 COMPUTING SITE-WIDE FALSE POSITIVE RATES [SWFPR]	19-4
      19.2.1 Basic Subdivision Principle	 19-7
   19.3 PARAMETRIC PREDICTION LIMITS WITH RETESTING	19-11
      19.3.1 Testing Individual Future Values	19-15
      19.3.2 Testing Future Means	  19-20
   19.4 NON-PARAMETRIC PREDICTION LIMITS WITH RETESTING	  19-26
      19.4.1 Testing Individual Future Values	19-30
      19.4.2 Testing Future Medians	  19-31

CHAPTER 20.    MULTIPLE COMPARISONS USING CONTROL CHARTS

   20.1 INTRODUCTION TO CONTROL CHARTS	 20-1
   20.2 BASIC PROCEDURE	 20-2
   20.3 CONTROL CHART REQUIREMENTS AND ASSUMPTIONS	  20-6
      20.3.1 Statistical Independence and Stationarity	 20-6
      20.3.2 Sample Size, Updating Background.	  20-8
      20.3.3 Normality and Non-Detect Data	  20-9
   20.4 CONTROL CHART PERFORMANCE CRITERIA	20-11
      20.4.1 Control Charts with Multiple Comparisons	  20-12
      20.4.2 Retesting in Control Charts	  20-14
                                           ix                                 March 2009

-------
                                                                  Unified Guidance
   PART IV.  COMPLIANCE/ASSESSMENT AND CORRECTIVE
ACTION TESTS

CHAPTER 21.    CONFIDENCE INTERVALS

   21.1 PARAMETRIC CONFIDENCE INTERVALS	21-1
      21.1.1 Confidence Interval Around a Normal Mean	 21-3
      21.1.2 Confidence interval Around a Lognormal Geometric Mean	 21-5
      21.1.3 Confidence Interval Around a Lognormal Arithmetic Mean	 21-8
      21.1.4 Confidence Interval Around an Upper Percentile	  21-11
   21.2 NON-PARAMETRIC CONFIDENCE INTERVALS	  21-14
   21.3 CONFIDENCE INTERVALS AROUND TREND LINES	  21-23
      21.3.1 Parametric Confidence Band Around Linear Regression	  21-23
      21.3.2 Non-Parametric Confidence Band Around a Trteil-Sen Line	  21-30

CHAPTER 22.    COMPLIANCE/ASSESSMENT AND CORRECTIVE ACTION TESTS

   22.1 CONFIDENCE INTERVAL TESTS FOR MEANS	22-1
      22.1.1 Pre-Specifying Power In Compliance/Assessment,	22-2
      22.1.2 Pre-Specifying False Positive Rates in Corrective Action	 22-9
   22.2 CONFIDENCE INTERVAL TESTS FOR UPPER PERCENTILES	22-18
      22.2.1 Upper Percentile Tests in Compliance/Assessment	 22-19
      22.2.2 Upper Percentile Tests in Corrective Action	 22-20

APPENDICES1

A.I   REFERENCES
A.2   GLOSSARY
A.3   INDEX
B     HISTORICAL NOTES
C.I   SPECIAL STUDY:  NORMAL VS.  LOGNORMAL PREDICTION LIMITS
C.2   CALCULATING STATISTICAL POWER
C.3   R SCRIPTS
D     STATISTICAL TABLES
 The full table of contents for Appendices A through D is found at the beginning of the Appendices

                                        x                                March 2009

-------
PART I	Unified Guidance
       PART  I.    STATISTICAL  DESIGN AND
                              PHILOSOPHY
     Chapter 1 provides introductory information, including the purposes and goals of the guidance, as
well as its potential applicability to other environmental programs. Chapter 2 presents a brief discussion
of the existing regulations and identifies key portions of these rules which need to be addressed from a
statistical standpoint, as well as some recommendations. In Chapter 3, fundamental statistical principles
are highlighted which play a prominent role in the Unified Guidance including the notions of individual
test false positive and negative decision errors and the accumulation of such errors across multiple tests
or comparisons. Chapter 4 sets the groundwater monitoring program context, the nature of formal
statistical  tests for  groundwater and  some caveats in identifying statistically significant increases.
Typical groundwater monitoring scenarios also are described in this chapter. Chapter 5 describes how
to establish background and how to periodically update it. Chapters 6 and 7 outline various factors to be
considered when designing a  reasonable  statistical strategy for  use  in detection  monitoring,
compliance/assessment monitoring,  or  corrective  action.  Finally,  Chapter  8  summarizes  the
recommended statistical tests and methods, along with  a concise review  of assumptions, conditions of
use, and limitations.
                                                                           March 2009

-------
PART I                                                            Unified Guidance
                    This page intentionally left blank
                                                                        March 2009

-------
Chapter 1.  Objectives	Unified Guidance

 CHAPTER 1.   OBJECTIVES AND  POTENTIAL USE OF THIS
                                   GUIDANCE
       1.1   OBJECTIVES	1-1
       1.2   APPLICABILITY TO OTHER ENVIRONMENTAL PROGRAMS	1-3
1.1 OBJECTIVES

      The  fundamental  goals  of  the  RCRA  groundwater monitoring  regulations  are  fairly
 straightforward. Regulated parties are to accurately characterize existing groundwater quality at their
 facility, assess whether a hazardous constituent release has occurred and, if so, determine whether
 measured levels meet the compliance standards. Using  accepted  statistical testing,  evaluation of
 groundwater quality should have a high probability of leading to correct decisions about  a facility's
 regulatory status.

       To implement these goals,  EPA first promulgated regulations in 1980  (for interim  status
 facilities) and 1982 (permitted facilities) for detecting contamination of groundwater at hazardous waste
 Subtitle C land disposal facilities.  In 1988, EPA revised portions of those regulations found at 40 CFR
 Part 264,  Subpart F. A similar set of regulations applying to Subtitle  D municipal and industrial waste
 facilities was adopted in 1991 under 40  CFR Part 258. In April 2006, certain modifications were made
 to the 40 CFR Part 264 groundwater monitoring regulations affecting statistical testing and decision-
 making.

       EPA  released the Interim Final Guidance [IFG] in  1989  for implementing the statistical
 methods and sampling procedures identified in the 1988 rule. A second guidance document followed in
 July  1992  called Addendum to Interim Final  Guidance  [Addendum], which  expanded certain
 techniques and also served as guidance for the newer Subpart D regulations.

       As the RCRA groundwater monitoring program has matured, it became apparent that the existing
 guidance needed to be updated to adequately cover statistical methods and issues important to detecting
 changes  in  groundwater.1 Research conducted in  the area of groundwater statistics since 1992 has
 provided a number of improved statistical techniques. At the same time, experience gained in applying
 the regulatory  statistical  tests in groundwater monitoring  contexts has identified certain constraints.
 Both needed  to be factored  into the guidance. This Unified Guidance document addresses these
 concerns and supercedes both the earlier IFG and Addendum.

       The Unified Guidance offers  guidance to  owners and operators, EPA  Regional  and  State
 personnel, and  other interested parties in selecting, using,  and interpreting appropriate  statistical
 methods for evaluating  data under the RCRA groundwater monitoring regulations. The guidance
1  Some recommendations in EPA's Statistical Training Course on Groundwater Monitoring were developed to better
  reflect the reality of groundwater conditions at many sites, but were not generally available in published form. See RCRA
  Docket #EPA\530-R-93-003, 1993

                                             iTl                                   March 2009

-------
Chapter 1. Objectives	Unified Guidance

 identifies recent approaches and recommends a consistent framework for applying these methods. One
 key aspect of the Unified Guidance is providing a systematic application of the basic statistical principle
 of balancing false positives and negative errors in designing good testing procedures (i.e., minimizing
 both the risk of falsely declaring a site to be out-of-compliance and of missing  real  evidence of an
 adverse change in the groundwater). Topics addressed in the guidance include basic statistical concepts,
 sampling design and sample sizes, selection of appropriate statistical approaches, how to check data and
 run statistical tests, and the interpretation of results. References for the suggested procedures and to
 more general statistical texts are provided. The guidance notes when expert statistical consultation may
 be advisable. Such guidance may also have applicability to other remedial activities as well.

       Enough commonality exists in sampling, analysis, and evaluation under the RCRA regulatory
 requirements that the Unified Guidance often suggests relatively general strategies. At the same time,
 there may be  situations where site-specific considerations for sampling and  statistical  analysis are
 appropriate or needed.  EPA policy has been  to  promulgate regulations that are specific enough to
 implement, yet flexible in accommodating a wide variety of site-specific environmental factors.  Usually
 this is accomplished by specifying criteria appropriate for the majority of monitoring situations,  while at
 the same time allowing alternatives that are also protective of human health and the environment.
                                                                                        r\
       40 CFR Parts 264 and 258 allow the use of other sampling procedures and test methods  beyond
 those explicitly identified in the regulations,3 subject to approval by the Regional Administrator or state
 Director. Alternative test methods must be able to meet the performance standards at §264.97(i) or
 §258.53(h). While these performance standards  are occasionally specific, they are much less so  in other
 instances.  Accordingly,  further guidance is provided concerning the types of procedures that should
 generally satisfy such performance standards.

       Although the Part 264  and 258  regulations explicitly  identify  five basic formal statistical
 procedures for testing two- or multiple-sample comparisons characteristic of detection  monitoring, the
 rules are silent on specific tests under compliance or corrective action monitoring when a groundwater
 protection standard is fixed (a  one-sample comparison). The rules also  require consideration of data
 patterns (normality, independence, outliers, non-detects, spatial and temporal dependence), but do not
 identify specific tests.  This  document expands  the potential  statistical procedures  to  cover these
 situations  identified in  earlier  guidance, thus  providing a comprehensive single EPA  reference  on
 statistical methods  generally recommended for RCRA groundwater monitoring programs. Not every
 technique will be appropriate in a given situation, and in many cases more than one statistical approach
 can be used. The Unified Guidance is meant to be broad enough in scope to cover a high percentage of
 the potential situations a user might encounter.

       The Unified  Guidance is not designed  as a treatise for statisticians; rather it is  aimed at the
 informed groundwater professional with a limited background in statistics. Most methods discussed are
 well-known to statisticians, but  not necessarily to regulators, groundwater engineers or scientists. A key
 thrust  of the Unified Guidance has been to tailor the standard statistical techniques to the RCRA
 groundwater arena and its unique constraints. Because of this emphasis, not every variation of each test
2 For example, §264.97(g)(2), §264.97(h)(5) and §258.53(g)(5)

3 §264.97(g)(l), §264.97(h)(l-4), and §258.53(g)(l-4) respectively
                                               1-2                                     March 2009

-------
Chapter 1.  Objectives	Unified Guidance

 is discussed in  detail. For example, groundwater monitoring in a detection monitoring program  is
 generally concerned  with increases  rather than decreases  in  concentration levels  of monitored
 parameters. Thus, most detection monitoring tests in the Unified Guidance are presented as one-sided
 upper-tailed tests. In the sections covering compliance and corrective action monitoring (Chapters 21
 and 22 in Part IV), either one-sided lower-tail or upper-tail tests are recommended depending on the
 monitoring  program.  Users requiring two-tailed tests  or additional information may need to consult
 other guidance or the statistical references listed at the end of the Unified Guidance.

       The Unified Guidance is not intended to cover all statistical methods that might be applicable to
 groundwater. The technical literature is even more extensive, including other published frameworks for
 developing  statistical  programs at RCRA facilities. Certain statistical methods and general strategies
 described in the Unified Guidance are outlined in American Society for Testing and Materials [ASTM]
 documents  entitled  Standard  Guide for  Developing Appropriate  Statistical  Approaches for
 Groundwater Detection Monitoring Programs (D6312-98[2005]) (ASTM, 2005) and Standard Guide
for Applying Statistical Methods for Assessment and Corrective Action Environmental Monitoring
 Programs (D7048-04) (ASTM, 2004).

      The first  of these  ASTM guidelines  primarily  covers  strategies for detection  monitoring,
 emphasizing the use of prediction limits and control charts. It also contains a series of flow diagrams
 aimed at guiding the user to an appropriate statistical approach. The second guideline covers statistical
 strategies useful in compliance/assessment monitoring and corrective action. While not identical  to
 those described in the Unified Guidance, the ASTM guidelines do provide an alternative framework for
 developing  statistical programs at RCRA facilities and are worthy of careful consideration.

       EPA's primary consideration in  developing the Unified Guidance was to select methods both
 consistent with the RCRA regulations, as well as straightforward to implement. We believe the methods
 in the guidance are not only effective, but also understandable and easy to use.

1.2 APPLICABILITY TO  OTHER ENVIRONMENTAL PROGRAMS

       The  Unified Guidance is tailored to the  context of the RCRA groundwater monitoring
 regulations. Some of the techniques  described are  unique  to  this guidance.    Certain regulatory
 constraints  and the nature of groundwater monitoring limit how statistical procedures are likely  to be
 applied. These include typically small  sample sizes  during a given evaluation period, a minimum  of
 annual monitoring and evaluation and typically at least semi-annual, often a large number of potential
 monitoring  constituents, background-to-downgradient well comparisons,  and a limited set of identified
 statistical methods. There are also  unique regulatory  performance constraints such as  §264.97(i)(2),
 which requires a minimum single test false positive a level  of 0.01 and a minimum  0.05  level for
 multiple comparison procedures such as analysis of variance [ANOVA].

      There are  enough commonalities with other regulatory groundwater monitoring programs  (e.g.,
 certain distributional features of routinely monitored background groundwater constituents) to allow for
 more general use of the tests and methods in the Unified Guidance. Many of these test methods and the
 consideration  of false positive and  negative errors in  site design are directly applicable to corrective
 action  evaluations of solid waste  management units under  40 CFR  264.101  and Comprehensive
                                             1-3                                   March 2009

-------
Chapter 1. Objectives	Unified Guidance

 Environmental  Response,  Compensation,  and Liability  Act [CERCLA]  groundwater monitoring
 programs.

      There are also comparable situations involving other environmental media to which the Unified
 Guidance  statistical methods might be applied. Groundwater detection  monitoring involves either a
 comparison between different monitoring stations  (i.e., downgradient compliance wells vs. upgradient
 wells) or a contrast between past and present data within a given station (i.e., intrawell comparisons).
 To the extent that an environmental monitoring station is essentially fixed in location (e.g., air quality
 monitors,  surface water stations) and measurements are made over time, the same statistical methods
 may be applicable.

       The Unified Guidance also details methods to compare background data against measurements
 from regulatory compliance  points. These  procedures  (e.g., Welch's Mest, prediction limits with
 retesting, etc.) are designed to contrast multiple groups of data. Many environmental problems involve
 similar comparisons, even if the groups of data  are not collected at fixed monitoring stations (e.g., as in
 soil sampling). Furthermore, the guidance describes diagnostic techniques for checking the assumptions
 underlying many statistical  procedures. Testing of normality is ubiquitous in environmental statistical
 analysis. Also common are checks of statistical independence in time series data, the assumption of
 equal variances across  different populations, and the need to identify outliers. The Unified Guidance
 addresses each of these topics, providing useful  guidance and worked out examples.

       Finally, the  Unified  Guidance discusses  techniques for  comparing datasets against  fixed
 numerical standards (as in compliance monitoring or corrective action). Comparison of data against a
 fixed standard is encountered in many regulatory programs. The methods described in Part IV of the
 Unified Guidance could therefore have wider applicability, despite being tailored to the  groundwater
 monitoring data context.

       EPA recognizes that many guidance users  will make use of either commercially available or
 proprietary statistical software in applying these statistical methods. Because of their wide range of
 diversity and coverage, the Unified Guidance does not evaluate software usage or applicability. Certain
 software is provided with the guidance. The guidance  limits itself to describing the basic statistical
 principles underlying the application of the  recommended tests.
                                              1-4                                    March 2009

-------
Chapter 2. Regulatory Overview                                           Unified Guidance

               CHAPTER 2.   REGULATORY  OVERVIEW
       2.1   REGULATORY SUMMARY	2-1
       2.2   SPECIFIC REGULATORY FEATURES AND STATISTICAL ISSUES	2-6
         2.2.1   Statistical Methods Identified Under §264.97(h) and §258.53(g)	2-6
         2.2.2   Performance Standards Under §264.97(i) and §258.53(h)	2-7
         2.2.3   Hypothesis Tests in Detection, Compliance/Assessment, and Corrective Action Monitoring	2-10
         2.2.4   Sampling Frequency Requirements	2-10
         2.2.5   Groundwater Protection Standards	2-12
       2.3   UNIFIED GUIDANCE RECOMMENDATIONS	2-13
         2.3.1   Interim Status Monitoring	2-13
         2.3.2   Parts 264 and 258 Detection Monitoring Methods	2-14
         2.3.3   Parts 264 and 258 Compliance/assessment Monitoring	2-15
      This chapter generally summarizes the RCRA groundwater monitoring regulations under 40 CFR
 Parts 264, 265 and 258 applicable to this guidance. A second section  identifies the most critical
 regulatory statistical issues and how they  are addressed by this guidance. Finally, recommendations
 regarding interim status facilities and certain statistical methods in the regulations are presented at the
 end of the chapter.

2.1 REGULATORY SUMMARY

       Section 3004 of RCRA directs EPA to establish regulations applicable to owners and operators
 of facilities that treat, store, or dispose of hazardous waste as may be necessary to protect human health
 and the environment. Section 3005 provides for the  implementation of these standards under permits
 issued to owners and operators by EPA or authorized States. These regulations are codified in 40 CFR
 Part 264. Section 3005 also provides that owners and operators of facilities in existence at the time of
 the  regulatory or statutory requirement for a permit, who apply for and  comply with applicable
 requirements, may operate until a permit determination is made. These  facilities are commonly known
 as interim status facilities, which must comply with the standards promulgated in 40 CFR Part 265.

      EPA first promulgated the groundwater monitoring regulations under Part 265 for interim status
 surface impoundments, landfills and land treatment  units ("regulated units") in  1980.1 Intended as a
 temporary system for units awaiting full permit requirements, the rules set out a minimal detection and
 assessment monitoring system consisting of at least a single upgradient and three downgradient wells.
 Following collection of the minimum number of samples prescribed in the rule for four  indicator
 parameters — pH, specific conductance, total organic carbon (TOC) and total organic halides (TOX) —
 and certain constituents  defining overall groundwater quality, the owner/operator of a land disposal
 facility is  required to implement a detection monitoring program.  Detection monitoring consists of
 upgradient-to-downgradient comparisons using the Student's t-test of the four indicator parameters at
 no less than  a  .01 level of significance (a). The regulations refer to the use of "replicate" samples for
 contaminant indicator comparisons. Upon failure  of a single detection-level test,  as well as a repeated
1  [45 FR 33232ff, May 19, 1980] Interim status regulations; later amended in 1983 and 1985

                                              2^1                                    March 2009

-------
Chapter 2. Regulatory Overview                                            Unified Guidance

 follow-up test, the facility is required to conduct an assessment program identifying concentrations of
 hazardous waste constituents from the unit in groundwater. A facility can return to detection monitoring
 if none of the latter constituents are detected. These regulations are still in effect today.

       Building on the interim status rules,  Subtitle C regulations for Part 264 permitted hazardous
 waste facilities followed in  1982,  where  the  basic elements of the  present RCRA groundwater
 monitoring program are defined. In §264.91, three monitoring programs —  detection  monitoring,
 compliance monitoring,  and corrective  action  — serve  to protect  groundwater  from  releases  of
 hazardous waste constituents at  certain regulated land disposal units (surface  impoundments, waste
 piles, landfills,  and land  treatment). In  developing permits,  the Regional Administrator/State Director
 establishes groundwater protection  standards [GWPS] under §264.92 using concentration limits
 [§264.94] for certain monitoring constituents [§264.93]. Compliance well monitoring locations are
 specified in the permit following the rules in §264.95 for the required compliance  period [§264.96].
 General monitoring requirements were established in §264.97, along with specific detection [§264.98],
 compliance [§264.99], and corrective action  [§264.100] monitoring requirements. Facility owners and
 operators are  required to  sample groundwater at specified intervals and to use a statistical procedure to
 determine whether  or not hazardous wastes  or  constituents from  the facility are contaminating the
 groundwater.

       As found in §264.91, detection monitoring is the first stage of monitoring when no or minimal
 releases have been identified, designed to allow identification of significant changes in the groundwater
 when compared to background  or established baseline levels. Downgradient well observations are
 tested against established background data, including measurements from upgradient wells. These are
 known as two- or multiple-sample tests.

       If there is statistically significant evidence  of a release of hazardous constituents [§264.91(a)(l)
 and  (2)], the  regulated unit  must  initiate compliance  monitoring,  with  groundwater  quality
 measurements compared to   the groundwater protection standards [GWPS]. The owner/operator is
 required to conduct a more extensive Part 261 Appendix VIE (later Part 264 Appendix IX)3 evaluation
 to determine if additional hazardous constituents must be added to the compliance monitoring list.

       Compliance/assessment as well as corrective action monitoring differ from detection monitoring
 in  that groundwater well data are tested against the groundwater protection standards  [GWPS]  as
 established in the permit. These may  be fixed health-based standards such as Safe Drinking Water Act
 [SDWA] maximum concentration limits [MCLs],  §264.94 Table  1 values,  a value  defined from
 background,  or alternate-concentration limits as  provided  in  §264.94(a).  Statistically, these are
 considered single-sample tests against a fixed limit (a background limit can either be a single- or  two-
 sample test depending on how the limit is defined).  An exceedance occurs when a constituent level is
 shown to be significantly greater than the GWPS or compliance standard.

       If a hazardous monitoring constituent under compliance monitoring statistically exceeds the
 GWPS  at any  compliance well, the facility is  subject to corrective action  and monitoring under
 §264.100. Following remedial action, a return to compliance consists of a statistical demonstration that
2 [47 FR 32274ff, July 26, 1982] Permitting Requirements for Land Disposal Facilities
3 [52 FR 25942, July 9, 1987] List (Phase I) of Hazardous Constituents for Groundwater Monitoring; Final Rule

                                              2^2                                    March 2009

-------
Chapter 2. Regulatory Overview                                            Unified Guidance

 the concentrations of all relevant hazardous constituents lie below their respective standards. Although
 the rules  define a three-tiered approach, the Regional Administrator or  State Director can assess
 available information at the  time of permit development to  identify which monitoring program is
 appropriate [§264.9l(b)].

       Noteworthy features of the 1982  rule included retaining  use  of  the four Part 265  indicator
 parameters, but allowing for additional constituents in detection monitoring. The number of upgradient
 and downgradient wells was  not specified;  rather the requirement is  to  have a  sufficient number of
 wells  to  characterize  upgradient  and downgradient  water quality passing beneath a regulated  unit.
 Formalizing the "replicate" approach in  the 1980 rules and the  use  of  Student's Mest, rules under
 §264.97 required the use of aliquot replicate samples, which  involved analysis of at least four physical
 splits of a single volume of water. In addition, Cochran's Approximation to the Behrens-Fisher [CABF]
 Student's  t-test was specified for detection monitoring at no less  than a  .01 level of significance (a).
 Background sampling was specified  for a one-year period consisting  of  four quarterly samples  (also
 using the aliquot approach). The rules allowed use of a repeated, follow-up test subsequent to failure of
 a detection monitoring test. A minimum of semi-annual sampling was required.

       In  response to a number of concerns  with  these regulations, EPA  amended portions of the 40
 CFR Part 264 Subpart F regulations including statistical methods and sampling procedures on October
 11, 1988.4 Modifications to the regulations included requiring (if  necessary)  that owners  and/or
 operators more accurately characterize the hydrogeology and  potential contaminants at the facility. The
 rule also identifies specific performance standards in  the regulations that all  the statistical methods and
 sampling procedures  must meet  (discussed  in  a  following  section).  That  is,  it is  intended that the
 statistical methods and sampling procedures meeting these performance  standards defined in §264.97
 have a low probability both of indicating contamination when it is not present (Type I error),  and of
 failing to detect contamination that actually is present (Type  n error). A facility owner and/or operator
 must demonstrate that a procedure is appropriate for the site-specific conditions at the  facility, and
 ensure that it meets  the performance standards. This demonstration applies to  any of the statistical
 methods  and  sampling  procedures outlined in the  regulation as well as  any alternate methods or
 procedures proposed by facility owners and/or operators.

       In addition, the amendments removed the required use of the CABF Student's t-test, in favor of
 five different statistical methods deemed to be more appropriate for analyzing groundwater monitoring
 data (discussed in a following section).  The CABF procedure is still retained in Part 264, Appendix IV,
 as  an  option, but there  are no longer specific citations in the regulations  for this test.  These newer
 procedures offer greater flexibility in designing a groundwater statistical  program appropriate to site-
 specific conditions.  A sixth option allows the use of alternative statistical  methods, subject to approval
 by  the Regional  Administrator. EPA  also  instituted new   groundwater  monitoring   sampling
 requirements,  primarily aimed  at ensuring  adequate statistical sample  sizes for use in analysis of
 variance  [ANOVA] procedures,  but  also allowing alternative sampling  plans to be approved  by the
 Regional Administrator. The requirements identify the need for statistically  independent samples to be
 used during evaluation. The  Agency further recognizes  that the selection of appropriate hazardous
4 [53 FR 39720, October 11, 1988] 40 CFR Part 264: Statistical Methods for Evaluating Groundwater Monitoring From
  Hazardous Waste Facilities; Final Rule

                                              2^3                                     March 2009

-------
Chapter 2. Regulatory Overview                                           Unified Guidance

 constituent monitoring parameters is an essential part of a reliable statistical evaluation. EPA addressed
 this issue in a 1987 Federal Register notice.5

       §264.101 requirements for corrective action at non-regulated units were added in 1985 and later.6
 The Agency determined that since corrective action at non-regulated units would work under a different
 program, these units are not required to follow the detailed steps of Subpart F monitoring.

       In 1991, EPA promulgated Subtitle D groundwater monitoring regulations for municipal solid
 waste landfills in 40 CFR Part 258.7 These rules also incorporate a three-tiered groundwater monitoring
 strategy (detection monitoring, assessment monitoring, and corrective action), and describe statistical
 methods for determining whether background concentrations or the groundwater protection standards
 [GWPS] have been exceeded.

       The statistical methods and related performance standards in 40 CFR Part 258 essentially mirror
 the requirements found as  of 1988  at 40 CFR Part 264 Subpart F, with certain differences. Minimum
 sampling frequencies are different than in the Subtitle C regulations.  The rules also specifically provide
 for the  GWPS using either current MCLs or  standardized risk-based limits  as well  as background
 concentrations.  In addition, a specific list of hazardous constituent analytes is identified in 40 CFR Part
 258, Appendix I for detection-level monitoring,  including the use of unfiltered (total) trace elements.

       The  1988  and  1991  rule  amendments identify certain statistical  methods  and sampling
 procedures believed appropriate for  evaluating groundwater monitoring  data under a variety  of
 situations. Initial guidance to implement these methods was released  in 1989 as: Statistical Analysis of
 Groundwater Monitoring Data at RCRA Facilities: Interim Final Guidance [IFG]. The IFG covered
 basic  topics such as checking distributional assumptions, selecting  one of the methods and sampling
 frequencies.  Examples  were  provided  for  applying  the recommended statistical procedures and
 interpreting the results.  Two types of compliance tests were provided for comparison to the GWPS —
 mean/median confidence intervals and upper limit tolerance intervals.

       Given additional interest from users of the comparable regulations adopted for Subtitle D solid
 waste facilities in 1991, and with experience gained in implementing various tests, EPA actively sought
 to improve existing groundwater statistical guidance. This culminated in a July 1992  publication of:
 Statistical Analysis of Groundwater Monitoring Data  at RCRA Facilities: Addendum to  Interim
 Final Guidance [Addendum].

       The 1992 Addendum included a chapter  devoted to retesting strategies, as well as new guidance
 on several non-parametric techniques not covered within the IFG. These included the Wilcoxon rank-
 sum test, non-parametric tolerance  intervals, and non-parametric prediction intervals.  The Addendum
 also included a reference approach  for evaluating statistical power to ensure that contamination could
 be adequately  detected.  The Addendum  did not replace the IFG  — the two documents contained
 overlapping material but were mostly intended to complement one another based on newer information
5  [52 FR 25942, July 9, 1987] op. cit.
6  [50 FR 28747, July 15, 1985] Amended in 1987, 1993, and 1998
7  [56 FR 50978, October 9, 1991] 40 CFR Parts 257 & 258: Solid Waste Disposal Facility Criteria: Final Rule, especially
  Part 258 Subpart E Groundwater Monitoring and Corrective Action

                                              2^4                                    March 2009

-------
Chapter 2. Regulatory Overview                                          Unified Guidance

 and comments from statisticians and users of the guidance. However, the Addendum changed several
 recommendations within the IFG and replaced certain test methods first published in the IFG. The two
 documents provided contradictory guidance on several points, a concern addressed by this guidance.

       More recently in April 2006, EPA promulgated further  changes to  certain  40 CFR Part 264
                                                                             o
 groundwater monitoring provisions as part of the Burden Reduction Initiative Rule.   A brief summary
 of the regulatory changes and the potential effects on existing RCRA groundwater monitoring programs
 is provided. Four items of specific interest are:

        »»»  Elimination of the requirements to sample four successive times per statistical evaluation
           under §264.98(d) and §264.99(f) in favor of more flexible, site-specific options as identified
           in §264.97(g)(l)&(2);

        »«»  Removal  of  the  requirements in §264.98(g)  and  §264.99(g) to  annually  sample  all
           monitoring wells  for Part 264 Appendix IX constituents in favor of a  specific subset of
           wells;

        »»»  Modifications of these provisions to allow for a specific subset of Part 264 Appendix IX
           constituents tailored to site needs; and

        »«»  A change in the resampling requirement in §264.98(g)(3) from "within a month" to a site-
           specific schedule.

       These changes to the  groundwater monitoring provisions require coordination between the
 regulatory agency and owner/operator with final  approval by the agency. Since the regulatory changes
 are not issued under the 1984 Hazardous and Solid Waste Amendments [HSWA] to RCRA, authorized
 State RCRA program adoption  of these rules is discretionary.  States may choose  to maintain  more
 stringent requirements, particularly if already  codified  in existing regulations. Where EPA has direct
 implementation authority, the provisions would go into effect following promulgation.

       The first provision reaffirms the  flexible  approach in  the Unified Guidance for detection
 monitoring sampling frequencies and testing options. State RCRA programs using the four-successive
 sampling requirements can still  continue  to do  so under §264.97(g)(l), but the rule now allows  for
 alternate sampling frequencies under §264.97(g)(2) in  both detection and compliance monitoring. The
 second and third provisions provide more site- and waste-specific options for Part  264 Appendix IX
 compliance monitoring. The final provision provides more  flexibility when resampling these Appendix
 IX constituents.

       Since portions of the earlier and the most recent rules are still operative,  all are considered in the
 present Unified Guidance. The effort to create  this guidance began in 1996, with a draft release in
 December 2004, a peer review in 2005, and a final version completed in 2009.
  [71 FR 16862-16915] April 4, 2006
                                              2-5                                    March 2009

-------
Chapter 2.  Regulatory Overview                                         Unified Guidance

2.2 SPECIFIC REGULATORY  FEATURES AND STATISTICAL ISSUES

       This section describes critical portions of the RCRA groundwater monitoring regulations which
 the present guidance addresses. The regulatory language is provided below in bold and italics.9 A brief
 discussion of each issue is provided in statistical terms and how the Unified Guidance deals with it.

2.2.1  STATISTICAL METHODS IDENTIFIED  UNDER §264.97(h) AND §258.53(g)

       The owner or operator will specify one of the following statistical methods to be  used in
       evaluating groundwater monitoring data for each  hazardous constituent which,  upon
       approval by the Regional Administrator, will be specified in the unit permit. The statistical test
       chosen shall be conducted separately for each hazardous constituent in each well...

       1. A parametric analysis of variance (ANOVA) followed by multiple comparison procedures
          to identify statistically  significant evidence of contamination. The method must include
          estimation and testing of the contrasts  between each compliance well's mean  and the
          background mean levels for each constituent.

       2. An analysis of variance  (ANOVA)  based on ranks followed by multiple  comparison
          procedures to identify statistically significant  evidence of contamination. The  method
          must include estimation and testing of the contrasts between  each compliance well's
          median and the background median levels for each constituent.

       3. A tolerance interval or prediction interval procedure in which an interval for each
          constituent is established from the distribution of the background data, and the level of
          each constituent in each compliance well is compared to the upper tolerance or prediction
          limit.

       4. A control chart approach that gives control limits for each  constituent.

       5. Another statistical  method  submitted by  the owner or operator and approved by  the
          Regional Administrator.

     Part III of the Unified  Guidance addresses these  specific  tests, as  applied to a detection
monitoring program.  It is assumed that  statistical testing will be conducted separately for each hazardous
constituent in each monitoring well.  The recommended non-parametric ANOVA method based on ranks
is identified  in this guidance as the Kruskal-Wallis test. ANOVA tests are discussed in Chapter 17.
Tolerance interval and prediction limit  tests are discussed  separately in Chapters  17 and  18, with
particular attention given  to implementing prediction limits with  retesting when  conducting  multiple
comparisons in Chapter 19. The recommended type of control chart is the combined Shewhart-CUSUM
control chart test, discussed  in Chapter 20. Where a groundwater protection standard  is based on
background levels, application of these  tests is discussed in Part I, Chapter 7 and Part IV, Chapter 22.
  The following discussions somewhat condense the regulatory language for ease of presentation and understanding. Exact
  citations for regulatory text should be obtained from the most recent Title 40 Code of Federal Regulations.

                                            2^6                                   March 2009

-------
Chapter 2.  Regulatory Overview                                         Unified Guidance

     If a groundwater protection standard involves a fixed limit, none of the listed statistical methods in
these regulations directly apply. Consequently, a number of other single-sample tests for comparison
with a  fixed limit are recommended in Part IV. Certain statistical limitations encountered when using
ANOVA and tolerance level tests in detection and compliance monitoring are also discussed in these
chapters. Additional use  of ANOVA tests for diagnostic identification of spatial variation or temporal
effects  is discussed in Part II, Chapters 13 and 14.

2.2.2  PERFORMANCE STANDARDS  UNDER §264.97(i) AND §258.53(h)

       Any statistical method chosen under §264.97(h) [or §258.53(g)] for specification  in the unit
       permit shall comply with the following performance standards, as appropriate:

       1.  The statistical method used to evaluate ground-water monitoring data shall be appropriate
         for the distribution of chemical parameters or hazardous constituents. If the distribution of
          the chemical parameters or hazardous constituents is shown by the owner or operator to be
          inappropriate for a normal theory  test,  then  the  data should be  transformed or a
          distribution-free test should be used. If the distributions for  the constituents differ, more
          than one statistical method may be needed.

       2.  If an individual well comparison procedure is used to compare an individual compliance
          well  constituent  concentration   with  background  constituent  concentrations  or a
          groundwater protection standard, the test shall be done at a Type I error level no less than
          0.01 for each testing period. If a multiple comparisons procedure is used, the Type I
          experiment-wise error rate for each testing period shall be no less than 0.05; however, the
          Type I error of no less than 0.01 for individual well comparisons must be maintained. This
         performance standard does not apply to  control charts, tolerance intervals,  or prediction
          intervals.

       3.  If a control chart approach is used to evaluate groundwater monitoring data, the specific
          type of control chart and its associated parameter values shall be proposed by the owner or
          operator and approved by the Regional Administrator if he or she finds it to be protective
          of human health and the environment.

       4.  If a tolerance interval or a prediction interval is used to evaluate groundwater monitoring
          data, the levels of confidence, and for tolerance intervals, the percentage of the population
          that the interval must contain, shall be proposed by the owner or operator and approved by
          the Regional Administrator  if he or she finds it protective of human health  and the
          environment.  These parameters  will be determined after  considering the number of
          samples in the background data base, the data distribution, and  the range of the
          concentration values for each constituent of concern.

       5.  The statistical method shall account for data below the limit of detection with one or more
         procedures  that are protective  of human health and the  environment. Any practical
          quantification limit  (pql)  approved by the Regional Administrator under §264.97(h) [or
          §258.53(g)] that is used in the statistical method shall be the lowest concentration level that
          can be reliably achieved within specified limits  of precision and accuracy during routine
          laboratory operating conditions available  to the facility.

                                            2^7                                   March 2009

-------
Chapter 2. Regulatory Overview                                            Unified Guidance

       6.  If necessary,  the statistical method shall include procedures to control or correct for
          seasonal and spatial variability as well as temporal correlation in the data.

     These performance  standards pertain to both the listed tests as well as others (such as those
recommended in Part IV of the guidance for comparison to fixed standards). Each of the performance
standards is addressed in Part I of the guidance for designing statistical monitoring programs and in
Part II of the guidance covering diagnostic testing.

     The first performance standard considers distributional properties of sample data; procedures for
evaluating normality, transformations to normality, or use of non-parametric (distribution-free) methods
are found in Chapter 10. Since some  statistical tests also require an assumption of equal variances
across  groups, Chapter 11 provides the relevant diagnostic tests. Defining an appropriate distribution
also requires  consideration of possible  outliers. Chapter 12 discusses techniques useful  in outlier
identification.

     The second performance standard identifies minimum  false positive error rates required when
conducting certain  tests.  "Individual well comparison procedures" cited  in  the  regulations include
various ANOVA-type tests, Student's Mests, as well  as one-sample compliance monitoring/corrective
action  tests against a fixed standard. Per the regulations, these significance level (a) constraints do not
apply to the other listed statistical methods — control charts, tolerance intervals, or prediction intervals.

     When comparing an individual  compliance well against background, the probability of the test
resulting in a false positive or Type I  error should be no less  than  1 in 100 (1%). EPA required a
minimum Type I error level for a given test and fixed sample size because false positive and negative
rates are inversely related. By limiting Type I error rates to  1%, EPA felt that the risk of incurring false
positives would be sufficiently low, while providing sufficient  statistical power (i.e., the test's ability to
control the false negative rate, that is, the rate of missing or not  detecting true changes in groundwater
quality).

     Though a procedure to test an individual well like the Student's  ^-test may be appropriate for the
smallest  of facilities,  more  extensive  networks  of  groundwater monitoring wells and monitoring
parameters will generally require a multiple comparisons procedure. The 1988 regulations recognized
this need in specifying a one-way analysis of variance  [ANOVA]  procedure as the method of choice for
replacing the CABF Student's Mest. The F-statistic in an analysis of variance [ANOVA]  does indeed
control the  site-wide or experiment-wise  error rate  when  evaluating  multiple upgradient and
downgradient  wells,  at  least for a  single  constituent.  Using  this  technique allowed  the Type I
experiment-wise error rate for each constituent to be controlled to  about 5% for each testing period.

      To maintain adequate statistical power, the regulations also mandate that the ANOVA procedure
be run at a minimum 5% false positive rate per constituent.  But  when  a full set of well-constituent
combinations  are considered (particularly large suites of detection monitoring analytes at  numerous
compliance wells), the site-wide false positive rate can be much greater than 5%. The one-way ANOVA
is  inherently an interwell technique, designed to simultaneously compare datasets from different well
locations.  Constituents with significant natural spatial variation are likely to  trigger the ANOVA  F-
statistic even in the absence of real contamination, an issue discussed in Chapter 13.
                                              2-8                                     March 2009

-------
Chapter 2. Regulatory Overview                                            Unified Guidance

     Control charts, tolerance intervals, and prediction intervals provide alternate testing strategies for
simultaneously controlling false positive rates while maintaining adequate power to detect contamination
during detection monitoring. Although the rules do not require a minimum nominal false positive rate as
specified in the second performance standard,  use of tolerance or prediction intervals combined with a
retesting strategy can result in sufficiently low experiment-wise Type I error rates and the ability to
detect  real  contamination. Chapters 17, 18 and 20 consider how tolerance limits, control charts, and
prediction limits can be designed to meet the third and fourth performance standards specific to these
tests considering the number of samples  in background, the  data distribution,  and  the  range  of
concentration  values for each  constituent of concern  [COC]. Chapters  19  and  20  on multiple
comparison procedures using prediction limits or control charts identify how retesting can be used to
enhance power and meet the specified false positive objectives.

     The fifth performance standard requires statistical tests to account for non-detect data. Chapter 15
provides some alternative approaches for either adjusting or modeling sample data in the presence of
reported non-detects. Other chapters include modifications of standard tests to properly account for the
non-detect portion of data sets.

     The sixth performance standard requires  consideration of spatial or temporal (including seasonal)
variation in the  data. Such patterns can have  major statistical consequences and need to be carefully
addressed. Most classical statistical tests in this guidance require assumptions of data independence and
stationarity. Independence roughly means that observing a given sample  measurement does not allow a
precise prediction of other sample measurements.  Drawing colored balls  from an urn at random
illustrates and fits this requirement; in groundwater, sample volumes are assumed to be drawn more or
less at random from the population  of possible same-sized volumes comprising the underlying aquifer.
Stationarity assumes that the population being sampled has a constant mean and variance across time
and space.  Spatial  or temporal variation in the  well means and/or variances  can negate these test
assumptions.  Chapter 13 considers the use of ANOVA techniques to establish evidence of spatial
variation.  Modification  of  the statistical  approach  may  be necessary  in  this case;   in  particular,
background levels will need to be established  at each compliance well for future comparisons (termed
intrawell tests).  Control chart, tolerance limit, and prediction limit tests can  be  designed for intrawell
comparisons; these topics are considered in Part III of this guidance.

     Temporal variation can occur for a number of reasons — seasonal fluctuations, autocorrelation,
trends  over time, etc. Chapter 14 addresses these forms of temporal variation, along with recommended
statistical procedures. In order to achieve stationarity and independence, sample data may need to be
adjusted to  remove trends or other forms of temporal dependence. In these cases, the residuals remaining
after trend  removal  or  other  adjustments  are used for formal testing  purposes.  Correlation  among
monitoring constituents  within and  between compliance wells can occur, a subject also treated in this
chapter.

     When evaluating statistical methods by these performance standards, it is important to recognize
that the ability of a particular  procedure to operate correctly in  minimizing unnecessary  false positives
while detecting possible contamination depends on several factors. These include  not only the choice of
significance level and test hypotheses, but  also the statistical test itself,  data distributions, presence or
absence of  outliers and non-detects, the presence or absence of spatial and temporal variation, sampling
requirements, number of samples and comparisons to be made, and frequency of sampling. Since all of
these  statistical  factors  interact to  determine  the procedure's effectiveness, any proposed statistical
                                               2^9                                     March 2009

-------
Chapter 2. Regulatory Overview                                         Unified Guidance

procedure  needs  to be evaluated in its entirety, not by individual components. Part I, Chapter 5
discusses evaluation of potential background databases considering all of the performance criteria.

2.2.3 HYPOTHESIS  TESTS  IN   DETECTION,  COMPLIANCE/ASSESSMENT,   AND
       CORRECTIVE ACTION MONITORING

     The Part 264 Subpart F groundwater monitoring regulations do not specifically identify the test
hypotheses to be  used  in detection monitoring (§264.98),  compliance monitoring  (§264.99),  and
corrective  action  (§264.100). The same is true  for the parallel  Part 258 regulations for  detection
monitoring (§258.54), assessment  monitoring  (§258.55),  and  assessment of corrective  measures
(§258.56),  as well as for evaluating interim  status indicator parameters (§265.93) or Appendix in
constituents. However, the language of these regulations as well as accepted statistical principles allow
for clear  definitions  of the  appropriate  test  hypotheses.  Two-  or multiple-sample  comparisons
(background vs. downgradient well data) are usually involved in detection monitoring (the comparison
could also be made against an ACL limit based on background data). Units under detection monitoring
are initially presumed not to be contributing a release to the groundwater unless demonstrated otherwise.
From a statistical  testing standpoint, the population  of downgradient well measurements is assumed to
be equivalent to or no worse than those of the background population; typically this translates into an
initial or null hypothesis that the downgradient population mean is equal to or less than the background
population  mean.  Demonstration  of a release is triggered when one or more well constituents indicate
statistically significant levels above background.

     Compliance and corrective action tests generally compare single sets of sample data to a fixed limit
or a background standard. The language of §264.99 indicates that a significant increase above a GWPS
will demonstrate the need for corrective action. Consequently, the null hypothesis is that the compliance
population  mean  (or  perhaps an  upper percentile) is  at or  below a given standard. The  statistical
hypothesis  is thus quite similar to that of detection monitoring. In contrast, once an exceedance has been
established and  §264.100 is triggered, the null hypothesis  is  that  a site is contaminated  unless
demonstrated to be significantly below the GWPS.  The same principles apply to Part 258 monitoring
programs. In Part 265, the detection monitoring hypotheses apply to an evaluation of the contaminant
indicator parameters. The general subject of hypothesis testing is discussed in Chapter 3, and specific
statistical hypothesis formulations are found in Parts III and IV of this guidance.

2.2.4 SAMPLING FREQUENCY REQUIREMENTS

     Each  of the RCRA  groundwater  monitoring  regulations defines somewhat different minimum
sampling requirements. §264.97(g)(l) & (2) provides two main options:

        1.  Obtaining a sequence of at least four samples taken at an interval that ensures, to the
           greatest extent technically feasible, that a statistically independent sample is obtained, by
           reference  to the  uppermost  aquifer  effective  porosity, hydraulic  conductivity,   and
           hydraulic gradient, and the fate and transport characteristics of potential contaminants;
           or

        2.  An alternate sampling procedure proposed by the owner or operator and approved by the
           Regional Administrator if protective of human health and the environment.
                                             2-10                                   March 2009

-------
Chapter 2.  Regulatory Overview                                          Unified Guidance

       Additional  regulatory  language  in  detection  [§264.98(d)]  and  compliance  [§264.99(f)]
monitoring reaffirms the first approach:

          [A] a sequence of at least four samples from each well (background and compliance wells)
          must be collected at least send-annually during detection/compliance monitoring...

       Interim status sampling requirements under §265.92[c] read as follows:

          (1)  For all monitoring wells, the owner or operator must establish initial background
          concentrations or values of all parameters specified in paragraph (b) of this section. He
          must do this quarterly for one year;

          (2)  For each  of the indicator parameters specified in paragraph (b)(3) of this section, at
          least four replicate  measurements must  be obtained for each  sample and the  initial
          background arithmetic mean and variance must be determined by pooling  the replicate
          measurements for the respective parameter concentrations or values  in samples obtained
          from upgradient wells during the first year.

       The requirements under Subtitle D §258.54(b) are somewhat different:

          The monitoring frequency for all constituents listed in Appendix I to this part,... shall be at
          least semi-annual during the active life of the facility.... A minimum  of four independent
          samples from each well (background and downgradient) must be collected and analyzed
          for  the Appendix I constituents...  during the first semi-annual event.  At least one sample
          from each well (background and downgradient)  must be collected and analyzed during
          subsequent semi-annual events...

     The  1980 and 1982 regulations required four analyses of essentially a single physical sample for
certain  constituents, i.e.,  the  four contaminant  indicator  parameters. The need  for statistically
independent  data was recognized  in the 1988 revisions to Part  264 and in the Part 258 solid waste
requirements.  In the latter rules, only a minimum single sample is required in successive semi-annual
sampling events. Individual Subtitle C programs have also made use of the provision in §264.97(g)(2) to
allow for fewer than four  samples collected during  a  given semi-annual  period, while other State
programs require the four successive sample measurements. As noted, by the recent changes in the April
2006 Burden Reduction Rule, the  explicit requirements to obtain at least four samples during the next
evaluation period under 40 CFR §264.98(d) and §264.99(f) have been removed, allowing  more general
flexibility under the §264.97(g) sampling options. Individual State RCRA programs should be consulted
as to whether these recent rule changes may be applicable.

     The  requirements of Parts 264 and 258 were generally intended to provide sufficient data for
ANOVA-type tests in detection monitoring. However, control chart, tolerance limit, and prediction limit
tests can be applied with  as few as one new sample per evaluation, once background data are established.
The guidance provides maximum flexibility in offering a range of prediction limit options in Chapter
18 in order to address these various sample size requirements. Although not discussed in detail, the same
conclusions pertain to the use of control charts or tolerance limits.

     The  use  of the term  "replicate" in the Part 265 interim status regulations  can be a significant
problem, if interpreted to mean repeat analyses of splits (or aliquots) of a single physical sample. The
                                             2^11                                   March 2009

-------
Chapter 2.  Regulatory Overview                                          Unified Guidance

regulations indicate the need for statistical independence among sample data for testing purposes. This
guidance discusses the technical statistical problems that arise if replicate (aliquot) sample data are used
with the required Student's Mest in Part 265.  Thus, the guidance recommends, if possible, that interim
status statistical evaluations be based on independent sample data as discussed in Chapters 13 and 14
and at the end of this chapter. A more  standardized Welch's version of the Student-t test for unequal
variances is provided as an alternative to the CABF Student's Mest.

2.2.5 GROUNDWATER  PROTECTION STANDARDS

     Part 265 does not use the term groundwater protection standards.  A first-year requirement under
§265.92(c)(l)is:

          For  all  monitoring  wells,   the  owner  or  operator  must  establish  background
          concentrations or values of all parameters specified in paragraph (b) of this section. He
          must do this quarterly for one year.

      Paragraph (b)  includes water supply parameters listed  in Part 265 Appendix HI,  which also
provides  a Maximum Level for each constituent. If a facility owner or operator  does not develop and
implement an assessment plan under §265.93(d)(4), there is a requirement in §265.94(a)(2) to report the
following information to the Regional Administrator:

          (i) During the first year when initial background concentrations are being established for
          the facility: concentrations or values  of  the parameters listed in §265.92(b)(l) for each
          groundwater monitoring well within 15 days after completing each quarterly analysis. The
          owner or operator must separately identify for each  monitoring well any parameters whose
          concentrations or  value has been found to exceed the maximum contaminant levels in
          Appendix III.

     Since the Part  265 regulations are explicit in requiring a one-to-one comparison, no statistical
evaluation is needed or possible.

     §264.94(a) identifies the permissible concentration limits as a GWPS under §264.92:

          The Regional Administrator will specify in the facility permit concentrations limits  in the
          groundwater for hazardous constituents established under §264.93. The concentration of a
          constituent:

          (1) must not exceed the background level of that constituent in the groundwater at the time
          the limit is specified in the permit; or

          (2) for any of the constituents listed in Table 1, must not exceed the respective value given
          in that table if the background level is below the value given in Table 1; or

          (3) must not exceed an alternate limit established by the Regional Administrator  under
          paragraph (b) of this section.
                                             2-12                                   March 2009

-------
Chapter 2.  Regulatory Overview                                          Unified Guidance

       The RCRA Subtitle D regulations establish the following standards under §258.55(h) and (i):

          (h)  The owner or operator must establish a groundwater protection standard for  each
          Appendix II constituent detected in groundwater. The groundwater protection standard
          shall be:

             (1)  For  constituents for which a maximum contaminant level (MCL) has  been
             promulgated under Section 1412 of the Safe Drinking Water Act (codified) under 40
             CFR Part 141, the MCL for that constituent;

             (2) for constituents for which  MCLs have  not  been promulgated, the background
             concentration  for  the  constituent  established from  wells  in  accordance  with
             §2 5 8.51 (a) (1); or

             (3) for constituents for which the background level is higher than the MCL identified
             under paragraph  (h)(l)  of this  section  or  health  based levels identified under
             §258(i)(l), the background concentration.

          (i) The Director of an approved State program may establish an alternative groundwater
          protection standard for constituents for which MCLs have not been established. These
          groundwater protection standards shall be appropriate health based levels that satisfy the
          following criteria:

             (1) the level is derived in a manner consistent with Agency guidelines for assessing
             health risks or environmental pollutants [51 FR 33992, 34006, 34014, 34028, Sept. 24,
             1986]

             (2) to (4)... [other detailed requirements for health risk assessment procedures]

       The two principal alternatives for defining a groundwater protection standard [GWPS] are either
a limit based on background data or a fixed health-based value (e.g., MCLs, §264.94 Table 1 values, or a
calculated risk limit). The Unified Guidance discusses these two basic kinds of standards in Chapters 7
and 21.  If a background limit is applied, some definition of how the limit is constructed from  prior
sample data is required at the time of development.  For fixed health-based limits, the regulatory program
needs to consider the statistical characteristic of the data (e.g., mean, median, upper percentile)  that best
represents the  standard in order to conduct appropriate  statistical comparisons. This subject is also
discussed in Chapter 21; the guidance provides a number of testing options in this regard.

2.3  UNIFIED GUIDANCE RECOMMENDATIONS

2.3.1 INTERIM STATUS MONITORING

     As discussed in Chapter 14, replicates required for the four contaminant indicator parameters are
not statistically independent when analyzed  as aliquots or splits from a single physical sample. This
results in incorrect estimates of variance and the degrees of freedom when used in a Student's t-tesi. One
of the most  important revisions in the 1988 regulations  was to require that  successive  samples  be
independent.   Therefore, at  a minimum, the Unified Guidance recommends that only independent
water quality sample data be applied to the detection monitoring Student's t-tesis in Chapter 16.

                                            2^13March 2009

-------
Chapter 2. Regulatory Overview                                            Unified Guidance

     There are other considerations limiting the application of these tests as well. As noted in Chapter
5, at least two of the indicator parameters (pH and specific conductance) are likely to exhibit natural
spatial differences among monitoring wells. Depending on site groundwater characteristics, TOC and
TOX may also vary spatially. TOX analytical limitations described in SW-84610  also note that levels of
TOX are affected by inorganic chloride levels, which themselves can vary spatially by well. In short, all
four indicator parameters may need to be evaluated on an intrawell basis, i.e., using historical data from
compliance monitoring wells.

     Since  this option  is  not  identified in existing Part  265 regulations for  indicator detection
monitoring, a  more appropriate strategy is to develop an alternative groundwater quality assessment
monitoring plan under §265.90(d)(3)  and  (4)  and §265.93(d)(3) and  (4).  These  sections of  the
regulations require evaluation of hazardous waste constituents  reasonably derived from the regulated
unit (either those which served as a basis  for listing in  Part  265  Appendix VII or which are found in
§261.24  Table  1).  Interim status  units  subject to a  permit  are  also subject  to  the groundwater
contaminant information collection provisions under §270.14[c], which potentially include all hazardous
constituents (a wider range of contaminants, e.g., Part 264 Appendix IX) reasonably expected from the
unit. While an interim status facility can return  to indicator  detection monitoring if no hazardous
constituent releases have been identified, such a return is itself optional.

     EPA  recommends that  interim status facilities develop  the §265.90(d)(3) &  (4)  alternative
groundwater quality assessment monitoring plan, if possible, using  principles and procedures found in
this guidance for monitoring design and statistical evaluation. Unlike Part 264 monitoring, there are no
formal compliance/corrective action steps associated with statistical testing. A regulatory agency may
take appropriate  enforcement  action if data indicate  a release or significant  adverse  effect.  The
monitoring plan can be applied for an indefinite period until permit  development. Multi-year collection
of semi-annual or quarterly hazardous constituent data is more determinative of potential releases.  The
facility or the  regulatory agency may also wish to continue evaluation of some or all of the Part 265
water quality indicators. Eventually these groundwater data can be used to establish which  monitoring
program(s) may be appropriate at the time of permit development under §264.91(b).

2.3.2 PARTS 264 AND 258  DETECTION  MONITORING METHODS

     As described in Chapter 13, many of the commonly monitored inorganic analytes exhibit natural
spatial variation among wells. Since the two ANOVA techniques in  §264.97(h) and §258.53(g) depend
on an assumption of a single common  background population, these tests may not be  appropriate in
many situations. Additionally, at least 50% of the data should be  detectable in order to compare either
well means or medians. For many hazardous trace elements, detectable percentages are considerably
lower.  Interwell ANOVA techniques would also not be generally useful in these cases.  ANOVA may
find limited applicability in detection monitoring  with  trace  organic  constituents,  especially where
downgradient levels are considerably higher than background  and there is a high percentage of detects.
Based on ranks alone,  it may be possible to determine that compliance well(s) containing one or more
hazardous  constituents exceed background.  However, the Unified Guidance  recommends avoiding
ANOVA techniques in the limiting situations just described.
10 Test Methods for Evaluating Solid Waste (SW-846), EPA OSWER, 3rd Edition and subsequent revisions, Method 9020B,
  September 1994

                                              2^14                                   March 2009

-------
Chapter 2.  Regulatory Overview                                           Unified Guidance

      Another detection monitoring method receiving less emphasis in this guidance is the tolerance
limit. In previous guidance, an upper tolerance limit based on background was suggested to identify
significant increases in downgradient well  concentration levels. While still acceptable by regulation
(e.g., under existing RCRA permits), use of  prediction limits are preferable to tolerance limits in
detection monitoring for the following reasons. The construction of a tolerance limit is nearly identical
to that of a prediction limit. In parametric normal distribution applications, both methods use the general
formula: x + KS . The  kappa (K) multiplier varies depending on  the  coverage  and confidence levels
desired, but in both cases some multiple of the standard deviation (s) is added  or subtracted from the
sample mean (x). For non-parametric limits, the similarity is even more apparent. Often the identical
statistic (e.g.., the maximum observed value in background) can either be used  as an upper prediction
limit or an upper tolerance limit, with only a difference in statistical interpretation.

     More fundamentally, given the wide variety of circumstances in which retesting strategies are now
encouraged and  even necessary, the mathematical underpinnings of retesting with prediction limits are
well established while those for retesting with  tolerance limits are not. Monte Carlo simulations were
originally conducted for the  1992  Addendum to develop appropriate retesting strategies  involving
tolerance limits.  Such simulations were found insufficient for the Unified Guidance.11

     While the  simultaneous  prediction limits presented in the Unified Guidance consider the actual
number of comparisons in defining exact false positive error rates,  some tolerance limit approaches
(including past guidance) utilized an approximate and less precise pre-selected low level of probability.
On balance,  there is little  practical need for  recommending two highly similar (but  not identical)
methods in the Unified Guidance, both for the reasons just provided and to  avoid confusion of which
method to use.  The final regulation-specified  detection monitoring  method — control charts  — is
comparable to prediction limits, but possesses some unique benefits and so is also recommended in this
guidance.

2.3.3 PARTS  264 AND 258  COMPLIANCE/ASSESSMENT  MONITORING

     A  second  use  of tolerance  limits  recommended  in  earlier  guidance was for  comparing
downgradient monitoring well data to a fixed  limit  during compliance/assessment monitoring. In this
case, an upper tolerance limit constructed on each compliance well  data set could be used to identify
non-compliance with a  fixed GWPS limit. Past guidance also used upper confidence limits around an
upper proportion in defining these tolerance limits.  A number of problems were identified using this
approach.

     A tolerance limit  makes statistical sense  if the limit represents an upper percentile, i.e., when a
limit is not  to  be exceeded  by more than, for  instance,  1% or  5% or  10% of  future individual
concentration values. However, GWPS limits can also be interpreted as long-term averages, e.g., chronic
risk-based values, which are better approximated by  a statistic like the mean or median. Chapters 7 &
  1) there were minor errors in the algorithms employed; 2) Davis and McNichols (1987) demonstrated how to compute exact
  kappa multipliers for prediction limits using a numerical algorithm instead of employing an inefficient simulation strategy;
  and 3) further research (as noted in Chapter 19) done in preparation of the guidance has shown that repeated prediction
  limits are more statistically powerful than retesting strategies using tolerance limits for detecting changes in groundwater
  quality.

                                              2^15March 2009

-------
Chapter 2. Regulatory Overview                                           Unified Guidance

22  discuss  important  considerations when identifying the  appropriate statistical parameter  to  be
compared against a fixed GWPS limit.

     More importantly,  since the upper  confidence level of tolerance limit overestimates the true
population  proportion  by design, demonstrating an  exceedance of a GWPS  by this limit  does not
necessarily indicate that the corresponding population proportion also exceeds the standard, leading to a
high false positive rate. Therefore, the Unified Guidance recommends that the compliance/assessment
monitoring null hypothesis be structured so that the compliance population characteristic (e.g.,  mean,
median, upper percentile) is assumed to be less than or equal to the fixed standard unless demonstrated
otherwise.  The correct test statistic  in this situation is then the lower confidence limit. The  upper
confidence limit is used in corrective action to identify whether a constituent has returned to compliance.

     To ensure consistency  with the underlying  statistical presumptions  of compliance/assessment
monitoring (see Chapter 4)  and to  maintain control  of  false positive rates, the Unified Guidance
recommends that this tolerance interval approach be replaced with a more coherent and comprehensive
strategy based on the use  of confidence intervals (see Chapters 21 and 22). Confidence intervals can be
applied in a consistent  fashion to GWPS concentration limits representing either long-term averages or
upper percentiles.
                                              2-16                                    March 2009

-------
Chapter 3. Key Statistical Concepts	Unified Guidance

            CHAPTER  3.  KEY STATISTICAL CONCEPTS
        3.1  INTRODUCTION TO GROUNDWATER STATISTICS	3-2
        3.2  COMMON STATISTICAL ASSUMPTIONS	3-4
          3.2.1   Statistical Independence	3-4
          3.2.2   Stationarity	3-5
          3.2.3   Lack of Statistical Outliers	3-7
          3.2.4   Normality	3-7
        3.3  COMMON STATISTICAL MEASURES	3-9
        3.4  HYPOTHESIS TESTING FRAMEWORK	3-12
        3.5  ERRORS IN HYPOTHESIS TESTING	3-14
          3.5.1   False Positives and Type I Errors	3-15
          3.5.2   Sampling Distributions, Central Limit Theorem	3-16
          3.5.3   False Negatives, Type II Errors, and Statistical Power	3-18
          3.5.4   Balancing Type I and Type II Errors	3-22
     The success of any discipline rests on its ability to accurately model and explain real problems.
Spectacular successes have been registered during the past four centuries by the field of mathematics in
modeling fundamental processes in mechanics and physics. The last century,  in turn, saw the rise of
statistics and its fundamental theory of estimation and hypothesis testing. All of the tests described in the
Unified Guidance are based upon this theory and involve the same key concepts. The purpose of this
chapter is to  summarize the statistical  concepts underlying  the  methods presented in  the Unified
Guidance, and to consider each  in the practical context of groundwater monitoring. These include:

    *»*  Statistical inference: the difference between samples and populations; the concept of sampling.
    »«»  Common  statistical  assumptions used  in  groundwater monitoring: statistical independence,
       Stationarity, lack of outliers, and normality.
    »»»  Frequently-used  statistical  measures:  mean,   standard  deviation,   percentiles,  correlation
       coefficient, coefficient of variation, etc.
    *»*  Hypothesis testing: How probability distributions are used to model the behavior of groundwater
       concentrations and how the  statistical evidence is used to "prove" or "disprove" the validity of
       competing models.
    »«»  Errors  in hypothesis testing: What false positives  (Type I errors) and  false negatives (Type II
       errors) really represent.
    *»*  Sampling  distributions  and  the Central Limit Theorem: How  the statistical behavior of test
       statistics differs from that of individual population measurements.
    »»»  Statistical power and power curves: How the ability to detect real contamination depends on the
       size or degree of the concentration increase.
    *»*  Type I vs. Type II  errors: The tradeoff between false positives  and false negatives;  why it is
       generally impossible to minimize both kinds of error simultaneously.
                                              3-1                                    March 2009

-------
Chapter 3. Key Statistical Concepts	Unified Guidance

3.1 INTRODUCTION TO GROUNDWATER STATISTICS

     This section briefly covers some basic statistical terms and principles used in this guidance. All of
these topics are more thoroughly discussed in standard textbooks. It is presumed that the user already has
some familiarity with the following terms and discussions.

     Statistics is a branch  of applied mathematics, dealing with the description, understanding,  and
modeling of data. An integral part of statistical analysis is the testing of competing mathematical models
and  the management  of data  uncertainty. Uncertainty  is present because  measurement  data exhibit
variability, with limited knowledge of the medium being sampled. The fundamental aim of almost every
statistical analysis is to draw inferences. The data analyst must infer from the observed data something
about the physical world without knowing or seeing all the possible facts or evidence.  So the question
becomes: how closely do the measured data mimic reality, or put another way, to what extent do the data
correctly identify a physical truth  (e.g., the compliance  well  is contaminated with  arsenic above
regulatory limits)?

     One way to ascertain whether an aquifer is contaminated with certain chemicals would  be to
exhaustively sample and measure every physical volume of groundwater underlying the site of interest.
Such a collection of measurements would be impossible to procure in practice and would be infinite in
size, since sampling would have to be continuously conducted over time at a huge number of wells and
sampling depths.  However,  one would possess the entire population of possible measurements at that
site and the exact statistical distribution of the measured concentration values.

     A statistical distribution is an organized  summary  of a set of data values, sorted into the relative
frequencies of occurrence of different measurement levels (e.g., concentrations of 5 ppb or less  occur
among 30 percent of the values,  or levels of 20 ppb or  more only occur 1  percent  of the time).  More
generally, a distribution may refer to a mathematical model (known as ^probability distribution) used to
represent the  shape and statistical characteristics of a given population and chosen according to  one's
experience with the type of data involved.

     By contrast to the population,  a statistical sample is  a finite subset of the population,  typically
called a data  set. Note that  the statistical definition of sample is usually different from a geological or
hydrological definition of the same term. Instead of a physical volume or mass, a statistical sample is a
collection of measurements, i.e., a set of numbers. This collection might contain only a single value, but
more generally has a number of measurements denoted as the sample size, n.

     Because a sample is only a partial representation of the population, an inference is usually desired
in order to conclude something from the observed data  about the underlying population. One or more
numerical characteristics of the population might be of  interest, such as the true average contaminant
level or the upper 95thpercentile of the concentration distribution. Quantities computed from the sample
data are known as statistics, and can be used to reasonably estimate the desired but unknown population
characteristics. An  example is when testing sample data  against  a regulatory standard such  as  a
maximum concentration limit [MCL] or background level.   A mean sample estimate of the average
concentration can be used to judge whether the corresponding population characteristic — the true mean
concentration (denoted by the Greek letter u) — exceeds the MCL or background limit.

     The accuracy of  these estimates depends on how representative the sample measurements of the
underlying  population  are.  In a representative sample, the distribution of sample values have the best
                                             3-2                                    March 2009

-------
Chapter 3. Key Statistical Concepts	Unified Guidance

chance of closely matching the population distribution. Unfortunately, the degree of representativeness
of a given sample is almost never known.  So it quite important to understand precisely how the sample
values were obtained from the population and to  explore whether or not they appear representative.
Though there  is no guarantee that  a sample  will  be adequate, the  best  protection  against an
unrepresentative sample is to select measurements from the population at random. A random sample
implies that each potential population value has an equivalent chance of being selected depending only
on  its likelihood of  occurrence.  Not only  does random sampling  guard against selection of an
unrepresentative portion of the population distribution, it also enables a mathematical estimate to be
drawn of the statistical uncertainty associated with the ability of a given sample to represent the desired
characteristic of the population. It  can be  very difficult to gauge the uncertainty surrounding a sample
collected haphazardly or by means of professional judgment.

     As a simple example, consider an urn filled with red and green balls. By thoroughly mixing the urn
and blindly sampling (i.e., retrieving) 10  percent of the balls,  a very nearly random sample  of the
population of balls will be obtained, allowing a fair estimate of the true overall proportion of one color
or the other. On the other hand, if one looked into the urn while sampling and only picked red balls or
tried  to  alternate between red  and  green,  the   sample would  be  far from random  and  likely
unrepresentative of the true proportions.

     At first glance, groundwater measurements obtained during routine monitoring would not seem to
qualify as random samples. The well points are generally not placed in random  locations or at random
depths, and the physical samples are usually collected at regular, pre-specified intervals.  Consequently,
further  distinctions  and  assumptions are necessary  when performing  statistical evaluations  of
groundwater data. First, the distribution  of a given  contaminant may not be spatially uniform  or
homogeneous.  That is, the local distribution of measured values at one  well may not be the same  as at
other  wells. Because this  is often  true for naturally-occurring groundwater constituents, the statistical
population(s) of interest may be well-specific.  A statistical sample gathered from a particular well must
then be treated as potentially representative only of that well's local population.  On the other hand,
samples drawn from a number of  reference background wells for which no  significant differences are
indicated, may permit the pooled data to serve as an estimate of the overall well  field behavior for that
particular monitoring constituent.

     The distribution of a contaminant may also not be temporally uniform  or stationary over time. If
concentration values indicates a trend,  perhaps because a plume intensifies or dissipates or natural in-situ
levels rise  or fall due to drought conditions, etc., the distribution is said to be  non-stationary.  In this
situation, some of the measurements collected over time may not be representative of current conditions
within the aquifer.  Statistical  adjustments might be needed or the data  partitioned into  usable and
unusable values.

     A similar difficulty is posed by cyclical or seasonal trends. A long-term constituent concentration
average at a well location or the entire site may  essentially be constant over time, yet temporarily
fluctuate up and down on a seasonal basis. Given a fixed interval between sampling  events, some of this
fluctuation may go unobserved due to the non-random nature of the sampling times.  This could result in
a sample that is unrepresentative of the population variance and possibly of the population mean as well.
In such settings, a shorter (i.e., higher frequency) or staggered sampling interval may be needed to better
capture key characteristics of the population as a part of the distribution of sample measurements.


                                              3-3                                     March  2009

-------
Chapter 3.  Key Statistical Concepts	Unified Guidance

     The difficulties in identifying a valid statistical framework for groundwater monitoring highlight a
fundamental assumption governing almost every statistical procedure and test.  It is the presumption that
sample data from a given population should be independent and identically distributed, commonly
abbreviated as i.i.d.  All of the mathematics and statistical formulas contained  in this guidance are built
on this basic assumption. If it is not satisfied, statistical conclusions and test results may be invalid or in
error. The associated statistical uncertainty may be different than expected from a given test procedure.

     Random sampling of a single, fixed, stationary population will guarantee independent, identically-
distributed sample data. Routine groundwater sampling  typically does not. Consequently, the Unified
Guidance discusses  both below and in later chapters what assumptions about the sample data must be
routinely or periodically checked. Many but not all of these assumptions are a simple consequence of the
i.i.d. presumption. The guidance also discusses how sampling ought to be conducted and designed to get
as close as possible to the i.i.d. goal.

3.2 COMMON  STATISTICAL ASSUMPTIONS

     Every statistical test or procedure makes certain assumptions about the  data used to compute the
method. As noted above, many of these assumptions flow as a natural consequence of the presumption
of independent, identically-distributed data (i.i.d.). The most common assumptions are briefly  described
below:

3.2.1  STATISTICAL IN DEPENDENCE

     A major advantage of truly random sampling of a population is that the measurements will be
statistically independent. This means that observing or knowing the value of one measurement does not
alter or influence the probability of observing any other measurement in the population. After  one value
is selected, the next  value is sampled again at random without regard to the previous measurement, and
so on. By contrast, groundwater samples are not chosen at  random times or  at random locations. The
locations are fixed and typically few in number. The intervals between sampling events are  fixed and
fairly regular.  While samples  of independent data exhibit no pairwise correlation  (i.e., no  statistical
association of similarity or dissimilarity between pairs of sampled measurements), non-independent or
dependent data do exhibit pairwise correlation and often other, more complex forms of correlation.
Aliquot split sample pairs are generally not independent because of the positive correlation induced by
the  splitting  of the same physical groundwater sample. Split measurements tend to be highly similar,
much more so than the random pairings of data from distinct sampling events.

     In a similar vein, measurements  collected close together in time from the same well tend to be
more highly correlated than pairs collected  at longer  intervals.   This is especially true  when the
groundwater is so slow-moving  that the  same general volume of groundwater is  being sampled on
closely-spaced consecutive sampling events. Dependence may also be exhibited spatially across a well
field. Wells  located  more closely in space and screened in the same hydrostratigraphic zone may show
greater similarity in concentration patterns than wells that are farther apart. For both of these temporal or
time-related  and spatial dependencies, the observed correlations are a result not only  of the non-random
nature of the sampling but also the fact that many groundwater populations are not uniform throughout
the  subsurface. The aquifer may instead exhibit pockets  or sub-zones of higher or lower concentration,
perhaps due to location-specific  differences in natural geochemistry or the dynamics of contaminant
plume behavior over time.

                                             3-4                                   March 2009

-------
Chapter 3. Key Statistical Concepts	Unified Guidance

     As a mathematical construct, statistical independence is essentially impossible to check directly in
a set of sample data — other than by ensuring ahead of time that the measurements were collected at
random. However, non-zero pairwise correlation, a clear sign of dependent data, can be checked and
estimated in a variety of ways.  The Unified Guidance describes two methods for identifying temporal
correlation  in Chapter 14: the  rank von Neumann ratio test and the sample autocorrelation function.
Measurable correlation  among consecutive sample pairs  may  dictate the need for  decreasing  the
sampling frequency or for a more complicated data adjustment.

     Defining and  modeling wellfield spatial  correlation is beyond the  scope of this guidance, but is
very much  the purview of the field of geostatistics. The Unified Guidance instead looks for evidence of
well-to-well spatial variation,  i.e., statistically identifiable  differences in mean and/or variance levels
across the well field. If evident, the statistical approach would need to be modified so that distinct wells
are treated  as individual populations with statistical testing being conducted separately at each one (i.e.,
intrawell comparisons).

3.2.2 STATIONARITY

     A stationary statistical distribution is one  whose population characteristics do not change over time
and/or space. In  a  groundwater context, this  means that the  true population  distribution of a given
contaminant is the same no matter where or when it is sampled. In the strictest form  of stationarity, the
full distribution must be exactly the same at every time and location. However, in  practice, a weaker
form is usually assumed: that the population mean (u) and  variance (denoted by the Greek symbol o )
are the same over time and/or space.

     Stationarity is important to  groundwater statistical analysis because of the way that monitoring
samples must be collected. If a sample set somehow represented the entire population  of possible aquifer
values, stationarity would not be an issue in theory. A limited number of physical groundwater samples,
however, must be individually collected from  each sampled location. To generate a statistical sample,
the individual measurements must be pooled together over time from multiple sampling events within a
well, or pooled together across space by aggregating data from multiple wells, or both.

     As long as the contaminant distribution is stationary, such pooling poses no statistical problem. But
with a non-stationary distribution, either the mean and/or variance  is changing over time in any given
well, or the means and variances differ at distinct locations.  In either case, the pooled measurements are
not identically-distributed even if they may be statistically independent.

     The effects of non-stationarity are commonly seen in four basic ways in the groundwater context:
1) as spatial variability, 2) in the existence of trends and/or seasonal variation,  3) via other forms of
temporal variation,  and 4) in the lack of homogeneity of variance.  Spatial variability (discussed more
extensively in Chapter 13) refers to statistically identifiable differences in mean  and/or variance levels
(but usually means) across the well field (i.e.,  spatial non-stationarity). The existence of such variation
often  precludes the pooling of data  across multiple background wells  or the  proper  upgradient-to-
downgradient comparison of background wells against distinct  compliance wells.  Instead,  the usual
approach is to perform intrawell comparisons, where well-specific background data  is culled from the
early sampling history at  each well. Checks  for spatial variability are conducted graphically with the aid
of side-by-side box plots (Chapter 9) and through the use  of analysis of variance [ANOVA, Chapter
13].

                                              3-5                                    March 2009

-------
Chapter 3. Key Statistical Concepts	Unified Guidance

     A trend over time at a given well location indicates that the mean level is not stationary but is
instead rising or falling. A seasonal trend is similar in that there are periodic increases and decreases.
Pooling  several  sampling events together  thus mixes  measurements with  differing  statistical
characteristics.  This can violate the identically-distributed presumption of almost all statistical tests and
usually leads to an inflated estimate of the current population variance. Trends  or seasonal variations
identified in (upgradient) background wells or in intrawell background data from compliance wells can
severely impact the accuracy and effectiveness  of statistical procedures described in this guidance if data
are pooled  over time to establish background limits. The approach that should be taken will vary with
the circumstance. Sometimes the trend component might need to be estimated and removed  from the
original data, so that what gets tested are the data residuals (i.e., values that result from subtracting the
estimated trend from the original data) instead of the raw measurements.  In other cases, an alternate
statistical approach might be needed such as a test for (positive) trend or construction of a confidence
band around an estimated trend. More discussion of these options is presented in Chapters 6, 7, 14, and
21

     To identify a linear trend, the Unified Guidance describes simple linear regression and the Mann-
Kendall test in Chapter 17. For seasonal patterns or a combination of linear and  seasonal trend effects,
the guidance discusses the seasonal Mann-Kendall test and the use of ANOVA tests to identify seasonal
effects. These diagnostic procedures are also presented in Chapter 14.

     Temporal variations are distinguished in this guidance from trends or seasonal effects by the lack
of a regular or identifiable pattern. Often a temporal effect will be observed as a temporary shift in
concentration levels that is similar in magnitude and direction at multiple wells. This can occur at some
sites,  for instance,   due  to rainfall  or recharge events. Because  the  mean level  changes  at  least
temporarily, pooling data over time again violates the assumption of identically-distributed data. In this
case, the temporal effect can be identified by looking for parallel traces on a time series plot of multiple
wells and then more formally by performing a one-way ANOVA for temporal effects. These procedures
are described  in Chapter  14.  Once  identified, the  residuals from the  ANOVA  can be  used for
compliance testing, since the common temporal effect has been removed.

     Lastly, homogeneity of variance is important in ANOVA tests, which simultaneously evaluates
multiple groups of data each representing a sample from a distinct statistical population.  In the latter
test, well means need not be the  same; the reason for performing the test in the first place is to find out
whether the means do indeed differ. But the procedure assumes that all the group variances are equal or
homogeneous. Lack  of homogeneity or stationarity in the variances  causes the test  to be much less
effective at discovering differences in the well means.  In extreme cases, the concentration levels would
have to differ by large amounts before the ANOVA would correctly register a statistical difference. Lack
of homogeneity of variance can be identified graphically via the use of side-by-side box plots and then
more formally with the use of Levene's test. Both these methods are discussed further in Chapter 11.
Evidence of unequal  variances may necessitate the use of a transformation to stabilize the variance prior
to running the ANOVA. It might also preclude  use of the ANOVA altogether for compliance testing, but
require intrawell approaches to be considered instead.

     ANOVA is not the only  statistical procedure which assumes homogeneity of variance. Prediction
limits and  control charts require a similar assumption between background and  compliance well data.
But if only one new  sample  measurement is collected per well per evaluation period (e.g.,  semi-
annually) it can be difficult to formally test this assumption with the diagnostic methods cited above. As
                                              3-6                                     March 2009

-------
Chapter 3.  Key Statistical Concepts	Unified Guidance

an alternative, homogeneity of variance can be periodically tested when a sufficient sample size has been
collected for each compliance well (see Chapter 6).

3.2.3 LACK OF STATISTICAL OUTLIERS

     Many authors have noted that outliers — extreme, unusual-looking measurements — are a regular
occurrence among groundwater data (Helsel and Hirsch, 2002; Gibbons and Coleman, 2001). Sometimes
an outlier results from nothing more than a typographical error on a laboratory data sheet or file.  In
others, the fault is an incorrectly calibrated measuring device or a piece of equipment that was not
properly decontaminated. An unusual measurement might also reflect the sampling of a temporary, local
'hot spot' of higher concentration. In each of these situations, outliers in a statistical context represent
values that are inconsistent with the distribution of the remaining measurements. Tests for outliers thus
attempt to  infer  whether the suspected outlier could have reasonably been  drawn from the same
population as the other measurements, based on the sample data  observed up to that point. Statistical
methods to help identify potential outliers are discussed in Chapter 12, including both Dixon 's and
Rosner 's tests, as well as references to other methods.

     The basic problem with including statistical outliers in analyzing groundwater data is that they do
not come from the same distribution as the other measurements in the sample and so fail the identically-
distributed presumption of most tests.  The consequences can be dramatic, as can be seen for instance
when considering non-parametric prediction limits. In this testing method, one of the largest values
observed in the background data such  as the maximum, is often the statistic selected as the prediction
limit. If a large outlier is present among the background measurements, the prediction limit may be set to
this value despite being unrepresentative of the background population. In effect, it arises from another
population, e.g., the 'population' of typographical errors. The prediction limit could then be much higher
than warranted based on the observed background data and may provide little if any probability that truly
contaminated  compliance wells will be identified. The test will then have lower than expected statistical
power.

     Overall, it pays to try to identify  possible outliers and to either correct the value(s) if possible, or
exclude known outliers from subsequent statistical analysis. It is  also possible to select a statistical
method that is resistant to the presence of outliers,  so that the test results are still likely to be accurate
even if one or more outliers is unidentified. Examples of this last strategy include setting non-parametric
prediction limits to values other than the background maximum using repeat testing (see Chapter 18) or
using Sen's slope procedure to estimate the rate of change in a linear trend (Chapter 17).

3.2.4 NORMALITY

     Probability distributions introduced in Section 3.1 are  mathematical models used to approximate
or represent the statistical characteristics of populations. Knowing  the exact form and defining equation
of a probability distribution allows one to assess how likely or unlikely it will be to observe particular
measurement values (or ranges of values) when selecting or drawing independent, identically distributed
[i.i.d] samples from the associated population. This can be done as follows. In the case of a continuous
distributional model,  a curve  can be drawn to  represent the  probability distribution by plotting
probability values along the_y-axis and measurement or concentration values along the x-axis. Since the
continuum of x-values along this curve is infinite,  the probability of occurrence of any single possible
value is negligible (i.e., zero), and does not equal the height  of the curve. Instead, positive probabilities
can be computed for ranges of possible values by summing the area under the distributional curve
                                              3-7                                     March 2009

-------
Chapter 3.  Key Statistical Concepts	Unified Guidance

associated with the desired range. Since by definition the total area under any probability distribution
curve sums to unity, all probabilities are then numbers between 0 and 1.

     Probability distributions form the basic building blocks of all statistical testing procedures. Every
test  relies on comparing  one  or more  statistics computed from  the sample data against a reference
distribution.  The reference distribution is in turn a probability distribution summarizing the expected
mathematical behavior of the statistic(s) of interest.   A formal  statistical test utilizes this reference
distribution  to make  inferences about the  sample statistic in terms of two contrasting conditions or
hypotheses.

     In any event, probability distributions used in statistical testing make differing assumptions about
how the underlying  population  of  measurements  is  distributed. A  case  in  point  is  simultaneous
prediction limits using retesting (Chapter  19). The first  and most common version of this test (Davis
and  McNichols,  1987)  is based on an  assumption that the sample  data  are  drawn  from a normal
probability distribution. The normal distribution is the well-known bell-shaped curve, perhaps the single
most important and frequently-used distribution in statistical analysis.  However, it is not the only one.
Bhaumik  and Gibbons (2006) proposed  similar  prediction limits  for data drawn  from  a gamma
distribution  and Cameron (2008) did the same for Weibull-distributed measurements. This more recent
research demonstrates that prediction limits with similar  statistical decision error rates can vary greatly
in magnitude, depending on the type of data distribution assumed.

     Because many tests  make an explicit assumption concerning the distribution represented by the
sample data, the form and exact type of distribution often has to be checked using a goodness-of-fit test.
A goodness-of-fit test assesses how closely the observed sample data resemble a proposed distributional
model. Despite the wide variety of probability distributions identified in the statistical  literature, only a
very few goodness-of-fit tests generally are needed in practice. This is because most tests are based on an
assumption of normally-distributed or normal data. Even  when an  underlying distribution is not normal,
it is often possible to use a mathematical transformation of the raw measurements  (e.g., taking the
natural logarithm or log of  each value) to normalize the data set.   The  original values can  be
transformed into a set of numbers that behaves as if drawn from a normal distribution.  The transformed
values can then be utilized in and analyzed with a normal-theory test (i.e., a procedure that assumes the
input data are normal).

     Specific goodness-of-fit tests for checking and identifying data distributions are found in Chapter
10 of this guidance. These methods  all are designed to check the fit to normality of the sample data.
Besides the normal, the lognormal distribution is also commonly used as a model for groundwater data.
This distribution is not symmetric in shape like the  bell-shaped normal curve, nor does it have similar
statistical  properties.  However, a simple  log transformation of lognormal  measurements works to
normalize such a data set. The transformed values can be tested using one of the standard goodness-of-
fit tests of normality to confirm that the original data were indeed lognormal.

     More generally, if a  sample shows evidence of non-normality using the techniques in Chapter 10,
the initial remedy is  to try  and find  a suitable normalizing transformation.  A set of useful possible
transformations in this  regard  has been termed the ladder of powers (Helsel and Hirsch,  2002).  It
includes not  only the natural logarithm, but also other mathematical power transformations  such as the
square root,  the  cube root, the square, etc. If none of these  transformations creates an adequately
normalized data set, a second approach is to consider what are known as non-parametric tests. Normal-

                                              3-8                                     March 2009

-------
Chapter 3. Key Statistical Concepts	Unified Guidance

theory and other similar parametric statistical  procedures  assume that the  form of the underlying
probability distribution  is  known.  They  are called  parametric  because the  assumed  probability
distribution is generally characterized by a small set of mathematical parameters. In the case of the
normal distribution, the general formula describing its shape and properties is completely specified by
two parameters: the population mean  (u) and the population variance (o ). Once  values for these
quantities are known, the exact  distribution representing a particular normal population can be computed
or analyzed.

     Most parametric tests do not require knowledge of the exact distribution represented by the sample
data, but rather just the type of distribution (e.g., normal, lognormal, gamma, Weibull, etc). In more
formal terms, the test assumes  knowledge of the family of distributions indexed  by the characterizing
parameters. Every different combination of population mean and variance  defines a different normal
distribution, yet all belong  to the normal family. Nonetheless, there are many data  sets for which a
known distributional family cannot be  identified. Non-parametric methods may then be appropriate,
since a known distributional form is not  assumed. Non-parametric tests are discussed in various chapters
of the Unified Guidance. These tests are typically based on either a ranking or an ordering of the sample
magnitudes in order to assess their statistical performance and accuracy.   But even non-parametric tests
may make use of a normal approximation to define how expected rankings are distributed.

     One other common difficulty in checking for normality among groundwater measurements is the
frequent presence of non-detect values,  known in statistical terms as left-censored measurements. The
magnitude  of these sample concentrations is known only  to lie  somewhere between zero  and  the
detection or reporting limit; hence the true concentration is  partially 'hidden' or  censored on the left-
hand side of the numerical concentration scale. Because the most effective normality tests assume that
all the sample measurements are known  and quantified and not censored, the Unified Guidance suggests
two possible approaches in this circumstance. First, it is usually possible to simply assume that the true
distributional form  of the underlying  population cannot be identified, and to instead apply a non-
parametric test alternative. This solution is not always ideal, especially when using prediction limits  and
the background sample size is  small, or when using control  charts (for which there is no current non-
parametric alternative to the Unified Guidance recommended test).

     As a second alternative, Chapter 10 discusses  methods for assessing approximate normality in the
presence  of non-detects. If normality can be established,  perhaps through a normalizing transformation,
Chapter 15 describes methods for estimating the mean and variance parameters of the specific normal
distribution needed for constructing tests (such as prediction limits or control  charts), even though the
exact value of each non-detect is unknown.

3.3  COMMON  STATISTICAL MEASURES

     Due to the variety of statistical tests and other methods presented in the Unified Guidance, there
are  a  large number of equations and formulas of relevance  to specific situations. The most common
statistical measures used in many settings are briefly described below.

     Sample mean and standard deviation  —  the mean of a set of measurements of sample size n is
simply the arithmetic average of each of the numbers in the sample (denoted by X[), described by formula
[3.1] below. The sample mean is a common estimate of the center or middle of a statistical distribution.
That is, x is an estimate of  u, the population mean. The basic formula for the sample standard deviation

                                             3-9                                    March 2009

-------
Chapter 3. Key Statistical Concepts	Unified Guidance

is  given in equation [3.2]. The sample standard deviation is an estimate of the degree of variability
within a distribution, indicating how much the values typically vary from the average value or mean.
Thus, the  standard deviation s is an estimate of the population standard deviation o. Note that another
measure of variability,  the sample variance, is simply the square of the standard deviation (denoted by
s2) and serves as an estimate of the population variance a2.


                                           x=-Yx                                       [3.1]
                                               ,„ "<-—I  '                                       L   J
                                                                                           [3.2]
                                                                                           L   J

     Coefficient of Variation —  for  positively-valued  measurements,  the  sample coefficient of
variation provides a quick and useful indication of the relative degree of variability within a data set. It is
computed as s/x and so indicates whether the amount of 'spread' in the sample is small or large relative
to the average observed magnitude.  Sample coefficients of variation can also be calculated for other
distributions such as the logarithmic (see discussion on logarithmic statistics below and Chapter 10,
Section 10.4)

     Sample percentile — the pih percentile of a sample  (denoted as xp) is  the  value such that
p x 100 % of the measurements are no greater than xp, while (l - p)x 100 % of the values are no less
than xp. Sample percentiles are computed by making an ordered  list of the measurements (termed the
order statistics of the sample) and either selecting an observed value from  the sample that comes closest
to satisfying  the above definition or interpolating between the pair  of  sample values closest  to the
definition if no single value meets it.

     Slightly different estimates  of the sample  percentile are  used to perform the interpolation
depending on the software package or statistics textbook. The Unified Guidance follows Tukey's (1977)
method for computing  the lower and upper quartiles (i.e., the 25th and 75th sample percentiles, termed
hinges by Tukey) when constructing box plots (Chapter 9). In that setting, the pair of sample values
closest  to the desired percentile  is simply  averaged. Another popular method for more generally
computing sample percentiles is to set the rank of the desired order statistic as k = («+l) * p. If & is not
an integer, perform linear interpolation between the pair of ordered sample values with ranks just below
and just above k.

     Median  and  interquartile  range —  the  sample median  is the  50th percentile  of  a  set of
measurements, representing the midpoint of an ordered list of the  values.  It is usually denoted as x or
x5, and represents an alternative estimate of the center of a distribution. The interquartile range [IQR] is
the difference between  the 75th and 25th sample percentiles, thus equal to (x 75 -x25). The IQR offers an
alternative estimate of variability  in a population, since it represents the  measurement range  of the
middle 50% of the ordered sample values. Both the median and the interquartile range are key statistics
used to construct box plots (Chapter 9).
                                              3-10                                   March 2009

-------
Chapter 3. Key Statistical Concepts
Unified Guidance
     The median and interquartile range can be very useful as alternative estimates of data centrality and
dispersion to the mean and standard deviation, especially when samples are drawn from a highly skewed
(i.e., non-symmetric) distribution or when one or more outliers is present. The table below depicts two
data sets, one with an obvious outlier, and demonstrates how these statistical measures compare.

     The median and interquartile ranges are not  affected by the inclusion of an outlier (perhaps  an
inadvertent reporting of units in terms of ppb rather than ppm). Large differences between the mean and
median, as well as between the standard deviation and interquartile range in the second data set can
indicate that an anomalous data point may be present.
Data Set #1
5
10
15
15
15
20
25
X = 15
x = 15
s = 6.5
IQR = 10
Data Set #2
5
10
15
15
15
20
25,000
X > 3,500
x = 15
s > 9,000
IQR = 10
     Log-mean, log-standard deviation and Coefficient of Variation — The lognormal distribution
is  a  frequently-used model in  groundwater  statistics.   When  lognormally  distributed  data  are
transformed, the normally-distributed measurements  can then  be input into normal-theory  tests. The
Unified Guidance frequently makes use of quantities computed on log-transformed values. Two of these
quantities,  the  log-mean and  the log-standard deviation, represent the  sample mean  and standard
deviation computed using log-transformed values instead of the raw measurements.  Formulas for these
quantities — denoted y and sy to distinguish them from the measurement-scale mean (x ) and standard
deviation (s) — are given below.  Prior to calculating the logarithmic mean and standard deviation, the
measurement scale data must first be log-transformed. Taking logarithms  of the sample mean (x ) and
the sample  standard deviation (s) based on the original measurement-scale data, will not give the correct
result.
                                                                                          [3.3]
                                                                                          [3.4]
     A population logarithmic  coefficient of  variation  can be  estimated from the logarithmically

transformed data as:  CFlog =yeSy -1.  It is based solely on the logarithmic standard deviation, sy, and
represents the intrinsic variability of the untransformed data.
                                             3-11
        March 2009

-------
Chapter 3. Key Statistical Concepts _ Unified Guidance

     Sample correlation coefficient — correlation is a common numerical measure of the degree of
similarity or linear association between two random variables, say x andy. A variety of statistics are used
to estimate the correlation depending  on the setting and how much is known about  the underlying
distributions of x andy. Each measure  is typically designed to take on values in the range  of-1 to +1,
where -1 denotes perfect inverse correlation (i.e., as x increases, y decreases,  and vice-versa), while +1
denotes perfect correlation (i.e., x andy increase or decrease together), and 0 denotes no correlation (i.e.,
x and y behave independently of one another).  The most popular measure  of linear correlation  is
Pearson's correlation coefficient (r), which can be computed for a set of n sample pairs (x\, y\) as:
                                r=  .       ,                                               [3.5]
3.4  HYPOTHESIS TESTING  FRAMEWORK

     An important component of statistical analysis involves the testing of competing mathematical
models, an activity known as hypothesis testing. In hypothesis testing,  a formal comparison is made
between two mutually exclusive possible statements about reality.  Usually these statements concern the
type or form of underlying statistical population from which the sample  data originated, i.e., either the
observed data  came from one statistical population or from another, but  not both. The sample data are
used to judge which statistical model identified by the two hypotheses is  most consistent with the
collected observations.

     Hypothesis testing is similar in nature to what takes place in a criminal trial. Just as one of the two
statements in an hypothesis test is judged true and the other false, so the defendant is declared either
innocent or guilty. The opposing lawyers each develop their theory  or model of the crime and what really
happened. The jury must then decide whether the available evidence better supports the prosecution's
theory or the defense's  explanation.  Just as  a strong presumption of innocence is given to a criminal
defendant, one of the  statements  in a statistical hypothesis is initially favored over the other. This
statement, known as the null hypothesis [Ho], is only rejected as false if the sample evidence strongly
favors the other side of the hypothesis, known as the alternative hypothesis [HA].

     Another  important parallel is that the same mistakes  which can occur in statistical hypothesis
testing are made in criminal trials. In a criminal proceeding, the innocent can falsely be declared guilty or
the guilty can wrongly be judged innocent. In the same way, if the null hypothesis [Ho] is a  true
statement about reality  but is rejected in  favor  of the  alternative hypothesis [//A], a mistake akin to
convicting the innocent has occurred.  Such a mistake is known in statistical terms as a false positive or
Type I error. If the alternative hypothesis [//A] is true but is rejected in favor of HO, the mistake is akin to
acquitting the guilty.  This mistake is known as a false negative or Type II error.

     In a criminal investigation, the test hypotheses can be reversed.  A detective investigating a crime
might consider a list of probable suspects as potentially guilty (the null hypothesis [Ho]), until substantial
evidence is  found to exclude  one or  more  suspects  [//A].   The burden of proof for accepting the
alternative hypothesis and the kinds of errors which can result are the opposite from a legal trial.

                                              3-12                                    March 2009

-------
Chapter 3.  Key Statistical Concepts _ Unified Guidance

      Certain steps are involved in conducting any statistical hypothesis test.  First, the null hypothesis
HO must be specified and is given presumptive weight in the hypothesis testing framework. The observed
sample (or a statistic derived from these data) is assumed to follow a known statistical distribution,
consistent with the distributional model used to describe reality under HQ. In groundwater monitoring, a
null hypothesis might posit that concentration measurements of benzene, for  instance, follow a normal
distribution with zero  mean. This statement is contrasted against the alternative hypothesis, which is
constructed as a competing model of reality. Under HA, the observed data or statistic follows a different
distribution, corresponding to a different distributional model. In the simple example above, HA might
posit that benzene concentrations follow a normal distribution, but this time with a mean no less than 20
ppb, representing a downgradient well that has been contaminated.

      Complete  descriptions  of statistical  hypotheses are usually not  made. Typically, a shorthand
formula is  used for the two competing statements. Denoting the true population mean as the Greek letter
[j, and a possible value of this mean as uo, a common specification is:

                                    HQ: JUjU0                                [3.6]
This formulation clearly distinguishes between the location (i.e., magnitude) of the population mean (I
under the two competing models, but it does not specify the form of the underlying population itself. In
most parametric tests, as explained in  Section 3.2,  the underlying model is assumed to be the normal
distribution, but this is not a necessary condition or the basic assumption in all tests. Note also that a
family of distributions  is specified by the hypothesis, not two individual, specific distributions. Any
distribution with a true mean no greater than uo satisfies the null hypothesis, while any distribution from
the same family with true mean larger than uo satisfies the alternative hypothesis.

     Once the statistical hypothesis has been  specified, the next step is to actually collect the data and
compute whatever test  statistic is required based on the observed measurements and the kind of test.
The pattern of the observed measurements or the  computed test statistic is then compared with the
population model predicted or described under HQ. Because this  model is specified as a  statistical
distribution, it can be used to assign probabilities to different results. If the observed result or pattern
occurs with very low probability under the null hypothesis model (e.g., with at most a 5%  or 1%
chance), one of two outcomes is assumed to have occurred.  Either the result is a "chance" fluctuation in
the data representing a real but unlikely outcome under HQ, or the null  hypothesis was an incorrect
model to begin with.

     A low probability of occurrence under HQ is cause for rejecting the null hypothesis in favor of H&,
as long as the  probability of occurrence under the latter alternative is also not too small. Still, one should
be careful to understand that statistics involves the art of managing uncertainty.  The null hypothesis may
indeed be true, even if the measured results seem unlikely to have arisen under the HQ model. A small
probability of occurrence is not the  same as no possibility of occurrence. The judgment in favor of HA
should be made with full recognition that & false positive mistake is always possible even if not very
likely.

     Consider the measurement  of benzene  in groundwater in  the  example  above.  Given natural
fluctuations in groundwater composition from week-to-week or month-to-month and  the  variability
introduced in the lab during the measurement process, the fact that one or two samples show either non-
detect  or very low levels of benzene does not guarantee that the true mean benzene concentration at the
                                              3-13                                    March 2009

-------
Chapter 3. Key Statistical Concepts
Unified Guidance
well is essentially zero. Perhaps the true mean is higher, but the specific sample values collected were
gotten from the "lower tail" of the benzene distribution just by chance or were measured incorrectly in
the lab. Figure 3-1 illustrates this possibility, where the full benzene distribution is divided into a lower
tail portion that has  been sampled and a remaining portion that has not so far been observed. The
sampled values are not representative of the entire population distribution, but only of a small part of it.

     Along a similar vein, if the observed result or pattern can occur with moderate to high probability
under the null  hypothesis, the  model represented by HO is  accepted  as consistent with the sample
measurements.  Again, this does not mean the null  hypothesis is necessarily true. The alternative
hypothesis could be true instead, in which case the judgment to accept HO would be considered a false
negative. Nevertheless the sample data do not provide  sufficient evidence or justification to reject the
initial presumption.

          Figure 3-1.  Actual, But Unrepresentative Benzene Measurements
                  0.4
                  0,3 -
                  0.2 H
               5J
               JD
               8
               OH

                  0.1  -I
                  0.0
                                                                True Population
                                                                Distribution
                           Sampled Values
                                        Benzene Concentration.
3.5  ERRORS IN HYPOTHESIS TESTING

     In order to properly interpret the results of any statistical test, it is important to understand the risks
of making a wrong decision. The risks of the two possible errors or mistakes mentioned above are not
fixed quantities; rather, false positive and false negative risks are best thought of as statistical parameters
that can  be adjusted when performing a  particular test.  This flexibility allows one, in  general, to
"calibrate" any test to meet specific risk or error criteria. However, it is important to recognize what the
different  risks represent. RCRA groundwater regulations stipulate that any test procedure maintain a
"reasonable balance" between the risks of false positives and false negatives. But  how does one decide
on a reasonable balance? The answer lies in a proper understanding of the real-life  implications attached
to wrong judgments.
                                             3-14
        March 2009

-------
Chapter 3.  Key Statistical Concepts                                        Unified Guidance
3.5.1 FALSE POSITIVES AND TYPE  I ERRORS

     P± false positive or Type I error occurs whenever the null hypothesis [Ho]  is falsely rejected in
favor of the alternative hypothesis [//A]. What this means in terms of the underlying statistical models is
somewhat different for every test. Many of the tests in the Unified Guidance are designed to address the
basic groundwater detection monitoring framework, namely, whether the concentrations at downgradient
wells are significantly greater than background. In this case, the null hypothesis is that the background
and  downgradient wells  share the same underlying distribution  and that downgradient concentrations
should be consistent with background in the absence  of any contamination. The alternative hypothesis
presumes  that downgradient well  concentrations are  significantly  greater than background and come
from a distribution with an elevated concentration.

     Given this  formulation of HQ and HA,  a Type I error occurs whenever one decides that the
groundwater at downgradient locations is significantly higher than  background when in reality it is the
same in distribution. A judgment of this sort concerns the underlying statistical populations and not the
observed sample data. The measurements  at a downgradient well may indeed be higher than those
collected in background.  But the  disparity must be great enough  to decide  with confidence that the
underlying populations also differ. A proper statistical test must account for  not just the difference in
observed mean levels but also  variability in the data likely to be  present in the  underlying statistical
populations.

     False positive mistakes can cause regulated  facilities to incur substantial unnecessary costs and
oversight  agencies to  become unnecessarily  involved.  Consequently, there is  usually  a desire by
regulators and the regulated community alike to minimize the false positive rate (typically denoted by
the Greek letter a). For reasons that will become clear below, the false positive rate is inversely related
to the false negative rate for a fixed sample size n.  It is impossible to completely eliminate the  risk of
either Type I or  Type n errors, hence the regulatory mandate  to minimize the  inherent tradeoff by
maintaining a "reasonable balance" between false positives and false negatives.

     Type I errors  are strictly defined in terms  of the hypothesis  structure of the test. While the
conceptual groundwater detection monitoring framework assumes that false positive errors are incorrect
judgments  of a release when there is none,  Type I  errors  in other statistical tests  may  have  a very
different meaning.  For instance,  in  tests  of normality (Chapter  10) the null hypothesis is that the
underlying population is normally-distributed, while the alternative is that the population follows some
other, non-normal pattern. In this setting, a false positive represents the mistake of falsely  deciding the
population to be non-normal, when in fact it is normal  in distribution. The implication of such an error is
quite different, perhaps  leading  one to  select  an alternate test method or to needlessly attempt  a
normalizing transformation of the data.

     As a matter of terminology, the  false positive rate a is also known as the significance level of the
test. A test conducted at the a = .01 level of significance means there is  at most a 1% chance  or
probability that a  Type I error will occur in the results. The test is likely to lead to a false rejection of the
null hypothesis at most about 1 out of every  100 times the  same test is performed. Note that this last
statement says nothing about how well the test will work if HA is true, when HQ shouldbe rejected. The
                                              3-15                                    March 2009

-------
Chapter 3. Key Statistical Concepts	Unified Guidance

false positive rate strictly concerns those cases where HQ is an accurate reflection of the physical reality,
but the test rejects HO anyway.

3.5.2 SAMPLING DISTRIBUTIONS, CENTRAL LIMIT THEOREM

     The false positive rate of any statistical test can be calibrated to meet a given risk criterion. To see
how this is done, it helps to understand the concept of sampling distribution. Most statistical test
decisions are  based on the magnitude  of a  particular test  statistic computed from the sample data.
Sometimes the test statistic is relatively simple, such as the sample mean (x ), while in other instances
the statistic is  more complex and non-intuitive. In every case, however, the test statistic is formulated as
it is for a specific purpose: to enable the analyst to identify the distributional behavior of the test statistic
under the null hypothesis. Unless one knows the expected behavior of a test statistic, probabilities cannot
be assigned to specific outcomes for deciding when the probability is too low to be a chance fluctuation
of the data.

     The distribution of the test statistic is known as its sampling distribution. It is given a special
name, in part,  to distinguish the behavior of the test statistic from the potentially different distribution of
the individual observations or measurements used to calculate the  test. Once identified, the sampling
distribution can  be used to establish critical points of the test associated with specific  maximal false
positive rates  for any given a level  of significance.  For most tests, a single level of  significance is
generally chosen.

     An example  of this  idea can be illustrated via the F-test. It is used  for instance in parametric
analysis  of variance [ANOVA]  to  identify differences in the population means at  three or  more
monitoring wells.  Although ANOVA assumes that the individual measurements input  to the test are
normally-distributed, the test statistic under  a null hypothesis  [Ho]  of no differences between  the true
means follows an F-distribution. More specifically, it applies  to one member of the F-distribution family
(an example using 5 wells and 6 measurements per well is pictured in Figure 3-2). As seen in the right-
hand tail of this  distribution by summing the area under the  distributional curve,  large values of the F-
statistic become  less and less probable as they increase in magnitude. For a given  significance level (a),
there is a corresponding F-statistic value such that the probability of exceeding this cutoff value is a or
less. In such situations, there is at most an a  x 100% chance of observing an  F-statistic under HQ that is
as large or  larger than the cutoff (shaded area in Figure 3-2). If a is quite small  (e.g., 5% or 1%), one
may then judge  the null hypothesis to be an untenable model  and accept HA- As a consequence, the
cutoff value can be defined as an a-level critical point for the F-test.

     Because  test  statistics can be quite complicated, there is no easy rule for determining the sampling
distribution of a  particular test. However, the sampling behavior of some statistics is a consequence of a
fundamental result known as the  Central Limit Theorem. This theorem roughly states that averages or
sums  of identically-distributed random variables will follow  an approximate  normal  distribution,
regardless of the distributional behavior of the individual measurements.  This averaged distribution will
have the same mean \i as the population of individual measurements and whose variance, compared to
the underlying population variance a2 , is scaled by a factor of the sample size n on which the average or
sum is based.  Specifically, the variance is greater by  a factor of n in the case  of a sum(«-<72) and
smaller by a factor of n in the case of an average  (<72/w). The approximation  of the averages or sums to
the normal  distribution improves as sample size increases (also see the power discussion on page 3-21).

                                              3-16                                    March 2009

-------
Chapter 3. Key Statistical Concepts
Unified Guidance
                  Figyre 3-2. F-Distribution with 4 and 25 Degrees of Freedom
              *"".
              d
              if)
              d
          -Q
           a
           2
           a.
              eo
              d
              OJ
              d
              o ~
              o
              d -
                                             F-statistic value
     Because of the Central Limit Theorem, a number of test statistics at least approximately follow the
normal distribution.  This allows critical points for these tests to be  determined from a table of the
standard normal distribution. The Central Limit Theorem also explains why sample means provide a
better estimate of the true  population  mean than  individual  measurements  drawn  from the same
population (Figure 3-3). Since the sampling distribution of the mean is centered on the true average (|l)
of the underlying population and the variance is lower by a factor of n, the sample average x will tend to
be much closer to (I than a typical individual measurement.
                                             3-17
        March 2009

-------
Chapter 3. Key Statistical Concepts
Unified Guidance
                       Figure 3-3.  Effect of Central Limit Theorem
                                         Sampling Distribution of Mean
                      Underlying Population
3.5.3 FALSE NEGATIVES, TYPE II ERRORS, AND  STATISTICAL POWER

     False negatives or Type II errors are the logical opposites of false positive errors.  An error of this
type occurs whenever the null hypothesis [Ho] is accepted, but instead the alternative hypothesis [//A] is
true. The false negative rate is denoted by the Greek letter (3. In terms of the groundwater detection
monitoring  framework,  a Type n  error  represents a  mistake of judging the compliance  point
concentrations to be consistent with  background, when in reality the compliance point distribution is
higher on average. False negatives  in this context describe the risk of missing or not identifying
contaminated groundwater when  it really exists.  EPA has traditionally been more concerned with such
false negative errors, given its mandate to protect human health and the environment.

     Statistical power is an alternate way  of describing false negative  errors.  Power is merely the
complement of the false negative rate.  If (3 is the probability of a false negative, (l-(3) is the statistical
power of a particular test. In terms of the hypothesis structure, statistical power represents the probability
of correctly rejecting the null hypothesis. That is, it is the minimum chance that one will decide to accept
//A, given that H\ is true. High power translates into a greater probability of identifying contaminated
groundwater when it really exists.

     A convenient way to keep track of the differences between false positives, false negatives, and
power is via a Truth Table (Figure 3-4). A truth table distinguishes between the underlying truth of each
hypothesis HQ or H\ and the  decisions made on the basis of statistical testing.  If HQ is true, then a
decision to accept the alternative  hypothesis  (//A) is a false positive error  which will occur with a
                                              3-18
        March 2009

-------
Chapter 3. Key Statistical Concepts
                                    Unified Guidance
probability of at most a. Because only one of two decisions is possible, HQ will also be accepted with a
probability of at least (1-a). This is also known as the confidence probability or confidence level of the
test, associated with making a 'true negative' decision. Similarly if HA is actually true, making a false
negative decision error by accepting the null hypothesis (Ho) has at most a probability of p.  Correctly
accepting HA when true then has a probability of at least (1- P) and is labeled a 'true positive' decision.
This probability is also known as the statistical power of the test.

     For any application of a test to a particular sample, only one of the two types of decision errors can
occur. This is because only one of the two mutually exclusive hypotheses will be a true statement. In the
detection monitoring context, this  means that if a well is uncontaminated (i.e., HQ is true), it may be
possible to commit a Type I false positive mistake, but it is not possible to make a Type II false negative
error. Similarly, if a contaminated well is tested (i.e.,  HA is true), Type I false positive errors cannot
occur, but a Type  II false negative error might occur.
                      Figure 3-4. Truth Table in Hypothesis Testing

                                                DECISION

                                              H                    H
                      s
                      H
                      P
                            H
                                         OK
                                     (True

                                        (1-a)
II


(P)
                     I
               (False Positive)

                    (a)
                                                              OK
                                                          (True Positive}

                                                              (1-P)
     Since the false positive rate can be fixed in advance of running most statistical tests by selecting a,
one might think the same could be done with statistical power. Unfortunately, neither statistical power
nor the false negative rate can be fixed in advance for a number of reasons. One is that power and the
false negative rate depends on the degree to which the true  mean concentration level is elevated with
respect to the  background null condition. Large concentration increases  are easier to detect than small
increments.  In fact, power can be graphed as an increasing function of  the true concentration level in
what is termed & power curve (Figure 3-5).  A power curve  indicates the probability of rejecting HQ in
favor of the alternative HA for any given  alternative to the null hypothesis (i.e.,  for a range of possible
mean-level increases above background).
                                              3-19
                                            March 2009

-------
Chapter 3. Key Statistical Concepts
Unified Guidance
     In interpreting  the  power  curve  below, note that  the  x-axis is labeled  in  terms  of relative
background standard deviation units (o) above the true background population mean (u). The zero point
along the x-axis is associated with the background mean itself, while the kth positive unit along the axis
represents a 'true' mean concentration in the compliance well being tested equal toju + ka. This mode of
scaling the graph allows the same power curve to be potentially applied to any constituent of interest
subject to the same test conditions. This is true no matter what the typical background concentration
levels of a chemical typically found in groundwater may be. But it also means that the same point along
the power curve will represent different absolute concentrations for different constituents.  Even if the
background means are the same, a two standard deviation increase in a chemical with highly variable
background concentrations will correspond to a  larger population mean increase at a compliance well
than the same relative increase in a less variable constituent.

     As  a simple example, if the background  population averages for arsenic  and manganese both
happen to be 10 ppb, but the arsenic standard deviation is 5 ppb while that for manganese is only 2 ppb,
then a compliance well with a mean equivalent to a three  standard deviation increase over background
would have an average arsenic level of 25 ppb, but an average manganese level of only 16 ppb. For both
constituents, however,  there  would be  approximately a  50% probability of detecting a difference
between the compliance well and background.

                            Figure 3-5.  Example Power Curve
                           100
                        o
                        a.
                           »
                           40
                                     1234
                                       SDs Above Background
     Because the power probability depends on the relative difference between the actual downgradient
concentration level and background, power cannot typically be fixed ahead of time like the critical false
positive rate for a test.  The true concentration level (and  associated power) in  a compliance well is
unknown. If it were known, no hypothesis test would be needed.  Additionally, it is often not clear what
specific magnitude of increase  over background is environmentally significant.   A two  standard
deviation increase  over the background  average might  not be protective of human health and/or the
                                             3-20
        March 2009

-------
Chapter 3. Key Statistical Concepts
Unified Guidance
environment for some monitoring situations. For others, a four standard deviation increase or more may
be tolerable before any threat is posed.

     Since the exact ramifications of a particular concentration increase are uncertain, it points to the
difficulty in setting a minimum power requirement (or a maximum  false negative rate) for a given
statistical test.  Some  State statutes contain water quality non-degradation provisions, for which  any
measurable increase might be of concern. By emphasizing relative power as in Figure 3-5, all detection
monitoring constituents  can be evaluated for significant concentration  increases on a common footing,
subject only to differences in measurement variability.

Another key factor affecting statistical power is sample size. All other test conditions being equal, larger
sample sizes provide higher statistical power and the lower the false negative rate ((3).  Statistical tests
perform more accurately with larger data sets, leading to greater power and fewer errors in the process.
The Central Limit Theorem illustrates why  this is true. Even if a downgradient well mean level is only
slightly greater than background, upgradient and  downgradient well sample means will  have so little
variance in their  sampling distributions with enough measurements that they will tend  to hover very
close to their respective  population means.  True mean differences in the underlying populations can be
distinguished with higher probability as sample sizes increase. In Figure 3-6, the sampling distributions
of means of size 5 and 10 between two different normal populations are provided  for illustration. The
narrower width of the  distribution for the n = 10 sample means are more clearly distinguished from each
other than for means  of sample  size n = 5. This implies higher probability and power  to distinguish
between the two population means.

            Figure  3-6. Why Statistical  Power Increases with Sample Size
                          Sampling Distits of Mean (n =10)
                                                     Sampling Distils of
                                                     Mean (n = 5)
        Actual Populations
-I «
>1234S67S
Pi pi
> 19 11 12
                                              3-21
        March 2009

-------
Chapter 3. Key Statistical Concepts
                    Unified Guidance
3.5.4 BALANCING TYPE I AND TYPE II  ERRORS

     In maintaining an appropriate balance between false positive and false negative error rates, one
would ideally like to simultaneously minimize both kinds of errors.  However, both risks are inherent to
any statistical test procedure, and the risk of committing a Type I error is indirectly but inversely related
to the risk of a Type n error unless the sample  size can be increased. It is necessary to find a balance
between  the  two error rates.   But given that the false negative  rate depends largely  on the true
compliance point concentrations, it is first necessary to designate what specific mean difference (known
as an effect size} between  the  background and compliance  point populations  should be considered
environmentally  important.   A minimum power  requirement can  be  based on this difference (see
Chapter 6)

       ^EXAMPLE 3-1

     Consider a  simple example of using the downgradient sample mean to test the proposition that the
downgradient population  mean is 4 ppb larger than background.  Assume that extensive sampling has
demonstrated that the  background population mean is equal to 1 ppb. If the true downgradient mean
were the same as the  background level, curves of the two sampling distributions would  coincide (as
depicted  in Figure 3-7). Then a critical point (e.g., CP = 4.5  ppb) can be selected so that  the risk of a
false positive mistake is a. The critical  point establishes the decision criteria for the test. If the observed
sample mean based on randomly selected data from the downgradient sampling distribution exceeds the
critical point, the downgradient population will be declared higher in concentration than the  background,
even though this  is not the case. The frequency that such a wrong decision will be made is  just the area
under the sampling distribution to the right of the critical point  equal to a.

         Figure 3-7.  Relationship Between Type I and Type II Errors, Part  A
                         : p = I ppb
                       HA ; fi = 5 ppb
Under H(j;
Background and Compliance Point
Populations Completely Overlap
                                                     CP  5
                         CP = Critical Point
     If the true downgradient mean is actually 5 ppb, the sampling distribution of the mean will instead
be centered over 5 ppb  as in the right-hand curve (i.e., the downgradient population) in Figure 3-8.
Since there really is a difference between the two populations, the alternative hypothesis and not the null
                                             3-22
                            March 2009

-------
Chapter 3. Key Statistical Concepts
Unified Guidance
hypothesis is true. Thus,  any observed sample  mean drawn from  the downgradient population  then
falling below the critical point is a false negative mistake.  Consequently, the area under the right-hand
sampling distribution in Figure 3-8 to the left of the critical point represents the frequency of Type II
errors ((3).

     The false negative rate ((3) in Figure 3-8 is obviously larger than  the false positive rate (a) of
Figure 3-7. This need not be the  case in general, but the key point is to understand that for a fixed
sample size, the Type I and Type II error rates cannot be simultaneously minimized. If a is increased, by
selecting a lower critical point in Figure 3-7,  the false negative rate will also be lowered in Figure 3-8.
Likewise, if a is decreased by selecting a higher critical point,  (3 will be enlarged. If the false positive
rate is indiscriminately lowered, the false negative rate (or reduced power)  will likely reach unacceptable
levels  even for mean  concentration levels of environmental importance. Such  reasoning lay behind
EPA's decision to mandate minimum false positive rates for Mests and ANOVA procedures in both the
revised 1988 and 1991 RCRA rules.

          Figure 3-8. Relationship Between Type  I  and Type II  Errors, Part B
                                                       Under HA:
                                                       Background and Compliance Point
                                                       Populations Differ
                   Background Population
                          CP = Critical Point
                                              3-23
        March 2009

-------
Chapter 3.  Key Statistical Concepts                                   Unified Guidance
                     This page intentionally left blank
                                        3-24                                March 2009

-------
Chapter 4.  Groundwater Monitoring Programs                           Unified Guidance

  CHAPTER 4.   GROUNDWATER MONITORING PROGRAMS
                     AND  STATISTICAL ANALYSIS
       4.1   THE GROUNDWATER MONITORING CONTEXT	4-1
       4.2   RCRA GROUNDWATER MONITORING PROGRAMS	4-3
       4.3   STATISTICAL SIGNIFICANCE IN GROUNDWATER TESTING	4-6
         4.3.1  Statistical Factors	4-8
         4.3.2  Well System Design and Sampling Factors	4-8
         4.3.3  Hydrological Factors	4-9
         4.3.4  Geochemical Factors	4-10
         4.3.5  Analytical Factors	4-10
         4.3.6  Data or Analytic Errors	4-11
     This chapter provides an overview of the basic groundwater monitoring framework, explaining the
intent of the federal groundwater statistical regulations and offering insight into the key identification
mechanism of groundwater monitoring, the statistically significant increase [SSI]:

   »«»  What are statistically significant increases and how should they be interpreted?
   »«»  What factors, both statistical and non-statistical can cause SSIs?
   »«»  What factors should be considered when demonstrating that an SSI does not represent evidence
       of actual contamination?

4.1 THE GROUNDWATER MONITORING CONTEXT

     The RCRA regulations frame a  consistent approach to groundwater monitoring, defining  the
conditions under which  statistical  testing takes place. Upgradient and  downgradient wells must be
installed to monitor the uppermost aquifer in order to identify releases or changes in existing conditions
as expeditiously as possible.  Geological  and hydrological expertise is  needed to properly locate  the
monitoring wells in the aquifer passing beneath the monitored unit(s). The regulations identify a variety
of design and sampling requirements for groundwater monitoring (such as measuring well piezometric
surfaces and identifying flow directions) to assure that this basic goal is achieved. Indicator or hazardous
constituents are measured in these wells at regular time intervals; these sample data serve as the basis for
statistical comparisons. For identifying releases under detection monitoring, the regulations generally
presume  comparisons of observations from downgradient wells against those  from upgradient wells
(designated as  background). The rules also recognize certain situations  (e.g.,  mounding  effects) when
other means to define background may be necessary.

     The Unified Guidance may apply to facility groundwater monitoring programs straddling a wide
range of conditions. In addition to units regulated under Parts 264 and 265 Subpart F and Part 258 solid
waste landfills, other non-regulated units  at Subtitle C facilities or CERCLA sites  may utilize similar
programs. Monitoring can vary from a regulatory minimum of one upgradient and three  downgradient
wells, to very  large facilities with multiple units, and perhaps 50-200 upgradient  and  downgradient
wells. Although the rules presume that monitoring will occur in the  single uppermost aquifer likely to be
affected by a release, complex geologic conditions may require sampling and evaluating a number of
aquifers or strata.
                                            4^1                                   March 2009

-------
Chapter 4. Groundwater Monitoring Programs                             Unified Guidance

     Detection monitoring constituents may include indicators  like common ions and other general
measures  of water  quality, pH, specific conductance, total organic carbon [TOC] and total organic
halides  [TOX].  Quite often, well monitoring data  sets are  obtained  for filtered or unfiltered trace
elements (or both) and sizeable suites of hazardous trace organic constituents, including volatiles, semi-
volatiles, and pesticide/herbicides. Measurement and analysis of hazardous constituents using standard
methods (in SW-846  or elsewhere) have become fairly routine over time. A large number of analytes
may be potentially available as monitoring constituents for statistical testing, perhaps  50-100 or more.
Identification  of the  most  appropriate  constituents  for  testing depends  to  a great extent on  the
composition of the managed wastes (or their decomposition products) as measured in leachate analyses,
soil gas sampling, or from prior knowledge.

     Nationally,  enough groundwater monitoring  experience  has  been gained in using routine
constituent  lists and  analytical techniques to suggest some common  underlying patterns.  This is
particularly true  when defining background conditions in groundwater.  Sampling frequencies have also
been standardized enough (e.g., semi-annual or quarterly sampling) to enable reasonable computation of
the sorts of sample sizes  that can be used for statistical testing. Nevertheless, complications can and do
occur over time — in the form of changes in  laboratories, analytical methods, sampled wells,  and
sampling frequencies — which can affect the quality and availability of sample data.

     Facility status  can also affect what data are potentially available for evaluation and testing — from
lengthy regulated unit monitoring records under the Part 265 interim status requirements at sites awaiting
either operational or post-closure 264 permits or permit re-issuance, to a new solid waste facility located
in a zone  of  uncontaminated  groundwater with little  prior  data.  Some combined RCRA/CERCLA
facilities  may  have  collected  groundwater information  under  differing   program  requirements.
Contamination from offsite or non-regulated units (or  solid waste management units) may complicate
assessment of likely contaminant sources or contributions.

     Quite  often, regulators  and regulated parties find themselves  with considerable  amounts of
historical constituent-well monitoring data that must be assessed for appropriate action, such as a permit,
closure, remedial action  or enforcement decision. Users will  need to  closely consider the diagnostic
procedures in Part II  of the Unified Guidance, with an eye towards selection of one or more appropriate
statistical tests in Parts III and IV. Selection will depend on key factors such as the number  of wells  and
constituents, statistical characteristics of the observed data, and historical patterns of contamination (if
present), and  may  also  reflect preferences for  certain types of tests. While the Unified Guidance
purposely identifies  a  range of tests which might fit a situation, it is generally recommended that one set
of tests be selected for final implementation, in order to avoid "test-shopping" (i.e., selecting tests during
permit implementation based on the most favorable  outcomes).  EPA recognizes that the  final permit
requirements are approved by the regulatory agency.

     All of the above situations share some features in  common. A certain number of facility wells will
be designated as compliance points, i.e., those locations  considered as  significant from a regulatory
standpoint for assessing potential releases.  Similarly, the most appropriate and critical indicator and/or
hazardous constituents for monitoring will be identified. If  detection monitoring (i.e., comparative
evaluations of compliance wells against background) is deemed  appropriate for some or all wells  and
constituents, definitions  of background or reference comparison levels  will  need to be  established.
Background data can be obtained  either from the upgradient wells or  from the  historical  sampling
database as described  in Chapter 5. Choice of background will depend on how statistically comparable
                                              4^2                                     March 2009

-------
Chapter 4. Groundwater Monitoring Programs                             Unified Guidance

the compliance point data are with respect to background and whether individual  constituents exhibit
spatial or temporal variability at the facility.

     Compliance/assessment or corrective action monitoring may be appropriate choices when there is a
prior or historical indication of hazardous constituent releases from a regulated unit. In those situations,
the regulatory agency will establish GWPS limits. Typically, these limits are found in established tables,
in SDWA drinking water MCLs, through risk-based calculations or determined from background data.
For remedial actions,  site-specific levels may be developed which account  not only  for risk, but
achievability and implementation costs as well. Nationally, considerable experience has been gathered in
identifying cleanup targets which might be applicable at a given facility, as well as how practical those
targets are likely to be.

     Use  of the Unified Guidance should thus be viewed in an overall context.  While  the guidance
offers important considerations and suggestions in selecting and designing a statistically-based approach
to monitoring, it is important to realize that it is only a part of the overall decision  process at a facility.
Geologic and hydrologic expertise, risk-based decisions, and legal and practical considerations by the
regulated  entity  and  regulatory agency  are  fundamental  in developing   the final   design  and
implementation. The guidance does not attempt to address the many other relevant decisions which
impact the full design of a monitoring system.

4.2  RCRA  GROUNDWATER MONITORING PROGRAMS

     Under the RCRA regulations,  some form of statistical testing of sample data  will generally be
needed to determine whether there has been a release, and if so, whether concentration levels lie below
or  above  a  protection  standard.   The  regulations  frame  the   testing programs  as  detection,
compliance/assessment, and corrective action monitoring.

     Under RCRA permit  development and during routine evaluations, all three monitoring program
options may need to be simultaneously considered. Where sufficient hazardous constituent data from site
monitoring or other evidence of a release exists, the regulatory agency can evaluate which monitoring
program(s) are  appropriate under §264.91. Statistical principles and testing provided in the  Unified
Guidance can be used to develop presumptive evidence for one program over another.

     In some applications, more than one monitoring program may be appropriate.  Both the number of
wells  and constituents to be tested can vary among the three monitoring programs at  a given site.  The
types  of non-hazardous indicator constituents used for detection monitoring might not be applied  in
compliance or corrective action monitoring. The latter focus is on hazardous constituents. Only a few
compliance well constituents may exceed their respective GWPSs.  The  focus in a corrective action
monitoring program might then be placed on the latter, with the remaining well constituents evaluated
under the other  monitoring schemes. But following the general regulatory structure, the three monitoring
systems are presented below and elsewhere in the guidance as an ordered sequence:

      Detection monitoring  is appropriate either when there is  no  evidence of  a release from a
regulated unit, or when the unit situated in a historically contaminated area is  not impacted by current
RCRA waste management  practices. Care must be taken to avoid  a situation where the constituents
might reasonably have originated offsite or from units not subject to testing, since any  adverse change in
groundwater quality would be attributed to on-site causes. Whether an observed change in groundwater

                                              4^3                                     March 2009

-------
Chapter 4. Groundwater Monitoring Programs                             Unified Guidance

quality is in fact due to a release from on-site waste activities at the facility may be open to dispute
and/or further demonstration. However, this basic framework underlies each of the statistical methods
used in detection monitoring.

     A crucial step in  setting up a detection monitoring program is to establish a set of background
measurements, a baseline or reference level for statistical  comparisons (see Chapter 5). Groundwater
samples  from  compliance wells are then compared against  this baseline to measure  changes in
groundwater  quality. If at least  one chemical parameter on the monitoring  indicates a statistically
significant increase above the baseline [SSI, see  Section 4.3], the facility or regulated unit moves into
the next phase: compliance or assessment monitoring.

       Compliance or assessment monitoring1 is appropriate when there is reliable statistical evidence
that  a concentration increase over the baseline has occurred.  The purpose of compliance/assessment
monitoring is two-fold: 1) to assess the extent  of contamination (i.e., the size of the increase, the
chemical parameters involved, and the locations on-site where contamination is  evident); and 2) to
measure  compliance with pre-established  numerical  concentration  limits generally  referred  to as
GWPSs.  Only the  second purpose is fully addressed using formal statistical tests. While important
information can be gleaned  from  compliance well  data,  more complex analyses (e.g.,  contaminant
modeling) may be needed to address the first goal.

     GWPSs  can be fixed health- or risk-based limits, against which  single-sample tests are made. At
some sites, no specific  fixed concentration  limit  may be assigned or readily available for one or more
monitoring parameters. Instead, the comparison is made against a limit developed from background data.
In this case,  an appropriate statistical approach might be to  use  the background measurements to
compute a statistical limit and set it as  the GWPS.  See Chapter  7 for further  details.  Many of the
detection monitoring design principles (Chapter 6) and statistical tests (Part III) can also be applied to
a set of constituents defined by a background-type GWPS.

     The RCRA Parts  264 and  258 regulations require an expanded analysis of potential hazardous
constituents (Part 258 Appendix n for municipal landfills or Part 264 Appendix IX for hazardous waste
units) when detection monitoring  indicates a release and compliance monitoring is potentially triggered.
The purpose is to better gauge which hazardous constituents have actually impacted groundwater. Some
detection monitoring programs may require  only limited testing of indicator parameters. This additional
sampling can be used to determine which wells have been impacted and provide some understanding of
the on-site distribution  of hazardous constituent concentrations  in groundwater. . The course of action
decided by the Regional Administrator or State Director will depend on the number of such chemicals
that are present in quantifiable levels and the actual concentration levels.
  The terms compliance monitoring (§264.99 & 100) and assessment monitoring (§258.55 & 56) are used interchangeably in
  this document to refer to RCRA monitoring programs. Compliance monitoring is generally used for permitted hazardous
  waste facilities under RCRA Subtitle C, while assessment monitoring is applied to municipal solid waste landfills regulated
  under RCRA Subtitle D. The term "assessment" is also used in 40 CFR 265 Subpart F for a second phase of additional
  analyte testing. Occasional use is also made of the term "compliance wells," which refers to downgradient monitoring wells
  located at the point(s) of compliance under §264.95 (any of the three monitoring programs may apply when evaluating
  these wells).

                                               4^4                                     March 2009

-------
Chapter 4. Groundwater Monitoring Programs                            Unified Guidance

     Following the occurrence of a valid statistically  significant increase [SSI] over baseline during
detection monitoring, the statistical presumption in compliance/assessment monitoring is quite similar to
the detection stage. Given G as a fixed compliance or background-derived GWPS, the null hypothesis is
that true concentrations (of the underlying compliance point population) are no greater than G. This
compares to the detection monitoring presumption that  concentration levels do not exceed background.
One reason for the similarity is that compliance limits  may be higher than background levels in  some
situations. An  increase over background in these situations does not necessarily imply an increase over
the compliance limit, and the latter must be formally tested. On the other hand, if a health- or risk-based
limit is below a background level, the RCRA regulations provide that the GWPS should be based on
background.
     	                                                                            9
     The Subtitle D regulations for municipal solid waste landfills [MSWLF] stipulate  that if "the
concentrations of all Appendix II constituents are shown to be at or below background values, using the
statistical procedures in §258.53(g), for two consecutive sampling events, the owner or operator... may
return  to detection  monitoring." In other words, assessment monitoring may be  exited  in  favor of
detection monitoring when concentrations at the compliance wells are statistically indistinguishable from
background for two consecutive sampling periods. While a demonstration that concentration levels are
below  background would generally not be realistic, it may be possible to show that compliance  point
levels of contaminants  do not exceed an upper limit computed from the background data. Conformance
to the limit would then indicate  an  inability to  statistically distinguish between background and
compliance point  concentration levels.

     If a hazardous constituent under compliance  or assessment monitoring statistically exceeds a
GWPS, the facility is  subject to corrective action.  Remedial activities must be undertaken to remove
and/or prevent the further  spread  of contamination  into groundwater.  Monitoring under corrective
action  is used to track the progress of remedial activities and to determine if the facility has returned to
compliance. Corrective action is usually preceded or accompanied by  a formal Remedial Investigation
[RI] or RCRA Facility  Investigation [RFI] to further delineate  the nature and extent of the contaminated
plume.  Corrective action may be confined to a single regulated unit if only that unit exhibits SSIs above
a standard during  the detection and  compliance/assessment monitoring phases.

     Often, clean-up  levels are established by the  Regional Administrator or State  Director during
corrective action.  Remediation must continue until these clean-up levels are met. The focus of remedial
action  and monitoring would be  on those hazardous  constituents  and well locations exceeding the
GWPSs. If specific clean-up levels have not been met, corrective action must continue until there is
evidence of a statistically significant decrease [SSD] below the compliance limit for three  consecutive
years.  At this point,  corrective action may be exited and  compliance monitoring re-started.   (As
described above  and in  Chapter  7, the protocol  for  assessing corrective action compliance with a
background-type  standard can differ).  If subsequent concentrations are statistically indistinguishable
from background  or no detectable concentrations can be demonstrated for three consecutive  years in any
of the contaminants that triggered corrective measures in the first place, corrective action may be exited
in favor of detection monitoring.
2 [56 FR 51016] October 9, 1991
                                              4-5                                    March 2009

-------
Chapter 4. Groundwater Monitoring Programs                            Unified Guidance

4.3  STATISTICAL SIGNIFICANCE IN GROUNDWATER TESTING

     The outcome of any statistical test is judged either to be statistically significant or non-significant.
In groundwater monitoring, a valid statistically significant result can force a change in the monitoring
program, perhaps even leading to remedial  activity. Consequently, it is important to understand what
statistically significant results represent and what they do not. In the language of groundwater hypothesis
testing (Chapter 3),  a statistically significant test result is a decision to reject the null hypothesis (Ho)
and to accept the alternative hypothesis (//A), based on the observed pattern of the sample data. At the
most elementary level, a statistically significant increase [SSI] (the kind  of result typically of interest
under RCRA detection and compliance monitoring) represents an observed increase in concentration at
one or more compliance wells. In order to be declared an SSI, the change in concentration must be large
enough after accounting for variability in the sample data, that the result is unlikely  to have occurred
merely by chance. What constitutes a statistically  significant result depends on the phase  of monitoring
and the type of statistical test being employed.

     If the detection monitoring statistical test being used is a t-test  or Wilcoxon rank-sum test
(Chapter 16), an SSI occurs whenever the ^-statistic or  fF-statistic is larger than an a-level critical point
for the test. If a retesting procedure is chosen using a prediction limit (Chapter 19), an SSI occurs only
when both  the initial compliance sample or  initial mean/median and one or more resamples all exceed
the upper prediction limit. For control charts (Chapter 20), an SSI occurs whenever either the CUSUM
or Shewhart portions of the chart exceed their respective control limits. In another variation, an SSI only
occurs if  one or another of the CUSUM or Shewhart statistics  exceeds the control limits when
recomputed using one or more resamples.  For tests of trend (Chapter 17), an SSI is declared whenever
the slope is significantly greater than zero at some significance level a.

     In compliance/assessment monitoring, tests are often made against a fixed compliance limit or
GWPS. In this setting, one can utilize a confidence interval around a mean, median, upper percentile or a
trend  line (Chapter  21). A confidence interval is an  estimated concentration or measurement range
intended to contain a given statistical characteristic of the population from which the sample is drawn. A
most common formulation is  a two-way confidence interval around a normally-distributed mean |i, as
shown below:


                          (x-Wi-7=   *   f*    *   * + Wi-7=]                       t4-1]
                          ^          V»                        -Jn)

where  x is the mean of a sample of size n,  s is the sample standard  deviation, and t\_^ n_\ is an upper
percentile selected from a Student's ^-distribution.  By constructing a range around the sample mean (x ),
this confidence interval is  designed to locate the  true population  mean (u) with  a high degree  of
statistical  confldence(l-2a) or conversely, with a low probability of error (2a).  If  a one-way lower
confidence interval is used, the right-hand term in  equation [4.1] would be replaced by  +00 at confidence
level 1-a.  In a similar fashion, the upper 1-a confidence interval would be defined in the range from -co
for the left-hand term to the right hand term in equation [4.1].

     When using a lower confidence interval on the mean, median,  or upper percentile,  an SSI occurs
whenever the lower edge of the confidence interval range exceeds the GWPS. For a confidence interval
around a trend line, an SSI is  declared whenever the lower confidence limit around the estimated trend

                                              4^6                                    March 2009

-------
Chapter 4. Groundwater Monitoring Programs
Unified Guidance
line first exceeds the GWPS at some point in time. By requiring that a lower confidence limit be used as
the basis of comparison, the statistical test will account for data variability  and ensure that the apparent
violation is unlikely to have occurred by chance.  Figure 4-1 below visually depicts a comparison to a
fixed GWPS for both lower confidence intervals for a stationary test like a  mean, and around  an
increasing trend. Where the confidence interval straddles the limit, the test results are inconclusive.  In
similar fashion, an SSD can be identified by using upper confidence intervals.

     Figure 4-1. Confidence Intervals Around  Means, Percentiles, or Trend Lines
                  Means, Percentiles
                                                                            GWPS
                                                                   Out-of-Compliance
                                      Time
                                                                            GWPS
               Increasing Trend
                                      Time
     SSIs offer the primary statistical justification for moving from detection monitoring to compliance
monitoring, or from compliance/assessment monitoring to corrective action.  However, it is important
that an SSI be  interpreted correctly. Any SSI at a compliance well represents a probable increase in
concentration level,  but it does not automatically imply or prove that contaminated groundwater from
the facility is the cause of the increase. Due to the complexities of the groundwater medium and the
nature of statistical testing, there are numerous reasons why a test may exhibit a statistically significant
result. These may or may not be indications of an actual release from a regulated unit.
                                             4-7
        March 2009

-------
Chapter 4. Groundwater Monitoring Programs                            Unified Guidance

     It is always reasonable to allow for a separate demonstration once an SSI occurs, to determine
whether or not the increase is actually due to a contaminant release. Such a demonstration will rely
heavily on hydrological and geochemical  evidence from the site, but could include additional  statistical
factors. Key questions and factors to consider are listed in the following sections.

4.3.1 STATISTICAL FACTORS

    *»*  Is  the result a false positive?  That is, were the data tested simply an unusual sample  of the
       underlying population triggering an SSI? Generally, this can be evaluated with repeat sampling.
    »»»  Did the test correctly identify an actual release of an indicator or hazardous constituent?
    »»»  Are there corresponding SSIs in upgradient or background wells? If so, there may be evidence of
       a natural in-situ concentration increase, or perhaps migration from an off-site source.
    *»*  Is  there  evidence  of significant  concentration  differences between separate upgradient or
       background  wells, particularly for inorganic constituents? If so,  there may  be natural spatial
       variations between distinct  well  locations that  have not been  accounted for. These spatial
       differences could  be local or systematic  (e.g..,  upgradient  wells in one formation  or zone;
       downgradient wells in another).
    *»*  Could observed SSIs for naturally occurring analytes be due to longer-term (i.e.,  seasonal or
       multi-year) variation? Seasonal or other cyclical  patterns should be observable in upgradient
       wells. Is this change occurring in both upgradient and  downgradient wells?  Depending on the
       statistical test and frequency of sampling involved, an observed SSI may be entirely due to
       temporal variation not accounted for in the sampling scheme.
    *»*  Do time series plots of the sampling data show  parallel "spikes" in concentration  levels from
       both background  and compliance well  samples  that were analyzed  at about the  same time?
       Perhaps there was an analytical problem or change in lab methodology.
    »»»  Are  there substantial correlations  among  within-well  constituents  (in  both upgradient and
       downgradient wells)? Highly correlated analytes treated as independent monitoring constituents,
       may generate incorrect significance levels for individual tests.
    *»*  Were trends properly accounted for, particularly in the background data?
    *»*  Was a  correct assumption made  concerning the underlying  distribution  from  which the
       observations were drawn (e.g., was a normal assumption applied to lognormal data)?
    »»»  Was the test computed correctly?
    »»»  Were the data input to the test of poor quality? (see various factors below)
4.3.2 WELL SYSTEM  DESIGN AND SAMPLING FACTORS

    »»»  Were early  sample data following  well installation utilized in statistical testing?  Initial well
       measurements are  sometimes highly variable during a 'break in' sampling and analysis period
       and potentially less trustworthy.
    »»»  Was there an effect attributable to recent well development, perhaps due to the use of hazardous
       constituent chemicals during development or present in drilling muds?
    »»»  Are there multiple geological formations at the site, leading to incorrect well placements?

                                              4^8                                    March 2009

-------
Chapter 4. Groundwater Monitoring Programs                             Unified Guidance

   *»*  Has there been degradation of the well casings and screens (e.g., PVC pipe)? Deteriorating PVC
       materials can release organic constituents under certain conditions. Occasionally, even stainless
       steel can corrode and release a number of metallic trace elements.
   *»*  Have there been changes in well performance over time?
   *»*  Were there excessive holding times or incorrect use of preservatives, cooling, etc.
   *»*  Was there incorrect calibration  or  drift in the field  instrumentation? This effect should be
       observable in both upgradient and downgradient data and possibly over a number of sample
       events. The data itself may be compromised or useless.
   *»*  Have there been  'mid-stream' changes in sampling procedures, e.g., increased or decreased well
       purging?  Have sampling or purging techniques been consistently applied from well to well or
       from sampling event to sampling event?

4.3.3 HYDROLOGICAL FACTORS

   *»*  Does the site have a history of previous waste management activity (perhaps prior to RCRA), and
       is there any evidence of historical groundwater contamination? Previous contamination or waste
       management contaminant levels can limit the ability to distinguish releases from the regulated
       unit, particularly  for those analytes found in historical contamination.
   »»»  Is there evidence of groundwater mounding or other anomalies that could lead to the lack of a
       reliable, definable  gradient? Interwell  statistical tests  assume that  changes  in downgradient
       groundwater quality only affect  compliance wells and  not upgradient (background)  wells.
       Changes that impact background wells also, perhaps in a complex manner involving seasonal
       fluctuations,  are often best resolved by running intrawell tests instead.
   *»*  Is there hydrologic evidence of any migration of contaminants (including DNAPL) from off-site
       sources or from other non-regulated units? Are any of these contaminants observed upgradient of
       the regulated units?
   »»»  Have there been other prior human or site-related  waste management activities which  could
       result in  the observed SSI changes for certain well  locations (e.g.,  buried  waste  materials,
       pipeline leaks, spills, etc.)?
   »»»  Have there been  unusual changes in groundwater directions and depths? Is there confidence that
       the  SSI did indeed  correspond  to  a potential  unit release based on observed groundwater
       directions, distance of the well from the unit, other well information, etc.?
   *»*  Is there evidence of migration of landfill gas affecting one or more wells?
   »»»  Have there been increases in well  turbidity and  sedimentation,  which  could  affect observed
       contaminant  levels?
   »»»  Are there preferential flow paths in the aquifer that could affect where contaminants are likely to
       be observed or not observed?
   »»»  Are the detected contaminants consistent with those found in the waste  or  leachate  of the
       regulated unit?
   »»»  Are there other nearby well pumping or extraction activities?


                                              4^9                                    March 2009

-------
Chapter 4. Groundwater Monitoring Programs                            Unified Guidance

4.3.4 GEOCHEMICAL FACTORS

   *»*  Were the measurements that triggered the SSI developed from unfiltered or filtered trace element
       sample data? If unfiltered,  is there any information regarding associated  turbidity  or  total
       suspended  solid measurements?  Unusual increases in well turbidity  can  introduce excess
       naturally occurring trace elements into the samples. This can be a particularly difficult problem in
       compliance monitoring when comparing data to a fixed standard, but can also affect detection
       monitoring well-to-well comparisons if turbidity levels vary.
   »»»  Were  there changes  in  associated  analytes at the  "triggered"  well  consistent  with  local
       geochemistry? For  example, given an SSI  for total dissolved  solids  [TDS], did measured
       cations/anions and  pH also show a consistent  change?  As another  example,  slight  natural
       geochemical changes can result in large specific conductance changes. Did  other constituents
       demonstrate a consistent change?
   »»»  Is there evidence  of  a simultaneous  release of more than one analyte, consistent with the
       composition of the waste or leachate? In particular, is there corollary evidence of degradation or
       daughter products for constituents like halogenated organics? For groundwater constituents with
       identified  SSIs,  is there a probable relationship to measured concentrations in waste or waste
       leachate? Are leachate concentrations high enough to be detectable in groundwater?
   »»»  If an  SSI is  observed in  one or more naturally occurring species, were organic  hazardous
       constituents not normally present in background  and found in  the  waste or  leachate also
       detected?  This could be an important factor in assessing the source of the possible release.
   »»»  Have  aquifer mobility factors  been  considered?  Certain  soluble constituents  like  sodium,
       chloride, or conservative volatile  organics might be expected to move through the aquifer much
       more  quickly than easily adsorbed  heavy  metals  or 4-5  ring polynuclear  aromatic [PNA]
       compounds.
   *»*  Do  the observed  data patterns (particularly  for naturally occurring constituents  in upgradient
       wells  or  other  background  conditions) make  sense in  an overall site geochemical  context,
       especially as compared with  other available local or  regional site data and published  studies? If
       not, suspect background data may need to be further evaluated for potential errors prior to formal
       statistical comparisons.
   *»*  Do  constituents exhibit correlated behavior among both upgradient and downgradient wells due
       to overall changes in the aquifer?
   »»»  Have there been natural changes in groundwater constituents over time and space due to multi-
       year, seasonal, or cyclical variation?
   »»»  Are there different geochemical regimes in upgradient vs. downgradient wells?
   *»*  Has there been a release of soil trace elements due to changes in pH?

4.3.5 ANALYTICAL  FACTORS

   *»*  Have  there been  changes  in laboratories, analytical methods, instrumentation, or procedures
       including specified detection limits that could cause  apparent jumps in concentration levels? In
       some  circumstances, using different values for  non-detects with different reporting limits has
       triggered  SSIs. Were inexperienced technicians involved in any of the analyses?

                                             4^10March  2009

-------
Chapter 4. Groundwater Monitoring Programs                            Unified Guidance

   *»*  Was more than one analytical  method used  (at different points in time) to generate the
       measurements?
   »»»  Were there changes in detection/quantification limits for the same constituents?
   »»»  Were there calibration problems, e.g.., drift in instrumentation?
   »»»  Was solvent or other laboratory contamination (e.g.,  phthalates, methylene chloride extractant,
       acetone wash) introduced into any of the physical samples?
   *»*  Were there known or probable interferences among the analytes being measured?
   *»*  Were there  "spikes" or unusually high values on  certain sampling  events  (either for one
       constituent among many wells or related analytical constituents) that would  suggest laboratory
       error?

4.3.6 DATA OR ANALYTIC ERRORS

   *»*  Were there data transcription  errors (incorrect  decimal  places, analyte units,  or data column
       entries)? These data can often be identified as being highly improbable.
   »»»  Were there  calculation  errors in either the analytical (e.g.., incorrect trace element valence
       assumptions or  dilution  factors) or in the statistical portions (mathematical mistakes, incorrect
       equation terms)  of the analysis?
                                             4-11                                    March 2009

-------
Chapter 4. Groundwater Monitoring Programs                         Unified Guidance
                    This page intentionally left blank
                                       4-12                              March 2009

-------
Chapter 5.  Background	Unified Guidance

          CHAPTER 5.   ESTABLISHING AND  UPDATING
                                BACKGROUND
       5.1   IMPORTANCE OF BACKGROUND	5-1
         5.1.1  Tracking Natural Groundwater Conditions	5-2
       5.2   ESTABLISHING AND REVIEWING BACKGROUND	5-2
         5.2.1  Selecting Monitoring Constituents and Adequate Sample Sizes	5-2
         5.2.2  Basic Assumptions About Background	5-4
         5.2.3  Outliers in Background	5-5
         5.2.4  Impact of Spatial Variability	5-6
         5.2.5  Trends in Background	5-7
         5.2.6  Expanding Initial Background Sample Sizes	5-8
         5.2.7  Review of Background.	5-10
       5.3   UPDATING BACKGROUND	5-12
         5.3.1  When to Update	5-12
         5.3.2  How to Update	5-12
         5.3.3  Impact of Retesting	5-14
         5.3.4  Updating When Trends are Apparent	5-14
     This chapter discusses the importance and use of background data in groundwater monitoring.
Guidance is provided for the proper identification, review, and periodic updating of background. Key
questions to be addressed include:

   »«»  How should background be established and defined?
   »«»  When should existing background data sets be reviewed?
   »«»  How and when should background be updated?
   »«»  What impact does retesting have on background updating?


5.1 IMPORTANCE OF BACKGROUND

     High  quality background  data  is the  single most  important  key to a  successful  statistical
groundwater monitoring program, especially for detection monitoring. All of the statistical tests listed in
the RCRA regulations are  predicated  on  having  appropriate  and representative  background
measurements. As indicated in Chapter 3, a statistical sample is representative if the distribution of the
sample measurements best  follows the distribution of the population from which the sample is drawn.
Representative background data has a similar but slightly  different connotation. The most  important
quality of background is that it reflects the historical conditions unaffected by the activities it is designed
to be compared to.  These conditions could range from an uncontaminated aquifer to an historically
contaminated site baseline unaffected by recent RCRA-actionable contaminant releases. Representative
background data will therefore have numerical characteristics closely matching those arising from the
site-specific aquifer being evaluated.

     Background must also be appropriate  to the statistical test.  All RCRA detection monitoring tests
involve comparisons of compliance point data against background.  If natural groundwater conditions
                                             
-------
Chapter 5.  Background	Unified Guidance

have changed over time — perhaps due to cycles of drought and recharge — background measurements
from five or ten years  ago may  not  reflect current  uncontaminated conditions.   Similarly,  recent
background data obtained using improved analytical methods may not be comparable to older data. In
each case, older background data may  have to be discarded in favor of more recent measurements in
order to construct an appropriate comparison. If intrawell tests are utilized due to strong evidence of
spatial  variability, traditional upgradient well  background data will not provide  an appropriate
comparison.  Even if the  upgradient measurements are reflective  of uncontaminated  groundwater,
appropriate background data must be obtained from each compliance point well.  The main point is that
compliance samples should be tested against data which best can represent background  conditions now
and those likely to occur in the future.

5.1.1  TRACKING NATURAL GROUNDWATER CONDITIONS

     Background measurements, especially from upgradient wells, can provide essential  information for
other than formal statistical testing. For one, background data can be used to gauge mean levels and
develop estimates of variability in naturally occurring groundwater constituents. They can also be used
to confirm the presence or absence of anthropogenic or non-naturally occurring constituents in the site
aquifer.  Ongoing sampling of  upgradient background  wells  provides a means of tracking  natural
groundwater conditions.  Changes that occur in parallel between the  compliance point and background
wells may signal site-wide aquifer changes in groundwater quality not specifically attributable to onsite
waste management.  Such observed changes may also  be indicative  of analytical  problems  due to
common artifacts of laboratory analysis (e.g., re-calibration of lab equipment,  errors in batch  sample
handling, etc), as well as indications of groundwater mounding, changes in groundwater gradients and
direction, migration of contaminants from other locations or offsite, etc.

     Fixed  GWPS  like  maximum   contaminant  levels   [MCLs]   may  be contemplated  for
compliance/assessment monitoring or corrective action. Background data analysis is important  if it is
suspected that naturally occurring levels of the constituent(s) in question are higher than the standards or
if a given hazardous constituent does not have  a  health- or  risk-based standard.  In the first case,
concentrations in  downgradient wells may indeed exceed the standard, but may not  be attributable to
onsite waste management if natural background levels also exceed the standard.  The Parts 264 and 258
regulations recognize these possibilities, and allow for GWPS to be based on background levels.

5.2 ESTABLISHING AND  REVIEWING BACKGROUND

     Establishing appropriate background  depends on  the  statistical approach contemplated  (e.g.,
interwell vs. intrawell). This section outlines the major  considerations concerning how to select and
develop background data including monitoring constituents and sample sizes,  statistical  assumptions,
and the presence of data outliers, spatial variation or trends.  Expanding and reviewing background data
are also discussed.

5.2.1  SELECTING MONITORING  CONSTITUENTS AND ADEQUATE SAMPLE SIZES

     Due to the  cost of management, mobilization,  field labor, and especially laboratory analysis,
groundwater monitoring  can be an expensive endeavor. The most efficient way to limit costs and still
meet environmental performance requirements is  to minimize the total number of samples which must
be sampled and analyzed.  This  will require tradeoffs  between  the number of monitoring constituents

                                             
-------
Chapter 5. Background	Unified Guidance

chosen, and the frequency of background versus compliance well testing. The number of compliance
wells  and  annual  frequency of testing also affect  overall  costs, but are  generally  site-specific
considerations. By limiting the number of constituents and ensuring adequate background sample sizes,
it is possible to select certain statistical tests which help minimize future compliance (and total) sample
requirements.

     Selection of an appropriate number of detection monitoring constituents should be dictated by the
knowledge of waste or waste leachate composition and the corresponding groundwater concentrations.
When historical background data are available, constituent choices may be influenced by their statistical
characteristics. A few representative constituents or analytes may serve to accurately assess the potential
for a release. These constituents should stem from the regulated wastes, be  sufficiently mobile,  stable
and occur at high enough concentrations to be readily detected in the groundwater. Depending on the
waste composition, some non-hazardous organic or inorganic  indicator analytes may serve the same
purpose. The  guidance suggests that between 10-15 formal  detection  monitoring constituents should be
adequate for most site conditions.  Other constituents can still be reported but not directly incorporated
into formal  detection monitoring, especially when large simultaneously analyzed suites like TCP-trace
elements, volatile or semi-volatile organics data are run.  The focus of adequate background and  future
compliance test sample sizes can then be limited to the selected monitoring constituents.

      The RCRA regulations do not consistently specify how many  observations must be collected in
background. Under  the Part 265 Interim  Status regulations, four quarterly background measurements are
required during the first year of monitoring.  Recent modifications to Part 264 for Subtitle C facilities
require a  sequence of at least four observations  to be collected in background during an interval
approved by the Regional Administrator.   On the other hand, at least four measurements must be
collected from each background well  during the  first semi-annual period  along with  at least one
additional observation during each subsequent period, for Subtitle D facilities under Part 258.  Although
these are minimum requirements in the regulations, are they  adequate sample sizes  for background
definition and  use?

     Four observations from a population are rarely enough  to adequately characterize its statistical
features; statisticians generally consider sample sizes  of n <  4 to be insufficient for good statistical
analysis. A decent  population survey, for example, requires several hundred and often a few to several
thousand participants to generate  accurate  results. Clinical trials of medical  treatments are usually
conducted on  dozens to hundreds of patients. In groundwater tests, such large sample sizes are  a rare
luxury.  However, it is feasible to obtain small sample  sets of up to  n = 20 for individual background
wells, and potentially larger sample sizes if the data characteristics allow for pooling of multiple well
data.

     The Unified Guidance recommends that a minimum of at least 8  to 10 independent background
observations be collected before running most statistical tests. Although still a small sample size by
statistical standards, these levels allow for minimally acceptable estimates of variability and evaluation
of trend and  goodness-of fit.   However, this  recommendation should be considered a temporary
minimum until additional background  sampling can be conducted  and the background sample size
enlarged (see further discussions below).

     Small  sample sizes in background can be  particularly  troublesome,  especially in controlling
statistical test  false positive and negative rates. False  negative rates in detection monitoring, i.e., the
                                              5^3                                    March 2009

-------
Chapter 5. Background	Unified Guidance

statistical  error of failing to identify a real concentration  increase above background,  are in part a
function of sample size. For a fixed false positive test rate, a smaller sample size results in a higher false
negative rate.   This  means  a decreased probability (i.e., statistical power) that real increases above
background will be detected.  With certain parametric tests,  control of the false positive rate using very
small sample sets comes at the price of extremely low power.  Power may be adequate using a non-
parametric test, but control of the false positive can be lost. In both cases, increased background sample
sizes result in better achievable false positive and false negative errors.

     The  overall recommendation of the guidance is to  establish background sample sizes as large as
feasible.  The final tradeoff comes in the selection of the type of detection tests to be used. Prediction
limit, control chart, and tolerance limit tests can utilize very small future sample sizes per compliance
well (in some cases a single initial sample), but require larger background sample sizes to have sufficient
power.   Since background samples generally are obtained from historical  data  sets  (plus future
increments as needed), total annual sample sizes (and costs) can be somewhat minimized in the future.

5.2.2 BASIC ASSUMPTIONS ABOUT BACKGROUND

     Any background sample should satisfy the key statistical  assumptions described in Chapter 3.
These  include statistical independence of  the  background  measurements, temporal  and  spatial
stationarity, lack of statistical outliers, and  correct distribution assumptions of the background sample
when a parametric statistical  approach is selected.  How independence and autocorrelation impact the
establishment  of background is  presented  below, with additional discussions on outliers,  spatial
variability and  trends in the following sections.   Stationarity assumptions are considered both in the
context of temporal and spatial variation.

     Both the Part 264 and 258 groundwater regulations  require statistically independent measurements
(Chapter  2).   Statistical independence is  indicated by random data sets.  But randomness is  only
demonstrated by the presence of mean and variance stationarity and the lack of evidence for effects such
as autocorrelation, trends, spatial and temporal variation.   These tests (described  in Part II of this
guidance)  generally require at least 8 to 10 separate background measurements.

     Depending on site groundwater velocity, too-frequent sampling at any given background well can
result in highly autocor related, non-independent data. Current or proposed sampling frequencies can be
tested for autocorrelation or other  statistical  dependence using the diagnostic procedures in Chapter 14.
Practically speaking, the best way  to ensure  some degree  of statistical independence is to allow as much
time as possible to elapse between sampling events. But a balance must be drawn between collecting as
many measurements as possible from a given well over  a specified time period, and ensuring that the
sample measurements are statistically independent. If significant  dependence  is identified in already
collected background, the interval between sampling events may  need to be lengthened to minimize
further autocorrelation. With fewer sampling  events per evaluation period,  it is also possible that a
change in  statistical method may be needed, say from analysis of variance [ANOVA], which requires at
least 4 new background measurements per evaluation, to prediction limits or control  charts, which may
require new background only periodically (e.g.,  during a biennial update).
                                              5-4                                    March 2009

-------
Chapter 5. Background	Unified Guidance

5.2.3 OUTLIERS IN  BACKGROUND

     Outliers or observations not derived from the same population as the rest of the sample violate the
basic statistical assumption of identically-distributed measurements. The Unified Guidance recommends
that testing of outliers be performed on background data, but they generally not be removed unless some
basis for a likely error or discrepancy can be identified.  Such possible errors or  discrepancies could
include data recording errors, unusual sampling and laboratory procedures or conditions, inconsistent
sample turbidity, and values significantly outside the historical ranges of background data. Management
of potential outliers carries both positive and negative risks, which should be carefully understood.

      If an outlier value with much higher concentration  than other background  observations is not
removed from  background prior to statistical testing, it will tend to increase both the background sample
mean and  standard deviation.  In turn,  this may  substantially raise the magnitude  of a parametric
prediction limit or control limit calculated from that sample. A subsequent compliance well test against
this background  limit will be much less likely to identify an exceedance. The  same is true with non-
parametric prediction limits, especially when the maximum background value is taken as the prediction
limit. If the  maximum  is an  outlier  not representative  of the background population, few truly
contaminated compliance wells are likely to be identified by such a test, lowering the statistical power of
the method and the overall quality of the statistical monitoring program.

     Because  of these concerns, it may  be advisable at times to remove high-magnitude outliers in
background even if the reasons for these apparently extreme observations are  not  known. The overall
impact of removal will tend to improve the power of prediction limits and control charts, and thus result
in a more environmentally protective program.

     But strategies that involve automated evaluation and removal of outliers may unwittingly eliminate
the evidence of real and  important changes to background conditions.  An example  of this phenomenon
may have occurred during the  1970s in some early ozone  depletion measurements over Antarctica
(http://www.nas.nasa.gov/About/Education/Ozone/history.html).  Automated  computer   routines for
outlier detection  apparently  removed several  measurements  indicating a sharp  reduction in ozone
concentrations, and thus prevented  identification of an enlarging ozone  hole by  many  years.   Later
review of the raw observations revealed  that these  automated  routines had statistically classified
measurements  as outliers, which were more extreme than most of the data from that time period. Thus,
there is some merit in saving  and revisiting apparent 'outliers' in future investigations,  even if removed
from present databases.

     In groundwater data collection and testing, background conditions may not be static over time.
Caution should be observed in removing observations which may signal a change in natural groundwater
quality. Even when conditions have not  changed, an apparently extreme  measurement may represent
nothing more  than a portion of the background  distribution that has  yet  to  be  observed.   This is
particularly true if the background data set contains fewer than 20 samples.

     In balancing these contrasting risks in retaining or removing one or more outliers, analyses of
historical data patterns can sometimes provide more definitive information depending on the types of
analytes and methods.  For example, if a potential order-of magnitude higher outlier is identified in a
sodium data set used as a monitoring constituent, cation-anion  balances can help  determine if this
change is geochemically probable.  In this  case,  changes  to other intrawell ions or TDS should be

                                             5^5                                     March 2009

-------
Chapter 5. Background	Unified Guidance

observed.  Similarly, if a trace element outlier is identified in a single well sampling event and occurred
simultaneously with other trace element maxima measured using the same analytical method (e.g., ICP-
AES) either in the same well or groups of wells, an analytical  error should be strongly suspected.  On
the other hand, an isolated increase without any other evidence could be a real but extreme background
measurement.  Ideally,  removal of one or more statistically identified outliers should be based on other
technical information or knowledge which can support that decision.

5.2.4 IMPACT  OF SPATIAL VARIABILITY

     In the absence of contamination, comparisons made between upgradient-to-downgradient wells
assume that the concentration distribution is spatially stationary across the well field (Chapter 3). This
implies that every well should have the same population mean and variance, unless a release occurs to
increase the concentration levels at  one or more compliance wells. At many sites, this is not the case for
many naturally occurring constituents. Natural or man-made differences in mean levels — referred to as
spatial variability or spatial variation — impact how background must be established.

     Evidence of spatial variation should drive the selection of an intrawell statistical  approach  if
observed among wells known to be uncontaminated (e.g., among a group of upgradient  background
locations).  Lack  of spatial  mean differences and a common variance allow for interwell comparisons.
Appropriate background differs between the two approaches.

     With  interwell tests,  background is derived from distinct, initially upgradient background wells,
which may be enhanced by data from historical compliance wells also shown not to exhibit significant
mean and variance differences. Future data from each of these compliance wells are then tested against
this common background.  On  the other hand, intrawell background is derived  from and represents
historical groundwater conditions in each individual compliance well. When the population mean levels
vary across a well field,  there is little  likelihood  that the upgradient background will provide an
appropriate comparison by which to judge any given compliance well.

     Although spatial variability impacts the choice of background, it does so only for those constituents
which evidence  spatial differences across the well  field.  Each  monitoring constituent should be
evaluated on its own statistical merits.  Spatial  variation in some constituents (e.g., common ions and
inorganic parameters) does not preclude the use of interwell background for other infrequently detected
or non-naturally  occurring analytes.  At  many  sites,  a mixture of  statistical  approaches may be
appropriate: interwell tests for part of the monitoring list and intrawell tests for another portion. Distinct
background observation sets will need to be developed under such circumstances.

     Intrawell background  measurements should be selected  from the available historical samples  at
each  compliance  well  and should include  only those  observations  thought  to  be uncontaminated.
Initially, this might result in very few measurements (e.g., 4 to 6). With such a small background sample,
it can be very difficult to develop an adequately  powerful intrawell prediction limit or control chart, even
when retesting is employed (Chapter 19).  Thus, additional background data will be needed to augment
the testing power.  One option is to periodically augment the existing background data base with recent
compliance well  samples  (discussed in  a further section below).  Another possible remedy  is to
statistically  augment  the   available  sample data  by  running  an analysis  of  variance [ANOVA]
simultaneously on all the sets of intrawell background from the  various upgradient and compliance wells
(see Chapter 13). The root mean squared error [RMSE] from this procedure can be used in place of the

                                              5^6                                    March 2009

-------
Chapter 5. Background	Unified Guidance

background standard deviation in parametric prediction and control limits to substantially increase the
effective background sample size of such tests, despite the limited number of observations available per
well.

     This strategy will only work if the key assumptions of ANOVA can be satisfied (Chapter 17),
particularly the requirement of equal variances  across wells. Since natural differences in mean levels
often correspond to similar differences in variability, a transformation of the data will often be necessary
to homogenize the variances prior to running the ANOVA. For  some constituents, no transformation
may work  well enough to  allow the  RMSE to be used as a  replacement estimate for the  intrawell
background standard deviation. In that case, it  may not be possible to construct reasonably  powerful
intrawell background limits  until background has been updated once or twice (see Section 5.3).

5.2.5 TRENDS IN BACKGROUND

     A key implication of the independent and identically distributed assumption [i.i.d] is that a series
of sample measurements  should be stationary over time (i.e., stable  in mean level and variance). Data
that are trending upward or downward violate this assumption since  the  mean level is changing.
Seasonal fluctuations also violate this assumption since both the mean and variance will likely oscillate.
The proper handling of trends  in background depends on the statistical approach and the cause of the
trend.  With interwell tests and  a common  (upgradient) background,  a trend can  signify several
possibilities:

   »»»  Contaminated background;
   »»»  A 'break-in' period following new well installation;
   »»»  Site-wide changes in the aquifer;
   »»»  Seasonal fluctuations, perhaps on the order of several months to a few years.
     If upgradient well background becomes contaminated, intrawell testing may be needed to avoid
inappropriate comparisons.  Groundwater flow  patterns should also be  re-examined to determine if
gradients are properly defined or if groundwater mounding might be occurring. With newly-installed
background wells, it may be necessary to discard  initially collected observations  and to wait several
months for  aquifer disturbances due to  well construction  to stabilize.  Site-wide changes  in  the
underlying  aquifer should be identifiable as similar trends in both upgradient and compliance wells. In
this case, it might be possible to  remove a common  trend from both the background and compliance
point  wells and to perform interwell testing on the trend residuals. However, professional statistical
assistance may be needed to do this correctly. Another option would be to switch to intrawell trend tests
(Chapter 17)

     Seasonal fluctuations in interwell background which are also observed in compliance wells, can be
accommodated by modeling the  seasonal trend  and removing it from all background and compliance
well data.  Data  seasonally-adjusted in this way (see Chapter 14  for details) will  generally be less
variable than the unadjusted measurements and lead to more powerful tests than if the seasonal patterns
had been ignored. For this  adjustment to work  properly, the same seasonal trend  should be  observed
across the well field and not be substantially different from well to well.
                                              5-7                                    March 2009

-------
Chapter 5. Background	Unified Guidance

     Roughly linear trends in intrawell background usually signify the need to switch from an intrawell
prediction limit or control chart to an explicit trend test, such as linear regression or the Mann-Kendall
(Chapter 17). Otherwise the background variance will be overestimated and biased on the high side,
leading to  higher than expected and ultimately less powerful prediction and control  limits. Seasonal
fluctuations in intrawell background can be treated in one of two ways. A seasonal Mann-Kendall trend
test built to accommodate such fluctuations can be employed (Section 14.3.4).  Otherwise, the seasonal
pattern can be estimated and removed from the background data, leaving a  set of seasonally-adjusted
data to be  analyzed with  either  a prediction limit or control chart.  In  this  latter approach, the same
seasonal pattern needs to be extrapolated beyond the current background to more recent measurements
from the compliance well being tested. These later observations also need to be seasonally-adjusted prior
to comparison against the adjusted background, even if there is not enough compliance data yet collected
to observe the same seasonal cycles.

     When trends are apparent in background, another option is to modify the groundwater monitoring
list to include only those constituents that appear to be temporally stable. Only certain analytes may
indicate evidence of trends or seasonal fluctuations. More powerful statistical  tests might be constructed
on constituents that appear to be stationary. All such changes to the monitoring list and method of
testing may require approval of the Regional Administrator or State Director.

5.2.6 EXPANDING  INITIAL BACKGROUND SAMPLE SIZES

       In the initial development of a detection monitoring statistical program under a permit or other
legal mechanism, a period of review will identify the appropriate monitoring constituents. For new sites
with no  prior data, plans for initial background definition need to be developed as part of  permit
conditions.  A more typical situation occurs for  interim status or older facilities which have already
collected substantial historical data in site monitoring wells.  For the most part, the  suggestions below
cover ways of expanding background data sets from existing information.

       Under  the  RCRA interim status regulations, only a single upgradient well is required as a
minimum.   Generally speaking, a  single  background well  will not generate observations that are
adequately representative of the underlying aquifer.  A single background well draws groundwater from
only one possible background location. It is accordingly not possible to determine if spatial variation is
occurring in the upgradient aquifer.  In addition, a single background well can only be sampled so often
since measurements that are collected too frequently run the risk of being autocorrelated. Background
observations  collected from a single well  are typically  neither representative nor constitute a large
enough sample to construct powerful, accurate statistical tests. One way to expand background is to
install at least 3-4 upgradient wells and collect additional data under permit.

       The early RCRA regulations  also allowed for aliquot replicate sampling as a means of expanding
background and other well sample  sizes.   This approach consisted of analyzing  splits or  aliquots of
single water quality samples.  As indicated in Chapter 2, this approach is not recommended in the
guidance. Generally limited analytical variability does not adequately capture  the overall variation based
on independent water quality sample data, and results in incorrect estimates of variability and degrees of
freedom (a function of sample size).

       Existing historical groundwater well data under consideration will need to meet the assumptions
discussed earlier  in this  chapter- independence,  stationarity, etc.,  including using statistical methods

                                               5^8                                     March 2009

-------
Chapter 5.  Background	Unified Guidance

which can  deal with  outliers,  spatial and temporal  variation including  trends.   Presuming these
conditions are met, it is statistically desirable to develop as large a background sample size as practical.
But no matter how many measurements are utilized, a larger sample size is advantageous only if the
background samples are both appropriate to the tests selected and representative of baseline conditions.

      In limited  situations, upgradient-to-downgradient,  interwell comparisons may be determined to be
appropriate using ANOVA testing of well mean differences.  To ensure appropriate and representative
background,  other conditions may also need to be satisfied when data from separate wells are pooled.
First, each  background well  should be  screened at the same hydrostratigraphic position as other
background wells.  Second, the groundwater chemistry at each of these wells should be similar. This can
be checked via  the use of standard geochemical bar charts, pie charts, and tri-linear diagrams  of the
major constituent groundwater ions and cations (Hem, 1989).  Third, the statistical characteristics of the
background wells should be similar — that is, they should be spatially stationary.,  with approximately
the same means and variances.   These conditions are particularly important for major water  quality
indicators, which generally reflect aquifer-specific characteristics.   For infrequently detected analytes
(e.g.,  filtered trace  elements like  chromium, silver,  and zinc), even data collected from wells from
different aquifers and/or geologic strata  may be statistically  indistinguishable  and  also eligible for
pooling on an interwell basis.

      If a one-way ANOVA (Chapter 13) on the set of background wells finds significant differences in
the mean  levels  for  some constituents,  and hence,  evidence of spatial variability, the guidance
recommends using intrawell tests. The data gathered from the  background wells will generally not be
used  in  formal  statistical  testing,  but are  still invaluable in ensuring that appropriate  background  is
selected.1 As indicated in the discussions above and Chapter 13, it may be possible to pool constituent
data from a number of upgradient and/or compliance wells having a common variance when parametric
assumptions  allow, even if mean differences exist.

      When larger historical databases are available, the data can be reviewed  and diagnostically tested
to determine which observations best represent  natural groundwater conditions  suitable  for  future
comparisons. During this  review, all historical well data collected from both upgradient and compliance
wells can be evaluated for potential inclusion into background.  Wells suspected of prior contamination
would need to be excluded, but otherwise each uncontaminated data point adds to the overall statistical
picture of background conditions at the  site and can be used to enlarge the background database.
Measurements can be preferentially selected to establish background samples, so long as a consistent
rationale is used (e.g., newer  analytical methods, substantial outliers in a portion of a data set, etc.)
Changes to an aquifer over time may require selecting newer data representing current groundwater
quality over earlier results even if valid.
  If the spatial variation is ignored and data are pooled across wells with differing mean levels (and perhaps variances) to run
  an interwell parametric prediction limit or control chart test, the pooled standard deviation will tend to be substantially
  larger than expected.  This will result in a higher critical limit for the test. Using pooled data with spatial variation will also
  tends to increase observed maximum values in background, leading to higher and less powerful non-parametric prediction
  limit tests. In either application, there will be a loss of statistical power for detecting concentration changes at individual
  compliance wells.  Compliance wells with naturally higher mean levels will also be more frequently determined to exceed
  the limit than expected, while real increases at compliance wells with naturally lower means will go undetected more often.

                                                
-------
Chapter 5. Background	Unified Guidance

5.2.7 REVIEW OF BACKGROUND

     As mentioned above, if a large historical database is available, a critical review of the data can be
undertaken  to  help  establish  initially  appropriate  and  representative  background samples.  We
recommend that other reviews of background also take place periodically. These include the following
situations:

    »»»  When periodically updating background, say every 1-2 years (see Section 5.3)
    »«»  When performing a 5-10 year permit review
     During these reviews, all observations designated as background should be evaluated to ensure that
they still  adequately reflect current natural or baseline groundwater  conditions.  In particular, the
background samples should be investigated for apparent trends or outliers. Statistical outliers may need
to be removed, especially if an error or discrepancy can be identified, so that subsequent compliance
tests can be improved. If  trends are indicated, a change in the statistical method or approach may be
warranted (see earlier section on "Trends in Background").

     If background has been updated or enlarged since  the last review,  and is being utilized in
parametric tests, the assumption of normality (or other distributional fit) should be re-checked to ensure
that the augmented background data are still consistent with a parametric approach. The presence of non-
detects and multiple reporting limits (especially with changes in analytical methods over time) can prove
particularly troublesome in checking distribution  assumptions. The methods of Chapters 10 "Fitting
Distributions" and Chapter 15 "Handling Non-Detects" can be consulted for guidance.

     Other periodic checks of the  revised background should also be conducted, especially in relation to
accumulated knowledge from other sites regarding analyte concentration patterns in groundwater. The
following are potential sources for comparison and evaluation:

    *»*  reliable regional groundwater data studies or investigations from nearby sites;
    *»*  published literature; EPA or other agency groundwater databases like STORET;
    »«»  knowledge of typical patterns for background  inorganic constituents and trace  elements. An
       example is found in Table 5-1 at the end of this chapter. Typical surface and groundwater levels
       for filtered trace elements can also be found in the published literature (e.g., Hem, 1989).
       Certain common features of routine groundwater monitoring analytes  summarized  in Table 5-1
have been observed in  Region 8  and  other background data sets, which can have implications for
statistical  applications.  Common water quality indicators like cations and anions, pH, TDS, specific
conductance  are almost  always  measurable  (detectable) and  generally  have  limited within-well
variability.  These  would be more amenable to  parametric applications;  however, these measurable
analytes are also most likely to exhibit well-to-well spatial variation and various kinds of within- and
between-well temporal  variation  including  seasonal and annual trends.  Many of these within-well
analytes are highly  correlated, and would not meet the criterion for independent data if simultaneously
used as monitoring constituents.

       A  second level of  common indicator analytes- nitrate/nitrite species, fluoride, TOC and TOX-
are less frequently detected and  subject to more  analytical  detection  instability (higher and  lower


                                             
-------
Chapter 5. Background	Unified Guidance

detect!on/quantitation limits).  As such, these analyte data are somewhat less reliable.   There is less
likelihood of temporal variation, although they can exhibit spatial well differences.

       Among routinely monitored .45u-filtered trace elements, different groups stand out.  Barium is
routinely detected with limited variation within most wells, but does exhibit spatial variation.  Arsenic
and selenium  commonly occur in groundwater as oxyanions, and data can range from virtually non-
detectable to  always detected in  different site  wells.   The largest group  of trace elements can be
considered colloidal  metals (Sb, Al, Be, Cd, Cr, Co, Fe, Hg, Mn, Pb, Ni, Sn, Tl, V and Zn).  While Al,
Mn and Fe are more commonly detected, variability is often quite high; well-to-well spatial  variability
can occur at  times.   The remaining colloidal metals  are  solubility-limited  in most background
groundwater, generally <1 to < 10 ug/1. But even with filtration,  some natural colloidal geologic solid
materials can often be detected in individual samples. Since naturally occurring Al, Mn and Fe soil solid
levels are much higher,  the effects on measured groundwater levels  are more pronounced and variable.
For most of the analytically and solubility-limited colloidal metals, there may not be any discernible well
spatial differences. Often these data can be characterized by a site-wide lognormal distribution, and may
be possible to pool individual well data to form larger background sizes.

       With unfiltered trace element data, it is more difficult to generalize even regarding background
data.  The method of well sample extraction and the aquifer characteristics will determine how much
solids material may be present in the samples.  Excessive amounts of sample solids can result in higher
levels  of  detection but  also elevated average values  and variability even for solubility-limited trace
elements.  The effect is most clearly  seen when TSS is simultaneously collected with unfiltered data.
Increases are proportional  to the amount of TSS and the natural background levels for trace elements in
soil/solid materials.  It is  recommended that TSS always be simultaneously monitored with unfiltered
trace elements.

       Most trace organic monitoring constituents are absent or non-detectable under clean background
conditions. However, with existing up-gradient sources, it is more difficult to generalize.  More soluble
constituents like benzene or chlorinated hydrocarbons may be amenable to parametric distributions,  but
changes in groundwater levels or direction can drastically affect observed levels.  For sparingly soluble
compounds like polynuclear aromatics (e.g., naphthalene), aquifer effects can result in highly  variable
data less amenable to statistical applications.

       Table 5-1  was based on the use of analytical  methods common in the 1990's to the  present.
Detectable filtered trace element data for the most part were limited by the available analytic techniques,
generally  SW-846 Method  6010 ICP-AES and select AA (atomic absorption) methods with  lower
detection limits in the 1-10 ppb range. As newer methods are  incorporated (particularly Method 6020
ICP-MS  capable of parts-per-trillion  detection  limits  for  trace  elements), higher  quantification
frequencies may result in data demonstrating more complex spatial and temporal characteristics.  Table
5-1 merely provides  a rough guide to  where various data patterns  might occur. Any extension  of these
patterns to other facility data sets should be determined by the formal guidance tests in Part II.

     The background database can also be specially organized and summarized to examine common
behavior among related  analytes (e.g.., filtered trace elements using ICP-AES) either over time or across
wells  during common sampling events. Parallel time  series plots (Chapter 9) are very useful in this
regard.  Groups of related analytes can be graphed on the same set of axes, or groups of nearby wells for
the same analyte. With either plot, highly suspect sampling events can be identified if a similar spike in
                                              
-------
Chapter 5. Background	Unified Guidance

concentration or other unusual pattern occurs  simultaneously at  all the wells or in all the analytes.
Analytical measurements that appear to be in error might be removed from the background database.

       Cation-anion balances and other more sophisticated geochemical analysis programs can also be
used to evaluate the reliability of existing water quality background data. A suite of tests like linear or
non-parametric  correlations, simple or non-parametric ANOVA described in later chapters offer overall
methods for evaluating historical data for background suitability.

5.3  UPDATING  BACKGROUND

     Due both to the complex behavior of groundwater and the need for sufficiently large sample sizes,
background once obtained should not  be regarded as a single fixed quantity.  Background  should be
sampled regularly throughout the  life of the facility, periodically reviewed and revised as necessary. If a
site  uses traditional,  upgradient-to-downgradient  comparisons,  it might seem  that updating of
background is  conceptually simple: collect new measurements from each background well at each
sampling event  and add these to the overall  background sample. However, significant trends or changes
in one or more upgradient wells might indicate problems with individual wells, or be part of a larger site-
wide groundwater change. It is worthwhile to consider the following principles for updating, whether
interwell or intrawell testing is used.

5.3.1 WHEN  TO  UPDATE

     There are  no firm rules on how often to update background data. The Unified Guidance  adopts the
general principle that  updating should occur when enough new measurements have been collected to
allow a two-sample statistical comparison between the existing background data and a potential set of
newer data. As mentioned in the following section, trend testing might also be used. With quarterly
sampling, at least 4 to 8 new measurements should be  gathered to enable such a test; this implies that
updating would take place every  1-2 years. With semi-annual sampling, the same principle would call
for updating every 2-3 years.

     Updating  should generally not  occur more frequently,  since adding  a  new observation to
background every one or two sampling rounds does not allow a  statistical evaluation of whether the
background mean is stationary over time. Enough new data needs to be collected to ensure that a test of
means (or medians in the case of non-normal data) can be conducted. Adding individual observations to
background can introduce subtle trends that might go  undetected  and ultimately reduce the statistical
power of formal monitoring tests.

     Another practical aspect is that when background is updated,  all statistical background limits (e.g.,
prediction and control limits) needs to be recomputed to account for the revised background sample. At
complex  sites,  updating the limits at  each well and constituent  on the  monitoring list may require
substantial  effort. This includes resetting the cumulative sum [CUSUM] portions of control charts to
zero after re-calculating the control limits and prior to additional testing against those limits.  Too-
frequent updating could thereby reduce the efficacy of control chart  tests.

5.3.2 HOW TO UPDATE

     Updating  background is primarily a concern for intrawell tests, although some of the  guidelines
apply to interwell data. The common (generally upgradient) interwell background pool can be tested for
                                             
-------
Chapter 5. Background	Unified Guidance

trends  and/or changes at intervals depending  on the sampling frequencies identified above.  Those
recently collected measurements from the background well(s) can be added to the existing pool if a
Student's t-test or Wilcoxon rank-sum test (Chapter 16) finds no significant difference between the two
groups at the a = 0.01 level  of significance. Individual background wells should also be evaluated in the
same manner for their respective newer data.  Two-sample tests of the interwell background data are
conducted to gauge whether or not background groundwater conditions have changed substantially since
the last update,  and are not tests  for indicating a potential  release  under detection monitoring.   A
significant  Mest or Wilcoxon rank-sum result should spur a closer  investigation and review of the
background sample,  in order to determine which observations are most representative of the current
groundwater conditions.

     With  intrawell tests using prediction limits or control charts, updating is performed both to enlarge
initially  small  well-specific  background  samples  and  to  ensure   that  more  recent  compliance
measurements are not already impacted by  a  potential  release (even if not  triggered  by the  formal
detection monitoring tests). A finding of significance using the above two-sample tests means that the
most recent data  should not be added to intrawell background.  However,  the same caveat as above
applies: these are not formal  tests  for determining a  potential release and the  existing tests and
background should continue to be used.

     Updating intrawell background should  also not occur  until at least 4 to 8 new compliance
observations have been collected. Further, a potential update is predicated on there being no statistically
significant  increase  [SSI] recorded for that well constituent, including since the last update.  Then a t-
test  or Wilcoxon rank-sum comparison can  be conducted at each compliance well between existing
intrawell background and the potential set of newer background.  A non-significant result implies that
the newer compliance data  can be re-classified as background measurements and added to the existing
intrawell background sample. On  the other hand, a  determination of significance  suggests that  the
compliance observations should be reviewed to determine whether a gradual trend or other change has
occurred that was missed by the intervening prediction limit or control  chart tests.  If intrawell tests
make use of a common pooled variance, the assumption of equal variance in the pooled wells should
also be checked with the newer data.

     Some users may wish to evaluate historical and future background data for potential trends.  If
plots of data versus time suggest either an overall trend in the combined data sets or distinct differences
in the respective  sets, linear or non-parametric trend tests covered in Chapter 17 might  be used.  A
determination of a  significant trend  might occur even if the  two-sample tests are  inconclusive, but
individual group sample sizes should be large enough to avoid identifying a significant trend based  on
too few samples and perhaps randomly occurring.  A trend in the newer data may reflect or depart from
the historical data conditions.  Some  form of statistical adjustments may  be necessary, but see Section
5.3.4 below.
                                             5-13                                    March 2009

-------
Chapter 5. Background	Unified Guidance

5.3.3 IMPACT OF RETESTING
                                                                                           r\
     A key question when updating intrawell background is how to handle the results of retesting. If a
retest confirms an  SSI, background should not be updated.  Rather, some regulatory action at the site
should be taken.  But what if an initial exceedance of a prediction or control limit is disconfirmed by
retesting? According to the logic of retesting (Chapter 19), the well passes the compliance test for that
evaluation and monitoring should continue as usual. But what should be done with the initial exceedance
when it comes time to update background at the well?

     The initial exceedance may be due  to a laboratory  error or other anomaly that has caused the
observation to be an outlier.  If so, the error should be documented and not included in the updated
background sample.  But if the exceedance is not explainable as an outlier or error,  it may represent a
portion of the background population that  has heretofore not been sampled. In that case, the data  value
could be included in the updated background sample (along with the repeat sample)  as evidence of the
expanded but true range of background  variation. Ultimately,  it is important to characterize the
background conditions at the site as completely and accurately as possible, so as  to minimize both false
positive and false negative decision errors in compliance testing.

     The severity and classification of the initial  exceedance will depend on the  specific  retesting
strategy that has been implemented (Chapter 19).  Using the same background data in a parametric
prediction limit or control chart test, background limits  are proportionately lower as the l-of-m  order
increases (higher m). Thus, a l-of-4 prediction limit will be lower than a l-of-3  limit, and similarly the
l-of-3  limit lower than  for a l-of-2 test.   An initial  exceedance triggered by a l-of-4 test limit and
disconfirmed by a repeat sample, might not trigger a lower order prediction limit test.  The initial sample
value may represent an upper tail value from the true distribution. Retesting schemes derive much  of
their statistical power by allowing more frequent initial  exceedances,  even if some  of these represent
possible  measurements  from background. The  initial and subsequent resamples taken together are
designed to identify which initial exceedances truly  represent SSIs and which do not.   These tests
presume that occasional  excursions beyond the background limit will occur. Unless the exceedance can
be documented as  an outlier or other anomaly, it should probably be included in the updated intrawell
background sample.

5.3.4 UPDATING WHEN TRENDS ARE APPARENT

     An increasing or decreasing trend may be apparent between the existing background and the newer
set  of candidate background values,  either using a time series  plot or applying Chapter 17  trend
analyses.   Should  such  trend  data  be added  to the existing background sample? Most detection
monitoring tests assume that background is stationary over time, with no discernible trends or seasonal
variation.  A mild trend will probably make very little difference, especially if a  Student-^ or Wilcoxon
rank-sum  test between the existing and candidate background data sets is non-significant. More severe
or continuing trends are likely to be flagged as SSIs by formal intrawell prediction limit or control chart
tests.
2 With interwell tests, the common (upgradient) background is rarely affected by retests at compliance point wells (unless the
  latter were included in the common pool). Should retesting fail to confirm an initial exceedance , the initial value can be
  reported alongside the disconfirming resamples in statistical reports for that facility.

                                              
-------
Chapter 5.  Background	Unified Guidance

      With interwell tests, a stronger trend in the common upgradient background may signify a change
in natural groundwater quality across the aquifer or an incomplete characterization of the full range of
background  variation. If a change  is evident,  it may be necessary to  delete  some of the earlier
background values from the updated background sample, so as to ensure that compliance testing is based
on current groundwater conditions and not on outdated measures of groundwater quality.
                                            5-15                                   March 2009

-------
Chapter 5.  Background
Unified Guidance
Table 5-1. Typical Background
Analyte
Groups
Detection Rates
Frequency of Multiple
Detection Reporting
by Well Limiits
Data Patterns for Routine Groundwater Monitoring Ana lytes
• . • . * • ' . \ • • • . • ' • , , '/•'.'''
•••..• ' .
Between Within Well Between Outlier
Well Variability Well Problems
Mean (CVs) Equal
Differ- Variances
ences
, •
Temporal Variation
Between Within Within Within Within
Well by Well Well Well Well
Analyte Among Auto- Seasonal Time
Group correl. Variation Correl.
Typical
Distribution
within well
Data
Grouping
Inorganic Constituents and Indicators
Major ions, pH,
TDS, Specific
Conductance
COS, F,
NO2,NO3
High to 100%

Some to most
detectable
'"

yv
Generally
low
(.1-.5)
Moderate Variable ^^
(.2-1.5)
"

,
Normal

Norm, Log or
NPM
Intrawell

Intrawell/
Interwell
 .45|i Filtered Trace Elements
Ba

As, Se




Al, Mn, Fe

Sb, Be, Cd, Cr,
Cu, Hg, Pb, Ni,
Ag, Tl, V, Zn
High to 100% ^^

Some wells
high, others

low to zero

Low to
Moderate
Zero to low


Low
(.1-.5)
Moderate Variable
(.2-1.5)
(some

wells)
Moderate
to high
Moderate
to high
(.5->2.0)
^ ^

^ ^




^ ^

V /V V


Normal

Normal, Log
or NPM



Log or NPM

Log or NPM


Intrawell

Intrawell/
Interwell



Intrawell/
Interwell
Interwell
or NDC

 Trace Organic and Indicator Ana lytes (patterns at sites with prior contamination; generally absent in clean sites)
VOA's-BETX and
CI-Hyd rocarbons

BNAs, Other
Trace Organics
Indicators: TOX,
TPH, TOC, sulfide
Variable, can
be high

Generally
low-mod
Variable

Variable by site and wells j


	 ,

jss

Variable by site and specific wells


" " " "

" " " "

Normal, Log
or NPM

„ „

„ „

Intrawell,
Interwell or
NDC
„ „

„ „

     NPM non-parametric methods;  NDC- never-detected constituents
      Checks:  None- unknown, absent or infrequently occurring;   •/ - Occasionally;  •/ •/ - Frequently;   •/•/•/- Very Frequently
                                                                   5-16
        March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

CHAPTER  6.  DETECTION  MONITORING PROGRAM DESIGN
        6.1  INTRODUCTION	6-1
        6.2  ELEMENTS OF THE STATISTICAL PROGRAM DESIGN	6-2
          6.2.1  The Multiple Comparisons Problem	6-2
          6.2.2  Site-Wide False Positive Rates [SWFPR]	6-7
          6.2.3  Recommendations for Statistical Power	6-13
          6.2.4  Effect Sizes and Data-Based Power Curves	6-18
          6.2.5  Sites Using More Than One Statistical Method	6-21
        6.3  How KEY ASSUMPTIONS IMPACT STATISTICAL DESIGN	6-25
          6.3.1  Statistical Independence	6-25
          6.3.2  Spatial Variation: Interwellvs. Intrawell Testing	6-29
          6.3.3  Outliers	6-34
          6.3.4  Non-Detects	6-36
        6.4  DESIGNING DETECTION MONITORING TESTS	6-38
          6.4.1  T-Tests	6-38
          6.4.2  Analysis Of Variance [ANOVA]	6-38
          6.4.3  Trend Tests	6-41
          6.4.4  Statistical Intervals	6-42
          6.4.5  Control Charts	6-46
        6.5  SITE DESIGN  EXAMPLES	6-46
6.1  INTRODUCTION

      This chapter addresses the initial statistical design of a detection monitoring program, prior to
routine implementation.   It considers what important elements should be  specified  in  site permits,
monitoring development plans or during periodic reviews.  A good statistical design can be critically
important for ensuring that the routine process of detection monitoring meets the broad objective of the
RCRA regulations:  using statistical testing to accurately evaluate whether or not there is a release to
groundwater at one or more compliance wells.

      This guidance recommends a comprehensive detection monitoring program  design, based on two
key performance characteristics: adequate  statistical power and a low  predetermined site-wide false
positive rate [SWFPR].  The design approach presented in Section 6.2 was developed in response to the
multiple comparisons problem affecting RCRA and other groundwater detection programs, discussed in
Section 6.2.1. Greater detail in applying design cumulative false positives and assessing power follows
in the next three sub-sections.  In Section 6.3, consideration is given to data features that impact proper
implementation of statistical testing, such as  outliers and non-detects, using interwell  versus intrawell
tests,  as well as the presence of spatial variability or trends.  Section 6.4 provides a general discussion of
specific detection testing methods listed in the regulations and their appropriate use. Finally, Section 6.5
applies the design concepts to three hypothetical site examples.

      The principles  and statistical tests which this chapter covers for a  detection monitoring program
can also  apply  to compliance/corrective  action monitoring when a background  standard  is used.
Designing a background standards compliance program is discussed in Chapter 7 (Section 7.5).
                                              6-1                                    March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

6.2  ELEMENTS OF THE STATISTICAL PROGRAM DESIGN

6.2.1 THE  MULTIPLE COMPARISONS PROBLEM

     The foremost goal in detection monitoring  is to identify a real release to  groundwater when it
occurs.    Tests must have adequate  statistical power  to  identify  concentration  increases  above
background.  A second critical goal is to avoid false positive decision errors, evaluations where one or
more wells are falsely declared to be contaminated when in fact their concentration distribution is similar
to background. Unfortunately, there is a trade-off (discussed in Chapter 3) between maximizing power
and minimizing the false positive rate in designing a statistical testing protocol.  The statistical power of
a given test procedure using a fixed background sample size (ri) cannot be improved without increasing
the risk of false positive error (and vice-versa).

     In RCRA and other groundwater detection monitoring programs, most facilities must monitor and
test for multiple constituents at all  compliance wells one or more times per year.  A separate statistical
test1 for each monitoring constituent-compliance well pair is  generally conducted semi-annually.  Each
additional  background  comparison test increases the  accumulative risk of making a false positive
mistake, known statistically as the multiple comparisons problem.

     The false positive rate a (or Type I error) for an individual test is the probability that the test will
falsely indicate an exceedance of background. Often, a single fixed low false positive error rate typically
found in textbooks or regulation, e.g.,  a = .01 or .05,  is applied to each statistical test performed for
every well-constituent pair at a facility.  Applying such a common false positive rate (a) to each of
several  tests  can result in an acceptable cumulative false positive error if the number of tests is  quite
small.

      But as  the number of tests increases, the false positive rate associated with the testing network as a
whole  (i.e., across all well-constituent pairs) can be surprisingly high.  If enough tests are  run, at least
one test is likely to indicate potential contamination even if a release has not occurred. As an example, if
the testing network consists of 20 separate well-constituent pairs and a 99% confidence upper prediction
limit is used  for each test (a = .01), the expected overall network-wide false positive rate is about  18%.
There is nearly a 1 in 5 chance that one or more  tests  will falsely identify a  release to groundwater at
uncontaminated wells. For 100 tests and the same statistical procedure, the overall network-wide false
positive rate  increases to more than 63%, creating additional steps to verify the lack of contamination at
falsely triggered wells.  This cumulative  false positive  error is also indicative of at least one well
constituent false positive error, but there could be more.  Controlling this cumulative false positive error
rate is essential in addressing the multiple comparisons problem.
1   The number of samples collected may not be the same as the number of statistical tests (e.g., a mean test based on 2
  individual samples). It is the number of tests which affect the multiple comparisons problem.
2  To minimize later confusion, note that the Unified Guidance applies the term "comparison" somewhat differently than most
  statistical literature. In statistical theory, multiple tests are synonymous with multiple comparisons, regardless of the kind of
  statistical test employed. But because of its emphasis on retesting and resampling techniques, the Unified Guidance uses
  "comparison" in referring to the evaluation of a single sample value or sample statistic against a prediction or control chart
  limit. In many of the procedures described in Chapters 19 and 20, a single statistical test will involve two or more such
  individual comparisons, yet all the comparisons are part of the same (individual) test.

                                               6-2                                     March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

     Three main strategies (or their combination) can be used to counter the excessive cumulative false
positive  error rate- 1) the number of tests can be reduced; 2) the individual test false positive rate can
be lowered, or 3) the type of statistical test can be changed.  A fourth strategy to increase background
sample sizes may also be appropriate.  Under an initial monitoring design, one usually works with fixed
historical sample sizes. However, background data can later be updated in periodic program reviews.

     To make use of these strategies, a sufficiently low target cumulative SWFPR needs to be initially
identified for design purposes. The target cumulative error applies to a certain regular time period.  The
guidance recommends and uses a value of 10% over a year period of testing.  Reasons for this particular
choice are  discussed in Section 6.2.2. These strategies have consequences for the overall test power of a
well monitoring network, which are considered following control of the false positive error.

     The number  of tests  depends on the number  of monitoring constituents, compliance wells and
periodic  evaluations.  Statistical testing  on a  regular basis can be limited to constituents shown to be
reliable  indicators of a contaminant release  (discussed further in  Section 6.2.2). Depending  on site
conditions, some constituents may  need to be tested only at wells for a single regulated waste unit, rather
than across the entire facility well network.  The frequency of evaluation  is a program decision, but
might be modified in certain circumstances.

     Monitoring data for other parameters should still be routinely collected and reported to trace the
potential arrival of new chemicals into the groundwater, whether from changes  in waste management
practices or degradation over time into hazardous daughter products. By limiting statistically  evaluated
constituents to the most useful  indicators, the overall number of statistical tests can be reduced to help
meet the SWFPR objective.  Fewer tests also imply a  somewhat higher single test false positive error
rate, and therefore an improvement in power.

     As a second strategy, the  Type I error rate (atest) applied to each individual test can be lowered to
meet the SWFPR. Using the Bonferroni adjustment  (Miller, 1981), the individual test error is designed
to limit  the  overall (or experiment-wise} false positive rate a associated with n individual tests by
conducting each individual test at an adjusted significance level  of octest = o/w. Computational details for
this approach are provided in a later section.

      A  full Bonferroni adjustment strategy was neither implemented in previous guidance3 nor allowed
by regulation.  However, the principle  of partitioning individual test error rates to meet an overall
cumulative false positive error target is highly  recommended as a design  element in this  guidance.
Because  of RCRA regulatory limitations, its application  is  restricted to certain  detection monitoring
3 A Bonferroni adjustment was recommended in the 1989 Interim Final Guidance [IFG] as a post-hoc (i.e., 'after the fact')
  testing strategy for individual background-to-downgradient well comparisons following an analysis of variance [ANOVA],
  However, the adjustment does not always effectively limit the risks to the intended 5% false positive error for any ANOVA
  test. If more than 5 compliance wells are tested, RCRA regulations restrict the single  test error rate to a minimum of a =
  1% for each of the individual post-hoc tests following the F-test. This in effect raises the cumulative ANOVA test risk
  above 5% and considerably higher with a larger number of tested wells. At least one contaminated well would typically be
  needed to trigger the initial F-test prior to post-hoc testing. This fact was also noted in the 1989 IFG. Additionally, RCRA
  regulations  mandate a minimum a error rate  of 5% per constituent tested with this strategy.  For sites with extensive
  monitoring  parameter lists, this  means  a substantial risk of at least one false positive test result  during any statistical
  evaluation.

                                                 6-3                                      March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

tests— prediction and tolerance limits along with control charts. Where not restricted by regulation, the
Bonferroni approach could be used to design workable single-test or post-hoc testing for ANOVAs to
meet the overall SWFPR criterion.

        Using this strategy of defining individual false positive test rates to meet a cumulative error
target, the effect on  statistical power is direct.  Given a statistical test and fixed sample  size, a lower
false positive rate coincides with lower power of the  test to detect contamination at the well.  Some
improvement in single test power can be gained by increasing background sample sizes at a fixed test
error rate.  However, the third strategy of utilizing a  different or modified  statistical test is generally
necessary.

      This strategy involves choices among certain detection monitoring tests- prediction limits, control
charts and tolerance intervals— to enhance both power and false positive error control. Except for small
sites with  a very limited  number of tests,  any of  the three detection monitoring options should
incorporate  some manner of retesting. Through proper design, retesting can simultaneously achieve
sufficiently high statistical power while maintaining control of the SWFPR.

       RECOMMENDED GUIDANCE CRITERIA

      The design of all testing strategies should specifically address the multiple comparisons problem in
light of these two fundamental  concerns— an acceptably low false positive site-wide error rate and
adequate power.  The  Unified Guidance  accordingly recommends two statistical performance criteria
fundamental to good design of a detection monitoring program:

      1.  Application of an annual cumulative SWFPR design target, suggested at 10% per year.

      2.  Use of EPA reference power curves [ERPC] to gauge the cumulative, annual ability of any
         individual  test to detect contaminated groundwater when it exists. Over the course of a
         single year assuming normally-distributed background data,  any single test performed at
         the site should have the ability to detect  3 and 4 standard deviation increases  above
         background at specific power levels at least as high as the reference curves.

     False positive  rates  (or errors)  apply  both to  individual tests and  cumulatively to all tests
conducted in some time period.  Applying the SWFPR annual 10% rate places different sites and state
regulatory programs  on an  equal footing, so that no facility is  unfairly burdened by false positive test
results. Use of a single overall target allows  a proper comparison to be made between alternative test
methods in  designing  a statistical program.  Additional  details  in applying the  SWFPR include the
following:

    »»» The SWFPR  false positive rate should be measured on a site-wide basis, partitioned among the
       total  number of annual statistical tests.

    »»» The SWFPR applies to all statistical tests conducted in an annual or calendar year period.

    »«» The total number of annual statistical tests used in SWFPR calculations depends on the number
       of valid monitoring constituents, compliance wells and evaluation periods per year.  The number
       of tests  may or may not coincide with the number of annual sampling events, for example,  if data
       for a future mean test are collected quarterly and tested semi-annually.
                                              6-4                                     March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

    »«»  The Unified Guidance recommends a uniform approach for dealing with monitoring constituents
       not historically detected in background (e.g., trace organic compounds routinely analyzed in large
       analytical suites).   It  is  recommended that  such constituents  not  be included  in  SWFPR
       computations, and an  alternate evaluation protocol be used  (referred to  as the Double
       Quantification rule) discussed in Section 6.2.2.

     Statistical power refers to the ability of a test to identify real  increases in  concentration levels
above  background  (true  SSIs).  The power  of a test is  evaluated on population characteristics and
represents average behavior defined by repeated or an infinitely large number of samples.  Power is
reported as a fraction between 0 and 1, representing the probability that the test will identify a specific
level or degree  of increase above  background. Statistical power varies with the size of the average
population concentration  above background— generally fairly low  power to detect  small  incremental
concentrations and substantially increasing power at higher concentrations.

     The  ERPC describe the cumulative,  annual  statistical  power to detect  increasing  levels  of
contamination above a true background mean.  These curves are based on  specific normal detection
monitoring prediction limit tests of single future samples against background  conducted once, twice, or
four times in a year. Reference curve power is linked to relative, not absolute, concentration levels.
Actual  statistical  test power  is  closely  tied  to the underlying  variability  of  the  concentration
measurements. Since individual data set variability will differ by site, constituent, and often by well, the
EPA reference power curves provide a generalized  ability to estimate power by standardizing variability.
By convention, all background concentration  data are assumed to follow  a standard normal distribution
(occasionally referred to  in this document as  a Z-normal distribution)  with a true mean p,  =  0 and
standard deviation o = 1.0. Then, increases above background are measured in increasing the k standard
deviation units corresponding to &a mean units above baseline. When the background population can be
normalized via a transformation, the same normal-based ERPC can be used without loss of generality.

       Ideally, actual test power should be assessed using the original  concentration data and associated
variability, referred to as  effect size power analysis.  The power of any statistical test can be readily
computed and compared to the appropriate reference  curve, if not analytically, then by Monte Carlo
simulation. But the reference power curves laid out in the Unified Guidance offer an important standard
by which to judge the adequacy of groundwater statistical programs and  tests. They  can be universally
applied to all RCRA sites and offer a uniform way to assess the environmental and health protection
afforded by a particular statistical detection monitoring program.4

     Consequently,  it is  recommended that design of any detection monitoring statistical  program
include an assessment of  its ability to meet the power standards set out  in the Unified Guidance. The
reference  power curve  approach does not place  an  undue statistical burden on facility owners  or
operators, and is believed to be generally protective of human health and the environment.
4 The ERPCs are specifically intended for comparing background to compliance data in detection monitoring. Power issues
  in compliance/assessment monitoring and corrective action are considered in Chapters 7 and 22.

                                               6-5                                     March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

     Principal features of the ERPC approach include the following:

   »»»  Reference curves are based on upper 99% prediction limit tests of single future samples against
       background.  The background sample consists of n = 10 measurements, a minimally adequate
       background sample size typical of RCRA applications.  It is assumed that the background sample
       and compliance well data are normally distributed and from the same population.

   »»»  The three reference curves described below are matched to the annual frequency of statistical
       evaluations: one each for quarterly, semi-annual, and annual evaluations. The annual cumulative
       false positive testing error is maintained at 1%, testing 1, 2, or 4 single future samples annually
       against the same background.  This represents the ability to identify a release to groundwater in at
       least one of the 1, 2 or 4 tests over the course of a year. Reporting power on an annual basis was
       chosen to correspond with the application of a cumulative annual SWFPR.

   »»»  In  the  absence of an acceptable effect size  increase (Section 6.2.4), the Unified Guidance
       recommends that any statistical test provide at least 55-60% annual power to detecting a 3a(i.e.,
       3 standard deviation) increase above the true background  mean and at least  80-85%  annual
       power for detecting increases of4a.  The percent power criteria change slightly for the respective
       reference power curves, depending on the annual frequency of statistical evaluations. For normal
       populations, a 3a increase above the  background average approximately corresponds to the upper
       99th percentile of the background distribution, implying better than a 50% chance of detecting
       such an increase.  Likewise, a 4o increase corresponds  to a true mean greater than the upper
       99.99th percentile of the background distribution, with better than a 4-in-5 chance of detecting it.

   »»»  A  single statistical  test is not adequately powerful  unless  its  power matches or betters the
       appropriate reference curve, at least  for mean-level  increases of 3 to 4 standard deviation units.
       The same concept can be applied to the overall detection monitoring test design.  It is assumed
       for statistical  design  purposes that each individual  monitoring well and constituent is of equal
       importance, and assigned a common test false positive error.  Effective power then measures the
       overall ability of the statistical program to identify any  single constituent release in any well,
       assuming all remaining constituents and wells are at background levels. If a number of different
       statistical methods are employed  in a single design, effective power can be defined with respect
       to  the least powerful of the methods being employed.  Applying effective power in this manner
       would  ensure that every well and  constituent is evaluated with adequate statistical power to
       identify potential contamination, not just those where more powerful tests are applied.

   »»»  While the Unified  Guidance  recommends  effective power as a  general  approach, other
       considerations  may  outweigh statistical  thoroughness. Not all  wells and  constituents are
       necessarily of equal practical importance.  Specific site circumstances may also result in some
       anomalous weak test power (e.g., a number of missing samples in a background data  set for one
       or  more constituents), which might be remedied by eventually increasing background size. The
       user  needs to consider all factors including effective statistical  power criteria in assessing the
       overall strength of a detection monitoring program.
                                              6-6                                    March 2009

-------
Chapter 6. Detection Monitoring Design _ Unified Guidance

6.2.2 SITE-WIDE FALSE POSITIVE RATES [SWFPR]

     In this section, a number of considerations in developing and applying the SWFPR are provided.
Following a brief discussion of SWFPR computations, the next section explains the rationale for the
10% design target SWFPR. Additional detail regarding the selection of monitoring constituents follows,
and a final discussion of the Double Quantification rule for never-detected constituents is included in the
last section.

       For cumulative false positive error and SWFPR computations, the following approach is used. A
cumulative false positive  error rate acum is calculated as  the probability of at least one statistically
significant outcome for a total  number of tests  «r in a calendar year at a single  false  positive error rate
   t using the properties of the Binomial distribution:
       By rearranging to solve for atest , the 10% design SWFPR (. 1) can be substituted for acum and the
needed per-test false positive error rate calculated as:
                                      =1_(9y/T
                                  "'test   1  V y)

       Although these calculations are relatively straightforward and were used to develop certain K-
factor  tables in the Unified Guidance (discussed in Section  6.5  and in later chapters), a further
simplification is possible using the Bonferroni approximation.  This assumes that cumulative, annual
SWFPR is roughly the additive sum of all the individual test errors. For low false positive rates typical
of guidance  application, the Bonferroni results are satisfactorily close to the Binomial formula for most
design considerations.

     Using  this principle,  the  design 10%  SWFPR can be  partitioned  among  the  potential annual
statistical  tests  at  a facility in a number of ways.  For facilities  with  different annual  monitoring
frequencies,  the SWFPR can be divided among quarterly or semi-annual period tests. Given (XSWFPR = .1
and WE evaluation periods, the quarterly cumulative false positive target rate (XE at a facility conducting
quarterly testing would be (XE = (XSWFPR/WE = .1/4 = .025 or 2.5% (and similarly for semi-annual testing).
The  total or sub-divided  SWFPR can likewise be  partitioned among dedicated monitoring well
groupings at a multi-unit facility or among individual monitoring constituents as needed.

       DEVELOPMENT AND RATIONALE FOR THE SWFPR

     The  existing RCRA Part 264 regulations  for parametric or non-parametric analysis  of variance
[ANOVA] procedures mandate a Type I error of at least 1% for any individual  test, and  at least  5%
overall. Similarly, the RCRA Part 265 regulations require a minimum 1% error for indicator parameter
tests.   The rationale for minimum false positive requirements is motivated by statistical power. If the
Type I error is set too low, the power of the test will be unacceptably low  for any  given test. EPA was
historically not  able to specify a minimum level of acceptable power within the RCRA regulations. To
do so would require specification of a minimum difference of environmental concern between the null
and alternative  test hypotheses.  Limits on current knowledge about the  health and/or environmental
effects associated with incremental changes in concentration levels of Part 264 Appendix IX or Part 258
Appendix II constituents greatly complicate this task. Tests of non-hazardous  or low-hazard indicators
                                              6-7                                     March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

might have different power requirements than for hazardous constituents.  Therefore, minimum false
positive rates  were  adopted  for  ANOVA-type  procedures until  more  specific guidance  could  be
recommended.  EPA's main  concern was adequate statistical  power to  detect real contamination of
groundwater, and not enforcing commonly-used false positive test rates.

      This  emphasis is  evident  in §264.98(g)(6)  and  §258.54(c)(3)  for  detection monitoring  and
§264.99(i) and §258.55(g)(2)  for compliance monitoring. Both pairs of provisions allow the owner or
operator to demonstrate that any statistically significant difference between background and compliance
point wells or between  compliance point wells  and  the GWPS is an  artifact caused by an error in
sampling, analysis,  statistical evaluation, or natural  variation  in  groundwater chemistry.  The rules
clearly expect that there will be occasional false positive errors, but existing rules are silent regarding the
cumulative frequency of false positives at regulated facilities.

      As previously noted, it is  essentially impossible to maintain a low cumulative  SWFPR for
moderate to large monitoring networks if the Type I errors for individual tests must be kept at or above
1%.  However, the RCRA regulations do not impose similar false positive error requirements on the
remaining  control  chart, prediction  limit and tolerance interval  tests.   Strategies that incorporate
prediction limit or control chart retesting can achieve very low individual  test false positive rates while
maintaining adequate power to detect contamination.  Based on prediction limit research in  the  1990's
and after, it became clear that these alternative methods with suitable retesting could also control the
overall cumulative false positive error rate to manageable levels.

       This guidance suggests the use of an annual SWFPR of .1 or 10% as a fundamental element of
overall detection monitoring design.  The choice  of a 10% annual SWFPR was made in light  of the
tradeoffs between false positive control and testing power. An annual period was chosen to put different
sized facilities on a common footing regardless of variations in scheduled testing. It is recognized that
even with such a limited error rate, the probability of false positive outcomes over a number of years
(such  as in the lifetime of  a 5-10 year permit) will be higher.   However,  such relatively limited
eventualities can be identified and adjusted for, since the  RCRA regulations do allow for demonstration
of a false positive error.  State programs may choose to use a different annual rate such as 5%  depending
on  the circumstances.   But  some predefined  SWFPR in a given  evaluation  period  is  essential for
designing a detection monitoring program, which can  then be translated into target individual test rates
for any alternative statistical testing strategy.

      To implement this  recommendation, a given facility should identify its yearly evaluation schedule
as  quarterly,  semi-annual, or  annual. This  designation is used both  to select an appropriate EPA
reference power curve by which to gauge acceptable  power, and to select prediction limit and control
chart multipliers useful in  constructing detection  monitoring tests. Some of the strategies described in
the Unified Guidance in later chapters require that more than one observation per compliance well be
collected prior to  statistical testing. The  cumulative, annual false positive rate is linked not  to the
frequency  of sampling  but  rather to the frequency of statistical evaluation.  When resamples (or
verification resamples)  are incorporated into a statistical procedure  (Chapter 19), the  individual
resample comparisons comprise  part  of a  single test. When  a single future  mean of m  individual
observations is evaluated against a  prediction  limit,  this constitutes  a  test based on one  mean
comparison.
                                               6-8                                    March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

NUMBER OF TESTS AND CONSTITUENTS

     In designing a detection monitoring program to achieve the target SWFPR, the number of annual
statistical  tests to be conducted needs to be identified.  This number is calculated as the number  of
distinct monitoring  constituents  x the number of compliance wells in  the network x the number  of
annual  evaluations.  Five constituents and  10  well  locations statistically  evaluated  semi-annually
constitute 100 annual tests (5x10x2), since each distinct well-constituent pair represents a different
statistical  test  that must be evaluated against their respective backgrounds. Even smaller facilities are
likely to have a substantial number of such tests, each incrementally adding to the SWFPR.

     While the retesting strategies outlined in Chapters 19 and 20 can aid tremendously in limiting the
SWFPR and ensure adequate statistical power, there are practical limits to meeting these goals due to the
limited number of groundwater observations that can be collected and/or the number of retests which can
feasibly be run.  To help  balance the risks of false positive  and false negative errors, the number  of
statistically-tested monitoring parameters should be limited to constituents thought  to be  reliable
indicators of a contaminant release.

     The  guidance assumes that data from large suites of trace elements and organics along with a set  of
inorganic  water  quality indicators (pH, TDS, common  ions, etc.) are  routinely collected as part  of
historical  site groundwater monitoring.  The number of constituents potentially available for testing can
be quite large,  perhaps as many as 100 different analytes.  At some sites, the full monitoring lists are too
large to feasibly limit the SWFPR while maintaining sufficiently high power.

     Non-naturally  occurring chemicals such as volatile organic compounds  [VOC]  and semi-volatile
organic compounds  [SVOC]  are often viewed as excellent indicators of groundwater contamination, and
are thereby included in the monitoring programs of many facilities. There is  a common misperception
that the greater the number of VOCs and SVOCs on the monitoring list, the greater the statistical power
of the monitoring program. The reasoning is that if none of these chemicals should normally be detected
in groundwater —  barring a release —  testing for more  of them ought to improve the  chances  of
identifying contamination.

     But  including a large suite of VOCs and/or SVOCs among the mix of monitoring parameters can
be counterproductive to the  goal of maintaining adequate effective  power  for the site as a whole.
Because of the trade-off between statistical power and false positive  rates (Chapter 3), the power  to
detect groundwater contamination in one  of these wells even with a retesting strategy in place may be
fairly low unless background  sample  sizes are quite large.  This is  especially  true if the regulatory
authority only  allows for a  single retest.

     Suppose  40  VOCs  and certain  inorganic  parameters  are  to  be  tested  semi-annually at 20
compliance wells totaling  1600 annual statistical tests.  To maintain a  10% cumulative annual SWFPR,
the per-test false positive rate would then need to be set at  approximately octest = .0000625.  If only 10
constituents were selected  for formal testing,  the per-test rate would be increased to octest = .00025.  For
prediction limits  and other detection tests, higher false positive test rates translate to lower K-factors and
improved  power.

      Some means of reducing the number  of tested constituents is  generally necessary to design an
effective detection monitoring system.  Earlier discussions have already suggested one obvious first  step,

                                              6-9                                     March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

by eliminating historically non-detected constituents in background  from the formal list of deletion
monitoring  constituents  (discussed  further in the following section).   These constituents  are  still
analyzed and informally tested, but do not count against the SWFPR.

     Results of waste and leachate testing and possibly soil gas analysis should serve as the initial basis
for designating constituents that are reliable leak detection indicators. Such specific constituents actually
present in, or derivable from, waste or soil gas samples, should be further evaluated to determine which
can be analytically detected a reasonable proportion of the time.  This  evaluation should include
considerations of how soluble and mobile a constituent may be in the underlying aquifer. Additionally,
waste or leachate concentrations should be high enough relative  to the groundwater levels to allow for
adequate detection.  By limiting  monitoring  and statistical  tests to fewer parameters with reasonable
detection frequencies and that are significant components of the  facility's waste, unnecessary statistical
tests can be avoided while focusing on the reliable identification of truly contaminated groundwater.

     Initial leachate testing should not serve as  the sole basis for designating monitoring parameters.
At many active hazardous waste facilities and solid waste landfills, the composition of the waste may
change over time. Contaminants  that initially were all non-detect may not remain so.  Because of this
possibility, the Unified Guidance recommends that the list of monitoring  parameters subject to formal
statistical evaluation be periodically reviewed, for example, every three to five years. Additional leachate
compositional  analysis and testing may be necessary, along with the measurement of constituents not on
the monitoring list but of potential health or environmental concern. If previously undetected parameters
are discovered in this evaluation, the permit authority should consider revising the monitoring  list to
reflect those analytes that will best identify potentially contaminated groundwater in the future.

     Further reductions are possible  in the number of constituents used for formal detection monitoring
tests,  even among constituents  periodically or always detected.  EPA's  experience at hazardous waste
sites and landfills across the country has shown that VOCs and  SVOCs detected in a release generally
occur in clusters; it is less common to detect only a single constituent at a given location. Statistically,
this implies that groups of detected VOCs or SVOCs are likely to be correlated. In  effect, the correlated
constituents are measuring a release in similar fashion and  not providing fully independent measures.
At petroleum  refinery sites, benzene, toluene, ethylbenzene and xylenes measured in a VOC  scan  are
likely to be detected together  Similarly at sites having releases of 1,1,1-trichloroethane, perhaps 10-12
intermediate chlorinated hydrocarbon degradation compounds  can  form in the aquifer over time.
Finally, among water quality indicators like common ions and TDS, there is a great deal of geochemical
inter-relatedness.   Again,  two or three indicators from  each of these  analyte groups may suffice as
detection monitoring constituents.

     The overall goal should be to select only the most reliable monitoring constituents for detection
monitoring test purposes.  Perhaps 10-15 constituents may  be a reasonable target, depending on site-
specific needs.  Those analytes not  selected should  still continue  to be  collected and evaluated.  In
addition to using the informal test to identify previously undetected constituents described in the next
section, information on the remaining constituents (e.g., VOCs, SVOCs  and trace elements) can still be
important in assessing groundwater conditions, including additional confirmation of a detected release.
                                              6-10                                    March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

DOUBLE  QUANTIFICATION RULE

     From the previous discussion, a full set of site historical monitoring parameters can be split into
three distinct groups: a) those reliable indicators and hazardous constituents selected for formal detection
monitoring testing and contributing to the SWFPR; b) other analytes which may be occasionally or even
frequently detected and will be monitored for general groundwater quality information but not tested;
and c) those  meeting the "never-detected" criteria. The last group may still be of considerable interest
for eventual formal testing,  should site or waste management conditions change and new compounds be
detected. All background measurements in the "never-detected"  group should be non-detects, whether
the full historical set or a subgroup considered most  representative (e.g., recently collected background
measurements using an  improved analytical method.5).   The following rule is suggested to provide a
means of evaluating "never-detected"  constituents.

     The  Double Quantification rule implies  that statistical tests should be designed for each of the
constituents in the first group.  Calculations involving the SWFPR should cover these constituents, but
not include constituents in second and the third '100% non-detect' categories.  Any constituent in this
third group should be evaluated by the following simple, quasi-statistical rule6:

          A confirmed exceedance is registered if any well-constituent pair in the '100%
          non-detect' group exhibits quantified measurements (i.e., at or above the
          reporting limit [RL]) in two consecutive sample and resample events.
     It is assumed when estimating an SWFPR using the Bonferroni-type adjustment, that each well-
constituent test is at equal risk for a specific,  definable false positive error. As a justification for this
Double Quantification rule,  analytical procedures involved  in identifying a reported non-detect value
suggest that  the error  risk is probably  much lower  for most chemicals analyzed as "never-detected."
Reporting limits are set high enough  so that if a chemical is not present at all in the  sample, a detected
amount will  rarely be recorded on the  lab sheet. This is particularly the case since method detection
limits  [MDLs] are often  intended as  99%  upper  prediction  limits  on the measured signal  of  an
uncontaminated laboratory sample. These limits are then commonly multiplied by a factor of 3 to 10 to
determine the RL.

     Consequently,  a  series of measurements for VOCs or SVOCs on samples  of uncontaminated
groundwater  will tend to be listed as a string of non-detects with possibly a very occasional low-level
detection.  Because the observed measurement levels (i.e., instrument signal levels)  are usually  known
only to the chemist, an approximate prediction limit  for the  chemical basically has to be set at the RL.
However, the true measurement distribution is likely to be clustered much more closely around zero than
the RL (Figure 6-1), meaning that the false positive rate associated with setting the RL as the prediction
5 Note: Early historical data for some constituents (e.g., certain filtered trace elements) may have indicated occasional and
  perhaps unusual detected values using older analytical techniques or elevated reporting limits. If more recent sampling
  exhibits no detections at lower reporting limits for a number of events, the background review discussed in Chapter 5 may
  have determined that the newer, more reliable recent data should be used as background.  These analytes could also be
  included in the '100% non-detect' group.

   The term "quasi-statistical" indicates that although the form is a statistical prediction limit test, only an approximate false
  positive error rate is implied for the reporting limit critical value.  The test form follows l-of-2 or l-of-3 non-parametric
  prediction limit tests using the maximum value from a background data set (Chapter 19).

                                               6-11                                    March  2009

-------
Chapter 6. Detection Monitoring Design
Unified Guidance
limit is likely already much lower than the Bonferroni-adjusted error rate calculated above. A similar
chain of reasoning would apply to site-specific chemicals that may be on the monitoring list but have
never been detected at the facility. Such constituents would also need a prediction limit set at the RL.

   Figure 6-1. Hypothetical Distribution of Instrument Signals  in  Uncontaminated
                                       Groundwater
                                       Measured Concentration
     In general,  there  should be  some  minimally sufficient sample  numbers  to  justify  placing
constituents in the "never-detected" category.  Even such a recommendation needs to consider individual
background well versus pooled well data.  Depending on the number of background wells (including
historical compliance well data used as background which reflect the same non-detect patterns), certain
risks may have to be taken to implement this strategy. With the same total number of non-detects (e.g.,
4 each in 5 wells versus 20 from a single well), the relative risk can change. Certain non-statistical
judgements may be needed, such as the likelihood of particular constituents arising from the waste or
waste management unit.  At a minimum, we recommend that at least 6 consecutive non-detect values
initially be present in each well of a pooled group,  and  additional background well sampling should
occur to raise this number to 10-15.

     Having  10-15 non-detects as a basis, a maximum worst-case probability of a future false  positive
exceedance under Double Quantification rule testing could be estimated.  But it should be kept  in mind
that the true individual comparison false positive rates based on analytical considerations are likely to be
considerably lower.  The number of non-detect constituents evaluated under the rule will also play a role.
There will  be some cumulative false positive error based  on the number  of comparisons at some true
false positive single test error or errors.  Since the true false positive test rates cannot be known (and may
vary considerably among analytes), it is somewhat problematic to make this cumulative false  positive
error estimate. Yet there is some likelihood that occasional false positive exceedances will occur under
this rule.
                                             6-12
        March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

     Some flexibility will be required in evaluating such outcomes, particularly if there is doubt that a
confirmed exceedance is actually due to a release from the regulated unit. In this circumstance, it might
be appropriate to allow for a second resample as more definitive confirmation.

     In implementing the Double Quantification rule, consideration should be given to how soon a
repeat  sample  should be taken.   Unlike  detectable parameters, the  question of autocorrelation is
immaterial since the compound should not be present in the background aquifer.  A sufficiently long
interval should occur between the initial and repeat samples to minimize the possibility of a systematic
analytical error.  But the time interval should be short enough to avoid  missing a subsequent real
detection due to seasonal changes in the aquifer depth or flow direction.  It is suggested that 1-2 months
could be appropriate, but will  depend on site-specific hydrological conditions.

     Using this rule, it should be possible to construct adequately powerful prediction and control limits
for naturally-occurring and detectable inorganic and organic chemicals in almost every setting.  This is
especially helpful at larger sites, since the total number of tests  on which the per-test false positive rates
(atest) are based will be significantly reduced. Requiring a verified quantification for previously non-
detected constituents should ensure that spurious  lab  results do not falsely trigger  a facility into
compliance/assessment monitoring, and will more reliably  indicate the presence of chemicals  that have
heretofore not been found in background.

6.2.3 RECOMMENDATIONS  FOR STATISTICAL POWER

     The second  but  more  important regulatory goal  of a testing  strategy  is to ensure  sufficient
statistical power for detecting contaminated groundwater.  Technically, in the context of groundwater
monitoring, power refers to  the probability that  a statistical  test will  correctly identify  a significant
increase in concentration above background. Note that power is typically defined with respect to a single
test, not a network of tests.  In this guidance, cumulative power is assessed for a single test over an
annual period, depending on the frequency  of the evaluation.  Since some testing procedures may
identify contamination more readily when several wells in  the network are contaminated as opposed to
just one or two,  the Unified Guidance  recommends that all testing strategies be compared on the
following more stringent common basis.

      The effective power of a testing protocol across a network of well-constituent pairs is defined as
the probability of detecting contamination in the monitoring  network when one  and only  one well-
constituent pair is contaminated. Effective power is a conservative measure of how a testing regimen
will perform across the network, because the set of statistical tests must uncover one contaminated well
among many clean ones (i.e., like 'finding a  needle in a haystack').  As mentioned above, this initial
judgment may need to be qualified with effect size and other site-specific considerations.

       INTRODUCTION TO  POWER CURVES

     Perhaps the best way to describe the power function associated with a particular testing procedure
is via a graph, such as the example below of the power of a  standard normal-based upper prediction limit
with 99%  confidence (Figure  6-2). The power in percent  is plotted along the _y-axis against the
standardized mean level of contamination along the x-axis. The standardized contamination levels are
presented in units of standard  deviations above the baseline  (defined as the true background mean). This
                                              6-13                                    March 2009

-------
Chapter 6.  Detection Monitoring Design _ Unified Guidance

allows different power curves to be compared across constituents, wells, or well-constituent pairs. These
standardized units A in the case of normally-distributed data may be computed as:

                        (Mean Contamination Level) - (Mean Background Level)
                    A = -                [6.1]
                                    (SD of Background Population)

     In some  situations, the probability that  contamination will be detected by a particular testing
procedure may be difficult if not impossible to derive analytically and will have to be simulated using
Monte Carlo analysis on a computer. In these cases, power is typically estimated by generating normally-
distributed random values at different mean contamination levels and repeatedly simulating the test
procedure. With enough repetitions a reliable power curve can be plotted.

     In the case of the normal power curve in Figure 6-2, the power values were computed analytically,
using properties of the non-central t-distribution. In particular, the statistical  power of a normal 99%
prediction limit for the next single future value can be calculated as
                                                                                             [6.2]
where A is the number of standardized (i.e., standard deviation) units above the background population
mean, (l-(3) is the fractional power, 8 is a non-centrality parameter, and:


                                                                                             [6.3]
represents  a non-central  ^-variate with  («-l)  degrees of freedom and non-centrality parameter 8.
Equation [6.2] was used with n = 10 to generate Figure 6-2.7

     On a general power curve,  the power at A = 0 represents the false positive rate  or size of the
statistical test, because at that point no contamination is actually present (i.e., the background condition),
even though the curve indicates how often a significant concentration increase will be detected.  One
should be careful to distinguish between the SWFPR across many statistical tests and the false positive
rate represented on a curve measuring effective power. Since the effective power is defined as the testing
procedure's ability to identify a single contaminated well-constituent  pair, the  effective power curve
represents an individual test, not a network of tests.  Therefore, the value of the curve at A = 0 will  only
indicate the false  positive rate associated with an  individual test (atest),  not across the network  as  a
whole.  For many of the  retesting strategies discussed in  Chapters  19 and 20,  the individual  per-test
false positive rate will be quite small and may appear to be nearly zero on the effective power curve.
  For users with access to statistical software containing the non-central T-distribution, this power curve can be duplicated.
  For example, the A = 3o fractional power can be obtained using the following inputs: a central t-value of t.99j 9 = 2.821, 9 df,
  and 8  = 3/yl + (l/lOj = 2.8604 .  The fractional power is .5414. It should be noted that the software may report the
                                               6-14                                    March 2009

-------
Chapter 6. Detection Monitoring Design
                                                                 Unified Guidance
       Figure 6-2.  Normal Power Curve (n = 10) for 99% Prediction Limit Test

                 1.00
CD
£
o
o_
                 0.75 -
                 0.50 H
                 0.25 -
                 0.00
                        0
                                  \
                                 2
 \
3
 \
4
                                       A(SDs above Background)

     To properly interpret a power curve, note that not only is the probability greater of identifying a
concentration increase above background (shown as a decimal value between 0 and 1 along the vertical
axis) as the magnitude of the increase gets bigger (as measured along the horizontal axis), but one can
determine the probability of identifying certain kinds of increases. For instance, with effective  power
equivalent  to that in  Figure 6-2, any mean concentration increase of at least 2 background  standard
deviations will be detected about 25% percent of the time, while  an increase of 3  standard deviations
will be  detected with approximately 55% probability or better than 50-50 odds. A mean increase of at
least 4 standard deviations will be detected with about 80% probability.

     An increase of 3  or 4 standard  deviations above the  baseline may or may not have practical
implications for human health or the environment. That will  ultimately depend on  site-specific factors
such  as the  constituents  being monitored, the  local  hydrogeologic environment,  proximity to
environmentally sensitive populations, and the observed  variability in background concentrations. In
some circumstances, more sensitive testing procedures might be warranted. As a general guide especially
in the  absence of direct site-specific information,  the  Unified  Guidance  recommends  that when
background is approximately normal in distribution,8 any statistical test should be able to detect a 3
  probability as (P) rather than (1-P). For more complex power curves involving multiple repeat samples or multiple tests,
  integration is necessary to generate the power estimates.
  If a non-parametric test is performed, power (or more technically, efficiency) is often measured by Monte Carlo simulation
  using normally distributed data. So these recommendations also apply to that case.
                                              6-15
                                                                        March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

standard deviation increase at least 55-60% of the time and a 4 standard deviation increase with at least
80-85% probability.

       EPA REFERENCE POWER CURVES

       Since effect sizes discussed in the next section often cannot or have not been quantified, the
Unified Guidance recommends using the ERPC as a suitable basis of comparison for proposed testing
procedures. Each reference power curve corresponds to one of three typical yearly statistical evaluation
schedules — quarterly, semi-annual, or annual —  and represents  the cumulative  power achievable
during a single year at one  well-constituent pair by a 99% upper (normal) prediction limit based on n =
10 background measurements and one new measurement from the compliance well (see Chapter 18 for
discussion of normal prediction limits). The ERPC are pictured in Figure 6-3 below.
     Any proposed  statistical test procedure with effective power  at least as high as the appropriate
ERPC, especially in the range of three or more  standard deviations above the background mean, should
be considered to have reasonable power.9 In particular, if the effective power first exceeds the ERPC at a
mean concentration increase no greater than 3 background standard deviations (i.e., A < 3), the power is
labeled 'good;'  if the effective power first exceeds the ERPC at  a mean increase between  3  and 4
standard deviations (i.e., 3 < A < 4), the power is considered 'acceptable;' and if the first exceedance of
the ERPC does not occur until an increase greater than 4 standard deviations  (i.e., A > 4), the power is
considered 'low.'
     With respect to the ERPCs, one should keep the following considerations in mind:

  1.    The effective power of any testing method applied to a groundwater monitoring network can be
       increased merely by relaxing the SWFPR guideline, letting the SWFPR become larger than 10%.
       This is why a maximum annual SWFPR of 10% is suggested as standard guidance, to ensure fair
       power comparisons  among competing tests and to limit the overall network-wide false positive
       rate.

  2.    The ERPCs are based on cumulative power  over a one-year period.  That is, if a single well-
       constituent pair is contaminated at standardized  level A during each of the yearly evaluations, the
       ERPC  indicates the  probability that  a  99%  upper  prediction limit test  will  identify the
       groundwater  as impacted during  at least one of those evaluations. Because the number of
       evaluations not only varies by facility, but also impacts the cumulative one-year power, different
       reference power curves  should be employed  depending on a  facility's  evaluation schedule.
       Quarterly evaluators  should  utilize the  quarterly reference power curve  (Q);  semi-annual
       evaluators the semi-annual curve (S); and annual evaluators the annual curve (A).

  3.    If Monte Carlo simulations are used to evaluate  the power of a proposed testing method,  it
       should incorporate  every aspect of the procedure, from initial screens  of the data  to final
  When using a retesting strategy in a larger network, the false positive rate associated with a single contaminated well (used
  to determine the effective power) will tend to be much smaller than the targeted  SWFPR. Since the point at which the
  effective power curve intersects A = 0 on the standardized horizontal axis represents the false positive rate for that
  individual test, the effective power curve by construction will almost always be less than the EPA reference power curve for
  small concentration increases above background. Of more concern is the relative behavior of the effective power curve at
  larger concentration increases, say two or more standard deviations above background.

                                              6-16                                    March 2009

-------
Chapter 6.  Detection Monitoring Design
Unified Guidance
       decisions concerning the presence of contamination. This is especially applicable to strategies
       that involve some form of retesting at potentially contaminated wells.

       Although monitoring networks incorporate multiple well-constituent pairs, effective power can
       be gauged by simulating contamination in one and only one constituent at a single well.

       The ERPCs should be considered a minimal power standard. The prediction limit test used to
       construct these reference curves  does not incorporate retesting  of any sort,  and is based on
       evaluating a single new measurement from the contaminated well-constituent pair. In  general,
       both retesting and/or the evaluation of multiple compliance point measurements tend to improve
       statistical power, so proposed tests that include such elements should be able to match the ERPC.

       At sites employing multiple types of test procedures (e.g., non-parametric prediction limits for
       some constituents, control charts for other constituents), effective power should be computed for
       each type of procedure to  determine which type  exhibits the least statistical  power. Ensuring
       adequate power across the site implies that the least powerful procedure should match or exceed
       the appropriate ERPC, not just the most powerful procedure.

                        Figure 6-3.  EPA Reference  Power Curves
                         Annual
                         Semi-Annual
                        - Quarterly
                    ° t_
                                           2         3

                                           SO Units Above BG
                                             6-17
        March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

6.2.4 EFFECT SIZES AND DATA-BASED POWER CURVES

       EFFECT SIZES

     If site-specific or chemical-specific risk/health information is available particularly for naturally-
occurring constituents, it can be used in some circumstances to develop an effect size of importance. An
effect size  (cp) is simply the smallest concentration increase above the mean background level that is
presumed or known to have a measurable, deleterious impact on human health and/or the environment,
or that would clearly signal the presence of contamination.

     When an effect size can be quantified for a given constituent and is approved by the regulating
authority, the acceptable power of the statistical test can be tailored to that amount. For instance, if an
effect size for lead in groundwater at a particular site is 9 = 10 ppb, one might require that the statistical
procedure have an 80% or 95% chance of detecting such an increase. This would be true regardless of
whether the power curve for lead at that site matches the ERPC. In some cases, an agreed-upon effect
size will result in a more stringent power requirement compared to the ERPCs. In other cases, the power
standard might be less stringent.

     Effect sizes are not known or have not been  determined  for many groundwater constituents,
including many inorganic parameters that have detection frequencies high  enough to be amenable to
effect size calculations. Because of this, many users will routinely utilize the relative power guidelines
embodied in the ERPC. Even if a specific effect size cannot be determined, it is helpful to consider the
site-specific and test-specific  implications of a three or four standard deviation concentration  increase
above background. Taking the background sample mean (x )  as the estimated baseline, and estimating
the underlying population variability by using the sample background standard deviation (s),  one can
compute the approximate actual concentrations associated with a three, four, five, etc. standard deviation
increase above the baseline (as would be done in computing a data-based power curve; Section 6.2.4).
These concentration values will only be approximate,  since the true background mean (|i) and  standard
deviation (a) are unknown. However, conducting this  analysis can be useful in at least two ways. Each
is illustrated by a simple example.

       By associating the standardized units on a reference power curve with specific but approximate
concentration levels, it is possible to evaluate whether the anticipated power characteristics of the chosen
statistical method are adequate for the site in question. If not, another method with better power might
be needed.  Generally, it is useful to discuss  and report statistical power in terms of concentration
levels rather than theoretical units.

       ^EXAMPLE 6-1

       A potential permit GWPS for lead is 15 ppb, while natural background lead levels are normally
distributed  with  an average of 6  ppb and a standard  deviation of 2  ppb.   The regulatory agency
determines  that a statistical test should be able to identify an exceedance of this GWPS with high power.
Further assume that the power curve for a particular statistical test indicated  40% power at 3  standard
deviations and 78% power at 4o above background (a low power rating).

       By comparing the actual standard deviation estimate to the required target increase q> = (15-6)72 =
4.5 standard units,  the power at the critical effect size would be 80% or higher using Figure  6-2 as a

                                             6-18                                   March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

rough guide. This might be sufficient for monitoring needs even though the test did not meet the EPA
reference criteria. Of course, the results apply only to this specific well-constituent test. -^

       For a given background  sample, one can consider the regulatory and environmental impact of
using that particular background as the basis of comparison in detection monitoring. Especially when
deciding between interwell and intrawell tests at the same  site,  it is not unusual for the intrawell
background from an individual well to exhibit much  less variability than a larger set of observations
pooled from multiple upgradient wells. This difference can be important since  an intrawell  test and an
interwell test applied to the same site —  using identical relative power criteria — might be associated
with different risks to human health and the environment. A similar type of comparison might also aid in
deciding whether the degrees of freedom  of an intrawell test ought to be enlarged via a pooled estimate
of the intrawell standard deviation (Chapter 13), whether a non-adjusted intrawell test is adequate, or
whether more background sampling ought to be conducted prior to running intrawell tests.

       ^EXAMPLE 6-2

     The  standard deviation  of an intrawell background population is  Ointra = 5 ppb,  but  that of
upgradient, interwell background is Ointer = 10 ppb. With the increased precision of an intrawell method,
it may be possible to detect a 20 ppb increase with high probability  (representing a A = 4Ointra increase),
while the corresponding probability for an interwell test is much lower (i.e., 20 ppb = 2ointer = A). Of
course, even if the  intrawell test meets the ERPC target at four standardized units above background,
consideration should be given as to whether or not 20 ppb is a meaningful increase. -4

     One caveat is that calculation of either effect sizes or data-based power curves (see below) requires
a reasonable estimate of the background standard deviation (a). Such calculations may often be possible
only for naturally-occurring inorganics or other constituents with  fairly high detection frequencies in
groundwater. Otherwise, power computations based on an effect size or the estimated standard deviation
(s) are likely to be unreliable due to the presence of left-censored measurements (i.e., non-detects).

     A type of effect  size  calculation  is  presented  in  Chapter  22 regarding  methods for
compliance/assessment and corrective  action monitoring. A  comparable  effect size is computed by
considering changes in mean concentration levels  equal  to  a multiple of a  fixed GWPS or clean-
up/action level. While  the mean level changes are multiples of the concentration limit and in that sense
still relative, because they are tied to a fixed concentration standard, the power  of the test can be linked
to specific concentration levels.

       DATA-BASED POWER CURVES

     Even if basing power on a specific effect size is impractical for a given facility or constituent, it is
still possible to relate power to absolute concentration  levels rather  than to  the standardized units of the
ERPC. While exact statistical power depends on the unknown population standard deviation (a), an
approximate power curve can be constructed based on the estimated background standard deviation (s).
Instead of an estimate of power at a single effect size (depicted in Example  6-1), the actual power over a
range of effect sizes can be evaluated.  Such a graph is denoted in the Unified Guidance as  a data-based
power curve, a term first coined by Davis  (1998).
                                              6-19                                   March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

     Since the sample standard deviation (s) is calculated from actual groundwater measurements, this
in turn  changes  an abstract power curve based on relative concentrations (i.e., ka units above  the
baseline  mean) into one displaying approximate,  but absolute, concentrations (i.e., ks units above
baseline). The advantages of this approach include the following:

    »»»  Approximate data-based power curves allow the user to determine statistical  power  at any
       desired effect size ((])).

    »»»  Even if the effect size ((])) is unspecified, data-based power curves tie the performance of the
       statistical test back to actual concentration levels of the population being tested.

    »»»  Once the theoretical power curve of a particular statistical test is known,  a  data-based power
       curve  is extremely easy to construct. One merely substitutes the observed background standard
       deviation (s) for a and multiply by k to determine concentration values along the horizontal axis
       of the power curve. Even if the theoretical power curve is unknown, the same calculations can be
       made on the reference curve to derive an approximate site-specific, data-based power curve for
       tests roughly matching the performance of the ERPCs.

    »»»  If the choice between  an interwell test  and an intrawell approach is a  difficult one  (Section
       6.3.2), helpful power comparisons can  be made between intrawell and interwell tests at the same
       site using data-based power curves. Even if both tests meet the ERPC criteria, they may be based
       on different sets of background measurements,  implying that the  interwell standard deviation
       Center) might  differ from the intrawell standard deviation (smtra). By plotting both data-based
       power curves on the same set of axes, the comparative performance of the tests  can be gauged.

       ^EXAMPLE 6-3

     The following  background sample is used to construct a  test with theoretical  statistical power
similar to the ERPC for annual evaluations  (see Figure 6-2). What will an approximate data-based
power curve look like, and  what is the approximate power for detecting a concentration  increase of 75
ppm?
Quarter
1/95
4/95
7/95
10/95
1/96
Mean
SD
Sulfate Concentrations (ppm)
BW-1 BW-2
560
530
568
490
510
545.0 ppm
29.7 ppm
550
570
540
542
590

                                             6-20                                    March 2009

-------
Chapter 6. Detection Monitoring Design
        Unified Guidance
       SOLUTION
     The  sample standard deviation of the pooled background sulfate concentrations is 29.7 ppm.
Multiplying this amount by the number of standard deviations above background along the x-axis in
Figure 6-2 and re-plotting, the approximate data-based power curve of Figure 6-3 can be generated.
Then the statistical power for detecting an increase of 75 ppm is almost 40%.

               Figure 6-3. Approximate s-Based  Power Curve for Sulfate
                         o.o
                                                100        150

                                      Sulfate Cone. Increase (ppm)
200
     Had the pooled sample size been n = 16 using the same test and sample statistics, a different and
somewhat more powerful theoretical power curve would result.  This theoretical curve can be generated
(for a 1-of-l prediction limit test) using the non-central T-distribution described earlier, if a user has the
appropriate statistical  software package.  The power for a 75 ppm increase can be calculated using
S = 75/A/l + (l/16) = 2.45 and t.99, 15 = 2.602, as closer to 46%.  The larger background sample  size
makes for a more powerful test.  -^

6.2.5 SITES USING MORE THAN ONE STATISTICAL METHOD

     There is no requirement that a facility apply one and only one statistical method to its groundwater
monitoring program.  The RCRA  regulations explicitly  allow for the  use of multiple  techniques,
depending on the distributional properties of the constituents being monitored and the characteristics of
the site.  If some constituent data contain a high percentage of non-detect values,  but  others can be
normalized, the statistical approach should vary by constituent.

     With interwell testing, parametric prediction limits might be  used with certain constituents and
non-parametric prediction limits for other highly non-detect parameters. If intrawell testing is used, the
most appropriate statistical technique for one constituent might differ at certain groups of wells than for
others. Depending on the monitoring constituent, available individual  well background, and other  site-
specific factors, some combination of intrawell prediction limits, control charts, and Wilcoxon rank-sum
tests might come into play. At other sites, a mixture of intrawell and interwell tests might be conducted.

     The Unified Guidance offers a range of possible methods which can be matched to the statistical
characteristics of the observed data. The primary goal is that the statistical program should maximize the
                                             6-21
               March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

odds  of making correct judgments about groundwater quality. The guidance  SWFPR and  ERPC
minimum power criteria serve as comprehensive guides for assessing any of the statistical methods.

     One major concern  is how statistical power should be compared when multiple methods  are
involved. Even if each method is so designed as not to exceed the recommended SWFPR, the effective
power for identifying contaminated  groundwater may vary considerably by technique and specific type
of test. Depending on the well network and statistical characteristics of available data, a certain control
chart test may or may not be as powerful as normal prediction limits.  In turn, a specific non-parametric
prediction limit test may be more powerful than some parametric versions.  It is important that effective
power be defined consistently, even at sites where more than one statistical method is employed.

     The guidance encourages employing the effective power concept in assessing  the ability of the
statistical  program to correctly identify and  flag real concentration increases above background.  As
already  defined, effective power is the probability that such an increase will be identified even  if only
one well-constituent pair is contaminated. Each well-constituent pair being tested should be considered
equally  at risk of containing a true increase above background. This also implies that the effective power
of each  statistical test in use should meet the criteria of the EPA reference curves.  That is, the test with
the least power should still have adequate power for identifying mean concentration increases.

     The Unified Guidance does not recommend that a single composite measure of effective power be
used  to gauge  a program's ability to identify  potential contamination.   To understand  this last
recommendation, consider the following hypothetical example.  Two  constituents  exhibiting different
subsurface travel times and diffusive potentials in the underlying aquifer are monitored with different
statistical techniques. The constituent  with the  faster travel time might be measured using a test with
very low effective power (compared to the ERPC), while the slower moving parameter is measured with
a test having very high effective power. Averaging the separate power results  into a single composite
measure might result in an effective power roughly equivalent to the ERPC. Then the chances of
identifying a release in a timely manner would  be diminished unless rather large concentrations of the
faster constituent began appearing  in compliance wells.  Smaller mean increases — even if 3 or 4
standard deviation units above background levels — would have little chance of being detected, while
the time it took for more readily-identified levels of the slower constituent to arrive at compliance wells
might be too long  to be  environmentally protective.  Statistical power  results   should be reported
separately,  so that the effectiveness of each distinct test can be adequately judged.  Further data-specific
power evaluations could still be necessary to identify the appropriate test(s).

     The following basic  steps are  recommended for  assessing effective power at sites using multiple
statistical methods:

  1.   Determine the number and assortment of distinct statistical tests. Different power characteristics
      may be  exhibited by  different statistical techniques.  Specific control  charts,  Mests, non-
      parametric prediction limits, etc. all tend to vary in their performance. The  performance of a
      given technique is  also strongly affected by the data characteristics.  Background sample sizes,
      interwell versus intrawell choices, the number of retests and type of retesting plan, etc., all affect
       statistical power. Each distinct data configuration and retesting plan will  delineate a slightly
      different statistical test method.
                                              6-22                                   March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

  2.    Once the various methods have been identified, gauge the effective power of each.10 Often the
       easiest way to measure power is via Monte Carlo simulation. Effective power involves a single
       well-constituent pair, so the simulation needs to incorporate only one population of background
       measurements representing  the baseline condition  and  one  population of compliance point
       measurements.

  3.    To run a Monte Carlo simulation, repeat the following algorithm a large number of times (e.g., N
       = 10,000). Randomly generate a set of measurements from the background population in order to
       compute either a comparison limit for a control chart or some type of prediction limit test, or the
       background portion for a ^-test or Wilcoxon rank-sum calculation, etc. Then generate compliance
       point samples  at successively higher  mean  concentration levels,  representing increases  in
       standard deviation units above the baseline average.  Perform each distinct test on the simulated
       data,  recording the result of each iteration. By determining how frequently the concentration
       increase is  identified at each successive mean level (including retests if necessary), the effective
       power for each distinct method can be estimated and compared.

       ^EXAMPLE 6-4

     As a  simple  example of measuring effective power, consider a site using two different statistical
methods. Assume that most of the constituents will be tested interwell with a l-of-3 parametric normal
prediction  limit retesting plan for  individual observations  (Chapter  19).  The remaining  constituents
having low detection rates and small well sample sizes will be tested  intrawell with a Wilcoxon rank-
sum test.

     To measure  the effective power of the normal prediction  limits, note that the same number of
background measurements (n = 30) is likely to be available for each of the relevant constituents. Since
the per-constituent false positive rate (ac) and the number of monitored wells (w) will also be identical
for these chemicals, the same K multiplier can be used  for each prediction limit, despite  the fact that the
background mean and standard deviation will almost certainly vary by constituent.

     Because of these identical data and well configurations,  the  effective power  of each normal
prediction limit will also be the same,11  so that only one prediction limit  test need be simulated. It is
sufficient to assume the background population has a standard normal distribution.  The compliance
point population at the  single contaminated well also has a  normal distribution with the same standard
deviation but a mean (|i) shifted upward to reflect successive  relative concentration increases  of 1
standard deviation, 2 standard deviations, 3 standard deviations, etc.

     Simulate  the power by conducting  a large number of iterations (e.g., N =  10,000-20,000) of the
following algorithm: Generate 30 random observations from background and compute the sample mean
10 Since power is a property of the statistical method and not linked to a specific data set, power curves are not needed for all
  well-constituent pairs, but only for each distinct statistical method. For instance, if intrawell prediction limits are employed
  to monitor barium at 10 compliance wells and the intrawell background sample  size is the same for each well, only one
  power curve needs to be created for this group of tests.
11 Statistical power measures the likely performance of the technique used to analyze the data, and is not a statement about the
  data themselves.

                                               6^23                                    March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

and standard deviation. Calculate the prediction limit by adding the background mean to K times the
background standard deviation. For a l-of-3 retesting plan, generate 3 values from the compliance point
distribution (i.e., a normal distribution with unit standard deviation but mean equal to (l).  If the first of
these measurements does not exceed the prediction limit, record a score of zero and move on to the next
iteration. If, however, the first value is an exceedance, test  the second value and possibly the third. If
either resample does not exceed the prediction limit, record a  score of zero and move to the  next
iteration. But if both resamples are also exceedances, record a score of one. The fraction of iterations (TV)
with scores equal to one is an estimate  of the effective power at a concentration  level of (I standard
deviations above the baseline.

     In the case of the intrawell Wilcoxon rank-sum test, the power will depend on the number of
intrawell background samples available at each well and for each  constituent.12 Assume for purposes of
the example that  all the intrawell  background sizes  are  the same with n = 6  and that two  new
measurements will be collected at each well during the evaluation period. The power will also depend on
the frequency of non-detects in the underlying groundwater population. To simulate this  aspect of the
distribution for each separate  constituent, estimate  the proportion (p) of observed  non-detects across a
series of wells. Then set a RL for purposes of the simulation equal to zp, the/>th quantile of the standard
normal distribution.

     Finally, simulate the effective power by repeating a large number of iterations of the following
algorithm:  Generate  n =  6  samples from  a standard normal distribution to  represent intrawell
background.  Also generate two samples from a normal distribution with unit standard deviation and
mean equal to [j, to represent new compliance point measurements from a distribution with mean level
equal to [j, standard deviations above background. Classify any values as non-detects that fall below zp.
Then jointly rank the background  and  compliance values  and compute  the Wilcoxon rank-sum test
statistic, making any necessary adjustments for ties (e.g., the  non-detects). If this test statistic exceeds its
critical value, record a score of one for the iteration. If not, record a score of zero. Again estimate the
effective power at mean concentration level (I as the proportion of iterations (TV) with scores of one.

     As a last step, examine the  effective  power for each of the two techniques. As long as the power
curves of the normal prediction  limit and the Wilcoxon rank-sum test both meet the criteria of the
ERPCs, the statistical program taken as a whole should provide acceptable power. -4
12 Technically, since the Wilcoxon rank-sum  test will often be applied to non-normal data, power will also depend
  fundamentally on the true underlying distribution at the compliance well. Since there may be no way to determine this
  distribution, approximate power is measured by assuming the underlying distribution is instead normal.

                                              6-24                                     March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance



6.3  HOW KEY ASSUMPTIONS IMPACT STATISTICAL DESIGN

6.3.1 STATISTICAL IN DEPENDENCE

       IMPORTANCE OF INDEPENDENT, RANDOM MEASUREMENTS

     Whether a facility is in detection monitoring, compliance/assessment, or corrective action, having
an appropriate and valid sampling program is critical. All statistical procedures infer information about
the underlying population  from the observed sample measurements. Since these populations are only
sampled a few times a year, observations should be carefully chosen to provide accurate information
about the underlying population.

     As discussed in Chapter 3, the mathematical theory behind standard statistical tests assumes that
samples were randomly obtained from the underlying population.  This is necessary to insure that the
measurements are independent and identically distributed [i.i.d.]).  Random sampling means that each
possible concentration value in the population has an equal or known chance of being selected any time
a measurement is taken. Only random sampling guarantees with sufficiently high probability that a set of
measurements is adequately representative of the underlying population. It also ensures that human
judgment will not bias the sample results, whether by intention or accident.

     A number of factors make classical random sampling of groundwater virtually impossible.  A
typical small number  of wells represent only a very small portion of an entire well-field.  Wells are
screened at specific depths  and combine potentially different horizontal and vertical flow regimes.   Only
a minute portion of flow that passes  a  well is actually  sampled.  Sampling normally occurs  at fixed
schedules, not randomly.

     Since a typical aquifer cannot be sampled at random, certain assumptions are made concerning the
data from  the  available  wells.  It  is  first  assumed that the  selected well locations  will generate
concentration data  similar to a randomly distributed  set of  wells. Secondly, it is  assumed that
groundwater flowing through the well screen(s) has a concentration distribution identical to the aquifer
as a whole. This second assumption is unlikely to be valid unless groundwater is flowing through the
aquifer at a pace fast enough and in such a way as to allow adequate mixing of the  distinct water
volumes  over a relatively  short (e.g., every few months or so) period of time, so that groundwater
concentrations seen at an existing well could also have been observed at other possible well locations.

     Adequate sampling of aquifer concentration distributions cannot be accomplished unless enough
time elapses between sampling events to allow different portions of the aquifer to pass through the well
screen.    Most  closely-spaced  sampling events  will  tend   to  exhibit  a statistical dependence
(autocorrelation). This means that pairs of consecutive measurements taken in a series will be positively
correlated, exhibiting a stronger similarity in concentration levels than expected from pairs collected at
random times. This would be particularly true for overall water quality indicators which are continuous
throughout an aquifer and only vary slowly with time.

     Another form of statistical dependence is  spatial correlation. Groundwater concentrations  of
certain constituents exhibit natural spatial variability, i.e., a distribution that varies depending on the
location of the  sampling coordinates.  Spatially variable constituents exhibit mean and occasionally
                                             6^25                                   March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

variance differences from one well to another.  Pairs of spatially variable measurements collected from
the same or nearby locations exhibit greater similarity than those collected from distinct, widely-spaced,
or distant wells.

     Natural spatial variability can result from a number  of geologic and hydrological processes,
including varying soil composition  across an aquifer. Various geochemical,  diffusion, and adsorption
processes may dominate depending on the specific locations being measured. Differential flow paths
can also impact the spatial distribution of contaminants in groundwater, especially if there is limited
mixing of distinct groundwater volumes over the period of sampling.

     An adequate groundwater monitoring sampling program needs to account for not only site-specific
factors such as hydrologic characteristics, projected flow rates, and directional  patterns, but also meeting
data assumptions  such  as independence.   Statistical  adjustments are  necessary, such as  selecting
intrawell comparisons for spatially distinct wells or removing autocorrelation effects in the case of time
dependence.

       DARCY'S EQUATION AND AUTOCORRELATION

     Past EPA  guidance  recommended  the use of  Darcy's equation  as  a  means  of establishing  a
minimum time interval between samples.  When validly applied  as a basic estimate of groundwater
travel time in a given aquifer, the Darcy equation ensures that separate volumes of groundwater are
being sampled (i.e., physical independence).  This increases the probability that the samples will also be
statistically independent.

     The Unified Guidance in Chapter  14 also includes  a  discussion  on  applying Darcy's equation.
Caution is advised in its use, however, since Darcy's equation cannot guarantee temporal independence.
Groundwater travel  time  is only  one  factor that  can influence the  temporal pattern  of aquifer
constituents.   The measurement  process  itself can  affect time related dependency.  An imprecise
analytical method might impart  enough  additional variability to  make the  measurements essentially
uncorrelated even in a short sampling interval.  Changes in analytical methods or laboratories and even
periodic re-calibration of analytical  instrumentation can impart time-related dependencies in a data set
regardless of the time intervals between samples.

     The overriding interest is  in  the behavior of  chemical contaminants  in  groundwater, not  the
groundwater itself. Many  chemical compounds do  not travel at  the same  velocity  as  groundwater.
Chemical characteristics such as adsorptive potential,  specific gravity, and molecular size can influence
the way chemicals move in the subsurface.  Large molecules, for example, will tend to travel slower than
the average linear velocity of groundwater because of matrix interactions. Compounds that exhibit  a
strong adsorptive potential will undergo a similar fate, dramatically changing  time of travel predictions
using the Darcy equation.  In some  cases,  chemical interaction with the matrix material will alter the
matrix structure  and its associated hydraulic conductivity and may result in an increase in contaminant
mobility. This last effect has been observed, for instance, with certain organic  solvents in clay units (see
Brown and  Andersen, 1981).

     The Darcy  equation is also not valid in turbulent  and non-linear laminar flow regimes. Examples of
these particular hydrological environments include karst and  'pseudo-karst' (e.g.,  cavernous basalt and
extensively fractured rock) formations. Specialized methods have been investigated by Quinlan  (1989)

                                             6-26                                    March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

for developing alternative monitoring procedures. Dye tracing as described by Quinlan (1989) and Mull,
et al.  (1988)  can be  useful  for  identifying  flow paths and travel times  in  these  two particular
environments;  conventional groundwater monitoring wells are often of little value in designing an
effective monitoring system in these type of environments.

     Thus, we suggest that Darcy's equation not be exclusively relied upon to gauge statistical sampling
frequency.   At many sites, quarterly or semi-annual sampling often provides a reasonable balance
between maintaining statistical  independence  among  observations  yet  enabling early detection of
groundwater problems. The Unified Guidance recommends three tools to explore or test for time-related
dependence among groundwater measurements. Time series plots (Chapter 9) can be  constructed on
multiple wells to examine whether there is a time-related dependence in the pattern  of  concentrations.
Parallel traces on such a plot may indicate correlation across wells as part of a natural temporal, seasonal
or induced laboratory effect. For longer data series, direct estimates of the autocorrelation in a series of
measurements  from a single well can be made using either the sample autocorrelation function or the
rank von Neumann ratio (Section 14.2).

       DATA  MIXTURES INCLUDING ALIQUOT REPLICATE SAMPLES

     Some facility data sets may contain both single and aliquot replicate groundwater measurements
such as duplicate splits.  An entire data set may also consist of aliquot  replicates from a number of
independent water quality samples.  The guidance recommends against using aliquot data directly in
detection monitoring tests, since they are almost never statistically independent. Significant positive
correlation almost always exists between such duplicate samples or among aliquot sets.  However, it is
still possible to utilize some of the aliquot information within a larger water quality data set.

     Lab  duplicates  and field  splits can provide valuable information about the level of measurement
variability  attributable to  sampling and/or analytical techniques. However, to  use  them as  separate
observations in a prediction limit, control chart, analysis of variance [ANOVA] or other procedure, the
test must be specially structured to account for multiple data values per sampling event.

     Barring the use of these more  complicated methods, one suggested strategy has been to simply
average each set of field splits  and  lab duplicates and treat the resulting mean as a single observation in
the overall data set. Despite eliminating the dependence between field splits and/or lab duplicates, such
averaging  is not  an ideal solution. The  variability in means of two  correlated  measurements is
approximately  30% less than the variability associated with two single independent measurements. If a
data set consists of a mixture  of single measurements and lab  duplicates and/or field splits, the
variability of the averaged values  will be less than the  variability of the single  measurements.  This
would imply that the final data set is not identically distributed.

     When data are not identically  distributed, the  actual false positive and false negative rates of
statistical tests may be higher or lower than expected.  The effect of mixing single measurements and
averaged aliquot replicates might be balanced out in a two-sample t-test if sample sizes are roughly
equal.  However, the impact of non-identically distributed data can be substantial for an upper prediction
limit test of a future single sample where the background sample includes a mixture of aliquot replicates
and  single measurements.   Background variability  will be  underestimated, resulting in a  lowered
prediction limit and a higher false positive rate.
                                              6-27                                    March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

     One statistically defensible but expensive  approach is to perform the same number of aliquot
replicate measurements on all physical  samples collected from  background  and compliance  wells.
Aliquot replicates can be averaged,  and the same variance reduction will  occur in all  the final
observations. The statistical test degrees of freedom, however, are based on the number of independent,
averaged samples.

     Mixing single and averaged aliquot data is a serious problem if the component of variability due to
field sampling methods and laboratory measurement error is a substantial fraction of the overall sample
variance. When natural variability  in groundwater concentrations  is the largest component, averaging
aliquot replicate measurements will do little to weaken the assumption of identically-distributed data.
Even when variability due to sampling and analytical methods is a large component of the total variance,
if the percentage of samples with aliquot replicate measurements is fairly small (say, 10% or less), the
impact  of  aliquot replicate averaging should usually be negligible. However,  consultation  with  a
professional statistician is recommended.

     The simplest alternative is to  randomly select one value from each aliquot replicate set along with
all non-replicate individual measurements, for use in statistical testing. Either this approach  or the
averaged replicate method described above will result in smaller degrees of freedom than the strategy of
using all the aliquots, and will more accurately reflect the statistical properties of the data.

       CORRECTING FOR TEMPORAL  CORRELATION

     The Unified Guidance  recommends two general  methods  to  correct for observable  temporal
correlation.  Darcy's  equation is mentioned  above as a rough guide  to physical independence of
consecutive groundwater  observations. A more  generally applicable strategy for yet-to-be-collected
measurements involves adjusting  the  sampling  frequency  to  avoid autocorrelation  in consecutive
sampling events.   Where autocorrelation is a serious  concern, the  Unified  Guidance recommends
running  a pilot study at two or  three  wells and  analyzing the study  data  by using the  sample
autocorrelation function (Section 14.3.1). The autocorrelation function plots the strength of correlation
between consecutive measurements  against  the time  lag between  sampling events.  When the
autocorrelation  becomes insignificantly  different  from  zero  at a particular  sampling interval, the
corresponding sampling frequency is the maximum that will ensure uncorrelated sampling events.

     Two other strategies are recommended for  adjusting already collected data. First, a  longer data
series at a single well can be  corrected for seasonality by estimating and removing the seasonal trend
(Section 14.3.3). If both a linear trend and seasonal fluctuations are evident, the seasonal Mann-Kendall
trend test can be run to identify the trend despite the seasonal effects (Section 14.3.4).  A second strategy
is for sites where a temporal effect (e.g., temporal dependence, seasonality) is apparent across multiple
wells.  This involves estimating a  temporal  effect via a one-way ANOVA and  then creating adjusted
measurements using the ANOVA residuals (Section 14.3.3).  The  adjusted data can then be utilized in
subsequent statistical procedures.
                                             6-28                                    March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

6.3.2 SPATIAL VARIATION: INTERWELL VS.  INTRAWELL TESTING

       ASSUMPTIONS IN BACKGROUND-TO-DOWNGRADIENT COMPARISONS

     The  RCRA groundwater  monitoring  regulations initially presume  that  detection monitoring
background can be defined on the basis of a definable groundwater gradient. In a considerable number of
situations, this approach is problematic. No groundwater gradient may be measurable for identifying
upgradient and downgradient well locations around a regulated unit. The hydraulic gradient may change
in direction, depth or magnitude due  to seasonal fluctuations.   Groundwater mounding or other flow
anomalies can occur. At most locations, significant spatial variability among wells exists for certain
constituents. Where spatial variation is a natural artifact of the  site-specific geochemistry, differences
between upgradient and downgradient wells are unrelated to on-site waste management practices.

     Both the Subtitle C and Subtitle D RCRA regulations allow for a determination that background
quality may include sampling of wells not hydraulically upgradient of the waste management area. The
rules recognize that this can occur either when  hydrological information is unable to indicate which
wells  are  hydraulically  upgradient or when sampling other wells  will be "representative or more
representative than that provided by the upgradient wells."

     For upgradient-to-downgradient  well comparisons, a crucial detection monitoring assumption is
that downgradient well  changes  in groundwater quality are only caused by on-site waste management
activity.  Up- and down-gradient well measurements are also assumed to be comparable and equal on
average unless  some waste-related change  occurs.  If other factors  trigger significant increases in
downgradient well locations, it may be very difficult  to pinpoint the monitored unit as the source or
cause of the contaminated groundwater.

     Several  other  critical  assumptions apply  to the interwell approach.  It is  assumed that the
upgradient and downgradient well samples are drawn from the same aquifer and that wells are screened
at essentially the same hydrostratigraphic position. At some sites, more than one aquifer underlies the
waste site or landfill, separated by confining layers of clay or other less permeable material.  The fate
and transport characteristics of groundwater contaminants likely  will  differ in each aquifer, resulting in
unique concentration patterns. Consequently, upgradient  and downgradient observations may not be
comparable (i.e.., drawn from the same statistical population).

     Another  assumption  is that groundwater  flows in a definable pathway from  upgradient  to
downgradient wells beneath the regulated unit. If flow paths are  incorrectly determined or this does not
occur, statistical comparisons can be invalidated.  For example, a real release may be occurring at a site
known to have groundwater mounding beneath the monitored unit.  Since the groundwater may move
towards both the downgradient and upgradient wells, it may not be possible to detect the release if both
sets of wells become equally or similarly contaminated.  One exception to this might occur if certain
analytes are shown to exhibit uniform behavior  in both historical upgradient and downgradient wells
(e.g., certain infrequently detected trace elements). As long as  the flow pathway from the unit to the
                                            6-29                                   March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

downgradient wells is assured, then an interwell test based on this combined background could still
reflect a real exceedance in the downgradient wells.13

     Groundwater flow should also move at a  sufficient velocity beneath the site,  so that the  same
groundwater observed at upgradient well locations is subsequently monitored at downgradient wells in
the course of an evaluation period (e.g., six months or a year).   If groundwater flow is much slower,
measurements from upgradient and downgradient wells may be more akin to samples from two separate
aquifers. Extraneous factors may separately influence the downgradient and background populations,
confusing the determination of whether or not a release has occurred.

     While statistical testing can determine whether there are significant differences between upgradient
and downgradient well measurements, it cannot determine why such differences exist. That is primarily
the  concern  of  a  hydrologist who  has carefully reviewed  site-specific  factors.  Downgradient
concentrations may be greater than background  because contamination of the underlying aquifer has
occurred. The increase may be due to other factors,  including  spatially variable  concentration levels
attributable to changing soil  composition and geochemistry from one well location to another. It could
also be due to the migration  of contaminants from off-site sources reaching downgradient wells.  These
and  other factors (including those summarized in Chapter 4 on SSI Increases) should be considered
before deciding that statistically significant  background-to-downgradient differences  represent site-
related contamination.

     An example of how background-to-downgradient well differences can be misleading is illustrated
in Figure 6-4 below. At this Eastern coastal site,  a Subtitle D landfill was located just off a coastal river
emptying  into  the  Atlantic  Ocean  a short  distance downstream.   Tests of specific  conductance
measurements comparing the  single upgradient  well  to  downgradient  well data indicated significant
increases at all  downgradient wells, with one well indicating levels more  than an order of magnitude
higher than background concentrations.

     Based on  this analysis, it was initially concluded that waste management activities at the landfill
had impacted groundwater.  However, further hydrologic investigation  showed that nearby river water
also exhibited  elevated levels  of specific  conductance,  even  higher than  measurements at  the
downgradient wells.  Tidal fluctuations and changes in river discharge caused sea water to periodically
mix with the coastal river water at a location near the downgradient wells.  Mixed river and sea water
apparently seeped into the aquifer, impacting downgradient wells but not at the upgradient location.  An
off-site  source  as opposed to  the landfill itself was  likely responsible for the observed elevations in
specific conductance. Without this additional hydrological information, the naive statistical comparison
between upgradient and downgradient wells would have reached an incorrect conclusion.
13  The same would be true of the "never-detected" constituent comparison, which does not depend on the overall flow
  pathway from upgradient to downgradient wells.

                                              6-30                                    March 2009

-------
Chapter 6. Detection Monitoring Design
Unified Guidance
                          Figure 6-4.  Landfill Site Configuration
                                                                  CONDUCT \NC15
                                                                I (Mil: Urm SIM \\utoi
                                                                    daia i 1U vis)
                                                                Max - MUM)
                                                                \\c = 22.01*)
                                                        Sea Water
TRADEOFFS IN INTERWELL AND  INTRAWELL APPROACHES

     The choice  between  interwell and intrawell  testing  primarily  depends  on  the  statistical
characteristics of individual constituent data behavior in background  wells.  It  is presumed that  a
thorough background  study described in Chapter 5 has been completed.  This involves selecting the
constituents deemed appropriate for detection  monitoring, identifying distributional characteristics, and
evaluating  the constituent  data  for trends,  stationarity,  and mean spatial  variability among wells.
ANOVA tests can be used to assess both well mean  spatial variability and the potential for pooled-
variance estimates if an intrawell approach is needed.

     As discussed in Chapter 5, certain classes of potential monitoring constituents are more likely to
exhibit spatial variation.  Water quality indicator parameters are  quite frequently spatially variable.
Some authors,  notably Davis and McNichols  (1994) and Gibbons  (1994a), have  suggested that
significant  spatial variation is  a nearly ubiquitous feature at RCRA-regulated landfills and  hazardous
waste sites, thus  invalidating the use of interwell  test methods.  The Unified Guidance  accepts that
interwell tests still have an important role in groundwater monitoring, particularly for certain classes of
constituents like non-naturally occurring VOCs and some trace elements.  Many sites may best be served
by a statistical program which combines interwell and intrawell procedures.

     Intrawell testing is an appropriate and recommended alternative strategy for many constituents.
Well-specific backgrounds afford intrawell tests  certain advantages over the interwell approach.  One
key advantage is confounding results due to spatial variability are eliminated, since all data used in an
intrawell test are obtained from a single location.  If natural background levels change substantially from
                                             6-31
        March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

one well to the next, intrawell background provides the most accurate baseline for use in statistical
comparisons.

      At times, the variability in a set of upgradient background measurements pooled from multiple
wells can be larger than the variation in individual intrawell background wells.  Particularly if not
checked with ANOVA well mean testing, interwell variability could substantially increase if changes in
mean levels from one location to the next are also incorporated. While pooling should not occur among
well means determined to be significantly different using ANOVA, a more likely situation is that pooled
well true means  and variance may be slightly different at  each well.  The  ANOVA test might still
conclude that the mean differences were insignificant and satisfy the equal variance assumption. The net
result (as explained below) is that intrawell tests can be more statistically powerful than comparable
interwell tests using upgradient background, despite employing a smaller background sample size.

     Another  advantage using  intrawell  background is that a reasonable baseline for tests of future
observations can be established at historically contaminated wells. In this case, the intrawell background
can be used to track the onset of even more extensive contamination in the future.  Some compliance
monitoring wells  exhibit chronic elevated contaminant levels (e.g., arsenic) considerably above other site
wells which may not be clearly attributed to a regulated unit  release.  The regulatory agency has the
option of continuing detection monitoring  or  changing to  compliance/corrective action monitoring.
Unless the  agency has already determined that the pre-existing contamination is subject to compliance
monitoring or remedial action under RCRA, the detection monitoring option would be to test for recent
or  future  concentration increases above the  historical  contamination  levels by  using intrawell
background as a well-specific baseline.

     Intrawell tests are not preferable for all groundwater  monitoring scenarios.  It may be unclear
whether a given compliance well was historically contaminated prior to being regulated or more recently
contaminated.  Using intrawell  background to  set  a  baseline  of comparison may ignore  recent
contamination subject to compliance testing and/or  remedial action. Even more  contamination in the
future would then be required to trigger a statistically significant increase [SSI] using the intrawell test.
The Unified Guidance  recommends the use  of intrawell  testing only when  it is clear that  spatial
variability is not the result of recent contamination attributable to the regulated unit.

     A second concern is that intrawell tests typically utilize a smaller set  of background data than
interwell methods. Since statistical power depends significantly on background sample size, it may be
more difficult to  achieve comparable statistical power with intrawell tests than with interwell methods.
For the latter, background data can be collected from multiple wells when appropriate, forming a larger
pool of measurements than would be available at a  single well. However, it may also be possible to
enhance intrawell sample sizes for parametric tests using the pooled- variance approach.

     Traditional  interwell tests can be appropriate for certain constituents if the hydraulic assumptions
discussed earlier are verified and there is no evidence of significant spatial variability. Background data
from other  historical compliance wells not significantly different from upgradient wells using ANOVA
may also be used in some cases.   When these conditions are met, interwell tests can be preferable as
generally more powerful tests.  Upgradient groundwater quality can then be more easily monitored in
                                              6-32                                    March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

parallel to  downgradient locations.  Such upgradient monitoring can signal changes in natural in-situ
concentrations or possible migration from off-site sources. 14

       For most situations, the background constituent data patterns will determine which option is most
feasible. Clear indications of spatially distinct well means through ANOVA testing will necessitate
some form of intrawell  methods. Further choices are then which type of statistical  testing will provide
the best power.

       It may be possible to increase the effective sample size associated with a series of intrawell tests.
As explained in Chapters 13 & 19, the K-multipliers for intrawell prediction limits primarily depend on
the number of background measurements used to estimate the standard deviation.  It is first necessary to
determine that  the intrawell background in a series of compliance  wells is both uncontaminated  and
exhibits similar levels of variability from  well to well.  Background data from these wells can then be
combined to form a pooled intrawell  standard deviation estimate with larger degrees of freedom, even
though  individual well  means vary.   A  transformation may be needed to stabilize the well-to-well
variances.  If one or more of the compliance wells is already contaminated, these should  not be mixed
with uncontaminated well data in obtaining the pooled standard deviation estimate.

       A  site-wide  constituent pattern of  no  significant  spatial variation will generally favor  the
interwell testing approach.  But given the potential for hydrological and other issues discussed above,
further evaluation of intrawell methods may be appropriate.  Example 6-2 provided an illustration of a
specific intrawell constituent having a lower absolute standard deviation than an interwell pooled data
set, and hence greater relative and absolute power.  In making such  an  interwell-intrawell comparison,
the specific test and all  necessary design inputs  must be considered.  Even if a given intrawell data set
has a low background standard deviation compared to an interwell counterpart, the advantage in absolute
terms over the  relative power approach will change with differing design inputs. The simplest way to
determine if the intrawell approach might be advantageous is to calculate the  actual background limits of
a potential test using existing intra- and inter-well data sets. In a given prediction limit test, for example,
the actual lower limit will determine the more powerful test.

      If desired,  approximate data-based power curves (Section  6.2.4) can be constructed to evaluate
absolute power over a range of concentration level  increases.  In practice,  the method for comparing
interwell versus intrawell testing strategies with the same well-constituent pair involves the following
basic steps:

   1.  Given the interwell background sample  size (winter),  the statistical test method (including  any
       retesting), and the individual per-test a for that well-constituent pair, compute  or simulate the
       relative power of the test at multiples of fester above the baseline mean level.  Let k range from 0
       to  5 in  increments of 0.5, where  the interwell  population standard deviation (Omter) has been
       replaced by the sample background standard deviation (sinter).
14 The same can be accomplished via intrawell methods if upgradient wells continue to be sampled along with required
  compliance well locations. Continued tracking of upgradient background groundwater quality is recommended regardless
  of the testing strategy.

                                              6^33                                     March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

   2.  Repeat Step 1 for the intrawell test.  Use the intrawell background sample size («intra), statistical
       test method, background sample standard deviation (sintra), and the same individual per-test a to
       generate a relative power curve.

   3.  On  the same graph, plot overlays of the estimated data-based interwell and  intrawell  power
       curves  (as  discussed  in  Section 6.2.4).  Use  the  same  range  of (absolute, not  relative)
       concentration increases over baseline along the horizontal axis.

   4.  Visually inspect the data-based power curves to determine which method offers better  power
       over a wider range of possible concentration increases.

     The  Unified  Guidance  recommends that users  apply the most powerful statistical  methods
available in detecting and identifying contaminant releases for each well-constituent pair.  The  ERPC
identifies a minimum acceptable standard for judging the relative power of particular tests. However,
more powerful methods based on absolute power may be considered preferable in certain circumstances.

     As a final concern, very small individual well samples in the early stages of a monitoring program
may make it difficult to utilize an intrawell method having both sufficient statistical power and meeting
false positive design criteria.  One  option would be to temporarily defer tests on those well-constituent
pairs until additional background observations can be collected.  A second option is to use the intrawell
approach despite  its inadequate power, until the  intrawell background is sufficiently large via periodic
updates (Chapter 5).  A third option might be to use a more powerful intrawell test (e.g., a higher order
\-of-m parametric or non-parametric prediction limit test).  Once background  is increased, a lower order
test might suffice.  Depending on the type of tests, some control of power may be lost (parametric) or the
false positive (non-parametric tests).  These tradeoffs are considered more fully in Chapter 19. For the
first two options and the parametric test under the third  option,  there is some added risk that a release
occurring during the period of additional data collection might be missed.  For the non-parametric test
under the third option, there is an increased risk of a true false positive error. Any of these options might
be included as special permit conditions.

6.3.3 OUTLIERS

     Evaluation of outliers should begin with historical  upgradient and possibly compliance well data
considered  for defining initial background, as described in Chapter 5, Section 5.2.3.  The key goal is to
select the data most representative  of near-term and likely future background. Potentially discrepant or
unusual values can occur for many reasons including 1) a contaminant release that significantly impacts
measurements at compliance wells;  2)  true but extreme  background groundwater  measurements, 3)
inconsistent sampling or analytical chemistry methodology resulting in laboratory contamination or other
anomalies;  and 4) errors in the transcription of data values  or decimal points.  While the first two
conditions may appear to be discrepant values, they would not be considered outliers.

     When appraising extensive background  data sets with long periods of  acquisition and somewhat
uncertain quality, it is recommended that a formal statistical evaluation of outliers not be conducted until
a  thorough review of  data  quality  (errors,  etc.) has  been  performed.    Changes in analytical
methodologies,  the presence of sample interferences or  dilutions can affect  the historical data record.
Past and current treatment of non-detects should also  be investigated, including whether there  are
multiple reporting limits  in the data base. Left-censored values can impact whether or not the sample

                                              6-34                                    March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

appears normal  (Chapter 15),  especially if  the  data  need to be normalized via a transformation.
Techniques for evaluating censored data should be considered,  especially those  which can properly
account for multiple RLs.  Censored probability plots (Chapter 15) or quasi-nonparametric box plots
(Chapter 12) adapted by John Tukey (1977) can be used as methods to screen for outliers.

     The guidance also recommends that statistical testing of potential outliers also be performed on
initial background data, including historical compliance well data potentially considered as additional
background data.  Recognizing the potential risks  as discussed in Chapter 5, removal of significant
outliers may be appropriate even if no probable error or discrepancy can be firmly identified. The risk is
that  high values  registering  as  statistical  outliers  may reflect  an extreme, but real  value  from the
background population rather than a true outlier, thereby increasing the likelihood of a false positive
error. But the effect of removing outliers from the background data will usually be to improve the odds
of detecting upward changes in concentration levels at compliance wells, and thus providing further
protection of human health and the environment.  Automated screening and removal of background data
for statistical outliers is not recommended without some consideration of the likelihood of an outlier
error.

     A statistical outlier is defined as a value  originating from a different statistical population than the
rest of the sample.  Outliers or  observations not derived from the same population as the rest of the
sample violate the basic statistical assumption of identically-distributed measurements. If an outlier is
suspected,  an initial helpful step is to construct a probability plot of the ordered sample data versus the
standardized normal distribution (Chapter 12). A probability plot is designed to judge  whether the
sample data are consistent with a normal population model. If the data can be normalized, a probability
plot of the transformed observations should also be constructed.  Neither is a formal test, but can still
provide important visual evidence as to whether the suspected outlier(s) should be further evaluated.

     Formal testing for outliers should be done only if an observation seems particularly high compared
to the rest of the sample. The data can be evaluated with either Dixon's or Rosner's tests (Chapter 12).
These outlier tests assume that the rest of the  data except for the suspect observation(s), are normally-
distributed (Barnett and Lewis, 1994). It is recommended that tests also be conducted on transformed
data, if the original  data indicates  one or more  potential  outliers.    Lognormal and other skewed
distributions can  exhibit apparently  elevated  values in  the original concentration  domain, but still be
statistically indistinguishable when normalized via a transformation. If the latter is the case, the outlier
should be retained and the data set treated as fitting the transformed distribution.

     Future background  and compliance well data  may also be  periodically  tested for  outliers.
However, removal of outliers should  only take  place under certain conditions, since a true elevated value
may  fit the pattern of a release or a change in historical background  conditions.   If either Dixon's or
Rosner's test identifies an observation as a statistical outlier, the measurement should not be treated as
such until a specific physical reason for the  abnormal  value can be determined. Valid reasons might
include  contaminated sampling  equipment,  laboratory  contamination   of  the   sample,  errors in
transcription of the data values, etc. Records documenting the sampling and analysis of the measurement
(i.e., the  "chain of custody") should be thoroughly investigated. Based on this review, one of several
actions might be taken as a general rule:
                                              6-35                                    March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

    »«»  If an error in transcription, dilution, analytical procedure, etc. can be identified and the correct
       value recovered, the observation should be replaced by its corrected value and further statistical
       analysis done with the corrected value.

    »»»  If it can shown that the observation is in error but the correct value cannot be determined, the
       observation should be removed from the data set and further statistical analysis performed on the
       reduced data set. The fact that the observation was removed and the reason for its removal should
       be documented when reporting results of the analysis.

    »»»  If no error in the value can be documented, it should be assumed that the observation is a true but
       extreme value.  In this  case, it should not be altered or removed. However, it may helpful to
       obtain another observation in order to verify or confirm the initial measurement.

6.3.4 NON-DETECTS

     Statistically, non-detects are considered 'left-censored'  measurements because the concentration of
any non-detect is known or assumed only to fall within a certain range of concentration values (e.g.,
between 0 and the RL). The direct estimate has been censored by limitations of the measurement process
or analytical technique.

     As noted, non-detect values can affect evaluations of potential outliers. Non-detects and detection
frequency also impact  what detection  monitoring tests  are appropriate  for a given constituent. A low
detection frequency  makes it  difficult to  implement parametric statistical  tests, since it may not  be
possible to determine if the underlying population is normal or can be normalized.  Higher detection
frequencies  offer more options,  including simple  substitution  or estimating the mean and standard
deviation of samples  containing non-detects by means of a censored estimation technique (Chapter 15).

     Estimates of the background mean  and standard deviation are needed to construct parametric
prediction and control chart limits, as well as confidence intervals. If simple substitution is appropriate,
imputed values for individual  non-detects can be used  as  an alternate way to construct  mean and
standard deviation estimates.  These estimates are also needed to update the cumulative sum [CUSUM]
portion of control charts or to compute  means of order/?  compared against prediction limits.

     Simple substitution is not recommended in  the Unified Guidance unless no more than 10-15% of
the sample observations are non-detect. In those  circumstances, substituting half the RL for each non-
detect  is not  likely to substantially  impact the  results  of statistical testing.  Censored estimation
techniques like Kaplan-Meier or robust regression on order statistics [ROS] are recommended any time
the detection frequency is no less than 50% (see Chapter 15).

     For  lower  detection  frequencies,  non-parametric  tests  are recommended.  Non-parametric
prediction limits (Chapter 18) can be constructed as an  alternative to  parametric prediction limits or
control charts. The Tarone-Ware two-sample test  (Chapter 16) is specifically designed to accommodate
non-detects  and serves as an alternative  to the t-test. By  the  same token, the Kruskal-Wallis test
(Chapter 17)  is a non-parametric, rank-based alternative to the parametric ANOVA.  These latter tests
can be used when the non-detects and detects can be jointly sorted and partially ordered (except for tied
values).
                                              6-36                                    March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

     When all data are non-detect, the Double Quantification rule (Section 6.2.2) can be used to define
an approximate non-parametric prediction limit, with the RL as  an upper bound. Before doing this, it
should be determined whether chemicals never or not recently detected in groundwater should even be
formally tested. This will depend on whether the monitored constituent from a large analytical  suite is
likely to originate in the waste or leachate.

     Even if a data set contains only  a small proportion of non-detects,  care should be taken when
choosing between the method detection limit [MDL], the quantification  limit [QL], and the RL in
characterizing 'non-detect' concentrations.  Many non-detects are  reported with  one of three data
qualifier flags: "U," "J," or "E." Samples with a U data qualifier represent 'undetected' measurements,
meaning that the signal  characteristic  of that analyte could not be observed or distinguished  from
'background noise' during lab  analysis. Inorganic  samples with an E flag and organic samples with  a J
flag  may  or  may not be reported  with an estimated concentration.  If no concentration estimate is
reported, these samples represent 'detected,  but not quantified' measurements. In this case, the actual
concentration is assumed to be  positive, falling somewhere between zero and the QL or possibly the RL.

     Since the  actual  concentration  is  unknown, the suggested  imputation  when using  simple
substitution is to replace each non-detect having a qualifier of E or J by one-half the RL. Note, however,
that E and J samples reported with estimated concentrations should be treated as valid measurements  for
statistical purposes. Substitution of one-half the RL is not recommended for these measurements, even
though the degree of uncertainty associated with the estimated concentration is probably greater than that
associated with measurements above the RL.

     As a general rule, non-detect concentrations should not be  assumed to be bounded above by the
MDL. The MDL is usually estimated on the basis of ideal laboratory conditions  with physical  analyte
samples that  may or  may not account for matrix or other interferences encountered when analyzing
specific field  samples. For certain trace element analytical methods, individual laboratories may report
detectable limits closer to an MDL than a nominal QL.  So long as the laboratory has confidence in the
ability to  quantify  at its lab- or occasionally event-specific  detection level, this  RL  may also  be
satisfactory. The RL should typically be taken as a more reasonable upper bound for non-detects when
imputing estimated concentration values to these measurements.

     RLs are sometimes but not always equivalent to a particular laboratory's QLs. While analytical
techniques may  change  and  improve  over time leading  to  a lowering of the  achievable  QL, a
contractually negotiated RL might be much higher. Often a multiplicative factor is built into the RL to
protect  a contract lab against  particular liabilities. A  good practice is to periodically review a given
laboratory's capabilities and to encourage reporting non-detects with actual QLs whenever possible, and
providing standard qualifiers with all data measurements as well as estimated concentrations for E- and
J-flagged samples.

     Even when no estimate of concentration can  be made, a lab  should regularly report the distinction
between 'undetected'  and 'detected, but not quantified' non-detect measurements. Data sets with such
delineations can be used  to advantage in rank-based non-parametric procedures. Rather than  assigning
the same tied rank to all non-detects (Chapter 16), 'detected but not quantified' measurements should
be given larger ranks than those assigned to 'undetected' samples.  These two types of non-detects should
be treated as two distinct groups of tied observations for use in the non-parametric Wilcoxon rank-sum
procedure.
                                             6-37                                    March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

6.4 DESIGNING DETECTION  MONITORING TESTS

     In the following sections, the main formal detection monitoring tests covered in this guidance are
described in the context  of site  design  choices.   Advantages as well  as limitations are  presented,
including the use of certain methods as diagnostic tools in determining the appropriate formal test(s).

6.4.1  T-TESTS

     A statistical comparison between two sets of data is known as a two-sample test. When normality
of the sample data can be presumed, the parametric Student t-tesi is commonly used (Section 16.1). This
test compares two distinct populations, represented by two samples.  These samples  can either be
individual well data sets, or a common pooled background versus individual compliance well data. The
basic goal of the t-test is to determine whether there is any statistically significant difference between the
two population means. Regulatory requirements for formal use of two-sample t-tests are limited to the
Part 265 indicator parameters, and have generally been superseded in the Parts 264 and 258 rules by tests
which can account for multiple comparisons.

     When the sample data are non-normal and may contain non-detects, the Unified Guidance provides
alternative two-sample tests to the parametric t-test. The Wilcoxon rank-sum test (Section 16.2) requires
that the combined  samples be sorted and ranked. This test evaluates potential differences in population
medians rather than the means. The Tarone-Ware test (Section 16.3) is specially adapted to handle left-
censored measurements, and also tests for differences in population medians.

     The t-test or a non-parametric variant is recommended as a validation tool when updating intrawell
or other background data  sets (Chapter 5).   More recently collected data considered for background
addition are compared to the  historical data set.   A non-significant  test result  implies no mean
differences, and the newer data may be added to the original set. These tests are generally useful  for any
two-sample diagnostic comparisons.

6.4.2  ANALYSIS OF VARIANCE  [ANOVA]

     The parametric one-way ANOVA is an extension of the t-test to  multiple sample groups.  Like its
two-sample counterpart, ANOVA tests  for significant differences in one or more  group (e.g., well)
means. If an overall significant difference is found as measured  by the F-statistic, post-hoc statistical
contrasts  may be used to  determine where the differences lie among individual  group means.   In the
groundwater detection  monitoring context,  only  differences  of mean well increases relative  to
background are considered of importance. The ANOVA test also has wide applicability as a  diagnostic
tool.

       USE OF ANOVA IN FORMAL DETECTION MONITORING TESTS

     RCRA regulations under Parts 264 and  258 identify parametric and non-parametric ANOVA as
potential detection monitoring tests. Because of its flexibility and power, ANOVA can sometimes be an
appropriate  method  of statistical  analysis  when groundwater monitoring is  based on an  interwell
comparison of background and compliance well data.  Two types of ANOVA are presented  in the
Unified Guidance: parametric and non-parametric one-way ANOVA (Section 17.1).   Both methods
                                            6-38                                  March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

attempt to assess whether distinct monitoring wells  differ in average concentration  during  a given
evaluation period.15

     Despite the potential attractiveness of ANOVA tests, use in formal detection monitoring is limited
by these important factors:

         »«»  Many monitoring constituents exhibit significant spatial variability and cannot make use of
            interwell comparisons;

         »«»  The test can be confounded by a large number of well network comparisons;

         »«»  A minimum well sample size must be available for testing; and

         *»*  Regulatory false positive error rate restrictions limit the ability to effectively control the
            overall false positive rate.

     As  discussed in Section 6.2.3, many if not most inorganic monitoring constituents exhibit spatial
variability, precluding an interwell form of testing.  Since ANOVA is inherently an interwell procedure,
the guidance recommends against its use for these constituents and  conditions.   Spatial variability
implies that the average groundwater concentration levels vary from well to well because of existing on-
site conditions.   Mean differences of this  sort can be identified by ANOVA,  but the cause of the
differences cannot. Therefore, results of a statistically significant ANOVA might be falsely attributed as
a regulated unit release to groundwater.

     ANOVA  testing might be  applied to synthetic  organic and trace element constituent data.
However, spatial variation across a site is also likely to occur from  offsite or prior site-related organic
releases.    An  existing contamination  plume  generally  exhibits varying  average  concentrations
longitudinally, as well  as in cross-section  and depth.  For other organic  constituents never detected  at a
site, ANOVA testing would be unnecessary.   Certain trace elements like  barium, arsenic and selenium
do often exhibit some spatial  variability.  Other trace element data generally have low overall detection
rates, which may also preclude ANOVA applications. Overall, very few routine monitoring constituents
are measurable (i.e., mostly detectable) yet not spatially distinct to warrant  using ANOVA as a formal
detection monitoring test. Other guidance  tests better serve this purpose.

     ANOVA  has good power for  detecting real  contamination provided the network is  small to
moderate  in size.   But for  large  monitoring networks, it may  be difficult to  identify single well
contamination.  One explanation is that the ANOVA F-statistic simultaneously combines all compliance
well effects into a single number, so that many other uncontaminated  wells with their own variability can
mask  the test effectiveness to detect the contaminated well.   This might occur at larger sites with
multiple waste units, or if only the edge of a plume happens to intersect one or two boundary wells.

     The statistical power of ANOVA depends significantly on having  at least 4 observations per well
available for testing.  Since the measurements must be statistically independent, collection of four well
observations may necessitate  a wait of several months to a few years if the natural groundwater velocity
15 Parametric ANOVA assesses differences in means; the non-parametric ANOVA compares median concentration levels.
  Both statistical measures are a kind of average.

                                              6-39                                     March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

is low.  In this case, other strategies (e.g., prediction limits) might be considered that allow each new
groundwater measurement to be tested as it is collected and analyzed.

     The  one-way ANOVA test in the RCRA regulations is not designed to control the false positive
error rate for multiple constituents.  The rules mandate a minimum false positive error rate (a) of 5% per
test application.  With an overall false positive rate of approximately 5% per constituent, a potentially
very high  SWFPR can result as the number of constituents tested by ANOVA increases and if tests are
conducted more than once per year.

     For  these reasons, the  Unified Guidance does not generally recommend  ANOVA  for formal
detection monitoring. ANOVA might be applicable to a small number of constituents, depending on the
site.  Prediction limit and control chart strategies using retesting are usually more flexible and offer the
ability to  accommodate even  very large monitoring networks, while meeting the false positive and
statistical power targets recommended by the guidance.

       USE OF ANOVA IN  DIAGNOSTIC TESTING

     In contrast, ANOVA is a versatile tool for diagnostic testing, and is frequently used in the guidance
for that purpose.  Parametric or non-parametric one-way versions are the principal means of identifying
prior spatial variability among background monitoring wells (Chapter 13). Improving sample sizes
using intrawell pooled variances also makes use of ANOVA (Chapter 13). Equality of variances among
wells  is evaluated with ANOVA (Chapter 11).  ANOVA is  also applied when determining certain
temporal trends in parallel well sample constituent data (Chapter 14).

     Tests of natural spatial variability can be made by running ANOVA prior to any waste disposal at a
new facility located above an undisturbed aquifer (Gibbons, 1994a).  If ANOVA identifies significant
upgradient and downgradient well differences when wastes have not yet been managed on-site, natural
spatial variability is the likely cause.  Prior on-site contamination might also be revealed in the form of
significant ANOVA differences.

     Sites with multiple upgradient background wells  can initially conduct an ANOVA on historical
data from just these locations.  Where  upgradient wells are  not significantly different for a  given
constituent, ANOVA testing can be extended to existing historical compliance well data for evaluating
potential additions to the upgradient background data base.

     If intrawell tests are chosen because of natural spatial variation, the results of a one-way ANOVA
on background data from multiple wells can sometimes  be used to  improve intrawell background  limits
(Section 13.3). Though the amount of intrawell background  at  any given well may be small, the
ANOVA provides an estimate of the root mean  squared error [RMSE], which is very close  to  an
estimate of the average per-well standard deviation. By substituting the  RMSE for the usual  well-
specific standard deviation (s), a  more powerful and accurate intrawell limit can be constructed, at least
at those sites where intrawell background across the group of wells can be normalized and the variances
approximately equalized using a common transformation.

     Although the Unified Guidance primarily makes use of one-way ANOVA, many kinds of ANOVA
exist. The one-way ANOVA applications so far discussed— in formal detection monitoring or to assess
well mean differences— utilize data from spatial locations as the factor of interest.  In some situations,

                                             6-40                                  March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

correlated behavior may exist for a constituent among well  samples evaluated in different temporal
events.  A constituent measured in a group of wells may simultaneously rise or fall in different time
periods. Under these conditions, the data are no longer random and independent.  ANOVA can be used
to assess the significance of such systematic changes, making time the factor of interest. Time can also
play a role if the sample data exhibit cyclical seasonal patterns or if parallel upward or downward trends
are observed both in background and compliance point wells.

      If time is an important second factor, a two-way ANOVA is probably appropriate.  This procedure
is discussed in Davis (1994). Such a method can be used to test for and adjust data either for seasonality,
parallel trends, or changes  in  lab  performance that cause temporal (i.e., time-related)  effects. It is
somewhat more complicated to apply than a one-way test. The main advantage of a two-way ANOVA is
to separate components of overall data variation into three sources: well-to-well mean-level differences,
temporal effects, and random  variation  or statistical  error.  Distinguishing the sources of variation
provides a more powerful test of whether significant well-to-well differences actually exist compared to
using only a one-way procedure.

     A significant temporal factor does not necessarily mean that the one-way ANOVA will not identify
actual well-to-well spatial differences. It  merely does not have as strong a  chance of doing so. Rarely
will  the one-way ANOVA identify non-existent well-to-well differences. One situation where this can
occur is when there is a  strong statistical  interaction between the well-to-well factor and the time factor
in the two-way ANOVA.  This would imply that changes in  lab performance or seasonal cycles affect
certain wells (e.g., compliance point) to a different degree or in a different manner than other wells (e.g.,
background). If this  is  the  case, professional consultation is recommended before  conducting  more
definitive statistical analyses.

6.4.3 TREND  TESTS

     Most formal detection monitoring tests in the guidance compare background and compliance point
populations under the key assumption that the populations are stationary over time. The distributions in
each group or  well  are assumed to be  stable during the period of monitoring,  with only random
fluctuations around a constant mean level.  If a significant trend occurs in the background data, these
tests cannot be directly  used.  Trends can  occur for several reasons including natural cycles, gradual
changes in aquifer parameters or the effects of contaminant migration from off-site sources.

     Although not specifically provided for in  the RCRA regulations, the guidance necessarily includes
a number  of tests for evaluating potential trends.  Chapter  17, Section 17.3 covers three basic trend
tests. (1)  Linear regression is a parametric method requiring normal and independent trend  residuals,
and  can be used both to identify a linear trend and estimate its magnitude; (2) For non-normal  data
(including sample data with left-censored  measurements), the Mann-Kendall test offers a non-parametric
method for identifying trends; and (3) To gauge trend magnitude with non-normal  data, the  Theil-Sen
trend line can be used.

     Trend analyses  are primarily diagnostic tests, which should be applied to background data prior to
implementing formal detection  monitoring tests.  If a significant trend is uncovered, two options may
apply.   The particular monitoring constituent may be dropped in favor of alternate  constituents not
exhibiting non-stationary behavior.  Alternatively, prediction limit or control chart testing can make use
of stationary trend residuals for testing purposes. One limitation of the latter approach requires making

                                              6-41                                    March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

an assumption that the historical trend will continue into future monitoring periods.  In addition, future
data needs to be de-trended prior to  testing.   If a  trend happened to be  of limited duration, this
assumption may not be reasonable and could result in identifying a background exceedance when it does
not exist.  If a trend occurs in future data at a compliance well and prior background data was stationary,
other detection monitoring tests are likely to eventually identify it. Trend testing may also be applied to
once-future data considered for a periodic background update, although the guidance primarily relies on
t-testing of historical and future groups to assess data suitability.

       At historically contaminated compliance wells, establishing a proper baseline for a prediction
limit or control chart is problematic,  since uncontaminated concentration  data cannot be collected.
Depending on the pattern of contamination, an intrawell  background may either have a stable  mean
concentration level or exhibit an increasing or decreasing trend. Particularly when intrawell background
concentrations are rising, the assumption of a static baseline population required by prediction limits and
control charts will be violated.

     As  an alternative, the Unified Guidance recommends a test for trend to measure the extent and
nature of the apparent increase. Trend testing can determine if there is a  statistically significant positive
trend over the period of monitoring and can also determine the magnitude (i.e., slope) of the trend. In
identifying a positive trend, it might be possible to  demonstrate that the level of contamination has
increased relative to historical behavior  and indicate how rapidly levels are increasing.

     Trend  analyses can  be  used  directly as  an alternative test against a GWPS in compliance and
corrective action monitoring.  For typical compliance monitoring, data collected at each compliance well
are used to generate a lower confidence limit compared to the fixed standard (Chapters 7, 21 and 22).
A similar situation occurs when corrective action is triggered, but making use  of an upper confidence
interval for comparison.  For compliance well  data  containing a trend, the  appropriate  confidence
interval is constructed around a linear regression trend line (or its non-parametric alternative) in order to
better estimate the most current concentration levels.   Instead of a single confidence limit for stationary
tests, the confidence limit (or band) estimate changes with time.

6.4.4 STATISTICAL INTERVALS

     Prediction limits, tolerance limits, control chart limits and confidence limits belong to the class of
methods  known as  statistical intervals.  The first three are used  to define their respective detection
monitoring test limits, while  the last is used in fixed standard compliance and corrective action tests.
When using a background GWPS, either approach is possible (see Section 7.5).  Intervals are generated
as a statistic from reference sample data, and represent a probable range of occurrence either for a future
sample statistic or some parameter of the population (in the case of confidence intervals) from which the
sample was drawn.   A future sample statistic might be one or more single values, as well as a future
mean or median of specific size, drawn from one or more  sample sets to be compared with the interval
(generally an upper limit).   Both the  reference  and  comparison  sample populations are  themselves
unknown, with the latter initially presumed to be identical to the reference  set population.  In the
groundwater monitoring context, the initial reference sample is the background data set.
                                              6-42                                    March 2009

-------
Chapter 6.  Detection Monitoring Design	Unified Guidance

      The key difference in confidence limits16 is that a statistical interval based on a single sample is
used to estimate the probable range of a population parameter like the true mean, median or variance.
The three detection monitoring tests use intervals to identify ranges of future sample statistics likely to
arise from the background population based on the initial sample, and are hence two- or multiple-sample
tests.

     Statistical intervals are  inherently two-sided,  since they represent a finite range in which the
desired  statistic or population parameter is expected to occur.  Formally, an interval is associated with a
level of confidence (1-a); by construction, the error rate a represents the remaining likelihood that the
interval does not contain the appropriate statistic or parameter. In a two-sided interval, the a-probability
is  associated with ranges both above and below the statistical interval.  A  one-sided upper interval is
designed to contain the desired statistic or parameter at the  same (1-a) level of confidence, but the
remaining  error represents only the range above  the limit.   As a general  rule, detection monitoring
options discussed below use one-sided upper limits because of the nature of the test hypotheses.

     PREDICTION LIMITS

     Upper prediction limits (or intervals) are constructed to contain with (1-a) probability,  the next few
sample  value(s) or sample statistic(s) such as a mean from a background population.  Prediction limits
are exceptionally versatile, since they can be designed to accommodate a wide variety of potential site
monitoring  conditions.   They have  been  extensively  researched,  and  provide a  straightforward
interpretation of the test results.  Since this guidance strongly encourages use of a comprehensive design
strategy to account for both the cumulative SWFPR and effective power to identify real exceedances,
prediction  limit options offer a most  effective  means of accounting  for both criteria.  The guidance
provides test options in the form  of parametric normal  and non-parametric prediction limit methods.
Since a retesting strategy of some form is usually necessary to meet both criteria, prediction limit options
are constructed to formally include resampling as part of the overall tests.

     Chapters 18 and 19 provide nine parametric  normal prediction limit test options:   four tests of
future values (l-of-2, l-of-3,  l-of-4 or a modified California plan) and five future mean options (1-of-l,
l-of-2,  or  l-of-3 tests  of mean size 2, and  1-of-l  or l-of-2 tests of mean size 3).  Non-parametric
prediction limit options cover the  same future value test options as the parametric versions, as well as
two median tests of size 3 (1-of-l or l-of-2 tests).  Appendix D tables provide the relevant  K-factors for
each parametric normal test option, the achievable false positive rates for non-parametric tests, and a
categorical rating of relative test power for each set of input conditions. Prediction limits can be  used
both for interwell  and intrawell testing. Selecting from among these  options should allow the two site
design criteria to be addressed for most groundwater site conditions.

     The options provided in the guidance are based on a wider class  known in the statistical literature
as p-of-m prediction limit tests.  Except for the two modified California plan options, those selected are
l-of-m test varieties.  The number of future measurements to be predicted (i.e., contained) by the interval
is  also denoted in the Unified Guidance by m and can be as  small as m =  1.  To test for a release to
groundwater, compliance well measurements are  designated  as future observations.  Then a limit is
constructed on the background  sample, with the prediction limit formula based on the number  of m
16
  Confidence limits are further discussed in Chapters 7,21 and 22 for use in compliance and corrective action testing.
                                              6-43                                     March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

future values or statistics to be tested against the limit.  As long as the compliance point measurements
are similar to background, the prediction limit should contain all m of the future values or statistics with
high probability (the level  of confidence).   For  a l-of-m test, all m values must be larger than the
prediction limit to be declared an exceedance, as initial evidence that compliance point concentrations
are higher than background.

     Prediction limits  with retesting are presented in Chapter  19.   When retesting is part of the
procedure, there are significant and instructive differences in statistical performance between parametric
and non-parametric prediction limits.

     Parametric prediction limits are constructed using the general formula: PL = x + K-s, where x
and s are the background sample mean and standard deviation, and K is the specific multiplicative factor
for the type of test, background sample size, and the number of annual tests.  The number of tests made
against a common background is also an input factor for interwell comparison.  The  Appendix D K-
factors are specifically designed to meet the SWFPR objective, but power will vary. Larger background
sample sizes and higher order (m) tests afford greater power.

     When  background data cannot be normalized, a non-parametric prediction limit can be used
instead. A non-parametric prediction limit test makes use of one or another of the largest sample values
from the background data set as the limit. For a given background sample size and test type, the level of
confidence of that maximal value is fixed.

     Using the absolute maximum of a background data set affords the highest confidence and lowest
single-test false positive error. However, even this confidence level may not be adequate to meet the
SWFPR objective, especially for lower order l-of-m tests. A higher order future values test using the
same maximum and background  sample size will provide greater false positive confidence and hence a
lower false positive error rate. For a fixed background sample size, a l-of-4 retesting scheme will have a
lower achievable significance level (a) than a l-of-3 or  l-of-2 plan for  any specific maximal value.  A
larger background sample  size using a fixed maximal value for any test also has a  higher confidence
level (lower a) than a smaller sample.

     But for a fixed non-parametric limit of a given background sample size, the power decreases as the
test order increases. If the non-parametric prediction limit is set at the maximum, a l-of-2 plan will be
more powerful than a l-of-4 plan.  It is relatively easy to understand why this is the case. A verified
exceedance in a l-of-2 test occurs only if two values exceed the limit, but would require four to exceed
for the  l-of-4  plan.  As  a rule, even the  highest order non-parametric  test using some  maximal
background value will be powerful enough to meet the ERPC power criteria, but achieving a sufficiently
low single-test error rate to meet the SWFPR is more problematic.

     If the SWFPR objective can be  attained at a maximum value for higher order l-of-m tests, it may
be possible to utilize lower maxima from a large background data base.  Lower maxima will have greater
power and a somewhat higher false positive rate.  Limited comparisons of this type can be made when
choosing between the largest or  second-largest order statistics in the  Unified  Guidance Appendix D
Tables 19-19 to 19-24.  A more useful and flexible comparison for l-of-m  future value plans can be
obtained using the EPA Region  8 Optimal Rank Values  Calculator discussed in Chapter 19.   The
calculator identifies the lowest ranked maximal value of a background data set for 1-of-l to l-of-4 future

                                             6-44                                   March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

value non-parametric tests which can meet the SWFPR objective, while providing ERPC ratings and
fractional power estimates at 2, 3, and 4 standard deviations above background.

       TOLERANCE INTERVALS

     Tolerance intervals  are  presented  in Section  17.2.  A  tolerance interval is generated  from
background sample data to contain a pre-specified proportion of the underlying population (e.g., 99% of
all possible population measurements) at a certain level of confidence. Measurements falling outside the
tolerance interval can be judged to be statistically different from background.

     While tolerance intervals are  an acceptable  statistical technique under RCRA as  discussed in
Section 2.3, the Unified Guidance generally recommends prediction limits instead. Both  methods can
be used to  compare compliance point measurements to background in detection monitoring. The  same
general formula is used in both tests for constructing a parametric upper limit of comparison: x + Ks.
For non-parametric upper limit tests, both prediction limits and tolerance intervals use an observed  order
statistic in  background  (often  the background maximum). But prediction limits  are ultimately  more
flexible and easier to interpret than tolerance intervals.

     Consider a parametric upper prediction limit test for the next two compliance point measurements
with 95% confidence. If either measurement exceeds the limit, one of two conditions is true: either the
compliance point distribution is significantly different and higher than background, or a false positive
has been observed and the two distributions are similar. False positives in this case are expected to occur
5% of the time. Using an upper tolerance interval is not so straightforward. The tolerance interval has an
extra statistical parameter that must be specified — the coverage (y) — representing  the fraction of
background to be contained beneath the upper limit. Since the confidence level (1-a) governs how  often
a statistical interval contains its target population parameter (Section 7.4), the complement a does not
necessarily represent the false positive rate in this case.

     In fact, a tolerance interval constructed  with 95% confidence to cover 80%  of background is
designed so that as many as 20% of all background measurements will exceed the limit with  95%
probability. Here, a = 5% represents the probability that the true coverage will be less than 80%. But less
clear is the false positive rate  of a tolerance interval  test in which as many  as 1 in 5 background
measurements are expected to exceed the upper background limit.  Are compliance point values above
the tolerance interval indicative of contaminated groundwater  or merely representative of the upper
ranges of background?

     Besides a more confusing interpretation, there is an added concern.  Mathematically valid retesting
strategies can be computed for prediction limits, but not yet for tolerance intervals,  further limiting their
usefulness in groundwater testing. It is also  difficult to construct powerful intrawell tolerance intervals,
especially when the intrawell background sample size is small. Overall, there is little practical  need for
two similar (but not identical) methods in the Unified Guidance, at least in detection monitoring.

     If tolerance intervals are employed as an alternative to ^-tests  or ANOVA  when performing
interwell tests, the RCRA regulations allow substantial flexibility in the choice of a.  This  means that a
somewhat arbitrarily high confidence level (1-a) can be specified when constructing a tolerance interval.
However, unless the coverage coefficient (y) is also set to a high value (e.g., > 95%), the test is  likely to
incur a large risk of false positives despite a small a.

                                              6-45                                    March  2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

     One setting in which an upper tolerance interval is very appropriate is discussed in Section 7.5.
Some constituents  that must be evaluated under compliance/assessment or corrective action may not
have a fixed GWPS.  Existing background levels may also  exceed a fixed GWPS. In these cases, a
background standard can be constructed  using an upper tolerance interval on background with 95%
confidence and 95% coverage. The standard will then represent a reasonable upper bound on background
and an achievable target for compliance and remediation testing.

6.4.5 CONTROL CHARTS

     Control charts (Chapter 20) are a viable alternative to  prediction limits in detection monitoring.
One advantage of a control chart over a prediction limit is that control  charts allow compliance point
data to be viewed and assessed graphically over time. Trends and changes in concentration levels can be
easily seen, because the compliance measurements are consecutively plotted on the chart as they are
collected, giving the data analyst an historical overview of the concentration pattern. Standard prediction
limits allow only point-in-time comparisons between the most recent data and background, making long-
term trends more difficult to identify.

     The guidance recommends use of the combined Shewhart-CUSUM control chart.  The advantage
is   that  two  statistical quantities  are assessed  at  every  sampling event,  both the new individual
measurement and the cumulative sum [CUSUM] of past and current measurements. Prediction limits do
not incorporate a CUSUM, and this can give control charts comparatively greater sensitivity to gradual
(upward) trends  and shifts in concentration  levels.  To enhance  false positive error rate control and
power, retesting can also be incorporated into the Shewhart-CUSUM control chart.  Following the same
restrictions as for prediction limits, they may be applied either to interwell or intrawell testing.

     A disadvantage in applying control charts to groundwater monitoring data is that less is understood
about their statistical performance, i.e., false positive rates and power. The control limit used to identify
potential  releases to groundwater is not based  on a formula incorporating a desired false positive rate (a).
Unlike prediction limits, the control limit cannot be precisely set to meet a pre-specified SWFPR, unless
the behavior of the control chart is modeled via Monte Carlo simulation. The same is true for assessing
statistical power. Control charts usually provide less flexibility than prediction limits in designing a
statistical monitoring program for a network.

     In addition, Shewhart-CUSUM control charts are a parametric procedure with no existing non-
parametric counterpart.  Non-parametric  prediction  limit tests are  still generally needed when the
background data on which the control chart is constructed cannot be normalized. Control charts are
mostly appropriate  for analytes with a reasonably high detection frequency in monitoring wells.  These
include inorganic constituents (e.g., detectable trace elements and geochemical monitoring parameters)
occurring naturally  in groundwater, and other persistently-found, site-specific chemicals.

6.5  SITE DESIGN  EXAMPLES

     Three  hypothetical design examples consider a  small, medium and large facility, illustrating the
principles discussed in this chapter.  In each example, the goal is to determine what statistical method or
methods  should be chosen and how those  methods can be implemented in light of the two fundamental
design criteria.  Further design details are covered in respective Part III detection monitoring test
                                             6-46                                   March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

chapters, although very detailed site design is beyond the  scope of the guidance.  More detailed
evaluations and examples of diagnostic tests are found in Part II of the guidance.

       ^EXAMPLE 6-5 SMALL FACILITY

     A municipal landfill has  3 upgradient wells and  8  downgradient wells. Semi-annual statistical
evaluations are required for five inorganic constituents.  So far, six observations have been collected at
each  well.  Exploratory  analysis  has  shown  that  the concentration  measurements appear  to be
approximately  normal  in distribution.  However, each of the five monitored  parameters exhibits
significant  levels of natural spatial variation from well to well.  What statistical approach should be
recommended at this landfill?

       SOLUTION
     Since the inorganic monitoring parameters are measurable and have significant spatial variability,
it is recommended that parametric intrawell rather than interwell tests should be considered. Assuming
that none of the downgradient wells is recently contaminated, each well has n = 6 observations available
for its respective intrawell background.  Six background measurements may or may not be enough for a
sufficiently powerful test.

     To address the potential problem  of inadequate power, a one-way ANOVA should be run on the
combined set of wells  (including background locations). If the well-to-well variances are significantly
different, individual standard deviation  estimates should be made from the six observations at the eight
downgradient wells. If the variances are approximately equal, a pooled standard deviation estimate can
instead be  computed from the ANOVA table. With  11 total wells and 6 measurements per well, the
pooled standard deviation has df = 11x5 = 55 degrees of freedom, instead of df= 5 for each individual
well.

     Regardless of ANOVA results, the per-test false positive rate is approximately the design SWFPR
divided by the annual number of tests. For w = 8 compliance wells, c = 5 parameters monitored,  and nE
=  2  statistical  evaluations  per  year,  the  per-test false positive  rate is approximately  atest  =
SWFPR/(wxcxnE) = 0.00125.  Given normal distribution data, several different parametric prediction
limit retesting plans can be examined,17 using either the combined sample size ofdf+ 1= 56 or the per-
well sample size of n = 6.

     Explained in greater detail  in Chapter 19, K-multiples and power ratings for each test type (using
the inputs w = 8  and n = 6 or 56  are obtained from  the nine parametric Appendix D Intrawell tables
labeled '5 COC, Semi-Annual'.  The following K-factors were obtained for tests of future values at n = 6:
K = 3.46 (l-of-2 test); K = 2.41 (l-of-3); K = 1.81 (l-of-4); and K = 2.97 (modified California) plans. For
future means, the corresponding K-factors were:  K =  4.46  (1-of-l  mean size 2); K = 2.78 (l-of-2 mean
size 2); K = 2.06 (l-of-3 mean size 2); K = 3.85 (1-of-l mean size 3); and K = 2.51 (l-of-2 mean size 3).
In these tables, K-factors reported in Bold have good power, those Italicized have acceptable power and
Plain  Text  indicates low power. For single well intrawell tests, only l-of-3 or l-of-4 plans for future
values, l-of-2 or l-of-3 mean size 2 or l-of-2 mean size 3 plans meet the ERPC criteria.
17 Intrawell control charts with retesting are also an option, though the control limits associated with each retesting scheme
  need to be simulated.

                                              6-47                                   March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

     Although each of these retesting plans is adequately powerful, a final choice would be made by
balancing 1) the cost of sampling and chemical analysis at the site; 2) the ability to collect statistically
independent samples should the sampling frequency be increased; and 3) a comparison of the actual
power curves of the three plans.  The last can be used to assess how differences in power might impact
the rapid identification of a groundwater release.  Since a l-of-3 test for future observations has good
power, it is unnecessary to make use of a l-of-4 test.  Similarly, the l-of-3 test for mean size  2 and a 1-
of-2 test for mean size 3 might also be eliminated, since a l-of-2 test of a mean  size 2 is  more than
adequate.   This leaves the  l-of-3 future values and l-of-2 mean 2 tests as the final prediction limit
options to consider.

     Though prediction limits around future means are more powerful than plans for observations, only
3 independent measurements might be required for a l-of-3 test, while 4 might be necessary for the 1-of-
2 test for mean size 2.  For most tests at  background, a single sample might suffice for the l-of-3 test and
2 independent samples for the test using a l-of-2 mean size 2.

     Much greater flexibility is afforded if the pooled intrawell standard deviation estimate can be used.
For this example, any of the nine parametric intrawell retesting plans is sufficiently powerful, including a
l-of-2 prediction limit test on observations and a 1-of-l  test of mean size 2.  In order to  make this
assessment using the  pooled-variance  approach,  a careful  reading of Chapter 13, Section  13.3. is
necessary to generate comparative K-factors.

     Less overall sampling is needed with the l-of-2 plan on observations, since only a single sample
may be  needed for most background conditions.   Two observations are always required for  the 1-of-l
mean size 2 test. More prediction limit testing options are generally available for a small facility. -^

       ^EXAMPLE 6-6  MEDIUM FACILITY

     A  medium-sized hazardous waste  facility has 4 upgradient background wells and 20 downgradient
compliance wells. Ten  initial measurements have been  collected at each upgradient well and 8 at
downgradient wells. The permitted monitoring list includes  10 inorganic parameters and 30 VOCs.  No
VOCs have yet been detected in groundwater. The remaining 10 inorganic constituents are normal or can
be normalized, and five show evidence of significant spatial variation across  the  site.  Assume that
pooled-variances cannot be obtained  from the historical upgradient or downgradient well data.  If one
statistical  evaluation  must  be  conducted each  year,  what  statistical  method  and  approach  are
recommended?

       SOLUTION
     At this site, there are potentially 800 distinct  well-constituent pairs that might be tested.  But since
none of the VOCs has been detected in  groundwater in background wells, all 30 of the VOCs should be
handled using the double quantification  rule (Section 6.2.2).  A second confirmatory resample should be
analyzed at those compliance wells for any of  the 30  VOC constituents initially  detected.  Two
successive quantified  detections above the RL are considered  significant evidence of groundwater
contamination at that well and VOC constituent. To properly limit the SWFPR, the 30 VOC constituents
are excluded from further SWFPR calculations, which is now based onw  x c x «# = 20 x 10 x 1= 200
annual tests.
                                             6-48                                    March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

     The five inorganic constituent background data sets indicate insignificant spatial variation and can
be normalized.  The observations from the four upgradient wells can be pooled to form background data
sets with an n = 40 for each of these five constituents. Future samples from the 20 compliance wells are
then compared against the respective interwell background  data.  With one annual evaluation, c =  10
constituents, w = 20 wells and n = 40 background samples, the Interwell '10 COC, Annual' tables for
parametric prediction limits with retesting can be searched in Appendix D.  Alternatively, control chart
limits can be fit to this configuration via Monte Carlo simulations.  Even though only five constituents
will be tested this way,  all of the legitimate constituents (c) affecting the SWFPR calculation, are used in
applying the tables.

     Most  of the interwell prediction limit retesting plans, whether for observations or means, offer
good power relative to the annual evaluation ERPC.  The final choice of a plan may be resolved by a
consideration of sampling effort and cost, as well as perhaps a more detailed power comparison using
simulated curves.   For prediction limits, a l-of-2 test for observations (K = 2.18) and the 1-of-l
prediction limit for a mean of order 2 (K = 2.56) both offer good power.  These two plans also require the
least amount of sampling to identify a potential  release (as discussed in Example 6-6).  Beyond this
rationale, the more powerful 1-of-l test of a future mean size 2 might be selected.  Full power curves
could be constructed and overlaid for several competing plans.

     The remaining 5 inorganic constituents  must be managed using intrawell methods based on
individual compliance well  sizes of n = 8.  For the same c, w, and n£ inputs as above, the Appendix D
Intrawell '10 COC, Annual' tables should be used.  Only four of the higher order prediction limit tests
have acceptable or good power:  l-of-4 future values (K = 1.84); l-of-2 mean size 2 (K = 2.68); l-of-3
mean size 2 (K = 2.00); and l-of-2 mean size 3 (K = 2.39) tests.  The l-of-2 mean size  2 has only
acceptable power.  The first two tests require the fewest samples under most background conditions and
in total, with the l-of-4 test having superior power. -4

       ^EXAMPLE 6-7 LARGE FACILITY

     A larger solid waste facility must conduct two statistical evaluations per year at two background
wells and 30 compliance wells.  Parameters on the monitoring list include five trace metals with a high
percentage  of non-detect measurements, and five other inorganic  constituents.  While the inorganic
parameters  are either normal or can be normalized, a significant degree of spatial variation is present
from one well to the next.  If 12 observations  were collected from  each background well, but only 4
quarterly measurements from each compliance well, what statistical approach is recommended?

       SOLUTION
     Because the two groups of constituents evidence distinctly different statistical characteristics, each
needs to be separately considered.  Since the  trace metals have occasional  detections or  'hits,' they
cannot be excluded from the  SWFPR computation. Because of their high non-detect rates, parametric
prediction limits or control charts may not be appropriate or valid unless a non-detect adjustment such as
Kaplan-Meier or robust regression on order statistics is used (Chapter 15). Assuming for this example
that parametric tests cannot be applied, the trace  metals  should  be  analyzed using non-parametric
prediction limits. The presence  of frequent non-detects may substantially limit  the potential degree of
spatial  variation, making an interwell non-parametric test potentially feasible. The Kruskal-Wallis non-
parametric ANOVA (Chapter 17) could be used to test this assumption.

                                             6-49                                    March 2009

-------
Chapter 6. Detection Monitoring Design	Unified Guidance

     In this case, the number of background measurements is n = 24, and this value along with w = 30
compliance wells would be used to examine possible non-parametric retesting plans in the Appendix D
tables  for  non-parametric prediction limits. As these tables  offer achievable per-evaluation, per-
constituent false positive rates for each configuration of compliance wells and background levels, the
target a level must be determined. Given semi-annual evaluations, the per-evaluation false positive rate
is  approximately OLE = O.IO/WE =  0.05.  Then, with 10 constituents altogether, the approximate per-
constituent false positive rate for each trace metal becomes ocCOnst = 0.05/10 = 0.005.

     Only one retesting plan meets the target false positive rate, a l-of-4 non-parametric prediction limit
using the maximum value in background  as the comparison  limit.  This plan has 'acceptable' power
relative to the ERPC. Other more powerful plans all have higher-than-targeted false positive rates.

     For the remaining 5 inorganic constituents, the presence of significant spatial variation and the fact
that the observations can be  normalized, suggests the use of parametric intrawell prediction or control
limits.  As in the previous Example 6-6, interwell prediction limit  tables in Appendix D are used by
identifying K multipliers  and power ratings based  on  all 10 constituents  subject to the SWFPR
calculations. This is true even though these parametric options only pertain to 5  constituents.  The total
number of well-constituent pair tests per year is equal tow  x c x «# = 30 x 10^2 = 600 annual tests.

     Assuming none of the observed spatial variation is due to already contaminated compliance wells,
the number of measurements that can be used as intrawell  background per well is small (n = 4). A quick
scan of the intrawell  prediction limit retesting plans  in Appendix D '10COC, Semi-Annual' tables
indicates that none of the plans offer even acceptable power for identifying a potential release. A one-
way ANOVA should be run  on the combined set of w = 30 compliance wells to determine if a pooled
intrawell standard deviation estimate can be used.

     If levels of variance across these wells are  roughly  the same, the pooled standard deviation will
have df = w(n -1)= 30 x 3 = 90 degrees of freedom, making each intrawell prediction or control limit
much more powerful. Using the R script provided in Appendix C for intrawell prediction limits with a
pooled standard deviation estimate (see Section 13.3), based on n = 4 and df = 90,  all of the relevant
intrawell prediction limits are sufficiently powerful  compared  to the semi-annual ERPC.   With  the
exception of the l-of-2 future values test at acceptable power, the other tests have good power.  The final
choice of retesting plan can be made by weighing the costs of required sampling versus perhaps a more
detailed comparison of the full power curves.  Plans with lower sampling requirements may be the most
attractive. ^
                                             6-50                                    March 2009

-------
Chapter 7.  Compliance Monitoring Strategies	Unified Guidance

                   CHAPTER  7.     STRATEGIES FOR
  COMPLIANCE/ASSESSMENT AND CORRECTIVE ACTION
       7.1   INTRODUCTION	7-1
       7.2   HYPOTHESIS TESTING STRUCTURES	7-3
       7.3   GROUNDWATERPROTECTION STANDARDS	7-6
       7.4   DESIGNING A STATISTICAL PROGRAM	7-9
         7.4.7  Fafae Positives and Statistical Power in Compliance/Assessment	 7-9
         7.4.2  False Positives and Statistical Power In Corrective Action	 7-72
         7.4.3  Recommended Strategies	 7-73
         7.4.4  Accounting for Shifts and Trends	 7-14
         7.4.5  Impact of Sample Variability, Non-Detects, And Non-Normal Data	 7-77
       7.5   COMPARISONS TO BACKGROUND DATA	7-20
     This chapter covers the fundamental design principles for compliance/assessment and corrective
action statistical monitoring programs. One important difference between these programs and detection
monitoring is that a fixed external GWPS is often used in evaluating compliance. These GWPS can be
an MCL, risk-based or background limit as well as a remedial  action goal. Comparisons to a GWPS in
compliance/assessment and corrective action are generally one-sample tests as opposed to the two- or
multi-sample tests in detection monitoring. Depending on the  program design, two- or multiple-sample
detection  monitoring  strategies   can  be  used  with  well  constituents  subject to  background
compliance/corrective action testing.   While a general framework is presented in this chapter, specific
test applications  and strategies are presented  in Chapters 21 and 22 for fixed GWPS comparisons.
Sections 7.1 through 7.4 discuss comparisons to  fixed GWPSs, while Section 7.5 covers background
GWPS testing  (either as a  fixed limit or based on a background statistic). Discussions of regulatory
issues are generally limited to 40 CFR Part 264, although they also apply to corresponding sections of
the 40 CFR Part 258 solid waste rules.

7.1 INTRODUCTION

     The  RCRA regulatory structure for compliance/assessment and corrective action monitoring is
outlined in Chapter 2. In detection and compliance/assessment monitoring phases, a facility is presumed
not to be  'out  of compliance' until significant evidence of an impact or groundwater  release can be
identified. In corrective action monitoring, the presumption  is reversed since contamination of the
groundwater has already been identified and confirmed. The null hypothesis of onsite contamination is
rejected only when there is significant evidence that the clean-up or remediation  strategy has been
successful.

     Compliance/assessment monitoring is generally begun when statistically significant concentration
exceedances above background have been confirmed for one or more detection monitoring constituents.
Corrective action is undertaken when at least one exceedance of a hazardous constituent GWPS  has
been  identified  in compliance/assessment  monitoring.   The  suite   of constituents  subject  to
compliance/assessment monitoring is determined from Part 264 Appendix IX or Part 258 Appendix n
testing, along with prior hazardous constituent data evaluated  under the detection monitoring program.
Following a  compliance monitoring statistical  exceedance, only a few of these constituents may require
                                             7-1                                   March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

the change in hypothesis structure to corrective action monitoring.  This formal corrective action testing
will need to await completion of remedial activities, while continued monitoring can track progress in
meeting standards.

     The same general statistical  method  of confidence interval testing  against  a fixed  GWPS is
recommended in both compliance/assessment and corrective action programs.  As discussed more fully
below  and in Chapter  21, confidence intervals  provide a flexible and statistically accurate method to
test how a parameter estimated from a single sample compares to  a fixed numerical  limit.  Confidence
intervals explicitly account for variation and uncertainty in  the sample data used to construct them.

     Most decisions about  a statistical program under §264.98 detection  monitoring  are tailored to
facility conditions, other than selecting a target site-wide cumulative false positive rate and a scheme for
evaluating power.  Statistical design details are likely to be site-specific, depending on the available data,
observed distributions  and  the  scope  of the monitoring network.   For compliance/assessment and
corrective action testing under §264.99 and §264.100 or similar tests against fixed health-based or risk-
based  standards, the testing regimen is instead  likely to  be  determined in  advance by the regulatory
agency. The Regional Administrator or State Director is charged with defining the nature of the tests,
constituents to be  tested,  and the wells or  compliance points to be  evaluated.  Specific  decisions
concerning false positive rates and power may also need to  be defined at a regulatory program level.

     The  advantage of  a  consistent approach  for compliance/assessment  and  corrective  action
monitoring tests is that  it can be applied across all Regional or State facilities. Facility-specific input is
still needed, including  the observed  distributions of key constituents and  the  selection  of  statistical
power  and false positive criteria for permits. Because of  the asymmetric nature of the  risks  involved,
regulatory  agency and  facility perspectives  may  differ on  which statistical risks  are most critical.
Therefore,  we  recommend  that  the  following  issues be addressed for compliance/assessment  and
corrective action  monitoring (both §264.99 and §264.100),  as well  as  for other programs  involving
comparisons to fixed standards:

    »«»  What are  the appropriate hypothesis  testing structures for making comparisons to a fixed
       standard?
    »»»  What do fixed GWPS represent in statistical terms  and which population parameter(s)  should be
       tested against them?
    »«»  What is a desirable frequency of sampling and testing, which test(s), and for what constituents?
    »«»  What statistical  power requirements  should be included to ensure protection of health and  the
       environment?
    »»»  What confidence level(s) should be  selected to  control false positive  error rates,  especially
       considering sites with multiple wells and/or constituents?
     Decisions regarding these five questions are  complex and interrelated, and have not been fully
addressed by previous RCRA guidance or existing regulations. This chapter addresses each of these
points  for both §264.99 and §264.100 testing. By developing answers at a regulatory program level,  the
necessity of re-evaluating the same questions at each specific site may be avoided.
                                               7-2                                    March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

7.2  HYPOTHESIS TESTING STRUCTURES

     Compliance testing under §264.99 specifically  requires a determination that one or more well
constituents  exceeds   a   permit-specific   GWPS.   The   correct   statistical   hypothesis  during
compliance/assessment monitoring is that groundwater concentrations are presumed not to exceed the
fixed  standard unless sampling data from one or more well constituents indicates  otherwise. The null
hypothesis, HQ, assumes that downgradient well concentration levels are less than or equal to a standard,
while the alternative hypothesis, HA, is accepted only if the standard is significantly exceeded. Formally,
for some  parameter  (0)  estimated  from  sample  data and representing a standard  G, the relevant
hypotheses under §264.99 compliance monitoring are stated as:

                                    H0:®G                                [7.1]

      Once a positive determination has  been made that  at least  one compliance well constituent
exceeds the fixed standard (i.e.., GWPS), the facility is subject to corrective action  requirements under
§264.100.  At this point,  the regulations imply and statistical  principles  dictate  that the hypothesis
structure should be reversed (for  those compliance  wells  and constituents  indicating exceedances).
Other compliance constituents (i.e., those not exceeding their respective GWPSs)  may continue to be
tested using equation 7.1 hypotheses.  It is then assumed that contamination equal to or in excess of the
GWPS exists and is presumed to be the case unless demonstrated otherwise.  A positive determination
that  groundwater concentrations  are below the  standard  is  necessary  to  demonstrate regulatory
compliance for any wells and constituents under remediation. In statistical terms, the  relevant hypotheses
for §264.100 are:

                                    H0:®>Gvs.HA:®
-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

     Non-RCRA programs seeking to use methods presented in the Unified Guidance may also presume
a different statistical hypothesis structure from that presented here. The primary goal is to ensure that the
statistical approach matches the  appropriate hypothesis framework. It is also allowable under RCRA
regulations to define GWPS based on background data, discussed further in Section 7.5.

     Whatever the population parameter (0) selected as representative of the GWPS, testing consists of
a confidence interval derived from the compliance point data at some choice of significance level (a),
and then compared to the standard G. The confidence intervals describe the probable distribution of the
sample statistic, 0,  employed to estimate the true parameter©. For testing under  compliance/assessment
monitoring, a lower confidence limit around the true parameter — LCL(0)  — is utilized. If LCL(0)
exceeds the standard, there is statistically significant evidence in favor of the alternative hypothesis, H\.
0 > G, that the compliance standard has been violated. If not, the confidence limit test is inconclusive
and the null hypothesis accepted.

     When the corrective action  hypothesis of [7.2] is employed, an upper confidence limit UCL(0) is
generated from the compliance point data and compared to the  standard G.  In  this case, the UCL(0)
should lie  below the standard to accept  the alternative hypothesis  that concentration  levels are in
compliance, HA'.  &
-------
Chapter 7. Compliance Monitoring Strategies _ Unified Guidance

constituent. The statistic used to estimate (I is the sample mean (x). With this statistic and normally-
distributed data, the lower and upper confidence limits are symmetric:


                                                                                           [7.3]
                                                                                           [7.4]


for a selected significance level (a) and sample  size n.  Note in these formulas that s is the sample
standard deviation, and tl_an_l is a central Student's lvalue with n-\ degrees of freedom.

     The two hypothesis structures and tests are defined as follows:

    Case A. Test of non-compliance (§264.99) vs. a fixed standard (compliance/assessment monitoring):

       Test Hypothesis: HQ : jU < G vs. HA : jU > G

                                       g
       Test Statistic: LCLl_a = x- tl_a ^ —j=
                                       V«

       Rejection Region: Reject null hypothesis (Ho) if LCLl_a > G ; otherwise, accept null hypothesis

    Case B. Test of compliance (§264.100) vs. a fixed standard (corrective action):

       Test Hypothesis: HQ : jU > G vs. HA : jU < G
       Test Statistic: UCLl_a = x + tl_a ^ -=
       Rejection Region: Reject null hypothesis (Ho) if UCLl_a < G ; otherwise, accept null hypothesis

     For all  confidence  intervals and tests presented in Chapters 21 and 22, the test structures are
similar to those above. But not every pair of lower and upper confidence limits (i.e., LCL and UCL) will
be symmetric, particularly for skewed distributions and in non-parametric tests on upper percentiles. For
a non-parametric technique such as a confidence interval around the median, exact confidence levels will
depend  on the available sample size and which order statistics are  used to estimate  the  desired
population parameter. In these cases, an exact target confidence level may or may not be attainable.

     When calculating confidence intervals, assignment  of the false positive error (a)  differs between a
one-sided and two-sided  confidence interval test.  The symmetric upper and lower confidence intervals
are shown in Figure 7-1 largely  for illustration purposes.  If the lower confidence interval for some
tested parameter 0 is the critical  limit, all of the a error is assigned to the region below the LCL(0).
Hence, a 1-a confidence level covers the range from the lower limit to positive infinity. Similarly, all of
the a error for an upper confidence limit  UCL(0) is assigned to the region above this value.  For a two-

                                               7-5                                    March 2009

-------
Chapter 7.  Compliance Monitoring Strategies
                                     Unified Guidance
sided interval, the error rate is equally partitioned on both sides of the respective confidence interval
limits. A 95% lower confidence limit implies that a 5% chance of an error exists for values lying below
the limit.   In contrast, a two-sided 95% confidence interval implies a 2.5% chance above and a 2.5%
chance of an error below the confidence level.  Depending on how confidence intervals are defined, the
appropriate statistical adjustment (e.g., the lvalue in Equations 7-3 and 7-4) needs to take this into
account.

      Figure 7-1. Confidence Interval  on Mean vs.  Fixed Upper Percentile Limit
                    a
                    o
                   ••a
                    cS
                   3
                   60
                                                                 Population Values
                        GWPS
CI on 95thD
Percentile
                                                     CI on Mean
7.3 GROUNDWATER PROTECTION STANDARDS

     A second essential design step is to identify the appropriate population parameter and its associated
statistical  estimate.   This  is primarily a  determination of what a given fixed GWPS represents in
statistical  terms. Not all fixed  concentration standards are meant  to  represent the  same statistical
quantities.  A distinction is drawn between 1) those central tendency standards designed to represent a
mean or  average concentration level and 2)  those which represent  either  an upper percentile or the
maximum of the concentration distribution. If the fixed standard represents an average concentration, it
is assumed in the Unified Guidance that the mean concentration (or possibly the median concentration)
in groundwater should not exceed the limit. When a fixed standard  represents  an upper percentile or
maximum, no more than a small, specified fraction of the individual concentration measurements should
exceed the limit.

     The  choice of confidence interval should be based on the type of fixed  standard to which the
groundwater data will be compared.  A fixed  limit best representing  an upper percentile concentration
(e.g., the upper 95th percentile) should not be compared to a confidence interval  constructed around the
arithmetic mean. Such an interval only estimates the location of the population mean, but says nothing
about the  specific upper percentile of the concentration distribution. The average concentration level
                                             7-6                                    March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

could be  substantially less  than the standard even though a significant fraction  of the individual
measurements exceeds the standard (see Figure 7-1).

     There are a variety of fixed standards to which different statistical measures apply.  Alternative
GWPSs based on Agency risk-assessment protocols are cited as an option in the solid waste regulations
at §258.55(i)(l).  Many of the risk-assessment procedures identified in the CERCLA program make use
of chronic, long-term exposure models for ingestion or inhalation. These procedures are identified in the
(EPA,  1989b) Risk Assessment  Guidance for Superfund (RAGS) and the Supplemental Guidance for
Calculating the Concentration Term (EPA,  1992c), and serve as guidance for other EPA programs.  In
the  latter  document, the arithmetic mean  is  identified as  the appropriate  parameter for identifying
environmental exposure levels. The levels are intended to identify chronic, time-weighted concentration
averages based on lifetime exposure scenarios.

      The primary maximum contaminant levels [MCL] promulgated under the Safe Drinking Water Act
(SDWA) follow the same exposure evaluation principles. An MCL is typically based on 70-year risk-
exposure scenarios (for carcinogenic compounds), assuming an ingestion rate of 2 liters of water per day
at the average concentration over time. Similarly, long-term risk periods (e.g., 6-years) are used for non-
carcinogenic  constituents, assuming average  exposure concentrations.  The  promulgated levels  also
contain a  safety multiplicative factor and are applied at the end-user tap.  Calculations for ingestion
exposure risk to soil contaminants by an individual randomly traversing a contaminated site are based on
the  average estimated  soil concentration. It is expected that an exposed individual drinking  the water or
ingesting the soil is  not afforded any protection in the form of prior treatment.

     Other standards  which may represent a  population mean  include some RCRA site permits that
include comparisons against an alternate  concentration limit [ACL] based  on the average value of
background data. In addition,  some standards represent time-weighted averages used for carcinogenic
risk assessments such as the lifetime average daily dose [LADD].

     Fixed limits based explicitly on the median concentration include fish ingestion exposure factors,
used in testing fish tissue  for  certain  contaminants.  The  exposure  factors represent  the  allowable
concentration level  below which at  least half of the fish sample concentrations  should lie, the 50th
percentile of the  observed concentration distribution. If this distribution is  symmetric,  the mean and
median will be identical. For positively skewed populations, the mean concentration could exceed the
exposure factor even though the median (and hence, a majority of the individual concentrations) is below
the  limit. It would therefore not  be appropriate to compare such exposure factors  against a confidence
interval around the mean contaminant level, unless one could be certain the distribution was symmetric.

     Fixed standards  are sometimes based on upper percentiles. Scenarios of this  type include risk-
based standards designed to limit acute effects that result from short-term exposures to certain chemicals
(e.g., chlorine gas leaking from a rail car or tanker).  There is greater interest in possible acute effects or
transient exposures  having a significant short-term risk.  Such exposure events may not happen often, but
can be important to  track for monitoring and/or compliance purposes.

     When even short exposures can result  in deleterious health or environmental effects,  the fixed limit
can be  specified as a maximum allowable concentration.  From a statistical standpoint, the standard
identifies  a level which can  only  be exceeded  a small  fraction  of the time (e.g.,  the  upper 90th
percentile). If a larger than allowable fraction of the individual exposures exceeds the standard, action is

                                               7-7                                     March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

likely warranted, even if the average concentration level is below the standard. Certain MCLs are
interpreted in this same manner; the term 'maximum' in maximum contaminant level would be treated
statistically as an upper percentile limit. Examples include criteria for bacterial counts and nitrate/nitrite
concentrations, best regarded as upper percentile limits.

     As an example, exposure of infants to nitrate concentrations in excess of 10 mg/L (NC>3~ as N) in
drinking water is  a  case where  greater concern surrounds  acute effects  resulting  from short-term
exposure.  The flora in the intestinal tract of infant humans and animals does not fully develop until the
age of about six months.  This results in a lower acidity in the intestinal tract, which permits the growth
of nitrate  reducing  bacteria.  These  bacteria convert  nitrate  to nitrite.  When absorbed  into  the
bloodstream,  nitrite interferes with the absorption of oxygen.  Suffocation by oxygen starvation in this
manner produces a bluish skin discoloration — a condition known  as "blue  baby" syndrome (or
methemoglobinemia) — which can result in serious health problems, even  death.  In such a scenario,
suppose that acute effects resulting from short-term exposure above some critical level should normally
occur in no more than 10 percent of all exposure events. Then the critical level so identified would be
equivalent to the upper 90th percentile of all exposure events.

     Another example is the so-called 20-year flood recurrence interval for structural design. Flood
walls and  drainage culverts are designed to handle not just the average flood level, but also flood levels
that have a 1  in 20 chance of being equaled or exceeded in any single year. A 20-year flood recurrence
level is  essentially equivalent to estimating the upper 95th percentile of the  distribution of flood levels
(e.g., a flood of this magnitude is expected to occur only 5 times every 100 years).

     The  various limits identified as potential GWPS in Chapter 2 pose some interpretation problems.
§264.94 Table 1 values are identified as "Maximum Concentration[s] of Constituents for Groundwater
Protection" for 14 hazardous constituents,  originating  from  earlier Federal Water Pollution Control
Administration efforts.  While not  a definitive protocol for comparison, it was indicated that the limits
were intended to represent  a concentration level that should not be exceeded most of the time.  In an
early Water Quality Criteria report (USDI, 1968), the  language is as follows:

       "It is  clearly  not  possible to  apply  these  (drinking water) criteria solely as maximum single
       sample values. The criteria  should not be exceeded over substantial portions of time."

       Similarly, the more current MCLs promulgated  under the  SDWA are identified as "maximum
contaminant limits".  Even if the limits were derived from chronic, risk-based assessments, the same
implication is that these limits should not be exceeded.

       Individual EPA programs make sample data  comparisons to MCLs using  different approaches.
For small-facility systems  monitored  under the  SDWA, only one or  two samples a year might be
collected for comparison.  Anything other than direct  comparisons isn't possible.  Some Clean Water Act
programs  use arithmetic comparisons (means or medians) rather than  a  fully statistical  approach.
CERCLA typically utilizes these standards in mean statistical comparisons, consistent with other chronic
health-based levels derived from their program risk assessment equations.  In  short, EPA nationwide
does not have a single operational definition or measure for assessing MCLs with sample data.

       The Unified Guidance cannot directly  resolve these issues.  Since the regulations promulgated
under   RCRA  presume  the use  of fully statistical measures  for groundwater  monitoring program
                                              7-8                                    March 2009

-------
Chapter 7.  Compliance Monitoring Strategies	Unified Guidance

evaluations, the guidance provides a number of options for both centrality-based and upper limit tests.
It falls upon  State or Regional programs to determine which is  the most appropriate parameter for
comparison to a GWPS.  As indicated above, the guidance does recommend that any operational
definition of the appropriate parameter of comparison to GWPS's be applied uniformly across a program.

       If  a  mean- or  median-based centrality parameter  is chosen,  the guidance  offers fairly
straightforward confidence interval testing options.   For a parameter representing some infrequent level
of exceedance to address the "maximum"  or "most" criteria, the program would  need to identify a
specific upper proportion and confidence level that the GWPS represents.  Perhaps a proportion of 80 to
95% would be appropriate, at 90-95% confidence.  It is presumed that the same standard would apply to
both compliance and corrective action testing under §264.99 and  §264.100. If non-parametric upper
proportion tests must be used for certain data, very high proportions make for especially difficult tests to
determine a return to compliance (Chapter 22) because of the number of samples required.

7.4  DESIGNING A STATISTICAL PROGRAM

7.4.1 FALSE POSITIVES AND STATISTICAL POWER IN  COMPLIANCE/ASSESSMENT

     As  discussed in Chapters 3 and 6,  the twin criteria in   designing an acceptable detection
monitoring statistical program are the site-wide false positive rate [SWFPR] and the effective power of
the testing regimen. Both statistical measures are  crucial  to  good statistical design, although from a
regulatory perspective, ensuring adequate power to detect  contaminated  groundwater is of primary
importance.

     In compliance/assessment monitoring, statistical power  is also of prime concern to EPA. There
should be a high probability that  the statistical test will  positively identify concentrations that have
exceeded a fixed, regulatory standard. In typical applications  where a confidence interval is compared
against a fixed standard,  a low false positive error rate (a) is chosen without respect to the power of the
test. Partly this is due to a natural  desire to have high statistical  confidence in the test, where (l-oc)
designates the confidence level of the interval. But statistical confidence is not the same as power. The
confidence level merely indicates how often — in repeated applications — the interval will contain the
true population parameter (0); not how often the test will indicate  an exceedance of a fixed standard.  It
has historically been much easier to select a single value for the false positive rate (a) than to measure
power, especially since power is not a single number but  & function of the level of contamination (as
discussed in Section 3.5).

     The power to detect increases above  a fixed standard using a lower confidence limit can be
negligible when contaminant variability is high, the sample size is small and especially when a high
degree of confidence has been selected. To remedy this problem, the Unified Guidance recommends
reversing the usual sequence: first select a desired level of power for the test (1-P), and then compute the
associated (maximum) false positive rate (a). In this way, a pre-specified power can be maintained even
if the sample size is too low to simultaneously minimize the risks of both Type I and Type n errors (i.e.,
false positives and false negatives).

     Specific methods for choosing power and computing false positive rates with confidence interval
tests are  presented in  Chapter 22. Detailed applications of confidence  interval tests are provided in
Chapter 21. The focus here is on setting a basic framework and consistent strategies.

                                             7-9                                   March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

     As noted above, selecting false positive error rates in compliance or assessment testing (§264.99)
has traditionally been accomplished under RCRA by choosing a fixed, individual test a. This strategy is
attractive if only for the sake of simplicity. Individual test-wise false positive rates in the range of a =
.01 to a = .10  are traditional and easily understood. In addition, the Part 264 regulations in §264.97(i)(2)
require a minimum individual false positive rate of a = .01  in both compliance and corrective action
testing against a  fixed standard, as well as in those tests not specifically exempted under detection
monitoring.1

     Given a fixed sample size and constant level  of variation, the statistical power of a  test method
drops as the false positive rate decreases. A low false positive rate is often associated with low power.
Since statistical power is of particular concern to EPA in compliance/assessment monitoring, somewhat
higher false positive rates  than the minimum a = .01 RCRA requirement may be necessary to maintain a
pre-specified power over  the range of sample sizes and variability likely to be  encountered in RCRA
testing situations. The key is sample variability. When the true population coefficient of variation [CV]
is  no greater than 0.5 (whether the underlying distribution is normal or lognormal), almost all lower
confidence limit tests exhibit adequate power. When the variation is higher, the risk of false  negative
error is typically much greater (and thus the power is lower), which may necessitate setting a larger than
usual individual a.

     Based on the discussion regarding false positives  in detection monitoring in  Chapter  6,  some
might be concerned about the use of relatively high individual test-wise false positive rates  (a) in order
to  meet a pre-specified power, especially when considering the cumulative false positive error rate across
multiple wells and/or constituents (i.e., SWFPR).  Given that  a  number of compliance wells and
constituents might need to be tested, the likelihood of occurrence of at least one false positive error
increases dramatically. However, several factors specific to compliance/assessment monitoring need to
be considered. Unlike detection monitoring where the number of tests is  easily  identified,  the issue is
less obvious for  compliance/assessment or corrective action testing.  The RCRA  regulations do  not
clearly specify which wells and constituents must be compared to the GWPS in compliance/assessment
monitoring other than wells at the 'compliance point.' In some situations, this has been interpreted to
mean all compliance wells; in other instances, only at those wells with a documented exceedance.

     While all hazardous constituents including additional  ones detected in Part  264 Appendix IX
monitoring are potentially subject to testing, many may still be at concentration levels insignificantly
different from  onsite background. Constituents without health-based limits may or may not  be  included
in  compliance testing. The latter would be tested  against background levels, using perhaps  an ACL
computed as a tolerance limit on background (see Section 7.5). This also tends to complicate derivation
of SWFPRs in compliance testing. It was also noted in Section 7.2 that the levels at which contaminants
are released bear  no necessary relationship to fixed, health-based standards. In a typical release,  some
constituent levels from a suite of analytical parameters may lie orders of magnitude below their GWPS,
while certain carcinogenic compounds may easily exceed their standards.
1  In  some  instances, a test with "reasonable confidence"  (that is, having adequate statistical power) for identifying
  compliance violations can be designed even if a < 0.01.  This is particularly the case when the sample coefficient of
  variation is quite low, indicating small degrees of sample variability.

                                              7-10                                    March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

     The  simple example below illustrates typical low-level aquifer concentrations following a release
of four common petrochemical facility hazardous organic constituents often detected together:
Analyte

Benzene
Toluene
Ethylbenzene
Xylene
Aquifer Concentration
Mean
20
35
40
100
(ug/i)
SD
10
15
20
35
MCL(ug/l)

5
1,000
700
10,000
       While benzene as a carcinogen has a very low health standard, the remaining three constituents
have aquifer concentrations orders of magnitude lower than their respective MCLs. Realistically, only
benzene is likely to impact the cumulative false positive rate in LCL testing. Similar relationships occur
in releases measured by trace element and semi-volatile organic suites.

     Even  though the null hypotheses in detection and compliance/assessment monitoring are similar
(and compound)  in nature (see [7.1]), it is reasonable to presume  in detection monitoring that the
compliance wells have average concentrations no less than mean background levels.  Since it is these
background levels to which the compliance point data are compared in the absence of a release, the
compound null hypothesis in detection monitoring (Ho: (ic < M-BG) can be reformulated practically as (H^.
M-c = M-BG)-  In this framework, individual concentration  measurements are likely to occasionally exceed
the background average and at times cause false positives to be identified even when there has been no
change in average groundwater quality.

     In compliance/assessment  monitoring, the situation is  generally different.  The compound null
hypothesis (H0: MC < GWPS) will include some wells and  constituents where the sample mean equals or
nearly  equals the GWPS when testing begins. But many well-constituent  pairs may have true means
considerably less  than the standard, making false positives much less likely for those comparisons and
lowering the  overall SWFPR. How much  so  will depend on both the variability of each individual
constituent and the degree to which the true mean (or relevant statistical parameter 0) is lower than the
GWPS for that analyte.

     Because of this, determining the relevant number of comparisons with non-negligible false positive
error rates may be quite difficult. The SWFPR in this situation would be defined as the probability that at
least one or more lower confidence limits exceeded the fixed standard G,  when the true parameter 0
(usually the mean) was actually below the standard. However, the relevant number of comparisons will
depend on the nature and extent of the release.  For a more extensive release, there is greater likelihood
that the null hypothesis is no longer true at one or more wells. Instead of computing false positive rates,
the focus should shift to minimizing false negative errors  (i.e., the risk of missing  contamination above
the GWPS).
2 Note that background might consist of early intrawell measurements from compliance wells when substantial spatial
  variability exists.

                                              7-11                                    March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

     On  balance,  the  Unified  Guidance  considers  computation  of  cumulative  SWFPRs  in
compliance/assessment testing to be problematic, and  reliance  on individual test false positive rates
preferable.  The above arguments also suggest that flexibility in setting individual test-wise a levels may
be appropriate.

7.4.2 FALSE POSITIVES AND STATISTICAL POWER IN CORRECTIVE ACTION

     When contamination above a GWPS  is confirmed, corrective action is triggered.  Following a
period of remediation activity, formal statistical testing  will usually involve an upper confidence limit
around the mean or an upper percentile compared against a  GWPS. EPA's overriding  concern in
corrective action is that remediation efforts not be declared successful without sufficient statistical proof.
Since groundwater is now presumed to be  impacted at unacceptable levels, a facility should not exit
corrective action until there is sufficient evidence that contamination has been abated.

     Given the reversal of test hypotheses from compliance/assessment monitoring to corrective action
(i.e., comparing  equation [7.1] with [7.2]), there is an asymmetry in regulatory considerations of false
positive and false negative  rates  depending  on the stage of monitoring.  In  compliance/assessment
monitoring using tests of the lower confidence limit,  the principal regulatory concern is that a given test
has adequate statistical power to detect exceedances above the GWPS.

     Permitted RCRA monitoring is likely to involve small annual well sample sizes based on quarterly
or semi-annual sampling.  To meet a pre-specified level of power by controlling the false negative rate
(P) necessitates varying the false positive rate  (a) for individual  tests. Controlling an SWFPR for these
tests (using a criterion like the SWFPR) is usually not practical because of the ambiguity in identifying
the relevant number of potential  tests  and the difficulty of properly assigning  via the subdivision
principle (Chapter 19) individual fractions of a targeted  SWFPR.

     By contrast under corrective  action using an  upper confidence limit for testing,  the principal
regulatory and environmental concern is that one or more constituents might falsely be declared below a
GWPS in concentration. Under the corrective action  null hypothesis [7.2] this would be a, false positive
error, implying that a should be minimized during this sort of testing, instead of p. Specific methods for
accomplishing this goal are presented in Chapter 22.

     A remaining question is whether  SWFPRs should be controlled during corrective action. While
potentially  desirable, the number of well-constituent pairs exceeding their respective GWPS and subject
to corrective action testing is likely to be small relative to compliance testing. Not all compliance wells
or constituents may have been impacted, and some  may not be contaminated to levels exceeding the
GWPS, depending on the nature, extent, and intensity of the plume. Remediation efforts would focus on
those constituents exceeding their GWPS.

     As noted  in  Section   7.4.1, the tenuous  relationship  between  ambient  background  levels,
contaminant magnitudes, and risk-based health standards implies  that most GWPS exceedances are
likely to be carcinogens,  usually  representing  a  small portion of all monitored constituents. Some
exceedances may also be related compounds, for instance, chlorinated hydrocarbon daughter degradation
products.
                                             7-12                                    March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

     Statistically, the fact that some wells are contaminated while others may not be makes it  difficult
to define SWFPRs in corrective action. Instead, the Unified  Guidance attempts to limit the individual
test-wise a at those wells where exceedances have been confirmed and that are undergoing remediation.
Since the most important consideration is to ensure that the true population parameter (0) is actually
below the clean-up standard before declaring remediation a success, this guidance recommends the use
of a reasonably low, fixed test-wise false positive rate (e.g., a = .05 or .10). Under this framework, there
will  be a 5% to 10% chance of incorrectly declaring any  single  well-constituent pair of being in
compliance when its concentrations are truly above the remedial standard.

     The regulatory position in corrective action concerning  statistical  power is  one of relative
indifference. Although power under [7.2]  represents the probability that the confidence interval test will
correctly identify concentrations to be below the regulatory standard when in fact they  are, the onus of
proof for demonstrating that remediation has succeeded (e.g., (ic < GWPS) falls on the regulated facility.
As it is the facility's interest to demonstrate compliance, it may wish to develop statistical power criteria
which would enhance this possibility (including increasing test sample sizes).

7.4.3 RECOMMENDED STRATEGIES

     As noted in Section 7.1, the Unified Guidance recommends the use of confidence intervals in both
compliance/assessment and corrective action testing. In compliance/assessment, the lower confidence
limit is the appropriate statistic of interest, while in corrective action it is the upper confidence limit. In
either case, the confidence  limit is compared against a fixed,  regulatory standard as a one-sample test.
These recommendations are consistent with good statistical practice, as well as literature in the field,
such as Gibbons and Coleman (2001).

     The type of confidence  interval test will initially be determined by the choice of parameter(s) to
represent the GWPS (Section 7.2). While this discussion has  suggested that the mean may be the most
appropriate parameter for chronic, health-based limits, other choices are possible.  Chapter 21 identifies
potential test statistical tests of a mean, median or upper percentile as the most appropriate parameters
for comparison to a  GWPS.   In turn, data characteristics will determine whether parametric or non-
parametric test versions can be used.  Depending on whether normality can be assumed for the original
data  or following transformation, somewhat different approaches may be  needed.  Finally, the presence
of data trends affects how confidence interval testing can be applied.

     Some regulatory programs prefer to  compare each individual measurement against G, identifying a
well  as  out-of-compliance if  any  of the individual concentrations exceeds the standard. However, the
false positive rate associated with such strategies tends to be  quite high if the parameter choice has not
been clearly specified.   Using this  individual comparison   approach and  assuming  a  mean as  the
parameter of choice,  is of particular concern.  If the true mean is less than but close  to the standard,
chances are very high that one or more  individual measurements will be greater than the limit even
though the hypothesis in [7.1] has not been violated. Corrective action could then be initiated on a false
premise. To evaluate whether a limited number of sample data exceed a standard, a lower confidence
interval test would need to be based on a pre-specified upper percentile assumed to be the appropriate
parameter for comparison to the GWPS.

     Small  individual   well  sample  sizes   and data  uncertainty   can rarely  be   avoided  in
compliance/assessment and corrective action. Given the nature of RCRA permits, sampling frequencies
                                             7-13                                    March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

in compliance/assessment or corrective action monitoring are likely to be  established  in  advance.
Relatively small sample sizes per well-constituent pair each year are likely to be the rule; the Unified
Guidance assumes that quarterly and semi-annual sampling will be very typical.

     For small and highly variable sample data sets, compliance/assessment monitoring and corrective
action tests will have low statistical power either to detect exceedances above fixed  standards or to
demonstrate compliance in corrective action. One way to both enhance statistical power and control false
positive error rates  is through incremental or sequential pooling of compliance point  data over time.
Adding  more  data  into a test of non-compliance or  compliance will  generally result in  narrower
confidence intervals and a clearer decision with respect to a compliance standard.

     The  Unified Guidance recommends accumulating compliance data over time at each  well, by
allowing construction of confidence limits on overlapping as opposed to distinct or mutually  exclusive
data sets.  If the lower confidence limit [LCL] exceeds the GWPS in compliance/assessment, a clear
exceedance can be  identified.  If the upper confidence limit [UCL] is below the GWPS in corrective
action, remediation  at that well can be declared a success.  If neither of these respective events occurs,
further sampling should continue. A confidence interval can be recomputed after each additional 1 or 2
measurements and  a determination made whether the  position of the confidence limit has changed
relative to the compliance standard.

     Tests constructed  in  this  way at each  successive evaluation  period will not  be  statistically
independent; instead, the proposed testing strategy falls into the realm of sequential analysis. But it
should help to minimize the possibility that a small group of spurious values will either push  a facility
into needless corrective action or prevent a successful remedial effort from being identified.

     One  caveat with this approach is that it must be reasonable to assume that the population parameter
© is stable over time. If a release has occurred and a contaminant plume is spreading through the aquifer,
concentration shifts in the form of increasing trends over time may be more likely at contaminated wells.
Likewise under active remediation, decreasing trends for a period of time may be more likely. Therefore,
it is recommended that the sequential testing approach be used after aquifer conditions have stabilized to
some degree. While concentration levels are actively changing with time, use of confidence  intervals
around a trend line should be pursued (see Section 7.4.4 and Chapter 21).

7.4.4 ACCOUNTING FOR SHIFTS AND TRENDS

     While accumulating compliance point data over time and successively re-computing confidence
limits is appropriate for  stable (i.e., stationary) populations, it can give misleading or false results when
the underlying population is  changing. Should a release  create an expanding contaminant plume within
the aquifer, concentration levels at some or all of the compliance wells will tend to shift upward,  either
in discrete jumps (as illustrated in Figure 7-2) or an increasing trend over time. In these cases, a  lower
confidence limit constructed on  accumulated data will be overly wide (due to high sample variability
caused by  combining pre- and post-shift data) and not be reflective of the more recent upward shift in the
contaminant distribution.
                                             7-14                                    March 2009

-------
Chapter 7.  Compliance Monitoring Strategies
                                 Unified Guidance
      Figure 7-2. Effect on Confidence Intervals of Stable Contamination Level
                      65
                      60
                   3  55 —
                   1
                      50
                      45-
                          CJWPS
                              90
                                    91
                                                  \
                                           Compliance Monitoring Begins
                                         92
                                              93    94

                                               Year
                                                        95
                                                              96
                                                                  97
                                                                       98
     A similar problem  can arise  with  corrective  action  data.  Aquifer modifications as part of
contaminant removals are likely to result in observable declines in constituent concentrations during the
active  treatment phase. At some point following cessation of remedial action, a new steady-state
equilibrium may be established (Figure 7-3).

                Figure  7-3. Decreasing Trend During Corrective Action

                    500
                 ~ 450  -
                 E
                 CL
                 d
                 ^ 400  -
                 O
                 U
                 S 350  H
                 O
                 u 300  -
                    250
8      12      16

 Sampling Month
                                                                 20
24
     Until  then, it is inappropriate to use a confidence interval test  around the mean or an upper
percentile to evaluate remedial success with respect to a clean-up  standard.  During active treatment
phases and under non-steady state conditions, other forms of analysis such as confidence bands around a
trend (see below), are recommended and should be pursued.
                                            7-15
                                         March 2009

-------
Chapter 7. Compliance Monitoring Strategies
Unified Guidance
     The Unified Guidance considers two basic types of non-stationary behavior: shifts and (linear)
trends. A shift refers to a significant mean concentration increase or decrease departing from a roughly
stable mean level.  A trend refers to  a series of consecutive measurements that evidence successively
increasing or decreasing concentration levels. More complicated non-random data patterns are also
possible, but beyond the  scope of this guidance.   With these two  basic scenarios, the strategy for
constructing an appropriate confidence interval differs.

     An important preliminary step is to track the individual compliance point measurements on a time
series plot (Chapter 9). If a discrete shift in concentration level is evident, a confidence limit should be
computed on the most recent stable measurements.  Limiting the observations in this fashion to a
specific time period is often termed a 'moving window.' The reduction in sample size will often be more
than offset by the gain in statistical power.  More recent measurements may exhibit less variation around
the shifted mean value, resulting in a shorter confidence interval (Figure 7-4). The sample size included
in  the  moving  window   should  be   sufficient  to   achieve   the  desired  statistical  power
(compliance/assessment)  or  false positive rate (corrective  action). However, measurements that are
clearly unrepresentative of the newly shifted distribution should not be included, even if the sample size
suffers.  Once a stable mean can be assumed, the strategy of sequential pooling can be used.

           Figure 7-4. Effect on Confidence  Intervals of Concentration Shift
                    500
                    450
                 I  400
                 o.
                 C
                    350
                u
                    300
                    250
                              Compliance Monitoring (CM) begins

                          GWPS
                                                                  CI on all
                                                                  CM data
                           _,  !  ,!   ,(,j  ,  j  ,  !   |   ^  ,   j   ,_

                       89    90    91     92    93    94    95     96    9?     98
                                                  Year
       If well concentration levels exhibit an increasing or decreasing trend over time (such as the
example in Figure 7-5) and the pattern is reasonably linear or monotone, the trend can be identified
using the methods detailed in Chapter 17. To measure compliance or non-compliance, a confidence
band can be constructed around the estimated trend line, as described in Chapter 21. A confidence band
is essentially a continuous series  of confidence intervals estimated along every point of the trend. Using
this technique, the appropriate upper or lower confidence limits at one or more points in the most recent
                                              7-16
        March 2009

-------
Chapter 7.  Compliance Monitoring Strategies
Unified Guidance
portion of the end of the sampling record can be compared against the fixed standard.  The lower band is
used to determine whether or not an exceedance has occurred in compliance/assessment, and an upper
confidence band to determine if remedial success has been achieved in corrective action.

                Figure  7-5. Rising Trend During Compliance Monitoring

                    900
                  .  750 -
                 ""
                    600 -
                 u 450 -
                    300
                                                          Sulfate
                                                          LinearRegressiot
                            _,  j,  |	,	l  ,|   ,  |   ,  1  ,   |  ,   |  ,-
                         89   90   91    92   93   94   95   96   97   98
                                               Year

     By explicitly accounting for the trend, the confidence interval in Chapter 21 will adjust upward or
downward with the trend and thus more accurately estimate the current true concentration levels. Trend
techniques are not just used to track progress towards exceeding or meeting a fixed standard. Confidence
bands around the trend line can also provide an estimate of confidence  in the average concentration as it
changes over time. This subject is further covered in the Comprehensive Environmental Response,
Compensation, and Liability Act [CERCLA] guidance Methods for Evaluating the Attainment of
Cleanup Standards — Volume 2: Groundwater (EPA, 1992a).

     A final determination of remedial success should not solely be a statistical decision.  In many
hydrologic settings, contaminant concentrations tend to  rise after groundwater pumping wells are turned
off due to changes in well  drawdown patterns.   Concentration levels may exhibit more complicated
behavior than the two situations considered above. Thus, on balance, it  is recommended that determining
achievement of corrective action goals be done in consultation with the site manager, geologist, and/or
remedial engineer.

7.4.5 IMPACT OF  SAMPLE VARIABILITY,  NON-DETECTS, AND NON-NORMAL DATA

     Selection of hazardous constituents to be monitored in compliance/assessment or corrective action
is largely determined  by permit decisions. Regulatory requirements (e.g., Part 264, Appendix IX)  may
also dictate the number of constituents.  As a  practical  matter,  the most  reliable  indicators  of
contamination should be favored. Occasionally, constituents subject to degradation and transformation in
the aquifer  (e.g., chlorinated  hydrocarbon suites) may result  in additional,  related constituents  of
concern.
                                            7-17
        March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

      Since health-based considerations  are paramount in this type of monitoring, the most sensitive
constituents from a health risk standpoint could be selected. But even with population parameters (0),
sample sizes, and  constituents  determined,  selecting an  appropriate confidence interval test from
Chapter 21 can be problematic. For mildly variable sample data, measured  at relatively stable levels,
tests based on the normal distribution should be favored, whether constructed around a mean or an upper
percentile. With highly variable sample data,  selection of a test is less straightforward. If the observed
data happen to be lognormal, Land's confidence interval around the arithmetic mean is a valid option;
however, it has low  power to measure  compliance as the observations become more variable,  and
upward adjustment of the false positive rate (a) may be necessary to maintain sufficient power.

      In addition, the extreme variability of an upper confidence  limit using Land's technique  can
severely restrict its usage in tests of compliance during corrective action. Depending on the  data pattern
observed, degree of variability, and how  closely the sample mimics the lognormal model, consultation
with  a professional statistician should be considered to  resolve unusual  cases. When the lognormal
coefficient  of variation is quite high, one alternative is to construct an upper confidence limit around the
lognormal geometric mean (Chapter 21). Although such a confidence limit does not fully  account for
extreme  concentration values in the right-hand tail  of the lognormal distribution, a bound on the
geometric mean will account for the bulk of possible measurements. Nonetheless, use of  a geometric
mean  as a surrogate  for the population arithmetic  mean  leads to distinctly  different statistical  test
characteristics in terms of power and false positive rates.

      In  sum,  excessive sample  variability  can   severely limit the  effectiveness of traditional
compliance/assessment  and  corrective action testing. On the other hand, if  excessive variability  is
primarily due to trends observable in the data,  confidence bands around a linear trend can be constructed
(Section 7.4.4).

       LEFT-CENSORED SAMPLES

      For  compliance point data  sets  containing  left-censored   measurements  (i.e., non-detects),
parametric  confidence intervals  cannot be computed  directly  without some  adjustment.  All of the
parametric  confidence intervals described in Chapter 21 require estimates of the population  mean u^ and
standard deviation o. A  number of adjustment strategies are presented in Chapter 15. If the percentage
of non-detects is small — no more than 10-15% — simple substitution of half the reporting limit [RL]
for each non-detect will generally work to give an approximately correct confidence interval.

      For samples of at least  8-10 measurements and up to 50% non-detects, the Kaplan-Meier or robust
regression on order statistics [ROS] methods  can be used.  Data should first be assessed via a censored
probability plot whether the  sample can be normalized. If so, these techniques can be used to compute
estimates of the mean u^ and standard deviation o adjusted for the presence of left-censored values. These
adjusted estimates can be used in place of the sample mean (x ) and standard deviation (s)  listed in the
confidence interval formulas of Chapter 21  around either a mean or upper percentile.

      If none of these adjustments is appropriate, non-parametric confidence intervals on either the
median or an upper percentile (Section 21.2) can be calculated. Larger sample  sizes are needed than with
parametric  confidence interval counterparts, especially for intervals around an upper percentile, to ensure
a high level of confidence and a sufficiently narrow interval. The principal advantage of non-parametric


                                              7-18                                   March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

intervals is their flexibility.  Not only can large fractions of non-detects be accommodated, but non-
parametric confidence intervals can also be applied to data sets which cannot be normalized.

     For heavily censored small data sets of 4-6 observations, the options are limited. One approach is
to replace each non-detect by half its RL and compute the confidence interval  as if the sample were
normal. Though the resulting interval will be approximate, it can provide a preliminary indication of the
well's compliance with the standard until further sampling data can be accumulated and the confidence
interval recomputed.

     Confidence bands around a trend can be constructed  with censored  data using a bootstrapped
Theil-Sen non-parametric trend line  (Section  21.3.2).  In this method,  the Theil-Sen trend  is first
computed using the  sample  data,  accounting  for the non-detects.  Then  a large number bootstrap
resamples are  drawn from the original sample,  and an alternate Theil-Sen trend is conducted on each
bootstrap  sample. Variability in these alternate  trend estimates is then used to construct a confidence
band around the original trend.

       LOGNORMAL AND OTHER NORMALIZED DATA

     Lognormal data may require special treatment  when building a  confidence interval around the
mean. Land's method (Section  21.1.3) can offer a reasonable way to accommodate the transformation
bias associated  with  the  logarithm,  particularly when computing  a  lower  confidence  limit  as
recommended  in compliance/assessment monitoring. For data normalized by transformations other than
the logarithm,  one  option is to calculate a normal-based confidence interval  around the mean using the
transformed measurements, then  back-transform  the limits  to  the original concentration scale. The
resulting interval will not represent a confidence interval around the arithmetic mean of the original data,
but rather will  estimate the confidence intervals of the median and/or geometric mean.

     If the difference between the arithmetic mean and median is not considered important for a given
GWPS, this strategy will be the easiest to implement. A wide range of results can occur with Land's
method on highly skewed lognormal populations especially when computing an upper confidence limit
around the arithmetic mean (Singh et al, 1997).  It  may be better to either construct a confidence interval
around the lognormal geometric mean (Section 21.1.2) or to use the technique of bootstrapping (Efron,
1979; Davison and Hinkley, 1997) to create a non-parametric interval around the arithmetic mean.3

     For confidence intervals  around an upper percentile, no bias is  induced by data that have been
normalized via a transformation. Whatever the transformation used (e.g., logarithm, square root, cube,
etc.), a confidence interval can be constructed on the transformed data.  The resulting limits can then be
back-transformed to provide confidence limits around the desired upper percentile in the concentration
domain.
3 Bootstrapping is widely available in statistical software, including the open source R computing environment and EPA's
  free-of-charge ProlJCL package. In some cases, setting up the procedure correctly may require professional statistical
  consultation.

                                             7-19                                    March 2009

-------
Chapter 7.  Compliance Monitoring Strategies	Unified Guidance

7.5 COMPARISONS TO BACKGROUND  DATA

     Statistical tests in compliance/assessment and corrective action monitoring will often involve a
comparison between compliance point measurements and a promulgated fixed health-based limit or a
risk-based remedial action goal as the GWPS, described earlier. But a number of situations arise where
a GWPS must be based on a background limit.  The Part 264 regulations presume such a standard as one
of the options under §264.94(a); an ACL may also be determined from background under §264.94(b).
More recent Part 258 rules specify a background GWPS where a promulgated or risk-based standard is
not available or if the historical background is greater than an MCL [§258.55(h)(2) & (3)].

     Health-based risk standards bear no necessary relationship  to site-specific aquifer concentration
levels.  At many sites this poses no  problem,  since the observed  levels of many constituents may be
considerably  lower  than  their GWPS.   However,  either naturally-occurring or pre-existing  aquifer
concentrations of certain analytes can exceed promulgated standards.  Two commonly monitored trace
elements in particular- arsenic  and  selenium— are occasionally found at uncontaminated background
well  concentrations exceeding their respective MCLs. The regulations then provide that a GWPS based
on background levels is appropriate.

     A number of factors should be considered in designing a background-type GWPS testing program
for compliance/assessment or corrective action monitoring.  The most fundamental decision is whether
to base  such comparisons on two- (or multiple-) sample versus single-sample tests. For the first, many
of the design factors discussed  for detection monitoring in Chapter  6 will be appropriate; for single
sample comparisons to a fixed background GWPS, a confidence level approach similar to that discussed
earlier for testing fixed health standards in this Chapter 7 would be applied. This basic decision then
determines how the GWPS is defined, the appropriate test hypotheses,  types of statistical tests, what the
background GWPS represents in statistical  terms, and the relevance of individual test and  cumulative
false positive error rates. Such decisions may also be constrained by State groundwater anti-degradation
policies.  Other design factors to consider are the number of wells and constituents tested, interwell
versus  intrawell options, background sample sizes, and power. Unlike a single fixed standard like an
MCL, background GWPS's  may be uniquely  defined for  a  given monitoring well constituent by a
number of these factors.

     SINGLE- VERSUS TWO-SAMPLE TESTING

       One of two fundamental testing approaches can be used with site-specific background GWPSs.
Either 1) a GWPS is defined as the critical limit from a pre-selected detection-level statistical test (e.g., a
prediction limit) based on background measurements, or 2) background data are used to generate a fixed
GWPS somewhat elevated above current background levels. In both cases, the resulting GWPS will be
constituent-  and possibly compliance well- specific.  The first  represents  a two-sample test of two
distinct populations (or more if a multiple-sample test) similar  to those utilized in detection monitoring.
As such, the individual  test false positive rate, historical background  sample size, cumulative false
positive considerations, number of annual tests and desired future sample size will uniquely determine
the limit.  Whatever the critical value  for a selected background test, it becomes the  GWPS under
compliance/assessment or corrective action monitoring.
                                             7-20                                   March 2009

-------
Chapter 7.  Compliance Monitoring Strategies	Unified Guidance

     The only allowable hypothesis test structure for the two-sample approach follows that of detection
and compliance monitoring [7.1].  Once exceeded and in corrective action, a return to compliance is
through evidence that future samples lie below the GWPS using the same hypothesis structure.

     The second option uses a fixed statistic from the background data as the GWPS in a single-sample
confidence interval test.   Samples from a single population are  compared to  the fixed limit.  In other
respects, the strategy follows that outlined in Chapter 7 for fixed health- or risk-based GWPS tests. The
compliance/assessment test hypothesis structure also follows [7.1], but the hypotheses are reversed as in
[7.2] for corrective action testing.

     The  choice of the  single-sample GWPS deserves careful  consideration. In the past, many such
standards were simply computed as multiples of the background sample average (i.e., GWPS = 2-x).
However,  this approach  may  not fully account for natural variation in  background levels and lead to
higher than  expected false positive rates.  If the GWPS were to be set at the historical background
sample mean, even  higher false positive rates would occur during  compliance monitoring,  and
demonstrating corrective action compliance becomes almost impossible.

      In the recommendations which follow below, an upper tolerance limit based on both background
sample size  and sample variability is recommended for identifying the background GWPS at a suitably
high enough level above current background to allow for reversal of the test hypotheses.  Although a
somewhat arbitrary choice, a  GWPS based on this method allows for a variety of confidence interval
tests (e.g., a  one-way normal mean confidence interval identified in equations [7.3] and [7.4]).

      WHAT  A  BACKGROUND GWPS REPRESENTS

     If the testing protocol involves two-sample comparisons, the background GWPS is an upper limit
statistical  interval derived from a given set of background data  based on one or another detection
monitoring tests discussed in Chapter 6 and detailed in Part III. In these cases, the appropriate testing
parameter is the true mean for the parametric tests, and the true  median for non-parametric tests. This
would include \-of-m prediction  limit detection tests involving future values.   If a single-sample
comparison against a fixed background GWPS is used, the appropriate parameter will also depend upon
the type of confidence interval test to be used (Part IV).  Except for parametric or non-parametric upper
percentile  comparisons, the likely statistical parameter would again be a mean (arithmetic, logarithmic,
geometric) or  the median.  A background  GWPS could be defined  as  an upper percentile parameter,
making use  of normal test confidence  interval structures found in Section  21.1.4.  Non-parametric
percentile  options would likely require test sample sizes too  large for most applications. The Unified
Guidance  recommended approaches for defining single-sample GWPSs discussed later in this section
presume a central tendency test parameter like the mean or median.

     NUMBER  OF MONITORED WELLS AND CONSTITUENTS

     Compliance/assessment or corrective  action monitoring tests  against a fixed health- or risk-based
standard (including single-sample background GWPSs) are not affected in a significant manner by the
number of annual tests.   But this would not be true for two- or multiple-sample background GWPS
testing.   In  similar  fashion  to detection  monitoring, the  total number of tests is an  important
consideration in defining the appropriate false positive error test rate (atest).  The total number of annual
tests is determined by how many compliance wells, constituents and evaluations occur per year.

                                            7-21                                   March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

     Regulatory  agency interpretations  will  determine  the number  and location  of compliance
monitoring wells. These can differ depending on whether the wells are unit-specific, and if a reasonable
subset can be shown to be affected by  a  release. Perhaps  only those compliance wells containing
detectable levels of a compliance monitoring constituent  need be  included.  Formal annual tests are
generally required semi-annually, but other approaches may be applied.

     The number of constituents subject to two-sample background GWPS testing will also depend on
several factors. Only hazardous constituents not having a health- or risk-based standard are considered
here.   The basic criterion in interpreting required Part 264 Appendix IX or Part 258 Appendix II
analyses is to identify those hazardous constituents found in downgradient compliance wells.  Some
initially detected common laboratory or sampling contaminants might be eliminated following a repeat
scan. The remainder of the qualifying constituents will then require some form of background GWPS's.
Along with the number of wells and annual evaluations, the total annual number of background tests will
then be used in addressing an overall design cumulative design false positive rate.

     In corrective action testing (for either the one- or two-sample approaches),  the number of
compliance wells and  constituents may differ.  Only those  wells and constituents showing a significant
compliance test exceedance might be used.  However, from  a standpoint of eventually demonstrating
compliance under corrective  action, it might be  appropriate to still use the compliance/assessment
GWPS for two-sample tests.  With single-sample tests, the  GWPS is compared individually by well and
constituent as described.

     BACKGROUND SAMPLE SIZES and  INTERWELL vs. INTRAWELL TESTING

     Some potential constituents may already have been monitored during the detection phase, and have
a reasonable background  size.   Others identified under Part 264 Appendix IX or part 258 Appendix n
testing may have no historical background  data bases and require a period of background sampling.

     Historical constituent well data patterns and the results of this testing may help determine  if an
interwell  or intrawell  approach should be used  for a given  constituent.  For example, if arsenic and
selenium were historical constituents in detection monitoring, they might also be identified as candidates
for compliance background GWPS testing. There may already be indications that individual well spatial
differences will  need to be taken  into account  and an intrawell  approach followed.   In  this  case,
individual compliance well background GWPSs need to be established and tested. On the other hand,
certain hazardous  trace  elements and organics  may only be detected and  confirmed in one  or more
compliance wells with non-detects in background upgradient wells and possibly historical compliance
well data.   Under the latter conditions,  the simpler Double Quantification Rule (Section 6.2.2.) might be
used with the GWPS set at a quantification limit.  However,  this could pose  some interpretation
problems.  Subsequent testing against the background GWPS at the same compliance well concentration
levels causing the initial detection monitoring exceedance,  might very likely  result in further excursions
above the background GWPS.  The more realistic option would be  to  collect and  use  additional
compliance well data to establish a specific minimum intrawell background,  and only apply the Double
Quantification Rule at other wells not exhibiting detections. Even this approach might be unnecessarily
stringent if a contaminant plume were to expand in size and gradually  affect other compliance wells
(now subject to GWPS testing).
                                             7-22                                   March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

     CUMULATIVE & INDIVIDUAL TEST FALSE POSITIVE RATES

      Each of the independent two-sample tests against background standards will have a roughly equal
probability of being  exceeded by chance alone.  Since an exceedance in the  compliance monitoring
mode based on background can result in a need for corrective action, it is  recommended that the
individual test false positive rate be set sufficiently low.  Much of the discussion in Chapter 6, Section
6.2.2 is relevant here. An a priori, cumulative error design rate must first be identified.  To allow for
application of the  Unified Guidance  detection  monitoring strategies and  Appendix D  tables,  it is
suggested that the . 1 SWFPR value also be applied to two-sample background GWPS testing. In similar
fashion to Chapter 6 and Part III, this can be translated into individual test configurations.

     If  the single-sample confidence  interval  option  will  be used with an elevated  GWPS,  the
compliance level test will have a very low probability of being exceeded by truly background data.
Cumulative   false   positive   error   considerations   are   generally   negligible.  For  testing
compliance/assessment or corrective action hypotheses, there is still a need to identify an appropriately
low single test false positive rate which meets the regulatory goals.    Generally, a single test false
positive error rate  of .1 to  .05  will be  suitable with  the recommended approach  for defining the
background GWPS.

     UNIFIED GUIDANCE RECOMMENDATIONS

      Two-Sample GWPS Definition and Testing

     As indicated above, any of the detection monitoring tests described in Chapter 6 might be selected
for two- or multiple- sample background compliance testing.  One highly recommended statistical test
approach is a prediction limit. Either a parametric prediction limit for a future mean (Section 18.2.2) or
a non-parametric prediction limit for a future  median (Section 18.3.2) can be used,  depending on the
constituent being  tested and  its statistical  and  distributional characteristics (e.g., detection  rate,
normality, etc.).  It would be equally possible  to utilize one of the l-of-m future value prediction limit
tests, on an interwell  or intrawell basis.  Use of repeat samples as part of the selected test is appropriate,
although the expected number of annual compliance/corrective action  samples may dictate which tests
can apply.

     One parametric example is the 1-of-l future mean test. If the background data can be normalized,
background  observations are used to  construct a parametric  prediction  limit  with  (1-oc) confidence
around a mean of order/?, using the equation:

                                                     n   r
                                                                                         [7.5]


      The next/? measurements from each compliance well are averaged and the future mean compared
to the background  prediction limit, PL (considered the background GWPS). In compliance/assessment
monitoring, if any  of the means exceeds the limit, those well-constituent pairs are deemed to be out of
compliance. In corrective  action, if the future mean is no greater than  PL, it can be concluded that the
well-constituent pair is sufficiently similar to background to be within the remediation goal. In both
monitoring phases, the  prediction limit is constructed  to  represent a reasonable upper limit on the

                                             7-23                                    March 2009

-------
Chapter 7.  Compliance Monitoring Strategies	Unified Guidance

background  distribution.  Compliance point means above  this limit  are  statistically  different  from
background; means below it are similar to background.

     If the background sample cannot be normalized perhaps due to a large fraction of non-detects, two-
sample non-parametric upper prediction limit detection monitoring tests (Chapters 18 & 19) can be
used.   As an example, a maximal order statistic (often the highest or second-highest value) can be
selected from background as a non-parametric 1-of-l upper prediction limit test of the median. Table
18-2 is used to guide  the choice based on background sample size (n) and the achievable confidence
level (a). The median  of the next 3 measurements from each compliance well is compared to the upper
prediction limit. As with the parametric case in compliance/assessment, if any of the medians exceeds
the limit, those well-constituent pairs would be considered out of compliance. In corrective action, well-
constituent pairs with  medians no greater than the background prediction limit would be considered as
having met the standard.

       If background measurements for a particular constituent are all non-detect, the GWPS should be
set equal to  the highest RL. In similar fashion to detection monitoring, l-of-2 or l-of-3 future value
prediction limit tests can be applied (Section 6.2.2 Double Quantification rule).

             Single-Sample GWPS  Definition and Testing

     For single-sample testing, the Unified  Guidance recommendation is to  define a fixed GWPS or
ACL based on a background upper tolerance limit with 95% confidence and 95% coverage (Chapter
17). For normal background, the appropriate formula for the GWPS would be the same as that given in
Section 17.2.1, namely:

                                  GWPS = x + r(n,.95,.95]-s                             [7.6]

where n = number of  background  measurements, x and s represent the background sample mean and
standard deviation, and T is a tolerance factor selected from Table  17-3. If the background sample is a
mixture of detects and  non-detects, but the non-detect  fraction  is no  more than 50%,  a censored
estimation method such as Kaplan-Meier or robust regression on order statistics [ROS]  (Chapter 15)
can be attempted to compute adjusted estimates of the background mean [j,  and standard deviation o in
equation [7.5].

     For  larger  fractions  of non-detects,  a non-parametric  tolerance limit  can  be  constructed, as
explained  in Section 17.2.2. In this case, the GWPS median will often be set to the largest or second-
largest observed value in background. Table 17-4 can be used to determine the achieved confidence
level (1-a) associated  with a 95% coverage GWPS constructed in this way.  Ideally,  enough background
measurements  should  be used to set the tolerance limit as  close to the target of 95% coverage, 95%
confidence as possible. However, this could require very large background sample sizes (n > 60).

     Multiple independent measurements are used to form either a mean or median confidence interval
for comparison  with the  background  GWPS. Preferably  at  least  4  distinct  compliance point
measurements  should  be used to define the  mean confidence interval  in the  parametric case, and 3-7
values should be used with a non-parametric median test. The guidance does not recommend retesting in
single-sample background GWPS compliance/assessment monitoring.  An implicit kind of retesting is
built in to any test of a sample mean or median as explained in Section 19.3.2.
                                             7-24                                   March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

      In essence, the background tolerance limit is used to set a somewhat higher mean target GWPS
which can  accommodate both compliance and corrective action testing under background conditions.
The GWPS in equation [7.6] can be interpreted as an approximation to the upper 95th percentile of the
background distribution. It is designed to be a reasonable maximum on the likely range of background
concentrations. It is high enough that compliance wells exceeding the GWPS via a confidence interval
test (i.e.., LCL > GWPS) are probably impacted and  not mere  false positives.  At the same  time,
successful remedial efforts must show that concentrations at contaminated wells have decreased to levels
similar to background. The GWPS above represents an upper bound on background but is not so low as
to make proof of remediation via an upper confidence limit [GWPS] impossible.

     To ensure  that the  GWPS in equation  [7.6] sets a reasonable target,  the Unified  Guidance
recommends that at least 8 to 10 background measurements (n) be utilized, and more if available.  If the
background sample is not normal, but can be normalized via a transformation, the tolerance limit should
be computed on the transformed measurements and the result back-transformed to obtain a limit in the
concentration scale (see Chapter 17 for further details).

     TRADEOFFS  IN BACKGROUND GWPS TESTING METHODS

     A two-sample GWPS approach offers a stricter test of background  exceedances.  There is also
greater flexibility in designing tests for  a variety of future comparison values (single with  repeat,  small
sample means, etc.).  The true test parameter is explicitly defined by the type of test chosen.   Non-
parametric upper prediction limit tests also allow for greater flexibility when data sets include significant
non-detect values or are not transformable to a normal distribution assumption.  The approach suggested
in this section  accounts for the cumulative false positive error rate.

     One negative feature of two-sample GWPS testing is that the test hypotheses cannot be reversed
for correction action monitoring.  The trigger for compliance/assessment testing may also be quite small,
resulting in important consequences (the need to move to corrective action).  It may also be difficult to
demonstrate longer-term compliance following remedial activities, if the actual background is somewhat
elevated.

     Single-sample GWPS testing, by contrast, does allow for the reversal of test hypotheses. Using a
suitable definition of the somewhat elevated GWPS takes into account background sample variability
and size.   Cumulative false positive error rates for compliance or corrective action testing are not
considered, and standardized alpha error levels (.1 or .05) can be used.  Exceedances under compliance
monitoring also offer clear evidence of a considerable increase above background.

     But applying an arbitrary increase above background recommended for single-sample testing may
conflict with  State anti-degradation policy.  Defining the GWPS as a specific population parameter is
also somewhat arbitrary. Using the suggested  guidance  approach  for defining the GWPS in equation
[7.6] above, may result  in very high values if the data  are not normal (including logarithmic or non-
parametric  applications).  There is also less flexibility in identifying testing options,  especially with data
sets containing significant non-detect values.  Annual testing with quarterly sampling may be the only
realistic choice.

     A possible compromise might utilize both approaches.    That  is, initially apply the two-sample
approach for compliance/assessment testing. Then evaluate the single-sample approach with reversed

                                             7-25                                   March 2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

hypotheses.  Some of the initially significant increases under the two-sample approach may also meet the
upper confidence level limit when tested against the higher GWPS. Those well constituents that cannot
meet this limit can then be subjected to corrective action remediation and full post-treatment testing.
This implies that the background GWPS would be a range based on the two testing methods rather than
a single value.
     ^EXAMPLE  7-1

       A  facility has triggered a  significant increase under detection monitoring.   One hazardous
constituent (arsenic) was identified which must be tested against a background GWPS at six different
compliance wells, since background well levels were above  the appropriate arsenic MCL of 10 ug/1.
Two semi-annual tests are required for compliance/assessment monitoring.  Assume that arsenic had
been detected in both background and downgradient wells, but was significantly higher in one  of the
compliance wells.   It must be determined whether any of the compliance wells have exceeded their
background GWPS,  and might require corrective action.

       Design a background GWPS monitoring system for the following arsenic data from the elevated
Well #1, consisting  of eight hypothetical historical intrawell background samples and four future annual
values  for two different simulated data distribution cases shown in the table below.  Sample means and
standard deviations are provided in the bottom row:

                    	Compliance Well #1 Arsenic (ng/l)	
                        Historical Well Data        Case 1        Case2
74.1
10.8
32.8
25.0


41.5
41.0
30.8
40.0
X = 37.0
s = 18.16
61.5
58.7
76.8
81.3
x = 69.58
s = 11.15
95.0
73.4
73.3
90.0
x = 82.93
s = 11.24
       Background values were randomly generated from a normal distribution with a true mean of// =
40 and a population standard deviation of a = 16.  Case 1 future data were from a normal distribution
with a mean 1.5 times higher, while Case 2 data were from a normal distribution twice as high as the
background true mean.  Both cases used the same background population standard deviation.  The intent
of these simulated values is to allow exploration  of both of the  Unified  Guidance recommended
background GWPS methods when background increases are relatively modest and sample sizes small.

       The two-sample background GWPS approach is first evaluated. Assume that the background
data are normal and stationary (no evidence of spatial or temporal variation and other forms of statistical
dependence).   Given a likely limit  of future quarterly sampling and required  semi-annual evaluations,
two guidance prediction limit options would seem appropriate—either a l-of-2 future values or a 1-of-l
future mean size 2 test conducted twice a year.  The l-of-2 future values option  is  chosen.

        Since there are a total  of  6 compliance wells, one background constituent and two annual
evaluations, there are a total of 12 annual background tests to  be conducted.  Either the Unified
Guidance tables in Appendix D or R-script can be used to identify the appropriate prediction limit K-
                                             7-26                                   March  2009

-------
Chapter 7. Compliance Monitoring Strategies	Unified Guidance

factor. For the l-of-2 future values test, K = 1.83 (found by interpolation from the second table on page
D-118), based on w = 6, COC = 1, and two tests per year.   The calculated prediction limit using the
background data set statistics and K-factor is 70.2 ug/1, serving as the background GWPS.

       When the future values from the table above are tested against the GWPS, the following results
are obtained.  A "Pass"  indicates that the  compliance/assessment null hypothesis was achieved, while a
"Fail" indicates that the alternative hypothesis (the GWPS has been exceeded) is accepted.
                            Well #1 As Compliance Comparisons
                                l-of-2 Future Values Test (ng/l)
Case 1
(data)
61.5
58.7
76.8
81.3
Result

Pass

Fail
Case 2
(data)
95.0
73.4
73.3
90.0
Result

Fail

Fail
                                         GWPS = 70.2
       Both cases indicate at least one GWPS exceedance using the l-of-2 future values tests.   These
may be indications of a statistically significant increase above background, but the outcome for Case 1 is
somewhat troubling.  While a  50%  increase above background (based on the simulated population
parameters)  is potentially significant, more detailed power evaluations indicate that  such a detected
exceedance would only be expected about 24% of the time (using R-script power calculations with a Z-
value of 1.25 standard deviations above background for the l-of-2 future values test).  In contrast, the
2.5 Z-value  for Case 2 would be expected to be exceeded about 76% of the time.  In order to further
evaluate the  extent of significance of these results, the single-sample GWPS method is also considered.

       Following the guidance above, define the single-sample mean GWPS using equation [7.6] for the
upper 95%  confidence,  95% proportion tolerance limit.  Then apply upper  and lower normal mean
confidence intervals tests of the Case 1 and 2 n = 4 sample data using equations [7.3] and [7.4].

       From Table 21-9 on page D-246, a r-factor of 3.187 is used with the background mean  and
standard deviation to generate the GWPS = 94.9. One-way upper and lower mean confidence levels are
evaluated at  90 or 95% confidence for the tests and compared to the fixed background GWPS.

       LCL  test Pass/Fail results are the same as above for the two-sample compliance test. However, a
"Pass" for the UCL test implies that the alternative hypothesis (less than the standard) is accepted while
a "Fail" implies greater than or equal to the GWPS under corrective action monitoring hypotheses:
                                             7-27                                   March 2009

-------
Chapter 7. Compliance Monitoring Strategies                             Unified Guidance
           As Mean Confidence Interval Tests Against Background GWPS (jig/1)
90%
LCL
60.5
73.7
LCL Test
Result 95%
LCL
Pass 56.5
Pass 69.7
Result
Cas
Pass
Cas
Pass
90%
UCL
e 1 Data
78.7
e 2 Data
92.1
UCL Test
Result 95%
UCL
Pass 82.7
Pass 96.2
Result
Pass
Fail
                                         GWPS = 94.9
       For either chosen significance level, the Case 1 90% and 95% UCLs of 78.7 and 82.7 are below
the GWPS and the alternative corrective action hypothesis (the mean is less than the standard) can be
accepted.  For Case 2, the 90% UCL of 92.1 is below the GWPS, but the 95% UCL of 96.2 is above.  If
a higher level of test confidence is appropriate, the Case 2 arsenic values can be considered indicative of
the need for corrective action.

       If only the single-sample background GWPS approach were applied to the same data as above in
compliance/assessment monitoring tests, neither case mean  LCLs would exceed the standard,  and no
corrective action monitoring would be necessary.   However, it should be noted from the  example that
this approach does allow for a  significant increase  above the reference background level before any
action would be indicated. -4

       The approaches provided above presume that well constituent data subject to background GWPS
testing are stationary over time.  If sampling data show evidence of a trend, the situation becomes more
complicated in making compliance or corrective action test decisions. Two- and single-sample stationary
scenarios for identifying standards  may not be appropriate.  Trend behavior  can be determined by
applying one of the methods provided in Chapter 17 (e.g., linear  regression or Mann-Kendall trend
tests) to historical data. A significant increasing slope can be  indicative of a background exceedance,
although it should be clear that the increase  is not  due to natural conditions.  A decreasing or non-
significant  slope can be considered evidence for  compliance  with  historical background.  The  most
problematic standard would be setting an eventual background  target for  compliance testing under
corrective action.  To a great extent, it will depend on site-specific conditions including the behavior of
specific constituent subject  to remediation.  A background GWPS might be determined following the
period of remediation and monitoring when aquifer conditions have hopefully stabilized.

       Setting and applying background GWPSs have not received a great deal of attention in previous
guidance.  The discussions and example above help illustrate the somewhat difficult regulatory choices
that need to be made. A regulatory agency needs to determine what levels, if any, above background can
be considered acceptable.  A further consideration is the degree of importance  placed on background
GWPS  exceedances, particularly when tested along with constituents  having health-based limits.
Existing regulatory programs may have already developed procedures to deal with many  of the issues
discussed in this section.
                                             7-28                                   March 2009

-------
Chapter 8.  Methods Summary	Unified Guidance

   CHAPTER 8.   SUMMARY  OF  RECOMMENDED METHODS

       8.1   SELECTING THE RIGHT STATISTICAL METHODS 	 8-1
       8.2   TABLE 8.1 INVENTORY OF RECOMMENDED METHODS	8-4
       8.3   METHOD SUMMARIES	8-9
     This chapter provides a quick guide to the statistical procedures discussed within the Unified
Guidance. The first section is a basic road map designed to encourage the user to ask a series of key
questions. The other sections offer thumbnail sketches of each method and a matrix of options to help in
selecting the right procedure, depending on site-specific characteristics and constraints.

8.1 SELECTING THE RIGHT  STATISTICAL METHODS

     Choosing  appropriate statistical methods  is  important in developing a sound groundwater
monitoring statistical program. The statistical test(s) should be  selected  to match  basic site-specific
characteristics such as number and configuration of wells, the water quality constituents being measured,
and general hydrology. Statistical methods should  also be selected with reference to the statistical
characteristics of the monitored  parameters — proportion  of non-detects, type of concentration
distribution (e.g., normal, lognormal), presence or absence of spatial variability, etc.

     Because site conditions  and permit  requirements  vary  considerably,  no single  "cookbook"
approach is readily available to select the right statistical method. The best strategy is to consider site-
specific conditions and ask a series of questions. A table of recommended options (Table  8-1) and
summary descriptions is presented in Section 8.2 to help select an appropriate basic approach.

     The first question is: what stage of monitoring is required?  Detection monitoring is the first stage
of any  groundwater monitoring program and typically involves comparisons between measurements of
background and compliance point groundwater. Most of the methods described in this document (e.g.,
prediction limits, control charts, tests for  trend,  etc.) are designed  for facilities engaged in detection
monitoring. However, it must be determined whether an interwell (e.g., upgradient-to-downgradient) or
an intrawell test is warranted. This entails consideration of the site hydrology, constituent detection rates,
and deciding whether separate (upgradient) wells or past intrawell data serves  as the most appropriate
and representative background.

     Compliance/assessment monitoring is required for facilities  that no longer meet the requirements
of a detection monitoring program by  exhibiting statistically significant indications of a release to
groundwater. Once in compliance/assessment,  compliance point measurements are typically tested
against a fixed GWPS. Examples of fixed standards include Maximum Concentration Limits [MCL],
risk-derived limits  or a single limit derived from background data. The most appropriate statistical
method for tests against GWPS is a lower confidence limit. The type of confidence limit will depend on
whether the regulatory standard represents an average concentration; an absolute maximum, ceiling, or
upper percentile; or whether the compliance data exhibit a trend over time.

     In cases where no fixed GWPS is specified for  a particular constituent, compliance point data may
be directly compared against background data. In this situation, the most appropriate statistical method is

                                             JTl                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

one or another detection monitoring two- or multiple-sample tests using the critical design limit as the
GWPS (discussed in Section 7.5).

     Corrective action is reserved for facilities where evidence of a groundwater release is confirmed
above a GWPS.  In these situations, the facility is required to submit an appropriate remediation plan to
the Regional Administrator and to institute steps to insure adequate containment and/or clean-up of the
release. Remediation  of groundwater can be very costly and also difficult to measure. EPA has not
adopted a uniform approach in the setting of clean-up standards or how one should determine whether
those clean-up standards have been attained. Some guidance on this issue is given in the EPA document,
Methods for Evaluating the Attainment of Cleanup Standards, Volume II: Groundwater (EPA, 1992).

     The null  hypothesis  in corrective  action  testing  is reversed  from  that  of detection  and
compliance/assessment monitoring. Not only is it assumed that contamination is above the compliance
or clean-up standard, but corrective action should continue until the average concentration level is below
the clean-up limit for periods specified in the regulations. For any fixed-value standard (e.g., the GWPS
or a remediation goal) a  reasonable  and consistent statistical test for  corrective action is  an upper
confidence limit. The type of confidence limit will  depend on whether the data have a stable mean
concentration or exhibit a trend over time. For those well constituents requiring remediation,  there will
be a period of activity before formal testing can take place.  A number of statistical techniques (e.g. trend
testing) can be applied to the data collected in this interim period to gauge prospects for eventual GWPS
compliance. Section 7.5 describes corrective action testing limitations involving a two-sample GWPS.

     Another  major question involves the statistical  distribution most appropriate to the observed
measurements. Parametric tests are those which assume the underlying population follows a known and
identifiable distribution, the most common examples in groundwater monitoring being the normal and
the lognormal. If a specific distribution cannot be determined, non-parametric test methods can be used.
Non-parametric tests do not require a known statistical distribution and can be helpful when the  data
contain a substantial  proportion of non-detects. All of the parametric tests described in the Unified
Guidance, except  for  control charts, have non-parametric counterparts that  can be used  when the
underlying distribution is uncertain or difficult to test.

     A special consideration in fitting distributions is the presence of non-detects, also known as  left-
censored measurements. As long as a sample contains a small fraction of non-detects  (i.e., no more than
10-15%), simple substitution of half the reporting limit [RL] is generally adequate. If the proportion of
non-detects is  substantial,  it may be difficult or impossible to determine whether a specific parametric
distributional model provides a good fit to the data. For some tests, such as the t-test, one can switch to a
non-parametric test with little loss of power or accuracy. Non-parametric interval tests, however,  such as
prediction and tolerance  limits,  require  substantially  more  data before providing statistical  power
equivalent to parametric intervals. Partly because of this  drawback,  the Unified Guidance  discusses
methods to  adjust datasets with  significant fractions of non-detects so that parametric distributional
models may still be used (Chapter 15).

     The Unified Guidance now recommends  a single, consistent Double Quantification rule approach
for handling constituents that have either never been detected or have not been recently detected. Such
constituents are not  included in cumulative annual  site-wide  false positive error rate  [SWFPR]
computations;  and no special adjustment for non-detects is necessary. Any confirmed quantification (i.e.,

                                              8^2                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

two  consecutive  detections above  the  RL)  at a  compliance point provides sufficient evidence of
groundwater contamination by that parameter.

     A key question when picking a test for detection monitoring is whether traditional background-to-
downgradient interwell or single-well intrawell tests are appropriate. If intrawell testing is  appropriate,
historical measurements form the individual compliance well's own background while future values are
tested  against these  data. Intrawell tests eliminate any  natural spatial differences among monitoring
wells.  They  can also be used when the groundwater flow gradient is uncertain  or unstable, since all
samples being tested  are collected from the same well.

     Possible disadvantages to intrawell tests also need to be considered. First, if the compliance well
has already been impacted, intrawell background will also be impacted.  Such contaminated  background
may provide  a skewed comparison to later data  from the same well, making it difficult to identify
contaminated groundwater in the  future. Secondly, if intrawell background is constructed from only a
few  early measurements, considerable  time  may be needed to accumulate a sufficient number of
background observations (via periodic updating) to run a statistically powerful test.

     If a compliance well has already been impacted by previous contamination,  trend testing can still
indicate whether conditions  have deteriorated since intrawell background was  collected.  For sites
historically contaminated above background, the only way to effectively monitor compliance wells may
be to establish an historical intrawell baseline and measure increases  above this baseline.

     Besides  trend  tests, techniques  recommended  for intrawell  comparisons  include  intrawell
prediction limits, control charts, and sometimes the Wilcoxon rank-sum test. The best choice between
these methods is not always clear.  Since  there is no non-parametric counterpart to control charts, the
choice will depend on whether the data is normal or can be normalized  via a transformation.  New
guidance for control  charts shows they also can be designed to incorporate retesting. For  sites with a
large number of well-constituent pairs, intrawell prediction  limits can incorporate retesting to  meet
specific site-wide false positive rate and statistical  power characteristics. Parametric intrawell prediction
limits  can  be used with  background that is  normal or transformable to normality; non-parametric
versions can also be applied for many other data sets.

     If interwell,  upgradient-to-downgradient tests are  appropriate, the choice  of  statistical method
depends primarily  on the number of compliance wells and  constituents being monitored, the number of
observations available from each of these wells, and the detection rates and distributional properties of
these parameters. If a very small number of comparisons must be tested (i.e., two or three  compliance
wells versus  background, for one or two constituents), a ^-test or Wilcoxon rank-sum test may be
appropriate if there are a sufficient number  of compliance measurements (i.e., at least two per well).

     For other  cases, the Unified  Guidance recommends a prediction limit or control chart constructed
from background. Whenever more than  a  few  statistical  tests  must be  run,  retesting should be
incorporated into the procedure. If multiple observations per compliance well can be collected during a
given evaluation period, either a prediction limit for 'future'  observations,  a prediction limit for means
or medians, or  a control chart can be considered, depending  on which option best achieves statistical
power  and SWFPR targets, while balancing the site-specific costs and feasibility of sampling. If only one
observation per compliance well can  be  collected per evaluation, the only practical  choices are a
prediction limit for individual observations or a control chart.

                                              JTs                                    March 2009

-------
Chapter 8. Methods Summary
Unified Guidance
8.2 TABLE 8-1 INVENTORY OF RECOMMENDED METHODS
Chapter 9. Exploratory
Statistical Method

Time Series Plot
Box Plot
Histogram
Scatter Plot
Probability Plot
Tools
Chapter
§9.1
§9.2
§9.3
§9.4
§9.5

Use

Plot of measurement levels over time; Useful for assessing trends,
data inconsistencies, etc.
Graphical summary of sample distribution; Useful for comparing key
statistical characteristics in multiple wells
Graphical summary of sample distribution; Useful for assessing
probability density of single data set
Diagnostic tool; Plot of one variable vs. another; Useful for exploring
statistical associations
Graphical fit to normality; Useful for raw or transformed data
Chapter 10. Fitting Distributions
Statistical Method
Skewness Coefficient
Coefficient of Variation
Shapiro-WilkTest
Shapiro-Francia Test
Filliben's Probability
Plot Correlation
Coefficient
Shapiro-Wilk Multiple
Group Test
Chapter 11. Equality of
Statistical Method
Box Plots (side-by-
side)
Levene's Test
Mean-SD Scatter Plot
Chapter
§10.4
§10.4
§10.5.1
§10.5.2
§10.6
§10.7
Variance
Chapter
§11.1
§11.2
§11.3
Use
Measures symmetry/asymmetry in distribution; Screening level test
for plausibility of normal fit
Measures symmetry/asymmetry in distribution; Screening tool for
plausibility of normal fit; Only for non-negative data
Numerical normality test of a single sample; for n < 50
Numerical test of normality for a single sample; Supplement to
Shapiro-Wilk; Use with n > 50
Numerical test of normality for a single sample; Interchangeable with
Shapiro-Wilk; Use with n < 100; Good supplement to probability plot
Extension of Shapiro-Wilk test for multiple samples with possibly
different means and/or variances; Good check to use with Welch's t-
test

Use
Graphical test of differences in population variances; Good screening
tool for equal variance assumption in ANOVA
Numerical, robust ANOVA-type test of equality of variance for > 2
populations; Useful for testing assumptions in ANOVA
Visual test of association between SD and mean levels across group
of wells; Use to check for proportional effect or if variance-stabilizing
transformation is needed
Chapter 12. Outliers
Statistical Method
Probability Plot
Box Plot
Dixon's Test
Rosner's Test
Chapter
§12.1
§12.2
§12.3
§12.4
Use
Graphical fit of distribution to normality; Useful for identifying
extreme points not coinciding with predicted tail of distribution
Graphical screening tool for outliers; quasi-non-parametric, only
requires rough symmetry in distribution
Numerical test for single low or single high outlier; Use when n < 25
Numerical test for up to 5 outliers in single dataset; Recommended
when n > 20; User must identify a specific number of possible
outliers before running
                                8-4
     March 2009

-------
Chapter 8. Methods Summary
Unified Guidance
Chapter 13. Spatial Variation
Statistical Method
Box Plots (side-by-
side)
One-Way Analysis of
Variance [ANOVA] for
Spatial Variation
Chapter 14. Temporal
Statistical Method
Time Series Plot
(parallel)
One-way ANOVA for
Temporal Effects
Sample
Autocorrelation
Function
Rank von Neumann
Ratio
Darcy Equation
Seasonal Adjustment
(single well)
Temporally-Adjusted
Data Using ANOVA
Seasonal Mann-Kendall
Test
Chapter 15. Managing
Statistical Method

Simple Substitution
Censored Probability
Plot
Kaplan-Meier
Robust Regression on
Order Statistics
Cohen' Method and
Parametric Regression
on Order Statistics
Chapter
§13.2.1
§13.2.2
Variability
Chapter
§14.2.1
§14.2.2
§14.2.3
§14.2.4
§14.3.2
§14.3.3
§14.3.3
§14.3.4
Use
Quick screen for spatial variability; Look for noticeably staggered
boxes
Test to compare means of several populations; Use to identify spatial
variability across a group of wells and to estimate pooled (background)
standard deviation for use in intrawell tests; Data must be normal or
normalized; Assumption of equal variances across populations

Use
Quick screen for temporal (and/or spatial) variation; Look for parallel
movement in the graph traces at several wells over time
Test to compare means of distinct sampling events, in order to
assess systematic temporal dependence across wells; Use to get
better estimate of (background) variance and degrees of freedom in
data with temporal patterns; Residuals from ANOVA also used to
create stationary, adjusted data
Plot of autocorrelation by lag between sampling events; Requires
approximately normal data; Use to test for temporal correlation
and/or to adjust sampling frequency
Non-parametric numerical test of dependence in time-ordered data
series; Use to test for first-order autocorrelation in data from single
well or population
Method to approximate groundwater flow velocity; Use to determine
sampling interval guaranteeing physical independence of consecutive
groundwater samples; Does not ensure statistical independence
Method to adjust single data series exhibiting seasonal correlations
(i.e., cyclical fluctuations); At least 3 seasonal cycles must be evident
on time series plot
Method to adjust multiple wells for a common temporal dependence;
Use adjusted data in subsequent tests
Extension of Mann-Kendall trend test when seasonality is present; At
least 3 seasonal cycles must be evident
Non-Detect Data
Chapter
§15.2
§15.3
§15.3
§15.4
§15.5
Use

Simplest imputation scheme for non-detects; Useful when < 10-15%
of dataset is non-detect
Probability plot for mixture of non-detects and detects; Use to check
normality of left-censored sample
Method to estimate mean and standard deviation of left-censored
sample; Use when < 50% of dataset is non-detect; Multiple detects
and non-detects must originate from same distribution
Method to estimate mean and standard deviation of left-censored
sample; Use when < 50% of dataset is non-detect; Multiple detects
and non-detects must originate from same distribution
Other methods to estimate mean and standard deviation of left-
censored sample; Use when < 50% of dataset is non-detect; Detects
and non-detects must originate from same distribution and there
must be a single censoring limit
                                       8-5
       March 2009

-------
Chapter 8. Methods Summary
Unified Guidance
Chapter 16. Two-sample Tests
Statistical Method
Pooled Variance t-Test
Welch's t-Test
Wilcoxon Rank-Sum
Test
Tarone-Ware Test
Chapter 17. ANOVA,
Statistical Method
One-Way ANOVA
Kruskal-Wallis Test
Tolerance Limit
Non-parametric
Tolerance Limit
Linear Regression
Mann-Kendall Trend
Test
Theil-Sen Trend Line
Chapter
§16.1.1
§16.1.2
§16.2
§16.3
Use
Test to compare means of two populations; Data must be normal or
normalized, with no significant spatial variability; Useful at very small
sites in upgradient-to-downgradient comparisons; Also useful for
updating background; Population variances must be equal
Test to compare means of two populations; Data must be normal or
normalized, with no significant spatial variability; Useful at very small
sites in interwell comparisons; Also useful for updating background;
Population variances can differ
Non-parametric test to compare medians of two populations; Data
need not be normal; Some non-detects OK; Should have no
significant spatial variability; Useful at very small sites in interwell
comparisons and for certain intrawell comparisons; Also useful for
updating background
Extension of Wilcoxon rank-sum; non-parametric test to compare
medians of two populations; Data need not be normal; Designed to
accommodate left-censored data; Should have no significant spatial
variability; Useful at very small sites in interwell comparisons and for
certain intrawell comparisons; Also useful for updating background
Tolerance Limits, & Trend Tests
Chapter
§17.1.1
§17.1.2
§17.2.1
§17.2.2
§17.3.1
§17.3.2
§17.3.3
Use
Test to compare means across multiple populations; Data must be
normal or normalized; Should have no significant spatial variability if
used as interwell test; Assumes equal variances; Mandated in some
permits, but generally superceded by other tests; Useful for
identifying spatial variation; RMSE from ANOVA can be used to
improve intrawell background limits
Test to compare medians across multiple populations; Data need not
be normal; some non-detects OK; Should have no significant spatial
variability if used as interwell test; Useful alternative to ANOVA for
identifying spatial variation
Test to compare background vs. > 1 compliance well; Data must be
normal or normalized; Should have no significant spatial variability if
used as interwell test; Alternative to ANOVA; Mostly superceded by
prediction limits; Useful for constructing alternate clean-up standard
in corrective action
Test to compare background vs. > 1 compliance well; Data need not
be normal; Non-Detects OK; Should have no significant spatial
variability if used as interwell test; Alternative to Kruskal-Wallis;
Mostly superceded by prediction limits
Parametric estimate of linear trend; Trend residuals must be normal
or normalized; Useful for testing trends in background or at already
contaminated wells; Can be used to estimate linear association
between two random variables
Non-parametric test for linear trend; Non-detects OK; Useful for
documenting upward trend at already contaminated wells or where
trend already exists in background
Non-parametric estimate of linear trend; Non-detects OK; Useful for
estimating magnitude of an increasing trend in conjunction with
Mann-Kendall test
                                       8-6
       March 2009

-------
Chapter 8. Methods Summary
Unified Guidance
Chapter 18. Prediction
Statistical Method
Prediction Limit for m
Future Values
Prediction Limit for
Future Mean
Non-Parametric
Prediction Limit for m
Future Values
Non-parametric
Prediction Limit for
Future Median
Chapter 19. Prediction
Statistical Method
Prediction Limits for
Individual
Observations With
Retesting
Prediction Limits for
Means With Retesting
Non-Parametric
Prediction Limits for
Individual
Observations With
Retesting
Non-Parametric
Prediction Limits for
Medians With
Retesting
Limit Primer
Chapter Use
§18.2.1 Test to compare m measurements from compliance well against
background; Data must be normal normalized; Useful in retesting
schemes; Can be adapted to either intrawell or interwell tests; No
significant spatial variability allowed if used as interwell test
§18.2.2 Test to compare mean of compliance well against background; Data
must be normal or normalized; Useful alternative to traditional
ANOVA; Can be useful in retesting schemes; Most useful for interwell
(e.g., upgradient to downgradient) comparisons; No significant
spatial variability allowed if used as interwell test
§18.3.1 Non-parametric test to compare m measurements from compliance
well against order statistics of background; Non-normal data and/or
non-detects OK; Useful in non-parametric retesting schemes; Should
have no significant spatial variability if used as interwell test
§18.3.2 Test to comPare median of compliance well against order statistics of
background; Non-normal data and/or non-detects OK; Useful in non-
parametric retesting schemes; Most useful for interwell (e.g.,
upgradient to downgradient) comparisons; No significant spatial
variability allowed if used as interwell test
Limit Strategies with Retesting
Chapter Use
§19.3.1 Tests individual compliance point measurements against background;
Data must be normal or normalized; Assumes common population
variance across wells; No significant spatial variability allowed if used
as interwell test; Replacement for traditional ANOVA, extends
Dunnett's multiple comparison with control (MCC) procedure; Allows
control of SWFPR across multiple well-constituent pairs; Retesting
explicitly incorporated; Useful at any size site
§19.3.2 Tests compliance point means against background; Data must be
normal or normalized; Assumes common population variance across
wells; No significant spatial variability allowed if used as interwell
test; Replacement for traditional ANOVA, extends Dunnett's multiple
comparison with control (MCC) procedure; More flexible than a series
of intrawell t-tests if used as intrawell test; Allows control of SWFPR
across multiple well-constituent pairs; Must be feasible to collect >2
resamples per evaluation period to incorporate retesting; 1-of-l
scheme does not require explicit retesting
§19.4.1 Non-parametric test of individual compliance point observations
against background; Non-normal data and/or non-detects OK; No
significant spatial variability allowed if used as interwell test;
Retesting explicitly incorporated; Large background sample size
helpful
§19.4.2 Non-parametric test of compliance point medians against
background; Non-normal and/or non-detects OK; No significant
spatial variability allowed if used as interwell test; Large background
sample size helpful; Must be feasible to collect > 3 resamples per
evaluation period to incorporate retesting; 1-of-l scheme does not
require explicit retesting
                                       8-7
       March 2009

-------
Chapter 8. Methods Summary
Unified Guidance
Chapter 20. Control
Statistical Method

Shewhart-CUSUM
Control Chart
Charts
Chapter
§20.2

Use

Graphical test of significant increase above background; Data must
be normal or normalized; Some non-detects OK if left-censored
adjustment made; At least 8 background observations
recommended; Viable alternative to prediction limits; Retesting can
be explicitly incorporated; Control limits can be set via published
literature or Monte Carlo simulation
Chapter 21. Confidence Intervals
Statistical Method

Confidence Interval
Around Normal Mean
Confidence Interval
Around Lognormal
Geometric Mean
Confidence Interval
Around Lognormal
Arithmetic Mean
Confidence Interval
Around Upper
Percentile
Non-Parametric
Confidence Interval
around Median
Non-Parametric
Confidence Interval
Around Upper
Percentile
Confidence Band
Around Linear
Regression
Non-parametric
Confidence Band
Around Theil-Sen Line
Chapter
§21.1.1
§21.1.2
§21.1.3
§21.1.4
§21.2
§21.2
§21.3.1
§21.3.2
Use

Data must be normal; Some non-detects OK if left-censored
adjustment made; Used in compliance/assessment or corrective
action to compare compliance well against fixed, mean-based
groundwater standard; Should be no significant trend; 4 or more
observations recommended
Data must be lognormal; Some non-detects OK if left-censored
adjustment made; Used in compliance/assessment or corrective
action to compare compliance well against fixed, mean-based
groundwater standard; Should be no significant trend; 4 or more
observations recommended; Geometric mean equivalent to
lognormal median, smaller than lognormal mean
Data must be lognormal; Some non-detects OK if left-censored
adjustment made; Used in compliance/assessment or corrective
action to compare compliance well against fixed, mean-based
groundwater standard; Should be no significant trend; 4 or more
observations recommended; Lognormal arithmetic mean larger than
lognormal geometric mean
Data must be normal or normalized; Some non-detects OK if left-
censored adjustment made; Used in compliance/assessment to
compare compliance well against percentile-based or maximum
groundwater standard; Should be no significant trend
For non-normal, non-lognormal data; Non-detects OK; Used in
compliance/assessment or corrective action to compare compliance
well against fixed, mean-based groundwater standard; Should be no
significant trend; 7 or more observations recommended
For non-normal, non-lognormal data; Non-detects OK; Used in
compliance/assessment or corrective action to compare compliance
well against percentile-based or maximum groundwater standard;
Should be no significant trend; Large background sample size helpful
Use on data with significant trend; Trend residuals must be normal or
normalized; Used in compliance/assessment or corrective action to
compare compliance well against fixed groundwater standard; > 8
observations recommended
Use on data with significant trend; Non-normal data and/or non-
detects OK; Used in compliance/assessment or corrective action to
compare compliance well against fixed groundwater standard;
Bootstrapping of Theil-Sen trend line used to construct confidence
band
                                       8-8
       March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

8.3  METHOD SUMMARIES

       TIME SERIES PLOT (SECTIONS 9.1 AND  14.2.1)
Basic purpose:  Diagnostic  and exploratory tool. It is  a  graphical technique to display changes in
   concentrations at one or more wells over a specified period of time or series of sampling events.

Hypothesis tested:  Not a formal statistical  test. Time series plots can be used to informally gauge the
   presence of temporal  and/or spatial variability in a collection of distinct wells sampled during the
   same time frame.

Underlying assumptions: None.

When to use: Given a collection of wells with several  sampling events recorded at each well, a time
   series plot can provide information not only on whether the mean concentration level changes from
   well to well (an indication of possible spatial variation),  but also on whether there exists time-related
   or temporal dependence in the data.  Such temporal dependence can be seen in parallel movement on
   the time series plot, that is, when several wells exhibit the same pattern of up-and-down fluctuations
   over time.

Steps involved:  1) For each well, make a plot of concentration against time or date of sampling for the
   sampling events that occurred during the specified time  period; 2) Make sure each well is identified
   on the plot with a distinct symbol and/or connected line pattern (or trace); 3) To observe possible
   spatial  variation,  look  for  well traces that are substantially  separated  from one  another in
   concentration level; 4) To look for temporal dependence,  look for well traces that rise and fall
   together in roughly the same (parallel) pattern; 5) To ensure that artificial trends due to changing
   reporting limits are not reported, plot any non-detects with a distinct symbol, color, and/or fill.

Advantages/Disadvantages: Time  series  plots are an excellent tool for examining the behavior of one
   or more samples over time. Although,  they  do  not  offer the compact summary of distributional
   characteristics that, say, box plots do, time series plots display each and every data point and provide
   an excellent  initial indication  of temporal dependence. Since temporal  dependence affects the
   underlying variability  in the data, its identification is important so adjustments can be made to the
   estimated standard deviation.

       Box PLOT (SECTIONS 9.2,  12.2, AND 13.2.1)
Basic purpose: Diagnostic and exploratory tool. Graphical summary of data distribution; gives compact
   picture of central tendency and dispersion.

Hypothesis tested: Although not a formal statistical test, a side-by-side box plot of multiple datasets can
   be used as  a rough  indicator of either  unequal   variances or  spatial  variation  (via  unequal
   means/medians). Also serves as a quasi-non-parametric screening tool for outliers in a symmetric
   population.

Underlying assumptions: When used to screen outliers,  underlying population should be approximately
   symmetric.
                                              8-9                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

When to use: Can be  used as a  quick  screen  in  testing for unequal variances across multiple
   populations. Box lengths indicate the range of the central 50% of sample data values. Substantially
   different box  lengths suggest possibly  different  population  variances.  It is useful  as a rough
   indication of spatial variability across multiple well  locations. Since the median (and often the mean)
   are graphed on each box, significantly staggered medians and/or means on a multiple  side-by-side
   box plot can suggest possibly different population means at distinct well locations. Can  also be used
   to screen for outliers: values falling beyond the 'whiskers' on the box plot are labeled as potential
   outliers.

Steps  involved:  1)  Compute  the median,  mean,  lower and  upper  quartiles (i.e., 25th and 75th
   percentiles) of each dataset; 2) Graph each set of summary statistics side-by-side on the same set of
   axes.  Connect the lower and upper quartiles as the ends of a box, cut the box in two with a line at the
   median,  and use an  'X'  or other symbol  to represent the  mean.  3) Compute the  'whiskers' by
   extending lines below and above the box by an amount  equal to 1.5 times the interquartile range
   [IQR].

Advantages/Disadvantages:  The box plot is an excellent screening tool and  visual aid in diagnosing
   either unequal variances for testing the assumptions of ANOVA, the possible presence of spatial
   variability, or  potential outliers. It is not  a formal statistical test, however, and should  generally be
   used in conjunction with numerical test procedures.

       HISTOGRAM (SECTION 9.3)
Basic purpose: Diagnostic and exploratory tool. It is a graphical summary of an entire data distribution.

Hypothesis tested: Not a formal statistical test.

Underlying assumptions: None.

When to use: Can be used as a rough estimate of the  probability density of a single sample. Shape of
   histogram helps determine whether the distribution is symmetric or  skewed.  For larger data sets,
   histogram can be visually compared to a normal distribution or other known model to assess whether
   the shapes are similar.

Steps involved: 1) Sort and bin the data set into non-overlapping concentration segments that span the
   range of measurement values; 2)  Create a bar chart of the bins created in Step 1: put the height of
   each bar equal to the number or fraction of values falling into each bin.

Advantages/Disadvantages:  The histogram is a good visual aid in exploring possible distributional
   models that might be appropriate. Since it is not a formal test, there is no way to judge possible
   models solely  on the basis of the histogram; however, it provides a visual 'feel' for a data set.

       SCATTER PLOT (SECTION 9.4)
Basic purpose: Diagnostic tool. It is a graphical method to explore the association between two random
   variables or two paired statistical samples.

Hypothesis tested: None


                                             JTlO                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Underlying Assumptions: None.

When to use: Useful as an exploratory tool  for discovering or identifying statistical  relationships
   between pairs of variables. Graphically illustrates the degree of correlation or association between
   two quantities.

Steps involved: Using Cartesian pairs of the variables of interest, graph each pair on the scatter plot,
   using one symbol per pair.

Advantages/Disadvantages: A scatter plot is not a formal test, but rather an excellent exploratory tool.
   Helps identify statistical relationships.

       PROBABILITY PLOT (SECTIONS 9.5 AND 12.1)
Basic purpose: Diagnostic tool.  A graphical method to compare a dataset against a particular statistical
   distribution,  usually the normal. Designed  to  show how well the data match up to or 'fit' the
   hypothesized distribution.  An  absolutely straight line  fit indicates perfect  consistency with the
   hypothesized model.

Hypothesis tested: Although not a formal test,  the probability plot can be used to graphically indicate
   whether a dataset is normal. The straighter the plot, the more consistent the dataset with a  null
   hypothesis of normality; significant curves, bends, or other non-linear patterns suggest a rejection of
   the normal model as a poor fit.

Underlying Assumptions: All observations come from a single statistical population.

When to use: Can be used as a  graphical indication of normality on a set  of raw measurements or, by
   first making a transformation, as an indication of normality on  the transformed  scale.  It should
   generally be supplemented by a formal  numerical test of normality. It can be used  on  the residuals
   from  a one-way ANOVA to test the joint normality of the groups being compared. The test can  also
   be used to help identify  potential  outliers (i.e.,  individual  values not  part of  the  same basic
   underlying population).

Steps  involved:  1)  Order the dataset  and determine matching percentiles  (or quantiles)  from the
   hypothesized distribution (typically the standard normal); 2) Plot the ordered data values against the
   matching percentiles; 3) Examine the plot for a straight line fit.

Advantages/Disadvantages:  Not  a formal test of  normality;  however,  the probability plot is an
   excellent graphical supplement to any  goodness-of-fit test. Because each data value  is depicted,
   specific  departures from normality can be  identified (e.g.,  excessive  skewness, possible outliers,
   etc.).

       SKEWNESS COEFFICIENT (SECTION 10.4)
Basic purpose: Diagnostic tool. Sample statistic  designed  to measure the degree of symmetry  in a
   sample. Because the normal distribution is perfectly symmetric, the skewness coefficient can provide
   a quick indication of whether a given dataset is symmetric enough to be consistent with the normal
   model.  Skewness coefficients close to zero are consistent with normality;  skewness values large in
   absolute value suggest  the underlying population is asymmetric and non-normal.

                                              JTll                                    March 2009

-------
Chapter 8. Methods Summary _ Unified Guidance

Hypothesis tested: The skewness coefficient is used in groundwater monitoring as a screening tool
   rather than a formal hypothesis test. Still, it can be used to roughly test whether a given sample is
   normal  by using the following rule of thumb: if the skewness coefficient is no greater than one in
   absolute value, accept a null hypothesis of normality; if not, reject the normal model as ill-fitting.

Underlying Assumptions: None

Steps involved:  1) Compute skewness coefficient; 2) Compare to cutoff of 1;  3) If skewness is greater
   than 1, considering running a formal test of normality.

Advantages/Disadvantages: Fairly simple calculation, good screening tool. Skewness coefficient can
   be  positive or negative,  indicating positive  or negative  skewness in the dataset, respectively.
   Measures symmetry rather than normality,  per se; since other non-normal  distributions can also be
   symmetric, might give a misleading result.  Not as powerful or accurate a test of normality as either
   the Shapiro-Wilk or Filliben tests, but a more accurate indicator than  the coefficient of variation,
   particularly for data on a transformed scale.

       COEFFICIENT OF VARIATION  [CV] (SECTION 10.4)
Basic purpose: Diagnostic tool. Sample statistic used to measure skewness in a sample of positively-
   valued measurements. Because the CV of positively-valued normal measurements must be close to
   zero,  the CV provides an easy indication  of whether a given sample  is symmetric enough to be
   normal. Coefficients  of variation close to zero are consistent with normality;  large CVs indicate a
   skewed, non-normal population.

Hypothesis tested: The coefficient of variation is not a formal hypothesis test. Still, it can be used to
   provide a 'quick and easy' gauge of non-normality: if the CV exceeds 0.5, the population is probably
   not normal.

Underlying Assumptions: Sample must be positively-valued for CV to have meaningful interpretation.

Steps involved:  1) Compute  sample mean and standard deviation; 2) Divide standard deviation by mean
   to get coefficient of variation.

Advantages/Disadvantages:  Simple calculation,  good  screening tool.  It  measures  skewness  and
   variability in positively-valued data. Not an accurate a test of normality, especially if data have been
   transformed.

       SHAPIRO-WILK AND  SHAPIRO-FRANCIA TESTS (SECTION 10.5)
Basic purpose:  Diagnostic  tool and a formal numerical goodness-of-fit  test of normality.  Shapiro-
   Francia test is a close variant of the Shapiro-Wilk useful when  the sample size is larger than 50.
Hypothesis tested: HQ — the dataset being tested comes from an underlying normal population.
   the underlying population is non-normal (note that the form  of this alternative  population is  not
   specified).

Underlying assumptions: All observations come from a single normal population.
                                             8-12                                   March 2009

-------
Chapter 8. Methods Summary _ Unified Guidance

When to use: To test normality on a set of raw measurements or following transformation of the data. It
   can also be used with the residuals from  a one-way ANOVA to test the joint normality of the groups
   being compared.

Steps involved (for  Shapiro-Wilk):  1) Order the dataset and compute successive differences between
   pairs of extreme values (i.e., most extreme pair = maximum - minimum, next most extreme pair =
   2nd  largest - 2nd smallest, etc.);  2) Multiply the pair differences by the Shapiro-Wilk coefficients
   and  compute the  Shapiro-Wilk test statistic;  3) Compare the test statistic against an a-level critical
   point; 4) Values higher than the critical point are consistent with the null hypothesis of normality,
   while values  lower than the critical point suggest a non-normal  fit.

Advantages/Disadvantages: The Shapiro-Wilk procedure is considered  one of the very best tests of
   normality. It  is much more powerful than the skewness coefficient or chi-square goodness-of-fit test.
   The  Shapiro-Wilk and Shapiro-Francia test statistics will tend to be large (and more indicative of
   normality)  when a probability plot  of the  same data exhibits a  close-to-linear pattern. Special
   Shapiro-Wilk coefficients are available for sample  sizes up to 50.  For larger sample sizes, the
   Shapiro-Francia test does not require a table of special coefficients, just the ability to compute
   inverse normal probabilities.

       FILLIBEN'S PROBABILITY PLOT CORRELATION COEFFICIENT TEST (SECTION 10.6)
Basic purpose: Diagnostic tool and a formal numerical goodness-of-fit procedure to test for normality.
Hypothesis tested: HQ — the dataset being tested comes from an underlying normal population.
   the underlying population is non-normal (note that the form of this alternative  population is  not
   specified).

Underlying assumptions: All observations come from a single normal population.

When to use: To test normality  on a set of raw measurements or following transformation of the data on
   the transformed scale. It can also be used on the residuals from a one-way ANOVA to test the joint
   normality of the groups being compared.

Steps involved: 1) Construct a normal probability plot of the  dataset; 2) Calculate the correlation
   between the pairs on the probability plot; 3) Compare the test statistic against an a-level critical
   point; 4) Values higher than the critical point are consistent with the null hypothesis of normality,
   while values lower than the critical point suggest a non-normal  fit.

Advantages/Disadvantages: Filliben's procedure is an  excellent test of normality,  with very similar
   characteristics to the Shapiro-Wilk test. As a correlation on  a  probability plot, the Filliben's test
   statistic will tend to be close to one (and more indicative of normality) when a probability plot of the
   same data exhibits a close-to-linear pattern. Critical points for Filliben's test are available for sample
   sizes up to 100. A table of special coefficients is not needed to run Filliben's test,  only the ability to
   compute inverse normal probabilities.

       SHAPIRO-WILK  MULTIPLE GROUP TEST (SECTION 10.7)
Basic purpose: Diagnostic tool and a formal normality goodness-of-fit test for multiple groups.

                                              JTl3                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Hypothesis tested: HO — datasets being tested all come from underlying normal populations, possibly
   with different means and/or variances. HA — at least one underlying population is non-normal (note
   that the form of this alternative population is not specified).

Underlying assumptions: The observations in each group all come from,  possibly different, normal
   populations.

When to use:  Can be used to test normality on multiple sets of raw measurements or, by first making a
   transformation, to test normality of the  data groups on  the transformed scale.  It is particularly
   helpful when used in conjunction with Welch's t-test.

Steps  involved:  1)  Compute  Shapiro-Wilk statistic (Section 10.5) on  each group  separately;  2)
   Transform  the Shapiro-Wilk  statistics  into z-scores  and combine into an omnibus z-score;  3)
   Compare the test  statistic against an a-level critical point; 4) Values higher than the critical point are
   consistent with the null hypothesis of normality for all the populations, while values lower than the
   critical point suggest a non-normal fit of one or more groups.

Advantages/Disadvantages:  As an extension of the Shapiro-Wilk test, the  multiple group test shares
   many  of its  desirable properties. Users  should  be careful, however, not to assume that a  result
   consistent  with the hypothesis of normality implies that  all  groups  follow the same  normal
   distribution.  The multiple group test does not  assume that all  groups have  the same means or
   variances.  Special coefficients are needed to convert Shapiro-Wilk statistics into z-scores, but once
   converted,  no other special tables needed to run test besides a standard normal table.

       LEVENE'STEST (SECTION 11.2)
Basic purpose: Diagnostic tool. Levene's test is a formal numerical test of equality of variances across
   multiple populations.

Hypothesis tested: HQ — The population variances across all the datasets being tested are equal. HA —
   One or more pairs of population variances are unequal.

Underlying assumptions: The data set  from each  population is assumed to be roughly normal in
   distribution. Since Levene's test is designed to work well even with somewhat non-normal data (i.e.,
   it is fairly robust to non-normality), precise normality is not an overriding concern.

When to use:  Levene's method can be used to test the equal variance assumption underlying one-way
   ANOVA for a group of wells. Used in this way, the test is run on the absolute values of the residuals
   after first subtracting the  mean of each group being compared. If Levene's test is significant,  the
   original data may need to be transformed to stabilize the variances before running an ANOVA.

Steps involved: 1) Compute  the residuals of each group by subtracting the group mean; 2) conduct a
   one-way ANOVA on the absolute values of the residuals; and 3) if the  ANOVA F-statistic is
   significant  at the  5% a-level, conclude  the underlying population variances  are unequal. If not,
   conclude the data are consistent with the null hypothesis of equal variances.

Advantages/Disadvantages:  As a test of equal variances, Levene's test is  reasonably robust to non-
   normality.  It is much more so than for Bartlett's  test (recommended within the 1989 Interim  Final

                                             JTl4                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   Guidance [IFG]). In addition, Levene's method uses the same basic equations as those needed to run
   a one-way ANOVA.

       MEAN-STANDARD DEVIATION SCATTER PLOT (SECTION  11.3)
Basic purpose: Diagnostic tool. It is a graphical method to examine degree of association between mean
   levels and standard deviations at a series of wells. Positive correlation or association between these
   quantities is known as a 'proportional effect' and is characteristic of skewed distributions such as the
   lognormal.

Hypothesis tested: Though not a formal test, the mean-standard deviation scatter plot provides a visual
   indication of whether variances are roughly equal from well to well, or whether the variance depends
   on the well mean.

Underlying Assumptions: None.

When to use: Useful  as a graphical indication of 1) equal variances or 2) proportional effects between
   the standard deviation and mean levels. A positive correlation between well means and standard
   deviations may signify that a transformation is needed to stabilize the variances.

Steps involved: 1) Compute the sample mean and  standard deviation for each well; 2) plot the mean-
   standard deviation pairs on a scatter plot; and 3) examine the plot for any association between the
   two quantities.

Advantages/Disadvantages: Not a formal test of homoscedasticity (i.e., equal variances). It is helpful in
   assessing whether a transformation might be warranted to stabilize unequal variances.

       DIXON'S TEST (SECTION  12.3)
Basic purpose: Diagnostic tool. It is used to identify (single) outliers within smaller datasets.

Hypothesis tested: HO — Outlier(s) comes from  same normal distribution as rest of the dataset. HA —
   Outlier(s) comes from different distribution than rest of the dataset.

Underlying assumptions: Data without  the suspected  outlier(s)  are normally distributed.  Test
   recommended only for sample  sizes up to 25.

When to use:  Try Dixon's test when one  value in a  dataset appears anomalously low or anomalously
   high when compared to the other data values. Be cautious about screening apparent high outliers in
   compliance  point wells. Even if found to be statistical  outliers,  such extreme concentrations may
   represent contamination events. A safer application of outlier tests is  with background or baseline
   samples. Even then, always try to establish a physical reason for the outlier if possible (e.g..,
   analytical error, transcription mistake, etc.).

Steps involved: 1) Remove the suspected  outlier and test remaining data for normality. If non-normal,
   try a transformation to achieve  normality; 2) Once remaining data are  normal, calculate Dixon's
   statistic, depending on the sample size «; 3)  Compare Dixon's statistic against an a-level  critical
   point;  and 4) If Dixon's statistic exceeds the critical  point, conclude the suspected value is  a
   statistical outlier. Investigate this measurement further.

                                              JTl5                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Advantages/Disadvantages: Dixon's test is only recommended for sample sizes up to 25. Furthermore,
   if there is more than one outlier, Dixon's test may lead to masking (i.e., a non-significant result)
   where two or more outliers close in value 'hide' one another. If more than one outlier is suspected,
   always test the least extreme value first.

       ROSNER'S TEST (SECTION  12.4)
Basic purpose: Diagnostic tool. It is used to identify multiple outliers within larger datasets.

Hypothesis tested: HO — Outliers come from same normal distribution as the rest of the dataset. HA —
   Outliers come from different distribution than the rest of the dataset.

Underlying assumptions:  Data  without the  suspected  outliers  are  normally distributed. Test
   recommended for sample sizes of at least 20.

When to use: Try Rosners's  test when multiple values in  a  dataset appear anomalously low or
   anomalously high when compared to  the other data  values.  As Dixon's  test, be cautious about
   screening apparent high outliers in  compliance point wells. Always try to establish a physical reason
   for an outlier if possible (e.g., analytical error, transcription mistake, etc.).

Steps involved: 1) Identify the maximum number of possible outliers (ro <  5)  and the  number of
   suspected outliers (r < ro). Remove the suspected outliers and test the remaining data for normality.
   If non-normal,  try a transformation to achieve normality; 2) Once remaining data are  normal,
   successively compute the mean  and standard deviation, removing the next most extreme value each
   time  until  ro possible outliers  have been removed; 3) Compute Rosner's statistic based on the
   number (r) of  suspected  outliers;  and 4) If Rosner's statistic exceeds an a-level critical point,
   conclude there are r statistical outliers. Investigate these measurements further.  If Rosner's statistic
   does  not  exceed the critical point, recompute the test for (r-1) possible  outliers, successively
   reducing r until either the critical point is exceeded or r = 0.

Advantages/Disadvantages: Rosner's test is only recommended for sample  sizes of 20 or more, but can
   be used to identify up to 5 outliers per use. It is more complicated to use than some other outlier
   tests, but does not require special tables other than to determine a-level critical points.

       ONE-WAY ANALYSIS OF VARIANCE [ANOVA] FOR SPATIAL VARIATION  (SECTION 13.2.2)
Basic purpose: Diagnostic  tool. Test to compare population means at multiple wells, in order to gauge
   the presence of spatial variability.

Hypothesis tested: HO — Population means across all tested wells are equal. HA — One or more pairs
   of population means are unequal.

Underlying assumptions:  1) ANOVA residuals at each  well  or group must be normally distributed
   using  the original data  or after transformation. Residuals should be tested for normality using a
   goodness-of-fit procedure; 2) population variances across  all wells must be equal. This assumption
   can be tested with box plots and Levene's test; and 3) each tested well should  have at least 3 to 4
   separate observations.
                                             8-16                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

When to  use: The one-way ANOVA  procedure  can be used to identify significant spatial variation
   across a group of distinct well locations. The method is particularly useful for a group of multiple
   upgradient wells, to determine whether or not there are large average concentration differences from
   one location to the next due to natural groundwater fluctuations and/or differences in geochemistry.
   If downgradient wells are included in an ANOVA,  the downgradient groundwater should not be
   contaminated, at least if a test of natural spatial variation is  desired. Otherwise, a significant
   difference in population means could reflect the presence of either recent or historical contamination.

Steps involved:  1) Form the ANOVA residuals by subtracting from  each measurement its sample well
   mean;  2) test the ANOVA residuals for normality and equal variance. If either of these assumptions
   is violated, try a transformation of the data and retest the assumptions; 3)  compute the one-way
   ANOVA F-statistic;  4)  if  the F-statistic exceeds  an  a-level  critical  point,  conclude  the  null
   hypothesis of equal population means has been violated and that there is  some (perhaps substantial)
   degree of spatial variation; 5) if the F-statistic does not exceed the critical point, conclude that the
   well averages are close  enough to treat the combined data as  coming from the  same statistical
   population.

Advantages/Disadvantages: One-way ANOVA is an excellent technique for identifying differences in
   separate well  populations,  as long  as the assumptions are generally met. However, a finding  of
   significant spatial variability does not specify the reason for the well-to-well differences. Additional
   information or investigation  may be necessary to determine why the spatial differences exist. Be
   especially  careful when (1) testing  a combination of upgradient and  downgradient wells that
   downgradient contamination  is not the source  of the difference found with ANOVA; and 2) when
   ANOVA identifies significant spatial variation and intrawell tests are called for. In the latter case, the
   ANOVA results can sometimes be used to estimate more powerful intrawell  prediction and control
   limits. Such an adjustment comes directly from the ANOVA computations, requiring no additional
   calculation.

       ANALYSIS OF VARIANCE  [ANOVA] FOR TEMPORAL EFFECTS (SECTIONS 14.2.2 & 14.3.3)
Basic purpose: Diagnostic tool. It is a test to compare population means at multiple sampling events,
   after pooling the event data across wells. The test can also used to adjust data across multiple wells
   for common temporal dependence.

Hypothesis tested: HO — Population means across all sampling events are equal. H\ — One  or more
   pairs of population means are unequal.

Underlying assumptions: 1) ANOVA residuals from the population at each sampling event must be
   normal or normalized. These  should be tested for normality using  a goodness-of-fit procedure; 2) the
   population variances across all sampling events must be equal. Test this  assumption with box plots
   and Levene's test; and 3) each tested well should have at least  3 to 4  observations per sampling
   event.

When to  use: 1) The ANOVA procedure for temporal effects should be used  to identify significant
   temporal variation  over a series of  distinct sampling events.  The method assumes that spatial
   variation by well location is not a significant factor (this should have already  been tested). ANOVA
   for temporal  effects should  be used when a time series plot of  a group of wells exhibits  roughly
   parallel traces over time, indicating a time-related phenomenon affecting all the wells in a similar
                                             JTl7March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   way on any given sampling event. If a significant temporal effect is found, the results of the ANOVA
   can  be employed to adjust the  standard deviation estimate and the degrees of freedom quantities
   needed for further upgradient-to-downgradient comparisons; 2) compliance wells can be included in
   ANOVA for temporal effects, since the temporal pattern is assumed to affect all the wells on-site,
   regardless of gradient; and 3) residuals from ANOVA for temporal effects can be used to create
   adjusted, temporally-stationary measurements in order to eliminate the temporal dependence.

Steps involved: 1) Compute the mean (across wells)  from data collected on  each  separate sampling
   event; 2)  form the ANOVA residuals by subtracting  from each measurement its sampling event
   mean; 3) test the ANOVA residuals for normality and equal variance. If either of these assumptions
   is violated,  try a transformation of the data and retest the assumptions; 4) compute the one-way
   ANOVA  F-statistic;  5)  if the F-statistic  exceeds an a-level critical point,  conclude the null
   hypothesis of equal population means has been violated and that there is some (perhaps substantial)
   degree  of temporal  dependence;  6) compute  the degrees  of freedom adjustment factor and the
   adjusted standard deviation for use in interwell comparisons; 7)  if the F-statistic does not exceed the
   critical point, conclude that the sampling event averages are close enough to treat the combined data
   as if there were no  temporal dependence; and use  the residuals, if necessary, to create adjusted,
   temporally-stationary measurements, regardless of the significance of theF-test (Section  14.3.3).

Advantages/Disadvantages:  1)  One-way ANOVA for  temporal effects is  a good technique for
   identifying time-related effects among a group  of wells. The procedure should be employed when a
   strong temporal dependence is indicated by parallel traces in time series plots; 2) if there is both
   temporal dependence and strong spatial variability, the ANOVA for temporal effects may be non-
   significant due to the added spatial variation.  A two-way ANOVA for temporal and spatial effects
   might be considered instead;  and 3) even if the ANOVA is non-significant, the ANOVA residuals
   can still be used to adjust data for apparent temporal  dependence.

       SAMPLE AUTOCORRELATION FUNCTION (SECTION 14.2.3)
Basic purpose:  Diagnostic tool. This is a parametric  estimate and test of autocorrelation (i.e., time-
   related dependence) in a data series from a single population.

Hypothesis tested: HO — Measurements from the population are independent of sampling  events (i.e..,
   they  are not influenced  by  the  time  when the data were collected). HA  — The distribution of
   measurements is impacted by the time  of data collection.

Underlying assumptions:  Data should be  approximately  normal,  with few non-detects. Sampling
   events represented in the sample should be fairly regular and evenly spaced in time.

When to use: When testing a data series from a single  population (e.g., a single well),  the sample
   autocorrelation function (also known as the correlogram) can determine whether there is a significant
   temporal dependence in the data.

Steps involved: 1) Form overlapping ordered pairs from  the data series by pairing measurements
   'lagged' by  a certain number of sampling events (e.g.,  all  pairs  with measurements spaced by k = 2
   sampling  events); 2) for each  distinct lag (K), compute the sample  autocorrelation;  3)  plot the
                                             8-18                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   autocorrelations from Step  2 by lag (k) on a scatter plot;   and 4) count any autocorrelation as
   significantly different from zero if its absolute magnitude exceeds 2/^1 n , where n is the sample size.

Advantages/Disadvantages:  1) The  sample  autocorrelation function provides a  graphical test of
   temporal dependence. It can be used not only to identify autocorrelation, but also as a planning tool
   for adjusting the sampling interval between events. The smallest lag (k) at which the autocorrelation
   is insignificantly  different  from zero  is  the minimum sampling interval  ensuring  temporally
   uncorrelated data; 2) the test only applies to a single population at a time and cannot be used to
   identify temporal effects that span across groups of wells simultaneously. In that scenario, use a one-
   way ANOVA for temporal  effects; and 3) tests for significant autocorrelation depend on the data
   being approximately normal; use the rank von Neumann ratio for non-normal samples.

       RANK VON  NEUMANN  RATIO (SECTION 9.4)
Basic purpose: Diagnostic tool.  It is  a non-parametric test of first-order autocorrelation (i.e., time-
   related dependence) in a data series from a single population.

Hypothesis tested: HQ — Measurements from the population are independent of sampling events (i.e..,
   they are not influenced by the time when the  data  were collected). H& — The distribution of
   measurements is impacted by the time of data collection.

Underlying assumptions: Data need not be normally distributed. However, it is assumed that the data
   series can be uniquely ranked according to concentration level. Ties in the data (e.g., non-detects) are
   not technically allowed. Although a mid-rank procedure (as used in the Wilcoxon rank-sum test) to
   rank tied values might be considered, the available critical points for the rank von Neumann ratio
   statistic only directly apply to cases where a unique ranking is possible.

When to  use:  When testing a data series from a single population (e.g.,  a single well)  for use in,
   perhaps, an  intrawell prediction limit, control chart, or test of trend, the rank von Neumann ratio can
   determine whether there is a significant temporal dependence in the data. If the dependence is
   seasonal, the data may be adjusted using a seasonal correction (Section 14.3.3). If the dependence is
   a  linear trend,  remove the  estimated trend and re-run the  rank von Neumann ratio on the trend
   residuals before concluding there are additional time-related effects.  Complex dependence may
   require consultation with a professional statistician.

Steps involved: 1) Rank the measurements by concentration level, but then list the ranks in the order the
   samples were collected; 2)  using the ranks, compute the von Neumann ratio;  3) if the  rank von
   Neumann ratio exceeds an a-level critical  point,  conclude the data exhibit no significant temporal
   correlation.  Otherwise, conclude that a time-related pattern does exist. Check for seasonal cycles or
   linear trends using time series plots.  Consult a professional statistician regarding possible statistical
   adjustments if the pattern is more complex.

Advantages/Disadvantages:  The rank von Neumann ratio,  as opposed to other common  time  series
   methods for determining autocorrelation, is a non-parametric test based on using the ranks of the
   data instead of the actual concentration measurements.  The test is simple to compute and can be used
   as a formal  confirmation of temporal dependence, even if the autocorrelation appears fairly obvious
   on a time series plot. As a limiting feature, the test only applies to a single population at a time and

                                              JTl9                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   cannot be used to identify temporal effects that span across groups of wells simultaneously. In that
   scenario, a one-way ANOVA for temporal effects is a better diagnostic tool. Because critical points
   for the rank von Neumann ratio have not been developed for the presence of ties, the test will not be
   useful for datasets with substantial portions of non-detects.

       DARCY EQUATION (SECTION 14.3.2)
Basic purpose: Method  to determine a sampling  interval ensuring that distinct physical volumes of
   groundwater are sampled on any pair of consecutive events.

Hypothesis tested: Not a statistical test or formal procedure.

Underlying assumptions: Flow regime is one in which Darcy's equation is approximately valid.

When to use: Use Darcy's equation to gauge the minimum travel time necessary for distinct volumes of
   groundwater to pass through each well screen. Physical independence of samples does not guarantee
   statistical  independence, but it increases the likelihood of statistical independence. Use to design or
   plan for a site-specific sampling frequency, as well  as what formal statistical tests and retesting
   strategies  are possible given the amount of temporally-independent data that can be collected each
   evaluation period.

Steps involved: 1) Using knowledge of the site hydrogeology, calculate the horizontal and vertical
   components  of average groundwater velocity with  Darcy's equation; 2) Determine the minimum
   travel time needed  between field samples to ensure physical independence; 3) Specify a sampling
   interval during monitoring no less than the travel time obtained via the Darcy computation.

Advantages/Disadvantages:  Darcy's equation is  relatively  straightforward, but is not a statistical
   procedure. It is not  applicable to certain hydrologic environments. Further, it is not a substitute for a
   direct estimate of autocorrelation. Statistical independence is not assured using Darcy's equation, so
   caution is advised.

       SEASONAL CORRECTION (SECTION 14.3.3)
Basic purpose: Method to adjust a longer data series from a single population for an obvious seasonal
   cycle or fluctuation pattern. By removing the seasonal pattern, the remaining residuals can be used in
   further statistical procedures (e.g., prediction limits, control charts) and treated as independent of the
   seasonal correlation.

Hypothesis tested: The  seasonal  correction is not a formal statistical test. Rather, it is a statistical
   adjustment to data for which a definite seasonal pattern has been identified.

Underlying assumptions:  There should be enough data so that  at least 3  full  seasonal cycles are
   displayed on a time series plot. It is also assumed that the seasonal component has a stationary (i.e.,
   stable) mean and variance during the period of data collection.

When to use: Use the seasonal correction when a  longer series of data must be examined, but a time
   series plot indicates a clearly recurring, seasonal fluctuation of concentration levels. If not removed,
   the seasonal dependence will tend to upwardly  bias the estimated variability and  could  lead to
   inaccurate or insufficiently powerful tests.

                                             8^20                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Steps involved: 1) Using a time series plot of the data series, separate the values into common sampling
   events for each year (e.g., all January measurements, all third quarter values, etc.); 2) compute the
   average of each subgroup and the overall mean of the dataset; and 3) adjust the data by removing the
   seasonal pattern.

Advantages/Disadvantages: The seasonal correction described in the Unified Guidance  is relatively
   simple  to perform and offers  a more accurate standard  deviation estimates  compared to  using
   unadjusted data. Removal of the seasonal  component may reveal other previously unnoticed features
   of the data, such  as  a slow-moving trend.   A fairly long data series is required to  confirm the
   presence of a recurring seasonal cycle. Furthermore, many complex time-related patterns cannot be
   handled by this simple correction. In such cases, consultation with a professional statistician may be
   necessary.

       SEASONAL MANN-KENDALL TEST FOR TREND (SECTION 14.3.4)
Basic purpose: Method  for detection monitoring. It is used to identify the presence  of a significant
   (upward) trend at a compliance point when data also exhibit seasonal fluctuations. It may also be
   used  in  compliance/assessment and corrective action  monitoring to track upward or downward
   trends.

Hypothesis tested: HQ — No discernible linear trend exists in the concentration data over time. HA — A
   non-zero, (upward) linear component to the trend does exist.

Underlying assumptions: Since the seasonal Mann-Kendall trend  test is a non-parametric method, the
   underlying data need not be normal or follow a particular distribution. No special adjustment for ties
   is needed.

When to use: Use when  1) upgradient-to-downgradient comparisons are inappropriate so that intrawell
   tests are called for; 2) a control chart or intrawell prediction limit cannot be used because of possible
   trends in the intrawell background, and  3) the data also exhibit seasonality. A trend test can be
   particularly  helpful  at sites with  recent or historical contamination  where  it  is uncertain if
   background is already contaminated. An  upward trend in these cases will document the changing
   concentration levels more accurately than either a control chart  or intrawell prediction limit, both of
   which assume a stationary background mean concentration.

Steps involved:  1) Divide the data into separate groups representing common sampling events from
   each year; 2) compute the Mann-Kendall  test statistic (5) and its standard deviation (SD[,S]) on each
   group; 3) sum the separate  Mann-Kendall statistics into an overall test statistic; 4) compare this
   statistic against an a-level critical point;  and 5) if the statistic exceeds the critical  point, conclude
   that a significant upward trend exists. If not, conclude there is insufficient evidence for identifying a
   significant, non-zero trend.

Advantages/Disadvantages: 1) The seasonal Mann-Kendall test does not require any special treatment
   for non-detects, only  that all non-detects  be set to a common value lower than any of the detected
   values;  and 2) the test is easy to compute and reasonably efficient for detecting (upward) trends in
   the presence of seasonality.  Approximate critical  points  are  derived  from the standard  normal
   distribution.

                                             8^21                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

       SIMPLE SUBSTITUTION (SECTION 15.2)
Basic purpose: A simple adjustment for non-detects in a dataset. One-half the reporting limit [RL] is
   substituted for  each  non-detect  to  provide a numerical  approximation to the  unknown true
   concentration.

Hypothesis tested: None.

Underlying assumptions: The true non-detect concentration is assumed to lie somewhere between zero
   and the reporting limit. Furthermore, that the probability of the true concentration being less than
   half the RL is about the same as the probability of it being greater than half the RL.

When to use:  In general, simple substitution should be used when the dataset contains a relatively small
   proportion of non-detects, say no more than 10-15%. Use with larger non-detect proportions can
   result in biased estimates, especially if most of the detected concentrations are recorded at low levels
   (e.g.., at or near RL).

Steps involved:  1) Determine the reporting limit; and 2) replace each non-detect with one-half RL as a
   numerical approximation.

Advantages/Disadvantages: Simple substitution of half the RL is the easiest adjustment available for
   non-detect data. However, it can lead to biased estimates of the mean and particularly the variance if
   employed when more than 10-15% of the data are non-detects.

       CENSORED PROBABILITY PLOT (SECTIONS 15.3 AND 15.4)
Basic purpose: Diagnostic tool. It is a graphical  fit to normality of a mixture of detected and non-detect
   measurements.  Adjustments are made to the plotting  positions of the detected data  under the
   assumption that all measurements come from a common distributional model.

Hypothesis tested: As a graphical tool, the censored  probability  plot is not a formal  statistical test.
   However, it can provide an indication as to whether a dataset is consistent with the hypothesis that
   the mixture of detects and non-detects come from the same distribution and that the non-detects
   make up the lower tail of that distribution.

Underlying assumptions: Dataset consists of a mixture of detects  and non-detects, all arising from a
   common distribution. Data must be normal or normalized.

When to use: Use the censored  probability plot to check the viability of the Kaplan-Meier or robust
   regression  on  order statistics [ROS] adjustments for non-detect measurements. If the plot is linear,
   the data are consistent with a model in which the unobserved non-detect concentrations comprise the
   lower tail of the underlying distribution.

Steps involved: 1) Using either Kaplan-Meier or ROS, construct a partial ranking of the detected values
   to account for the presence of non-detects; 2) determine  standard normal quantiles  that match the
   ranking of the detects; and 3) graph the detected values against their matched normal quantiles on a
   probability plot and examine for a linear fit.
                                             8-22                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Advantages/Disadvantages: The censored probability plot  offers a visual indication  of whether a
   mixture of detects and non-detects come from the same (normal) distribution. There are, however, no
   formal critical points to aid in deciding when the fit is 'linear enough.' Correlation coefficients can
   be computed to informally aid the assessment. Censored probability plots can also be constructed on
   transformed data to help select a normalizing transformation.

       KAPLAN-MEIER ADJUSTMENT (SECTION 15.3)
Basic purpose: Diagnostic tool.  It is used to adjust a mixture of detected and non-detect data for the
   unknown concentrations of  non-detect values.  The Kaplan-Meier procedure leads to adjusted
   estimates for the mean and standard deviation of the underlying population.

Hypothesis tested: As a statistical  adjustment procedure,  the Kaplan-Meier method  is not a formal
   statistical test.  Rather, it allows estimation of characteristics of the population by  assuming the
   combined group of detects and non-detects come from a common distribution.

Underlying assumptions: Dataset consists of a mixture of detects and non-detects, all  arising from the
   same distribution.  Data must be normal or normalized in the context of the Unified  Guidance.
   Kaplan-Meier should not be used when more than 50% of the data are non-detects.

When  to use: Since the Kaplan-Meier adjustment assumes all the measurements arise from the same
   statistical process,  but that some of these measurements (i.e.., the non-detects) are unobservable due
   to limitations in analytical technology,  Kaplan-Meier should be used when this model is the most
   realistic or reasonable choice.  In particular, when constructing prediction limits, confidence limits, or
   control  charts, the  mean and  standard deviation of the underlying population must be estimated. If
   non-detects  occur  in the dataset (but do not account for  more than half of the observations), the
   Kaplan-Meier adjustment can be used to determine these estimates, which in turn can be utilized in
   constructing the desired statistical test.

Steps involved: 1) Sort the detected values and compute the 'risk set'  associated with each detect; 2)
   using the risk  set, compute  the Kaplan-Meier cumulative distribution  function  [CDF] estimate
   associated with each detect;  3) calculate adjusted  estimates of the population mean and standard
   deviation using the Kaplan-Meier CDF values; and 4)  use these adjusted population estimates in
   place of the sample mean and standard  deviation in prediction limits, confidence limits, and control
   charts.

Advantages/Disadvantages: Kaplan-Meier offers a way to adjust for significant fractions of non-detects
   without having to  know the actual non-detect concentration values. It is more difficult to use than
   simple substitution, but avoids the biases inherent in that method.

       ROBUST REGRESSION ON ORDER STATISTICS [ROS] (SECTION 15.4)
Basic purpose: Diagnostic tool.  It is  a method to adjust  mixture of detects and non-detects for the
   unknown concentrations of non-detect values. Robust ROS leads to adjusted estimates for the mean
   and standard deviation of the  underlying population by imputing a distinct estimated value for each
   non-detect.
                                             8-23                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Hypothesis tested: As a statistical  adjustment procedure, robust ROS is not a formal statistical test.
   Rather, it allows estimation of characteristics of the population by assuming the combined group of
   detects and non-detects come from a common distribution.

Underlying assumptions: Dataset consists of a mixture of detects and non-detects, all arising from the
   same distribution. Data must be normal or normalized in the context of the Unified Guidance.
   Robust ROS should not be used when more than 50% of the data are non-detects.

When to use: Since robust regression on order statistics assumes all the measurements arise from the
   same statistical process, robust ROS should be used when this model is reasonable.  In particular,
   when constructing prediction  limits, confidence limits, or control charts, the mean  and standard
   deviation of the underlying population must be estimated. If non-detects occur in the dataset (but do
   not account for more than half  of the observations), robust ROS can be used to determine these
   estimates, which in turn can be utilized to construct the desired statistical test.

Steps involved: 1) Sort the distinct reporting limits [RL] for non-detect values and compute 'exceedance
   probabilities'  associated with each RL; 2) using the exceedance probabilities, compute 'plotting
   positions' for the non-detects, essentially representing CDF estimates associated  with each RL; 3)
   impute values for individual non-detects based on their RLs and plotting positions; 4) compute
   adjusted mean and standard  deviation estimates via the sample mean and standard deviation of the
   combined set  of detects and imputed non-detects;  and 5) use these adjusted population estimates in
   place of the (unadjusted) sample mean and standard deviation in prediction limits,  confidence limits,
   and control charts.

Advantages/Disadvantages: Robust ROS offers an alternative to Kaplan-Meier to adjust for significant
   fractions of non-detects without having to know the  actual non-detect  concentration values. It is
   more difficult to use than simple  substitution, but avoids the biases inherent in that  method.

       COHEN'S  METHOD AND PARAMETRIC ROS (SECTION  15.5)
Basic purpose: Diagnostic tools.  These are other methods to adjust mixture of detects and non-detects
   to obtain the unknown mean and  standard deviation for the entire data set

Hypothesis tested:   Neither technique is a formal statistical test. Rather, they allow  estimation of
   characteristics of the population  by assuming the combined group of detects and  non-detects come
   from a common distribution.

Underlying assumptions: Dataset consists of a mixture of detects and non-detects, all arising from the
   same distribution. Data must be normal or normalized in the context of the Unified Guidance.
   Neither should be used when more than 50% of the data  are non-detects  nor when data contain
   multiple non-detect levels.

When to  use:  Since these methods assume that all the  measurements arise from the  same statistical
   process, they should be used when  this model is  reasonable.  In  particular, when constructing
   prediction limits, confidence  limits, or control charts, the mean and  standard deviation  of the
   underlying population must be estimated. If non-detects occur in the dataset (but do not account for
   more than half of the observations), they can be used to determine these estimates, which in turn can
   be utilized to construct the desired statistical test.
                                             8^24                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Steps involved: Cohen's Method:  1) data are sorted into non-detect and detected portions; 2) detect
   mean and standard deviation estimates are calculated; 3) intermediate quantities of the ND% and a
   factor y are calculated and used to locate the appropriate X value from a table; and 4) full data set
   mean and standard deviation estimates are then obtained using formulas based on the detected mean,
   standard deviation, the detection limit and X. Parametric ROS:   1)  detected data are  sorted in
   ascending order; 2) standardized normal distribution Z-values are generated from the full set of
   ranked  values.  Those corresponding to the sorted detected values are retained;   3)  the detected
   values are then regressed against the Z-values;  and 4) the resulting regression intercept and slope are
   the estimates of the mean and standard deviation for the full data set.

Advantages/Disadvantages:   These two methods offer alternatives to Kaplan-Meier and robust ROS.
   The key limitation is  that only data containing  a single censoring limit can be used.   In some
   situations using logarithmic data, their application can lead to biased estimates  of the mean and
   standard deviation. Where appropriate, these methods are less computationally intensive that either
   Kaplan-Meier or robust ROS.

       POOLED VARIANCE T-TEST (SECTION 16.1.1)
Basic purpose: Method for detection monitoring. This test compares the means of two populations.

Hypothesis tested: HQ — Means of the two populations are equal; HA — Means of the two  populations
   are unequal (for the usual one-sided alternative,  the hypothesis would state that the mean of the
   second  population is greater than the mean of the first population).

Underlying assumptions:  1) The data from  each population must be normal or normalized; 2) when
   used for interwell  tests, there  should be no significant spatial variability; 3)  at least 4 observations
   per well should be available before applying the test; and 4) the two group variances are equal.

When to use: The pooled variance t-test can be used to test for groundwater contamination at very small
   sites, those  consisting of maybe 3  or 4 wells  and monitoring for 1 or 2  constituents.  Site
   configurations with larger combinations of wells and constituents should employ a retesting scheme
   using either prediction limits or control charts. The pooled variance t-tesi can also be  used to test
   proposed updates to intrawell background. A non-significant t-test in this latter case suggests the two
   sets of data are sufficiently similar to allow the initial background to be updated by augmenting with
   more recent measurements.

Steps involved:  1) Test the  combined residuals from each population for normality. Make a  data
   transformation if necessary; 2) test for equal  variances, and if equal, compute a pooled variance
   estimate; 3) compute the pooled variance ^-statistic and the  degrees of freedom; 3) compare the t-
   statistic against a critical point based on both the a-level and the degrees of freedom; and 4) if the t-
   statistic exceeds the critical point, conclude the null hypothesis of equal means has been violated.

Advantages/Disadvantages:  1)  The pooled variance t-test is one  of the  easiest  to compute  t-tesi
   procedures, but requires an assumption of equal variances across both populations; 2) because the t-
   test is a well-understood statistical procedure, the Unified  Guidance recommends its  use at very
   small groundwater monitoring facilities.  For larger sites, however, repeated use  of the t-test at a
   given a-level will  lead to an unacceptably high  risk of false positive error; and  3) if substantial
   spatial variability exists, the use of any t-test for upgradient-to-downgradient comparisons may lead
                                             8^25                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   to inaccurate conclusions. A significant difference in the population averages could also indicate the
   presence of natural geochemical factors differentially affecting the concentration levels  at different
   wells. In these situations, consider an intrawell test instead.

       WELCH'S T-TEST (SECTION 16.1.2)
Basic purpose: Method for detection monitoring. This test compares the means of two populations.

Hypothesis tested: HO — Means of the two populations are equal; HA — Means of the two populations
   are unequal  (for  the usual one-sided alternative, the hypothesis would state that the mean of the
   second population is greater than the mean of the first population).

Underlying assumptions:  1) The data from each population must be normal or normalized; 2) when
   used  for  interwell tests, there should be  no  significant spatial variability;  and 3)  At least 4
   observations per well should be available before applying the test.

When to use: Welch's t-tesi can be used to test for groundwater contamination at very small sites, those
   consisting of maybe 3 or 4 wells and monitoring for 1 or 2 constituents. Site configurations with
   larger  combinations of wells  and constituents  should employ  a retesting scheme using  either
   prediction limits or control  charts. Welch's t-test  can also  be used to test proposed  updates to
   intrawell background data. A non-significant t-test in this latter case suggests the two sets of data are
   sufficiently similar to allow the initial background to be updated by augmenting with the more recent
   measurements.

Steps involved:  1)  Test the combined  residuals from each population for normality. Make  a data
   transformation if necessary; 2) compute Welch's ^-statistic and approximate degrees of freedom; 3)
   compare the  ^-statistic against a critical point based  on both the a-level and the estimated degrees of
   freedom; and 4)  if the  ^-statistic exceeds the critical point, conclude the null  hypothesis of equal
   means has been violated.

Advantages/Disadvantages: 1) Welch's  t-test is slightly more difficult to compute than other common
   t-test procedures, but has the advantage of not  requiring equal variances across both populations.
   Furthermore, it has been shown to perform statistically as well or better than other ^-tests;  2) it can be
   used at  very  small groundwater monitoring facilities, but should be avoided at larger sites. Repeated
   use of the t-test at a given a-level will lead to an unacceptably high risk of false positive error; and 3)
   if there is substantial  spatial  variability, use  of Welch's t-test  for interwell tests  may lead to
   inaccurate conclusions.  A significant difference  in the population averages may reflect the presence
   of natural geochemical  factors  differentially affecting the  concentration  levels  at different wells. In
   these situations, consider an intrawell test instead.

       WILCOXON RANK-SUM TEST (SECTION  16.2)
Basic purpose: Method for detection monitoring. This test compares the medians of two populations.

Hypothesis tested:  HQ — Both populations  have equal medians  (and, in  fact, are identical  in
   distribution). H^ —  The two population medians are unequal  (in the usual one-sided alternative, the
   hypothesis would state  that the median of the second population  is larger than the median of the
   first).

                                              8^26                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Underlying assumptions: 1) While the Wilcoxon rank-sum test does not require normal data, it does
   assume both populations have the same distributional form and that the variances are equal.  If the
   data are non-normal but there at most a few non-detects, the equal variance assumption may be
   tested through the use of box plots and/or Levene's test. If non-detects  make-up a large fraction of
   the observations, equal variances may have to be assumed rather than formally verified; 2) use  of the
   Wilcoxon rank-sum procedure for interwell tests assumes there is no significant spatial variability.
   This is  more likely to be the case in precisely those circumstances where the Wilcoxon procedure
   might be used: when there are high fractions  of non-detects, so  that most  of the concentration
   measurements at any location are at low levels;  and 3) there  should be at least 4  background
   measurements and at least 2-4 compliance point values.

When to use: The Wilcoxon rank-sum test can be used to test for groundwater contamination at very
   small sites, those  consisting of maybe 3 or  4 wells and monitoring for 1 or 2  constituents. Site
   configurations with larger combinations of wells and constituents should employ a retesting scheme
   using non-parametric prediction limits. Note, however, that non-parametric prediction limits often
   require  large background sample sizes to be effective. The Wilcoxon rank-sum  can be useful when a
   high percentage of the data is non-detect, but the amount of available background data is limited.
   Indeed, an intrawell Wilcoxon procedure may be helpful in some situations where the false positive
   rate would otherwise be too high to run intrawell prediction limits.

Steps involved: 1) Rank the combined set of values from the two datasets, breaking ties if necessary by
   using midranks; 2) compute the sum of the ranks from the compliance point well and calculate the
   Wilcoxon test statistic; 3) compare the Wilcoxon test statistic against an a-level critical point; and 4)
   if the test statistic  exceeds the critical point, conclude that the null hypothesis of equal medians  has
   been violated.

Advantages/Disadvantages:  1) The  Wilcoxon rank-sum test is an excellent technique for small sites
   with constituent non-detect data. Compared to other possible methods such as the test of proportions
   or exact binomial  prediction limits, the Wilcoxon rank-sum  does a better job overall  of correctly
   identifying elevated groundwater concentrations while limiting false positive error; 2) because the
   Wilcoxon rank-sum is easy to compute and understand, the Unified Guidance recommends its  use at
   very small groundwater monitoring facilities. For larger sites, repeated use of the Wilcoxon rank-
   sum at a given a-level will lead to  an unacceptably high risk of false positive error; and 3) if
   substantial spatial variability exists, the use of the Wilcoxon rank-sum for interwell tests may lead to
   inaccurate conclusions.  A significant difference in the population medians may signal the presence
   of natural geochemical  differences  rather  than  contaminated groundwater.  In  these situations,
   consider an intrawell test instead.

       TARONE-WARE TEST (SECTION  16.3)
Basic purpose: Non-parametric method  for detection monitoring. This is an extension of Wilcoxon
   rank-sum,  an alternative test  to  compare the medians  in two populations when non-detects  are
   prevalent.

Hypothesis tested: HQ — Both populations  have equal  medians (and, in fact,  are  identical  in
   distribution). HA — The two population medians are unequal (in the usual one-sided alternative, the
   hypothesis would  state that the median of the second population is larger than the median of the
   first).	
                                              8-27                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Underlying assumptions: 1) The Tarone-Ware test does not require normal data, but does assume both
   populations have the same distributional form and that the variances are equal;  and 2) use of the
   Tarone-Ware procedure for interwell tests assumes there is no significant spatial variability. This is
   more likely to be the case when there are high fractions of data non-detects, so that most of the
   concentration measurements at any location are at low and similar levels.

When to use: The Tarone-Ware test can be used to test for groundwater contamination at very small
   sites, those consisting  of perhaps 3 or  4  wells  and  monitoring  for  1  or 2  constituents. Site
   configurations with larger combinations of wells and  constituents should employ a retesting scheme
   using non-parametric prediction limits. Note, however, that non-parametric prediction limits often
   require  large background sample sizes to be effective. The Tarone-Ware test can be useful when a
   high percentage of the data is non-detect, but the amount of available background data is limited.
   The Tarone-Ware test is also an  alternative to the  Wilcoxon  rank-sum when there are multiple
   reporting limits and/or it is unclear how to fully rank the data as required by the Wilcoxon.

Steps involved: 1) Sort the distinct detected values in the combined data set; 2) count the  'risk set'
   associated  with each distinct value from Step 1 and compute the expected number of compliance
   point detections within each risk set; 3) form the Tarone-Ware test statistic from the expected counts
   in Step 2;  4) compare the test statistic against a standard normal a-level critical point; and 5) if the
   test statistic exceeds the critical  point, conclude that the null hypothesis of equal  medians has been
   violated.

Advantages/Disadvantages:  The  Tarone-Ware test is  an excellent technique  for  small sites  with
   constituent non-detect data having multiple reporting limits.  If substantial spatial variability exists,
   use of the  Tarone-Ware  test for interwell tests may  lead to inaccurate conclusions.  A significant
   difference  in the population medians may signal the presence  of natural geochemical differences
   rather than contaminated groundwater. In these situations, consider an intrawell test instead.

       ONE-WAY ANALYSIS OF VARIANCE [ANOVA] (SECTION 17.1.1)
Basic purpose: Formal interwell detection monitoring test and diagnostic tool.  It compares population
   means  at  multiple  wells,  in  order  to  detect  contaminated  groundwater when  tested against
   background.

Hypothesis tested: HQ — Population means across all tested wells are equal. HA — One or more pairs
   of population means are unequal.

Underlying assumptions: 1) ANOVA residuals at each well or population must be normally distributed
   or  transformable  to  normality. These  should be tested for  normality using  a goodness-of-fit
   procedure;  2) the population variances across all wells must be equal.  This assumption can be tested
   with box plots and Levene's test;  and  3) each tested well should have at least 3  to 4  separate
   observations.

When to use: The one-way ANOVA can sometimes be  used to  identify to  simultaneously test for
   contaminated groundwater across a group  of distinct  well locations. As an inherently interwell test,
   ANOVA should be utilized only on constituents exhibiting little to no spatial variation. Most uses of
   ANOVA have been superseded by prediction limits and control charts, although it is commonly
   employed to identify spatial variability or temporal dependence across a group of wells.
                                              8^28                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Steps involved: 1) Form the ANOVA residuals by subtracting from each measurement its sample well
   mean; 2) test the ANOVA residuals for normality and equal variance. If either of these assumptions
   is violated, try a transformation of the data and retest the assumptions; 3) compute the one-way
   ANOVA  F-statistic;  4) if  the  F-statistic exceeds  an a-level critical  point, conclude the null
   hypothesis of equal population means has been violated and that at least  one pair of wells shows a
   significant difference in concentration levels;  and 5) test each compliance well individually to
   determine which one or more exceeds background.

Advantages/Disadvantages: ANOVA is  only likely to be infrequently  used  to make upgradient-to-
   downgradient comparisons in formal detection monitoring testing. The  regulatory restrictions for
   per-constituent a-levels using ANOVA make it difficult to adequately control site-wide false positive
   rates [SWFPR].  Even if spatial variability is not a significant problem, users are advised to consider
   interwell prediction limits or control charts, and to incorporate some form of retesting

       KRUSKAL-WALLIS TEST (SECTION 17.1.2)
Basic purpose: Formal interwell  detection monitoring test and diagnostic tool. It compares population
   medians at multiple  wells,  in  order to detect  contaminated groundwater  when tested  against
   background.  It is  also  useful as a non-parametric alternative to  ANOVA for identifying spatial
   variability in constituents with non-detects or for data that cannot be normalized.

Hypothesis tested: HQ — Population medians across all tested wells are equal. HA — One or more pairs
   of population medians are unequal.

Underlying assumptions:  1) As a non-parametric alternative to ANOVA, data need not be normal; 2)
   the population variances across all wells must be equal. This assumption can be tested with box plots
   and Levene's test if the non-detect proportion is not too high; and 3) each tested well should have at
   least 3 to 4 separate observations.

When to  use:  The Kruskal-Wallis  test can sometimes be used to identify to simultaneously  test for
   contaminated groundwater across a group of distinct well locations. As an inherently interwell test,
   Kruskal-Wallis  should  be utilized for this purpose  only  with constituents exhibiting little to  no
   spatial variation. Most  uses of the Kruskal-Wallis (similar to ANOVA) have been superseded by
   prediction limits, although it can be used to identify spatial variability and/or temporal dependence
   across a group of wells when the  sample data  are non-normal or  have higher proportions  of non-
   detects.

Steps involved: 1)  Sort and form the ranks of the combined measurements; 2) compute the rank-based
   Kruskal-Wallis test statistic (//); 3) if the //-statistic exceeds an a-level critical point, conclude the
   null hypothesis  of equal population medians has been violated and that  at least one pair of wells
   shows a significant difference in concentration levels; and  5) test each compliance well individually
   to determine which one or more exceeds background.

Advantages/Disadvantages: 1) The Kruskal-Wallis test is only likely to be infrequently used to make
   upgradient-to-downgradient  comparisons in  formal  detection  monitoring  testing. The regulatory
   restrictions for per-constituent a-levels using ANOVA make  it difficult to adequately control the
   SWFPR.   Even if spatial variability is  not  a  significant  problem, users are advised to consider

                                              8^29                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   interwell prediction limits, and to incorporate some form of retesting; and 2) the Kruskal-Wallis test
   can be used to test for spatial variability in constituents with significant fractions of non-detects.

       TOLERANCE LIMIT (SECTION 17.2.1)
Basic  purpose: Formal interwell detection monitoring test of background versus  one  or  more
   compliance wells. Tolerance limits can be used as  an alternative to one-way ANOVA. These can
   also be used in corrective action as an alternative clean-up limit.

Hypothesis tested: HQ — Population means across all tested wells are equal. HA — One or more pairs
   of population means are unequal.

Underlying assumptions: 1) Data should be normal or normalized;  2) the population variances across
   all wells are assumed to be equal.  This assumption can be difficult to test when comparing a single
   new observation from each compliance well against a tolerance limit based on background; and  3)
   there should be a minimum of 4 background measurements, preferably 8-10 or more.

When  to  use: A tolerance limit can  be used  in place of ANOVA  for  detecting contaminated
   groundwater. It  is more flexible  than ANOVA since 1) as few  as  one  new measurement per
   compliance well is needed to run  a tolerance limit  test, and 2) no post-hoc testing is necessary  to
   identify which compliance wells are elevated over background. Most uses of tolerance limits (similar
   to ANOVA) have been superseded by prediction limits, due to difficulty of incorporating retesting
   into tolerance limit schemes. If a  hazardous  constituent requires  a background-type standard  in
   compliance/assessment or corrective action, a tolerance limit can be computed on background and
   used as a fixed GWPS.

Steps involved: 1) Compute background sample mean and  standard deviation; 2) calculate upper
   tolerance limit on background with  high  confidence and  high  coverage;  3) collect one or more
   observations from each compliance well and test each against the tolerance limit;  and 4) identify a
   well as contaminated if any of its observations exceed the tolerance limit.

Advantages/Disadvantages: Tolerance limits are likely to be used only infrequently to be used as either
   interwell or intrawell tests.  Prediction limits or control charts offer better control of false positive
   rates, and less is known about the impact of retesting on tolerance  limit performance.

       NON-PARAMETRIC TOLERANCE LIMIT (SECTION 17.2.2)
Basic  purpose: Formal interwell detection monitoring test of background versus  one  or  more
   compliance wells. Non-parametric tolerance limits can be used as an alternative to the Kruskal-
   Wallis  test. They may  also be used in  compliance/assessment or corrective action to define a
   background GWPS.

Hypothesis tested: HQ — Population medians across all tested wells are equal. HA — One or more pairs
   of population medians are unequal.

Underlying assumptions: 1) As a non-parametric  test, non-normal data with non-detects can be used;
   and 2) there should be a minimum of 8-10 background measurements and preferably more.
                                             8-30                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

When to  use:  A non-parametric tolerance limit can be used in place of the Kruskal-Wallis test for
   detecting contaminated groundwater. It is more flexible than Kruskal-Wallis since 1) as few as one
   new measurement per compliance well is needed to run a tolerance limit test, and 2)  no post-hoc
   testing is necessary to identify which compliance wells are elevated over background. Most uses of
   tolerance limits have been superseded by prediction limits, due to difficulty of incorporating retesting
   into tolerance limit schemes. However, when a clean-up limit cannot or has not been  specified in
   corrective action,  a tolerance limit can be computed on background and used as  a  site-specific
   alternate concentration limit [ACL].

Steps involved: 1) Compute a large order statistic from background and set this value as the upper
   tolerance limit;  2) calculate the confidence and coverage associated with the tolerance limit; 3)
   collect one or more observations from each compliance well and test each against the tolerance limit;
   and 4) identify a well as contaminated if any of its observations exceed the tolerance limit.

Advantages/Disadvantages:  1) Tolerance limits are likely to be used only infrequently to be used as
   either  interwell  or intrawell tests.   Prediction limits or control charts offer better control of false
   positive rates, and less is known about the impact of retesting on tolerance limit performance; and 2)
   non-parametric tolerance limits have the added disadvantage of generally requiring large background
   samples to ensure adequate confidence and/or coverage. For this reason, it is strongly recommended
   that a parametric tolerance limit be constructed whenever possible.

       LINEAR  REGRESSION  (SECTION 14.4)
Basic purpose: Method for detection monitoring and diagnostic tool.  It is used to identify the presence
   of a significantly increasing trend at a compliance point or any trend in background data sets.

Hypothesis tested: HQ — No discernible linear trend exists in the concentration data over time. HA — A
   non-zero, (upward) linear component to the trend does exist.

Underlying assumptions: Trend residuals should be  normal or normalized, equal  in variance, and
   statistically independent. If a small fraction of non-detects exists (<10-15%), use simple substitution
   to replace each non-detect by half the reporting limit [RL]. Test homoscedasticity of residuals with a
   scatter plot (Section 9.1).

When to use: Use a test for trend when 1) upgradient-to-downgradient comparisons are inappropriate so
   that intrawell tests are called for, and 2) a control chart or intrawell prediction limit cannot be used
   because of possible  trends in the intrawell background. A trend test can be particularly helpful at
   sites  with recent or  historical contamination  where it is  uncertain to what  degree  intrawell
   background is already contaminated. The presence of an upward trend in these cases will document
   the changing nature of the concentration data much more accurately than either a control chart or
   intrawell prediction limit, both of which assume a stable baseline concentration.

Steps involved: 1) If a linear  trend is evident on a time series plot, construct the linear regression
   equation; 2) subtract  the estimated trend  line  from each observation to form residuals;  3) test
   residuals for assumptions  listed  above;  and 4) test regression slope to  determine whether it is
   significantly different from  zero.  If so and the slope is positive, conclude there is evidence  of a
   significant upward trend.

                                             JTsi                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Advantages/Disadvantages: Linear regression is a standard statistical method for identifying trends and
   other linear associations between pairs of random variables.  However, it requires approximate
   normality  of  the  trend residuals.  Confidence bands  around regression trends  can be used in
   compliance/assessment and corrective action to determine compliance with fixed standards  even
   when concentration levels are actively changing (i.e.., when a trend is apparent).

       MANN-KENDALL TEST  FOR TREND (SECTION 17.3.2)
Basic purpose: Method for detection monitoring and diagnostic tool. It is used to identify the presence
   of a significant (upward) trend at a compliance point or any trend in background data.

Hypothesis tested: HO — No discernible linear trend exists in the concentration data over time. HA — A
   non-zero, (upward) linear component to the trend does exist.

Underlying assumptions: Since the Mann-Kendall  trend  test  is  a  non-parametric method,  the
   underlying data need not be normal or follow any particular distribution. No special  adjustment for
   ties is needed.

When  to use: Use a test for trend when 1) interwell tests are inappropriate  so that intrawell tests are
   called for,  and 2)  a control chart or intrawell prediction limit cannot be  used because  of possible
   trends in intrawell background. A trend  test can be  particularly  helpful at sites with recent or
   historical contamination where it is uncertain if intrawell background is already contaminated. An
   upward trend in these cases documents changing concentration levels more accurately than either a
   control  chart or intrawell  prediction  limit, both of which assume a  stationary background mean
   concentration.

Steps involved:  1) Sort the data values by time of sampling/collection; 2) consider all possible pairs of
   measurements from different sampling events; 3) score each pair depending on whether the later data
   point is higher or lower in concentration than the earlier  one, and sum the scores to get Mann-
   Kendall statistic; 4) compare  this statistic against an a-level critical  point;  and 5)  if the statistic
   exceeds the critical point, conclude that a significant upward trend exists. If not, conclude there is
   insufficient evidence for identifying a significant, non-zero trend.

Advantages/Disadvantages: The Mann-Kendall test does not require any special treatment for  non-
   detects, only that all non-detects can be set to a common value lower than any of the detects.   The
   test is easy to compute and reasonably efficient for detecting (upward) trends. Exact critical points
   are provided in the Unified Guidance for n < 20; a normal approximation can be used for n > 20. 3)
   A version of the Mann-Kendall test (the seasonal Mann-Kendall, Section 14.3.4) can be used to test
   for trends in data that exhibit seasonality.

       THEIL-SEN TREND LINE (SECTION 17.3.3)
Basic  purpose: Method  for  detection monitoring.  This is  a non-parametric  alternative to linear
   regression for estimating a linear trend.

Hypothesis tested: As  presented in the Unified Guidance,  the Theil-Sen trend line is not a formal
   hypothesis test but rather an estimation procedure. The algorithm can be  modified to formally test
   whether the true slope is significantly different from zero, but this question will already be answered
   if used in conjunction with the Mann-Kendall procedure.
                                              8^32                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Underlying assumptions: Like the Mann-Kendall trend test, the Theil-Sen trend line is non-parametric,
   so the underlying data need not be normal or follow a particular distribution. Furthermore, data ranks
   are not used, so no special adjustment for ties is needed.

When to use: It is particularly helpful when used in conjunction with the Mann-Kendall test for trend.
   The latter test offers information about whether a trend exists, but does not estimate the trend line
   itself. Once a  trend is identified, the Theil-Sen procedure indicates how quickly the concentration
   level is  changing with time.

Steps involved: 1) Sort the data set by date/time of sampling; 2)  for each pair of distinct sampling
   events,  compute the simple pairwise slope; 3) sort the list of pairwise slopes and set the overall slope
   estimate (Q) as the median slope in this list; 4) compute the median concentration and  the median
   date/time of sampling; and 5) construct the Theil-Sen trend as the line passing through  the median
   scatter point from Step 4 with slope Q.

Advantages/Disadvantages: Although non-parametric, the Theil-Sen slope estimator does not use data
   ranks but rather the concentrations themselves. The method is non-parametric because  the median
   pairwise slope is utilized, thus ignoring extreme values that might otherwise skew the slope estimate.
   The Theil-Sen trend line is as easy to compute as the Mann-Kendall test and does not require any
   special adjustment for ties (e.g., non-detects).

       PREDICTION LIMIT FOR M FUTURE VALUES (SECTION 18.2.1)
Basic purpose: Method for detection monitoring. This technique estimates numerical bound(s) on a
   series of m independent future values. The prediction limit(s) can be used to test whether the mean of
   one or more compliance well populations are equal to the mean of a background population.

Hypothesis tested: HQ — The true mean of TO future observations arises from the same population as the
   mean of measurements used to construct the prediction limit. HA — The m future observations come
   from a  distribution with a different mean than the population  of measurements. Since an upper
   prediction limit is of interest in detection monitoring, the alternative hypothesis would state that the
   future observations are distributed with a larger mean than the background population.

Underlying assumptions: 1) Data used to construct the prediction limit must be normal or normalized.
   Adjustments for small to moderate fractions of non-detects can be made, perhaps using Kaplan-
   Meier or robust ROS; 2) although the variances of both populations (background and future values)
   are assumed to be equal, rarely will there be  enough data from the future population to verify this
   assumption except during periodic  updates to background; and 3) if used for  upgradient-to-
   downgradient comparisons, there should be no significant spatial variability.

When to use: Prediction  limits  on individual observations can be  used as an alternative in detection
   monitoring  to either  one-way  ANOVA  or Dunnett's  multiple comparison with control  [MCC]
   procedure. Assuming there is  insignificant natural spatial variability, an interwell prediction limit can
   be  constructed using  upgradient or other representative background data.  The  number of future
   samples (m) should be chosen to reflect a single new observation collected from each downgradient
   or compliance well prior to the next  statistical evaluation, plus a fixed number (m-l)  of possible
   resamples. The  initial  future observation at each compliance point is then  compared  against the
   prediction limit. If it  exceeds the  prediction  limit,  one or more resamples are collected from the
                                             JTslMarch 2009

-------
Chapter 8. Methods Summary	Unified Guidance

    'triggered' well and also tested against the prediction limit. If substantial spatial variability exists,
    prediction  limits for individual values can be constructed on a well-specific basis using intrawell
    background. The larger the intrawell background size, the better. To incorporate retesting, it must be
    feasible to collect up to (m-\) additional, but independent, resamples from each well.

Steps involved: 1)  Compute the estimated mean and standard deviation of the background data; 2)
    considering the type of prediction limit (i.e., interwell or intrawell), the number of future samples m,
    the desired site-wide false  positive rate, and the number of wells and monitoring parameters,
    determine the prediction limit multiplier (K); 3) compute the prediction limit as the background mean
    plus K times the background standard deviation; and 4)  compare each initial future observation
    against  the prediction limit. If both the  initial  measurement  and  resample(s) exceed the limit,
    conclude the null hypothesis of equal means has been violated.

Advantages/Disadvantages: Prediction  limits for individual values offer several advantages compared
    to the traditional one-way  ANOVA and Dunnett's multiple comparison  with control  [MCC]
    procedures. Prediction limits are not bound to a minimum 5% per-constituent false positive rate and
    can be  constructed  to meet a target site-wide  false positive rate  [SWFPR]  while  maintaining
    acceptable statistical power. Unlike the one-way ANOVA F-test, only the comparisons of interest
    (i.e., each  compliance point against  background) are tested.  This gives the prediction limit more
    statistical power.  Prediction limits can be designed for intrawell as well as interwell comparisons.

       PREDICTION LIMIT FOR  FUTURE MEAN (SECTION 18.2.2)
Basic purpose: Method for  detection monitoring or  compliance  monitoring. It  is used to  estimate
    numerical limit(s) on an independent mean constructed from/? future values. The prediction limits(s)
    can be  used to test whether the  mean  of one population is equal to  the  mean  of a  separate
    (background) population.

Hypothesis tested: HO — The true mean of p future observations arise from the same population as the
    mean of measurements used to construct the prediction limit. HA — The/? future observations come
    from a distribution with a different mean than the population of background measurements.  Since an
    upper prediction limit is of interest in both detection and compliance  monitoring, the alternative
    hypothesis would state that the future observations are distributed with a larger mean than that of the
    background population.

Underlying assumptions: 1) Data used to construct the prediction  limit must be normal or normalized.
    Adjustments for small to moderate fractions of  non-detects can be made, perhaps using Kaplan-
    Meier or robust ROS; 2) although the variances of both populations (background and future values)
    are assumed to be equal, rarely will  there  be  enough data  from the future population to verify this
    assumption; and 3)  if used  for  upgradient-to-downgradient  comparisons, there should be  no
    significant spatial variability.

When to use:  Prediction limits on means can be used as an alternative in detection  monitoring to either
    one-way ANOVA or Dunnett's multiple comparison with control [MCC] procedure. Assuming there
    is insignificant natural spatial  variability, an interwell prediction limit can be constructed using
    upgradient or  other representative background data. The  number of future samples p should be
    chosen to reflect the number of samples that will be collected at each compliance well prior to the
    next statistical evaluation  (e.g., 2, 4, etc.). The average of these/? observations at each compliance
                                             8^34                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   point is then compared against the prediction limit. If it is feasible to collect at least/? additional, but
   independent, resamples from each well, retesting can be incorporated into the procedure by using
   independent mean(s) of/? samples as confirmation value(s).

    If substantial spatial variability exists, prediction limits for means can be constructed on a well-
   specific basis using intrawell background. At least  two future values must be available per well.
   Larger intrawell background size  are preferable. To incorporate retesting, it must be feasible to
   collect at least/? independent resamples from each well,  in addition to the initial set ofp samples. A
   prediction limit can also be used in some compliance monitoring settings when a fixed compliance
   health based limit cannot be use and the compliance  point data must be compared  directly to a
   background GWPS. In this case, the compliance point mean concentration is tested against an upper
   prediction limit computed from background.  No retesting would be employed for this latter kind of
   test.

Steps involved: 1) Compute  the background sample mean and standard deviation;  2) considering the
   type of prediction limit (i.e.,  interwell or intrawell), the number of future samples/?, use of retesting,
   the desired site-wide  false  positive rate,  and the  number of wells  and  monitoring parameters,
   determine the prediction limit multiplier (K); 3) compute the prediction limit as the background mean
   plus K times the background  standard deviation; 4) compare each future mean of orderp (i.e.., a mean
   constructed from p values) against the prediction limit; and 5) if the future mean exceeds the limit
   and retesting is not feasible (or if used  for compliance monitoring), conclude the null hypothesis of
   equal means has been violated. If retesting is  feasible, conclude the null hypothesis has been violated
   only when the resampled mean(s) of order/? also exceeds the prediction limit.

Advantages/Disadvantages:  Prediction limits  on means  offer several  advantages compared to the
   traditional  one-way ANOVA and Dunnett's multiple comparison with control  [MCC] procedure:
   Prediction  limits are not  bound to a minimum 5% per-constituent  false  positive rate.  As such,
   prediction limits can be constructed to meet a target SWFPR, while maintaining acceptable statistical
   power.  Unlike the  one-way F-test, only the comparisons  of interest (i.e.., each compliance point
   against background) are tested,  giving the prediction limit more statistical power. Prediction limits
   can be designed  for intrawell  as well  as  interwell  comparisons. One slight  disadvantage is that
   ANOVA combines compliance point data with background  to  give  a somewhat better  per-well
   estimate of variability. But  even this disadvantage can be  overcome when using  an interwell
   prediction limit by first running ANOVA on the combined background and compliance point data to
   generate a  better variance estimate with a larger degree of freedom.  A disadvantage compared to
   prediction limits on individual future values is that two  or more new  compliance point observations
   per well must be available to run the prediction  limit on means.  If only one new measurement per
   evaluation  period can be collected, the user should instead construct a prediction  limit on individual
   values.

       N ON-PARAMETRIC PREDICTION LIMIT FOR M FUTURE VALUES (SECTION  18.3.1)
Basic purpose: Method for detection monitoring. It is a  non-parametric technique to estimate numerical
   limits(s) on a series of m independent future values. The  prediction limit(s) can be used to test
   whether two samples are drawn  from the same or different populations.

Hypothesis tested:  HQ  — The m  future observations  come  from  the same distribution as the
   measurements used to construct the prediction limit.  HA — The m future observations  come from a
                                             8^35                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   different distribution than the population of measurements used to build the prediction limit. Since
   an upper prediction limit is of interest in detection monitoring, the alternative hypothesis is that the
   future observations are distributed with a larger median than the background population.

Underlying assumptions: 1) The  data  used  to construct the  prediction limit need not be normal;
   however, the forms of the both the background distribution and the future distribution are assumed to
   be  the  same. Since the non-parametric prediction  limit is constructed  as an order statistic of
   background,  high fractions  of non-detects  are  acceptable; 2)  although  the variances  of both
   populations (background and future values) are assumed to be equal, rarely will there be enough data
   from the future population to verify this assumption; and 3)  if used for upgradient-to-downgradient
   comparisons, there should be no significant spatial variability. Spatial variation is less likely to be
   significant in many cases where  constituent data are primarily non-detect, allowing the use of a non-
   parametric interwell prediction limit test.

When  to use: Prediction  limits on individual values can be used as a non-parametric alternative in
   detection  monitoring to either one-way ANOVA or Dunnett's multiple comparison  with  control
   [MCC]  procedure. Assuming there is  insignificant natural spatial variability, an interwell prediction
   limit can be  constructed using upgradient or other representative background data. The number of
   future samples  m should  be chosen to  reflect  a  single  new  observation collected from each
   compliance well prior  to the next statistical evaluation, plus a  fixed number (m-\) of possible
   resamples. The  initial  future  observation at each compliance point  is then compared against the
   prediction  limit. If it exceeds the prediction limit, one or more  resamples  are collected from the
   'triggered' well and also compared to the prediction limit.

Steps involved:  1) Determine the maximum, second-largest, or other highly ranked value in background
   and set the non-parametric prediction limit equal  to this level; 2) considering the  number of future
   samples m, and the number of wells and monitoring parameters, determine the achievable site-wide
   false positive rate  [SWFPR]. If the error rate is  not acceptable, consider possibly enlarging the pool
   of background data used to  construct the limit or increasing the number of future samples m; 3)
   compare each  initial future  observation against the prediction  limit;  and 4) if both the initial
   measurement and  resample(s) exceed the limit, conclude the null hypothesis of equal distributions
   has been violated.

Advantages/Disadvantages: Non-parametric  prediction  limits on  individual  values  offer  distinct
   advantages compared to the Kruskal-Wallis non-parametric ANOVA test.  Prediction limits are not
   bound to  a minimum  5% per-constituent false  positive rate. As such, prediction limits  can be
   constructed to  meet a target  SWFPR, while maintaining acceptable  statistical power. Unlike the
   Kruskal-Wallis  test,  only  the  comparisons   of interest  (i.e..,  each compliance point  against
   background)  are tested,  giving the prediction limit more statistical power.  Non-parametric prediction
   limits have the disadvantage of generally requiring fairly large background samples to effectively
   control false positive error and ensure  adequate power.

       PREDICTION LIMIT FOR FUTURE MEDIAN (SECTION 18.3.2)
Basic purpose: Method for detection monitoring and compliance monitoring.  This is a non-parametric
   technique  to estimate  numerical limits(s)  on  the  median  of p independent future values.  The
   prediction limit(s) is used to test whether the median of one or more compliance well populations is
   equal to the median of the background population.
                                             8^36                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Hypothesis tested: HO — The true median of p future observations arise from the same population as
   the median of measurements used to construct the prediction limit. HA — The/? future observations
   come from a distribution with a different median than the background population of measurements.
   Since  an upper prediction limit  is of interest  in  both  detection  monitoring  and compliance
   monitoring, the alternative hypothesis is that the future observations are distributed with a larger
   median than the background population.

Underlying assumptions: 1) The data used to construct the  prediction limit need  not be normal;
   however, the forms of the both the background distribution and the future distribution are assumed to
   be  the  same. Since  the non-parametric prediction  limit is constructed as  an  order statistic of
   background,  high  fractions of non-detects  are  acceptable: 2) although  the variances  of both
   populations (background and future values) are assumed to be equal, rarely will there be enough data
   from the future  population to verify this assumption; and 3)  if used for upgradient-to-downgradient
   comparisons, there should be no significant spatial variability.

When  to use: Prediction limits on medians can be used as a non-parametric alternative in detection
   monitoring to  either one-way  ANOVA or Dunnett's multiple comparison with  control  [MCC]
   procedure. Assuming there is insignificant natural spatial variability, an interwell  prediction limit
   can be constructed  using upgradient or other representative background data. The number of future
   samples/* should be odd and chosen to reflect the number of samples that will be collected at each
   compliance well prior to the next statistical evaluation (e.g., 3).  The median of these p observations
   at each compliance point is then compared against the prediction limit. If it is feasible to collect at
   least/? additional, but independent, resamples from each well, retesting can be incorporated into the
   procedure by using independent median(s) of p samples as confirmation value(s). A prediction limit
   for a compliance point median can also be constructed in certain compliance monitoring settings,
   when no fixed health-based compliance limit can be used and  the compliance point data must be
   directly  compared  against  a  background GWPS.  In this  case,  the  compliance point  median
   concentration is compared to an upper prediction limit computed from background. No retesting is
   employed for this latter kind of test.

Steps involved:  1) Determine the maximum, second-largest, or other highly ranked value in background
   and set the non-parametric prediction limit equal  to this level; 2) considering the number of future
   samples p, whether or not retesting will be incorporated, and the number of wells and monitoring
   parameters, determine the achievable SWFPR. If the error rate is  not acceptable,  increase the
   background sample size or  consider a non-parametric prediction limit on individual future values
   instead; 3) compare each future median of order/? (i.e., a median of/? values) against the prediction
   limit; and 4) if the future median exceeds the limit and retesting is not feasible (or if the test is used
   for compliance  monitoring), conclude the null hypothesis of equal medians has been violated. If
   retesting is feasible,  conclude the null  hypothesis  has been violated  only  when the  resampled
   median(s) of order/? also exceeds the prediction limit.

Advantages/Disadvantages: Non-parametric prediction  limits  on medians offer distinct advantages
   compared to the Kruskal-Wallis test (a non-parametric one-way ANOVA). Prediction limits are not
   bound to a minimum 5% per-constituent false positive rate.  As such, prediction limits  can be
   constructed to meet a target SWFPR, while  maintaining acceptable  statistical power.  Unlike the
   Kruskal-Wallis  test,  only  the  comparisons  of  interest  (i.e., each  compliance point  against
   background)  are tested, giving the  prediction limit  more  statistical power.  A disadvantage in
                                             JTs?                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   detection monitoring compared to non-parametric prediction limits on individual future values is that
   at least three new compliance point observations per well must be available to run the prediction
   limit on medians. If only one new observation per evaluation period can be collected,  construct
   instead a non-parametric prediction limit for individual values. All non-parametric prediction limits
   have the disadvantage of usually requiring fairly large background  samples  to effectively  control
   false positive error and ensure adequate power.

       SHEWHART-CUSUM CONTROL CHART (SECTION 20.2)
Basic purpose: Method for detection monitoring. These are used to quantitatively and visually track
   concentrations at a given well over time to determine whether they exceed a critical threshold (i.e..,
   control limit), thus implying a significant increase above background conditions.

Hypothesis tested: HQ — Data plotted on the  control chart  follow  the  same distribution  as the
   background data used  to compute the baseline chart parameters. HA —  Data plotted  on the chart
   follow a different distribution with higher mean level than the baseline data.

Underlying assumptions: Data used to construct the  control chart must be approximately normal or
   normalized. Adjustments for small to moderate  fractions of non-detects, perhaps using Kaplan-Meier
   or ROS, can be acceptable. There should be no discernible trend in the baseline data used to calculate
   the control limit.

When to use: Use control charts as an alternative to  parametric prediction limits, when 1) there are
   enough uncontaminated baseline data to compute an accurate control limit, and 2) there are no trends
   in intrawell background. Retesting can be incorporated into  control charts by judicious choice of
   control limit. This may need to be estimated using Monte Carlo simulations.

Steps  involved: 1)  Compute the intrawell  baseline  mean  and standard deviation;  2)  calculate  an
   appropriate control limit from these baseline parameters, the desired retesting strategy and number of
   well-constituent pairs  in  the network; 3)  construct the chart,  plotting the control  limit,  the
   compliance point observations, and the cumulative  sums [CUSUM];  and  4) determine that the null
   hypothesis is violated when either an individual concentration measurement or the  cumulative sum
   exceeds the control limit.

Advantages/Disadvantages: Unlike prediction limits, control charts offer an explicit visual tracking of
   compliance point values over time and provide  a method to judge whether these concentrations have
   exceeded  a critical threshold. The Shewhart portion of the  chart is especially good at  detecting
   sudden concentration increases, while the CUSUM  portion is preferred for detecting slower, steady
   increases over time. No non-parametric version of the combined Shewhart-CUSUM control chart
   exists, so non-parametric prediction limits should be considered if the data cannot be normalized.

       CONFIDENCE INTERVAL AROUND NORMAL MEAN (SECTION 21.1.1)
Basic purpose: Method for compliance/assessment monitoring or corrective action. This is a technique
   for estimating a range of concentration values from  sample data, in which the true mean of a  normal
   population is expected to occur at a certain probability.

Hypothesis tested: In compliance monitoring, HO — True mean concentration at the compliance point is
   no greater than the  predetermined groundwater protection standard [GWPS]. HA — True mean
                                             8^38                                    March 2009

-------
Chapter 8.  Methods Summary	Unified Guidance

   concentration is greater than the GWPS.  In corrective action, HQ — True mean concentration at the
   compliance point is greater than or equal to the fixed GWPS. HA — True mean concentration is less
   than or equal to the fixed standard.

Underlying  assumptions:  1) Compliance  point  data  are approximately  normal  in distribution.
   Adjustments for small to moderate fractions of non-detects, perhaps using Kaplan-Meier or ROS, are
   encouraged;  2)  data do not exhibit any significant trend over time;  3) there are a minimum of 4
   observations for testing. Generally, at least 8 to 10 measurements are recommended; and 4) the fixed
   GWPS is assumed to represent a true mean average concentration, rather than a maximum or upper
   percentile.

When to use: A mean confidence interval can be used for normal data  to determine whether there is
   statistically significant evidence that the average is either  above a  fixed GWPS (in compliance
   monitoring) or below the fixed standard (in corrective action). In either case, the null hypothesis is
   rejected only when the entire confidence interval lies on one or the  other side of the GWPS.  The key
   determinant in compliance monitoring is whether the lower confidence limit exceeds the GWPS,
   while in  corrective action  the upper confidence limit lies below the clean-up standard. Because of
   bias introduced by transformations when estimating a mean, this  approach should not be used for
   highly-skewed or non-normal  data. Instead consider a confidence interval around a lognormal mean
   or a non-parametric confidence interval. It is also not recommended for use when the data exhibit a
   significant trend. In that case,  the estimate  of variability  will likely be too high,  leading to an
   unnecessarily wide interval and possibly little chance of deciding the hypothesis. When a trend is
   present, consider instead a confidence interval around a trend line.

Steps involved: 1) Compute the sample mean and  standard deviation; 2) based on the sample size and
   choice of a confidence level (1-oc), calculate either the lower confidence limit (for use in compliance
   monitoring) or the upper confidence limit (for use in corrective action); 3) compare the confidence
   limit against the GWPS or clean-up standard; and 4) if the lower confidence limit exceeds the GWPS
   in compliance monitoring or the upper confidence limit is below the clean-up standard, conclude that
   the null hypothesis should be rejected.

Advantages/Disadvantages: Use of a confidence interval instead of simply the sample mean for
   comparison to a fixed standard accounts for both the level of statistical variation in the data and the
   desired or targeted  confidence level.  The same basic  test  can  be used  both  to document
   contamination above the compliance standard in compliance/assessment  and to show a sufficient
   decrease in concentration levels below the clean-up standard in corrective action.

       CONFIDENCE INTERVAL ON  LOGNORMAL  GEOMETRIC MEAN (SECTION 21.1.2)
Basic purpose: Method for compliance/assessment monitoring or corrective action. It is  a technique to
   estimate the range  of concentration values from sample data, in which the  true geometric mean of a
   lognormal population is expected to occur at a certain probability.

Hypothesis tested: In  compliance monitoring, HQ — True mean concentration at the compliance point is
   no greater than the fixed compliance or groundwater protection standard [GWPS]. HA — True mean
   concentration is greater than the GWPS.  In corrective action, HO — True mean concentration at the
   compliance point is greater than the  fixed compliance  or  clean-up standard. HA — True mean
   concentration is less than or equal to the fixed standard.
                                             8^39                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Underlying assumptions:  1)  Compliance  point  data  are approximately lognormal  in  distribution.
   Adjustments for small to moderate fractions of non-detects, perhaps using Kaplan-Meier or ROS, are
   encouraged; 2) data do not exhibit any  significant trend over time;  3) there are a minimum of 4
   observations. Generally, at least 8 to 10 measurements are recommended; and 4) the fixed GWPS is
   assumed to  represent a  true  geometric mean  average  concentration  following  a  lognormal
   distribution, rather than a maximum or upper percentile. The GWPS also represents the true median.

When to use: A confidence interval on the geometric mean can be used for lognormal data to determine
   whether there  is statistically significant evidence that the geometric average is either above a fixed
   numerical standard (in compliance monitoring) or below a fixed standard (in corrective  action). In
   either case, the null hypothesis is rejected only when the entire confidence interval is to one side of
   the compliance or clean-up standard. Because of this fact, the key question in compliance monitoring
   is whether the lower confidence limit exceeds the GWPS, while in corrective action the user must
   determine whether the upper confidence  limit is below  the clean-up standard. Because of bias
   introduced  by  transformations  when estimating the arithmetic  lognormal  mean, and  the  often
   unreasonably  high upper confidence limits  generated by Land's  method for lognormal  mean
   confidence intervals (see  below), this approach is an alternative approach for lognormal data. One
   could also consider a non-parametric confidence interval. It is also not recommended for use when
   data exhibit a significant trend. In that case, the estimate of variability will likely be too high, leading
   to an unnecessarily wide interval and possibly little chance of deciding the hypothesis. When a trend
   is present, consider instead a confidence interval around a trend line.

Steps involved: 1) Compute the  sample log-mean and log-standard deviation; 2) based on the sample
   size and choice of confidence level  (1-oc), calculate either the lower confidence limit (for use in
   compliance monitoring) or the upper confidence limit (for use in corrective action) using the logged
   measurements and exponentiate the result; 3) compare the confidence limit against the  GWPS or
   clean-up standard;  and 4)  if the lower confidence limit exceeds the GWPS in compliance monitoring
   or the upper confidence  limit is below the clean-up standard, conclude that the  null hypothesis
   should be rejected.

Advantages/Disadvantages: Use of a confidence interval instead of simply the sample geometric mean
   for comparison to a fixed standard accounts for both statistical variation in the data and the targeted
   confidence level.  The  same basic test  can  be used both to document contamination  above the
   compliance  standard in compliance/assessment and to  show a sufficient decrease in concentration
   levels below the clean-up  standard in corrective action.

       CONFIDENCE INTERVAL ON LOGNORMAL ARITHMETIC MEAN (SECTION 21.1.3)
Basic purpose: Test for compliance/assessment monitoring or corrective action. This  is a method by
   Land (1971) used to estimate  the range of concentration values from sample data, in which the true
   arithmetic mean of a lognormal population is expected to occur at a certain probability.

Hypothesis tested: In  compliance monitoring, HO — True mean concentration at the compliance point is
   no greater than the fixed compliance or groundwater protection standard [GWPS]. HA — True mean
   concentration is greater than the GWPS.  In corrective action, HO — True mean concentration  at the
   compliance  point  is  greater than the fixed  compliance or clean-up  standard. HA — True  mean
   concentration is less than or equal to the fixed standard.

                                             8^40                                  March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Underlying assumptions:  1) Compliance point data  are  approximately  lognormal in distribution.
   Adjustments for small to moderate fractions of non-detects, perhaps using Kaplan-Meier or ROS, are
   encouraged; 2) data do not exhibit any significant trend over time; 3)  there are a minimum of 4
   observations. Generally, at least 8 to 10 measurements are strongly recommended; and 4) the fixed
   GWPS  is  assumed  to  represent  the true  arithmetic  mean  average  concentration,  rather than a
   maximum or upper percentile.

When to use: Land's confidence interval procedure can be used for lognormally-distributed data to
   determine whether there is statistically significant evidence that the average is either above a fixed
   numerical standard (in compliance monitoring) or below a fixed standard (in corrective action). In
   either case, the null hypothesis is rejected only when the entire confidence interval is to one side of
   the compliance or clean-up standard. Because of this fact, the key question in compliance monitoring
   is whether the lower confidence limit exceeds the GWPS, while in corrective action the user must
   determine  whether the upper  confidence  limit is  below  the  clean-up  standard.   Because the
   lognormal distribution can have a highly skewed upper tail, this approach should only be used when
   the data fit the lognormal model rather closely, especially if used in corrective action. Consider
   instead  a confidence interval around the lognormal geometric mean or a non-parametric confidence
   interval if this is not the case. It is also not recommended for data that exhibit a significant trend. In
   that situation,  the  estimate of variability will likely  be too high, leading to an unnecessarily wide
   interval and possibly little chance of deciding the hypothesis. When a trend is  present,  consider
   instead  a confidence interval around a trend line.

Steps involved: 1) Compute the sample log-mean and log-standard deviation; 2) based on the sample
   size, magnitude of the log-standard deviation and choice of confidence level (1-a), determine Land's
   adjustment  factor;  3)  then calculate either the lower  confidence limit (for use in compliance
   monitoring) or the upper confidence limit (for use in corrective action);  4) compare the confidence
   limit against the GWPS  or clean-up standard; and 5) if the lower confidence limit exceeds the GWPS
   in compliance montoring or the upper confidence limit  is below the clean-up standard, conclude that
   the null hypothesis should be rejected.

Advantages/Disadvantages:  Use of a  confidence interval instead  of simply the  sample mean for
   comparison to a fixed standard accounts for both statistical  variation in the data and the targeted
   confidence level.  The same  basic test can be  used both to document contamination above the
   compliance standard in compliance/assessment and to show a sufficient decrease in concentration
   levels below the clean-up standard  in  corrective action. Since the upper confidence limit on a
   lognormal mean can be extremely high for some populations, the user may need to consider a non-
   parametric upper confidence limit on the median concentration as an alternative  or use a program
   such as  Pro-UCL to determine an alternate upper confidence limit.

       CONFIDENCE INTERVAL ON UPPER PERCENTILE (SECTION 21.1.4)
Basic purpose: Method for compliance monitoring. It is  used to estimate  the range of concentration
   values from sample data in which a pre-specified true proportion of a normal population is expected
   to occur at a certain probability.  The test can also be used to identify the range of a true proportion
   or percentile (e.g., the 95th) in population data which can be normalized.
                                             8-41                                   March 2009

-------
Chapter 8.  Methods Summary	Unified Guidance

Hypothesis tested: HO — True upper percentile concentration at the compliance point is no greater than
   the fixed  compliance or groundwater protection standard [GWPS]. HA — True upper percentile
   concentration is greater than the fixed GWPS.

Underlying  assumptions:  1)  Compliance  point  data  are  either  normal in distribution or can be
   normalized. Adjustments for small to moderate fractions of non-detects, perhaps using Kaplan-Meier
   or ROS, are encouraged;  2) data do not exhibit any significant trend over  time; 3) there  are a
   minimum of at least 8  to  10 measurements; and 4) the fixed GWPS is assumed to represent a
   maximum or upper percentile, rather than an average concentration.

When to use:  A confidence interval around an upper percentile can be used to determine whether there
   is statistically significant evidence that the percentile is above a fixed numerical standard.  The null
   hypothesis is rejected only when the entire confidence interval  is greater than the compliance
   standard. Because of this fact, the  key  question in compliance monitoring is whether the lower
   confidence limit exceeds the GWPS. This  approach is  not recommended for use when  the data
   exhibit a significant trend. The  estimate of variability will likely be  too  high, leading to an
   unnecessarily wide interval and possibly little chance of deciding the hypothesis.

Steps involved: 1) Compute the sample mean and standard deviation; 2) based on the sample size, pre-
   determined true proportion  and test confidence level  (1-a),  calculate the lower confidence limit; 3)
   compare the confidence limit against the GWPS; and 4) if the lower confidence limit exceeds the
   GWPS, conclude that the true upper percentile is larger than the compliance standard.

Advantages/Disadvantages: If a fixed GWPS is intended to represent a 'not-to-be-exceeded' maximum
   or an upper percentile, statistical comparison requires the prior definition of a true or expected upper
   percentile against which sample data can be compared. Some standards may explicitly identify the
   expected percentile.  The appropriate test then must estimate the confidence interval in which this
   true proportion is expected to lie.    Either an upper or  lower confidence limit can be generated,
   depending on whether compliance or corrective action hypothesis testing is appropriate. Whatever
   the interpretation of a given limit used as a GWPS, it should be determined in advance what a  given
   standard represents before choosing which type of confidence interval to construct.

       N ON-PARAMETRIC CONFIDENCE INTERVAL ON MEDIAN (SECTION  21.2)
Basic purpose: Test for compliance/assessment monitoring or corrective action.  It is a non-parametric
   method used to estimate the range of concentration values from sample data in  which the true
   median of a population is expected to occur at a certain probability.

Hypothesis tested: In compliance monitoring, HO — True median concentration at the compliance point
   is no greater than the fixed compliance  or groundwater  protection standard [GWPS].  HA — True
   median concentration  is  greater than  the  GWPS.  In  corrective action, HO — True  median
   concentration at the compliance point is greater than the fixed compliance or clean-up standard. HA
   — True median concentration is less than or equal to the fixed standard.

Underlying  assumptions: 1) Compliance data need not  be normal  in distribution; up to 50% non-
   detects are acceptable; 2) data do not exhibit any significant trend over time; 3) there are a minimum
   of at least 7 measurements; and 4) the fixed GWPS is assumed to represent a true median average
   concentration, rather than a maximum or upper percentile.
                                             8^42                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

When to use:  A confidence interval on the median can be used for non-normal data (e.g., samples with
   non-detects)  to determine whether there is statistically significant evidence that the average (i.e.,
   median) is either above a fixed numerical standard  (in compliance monitoring) or below a fixed
   standard (in  corrective action). In either case, the null hypothesis is rejected only when the  entire
   confidence interval is to one side of the compliance  or clean-up  standard.  Because of this fact, the
   key question in compliance monitoring is whether the lower confidence limit exceeds the GWPS,
   while in corrective  action the user must determine whether the upper confidence limit is below the
   clean-up standard.  This approach is not recommended for use when data exhibit a significant trend.
   In that case, the variation in the data will likely be too high, leading to an unnecessarily wide interval
   and possibly little chance of deciding the hypothesis.  It is also possible that the apparent trend is an
   artifact of differing detection  or reporting  limits that have changed over time.  The trend may
   disappear if  all non-detects are imputed at a common value or RL. If a trend is still present after
   investigating this possibility, but a significant portion of the data are non-detect,  consultation with a
   professional statistician is recommended.

Steps involved: 1)  Order and rank the data values; 2)  pick tentative interval endpoints close to the
   estimated median concentration; 3) using the selected endpoints, compute the achieved confidence
   level of the lower confidence limit for use in compliance monitoring or that of the upper confidence
   limit  for corrective action;  4) iteratively expand the interval until either the  selected endpoints
   achieve the targeted confidence level  or  the maximum or minimum  data value is chosen as the
   confidence limit; and 5) compare the confidence limit against the GWPS or clean-up standard. If the
   lower confidence limit exceeds the GWPS in compliance monitoring or the upper confidence limit is
   below the clean-up standard, conclude that the null hypothesis should be rejected.

Advantages/Disadvantages: Use of a confidence  interval instead  of simply the  sample median for
   comparison to a fixed  limit accounts  for both statistical variation in the  data and the targeted
   confidence level. The same  basic test can be used both to document contamination  above the
   compliance standard in compliance/assessment and to show a sufficient decrease in concentration
   levels below the clean-up standard in corrective action. By not requiring normal  or normalized data,
   the non-parametric confidence interval  can accommodate a substantial fraction of non-detects.  A
   minor disadvantage is that a non-parametric confidence interval estimates the location of the median,
   instead of the mean. For symmetric populations, these quantities will be the same, but for skewed
   distributions  they will differ. So if the compliance or clean-up standard is designed to represent a
   mean concentration, the non-parametric interval around the median may not provide a completely
   fair and/or accurate comparison. In some cases, the non-parametric confidence limit will not achieve
   the desired confidence level even if set to the maximum or minimum data value, leading to a higher
   risk of false positive error.

       N ON-PARAMETRIC CONFIDENCE INTERVAL ON UPPER PERCENTILE (SECTION  21.2)
Basic purpose: Non-parametric method for compliance  monitoring.  It is used to estimate the range of
   concentration values from sample data in which a pre-specified  true proportion of a population is
   expected to occur at a certain probability. Exact probabilities will depend upon sample data ranks.

Hypothesis tested: HQ  — True upper percentile concentration at the compliance point is no greater than
   the fixed compliance or groundwater  protection standard [GWPS]. HA — True  upper percentile
   concentration is greater than the GWPS.

                                              8^43                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Underlying assumptions: 1) Compliance point data need not be normal; large fractions of non-detects
   can be acceptable; 2) data do not exhibit any significant trend over time; 3) there are a minimum of
   at least 8 to 10 measurements; and 4) the fixed GWPS is assumed to represent a true upper percentile
   of the population, rather than an average concentration.

When to use: A confidence interval on an upper percentile can be used to determine whether there is
   statistically significant evidence that the percentile is above a fixed numerical standard. The null
   hypothesis is  rejected only when the entire confidence interval is greater than the compliance
   standard. Because of this fact, the key determinant in compliance/assessment monitoring is whether
   the lower confidence limit exceeds the GWPS. This approach is not recommended for use when data
   exhibit a significant trend. In that case, the estimate of variability will likely be too high, leading to
   an unnecessarily wide interval and possibly little chance of deciding the hypothesis.

Steps involved: 1)  Order and rank the data values; 2) select tentative  interval endpoints close to the
   estimated upper percentile concentration; 3)  using the  selected endpoints, compute the achieved
   confidence level  of the lower confidence limit; 4) iteratively  expand the interval  until either the
   selected lower endpoint achieves the targeted confidence level or the minimum data value is chosen
   as the  confidence limit; and 5) compare the  confidence limit against the GWPS. If the lower
   confidence limit exceeds the GWPS, conclude that the population upper percentile is larger than the
   compliance standard.

Advantages/Disadvantages: If a fixed GWPS is intended to represent a 'not-to-be-exceeded'  maximum
   or an upper percentile, statistical comparison requires the prior definition of a true or expected upper
   percentile against which sample data can be compared. Some standards may explicitly identify the
   expected percentile.  The appropriate test then must  estimate the confidence interval in which this
   true  proportion is expected to lie.   Either an  upper or lower confidence limit can be generated,
   depending  on whether compliance or corrective action hypothesis testing is appropriate.  Whatever
   the interpretation of a given limit used as a GWPS, it should be determined in advance what a given
   standard represents  before choosing which type of confidence interval  to construct.   However,
   precise non-parametric estimation of upper percentiles often requires much larger sample sizes than
   the parametric option (Section 21.1.4). For this reason, a, parametric confidence interval for upper
   percentile tests is  recommended whenever possible,  especially if a  suitable transformation can be
   found or adjustments made for non-detect values.

      CONFIDENCE BAND AROUND LINEAR REGRESSION (SECTION 21.3.1)
Basic purpose: Method for compliance/assessment monitoring or  corrective action when  stationarity
   cannot be assumed.  It is used to estimate ranges of concentration values  from sample data around
   each point  of a predicted linear regression line at a specified probability. The prediction line (based
   on regression of concentration values against time) represents the best estimate of gradually changing
   true mean levels over the time period.

Hypothesis tested: In compliance monitoring, HO — True mean concentration at the compliance point is
   no greater than the fixed compliance or groundwater protection standard [GWPS]. HA — True mean
   concentration is  greater than the GWPS. In corrective action, HO — True mean concentration at the
   compliance point  is greater  than the fixed compliance or clean-up standard. HA — True mean
   concentration is less than or equal to the fixed standard.

                                             8^44                                    March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

Underlying assumptions: 1) Compliance point values exhibit a linear trend with time, with normally
   distributed residuals. Use simple substitution with small (<10-15%) fractions of non-detects. Non-
   detect adjustment methods  are not recommended;  2)  there are a minimum of 4  observations.
   Generally,  at least 8 to 10 measurements are recommended; and 3) the fixed GWPS is assumed to
   represent an average concentration, rather than a maximum or upper percentile.

When to use:  A confidence interval around a trend line  should be used in cases where a linear trend is
   apparent on a time series  plot of the compliance point data. Even if observed well concentrations are
   either increasing under  compliance monitoring  or  decreasing  in corrective action,  it does  not
   necessarily imply that the true mean concentration at the current time is either above or below the
   fixed GWPS. While the trend line properly accounts for the fact that the mean is  changing with
   time, the null hypothesis is rejected only when the entire confidence interval is to  one side of the
   compliance or  clean-up  standard at the most recent point(s) in time. The key  determinant in
   compliance monitoring is whether the lower confidence limit at a specified point in time exceeds the
   GWPS, while in corrective action the upper confidence limit at a specific time must lie below the
   clean-up standard to be considered in compliance.

Steps involved:  1) Check for presence of a trend on a time series plot; 2) estimate the coefficients of the
   best-fitting linear regression line; 3) compute the  trend line residuals and check for normality; 4) if
   data are non-normal, try  re-computing the regression and residuals after transforming the  data; 5)
   compute the lower confidence limit band around the trend line for compliance monitoring or the
   upper confidence  limit band around the trend line for corrective action;  and 6) compare  the
   confidence limit  at each sampling event  against the GWPS  or clean-up standard. If the lower
   confidence limit exceeds the GWPS in compliance/assessment or the upper confidence limit is below
   the clean-up standard on one  or more  recent sampling  events, conclude  that the null hypothesis
   should be rejected.

Advantages/Disadvantages: Use of a confidence interval around the trend line instead of simply the
   regression line itself for comparison to a fixed standard accounts for both statistical  variation in the
   data  and the targeted confidence level. The same basic  test can  be used both to document
   contamination above the compliance standard in compliance/assessment and to  show a sufficient
   decrease in concentration levels below the clean-up standard in corrective action.  By estimating the
   trend line first and then using the residuals to construct the confidence interval, variation due to the
   trend itself is removed, providing a more powerful test (via a narrower interval) of whether or not the
   true mean is on one side of the fixed standard.  This technique can only be used when the identified
   trend is reasonably linear  and the trend residuals are approximately normal.

       N ON-PARAMETRIC CONFIDENCE BAND AROUND THEIL-SEN TREND (SECTION 21.3.1)
Basic  purpose:  Non-parametric method  for  compliance/assessment   or  corrective action  when
   stationarity cannot be assumed. It is used to estimate ranges of concentration values from sample
   data around each point of a predicted Theil-Sen trend line at a specified probability.  The prediction
   line represents the best estimate of gradually changing true median levels over the time period.

Hypothesis tested: In compliance monitoring, HQ — True mean concentration at the compliance point is
   no greater than the fixed compliance or groundwater protection standard [GWPS]. HA — True mean
   concentration is greater than the GWPS. In corrective action, HQ — True mean concentration at the

                                             8^45                                   March 2009

-------
Chapter 8. Methods Summary	Unified Guidance

   compliance point is  greater than the fixed compliance or  clean-up standard. HA — True  mean
   concentration is less than or equal to the fixed standard.

Underlying assumptions: 1) Compliance point values exhibit a linear trend with time; 2) non-normal
   data and substantial levels of non-detects up to 50% are acceptable; 3) there are a minimum of 8-10
   observations available to construct the confidence band; and 4) the fixed GWPS is assumed to
   represent a median average concentration, rather than a maximum or upper percentile.

When to use:  A confidence interval around a trend line should be used in cases where a linear trend is
   apparent on a time series plot of the compliance point data. Even if observed well concentrations are
   either  increasing under compliance  monitoring  or decreasing  in  corrective  action,  it does  not
   necessarily imply that the true mean  concentration at  the current time is either above or below the
   fixed GWPS.  While the trend line properly accounts for the fact that the mean is changing with
   time, the null  hypothesis is rejected only when the entire confidence interval is to one side of the
   compliance or clean-up standard  at the  most recent point(s)  in time. The  key determinant in
   compliance monitoring is whether the lower confidence limit at a specified point in time exceeds the
   GWPS, while  in corrective action the upper confidence limit at  a specific time must lie below the
   clean-up standard to be considered in compliance.

Steps involved: 1) Check for presence of a trend on a time series plot; 2) construct a Theil-Sen trend
   line; 3) use bootstrapping to create a large number of simulated Theil-Sen trends on the sample data;
   4) construct a confidence band by selecting lower and upper percentiles from the set of bootstrapped
   Theil-Sen trend estimates; and 5) compare the confidence band at each  sampling event against the
   GWPS  or clean-up   standard.   If  the   lower  confidence  band   exceeds   the  GWPS   in
   compliance/assessment or the upper confidence band is below the clean-up standard on one  or more
   recent sampling events, conclude that the null hypothesis should be rejected.

Advantages/Disadvantages: Use of a confidence band around the trend line instead of simply the Theil-
   Sen  trend line itself for comparison to a fixed standard accounts for both statistical variation in the
   data  and  the  targeted  confidence  level.   The  same  basic  test  can   be  used both   in
   compliance/assessment and in corrective action.  By estimating  the trend  line  first and then  using
   bootstrapping  to  construct the confidence band, variation due to the trend itself is removed,
   providing a more powerful  test (via a narrower interval) of whether or not the  true mean is on  one
   side of the fixed standard.  This technique  can only be used when the identified trend is reasonably
   linear.  The Theil-Sen trend estimates the change in median level rather than the mean. For roughly
   symmetric  populations, this will make little difference; for highly skewed populations, the  trend in
   the median may not accurately reflect  changes in mean concentration levels.
                                             8-46                                   March 2009

-------
PART II. DIAGNOSTIC METHODS	Unified Guidance
     PART II:  DIAGNOSTIC  METHODS AND
                                 TESTING
      Part II covers diagnostic evaluations of historical facility data for checking key assumptions
implicit in the recommended statistical tests and for making appropriate adjustments to
the data (e.g., consideration of outliers, seasonal autocorrelation, or non-detects). Also included is a
discussion of groundwater sampling and how hydrologic factors such as flow and gradient can
impact the sampling program.

      Chapter 9 provides a number of exploratory data tools and examples, which can generally be
used in data evaluations. Approaches for fitting data sets to normal and other parametric distributions
follows in Chapter 10. The importance of the normal distribution and its potential uses is also
discussed. Chapter 11 provides methods for assessing the equality of variance necessary for some
formal testing. The subject of outliers and means of testing for them is covered in Chapter 12.
Chapter 13 addresses spatial variability, with particular emphasis on ANOVA means testing. In
Chapter 14, a number of topics concerning temporal variation are provided. In addition to providing
tests for identifying the presence of temporal variation, specific adjustments for certain types of temporal
dependence are covered. The final Chapter 15 of Part II discusses non-detect data and offers several
methods for estimating missing data. In particular, methods are provided to deal with data containing
multiple non-detection limits.
                                                                          March 2009

-------
PART II. DIAGNOSTIC METHODS                                  Unified Guidance
                   This page intentionally left blank
                                                                    March 2009

-------
Chapter 9. Exploratory Tools                                              Unified Guidance
          CHAPTER  9.   COMMON  EXPLORATORY TOOLS

       9.1   TIME SERIES PLOTS	9-1
       9.2   Box PLOTS	9-5
       9.3   HISTOGRAMS	9-8
       9.4   SCATTER PLOTS	9-13
       9.5   PROBABILITY PLOTS	9-16
     Graphs are an important tool for exploring and understanding patterns in any data set.  Plotting the
data visually depicts the structure and helps unmask possible relationships between variables affecting
the data set. Data plots which accompany quantitative statistical tests can better demonstrate the reasons
for the results of a formal test. For example, a Shapiro-Wilk test may conclude that data are not normally
distributed. A probability plot or histogram of the data can confirm this conclusion graphically to show
why the data are not normally distributed (e.g., heavy skewness, bimodality, a single outlier, etc.).

     Several  common exploratory  tools are presented in Chapter 9. These graphical techniques are
discussed in statistical texts,  but are presented here in detail for easy reference for the data analyst. An
example data set is used to demonstrate how each of the following plots is created.

    »«»  Time series plots (Section 9.1)
    *  Box plots (Section 9.2)
    *»*  Histograms (Section 9.3)
    *  Scatter plots (Section 9.4)
    *  Probability plots (Section 9.5)

9.1 TIME SERIES PLOTS

     Data  collected over specific time intervals (e.g., monthly, biweekly, or hourly) have a temporal
component. For example, air monitoring measurements of a pollutant may be collected once a minute or
once a day. Water quality monitoring measurements may be collected weekly  or monthly.  Typically,
groundwater sample data are collected quarterly from the same monitoring wells, either for detection
monitoring testing or demonstrating compliance to a  GWPS. An analyst examining temporal data may
be  interested in the  trends  over time, correlation  among time  periods,  or  cyclical patterns.  Some
graphical techniques specific to temporal data are  the time plot, lag plot,  correlogram, and variogram.
The degree to which some of these techniques can be used will depend in part on the frequency and
number of data collected over time.

     A data sequence collected at regular time intervals is called a time series. More sophisticated time
series data  analyses are beyond  the scope of this guidance. If needed, the interested user should consult
with a statistician or appropriate statistical texts. The graphical representations presented in this section
are recommended for any data set that includes a temporal component. Techniques described below will
help identify temporal patterns that need to be accounted for in any analysis  of the data. The analyst
examining  temporal environmental  data may be interested in seasonal trends,  directional trends, serial
correlation, or stationarity. Seasonal trends are patterns in the data that repeat over time, i.e., the data

                                             9^1                                   March 2009

-------
Chapter 9. Exploratory Tools	Unified Guidance

rise and fall regularly over one or more time periods. Seasonal trends may occur over long periods of
time (large scale), such as a yearly cycle where the data show the same pattern of rising and falling from
year to year, or the trends may be over a relatively short period of time (small scale), such as a daily
cycle. Examples of seasonal trends  are quarterly seasons (winter, spring, summer and fall),  monthly
seasons, or even hourly (e.g., air temperature rising and falling over the course of a  day). Directional
trends are increasing or decreasing patterns over time in monitored constituent data,  which may be of
importance in assessing the levels of contaminants. Serial correlation is a measure of the strength in the
linear relationship of successive observations. If successive observations are related, statistical quantities
calculated without accounting for the serial correlation may be biased. A time series is stationary if there
is no systematic change in the mean (i.e., no trend) and variance across time. Stationary data look the
same over all time periods except for random behavior. Directional trends or a change in the variability
in the data imply non-stationarity.

       A time series plot of concentration data versus time makes it easy to identify lack of randomness,
changes in location, change  in scale, small scale trends, or large-scale trends over time.  Small-scale
trends are displayed as fluctuations over smaller time periods. For example, ozone levels over the course
of one day typically rise until the afternoon, then decrease, and this process is repeated  every day. Larger
scale trends such as seasonal fluctuations  appear as regular rises and drops in the graph.  Ozone levels
tend to be higher in the summer than in the winter, so ozone data tend to show both a daily trend and a
seasonal trend. A time plot can also show directional trends or changing variability over time.

       A time plot is constructed by plotting the measurements on the vertical axis  versus the actual
time of observation or the  order of observation on the  horizontal  axis. The points plotted may be
connected by lines, but this may create an unfounded sense of continuity. It is important to use the actual
date, time or number at which the observation was made. This can create discontinuities in the plot but
are needed as the data that should have been collected now appear as "missing values" but do not disturb
the integrity of the plot. Plotting the data at equally spaced intervals when in reality there were  different
time periods between observations is  not advised.

       For environmental data, it is also important to use a different symbol or color to distinguish non-
detects from detected data. Non-detects are often reported by the analytical laboratory with  a "U" or "<"
analytical  qualifier  associated with the reporting limit  [RL]. In statistical terminology, they  are left-
censored data, meaning the actual concentration of the chemical is known only to be below the RL. Non-
detects contrast with detected data, where the laboratory reports the result as a known concentration that
is statistically  higher than the analytical limit of detection.  For example, the laboratory may report a
trichloroethene concentration in groundwater of "5 U" or "< 5" |ig/L, meaning the actual trichloroethene
concentration is unknown, but is bounded between zero and 5 |ig/L. This result is different  than a
detected concentration of 5 jig/L which is unqualified by the laboratory or data validator. Non-detects
are handled differently than detected data when  calculating summary statistics. A statistician should be
consulted  on the proper use  of non-detects in statistical analysis. For  radionuclides negative and zero
concentrations should be plotted as reported by the laboratory, showing the detection status.

       The scaling of the vertical axis of a time plot is of some  importance. A wider scale tends to
emphasize large-scale trends, whereas a narrower scale  tends to emphasize small-scale trends. A wide
scale would emphasize the  seasonal component of the data, whereas a  smaller scale would tend to
                                              9-2                                    March 2009

-------
Chapter 9.  Exploratory Tools
Unified Guidance
emphasize the daily fluctuations.  The scale needs to contain the full range of the data.  Directions for
constructing a time plot are contained in Example 9-1 and Figure 9-1.

     ^EXAMPLE 9-1

     Construct a time series plot using trichloroethene groundwater data in Table 9-1  for each well.
Examine the time series for seasonality, directional trends and stationarity.

              Table 9-1. Trichloroethene  (TCE) Groundwater Concentrations
Date
Collected
1/2/2005
4/7/2005
7/13/2005
10/24/2005
1/7/2006
3/30/2006
6/28/2006
10/2/2006
10/17/2006
1/15/2007
4/10/2007
7/9/2007
10/5/2007
10/29/2007
12/30/2007

Welll
TCE Data
(mg/L) Qualifier
0.
0.
0.
0.
0.
0.
0.
0.
0
0
0
0
005 U
005 U
004 J
006
004 U
009
017
045
.05
.07
.12
.10
NA
0
0
.20
.25
Well 2
TCE Data
(mg/L) Qualifier
0.10 U
0.12
0.125
0.107
0.099 U
0.11
0.13
0.109
NA
0.10 U
0.115
0.14
0.17
NA
0.11
                   NA = Not available (missing data).
                   U denotes a non-detect.
                   J denotes an estimated detected concentration.

       SOLUTION
Step 1.   Import the data into data analysis software capable of producing graphics.

Step 2.   Sort the data by date collected.

Step 3.   Determine the range of the data by calculating the minimum and maximum concentrations for
         each well, shown in the table below:
                                            9-3
        March 2009

-------
Chapter 9. Exploratory Tools
Unified Guidance

Welll
TCE
(mg/L)
Min 0.004
Max 0.25
Data
Qualifier
Well 2
TCE
(mg/L)
U 0.099
0.17
Data
Qualifier
U
Step 4.   Plot the data using a scale from 0 to 0.25 if data from both wells are plotted together on the
         same time series plot. Use separate symbols for non-detects and detected concentrations.  One
         suggestion is to use "open"  symbols (whose centers are white) for non-detects and "closed"
         symbols for detects.

Step 5.   Examine each  series  for directional trends, seasonality  and stationarity.  Note that  Well 1
         demonstrates a positive directional trend across time, while Well 2 shows seasonality within
         each year. Neither well exhibits stationarity.

Step 6.   Examine each  series  for missing values. Inquire from the project laboratory why data are
         missing or collected at unequal time intervals. A response from the laboratory for this  data set
         noted that on 10/5/2007 the sample was accidentally broken in the laboratory from Well 1, so
         Well 1 was resampled on 10/29/2007. Well 1 was  resampled on 10/17/2006  to confirm the
         historically high concentration collected on 10/2/2006. Well 2 was not sampled on 10/17/2006
         because the  data collected on 10/2/2006 from Well 2 did not merit a resample, as did Well 1.

Step 7.   Examine each series for elevated detection limits. Inquire why the detection limits for Well 2
         are much larger than detection limits for Well 1. A reason may be that different laboratories
         analyzed the samples from the two wells. The laboratory analyzing samples from Well 1 used
         lower detection limits than did the laboratory analyzing samples from Well 2. -^
                                              9-4
        March 2009

-------
Chapter 9. Exploratory Tools
                                                                  Unified Guidance
        Figure 9-1. Time Series Plot of Trichloroethene Groundwater for Wells 1 and
                                         2 from 2005-2007.
           (U

           I
           "5
           o
           H
                 0.25-
                 0.20-
                 0.15 -
                 0.10-
                 0.05 -
• Well 1
• Well 2
                                         -o-
                     Jan
                     2005
                     Jul       Jan       Jul
                              2006

Open symbols denote non-detects. Closed symbols denote detected concentrations.
 Jan
2007
Jul
 Jan
2008
9.2  BOX PLOTS

      Box plots (also known as Box and Whisker plots) are useful in situations where a picture of the
distribution is desired, but it is not necessary or feasible to portray all the details of the data. A box plot
displays several percentiles of the data set. It is  a simple plot, yet provides insight into the location,
shape, and spread of the data and underlying distribution. A  simple box plot contains only the 0th
(minimum data value), 25 , 50l , 75  and  1001 (maximum data value) percentiles. A box-plot divides
the data into 4 sections, each containing 25% of the data. Whiskers are the lines drawn to the minimum
and maximum data values  from the 25   and 75l percentiles.  The box shows the interquartile range
(IQR) which is defined as the difference between the 75th and the 25th  percentiles. The length of the
central box indicates the spread of the data (the central 50%), while the length of the whiskers shows the
breadth of the tails of the distribution.  The 50th percentile (median) is the line  within the box. In
addition, the mean and the 95% confidence  limits around the  mean are shown. Potential outliers are
categorized into two groups:

         »»» data points between 1.5 and 3 times the IQR above the 75th percentile or between 1.5 and 3
                                     th
            times the IQR below the 25  percentile, and
            data points that exceed 3 times the IQR above the 75™ percentile or exceed 3 times the IQR
                                                 th
                        th
            below the 25  percentile.
                                             9-5
                                                                          March 2009

-------
Chapter 9. Exploratory Tools	Unified Guidance

       The mean is shown as a star, while the lower and upper 95% confidence limits around the mean
are shown as bars. Individual data points between 1.5 and 3 times the IQR above the 75th percentile or
below the 25l  percentile are shown as circles. Individual data points at least 3 times the IQR above the
75th percentile  or below the 25th percentile are shown as squares.

       Information from box plots can assist in identifying potential data distributions. If the upper box
and whisker are approximately the same length as the lower box and whisker, with the mean and median
approximately  equal, then the data are distributed symmetrically.  The normal  distribution  is one of a
number that is  symmetric. If the upper box and whisker are longer than the lower box and whisker, with
the mean  greater than the median, then the data are right-skewed (such  as lognormal or square root
normal  distributions in original units). Conversely, if the upper box and  whisker are shorter than the
lower box and  whisker with the mean less than the median, then the data are left-skewed.

       A box  plot showing a normal distribution will have the following characteristics: the mean and
median will be in the center of the box, whiskers to the minimum and maximum values are the same
length, and there would be no potential outliers. A box plot showing a lognormal distribution (in original
units) typical of environmental applications will have the following characteristics: the mean will  be
larger than the median, the whisker above the 75l percentile will be longer than the whisker below the
25th percentile, and extreme upper values may be indicated as potential outliers. Once the data have been
logarithmically transformed, the pattern  should follow that described for a normal distribution.  Other
right-skewed distributions transformable to normality would indicate similar patterns.

       It is often helpful to show box plots of different sets of data  side by side to show  differences
between monitoring stations (see Figure 9-2).  This allows a simple method to compare the locations,
spreads and shapes of several data sets or different groups within a single  data set. In this situation, the
width of the box can be proportional to the sample size of each data set. If the data will be compared to a
standard, such  as a preliminary remediation goal (PRO) or maximum contaminant level (MCL), a line on
the graph can be drawn to show if any results exceed the criteria.

       It is important to plot the  data as reported by the laboratory  for non-detects  or negative
radionuclide data. Proxy values for non-detects should  not  be plotted  since  we want  to  see the
distribution of the original data. Different symbols can be used to display non-detects, such as the open
symbols described in Section 9.1. The mean will be biased high if using  the RL of non-detects  in the
calculation,  but the purpose of the box plot is  to assess the  distribution of the data, not quantifying a
precise estimate of an unbiased mean. Displaying the frequency of detection (number of detected values /
number of total samples) under the station name is also useful. Unlike time series plots, box plots cannot
use missing data, so missing data should be removed before producing a box plot.

       Directions for generating a box plot are contained in Example 9-2, and an example is shown in
Figure 9-2. It  is important to remove lab and field duplicates from the data before calculating summary
statistics such  as the  mean and UCL  since these statistics assume  independent data. The box plot
assumes the data are statistically independent.
                                              9-6                                    March 2009

-------
Chapter 9. Exploratory Tools	Unified Guidance

     ^EXAMPLE 9-2

     Construct a box plot using the trichloroethene groundwater data in Table 9-1 for each  well.
Examine the box plot to assess how each well is distributed (normal, lognormal, skewed, symmetric,
etc.). Identify possible outliers.

     SOLUTION

Step 1.   Import the data into data analysis software capable of producing box plots.

Step 2.   Sort the data from smallest to largest results by well.

Step3.   Compute the  0th (minimum value), 25th, 50th (median), 75th and 100th  (maximum value)
         percentiles by well.

Step 4.   Plot these points vertically. Draw a box around the 25th and 75th percentiles and add a line
         through the box at the 50l  percentile. Optionally, make the width of the box proportional to
         the sample size. Narrow boxes reflect smaller sample sizes, while wider boxes reflect larger
         sample sizes.

Step 5.   Compute the mean and the lower and upper 95% confidence limits. Denote the mean with a
         star  and  the confidence limits as bars. Also, identify potential  outliers between l.SxIQR and
         3>
-------
Chapter 9. Exploratory Tools
             Unified Guidance
           Figure 9-2.  Box Plots of Trichloroethene Data for Wells 1 & 2

                0.25 -              =
                      PRO = 0.23 mg/L
           1)
           "u
           o
           o
          H
                0.20-
                0.15-
                0.10-
                0.05-
                o.oo-
                                               outlier >3xIQR
                                               outlier> l.SxIQR
                                               mean
                                               95% LCL and UCL
                                    Welll
                                   FOD 11/14
 Well 2
FOD 10/13
9.3  HISTOGRAMS

      A histogram is a visual representation of the data collected into groups. This graphical technique
provides a visual method of identifying the underlying distribution of the data. The data range is divided
into several bins or classes and the data is sorted into the bins. A histogram is a bar graph conveying the
bins and the frequency of data points in  each bin. Other forms of the histogram use a normalization of
the bin frequencies for the heights of the bars.  The two  most common normalizations  are relative
frequencies (frequencies  divided by sample  size) and densities (relative frequency  divided by the bin
width). Figure 9-3 is an example of a histogram using frequencies and Figure 9-4 is a histogram of
densities. Histograms provide a visual method of accessing location, shape and spread of the data. Also,
extreme values and multiple modes can be identified.  The details  of the data  are lost, but an overall
picture of the data is obtained. A stem and leaf plot offers the same insights into the data as a histogram,
but the data values are retained.

      The visual impression of a histogram is sensitive to the number of bins selected. A large number of
bins will increase data detail, while fewer bins will increase the smoothness of the  histogram. A good
starting point when choosing the number of bins is the  square root of the sample size n. The minimum
number of bins for any histogram should be at least 4. Another factor in choosing bins is the choice of
endpoints. When feasible,  using  simple bin endpoints can improve the readability of the histogram.
Simple bin endpoints include multiples of 5k units for some integer k > 0 (e.g., 0 to <5,  5 to <10, etc. or
1  to  <1.5,  1.5  to <2, etc.). Finally, when plotting  a histogram for a  continuous variable (e.g.,
                                              9-8
                     March 2009

-------
Chapter 9. Exploratory Tools	Unified Guidance

concentration), it is necessary to decide on an endpoint convention; that is, what to do with data points
that fall on the boundary of a bin. Also, use the data as reported by the laboratory for non-detects and
eliminate any missing values, since histograms cannot include  missing data.  With discrete variables,
(e.g., family size) the intervals can be centered in between the variables. For the family size data, the
intervals can span between 1.5 and 2.5, 2.5 and 3.5, and so on. Then the whole numbers that relate to the
family size can  be centered within the box. Directions for generating a histogram are contained in
Example 9-3

     ^EXAMPLE 9-3

     Construct a histogram using the  trichloroethene groundwater data in Table 9-1  for each well.
Examine the histogram to assess how each well is distributed (normal, lognormal, skewed, symmetric,
etc.).

     SOLUTION

Step 1.   Import the data into data analysis software capable of producing histograms.

Step 2.   Sort the data from smallest to  largest results by well.

Step 3.   With n = 14 concentrations for Well 1, a rough estimate of the number of bins isvl4  = 3.74
         or 4 bins. Since the data from Well 1 range from 0.004 to 0.25, the suggested bin width is
         calculated as (maximum concentration - minimum concentration) / number of bins = (0.25 -
         0.004) /  4 = 0.0615. Therefore, the bins for Well  1  are 0.004 to <0.0655, 0.0655 to <0.127,
         0.127 to  <0.1885, and 0.1885  to 0.25 mg/L.

         Similarly, with n = 13 concentrations for Well 2, the number of bins isv!3 =  3.61 or 4 bins.
         Since the data from Well 2 range from 0.099 to 0.17, the suggested bin width is calculated as
         (maximum concentration - minimum concentration) / number of bins  = (0.17 - 0.099) 74 =
         0.01775. Therefore, the bins for Well 2 are 0.099 to <0.11675, 0.11675 to O.1345, 0.1345 to
         O.15225, and 0.15225 to 0.17 mg/L.

Step 4.   Construct a frequency table using the bins defined in Step 3. Table 9-2 shows the frequency or
         number of observations within each bin defined in  Step 3 for Wells 1 and 2. The third column
         shows the relative frequency  which is the  frequency divided by the  sample size n. The  final
         column of the table gives the densities or the relative frequencies divided by  the bin widths
         calculated in Step 3.

Step 5.   The horizontal axis for the data is from 0.004 to 0.25 mg/L for Well 1 and 0.099 to 0.17 for
         Well 2. The vertical axis for the histogram of frequencies is from  0 to 9  and the  vertical axis
         for the histogram of relative frequencies is from 0% - 70%.

Step 6.   The histograms  of frequencies  are shown in Figure 9-3. The  histograms  of  relative
         frequencies or densities are shown in Figure 9-4. Note that frequency,  relative frequency and
         density histograms all show the same shape since the  scale of the vertical axis is divided  by
                                              9-9                                    March 2009

-------
Chapter 9. Exploratory Tools	Unified Guidance

        the sample size  or the bin width. These histograms confirm the  data are not  normally
        distributed for either well, but are closer to lognormal.
             Table 9-2. Histogram Bins for Trichloroethene Groundwater Data

                                                     Relative
             	Bin	Frequency    Frequency (%)	Density
                                            Well 1
0.0040 to O.0655 mg/L
0.0655 to <0. 1270 mg/L
0. 1270 to <0. 1885 mg/L
0.1 885 to 0.2500 mg/L
9
3
0
2
64.3
21.4
0
14.3
10.5
3.5
0
2.3
                                            Well 2
            0.099 to <0.11675 mg/L         8             61.5               34.7
            0.11675 to <0.1345 mg/L        3             23.1               13.0
            0.1345 to <0.15225 mg/L        1              7.7               4.3
            0.15225 to 0.17 mg/L	1	11	4.3
                                         9-10                                 March 2009

-------
Chapter 9.  Exploratory Tools
                                                           Unified Guidance
               Figure 9-3. Frequency Histograms of Trichloroethene by Well.
          §
9-

8-

7-

6-

5-

4-
                9-
                8-

                7-
                6-
                5 -

                4
                3 -

                1-
                1 -

                o-
                     0.004
                  0.0655        0.127        0.1885
                     Trichloroethene (mg/L) in Well 1
0.25
                       0.099        0.11675       0.1345       0.15225
                                     Trichloroethene (mg/L) in Well 2
                                                          0.17
                                           9-11
                                                                   March 2009

-------
Chapter 9.  Exploratory Tools
                                       Unified Guidance
          Figure 9-4.  Relative Frequency Histograms of Trichloroethene by Well.
                70-

                60-

           ^   50-
           S
           §   40-
           aT
           ^   30-

                20-

                10-
                       0.004
0.0655        0.127        0.1885
   Trichloroethene (mg/L) in Well 1
0.25
               70-
               60-
               50-
           o
           S   40
           is
               30-
           Pi
               20-
               10-
                o-
                       0.099       0.11675       0.1345       0.15225
                                     Trichloroethene (mg/L) in Well 2
                                      0.17
                                           9-12
                                              March 2009

-------
Chapter 9. Exploratory Tools
Unified Guidance
9.4  SCATTER PLOTS

      For data sets consisting of multiple observations per sampling point, a scatter plot is one of the
most powerful graphical tools for analyzing the relationship between two or more variables. Scatter plots
are easy to construct for two variables, and many software packages can construct 3-dimensional scatter
plots. A scatter plot  can clearly show the  relationship between two variables if the data range is
sufficiently large. Truly linear relationships can always be identified in scatter plots, but truly nonlinear
relationships may appear linear (or some other form) if the data range is relatively small. Scatter plots of
linearly correlated variables cluster about a straight line.

      As an  example of a nonlinear relationship,  consider two  variables  where one variable is
approximately equal to the square of the other. With an adequate range in the data, a scatter plot of this
data would display a partial parabolic curve. Other important modeling relationships that may appear are
exponential or logarithmic. Two additional uses of scatter plots are the identification of potential outliers
for a single variable or for the paired variables and the identification of clustering in the data. Directions
for generating a scatter plot are contained in Example 9-4.

      ^EXAMPLE 9-4

     Construct a scatter plot using the groundwater data in Table 9-3 for arsenic and mercury from a
single well collected approximately quarterly across time. Examine the scatter plot for linear or quadratic
relationships between  arsenic and mercury, correlation, and for potential outliers.

                     Table 9-3. Groundwater Concentrations from Well 3
Date
Collected
1/2/2005
4/7/2005
7/13/2005
10/24/2005
1/7/2006
3/30/2006
6/28/2006
10/2/2006
10/17/2006
1/15/2007
4/10/2007
7/9/2007
10/5/2007
10/29/2007
12/30/2007
Arsenic Mercury
Cone. Data Cone. Data
(mg/L) Qualifier (mg/L) Qualifier
0.01 U 0.02 U
0.01 U 0.03
0.02 0.04 U
0.04 0.06
0.01 0.02
0.05 0.07
0.09 0.10
0.07 0.08
0.10 NA
0.02 U 0.03 U
0.15 0.11
0.12 0.08
0.10 0.07
0.30 0.29
0.25 0.23
Strontium
Cone.
(mg/L)
0.10
0.02
0.05
0.11
0.05
0.07
0.03
0.04
0.02
0.15
0.03
0.10
0.09
0.05
0.22
Data
Qualifier

U
U





U






         NA = Not available (missing data).
         U denotes a non-detect.
                                             9-13
        March 2009

-------
Chapter 9. Exploratory Tools
                                                          Unified Guidance
       SOLUTION
Step 1.   Import the data into data analysis software capable of producing scatter plots.

Step 2.   Sort the data by date collected.

Step 3.   Calculate the range of concentrations for each constituent. If the range of both constituents are
         similar, then scale both the X and Y axes from the minimum to the maximum concentrations
         of both constituents. If the range of concentrations are very different (e.g., two or more orders
         of magnitude), then perhaps the scales for both axes should be logarithmic (logic). The  data
         will be plotted as pairs from (Xi, YI) to (Xn, Y«) for each sampling date, where n = number of
         samples.

Step 4.   Use separate symbols to distinguish detected from non-detected concentrations. Note that the
         concentration for one constituent may  be detected, while the concentration for the other
         constituent  may  not be detected for the same sampling date. If the concentration for one
         constituent  is missing, then the pair (X;, Y;) cannot be plotted since both concentrations are
         required.  Figure  9-5  shows a linear correlation between arsenic and mercury  with  two
         possible outliers. The Pearson correlation coefficient is 0.97, indicating a significantly high
         correlation.  The  linear  regression line is displayed to show  the linear correlation between
         arsenic and  mercury. -4

                Figure 9-5. Scatter  Plot of Arsenic with Mercury from Well 3
        on
              0.30-
              0.25 -
              0.20-
              0.15 -
              0.10-
              0.05 -
• both detected
O both non-detects
O arsenic non-detect only
O mercury non-detect only
                              0.05        0.10        0.15       0.20
                                               Arsenic (mg/L)
                                                     0.25
0.30
                                              9-14
                                                                  March 2009

-------
Chapter 9.  Exploratory Tools	Unified Guidance

      Many software packages can extend the 2-dimensional scatter plot by constructing a 3-dimensional
scatter plot for 3  constituents. However, with more than 3 variables, it is difficult to construct and
interpret a  scatter plot. Therefore, several graphical representations have been developed that extend the
idea of a scatter plot for  data consisting of more than 2  variables.  The simplest of these graphical
techniques is a coded scatter plot.  All possible two-way combinations  are given a symbol and the pairs
of data are  plotted  on  one  2-dimensional scatter plot.  The  coded scatter plot does not  provide
information on three  way  or higher interactions between the variables since  only two dimensions are
plotted. If the data ranges for the variables are comparable,  then a single set of axes may suffice. If the
data ranges are too dissimilar (e.g., at least two orders of magnitude), different scales may be required.

      ^EXAMPLE 9-5

      Construct a coded scatter plot using the groundwater data in Table 9-3 for arsenic, mercury, and
strontium from Well 3 collected approximately quarterly across time. Examine the scatter plot for linear
or quadratic relationships between the three inorganics, correlation, and for potential  outliers.

      SOLUTION

Step 1.   Import the data into data analysis software capable of producing scatter plots.

Step 2.   Sort the data by date collected.

Step 3.   Calculate the range of concentrations for each constituent. If the  ranges of both constituents
         are  similar, then  scale  both  the X  and  Y  axes from the  minimum to  the maximum
         concentrations of all three constituents. Since the ranges of concentrations  are very similar, the
         minimum to the maximum concentrations of all three constituents will be used for both axes.

Step 4.   Let each arsenic concentration be denoted by X;, each mercury concentration  be denoted by
         Y;, and  each strontium concentration be denoted by Z;. The arsenic and mercury paired data
         will be plotted as pairs (X;, Y;) with solid blue circles for !
-------
Chapter 9. Exploratory Tools
                                      Unified Guidance
      Figure 9-6.  Coded Scatter Plot of Well 3 Arsenic, Mercury, and  Strontium
             0.30-
             0.25 -
             0.20-
             0.15 -
             0.10-
             0.05 -
                     Arsenic (X) vs. Mercury (Y)
                     Arsenic (X) vs. Strontium (Y)
                     Mercury (X) vs. Strontium (Y)
                            0.05
0.10
0.15
mg/L
0.20
0.25
0.30
9.5  PROBABILITY PLOTS

     A simple, but extremely useful visual assessment of normality is to graph the data as a probability
plot. The ^-axis is scaled to represent quantiles or z-scores from a standard normal distribution and the
concentration measurements are arranged in increasing order along the x-axis. As each observed value is
plotted on the x-axis, the z-score corresponding to the proportion of observations less than or equal to
that measurement is plotted as the _y-coordinate. Often, the _y-coordinate is computed by the following
formula:
                                                                                          [9.1]
where  <£> : denotes the inverse of the cumulative standard normal distribution, n represents the sample
size, and /' represents the rank position of the /'th ordered concentration. The plot is constructed so that, if
the data are normal, the points when plotted will lie on a straight line. Visually apparent curves or bends
indicate that the data do not follow a normal distribution.

     Probability plots are particularly useful for spotting irregularities within the data when compared to
a specific distributional  model (usually, but not always, the normal). It is easy to determine whether
departures from normality are occurring more or less in the middle ranges of the data or in the extreme
                                             9-16
                                              March 2009

-------
Chapter 9. Exploratory Tools	Unified Guidance

tails. Probability plots can also indicate the presence of possible outlier values that do not follow the
basic pattern of the data and can show the presence of significant positive or negative skewness.

     If a (normal) probability plot is constructed on the combined data from several wells and normality
is  accepted, it suggests — but  does not prove — that  all of the  data came from the same normal
distribution. Consequently, each subgroup of the  data  set (e.g.,  observations from distinct wells)
probably has the same mean and  standard deviation. If  a  probability  plot is constructed on the data
residuals (each value minus its  subgroup mean) and is not a straight  line, the interpretation  is more
complicated. In this case, either the residuals are not normally-distributed, or there is a subgroup of the
data with a normal distribution but a different mean or standard deviation than the other subgroups. The
probability plot will indicate  a deviation from the  underlying assumption of  a  common  normal
distribution in either case. It would be prudent to examine normal probability plots by well  on the same
plot if the ranges of the data are  similar.  This would show how the  data are distributed by well to
determine which wells may depart from normality.

     The  same probability plot technique may be used to investigate whether a set of data or residuals
follows a lognormal distribution. The procedure is generally the same, except that one first replaces each
observation by its natural logarithm. After the data have been transformed to their natural logarithms, the
probability plot  is constructed as before.  The only difference  is that the natural logarithms of the
observations  are used  on the x-axis. If the data are  lognormal, the  probability plot of the logged
observations will approximate a straight line.

     ^EXAMPLE 9-6

     Determine whether the dataset in Table 9-4 is normal by using a probability plot.

     SOLUTION

Step 1.   After combining the data into a single group, list the measured nickel concentrations in order
         from lowest to highest.

Step 2.   The cumulative probabilities, representing for each observation (xi) the  proportion of values
         less than or equal to X[, are given in the third column of the table below. These are computed
         as /'/(«+ 1) where n is the total number of samples (n = 20).

Step 3.   Determine the quantiles or z-scores from the standard normal distribution corresponding to the
         cumulative probabilities in Step 2. These can be  found by successively letting P equal each
         cumulative probability and then looking  up  the  entry in Table  10-1  (Appendix  D)
         corresponding to P.  Since the standard  normal  distribution is  symmetric about zero,  for
         cumulative probabilities P < 0.50, look up the entry for (1-P) and give this value a negative
         sign.

Step 4.   Plot the normal quantile (z-score) versus the ordered concentration for each sample, as in the
         plot below (Figure 9-7).  The curvature found  in  the probability plot indicates that  there is
         evidence of non-normality in the data. -4
                                              9-17                                    March 2009

-------
Chapter 9.  Exploratory Tools
Unified Guidance
                    Table 9-4.  Nickel Concentrations from a Single Well
Nickel
Concentration
(PPb)
1.0
3.1
8.7
10.0
14.0
19.0
21.4
27.0
39.0
56.0
58.8
64.4
81.5
85.6
151.0
262.0
331.0
578.0
637.0
942.0
Order
(0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Cumulative
Probability
[//(»+!)]
0.048
0.095
0.143
0.190
0.238
0.286
0.333
0.381
0.429
0.476
0.524
0.571
0.619
0.667
0.714
0.762
0.810
0.857
0.905
0.952
Normal
Quantile
(z-score)
-1.668
-1.309
-1.068
-0.876
-0.712
-0.566
-0.431
-0.303
-0.180
-0.060
0.060
0.180
0.303
0.431
0.566
0.712
0.876
1.068
1.309
1.668
       PROBABILITY PLOTS FOR LOG TRANSFORMED DATA

Step 1.   List the natural logarithms of the measured nickel concentrations in Table 9-4 in order from
         lowest to highest. These are shown in Table 9-5.

Step 2.   The cumulative probabilities representing the proportion of values less than or equal to xt for
         each observation (x;), are given in the third column of Table 9-4. These are computed as / / (n
         + 1) where n is the total number of samples (n = 20).

Step 3.   Determine the quantiles or z-scores from the standard normal distribution corresponding to the
         cumulative probabilities in  Step 2. These can be  found by successively letting P equal each
         cumulative  probability and  then  looking  up   the  entry  in  Table  10-1 Appendix  D
         corresponding to P. Since the standard normal  distribution is  symmetric about  zero, for
         cumulative probabilities P < 0.50, look up the entry for (l-P) and give this value a negative
         sign.
                                            9-18
        March 2009

-------
Chapter 9.  Exploratory Tools                                             Unified Guidance
                  Table 9-5. Nickel Log Concentrations from a Single Well
Order
(0
1
2
O
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Log Nickel
Concentration
log(ppb)
0.00
1.13
2.16
2.30
2.64
2.94
3.06
3.30
3.66
4.03
4.07
4.17
4.40
4.45
5.02
5.57
5.80
6.36
6.46
6.85
Cumulative
Probability
[//(»+!)]
0.048
0.095
0.143
0.190
0.238
0.286
0.333
0.381
0.429
0.476
0.524
0.571
0.619
0.667
0.714
0.762
0.810
0.857
0.905
0.952
Normal
Quantile
(z-score)
-1.668
-1.309
-1.068
-0.876
-0.712
-0.566
-0.431
-0.303
-0.180
-0.060
0.060
0.180
0.303
0.431
0.566
0.712
0.876
1.068
1.309
1.668
Step 4.   Plot the normal quantile (z-score) versus the ordered logged concentration for each sample, as
         in the plot below (Figure 9-8). The reasonably linear trend found in the probability  plot
         indicates that the log-scale data closely follow a normal pattern, further suggesting that the
         original data closely follow a lognormal distribution.
                                            9-19                                  March 2009

-------
Chapter 9.  Exploratory Tools
Unified Guidance
            CD
            S   o
            CO
            r\i
               -1
               -2
                     Figure 9-7. Nickel Normal Probability Plot
                       0       200      400      600      800     1000
                               Nickel Concentration (ppb)
            Figure 9-8. Probability Plot of Log Transformed Nickel Data
                 2
             CD
             S   o
             IM
                -1
                -2
                         0246
                             Log(nickel) Concentration log(ppb)
  8
                                       9-20
       March 2009

-------
Chapter 10. Fitting Distributions	Unified Guidance

             CHAPTER  10.   FITTING  DISTRIBUTIONS
       10.1   IMPORTANCE OF DISTRIBUTIONAL MODELS	10-1
       10.2   TRANSFORMATIONS TO NORMALITY	10-3
       10.3   USING THE NORMAL DISTRIBUTION AS A DEFAULT	10-5
       10.4   COEFFICIENT OF VARIATION AND COEFFICIENT OF SKEWNESS	10-9
       10.5   SHAPIRO-WILK AND SHAPIRO-FRANCIA NORMALITY TESTS	10-13
          10.5.1   Shapiro-Wilk Test (n< 50)	10-13
          10.5.2   Shapiro-Francia Test (n > 50)	10-15
       10.6   PROBABILITY PLOT CORRELATION COEFFICIENT	10-16
       10.7   SHAPIRO-WILK MULTIPLE GROUP TEST OF NORMALITY	10-18
     Because a statistical or mathematical model is at best an approximation of reality, all statistical
tests and procedures require certain assumptions for the methods to be used correctly and for the results
to be properly interpreted. Many tests make an assumption regarding the underlying distribution of the
observed data; in particular, that the original or transformed sample  measurements follow a normal
distribution. Data transformations are discussed in Section 10.2 while considerations as to whether the
normal distribution should be used as a 'default'  are explored in Section 10.3. Several techniques for
assessing normality are also examined, including:

    »«»  The skewness coefficient (Section 10.4)
    »«»  The Shapiro-Wilk test of normality and its close variant, the Shapiro-Francia test (Section 10.5)
    »»»  Filliben's probability plot correlation coefficient test (Section 10.6)
    »«»  The Shapiro-Wilk multiple group test of normality (Section 10.7)
10.1 IMPORTANCE OF DISTRIBUTIONAL  MODELS

     As introduced in Chapter 3, all statistical testing relies on the critical assumption that the sample
data are representative of the population from which they are selected. The statistical distribution of the
sample is  assumed to be similar to the distribution of the mostly unobserved population  of possible
measurements. Many parametric testing methods make a further assumption: that the form or type of the
underlying population is at least approximately known or can be identified through diagnostic testing.
Most  of these parametric tests assume that the population is normal in distribution; the validity or
accuracy of the test results may be in question if that assumption is violated.

     Consequently, an important facet of choosing among appropriate test  methods is determining
whether a commonly-used statistical distribution such as the normal, adequately models the observed
sample data. A large variety of possible distributional models exist in the statistical literature; most are
not typically applied to groundwater measurements and  often introduce  additional  statistical or
mathematical complexity in working with them.  So groundwater statistical models  are usually confined
to the gamma distribution, the Weibull distribution, or distributions that are normal or can be normalized
via a transformation (e.g., the logarithmic or square  root).

                                             10-1                                    March 2009

-------
Chapter 10.  Fitting Distributions	Unified Guidance

     Although the Unified Guidance will occasionally reference procedures that assume an underlying
gamma or Weibull distribution, the presentation in this guidance will focus on distributions that can be
normalized  and diagnostic tools  for  assessing  normality.  The  principal reasons for limiting  the
discussion in this manner are: 1) the same tools useful for testing normality can be utilized with  any
distribution  that can be normalized— the  only change needed is perform  the normality test after first
making a data transformation; 2) if no transformation works to adequately  normalize the sample data, a
non-parametric test can often be used as an alternative statistical approach; and  3) addressing more
complicated scenarios  is outside the scope of the guidance and  may require  professional statistical
consultation.

     Understanding the statistical behavior of groundwater measurements can be very challenging. The
constituents of interest may occur  at  relatively  low  concentrations  and frequently be left-censored
because of current  analytical  method limitations.  Sample data are  often positively skewed  and
asymmetrical in distributional pattern, perhaps due to the presence of outliers, inhomogeneous mixing of
contaminants in the  subsurface, or spatially variable  soils deposition affecting the local groundwater
geochemistry.  For some constituents, the  distribution  in groundwater  is not stationary over time (e.g.,
due to linear or seasonal trends) or not stationary across space (due to spatial variability in mean levels
from well to well). A set of these measurements pooled over time and/or space may appear highly non-
normal, even if the underlying population at any fixed point in time or space is normal.

     Because of these complexities, fitting a distributional model to a set of sample data cannot be done
in isolation from checks of other key  statistical assumptions. The data must  also be evaluated for outliers
(Chapter  12), since the  presence of even one extreme outlier may cause an otherwise recognizable
distribution  from being correctly identified. For data  grouped  across wells,  the possible presence of
spatial variability must be considered (Chapter 13). If identified, the Shapiro-Wilk multiple group  test
of normality may  be needed to account for differing means and/or variances  at  distinct wells. Data
pooled across  sampling events (i.e., over time) must be examined for the presence of trends or seasonal
patterns (Chapter 14). A clearly identified pattern may need to be removed and the data residuals tested
for normality, instead of the raw measurements.

     A frequently encountered problem involves testing normality on data sets containing non-detect
values. The best goodness-of-fit tests attempt to assess whether the  sample data closely resemble the
tails of the candidate distributional model. Since non-detects represent left-censored observations where
the  exact concentrations are unknown for the lower tail of the sample distribution, standard normality
tests cannot be run without some estimate or imputation of these unknown values. For a small fraction of
non-detects in a sample (10-15% or less) censored at a single reporting limit, it may be possible to apply
a normality test by simply replacing each non-detect with an imputed value of half the RL. However,
more complicated situations arise when there  is  a  combination of multiple  RLs (detected values
intermingled with different non-detect levels), or the proportion of non-detects  is larger. The Unified
Guidance recommends different strategies in these circumstances.

     Properly ordering  the sample observations  (i.e.,  from least  to  greatest) is critical to any
distributional goodness-of-fit test. Because the concentration of a non-detect measurement is only known
to be in the range from zero to the  RL, it is generally impossible to construct a  full  ordering of the
                                              10-2                                    March 2009

-------
Chapter 10.  Fitting Distributions	Unified Guidance

sample.1 There  are  methods, however,  to  construct partial orderings of the data that  allow the
assignment of relative rankings to each of the detected measurements and which account for the
presence of censored values. In turn, a partial ordering enables construction of an approximate normality
test. This subject is covered in Chapter 15.

10.2 TRANSFORMATIONS TO NORMALITY

     Guidance users will often encounter data sets indicating significant evidence of non-normality.
Due to the presumption of most parametric  tests that the underlying population is normal, a common
statistical strategy for apparently non-normal observations is to search for a normalizing mathematical
transformation. Because of the complexities associated with interpreting statistical results from data that
have been transformed to another scale, some  care must be taken in applying statistical procedures to
transformed measurements. In questionable or  disputable  circumstances,  it may be wise to analyze the
same  data with an equivalent non-parametric version of the same test (if it exists) to see if the same
general  conclusion is reached. If not,  the data transformation and its interpretation may need further
scrutiny.

     Particularly with prediction limits, control charts, and some of the confidence intervals described in
Chapters 18, 20, and 21, the parametric versions of these procedures are especially advantageous. Here,
a transformation may be warranted to  approximately normalize the statistical sample. Transformations
are also often useful when combining or pooling intrawell background from several wells in  order to
increase the degrees  of freedom available for intrawell  testing (Chapter 13).  Slight differences in the
distributional  pattern from well to well  can skew  the  resulting pooled  dataset,  necessitating  a
transformation to bring about approximate normality and to equalize the variances.

     The  interpretation  of transformed data is straightforward  in  the  case  of prediction  limits for
individual observations or when building a confidence interval around an upper percentile. An interval
with limits constructed from the transformed data and then re-transformed (or back-transformed) to the
original measurement domain will retain its original probabilistic interpretation. For instance, if the data
are approximately normal under a square  root transformation  and a 95% confidence prediction limit is
constructed on the square roots of the original measurements, squaring the resulting prediction limit
allows for a 95% confidence level when applied to the original  data.

     The  same ease of interpretation does not apply to prediction limits for a future arithmetic mean
(Chapter  18) or to confidence intervals around  an arithmetic mean  compared  to  a  fixed  GWPS
(Chapter 21). A back-transformed confidence  interval constructed around the mean of log-transformed
data (i.e..,  the  log-mean) corresponds to a confidence interval around the geometric mean of the raw
(untransformed) data. For the lognormal distribution, the geometric mean is equal  to the median, but it is
not the same as the arithmetic mean. Using this back-transformation to bracket the location of the true
arithmetic population mean will result in an incorrect interval.

     For these particular applications, a similar  problem  of scale  bias occurs with other potential
normality transformations. Care is needed when applying  and interpreting transformations to a data set
1  Even when all the non-detects represent the lowest values in the sample, there is still no way to determine how this subset is
  internally ordered.

                                              10-3                                    March 2009

-------
Chapter 10.  Fitting Distributions _ Unified Guidance

for which either a confidence interval around the mean or a prediction limit for a future mean is desired.
The interpretation depends on which statistical parameter is being estimated or predicted. The geometric
mean or median in some situations may be a satisfactory alternative as a central tendency parameter,
although that decision must be weighed carefully when making comparisons against a GWPS.

     Common normalizing transformations include the natural logarithm, the square root, the cube root,
the square, the cube, and the reciprocal functions, as well as a few others.  More generally, one might
consider the "ladder of powers" (Helsel and Hirsch, 2002) technically known  as the set of Box-Cox
transformations (Box and Cox,  1964). The heart of these transformations is a power transformation of
the original data, expressed by the equations:
                                                    for 1^0
                                                                                         [10.1]
                                           logx     for/l = 0

     The goal of a Box-Cox analysis is to find the value X that best transforms the data to approximate
normality, using a procedure such as maximum likelihood. Such algorithms are beyond the scope of this
guidance, although  an excellent discussion  can be found in Helsel and Hirsch (2002).  In  practice,
slightly different equation formulations can be used:


                                     *=P   f°rl*°                                [10.2]
                                          [log*  for/l = 0

where the parameter X can generally be limited to the choices 0, -1, 1/4, 1/3, 1/2,  1, 2, 3, and 4, except
for unusual cases of more extreme powers.

     As  noted in  Section  10.1, checking  normality  with  transformed data does not  require any
additional tools. Standard normality tests can be applied using the transformed  scale measurements.
Only the interpretation of the test changes. A goodness-of-fit test can assess the  normality of the raw
measurements. Under a transformation, the same test checks for normality on the transformed scale. The
data will still follow the non-normal distribution in the original  concentration domain. So if a cube root
transformation is attempted and the transformed data are found to be approximately normal, the original
data are not normal but rather cube-root normal in distribution. If a log transformation is  successfully
used, the original measurements  are not normal but lognormal  instead. In sum, a  series of non-normal
distributions can be  fitted to data with the goodness-of-fit tests described in this chapter without needing
specific tests for other potential distributions.

     Finding a reasonable transformation in practice amounts to systematically 'climbing' the "ladder of
powers" described above. In other words, different choices of the power parameter X would be attempted
— beginning with X = 0 and working upward from -1  toward more extreme power transformations —
until a specific X normalizes the data or all choices have been attempted.  If no transformation  seems to
work, the user should instead consider a non-parametric test alternative.
                                             10-4                                   March 2009

-------
Chapter 10.  Fitting Distributions                                          Unified Guidance
10.3 USING THE  NORMAL DISTRIBUTION AS A DEFAULT

     Normal and lognormal distributions are frequently applied models in groundwater data because of
their general utility.  One or the other of these models might be chosen as a default distribution when
designing a statistical approach, particularly when relatively little data has been collected at a site. Since
the statistical behavior of these two models is  very different  and can lead to substantially different
conclusions, the choice  is not arbitrary. The type of test involved, the monitoring program, and the
sample  size can all affect the decision. For many data sets  and  situations, however, the normal
distribution can be  assumed as a default unless and until a  better model can be pinpointed through
specific goodness-of-fit testing provided in this chapter.

     Assumptions of normality are most easily made with regard to naturally-occurring and measurable
inorganic parameters, particularly under background conditions. Many ionic and other inorganic water
quality analyte measurements exhibit decent symmetry and low variability within a given well data set,
making these data amenable to assumptions of normality.  Less frequently detected analytes (e.g., certain
colloidal trace elements) may be better fit either by a site-wide lognormal or another distribution that can
be normalized, as well as evaluated with non-parametric methods.

     Where contamination in groundwater is known to exist a priori (whether in background or
compliance wells),  default distributional assumptions become more  problematic.  At a given well,
organic or inorganic contaminants may exhibit high or low variability,  depending on local hydrogeologic
conditions, the pattern of release from the source, the degree of solid phase absorption, degradability of a
given constituent, and the variation in groundwater flow direction and depths. Non-steady state releases
may result in a historical, occasionally non-linear pattern of trend increases or decreases. Such data
might be fit by an apparent lognormal distribution, although removal of the trend may lead to normally-
distributed residuals.

     Sample size is also a consideration. With fewer than 8 samples in a data set, formal goodness-of-fit
tests are often of limited value. Where larger sample sizes are available, goodness-of-fit tests should be
conducted. The Shapiro-Wilk multiple group well test (Section 10.7) — even with small sample sizes —
can sometimes be used to identify individual anomalous  wells which might otherwise be presumed to
meet the criterion of normality. Under compliance/assessment or corrective action monitoring, one might
anticipate only four samples per well in the first year after  instituting such monitoring. Under these
conditions, a default assumption of normality for  testing of the mean against a fixed standard is probably
necessary. Aggregation of multi-year data when conducting compliance tests (see Chapter 7) may allow
large enough sample sizes to warrant formal goodness-of-fit testing. With 8 (or more) samples,  it may be
possible to determine that a lognormal distribution is an appropriate fit for the  data. Even in this latter
approach, caution may be needed in applying Land's confidence interval  for a lognormal mean (Chapter
21) if the sample  variability is  large and especially if the upper  confidence  limit is used in the
comparison (i.e.., in corrective action monitoring).

     The normal distribution may also serve as a reasonable default when it is not critical to ensure that
sample data closely follow a specific distribution. For example, statistical tests on the mean are  generally
considered more robust with respect to departures from normality than procedures which involve upper
or lower limits of an assumed distribution. Even  if the data are not quite normal, tests on the mean such
                                             10-5                                   March 2009

-------
Chapter 10.  Fitting Distributions _ Unified Guidance

as a Student's Mest will often still provide a valid result.  However,  one might  need to consider
transformations of the  data for other reasons. Analysis of variance [ANOVA]  can be run with small
individual well samples (e.g., n = 4),  and as a comparison of means, it is fairly robust to departures from
normality. A logarithmic or other transformation may be needed to stabilize or equalize the well-to-well
variability (i.e., achieve homoscedasticity), a separate and more critical assumption of the test.

     Given their importance in statistical testing and the risks that sometimes occur in trying to interpret
tests on other data transformation  possibilities, it is  useful to briefly  consider the logarithmic
transformation in more detail. As noted in Section 10.1, groundwater data can frequently be normalized
using a logarithmic distribution model.  Despite this, objections  are  sometimes  raised that  the  log
transformation is merely used to "make large numbers look smaller."

     To better understand the log  transformation, it  should  be recognized that logarithms are, in fact,
exponents to some unit base. Given a concentration-scale variable x, re-expressed as  x = ICPor x = ey ,
the logarithm y is the exponent of that base (10 or the natural base e). It is the behavior of the resultant^
values that is assessed when data are log-transformed. When data relationships are multiplicative in the
original arithmetic domain (xl xx2\  the relationships between exponents (i.e., logarithms) are additive
(yl + y2).  Since the logarithmic  distribution by mathematical definition is normal in  a log-transformed
domain, working with  the logarithms instead of the  original concentration measurements may offer a
sample distribution much closer to normal.

     Similar to  a  unit scale transformation (ppm to ppb or  Fahrenheit to Centigrade), the  relative
ordering of log-transformed measurements does not change.  When  non-parametric tests based on ranks
(e.g., the Wilcoxon rank-sum test)  are applied to  data transformed  either to a different unit scale or by
logarithms, the outcomes are identical. However, other relationships among the log-transformed data do
change, so that the log-scale numerical 'spacing' between lower values is more similar to the log-scale
spacing between higher values. While parametric tests like prediction limits, ^-tests, etc., are not affected
by unit scale  transformations,  these tests may have different outcomes depending on whether raw
concentrations or log-transformed measurements are used. The justification for utilizing log-transformed
data is that the transformation helps to normalize the data so that these tests can be properly applied.

     There is  also a plausible physical explanation as to why pollutant concentrations  often follow a
logarithmic pattern  (Ott, 1990).  In Ott's  model, pollutant sources  are randomly  dispersed through the
subsurface or  atmosphere in a  multiplicative fashion through repeated  dilutions when mixing with
volumes of (uncontaminated) water  or air,  depending on  the  medium.  Such random and repeated
dilutions can mathematically lead to a lognormal distribution. In particular, if a final concentration (c0)
is the product of several random dilutions  (c. ) as suggested by the following equation:
the logarithm of this concentration is equivalent to the sum of the logarithms of the individual dilutions:
                                              10-6                                    March 2009

-------
Chapter 10.  Fitting Distributions	Unified Guidance

     The Central Limit Theorem (Chapter 3) can be applied to conclude that the logged concentration
in equation [10.4] should be approximately normal, implying that the original concentration (c0) should
be approximately lognormal in distribution. Contaminant fate-and-transport models more or less follow
this  same approach, using  successive multiplicative  dilutions (while accounting for absorption  and
degradation effects) across grids in time and space.

     Despite the mathematical elegance of the Ott model, experience with groundwater monitoring data
has shown that the lognormal model alone is not adequate to account for observed distribution patterns.
While contaminant modeling might predict a lognormal contaminant distribution in space (and often in
time at a fixed point during transient phases), individual well location points fixed in space  and at rough
contaminant equilibrium are more likely to be  subject to a variety of local hydrologic and other factors,
and the observed distributions can be almost limitless in form. Since most of the tests within the Unified
Guidance presume a stationary population over time at a given well location (subject to identification
and removal of trends),  the resultant distributions may  be other than lognormal in character. Individual
constituents may also exhibit varying aquifer-related distributional characteristics.

     A practical issue  in  selecting a default transformation is ease of  use. Distributions like the
lognormal usually entail more  complicated  statistical adjustments or calculations than the normal
distribution.  A confidence interval  around the  arithmetic  mean of a lognormal distribution utilizes
Land's //-factor, which  is a function of both log sample data variability and sample size, and is only
readily available for specific confidence levels.  By contrast, a normal confidence interval around the
sample mean based on the ^-statistic can easily be defined for virtually any  confidence level. As noted
earlier, correct use  of these confidence intervals depends on  selecting the appropriate  parameter and
statistical measure (arithmetic mean versus the geometric mean).

     While a transformation does not always necessitate using a different statistical  formula to ensure
unbiased results, use of a transformation does assume that the  underlying population is  non-normal.
Since the true population will almost never be  known with certainty, it may not be advantageous to
simply default to a lognormal assumption for a variety of reasons. Under detection monitoring, the
presumption is made that a statistically significant increase above background concentrations will trigger
a monitoring  exceedance.  But the larger the prediction limit computed  from background, the  less
statistical power the test will have for detecting true increases.  An important question to answer is what
the consequences are  when incorrectly applying statistical techniques based  on one distributional
assumption  (normal or lognormal), when  the underlying distribution is in fact the  other.  More
specifically,  what is the impact  on statistical  power and accuracy  of assuming  the wrong underlying
distribution? The general effects of violating underlying test assumptions can be measured in terms of
false positive and negative  error rates (and therefore power). These questions are particularly pertinent
for prediction limit and control  chart tests in  detection monitoring. Similar questions could be  raised
regarding the application of confidence  interval tests on  the  mean when compared against fixed
standards.

     To  answer these questions, a series of Monte Carlo  simulations was generated for the Unified
Guidance to evaluate the impacts on prediction limit false positive error rates and statistical power of
using normal and lognormal   distributions   (correctly and  incorrectly  applied to the underlying
distributions). Detailed results of this study are provided in Appendix C, Section C.I.
                                              10-7                                    March 2009

-------
Chapter 10.  Fitting Distributions	Unified Guidance

     The conclusions of the Monte Carlo study are summarized as follows:

    »«»  If  an underlying population  is truly normal,  treating the  sample data  as lognormal in
       constructing a  prediction limit  can have  significant  consequences. With  no retesting, the
       lognormal prediction limits were in every case considerably larger and thus less powerful than
       the normal prediction limits.  Further, the lognormal limits consistently exhibited less than the
       expected (nominal) false positive rate, while the normal prediction limits tended to have slightly
       higher than nominal error rates.
    »«»  When retesting was added to the procedure,  both types of prediction limits improved. While
       power uniformly improved compared to no retest, the normal limits were still on average about
       13% shorter than the lognormal limits, leading again to  a measurable loss of statistical power in
       the lognormal case.
    »«»  On  balance, misapplication  of logarithmic prediction  limits  to  normally-distributed  data
       consistently resulted in (often  considerably) lower power and false positive rates that were lower
       than expected. The results argue against presuming the underlying data to be lognormal without
       specific goodness-of-fit testing.
    »«»  The  highest penalties from  misapplying lognormal  prediction limits  occurred for smaller
       background sizes. Since goodness-of-fit tests  are least able to  distinguish between normal and
       lognormal data with small samples, small background  samples should not be  presumed to be
       lognormal  as a default  unless other evidence from  the  site suggests  otherwise. For  larger
       samples, goodness-of-fit  tests  have  much  better  discriminatory power,  enabling a  better
       indication of which model to use.
    »«»  If the underlying population is truly lognormal but the  sample data are treated as normal, the
       penalty in overall statistical performance is substantial only if no retesting is conducted. With no
       retesting, the false positive rates of normal-based limits were often substantially higher than the
       expected rate.   Under conditions  of no  retesting, misapplying normal prediction  limits to
       lognormal data would result in an excessive site-wide false positive rate (SWFPR).
    »»»  If at least one retest was added, the  achieved false positive rates for the misapplied normal limits
       tended to be less than the expected rates, especially for  moderate to larger sample sizes. Except
       for highly skewed lognormal  distributions, the power of the normal limits was comparable or
       greater than the power of the lognormal limits.
     Overall, the Monte Carlo study  indicated that adding a retest to the  testing procedure significantly
minimized the penalty of misapplying normal prediction limits to lognormal data, as long as the sample
size was at least 8 and the distribution was not too skewed. Consequently, there is less penalty associated
with making a  default assumption of normality than in making a default assumption  of lognormality
under most situations. With highly skewed  data, goodness-of-fit tests tend to better discriminate between
the normal and lognormal models. The Unified Guidance therefore recommends that  such diagnostic
testing be done explicitly rather than simply assuming the data to be normal or lognormal.

     The most problematic cases in the study occurred for very small background sample sizes, where a
misapplication of prediction limits in either direction often resulted in poorer statistical performance,
even with retesting. In  some situations, compliance testing may need to be conducted on an interim
basis until enough  data has been collected to accurately identify  a distributional model.  The Unified
Guidance does not recommend an automatic default assumption of lognormality.
                                             10-8                                    March 2009

-------
Chapter 10.  Fitting Distributions	Unified Guidance

     In summary, during detection and compliance/assessment monitoring, data sets should be treated
initially as normal in distribution unless a better model can be pinpointed through specific testing. The
normal distribution is a fairly  safe assumption for background distributions,  particularly for naturally
occurring, measurable constituents and when sample sizes are small. Goodness-of-fit tests provided in
this chapter can be used to more closely identify the appropriate distributions for larger sample sizes. If
the initial assumption of normality is not rejected, further statistical analyses should be performed on the
raw observations. If the normal distribution is rejected by a goodness-of-fit test, one should generally test
the normality of the logged data,  in order to check for lognormality of the  original observations. If this
test also fails, one can  either  look for an alternate transformation  to achieve approximate normality
(Section 10.2) or use a non-parametric technique.

     Since tests of normality have low power for rejecting the null hypothesis when the data are really
lognormal but the sample size and degree of skewness are small, it is reassuring that a "wrong" default
assumption of normality will infrequently lead to an incorrect statistical conclusion. In fact, the statistical
power for detecting real concentration increases will generally be better than if the data were assumed to
be lognormal. If the data are truly lognormal, there is a risk of greater-than-expected  site-wide false
positive error rates.

     When the population is more skewed,  normality tests in the Unified Guidance have much greater
power for correctly rejecting the normal model in favor of the lognormal distribution. Consequently, an
initial assumption of normality will not,  in most cases, lead to an incorrect final conclusion, since the
presumed normal model will tend to be rejected before further testing is conducted.

     These recommendations do not apply to corrective action monitoring or other programs where it
either known  or reasonable to presume that  groundwater is already impacted  or has  a non-normal
distribution. In such settings,  a  default  presumption of lognormality could  be  made,  or a series of
normalizing transformations could be attempted until a suitable fit is determined. Furthermore, even in
detection monitoring,  there  are situations that often require the use of alternate transformations, for
instance when pooling intrawell  background across  several wells to increase the degrees  of freedom
available for intrawell testing (Chapter 13).

     Whatever the  circumstance, the Unified  Guidance recommends whenever possible that site-
specific data be used to test the distributional  presumption. If no data are initially available to do this,
"referencing"  may be  employed to justify the use of, say, a  normal or lognormal  assumption in
developing statistical tests at a particular site. Referencing involves the use of historical data  or data
from sites in similar hydrologic  settings to  justify the assumptions  applied to the proposed statistical
regimen. These initial  assumptions should be checked when  data from the  site become available, using
the procedures described in the Unified Guidance. Subsequent changes to the initial assumptions should
be made if goodness-of-fit testing contradicts the initial hypothesis.

10.4 COEFFICIENT OF VARIATION AND COEFFICIENT  OF SKEWNESS

       PURPOSE AND BACKGROUND

     Because the normal  distribution has  a  symmetric 'bell-shape,'  the normal mean and median
coincide and random observations drawn from a normal population are just as likely to occur below the
mean as  above it. More generally, in any symmetric distribution the distributional pattern below the

                                             10-9                                   March 2009

-------
Chapter 10.  Fitting Distributions	Unified Guidance

mean is a mirror-image of the pattern above the mean. By definition, such distributions have no degree
of skewness or asymmetry.

      Since the normal distribution has zero skewness, one way to look for non-normality is to estimate
the  degree  of skewness. Non-zero values of this measure imply that the population is asymmetric and
therefore something different from normal.  Two exploratory screening tools useful for this task are the
coefficient of variation and the coefficient of skewness.

      The coefficient of variation [CV] is  extremely easy to compute, but only indirectly  offers an
estimate of skewness and hence normality/non-normality. A more direct estimate can be determined via
the  coefficient of skewness. Furthermore, better, formal tests can be used instead of either coefficient to
directly assess normality. Nevertheless, the CV provides a  measure of intrinsic variability in positive-
valued data sets. Although approximate, CVs can  indicate the relative  variability of  certain  data,
especially with small sample sizes and in the absence of other formal tests (e.g., see Chapter 22, when
comparing  confidence limits on the mean to  a fixed standard in compliance monitoring).

      The CV is also a valid measure of the multiplicative relationship between the population mean and
the  standard deviation for positively-valued  random variables. Using sample statistics for the mean (x)
and standard deviation (s), the true CV for non-negative normal populations can be reasonably estimated
as:

                                         CV=slx                                       [10.5]

      In lognormal populations, the CV is also used in evaluations of statistical  power.  In this latter
case, the population CV works out to be:
                                                                                          [10.6]

where oy is the population log-standard deviation. Instead of a ratio between the original scale standard
deviation and the mean, the lognormal CV is estimated with the equation:
                                                                                          [10.7]


where s  is the sample log-standard deviation. The estimate in equation [10.7] is usually more accurate
than the simple CV ratio of the arithmetic standard deviation-to-mean, especially when the underlying
population coefficient of variation is high. Similar to using the normal CV as a formal indicator of
normality, the lognormal coefficient of variation estimator in equation [10.7] will have little relevance as
a test of lognormality of the data.   Using it for that  purpose is  not recommended  in the Unified
Guidance.  But it can provide a sense of how variable a data set is and whether a lognormal assumption
might need to be tested.

     While others have  reported  a ratio CV on  logged measurements as  CV=syly   for the
transformation y = log x, the result is essentially meaningless. The actual logarithmic CV in  equations
[10.6] and [10.7] is solely determined by the logarithmic variability of oy or sy. Negative logarithmic
mean values are  always  possible, and the  log ratio  statistic  is  not invariant  under  a unit  scale
                                             10-10                                   March 2009

-------
Chapter 10.  Fitting Distributions _ Unified Guidance

transformation (e.g., ppb to ppm or ppt). Similar problems in interpretation occur when CV estimators
are applied to any variable which can be negatively valued, such as following a z-transformation to a
standard normal distribution.  This log ratio  statistic is not recommended for any  application in the
guidance.

     The  coefficient  of skewness  (YJ) directly indicates to  what degree a  dataset is skewed  or
asymmetric with respect to the mean.  Sample data from a normal distribution will have a  skewness
coefficient  near zero, while data from an  asymmetric distribution  will have a positive or negative
skewness depending on whether the right- or left-hand tail of the distribution is longer and  skinnier than
the opposite tail.

     Since groundwater monitoring concentrations are inherently non-negative, such data  often exhibit
skewness.  A small degree  of skewness is not  likely to affect the results of statistical tests that  assume
normality. However, if the  skewness coefficient is larger than 1 (in absolute value) and the sample size is
small (e.g., n  < 25), past  research has shown that standard normal theory-based tests are much less
powerful than when the absolute skewness is less than 1 (Gayen,  1949).

     Calculating the skewness coefficient is useful  and only slightly more difficult than computing the
CV. It provides a quick indication of whether the skewness is minimal enough to assume that the data
are roughly symmetric and hopefully normal in distribution. If the original data exhibit a high skewness
coefficient, the normal distribution will provide a poor approximation to the dataset. In that case — and
unlike the CV — YI can be computed on the  log-transformed data to test for symmetry of the logged
measurements, or similarly for other transformations.

       PROCEDURE

     The CV  is calculated simply  by taking the ratio  of the sample standard deviation to the sample
mean, CV = s/x or its corresponding logarithmic version CV = Jexp (s2 V- 1 .

     The skewness coefficient may be computed using the following equation:
where the numerator represents the average cubed residual after subtracting the sample mean.

       ^EXAMPLE 10-1

     Using the following data, compute the CVs and the coefficient of skewness to test for approximate
symmetry.
                                             10-11                                   March 2009

-------
Chapter 10. Fitting Distributions                                          Unified Guidance
Month
Jan
Mar
Jun
Aug
Oct
Year 1
58.8
1.0
262
56
8.7
Nickel Concentration (ppb)
Year 2 Year 3
19
81.5
331
14
64.4
39
151
27
21.4
578
Year 4
3.1
942
85.6
10
637
       SOLUTION
Step 1.   Compute the mean, standard deviation (s\ and sum of the cubed  residuals for the nickel
         concentrations:

                            x = — (58.
           = I— [(58.8-169.52)2+(l-169.52)2 + ... + (637- 169.52)2]  =  259.7175 ppb


                   3      r                                  1
               -x)   =   [(58.8-169.52)3 +... + (637 -169.52)3J = 5.97845791xl08 ppb3
Step 2.   Compute  the  arithmetic  normal   coefficient  of variation  following  equation  [10.5]:
         CV = 259.7175/169.52 = 1.53

Step 3 .   Calculate the coefficient of skewness using equation [10.8]:

                     Yl = (20 j/2 (5. 97845791 x 108 ^/foj2 (259.7175)' = 1.84

         Both the CV and the coefficient of skewness are much larger than 1, so the data appear to be
         significantly positively skewed. Do not assume that the underlying population is normal.

Step 4.   Since the  original  data evidence a high degree of skewness, one can instead compute the
         skewness coefficient and corresponding  sample CV with equation [10.7] on the logged nickel
         concentrations.   The  logarithmic CV  equals 4.97,  a  much more variable  data  set than
         suggested  by the arithmetic CV. The skewness coefficient works out to be |Yj|= 0.24 <  1,
         indicating that the logged data values are slightly skewed but not enough to clearly reject an
         assumption of normality in the logged data. In other words, the original nickel values may be
         lognormally distributed. -^
                                            10-12                                  March 2009

-------
Chapter 10. Fitting Distributions	Unified Guidance

10.5 SHAPIRO-WILK AND SHAPIRO-FRANCIA  NORMALITY TESTS

       10.5.1  SHAPIRO-WILK TEST (N <  50)

       PURPOSE AND BACKGROUND

     The Shapiro-Wilk test is based on the premise that if a data set is normally distributed, the ordered
values should be highly correlated with corresponding quantiles (z-scores) taken from  a normal
distribution (Shapiro  and Wilk, 1965).  In particular, the  Shapiro-Wilk test gives substantial weight to
evidence of non-normality in the tails of a distribution, where the robustness of statistical tests based on
the normality assumption is most severely affected.  A variant of  this test, the  Shapiro-Francia test, is
useful for sample sizes greater than 50 (see Section 10.5.2).

     The Shapiro-Wilk test statistic (SW) will tend to be large  when a probability plot of the data
indicates a nearly straight line. Only when the plotted data show significant bends or curves will the test
statistic be  small. The  Shapiro-Wilk test is considered  one  of the best tests of normality available
(Miller, 1986; Madansky, 1988).

       PROCEDURE

Step 1.   Order and rank the dataset from least to greatest, labeling the observations as xt for rank /' =
         1.. .n.  Using the notation XQ, let the /'th rank statistic from a data set represent the rth smallest
         value.
Step 2.   Compute differences  x,_.+ , -x,..  for each i=\...n. Then determine k as the greatest integer

         less than or equal to (n/2).

Step 3.   Use Table 10-2 in Appendix D to determine the Shapiro-Wilk coefficients, an-i+i , for / =
         \...k. Note that while these coefficients depend only on the sample size («), the order of the
         coefficients must be preserved when used in Step 4. The coefficients can be determined for
         any sample size from n = 3 up to n = 50.

Step 4.   Compute the quantity b given by the following equation:

                                   k      k
                                b=Tb =ya  . ,(v     -*„)                           [10.9]
                                   Z_^ i   Z_^  «-z+lv («-z + l)   (i)'                           L    J
                                   z=l    z=l

         Note that the values bt are simply intermediate quantities represented by the terms in the sum
         of the right-hand expression in equation [10.9].

Step 5.   Calculate  the standard  deviation (s) of  the dataset.  Then compute the Shapiro-Wilk  test
         statistic using the equation:
                                      SW =
[10.10]
                                            10-13                                  March 2009

-------
Chapter 10. Fitting Distributions
Unified Guidance
Step 6.   Given the significance level (a) of the test, determine the critical point of the Shapiro-Wilk
         test with n observations using Table 10-3 in Appendix D. To maximize the utility and power
         of the test, choose a = .10 for very small data sets (n < 10), a = .05 for moderately sized data
         sets (10 < n < 20), and a = .01 for larger sized data sets (n > 20). Compare the SW against the
         critical  point (swc). If the test statistic exceeds the  critical point, accept normality as a
         reasonable model for the underlying population. However,  if SW < swc, reject the null
         hypothesis of normality at the a-level and decide that  another distributional model might
         provide a better fit.

       ^EXAMPLE 10-2

     Use the nickel data of Example 10-1 to compute the Shapiro-Wilk test of normality.

     SOLUTION
Step 1.   Order the data from smallest to largest, rank in ascending order and list, as shown in columns
         1 and 2  of the table below. Next list the data in reverse order in a third column.
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
x(f)
1.0
3.1
8.7
10.0
14.0
19.0
21.4
27.0
39.0
56.0
58.8
64.4
81.5
85.6
151.0
262.0
331.0
578.0
637.0
942.0
x(n-i + l)
942.0
637.0
578.0
331.0
262.0
151.0
85.6
81.5
64.4
58.8
56.0
39.0
27.0
21.4
19.0
14.0
10.0
8.7
3.1
1.0
x(n-i+1) - x(i)
941.0
633.9
569.3
321.0
248.0
132.0
64.2
54.5
25.4
2.8
-2.8
-25.4
-54.5
-64.2
-132.0
-248.0
-321.0
-569.3
-633.9
-941.0
an-i + l
.4734
.3211
.2565
.2085
.1686
.1334
.1013
.0711
.0422
.0140










b.
445.47
203.55
146.03
66.93
41.81
17.61
6.50
3.87
1.07
0.04
b = 932.88









Step 2.   Compute the differences  */ .+1-> - */.•>  in column 4 of the table by subtracting column 2 from

         column 3. Since the total sample size is n = 20, the largest integer less than or equal to (w/2) is
         £=10.

Step 3.   Look up the coefficients an_i+i from Table 10-2 in Appendix D and list in column 4.
                                             10-14
        March 2009

-------
Chapter 10. Fitting Distributions
                                     Unified Guidance
Step 4.   Multiply the differences  in column 3 by the coefficients in column 4 and add the first k
         products (b[) to get quantity 6, using equation [10.9].

                     b =[.4734(941.0)+.3211(633.9)+... + .0140(2.8)] = 932.88

Step 5.   Compute the standard deviation of the  sample, s = 259.72. Then use equation [10.10] to
         calculate the SW:
                                  SW =
                                           932.88
                                         259.72
       A/19.
              = 0.679
Step 6.   Use Table 10-3 in Appendix D to determine the 0.01-level critical point for the Shapiro-Wilk
         test when n = 20. This gives swc = 0.868. Then compare the observed value of SW= 0.679 to
         the 1% critical point.  Since SW < 0.868,  the sample  shows significant evidence  of non-
         normality by the Shapiro-Wilk test. The data should be transformed using logarithms or
         another transformation  on the ladder of powers and re-checked using the Shapiro-Wilk test
         before proceeding with further statistical analysis. -^

       10.5.2   SHAPIRO-FRANCIA TEST (N > 50)

     The Shapiro-Wilk test of normality can be used  for sample sizes up to 50. When n is larger than
50, a slight modification of the procedure called the Shapiro-Francia test (Shapiro  and Francia,  1972)
can be used instead. Like the Shapiro-Wilk test, the Shapiro-Francia test statistic (SF) will tend to be
large when a probability plot of the data indicates a nearly straight line. Only when the plotted data show
significant bends or curves will the test statistic be small.

     To calculate the test statistic SF, one can use the following equation:
                               SF =
                                       m.x
                                          (0
                                                [10.11]
where XQ represents the /'th ranked value of the sample and where mi denotes the approximate expected
value of the /'th rank normal quantile (or z-score). The values for m\ are approximately equal to
       /  i  }
m = <£   	
        U+iJ
                                                                                       [10.12]
where  <£> : denotes the inverse of the standard normal distribution with zero mean and unit variance.
These values can be computed by hand using the normal distribution in Table 10-1 of Appendix D or
via simple commands found in many statistical computer packages.

     Normality of the data should be rejected if the Shapiro-Francia statistic is too low when compared
to the critical points provided in Table 10-4 of Appendix D. Otherwise one can assume the data are
approximately normal for purposes of further statistical analysis.
                                            10-15
                                            March 2009

-------
Chapter 10. Fitting Distributions	Unified Guidance

10.6 PROBABILITY PLOT CORRELATION  COEFFICIENT

       BACKGROUND AND PURPOSE

     Another test for normality that is essentially equivalent to the Shapiro-Wilk and Shapiro-Francia
tests is the probability plot correlation coefficient test described by Filliben (1975). This test meshes
perfectly with the use of probability plots, because the  essence of the test is to compute the usual
correlation coefficient for points on a probability plot. Since the correlation coefficient is a measure of
the linearity of the points on a scatterplot,  the probability plot correlation coefficient, like the SW test
statistic, will be high when the plotted points fall along a straight line and low when there are significant
bends  and curves in the probability plot.  Comparison of  the  Shapiro-Wilk and  probability  plot
correlation coefficient tests has indicated  very similar statistical power for detecting  non-normality
(Ryan and Joiner, 1990).

     It should be noted that although some statistical software may not compute Filliben's test directly,
the usual Pearson's correlation coefficient computed on the data pairs used to construct a probability plot
will  provide  a  very close approximation  to the Filliben  statistic.  Some users  may find this latter
correlation easier to compute or more accessible in their software.

       PROCEDURE

Step 1.   List the observations in  order from smallest  to largest, denoting XQ as the rth smallest rank
         statistic  in the  data set. Then let n = sample size and compute the sample mean (x ) and the
         standard deviation (s).

Step 2.   Consider a random sample drawn from a standard normal distribution. The rth rank statistic of
         this sample is fixed once the sample is drawn, but beforehand it can be considered a random
         variable, denoted as XQ.  Likewise, by considering all possible datasets of size n that might be
         drawn from the normal distribution, one can think of the sampling distribution  of the statistic
         X(i). This  sampling distribution has its  own  mean and variance, and, of importance to the
         probability plot correlation coefficient, its own median, which can be denoted M;.

         To compute the median of the rth rank statistic, first compute intermediate probabilities m\ for
         7 = 1.. .n using the equation:

                                              !/„
                                                        for 7 = 1
                            m =
forl<7<»                      [10.13]
         Then compute the medians M\ as the standard normal quantiles or z-scores associated with the
         intermediate probabilities m^. These can be determined from Table 10-1 in Appendix D or
         computed according to the following equation, where $~: represents the inverse of the
         standard normal distribution:

                                                                                       [10.14]

                                            10-16                                  March 2009

-------
Chapter 10.  Fitting Distributions	Unified Guidance

Step 3.   With the rank statistic medians in hand, calculate the arithmetic mean of theM's, denoted M,
         and the intermediate quantity Cn, given by the equation:
                                      C =JyM2-nM2                                [10.15]
                                        n   \i ^—^  i                                        L      j
                                           V 2 = 1

         Note that when the dataset is "complete" (meaning it contains no non-detects, ties, or censored
         values), the mean of the order statistic medians reduces to M = 0. This in turn reduces the
         calculation of Cn to:
                                                                                         [10.16]


     Step 4.   Finally compute Filliben's probability plot correlation coefficient:


                                                  - nxM
                                                                                         [10.17]
         When the dataset is complete, the equation for the probability plot correlation coefficient also
         has a simplified form:
Step 5.   Given the level of significance (a), determine the critical point (rcp) for Filliben's test with
         sample size n  from  Table  10-5  in Appendix D. Compare the probability plot  correlation
         coefficient (r) against the critical point (rcp). If r > rcp, conclude that normality is a reasonable
         model for the underlying population at the  a-level of significance. If, however, r < rcp, reject
         the null hypothesis and conclude that another distributional model would provide a better fit.

       ^EXAMPLE 10-3

     Use the data of Example 10-1 to compute Filliben's probability plot correlation coefficient test at
the a = .01 level of significance.

       SOLUTION
Step 1.   Order and  rank the nickel data from  smallest to largest and list, as in the table below. The
         sample size is n = 20, with sample mean x = 169.52 and the standard deviation s =  259.72.

Step 2.   Compute the intermediate probabilities m\ from equation [10.13] for each /' in  column 3 and
         the rank statistic medians, M;, in  column 4 by applying the inverse normal transformation to
         column 3 using equation [10.14] and Table  10-1 of Appendix D.
                                             10-17                                   March 2009

-------
Chapter 10.  Fitting Distributions
                                                                Unified Guidance
Step3.
Step 4.
Since this sample contains no non-detects or ties, the simplified equations for Cn in equation
[10.16] and for r in equation [10.18] may be used. First compute Cn using the squared order
statistic medians in column 5:
                Ctt = Vl.3.328 + 1.926 + ... + 3.328]  =  4.138

Next compute the  products x^xM.'m  column 6 and  sum to get the  numerator of the

correlation coefficient (equal to 3,836.81 in this case). Then compute the final correlation
coefficient:
                            = 3,836.81/[4.138 x259.72>/19l = 0.819
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
x(f)
1.0
3.1
8.7
10.0
14.0
19.0
21.4
27.0
39.0
56.0
58.8
64.4
81.5
85.6
151.0
262.0
331.0
578.0
637.0
942.0
m,
.03406
.08262
.13172
.18082
.22993
.27903
.32814
.37724
.42634
.47545
.52455
.57366
.62276
.67186
.72097
.77007
.81918
.86828
.91738
.96594
M,
-1.8242
-1.3877
-1.1183
-0.9122
-0.7391
-0.5857
-0.4451
-0.3127
-0.1857
-0.0616
0.0616
0.1857
0.3127
0.4451
0.5857
0.7391
0.9122
1.1183
1.3877
1.8242
(M,)2
3.328
1.926
1.251
0.832
0.546
0.343
0.198
0.098
0.034
0.004
0.004
0.034
0.098
0.198
0.343
0.546
0.832
1.251
1.926
3.328
x(i) x Mj
-1.824
-4.302
-9.729
-9.122
-10.347
-11.129
-9.524
-8.444
-7.242
-3.448
3.621
11.959
25.488
38.097
88.445
193.638
301.953
646.376
883.941
1718.408
Step 5.   Compare Filliben's test statistic of r = 0.819 to the 1% critical point for a sample of size 20 in
         Table 10-5 of Appendix D, namely rcp = 925. Since r < 0.925, the sample shows significant
         evidence of non-normality by the probability  plot correlation coefficient. The data should be
         transformed and  the  correlation coefficient re-calculated  before proceeding  with further
         statistical analysis. A
10.7  SHAPIRO-WILK MULTIPLE GROUP TEST OF NORMALITY

       BACKGROUND AND PURPOSE

     The main purpose for including the multiple group test normality (Wilk and Shapiro, 1968) in the
Unified Guidance is to serve as a check for normality when using a Student's Mest (Chapter 16) or
                                           10-18
                                                                        March 2009

-------
Chapter 10.  Fitting Distributions	Unified Guidance

when assessing the joint normality of multiple intrawell data sets. The multiple group test is an extension
of the Shapiro-Wilk procedure for assessing the joint normality of several independent samples. Each
sample may have a different mean and/or variance, but as long as the underlying distribution of each
group is normal, the multiple group test statistic will tend to be non-significant. Conversely, the multiple
group test is designed to identify when at least one of the groups being tested is definitely non-normal.

     This test extends the  Shapiro-Wilk procedure for a single sample, using individual SW test
statistics computed separately  for each  group  or sample.  Then  the  individual SW  statistics  are
transformed and combined into an overall or "omnibus" statistic (G). Like the single sample procedure
— where non-normality is indicated when the test statistic SW is too low — non-normality in one or
more groups is indicated when G is too low. However, instead of a special table of critical points, G is
constructed to follow a standard normal distribution under the null hypothesis of normality. The value of
G can simply be compared to  an a-level z-score or  normal quantile to decide whether the null  or
alternative hypothesis is better supported.

     Since it  may be unclear which one  or more of the groups  is actually non-normal when the G
statistic is significant, Wilk and Shapiro recommend that a probability plot (Chapter 9) be examined on
the intermediate quantities, G; (at least for the case where several groups are being simultaneously
tested). One of these statistics is computed for each separate sample/group and  is designed to follow a
standard normal distribution undergo- Because of this, the G; statistics for non-normal groups will tend
to look like outliers on a normal probability plot (see Chapter 12).

     The multiple group test can also be  used to check normality when performing Welch's  Mest,  a
two-sample  procedure in  which the underlying data of both groups are assumed to be normal, but no
assumption is  made that the means or variances are the same. This is different from either the pooled
variance t-test or the one-way analysis of variance [ANOVA], both of which assume homoscedasticity
(i.e., equal variances across groups). If the  group variances can be shown to be equal, the  single sample
Shapiro-Wilk test can be run on the combined residuals, where  the residuals of each group are formed by
subtracting off the  group mean from each  of the individual measurements. However,  if the group
variances are possibly different, testing the  residuals as  a single group using the SW statistic may give an
inaccurate or  misleading result.  Consequently,  since  a test  of homoscedasticity is not required for
Welch's t-tesl, it is suggested to first use the multiple group test to check normality.

     Although the Shapiro-Wilk multiple  group method is an attractive procedure for  accommodating
several groups of data at once, the user is cautioned against indiscriminate use.  While many of the
methods described in the Unified  Guidance  assume  underlying  normality,  they  also  assume
homoscedasticity. Other parametric multi-sample methods recommended for detection monitoring —
prediction limits in Chapter 18 and control charts in Chapter  20 — all assume  that each  group has the
same variance. Even if normality of the joint data can be demonstrated using the Shapiro-Wilk multiple
group test, it says nothing about whether the assumption of equal variances is also satisfied. Generally
speaking, except for Welch's Mest,  a separate test of homoscedasticity may also be needed. Such tests
are described in Chapter  11.

       PROCEDURE

Step 1.   Assuming there are K groups to be tested, let the sample size of the /'th group be  denoted n\.
         Then compute the SW\ test statistic for each of the K groups using equation [10.10].

                                             10-19                                    March 2009

-------
Chapter 10.  Fitting Distributions _ Unified Guidance

Step 2.   Transform the SWi statistics to the intermediate quantities (G;). If the sample size («;) of the /'th
         group is at least 7, compute G; with the equation:

                                    G=y+d\n\ — '— -I                               [10.19]
                                      1           l-SW
         where the quantities y, 8, and e can be found in Table 10-6 of Appendix D for 7 < n\ < 50. If
         the sample size («;) is less than 7, determine G; directly from Table 10-7 in Appendix D by
         first computing the intermediate value

                                             ( SW.-e\
                                       «,=la|—^1                                 [10-2°]
         (obtaining £ from the top of Table 10-7), and then using linear interpolation to find the closest
         value G; associated with u\.

Step 3.   Once the G; statistics are derived, compute the Shapiro-Wilk multiple group statistic with the
         equation:
Step 4.   Under the null hypothesis that all K groups are normally-distributed, G will follow a standard
         normal distribution. Given the significance level (a), determine an a-level critical point from
         Table 10-1 of Appendix D as the lower a x 100th normal quantile (za).  Then compare G to
         za. If G < za, there is significant evidence of non-normality  at the a level.  Otherwise, the
         hypothesis of normality cannot be rejected.

       ^EXAMPLE 10-4

     The previous examples in this chapter pooled the data of Example  10-1 into a single group before
testing for normality. This time, treat each well separately and compute the Shapiro-Wilk multiple group
test of normality at the a = .05 level.

       SOLUTION
Step 1.   The nickel data in Example 10-1  come from K = 4 wells with n\ = 5 observations per well.
         Using equation [10.10], the SWt individual well test statistics are calculated as:
             Welll:       SWi = 0.6062

             Well 2:       SW2 = 0.5917

             Well 3:       SW3 = 0.5652

             Well 4:       ^ = 0.6519
                                             10-20                                   March 2009

-------
Chapter 10. Fitting Distributions
Unified Guidance
Step 2.   Since n\ = 5 for each well, use Table 10-7 of Appendix D to find e = .5521. First calculating
         MI with equation [10.20]:

                                     , (.6062-.5521^
                                M, = In I 	I = -1.985
                                  1    I   1-.6062  )

         The performing this step for each well group and using linear interpolation on M in Table 10-7,
         the approximate G; statistics are:

             Welll:       MI =-1.985   GI =-3.238

             Well 2:       M2 =-2.333   G2 =-3.488

             Well 3:       M3 = -3.502   G3 = -4.013 (taking the last and closest entry)

             Well 4:       w4 = -1.249   G4 =-2.755

Step 3.   Compute the multiple group test statistic using equation [10.21]:

                    G = -^=[(-1.985)+ (-2.333)+ (-4.013)+ (-2.755)] = -5.543
                        V4

Step 4.   Since a = 0.05, the lower a  x  100th critical  point from the standard normal distribution in
         Table 10-1 of Appendix D is z.os = -1.645. Clearly, G < z.os  ; in fact G is smaller than just
         about any a-level critical point that might be  selected. Thus, there is significant evidence of
         non-normality in at least one of these wells (and probably all of them). -4

       ^EXAMPLE 10-5

     The data in Example 10-1 showed significant evidence of non-normality. In this example, use the
same nickel data applying the coefficient of skewness,  Shapiro-Wilk and the Probability Plot Correlation
Coefficient tests to determine whether the combined well measurements  better follow a  lognormal
distribution by first log-transforming the measurements.  Computing the  natural logarithms of the data
gives the table below:
Month
1
2
3
4
5
Well 1
4.07
0.00
5.57
4.03
2.16
Logged Nickel Concentrations log(ppb)
Well 2 Well 3
2.94 3.66
4.40 5.02
5.80 3.30
2.64 3.06
4.17 6.36
Well 4
1.13
6.85
4.45
2.30
6.46
                                             10-21
        March 2009

-------
Chapter 10. Fitting Distributions	Unified Guidance

       SOLUTION

             METHOD 1.  COEFFICIENT OF SKEWNESS

Step 1.   Compute the log-mean (]7), log-standard deviation (sy), and sum of the cubed residuals for the
         logged nickel concentrations (y;):

                       J7 = — (4.07 + 0.00 + ... + 6.46) =  3.918 l
      Sy
=   I— [(4.07-3.918)2 +(0.00-3.918)2 + ... + (6.46 - 3.918)2]  =  lWU\og(ppb)


               =  [(4.07-3.918)3 +... + (6.46-3.918)3]=-26.5281og3(^*)
Step 2.   Calculate the coefficient of skewness using equation [10.8] with Step 1 values as:

                         yl = (20 j/2 (-26.528)/(l9]3/2 (1.8014 J = -0.245
         Since the  absolute value of the skewness is less than 1, the data do not show evidence of
         significant skewness.  Applying  a normal  distribution  to  the  log-transformed  data  may
         therefore be appropriate, but this model should be further checked. The logarithmic CV of
         4.97 computed in Example  10-1 was also suggestive of a highly skewed distribution, but can
         be  difficult to  interpret in determining if measurements,  in  fact, follow  a logarithmic
         distribution.

              METHOD 2.  SHAPIRO-WILK TEST

Step 1.   Order and rank the data from smallest to largest and list, as in the table below. List the data in
         reverse order alongside the first column. Denote the rth logged observation by_y; = log(x;).

Step 2.   Compute differences  >Yn_!+1)->YA   m column 4 of the table by subtracting column 2 from

         column 3. Since n = 20, the largest integer less than or equal to (n/2) is k = 10.

Step 3.   Look up the coefficients an_;+i from Table 10-2 of Appendix D and list in column 5.

Step 4.   Multiply the differences in column 4 by the coefficients in column 5 and add the first k
         products (b[) to get quantity b, using equation [10.9].

                      b =[.4734(6.85)+. 321 1(5. 33) +... + .0140(. 04)] = 7.77
                                            10-22                                   March 2009

-------
Chapter 10. Fitting Distributions                                         Unified Guidance

i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Yd)
0.00
1.13
2.16
2.30
2.64
2.94
3.06
3.30
3.66
4.03
4.07
4.17
4.40
4.45
5.02
5.57
5.80
6.36
6.46
6.85
V(n-i + l)
6.85
6.46
6.36
5.80
5.57
5.02
4.45
4.40
4.17
4.07
4.03
3.66
3.30
3.06
2.94
2.64
2.30
2.16
1.13
0.00
Y(n-l + l) - Y(l)
6.85
5.33
4.20
3.50
2.93
2.08
1.39
1.10
0.51
0.04
-0.04
-0.51
-1.10
-1.39
-2.08
-2.93
-3.50
-4.20
-5.33
-6.85
3n-i + l
.4734
.3211
.2565
.2085
.1686
.1334
.1013
.0711
.0422
.0140










b,
3.24
1.71
1.08
0.73
0.49
0.28
0.14
0.08
0.02
0.00
b = 7.77









Step 5.   Compute the log-standard deviation of the sample, sy = 1.8014. Then use [10.10] to calculate
         the SWtest statistic:
                                 SW =
                                           7.77
                                        1.8014
Vl9
      = 0.979
Step 6.   Use Table 10-3 of Appendix D to determine the .01-level critical point for the Shapiro-Wilk
         test when n = 20. This gives swcp = 0.868. Then compare the observed value of SW= 0.979 to
         the 1% critical point. Since SW> 0.868, the  sample shows no significant evidence of non-
         normality by the Shapiro-Wilk test.  Proceed  with further  statistical analysis using the log-
         transformed data or by assuming the underlying population is lognormal.

             METHOD 3. PROBABILITY PLOT CORRELATION COEFFICIENT

Step 1.   Order and rank the logged nickel data from smallest to largest and list, as in the table below.
         Again let the rth logged value be denoted by y\ = log(x;). The sample size is n = 20, the log-
         mean is y = 3.918, and the log-standard deviation is sy = 1.8014.

Step 2.   Compute the intermediate probabilities m\ from  equation [10.13] for each /' in column 3 and
         the rank statistic medians, M\ , in column 4 by applying the inverse normal transformation to
         column 3 using equation [10.14] and Table 10-1  of Appendix D.
                                            10-23                                  March 2009

-------
Chapter 10.  Fitting Distributions
                                                                             Unified Guidance
                       Yd)
                                                                         V(i) x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.00
1.13
2.16
2.30
2.64
2.94
3.06
3.30
3.66
4.03
4.07
4.17
4.40
4.45
5.02
5.57
5.80
6.36
6.46
6.85
.03406
.08262
.13172
.18082
.22993
.27903
.32814
.37724
.42634
.47545
.52455
.57366
.62276
.67186
.72097
.77007
.81918
.86828
.91738
.96594
-1.8242
-1.3877
-1.1183
-0.9122
-0.7391
-0.5857
-0.4451
-0.3127
-0.1857
-0.0616
0.0616
0.1857
0.3127
0.4451
0.5857
0.7391
0.9122
1.1183
1.3877
1.8242
3.328
1.926
1.251
0.832
0.546
0.343
0.198
0.098
0.034
0.004
0.004
0.034
0.098
0.198
0.343
0.546
0.832
1.251
1.926
3.328
0.000
-1.568
-2.416
-2.098
-1.951
-1.722
-1.362
-1.032
-0.680
-0.248
0.251
0.774
1.376
1.981
2.940
4.117
5.291
7.112
8.965
12.496
Step 3.
         Since this sample contains no non-detects or ties, the simplified equations for Cn in [10.16]
         and for r in [10.18] may be used. First compute Cn using the squared order statistic medians in
         column 5:
                          Ctt = Vl.3.328 + 1.926 + ... + 3.328]  =  4.138

Step 4.   Next compute  the products  y^ x M.  in  column  6 and sum to get the  numerator of the
Step 5.
         correlation coefficient (equal to 32.226  in  this case). Then compute the final correlation
         coefficient:

                             r = 32.226/[4.138 x l.SQl4-Jl9~\ = 0.992
         Compare the Filliben's test statistic of r = 0.992 to the 1% critical point for a sample of size 20
         in Table  10-5  in Appendix D, namely rcp = 925. Since r > 0.925, the sample shows no
         significant evidence of non-normality by  the probability  plot correlation coefficient test.
         Therefore, lognormality  of the  original  data can  be assumed  in  subsequent statistical
         procedures.

         Note: the  Shapiro-Wilk and  Filliben's Probability  Plot  Correlation Coefficient  tests for
         normality on a single data set perform quite comparably. Only one of these tests need be run in
         routine applications. -^
                                             10-24
                                                                                     March 2009

-------
Chapter 11. Testing Equality of Variance	Unified Guidance

       CHAPTER 11.  TESTING EQUALITY OF VARIANCE
       11.1   Box PLOTS	11-2
       11.2   LEVENE'STEST	 11-4
       11.3   MEAN-STANDARD DEVIATION SCATTER PLOT	11-8
     Many of the methods described in the Unified Guidance assume that the different groups under
comparison  have the  same variance (i.e., are homoscedastic). This chapter covers  procedures for
assessing homoscedasticity and its counterpart, heteroscedasticity (i.e., unequal variances). Equality of
variance  is  assumed, for instance,  when  using  prediction limits  to  make either upgradient-to-
downgradient or intrawell comparisons. In the former case,  the method  assumes that the upgradient
variance is equal to the variance in each downgradient well. In the latter case, the presumption is that the
well variance is  stable over time (i.e., stationary) when comparing intrawell background versus  more
recent measurements.

     If a prediction limit is constructed on a single new measurement at each downgradient well, it isn't
feasible to test the variance equality assumption prior to each statistical evaluation. Homoscedasticity
can  be tested after several new rounds  of compliance sampling  by pooling collected compliance
measurements within a well. The Unified Guidance recommends periodic testing of the presumption of
equal variances by comparing newer data to historical background (Chapter 6).

     Equality of variance  between  different  groups  (e.g., different wells) is also an important
assumption for an analysis of variance [ANOVA]. If equality of variance does not hold, the power of the
F-test (its ability to detect differences among the group means) is reduced. Mild differences in variance
are generally acceptable. But the  effect  becomes noticeable when the largest and  smallest group
variances differ by a ratio of about 4, and becomes quite severe when the  ratio is  10 or more (Milliken
and Johnson, 1984).

     Three procedures for assessing or testing homogeneity  of variance  are described in the Unified
Guidance, two of which  that are more robust to departures from normality (i.e., less sensitive to non-
normality). These include:

  1.    The box plot (Chapter 9), a graphical method useful not only for checking equality of variance
       but also as an exploratory tool for visualizing the basic statistical characteristics of data sets.  It
       can also provide a rough indication of differences in mean or median concentration levels across
       several wells;
  2.    Levene's test (Section  11.2), a formal ANOVA-type procedure for testing variance inequality;
       and
  3.    The mean-standard deviation scatter plot (Chapter  9  and Section  11.3), a  visual  tool for
       assessing whether the degree of variability in a set of data groups or wells is correlated with the
       mean levels for those groups.  This could potentially indicate whether a variance stabilizing
       transformation might be needed.

                                             11-1                                  March 2009

-------
Chapter 11. Testing Equality of Variance	Unified Guidance

11.1 BOX PLOTS

       PURPOSE AND BACKGROUND

     Box plots are described in Chapter 9. In the context of variance testing, one can construct a box
plot for each well group and compare the boxes to see if the assumption of equal variances is reasonable.
The comparison  is not a formal  test procedure, but is easier to perform and is often sufficient for
checking the group variance assumption.

     Box plots  for each  data group  simultaneously graphed  side-by-side  provide a direct visual
comparison of the dispersion in each group. As a rule of thumb, if the box length for each group is less
than 1.5-2 times the length of the shortest box, the sample variances  may be close enough to assume
equal  group variances. If the box length for any group is greater than 1.5-2 times the length of the box
for another group, the variances may be significantly different. A  formal test such as Levene's might be
needed to more  accurately decide. Sample data  sets with unequal variances may need a variance
stabilizing transformation.,  i.e., one in which the transformed measurements have approximately equal
variances.

     Most statistical software packages will calculate the statistics needed to draw a box plot, and many
will construct side-by-side box plots directly. Usually a box plot will also be shown with two "whiskers"
extending from the edges of the box. These lines indicate either  the positions of extreme minimum or
maximum values in the data set.  In Tukey's original formulation (Tukey, 1977), they indicate the most
extreme lower and upper data points outside  the box but falling within a distance of 1.5 times the
interquartile range (that is, the length of the box) from either edge. The whiskers should generally not be
used to approximate the overall variance under either formulation.

     A convenient tactic when using box plots to screen for heteroscedasticity is to plot the residuals of
each data group  rather than the measurements themselves.  This will  line the boxes up at roughly a
common level (close to zero), so that a visual comparison of box lengths is easier.

       REQUIREMENTS AND ASSUMPTIONS

     The requirements and assumptions for box plots are discussed in Section 9.2.

       PROCEDURE

Step 1.   For each of j wells or data groups, compute the sample mean of that group x.. Then compute
         the residuals  (r;j) for each  group  by subtracting the group mean from  each individual
         measurement:r = x  -x .
                      y   v    i

Step 2.   Use the procedure outlined in Section 9.2  to create side-by-side box plots of the residuals
         formed in Step 1. Then compare the box lengths to check for possibly unequal variances.

       ^EXAMPLE 11-1

     Construct  box plots  on the  residuals  for  each of the  following  well  groups  to check for
homoscedasticity.

                                            11-2                                  March 2009

-------
Chapter 11. Testing Equality of Variance
Unified Guidance
Month
1
2
3
4
Well 1
22.9
3.1
35.7
4.2
Well 2
2.0
1.2
7.8
52
Arsenic Concentration (ppb)
Well 3 Well 4
2.0
109.4
4.5
2.5
7.8
9.3
25.9
2.0
Well 5
24.9
1.3
0.8
27
Well 6
0.3
4.8
2.8
1.2
       SOLUTION
Step 1.   Form the residuals for each well by subtracting the sample well mean from each observation,
         as shown in the table below.
Month
1
2
3
4
Mean
Well 1
6.43
-13.38
19.22
-12.28
16.48
Well 2
-13.75
-14.55
-7.95
36.25
15.75
Arsenic Residuals (ppb)
Well 3 Well 4
-27.6 -3.45
79.8 -1.95
-25.1 14.65
-27.1 -9.25
29.6 11.25
Well 5
11.4
-12.2
-12.7
13.5
13.5
Well 6
-1.98
2.52
0.52
-1.08
2.28
Step 2.   Follow the procedure in Section 9.2 to compute a box plot of the residuals for each well. Line
         these up side by side on the same graph, as in Figure 11-1.

Step 3.   Compare the box lengths. Since the box length for Well 3 is more than three times the box
         lengths of Wells 4 and 6, there is informal evidence that the population group variances may
         be different. These data should be further checked using a formal test and perhaps a variance
         stabilizing transformation attempted. -^
                                            11-3
        March 2009

-------
Chapter 11. Testing Equality of Variance
Unified Guidance
               Figure 11-1. Side-by-Side Box Plots of Arsenic Residuals
                      8 -
                                   Wsll2
                                          WelB
                                                 We!l4
                                                        WollS
                                                               Wane
 11.2 LEVENE'S TEST

       PURPOSE AND  BACKGROUND

     Levene's test is a formal procedure for testing homogeneity of variance that is fairly robust (i.e.,
 not overly sensitive) to non-normality in the data. It is based on computing the new variables:

                                                                                      [11.1]

 where xy represents they'th sample value from the rth group (e.g.., well) and x.. is the rth group sample
 mean.  The symbol (•)  in the notation for the group sample mean represents an averaging over subscript
j.   The values zt] then  represent the absolute values of the residuals.  Levene's test involves running a
 standard one-way ANOVA (Chapter 17) on the variablesz.;. If the F-test is significant, reject the
 hypothesis of equal group variances and perhaps seek a variance stabilizing transformation. Otherwise,
 proceed with analysis of the original xt 's.
     Levene's test is based on a one-way ANOVA and contrasts the means of the groups being tested.
This implies a comparison between averages of the form:
                                            11-4
       March 2009

-------
Chapter 11. Testing Equality of Variance                                  Unified Guidance
                                      z,. =•
Such averages of the z^. 's are very similar to the standard deviations of the original data groups, given
by the formula:
       In both cases, the statistics are akin to an average absolute residual. Therefore, the comparison of
means in Levene's test is closely related to a direct comparison of the group standard deviations, the
underlying aim of any test of variance equality.

       REQUIREMENTS  AND ASSUMPTIONS

     The requirements and assumptions  for  Levene's test are essentially the  same as the one-way
ANOVA in Section 17.1, but applied to the absolute residuals instead of the raw measurements.

       PROCEDURE

Step 1.   Suppose there are p data groups to be compared.  Because there may be different numbers of
         observations per well, denote the sample  size of the /'th group by n\ and the total number of
         data points across  all groups by N.

         Denote the observations in the /'th group by xi}.  for /' = 1 . . .p and/' = 1 . . .n\. The first subscript
         then designates the well, while the second denotes the /th value in the  /'th well. After
         computing the sample mean  (x. ) for each group,  calculate the absolute residuals (zjy. ) using
         equation [11.1].

Step 2.   Utilizing the  absolute residuals  — and not the original data — compute the mean of each
         group along with the overall (grand) mean of the combined data set using the formula:
Step 3.   Compute the sum of squares of differences between the group means and the grand mean,
         denoted SSgrps:


                             SS   =Tn.(z. -z  t =T«J2-7VF2                        [11.5]
                               grps  1—t i\ i»    •• /   Z_^ i i*     ••                        L    J
                                     i=l              i=l

         The  formula on the far right is usually the most convenient for calculation. This sum of
         squares has (p-l) degrees of freedom associated with  it and is a measure of the variability
         between groups. It constitutes the numerator of theF-statistic.
                                             11-5                                   March 2009

-------
Chapter 11. Testing Equality of Variance	Unified Guidance

Step 4.   Compute the corrected total sum of squares, denoted by -SIStotai:
                            ss  , = yyu -z  )=yvz2-7VF2                       [n.e]
                              total   /—I /—I \ ij   •• J   Z_f Z_f  7;     ••                       L    J
                                   z=l ;=1            i=l ;=1

         Again, the formula on the far right is usually the most computationally convenient. This sum
         of squares has (A/-1) associated degrees of freedom.

Step 5.   Compute the sum  of squares  of differences between the absolute residuals  and the group
         means. This is known as the within-groups  component of  the  total  sum  of  squares or,
         equivalently, as the sum of squares due to error. It is easiest to obtain by subtracting SS^pS
         from XS'totai and is denoted SSenor:

                            P "' ,       N2                  P  "i      P
                   ss   =yyfc..-z.) =ss  ,-ss  =yy>2-y>.z2              [n.?]
                      error    ' ' '  ' \ 77   7» /      total     frrss   ' ' ' ' 77   ' ' 7  7»              L    J
                                              total     grps  4-^ 4-^  ij
                                                         7 = 1 j=\     7 = 1

               is associated with (N-p) degrees of freedom and  is a measure of the variability within
         groups. This quantity goes into the denominator of the F-statistic.

Step 6.   Compute  the  mean sum of  squares  for both the  between-groups and  within-groups
         components of the total sum of squares, denoted by MS grps  and MSenor. These quantities are
         obtained by dividing each sum of squares by its corresponding degrees of freedom:

                                    MS   =SS  /(p-l)                              [11.8]
                                       grps     grps I \f   /                              L    J

                                   MS   =SS   /(N-p)                              [11.9]
                                       error     error /  \    r /                              L    J

Step 7.   Compute the F-statistic by forming the ratio between the mean sum of squares for wells and
         the mean sum of squares due to error, as in Figure 11-2 below. This layout is known as the
         one-way parametric ANOVA table and illustrates each sum of squares component of the total
         variability, along with the corresponding degrees of freedom, the mean squares  components,
         and the final F-statistic calculated as F = M$rgrpS/M$'error. Note that the first two rows of the
         one-way table sum to the last row.

Step 8.   Figure 11-2 is a generalized ANOVA table for Levene's test. To test the hypothesis of equal
         variances across all p well groups, compare the F-statistic in Figure 11-2 to the a-level critical
         point found from the F-distribution with (p-l) and (N-p) degrees of freedom in Appendix D
         Table 17-1. When testing variance equality, only severe levels of difference typically impact
         test performance in a substantial way. For this reason, the Unified Guidance recommends
         setting a = .01 when screening multiple wells and/or constituents using Levene's test. In that
         case, the  needed critical point equals the upper 99th  percentage point of the F-distribution. If
         the observed F-statistic exceeds the critical point CF.99,P-i,N-p),  reject the hypothesis of equal
         group population  variances.  Otherwise, conclude that there  is insufficient  evidence  of  a
         significant difference between the variances.
                                            11-6                                   March 2009

-------
Chapter 11.  Testing Equality of Variance
Unified Guidance
                     Figure 11-2. ANOVA Table for Levene's Test
Source of Variation
Between Wells
Error (within wells)
Total
Sums
of Squares
-'-'grps
•^•^error
SStotal
Degrees of
Freedom
p-1
A/-p
A/-1
Mean Squares
MSgrps= SSgrps/(p-l)
MSenor = SSerror/(/V-p)
F-Statistic
F=MS"/MS™
       ^EXAMPLE 11-2

     Use the data from Example 11-1 to conduct Levene's test of equal variances at the a = 0.01 level
of significance.

       SOLUTION
Step 1.    Calculate the group arsenic mean for each well (x.t):

             Well  1 mean = 16.47 ppm          Well 4 mean = 11.26 ppm

             Well  2 mean = 15.76 ppm          Well 5 mean = 13.49 ppm

             Well  3 mean = 29.60 ppm          Well 6 mean = 2.29 ppm

         Then compute the absolute  residuals zy- in each well using equation [11.1]  as in the table
         below.
Month

Well
1
2
3
4
Mean (z..)
Overall Mean (F, )
Well 1
6.43
13.38
19.23
12.29
12.83
15.36
Absolute Arsenic
Well 2 Well 3
13.76
14.51
7.96
36.24
18.12

27.6
79.8
25.1
27.1
39.9

Residuals (;
Well 4
3.42
1.96
14.64
9.26
7.32

Well 5
11.41
12.19
12.74
13.51
12.46

Well 6
1.95
2.49
0.56
1.09
1.52

Step 2.   Compute the mean absolute residual (z..) in each well and then the overall grand mean using
         equation [11.4]. These results are listed above.

Step 3.   Compute the between-groups sum of squares for the absolute residuals using equation [11.5]:
                                           11-7
       March 2009

-------
Chapter 11. Testing Equality of Variance	Unified Guidance


              SSgrps =[4(l2.83)2 +4(18.12)2 + ... + 4(l.52)2J-24-(15.36)2 =3,522.90


Step 4.   Compute the corrected total sum of squares using equation [11.6]:

                 SStotal =[(6.43)2 +(13.38)2 + ... + (l.09)2J- 24- (15.36)2 =6,300.89

Step 5.   Compute the within-groups or error sum of squares using equation [11.7]:

                             SS   =6,300.89-3,522.90 = 2,777.99
                               error   '          ?         '

Step 6.   Given that the number of groups is p = 6 and the total sample size is N = 24, calculate the
         mean squares for the between-groups and error components using formulas [11.8] and [11.9]:

                               MS   =3,522.90/(6-l)= 704.58
                                  grps    '      / \   /

                              MS   =2,777.99/(24-6)=154.33
                                 error    7      /  \      J

Step 7.   Construct an ANOVA table following Figure 11-2 to calculate the F-statistic. The numerator
         degrees of freedom \df\ is computed as (p-1) = 5,  while the denominator dfis equal to (N-p) =
         18.
Source of Variation
Between Well Grps
Error (within grps)
Total
Sums of Squares
3,522.90
2,777.99
6,300.89
Degrees of
Freedom
5
18
23
Mean Squares
704.58
154.33
F-Statistic
4.56
Step 8.   Determine the .01-level critical point for the F-test with 5 and 18 degrees of freedom from
         Table 17-1. This gives F,99^^ = 4.25. Since the F-statistic of 4.56 exceeds the critical point,
         the assumption of equal variances should be rejected. Since the original concentration data are
         used in this example, a transformation such as the natural logarithm might be tried and the
         transformed data retested.  -4
11.3 MEAN-STANDARD DEVIATION  SCATTER PLOT

       BACKGROUND AND PURPOSE

     The mean-standard deviation scatter plot is described in Chapter 9. It is useful as an exploratory
tool for multiple groups of data (e.g., wells) to aid in identifying relationships between mean levels and
variability.  It is also helpful in providing  a visual assessment of variance homogeneity across data
groups. Like side-by-side box plots, the mean-standard  deviation scatter plot graphs a measure  of
variability for each well. In the latter, however,  the  standard deviation is plotted  rather  than the
interquartile range, so a more  direct assessment of variance equality can be made.  Since standard

                                            11-8                                  March 2009

-------
Chapter 11. Testing Equality of Variance	Unified Guidance

deviations (and consequently variances) are often  positively correlated with sample mean levels in
skewed populations, the observed pattern on the mean-standard deviation scatter plot can offer valuable
clues as to what sort of variance stabilizing transformation if any might work.

      REQUIREMENTS AND ASSUMPTIONS

     The requirements for the mean-standard deviation scatter plot are listed in Section 9.4.

      PROCEDURE

     See Section 9.4.

      ^EXAMPLE 11-3

     Use the data from Example 11-1 to construct a mean-standard deviation scatter plot.

      SOLUTION
Step 1.   First compute the sample mean (x ) and standard deviation (s) of each well, as listed below.
Well
1
2
3
4
5
6
Mean
16.468
15.762
29.600
11.260
13.488
2.292
Std Dev
15.718
24.335
53.211
10.257
14.418
1.958
Step 2.   Plot the well means versus the standard deviations as in Figure 11-3 below. Note the roughly
         linear relationship between the magnitude of the standard deviations and their corresponding
         means. The data suggest unequal variances among the wells, as indicated by the large range in
         the standard deviations.
                                            11-9                                  March 2009

-------
Chapter 11.  Testing Equality of Variance
                         Unified Guidance
                               Figure 11-3, Arsenic Mean-Standard Deviation Plot
                      <
                      «
                                        10
                                               —i—
                                                IS
—I—
 20
—i—
 25
                                                                       30
                                            Mean Afsenic (ppb)
Step 3.   Because lognormal  data groups will tend to show a linear association between the sample
         means  and  standard  deviations,  apply  a log  transformation  to the  original  arsenic
         measurements  and  reconstruct the mean-standard deviation  scatter plot  on the log scale.
         Computing the log-means and log-standard deviations and then re-plotting gives Figure 11-4.
         Now the apparent trend between the means and standard deviations is gone. Further, on the
         log scale, the standard deviations are much more similar in magnitude, all with values between
         1 and 2. The log transformation thus appears to roughly stabilize the arsenic variances. -4
                        KJ
                        cj
                      Q,  O
                      &  «M
                      O>
                      3
                      u
                     
-------
Chapter 12. Identifying Outliers	Unified Guidance

              CHAPTER 12.   IDENTIFYING  OUTLIERS
       12.1   SCREENING WITH PROBABILITY PLOTS	12-1
       12.2   SCREENING WITH Box PLOTS	12-5
       12.3   DKON'STEST	12-8
       12.4   ROSNER'STEST	 12-10
     This chapter discusses  screening tools and formal tests for identifying statistical  outliers.  Two
screening tools are first presented: probability plots (Section 12.1) and box plots (Section 12.2). These
are followed by two formal outlier tests:

   »»»  Dixon's test (Section 12.3) for a single outlier in smaller data sets, and
   *»*  Rosner's test (Section 12.4) for up to five separate outliers in larger data sets.
     A statistical  determination  of one or  more  statistical  outliers  does   not  indicate  why the
measurements are discrepant from the rest of the data set. The Unified Guidance does not recommend
that  outliers  be  removed solely on a statistical  basis.   The  outlier  tests can provide supportive
information, but  generally a reasonable rationale needs to be identified  for removal of suspect outlier
values  (usually limited to background data).  At the same time there must be some level of confidence
that the data are representative of ground water quality. A number  of factors and considerations in
removing outliers from potential background data are discussed in Section 5.2.3.

12.1  SCREENING WITH PROBABILITY  PLOTS

       BACKGROUND AND PURPOSE

     Probability plots (Chapter 9) are helpful in identifying outliers in at least two ways. First, since the
straightness of the plot indicates how closely the data fit the pattern of a normal distribution, values that
appear "out of line" with the remaining data can be visually identified as possible outliers. Secondly, the
two formal outlier tests presented in the Unified Guidance assume that the underlying population minus
the suspected outlier(s) is  normal. Probability plots can provide visual evidence  for this assumption.
Data that appear non-normal after the suspected outliers have been removed from the probability plot
may need to be transformed (e.g.,  via the natural logarithm) and re-examined on the transformed scale to
see if potential outliers are still apparent.

     As an aid to the interpretation of a given probability plot, the Unified Guidance recommends
computation of the probability plot correlation coefficient,  using either Filliben's  procedure (Chapter
10) or  the simple (Pearson) correlation (Chapter 3) between the numerical pairs plotted on the graph.
The higher the correlation, the more linear the pattern is on the probability plot and therefore a better fit
to normality. Note that while the Filliben correlation coefficient can be compared to  critical points
derived for that test of normality (Chapter 10), a low correlation may be related  to other causes of non-
normality besides the presence of outliers. The correlation coefficient is not a  substitute for a formal
outlier test, but can be useful as a screening tool.
                                             12-1                                   March 2009

-------
Chapter 12.  Identifying Outliers	Unified Guidance

       REQUIREMENTS AND ASSUMPTIONS

     Probability plots are primarily a tool to assess normality, and not to identify outliers per se. It is
critical that the remaining data without potential outliers is  either normal in distribution  or  can be
normalized via a transformation. Otherwise, the probability plot may appear non-linear and non-normal
for reasons unrelated to the presence of outliers. Right-skewed lognormal distributions can  appear to
have one or more outliers on a probability plot unless the original data are first log-transformed. As a
general rule, probability plots should be constructed on the original (or raw) measurements and one or
more transformed data sets (e.g., log or square root), in order to avoid mistaking inherent data skewness
for outliers.

     If the raw and transformed-data probability plots both indicate one or more values inconsistent
with the pattern of the remaining values, continue with a second level of screening by temporarily
removing the  suspected  outlier(s) and  re-constructing the probability  plots.  If the raw-scale  plot is
reasonably linear, consider running  a formal  outlier test on the original measurements. On the  other
hand, if the raw-scale  plot is skewed but the transformed-scale plot is linear, consider conducting a
formal outlier test on the transformed measurements.

     A related difficulty occurs when sample data includes  censored or non-detect values.  If simple
substitution is used to estimate a value for each non-detect prior to plotting, the resulting probability plot
may appear non-linear simply because the censored observations were not properly handled. In this case,
a censored probability plot (Chapter 15) should be constructed instead of an uncensored,  complete
sample plot (Chapter 9). The same caveats apply to normalizing the sample data, perhaps by attempting
at least  one transformation.  The  only  difference is that  each  probability plot  constructed  must
appropriately account for the observed censoring in the sample.

     PROCEDURE

Step 1.   After identifying one or more possible outliers (e.g., values much higher in concentration than
         the remaining measurements), construct  a probability plot on the  entire  sample  using  the
         procedure described in Section 9.5.  Construct a censored probability plot from Section 15.3
         if the sample contains  non-detects. If the data including the suspected outlier(s) follow a
         reasonably linear pattern, a formal outlier test is probably unnecessary. However, if one or
         more values are out of line compared to the pattern of the remaining data, construct a similar
         probability plot after applying one or more transformations. If one or more suspected outliers
         is still inconsistent, proceed to Step 2.

Step 2.   Compute  a probability plot correlation coefficient for each plot constructed in Step 1. Use
         these correlations as an aid to interpreting the degree of linearity in each probability plot.

Step 3.   Reconstruct the  probability plots  from  Step  1  after removing  the  suspected   outlier(s).
         Recompute the correlation coefficients from Step 2 on this reduced sample.

Step 4.   If the 'outlier-deleted' probability plot on the raw concentration scale indicates a linear pattern
         with  high correlation, consider running a formal outlier test on the original measurements.
         When the pattern  is distinctly  non-linear but the corresponding  probability  plot  on  the
         transformed-scale is fairly linear (and  higher in correlation),  conduct the outlier test on  the
         transformed values.
                                              IF2                                    March 2009

-------
Chapter 12. Identifying Outliers
                                                                   Unified Guidance
       ^EXAMPLE 12-1

     The table below contains data from five background wells measured over a four month period. The
value 7,066 is found in the second month at Well 3. Use probability plots on the combined sample to
determine whether or not a formal outlier test is warranted.
                 Well 1
                  Carbon Tetrachloride Concentrations (ppb)
                      Well 2        Well 3         Well 4
         Well 5
1.7
3.2
7.3
12.1
302
35.1
15.6
13.7
16.2
7066
350
70.1
199
41.6
75.4
57.9
275
6.5
59.7
68.4
       SOLUTION
Step 1.   Examine the probability plots of the entire sample first using the raw measurements and then
         log-transformed values (Figures 12-1 and 12-2). Both these plots indicate that the suspected
         outlier does not follow the pattern of the remaining observations, but seems 'out of line.' The
         Pearson correlation coefficients for these probability  plots are, respectively, r = 0.513 and
         0.975,  indicating that  the fit  to normality overall is much  closer using log-transformed
         measurements.

            Figure 12-1.  Probability Plot on Raw Concentrations (r =  .513)
Step 2.
                   2.50
                   1.25
               OJ

               8   o.oo
               CO
               r\i
                  -1.25
                  -2.50
                                        2000
                                          4000
6000
8000
                                     Carbon tetrachloride (ppb)
Next remove the suspected outlier and reconstruct the probability plots on both the original
and logged observations (Figures 12-3 and 12-4).  The plot on the original scale indicates
heavy positive (or right-) skewness and a non-linear pattern, while the plot on the log-scale
exhibits a fairly linear pattern. The respective correlation coefficients now become r = 0.854
and 0.985, again favoring the  log-transformed sample. On the  basis  of these plots, the
                                             12-3
                                                                          March 2009

-------
Chapter 12. Identifying Outliers
    Unified Guidance
        underlying data should be modeled as lognormal and the observations logged prior to running
        a formal outlier test. -^


          Figure 12-2. Probability Plot on Logged Observations (r = .975)


                   2
                  -2
                     0.0         2.5         5.0          7.5


                              Log(carbon tetrachloride) log(ppb)
 10.0
      Figure 12-3. Outlier-Deleted Probability Plot on Original Scale (r = .854)

                     2
                 CD


                 8   o
                 OT
                 I
                 M
                    -1
                    -2
                            0        100       200       300


                                  Carbon Tetrachloride (ppb)
400
                                          12-4
           March 2009

-------
Chapter 12. Identifying Outliers
                                           Unified Guidance
    Figure 12-4. Outlier-Deleted Probability Plot on Logarithmic Scale (r =  .985)

                    2
                CD
                O
                (J
                (f)
                I
                N
                   -1
                   -2
                     0.00
1.25
2.50
3.75
5.00
6.25
                               Log(carbon tetrachloride) log(ppb)
12.2 SCREENING WITH BOX PLOTS

       BACKGROUND AND PURPOSE

     Probability plots as described  in  Section  12.1  require the  remaining  observations following
removal of one or  more suspected  outliers to be  either approximately normal  or  normalized  via
transformation.  Box plots (Chapter  9) provide an alternate method to perform outlier screening, one
not dependent on normality of the underlying measurement population. Instead of looking for points
inconsistent with a linear pattern on a probability plot, the box plot flags as possible outliers values that
are located in either or both of the extreme tails of the sample.

     To define the extreme tails, Tukey (1977) proposed the concept of 'hinges' that would 'swing' off
either end of a box plot, defining the  range of concentrations consistent with the bulk of the data. Data
points outside this concentration range could then be identified as potential outliers. Tukey defined the
hinges, i.e., the lower and upper edges of the box plot, essentially as the lower and upper quartiles of the
data set. Then multiples of the interquartile range [IQR]  (i.e., the range represented by the middle half of
the sample)  were added  to  or subtracted  from these hinges as  potential outlier boundaries. Any
observation from 1.5 x IQR to 3 x IQR below the lower edge of the box plot was labeled a 'mild' low
outlier; any value more than 3 x IQR below the lower edge of the box plot was labeled an 'extreme' low
outlier. Similarly, values greater than the upper edge of the box plot in the range of 1.5 to 3 times the
IQR were labeled 'mild' higher outliers, and  'extreme' high outliers if more than 3  times the IQR
beyond the upper box plot edge.

       REQUIREMENTS AND ASSUMPTIONS

     By using hinges and multiples of the interquartile range, Tukey's box plot method utilizes statistics
(i.e., the lower and upper quartiles) that are generally not or minimally affected by one or a few outliers
                                            12-5
                                                  March 2009

-------
Chapter 12. Identifying Outliers _ Unified Guidance

in the sample. Consequently, it isn't necessary to first delete possible outliers before constructing the
box plot.

     Screening for outliers with box plots is a very simple technique. Since no assumption of normality
is needed, Tukey's procedure can be considered quasi-non-parametric. But note that rough symmetry of
the underlying distribution  is  implicitly  assumed. Legitimate observations from  highly  skewed
distributions could be flagged as potential outliers on a box plot if no transformation of the data is first
attempted. It may be necessary to first conduct multiple  data  transformations in order  to  achieve
approximate symmetry before applying and evaluating potential outliers with box plots.

PROCEDURE

Step 1 .   Construct a box plot on the sample using the method given in Section 9.2. Using the IQR from
         that calculation, along with the lower and upper quartiles (  x25 and x75), compute the first pair
         of lower and upper boundaries as:

                                     LB1 = x25- \.5xIQR                                (12.1)


                                     UB1 = x75 + \5xIQR                                (12.2)

Step 2.   Construct the second pair of lower and upper boundaries as:

                                     LB2  = x25-3xIQR                                (12.3)


                                     UB2  = x_75 + 3 x IQR                                (12.4)
Step 3.   Label any sample measurement lower than the first lower boundary (£#1) but no less than the
         second lower boundary (LB^) as a mild low outlier. Label any measurement greater than the
         first upper boundary (UB\) but no greater than the second upper boundary (UB^) as a mild high
         outlier.

Step 4.   Label any sample measurement lower than the second lower boundary (LB^)  as an extreme
         low outlier. Label any value higher than the second upper boundary (UB2) as an extreme high
         outlier.

       ^EXAMPLE 12-2

     Use the carbon tetrachloride data from Example 12-1 to screen for possible outliers using Tukey's
box plot.

       SOLUTION
Step 1.   Using the procedure described in  Section 9.2, the upper and  lower quartiles of carbon
         tetrachloride sample are found to be  x25  = 12.9 and x75 = 137.2, leading to an  IQR = 124.3.

Step 2.   Compute the two pairs  of lower and upper boundaries using equations (12.1), (12.2), (12.3),
         and (12.4):
                                            IF6                                  March 2009

-------
Chapter 12. Identifying Outliers
             Unified Guidance
                               LBl = 12.9 -1.5 x 124.3 = -173.55
                               UBl = 137.2 +1.5 x 124.3 = 323.65
                               LB2 = 12.9- 3x124.3 = -360
                               UB2 = 137.2 + 3x124.3 = 510.1

Step 3.   Scan the list of carbon tetrachloride  measurements and compare against the boundaries of
         Step 2. It can be seen that the value of 350 from Well 3 is greater than UB\ but lower than
         UB2, thus qualifying as a mild high outlier. Also, the measurement 7,066 from the same well
         is higher than UBi and so qualifies as an extreme high outlier.

Step 4.   Because the box plot outlier screening method assumes roughly symmetric data, recompute
         the  box plot on the log-transformed  measurements (as shown in Figure 12-5 alongside a
         similar box plot of the raw concentrations). Transforming the sample to the log-scale does
         result in much greater symmetry compared to the original  measurement scale.   This can be
         seen in the close similarity between the mean and median  on the log-scale box plot. With a
         more symmetric data set, the mild high outlier from Step 3 disappears, but the  extreme high
         value is still classified as an outlier. -4

    Figure 12-5. Comparative Carbon Tetrachloride Box  Plots Indicating Outliers

                 I
                                #
#
                                            12-7
                     March 2009

-------
Chapter 12.  Identifying Outliers	Unified Guidance



12.3 DIXON'STEST

       BACKGROUND AND PURPOSE

     Dixon's test is helpful for documenting statistical outliers in smaller data sets (i.e., n < 25). The
test is particularly designed for cases where there is only a single high or low outlier, although it can also
be adapted to test for multiple outliers. The test falls in the general class of tests for discordancy (Barnett
and  Lewis,  1994).  The test statistic for such procedures is  generally a ratio: the numerator is  the
difference between the suspected outlier and  some  summary  statistic  of the data set, while  the
denominator is always a measure of spread within the data. In this version of Dixon's test, the  summary
statistic in the numerator is an order statistic nearby to the potential outlier (e.g., the second or third most
extreme value). The measure of spread is essentially the observed sample range.

     If there is more than  one  outlier in the data set, Dixon's test can be vulnerable to masking, at least
for very small samples.  Masking in the statistical literature refers to the problem  of an extreme outlier
being missed because one  or more additional extreme outliers  are also present. For instance, if the data
consist of the values (2, 4, 10,  12,  15, 18, 19, 22, 200, 202}, identification of the maximum value (202)
as an outlier might fail since the maximum by itself is not extreme with respect to the next highest value
(200). However, both of these values are clearly much higher than the rest of the data set and might
jointly be considered outliers.

     If more than one outlier is suspected, the user is encouraged to consider Rosner's test (Section
12.4) as an alternative to Dixon's test, at least if the sample size is 20 or more. If the data set is smaller,
Dixon's test should be modified so that the least extreme of the suspected outliers is tested first. This
will  help avoid the risk  of masking. The same equations given below can be used, but the data set and
sample size  should be temporarily reduced to exclude any suspected outliers that are more extreme than
the one being tested. If a less extreme value is found to be an outlier, then that observation and any more
extreme values can also be regarded as outliers. Otherwise, add back the next most extreme value and
test it in the  same way.

       REQUIREMENTS AND  ASSUMPTIONS

     Dixon's test is only recommended for sample  sizes n < 25.  It assumes that the data set (minus the
suspected outlier) is normally-distributed. This assumption should be checked prior to running Dixon's
test using a goodness-of-fit technique such as  the probability plots described in Section 12.2.

       PROCEDURE

Step 1.   Order the data set and label the ordered values, XQ.

Step 2.   If a  "low" outlier is suspected (i.e., X(i)), compute the test statistic C using the appropriate
         equation [12.5] depending on the sample size (n):
                                             12-8                                    March 2009

-------
Chapter 12. Identifying Outliers
                                                                    Unified Guidance
                          C =
                                                                                [12.5]
Step 3.   If a "high" outlier is suspected (i.e., X(n)), and again depending on the sample size (n), compute
         the test statistic C using the appropriate equation [12.6] as:
                          C =
                                                                                [12.6]
Step 4.
In either case, given the significance level (a), determine a critical point for Dixon's test with
n observations from Table 12-1 in Appendix D. If C exceeds this critical point, the suspected
value should be  declared a  statistical outlier  and investigated further (see discussion  in
Chapter 6)
       ^EXAMPLE 12-3

     Use the data from Example 12-1 in Dixon's test to determine if the anomalous high value is a
statistical outlier at an a = 0.05 level of significance.

       SOLUTION
Step 1.   In Example 12-1, probability plots of the carbon tetrachloride data indicated that the highest
         value might be an  outlier, but that the distribution of the measurements was more nearly
         lognormal than normal. Since the sample size n = 20, Dixon's test can be used on the logged
         observations. Logging the values and ordering them leads to the following table:
                                             12-9
                                                                            March 2009

-------
Chapter 12. Identifying Outliers                                          Unified Guidance
Order
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Concentration
(ppb)
1.7
3.2
6.5
7.3
12.1
13.7
15.6
16.2
35.1
41.6
57.9
59.7
68.4
70.1
75.4
199.0
275.0
302.0
350.0
7066.0
Logged
Concentration
0.531
1.163
1.872
1.988
2.493
2.617
2.747
2.785
3.558
3.728
4.059
4.089
4.225
4.250
4.323
5.293
5.617
5.710
5.878
8.863
Step 2.   Because a high  outlier is suspected  and n = 20, use the  last option of equation [12.6] to
         calculate the test statistic C:

                                       8.863-5.710
                                       8.863-1.872

Step 3.   With n = 20 and a = .05, the critical point from Table 12-1 in Appendix D is equal to 0.450.
         Since the test statistic C exceeds this critical point, the extreme high value can be declared a
         statistical outlier.  Before excluding this value from  further  analysis, however,  a valid
         explanation for this unusually high value should be sought. Otherwise, the outlier may need to
         be treated as an extreme but valid concentration measurement. -^
12.4 ROSNER'STEST

       BACKGROUND AND PURPOSE

     Rosner's test (Rosner,  1975) is a useful method for identifying multiple outliers in moderate to
large-sized data sets. The approach developed in Rosner's method is known as a block-style test. Instead
of testing for outliers one-by-one in a consecutive manner from most extreme to least extreme (i.e., most
to least suspicious), the data are examined first to identify the total number of possible outliers, k. Once k
is determined, the set of possible outliers is tested together as a block. If the test is significant, all k
measurements are regarded as statistical outliers. If not, the set of possible outliers is reduced by one and
the test repeated on the smaller block. This procedure is iterated until either a set of outliers is identified
                                             12-10                                  March 2009

-------
Chapter 12.  Identifying Outliers _ Unified Guidance

or none of the observations are labeled an outlier. By testing outliers in blocks instead of one-by-one,
Rosner's test largely avoids the problem of masking of one outlier by another (as discussed in Section
12.3 regarding Dixon's test).

     Although Rosner's test avoids the problem of masking when multiple outliers are present in the
same data set, it is not immune to the related problem of swamping.   A good discussion is found in
Barnett and Lewis, 1994, Outliers in Statistical Data (3rd Edition), p. 236.  Swamping refers to a block
of measurements all being labeled as  outliers even though only some  of the observations are actually
outliers. This can occur with Rosner's test especially if all the outliers tend to be at one end of the data
set (e.g., as upper extremes). The difficulty is in properly identifying the total number of possible outliers
(&), which can be low outliers, high outliers, or some combination of the two extremes. If k is made too
large, swamping may occur.  Again, the  user  is reminded to always  do a preliminary screening for
outliers via box plots (Section 12.2) and probability plots (Section 12.1).

       REQUIREMENTS AND ASSUMPTIONS

     Rosner's test is recommended when the sample size (n) is 20 or larger. The critical points provided
in Table 12-2 in Appendix D can  be used  to identify from 2 to 5 outliers in a given data set. Like
Dixon's test,  Rosner's  method assumes the  underlying data set (minus any outliers)  is  normally
distributed. If a probability plot of the  data exhibits significant bends or curves,  the data should first be
transformed (e.g.,  via a logarithm)  and then re-plotted. The formal  test for outliers should only be
performed on (outlier-deleted) data sets that have been approximately normalized.

     A potential drawback of Rosner's test is that the user must first identify the maximum number of
potential outliers (k) prior to running the test. Therefore, this requirement makes the test ill-advised as an
automatic outlier screening tool, and somewhat reliant on the user to identify candidate outliers.

       PROCEDURE

Step 1 .   Order the data set and denote the ordered values XQ. Then by simple  inspection, identify the
         maximum number of possible outliers, r$.

Step 2.   Compute the sample mean and standard deviation of all the data; denote these values by x^ '
         and 5^. Then  determine the measurement furthest from  x^' and denote ity^°\ Note thaty0-*
         could be either a potentially low or a high outlier.

Step 3.   Delete y^ from the data set and compute the sample mean and standard deviation from the
         remaining  observations. Label these new  values x^'  and s^\ Again find the value in  this
         reduced data set furthest from x^' and label it_y(1).

Step 4.   Delete _y(1), recompute the mean and standard deviation, and continue  this process until all r0
         potential outliers have been removed.  At this point, the following set of  statistics will be
         available:
                                                                                         [12.7]

Step 5.   Now test for r outliers (where r < r0) by iteratively computing the test statistic:
                                             12-11                                   March 2009

-------
Chapter 12.  Identifying Outliers	                                    Unified Guidance


                                    ^-1 =
|/sH)                               [12.8]
         First test for r0 outliers. If the test statistic Rr_l  in equation [12.8] exceeds the first critical

         point from Table 12-2 in Appendix D based on sample size  («) and the Type I error (a),
         conclude there are TO outliers. If not, test for ro-l outliers in the same fashion using the next
         critical  point, continuing  until a certain number of outliers have either been identified or
         Rosner's test finds no outliers at all.

       ^EXAMPLE  12-4

     Consider the following series  of 25 background napthalene measurements (in ppb). Use Rosner's
test to determine whether any of the  values should be deemed statistical outliers.
Qtr
1
2
3
4
5
BW-1
3
5
5
6
5
34
39
74
88
85
Naphthalene
BW-2
5.59
5.96
1.47
2.57
5.39
Concentrations (ppb)
BW-3 BW-4
1.91
1.74
23.23
1.82
2.02
6
6
5
4
1
12
05
18
43
00
BW-5
8
5
5
4
35
64
34
53
42
45
       SOLUTION
Step 1 .   Screening with probability plots of the combined data indicates a less than linear fit with both
         the raw measurements  and log-transformed  data (see Figures 12-6  and 12-7);  two points
         appear rather discrepant from the rest. Correlation coefficients for these plots are 0.740 on the
         concentration  scale and 0.951 on the log-scale. Re-plotting after removing the two possible
         outliers gives a substantially improved correlation on the concentration  scale of 0.958 but
         reduces the log-scale correlation to 0.929. Normality appears to be a slightly better default
         distribution for the outlier-deleted data set. Run Rosner's test on the original data with k = 2
         possible outliers.

Step 2.   Compute the  mean and  standard deviation  of the  complete data  set.  Then identify the
         observation farthest from the mean. These results are listed, along with the ordered data, in the
         table below. After removing the farthest value (35.45), recompute the mean and standard
         deviation on the remaining values and again identify the most discrepant observation (23.23).
         Repeat  this process one more time so that both suspected outliers have been removed (see
         table below).
Step 3.   Now test for 2 joint outliers by computing Rosner's statistic on subset SS^-i  = SS\ using
         equation [12.8]:

                                         23 23 - 5 23
                                     R=            =4.16
                                      ^     4.326
                                              12-12                                    March 2009

-------
Chapter 12.  Identifying Outliers
                                   Unified Guidance
                      Figure 12-6. Napthalene Probability Plot
               o
               o
               N
                  0
                 -1
                 -2
                     0
10         20          30


     Napthalene (ppb)
40
                    Figure 12-7.  Log Napthalene Probability Plot
              OJ



              S  o
              CO
                 -1
                 -2
                                 1            2           3



                                 Log Napthalene log(ppb)
                                       12-13
                                          March 2009

-------
Chapter 12.  Identifying Outliers                                            Unified Guidance
Successive
SS0
1.00
1.47
1.74
1.82
1.91
2.02
2.57
3.34
4.42
4.43
5.18
5.34
5.39
5.39
5.53
5.59
5.74
5.85
5.96
6.05
6.12
6.88
8.64
23.23
35.45
XQ = 6.44
S0 = 7.379
Yo = 35.45
Naphthalene Subsets
SSx
1.00
1.47
1.74
1.82
1.91
2.02
2.57
3.34
4.42
4.43
5.18
5.34
5.39
5.39
5.53
5.59
5.74
5.85
5.96
6.05
6.12
6.88
8.64
23.23

JCj = 5.23
Si = 4.326
Xi = 23.23
(SSj)
SS2
1.00
1.47
1.74
1.82
1.91
2.02
2.57
3.34
4.42
4.43
5.18
5.34
5.39
5.39
5.53
5.59
5.74
5.85
5.96
6.05
6.12
6.88
8.64


X2 = 4.45
S2 = 2.050
y2 = 8.64
Step 4.   Given a = 0.05, a sample size of n = 25, and k = 2, the first critical point in Table 12-2 in
         Appendix D equals 2.83 for n = 20 and 3.05 for n = 30. The value R\ in Step 3 is larger than
         either of these critical points,  so both suspected values may be declared statistical outliers by
         Rosner's test at the 5% significance level. Before excluding these values from further analysis,
         however,  a valid  explanation for them should be  found. Otherwise, treat the outliers as
         extreme but valid concentration measurements.

         Note: had R\ been less than these values, a test could still be run for a single outlier using the
         second critical point for each sample size (or a critical point interpolated between them). -^

       The  guidance  considers  Dixon's  and Rosner's  outlier evaluation methods  preferable for
groundwater monitoring data situations, when  assumptions of normality  are reasonable and data are
quantified.   We did  not include the older method found in the 1989 guidance  based on ASTM paper
El78-75, which can  still be used as an  alternative.  Where data do not appear to be fit by a normal or
transformably  normal distribution, other robust outlier evaluation methods can be considered from the
wider statistical literature.  The literature will also need to be consulted when data contains non-detect
values along with potential outliers.
                                              12-14                                    March 2009

-------
Chapter 13. Spatial Variability	Unified Guidance

                CHAPTER  13.  SPATIAL VARIABILITY
        13.1   INTRODUCTION TO SPATIAL VARIATION	13-1
        13.2   IDENTIFYING SPATIAL VARIABILITY	13-2
          13.2.1  Side-by-Side Box Plots	13-2
          13.2.2  One-Way Analysis of Variance for Spatial Variability	13-5
        13.3   USING ANOVA TO IMPROVE PARAMETRIC INTRAWELL TESTS	13-8
     This chapter discusses a type of statistical dependence in groundwater monitoring data known as
spatial  variability.  Spatial  variability  exists when  the distribution  or pattern of  concentration
measurements changes from well location to well  location (most typically in the form of differing mean
levels).  Such variation may be natural or synthetic, depending on whether it is caused by natural or
anthropogenic factors. Methods for identifying spatial variation  are detailed  via the use of box  plots
(Section 13.2.1) and analysis of variance [ANOVA] (Section 13.2.2). Once identified, ANOVA can
sometimes be employed to construct more powerful intrawell background limits. This topic is addressed
in Section 13.3.

13.1 INTRODUCTION  TO  SPATIAL  VARIATION

     Spatial dependence,  spatial variation or variability, and spatial correlation are closely related
concepts. All refer to the notion of measurement levels that vary in a structured way as a function of the
location  of sampling.  Although  spatial  variation  can apply to any statistical  characteristic of the
underlying population  (including the  population variance or upper percentiles),  the usual sense in
groundwater monitoring is that mean levels of a given constituent vary from one well to the next.

     Standard geostatistical models posit that an area exhibits positive spatial  correlation  if any two
sampling locations share a greater similarity in concentration level the closer the distance between them,
and more dissimilarity the further apart they are.  Such models have been applied to both groundwater
and soil  sampling problems,  but are  not applicable  in  all  geological configurations. It may be, for
instance, that mean concentration levels differ across wells but vary in a seemingly random way with no
apparent connection to the distance between the  sampling points. In that  case, the concentrations
between  pairs of wells are not correlated with distance, yet the measurements  within  each well are
strongly  associated  with the mean level  at that  particular location,  whether due to  a change in soil
composition, hydrological characteristics  or some other  factor. In other words,  spatial variation may
exist even when spatial correlation does not.

     Spatial variation is  important in the guidance  context since substantial  differences in mean
concentration  levels between  different wells can invalidate  interwell,  upgradient-to-downgradient
comparisons and point instead toward intrawell tests (Chapter 6).  Not all spatial variability is natural.
Average concentration levels can vary from well to well for a variety of reasons.

     In this guidance, a  distinction is occasionally made between natural versus  synthetic spatial
variation. Natural  spatial  variability  refers to a pattern of changing mean levels  in groundwater
associated with normal geochemical behavior unaffected by human activities. Natural spatial variability

                                              iTl                                   March 2009

-------
Chapter 13.  Spatial Variability	Unified Guidance

is not an indication of groundwater contamination, even if concentrations at one or more compliance
wells exceed (upgradient) background.  In contrast, synthetic spatial variability is  related  to human
activity. Sources can include recent releases affecting compliance wells, migration of contaminants from
off-site sources,  or historic contamination  at certain wells due to past industrial activity or pre-RCRA
waste disposal.  Whether natural or synthetic, techniques  and  test methods for dealing with  spatial
variation will still be identical from a purely statistical standpoint. It is interpreting the testing outcomes
which will necessitate a consideration of why the spatial variation occurs.

     The goal of groundwater analysis is not simply to identify significant concentration differences
among monitoring wells  at compliance point locations.  It is also to determine why those differences
exist. Especially with prior groundwater contamination, regulatory decisions outside the scope  of this
guidance need to address the problem. In  some cases, compliance/assessment monitoring or remedial
action may be warranted.  In other cases, chronic contamination from offsite sources may simply have to
be considered as the current background condition at a given location. At least the ability to attribute
certain mean differences to natural spatial variation allows the range of potential  concerns to be
somewhat narrowed. Of course, deciding that an observed pattern of spatial variation is natural and not
synthetic may not be easy. Ultimately, expert judgment and knowledge concerning  site  hydrology,
geology and geochemistry are important in providing more definitive answers.

     One statistical approach to use when a site has multiple,  non-impacted background wells is to
conduct a one-way ANOVA for inorganic constituents on those wells. Substantial differences among the
mean levels  at a set of uncontaminated sampling locations are suggestive of natural spatial variability. At
a true 'greenfield' site, ANOVA can be run on all the wells — both background and compliance — after
a few preliminary sampling rounds have been collected.

     The Unified Guidance offers two basic tools to explore and test for spatial correlation. The first,
side-by-side box plots (Section 13.2.1), provides a quick screen for possible spatial variation. When
multiple well data are plotted on the same concentration axis, noticeably staggered boxes are often an
indication of significantly different mean levels.

     A more formal test of spatial variation is the one-way ANOVA (Section 13.3.2).  When significant
spatial variation  exists and an intrawell test strategy is pursued,  one-way ANOVA can  also be used to
adjust the standard deviation estimate used in forming intrawell prediction and control chart limits, and
to increase the effective sample size of the  test, via the degrees of freedom. This is discussed in Section
13.3

13.2 IDENTIFYING  SPATIAL VARIABILITY

13.2.1       SIDE-BY-SIDE  BOX  PLOTS

       BACKGROUND AND PURPOSE

     Box plots  for graphing side-by-side  statistical summaries  of multiple wells were introduced in
Chapter 9.  They are also discussed in Chapter 11 as an initial screen for differences in population
variances and as a tool to check the assumption of equal variances in ANOVA. They can  further be
employed to screen for possible spatial variation in mean levels. While variability in a sample from a
given well is roughly indicated by the length of the box, the average concentration level is indicated by
the  position of the box relative to the concentration axis. Many standard box plot  software routines
                                              13-2                                   March 2009

-------
Chapter 13. Spatial Variability	Unified Guidance

display both  the sample median value and  the sample mean  on each box, so these values may be
compared from well to well. A high degree of staggering in the box positions is then indicative of
potentially significant spatial variation.

     Since side-by-side box plots provide a picture of the variability at each well, the extent to which
apparent  differences in mean levels seem  to be real rather than chance fluctuations  can be examined. If
the boxes are staggered but there is substantial overlap between them, the degree of spatial variability
may not be significant. A more formal ANOVA might still be warranted as a follow-up test, but side-by-
side box plots will offer a initial sense of how spatially variable the groundwater data appear.

       REQUIREMENTS,  ASSUMPTIONS AND PROCEDURE

     Requirements, assumptions and the procedure for box plots are outlined in Chapter 9, Section 9.2.

       ^EXAMPLE 13-1

     Quarterly dissolved iron concentrations measured at each of six upgradient wells are listed below.
Construct side-by-side box plots to initially screen for the presence of spatial variability.
Date
Jan 1997
Apr 1997
Jul 1997
Oct 1997
Mean
Median
Well 1
57.97
54.05
29.96
46.06
47.01
55.06
Well 2
46.06
76.71
32.14
68.03
55.71
57.04
Iron Concentrations (ppm)
Well 3 Well 4
100.48
170.72
39.25
52.98
90.86
76.73
34.12
93.69
70.81
83.10
70.43
76.96
Well 5
60.95
72.97
244.69
202.35
145.24
137.66
Well 6
83.10
183.09
198.34
160.77
156.32
171.93
       SOLUTION
Step 1.   Determine the median, mean, lower and upper quartiles of each well. Then plot these against a
         concentration  axis to form side-by-by side box plots  (Figure  13-1) using the procedure in
         Section 9.2

Step 2.   From this plot, the means and  medians at the last two  wells (Wells 5 and 6) appear elevated
         above the rest. This is a possible indication of spatial variation. However, the variances as
         represented by the box lengths also appear to differ, with the highest means associated with
         the largest boxes.  A transformation of the data should be  attempted and the  data re-plotted.
         Spatial variability is only a significant problem if it is apparent on the scale of the data actually
         used for statistical analysis.

Step 3.   Take  the logarithm of each measurement as in the table below. Recompute the mean, median,
         lower and upper quartiles, and then re-construct the box plot as in Figure 13-2.
                                             13-3                                   March 2009

-------
Chapter 13. Spatial Variability
Unified Guidance
Date
Jan 1997
Apr 1997
Jul 1997
Oct 1997
Mean
Median
Well 1
4.06
3.99
3.40
3.83
3.82
3.70
Log
Well 2
3.83
4.34
3.47
4.22
3.96
4.02
Iron Concentrations log(ppm)
Well 3 Well 4 Well 5
4.61
5.14
3.67
3.97
4.35
4.29
3.53
4.54
4.26
4.42
4.19
4.34
4.11
4.29
5.50
5.31
4.80
4.80
Well 6
4.42
5.21
5.29
5.08
5.00
5.14
                      Figure 13-1. Side-by-Side Iron Box Plots
                    •
              200,00 -
              150,00 -
          E
          a.
          a.
          o   100,00 •
               50.00 -
                 .00'
                                      o
                                                    O
                            II       I       I       I

                      Weil 1  Well 2 Well 3  Weil 4      5      6
                                         13-4
       March 2009

-------
Chapter 13. Spatial Variability
                                                          Unified Guidance
                     Figure 13-2. Side-by-Side  Log(Iron) Box Plots
                    8,00-
                    5,50-
£ 5.00'
a
a
at
o
- 4,50 H
o
                  5 4.00-
                    3.50-
                    3.00-
          T
                                         i
                                          1
                                                        J_
                                 I      I       I       I       I
                          Well 1 Weil 2 Well 3 Well 4  Well 5  Well 6
Step 4.   While more nearly similar on the log-scale, the means and medians are still elevated in Wells
         5 and  6.  Since  the  differences in  box lengths  are much  less on  the  log-scale, the log
         transformation has worked to  somewhat stabilize  the variances. These data should be tested
         formally for significant spatial  variation using an ANOVA, probably on the log-scale. -^

13.2.2      ONE-WAY ANALYSIS OF  VARIANCE FOR SPATIAL VARIABILITY

       PURPOSE AND BACKGROUND

     Chapter 17 presents Analysis of Variance [ANOVA] in greater  detail.  When using ANOVA to
check for spatial variability,  the observations from each well  are taken as a single group. Significant
differences between data groups represent monitoring wells with different mean concentration levels.
The lack of significant well mean  differences may afford  an opportunity to pool the data  for larger
background sizes and conduct interwell detection monitoring tests.

     ANOVA used for this purpose should be performed either on a set of  multiple non-impacted
upgradient  wells,  or  on historically uncontaminated  compliance and upgradient background wells. If
significant mean differences exist among naturally occurring  constituent data at upgradient wells, natural
spatial variability is the likely  reason.  Synthetic consitituents in upgradient  wells might also exhibit
spatial differences if affected by an  offsite- plume. Presumably, if the flow gradient has been correctly
                                             13-5
                                                                 March 2009

-------
Chapter 13. Spatial Variability	Unified Guidance

assessed and no migration of contaminants from off-site has occurred, differences in mean levels across
upgradient wells ought to signal the influence of factors not attributable to  a  monitored release. A
similar, but  potentially  weaker,  argument  can be  made  if  spatial  differences  exist  between
uncontaminated  historical  data  at compliance  wells.  The  lack of  spatial  differences  between
uncontaminated compliance  and  upgradient background well  data, may again allow for even larger
background sample sizes.

       REQUIREMENTS AND ASSUMPTIONS

     The basic assumptions and data requirements for one-way ANOVA are presented in Section 17.1.
If the assumption that the observations are statistically independent over time is not met, both identifying
spatial  variability using ANOVA as well as improving intrawell prediction limits and control charts can
be impacted.  It is usually difficult to verify that the measurements are temporally independent with only
a limited number of observations per well.  This potential problem can be somewhat minimized by
collecting  samples far  enough apart in time to guard  against autocorrelation.  Another option is to
construct a parallel time series plot (Chapter 14) to  look for time-related effects or dependencies
occurring simultaneously across the  set of wells.

     If a  significant temporal dependence or autocorrelation  exists, the one-way ANOVA can still
identify well-to-well mean level differences. But the power of the test to do so is lessened. If a parallel
time series plot indicates a potentially strong time-related effect, a two-way ANOVA including temporal
effects  can be performed to test and correct for a significant  temporal  factor. This slightly  more
complicated procedure is discussed in Davis (1994).

     Another key  assumption of  parametric ANOVA is  that the residuals are normal or  can be
normalized. If a normalizing transformation cannot be found, a test for spatial variability can be made
using the Kruskal-Wallis non-parametric ANOVA (Chapter 17).  As long as the measurements can be
ranked, average ranks that differ significantly across wells provide evidence of spatial variation.

       PROCEDURE

Step 1.   Assuming there are p distinct wells to test, designate the measurements from each well as a
         separate group for purposes of computing the ANOVA. Then follow Steps 1 through 7 of the
         procedure in Section 17.1.1  to  compute the overall F-statistic and  the  quantities of the
         ANOVA table in Figure 13-3 below.
                    Figure 13-3.  One-Way Parametric ANOVA Table

 Source of Variation   Sums of   Degrees of      Mean Squares          F-Statistic
                       Squares    Freedom
Between Wells
Error (within wells)
Total
SSWells
•^•^error
SStotal
P-1
n-p
n-l
/V7 C ,, — QQ 1 1 / f n 	 1 ^ F — /V7 Q • • //V7 Q
* * Dwells — *J*J wslls/ vA^ •'• / * — * * *-^ wslls/ * * *"^srTor
MSerror = SSerror/(A7-p)

                                             13-6                                   March 2009

-------
Chapter 13.  Spatial Variability	Unified Guidance

Step 2.   To test the hypothesis of equal means for all p wells, compare the F-statistic from Step 1 to the
         a-level critical point found from the F-distribution with (p-\) and (n-p) degrees of freedom in
         Table 17-1 of Appendix D. Usually a is taken to be 5%, so that the needed comparison value
         equals the upper 95th percentage point of the F-distribution. If the observed F-statistic exceeds
         the critical  point  (F.95,P-i,n-p),  reject the  hypothesis  of equal  well  population means  and
         conclude there is  significant spatial variability. Otherwise, the evidence is insufficient to
         conclude there are  significant differences between the means at the/? wells.

       ^EXAMPLE  13-2

     The iron concentrations in Example 13-1 show evidence of spatial variability in side-by-side box
plots. Tested for equal variances and normality, these same data are best fit by a lognormal distribution.
The  statistics for natural logarithms of the iron measurements are shown below;  individual log data are
provided in the Example 13-1 second table. Compute  a one-way parametric ANOVA to  determine
whether there is significant spatial variation at the a = .05 significance level.
Date
N
Mean
SD

Well 1

3
0

4
820
296

Log
Well 2

3
0

4
965
395

Iron Concentration
Well 3
4
4.348
0.658
Grand Mean
Statistics
Well 4
4
4.188
0.453
= 4.354
log(ppm)
Well 5
4
4.802
0.704

Well 6

5
0

4
000
396

       SOLUTION
Step 1.   With 6 wells and 4 observations per well, n; = 4 for all the wells. The total sample size is n =
         24 and/? = 6. Compute the (overall) grand mean and the sample mean concentrations in each
         of the well groups using equations [17.1] and [17.2]. These values are listed (along with each
         group's standard deviation) in the above table.

Step 2.   Compute the sum of squares due to well-to-well differences using equation [17.3]:

            ss*eih  =  [4(3.820)2 +4(3.965)2 +  ...  +4(5.000)2]-24(4.354)2  =4.294

         This quantity has (6 - 1) = 5 degrees of freedom.

Step 3.   Compute the corrected total sum of squares using equation [17.4] with («-!) = 23 df:

                    SS^  =   [(4.06)2+   ...  +(5.08)2]-24(4.354)2  =8.934

Step 4.   Obtain the within-well or error sum of squares by subtraction using equation [17.5]:

                                  SS    =8.934-4.294 = 4.640
                                    error

         This quantity has (n -p) = 24-6 = 18 degrees of freedom.
                                              IF?                                   March 2009

-------
Chapter 13. Spatial Variability	Unified Guidance

Step 5.   Compute the well and error mean sum of squares using equations [17.6] and [17.7]:

                                   MS  „ =4.294/5 = 0.859
                                      wells       '

                                  MS    =4.640/18 = 0.258
                                     error       '

Step 6.   Construct the F-statistic and the one-way ANOVA table, using Figure 13-3 as a guide:
Source of Variation
Between Wells
Error (within wells)
Total
Sums of Squares
4.294
4.640
8.934
Degrees of
Freedom
5
18
23
Mean Squares
0.859
0.258
F-Statistic
F= 0.859/0.258=3.33
Step 7.   Compare the observed F-statistic of 3.33 against the critical point taken as the upper 95th
         percentage point from the F-distribution with 5 and  18 degrees of freedom. Using Table 17-1
         of Appendix D, this gives a value of Fgs^jg = 2.77. Since the F-statistic exceeds the critical
         point, the null hypothesis of equal  well means can be rejected, suggesting the presence of
         significant spatial variation. -^

13.3 USING ANOVA TO IMPROVE PARAMETRIC  INTRAWELL TESTS

     BACKGROUND AND PURPOSE

     Constituents that exhibit  significant spatial variability usually should be formally tested with
intrawell procedures such as a prediction limit or control chart. Historical data from each compliance
well are used  as background for these tests  instead of from upgradient wells. At an early stage of
intrawell testing,  there may only be  a few  measurements  per well which can be designated as
background. Depending on the number of  statistical tests that need to be performed across the
monitoring network, available intrawell background at individual compliance wells may not provide
sufficient statistical power or meet the false positive rate criteria (Chapter 19).

     One remedy first suggested by Davis  (1998) can increase the degrees  of freedom of the test by
using one-way ANOVA results (Section 13.2) from a number of wells to  provide an alternate estimate
of the  average intrawell variance.  In constructing a parametric intrawell  prediction limit for a single
compliance well, the intrawell background of sample size n is used to compute a well-specific sample
mean (x). The intrawell standard deviation (s)  is replaced by the root mean squared error [RMSE]
component from an ANOVA of the intrawell background associated with a series of compliance and/or
background wells.1 This raises the degrees of freedom from (n-\) to (N-p\ where TV is the total sample
size across the group of wells input to the ANOVA and/? is the number of distinct wells.
1  RMSE is another name for the square root of the mean error sum of squares (MS^^) in the ANOVA table of Figure 13-3.
                                                                                   March 2009

-------
Chapter 13.  Spatial Variability	Unified Guidance

     As an example of the difference this adjustment can make, consider a site with 6 upgradient wells
and 15 compliance wells. Assuming n =  6 observations per well that have been collected over the last
year, a total of 36 potential background measurements are available to construct an interwell test. If there
is significant natural spatial variation in the mean levels from well to well, an interwell test is probably
not appropriate.  Switching to an  intrawell  method is  the next  best solution, but  with only  six
observations  per compliance  well,  either  the  power of an intrawell test  to identify contaminated
groundwater is likely to be quite low (even with retesting) or the site-wide false positive rate [SWFPR]
will exceed the recommended target.

     If the six upgradient wells were tested for spatial variability using a one-way ANOVA (presuming
that the  equal variance assumption is met),  the degrees of freedom [df\ associated with the mean error
sum of  squares term is (6 wells x 5 df per well) = 30 df (see Section 13.2). Thus by substituting  the
RMSE in place of each compliance well's intrawell standard deviation (s), the degrees of freedom  for
the modified intrawell prediction or control chart limit is 30 instead of 5.

     ANOVA can  be usefully employed in this  manner since the RMSE is very close to being a
weighted average of the individual well  sample standard deviations. As  such, it  can be  considered a
measure of average within-well variability across the wells input to the ANOVA. Substituting the RMSE
for s at  an individual well consequently provides a better estimate of the typical within-well variation,
since the RMSE is based on levels of fluctuation averaged across several wells. In addition, the number
of observations used to construct the RMSE is much greater than  the n values used to  compute  the
intrawell sample  standard deviation (s).  Since  both statistical measures  are estimates of within-well
variation, the  RMSE with its larger degrees of freedom is generally a superior estimate if certain
assumptions are met.

     REQUIREMENTS AND ASSUMPTIONS

     Using ANOVA to bolster parametric intrawell prediction or control chart limits will not work at
every site or for every constituent. Replacement of the well-specific, intrawell sample standard deviation
(s) by the RMSE from ANOVA assumes  that the true within-well variability is approximately the same
at  all the  wells for which  an intrawell  background limit (i.e., prediction  or control chart) will  be
constructed, and not just those wells tested in the ANOVA procedure. This last assumption can be
difficult to verify if the ANOVA includes only background or upgradient wells. But to the extent that
uncontaminated intrawell background measurements from compliance point wells can be  included,  the
ANOVA should be run on all  or a substantial fraction of the site's wells (excluding those which might
already  be  contaminated). Whatever mix of upgradient  and downgradient wells are  included  in  the
ANOVA, the  purpose of the  procedure  is not to identify groundwater contamination, but rather to
compute a better and more powerful estimate of the  average intrawell standard deviation.

     For the ANOVA to be valid and the  RMSE to be  a reasonable estimate of average within-well
variability,  a formal check of the equal variance assumption should be conducted using Chapter 11
methods. A spatially variable constituent  will often  exhibit well-specific  standard deviations that
increase with the well-specific mean concentration.  Equalizing the variances in these cases will require a
data transformation, with an ANOVA conducted on the transformed data. Ultimately, any transformation
applied  to the wells in the ANOVA also  need to be applied to intrawell background before computing
intrawell prediction or control  chart limits. The same transformation has to be appropriate for both sets

                                              13^9                                    March 2009

-------
Chapter 13. Spatial Variability	Unified Guidance

of data (i.e.,  wells included in ANOVA and intrawell background at wells for which background limits
are desired).

     Even when the  ANOVA procedure described in this section is utilized, the resulting intrawell
limits  should also  be designed  to incorporate retesting. When intrawell background  is employed to
estimate both  a well-specific background  mean  (x)  and well-specific  standard deviation (s), the
Appendix D tables associated with Chapters 19 and 20 can be used to look up the intrawell sample size
(n) and number of wells (w) in the network in order to find a prediction or control  chart multiplier that
meets the targeted SWFPR and has acceptable statistical power. However, these tables implicitly assume
that the degrees of freedom [df\  associated with the test  is equal to (w-1). The ANOVA method of this
section results in a much larger df, and more importantly, in a df that does not 'match' the intrawell
sample size (n).

     Consequently, the parametric multipliers in the Appendix D tables cannot be directly used when
constructing  prediction or control chart limits with retesting. Instead, a multiplier must be computed for
the specific  combination of n  and df computed  as a  result of the ANOVA.  Tabulating  all  such
possibilities would  be prohibitive.  For prediction limits, the Unified Guidance recommends the free-of-
charge, open source  R statistical  computing  environment.   A pre-scripted program  is included in
Appendix C that can  be run in R to calculate appropriate prediction limit multipliers, once the user has
supplied an intrawell sample size («), network size (w), and type of retesting scheme.

     If guidance users are unable to utilize the R-script approach, the following approximation for the
well-specific prediction limit K-factors is suggested based on EPA Region 8 Monte Carlo evaluations.
Given  a per- test confidence level of 1- a , r total tests ofw -c well-constituents, an individual well size
«,, a pooled variance sample size of njf =  df+  1, and  Kndfj-a obtained from annual intrawell Unified
Guidance tables, the individual well Knij.a factor can be estimated using the following equation:
                                    ~    Kndf,\-a
                                                                       m*=-
                                                                           A.^
     where // =1  for future l:m  observations  or ju is the size  of a future mean.   The value of m* is
specific to each of the nine parametric prediction limit tests and is a function of the three coefficients A,
b and c, individual well sample size n\ and r tests. For a 1:1 test of future means or observations,  the
equation is exact;  for higher order \:m tests, the results are approximate.2 The equation is also useful in
        For each of the nine prediction limit tests, the following coefficients (A, b & c) are recommended:  a 1:2 future
values test (1.01, .0524 & .0158);  a 1:3 test (1.63, .108 & .0407); a 1:4 test (2.41, .157 & .0668); the modified California
plan (1.36, .103 & .0182); a 1:1 mean size 2 test (.5, 0 & 0); a 1:2 mean size 2 test (.898, .0856 &  .0172); a 1:3 mean size 2
test (1.27, .168 & .0363); a 1:1 mean size 3 test (.5, 0 & 0); and a 1:2 mean size 3 test (.817, .108 & .0158). %.  The
coefficients were obtained from regression analysis; approximation values were compared with R-script values for K-factors.
In 1260 comparisons of the seven tests using repeat values (m > 1), 86% of the approximations lay within or equal to + 1% of
the true value and 96% within or equal to + 2%. The 1:4 test had the greatest variability, but all values lay within +_4%.  81%
of the values lay within or equal to + .01  K-units and 93% less than or equal to + .02 units.
                                               13-10                                     March 2009

-------
Chapter 13. Spatial Variability
Unified Guidance
gauging R-script method results.  Another virtue of this equation is that it can be readily applied to
different individual well sample sizes based on the common  Kndf,i-a for pooled variance data.

     A less elegant solution is available for intrawell control charts. Currently, an appropriate multiplier
needs to be  simulated via Monte Carlo  methods.  The approach is to simulate  separate normally-
distributed  data sets for the background mean based on n measurements, and the background standard
deviation based on df+ 1 measurements. Statistical independence of the sample mean (x ) and standard
deviation (s)  for  normal populations allows this to work. With the background mean and standard
deviation available, a series of possible multipliers (h) can be investigated in simulations of control chart
performance.  The multiplier which meets the targeted SWFPR and provides acceptable power should be
selected. Further detail is presented in Chapter 20.  R can also be used to conduct these simulations.

       ^EXAMPLE 13-3

     The  logged iron concentrations from  Example  13-2  showed  significant evidence of spatial
variability.  Use the results of the one-way ANOVA to compute adjusted intrawell prediction limits
(without retesting) for each of the wells in that example and compare them to the unadjusted prediction
limits.

     SOLUTION
Step 1.   Summary statistics by well for the logged iron measurements are listed in the table below.
         With n = 4 measurements per well, use equation [13.1] and ^-a,n-i = ^.99,3 = 4.541 from Table
         16-1 in Appendix D to compute at each well an unadjusted 99% intrawell  prediction limit for
         the  next single measurement, based on lognormal data:
                                                                                        [13.1]

Log-mean
Log-SD
n
t.99,3
99% PL
Well 1
3.820
0.296
4
4.541
204.9
Unadjusted
Well 2
3.965
0.395
4
4.541
391.6
99% Prediction
Well 3
4.348
0.658
4
4.541
2183.0
Limits for Iron
Well 4
4.188
0.453
4
4.541
657.0
(ppm)
Well 5
4.802
0.704
4
4.541
4341.5
Well 6
5.000
0.396
4
4.541
1108.1
Step 2.   Use the RMSE (i.e., square root of the mean error sum of squares [Mirror] component) of the
         ANOVA in Example 13-2 as an estimate of the adjusted, pooled standard deviation, giving
         •^MSerror = V.258 = .5079 . The degrees of freedom (df) associated with this pooled standard
         deviation isp(n-1)= 6^3j= 18, the same as listed in the ANOVA table of Example 13-2.
                                             13-11
        March 2009

-------
Chapter 13.  Spatial Variability
                                              Unified Guidance
Step 3.   Use equation [13.2], along with the adjusted pooled standard deviation and its associated df, to
         compute an adjusted 99% prediction limit for each well, as given in the table below. Note that
         the adjusted t-value based on the larger dfis ^i-a,df = ^.99,18 = 2.552.
                                                                                          [13.2]
                  Well 1
 Adjusted 99% Prediction Limits for Iron (ppm)
Well 2         Well 3         Well 4         Well 5
Well 6
Log-mean
RMSE
df
t.99,18
99% PL
3.820
0.5079
18
2.552
194.3
3.965
0.5079
18
2.552
224.6
4.348
0.5079
18
2.552
329.4
4.188
0.5079
18
2.552
280.7
4.802
0.5079
18
2.552
518.7
5.000
0.5079
18
2.552
632.3
Step 4.   Compare the adjusted and unadjusted lognormal prediction limits. By estimating the average
         intrawell standard deviation using ANOVA, the adjusted prediction limits  are significantly
         lower and thus more powerful than the unadjusted limits, especially at Wells 3, 5, and 6.

         In this example, use of the R-script approach was unnecessary, since the corresponding K-
         multiple used in 1-of-l prediction limit tests can be directly derived analytically. -4
                                             13-12
                                                      March 2009

-------
Chapter 14.  Temporal Variability	Unified Guidance


               CHAPTER 14.   TEMPORAL VARIABILITY

        14.1   TEMPORAL DEPENDENCE	14-1
        14.2   IDENTIFYING TEMPORAL EFFECTS AND CORRELATION	14-3
          14.2.1   Parallel Time Series Plots	14-3
          14.2.2   One-Way Analysis of Variance for Temporal Effects	14-6
          14.2.3   Sample Autocorrelation Function	14-12
          14.2.4   Rank von Neumann Ratio Test	14-16
        14.3   CORRECTING FOR TEMPORAL EFFECTS AND CORRELATION	14-19
          14.3.1   Adjusting the Sampling Frequency and/or Test Method.	14-19
          14.3.2   Choosing a Sampling Interval Via Darcy's Equation	14-20
          14.3.3   Creating Adjusted, Stationary Measurements	14-28
          14.3.4   Identifying Linear Trends Amidst Seasonality: Seasonal Mann-Kendall Test	14-37
     This chapter discusses the importance of statistical independence in groundwater monitoring data
with respect to temporal variability. Temporal variability exists when the distribution of measurements
varies with the times at which sampling or analytical measurement occurs. This variation can be caused
by seasonal fluctuations in  the groundwater itself,  changes in the analytical  method used, the re-
calibration of instruments, anomalies in sampling method, etc.

     Methods to identify temporal variability are discussed for both groups of wells (parallel time series
plots; one-way analysis of variance  [ANOVA] for temporal effects) and single  data series  (sample
autocorrelation function; rank von Neumann ratio).  Procedures are also presented for  correcting or
accommodating temporal effects. These include guidance on adjusting the sampling frequency to avoid
temporal correlation, choosing a sampling interval using the Darcy  equation,  removing seasonality or
other temporal dependence, and finally testing for trends with seasonal data.

14.1 TEMPORAL DEPENDENCE

     A key assumption underlying most statistical tests is that the sample data are independent  and
identically distributed [i.i.d.]  (Chapter 3). In part, this means that measurements collected over a period
of time should not exhibit a clear time dependence  or  significant autocorrelation.  Time dependence
refers to the presence of trends or cyclical patterns when the observations are graphed on a time series
plot. The  closely related concept of autocorrelation  is essentially the degree  to which measurements
collected later in a series can be predicted from previous measurements. Strongly  autocorrelated data are
highly predictable from one value to the next.  Statistically independent values vary in a random,
unpredictable fashion.

     While temporal independence is a complex topic, there are several common  types of temporal
dependence. Some of these include: 1) correlation across wells over time in the concentration pattern of
a single constituent (i.e., concentrations tending to jointly rise or fall at each of the  wells on common
sampling events); 2) correlation across multiple constituents over time in their  concentration patterns
(i.e.., a parallel rise or fall  in concentration across  several parameters on common sampling events); 3)
seasonal cycles; 4) trends, linear or otherwise; and 5) serial  dependence or autocorrelation (i.e., greater
correlation between sampling events more closely spaced in time).
                                              14-1                                    March 2009

-------
Chapter 14.  Temporal Variability	Unified Guidance

     Any of these patterns can invalidate or weaken the results of statistical testing. In some cases, a
statistical  method  can be chosen that specifically accounts for temporal dependence (e.g.,  seasonal
Mann-Kendall trend test). In other instances, the sample data need to be adjusted for the dependence.
Future data might also need to be collected in a manner that avoids temporal correlation. The goal of this
chapter is to present straightforward tools that can be used to first identify temporal dependence and then
to adjust for this correlation.

     To better understand  why  most statistical  tests  depend  on  the  assumption of  statistical
independence, consider a hypothetical series of groundwater measurements exhibiting an obvious pattern
of seasonal fluctuation (Figure 14-1).  These data demonstrate regular and repeated cycles of higher and
lower values. Even though fluctuating predictably and highly dependent, the characteristics of the entire
groundwater population will be observed over a long period of monitoring. This provides an estimate of
the full range of concentrations and an accurate  gauge of total variability.

     The same is not true for data collected from the same population over a much shorter span, say in
five to six months. A much narrower range of sample concentrations would be  observed due to  the
cyclical pattern. Depending on when the sampling was conducted, the average concentration level would
either be much higher or much lower than the overall  average; no single sampling period is likely to
accurately estimate either the true population mean or its variance.

     From this example, an important lesson can be drawn about temporally dependent data. Variance
estimates in a sample of dependent, positively autocorrelated data are likely to be biased low. This is
important  because the guidance methods require and assume that an accurate and unbiased estimate of
the sample standard deviation be available. A case in point was the practice of using aliquot replicates of
a single physical sample for comparison with other combined replicate aliquot samples from a number of
individual physical water quality samples  (e.g., in a Student-^ test).  Aliquot replicate values  are much
more  similar to each  other than to measurements made on physically discrete groundwater  samples.
Consequently, the estimate of variance was too low and the Mest frequently registered false positives.

     Using physically discrete  samples is not always sufficient. If the sampling interval ensures that
discrete volumes of groundwater are being sampled on  consecutive sampling events, the observations
can be described as physically independent. However, they are not necessarily statistically independent.
Statistical  independence is based not on the physical characteristics of the sample data, but rather on the
statistical pattern of measurements.

     Temporally dependent and autocorrelated data generally contain both  a truly random  and non-
random component.  The relative strength of the  latter effect is a measured by one or more correlation
techniques.  The degree of correlation among dependent sample measurements lies on a continuum.
Sample pairs  can  be  mildly correlated or strongly correlated. Only strong  correlations are likely to
substantially impact the results of further statistical testing.
                                              14-2                                    March 2009

-------
Chapter 14.  Temporal Variability
                                                            Unified Guidance
                           Figure 14-1. Seasonal Fluctuations
              §

              I
              w
              o
              §
              o
                                            DATE
14.2 IDENTIFYING TEMPORAL EFFECTS AND CORRELATION
14.2.1
PARALLEL TIME SERIES PLOTS
       BACKGROUND AND  PURPOSE

     Time series plots were introduced in Chapter 9. A time series plot such as Figure 14-1 is a simple
graph of concentration versus time of sample collection. Such plots are useful for identifying a variety of
temporal patterns. These include identifying a trend over time, one or more sampling events that may
signal contaminant releases, measurement outliers resulting in anomalous 'spikes' due to field handling
or analytical problems, cyclical and seasonal fluctuations, as well as the presence of other time-related
dependencies.

     Time  series plots can be used in two basic ways to identify temporal dependence. By graphing
single constituent data from multiple wells together on a time series plot, potentially significant temporal
components of variability can be  identified.  For example, seasonal fluctuations can cause the mean
concentration levels at a number of wells to vary with the time of sampling events. This dependency will
show up in  the time series plot as a pattern of parallel traces., in which the individual wells will tend to
rise and fall together across the sequence of sampling dates. The parallel pattern may be the result of the
measurement process  such as mid-stream changes in field handling or sample collection procedures,
periodic re-calibration of analytical instrumentation, and changes in laboratory or analytical methods. It
could also be the result from significant autocorrelation present in the groundwater  population itself.
Hydrologic  factors such as drought, recharge patterns or regular (e.g., seasonal) water table fluctuations
may be responsible.  In these cases, it may be useful to test for the presence of a significant temporal
                                            14-3
                                                                    March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

effect by first constructing a parallel time series plot and then running a formal one-way ANOVA for
temporal effects (Section 14.2.2).

     The second way time series plots can be helpful is by plotting multiple constituents over time for
the same well, or averaging values for each constituent across wells on each sampling event and then
plotting the averages over time. In either case, the plot can signify whether the general concentration
pattern over time is simultaneously observed for different constituents. If so, it may indicate that a group
of constituents is highly correlated in groundwater or that the same  artifacts of sampling and/or lab
analysis impacted the results of several monitoring parameters.

       REQUIREMENTS AND ASSUMPTIONS

     The requirements for time  series plots were discussed in  Chapter  9.  Two  very  useful
recommendations follow from that discussion.  First, a different plot symbol should be used to display
any non-detect measurements (e.g.,  solid symbols for detected values, hollow symbols for non-detects).
This can help prevent mistaking a change over time in reporting limits as a trend, since detected and
non-detected data are clearly distinguished on the plot. It also allows one to determine whether  non-
detects are more prevalent during certain portions of the sample record  and less prevalent at other times.
Secondly, when multiple constituents are plotted on the same graph, it  may be necessary to standardize
each constituent prior to plotting to avoid trying to simultaneously visualize high-valued and low-valued
traces on the same j/-axis (i.e., concentration axis).  The goal of such a plot  is to identify parallel
concentration patterns over time. This can be done most readily by subtracting each constituent's sample
mean (x ) from the measurements for that constituent and dividing by the standard deviation (s), so that
every constituent is plotted on roughly the same scale.

       PROCEDURE  FOR MULTIPLE WELLS, ONE CONSTITUENT

Step 1.   For each well to be  plotted, form data pairs by matching each concentration value with its
         sampling date.

Step 2.   Graph the data pairs for each well on the same set of axes, the horizontal axis representing
         time and the  vertical axis representing concentration. Connect the points for each  individual
         well to form a 'trace' for that well.

Step 3.   Look for parallel movement in the traces across the wells. Even if all the well concentrations
         tend  to rise on a given sampling event, but not to the same magnitude  or degree,  this is
         evidence of a possible temporal effect.

       PROCEDURE  FOR MULTIPLE CONSTITUENTS, ONE OR MANY WELLS

Step 1.   For each constituent to be plotted, compute the  constituent-specific sample mean (x) and
         standard  deviation (s). Form  standardized  measurements (z;) by subtracting the mean  from
         each concentration (x;) and dividing by the standard deviation, using the equation:

                                              x - x
                                          z!=^—                                     [14.1]


         Form data pairs by matching each standardized concentration with its sampling event.

                                             14-4                                  March  2009

-------
Chapter 14. Temporal Variability
                                        Unified Guidance
Step 2.   If correlation is suspected in a group of wells, average the standardized concentrations for each
         given constituent across wells for each specific sampling event. Otherwise, form a multi-
         constituent time series plot separately for each well.

Step 3.   Graph  the data  pairs for each constituent on  the  same  set of axes,  the  horizontal axis
         representing time and the vertical  axis representing standardized concentrations. Connect the
         points for each constituent to form  a trace for that parameter.

Step 4.   Look for parallel movement  in  the  traces across  the constituents.  A  strong  degree  of
         parallelism indicates a high degree  of correlation among the monitoring parameters.

       ^EXAMPLE 14-1

     The  following well  sets  of manganese measurements were collected over a two-year period.
Construct a time series plot of these data to check for possible temporal effects.
Qtr
1
2
3
4
5
6
7
8
Manganese Concentrations
BW-1 BW-2 BW-3
28.14
29.33
30.45
32.42
34.37
33.25
31.02
28.50
31.41
30.27
32.57
32.77
33.03
32.18
28.85
32.88
27.15
30.24
29.14
30.59
34.88
30.53
30.33
30.42
(ppm)
BW-4
30.46
30.60
30.96
30.70
32.71
31.76
31.85
29.58
       SOLUTION
Step 1.   Graph each well's concentrations versus sampling event on the same set of axes to construct
         the following time series plot (Figure 14-2).
               40
            CD
            W
            CD
               35  H
               30  H
               25
                            \
                           2
\
4
\
6
10
                                  Sampling Event
                   Figure 14-2. Manganese Parallel Time Series Plot
                                             14-5
                                                March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

Step 2.   Examining the traces on the plot, there is some degree of parallelism in the pattern over time.
         Particularly for the fifth quarter, there is an across-the-board increase in the manganese level,
         followed by a general decline the next two quarterly events. Note, however, that there is little
         evidence of differences in mean levels by well location. ~4
14.2.2      ONE-WAY ANALYSIS OF VARIANCE FOR TEMPORAL EFFECTS

       PURPOSE AND BACKGROUND

     Parametric ANOVA is a comparison of means among a set of populations. The one-way ANOVA
for temporal effects is no exception. A one-way ANOVA for spatial variation (Chapter 13) uses well
data  sets to represent locations as the statistical factor of interest.  In contrast, a one-way ANOVA for
temporal effects considers multiple well  data sets  for individual sampling events or seasons  as  the
relevant statistical factor. A significant temporal factor implies that the average concentration depends to
some degree on when sampling takes place.

     Three common examples of temporal factors include: 1) an irregular, but consistent shift of
average concentrations  over  time  perhaps  due to changes in laboratories  or analytical method
interferences; 2) cyclical seasonal patterns; or 3) parallel upward or downward trends.  These can occur
in both upgradient and downgradient well data.

     If event-specific analytical differences or seasonality appear to be an important temporal factor, the
one-way ANOVA for temporal effects can be  used  to formally identify seasonality, parallel trends, or
changes in lab performance that affect other temporal effects. Results of the ANOVA can also be used to
create temporally stationary residuals, where the temporal effect has been 'subtracted from' the original
measurements.  These stationary residuals may be  used to replace  the original  data in  subsequent
statistical testing.

     The one-way ANOVA for a temporal factor described below can be used for an additional purpose
when interwell testing is appropriate. For this situation, there can be no significant spatial variability. If
a group of upgradient or other background wells indicates a significant temporal effect, an interwell
prediction limit can be designed  which  properly accounts for this  temporal  dependence.  A more
powerful interwell test  of upgradient-to-downgradient differences can be developed than otherwise
would  be possible.  This can occur because the ANOVA  separates  or 'decomposes'  the overall  data
variation into two sources: a) temporal effects and b) random variation or statistical error. It  also
estimates how  the background mean is  changing  from one sampling  event to the next. The final
prediction limit is formed by computing the background mean, using the separate structural and random
variation components of the ANOVA to better estimate the standard  deviation, and then adjusting the
effective sample size (via the degrees of freedom) to account for these factors.

       REQUIREMENTS AND ASSUMPTIONS

     Like the one-way ANOVA for spatial variation (Chapter 13), the one-way ANOVA for temporal
effects assumes that the data groups are normally-distributed with constant variance. This requirement
means  that the  group residuals should be tested for normality (Chapter 10) and  also for  equality of

                                             14-6                                    March 2009

-------
Chapter 14.  Temporal Variability	Unified Guidance

variance (Chapter 11). It is also assumed that for each of a series of background wells, measurements
are collected at each well on sampling events or dates common to all the wells.

     Two variations in the basic procedure are described below. For cases of temporal effects excluding
seasonally, each sampling event is treated as a separate population. The ANOVA residuals are grouped
and tested by sampling event to test for equality of variance. In cases of apparent seasonality,  each
season is treated as a distinct population. The difference is that seasons contain multiple sampling events
across a span of multiple years, with sampling events collected at the same time of year assigned to one
of the seasons (e.g., all January or first quarter measurements). Here, the ANOVA residuals are grouped
by season to test for homoscedasticity.

     If the assumption of equal variances or normal residuals is violated, a data transformation should
be considered. This should be followed by testing of the assumptions on the transformed scale. The one-
way  ANOVA for a non-seasonal effect should include a minimum  of four wells  and at least  4
observations (i.e., distinct sampling dates) per well. In the seasonal case, there should be a minimum of
3-4 sampling events per distinct season,  with the events thus spanning at least three years (i.e., one per
year per season). Larger numbers of both wells and observations are preferable. Sampling dates should
also be approximately the same for each well if a temporal effect is to be tested.

     If the data cannot be normalized, a similar test for a temporal or seasonal effect can be performed
using the Kruskal-Wallis test (Chapter 17). The only difference from the procedure outlined in Section
17.1.2 is that the roles of wells/groups and sampling events have to be reversed. That is, each sampling
event should  be treated as a separate 'well,' while each well is treated as a separate 'sampling event.'
Then the same equations  can be  applied to the reversed data set  to test for a significant temporal
dependence. If testing  for a seasonal  effect, the wells in the notation  of  Section 17.1.2 become the
groups of common sampling events from different years, while the sampling events are again the distinct
wells.

     Even when a temporal effect exists and is apparent on a time series plot, the variation between well
locations (i.e., spatial variability) may overshadow the temporal variability.   This could result in a non-
significant one-way ANOVA finding for the temporal factor. In these cases, a two-way ANOVA can be
considered where both well location and sampling event/season are treated as statistical factors.  This
procedure is described  in Davis (1994).  Evidence for a temporal effect can be documented using this
last technique, although the  two-way  ANOVA isn't  necessary if the goal is simply to  construct
temporally  stationary  residuals.  That  can  be  accomplished with  a  one-way  ANOVA even  when
significant spatial variability exists.

       PROCEDURE

Step 1.   Given a set of Wwells and measurements from each of T sampling events at each well on each
         of K years, label the observations as xp, for /' = 1 to W,j = 1 to  T, and k = 1  to K. Then xp
         represents the measurement from the rth well on theyth sampling event during the Mi year.

Step 2.   When testing for a non-seasonal temporal effect, form the set of event means (x   ) and the
         grand mean (x...) using equations [14.2] and [14.3] respectively:
                                              14-7                                    March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

                          *.*  =  if]***  fory=ltorand£=lto^                    [14.2]
                                      i=\
                                          W  T  K      I
                                    v   — ^~* ^~* ^~* v   /HTTTf                               F1 A 71
                                    x   — 2_i2_i2__i  i- /PYj.j\.                               [i^f.jj
                                         t=i j=\ k=\   I
Step 2a.  If testing for a seasonal effect common to all wells, form the seasonal means (xt ..) instead of
         the event means of Step 2, using the equation:

                                                 * for/ =1 toT                          [14.4]
Step 3.   Compute the set of residuals for each sampling event or season using either equation [14.5] or
         equation [14.6] respectively:

                                 r,]k  =   xi]k-X.]kfori=ltoW                           [14.5]

                          ri-k  =  xk ~ x..  for J=ltoWandk=ltoK                    [14.6]
Step 4.   Test the residuals for normality (Chapter 10). If significant non-normality is evident, consider
         transforming the data and re-doing the computations in Steps 1 through 4 on the transformed
         scale.

Step 5.   Test the sets of residuals grouped  either  by sampling event or season for equal variance
         (Chapter 11). If the variances are significantly  different, consider transforming the data and
         re-doing the computations in Steps 1 through 5 on the transformed data.

Step 6.   If testing for a non-seasonal temporal effect, compute the mean error sum of squares term
         (M$E) using equation:

                                         W  T K    I

                                        i=\  j=l k=l   I

         This term is associated with TK(W-\) degrees  of freedom. Also compute the mean sum of
         squares for the temporal effect (MSV) with degrees of freedom (7X-1), using equation:

                                       T K           \? /
                                       j=\ k=l           I

Step 6a.  If testing for a seasonal effect, compute the mean error sum of squares (M$E) using equation:
                                             14-8                                   March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

         This term is associated with T(WK-\) degrees of freedom. Also compute the mean sum of
         squares for the seasonal effect (MSr) with degrees of freedom (T-l), using equation:

                                                                                        [14.10]


Step 7.   Test for a significant event-to-event or seasonal effect by computing the ratio of the mean sum
         of squares for time and the mean error sum of squares:

                                        FT=MST/MSE                                  [14.11]

Step 8.   If testing for a non-seasonal temporal effect, the test statistic FT under the null hypothesis (i.e.,
         of no  significant  time-related  variability among the  sampling events) will follow an  F-
         distribution with (7X-1) and TK(W-\)  degrees of freedom. Therefore, using a significance
         level of a = 0.05, compare FT against  the critical point FOS, TK-I,TK(W-I) taken from the  F-
         distribution in Table 17-1 in Appendix D. If the critical point is  exceeded, conclude there is a
         significant temporal effect.

Step 8a.  If testing  for a  seasonal effect, the test statistic FT under the null hypothesis (i.e., of  no
         seasonal pattern) will follow an F-distribution with (T-l) and T(WK-V) degrees of freedom.
         Therefore, using a significance level of a = 0.05, compare FT against the critical point FQS, T-
         I,T(WK-I) taken from the F-distribution in Table 17-1  of Appendix D.  If the critical point is
         exceeded, conclude there is a significant  seasonal pattern.

Step 9.   If there is  no  spatial  variability but a significant temporal effect exists among a set  of
         background wells,  compute  an appropriate interwell  prediction or control chart limit  as
         follows.  First replace the background sample standard deviation (s) with the following
         estimate built from the one-way ANOVA table:
                                                                                        [14.12]

         Then calculate the effective sample size for the prediction limit as:
                                                                                        [14.13]

       ^EXAMPLE 14-2

     Some parallelism was found in the time series plot of Example 14-1. Test those same manganese
data for a significant, non-seasonal temporal effect using a one-way ANOVA at the 5% significance
level.
                                              14-9                                    March 2009

-------
Chapter 14. Temporal Variability
Unified Guidance
Qtr
1
2
3
4
5
6
7
8

Event
Mean
29.290
30.110
30.780
31.620
33.747
31.930
30.513
30.345

Manganese Concentrations
BW-1 BW-2
28.14
29.33
30.45
32.42
34.37
33.25
31.02
28.50
Grand
31.41
30.27
32.57
32.77
33.03
32.18
28.85
32.88
mean = 31.042
(ppm)
BW-3
27.15
30.24
29.14
30.59
34.88
30.53
30.33
30.42

BW-4
30.46
30.60
30.96
30.70
32.71
31.76
31.85
29.58

       SOLUTION
Step 1.   First compute the means for each sampling event and the grand mean of all the data. These
         values are listed in the table above. With four wells and eight quarterly events per well, W=4,
         T=4,andK = 2.

Step 2.   Determine the residuals for each sampling event by subtracting off the event mean. These
         values are listed in the table below.
Qtr
1
2
3
4
5
6
7
8
Manganese Event
BW-1 BW-2
-1.150
-0.780
-0.330
0.800
0.622
1.320
0.508
-1.845
2.120
0.160
1.790
1.150
-0.718
0.250
-1.662
2.535
Residuals (ppm)
BW-3 BW-4
-2.140 1.170
0.130 0.490
-1.640 0.180
-1.030 -0.920
1.132 -1.038
-1.400 -0.170
-0.182 1.338
0.075 -0.765
Step 3.   Test the residuals for normality. A probability plot of these residuals is given in Figure 14-3.
         An adequate fit to normality is suggested by Filliben's probability plot correlation coefficient
         test.
                                             14-10
        March 2009

-------
Chapter 14. Temporal Variability
                                        Unified Guidance
        Figure 14-3.  Probability Plot of Manganese Sampling Event Residuals

                    2
                0)
                8   o
                IM
                   -1
                   -2
     z
                      -4
-2
0
                                    Event Mean Residuals (ppm)
Step 4.   Next, test the groups of residuals for equal variance across sampling events.  Levene's test
         (Chapter 11) gives an F-statistic of 1.30, well below the 5% critical point with 7 and 24
         degrees of freedom of F,9sj^4 = 2.42. Therefore, the group variances test out as adequately
         homogeneous.

Step 5.   Compute the mean error sum of squares term using equation [14.7]:

           MSE  =   [(-1.150)2+(-.780)2+... + (!.338)2+(-.765)2J/(4-2)(7)   =   1.87

Step 6.   Compute the mean sum of squares term for the time effect using equation  [14.8]:

      MST  = 4[(29.290-31.042)2+(30.11-31.042)2+... + (30.345-31.042)2J/7  =   7.55

Step 7.   Test for a significant temporal effect, computing the F-statistic in equation [14.11]:

                                     FT  =7.55/1.87 = 4.04

         The degrees of freedom associated with the numerator and denominator respectively are (T-l)
         = 7 and T(W-\) = 24. Just as with Levene's test run earlier, the 5% level  critical point for the
         test is F.95;7;24 = 2.42. Since FT exceeds this value, there is evidence of a  significant temporal
         effect in the manganese background data.

Step 8.   Assuming a lack of spatial variation, the presence of a temporal effect can be used to compute
         a standard  deviation  estimate and effective background  sample size  appropriate  for an
                                            14-11
                                               March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

         interwell prediction limit test, using equations [14.12] and [14.13] respectively. The adjusted
         standard deviation becomes:
                                     a- ^[7.55+7(1.87)]
         while the effective sample size is:

                    n * = 1+ J8 • 7 • (4.04 + 4-1)2 ]/[s • (4.04)2 +7 • 3J}= 19.31 = 19

         If the background data had simply been pooled together and the sample standard deviation
         computed, s = 1.776 ppm with a sample size of n = 32.  So the adjustments based on the
         temporal effect alter the final  prediction limit by enlarging it to account for the additional
         component of variation. -4


14.2.3      SAMPLE AUTOCORRELATION FUNCTION

       BACKGROUND AND  PURPOSE

     The  sample autocorrelation function enables a test of temporal autocorrelation in a single data
series (e.g., from a single  well over time). When  a time-related  dependency affects  several wells
simultaneously, parallel  time series plots (Section 14.2.1) and one-way ANOVA  for temporal effects
(Section 14.2.2) should be considered. But when a longer data series is to be used for an intrawell test
such as a  prediction  limit or control chart,  the autocorrelation function does an excellent job  of
identifying temporal dependence.

     Given a  sequence of consecutively-collected  measurements, x\,  X2,..., xn, form  the set  of
overlapping pairs (*;, x;+i) for / =  1,..., n-l. The approximate  first-order  sample autocorrelation
coefficient is then computed from these pairs as (Chatfield, 2004):
                                    r=^-	                              [14.14]
Equation [14.14]  estimates the first-order autocorrelation, that is, the correlation between pairs of
sample measurements collected one event apart (i.e., consecutive events). The number  of sampling
events separating each pair is called  the lag, representing the temporal distance between the pair
measurements.

     Autocorrelation can also be computed at other lags. The general approximate equation for the Mi
lag is given by:
                                            14-12                                   March 2009

-------
Chapter 14.  Temporal Variability _ Unified Guidance
which estimates the Mi-order autocorrelation for pairs of measurements separated in time by k sampling
events. Note that the number of pairs used to compute r^ decreases with increasing k due to the fact that
fewer and fewer sample pairs can be formed which are separated by that many lags.

     By computing the first  few sample autocorrelation  coefficients and defining TO = 1,  the sample
autocorrelation function can be formed by plotting r^ against k. Since the autocorrelation coefficients are
approximately normal in distribution with zero mean and variance equal to l/n,  a test of significant
autocorrelation at the 95% significance level can be  made by  examining the sample autocorrelation
function to see if any coefficients exceed 2/\n in absolute value (±2/\n represent approximate upper
and lower confidence limits).

     The sample autocorrelation function is a valuable visual  tool for assessing different types of
autocorrelation (Chatfield, 2004). For instance, a stationary (i.e.,  stable, non-trending) but non-random
series of measurements will  often exhibit a large value  of r\ followed by perhaps one or two other
significantly non-zero coefficients. The remaining coefficients will be progressively smaller and smaller,
tending towards zero. A series with a clear seasonal pattern will exhibit a seasonal (i.e., approximately
sinusoidal) autocorrelation function.  If the series  tends to alternate between  high  and low  values, the
autocorrelation function  will  also  alternate,  with  r\  being negative  to  reflect that  consecutive
measurements tend to be on 'opposite sides'  of the sample mean. Finally, if the series contains a trend,
the sample autocorrelation function will not drop to zero as  the lag k increases.  Rather, there will a
persistent autocorrelation even at very large lags.

       REQUIREMENTS AND ASSUMPTIONS

     The approximate distribution of the sample autocorrelation coefficients is predicated on the sample
measurements following  a normal distribution. A test for significant autocorrelation may therefore be
inaccurate unless the sample measurements are roughly normal. Non-normal data series should be tested
for temporal autocorrelation using the non-parametric rank Von Neumann ratio (Section 14.2.4).

     Outliers  can drastically affect the sample  autocorrelation function (Chatfield, 2004). Before
assessing  autocorrelation, check the sample for possible outliers, removing those that are identified. A
series of  at least  10-12  measurements is minimally recommended to construct the autocorrelation
function.  Otherwise, the number of lagged data pairs  will be too small  to reliably estimate the
correlation, especially for larger lags. Sampling events should be regularly spaced so that pairs lagged by
the same number of events (k) represent the same approximate time interval.

       PROCEDURE

Step 1.    Given a series of n  measurements,  x\,..., xn,  form sets of lagged data pairs (x;, xj+k), i = 1,...,
          n-k, for k < [n/3], where the notation [c] represents the largest integer no greater than c. For
          longer series, computing lags to a maximum of k = 15 is generally sufficient.

                                             14-13                                   March 2009

-------
Chapter 14.  Temporal Variability
                                                                 Unified Guidance
Step 2.   For each set of lagged pairs from Step 1, compute the sample autocorrelation coefficient, r^,
         using equation [14.15]. Also define TO = 1.

Step 3.   Graph the sample autocorrelation function by plotting r^ versus k for k = 0,..., [«/3], generally
         up to a maximum lag of 15. Also plot horizontal lines at levels equal to: ±2/V« .

Step 4.   Examine the sample autocorrelation function. If any coefficient r^ exceeds 2/vn in absolute
         value, conclude that the sample has significant autocorrelation.

       ^EXAMPLE 14-3

     The following series of monthly total alkalinity  measurements were collected from  leachate at a
solid waste landfill during a four and a half year period. Use the sample autocorrelation function to test
for significant temporal dependence in this  series.
Date
01/26/96
02/20/96
03/19/96
04/22/96
05/22/96
06/24/96
07/15/96
08/21/96
09/15/96
10/15/96
11/11/96
12/10/96
01/22/97
02/11/97
03/04/97
04/07/97
05/01/97
06/09/97
Total
Alkalinity
(mg/L)
1400
1700
1900
1800
1300
2000
2300
2500
1700
1600
1400
1600
1800
1000
720
1400
1600
990
Date
07/01/97
08/15/97
09/15/97
10/15/97
11/15/97
12/15/97
01/15/98
02/15/98
03/15/98
04/15/98
05/08/98
06/15/98
07/15/98
08/15/98
09/02/98
10/06/98
11/03/98
12/15/98
Total
Alkalinity
(mg/L)
2400
3500
3100
3300
2100
2100
1500
710
1100
1900
2100
2000
2500
2700
2400
3000
2700
2680
Date
01/15/99
02/02/99
03/02/99
04/15/99
05/04/99
06/02/99
07/07/99
08/03/99
09/02/99
10/07/99
11/02/99
12/07/99
01/06/00
02/02/00
03/02/00
04/04/00
05/02/00
06/06/00
Total
Alkalinity
(mg/L)
1350
1560
1220
1390
1940
2160
1990
2540
2250
1630
1710
1210
1170
1330
1540
1670
1520
2080
       SOLUTION
Step 1.   Create a time series plot of the n = 54 alkalinity measurements, as in Figure 14-4. The series
         indicates an apparent seasonal fluctuation.

Step 2.   Form lagged data pairs from the alkalinity series for each lag k = 1,..., [n/3] = 18. The first
         two pairs for k = 1 (i.e., first order lag) are (1400,  1700) and (1700, 1900). For k = 2, the first
         two pairs are (1400, 1900) and (1700, 1800), etc.

Step 3.   At  each lag (&),  compute the sample autocorrelation coefficient using equation [14.15]. Note
         that the denominator of this equation equals (n-l)s2. For the alkalinity data, the sample mean
         and variance are x = 1865.93 and s = 392349.1 respectively. The lag-1 autocorrelation is thus:
     r,   =
(1400 - 1865.93)- (1700 -1865.93) + ... + (1520 - 1865.93)- (2080 - 1865.93)
                           (54-l)-392349.1
                                             14-14
                                                                                     =  .64
                                                                        March 2009

-------
Chapter 14. Temporal Variability
Unified Guidance
         Other lags are computed similarly.

Step 4.   Plot the  sample autocorrelation function as in Figure 14-5.  Overlay the  plot with  95%
         confidence limits (dotted lines) shown at ±2/v« = ±2/V54 = 0.27 .

Step 5.   The autocorrelation function indicates coefficients  at several lags that lie outside the  95%
         confidence limits,  confirming the presence of temporal dependence.  Further, the shape of
         autocorrelation function is sinusoidal, suggesting a strong seasonal fluctuation in the alkalinity
         levels. -4

                Figure 14-4.  Time Series Plot of  Total Alkalinity (mg/L)
   I
   I
   «2
         1996
                          1997
                                           19S8

                                            Sampling Date
                                                           1399
                                                                            2000
                                            14-15
        March 2009

-------
Chapter 14. Temporal Variability
                                                              Unified Guidance
           Figure 14-5. Sample Autocorrelation Function for Total Alkalinity
 C
 0
 1
    p
    o
                                                              ~TT
                                                       I
                                                      10
                                                               15
                                               Lag
14.2.4
RANK VON  NEUMANN RATIO TEST
       BACKGROUND AND PURPOSE

     The rank von Neumann ratio is a non-parametric test of first-order temporal autocorrelation in a
single data series (e.g., from  a single well over time). It can be used as an alternative to the sample
autocorrelation function (Section 14.2.3) for non-normal data, and is both easily computed and effective.

     The rank von Neumann ratio is based on the idea that a truly independent series of data will vary in
an unpredictable fashion as the list is examined sequentially. The first order or lag-1 autocorrelation will
be approximately zero. By  contrast, the  first-order autocorrelation in dependent data will tend to be
positive (or negative), implying that lag-1 data pairs in the series will tend to be more similar (or
dissimilar) in magnitude than would expected by chance.

     Not only will the concentrations of lag-1 data pairs tend to be similar (or dissimilar) when the
series is  autocorrelated,  but the ranks of lag-1  data pairs will  share that  similarity or  dissimilarity.
Although the test is non-parametric and rank-based, the ranks of non-independent data  still follow a
discernible pattern. Therefore, the rank von Neumann ratio is constructed from the sum of differences
between the ranks of lag-1 data pairs. When these differences are small, the ranks of consecutive data
measurements  need to  be  fairly similar,  implying that the pattern  of observations  is  somewhat
predictable.  Given the relative position and magnitude of one observation, the approximate relative
position and magnitude of the next sample  measurement can be predicted. Low values of the rank von
Neumann ratio are therefore indicative of temporally dependent data  series.
                                            14-16
                                                                      March 2009

-------
Chapter 14. Temporal Variability _ Unified Guidance

     Compared to other tests of statistical independence, the rank von Neumann ratio has been shown to
be more powerful than non-parametric methods such as the Runs up-and-down test (Madansky,  1988). It
is also a reasonable test when the data follow a normal distribution. In that case, the efficiency of the test
is always close to 90 percent when compared to the von Neumann ratio computed on concentrations
instead of the ranks. Thus, very little effectiveness is lost by using the ranks in place of the original
measurements. The rank von Neumann ratio will correctly detect dependent data and do so over a variety
of underlying data distributions.  The rank von Neumann ratio is also fairly robust to departures from
normality, such as when the data derive from a skewed distribution like the lognormal.

       REQUIREMENTS AND ASSUMPTIONS

     An unresolved problem with the  rank von Neumann ratio test is  the presence of a substantial
fraction of tied  observations.  Like the Wilcoxon  rank-sum test (Chapter 16), Bartels (1982)
recommends replacing each tied value by its mid-rank (i.e., the average of all the ranks that would have
been  assigned to that  set of ties).  However,  no explicit adjustment  of the ratio for ties has been
developed.  The rank von Neumann critical points may not be appropriate (or at  best very approximate)
when a large portion of the data consists of non-detects or other tied values. Especially in the case of
frequent non-detects, too much information is lost regarding the pattern of variability to use the rank von
Neumann ratio as an accurate indication of autocorrelation. In fact, no test  is likely to provide a good
estimate of temporal correlation, whether non-parametric or parametric.

     While the rank von Neumann ratio test is recommended in the Unified Guidance for its ease of use
and robustness when applied to either normal or non-normal  distributions, the literature on time series
analysis and temporal correlation is  extensive with respect to other potential tests. Many other tests of
autocorrelation are available, especially when either the original measurements  or the residuals  of the
data are normally distributed after a trend has been removed. Chatfield (2004) and (Madansky, 1988) are
two good references for some of these alternate tests.

       PROCEDURE

Step 1 .   Order the sample from least to greatest and assign a unique rank to each measurement. If some
         data values are tied, replace tied values with their mid-ranks as in the Wilcoxon rank-sum test
         (Chapter 16). Then list the observations and their corresponding ranks in the order that they
         were collected (i.e., by sampling event or time order).

Step 2.   Using the list  of ranks, R\, for the sampling events / = !...«, compute  the rank von Neumann
         ratio with the equation:
Step 3.   Given sample size (ri) and desired significance level (a), find the lower critical point of the
         rank von Neumann ratio in Table 14-1 of Appendix D. In most cases, a choice of a = .01
         should be sufficient, since only substantial non-independence is likely to affect subsequent
         statistical testing.  If the computed ratio, v, is smaller than this critical point, conclude that the
         data series is strongly autocorr elated. If not, there is insufficient evidence to reject the
                                             14-17                                   March 2009

-------
Chapter 14. Temporal Variability
                                                                    Unified Guidance
         hypothesis of independence; treat the data as temporally independent in subsequent statistical
         testing.

       ^EXAMPLE 14-4

     Use the rank von Neumann ratio test on the following series of 16 quarterly measurements  of
arsenic (ppb) to determine whether or not the data set should be treated as temporally independent  in
subsequent tests. Compute the test at the a = .01 level of significance.
Sample Date
Jan 1990
Apr 1990
Jul 1990
Oct 1990
Jan 1991
Apr 1991
Jul 1991
Oct 1991
Jan 1992
Apr 1992
Jul 1992
Oct 1992
Jan 1993
Apr 1993
Jul 1993
Oct 1993
Arsenic (ppb)
4.0
7.2
3.1
3.5
4.4
5.1
2.2
6.3
6.5
7.5
5.8
5.9
5.7
4.1
3.8
4.3
Rank (/?,)
5
15
2
3
8
9
1
13
14
16
11
12
10
6
4
7
       SOLUTION
Step 1.   Assign ranks to the data values as in the table above. Then list the data in chronological order
         so that each rank value occurs in the order sampled.
Step 2.
Step3.
Compute the von Neumann ratio using the set of ranks in column 3 using equation [14.16],
being sure to take squared differences of successive, overlapping pairs of rank values:
                           ]/ =
                                                     = 1.67
Look up the lower critical point (vcp) for the rank von Neumann ratio in Table 14-1 of
Appendix D. For n = 16 and a = .01, the lower critical point is equal to 0.93. Since the test
statistic v is larger than vcp, there is insufficient evidence of autocorrelation at the a = .01 level
of significance. Therefore, treat these data as statistically independent in subsequent testing. A
                                             14-18
                                                                           March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

14.3 CORRECTING FOR TEMPORAL EFFECTS AND CORRELATION

14.3.1      ADJUSTING THE SAMPLING FREQUENCY AND/OR TEST METHOD

     If a data series is temporally correlated, a simple remedy (if allowable under program rules) is to
change the sampling frequency and/or statistical method used to analyze the data.  In  some cases,
increasing the sampling  interval will effectively eliminate the statistical dependence exhibited by the
series. This may happen  because the longer time between  sampling events allows more groundwater to
flow  through  the  well  screen,  further differentiating  measurements of  consecutive  volumes  of
groundwater and lessening the impact of seasonal fluctuations or other time-dependent patterns in the
underlying concentration distribution.

     Many authors including Gibbons (1994a) and ASTM (1994) have recommended that sampling be
conducted no more  often than  quarterly to avoid temporal dependence. If the sampling  frequency is
reduced, there are obviously fewer measurements available for statistical  analysis  during any  given
evaluation period. A t-test or ANOVA cannot realistically be run with fewer than four measurements per
well. A prediction limit for a future mean requires at least two new observations, and a prediction limit
for a future median requires at least three measurements, not counting any resamples. Depending on the
length of the evaluation period (i.e., quarterly, semi-annual, annual), a change of statistical  method may
also be necessary when groundwater measurements are autocorrelated.

     When sufficient background data have been collected over a longer period of time,  a prediction
limit test for future values can be run with as few as one or two new measurements per compliance well.
The same is true for control  charts. Therefore, if a low groundwater flow velocity and/or evidence of
statistical dependence suggest a reduction in sampling frequency, certain prediction  limits and control
charts should be strongly considered as alternate statistical procedures.

       RUNNING A PILOT STUDY

     An optional approach to adjusting the sampling frequency is to run a site-specific pilot study of
autocorrelation.  Such a study can be conducted in several ways, but perhaps the easiest is to pick two or
three wells from the network (perhaps one background well and one or two compliance wells) and then
conduct weekly sampling at these wells over a one year period. For each well in the study, construct the
sample autocorrelation function (Section 14.2.3) for a variety of constituents, and determine from these
graphs the  smallest lagged  interval at which the autocorrelation  coefficient becomes insignificantly
different from zero for most of the study constituents.

     Since an autocorrelation of zero  is equivalent to temporal independence for practical purposes,
finding the smallest lag between sampling events with no correlation indicates the minimum sampling
frequency  needed to approximately ensure statistical independence.  If  the sample autocorrelation
function does not drop down to zero with increasing lag (&), there may be a strong seasonal component
or a trend involved.  In these circumstances, lengthening the sampling frequency may do little to lessen
the temporal dependence. A seasonal pattern may need to be estimated instead and regularly removed
from the data prior to  statistical  testing.  Likewise, any  apparent trends  should be investigated to
determine if there is evidence of increasing concentration levels indicative of a possible release.
                                            14-19                                  March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

14.3.2      CHOOSING A SAMPLING INTERVAL VIA DARCY'S EQUATION

     Another strategy for determining an appropriate sampling interval is to use Darcy's equation. The
goal of this approach is to  calculate groundwater flow velocity and the time needed to ensure that
physically independent or distinct volumes of groundwater are collected on each sampling trip. As noted
in Chapter 6, physical independence does not guarantee statistical independence. However, statistical
independence may be more likely if the same general volume of groundwater is not re-sampled on
multiple occasions.

     This  section discusses the  important hydrological  parameters  to  consider  when choosing a
sampling interval. The Darcy equation is used to determine the horizontal component of the average
linear velocity  of ground water  for  confined,  semi-confined,  and unconfined aquifers. This value
provides a good estimate of travel  time for most soluble constituents in groundwater, and can be used to
determine a minimal sampling interval. Example calculations are provided to further assist the reader.
Alternative methods should be employed to determine a sampling interval in groundwater environments
where Darcy's law is invalid. Karst, cavernous basalt, fractured rocks, and other 'pseudo-karst' terranes
usually require specialized monitoring approaches.

     Section 264.97(g) of 40 CFR Part 264 Subpart F allows the owner or operator of a RCRA facility
to choose a sampling procedure that will reflect site-specific concerns. It specifies that the owner or
operator shall obtain a sequence of at least four samples from each well  collected at least semi-annually.
The interval  is  determined after evaluating the  uppermost aquifer's  effective  porosity,  hydraulic
conductivity, and hydraulic gradient, and the fate and transport characteristics of potential contaminants.
The intent of this  provision is to set a sampling frequency that allows sufficient time between sampling
events to ensure, to the greatest extent technically feasible, that independent groundwater observations
are taken from each well.

     The  sampling frequency required  in Part 264  Subpart  F  can be based on estimates using the
average linear  velocity  of  ground water.  Two  forms of the Darcy equation stated  below relate
groundwater velocity (V) to effective porosity (Ne), hydraulic gradient (/'), and hydraulic conductivity
(K):
                                                     e                                  [14.17]

                                        Vv=(Kv-i)/Ne                                  [14.18]

where  Vh and Vv are the  horizontal and  vertical  components  of the average  linear velocity  of
groundwater,  respectively; Kh  and  Kv are  the  horizontal and  vertical components  of hydraulic
conductivity, respectively;  /' is the head gradient; and Ne is the effective porosity.

     In applying these equations to ground-water monitoring, the horizontal component of the average
linear  velocity (Vh)  can  be used to determine  an appropriate  sampling interval. Usually, field
investigations  will yield  bulk values for  hydraulic conductivity.  In most  cases, the bulk hydraulic
conductivity determined by a pump test, tracer test, or a slug test will be sufficient for these calculations.
The vertical component (Fv), however, should be considered in  estimating flow velocities in areas with
significant components of vertical velocity such as recharge and discharge zones.

                                             14-20                                   March 2009

-------
Chapter 14.  Temporal Variability
                                                              Unified Guidance
     To apply the Darcy equation to groundwater monitoring, the parameters K, /', and Ne need to be
determined. The hydraulic conductivity, K, is the volume of water at the existing kinematic viscosity that
will move in unit time under a unit hydraulic gradient through a unit area measured at right angles to the
direction of flow. "[E]xisting kinematic viscosity" refers to the fact that hydraulic conductivity is not
only determined by the media  (aquifer),  but  also by  fluid properties (groundwater or potential
contaminants). Thus, it is possible to have several hydraulic conductivity values for different chemical
substances present in the same aquifer. The lowest velocity value calculated using  the Darcy equation
should be used to determine sampling intervals, ensuring physical independence  of  consecutive sample
measurements.

                 Figure 14-6.  Hydraulic Conductivity of  Selected  Rocks
                   IGNEOUS AND  METAMORPHIC   ROCKS
               Unf roct u red
                                 Fractured
                                           BASALT
               Unf ractured
                             Fractured
                      SANDSTONE
                                                                    Lava flow
                                Fractured      Semiconsolidoted
                        SHALE
               Unfractured      Fractured
                                                  CARBONATE  ROCKS
                                      Fractured
                           CLAY              SILT,  LOESS
                                                    Cavernous
                                                    SILTY  SAND
                                                       CLEAN  SAND
                                                          Fine     Coarse
                               GLACIAL  TILL
                l _ i _ I _ i _ l _ l _ I _ I _ L
               I0~8  IO"7  IO"6   IO"5   10""  I0~3   IO"2  10"'     I
                                                     GRAVEL
                                              _l	I	I	I
                                               10   10 2  IO3   IO4
                  I _ I
IO"7  IO"6  I0~5   I0~4  IO"3   IO"2  10"'
                             ft d"1
                                                             10   10 2   10 3   10 *  10  5
            IO"7  IO"6  IO"5   IO"4  IO"3   IO"2  10"'    I     10    10 2   10 3  10 *   10 5
                                            gal d-1 ft-2
            Source: Heath, B.C. 1987. Basic Ground-Water Hydrology. U.S. Geological Survey Water Supply Paper, 2220, 13 pp.
                                              14-21
                                                                      March 2009

-------
Chapter 14.  Temporal Variability
Unified Guidance
     A range of hydraulic conductivities (the transmitted fluid is water) for various aquifer materials is
given in Figures  14-6 and  14-7. The conductivities are given in several  units.  Figure  14-8 lists
conversion factors to change between various permeability and hydraulic conductivity units.

     The hydraulic gradient, /', is the change in hydraulic head per unit of distance in a given direction. It
can be determined by dividing the difference in head between two points on a potentiometric surface
map by the orthogonal distance between those two points (see calculation in Example 14-5). Water level
measurements are normally used to determine the natural hydraulic gradient at a facility. However, the
effects of mounding in the event  of a release  may produce a steeper local hydraulic gradient in the
vicinity of the monitoring well. These local changes in hydraulic gradient should be accounted for in the
velocity calculations.
       Figure  14-7.  Range of Values of Hydraulic Conductivity and Permeability
Unconsolidated k k K K K
h!OCKS deposits ^ (darcy) (cma) (cm/s) (m/s) (gal/day/ft





j H_
5 ^
— 1^

-  o
Q_ OJ U
c'£
CT Q--O
-~ i- c
T3 O O















T




plO5 r10"3 r102


o
(.
5

5


o
C

1"


o

« £ 0)  In
3 a c — a) .•!
Itl
"B QJ
•og^ 11
^f 2 
-------
Chapter 14. Temporal Variability
Unified Guidance
 Figure 14-8. Conversion Factors for Permeability and Hydraulic Conductivity Units

cm2
ft2
darcy
m/s
ft/s
gal/day/ft2
cm2
1
9.29xl02
9.87xl(T9
1.02xl(T3
3.11xlO-4
5.42xl(T10
Permeability, k*
ft2
l.OSxlCT3
1
i.oexicr11
i.ioxicr5
3.35xl(T7
5.83xlQ-13
darcy
l.OlxlO8
9.42xl010
1
1.04xl05
3.15xl04
5.49xlQ-2
Hydraulic conductivity, K
m/s ft/s gal/day/ft2
9.80xl02
9.11xl05
9.66xlQ-5
1
s.osxicr1
4.72xlQ-7
3.22xl03
2.99xl05
3.17xlQ-5
3.28
1
i.ssxicr5
1.85xl09
1.71xl012
1.82X101
2.12xl05
6.46xl05
1
                                             -3
    *To obtain k in ft2, multiply k in cm2 by 1.08x10

    Source: Freeze, R.A., and J.A. Cherry (1979). Ground Water. Prentice Hall, Inc., Englewood Cliffs,
    New Jersey, p. 29.
     The effective porosity, Ne, is the ratio, usually expressed as a percentage, of the total volume of
voids available for fluid transmission  to the total volume of the porous medium de-watered.  It can be
estimated during a pump test by dividing the volume of water removed from an aquifer by the total
volume of aquifer dewatered (see calculation  in Example 14-5). Figure 14-9 presents approximate
effective porosity values for a  variety of aquifer materials. In cases where the effective porosity is
unknown, specific yield may be substituted into the equation. Specific yields of selected rock units are
given in Figure  14-10. In the absence  of measured  values,  drainable porosity  is often  used to
approximate effective porosity. Figure 14-11 illustrates representative values of drainable porosity and
total porosity as a function of aquifer particle size. If available, field measurements of effective porosity
are preferred.
                                             14-23
        March 2009

-------
Chapter 14.  Temporal Variability
              Unified Guidance
   Figure 14-9.  Default Values of Effective Porosity (A/e)  For Travel Time Analyses
                         Soil textural classes
   Effective porosity of
        saturation3
       Unified soil classification system

            GS, GP,  GM, GC,  SW, SP, SM,  SC
            ML, MH
            CL, OL, CH, OH, PT

       USDA soil textural classes

            Clays, silty clays, sandy clays
            Silts, silt loams, silty clay loams
            All others

       Rock units (all)

            Porous media (non-fractured rocks such as sandstone
            and some carbonates)
      	Fractured rocks (most carbonates, shales, granites, etc.)
 0.20 (20%)
 0.15 (15%)
 0.01 (l%)b
 0.01 (l%)b
 0.10 (10%)
 0.20 (20%)
 0.15 (15%)

 0.0001 (0.01%)
       Source: Barari, A., and L. S. Hedges (1985). Movement of Water in Glacial Till. Proceedings of
       the 17th International Congress of the International Association of Hydrogeologists, pp.  129-
       134.

       aThese values are estimates and there may be differences between similar units. For example,
       recent studies indicate that weathered and unweathered glacial till may have markedly
       different effective porosities (Barari and Hedges, 1985;  Bradbury et al., 1985).

       bAssumes de minimus secondary porosity. If fractures or soil structure are present, effective
       porosity should be 0.001 (0.1%).
              Figure 14-10. Specific Yield Values for Selected Rock Types
                   Rock Type
Specific Yield (%)
       Clay
       Sand
       Gravel
       Limestone
       Sandstone (semi-consolidated}
       Granite
       Basalt (young}	
        2
        22
        19
        18
        6
       0.09
        8
Source: Heath, R.C. (1983). Basic Ground-Water Hydrology. U.S. Geological Survey, Water Supply Paper
2220, 84 pp.
                                              14-24
                      March 2009

-------
Chapter 14.  Temporal Variability
                                                                 Unified Guidance
     Once the values for K, /', and Ne are determined, the horizontal component of average linear
groundwater velocity  can be calculated. Using the Darcy equation [14.17],  the  time required for
groundwater to pass through the complete monitoring well diameter can be determined by dividing the
well diameter by the horizontal component of the average linear groundwater velocity.  If considerable
exchange of water occurs during well purging, the diameter of the filter pack may be used rather than the
well diameter. This value represents  the minimum time  interval  required  between sampling  events
yielding  a physically  independent (i.e., distinct) ground-water sample.  Note that three-dimensional
mixing of groundwater in the vicinity of the monitoring well is likely to occur when the well is purged
before sampling. Partly for that reason, this method can only provide an estimated travel time.

  Figure 14-11. Total  Porosity and Drainable Porosity for Typical Geologic Materials
c

-------
Chapter 14. Temporal Variability _ Unified Guidance

       ^EXAMPLE 14-5

     Compute the effective porosity, Ne, expressed as a percent (%), using results obtained during a
pump test.

       SOLUTION
Step 1 .    Compute the effective porosity using the following equation:

                Ne = 100% x volume of water removed/ volume of aquifer dewatered         [14.19]

Step 2.    Based on a pumping rate of 50 gal/min and a pumping  duration  of 30  min, compute the
         volume of water removed as:

                   volume of water removed = 50 gal/min x 30 min = 1,500 gal

Step 3 .    To calculate the volume of aquifer de-watered, use the equation:
                                                                                     [14.20]


         where r is the radius (in ft) of the area affected by pumping and h (ft) is the drop in the water
         level. If, for example, h = 3 ft and r = 18 ft, then:
                                F = -(3.14x3xl82)=l,018ft3
                                    3

         Next, converting cubic feet of water to gallons of water,

                             V = 1,018 ft3 x 7.48  gal/ft3 = 7,615 gal

Step 4.   Finally, substitute the two volumes from Step 3 into equation [14.19] to obtain the effective
         porosity:

                         Ne = 100% x (l,500 gal/7,615 gal)= 19.7%  ^

       ^EXAMPLE 14-6

     Determine the hydraulic gradient, /', from a potentiometric surface map.

       SOLUTION
Step 1.   Consider the potentiometric  surface map in Figure 14-12. The hydraulic gradient  can be
         constructed as /' = Ah/1, where Ah is the difference measured in the gradient at piezometers Pzj
         and Pz2,and I is the orthogonal distance between the two piezometers.
                                            14-26                                  March 2009

-------
Chapter 14. Temporal Variability
Unified Guidance
  Figure 14-12.  Potentiometric Surface Map for Computation of Hydraulic Gradient
                                                                           29.2'
     N
Step 2.   Using the values given in Figure 14-12, the hydraulic gradient is computed as:

                         / = A/2// = (29.2 ft - 29.1 ft)/100 ft = 0.001  ft/ft

Step 3.   Note that this method provides only a very general estimate of the natural hydraulic gradient
         existing in the vicinity of the two piezometers. Chemical gradients are known to exist and may
         override the effects of the hydraulic gradient. A  detailed study of the effects of multiple
         chemical contaminants may be necessary to determine the actual average linear groundwater
         velocity (horizontal component) in the vicinity of the monitoring wells. ~4

       ^EXAMPLE 14-7

     Determine the horizontal component of the average linear groundwater velocity (Vh) at  a land
disposal facility which has monitoring wells screened in an unconfmed silty sand aquifer.

       SOLUTION
Step 1.   Slug tests, pump tests, and tracer tests conducted during a hydrologic site investigation have
         revealed that the  aquifer has a horizontal hydraulic conductivity (Kh) of 15  ft/day and an
         effective porosity (Ne) of 15%. Using a potentiometric map (as in Example 14-6), the regional
         hydraulic gradient (/') has been determined to be 0.003 ft/ft.

Step 2.   To estimate the minimum time interval between sampling events enabling the collection of
         physically independent  samples of ground water, calculate the horizontal component of the
         average linear groundwater velocity (Vh) using Darcy's equation [14.17]. With^/, = 15 ft/day,
         Ne = 15%, and / = 0.003 ft/ft, the velocity becomes:

                   Vh = (l5 ft/day x0.003 ft/ft)/15% = 0.3 ft/day  or 3.6 in/day

Step 3.   Based on  these calculations,  the horizontal component of the average linear groundwater
         velocity, Vh, is  equal to 3.6 in/day.  Since monitoring well  diameters at this particular facility
         are  4 inches, the minimum time interval between sampling events enabling a physically
                                            14-27
        March 2009

-------
Chapter 14. Temporal Variability
                                                             Unified Guidance
         independent groundwater sample can be computed by dividing the horizontal component into
         the monitoring well diameter:

                      Minimum time interval = (4 iny(3.6 in/day)= 1.1 days

         As a result, the facility could theoretically  sample  every other day. However, this may be
         unwise because velocity can seasonally vary with recharge rates. It is also emphasized that
         physical independence does not guarantee statistical independence. Figure 14-13 gives results
         for common  situations. The  overriding point  is that it may not  be necessary to set the
         minimum sampling frequency to quarterly at every site.  Some hydrologic environments may
         allow for more frequent sampling, some less. ~4

  Figure 14-13. Typical  Darcy Equation Results  in Determining a Sampling Interval
Unit
Gravel
Sand
Silty Sand
Till
Silty Sand (semi-consolidated)
Basalt
Kh (ft/day)
104
102
10
io-3
1
io-1
Ne (%)
19
22
14
2
6
8
Vh (in/mo)
9.6xl04
8.3xl02
l.SxlO2
9.1xlO-2
30
2.28
Sampling Interval
Daily
Daily
Weekly
Monthly
Weekly
Monthly
14.3.3
CREATING ADJUSTED, STATIONARY MEASUREMENTS
     When an  existing  data set exhibits temporal  correlation or other variability, it is sometimes
possible to remove the temporal pattern and thereby create a set of adjusted data which are uncorrelated
and stationary over time in mean level. As  long  as the same temporal pattern seems to  affect both
background and the compliance point data to be tested, the effect (e.g., regular seasonal fluctuation) can
be estimated  and removed from  both data sets prior to statistical testing. Testing the adjusted data
instead of the  raw measurements in this way results in a more powerful and accurate test. An extraneous
source of variation not related to identifying a contaminant  release has been removed from the sample
data.

     The general topic of stationary, adjusted data is  complex, contained within the extensive literature
on time series. The Unified Guidance discusses two  simple cases below: removing a seasonal pattern
from a single well and creating adjusted data from a one-way ANOVA for temporal effects across
several wells.  More complicated situations may need professional consultation.

14.3.3.1    CORRECTING FOR SEASONAL  PATTERN IN A SINGLE WELL
       BACKGROUND AND PURPOSE

     Sometimes an obvious cyclical seasonal pattern can be seen in a time series plot. Such data are not
statistically independent.  They do not fluctuate randomly but rather in a predictable way from one
sampling event to the next. Data from such patterns can be adjusted to correct for or remove the seasonal
fluctuation, but only if a longer series of data is available. This is also  known  as deseasonalizing the
data. Seasonal correction should be done both to minimize the chance of mistaking a seasonal effect for
                                           14-28
                                                                     March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

evidence of contaminated groundwater, and also to build  more powerful background-to-compliance
point tests.

     Problems can arise, for instance, from measurement variations associated with changing recharge
rates during different seasons. Compliance point concentrations can exceed a groundwater protection
standard [GWPS] for a portion of the year, but on average lie below.  If the long-term  average is of
regulatory concern, the data should first be de-seasonalized before comparing it against a GWPS.

     If point-in-time,  interwell  comparisons  are  being  made  between  simultaneously collected
background and downgradient data, a correction may not be necessary even when seasonal fluctuations
exist. A temporal cycle may  cover a period of  several  years  so that  both the background and
downgradient values are observed on essentially the same parts of the overall cycle.  In  this case, the
short-term  averages  in both  data  sets  will be  fairly stable and the seasonal  or  cyclical  effect  may
equivalently impact both sets of data.

     For intrawell tests, the data need to be collected sequentially at each well, with background formed
from the earliest measurements in the series. The  point-in-time  argument would not apply and the
presence of seasonality should be checked and accounted for.

     Even with interwell testing, it is sometimes difficult to verify whether or not a seasonal pattern is
impacting upgradient and compliance point wells similarly.  If the groundwater velocity is low, the lag
between the time groundwater passes through a background well screen and then travels downgradient
may create a noticeable shift  as to when corresponding portions of the  seasonal cycle are observed in
compliance point locations. It also may be the case that differences in geochemistry from well to well
may cause the  same seasonal pattern  to  differentially impact  concentration levels at distinct  wells
(Figure 14-14)
                                             14-29                                   March 2009

-------
Chapter 14. Temporal Variability
      Unified Guidance
    Figure 14-14.  Differential  Seasonal Effects:  Background vs.  Compliance Wells
                  H
                  H
                  O
                  O
                  o
                             10
                                  20
                                       30
                                            40    50    60
                                                WEEK
                                                            70    80    90    100
                                E5aekgroimd
Compliance
     If the timing of the cycle and the relative magnitude of the concentration swings are essentially the
same in upgradient and downgradient wells, both data sets should be deseasonalized prior to statistical
analysis. If the seasonal effects are ignored, real differences in mean levels between upgradient and
downgradient well data may not be observed, simply because the short-term seasonal fluctuations add
variability that can mask the difference being tested. In this case, the  non-independent nature of the
seasonal pattern adds unwanted noise to the observations, obscuring statistical evidence of groundwater
contamination.

       REQUIREMENTS AND ASSUMPTIONS

     Seasonal correction is only appropriate for wells where a cyclical pattern is clearly present and very
regular  over time. Many approaches to deseasonalizing data exist.  If the  seasonal pattern is highly
regular, it may be modeled with a sine or cosine function.  Often, moving averages and/or lag-based
differences (of order 12 for monthly data, for example) are used. General time series models may include
these and other more complicated methods for deseasonalizing the data.

     The  simple method  described in the Unified Guidance has  the advantage  of being  easy  to
understand and apply, and of providing natural estimates of the monthly or quarterly seasonal effects via
the monthly or quarterly  means.  The method can be applied to any seasonal or recurring cycle— perhaps
an annual cycle  for monthly  or  quarterly  data or a longer cycle for certain kinds  of geologic
environments.  In some cases, recharge rates are linked to  drought cycles that may be  on the order of
                                            14-30
              March 2009

-------
Chapter 14.  Temporal Variability	Unified Guidance

several years long. For these situations, the 'seasonal' cycle may not correspond to typical fluctuations
over the course of a single year.

     Corrections for  seasonality  should  be used cautiously,  as they  represent extrapolation into the
future. There  should  be a good  physical  explanation for the seasonal  fluctuation  as  well  as  good
empirical evidence for seasonality before corrections are made. Higher than average rainfall for two or
three Augusts in a row does not justify the belief that there will never  be a drought in August, and this
idea extends directly  to groundwater quality.  At least three  complete cycles  of the seasonal pattern
should be observed on a time series plot  before  attempting the adjustment below. If seasonality is
suspected but the pattern is complicated, the user should seek the help of a professional statistician.

       PROCEDURE

Step 1.   If a time series plot clearly shows at least 3 full cycles of the seasonal pattern, determine the
         length of time to complete one full cycle.  Since the correction  presumes a regular sampling
         schedule, determine the number of observations (K) in each full  cycle (this number should be
         the same for each cycle). Then, assuming that N complete cycles of data are available, let xy-
         denote the raw, unadjusted measurement for the /'th sampling event during they'th complete
         cycle. Note  that this could represent monthly data over an annual cycle, quarterly data over a
         biennial cycle, semi-annual data over a 10-year cycle, etc.

Step 2.   Compute the mean concentration for sampling event /' over the TV-cycle period:

                                x-   =   (xn+x!2  +... + XiN)/N                          [14.21]

         This is the average of all observations taken in different cycles, but during the same sampling
         event.  For instance, with monthly data over an annual cycle, one would calculate the mean
         concentrations for all Januarys, the  mean for all Februarys, and so on for each of the 12
         months.

Step 3.   Calculate the grand mean, x , of all Nxk observations:
                                               Ixk   ~* k

Step 4.   Compute seasonally-corrected, adjusted concentrations using the equation:

                                         z.. = x.. - x. + x                                   [14.23]

         Computing  x. -x.  removes the average seasonal effect of sampling event /' from the data
         series. Adding back the overall mean,  x , gives the adjusted zy values the same mean as the
         raw, unadjusted data. Thus, the overall mean  of the corrected values, z , equals the overall
         mean value, x .
                                             14-31                                   March 2009

-------
Chapter 14. Temporal Variability
Unified Guidance
       ^EXAMPLE 14-8

     Consider the monthly unadjusted  concentrations of an analyte over a 3-year period graphed in
Figure 14-15 and listed in the table below. Given the clear and regular seasonal pattern, use the above
method to produce a deseasonalized data set.
Unadjusted Concentrations
1983 1984 1985
January
February
March
April
May
June
July
August
September
October
November
December
Overall
1.99
2.10
2.12
2.12
2.11
2.15
2.19
2.18
2.16
2.08
2.05
2.08
3-year
2.01
2.10
2.17
2.13
2.13
2.18
2.25
2.24
2.22
2.13
2.08
2.16
average = 2.17
2.15
2.17
2.27
2.23
2.24
2.26
2.31
2.32
2.28
2.22
2.19
2.22

Monthly
Average
2.05
2.12
2.19
2.16
2.16
2.20
2.25
2.25
2.22
2.14
2.11
2.16

Adjusted Concentrations
1983 1984 1985
2.11
2.14
2.10
2.13
2.12
2.12
2.11
2.10
2.11
2.10
2.11
2.09

2.13
2.14
2.15
2.14
2.14
2.15
2.17
2.16
2.17
2.15
2.14
2.17

2.27
2.21
2.25
2.24
2.25
2.23
2.23
2.24
2.23
2.24
2.25
2.23

       SOLUTION
Step 1.   From Figure 14-15, there are N = 3 full cycles represented, each lasting approximately a year.
         With monthly data, the number of sampling events per cycle is k = 12.

Step 2.   Compute the monthly averages across  the 3 years for each of the 12 months using equation
         [14.21]. These values are shown in the fifth column of the table above.

Step 3.   Calculate the grand mean over the 3-year period using equation [14.22]:

                  x  =  —!—(1.99 + 2.01 + 2.15 + 2.10   +... + 2.22)  =  2.17
                         3-12

Step 4.   Within each month and year,  subtract the average monthly concentration for that month and
         add-in the grand mean, using equation [14.23]. As an example, for January 1983, the adjusted
         concentration becomes:

                                 z,, =1.99-2.05 + 2.17 = 2.11
                                            14-32
        March 2009

-------
Chapter 14. Temporal Variability
                                        Unified Guidance
            Figure 14-15. Seasonal Time Series Over a Three-Year Period
             Jan-83    May-83   Sep-83
                       Unadjusted
 Jan-84   May-84    Sep-84    Jan-85   May-85   Sep-85

      TIME (Month)

—     Adjusted 	+	       3-Year Mean 	
         The  adjusted concentrations are shown in the last three columns  of the table  above. The
         average of all 36  adjusted concentrations equals  2.17, the same  as the mean unadjusted
         concentration. Figure 14-15 shows the adjusted data superimposed on the unadjusted data.
         The raw data exhibit seasonality, as well as an upward trend. The adjusted data, on the other
         hand, no longer exhibit a seasonal pattern, although the upward trend still remains. From a
         statistical standpoint, the trend is much more easily identified by a trend test on the adjusted
         data than with the raw data. -4
14.3.3.2    CORRECTING FOR A TEMPORAL EFFECT ACROSS SEVERAL WELLS
       BACKGROUND AND PURPOSE

     When a significant temporal dependence or correlation is identified across a group of wells using
one-way ANOVA for temporal effects (Section 14.2.2), results of the ANOVA can be used to create
stationary adjusted data similar to the seasonal correction described in Section 14.3.3.1. The difference
is that the adjustment is not applied to a data series at a single well, but rather simultaneously to several
well sets.
                                            14-33
                                                March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

     The adjustment works in the same way as a correction for seasonality. First, the mean for each
sampling event or season (averaged across wells) is computed along with the grand mean. Then each
individual measurement is adjusted by subtracting off the event/seasonal mean and adding the overall or
grand mean. In practice, this process is identical to  adding the one-way ANOVA residual to the grand
mean, so the already-computed results of the ANOVA can be used.  By  removing or correcting for a
significant temporal effect, the adjusted  data will have a temporally stationary mean and less  overall
variation. This allows for more powerful  and accurate detection monitoring tests.

     Temporal  dependence (e.g., seasonality) is sometimes observed as parallel traces on a time series
plot across multiple wells (Section 14.2.1), although the one-way ANOVA for temporal effects  is non-
significant.  This can occur due to the simultaneous presence of strong spatial variability (Chapter 13).
Differences in mean levels from well to well can be large enough to 'swamp' the added variation due to
the temporal dependence. The one-way ANOVA for temporal effects will not identify the dependence
because the mean error sum  of squares will then include the spatial variation component and not just
random error.

     Two remedies are possible when the ANOVA for temporal effects  is non-significant. First, if a
strong parallelism is evident on time series plots, the residuals from the ANOVA can still be used to
create a set of adjusted, temporally-stationary measurements.  The adjustment will not eliminate  or
remove any  existing spatial variation, but it may not matter. Intrawell  tests are needed anyway when
such spatial variability is evident,  and those tests assume temporal independence of the measurements
collected at each well.

     A second remedy is to perform a two-way  ANOVA, testing for both spatial variation and temporal
effects.  This procedure is discussed in Davis (1994). Not only will a two-way ANOVA more  readily
identify a significant temporal effect even when there is simultaneous spatial variability,  but  the F-
statistic used to test for the temporal  dependence can be utilized to further adjust the appropriate degrees
of freedom in intrawell background limits, such as prediction limits and control charts.

       REQUIREMENTS AND ASSUMPTIONS

     The key requirement to correct for a temporal effect using ANOVA is that the same effect must be
present  in all wells to which the adjustment is applied. Otherwise, the adjustment will tend to skew or
bias measurements at wells with no observable temporal dependence. Parallel time series plots (Section
14.2.1)  should be examined  to determine whether all the wells under  consideration exhibit a  similar
temporal pattern.

     The parametric one-way ANOVA assumes the data are normal or can be normalized. If the data
cannot be normalized,  a Kruskal-Wallis non-parametric ANOVA can be conducted to  test for  the
presence of a temporal dependence. In this case, no residuals can be computed since the Kruskal-Wallis
test employs ranks of the data rather than the measurements themselves.  So the adjustment presented
below is only applicable for data sets that can be normalized.

     PROCEDURE

Step 1.   Given a set of Wwells and measurements from each of T sampling events at each well on each
         of K years, label the observations as xp,  for /' =  1 to W, j = 1 to T, and k = 1 to K. Then xp
         represents the measurement from the rth well on they'th sampling event during the Mi year.
                                             14-34                                   March 2009

-------
Chapter 14. Temporal Variability
Unified Guidance
Step 2.   Using the one-way ANOVA for temporal effects (Section 14.2.2), compute the sampling
         event or seasonal means (whichever is appropriate), along with the grand (overall) mean. Also
         construct the ANOVA residuals using either equation [14.5] or [14.6].

Step 3.   Add each residual to the  grand mean to  form  adjusted values z..k=xttt+r..k. Use these
         adjusted values in subsequent statistical testing instead of the original measurements.

       ^EXAMPLE 14-9

     The manganese data of Examples  14-1 and  14-2 were found to have a significant temporal
dependence using ANOVA for temporal effects. Adjust these data to remove the temporal pattern.
Qtr
1
2
3
4
5
6
7
8

Event
Mean
29.290
30.110
30.780
31.620
33.747
31.930
30.513
30.345

Manganese Residuals (ppm)
BW-1 BW-2 BW-3
-1.15
-0.78
-0.33
0.80
0.6225
1.32
0.5075
-1.845
Grand
2.12
0.16
1.79
1.15
-0.7175
0.25
-1.6625
2.535
mean = 31
-2.14
0.13
-1.64
-1.03
1.1325
-1.40
-0.1825
0.075
042
BW-4
1.17
0.49
0.18
-0.92
-1.0375
-0.17
1.3375
-0.765

       SOLUTION
Step 1.   The mean of each sampling event taken across the four background wells was computed in
         Example 14-2, along with the grand mean. These results are listed in the table above, along
         with the individual residuals which were also computed in that example.

Step 2.   Add the grand mean to each residual to form the adjusted manganese concentrations, as in the
         table below.
Qtr
1
2
3
4
5
6
7
8

Event
Mean
29.290
30.110
30.780
31.620
33.747
31.930
30.513
30.345

Adjusted
BW-1
29.89
30.26
30.71
31.84
31.66
32.36
31.55
29.20
Grand
Manganese
BW-2
33.16
31.20
32.83
32.19
30.32
31.29
29.38
33.58
(ppm)
BW-3
28.90
31.17
29.40
30.01
32.17
29.64
30.86
31.12
BW-4
32.21
31.53
31.22
30.12
30.00
30.87
32.38
30.28
mean = 31.042
                                            14-35
        March 2009

-------
Chapter 14. Temporal Variability
Unified Guidance
Step 3.   Plot a time  series of the adjusted manganese values, as in  Figure 14-16. The 'hump-like'
         temporal pattern evident in Figure 14-2 is no longer apparent. Instead, the overall mean is
         stationary across the 8 quarters. -4
   Figure 14-16. Parallel Time Series Plot of Adjusted Manganese Concentrations
          f °
          I
                                              Quarter
14.3.3.3    CORRECTING FOR LINEAR TRENDS

     If a data series exhibits a linear trend, the sample will exhibit temporal dependence when tested via
the sample autocorrelation function (Section  14.2.3), the rank von Neumann ratio (Section 14.2.4), or
similar procedure.  These data can be de-trended, much like the  data in the previous  example were
deseasonalized.  Probably the easiest way to de-trend observations with a linear trend is to compute a
linear regression on the data (Section 17.3.1) and then use the regression residuals instead of the original
measurements in subsequent statistical analysis.

     But no matter how tempting it may be to  automatically  de-trend data of this sort, the user is
strongly cautioned to consider what a linear trend may represent. Often, an upward trend is indicative of
changing groundwater conditions at a site, perhaps due to the increasing presence of contaminants
during a gradual waste release. The trend in this case may itself be statistically significant evidence of
groundwater contamination,   particularly  if  it occurs  at  compliance  wells but  not  at  upgradient
background wells. The  trend tests of Chapter 17 are useful for such determinations. Trends in
background may  signal  other important factors, including migration of contaminants from off-site
sources, changes in the regional aquifer, or possible groundwater mounding.
                                            14-36
        March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance

     The overriding point is that data should be deseasonalized when a cyclical pattern might obscure
the random deviations around an otherwise stable average concentration level, or to more clearly identify
an existing trend. However, a linear trend is inherently indicative of a changing mean level. Such data
should not be de-trended before it is determined what the trend likely represents, and whether or not it is
itself prima facie evidence of possible groundwater contamination.

     A similar trend both in direction and slope may be exhibited by background wells and compliance
wells, perhaps suggestive of sitewide changes in natural groundwater conditions. Residuals from a one-
way  ANOVA for temporal effects (Section 14.2.2) can be used to simultaneously create adjusted values
across the well network (Section 14.3.3.2). Linear trends are just as easily identified and adjusted in this
way  as are parallel seasonal fluctuations or other temporal effects.
14.3.4      IDENTIFYING  LINEAR  TRENDS  AMIDST  SEASONALITY:  SEASONAL
       MANN-KENDALL TEST

       BACKGROUND AND PURPOSE

     Corrections for seasonality or other cyclical patterns over time in a single well  are discussed in
Section 14.3.3.1. These adjustments work best when the long-term mean at the well is stationary. In
cases where a test for trend is desired and there are also seasonal fluctuations, Chapter 17 tests may not
be sensitive enough to detect a real trend due to the added seasonal variation.

     One possible remedy is to use the seasonal  correction in Section  14.3.3.1 and illustrated in
Example 14-8.  The seasonal component of the trend is removed prior to conducting a formal trend test.
A second option is the seasonal Mann-Kendall test (Gilbert, 1987).

     The seasonal Mann-Kendall is a simple modification to the  Mann-Kendall test for trend (Section
17.3.2) that accounts for apparent seasonal fluctuations. The basic idea is to divide a longer multi-year
data series  into subsets, each  subset representing the measurements collected on a common sampling
event (e.g., all January events or all fourth quarter events). These subsets then represent different points
along the regular seasonal cycle, some associated with peaks and others with troughs. The usual Mann-
Kendall test is performed on each subset separately and a Mann-Kendall test statistic S\  formed for each.
Then the separate S\ statistics are summed to get an overall Mann-Kendall statistic S.

     Assuming that the same basic trend impacts each subset,  the combined statistic S will be powerful
enough to identify a trend despite the seasonal fluctuations.

       REQUIREMENTS AND ASSUMPTIONS

     The basic requirements of the Mann-Kendall trend test are discussed in Section  17.3.2. The only
differences with the seasonal Mann-Kendall test are that  1) the  sample should be a multi-year series with
an observable seasonal  pattern each year; 2) each 'season' or subset of the overall series should include
at least three measurements in order to  compute the  Mann-Kendall  statistic;  and 3) a  normal
approximation to the overall Mann-Kendall test statistic  must be tenable. This will generally be the case
if the series has at least  10-12 measurements.

                                            14-37                                  March 2009

-------
Chapter 14.  Temporal Variability	Unified Guidance

       PROCEDURE

Step 1.   Given a series of measurements from each of T sampling events on each of K years, label the
         observations as .% for /' = 1 to T, andy = 1 to K. Then xy represents the measurement from the
         rth sampling event during they'th year.

Step 2.   For each distinct sampling event (/'), form a seasonal subset by grouping together observations
         xn, Xi2,...., x;K. This results in T separate seasons.

Step 3.   For each seasonal  subset, use the procedure in Section 17.3.2 to compute the Mann-Kendall
         statistic Si and its standard deviation SD[Si\. Form the overall seasonal Mann-Kendall statistic
         (S) and its standard deviation with the equations:


                                           S = i>,                                     [14.24]
Step 4.   Compute the normal approximation to the seasonal Mann-Kendall statistic using the equation:

                                                                                        [14.26]
Step 5.   Given significance level,  a,  determine the  critical point zcp  from the  standard  normal
         distribution in Table  10-1  of Appendix D. Compare Z against this critical point. If Z > zcp,
         conclude there is statistically significant evidence at the a-level of an increasing trend. If Z < -
         zcp, conclude  there is statistically  significant  evidence  of a decreasing trend.  If neither,
         conclude that the sample evidence is insufficient to identify a trend.

       ^EXAMPLE 14-10

     The data set in Example  14-8  replicated below indicated both clear seasonality and an apparent
increasing trend. Use the seasonal Mann-Kendall procedure to test for a significant trend with a = 0.01
significance.
                                             14-38                                   March 2009

-------
Chapter 14. Temporal Variability
                          Unified Guidance
                              Analyte Concentrations
                            1983       1984      1985
           Si
                                                           S= 35
                              SD[S|]
January
February
March
April
May
June
July
August
September
October
November
December
1.99
2.10
2.12
2.12
2.11
2.15
2.19
2.18
2.16
2.08
2.05
2.08
2.01
2.10
2.17
2.13
2.13
2.18
2.25
2.24
2.22
2.13
2.08
2.16
2.15
2.17
2.27
2.23
2.24
2.26
2.31
2.32
2.28
2.22
2.19
2.22
3
2
3
3
3
3
3
3
3
3
3
3
1.915
1.633
1.915
1.915
1.915
1.915
1.915
1.915
1.915
1.915
1.915
1.915
                   SD[S]= 6.558
       SOLUTION
Step 1.   Form a seasonal subset for each month by grouping all the January measurements, all the
         February measurements, and so on, across the 3 years of sampling. This gives 12 seasonal
         subsets with n  = 3 measurements  per season. Note there  are no  tied values  in any of the
         seasons except for February.

Step 2.   Use  equations [17.30] and [17.31]  in  Section 17.3.2 to compute the Mann-Kendall statistic
         (Si) for each subset. These values are listed in the table above. Also compute their sum to form
         the overall seasonal Mann-Kendall statistic, giving S= 35.

Step 3.   Use equation [17.28] from Section 17.3.2 for all months but February to compute the standard
         deviation of Si. Since n = 3 for each of these subsets, this gives
                                                   =— 3-2-11 = 1.915
         For the month of February, one pair of tied values exists. Use equation [17.27] to compute the
         standard deviation for this subset:
(t  -i
\,   A
t
                                                                                = 1.633
         List  all the subset  standard deviations  in  the table above. Then  use  equation  [14.25] to
         compute the overall  standard deviation:
                                                   . 915 )  + (1.633 ) =6.558
                                             14-39
                                  March 2009

-------
Chapter 14. Temporal Variability	Unified Guidance



Step 4.   Compute a normal approximation to S with equation [ 17.29]:

                                   Z = (35-l)/6.558 = 5.18

Step 5.   Compare Z against the 1% critical point from the standard normal distribution in Table 10-1
         of Appendix D, z.oi = 2.33. Since Z is clearly larger than z.oi, the increasing trend evidence in
         Figure 14-15 is highly significant. -4
                                            14-40                                  March 2009

-------
Chapter 15. Managing Non-Detect Data	Unified Guidance

     CHAPTER  15.  MANAGING  NON-DETECT DATA
       15.1   GENERAL CONSIDERATIONS FOR NON-DETECT DATA	 15-1
       15.2   IMPUTING NON-DETECT VALUES BY SIMPLE SUBSTITUTION	15-3
       15.3   ESTIMATION BY KAPLAN-MEIER	15-7
       15.4   ROBUST REGRESSION ON ORDER STATISTICS	15-13
       15.5   OTHER METHODS FOR A SINGLE CENSORING LIMIT	 15-21
             15.5.1 COHEN'S METHOD	15-21
             15.5.2 PARAMETRIC REGRESSION ON ORDER STATISTICS	15-23
       15.6   USEOFTHE15%/50%NON-DETECTSRULE	15.24
     This chapter considers strategies  for accommodating non-detect measurements in groundwater
data analysis. Five particular  methods are described  for incorporating non-detects into  parametric
statistical procedures. These include:

    »»»  Simple substitution (Section 15.2);
    »»»  Kaplan-Meier (Section 15.3);
    »»»  Robust Regression on Order Statistics (Section 15.4);
    *  Cohen's Method (Section 15.5.1); and
    »»»  Parametric Regression on Order Statistics (Section 15.5.2).
15.1 GENERAL CONSIDERATIONS FOR NON-DETECT DATA

     Non-detects  commonly reported  in groundwater monitoring are statistically known  as  "left-
censored" measurements, because the concentration of any non-detect either cannot be estimated or is
not reported directly. Rather, it is known or assumed only to fall within a certain range of concentration
values (e.g., between zero and the quantitation limit [QL]).  The direct estimate has been censored by
the limitations of the measurement process or analytical technique, and is deemed too uncertain to be
considered  reliable.  Groundwater non-detect data  are censored on the low or left end of a sample
concentration range. Other kinds of threshold data,  particularly survival rates in the medical literature,
are often reported as right-censored values.

     Historically, there has been inconsistent treatment of non-detects in groundwater analysis.  Often,
easily applied techniques have been favored over more sophisticated methods of handling non-detects.
This may primarily be  due to the lack of familiarity and difficulties with software that can incorporate
such methods. Even at present, most statistical  packages include  analysis routines for right-censored
values but not left-censored ones (Helsel, 2005).  Left-censored data needs to be converted to right-
censored data  for analysis and then back  again.   Despite these  limitations, the more sophisticated
methods are almost always superior to the methods of simple substitution.

     The past twenty  years has seen considerable research on statistical aspects  of  non-detect data
analysis.   Helsel  (2005)  provides  a detailed summary  of available methods for non-detects, and

                                            lifl                                  March 2009

-------
Chapter 15.  Managing Non-Detect Data	Unified Guidance

concludes that simple substitution usually leads to greater statistical bias and inaccuracy than with better
technical methods.  Gibbons (1994b) and Gibbons & Coleman (2001) offer a broad review of some of
the same research, not all of it directly relating to groundwater data.  Both Gibbons and McNichols &
Davis (1988) note that most of the existing studies focus  on an estimation of parameters such as the
mean and variance of an underlying population  from which the censored and detected data originate.
For these tasks, simple substitution methods tend  to perform poorly, especially when the non-detect
percentages are high (Gilliom & Helsel, 1986).

     Much less attention has been given to how left-censored data impact the results of statistical tests,
the actual data-based conclusions that are drawn when using detection, compliance, or corrective action
monitoring tests.  Closely  estimating  the  true  mean  and  variance  of the underlying background
population may be important, but does not directly answer how well a given test performs (in achieving
the nominal false positive error rate and correctly identifying true significant differences). McNichols &
Davis (1988) performed a limited study to address these concerns. They found that simple substitution
methods were among the best performers in simulated prediction limit tests even with fairly high rates of
censoring, so long as the prediction limit procedure incorporated a verification resample.

     Gibbons (1994b; also Gibbons and Coleman,  2001) conducted a similar limited simulation of
prediction limit testing performance incorporating a verification resample. They, too, found that a type
of simple substitution was one of the best performers when either an average of 20% or 50% of the data
was non-detect. The Gibbons study concluded that substituting zero for each non-detect worked better to
keep the false positive rate low than by substituting half the method detection limit [MDL].

     Both studies  primarily focused on the achievable false positive rate when censored  data are
present, rather than the statistical power of these tests to identify contaminated groundwater. In addition,
both only considered parametric prediction limits.  For data sets with fairly low  detection frequencies
(e.g., <50%), parametric prediction limits may not accurately accommodate left-censored measurements,
with or without retesting. The McNichols & Davis study in particular found that none of the simpler
methods for handling non-detects did well when the  underlying data came from a skewed distribution
and the non-detect percentage was over 50%.

     On balance, there are four general  strategies for handling non-detects:  1)  employing a  test
specifically designed to accommodate non-detects,  such as the Tarone-Ware two-sample alternative to
the t-test (Section 16.3);  2) using a rank-based, non-parametric test such  as the Kruskal-Wallis
alternative to analysis of variance [ANOVA] (Section 17.1.2) when the non-detects and detects can be
jointly sorted and ordered  (except for tied values); 3)  estimating the  mean and standard deviation of
samples  containing non-detects  by means of a censored estimation technique; and  4)  imputing an
estimated value for each non-detect prior to further statistical manipulation.

     The first two strategies mentioned  above are discussed in Chapters 16 and 17 of the Unified
Guidance as alternative testing procedures for evaluating left-censored data when parametric distribution
assumptions cannot be made. Tests that can accommodate non-detects are typically non-parametric and
thus carry  both the advantages  and disadvantages of  non-parametric methods. The third and fourth
strategies — presented in  this chapter —  are often  employed as an intermediate step in parametric
analyses. Estimates of the background mean and standard deviation are needed to construct parametric
prediction and control chart limits, as well  as confidence intervals. Imputed values for individual non-
detects can be used as an alternate way to construct mean and standard deviation estimates, which are


                                              15^2                                   March 2009

-------
Chapter 15. Managing Non-Detect Data	Unified Guidance

needed to update the cumulative sum [CUSUM] portion of control charts or to compute the means of
orderp that get compared against prediction limits.

     The guidance generally favors the use of the more sophisticated Kaplan-Meier or Robust ROS
methods  which  can address the problem of multiple detection limits.  Two older techniques— Cohen's
method and parametric ROS-- are also included as somewhat easier methods which can work in some
circumstances.  Applying any of the four estimation techniques as well as simple substitution does rely
on a fundamental underlying assumption.  Both the detectable and non-detect portions of a data  set are
assumed  to arise from a single distribution, and in particular this underlying population is expected to be
stable or stationary during the period of the sampling record.

     However,  if an underlying distribution is  subject to  a trend over time,  applying any of these
techniques including simple substitution is more problematic. If data indicating a decreasing trend also
happen to contain multiple detection limit data (perhaps the result of improved analytical methods), it
may be very difficult to determine whether there is truly a trend or analytical problems are the apparent
cause of  the observed decreases.  None of the techniques provided in this chapter can directly address
this issue.  As discussed in Chapter 5, careful exploratory review of the historical data sets, particularly
those which might serve as background, need to consider which data including non-detects are most
representative of present or near-term future conditions.   In some cases, removal  of the older, less
reliable data may also resolve multiple detection limit problems.  If non-detect values higher than other
quantified data  at  reasonable detection limits  are  included in a data set  (especially if dictated by
reporting policy rather than analytical considerations), these will almost invariably need to be removed.
Even sophisticated multiple detection  limit  techniques cannot  realistically address these particular
information-limited data values.  But presuming valid and reliable data are selected, the four estimation
techniques are provided to address the management of non-detects.

     A data set may also not be defined by a single distribution.  If observed data are the result  of two
or more different generative processes and indicate one or more separate peaks, it is referred to as a
mixture distribution.  One example might be trace organics data in a release subject to changes in the
flow direction of the  aquifer, which can result in very high to absent values.  The subject is a complex
one  and  generally  beyond the scope of this guidance.  Aitchison's method can  be used in limited
situations where detectable data form one discrete distribution, and the remainder are non-detect.  The
following discussion  also addresses when Aitchison's method might be appropriate.   The non-detect
data are simply considered as some single value, another form of simple substitution.

15.2 IMPUTING NON-DETECT VALUES  BY  SIMPLE  SUBSTITUTION

     The simplest approach in managing non-detects is to substitute an imputed value for each prior to
subsequent statistical  analysis. The imputation is intended to be a 'reasonable estimate' of the true, but
unknown concentration, usually a fraction (e.g., 0, Va, 1) of the reporting  limit [RL]. If non-detects
represent an absence  of the contaminant being measured, replacing a reported 'less than' value by zero
may make sense. If the true concentration is completely unknown, but believed to be between zero and
the RL, half the RL, or RL/2, may  be a reasonable substitution, since this choice is the maximum
likelihood estimate [MLE] of the mean or median for  a population of measurement values uniformly
                                             15-3                                   March 2009

-------
Chapter 15.  Managing Non-Detect Data
Unified Guidance
distributed along the interval [0, RL].1 In other cases, a conservative choice might be made to maximize
the possible concentration levels present in non-detects by selecting the RL itself as the imputation.

     Any of these substitution choices is  imperfect since they ignore two realities about left-censored
measurements.  First,  non-detects  are  a  product  of  both the underlying distribution  of actual
concentrations and the measurement process used to estimate these concentrations. In particular, the
measurement technique may impart random or not so random bias to the  'true'  concentration levels,
causing the reported values to be 'shifted away from' the true values. As an example, simple substitution
of zero for each non-detect ignores the fact that only the measurements can be observed and analyzed,
not the actual concentration levels. Physical groundwater samples that are completely devoid of a given
chemical may not receive measurements of zero, even if the actual amount is zero. Simple substitution
by zero thereby ignores the measurement distribution in favor of an a priori assumption about what non-
detects might represent.

     A second reality  is that  non-detects  must  be  considered  with  respect to  other,  detected
measurements, as well as the physical process  that generated the  data. In many cases, the entire sample
is drawn from a single statistical distribution (representing a common physical process) but some portion
of the  lower tail has been censored during measurement, as illustrated in Figure 15-1.  In this situation,
the overall distribution (and especially the shape of the lower tail) dictates how likely  it is that a given
non-detect would have an uncensored measurement close to zero or close to the RL. Substitution by half
the RL or by the RL itself ignores the larger distributional pattern, especially since this distribution will
rarely be uniform in the interval [0, RL].

        Figure 15-1.  Single Distributional Model For Detects and Non-Detects
                                                            Detects
                 Non-Detects
 The uniform distribution places equal probability along every point of a finite concentration or measurement range. This
  model implies that a true value close to zero is just as likely as a true value close to RL or any other point along the
  interval.
                                              15-4
        March 2009

-------
Chapter 15.  Managing Non-Detect Data	Unified Guidance

     These realities can lead to severe biases in statistical parameter estimates made from censored data
when simple substitution methods are used (Helsel, 2005). Even if only 20% of the data are censored,
Gibbons (1994b) found that the false positive rate of a prediction limit test was far above the nominal
(i.e., expected or targeted) rate of a = .05 when a simple imputation strategy was employed. For that
reason,  the Unified   Guidance recommends  imputation by simple  substitution  only  in  select
circumstances  described below:

    »«»  When the sample size is too small to do anything else.
     With only a handful of measurements (e.g., 5 or less), it will be almost impossible to accurately
apply a censored estimation technique, such as those described in Sections 15-3 to 15-5. Instead, simple
substitution of half the RL is recommended, perhaps until enough data has been collected to allow a
more sophisticated analysis. Three situations where  simple  substitution might commonly be needed
include:

     1.  Plotting cumulative sums [CUSUM] on control  charts (Chapter 20). While there  should  be
        enough background data to allow for a more sophisticated estimate  of the control limit, the
        CUSUM must be updated with each single  new compliance observation (n = 1). If the new
        measurement is a non-detect, the value must be imputed for the CUSUM to be calculated.

     2.  Constructing future means for prediction limits (Chapter 19). Again, if censored  data exist in
        background, the prediction limit for a future mean can be computed with the help of a censored
        estimation technique.  But with only 2 or 3 new measurements per compliance well (p = 2, 3),
        the same strategy will not work for computing a mean of order/?.

     3.  Construction of confidence intervals in compliance monitoring or corrective action. Especially
        in the early months or years after the onset of compliance monitoring or a corrective action
        plan, there may be too few compliance point measurements to allow for a statistically refined
        treatment of non-detects. Until  more  data  has been collected that  is representative of the
        conditions under which these phases of monitoring have been triggered, simple substitution of
        non-detects will probably be needed. Furthermore, if groundwater conditions are in a state of
        flux,  it may be impossible — even with a larger sample size — to postulate a single, stationary
        distributional  model  (similar to Figure 15-1)  on which to  base  a censored estimation
        technique.

    »«»  When non-detects comprise no more than 10-15% of the total sample.
     If  the percentage of non-detects is small enough, results of parametric ^-tests and ANOVA are
usually not significantly affected if non-detects are first replaced by half their reporting limits [RLs]. A
similar statement can be made for parametric prediction limits, tolerance limits, control charts, and
confidence intervals. However, because ^-tests and ANOVA involve a comparison of means utilizing
multiple data points per mean  estimate,  while prediction limits for  individual observations, tolerance
limits, and control charts focus on single measurements, it is important that retesting be included in the
statistical procedure whenever simple substitution is utilized with these latter methods.
2 Parametric confidence intervals around the mean also involve an estimate of the population average using multiple data
  points.

                                              15^5                                    March 2009

-------
Chapter 15.  Managing Non-Detect Data
Unified Guidance
    »«»  When non-detects are generated by a different physical process than the detected values, and
       thus represent a distinct statistical distribution.
     One non-detect treatment recommended in past EPA guidance — Aitchison's method (1955), as
applied to groundwater3 —  assumed  that non-detects  were actually free of the contaminant being
measured, so that all non-detects could be regarded as zero concentrations. In some cases, if an analyte
has been detected infrequently or not  at all in background measurements, and/or all non-detects are
qualified as "U" (i.e., undetected) values, this assumption may be practical, even if it cannot be directly
verified. Another example might be seasonal changes in groundwater elevation at wells located on the
edges of a contaminant plume. Parameters detectable at certain times of the year may be non-detect
during other  seasons, even within the  same well. Such non-detects may result from a different data-
generating mechanism, due to seasonal changes  in groundwater chemistry,  and so may not follow the
same distribution as detects.
 Figure 15-2.  Modified Delta Model For Mixture Distribution of Detects/Non-Detects
                                                                  Detects
                     Non-Detects
     More generally,  Aitchison's  original  model  posited  a 'spike'  of zero-valued measurements,
combined with a lognormal distribution governing the detected values. A modification to Aitchison's
model  known as the modified delta method4 (USEPA, 1993) has been found to be more practical and
realistic in many circumstances (Figure 15-2). Instead of assuming that all non-detects represent zero
  Aitchison's model was not originally applied to concentration data. More typical applications were in the fields of
  economics and demographics.
  The original Aitchison model was termed the delta-lognormal, so called because it presumed that the data consisted of a
  mixture of two distinct populations: 1) a lognormal distribution representing the observed continuous measurements, and
  2) a 'spike' of values, known as a delta function, located at zero.
                                              15-6
        March 2009

-------
Chapter 15.  Managing Non-Detect Data	Unified Guidance

concentrations, the  modified delta method  assumes that non-detects constitute a separate, discrete
distribution. When combined with the detected values, a mixture distribution is formed consisting of a
continuous detected portion (usually  the normal or  lognormal  distribution) and  a discrete non-detect
portion. Rather than assuming that all non-detects are zeros, the modified delta model assigns all non-
detects  at half the  reporting limit [RL]. (Note: this  might be a method detection  limit  [MDL], a
quantitation limit [QL], or  a contract RL).   This method can accommodate multiple reporting limits
since each non-detect is assigned half of its possibly sample-specific RL. It can also accommodate low-
valued detects  intermingled with  the non-detects, since the non-detects and detects  are modeled  by
distinct distributions.
15.3 ESTIMATION  BY KAPLAN-MEIER

       BACKGROUND AND PURPOSE

     When  a sample contains both detects  and non-detects generated by  a common process  and
governed by a single underlying distribution (Figure 15-1), a more reliable strategy is to attempt to fit
the sample to a known distribution (e.g., normal, lognormal) and then to estimate the mean and standard
deviation of this distribution via a censored estimation technique. These adjusted estimates can be input
into standard equations for parametric prediction, tolerance, and control chart limits, as well parametric
confidence intervals around the mean.

     Two censored estimation methods which can  address the multiple detection limit problem are
discussed in the Unified Guidance: the Kaplan-Meier estimator and robust regression on order statistics
[ROS] (Section 15.4). Both involve initially fitting a left-censored sample to a known distribution. After
that, the procedures differ. The Kaplan-Meier creates an estimate of the population mean and standard
deviation adjusted for data censoring, based on the fitted distributional model, whereas the Robust ROS
uses the fitted model to construct a model-based imputation for each non-detect. Once the imputations
are made, the adjusted mean  and standard deviation are estimated using  standard equations  for the
sample mean (x ) and standard deviation (s).

     The  key to either method is finding a single distributional model that adequately  fits the joint
sample of detects and non-detects. While each procedure does the fitting in  a slightly  different fashion,
both utilize the notion of partial ranking. As discussed in Section 16.2 on "Handling Non-Detects," the
presence  of  left-censored measurements, particularly  when  there  are  multiple  RLs and/or  an
intermingling of detects and  non-detects,  prevents a full and complete ranking of the sample. Both
Kaplan-Meier and ROS construct a partial ranking of the data, accounting for the non-detects  and
assigning explicit ranks to each of the detected values. These  detected values can then be graphed on a
censored probability plot and fitted against  a known distribution.

     The  Kaplan-Meier technique estimates the approximate proportion of concentrations below each
observed level by sorting and ordering the  distinct sample values, although  the exact concentrations of
non-detects are unknown.  In particular, the probability of observing a concentration no greater than a
given level (x[) depends on the relative proportion of the sample greater than X[. Any detects larger than
X[ obviously fall into this latter proportion, while non-detects with RLs of at most X[  do not.  On balance,
the proportion of the sample  greater than x; cannot be precisely calculated for every x;, but it can be
estimated.

                                             liT?                                   March 2009

-------
Chapter 15.  Managing Non-Detect Data	Unified Guidance

     The Kaplan-Meier estimator for left-censored data thus depends  on a  series  of conditional
probabilities, where the frequency of lower concentrations depends on how many larger concentrations
have already been observed. The final result is an estimate of the cumulative distribution function [CDF]
for each distinct concentration level in the sample.

     In mathematical notation, suppose there are m distinct values in the  sample  (out of a total of n
measurements), including distinct reporting limits. Order these values from least to greatest and denote
them as *(i), x^), ..., X(m). Let n\ for / = 1 to m denote the 'risk set' associated with value XQ. The risk set
represents the total  number of measurements — both detects and non-detects — no greater than XQ.
Since a non-detect with a RL larger than XQ is potentially (but not necessarily) larger than XQ, non-
detects with RL >  x(i)  are not included  in n\.  A further term d\ identifies the number of detected
measurements exactly equal to XQ.

     With these definitions in place and letting X denote a random variable concentration from the true
underlying distribution, the Kaplan-Meier estimator is constructed from the pair of probabilities:
                                        PrU
-------
Chapter 15.  Managing Non-Detect Data _ Unified Guidance

concern is  actually present in non-detect samples, but that the analytical  method cannot accurately
measure, or is not sufficiently sensitive to, concentrations lower than the RL.

     To construct a censored probability plot, a normal quantile or z-score needs to be computed for
each value of the Kaplan-Meier CDF (FKM)- Doing so is straightforward except for the CDF value of the
sample  maximum,  which  is assigned a  value  of one. The z-score  associated with a  cumulative
probability of  one is infinite.   To  surmount  this difficulty,  the Unified Guidance  recommends
temporarily setting the CDF value for the sample maximum equal to (n - .375)/(w + .25). This value is
the Blom plotting position often utilized in standard probability plots (Helsel, 2005).  It is close to one
for large n, but  allows for a finite z-score.

     Estimation of the Kaplan-Meier  mean and standard deviation using equations  [15.4] and [15.5]
below will tend to be slightly biased, typically with the mean on the high side and the standard deviation
on the low side. This occurs because the Kaplan-Meier CDF levels corresponding to distinct RLs are
treated as if they were known measurements rather than the upper bounds on possible values. As long as
the total proportion of censored measurements is not too high, the degree of bias will tend to be small.
Larger biases are more likely whenever the detection rate is less than 50%.

       PROCEDURE

Step 1.   Given a sample of size n containing left-censored measurements, identify and sort the m < n
         distinct values,  including distinct RLs. Label these as
Step 2.   For each /' = 1  to  m, calculate the risk set («;) as the total number of detects and non-detects
         no greater than XQ. Also compute d\ as the number of detected values exactly equal to XQ.

Step 3.   Using equation [15.3], compute the Kaplan-Meier CDF estimate Fm IJKV.N Jfor /' = 1, ..., m-l.
         Also let F „
                   KM

Step 4.   Construct  censored  probability  plots  using  the  estimated  CDF.  First  temporarily  set

         -^KM (x(Vi i  (w~-375y67 + .25j so that  a finite normal quantile (or z-score; see Chapter 9)
              XV ) S
         can be associated with X(m). Then compute normal quantiles (i.e.., z-scores) for each value of

             from Step 3 as  z^ = <&~l  FKM uc,-. \ , where O-1[-] is the inverse of the standard normal

         distribution function as discussed in the construction of probability plots in Chapter  9. Plot
         the values ZQ  against the  unique detected  concentrations XQ to form a normal censored
         probability plot. Plot the ZQ'S against a transformation of the XQ'S  (e.g., log,  square root,
         inverse, etc} to form a normalized censored probability plot.

Step 5.   For each attempted transformation XO including the unchanged observations as one option,
         compute  the  correlation  coefficient between the  pairs  [/(x(i)X  z(i)]  (Chapter 3).  The
         transformation  with  the highest correlation coefficient and also a  linear  appearance  on the
         censored probability  plot, is one that optimally normalizes the left-censored sample. Estimate
         the mean and standard deviation in Step  6 on the transformed scale and use these estimates in
         subsequent statistical analysis.


                                             15^9                                    March 2009

-------
Chapter 15.  Managing Non-Detect Data
                                                                             Unified Guidance
Step 6.
         If no transformation results in an adequately linear censored probability plot, conclude that the
         sample  cannot be  normalized. Mean  and  standard  deviation estimates of the  original
         concentrations can still be computed, but they will not correspond to a known probability
         distribution.

         If the raw concentration data are approximately normal, compute mean and standard deviation
         estimates adjusted for censoring using the equations:
                              1^ KM ~ / j
                                                \X(i) }   ^KM V^z-l) /J
[15.4]
                                    *(0 ~ ft™ I' ' \FKM (*(,-) ) - FKM (x(r-i) )]
                                                                                         [15.5]
         where x/0x = 0  and F^ Ix/x j= -^KM(o)=0  by definition.  Otherwise, compute the adjusted

         mean and  standard  deviation after applying the normalizing transformation _/(') with the
         equations:
                                                      - FKM
                                                                                         [15.6]
                                                      /   \       /    \n
                                                    M t(0 ) ~ FKM (^,--i) )J
                                                                                         [15.7]
         Estimates from equations [15.4] and [15.5] can then be used in place of the sample mean (x )
         and standard deviation (s) in parametric equations for prediction and control limits, and for
         confidence intervals. If a normalizing transformation is required, equations [15.6]  and [15.7]
         can be used to construct similar statistical limits and intervals on the transformed scale.

              ^EXAMPLE 15-1

     Use  the Kaplan-Meier technique on the  following manganese concentration data to construct
estimates of the population mean and standard deviation that are adjusted for censoring.
Sample
1
2
3
4
5
Manganese Concentrations (ppb)
Well 1 Well 2 Well 3
<5.0
12.1
16.9
21.6
<2.0
<5.0
7.7
53.6
9.5
45.9
<5.0
5.3
12.6
106.3
34.5
in Background
Well 4 Well 5
6.3
11.9
10.0
<2.0
77.2
17.9
22.7
3.3
8.4
<2.0
                                             15-10
                                                                                     March 2009

-------
Chapter 15. Managing Non-Detect Data                                  Unified Guidance
       SOLUTION
Step 1.   From the combined sample of n = 25 measurements, identify and sort the 21 distinct values
         including distinct RLs as in the table below. Compute the risk set («;) for each distinct level
         (XQ) as the total number of detects and non-detects no greater than XQ. Also calculate the exact
         number of detects (d[) equal to each level.

Step 2.   Compute the Kaplan-Meier estimate of the CDF using equations [15.1] and [15.3], shown in
         column 5 of the table below. Two example calculations are given by:
                                  2l      22      23      24      25
                     (3.3)=  ,-2  .  ,-i
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
X(i)
<2.0
3.3
<5.0
5.3
6.3
7.7
8.4
9.5
10.0
11.9
12.1
12.6
16.9
17.9
21.6
22.7
34.5
45.9
53.6
77.2
106.3
At Risk (rij)
3
4
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
d,
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
CDF
0.21
0.28
0.28
0.32
0.36
0.40
0.44
0.48
0.52
0.56
0.60
0.64
0.68
0.72
0.76
0.80
0.84
0.88
0.92
0.96
1.00
Step 3.   Compute normal quantiles or z-scores for each value of FKM in the above table. First re-set the
         last entry to (n - .375)/(w + .25) = 0.9752 so that a finite quantile can be associated with the
         sample maximum.

Step 4.   Plot the z-scores against the distinct manganese levels to form a normal censored probability
         plot (Figure 15-3). The probability plot correlation coefficient is r =  0.902. The plot itself
         shows substantial curvature, suggesting that the sample is non-normal.
                                            15-11                                  March 2009

-------
Chapter 15. Managing Non-Detect Data
Unified Guidance
Step 5.   Plot the z-scores against one or more transformations of the manganese levels. First attempt a
         log transformation, as shown in Figure 15-4. In this case, the correlation coefficient improves
         to r = 0.989 and the normalized censored probability plot looks fairly linear. Conclude that the
         sample is approximately normal  on the log-scale,  that is, the manganese concentrations are
         lognormal in distribution.

Step 6.   Compute Kaplan-Meier log-mean (jHyKM) and log-standard deviation (&yKM) estimates for
         the manganese data using equations [15.6] and [15.7], taking^') as the natural logarithm. This
         gives for the log-mean:

      fty-KM  = log(2).[.21-0] + log(3.3)-[.28-.2l] +  ... +  log(l06.3)-[l-.96] = 2.311og(flpi)

         and for the log-standard deviation:
        &y-KM =V(log(2)-2.3l)2- [.21-0] + ... + (log(106.3)-2.3l)2-[l-.96] = 1.18 log(ppb)

         These adjusted mean and standard deviation estimates can then be used in place of the sample
         log-mean and  log-standard deviation  in  parametric  prediction and control  limits,  or in
         parametric confidence intervals. -^
         Figure 15-3. Censored Probability Plot of Manganese Concentrations
                                         40       60
                                                        so
                                                                100
                                           15-12
        March 2009

-------
Chapter 15. Managing Non-Detect Data
Unified Guidance
        Figure 15-4. Censored Probability Plot of Logged Manganese Sample
                                       2         3

                                        log(mangantte} logfppbj
15.4 ROBUST REGRESSION  ON ORDER STATISTICS

       BACKGROUND AND PURPOSE

     Robust regression on order statistics [ROS] differs from Kaplan-Meier in that it uses the fitted
model to construct a model-based imputation for each non-detect. Once the imputations are made, the
adjusted mean and standard deviation are estimated using standard equations for the sample mean (x )
and standard deviation (s).

     The first step in using Robust ROS is to find a single distributional model that adequately fits the
joint sample of detects and  non-detects. Standard  probability plots (Chapter 9)  and normality tests
(Chapter 10) rely on a full ranking or ordering of the sample in order to fit candidate distributions. With
left-censored data, the true concentrations of non-detects  are unknown, so  only a partial ranking is
possible.  Like  Kaplan-Meier, the Robust ROS  technique constructs  a partial ranking of the data,
accounting  for the  non-detects  and assigning explicit ranks to each  of  the detected values.  These
detected values can be  graphed on a censored probability plot io check  the fit of possible distributional
models.

     Once  an  adequate  distribution is  found, Robust ROS  determines the approximate cumulative
probability associated with each distinct RL.  The method then arbitrarily distributes non-detects with a
common  RL so that each one accounts for an equal  share  of the estimated cumulative probability
                                           15-13
       March 2009

-------
Chapter 15.  Managing Non-Detect Data _ Unified Guidance

assigned to that RL. Once non-detects are ranked in this manner, the fitted distributional model is used
to impute a value for each non-detect. This last task is accomplished by conducting a linear regression
(Chapter 17) between the detected values  and the z-scores from the censored probability plot. The
parameters of the regression line (i.e., intercept and slope) can be used to estimate the mean and standard
deviation of the distributional model, which in turn will generate imputed values for the non-detects.

     The mathematics behind Robust ROS can be  expressed as follows.  First suppose there are  k
distinct RLs in the sample. Order these from least to greatest. Define A\ as the number of detected values
between the rth and (/+l)th RLs for / = 1 to k-\ . Let A^ = number of detects above the highest RL, and
take AQ = number of detects below the lowest RL.  Also define Bi as the total number of observations,
both detects and non-detects, with values below the rth RL.  Define ,80 = 0. Then the number of non-
detects below the rth RL can be written as:

                              Ct = Bt - Bt_, - 4_,     for i = \tok                         [15.8]

     With these definitions  in place, exceedance probabilities can be assigned to each of the k RLs,
representing the proportion of the sample greater than or equal to each distinct RL. These probabilities
can be written as:
where pej denotes the proportion of the sample exceeding the rth RL. Equation [15.9] can be interpreted
in the  following  manner.  The exceedance probability associated with a given RL is equal to the
exceedance probability assigned to the next highest RL combined with a fraction of the remaining, non-
exceedance probability (i.e., 1  - pe\+\ ).   The specific fraction depends on the relative occurrence of
detects between the rth and (/+l)th RLs. When /' = k, define pe\+\ = 0; when / = 0, define peo = 1.

     Once the exceedance probabilities  are  computed, plotting  positions  for the detects  — i.e.,
cumulative probabilities on a probability plot — can be calculated with the equation
              pdv=\-pei + -       pet-pe^      forj = ltoAi-   andi = 0tok       [15.10]


for each set of detected values falling between the rth and (/+l)th RLs.  Note that this equation also
applies to any detects below the lowest RL [/' = 0] or above the highest RL [/' = &]). Similarly, plotting
positions for each group of non-detects can be written as:


                                                                   = ltok               [15.11]
     With plotting positions for the detects,  a normal quantile or z-score can be computed for each
value ofpdij. Then censored probability plots can be constructed using either the detected concentrations
(xij) or some normalizing transformation of the detected values, say f(xy). If a linear probability plot can
be identified, a linear regression (Chapter 17)  can be calculated for the pairs (zy-, Xxy)) and used to
impute values for the non-detects in the sample.
                                             15-14                                   March 2009

-------
Chapter 15.  Managing Non-Detect Data	Unified Guidance

       REQUIREMENTS  AND ASSUMPTIONS

     Robust ROS was originally devised to account for non-detects in water quality data (Helsel, 2005).
Robust ROS is an extension of a technique termed regression on order statistics [ROS] (Gilliom and
Helsel, 1986),  described in  Section 15.5. That procedure assumes the joint sample of detects and non-
detects follows an underlying lognormal distribution. The  fitted lognormal  is used  to estimate  the
population mean and standard deviation as a parametric technique. Robust ROS by contrast only relies
on a parametric model to  impute values for  the non-detects. It can be applied to any normal  or
normalized distribution, rather than just the lognormal distribution.  It may also be regarded as quasi-
non-parametric since estimates for the sample are computed  from the combined group of observed
detects and imputed non-detects, rather than from the mean and standard deviation of the underlying
distributional model, as in the original formulation.

     In practice, because Robust ROS is not fully non-parametric, a known distribution must be fitted to
the entire sample in order to construct imputed values for the non-detects. Closely related to this, Robust
ROS assumes  that both detected and non-detect data arise from the same population,  with non-detect
values censored at their respective RLs. Like Kaplan-Meier, this implies that the contaminant of concern
is  present in  non-detect  samples,  but  that  the  analytical method  cannot accurately  measure
concentrations lower than the RL.

       PROCEDURE

Step 1.   Given a left-censored sample with a total of n measurements, identify and sort the k distinct
         RLs. Following the discussion above, count the number of detected  values below the lowest
         RL (Ao), the number of detected values at least as great as the highest RL (/4k), and the number
         of detects between the rth and (/'+l)th RLs (A\ for /  =  1 to k-\).  Also let BQ = 0 and count the
         total number of detects and non-detects below the rth  RL (B[ for /' = 1 to K). Then use equation
         [15.8] to calculate the number of non-detects (C; for /  = 1 to K) below the rth RL.

Step 2.   Letpeo = 1 and/?ek+1 = 0. For /' = 1 to k, compute the probability of exceeding the rth distinct
         RL (pc[) using equation [15.9].

Step 3.   With the exceedance probabilities from Step 2, sort each group of detects associated with A\
         and then compute plotting positions (i.e., cumulative probabilities) for these detects —pd^ —
         using equation [15.10].

Step 4.   Form normal  quantiles (i.e., z-scores) associated with the detected measurements and plotting
         positions pd^  by  computing 2* = 3>~l (pd.\ where  <&~l Q is the inverse standard normal
         CDF.

Step 5.   Construct censored probability plots using the z-scores from Step 4. Plot the values zd.. against

         the detected concentrations  xd..  to form a normal censored probability plot. Plot the z^'s

         against a transformation of the xd.. 's (e.g., log, square root, inverse, etc.) to form a normalized
         censored probability plot.
                                            15-15                                   March 2009

-------
Chapter 15. Managing Non-Detect Data	Unified Guidance

Step 6.   For each attempted transformation y(0 including the unchanged observations as one option,
         compute the correlation coefficient between  the  pairs  If\xf\zf.   (Chapter  3).   The

         transformation with the highest correlation coefficient and also  a linear appearance  on the
         censored probability plot, is the one that optimally normalizes the left-censored sample. If no
         transformation results in an adequately  linear  censored  probability plot, conclude that the
         sample  cannot be normalized  and  that  the  Robust ROS may not  provide  reasonable
         imputations for the non-detects.

Step 7.   If a normalizing transformation can be identified, compute a linear regression (Chapter 17) of
         the values f(xf J on the z-scores, zd.., to form the regression equation f(x) = a + b • Z . The
         slope and intercept can be estimated using the equations
                                        a = xd-b-zd                                  [15.13]

         where  ~zd is the mean of the z-scores  associated with the detected values, n& = number of
         detects, s2z  is the sample variance of the detected z-scores, and xd is the mean of the detected
         measurements. The regression intercept (a) is an estimate of the population mean of the
         normalized distribution, while  the  slope  (b)  is an  estimate of the population standard
         deviation.

Step 8.   Compute plotting positions (pc-^) for the non-detects (i.e., censored observations)  associated
         with each  distinct RL using equation [15.11]. Then form a second set of z-scores, this time
         associated with the non-detects, by computing zc.. = ~1 \pc. \ fory = 1 to Q; and /' = 1 to k.

Step 9.   Form imputed values f(x°.) = a + b-z°. using the slope  and intercept from Step  7  and the
         censored z-scores from Step  8.  Combine  these (transformed) imputed values for the non-
         detects with the (transformed) detected measurements  / (xf. J to get censored estimates of the
         population mean and standard deviation by computing the overall sample mean (ju = x) and
         sample standard deviation (a = s).

         These  censored estimates can be used in place of the  unadjusted  sample mean (x) and
         standard deviation (s) in  parametric equations for prediction and  control limits,  and for
         confidence intervals. If a normalizing transformation _/(') is needed, the censored estimates
         should be used to construct statistical limits and intervals on the transformed scale.

       ^EXAMPLE 15-2

     In Example 15-1, the Kaplan-Meier technique was used on a sample of background manganese
concentrations to  compute the log-mean and log-standard deviation, adjusted for the presence of non-
detects. Apply Robust ROS to these same  data to compare the estimates.
                                            15-16                                  March 2009

-------
Chapter 15. Managing Non-Detect Data	Unified Guidance

       SOLUTION
Step 1.   The n = 25 manganese observations include 2 distinct RLs (<2 and <5). Count the number of
         detected measurements below the lowest RL, above the highest RL, and between the two RLs,
         denoted by A\ in the table below. Also count the total number of measurements — both
         detected and non-detect — below each RL, denoted below by B\. Use equation [15.8] to count
         the number of non-detects associated with each RL, denoted below by C\.
                  j	RL	Aj	Bj	CL
                   0                         000
                   1            <2           1            3            3
                   2            <5           18           7            3
Step 2.   Compute the probability of exceeding each RL using equation [15.9] and noting thatpe^ = 0:

                                       A.   f      x    18
                           pe, = pe^ +	— (1 - pe. )=	= 0.72
                           ^2   ^3  4+52V     3'  18 + 7

                                 A                     l
                    pe. = pe,+   ^  (l-pe^)=0.72 +	(l-0.72)= 0.79
                    Pl     2   4+5, V   P 2j        1 + 3V       ;

Step 3.   Sort the detects associated with each A\ and compute plotting positions for these detects using
         equation [15.10], as listed in  the table below.  For instance, A\ =  1, corresponding to the
         detected value 3.3. The plotting position for this observation equals


               pdu = (l-pel)+1 —!— I (pel -pe2)= 0.21 + 0.5(o.79 - 0.72)= 0.245


         Also form the  normal quantiles (i.e., z-scores) associated  with the detected observations, as
         listed below:
                                            15-17                                  March 2009

-------
Chapter 15. Managing Non-Detect Data                                   Unified Guidance
Detected
Value (ppb)
3.3
5.3
6.3
7.7
8.4
9.5
10.0
11.9
12.1
12.6
16.9
17.9
21.6
22.7
34.5
45.9
53.6
77.2
106.3
Plotting
Position
0.245
0.318
0.356
0.394
0.432
0.469
0.507
0.545
0.583
0.621
0.659
0.697
0.735
0.773
0.811
0.848
0.886
0.924
0.962
z-score
-0.690
-0.474
-0.370
-0.270
-0.172
-0.077
0.018
0.114
0.210
0.308
0.410
0.515
0.627
0.748
0.880
1.030
1.207
1.434
1.776
Step 4.   Plot the z-scores against the detected manganese levels to form a normal censored probability
         plot (Figure 15-5). The probability plot correlation coefficient is r = 0.901, almost identical to
         the Kaplan-Meier censored probability plot constructed in Example 15-1. The plot also shows
         substantial curvature, suggesting that the sample is non-normal. Also plot the z-scores against
         a log transformation of the detected manganese values (Figure 15-6). Not only  does  the
         normalized probability plot appear linear, but the correlation coefficient increases to r = 0.994.
         Conclude as in Example 15-1 that the sample  is approximately normal  on the log-scale, so
         that the manganese concentrations are lognormal in distribution.

Step 5.   Compute a linear regression of the «d = 19 logged manganese detects against their
         corresponding z-scores using equations [15.12] and [15.13]. The sample mean and variance of
         the detected z-scores are zd = 0.3802 and s2z  = 0.4577 . Also, the log-mean of the detected
         observations equals \og(xd)= 2.80 . The slope and intercept of the resulting line are:


              b=	[l. 194- (-.690-.3802) + ... + 4.666(l.776-.3802)] = 1.372
                  18X.4577
                            = xd-b-zd = 2.80 -l.372x.3802 = 2.278
                                            15-18                                   March 2009

-------
Chapter 15. Managing Non-Detect Data
                                    Unified Guidance
 Figure 15-5. Robust ROS Censored Probability Plot of Manganese Concentrations
                 to
                 o
                 q
                 d
                 to
                 o -
                 I
                            I
                            20
      I
     40
        I
       60
                                                     80
                                                             100
                                  Manganese Concentration (ppb)
     Figure 15-6. Robust ROS Censored Probability Plot of Logged Manganese
                  a
                  o
                  iq
                  o
                   I
                          1
                         1,8
 I
2,0
 i
2,5
 I
3,0
 I
3.5
 I
4,0
 I
4,8
                                    log{Manganese) log{ppb)
                                       15-19
                                           March 2009

-------
Chapter 15. Managing Non-Detect Data	Unified Guidance

Step 6.   Compute plotting positions for the non-detects (i.e., censored observations) associated with
         each distinct RL using equation  [15.11], listed in the table below. Form a second set of z-
         scores, this time associated with the non-detects, also listed below. Note that each non-detect
         is given a distinct plotting position, even though they cannot be ordered. This is done to 'fill
         in' the unknown portion of the  underlying distribution, but should  not be interpreted as a
         legitimate 'estimate' for any particular non-detect observation. The positions for the first pair
         of the 3 non-detects with RLs of 2 (i.e., <2) are


                       ^cn= I  77^1(1-^)= ^(l-0.79)= 0.0525
RL
<2
<2
<2
<5
<5
<5
Plotting
Position
0.0525
0.1050
0.1575
0.0700
0.1400
0.2100
z-score
-1.621
-1.254
-1.005
-1.476
-1.080
-0.806
Imputed
Value
0.054
0.558
0.899
0.253
0.796
1.172
Step 7.   Form a second set of z-scores associated with the censored plotting positions from Step 6.
         These are listed in the table above. Then, using the regression parameters from Step 5, form a
         prediction for each non-detect using the equation  log(x^) = a + ft • z°..  Take these predictions
         as the imputed values for the set of non-detects, as listed above. The first two imputed values
         are computed as:

                            \og(xcu) = 2.278 + 1.372 • (-1.621) = 0.054


                            log(<2) = 2.278 + 1.372 • (-1.254) = 0.558

Step 8.   Combine the logged detected manganese values with the imputed values from Step 7. Then
         compute the  sample mean  and  standard  deviation using  the  adjusted sample. These
         calculations give jU = 2.28\og(ppb)  and  a = 1.26 \og(ppb).  By comparison,  the Kaplan-
         Meier method in Example 15-1 gives very similar corresponding estimates of 2.31 log(ppb)
         and l.lSlog(ppb). ^
                                             15-20                                   March 2009

-------
Chapter 15. Managing Non-Detect Data	Unified Guidance

SECTION 15.5  OTHER METHODS FORA  SINGLE CENSORING LIMIT

     The two preferred methods using Kaplan-Meier or Robust ROS  provided above for multiple
detection limits are computationally intensive.  Helsel (2005) indicates that public software is available
for the Robust ROS method.  Although the more common  situation encountered in evaluating data sets
is the presence of multiple detection limits (hence the UG  recommendations), two older techniques are
still applicable in some situations.  The Cohen method and the parametric ROS techniques are both
simpler to apply,  but depend on the use of a single censoring limit. One needs to evaluate the prospects
before applying them.  If detectable data sets are large enough (e.g., n > 50) and detection percentages
near or greater than 50%, most of these methods will work comparably.

15.5.1 COHEN'S ADJUSTMENT

     Cohen's adjustment (Cohen, 1959) can be useful when  a significant fraction (up to 50%) of the
observed measurements in a data set are reported as non-detects. The technique assumes that all the
measurements, detects and non-detects alike, arise  from  a common population, but that the lowest
valued observations have been censored at the QL. Using the  censoring point (i.e.., QL) and the pattern
in  the detected  values,  Cohen's method attempts  to reconstruct the key features of the  original
population,  providing explicit estimates of the population  mean and standard deviation.  These in turn
can be used in certain statistical interval estimates, where  Cohen's adjusted estimates are  used  as
replacements for the sample mean and sample standard deviation.

       REQUIREMENTS AND ASSUMPTIONS

     Cohen's adjustment assumes that the common underlying population  has a normal distribution.
The technique should only be used when the observed sample data approximately fit a normal model
including transformations to  normality. Because the presence of a large fraction of non-detects will
make explicit normality testing difficult, if not impossible, the most  helpful diagnostic aid  may be to
construct a  censored probability plot on the detected measurements. If the censored  probability plot is
clearly linear on the original measurement scale but not on  the log-scale, assume normality for purposes
of computing Cohen's adjustment. If, however, the censored probability plot is clearly linear on the log-
scale, but not on the original scale, assume instead that the common underlying population is lognormal.
Then  compute Cohen's  adjustment to the estimated mean  and standard  deviation on the log-scale
measurements  and construct  the desired statistical  interval using  the  algorithm   for lognormally-
distributed observations.

     When  the detection rate is less  than 50%,  the accuracy of Cohen's method worsens as the
percentage of non-detects increases. The guidance does not generally recommend the use of Cohen's
adjustment when more than half the data are non-detect. In such circumstances, one should consider an
alternate statistical  method, for instance a non-parametric  interval or perhaps the Wilcoxon rank-sum
test for small samples.

     One other requirement of Cohen's original method is that there  should be just  a single censoring
point. Data  sets with multiple RLs will usually require a more sophisticated treatment such as Kaplan-
Meier  or Robust ROS methods  or via maximum  likelihood  techniques (Cohen, 1963) or  perhaps  a
multiply-censored probability plot technique  (Helsel and Cohn,  1988). If only 2  or  3 RLs do not
substantially differ and few detected intermingled data are lost, the censoring point (QL) can be set to
                                            15-21                                  March 2009

-------
Chapter 15.  Managing Non-Detect Data
                                                                Unified Guidance
the highest RL.  Cohen's method requires explicit definition of the censoring limit, and is somewhat
sensitive to variation in this parameter.

       PROCEDURE

Step  1.   Divide the data set into two groups, detects and non-detects. If the total sample size equals «,
         let m represent the number of detects and (n-m) represent the number of non-detects. Denote
         the /'th detected measurement by x\. Then compute the mean and sample variance of the set of
         detects using the equations:
Step 2.
                          —
                          x, = —
                              and
                                                  1
                                                 m-\
      —
x - mx.
Denote the single censoring point by QL. Then compute the two intermediate quantities, h and
y, necessary to derive Cohen's adjustment via the following equations:
               = WO-(n-m)/n =
                                               and  y =
Step 3.   Use the intermediate quantities h and y to determine Cohen's adjustment parameter X from the
         table below.

                     Values of Lamba (X) for Cohen's Adjustment
y\ND%
.01
.05
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.50
4.00
4.50
5.00
5.50
6.00
1
.0102
.0105
.0110
.0116
.0122
.0128
.0133
.0137
.0142
.0146
.0150
.0153
.0162
.0170
.0177
.0184
.0191
.0197
.0203
.0209
.0219
.0229
.0239
.0248
.0256
.0264
5
.0530
.0547
.0566
.0600
.0630
.0657
.0681
.0704
.0726
.0747
.0766
.0785
.0828
.0868
.0905
.0940
.0973
.1005
.1035
.1063
.1118
.1168
.1216
.1262
.1305
.1346
10
.1111
.1143
.1180
.1247
.1306
.1360
.1409
.1455
.1499
.1540
.1579
.1617
.1705
.1786
.1861
.1932
.1999
.2062
.2123
.2182
.2292
.2395
.2492
.2585
.2673
.2757
15
.1747
.1793
.1848
.1946
.2034
.2114
.2188
.2258
.2323
.2386
.2445
.2502
.2636
.2758
.2873
.2981
.3082
.3179
.3272
.3361
.3529
.3687
.3836
.3977
.4111
.4240
20
.2443
.2503
.2574
.2703
.2819
.2926
.3025
.3118
.3206
.3290
.3370
.3447
.3627
.3793
.3948
.4093
.4231
.4363
.4489
.4609
.4838
.5052
.5253
.5445
.5628
.5803
25
.3205
.3279
.3366
.3525
.3670
.3803
.3928
.4045
.4156
.4261
.4362
.4459
.4687
.4897
.5094
.5279
.5454
.5621
.5781
.5935
.6226
.6498
.6755
.7000
.7233
.7456
30
.4043
.4130
.4233
.4422
.4595
.4755
.4904
.5046
.5180
.5308
.5430
.5548
.5825
.6081
.6321
.6547
.6761
.6965
.7161
.7348
.7704
.8038
.8353
.8653
.8938
.9212
35
.4967
.5066
.5184
.5403
.5604
.5791
.5967
.6133
.6291
.6441
.6586
.6725
.7053
.7357
.7641
.7909
.8164
.8407
.8639
.8863
.9287
.9685
1.0060
1.0418
1.0758
1.1085
40
.5989
.6101
.6234
.6483
.6713
.6927
.7129
.7320
.7502
.7676
.7844
.8005
.8385
.8738
.9069
.9382
.9679
.9962
1.0234
1.0495
1.0990
1.1455
1.1895
1.2312
1.2711
1.3094
45
.7128
.7252
.7400
.7678
.7937
.8179
.8408
.8625
.8832
.9031
.9222
.9406
.9841
1.0245
1.0625
1.0984
1.1325
1.1651
1.1963
1.2264
1.2835
1.3371
1.3878
1.4359
1.4820
1.5262
50
.8403
.8540
.8703
.9012
.9300
.9570
.9826
1.0070
1.0303
1.0527
1.0743
1.0951
1.1443
1.1901
1.2332
1.2739
1.3127
1.3498
1.3854
1.4197
1.4847
1.5458
1.6037
1.6587
1.7113
1.7617
                                           15-22
                                                                        March 2009

-------
Chapter 15.  Managing Non-Detect Data	Unified Guidance

Step 4.   Using the  adjustment parameter X  found in Step 3, compute adjusted estimates  of the
         population mean and standard deviation with the equations:
                     ju  =xd - A.(xd - QL)  and  d  = ^sd + A. • (xd -QL)2

Step 5.   Once the adjusted estimates for the population mean and standard deviation are derived, these
         values can be substituted for the sample mean and standard deviation in equations for the
         statistical intervals.
15.5.2  PARAMETRIC REGRESSION  ON ORDER STATISTICS (ROS)

     A second useful method (EPA, 2004) for estimating mean and standard deviation parameters for
data sets with non-detect values censored at a single limit is a parametric Regression on Order Statistics
(ROS).  The same assumptions apply as with Cohen's method.   Both the detected and non-detect
portions of the  data are presumed to arise from a single population.   That population should either be
normal or transformable to a normal distribution.  The parametric ROS method performs similarly to
Cohen's method, and offers two principal advantages.  The procedure can  easily be implemented on
almost any statistical software, and the method is not sensitive to the exact censoring limit.

     If variable X originates from a normal  distribution  with  mean ju  and  standard  deviation a
 [X >- jV(//,cr)]  and Z is the standard normal distribution  [Z >- 7V(0, l)], statistical theory indicates that
X = ju + a • Z when X and Z are at  the same percentiles in their respective  distributions.  For a given
observation or sample x above a detection limit, the order statistic (i.e., the  proportion of observations
less than x) can be estimated.  This order statistic is an estimate of the percentile.  The corresponding Z-
value can be obtained from reference tables or a computer algorithm.   For a list of ordered observations
above the  detection limit (xj, X2,	to x^ of m detectable samples out of a total n and a corresponding
set  of Z-values (Zi,   Z2,  	 to Zm) at the same percentiles, regression analysis of X against Z will
provide estimates of the  mean  and  standard deviation of distribution X.    The  intercept  is the mean
estimate and the slope of the regression is the standard deviation estimate.

     When sample data better fit a lognormal or other normal transformable  distribution, the regression
is performed on the transformed  data.   The mean  and standard deviation  estimates are  also for the
transformed data  (e.g., logarithmic  mean and standard deviation).  One may also use the regression
results to "fill in" or quantify the values below the detection limit.  When the  Z-distribution is developed
for  the full set of total n sample values, the Z-values for the detectable portion are separated from those
for  the remaining n - m non-detect percentiles.  Estimates for the non-detect values are obtained from
the  equationX = ju + d-Z,  using ju  the intercept  mean estimate,  a the slope  standard deviation
estimate and the non-detect Z-values.  These can then be aggregated with the sample detectable values
to obtain the overall mean and standard deviation estimate.

       PROCEDURE

Step 1.   Determine the appropriate normal transformation and convert the  data if necessary.  Divide
         the data set  into two groups, detects and non-detects. If the total sample size equals n, let m
         represent the number of detects and (n - m) represent the number  of non-detects. Denote the
         rth detected measurement by X[.  Order the m detected data from smallest to largest.
                                             15-23                                   March 2009

-------
Chapter 15. Managing Non-Detect Data	Unified Guidance

Step 2.   Define the normal percentiles for the total n sample set as follows.  For a set of/' values from 1
         to n, pt = (i - .375)/(« + .25). Then convert to Z-values using the inverse normal distribution
         Z. = <£>~l (pt).   Separate the Z; values into two groups: the larger m detected and n - m non-
         detected portions.

Step 3.   Use linear regression of the ordered m data values against the corresponding Z-values.  Obtain
         the intercept and slope of the regression as the  estimated mean and standard deviation
         estimates,  ju and <7.  These can be used directly as the distributional parameter estimates or
         Step 4 can be followed.

Step 4.   Using  equation  Xn_m = ju + d • Zn_m  with ju the intercept mean estimate,  d  the slope
         standard deviation estimate and the non-detect Zn.m values, calculate the remaining xn.m values
         and combine with the  xm detected  data.  Use the combined direct sample mean and standard
         deviation calculations as the final parameter estimates:
                             = — "V x,  and <7 =
                                                    w-1
     ^EXAMPLE 15-3
     Use Cohen's and the parametric ROS  methods for the data in Example  15-1 and compare the
results to the Kaplan-Meier and Robust ROS Methods. A single overall logarithmic distribution can be
assumed. In the example, it is possible to utilize the higher detection limit (<5) as the censoring limit,
with the loss of only a single detected point of information. The detection frequency is still 72%.

     For Cohen's method, h = .28 and y = .465 for the logarithmic data.  The adjustment parameter from
the above table is interpolated as X = .445.  The resulting mean and standard deviation estimates for the
full data set are // = 2.32 log(ppb) and d = 1.22 log(ppb).

     Mean and standard deviation estimates  for the parametric ROS method are ju= 2.33 log(ppb) and
d = 1. 21 log(ppb) following regression of the ordered detectable log values against the corresponding
Z- values of the  standard normal distribution.   With such few non-detects near the lowest end of the
sample distribution,  the results are quite similar to the Robust ROS and Kaplan-Meier methods. For
higher non-detect percentages and more heavily intermingled non-detect data, the results  using these
methods can differ considerably. ~4
15.6  USE OF THE  15% AND 50% NON-DETECT RULE

       In this chapter and elsewhere in the Unified Guidance, it is recommended that imputing arbitrary
values be  limited to  data sets with  10-15% or fewer non-detects and that parametric procedures be
applied when there are 50% or fewer non-detects.   The guidance continues to suggest this basic non-
detects rule for both  historical  and conservative reasons.  The same approach was  found in both the
earlier RCRA 1989 and 1992 RCRA statistical guidance documents, although it was recognized in the

                                            15-24                                  March 2009

-------
Chapter 15.  Managing Non-Detect Data	Unified Guidance

first as a guideline "based on judgment".  It was also noted that "there is no general procedure that is
applicable in all  cases."  The  10-15%  rule using direct  substitution of arbitrary values  is believed
adequate for many applications, but one of the censoring estimation techniques provided in this chapter
can be used  instead.   For a skewed distribution  like the lognormal, the latter  approach would  be
preferable. We have cited studies above by Davis and others indicating that parameter estimation and
test performance  can suffer when more than 50%  of the data are non-detects.  Most  of the common
parameters (i.e., mean, median, standard deviation, etc.)  can be estimated with tolerable bias and error
when no more than 50% of the values are originally non-detect and the  superior non-detect fitting
techniques used.  Statistical test performance using these limitations appears to be reasonable for most
applications. However, it should be recognized that they are only "rules of thumb", not absolute criteria.

       Other authors (e.g., Helsel 2005) have suggested that certain tests will perform adequately even
with higher non-detect rates in data.  The  criterion of non-detect percentage is not  the only factor. For
example  with very large data sets (e.g.,  100-300),  quite  reasonable fits can be made to the detectable
portion using techniques found in Chapter 15 even with non-detect percentages greater than 50%.
Having a sufficient number of detectable  data is also an important consideration,  applying equally to
small data sets. One should have a fairly good idea that the  detect data themselves follow one or another
parametric distributions.  To do so, one should have a sufficiently large number of detected data points
for comparison.

       A second factor is the potential application for fitted non-detect data.  As an example, fits of high
non-detect percentage larger data sets using the lognormal distribution can provide decent parameter
estimates (log mean and log standard deviation) for use with upper prediction limit detection monitoring
tests.  Generally, the fits accurately describe the upper portions of the observed data sets.  At the same
time,  these estimated logarithmic parameters may result  in considerably larger errors when estimating
the true arithmetic mean and standard deviation (the bias problem in transformations), such as with
compliance level tests.  In this case, the 50% rule is best followed.

       The guidance generally recommends non-parametric options when non-detect data exceed 50%.
However, even this suggestion comes with caveats.  For  example, if a number of wells to be compared
using Kruskal-Wallis non-parametric ANOVA had mostly or all well  data sets greater than 50% non-
detects, the outcome would be ambiguous.   This is because the test involves comparisons of medians,
which would lie  below the detection limit.  At very high non-detect percentages, fewer  options are
available.  Upper non-parametric prediction limits can work with very few detectable values, but the
assumption of any  distributional pattern is increasingly tenuous.   In some cases, a binomial test of
proportions (found in the 1989 guidance)  may be the only realistic option.  As  a final suggestion, we
recommend that users take these factors into account and consider recommendations of other statistical
literature in the field as well, when considering non-detect limitations to specific test procedures.
                                             15-25                                   March 2009

-------
Chapter 15.  Managing Non-Detect Data                              Unified Guidance
                    This page intentionally left blank
                                      15-26                              March 2009

-------
PART III.  DETECTION MONITORING TESTS	Unified Guidance
        PART  in.  DETECTION   MONITORING
                                     TESTS
     This  third part of the Unified  Guidance presents core  procedures recommended for  formal
detection monitoring at RCRA-regulated facilities. Chapter 16 describes two-sample tests appropriate
for some small facilities, facilities in interim status, or for periodic updating of background data. These
tests include two varieties of the t-test and two non-parametric versions- the Wilcoxon rank-sum and
Tarone-Ware procedures. Chapter 17  discusses one-way  analysis of variance [ANOVA],  tolerance
limits, and the application of trend tests during detection monitoring. Chapter 18 is a primer on  several
kinds of prediction limits, which are combined with retesting strategies in Chapter 19 to address the
statistical necessity of performing multiple comparisons during RCRA statistical evaluations.  Retesting
is also discussed in Chapter 20, which presents control charts as an alternative to prediction limits.

     As discussed  in  Section 7.5,  any  of these  detection-level  tests  may  also  be applied  to
compliance/assessment  and  corrective action monitoring, where a background groundwater protection
standard [GWPS] is defined as a critical limit using two- or multiple-sample comparison tests. Caveats
and limitations discussed for detection monitoring tests are  also relevant to this situation.  To maintain
continuity  of presentation, this additional application is presumed but  not repeated in the  following
specific test and procedure discussions.

     Although other users and programs may find these statistical tests of benefit due to their wider
applicability to other environmental media and types of data, the methods described in Parts III  and IV
are  primarily  tailored  to the  RCRA  setting  and  designed to  address formal  RCRA monitoring
requirements.  In particular,  the series of prediction limit tests  found in Chapter 18 is designed  to
address the range of interpretations of the  sampling rules in  §264.97(g),  §264.98(d) and  §258.54.
Further,  all of the regulatory tests listed in  §264.97(i) and §258.53(h) are  discussed, as well as the
Student's Mest requirements of §265.93(b).

     Taken as a whole, the set of detection monitoring methods presented in the Unified  Guidance
should be  appropriate for almost all the situations likely to be encountered in practice. Professional
statistical consultation is recommended for the rest.
                                                                               March 2009

-------
PART III. DETECTION MONITORING TESTS                           Unified Guidance
                    This page intentionally left blank
                                                                       March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

                 CHAPTER 16.  TWO-SAMPLE TESTS
        16.1   PARAMETRIC T-TESTS	16-1
          16.1.1  Pooled Variance T-Test	16-4
          16.1.2  Welch's T-Test	16-7
          16.1.3  Welch's T-Test andLognormalData	16-10
        16.2   WILCOXON RANK-SUM TEST	16-14
        16.3   TARONE-WARE TWO-SAMPLE TEST FOR CENSORED DATA	16-20
     This chapter describes statistical tests between two groups of data, known as two-sample tests.
These tests may be appropriate for the smallest of RCRA sites performing upgradient-to-downgradient
comparisons on a very limited number of wells and constituents. They may also be required for certain
facilities in interim status, and can be more generally used to compare older versus newer data when
updating background.

     Two versions of the classic Student's ^-test are first discussed: the pooled variance  t-test and
Welch's t-test. Since both these tests expect approximately normally-distributed data as input, two non-
parametric alternatives to the t-test are also described:  the Wilcoxon rank-sum test (also known as the
Mann-Whitney) and the Tarone-Ware test. The latter is particularly helpful when the sample data exhibit
a moderate to larger fraction of non-detects and/or multiple detection/reporting limits.
 16.1 PARAMETRIC T-TESTS

       BACKGROUND AND PURPOSE

     A statistical comparison between two sets of data is known as a two-sample test. While several
varieties of two-sample tests exist, the most common is the parametric t-tesi. This test compares two
distinct statistical populations. The goal of the two-sample ^-test is to determine whether there is any
statistically significant difference between the mean of the first population when compared  against the
mean of the second population, based on the results observed in the two respective samples.

     In groundwater monitoring, the typical hypothesis at issue is whether the average concentration at a
compliance point is the same as (or less than) the average concentration in background, or whether the
compliance point mean is larger than the background mean, as represented in equation [16.1] below:

                                #o : Vc ^ VBG vs- HA-JUc> VBG

     A natural statistic for comparing two population means is the difference between the sample
means,  (xc -XBGJ. When this difference is small, a real difference between the respective  population
means is considered unlikely. However, when the sample mean difference is large, the null hypothesis is
rejected, since in that case  a real difference between the populations seems plausible. Note that an
observed difference  between the sample  means  does not  automatically imply  a true  population
difference. Sample means can vary for many reasons even if the two underlying parent populations are
                                            16-1                                   March 2009

-------
Chapter 16. Two-Sample Tests
Unified Guidance
identical. Indeed, the  Student's Mest was invented precisely to determine when an observed sample
difference should be considered significant (i.e., more than a chance fluctuation), especially when the
sizes of the two samples tend to be small, as is the usual case in groundwater monitoring.

     Although the null hypothesis (Ho) represented in equation [16.1] allows for a true compliance point
mean to be less than background, the behavior of the Mest statistic is assessed at the point where HQ is
most difficult to verify — that is, when HO is true and the two population means are identical. Under the
assumption of equal population means, the test statistic in any Mest will tend to follow a Student's t-
distribution. This fact  allows the selection of critical points for the t-test based on a pre-specified Type I
error or  false positive rate (a). Unlike the similarly symmetric normal distribution, however, the
Student's ^-distribution also depends on the number of independent  sample values used in the test,
represented by the degrees of freedom [df\.

     The number of degrees of freedom impacts the  shape of the ^-distribution, and consequently the
magnitude  of the critical (percentage) points selected from the ^-distribution  to provide  a basis  of
comparison against the ^-statistic (see Figure 16-1). In general, the larger the sample sizes  of the two
groups being compared, the larger the corresponding degrees of freedom, and the smaller the critical
points  (in absolute value) drawn  from the Student's ^-distribution. In a one-sided  hypothesis test  of
whether  compliance point concentrations  exceed background concentrations, a smaller critical point
corresponds to a more powerful test. Therefore, all other things being equal, the larger the sample sizes
used in the two-sample Mest, the more protective the test will be of human health and the environment.
        Figure 16-1. Student's t-Distribution for Varying Degrees of Freedom
                0.0
                  -5.0
                                           t-value
     In groundwater monitoring, t-tests  can be useful  in  at least  two ways. First, a t-test can be
employed to compare background data from one or more upgradient  wells against a single compliance
                                             16-2
        March 2009

-------
Chapter 16.  Two-Sample Tests	Unified Guidance

well. If more than one background well is involved, all the upgradient data would be pooled into a single
group or sample before applying the test.

     Second,  a t-test can be used to assess whether updating of background data is appropriate (see
Chapter 5 for further discussion). Specifically, the two-sample t-test can be utilized to check whether
the more recently collected data  is consistent with the earlier data assigned initially  as the  background
data pool. If the t-test is non-significant, both the initial background and more recent observations may
be considered  part of the same statistical population, allowing the overall background data set to grow
and to provide more accurate information about the characteristics of the background population.

     The  Unified Guidance describes  two  versions of the parametric  t-test, the pooled  variance
Student's t-test and a modification to the Student's t-test known as Welch's t-test.  This guidance prefers
the latter  t-test to use of Cochran's Approximation to the Behrens-Fisher (CABF) Student's t-test.
Initially codified in the 1982 RCRA regulations, the CABF t-test is no longer explicitly cited in the 1988
revision to  those  regulations.  Both the  pooled  variance and Welch's t-tests are  more  standard in
statistical usage than the CABF  t-test. When the  parametric assumptions  of the two-sample t-test are
violated, the  Wilcoxon  rank-sum  or the Tarone-Ware  tests  are recommended  as non-parametric
alternatives.

       REQUIREMENTS AND ASSUMPTIONS

     The two-sample t-test has been widely used and carefully studied as a  statistical  procedure. Correct
application of the  Student's t-test depends on certain key assumptions. First, every t-test assumes that the
observations in each data set or group are statistically independent. This assumption can be difficult to
check in practice (see Chapter 14 for further discussion of statistical independence),  especially if only a
handful of measurements are available for testing. As noted in Chapter 5 in discussing data mixtures,
lab replicates  or  field duplicates  are  not statistically  independent  and should  not be  treated as
independent water quality samples.   That section discussed the  limited conditions under which certain
replicate data might be applicable for t- testing. Incorrect usage of replicate  data was one of the concerns
that arose in the application of the CABF t-test.

     Second,  all  ^-tests  assume that the underlying  data are  approximately  normal in distribution.
Checks of this assumption can be made using one of the tests of normality described in Chapter 10. The
t-test is a  reasonably robust statistical procedure,  meaning that  it will usually provide accurate results
even if the  assumption of normality is partially violated.  This  robustness of the  t-test  provides  some
insurance against incorrect test results if the underlying populations are non-normal. However, the robust
assumption  is  dubious when the parent population is heavily skewed.  For data that  are lognormal and
positively  skewed, the two-sample  t-test can give misleading  results  unless the  data are first log-
transformed. Similarly, a transformation may be needed  to first normalize data from other  non-normal
distributions.

     Another  assumption particularly relevant to the use  of ^-tests in groundwater monitoring is that the
population means  need to be stable or stationary over the time of data collection and testing. As
discussed in Part II of the guidance, many commonly monitored groundwater parameters exhibit  mean
changes in both space and time.  Consequently, correct application of the t-test in groundwater requires
an implicit  assumption  that the two populations being  sampled (e.g., a background well  and  a


                                              16-3                                     March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

compliance point well) have average concentrations that are not trending with time. Time series plots
and diagnostic trend tests (Chapter 14) can sometimes be used to check this assumption.

     The  t-test  does  an excellent job of identifying a stable mean level difference between two
populations. However, if one or both populations have trends observable in the sample measurements,
the t-test may have difficulty correctly identifying a difference between the two groups. For instance, if
earlier samples in a compliance well were uncontaminated but later samples are increasing with time, the
t-test may  still provide a non-significant result. With compliance point concentrations increasing relative
to background, the t-test may not be the appropriate method for identifying this change.  Some  form of
trend testing will provide a better evaluation.

     Another concern in applying the t-test to upgradient-downgradient interwell comparisons is that the
null hypothesis is assumed to be true unless the downgradient well becomes contaminated. Absent such
an impact, the  population means  are implicitly assumed to be  identical.  Spatial variability in
background and compliance well groundwater concentrations for certain monitoring constituents do not
allow clear conditions for comparisons intended to identify a release at a downgradient compliance well.
Natural  or pre-existing synthetic mean differences among background wells will  be confused with a
potential release.  In such cases, neither  the two-sample t-test nor any interwell procedure comparing
upgradient against downgradient measurements is likely to give a correct conclusion.

     One  final requirement for running any t-test is  that each group should have  an adequate sample
size. The  t-test will have minimal  statistical power to identify any but the largest of concentration
differences if the sample size in each group is less than four.  Four measurements per group should be
considered a minimum requirement,  and much greater power will  accrue from larger sample sizes. Of
course,  the attractiveness of larger  data  sets must be weighed against the need to have statistically
independent samples and the practical limitation of semi-annual or annual statistical evaluations. These
latter  requirements often constrain the frequency of  sampling so that it may be impractical to secure
more than  4 to 6 or possibly 8 samples during any annual period.
16.1.1       POOLED VARIANCE T-TEST

       BACKGROUND AND  PURPOSE

     In the case  of two independent samples from normal populations with common variance,  the
Student's t-test statistic is expressed by the following equation:
                                                                                        [16.2]
The first bracketed quantity in the denominator is known as the pooled variance, a weighted average of
the two sample variances. The entire denominator of equation [16.2] is labeled the standard error of the
difference (SEdtff).  It represents the probable chance fluctuation likely to be observed between  the
background and  compliance point sample means when the null hypothesis in equation [16.1] is true.
Note that the formula for SEdiff depends on both the pooled variance and the sample size of each group.

                                             16-4                                   March 2009

-------
Chapter 16.  Two-Sample Tests	Unified Guidance

     When the null hypothesis (Ho) is satisfied and the two populations are truly identical, the test
statistic in equation [16.2] behaves according to an exact Student's ^-distribution. This fact enables
critical points for the t-test to be selected based  on  a pre-specified  Type I error rate (a) and an
appropriate  degrees of freedom.  In  equation  [16.2],  the joint  degrees of freedom  is  equal to
(nBG +nc- 2j,  the sum of the background and compliance point sample sizes less two degrees of
freedom (one for each mean estimate).

       REQUIREMENTS AND ASSUMPTIONS

     Along with the general requirements for Mests, the pooled variance version of the test assumes that
the population variances are equal in both groups. Since only the sample variances will be known, this
assumption requires a formal statistical test of its own such as Levene's test described in Chapter 11.
An easier, descriptive method is to construct side-by-side box plots of both data sets. If the population
variances are equal, the interquartile ranges represented by the box lengths should also be comparable. If
the population variances are distinctly different, on the other hand, the box lengths should also tend to be
different, with one box much shorter than the other.

     When variances are  unequal, the Unified Guidance recommends Welch's t-test be run instead.
Welch's t-tesi does not require the assumption of equal variances across population groups. Furthermore,
the performance of Welch's t-test is almost always equal  or superior to that of the usual Student's t-test.
Therefore, one may  be able to skip the test of equal variances altogether before running Welch's t-tesi.
                                                                                       r\
     All t-tests require approximately normally-distributed data.  If a common variance (o )  exists
between the background and compliance point data  sets, normality in the pooled variance t-tesi can be
assessed by examining the combined set of background and compliance point residuals.  A residual can
be defined as the  difference between any individual value and its sample group mean (e.g., x. - XBG for
background values X[). Not only will the combined  set of residuals allow  for a more powerful test of
normality than if the two samples are checked separately, but it also avoids a difficulty that can occur if
the sample measurements are naively evaluated with  the Shapiro-Wilk multiple group test. The multiple
group normality test allows for populations with different means and different variances. If an equal
variance check has  not already been made, the multiple group test could register both populations as
being normal even though the two population variances are distinctly different. The latter would violate
a key assumption of the pooled variance t-test. To  avoid this potential problem, either always  check
explicitly for equal  variances before running the pooled variance t-tesi,  or consider  running Welch's t-
test instead.

       PROCEDURE

Step 1.   To conduct the two-sample Student's  t-test at an a-level of  significance, first compute the
         sample  mean (x ) and standard deviation (s) of each group. Check for equal variances using a
         test from  Chapter 11. If there is  no evidence  of heteroscedasticity, check normality in both
         samples, perhaps by calculating the residuals from each group  and running a normality test on
         the combined data set.
                                              16-5                                    March 2009

-------
Chapter 16. Two-Sample Tests
Unified Guidance
Step 2.   Once the key assumptions have been checked, calculate the two-sample ^-statistic in equation
         [16.2], making use of the sample mean, sample standard deviation, and sample size of each
         group.

Step 3.   Set the degrees of freedom to df = nBG + nc -2, and look up the (1-a) x 100th percentage
         point from the ^-distribution in Table 16-1 in Appendix D. Compare this a-level critical point
         against the ^-statistic. If the ^-statistic  does  not exceed the critical point, conclude there is
         insufficient  evidence of a significant difference between the two  population  means. If,
         however, the ^-statistic is greater than the  critical point, conclude that the compliance point
         population mean is significantly greater than the background mean.

       ^EXAMPLE 16-1

     Consider  the quarterly sulfate data in the  table below  collected from one upgradient and one
downgradient well during 1995-96. Use the Student's t-test to determine if the downgradient sulfate
measurements are significantly higher than the background values at an a = 0.01 significance level.
Quarter
1/95
4/95
7/95
10/95
1/96
4/96
7/96
10/96
Mean
SD
Background
560
530
570
490
510
550
550
530
536.25
26.6927
Sulfate Concentrations (ppm)
Background
Downgradient Residuals


600
590
590
630
610
630
608.33
18.3485
23.75
-6.25
33.75
-46.25
-26.25
13.75
13.75
-6.25


Downgradient
Residuals


-8.33
-18.33
-18.33
21.67
1.67
21.67


       SOLUTION
Step 1.   Compute the sample mean and standard deviation in each well, as listed in the table above.
         Then compute the sulfate residuals by subtracting the well mean from each individual value.
         These differences are also  listed  above.  Comparison of the sample  variances shows no
         evidence that the population variances are unequal. Further, a probability plot of the combined
         set of residuals  (Figure 16-2) indicates that the normal  distribution  appears to provide a
         reasonable fit to these data.
                                             16-6
        March 2009

-------
Chapter 16. Two-Sample Tests
                                                                  Unified Guidance
              Figure 16-2. Probability Plot of Combined Sulfate Residuals

                      2
                   CD
                   L_
                   O

                   (fl
                   I
                   IM
                     -1
                     -2
                        -50
                           -25           0          25

                             Sulfate residuals (ppm)
50
Step 2.
Compute the two-sample ^-statistic on the raw sulfate measurements using equation  [16.2].
Note that the background sample size is npo = 8 and the downgradient sample size is nc = 6.
               t = (608.33 -536.25
                              7 (26.6927 )f+5(l8.3485)f
                                                  6-2
                                                        - + -
                                                        8  6
Step3.
Compute the degrees of freedom as df= 8 + 6-2=12. Since a = .01, the critical point for the
test is the upper 99th percentile of the ^-distribution with 12 df.  Table 16-1 in Appendix D
then gives the value for tcp = 2.681. Since the ?-statistic is clearly larger than the critical point,
conclude the downgradient sulfate population mean is significantly larger than the background
population mean at the 0.01 level.  -4
16.1.2
    WELCH'S T-TEST
       BACKGROUND AND PURPOSE


     The pooled variance  Student's t-test in Section 16.1.1 makes the explicit assumption that both
                                        r\
populations  have a  common  variance,  a .  For many wells  and  monitoring constituents,  local
geochemical conditions can result in both different well means and variances.  A contamination pattern
at a compliance well can have very different variability than its background counterpart.


     Welch's t-test was designed as a modification to the Student's t-test when the population variances
might differ between the two groups. The Welch's t-test statistic is defined by the following equation:
                                            16-7
                                                                          March 2009

-------
Chapter 16. Two-Sample Tests                                            Unified Guidance
                                   t=(*r-*mv<^L+^L                              [16.3]
The denominator of equation [16.3] is also called the standard error of the difference (SE&s), similar to
the pooled variance t-test.   But it is  a different  weighted estimate based on the respective  sample
variances and sample sizes, reflecting the fact that the two population variances may not be the same.

     The  most difficult part of Welch's t-test is  deriving the correct degrees of freedom. Under the
assumption of a common variance, the pooled variance estimate incorporated into the usual Student's t-
test has df = (nBG + nc-2) degrees  of freedom,  representing the number of independent  "bits" of
sample information included in the variance estimate. In Welch's t-test, the derivation of the degrees of
freedom is more complicated, but can be approximately computed with the following equation:
                                      n
nr -1
     Despite its lengthier calculations, Welch's t-test has several practical advantages. Best and Rayner
(1987) found that among statistical tests specifically designed to compare two populations with different
variances, Welch's t-tesi  exhibited comparable statistical power (for df> 5) and was much easier to
implement in practice than other tests they examined.  Moser and Stevens (1992) compared Welch's t-
test against  the usual pooled variance t-tesi  and determined that Welch's procedure was the more
appropriate in almost every case.  The only advantage registered by the usual Student's t-test in their
study was in the case  where the sample  sizes in  the two groups were unequal and the population
variances were known to be essentially the same. In  practice, the population variances will almost never
be known in advance, so it appears reasonable to use Welch's t-tesi in the majority of cases where a two-
sample t-test is warranted.

       REQUIREMENTS  AND ASSUMPTIONS

      Welch's t-test  is  also a reasonably robust statistical procedure, and will usually provide accurate
results even if the assumption of normality is partially violated.  This robustness of the t-test provides
some insurance against incorrect test results if the underlying populations are non-normal. But heavily
skewed distributions do require normalizing  transformations.  Certain limitations apply when using
transformed data, discussed in the following section.

     Unlike the pooled variance t-test, Welch's procedure does not require that the population variances
be equal in both groups.  Other general requirements of ^-tests, however,  such as statistical independence
of the sample data, lack of spatial variability  when conducting an interwell  test, and stationarity over
time, are applicable to Welch's t-test and needs to be checked prior to running the procedure.

     Because the variances of the tested populations may not be equal, an assessment of normality
cannot be made under Welch's t-test by combining the residuals (as with the pooled variance t-test),
unless an explicit check for equal variances is first conducted. The reason is that the combined residuals
from  normal populations with different variances  may not test as normal, precisely because of the

                                             16-8                                    March 2009

-------
Chapter 16. Two-Sample Tests
                                                                            Unified Guidance
heteroscedasticity. Since this latter variance check is not required for Welch's test, it may be easier to
input the sample data directly into the multiple group test of normality described in Chapter 10.

       PROCEDURE

Step 1.   To run the two-sample Welch's Mest, first compute the sample mean  (x), standard deviation
                          r\
         (s), and variance (s ) in each of the background (BG) and compliance point (C) data sets.

         Compute Welch's ^-statistic with equation [16.3].
Step 2.

Step3.



Step 4.



Step 5.
         Compute the approximate degrees of freedom in equation [16.4] using the sample variance
         and sample  size from each group.  Since this quantity often results in a fractional amount,
         round the approximate df to the nearest integer.

         Depending on the a significance level of the test, look up an appropriate critical point (tcp) in
         Table 16-1 in Appendix D. This entails finding the upper  (j - ajx WOth percentage point of
         the Student's ^-distribution with ^degrees of freedom.

         Compare the ^-statistic against the critical point. If t < tcp, conclude there is no  statistically
         significant difference between the background and compliance  point population means.  If,
         however, t > tcp, conclude that the compliance point population mean is significantly  greater
         than the background mean at the a level of significance.

     ^EXAMPLE 16-2

     Consider the following series of monthly benzene measurements (in ppb) collected over 8 months
from one upgradient and one downgradient well. What significant difference, if any, does Welch's t-test
find between these populations at the a = .05 significance level?

Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
N
Mean
SD
Variance
Benzene
BG
0.5
0.8
1.6
1.8
1.1
16.1
1.6
0.6
8
3.0
5.31
28.204
(ppb)
DG
0.5
0.7
4.6
2.0
16.7
12.5
26.3
186.0
8
31.2
63.22
3997.131
                                             16-9
                                                                                    March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

Step 1.   Compute the sample mean, standard deviation, and variance of each group as in the table
         above.

Step 2.   Use equation [16.3] to compute Welch's ^-statistic:
                                           28.204  3997.131
Step 3.   Compute the approximate degrees of freedom using equation [16.4]:
              df =
                    28.204   3997.131
(28.204/8)2   (3997.131/8)2
                           = 7.1  =
Step 4.   Using Table 16-1 in Appendix D and given a = .05, the upper 95% critical point of the
         Student's ^-distribution with 7 dfis equal to 1.895.

Step 5.   Compare the ^-statistic against the  critical point, tcp. Since t < tcp, the test on the raw
         concentrations provides insufficient evidence of a true difference in  the population means.
         However, given the order of magnitude difference  in the sample means and the fact that
         several  of  the  downgradient measurements are substantially  larger than almost all  the
         background values, we might suspect that one or more of the t-test assumptions was violated,
         possibly invalidating the result.  -4
16.1.3      WELCH'S T-TEST AND LOGNORMAL DATA

     Users should recall that if the underlying populations are lognormal instead of normal and Welch's
^-test is run on the logged data, the procedure is not a comparison of arithmetic means but rather between
the population geometric  means.   In the case of a lognormal distribution, the  geometric means are
equivalent to the population medians.  In effect, a test of the log-means is equivalent to a test of the
medians in terms of the raw concentrations.  Both the population  geometric mean and the lognormal
median can be estimated from the logged measurements asexp(y), where  y = log*  represents a logged
value  and  y is the  log-mean. On the other hand, the (arithmetic) lognormal mean on the concentration
scale would be estimated as exp\v + s2y/2\ a quantity larger than the geometric mean or median due to

the presence of the term involving s1, the log-variance.
                                y
     Although a t-tesi conducted in the logarithmic domain is not a direct comparison of the arithmetic
means, there are situations where that comparison can be inferred from the test results. For instance,
consider using the pooled variance two-sample Student's t-tesi on logged data with a common (i.e..,
equal) population log-variance (
-------
Chapter 16. Two-Sample Tests	Unified Guidance

a larger compliance point geometric mean or median when testing the log-transformed data, even though
the compliance point population arithmetic mean is smaller than the background arithmetic mean.

     Fortunately, such a reversal can only occur in the unlikely situation that the background population
log-variance is  distinctly larger than the compliance point log-variance. Factors  contributing to an
increase in the log-mean concentration level in lognormal populations often serve, if anything, to also
increase the log-variance,  and almost never to decrease it.  Consequently, Mest results indicating  a
compliance  point geometric  mean  higher  than  background should  very rarely imply  a  less-than-
background compliance point log-variance. This in turn will generally ensure that the compliance point
arithmetic mean is also larger than the background arithmetic mean, so that a test of the log-transformed
measurements can be used to infer whether a difference exists in the population concentration means.

     One caution in this discussion is for cases where the Welch's Mest is not significant on the log-
transformed measurements. Because the log-variances (
-------
Chapter 16. Two-Sample Tests
Unified Guidance

Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
N
Mean
SD
Variance
Benzene
BG
0.5
0.8
1.6
1.8
1.1
16.1
1.6
0.6
8
3.0
5.31
28.204
(ppb)
DG
0.5
0.7
4.6
2.0
16.7
12.5
26.3
186.0
8
31.2
63.22
3997.131
Log(Benzene)
BG
-0.693
-0.223
0.470
0.588
0.095
2.779
0.470
-0.511
8
0.372
1.0825
1.1719
log(ppb)
DG
-0.693
-0.357
1.526
0.693
2.815
2.526
3.270
5.226
8
1.876
1.9847
3.9392
       SOLUTION
Step 1.   First check normality of the original  measurements. To do this, compute the Shapiro-Wilk
         statistic (SW) separately for  each well. SW= 0.505 for the background data, and SW= 0.544
         for the downgradient well. Combining these two values using the equations in Section 10.7,
         the multiple group Shapiro-Wilk statistic becomes G = -6.675, which is significantly less than
         the 5% critical point of-1.645 from the standard normal distribution.1 Thus, the assumption of
         normality was violated in Example 16-2.

 Step 2.  Compute the log-mean,  log-standard deviation, and log-variance of each group, as listed
         above. Then compute the multiple group Shapiro-Wilk test to check for (joint) normality on
         the log-scale. The respective SW statistics now increase to  0.818 for the background data and
         0.964 for the downgradient  well.  Combining these into an overall test, the multiple group
         Shapiro-Wilk statistic becomes -0.721  which now exceeds the a = 0.05 standard  normal
         critical point. A log transformation adequately normalizes the benzene data — suggesting that
         the underlying populations are lognormal in distribution — so that Welch's Mest can be run
         on the logged data.

Step 2.   Using the logged measurements and equation [16.3], the ^-statistic becomes:
                           t =  l.876-0.372
                                              II. 1719   3.9392 _
1  Note that a = 5% is used in this example because the total sample size (BG and DG) is n = 16. Nevertheless, the test would
  also fail at a = 1% or just about any significance level one might choose.
                                             16-12
        March 2009

-------
Chapter 16. Two-Sample Tests
                                                                  Unified Guidance
Step 3.   Again using the log-variances and equation [16.4], the approximate df'works out to:
Step 4.


Step 5.
                df =
                     1.1719   3.9392
                                 [1.1719/8]2   [3.9392/8J
                                                          = 10.8
Note that the approximate dfin Welch's Mest is somewhat less than the value that would be
computed for the two-sample pooled variance Student's t-tesi. In that case, with 8 samples per
data set, the df would have been 14 instead of 11. The reduction in degrees of freedom is due
primarily to the apparent difference in variance between the two groups.

Using Table 16-1 in Appendix D and given  a = .05, the upper 95% critical point of the
Student's ^-distribution with 11 dfis equal to 1.796.

Comparing t against tcp, we find that 1.88 exceeds 1.796, suggesting a statistically significant
difference between the background and downgradient population log-means, at least at the 5%
level of significance. This means that the downgradient geometric mean concentration — and
equivalently for lognormal  populations, the median concentration — is statistically greater
than the same statistical measure in background. Further, since the downgradient sample log-
variance is  over three times the magnitude of the background log-variance, it is also probable
that the downgradient arithmetic mean is larger than the background arithmetic mean.
                         Figure 16-3. Benzene Time Series Plot
                200
                150  -
             -O
             D.
             Q.

             1 100
             CD
              CD
             CD
                 50  -
                                               Month
                                            16-13
                                                                          March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

         A note of caution in this example is that the same test run at the a = 0.01 level would yield a
         non-significant result, since the upper 99% Student's t critical point in that case would be
         2.718. The fact that the conclusion differs based on a small change to the significance level
         ought to prompt review of other t-test assumptions. A check of the downgradient sample
         measurements indicates an upward (non-stationary) trend over the sample collection period
         (Figure 16-3). This reinforces the fact that the t-test can be ill-suited for measuring differences
         between  populations when trends over time cause instability in the underlying population
         means. It might be necessary to either perform a formal test  of trend at the downgradient well
         or to limit the compliance data included in the evaluation only to those most representative of
         current conditions at the downgradient well (e.g., the last four measurements). -^
16.2 WILCOXON  RANK-SUM  TEST

       BACKGROUND AND PURPOSE

     When the underlying  distribution of a data set is unknown and cannot be readily identified as
normal or normalized via a transformation, a  non-parametric alternative to the two-sample t-test is
recommended. Probably the best and most practical substitute is the Wilcoxon rank-sum test (Lehmann,
1975; also known as the two-sample Mann-Whitney U test), which can be used to compare a single
compliance well or data group against background. Like many non-parametric methods, the Wilcoxon
rank-sum test is based on the ranks of the  sample measurements rather than the actual concentrations.
Some statistical information contained in the original data is lost when switching to the Wilcoxon test,
since it only uses the relative magnitudes of data values.

     The benefit is that the ranks can be used to conduct  a statistical test even when the underlying
population has  an unusual  form and is non-normal.  The parametric t-test depends on the population
being at least approximately normal; when this is not the case, the critical points of the t-test can be
highly inaccurate. The Wilcoxon rank-sum test is also a statistically efficient procedure. That is, when
compared to the t-test using normally-distributed data especially  for larger sample  sizes, it performs
nearly as well as the t-test.  Because of this fact, some authors  (e.g., Helsel and Hirsch, 2002) have
recommended  routine use  of the  Wilcoxon  rank-sum  even when the parametric t-test might be
appropriate.

     Although  a reasonable strategy for larger data sets, one should be careful  about automatically
preferring the Wilcoxon over the t-test on samples as small as those often  available in groundwater
monitoring.  For instance,  a Wilcoxon rank-sum test  of four samples in each of a background and
compliance well and an a = 0.01 level of significance can never identify a significant difference between
the two populations. This  is true  no matter what the sample  concentrations are, even  if all  four
compliance measurements are larger than any of the background measurements. This Wilcoxon test will
require at least five samples in at least one of the groups, or  a higher level of significance (say a = 0.05
or 0.10) is needed.

     The Wilcoxon  test statistic  (W) consists of the sum  of  the  ranks  of the compliance  well
measurements. The rationale of the test is that if the ranks of the compliance data are quite large relative
to the background ranks, then the hypothesis that the compliance and background values came from the
same population ought  to  be rejected.  Large values  of the  W statistic give evidence  of possible
                                            16-14                                   March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

contamination in the compliance well. Small values of W, on the other hand, suggest there is little
difference between the background and compliance well measurements.

       REQUIREMENTS AND ASSUMPTIONS

     The  Wilcoxon rank-sum test assumes that both populations being compared follow a common,
though unknown,  parent distribution under the null hypothesis (Hollander and Wolfe, 1999). Such an
assumption is akin to that used in the two-sample pooled variance Student's Mest, although the form of
the common distribution need not be normal. The Wilcoxon test assumes that both population variances
are equal, unlike Welch's Mest.  Side-by-side box plots of the two  data  groups  can be  compared
(Chapter  9) to examine whether or not the level of variability appears to be approximately equal in both
samples. Levene's test (Chapter 11) can also be applied as a formal test of heteroscedasticity given its
relative robustness to  non-normality.  If there is  a  substantial difference  in  variance between the
background and compliance point populations,  one  remedy is the Fligner-Policello test (Hollander and
Wolfe, 1999), a more complicated rank-based procedure.

     The  Wilcoxon procedure as described in  the Unified Guidance is generally used as an interwell
test,  meaning  that it  should be avoided under  conditions of  significant  natural spatial variability.
Otherwise, differences between background and compliance point wells identified by the test may be
mistakenly attributed to possible contamination, instead of natural differences in geochemistry, etc. At
small sites, the Wilcoxon procedure can be adapted  for use as an  intrawell test, involving a comparison
between intrawell background and more recent measurements  from the same well. However, the per-
comparison false  positive rate in  this case should be raised to either a  = 0.05  or a = 0.10.  More
generally,  a  significance level of at  least 0.05  should be  adopted whenever the sample size of either
group is no greater than n = 4.

     In addition to spatial stationarity (i.e., lack of natural spatial variability), the Wilcoxon rank-sum
test assumes that the tested populations are stationary over time, so that mean levels are not trending
upward or downward. As with the ^-test, if trends are evident in  time series plots of the sample data, a
formal trend test might need to be employed instead of the Wilcoxon rank-sum, or the scope  of the
sample may need to be limited to only include data representative of current groundwater conditions.

       HANDLING  TIES

     When ties are present in a combined data  set, adjustments need to be made to the usual Wilcoxon
test  statistic.  Ties will occur in two  situations: 1) detected  measurements reported with the same
numerical  value and 2) non-detect measurements with a common RL. Non-detects are considered ties
because the actual concentrations are unknown; presumably, every non-detect has a concentration
somewhere between zero and the quantitation limit  [QL].  Since these measurements cannot be ordered
and ranked explicitly, the approximate remedy in the Wilcoxon rank-sum procedure is to treat such
values as ties.

     One  may be able to partially rank the  set of non-detects by making use of laboratory-supplied
analytical  qualifiers. As discussed in Section 6.3, there are probable concentration differences between
measurements labeled as undetected (i.e., given a "U" qualifier),  non-detect (usually reported without a
qualifier),  or as estimated concentrations (usually labeled with "J" or "E"). One reasonable strategy is to
group all U values as the lowest set of ties, other non-detects as a higher set of ties, and to rank all J

                                             16-15                                   March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

and/or E values according to their estimated concentrations. In situations where estimated values for J
and E samples  are not provided, treat these measurements as the highest group of tied  non-detects.
Always  give the highest ranks to explicitly quantified or estimated concentration measurements. In this
way, a more detailed partial ranking of the data will be possible.

     Tied observations in the Wilcoxon rank-sum test are handled as follows. All tied observations in a
particular group should receive the same rank.  This  rank called the midrank  (Lehmann,  1975) is
computed as the average of the ranks that would be assigned to a group of ties if the tied values actually
differed by  a tiny amount and could  be ranked uniquely.  For example, if the  first  four ordered
observations are all the same, the midrank given to each of these samples would be equal to (1 + 2 + 3 +
4)/4 = 2.5. If the next highest measurement  is a unique value, its rank would be 5,  and so on until all
observations are appropriately ranked. A more detailed example is illustrated in Figure 16-4.
           Figure 16-4. Computation of Midranks for Groups of Tied Values
                    Order   Concentration   Mid-Rank
                 [   i
1.5
1.5
                                 1.2
4
5
6

8
9
1.3
1.3
1.3
1.5
1.5
1.6
5
5
5
7.5
7.5
9
=> ^(4+5+6)

1
=* 2V7+8)

       HANDLING NON-DETECTS

     If either of the samples contains a substantial fraction of non-detect measurements (say more than
20-30%),  identification  of an appropriate  distributional  model  (e.g., normality) may be difficult,
effectively ruling out the use of parametric tests like the Mest. Even when a normal or other parametric
model can be fit to such left-censored data, a Mest cannot be run without imputing estimated values for
each non-detect. Past guidance has recommended the Wilcoxon rank-sum test as an alternative to the t-
test in the presence of non-detects, with all non-detects at a common RL being treated as tied values.

     If the combined data set contains  a single, common RL, that limit  is smaller than any of the
detected/quantified values, and the proportion of censored data is small (say no more than 10-15% of the
total), it may be reasonable to treat the non-detects as a set of tied values  and to apply the Wilcoxon
rank-sum  test adjusted for ties (described below). More generally, however, the statistical behavior of
the Wilcoxon statistic depends on a full and accurate ranking of all the measurements.  Groups of left-
censored values cannot be ranked with certainty, even if each such measurement possesses a common
RL. The problem is compounded in the presence of multiple RLs and/or quantified values less than the
                                            16-16                                  March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

RL(s). What is the relative ranking,  for instance, of the pair of measurements (<1, <5)? A higher RL
does not guarantee that the second observation is larger in magnitude than the first. A similar uncertainty
plagues the pair of values (4, <10). And there is no guarantee either that the pair (<2, <2) is actually tied.
One may be able to partially rank the set of non-detects by making use of laboratory-supplied analytical
qualifiers as described in the previous section.

     Because non-detects generally prevent a  complete ranking of the measurements, the Wilcoxon
rank-sum test is not recommended  for  most censored data  sets. Instead,  a  modified version of the
Tarone-Ware test (Hollander and Wolfe, 1999) is presented in Section 16.3.  The Tarone-Ware test is
essentially a generalization of the Wilcoxon test specifically designed to accommodate censored values.

       PROCEDURE

Step 1.   To conduct the Wilcoxon  rank-sum test,  first combine the compliance and background data
         into a single data set. Sort the combined values from smallest to largest, and — if there are no
         tied values  or non-detects with  a common  RL — rank the ordered values from 1 to N. Assume
         there are n  compliance well samples and m background samples so that N=m + n. Denote the
         ranks of the compliance samples by C\ and the ranks of the background samples by B\.

Step 2.   If there are groups of tied values (including non-detects with  a  common RL),  form the
         midranks of the combined  data set by assigning to each set of ties the average of the potential
         ranks the tied members would have been given if they could be uniquely ranked.

Step 3.   Sum the ranks of the compliance samples to get the Wilcoxon statistic W:

                                          W = Zl_lCi                                    [16.5]

Step 4.   Find the a-level critical point of the Wilcoxon test, making use of the fact that the sampling
         distribution of W under the null hypothesis, HO, can be approximated by a normal curve. By
         standardizing the statistic W (i.e., subtracting off its mean or expected value and  dividing by
         its standard deviation), the standardized  statistic or z-score, Z, can be approximated  by a
         standard normal distribution. Then an appropriate critical point (zcp)  can be determined as the
         upper (1-oc) x 100th percentage point of the  standard normal distribution, listed in Table 10-1
         in Appendix D.

Step 5.   To compute Z when there are no ties, first compute the expected value and standard deviation
         of W, given respectively by the  following equations:

                                              -n(N + l)                                [16.6]
                                                                                         [16.7]
         Then compute the approximate z-score for the Wilcoxon rank-sum test as:
                                             16-17                                   March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

                                      z__W-E(W)-ll2
                                             SD(W)

         The  factor of 1/2 in the numerator  serves as a continuity  correction since the  discrete
         distribution of the Wilcoxon  statistic W is being approximated  by a continuous normal
         distribution.

Step 6.   If there are tied values, compute the expected value  of W using [16.6] and the standard
         deviation of W adjusted for the presence of ties with the equation:
                                                       -
                                           12          '-N3-N

         where g equals the number of different groups of tied observations and  t\  represents  the
         number of tied values in the rth group.

         Then compute the approximate z-score for the Wilcoxon rank-sum test as:

                                         W-E(w}-\/2
                                     z=
                                             SD
Step 7.   Compare the approximate z-score against the critical point, zcp. If Z exceeds zcp, conclude that
         the compliance well concentrations are significantly greater than background at the a level of
         significance.  If  not,  conclude that  the null hypothesis of  equivalent background and
         compliance point distributions cannot be rejected.

       ^EXAMPLE 16-4

     The table below contains copper concentrations (ppb) found in groundwater samples at a Western
monitoring facility. Wells 1 and 2 denote background wells while Well 3 is a single downgradient well
suspected of being contaminated. Calculate the Wilcoxon rank-sum test on these data at the a = .01 level
of significance.
                                            16-18                                   March 2009

-------
Chapter 16. Two-Sample Tests
                                         Unified Guidance
                                      Copper Concentration (ppb)
                                      Background             Compliance
                     Month
Well 1
      Well 2
Well 3
1
2
3
4
5
6
4.2
5.8
11.3
7.0
7.0
8.2
5.2
6.4
11.3
11.5
10.1
9.7
9.4
10.1
14.5
16.1
21.5
17.6
       SOLUTION
Step 1.   Sort the N =  18  observations from least to greatest. Since there are 3 pairs of tied values,
         compute the midranks as in the table below. Note that m = 12 and n = 6.

Step 2.   Compute the Wilcoxon statistic by summing the compliance well ranks, so that W= 84.5.

Step 3.   Using a = .01, find the upper 99th percentage point of the standard normal distribution in
         Table 10-1 of Appendix D. This gives a critical value of zcp = 2.326.
                                   Midranks of Copper Concentrations
                                      Background             Compliance
                     Month
Well 1
      Well 2
Well 3
1
2
3
4
5
6
1
3
12.5
5.5
5.5
7
2
4
12.5
14
10.5
9
8
10.5
15
16
18
17
Step 4.   Compute the expected value and adjusted standard deviation of Wusing equations [16.6] and
         (16.10), recognizing there are 3 groups of ties with t\ = 1 measurements in each group:
               SD(w)=   — -12-6-fl8
                  V  '  ^12       V
1-3-
                                                 23-2 V
               183-18
                         = Vll3.647 =10.661
                                            16-19
                                                 March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

         Then compute the standardized statistic or z-score, Z, using equation (16.10):

                                  z=84.5-57-0.5
                                          10.661

Step 5.   Compare the  observed z-score against the critical  point zcp. Since Z = 2.533 > 2.326 = 2,99,
         there is statistically significant evidence of possible contamination in the compliance well at
         the a = .01 significance level. ~4
16.3 TARONE-WARE TWO-SAMPLE TEST FOR CENSORED DATA

       BACKGROUND

     In statistical terms, non-detect measurements represent left-censored values, in which the 'true'
magnitude is known only to exist somewhere between zero and the RL, i.e., within the concentration
interval [0, RL). The uncertainty introduced by non-detects impacts the applicability of other two-sample
comparisons like the t-test and Wilcoxon rank-sum test. Because the Student's t-test cannot  be run
unless a specific magnitude is  assigned to each  observation, estimated or imputed values need to be
assigned to the non-detects.  The Wilcoxon procedure requires that every observation be ranked in
relation to other values in the combined sample, even though non-detects allow at best only a partial
ranking, as discussed in Section 16.2.

     The  Tarone-Ware two-sample test  can  be utilized to overcome these limitations  for many
groundwater data with substantial fractions of non-detects along with multiple RLs. Tarone and Ware
(1977) actually proposed a family of tests to analyze censored data.  One variant of this family is the
logrank test, frequently used in survival analysis  for right-censored data.  Another variant is known as
Gehan's generalized Wilcoxon test  (Gehan, 1965).  The  Unified Guidance  presents  the  variant
recommended by Tarone and Ware, slightly modified to account for left-censored measurements.

     The key benefit of the Tarone-Ware procedure is that it is designed to provide a valid statistical
test, even with a large fraction of censored data. As a non-parametric test, it does not require normally-
distributed observations. In addition,  non-detects do not have to be imputed or  even fully ranked.
Instead, for each detected  concentration (c), a simple count needs to be made within each sample of the
number of detects and non-detects no greater in magnitude than c. These counts  are then combined to
form the Tarone-Ware statistic.

       REQUIREMENTS AND ASSUMPTIONS

     The null hypothesis (Ho) under the Tarone-Ware  procedure assumes  that the populations in
background and the compliance well being tested are identical. This implies that the variances in the two
distributions are the same, thus necessitating a check of equal variances. With many non-detect data sets,
it can be very difficult to formally test for  heteroscedasticity.  Often  the best remedy is to make an
informal, visual check of variability using side-by-side box plots (Chapter 9), setting each non-detect to
half its RL.
                                            16-20                                  March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

     The Tarone-Ware test will typically be used as an interwell test, meaning that it should be avoided
under conditions of significant natural spatial variability. In addition, the tested populations should be
stationary over time, so that mean levels are not trending upward or downward. Both assumptions can be
more difficult to verify with censored data. Spatial variation can sometimes be checked with a non-
parametric  Kruskal-Wallis  analysis of variance  (Chapter 17). Trends with censored data can  be
identified with the Mann-Kendall  test (Chapter 14).

     As with other two-sample tests, if a trend is identified in one or both samples, a formal trend test
may be needed instead of the Tarone-Ware, or the scope of the sample may need to be limited to only
include data representative of current groundwater conditions.

     Because  the Tarone-Ware  test presented  in  the  Unified  Guidance depends  on counts  of
observations with magnitudes no  greater than each detected concentration, and in that sense generalizes
the ranking process  used  by the Wilcoxon  rank-sum  procedure,  it is recommended  that estimated
concentrations (i.e., sample measurements assigned unique magnitudes but labeled with qualifiers "J" or
"E") be treated as detections for the purpose of computing the Tarone-Ware statistic. Such observations
provide valuable  statistical information about the relative ranking of  each  censored sample, even if
estimated concentrations possess larger measurement uncertainty than fully quantified values.

       PROCEDURE

Step 1.   To compare a background data set against a  compliance well using the Tarone-Ware test, first
         combine the two samples. Locate  and sort the k distinct detected values and label these as:



         Note that the set of w's will not include any RLs from non-detects. Also, if two  or more
         detects are tied, & will be less than the total number of detected measurements.

Step 2.   For the combined sample,  count the number of observations (described by Tarone & Ware as
         'at risk') for each distinct detected concentration. That is, for / = l,...,k, let n\ = the number of
         detected values no greater than WQ plus the number of non-detects with RLs no greater than
         W(i). Also let d\ = the  number of detects with concentration equal to WQ.  This value will equal
         1 unless there are multiple detected values with the same reported concentration.

Step 3.   For the compliance sample, count the observations 'at risk',  much as in Step  2. For / =  1 to k,
         let n\i = the number of detected  compliance values no greater than WQ plus  the number of
         compliance point non-detects with RLs no greater than WQ. Also let d^ = the number of
         compliance point detects with concentration  equal to WQ.  Note that d\i = 0 if WQ represents a
         detected value from background. Also compute n\\, the number 'at risk' in the background
         sample.

Step 4.   For /' = 1 to k, compute the expected number  of compliance point detections using the formula:

                                        Ei2=d.njn.                                  (16.11)

         Also compute the variance of the number of compliance point detections, using the equation:

                                            16-21                                   March 2009

-------
Chapter 16.  Two-Sample Tests _ Unified Guidance
                                           d  n -d
                                                                                        (16.12)
         Note in equation (16. 12) that if n\ = 1 for the smallest detected value, the numerator of V\i will
         necessarily equal zero (since d\ = 1 in that case), so compute Vii = 0.

Step 5.   Construct the Tarone-Ware statistic (TW) with the equation:


                                                                                        (,6.13)
Step 6.   Find  the  a-level critical point of the Tarone-Ware test, making  use  of the  fact that the
         sampling  distribution of TW under the null hypothesis, HQ, is  designed  to approximately
         follow a standard normal distribution. An appropriate critical point (zcp) can be determined as
         the upper (1-a) x 100th percentage point of the standard normal distribution, listed in Table
         10-1 of Appendix D.

Step 7.   Compare  TW against the critical point, zcp. If TW exceeds zcp, conclude that the compliance
         well concentrations are significantly greater than background at the a level of significance. If
         not, conclude  that the null hypothesis  of equivalent  background and compliance  point
         distributions cannot be rejected.

       ^EXAMPLE  16-5

     A heavily industrial site has been historically contaminated with tetrachloroethylene [PCE]. Using
the Tarone-Ware procedure at  an  a = .05 significance level, test the following PCE measurements
collected from one background and one compliance well.
                                           PCE (ppb)
                                    Background    Compliance
<4
1.5
<2
8.7
5.1
<5


6.4
10.9
7
14.3
1.9
10.0
6.8
<5
       SOLUTION
Step 1.   Combine the background and compliance point samples. List and sort the distinct detected
         values (as in the table below), giving k = 10. Note that the 4 non-detects comprise 28% of the
         combined data.

Step 2.   Compute the number of measurements (n\) in the combined sample 'at risk' for each distinct
	detected value (wg)\ indexed from  i = 1,...,  10, by adding the  number of detects and non-
                                             16-22                                  March 2009

-------
Chapter 16. Two-Sample Tests	Unified Guidance

         detects no greater than WQ, as listed in column 6 of the table below. Also list in column 3 the
         number of detected values (d[) exactly equal to WQ.

Step 3.   For the compliance point sample, compute the number (w;2) 'at risk' for each distinct detected
         value, as listed in column 5 below. Also compute the number («ii) 'at risk' for the background
         sample (column 4) and the number of compliance point measurements exactly equal to WQ
         (column 2).

Step 4.   Use equations  (16.11) and (16.12) to compute the expected value (E^) and variance (Vii) of
         the number of compliance point detections at each WQ (columns 7 and 8 below).
w(i)
1.5
1.9
5.1
6.4
6.8
7.0
8.7
10.0
10.9
14.3
di2
0
1
0
1
1
1
0
1
1
1
d,
1
1
1
1
1
1
1
1
1
1
riii
1
1
5
5
5
5
6
6
6
6
n,2
0
1
2
3
4
5
5
6
7
8
Hi
1
2
7
8
9
10
11
12
13
14
Ei2
0
0.5
0.2857
0.375
0.4444
0.5
0.4545
0.5
0.5385
0.5714
Vi2
0
0.25
0.2041
0.2344
0.2469
0.25
0.2479
0.25
0.2485
0.2449
Step 5.   Calculate the Tarone-Ware statistic (TW) using equation (16.13):

               = VT-(0-0) + A/2- (1-0.5)
Step 6.   Determine the 0.05 level critical point from Table 10-1 in Appendix D as the upper 95th
         percentage point from a standard normal distribution. This gives zcp = 1.645.

Step 7.   Compare the Tarone-Ware  statistic against the critical point. Since TW = 1.85 > 1.645 = zcp,
         conclude that the PCE concentrations are significantly greater at the compliance well than in
         background at the 5% significance level. -^
                                            16-23                                   March 2009

-------
Chapter 16. Two-Sample Tests                                      Unified Guidance
                     This page intentionally left blank
                                       16-24                              March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

  CHAPTER 17.   ANOVA, TOLERANCE  LIMITS,  AND TREND
                                      TESTS

       17.1  ANALYSIS OF VARIANCE [ANOVA]	17-1
         17.1.1  One-Way Parametric F-Test	17-1
         17.1.2  Kruskal-Wallis Test	17-9
       17.2  TOLERANCE LIMITS	17-14
         17.2.1  Parametric Tolerance Limits	17-15
         17.2.2  Non-Parametric Tolerance Intervals	17-18
       17.3  TREND TESTS	17-21
         17.3.1  Linear Regression	77-23
         77.3.2  Mann-Kendall Trend Test	77-30
         773.3  Theil-Sen Trend Line	77-34
     This chapter describes two statistical procedures — analysis of variance [ANOVA] and tolerance
limits — explicitly allowed within §264.97(h) and §258.53(g) for use in groundwater monitoring. The
Unified Guidance does not generally recommend either  technique for formally making regulatory
decisions about compliance wells or regulated units, instead focusing on prediction limits, control charts,
and confidence intervals. But both ANOVA and tolerance tests are standard statistical procedures that
can be adapted for a variety of uses.  ANOVA is particularly helpful in both identifying on-site spatial
variation and in sometimes aiding the computation of more effective and statistically powerful intrawell
prediction limits (see Chapters 6 and 13 for further discussion).

     This chapter also presents selected trend tests as an alternative statistical method that can be quite
useful  in groundwater detection monitoring,  particularly when  groundwater populations  are  not
stationary over time.  Although trend tests are not explicitly listed within the RCRA regulations, they
possess advantages in certain situations and can meet the  performance requirements  of §264.97(i) and
§258.53(h). They can also be helpful during diagnostic  evaluation and establishment  of historical
background (Chapter 5) and in verifying key statistical assumptions (Chapter 14).
17.1 ANALYSIS OF VARIANCE [ANOVA]

     17.1.1 ONE-WAY PARAMETRIC F-TEST

       BACKGROUND AND PURPOSE

     The  parametric one-way  ANOVA  is a  statistical  procedure  to  determine whether  there are
statistically significant differences in mean  concentrations  among  a set  of wells.  In groundwater
applications, the question of interest is  whether there is potential contamination at one or more
compliance wells when compared to background. By  finding a significant difference  in means  and
specifically higher  average concentrations at one or more  compliance wells, ANOVA results  can
sometimes be used to identify unacceptably high contaminant levels in  the absence of  natural spatial
variability.

     Like the two-sample Mest, the one-way ANOVA is a comparison of population means. However,
the one-way parametric ANOVA is  a  comparison of several populations,  not just two: one set of
                                            17-1                                  March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                       Unified Guidance

background data versus  at least two compliance wells. The F-statistic that forms the heart  of the
ANOVA procedure is actually an extension of the ^-statistic; an F-statistic formed in a comparison of
only two datasets reduces to the square of the usual pooled variance Student's ^-statistic. Like the t-
statistic, the F-statistic is a ratio  of two quantities.   The numerator is a measure of the average squared
difference observed between the pairs of sample means, while the denominator represents the average
variability found in each well group.

     Under the null hypothesis  that all the wells or groups have the same  population mean,  the F-
statistic follows the F-distribution. Unlike the ^-distribution with a single degrees of freedom df, there
are two df quantities associated  with F.  One is for the numerator and the other for the denominator.
When  critical points are needed from the F-distribution, one must  specify both degrees of freedom
values.

     Computation of the F-statistic is only the first step of the full ANOVA procedure, when used as a
formal compliance test. It can only determine whether any significant mean difference exists between the
possible pairs of wells or data groups, and not whether or what specific compliance wells differ from
background.  To accomplish  this latter task  when a significant F-test  is registered, individual tests
between each compliance well and background needs to be conducted,  known as individual post-hoc
comparisons or contrasts. These  individual tests are a specially constructed series of ^-tests, with critical
points chosen to limit the test-wise or experiment-wise false positive rate.

       REQUIREMENTS AND ASSUMPTIONS

     The  parametric  ANOVA  assumes that the data groups are normally-distributed  with  constant
variance. This means that the group residuals should be tested for normality (Chapter 10) and that the
groups have to be tested for equality of variance, perhaps with Levene's test (Chapter 11). Since the F-
test used in the one-way ANOVA is reasonably robust to small departures from normality,  the first of
these assumptions turns out to be less critical than the second. Research (Milliken and Johnson, 1984)
has shown that the statistical power of the F-test is strongly affected by inequality in the population
variances. A noticeable drop in power is seen whenever the ratio of the largest to smallest group variance
is at least 4. A severe drop in power is found whenever the ratio of the largest to smallest group variance
is at least a factor of 10. These ratios imply that the F-test will lose some statistical power if any of the
group population standard deviations is at least twice the size of any other group's standard deviation,
and that the power will be greatly curtailed if any standard deviation is at least 3 times as large as any
other group's.

     If the hypothesis of equal  variances  is rejected or if the group residuals are found to violate an
assumption  of normality  (especially  at the  .01  significance level  or  less), one should  consider a
transformation of the data, followed by testing of the ANOVA assumptions on the transformed scale. If
the residuals from the transformed data still do not satisfy normality or if there are too many non-detect
measurements to adequately test  the assumptions, a non-parametric ANOVA (called the Kruskal-Wallis
test) using the ranks of the observations is recommended instead (see Section 17.1.2).

     Since ANOVA is inherently an interwell statistical method,  a critical point in using ANOVA for
compliance testing is that the well field should exhibit minimal spatial variability.  Interwell tests also
require the groundwater well populations to be spatially  stationary,  so that absent  a release  the
population well means are stable over time. Because spatial variation is frequently observed in many

                                              17-2                                    March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

groundwater constituents, especially for common inorganic constituents and some metals, ANOVA may
not be usable as compliance testing tool. Yet it can be utilized on the same data sets to help identify the
presence of spatial variability.  In this  capacity, the same procedure  and formulas are utilized as
described below (with the exception  of the post-hoc contrasts, which are unnecessary for  assessing
spatial variation).  The results are then employed to guide the appropriate choice of a compliance test
(e.g., intrawell or interwell prediction limits).

     For formal ANOVA  testing  under  §264.97(i) and §258.53(h), the  experiment-wise or test-wise
false positive rate (a) needs to be at least 5% during any statistical evaluation for each constituent tested.
Furthermore,  the individual post-hoc contrasts used to test single compliance wells against background
need to be run at a significance level of at least a* = 1% per well. Combined, these regulatory constraints
imply that if there are more  than five post-hoc contrasts that need  to  be tested (i.e., more than 5
compliance wells are included in  the ANOVA test), the overall,  maximal  false  positive rate of the
procedure will  tend to be greater,  and perhaps substantially so, than  5%. Also, since  a  = 5% is the
minimum significance level  per monitoring constituent, running multiple  ANOVA procedures to
accommodate a list of constituents will lead to a  minimum site-wide false positive rate [SWFPR] greater
than the Unified Guidance recommended target of 10% per statistical evaluation.

     In addition, if a contaminated compliance well exists but too many uncontaminated wells are also
included in the same ANOVA, the F-statistic  may result in a non-significant outcome. Performing
ANOVA with more than 10 to 15 well groups can "swamp" the procedure, causing it to lose substantial
power. It therefore will be necessary to consider  one of the retesting strategies described in Chapters 18
and 20 as an alternative to ANOVA in the event that either the expected false positive rate is  too large,
or if more than a small number of wells need to be tested.

     Another drawback to the one-way  ANOVA is  that the F-test accounts for all possible paired
comparisons among the well groups. In some cases, the F-statistic may be  significant even though all of
the contrasts between compliance wells and background are non-significant. This  does not mean that the
F-test has necessarily registered a  false positive.  Rather, it may be that  two of the compliance wells
significantly differ  from each other,  but neither differs  from background.  This  could  happen, for
instance, if a compliance well has a lower mean concentration than  background while other compliance
wells have near background means. The F-test looks for all possible differences between pairs of well
groups, not just those comparisons against background.

     In order to run a valid one-way F-test, a minimum number of observations are needed.  Denoting
the number of data groups by p, at  least/? > 2 groups must be compared (e.g., two or more compliance
wells versus  background). Each group should have at  least  three to four  statistically  independent
observations and the total sample size, N,  should be large enough so that N-p > 5. As long asp > 3 and
there  are at least 3 observations per well, this last requirement  will always be met. But the  statistical
power of an ANOVA to identify differences in population means tends to be minimal unless there are at
least 4 or more observations  per  data group. It  is also  helpful to have at least 8 measurements in
background for the test.

     Similarly to the two-sample  Mest, it may be very difficult to verify that  the measurements are
statistically independent with  only  a handful of observations per well.  One should additionally ensure
that the samples are collected far enough apart in time to avoid significant autocorrelation (see Chapter
14 for further discussion). A periodic check  of statistical independence in each may be possible after a
                                              17-3                                    March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                       Unified Guidance

few testing periods, when enough data has been collected to enable  a statistical assessment of this
assumption.

     PROCEDURE

Step 1.   Combine all the relevant background data collected from multiple wells into one group.  These
         wells should have insignificant mean differences under prior ANOVA testing. If the regulated
         unit has (p-l) compliance wells, there will then be a total ofp data groups. Because there may
         be different  numbers of observations per well, denote the sample size of the /'th group by n\
         and the total number of data points across all groups by N.

Step 2.   Denote the observations in the /'th well group by Xy for /' = 1 to p and j = 1 to n\. The first
         subscript designates the well, while the second denotes they'th value in the /'th well.  Then
         compute the mean of each well group along with the overall (grand) mean of the combined
         dataset using the following formulas:

                                                 nt
                                         x  =— Yx..                                    [17.1]
                                          7»     / J  77                                    L   J
                                                                                         [n.2]
                                             A T *—> *—> '}                                   L    J
                                             7V i=l j=l

Step 3 .   Compute the sum of squares of differences between the well group means and the grand mean,
         denoted SSweiis:

                                     p           ^   P
                             SS  „  = yn(x  -x  )=ynx2-Nx2                        [17.3]
                               wells  4—1 i \ !•   •• /   4—1 i  :•     ••                        L    J
         The  formula on the far right  is usually the most convenient  for calculation.  This  sum of
         squares has (p-l) degrees of freedom associated with it and is a measure of the variability
         between wells. It constitutes the numerator of the F-statistic.

Step 4.   Compute the corrected total sum of squares, denoted by SS total'.
                                 ,
                               total
         The far right equation is convenient for calculation. This sum of squares has (N-l) degrees of
         freedom associated with it and is a measure of the variability in the entire dataset. In fact, if
               is divided by (7V-1), one gets the overall sample variance.
Step 5.   Compute the sum of squares  of differences between the observations and the well group
         means.  This is  known as the  wi thin-wells  component of the total  sum  of squares  or,
         equivalently, as the sum of squares due to error. It is easiest to obtain by subtraction using the
         far right side of equation [17.5] and is denoted SSerror:
                                             17-4                                   March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance


                                     P  ">  /       x2
                             SS   = yy(x..-x. )=SS  ,-SS „                        [17.5]
                               error   1—t 1—t \ ij    i* /     total     wells                       L     J
         SSerror is associated with (N-p) degrees of freedom and is a measure of the variability within
         well groups. This quantity goes into the denominator of the F-statistic.

Step 6.   Compute the mean sum of squares for both the between-wells and within-wells components of
         the total sum of squares, denoted by MSweus and MSermr. These quantities are simply obtained
         by dividing each sum of squares by its corresponding degrees of freedom:

                                    MS „  =SS ,,/(p-l)                               [17.6]
                                        wells     wells/ \f   /                               L     J

                                    MS   =SS   /(N-p)                              [17.7]
                                       error     error f \    r /                              L     J

Step 7.   Compute the F-statistic by forming the ratio between the mean sum of squares for wells and
         the mean sum of squares due to error, as in Figure 17-1. This layout is known as a one-way
         parametric  ANOVA table  and  illustrates the  sum  of  squares  contribution  to the total
         variability,  along with the corresponding degrees of freedom, the mean squares components,
         and the final F-statistic calculated as F = MSweiisIMSerror. Note that the first two rows of the
         one-way table sum to the last row.

                    Figure 17-1.  One-Way Parametric ANOVA Table
Source of Variation
Between Wells
Error (within wells)
Total
Sums of Squares
^ Dwells
•^•^error
sstotai
Degrees of
Freedom
p-1
N-p
A/-1
Mean Squares
WSwe//s = SSwe//s/(p-l)
MSermr = SSermr/(N-p)
F-Statistic
F = MSwells/MSermr
Step 8.   To test the hypothesis of equal means for all p wells, compare the F-statistic in Figure 17-1 to
         the  a-level critical  point found from the F-distribution with (p-V) and  (N-p)  degrees of
         freedom in Table 17-1 of Appendix D.  a is usually set at 5%, so that the needed comparison
         value equals the upper 95th percentage point of the F-distribution.  The numerator (p-1) and
         denominator (N-p) degrees of freedom for the F-statistic are obtained from the above table.  If
         the  observed F-statistic exceeds the critical point (F^,p-\,N-p), reject the hypothesis of equal
         well group population means.  Otherwise, conclude that there is insufficient  evidence of a
         significant difference between the concentrations at the/? well  groups and thus no evidence of
         potential contamination in any of the compliance wells.

Step 9.   In the case of a significant F-statistic that exceeds the critical point in Step 8, determine which
         compliance wells have elevated concentrations compared to background. This is done by
         comparing each compliance well individually against the background measurements. Tests to
         assess  concentration differences between a pair  of well groups are called contrasts in a
         multiple comparisons  ANOVA framework. Since the contrasts are a series of individual
                                             17-5                                    March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                       Unified Guidance

         statistical tests, each run at a fixed significance level a*, the Type I error accumulates across
         the tests as the number of contrasts increases.

         To keep the overall false positive rate close to the targeted rate of 5%, the individual contrasts
         should be set up as follows: Given (p-l) separate background-compliance contrasts, if (p-l) <
         5, run each contrast at a significance level equal to a* = .05/(p-l). However, if (p-l) > 5, run
         each contrast at a significance level equal to a* = .01. Note that when there are more than 5
         compliance wells, this last provision will tend to raise the overall false positive rate above 5%.

Step 10.  Denote the background data  set as the first well  group,  so that the number of background
         samples is equal to «&. Then for each of the remaining (p-l) well groups, compute the standard
         error of the difference between each compliance well and background:
                                   SEt=MSimr'—+-                               [17.8]
         Note thatMSerror is taken from the one-way ANOVA table in Figure 17-1. The standard error
         here is an extension of the standard error of the difference involving the pooled variance in the
         Student's f-test of Chapter 16.

Step 1 1 .  Treat the background data as the first well group with the average background concentration
         equal toxb . Compute the Bonferroni ^-statistic for each of the (p-l) compliance wells from /' =
         2 to p,  dividing the standard error in Step  10 into the  difference between the average
         concentration at the compliance well and the background average, as shown below:

                                       ti=(xi-xb)/SEi                                   [17.9]

Step 12.  The Bonferroni ^-statistic in equation [17.9] is a type of Mest. Since the estimate of variability
         used in equation  [17.8] has  (N-p) degrees of freedom, the  critical point can be determined
         from the Student's ^-distribution in  Table  16-1 of Appendix D.  Let the Bonferroni critical
         point (tcp) be equal to the upper (1-a*) x 100th percentage point of the ^-distribution with (7V-
         p) degrees of freedom.

Step 13.  If any of the Bonferroni ^-statistics (t[) exceed the critical point tcp, conclude that  these
         compliance wells have population mean concentrations significantly greater than background
         and thus exhibit evidence  of possible contamination. Compliance  wells  for  which the
         Bonferroni ^-statistic does not exceed tcp should be regarded as similar to background in mean
         concentration level.

       ^EXAMPLE 17-1

     Lead concentrations in ground water at two background and four compliance wells were tested for
normality and homoscedasticity.   These data were found to be best fit by a lognormal distribution with
approximately equal variances.   The  two  background wells  also indicated insignificant log mean
differences. The natural  logarithms of these lead values are shown in the table below. Use the one-way
parametric  ANOVA to determine whether there  are any significant concentration  increases over
background in any of the compliance wells.
                                             17-6                                   March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests
Unified Guidance
                                     Log(Lead)log(ppb)
Date
Jan 1995
Apr 1995
Jul 1995
Oct 1995
Well Mean
Well SD

Well
4.06
3.99
3.40
3.83
Background
1 Well 2
3.83
4.34
3.47
4.22
3.82 3.96
0.296 0.395
X = 3
BG
89 s = 0.333
BG
Well 3
4.61
5.14
3.67
3.97
4.35
0.658

Compliance
Well 4 Well 5
3.53
4.54
4.26
4.42
4.19
0.453
Grand Mean
4.11
4.29
5.50
5.31
4.80
0.704
= 4.35
Well 6
4.42
5.21
5.29
5.08
5.00
0.143

       SOLUTION
Step 1.   Combine the two background wells into one group, so that the background average becomes
         3.89 log(ppb). Then n\, = 8, while n\ = 4 for each of the other four well groups. Note that the
         total sample size is N = 24 and/? = 5.

Step 2.   Compute the (overall) grand mean and the sample mean concentrations in each of the well
         groups using equations [17.1] and [17.2]. These values are listed (along with each group's
         standard deviation)  in the above table.

Step 3.   Compute the sum of squares due to well-to-well differences using equation [17.3]:

                SSwells =[s-(3.89)2+4-(4.35)2 + ... +4- (5.00)2]- 24- (4.35)2 =4.289

         This quantity has (5-1) = 4 degrees of freedom.

Step 4.   Compute the corrected total sum of squares using equation [17.4] with (N-\) = 23 df.

                       SStotal =[(4.06)2 +...+(5.08)2]-24.(4.35)2 =8.934

Step 5.   Obtain the within-well or error sum of squares by subtraction using equation [17.5]:

                                 SS    =8.934-4.289 = 4.646
                                   error

         This quantity has (N-p) = 24-5 = 19 degrees of freedom.

Step 6.   Compute the mean sums of squares using equations [17.6] and [17.7]:

                                   MS  „  =4.289/4 = 1.072
                                      wells        '

                                  MS   =4.646/19 = 0.245
                                      error        '
                                            17-7
        March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                       Unified Guidance

Step 7.   Construct the F-statistic and the one-way ANOVA table, using Figure 17-1 in Appendix D as
         a guide:
Source of Variation
Between Wells
Error (within wells)
Total
Sums of Squares
4.289
4.646
8.934
Degrees of
Freedom
4
19
23
Mean Squares
1.072
0.245
F-Statistic
F = 1.072/0.245
= 4.39
Step 8.   Compare the observed F-statistic of 4.39  against the critical  point taken as the upper 95th
         percentage point from the F-distribution with 4 and 19 degrees  of freedom. Using Table 17-1,
         this gives a value of ^95^19 = 2.90.  Since the F-statistic exceeds the critical point,  the
         hypothesis of equal well means is rejected,  and post-hoc Bonferroni t-test comparisons should
         be conducted.

Step 9.   Determine the number of individual contrasts needed. With four compliance wells, (p-1) = 4
         comparisons need to be made against background. Therefore, run each Bonferroni t-test at the
         a* = .05/4 = .0125 level of significance.

Step 10.  Compute the standard error of the difference between each compliance well average and the
         background mean using equation [17.8]. Since the number of observations is the same in each
         compliance well, the standard error in all four cases will be equal to:
                                  SE =0.2451 - + -| =0.303
                                    1   \      U  4)

Step 11.  Compute the Bonferroni ^-statistic for each compliance well using equation [17.9]:

                             Well 3:  f2= (4.35-3.89yo.303 = 1.52
                             Well 4:  r3 = (4.19-3.89)/0.303 = 0.99
                             Well 5:  r4 = (4.80-3.89)/0.303 = 3.00
                             Well 6:  r5=(5.00-3.89)/0.303 = 3.66

         Note  that  because Wells  1  and  2 jointly  constitute  background, the  subscripts  above
         correspond to the well  groups  and not the actual well numbers. Thus, subscript 2  in  the
         Bonferroni ^-statistic corresponds to Well 3, subscript 3 corresponds to Well 4, and so forth.

Step 12.  Look up the critical point from the ^-distribution in  Table 16-1 of Appendix D using a
         significance level of a* = .0125  and (N-p) = 19 df. This gives tcp = 2.433.

Step 13.  Compare each Bonferroni ^-statistic  from Step 11 against the critical point from Step  12.
         Because the ^-statistics at compliance wells 5 and 6 both exceed 2.433, while those at wells 3
         and 4 do  not,  conclude that the population averages  in compliance wells  5  and 6  are
         significantly higher than background.  -^
                                             17-8                                   March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance



     17.1.2  KRUSKAL-WALLIS TEST

       BACKGROUND AND PURPOSE

     The parametric one-way  ANOVA makes a key assumption that the data residuals are normally-
distributed. If this assumption  is inappropriate  or cannot be tested because of a large fraction of non-
detects, a non-parametric ANOVA can be conducted using the ranks of the observations rather than the
original observations. In Chapter 16, the Wilcoxon rank-sum  test  is presented  as  a non-parametric
alternative to the  Student's t-test when comparing two groups. The Kruskal-Wallis test is  offered as a
non-parametric alternative to  the one-way  F-test  when several  groups need to be  simultaneously
compared, for instance when assessing patterns of  spatial variability.  Instead of a test of means, the
Kruskal-Wallis tests differences among average population ranks equivalent to the medians.

     The Kruskal-Wallis test statistic, H, does not have the intuitive form of the Student's t-tesi. Under
the null  hypothesis that all the sample measurements come from  identical parent populations, the
Kruskal-Wallis statistic follows the well-known chi-square statistical distribution.  Critical points for the
Kruskal-Wallis test can be found as upper percentage points of the chi-square (Zi-adf) distribution in
Table 17-2 of Appendix D

     If H indicates  a significant difference between the populations, individual post-hoc comparisons
between each compliance well  and background need to be conducted if the Kruskal-Wallis is being used
for formal compliance testing.  Post-hoc contrasts  are not  generally necessary for identifying spatial
variability. Rather than Bonferroni ^-statistics, contrasts are based on  the  data ranks and approximately
follow a standard normal  distribution. The critical points for these contrasts can be obtained from the
standard normal distribution in Table 10-1 of Appendix D.

       REQUIREMENTS AND ASSUMPTIONS

     While  the  Kruskal-Wallis test does  not require the  underlying  populations  to be normally-
distributed, statistical independence  of the  data is still assumed. Under  the null  hypothesis  of no
difference among the groups,  the observations are  assumed to arise from  identical  distributions with
equal population variances (Hollander and Wolfe, 1999). However, the form of the distribution need not
be specified.

     A non-parametric ANOVA can be used in any situation that the parametric ANOVA can be used.
The minimum data requirements are similar:  the sample  size for each  group  in the Kruskal-Wallis
procedure should  generally be  at least four to five observations per group. Despite this similarity, it is
often true that non-parametric tests require larger sample sizes than their parametric test counterparts to
ensure a similar level of statistical power or efficiency. Non-parametric tests make fewer assumptions
concerning the underlying data distribution and  so more observations are often needed to make the same
judgment that would be rendered by a parametric test.  However, the greater efficiency of parametric
tests is only achieved when the parent population follows certain known statistical distributions. When
the distribution is unknown, non-parametric  tests may have much greater power than their parametric
counterparts.


                                             17-9                                   March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

     Even when a known statistical distribution is considered, rank-based non-parametric tests like the
Wilcoxon rank-sum and Kruskal-Wallis often perform reasonably well compared to the t-test and
ANOVA. The relative efficiency of two procedures is defined as the ratio of the sample sizes needed by
each to achieve a certain level of power against a specified alternative hypothesis. As sample sizes get
larger, the efficiency of the Kruskal-Wallis test relative to the parametric ANOVA approaches a limit
that depends on the underlying distribution of the data, but is  always at least 86 percent. This means
roughly that, in the worst case, if 86 measurements are available for a parametric ANOVA, only 100
sample values are needed to have an  equivalently  powerful Kruskal-Wallis test. In many cases, the
increase in sample size necessary to match the power of a parametric ANOVA is much smaller or not
needed at all. The efficiency of the Kruskal-Wallis test is 95% if the underlying data are really normal,
and can be  much larger than  100% in other cases (e.g.,  it is  150% if the data residuals follow a
distribution  called the double exponential). When  the efficiency exceeds  100%, the Kruskal-Wallis
actually needs fewer observations than the parametric ANOVA to achieve a certain power.

     These results imply that the Kruskal-Wallis test is reasonably powerful for detecting concentration
differences despite the fact that the original data have been  replaced by their ranks. The test can be used
with fair success  even when the data are normally-distributed and the Kruskal-Wallis is not needed.
When  the data are not normal or  a normalizing transformation  cannot be found, the Kruskal-Wallis
procedure tends to be more powerful for detecting differences than the usual parametric approach.

       ADJUSTING FOR TIED  OBSERVATIONS

     The Kruskal-Wallis procedure will frequently be used when the sample data contain a significant
fraction of non-detects.  However, the presence of non-detects prevents a unique and complete ranking
of the concentration values since the exact values of non-detects are unknown.

     To address this problem, two steps are necessary.  Since they cannot be uniquely ranked, all non-
detects are to be  treated statistically as 'tied' values. This is an imperfect remedy, since  non-detects
represent left-censored values and  are not necessarily tied. Unfortunately,  there is no straightforward,
easily implemented alternative to the Kruskal-Wallis  for comparing three or more groups containing left-
censored observations, unlike the Tarone-Ware alternative  to the Wilcoxon rank-sum test discussed in
Chapter 16. So in the presence of ties (e.g., non-detects  or quantified  concentrations rounded to the
same  value), all tied observations should receive the same midrank (discussed in Section  16.3). This
rank is computed as the average of the ranks that would be given  to each group of ties if the tied values
actually differed by a tiny amount and could be ranked.

     To account for multiple reporting limits, all non-detects  should be treated as if censored at the
highest reporting limit [RL] in the overall sample. Thus, a non-detect reported as <5 would be treated as
'tied'  with a non-detect reported as <1, due  to the impossibility of knowing which  value is actually
larger. The only exception to this strategy  is when laboratory qualifiers can be used to rank some non-
detects as probably greater in magnitude than  others. A reasonable strategy discussed in Section 16.3 is
to group all "U" values as the lowest set of ties, other non-detects as a higher set of ties, and to rank all
"J" and/or "E" values according to their estimated concentrations. In situations where estimated values
for J and E samples are not provided, treat these measurements as the highest group of tied non-detects.
Always give the highest ranks to explicitly quantified or estimated concentration measurements.
                                             17-10                                   March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests
                                                                   Unified Guidance
     The  second step for handling ties is to compute the Kruskal-Wallis  statistic as described below,
using for each tied value its corresponding midrank. Then an adjustment to the Kruskal-Wallis statistic
needs to be made to account for the presence of ties. This adjustment requires computation  of the
formula:
                                  H  =HI
   (8
i-Z-
                                                    t3-l
                                                                               [17.10]
where g equals the number of distinct groups of tied observations, N is the total sample size across all
groups, and t\ is the number of observations in the rth tied group. Unless there are a substantial number
of ties in the overall dataset, the adjustment in equation [17.10] will tend to be small. Still, it is important
to properly account for the presence of tied values.

       PROCEDURE

Step 1.   To  run  the Kruskal-Wallis test,  denote the total  sample size across all well groups by N.
         Temporarily combine all the data into one group and rank the observations from smallest to
         largest.  Treat all non-detects as tied at the lowest possible concentration value, unless using
         lab  qualifiers to distinguish  between  'undetected'  and other non-detects. Combine  all
         background wells into a single group where appropriate.  Denote this set of background data as
         group 1. Then let E^  denote the yth rank from  the rth  well group, and let k equal the total
         number of groups (i.e., one group of  background values and (£-1) groups of compliance
         wells).

Step 2.   Compute the sum of the ranks and the average  rank in each well group, letting n\ equal  the
         sample size in the /'th group and using the following formulas:
                                          R =
                                                                               [17.11]
                                          R =
                                            "  n
                                                                               [17.12]
Step 3.   Calculate the Kruskal-Wallis test statistic H and the adjustment for ties, if necessary, using
         equation [17.10], where H is given by:
Step 4.
                                H =
                                        12
                                                                               [17.13]
Given the level of significance (a), determine the Kruskal-Wallis critical point (%2cp) as the
upper (1-oc) x 100th percentage point from the chi-square distribution with (k-V) degrees of
freedom (Table 17-2 in Appendix D). Usually a is set equal to 0.05, so that the upper 95th
percentage point of the chi-square distribution is needed.
                                             17-11
                                                                           March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

Step 5.   Compare the Kruskal-Wallis test statistic, H, against the critical point ^.  If H is no greater
         than the critical  point, conclude  there is insufficient  evidence  of significant  differences
         between any of the well group populations. If H>^p, however, conclude there is a significant
         difference between at least one  pair of the well  groups. Post-hoc comparisons  are  then
         necessary to determine whether any of the compliance wells significantly exceeds background
         (note that post-hoc comparisons are not necessary if using the Kruskal-Wallis test to merely
         identify spatial variability).

Step 6.   In the case of a significant //-statistic that exceeds the critical point in Step 5, determine which
         compliance wells  have  elevated concentrations compared to background.  This  is done by
         comparing each compliance well against background, using a set of contrasts (as described for
         the parametric one-way ANOVA in Section 17.1.1).

         To keep the test-wise  or  experiment-wise  false positive rate close  to the targeted  (i.e.,
         nominal) rate of 5%, the individual contrasts should be set up as follows: Given (£-1) separate
         background-compliance contrasts, if(£-l) < 5, run each contrast at a significance level equal
         to a* = .05/(&-l). However, if (&-1) > 5, run each contrast at a significance level equal to a* =
         .01. Note that when there are more than  5 downgradient wells, this last provision will tend to
         raise the overall false positive rate above 5%.

Step 7.   Since the background data is the first  well group, the number of background observations is
         equal to n\. For each of the remaining (£-1) well groups, compute the approximate rank-based
         standard error of the difference between each compliance well and background using equation
         [17.14]:
                                                                                        [\7.\4]
                                                                                        L     J
Step 8.   Let the average background rank be identified as/?fe. Compute the post-hoc Z-statistic for each
         of the (k-l) compliance wells for /' = 2 to k, dividing the  standard error in step 7 into the
         difference between the average rank at the compliance well and the background rank average,
         as shown below:

                                       Z,=(R,-Rb)/SE,                                [17.15]

Step 9.   The Z-statistic in equation [17.15] has an approximate standard normal distribution under the
         null hypothesis that the /th compliance well is identical in  distribution to background.  The
         critical point (zcp) can be found as the upper (1-oc) x 100th percentage point of the normal
         distribution in Table 10-1 of Appendix D

Step 10.  Compare the post-hoc Z-statistics for each of the (k-l) compliance wells against the critical
         point  (zcp). Any Z-statistic that  exceeds the critical point provides significant evidence of an
         elevation over background in that compliance well at the a level of significance.
                                             17-12                                   March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

     ^EXAMPLE 17-2

     Use the non-parametric Kruskal-Wallis test on the following data to determine whether there is
evidence of possible toluene contamination at a significance level of a = 0.05.
Month
1
2
3
4
5
Toluene Concentration (ppb)
Background Wells Compliance Wells
Well 1 Well 2 Well 3 Well 4 Well 5
<5
7.5
<5
<5
6.4
<5
<5
<5
<5
<5
<5
12.5
8.0
<5
11.2
<5
13.7
15.3
20.2
25.1
<5
20.1
35.0
28.2
19.0
       SOLUTION
Step 1.   Since non-detects account for 48% of these data, it would be very difficult to verify the
         assumptions of normality and equal variance necessary for a parametric ANOVA. Use the
         Kruskal-Wallis test instead, pooling both background wells into one group and treating each
         compliance well as a separate group. Note that N = 25 and k= 4.

         Compute ranks for all the data including tied  observations  (e.g., non-detects)  as  in the
         following table. Note that each non-detect is given the same midrank, equal to the average of
         the first 12 unique ranks.
Month
1
2
3
4
5
Group Size
Rank Sum

Rank Mean

Background Wells
Well 1 Well 2
6.5
14
6.5
6.5
13





6.5
6.5
6.5
6.5
6.5
H! = 10
R = 79
i.
R = 7.9
i.
Toluene Ranks
Compliance Wells
Well 3 Well 4 Well 5
6.5
17
15
6.5
16
n2 = 5
R =61
2.
R = 12.2
2.
6.5
18
19
22
23
n3 = 5
R = 88.5
3.
R = 17.7
3.
6.5
21
25
24
20
n4 = 5
R = 96.5
4.
R = 19.3
4.
Step 2.   Calculate the sum and average of the ranks in each group using equations [17.11] and [17.12].
         These results are given in the above table.

Step 3.   Compute the Kruskal-Wallis statistic//using equation [17.13]:
H- 12
25-26
792 612 i
I I
10 5
38. 52 96.52
I
5 5
(T. . 7f\ i n sfi

                                            17-13                                  March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

         Also compute the adjustment for ties with equation [17.10]. There is only one group of distinct
         tied observations — the non-detects — containing  12 samples. Thus, the adjusted Kruskal-
         Wallis statistic is given by:
                               H  = 10.56
1-
                                                253-:
= 11.87
Step 4.   Determine the critical point of the Kruskal-Wallis test: with a = .05, the upper 95th percentage
         point of the chi-square distribution with (£-1) = 4-1 = 3 degrees of freedom [df\ is needed.
         Table 17-2 of Appendix D gives ^ = zl5,3 = 7.81.

Step 5.   Since  the observed Kruskal-Wallis statistic of 11.87 is  greater than the  chi-square critical
         point, there is evidence of significant differences between the well groups. Therefore, post-hoc
         pairwise comparisons are necessary.

Step 6.   To determine the significance level appropriate for post-hoc comparisons, note there are three
         compliance wells that need to be tested against background. Therefore, each of these contrasts
         should be run at the a* = 0.05/3 = 0.0167 significance level.

Step 7.   Calculate the standard error of the difference  for the three contrasts using equation [17.14].
         Since  the sample size at each compliance well  is five, the SE will be  identical for each
         comparison, namely,
                                        125-26 f 1   ^_
                                          12  UO  5} ~

Step 8.   Form the post-hoc Z-statistic for each contrast using equation [17.15]:

                              Well 3: Z2 = (l2.2-7.9)/4.031 = 1.07
                              Well 4: Z3 = (l7.7 - 7.9)/4.031 = 2.43

                              Well 5: Z4 = (l9.3-7.9)/4.031 = 2.83

Step 9.   Find  the upper (1-a*)  x 100th percentage point  from the standard  normal distribution in
         Table 10-1  in  Appendix  D.  With a*  =  .0167,  this  gives  a critical  point  (by linear
         interpolation) of zcp = 2.9333 = 2.127.
Step 10.  Since the Z-statistics at wells 4 and 5 exceed the critical point, there is significant evidence of
         increased concentration levels at wells 4 and 5, but not at well 3.  -^

17.2 TOLERANCE LIMITS

     A tolerance interval is a concentration range designed to contain a pre-specified proportion of the
underlying  population from which the statistical  sample is  drawn  (e.g., 95  percent of  all  possible
population  measurements). Since the  interval is constructed from random sample  data,  a tolerance
interval is expected to contain the specified population proportion only with a certain level of statistical

                                             17-14                                   March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

confidence. Two coefficients are thus associated with any tolerance interval. One is the population
proportion that the interval is supposed to contain, called the coverage (y). The second is the degree of
confidence with which the  interval reaches the specified coverage. This is  sometimes known as the
tolerance coefficient or more simply, the confidence level (1-oc). A tolerance interval with 95% coverage
and a tolerance coefficient of 90 percent is constructed to contain, on average,  95% of the distribution of
all possible population measurements with a confidence probability of 90%.

     A tolerance limit is a one-sided tolerance interval. The upper  limit is typically of most interest in
groundwater monitoring. Tolerance limits  are  a standard statistical method that  can  be useful in
groundwater data analysis, especially as an alternative to ^-tests or  ANOVA for interwell testing.  The
RCRA regulations allow greater flexibility in the choice of a when using tolerance and prediction limits
and  control charts, so a larger  variety of data configurations may  be  amenable  to  one  of these
approaches. The Unified  Guidance  still recommends prediction limits or control charts  over tolerance
limits for formal compliance testing in detection monitoring, and  confidence intervals over tolerance
limits in compliance/assessment monitoring when a background standard is needed.

     An  interwell tolerance  limit constructed  on background  data is  designed to cover all but a small
percentage of the background population measurements.  Hence background observations should rarely
exceed the upper tolerance  limit. By the  same token, when testing a null  hypothesis  (Ho) that the
compliance point  population is identical to background, compliance point measurements also  should
rarely exceed the upper tolerance limit, unless HQ is false. The upper tolerance limit thus gauges whether
or not  concentration measurements sampled from compliance point wells  are too extreme relative to
background.
     17.2.1  PARAMETRIC TOLERANCE LIMITS

       BACKGROUND AND PURPOSE

     To test the null  hypothesis  (Ho)  that a compliance point  population  is identical to that  of
background, an  upper tolerance limit with high  coverage  (y) can  be constructed  on the sample
background data. Coverage of 95% is usually recommended.  In this case, random observations from a
distribution identical to background should exceed the upper  tolerance limit less than 5% of the time.
Similarly, a tolerance coefficient or confidence level of at least 95% is recommended. This gives 95%
confidence that the (upper) tolerance limit will contain at least  95% of the distribution of observations in
background  or in any distribution  similar to  background. Note that  a  tolerance coefficient of 95%
corresponds  to choosing a significance level (a) equal to 5%. Hence, as  with a one-way ANOVA, the
overall false positive rate for a tolerance interval is set to approximately  5%.

     Once the limit is constructed on background, each compliance point  observation (perhaps from
several different wells) is compared to the upper tolerance limit. This is  different from the comparison of
sample means in an ANOVA test. If any compliance point measurement exceeds the limit, the well from
which it was drawn is flagged as showing a significant increase over background. Note that the factors K
used to adjust the width of the tolerance interval (Table 17-3 in Appendix D) are designed to provide at
least 95% coverage of the parent population. Applied over many data sets, the average coverage of these
intervals will often be close to 98%  or more (see Guttman,  1970). Therefore,  it would be unusual to find

                                            17-15                                   March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

more than 2 or 3 samples out of every 100 exceeding the tolerance limit under the null hypothesis. This
fits with the purpose behind the use of a tolerance interval, which is to establish an upper limit on
background that will rarely be exceeded, unless some change in the groundwater causes concentration
levels to rise significantly at one or more compliance points.

      Testing a large number of compliance point samples against such a background tolerance limit
even under conditions of no releases practically ensures a few measurements will occasionally exceed
the limit. The Unified Guidance therefore recommends that tolerance limits be used in conjunction with
verification resampling of those wells suspected of possible contamination, in order to either verify or
disconfirm the initial round of sampling and to avoid false positive results.

       REQUIREMENTS AND ASSUMPTIONS

     Standard parametric tolerance limits assume normality of the sample background  data used to
construct the limit. This assumption is critical to the statistical validity of the method, since a tolerance
limit with high coverage can be viewed as an estimate of a quantile orpercentile associated with the  tail
probability of the underlying distribution.  If  the  background sample  is non-normal, a normalizing
transformation should be sought. If a suitable transformation is found, the limit should be constructed on
the transformed measurements and can then be  back-transformed to the raw concentration  scale prior to
comparison against individual compliance point values.

     If no transformation will work, a non-parametric tolerance limit should be  considered instead.
Unfortunately, non-parametric tolerance limits  generally require a much larger number of observations
to provide the same levels of coverage and confidence as a parametric limit.  It is recommended that a
parametric model be fit to the data if at all possible.

     A tolerance limit can be computed with as few as three observations from background.  However,
doing so results in a high upper tolerance limit with limited  statistical power for detecting increases over
background.  Usually,  a background  sample size of at least eight  measurements will be needed to
generate an  adequate tolerance  limit. If multiple  background wells  are  screened in  equivalent
hydrostratigraphic positions and the data can reasonably be combined (Chapter 5), one should consider
using pooled background data from multiple wells to increase the background sample size.

     Like many tests  described  in the Unified Guidance,  tolerance limits as applied  to groundwater
monitoring assume stationarity  of the well field  populations both temporally (i.e., over time) and
spatially. The data also needs to be statistically independent.  Since an adequately-sized background
sample  will have to  be  amassed over time (in part to maintain enough temporal spacing between
observations so that independence can be assumed), the background data should be checked for apparent
trends or seasonal effects. As long the background mean is stable over time, the amassed data from a
longer span  of sampling  will provide a better  statistical  description of the underlying background
population.

     As a primarily interwell technique, tolerance limits should only be utilized when there is minimal
spatial variability. Explicit  checks for spatial  variation should  be  conducted using box plots and/or
ANOVA.

     In the usual test setting, one new compliance point  observation from each distinct well is compared
against  the tolerance limit during  each statistical  evaluation. Under the  null hypothesis of identical
                                             17-16                                    March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

populations, the compliance point measurements are assumed  to  follow  the  same distribution  as
background. Further, the compliance data are assumed to be mutually statistically independent. Such
assumptions are almost impossible to check with only one new value per compliance well. However,
periodic checks of the key assumptions are recommended after accumulating several sampling rounds of
compliance data.

       PROCEDURE

Step 1 .   Calculate the mean x , and the standard deviation s, from the background sample.

Step 2.   Construct the one-sided upper tolerance limit as

                                                                                      [17.16]
         where K(n, y,l-a}is the one-sided normal tolerance factor found in Table 17-3 of Appendix D
         associated with a sample size of w, coverage coefficient of y, and confidence level of (1-a).

         Equation [17.16] applies to normal data. If a transformation is needed to normalize the sample,
         the tolerance limit needs  to be constructed on the transformed measurements and the limit
         back-transformed to the original concentration scale. If the  limit was constructed, for example,
         on the logarithms of the  original observations, where y  and sy are the log-mean and log-
         standard deviation, the tolerance limit can be back-transformed to the concentration scale by
         exponentiating the limit.  The tolerance limit is computed as:

                                                                                      [17.17]

Step 3.   Compare each observation from the compliance well(s) to the upper tolerance limit found in
         Step 2. If any observation exceeds the tolerance limit, there is statistically significant evidence
         that  the  compliance  well concentrations  are  elevated  above background.  Verification
         resampling should be conducted to verify or disconfirm the initial result.

       ^EXAMPLE 17-3

     The table below consists of chrysene concentration data (ppb) found in water samples obtained
from two background wells (Wells 1 and 2) and three compliance wells (Wells 3, 4, and 5). Compute the
upper tolerance limit on background for coverage of 95% with 95% confidence and determine whether
there is evidence of possible contamination at any of the compliance wells.
Month
1
2
3
4
Mean
SD
Well 1
19.7
39.2
7.8
12.8
19.88
13.78
Chrysene
Well 2
10.2
7.2
16.1
5.7
9.80
4.60
Concentration
Well 3
68.0
48.9
30.1
38.1
46.28
16.40
(ppb)
Well 4
26.8
17.7
31.9
22.2
24.65
6.10
Well 5
47.0
30.5
15.0
23.4
28.98
13.58
                                            17-17                                  March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                     Unified Guidance
       SOLUTION
Step 1.   A Shapiro-Wilk test of normality on the pooled set of eight background measurements gives
         W= 0.7978 on the original scale and W= 0.9560 after log-transforming the data, suggesting
         that the data are better fit by a lognormal distribution. Therefore, construct the tolerance limit
         on  the  logged  observations, listed below  along  with the  log-means and  log-standard
         deviations.
Month
1
2
3
4
Mean
SD
BG Mean
BG SD
Well 1
2.981
3.669
2.054
2.549
2.813
0.685


Well 2
2.322
1.974
2.779
1.740
2.204
0.452
2.509
0.628
Log Chrysene log(ppb)
Well 3
4.220
3.890
3.405
3.640
3.789
0.349


Well 4
3.288
2.874
3.463
3.100
3.181
0.253


Well 5
3.850
3.418
2.708
3.153
3.282
0.479


Step 2.   Compute the upper tolerance limit on the pooled background data using the logged chrysene
         concentration data. The tolerance factor for a one-sided upper normal tolerance limit with 95%
         coverage and 95% probability and n = 8  observations is equal to (from Table  17-3  of
         Appendix D) K=  3.187. Therefore, the upper tolerance limit is computed using equation
         [17.17] as:
                          TL = exp[2.509 + 3.187 x 0.628J = 90.96 ppb

Step 3.   Compare the measurements at each compliance well to the upper tolerance limit, that is TL =
         90.96 ppb. Since none of the original chrysene concentrations exceeds the upper TL, there is
         insufficient evidence of chrysene contamination in these data. ~4
     17.2.2 NO N-PARAMETRIC TOLERANCE INTERVALS

       BACKGROUND AND PURPOSE

     When an assumption of normality cannot be justified especially with a significant portion of non-
detect observations, the use  of non-parametric  tolerance intervals should be considered. The  upper
tolerance limit in a non-parametric setting is usually chosen as  an order statistic of the sample data
(Guttman, 1970), commonly the maximum value  or maybe the second or third largest value observed.

     Because the maximum observed background value is often taken  as the upper tolerance limit, non-
parametric tolerance intervals are easy to construct and use. The sample data needs to be ordered, but no

                                            17-18                                  March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                     Unified Guidance

ranks need be assigned to the concentration values other than to determine the largest measurements.
This also means that non-detects do not have to be uniquely ordered or handled in any special manner.

     One advantage to using a maximum concentration instead of assigning ranks to the data (Wilcoxon
rank-sum  or  Kruskal-Wallis tests) is that  non-parametric tolerance intervals are reflective  of actual
concentration magnitudes. Another advantage is that unless all the background data are non-detect, the
maximum value  will be a detected concentration leading to a well-defined upper tolerance limit. If all
the sample data are non-detect, an RL (e.g., the lowest achievable quantitation limit [QL]) may serve as
an approximate upper tolerance limit.

       REQUIREMENTS AND ASSUMPTIONS

     Unlike parametric tolerance intervals, the desired coverage (y) or confidence level (1- a) cannot be
pre-specified  using  a non-parametric limit.  Instead,  the  achieved  coverage and/or confidence level
depends entirely  on the background sample size (n) and the order statistic chosen as the upper tolerance
limit (e.g., the maximum value).  Guttman (1970) has shown that the coverage of the limit follows a beta
probability density with cumulative distribution:
                        ,          \   (•        w+           ,    ,m_i
                      It(n-m + \,m)=  \ —f - ^ - ^—r^""" U - « )   du               [17.18]
                       ^          J        -                ^    J
where n = sample  size and m =  [(w+l)-(rank of upper tolerance limit value)]. If the background
maximum is selected as the tolerance limit, its rank is equal to n and so m = 1. If the second largest value
is chosen as the limit, its rank would be equal to (w-1) giving m = 2.

      As a  non-parametric  procedure,  no  distributional  model  must  be fit  to  the background
measurements. It is  assumed, however, that the compliance point data follow the same distribution as
background — even if unknown — under the null hypothesis. Even though no distributional model is
assumed, order statistics of any random sample follow certain probability laws as noted above. Since the
beta distribution is closely related to the more familiar binomial distribution, Guttman showed that in
order to construct a non-parametric tolerance interval with at least y coverage and (1-a) confidence
probability, the number of (background) samples should be chosen such that:
If the background maximum is selected as the upper tolerance  limit, so that m = 1, this inequality
reduces to the simpler form

                                         \-y">\-a.                                 [17.20]

     Table 17-4 in Appendix D provides minimum coverage levels with 95% confidence for various
choices of n, using either the maximum sample value or the second largest measurement as the tolerance
limit. As an example, with n = 16 background measurements, the minimum coverage is y = 83% if the
background maximum is designated as the upper tolerance limit and y = 74% if the tolerance limit is

                                            17-19                                  March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

taken as the second largest background value. In general, Table 17-4 of Appendix D illustrates that if
the underlying distribution is  unknown, more background samples  are needed  compared to the
parametric setting in order to construct a tolerance interval with sufficiently high coverage. Parametric
tolerance intervals do not require  as many background samples precisely  because the  form of the
underlying distribution is assumed to be known.

     An alternate way  to  construct an  appropriate tolerance  limit is to calculate  the  maximum
confidence level for various choices of n guaranteeing at least 95% coverage. With  n = 8 background
measurements, the approximate confidence level is at most 34% when the largest value is taken as the
tolerance limit and only 6% if the second largest value is taken as  the tolerance limit. Clearly, it is
advantageous to fit a parametric distributional model to the data if at all possible unless n is fairly large.

     Although non-parametric tolerance  limits do  not require  an  assumption  of normality,  other
assumptions  of tolerance limits  apply  equally  to  the parametric  and  non-parametric  versions.
Specifically,  the  sample  data should  be  statistically  independent  and  show   no evidence  of
autocorrelation, trends, or seasonal effects in background. Applied as an interwell test, there should also
be minimal to no natural on-site spatial variation.

     By construction,  outliers in background can be a particular problem for non-parametric tolerance
limits, especially if the background maximum is chosen as the upper limit.  A limit based on a large,
extreme outlier will  result in  a test having  little power to  detect increases in compliance wells.
Consequently, the background sample should be screened ahead of time for possible  outliers (Chapter
12). Confirmed outliers should be removed from the data set before setting the tolerance limit.

     An important caveat to this advice is that almost all statistical outlier tests depend crucially on the
ability to fit the remaining data (minus the suspected outlier(s)) to a known statistical  distribution. In
those cases where a non-parametric tolerance limit is selected because of a large fraction of non-detects,
fitting the data to a distributional model may be difficult or impossible, negating formal outlier tests. As
an  alternative, the non-parametric upper tolerance limit could be set to a different order statistic in
background (i.e., other than the maximum), to  provide some insurance against possible large outliers.
This strategy will work provided there are enough background measurements to allow for adequately
high coverage and confidence in the resulting limit.

       PROCEDURE

Step 1.  Sort the  set of background data into ascending order and choose either the largest or second
         largest measurement  as  the upper  TL.  Use  Table 17-4 in Appendix  D  to determine the
         coverage y associated with 95% or 99% confidence. Note  also that if the largest or second
         largest measurement is a non-detect, the upper tolerance limit should be set to the RL  most
         appropriate to the data (e.g., the lowest achievable practicable quantification limit [PQL]).

Step 2.  Compare each compliance  point measurement against the upper tolerance limit. Identify
         significant evidence of possible contamination at any compliance well in which one or more
         measurements exceed the upper tolerance limit. If the upper tolerance limit equals the RL, a
         violation should be flagged anytime a detected value is quantified above the RL.

Step 3.  Because the  risk of false positive errors is greatly increased if either the confidence level or
         coverage drop substantially below 95%, both of these parameters should be routinely reported
                                             17-20                                   March  2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests
Unified Guidance
         and noted as being below the target levels. One should also strongly consider comparing one
         or more verification resamples against the upper tolerance limit before identifying a clear
         violation.

     ^EXAMPLE 17-4

     Use the following copper background data to establish a non-parametric upper tolerance limit and
determine if either compliance well shows evidence of copper contamination.

Month
1
2
3
4
5
6
7
8

Well 1
<5
<5
7.5
<5
<5
<5
6.4
6.0
Copper
Background Wells
Well 2
9.2
<5
<5
6.1
8.0
5.9
<5
<5
Concentration
Well 3
<5
5.4
6.7
<5
<5
<5
<5
<5
(ppb)
Compliance
Well 4




6.2
<5
7.8
10.4
Wells
Well 5




<5
<5
5.6
<5
       SOLUTION
Step 1.   The pooled background data in Wells 1, 2, and 3 have a maximum observed value of 9.2 ppb.
         Set the 95% confidence  upper tolerance limit equal to this value. Because 24 background
         samples are available, Table 17-4 in Appendix D indicates that the minimum coverage is
         equal to 88%. To increase either the coverage, more background samples would have to be
         collected.

Step 2.   Compare each sample in compliance Wells 4 and 5 to the upper tolerance limit. Since none of
         the measurements at Well 5 is above 9.2 ppb, while one sample from Well  4 is above the
         limit, conclude that there may be significant evidence of copper contamination at Well 4 but
         not Well 5.

Step 3.   Note that with only 88% coverage and 24 background  samples, the risk of a false positive
         result is more than 10%. Well 4 should be resampled to determine whether the exceedance is
         replicated. -4
17.3 TREND TESTS

     The Unified Guidance recommends trend testing as an intrawell alternative to prediction limits or
control charts when those methods are not suitable. Prediction limits and control charts (as well as Wests
and ANOVA)  all involve a  comparison of compliance and background populations  under  the key
assumption that the  underlying concentration  distributions  are  stationary  over time. That  is,  the
populations are presumed to have stable (i.e., roughly constant) means over the period of sampling prior
to statistical evaluation.
                                           17-21
        March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

     Unfortunately, there is no guarantee that groundwater populations will remain stable during long-
term monitoring.   Because sampling at many sites is generally done on a quarterly,  semi-annual, or
annual basis, it will generally take one to two years or more to collect enough background data to run the
statistical tests discussed in the Unified Guidance. Over this length of time, the statistical characteristics
of groundwater may or may not change in significant ways.

     If background groundwater conditions are in a state of flux, trend tests provide  a  significant
advantage over both intrawell prediction limits and control charts.  Both of the latter methods involve a
designation of some portion of the historical sampling record as the intrawell background for a given
compliance well.  Ideally, this intrawell background  should  consist  of  measurements  known to be
uncontaminated and which represent a random sample from a stable underlying population, just as with
^-tests and ANOVA. If the mean and/or standard deviation of the underlying population changes while
intrawell background is being compiled, results of either prediction  limit or control chart tests against
more recently collected data can be severely biased or altogether inaccurate.

     One drawback to the  Shewhart-CUSUM control charts presented in Chapter 20 is that  they are
somewhat sensitive to the  parametric  assumption of underlying  normality. If the measurements are
lognormal rather than normal,  for instance, the nominal performance characteristics (i.e., Type I error
rate and statistical  power) of control charts are significantly affected. By the same token,  control charts
are impacted if the intrawell background contains a large fraction of non-detects. Non-detect adjustments
can sometimes be made to the baseline data via methods discussed in Chapter 15, but if a normalizing
transformation or adjustment is not successful, no straightforward non-parametric control chart exists.

     Consequently, neither prediction limits nor control charts are appropriate for every circumstance
where  an  intrawell  comparison  may  be  warranted or necessary. Thus,  the Unified  Guidance
recommends that users consider trend testing as an alternative to prediction limits or control charts
when those methods are not suitable as intrawell techniques. Tests for trend are specifically designed
to identify those groundwater populations whose mean concentrations are not stationary over time, but
rather are increasing (or  decreasing) by measurable amounts. Ultimately, the goal of any reasonable
detection  or  compliance/assessment  monitoring  program  is to  determine  whether  or  not the
concentration levels of key contaminants or indicator parameters have significantly increased during the
period of monitoring and,  if so,  whether the  increase  is attributable to facility waste  management
practices.

     The detection of trends is a complex subject. Whole textbooks are devoted to the more general
topic of time series analysis, including the identification and modeling of time trends — step functions,
linear and quadratic trends,  exponential growth, etc. The Unified Guidance only attempts to identify the
simplest kind of linear increases, not the specification or testing of more complex models. The methods
described below are all designed to effectively test for (increasing) linear trends, though  they will also
identify simple increases over time when a trend is present but does not follow a strictly linear pattern.

     The Unified Guidance recommends using trend tests in detection monitoring to measure the extent
and nature of  an apparent concentration increase, especially to determine whether or not the  increase
occurs consistently over time. Two  questions  are of particular interest:  1)  is  there  a statistically
significant, (positive)  trend over the period  of monitoring? and 2) what  is the nature (i.e., slope and
intercept) of the trend? By identifying a positive trend, one can show that contaminant levels have gotten
worse compared to early measurements from the well being tested. Furthermore, by measuring the nature
                                             17-22                                   March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

of the trend, including the average rate of increase  per unit of time, one can estimate how rapidly
concentration levels are increasing and the current mean- or median-level magnitude of contamination.
Such information can provide an invaluable portrait of the changes occurring on-site and probably offers
the most compelling  evidence — under these conditions — for demonstrating that the basic null
hypothesis of detection monitoring has been violated.
     17.3.1  LINEAR REGRESSION

       BACKGROUND AND PURPOSE

     The most common way to measure a linear trend is to compute a linear regression of concentration
data when plotted against the  time or date of sample collection. By way of interpretation, each point
along a linear regression trend line is an estimate of the true mean concentration at that point in time.
Thus, a linear regression can be used to assess whether or not the population mean at a compliance well
has significantly increased or decreased.

     Linear regression is a standard technique in statistics textbooks and many data analysis software
packages. It is more generally applicable to linear relationships between any pair of random variables
and not simply to time trends. Good references for performing linear regression and for checking and
verifying its assumptions include Draper and Smith (1998) and Cook and Weisberg (1999).

     Unlike prediction limits or control charts which are constructed using only the background data,
trend tests including linear regression are computed with all available earlier and more recent data at the
compliance well  of interest. One then might incorrectly  assume that a  comparison against intrawell
background is not being conducted. But an intrawell comparison does occur with a trend test.  Statistical
identification of a structured pattern of increase from the first portion of the sampling record to more
recent  data indicates that concentration levels are no longer  similar to intrawell background, but have
risen more than expected by chance.

     Statistical identification of a positive trend involves testing the estimated slope coefficient from the
linear  regression  trend  line.  A  specially  constructed t-test is  used to  make this determination,  as
described below. If this test is significant, the slope is judged to be different from zero, indicating that a
change in concentration levels has occurred over the period of sampling represented by the data set.

       REQUIREMENTS AND ASSUMPTIONS

     Linear regression as a parametric statistical technique makes a number of underlying assumptions.
Among the most  important of these are that the regression residuals (i.e., the  difference between each
concentration measurement and  its  predicted value from the regression  equation) are approximately
normal in distribution, homoscedastic (i.e., equal in  variance at different times and for different mean
concentration levels), and statistically independent. Significant skewness or the presence of outliers can
bias or invalidate the results of a trend test based on linear regression. Furthermore, standard linear
regression methods do not account for non-detects or missing data values at selected sampling events.

     Because the key assumptions for linear  regression depend not  on the original measurements but
rather  on the regression  residuals, a  tentative trend line  needs to  first be constructed before its

                                             17-23                                   March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                       Unified Guidance

assumptions can be checked. Once a linear regression on time is fitted to the data, the residuals around
the trend line need to be computed and then tested for normality, apparent skewness, and equal variance
over time. This last assumption is particularly important to testing whether the slope of an apparent trend
is statistically different from zero (a zero slope indicating that well concentrations have not changed over
time).

     Inferences around a linear regression are generally appropriate when three conditions hold: 1) the
residuals from the regression are approximately normal  or at least reasonably symmetric in distribution;
2) a scatter plot of residuals  versus concentrations indicates  a scatter cloud of essentially uniform
vertical thickness or width (i.e.,  the  scatter cloud does not tend to increase in width with the level of
concentration which would suggest a proportional  effect between the underlying population mean and
variance); and 3) a scatter plot of residuals versus time also exhibits a uniformly thick scatter cloud. If
the thickness or width is substantially different at distinct time points, the assumption of equal variances
over time may not be true.

     If any of these conditions is substantially violated,  it may indicate that the basic trend is either non-
linear or the magnitude of the variance  is not independent of the mean concentration level and/or the
time of sampling. One possible remedy is to try  a transformation of the concentration  data and re-
estimate  the linear regression. This will change the interpretation of the estimated regression from a
linear trend of the form y = a + bt,  where y and t represent concentration and time  respectively, to a
non-linear pattern. As an example, if the concentration data are log-transformed, the regression equation
will  have the form  log>' = a + ^.  Back-transformed  to  the  original  concentration scale, the  trend
function will then have the form y = expfa + bt\

     In transforming the regression data this way, the estimated trend in the concentration domain (after
back-transforming) no longer represents the original mean.  Rather, the transformation induces a bias
when converted back to the raw-scale  data. If  a log transformation is used, for instance, the back-
transformed trend line will represent the raw-scale geometric mean and not the arithmetic mean. As with
Student's t-tests on lognormal data (Chapter 16), demonstrating that the geometric mean is increasing
also implies that the arithmetic mean has risen so long as the regression residuals are homoscedastic.

     A minimum of 8 to 10 measurements is generally  necessary to compute  a linear  regression,
especially to estimate the variance around the trend line (known as the mean squared error or MSB).
The  regression residuals should be  statistically independent, an assumption that  can be approximately
verified via one of the autocorrelation tests of Chapter 14.

     One last assumption is that there  should  be few if any non-detects when  computing a linear
regression. As a matter of common sense, a significant increasing or decreasing trend should be based on
reliably quantified measurements. If this  is not the case,  the user  should check to see whether the "trend"
may be  an artifact induced  by  changes in detection and/or quantitation  limits over  time.  The
concentration levels of a series of non-detects may appear to be decreasing, for instance, simply because
analytical methods have improved over  the years leading to lower RLs. Such artifacts of plotting and
data reporting should not be considered real trends.

     When the  assumptions  of linear  regression  cannot  be verified  at least approximately, a non-
parametric trend method should be considered instead.  Sections 17.3.2 and 17.3.3 discuss the Mann-

                                              17-24                                    March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

Kendall test for trend and the Theil-Sen trend line. These methods can be particularly valuable when
constructing trends on data sets containing non-detects.

       PROCEDURE

Step 1.   Construct a time  series plot of the compliance point measurements. If a discernible trend is
         evident, compute a linear regression of concentration against sampling date (time), letting x;
         denote the rth concentration value and t\ denote the rth sampling date. Estimate the linear slope
         b with the formula:

                                            -fl-xVCH-l)-*,2                             [17.21]
                                                  /

         This estimate then leads to the regression equation, given by:

                                        xt=x + b-(t-t)                                 [17.22]

         where t  denotes  the mean sampling date,  s2t is the variance of sampling dates, x is the mean
         concentration level, and xt represents the estimated mean concentration at time t.

         Note: though the variable  t above represents time, it could just  as  easily signify another
         variable, perhaps  a second constituent for which an association with x is estimated.

Step 2.   Compute the regression residual at each sampling event / with equation [17.23]:

                                           >;•=*,•-*,•                                    [17.23]

         Check the set of residuals for lack of normality and significant skewness using the techniques
         in Chapter 10. Also, plot the residuals against the estimated regression values (xi ) to check
         for non-uniform vertical thickness  in the scatter  cloud.  Make a similar check by plotting the
         residuals against the sampling dates (/;).

         If the residuals are non-normal and substantially skewed and/or  the scatter clouds  appear to
         have a definite pattern (e.g., funnel-shaped; "U"-shaped; or, residuals mostly positive on one
         end of graph and mostly negative on the  other end, instead of randomly scattered around the
         horizontal line r = 0), repeat Steps 1 and 2 after first attempting a normalizing transformation.

Step 3.   Calculate the estimated variance around the regression line (also  known as the mean squared
         error [MSB]) with  equation [17.24]:


                                        S2=—Yr2                                  [17.24]
                                          e           '
Step 4.   Compute the standard error of the linear regression slope coefficient using the s2e result from
         Step 3 in equation [17.25]:
                                             17-25                                   March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests
                                                                     Unified Guidance
 Step 5.
                                                                                          [17.25]
Test whether the trend is significantly different from zero by forming the ^-statistic ratio in
equation [17.26]:
                                             = b/se(b)
                                                                                [17.26]
         This ^-statistic (ft,) has n-2 degrees of freedom [df\. Given a level of significance (a), choose
         the critical point (tcp) for the test as the (1- a) * 100th percentage point of the Student's t-
         distribution with (n-2) df or tcp = ?i-a,n-2. Compare  4> against the critical point. If t\>  > tcp,
         conclude that the slope of the trend is both positive and significantly different from zero  at the
         a-level of significance. If 4> < -tcp, conclude there is  a significant decreasing trend. If neither
         exists, there is insufficient evidence of an increasing or decreasing trend.

       ^EXAMPLE 17-5

     The following groundwater chloride measurements (n= 19) were collected  over a five-year period
at a solid waste landfill. Test for a significant trend at the a = 0.01 level using linear regression.
Sample Date Chloride (ppm)
2002-03-18
2002-05-14
2002-08-22
2003-02-12
2003-05-29
2003-08-18
2003-11-20
2004-02-19
2004-04-26
2004-07-29
2004-11-09
2005-02-24
2005-06-14
2005-08-23
2005-10-17
2006-02-08
2006-04-27
2006-08-10
2006-10-26
11.5
12.6
13.8
12.3
12.8
13.2
14.1
13.3
13.1
13.2
15.3
15.0
15.2
15.8
16.1
15.1
16.4
17.7
17.7
Elapsed Days
76
133
233
407
513
594
688
779
846
940
1043
1150
1260
1330
1385
1499
1577
1682
1759
Residuals
-0.25
0.67
1.56
-0.48
-0.30
-0.15
0.45
-0.63
-1.04
-1.23
0.56
-0.08
-0.22
0.17
0.30
-1.06
0.00
0.98
0.74
       SOLUTION
Step 1.   Check for an apparent trend on a time series plot (Figure 17-2). Since the chloride values are
         increasing in reasonably linear fashion, compute the tentative regression line using equations
         [17.21] and [17.22]. To compute the slope estimate, first convert the sample dates to elapsed
         days using a starting date prior to the first event. In this case, choose an arbitrary starting date
         of 2002-01-01 as zero and compute the elapsed days as listed in the table above.

         Using elapsed days as the time variable, compute the sample mean and variance to get:
                                              17-26
                                                                             March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests
Unified Guidance
                                     t =941.79 days
                                     s2t = 279374.3 days2

         Then compute the tentative slope as:

            6 = [(76-941.79)-11.5+ ...+(! 759-941.79)-17.7]/[(l9-l)-279374.3] = .0031

         and the regression line itself as:

                         x, = x + b-(t-t) = 14.432 + .0031- (f-941.79)

         where the mean  chloride  value is  x = 14.432 ppm . The regression line is overlaid on the
         scatter plot in Figure 17-2.



  Figure 17-2. Time Series Plot of Chloride (ppm) Overlaid With Linear Regression
                                         2004

                                           Sampling Date
                                                            2006
                                                                     200?
Step 2.   Calculate the regression residual at each sampling event using equation [17.23]. This involves
         computing an estimated concentration along the regression line for each sampled time (f) and
         then subtracting from the observed concentration. For example, the residual at t = 407 is

                                 x, -x, =12.3-12.78 = -0.48

         All  the  residuals  are listed in the table  above. Then check the residuals  for  normality,
         homoscedasticity, and lack of association with the predicted values from the regression line.
                                           17-27
        March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                       Unified Guidance

         Figure 17-3 is a probability plot of the residuals, indicating good agreement with normality.
         Figure 17-4 is a scatter plot of the residuals versus sampling date and Figure 17-5 is a scatter
         plot of the residuals versus predicted values from the trend line. Both of these last plots do not
         exhibit any  particular trends or patterns with sampling date or the trend line predicted values;
         the residuals are fairly randomly  scattered.

Step 3.   Compute the MSB of the regression using the squared residuals in equation [17.24] to get

                   si =J-~±r? = -M(-.25)2 +(.67)2 +... +(.74)2] = 0.5628
                        n — 2  i=1     1 /

Step 4.   Calculate the standard error of the regression slope coefficient using equation [17.25]:
           se(b)= Js2e/^(t-t)2  =^/.5628/[(76-941.79)2 +... + (l759-941.79)2J = .00033
Step 5.   Form the ^-statistic ratio with formula [17.26] to get:

                               th =b/se(b)= 0.003 1/0.00033 = 9.39

         Since a = 0.01, compare this value to a critical point equal to the 99th percentile of a Student's
         ^-distribution with (n-2) =  17 degrees of freedom, that is, tcp = t,99^7 = 2.567. Since the t-
         statistic is substantially larger than the critical point, conclude the upward trend is  significant
         at the l%a-level. -^
                                              17-28                                    March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests
Unified Guidance
            Figure 17-3. Probability Plot of Chloride Regression Residuals
                N
                           I  	1 	1	[	I	I

                         -1.0     -0.5      0.0      0.5      1.0      1.5


                                     Chloride Residuals (ppm)
          Figure 17-4. Scatter Plot of Chloride Residuals vs. Sampling Date
                    s
                    6
                               2003
                                       2004     2005


                                         Sample Date
                                                      2006     800?
                                         17-29
       March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests
                                                       Unified Guidance
     Figure 17-5. Scatter Plot of Chloride Residuals vs. Predicted Regression Fits

                         1
          12      13      14      15      18

                  Predicted Regression Line Fit {pprn)
                                                                   17
       17.3.2
MANN-KENDALL TREND TEST
       BACKGROUND AND PURPOSE

     The Mann-Kendall test (Gilbert, 1987) is a non-parametric test for linear trend, based on the idea
that a lack of trend should correspond to a time series plot fluctuating randomly about a constant mean
level, with no visually apparent upward or downward pattern. If an increasing trend really exists, the
sample taken first from any randomly selected pair of measurements should on average have a  lower
concentration than the measurement collected at a later point. The Mann-Kendall statistic is computed
by examining all possible pairs of measurements in the data set and  scoring each pair  as follows.  An
earlier measurement less in  magnitude than a later one is assigned a value of 1.  If an earlier value is
greater in magnitude than a later sample, the pair is tallied as -1; two identical measurement values are
assigned  0.

     After scoring each pair in this way and adding up the total to get the Mann-Kendall statistic (S), a
positive value of S implies that a majority of the differences between earlier and later measurements are
positive,  suggestive of an upward trend  over time. Likewise,  a negative value for S implies that a
majority of the differences between  earlier and later values  are negative, suggestive  of a decreasing
trend. A value near zero indicates a roughly equal number of positive  and negative differences.   This
would be expected if the measurements were  randomly fluctuating about a constant mean with no
apparent trend.

     To account for randomness and inherent variability in the sample,  the Mann-Kendall test is  based
on the critical ranges of the statistic S likely to occur under stationary conditions. The larger the absolute
                                            17-30                                  March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                       Unified Guidance

value of S, the stronger the evidence for a real  increasing or decreasing trend. The critical points for
identifying a trend get larger as the level of significance (a) drops. Only if the absolute value of the test
statistic (S) is larger than the critical point is a statistically significant increasing or decreasing trend
indicated.

REQUIREMENTS AND ASSUMPTIONS

     As a non-parametric  procedure, the Mann-Kendall test  does  not require the underlying data to
follow a specific distribution. Ranks of the data are not explicitly used in forming the test statistic as
with the Wilcoxon rank-sum. Only the relative magnitudes of the  concentration values are needed to
compute S, not the actual concentrations themselves. Non-detects can be treated by assigning them a
common value lower than any of the detected measurements. Any pair of tied values or any pair of non-
detects is simply given a score of 0 in the calculation of the Mann-Kendall statistic S.

     This  treatment of non-detects is an  imperfect remedy  since it  is usually impossible to  know
whether censored values are actually tied in magnitude. Further complications are introduced when there
are multiple RLs and/or an  intermingling of detected values and RLs. Lab qualifiers may be used to aid
the scoring of pairs that involve non-detects or estimated concentrations. Instead of treating all non-
detects as tied, consider 'undetected or U' values as the lowest in magnitude,  other non-detects as higher
in magnitude than U's but lower than estimated concentrations (T or 'E' values).  In this way, a richer
scoring of the  sample pairs may be possible.

     When the  sample size  n becomes large, exact critical  values for the statistic S are  not readily
available. However, as a sum of identically-distributed random quantities, the behavior of S for larger n
tends to approximate  the  normal  distribution by  the Central Limit Theorem.  Therefore a normal
approximation to S can be used for n >  101. In this case, a standardized Z-statistic is  formed by first
computing the expected mean value and standard deviation of S. From the discussion above, when no
trend is present, positive  differences in randomly selected  pairs of measurements  should balance
negative differences, so the expected mean value of S under the null hypothesis of no trend is simply
zero. The standard deviation of S can be computed using equation [17.27]:
+ 5
                                                                       )
[17.27]
where n is the sample size, g represents the number of groups of ties in the data set (if any), and tj is the
number of ties in they'th group of ties. If no ties or non-detects are present, equation [17.27] reduces to
the simpler form:
1   Guidance Table 17-5 contains exact confidence levels up to n = 10. Exact confidence levels for n < 20 have been
  developed in (Hollander & Wolfe, 1999), Table A.30.  These might be preferentially used if sample sizes are fairly small
  and the data contain non-detect values.

                                             17-31                                    March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests                       Unified Guidance

     Once the standard deviation of S has been derived, the standardized Z-statistic for an increasing (or
decreasing) trend is formed using the equation:

                                                                                         [17.29]

Note that although the expected mean value of S is zero, applying the continuous normal to the discrete S
distribution is an approximation.  Therefore, a continuity correction is made to Z by first subtracting 1
from the absolute value of S. The final Z-statistic can then be compared to an a-level critical point taken
from Table 10-1 in Appendix D to complete the test.

       PROCEDURE

Step 1.   Order the data set by  sampling event or  time  of collection,  x\,  X2, to xn. Then  consider all
         possible  differences between distinct pairs of measurements, (Xj - x,) for 7 > /'. For each pair,
         compute the sign of the difference, defined by:
                                sgr
 0  if(x;-x)=0                         [17.30]

-'  if^-x-
         Pairs of tied values including non-detects, will receive scores of zero using equation [17.30].

Step 2.   Compute the Mann-Kendall statistic fusing equation [17.31]:
                                                                                         [17.31]
                                         2=1 j=j+\

         In equation [17.31] the summation  starts with a comparison of the very first sampling event
         against each of the subsequent measurements. Then the second event is compared with each of
         the samples taken after it (i.e., the third, fourth, fifth, etc.). Following this pattern is probably
         the most convenient way to ensure that all distinct pairs are tallied in forming S.  For a sample
         of size n, there will be w(w-l)/2 distinct pairs.

Step 3.   If n < 10, and given the level of significance (a), determine the critical point scp from Table
         17-5 of Appendix D.  If S > 0 and  S > sc ,  conclude there is statistically significant evidence

         of an increasing  trend at the a significance level.  If S < 0 and  \S  > s  , conclude there is
                                                                            cp •
         statistically significant evidence of a decreasing trend. Ifp  < s  , conclude there is insufficient
         evidence to identify a significant trend.

Step 4.   If n > 10,  determine the number of groups of ties (g) and the number of tied values in each
         group of ties (Yj). Then use equation [17.27] to compute the standard deviation of S and
         equation [17.29] in turn to compute the standardized Z-statistic.
                                             17-32                                   March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests
                   Unified Guidance
Step 5.   Given the significance level (a),  determine the critical point zcp from the standard normal
         distribution in Table 10-1 in Appendix D. Compare Z against this critical point. If Z > zcp,
         conclude there is statistically significant evidence at the a-level of an increasing trend. If Z < -
         zcp, conclude there is statistically significant evidence of a decreasing trend. If neither exists,
         conclude that the sample evidence is insufficient to identify a trend.

       ^EXAMPLE 17-6

     Test for a significant upward trend using the Mann-Kendall  procedure in the following set of
sulfate measurements (ppm) collected over several years.
Sample No.
1
2
3
4
5
6
7
8
9
10
11
12
Sampling Date
(yr.mon)
89.6
89.8
90.1
90.3
90.6
90.8
91.1
91.3
91.6
91.8
92.1
92.6
Sulfate Cone.
(ppm)
480
450
490
520
485
510
510
530
510
560
560
540
Sample No.
13
14
15
16
17
18
19
20
21
22
23

Sampling Date
(yr.mon)
93.1
93.6
94.1
94.6
95.1
95.6
95.8
96.1
96.3
96.6
96.8

Sulfate Cone.
(ppm)
590
550
600
700
570
610
650
620
830
720
590

SOLUTION
Step 1.   Construct a time series  plot of the sulfate observations to check for a possible trend as in
         Figure 17-6. A clearly  rising concentration pattern is seen, although the variability in the
         measurements appears greater toward the end of the sampling record than at the beginning.

Step 2.   Compute the difference between each distinct pair of measurements and determine the sign of
         the difference, using equation [17.30]. Then sum up the signs with equation [17.31]. Note that
         to make sure all the distinct pairs have been summed, begin with the first listed observation
         and compare it to each of values below it. Then take the second listed value and compare it to
         each of the remaining ones below it, etc. The Mann-Kendall statistic becomes:

                  S = sgn(450-480) + sgn(490-480) + ... + sgn(590-720) = 196

Step 3.   Since the sample size n = 23 >  10, form the normal approximation to the Mann-Kendall
         statistic.  Because there  are some ties in the data,  use  equation  [17.27] to compute the
         approximate  standard  deviation. Among the sulfate measurements, there are three groups of
         ties with 3, 2, and 2 tied values in each set respectively (at values 510, 560, and 590). The
         adjusted standard deviation is then:
SD[S]=  J— • [23- (23-l)(2-23 + 5) -{3- (3-
+...+2- (2-
                                                                             5)}] =37.79
         Finally, using equation [17.29], the normalized Mann-Kendall statistic is:
                                             17-33
                          March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests
                                                       Unified Guidance
                                  Z = ((l96 -^37.79 = 5.16
Step 4.   The Z statistic can be compared to a critical point from the standard normal distribution in
         Table 10-1 in Appendix D. As large as it is, the test statistic is bigger than the critical point
         for any usual significance level, suggesting that the trend appears to be real and not just a
         chance artifact of the sample. -4
            Figure  17-6. Time Series Plot of Sulfate Concentrations (ppm)

                875
                750  -
             Q.
             Q.
             o

             3  625
             CD
                500  -
                375
                    87.5
              90.0
92.5
95.0
97.5
                                          Sampling Date
       17.3.3
THEIL-SEN TREND LINE
       BACKGROUND AND PURPOSE

     The Mann-Kendall procedure is a non-parametric test for a significant slope in a linear regression
of the concentration values plotted against time of sampling.  But the Mann-Kendall statistic S does not
indicate the magnitude of the slope or estimate the trend line itself even when a trend is present. This is
slightly different from parametric linear regression, where a test for a significant slope follows naturally
from the estimate of the trend line. Even a relatively modest slope can be statistically distinguished from
zero with  a large  enough sample.  It  is best to first identify whether or not a trend exists, and then
determine  how  steeply  the concentration levels are increasing over time for a significant  trend. The
Theil-Sen trend line (Helsel, 2005) is a non-parametric alternative to linear regression which can be used
in conjunction with the Mann-Kendall test.

     The  Theil-Sen method handles  non-detects in almost exactly the same  manner as the Mann-
Kendall test.  It assigns each non-detect a common value less than any other detected measurement (e.g.,
                                            17-34
                                                              March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

half the RL). Unlike the Mann-Kendall test, however, the actual concentration values are important in
computing the slope estimate in the Theil-Sen procedure. The  essential idea is that if a simple slope
estimate is computed for every pair of distinct measurements in the sample (known as the set of pairwise
slopes), the  average of this series of slope values should approximate the true slope. The Theil-Sen
method is non-parametric because instead  of taking  an arithmetic average of the pairwise slopes, the
median slope value is determined. By taking the median pairwise slope instead of the mean, extreme
pairwise slopes — perhaps due to one or more outliers or other errors — are ignored and have little if
any impact on the final slope estimator.

     The Theil-Sen trend line is also  non-parametric because the median  pairwise slope is combined
with the median  concentration value and the median sample date to construct the final trend line. As a
consequence of this construction, the Theil-Sen line estimates the  change in median concentration over
time and not the mean as in linear regression.

       REQUIREMENTS  AND ASSUMPTIONS

     The  Theil-Sen procedure  does  not require normally-distributed trend  residuals as in a  linear
regression.   It is also not critical that the residuals be homoscedastic (i.e., having equal variance over
time and with increasing average concentration level). It is important to have at least 4 and preferably at
least 8 or more  observation on  which to construct the trend.   But trend residuals are assumed to be
statistically independent.  Approximate checks of this assumption can be made using the  techniques of
Chapter 14, once the estimated trend  has been removed  and the number of non-detect data  is limited.
Sampling events  should also be spaced far enough apart relative to the site-specific groundwater velocity
so that an assumption of physical independence of consecutive sample volumes is reasonable.

     A more difficult problem is encountered when a large fraction of the data is non-detect. As long as
less than half the measurements are non-detects occurring in the lower part of the observed  concentration
range, the median concentration value will be quantified and the median pairwise slope will generally be
associated with a pair of detects.  Larger proportions  of non-detect data make computation of the Theil-
Sen trend line more difficult and uncertain. The reason is that each time a non-detect is  paired with a
quantified measurement, the pairwise slope is known only within a range of values. One end of the range
results from supposing the true non-detect concentration is equal to zero; the other when the non-detect
concentration is equal to the RL.

       PROCEDURE

Step 1.   Order the data set by sampling event or time  of collection, x\, X2, to xn.  Then consider all
         possible distinct pairs  of measurements, (x;, Xj) for j > i. For each pair, compute the  simple
         pairwise slope estimate:
         With a sample size of w, there should be a total of N= n(n-\)!2 such pairwise estimates m^. If
         a given observation is a non-detect, use half the RL as its estimated concentration.
Step 2.   Order the N pairwise slope estimates (m-^) from least to greatest and rename them as
         ..., /H(N). Then determine the Theil-Sen estimate of slope (Q) as the median value of this list.
                                            17-35                                   March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests                      Unified Guidance

         Finding this value will depend on whether TV is even or odd, but the following equation can be
         used:

                                                     is odd
                                                                                       [17.33]
                                                           s even


Step 3.   Order the sample by concentration magnitude from least to greatest, X(\), x@), to *(„). Determine
         the median concentration with the formula:
                                         Xfn+iv?    n g
                               x=\(      (   }l (,                                      [17.34]
                                   [(xn/2 + x(n+2)/2)/2  if n is even

         Again replace each non-detect by half its RL during this calculation. Also find the median
         sampling date (7 ) using the ordered times t\, h, to tn by a similar computation.

Step 4.   Compute the Theil-Sen trend line with the equation:

                               x = x +Q-(t-7)=(x-Q-7) + Q-t                         [17.35]

         Using equation [17.35], an estimate  can be made at any time (f) of the expected median
         concentration (x).

       ^EXAMPLE 17-7

     Use the following sodium measurements to compute a Theil-Sen trend line. Note that the sample
dates are recorded as the year of collection (2-digit format) plus a fractional part indicating when during
the year the sample was collected. This allows an annual  slope estimate, since 1 unit = 1 year.
Sample
Date (yr)
89.6
90.1
90.8
91.1
92.1
93.1
94.1
95.6
96.1
96.3
Sodium Cone.
(ppm)
56
53
51
55
52
60
62
59
61
63
       SOLUTION
Step 1.   Compute the pairwise slopes for each distinct pair of measurements using equation [17.32].
         With n = 10 observations, there will be a total of 10(9)/2 = 45 such pairs. The first few are
         listed below:
                                            17-36                                  March 2009

-------
Chapter 17. ANOVA, Tolerance Limits & Trend Tests
                                                                  Unified Guidance
                              mu = (53 - 56)/(90.1 - 89.6)= -6

                              w13 = (51-56)/(90.8-89.6)=-4.17

                              ml4 = (55 -56)/(91.1-89.6)= -667
Step 2.
Step3.
Since the total number of distinct pairs is odd, sort the list of pairwise slopes as in the table
below and let Sen's estimated slope equal the middle or 23rd largest value in this list.  This
gives an estimate of Q = 1.33 ppm increase per year, an estimate in line with the time series
plot of Figure 17-7.

Compute the median concentration value x = 57.5 and the median sample date t = 92.6 from
the table above. Then calculate the Theil-Sen trend line using the slope estimate from Step 2:
                  x
                             = 57.5 + 1.333(^-92.6)= -65. 97 + \.333t
         This trend line can be used to estimate the predicted median concentration (x) at any desired
         time in years (i). For example, at the beginning of 1998 (t = 98), the trend line would predict a
         median sodium concentration of approximately x = 64.7 ppm. -^
Rank
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Pairwise
Slope
-6
-4.167
-3
-2.857
-2
-1.6
-0.667
-0.5
-0.5
-0.4
0.333
0.455
0.5
0.769
0.769
0.889
0.938
1.045
1.091
1.143
1.2
1.333
1.333
Rank
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45

Pairwise
Slope
1.538
1.613
1.667
1.887
2
2
2
2.182
2.25
2.25
2.333
2.333
2.5
2.619
3.333
3.913
4
5
5.714
8
10
13.333

                                            17-37
                                                                          March 2009

-------
Chapter 17.  ANOVA, Tolerance Limits & Trend Tests
Unified Guidance
          Figure 17-7. Time Series Plot of Sodium Concentrations (ppm)
          CL
                   90
                           91
                                     Year (2-Digit Format)
                                       17-38
       March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance
            CHAPTER  18.  PREDICTION LIMIT PRIMER

        18.1   INTRODUCTION TO PREDICTION LIMITS	18-1
          18.1.1  Basic Requirements for Prediction Limits	18-4
          18.1.2  Prediction Limits With Censored Data	18-6
        18.2   PARAMETRIC PREDICTION LIMITS	18-7
          18.2.1  Prediction Limit for m Future Values	18-7
          18.2.2  Prediction Limit for a Future Mean	18-11
        18.3   NON-PARAMETRIC PREDICTION LIMITS	18-16
          18.3.1  Prediction Limit for m Future Values	18-17
          18.3.2  Prediction Limit for a Future Median	18-20
     This chapter introduces  the  concept of statistical  intervals  and focuses on  several types of
prediction limits useful for detection monitoring. The requirements and common assumptions of such
limits are explained, as well as specific descriptions of:

    »»»  Prediction limits for m future values (Section 18.2.1)
    *»*  Prediction limits for future means (Section 18.2.2)
    »»»  Non-parametric prediction limits for m future values (Section 18.3.1)
    »«»  Non-parametric prediction limits for a future median (Section 18.3.2)
18.1 INTRODUCTION TO  PREDICTION LIMITS

     First discussed in Chapter 6, prediction limits belong to a class of methods known as statistical
intervals. Statistical intervals  represent concentration or measurement ranges computed from a sample
that  are designed to estimate one or more  characteristics of the parent population. In groundwater
monitoring,  statistical intervals offer  a convenient  and  statistically valid way to test for significant
differences between background versus compliance point groundwater measurements.

     The statistical interval accounts for variability inherent not only in future measurements, but also
additional uncertainty in the  prediction limit itself.    The latter is derived from a relatively small
background  sample with an associated level  of variability in estimating the true characteristics of the
underlying groundwater population.

     Prediction  limits are generally  easy to construct and have a straightforward  interpretation.
Background data are used to construct a concentration limit PL, which is then compared to one or more
observations from a compliance point population. The acceptable range of concentrations includes all
values no greater than the prediction limit. The appropriate prediction interval will generally have the
form  [0, PL], with the upper limit PL as the  comparison of importance.  Unless pH or  a similar
parameter is being monitored,  a one-sided upper prediction limit is used in detection monitoring.

     A significant advantage to prediction limits is their flexibility, which can accommodate a wide
variety of groundwater monitoring networks.  Prediction limits can be constructed so that as few as one

                                             IJTl                                   March 2009

-------
Chapter 18.  Prediction Limit Primer                                        Unified Guidance

new measurement per compliance well may suffice for a test. Prediction limits may be based on a
comparison of means,  medians, or individual  compliance  point  measurements, depending on the
characteristics of the monitoring network and the constituents being tested.

     Prediction limits can  also be  designed to accommodate  a wide  range  of multiple statistical
comparisons or tests. Each periodic statistical evaluation (e.g., semi-annually) under RCRA and other
regulations involves separate tests at all compliance well locations for each monitoring constituent.
Often, the number of separate statistical tests can be quite sizeable.  Prediction limits can be constructed
to precisely account for the  number of tests to be conducted,  so as to limit the site-wide false positive
rate [SWFPR] and ensure an adequate level of statistical power (see discussion in Chapter 6).

     This and the following chapter present basic concepts and procedures for using prediction limits as
detection monitoring tests. The intent is to provide a relatively simple framework for using prediction
limits  in RCRA or CERCLA groundwater monitoring.  Chapter 18  describes the construction of
prediction limits for tests involving a single constituent at one well. It describes the basic mechanics of
each type of prediction limit  and how they differ from one another.

     Chapter 19 expands this  discussion to cover multiple simultaneous prediction limit tests (i.e., all
occurring  during a single statistical  evaluation or during a  single  year  of monitoring).   Cumulative
SWFPRs  and  statistical power are considered,  including how  these  criteria  impact the  expected
performance of a given prediction limit strategy. Examples are provided to illustrate these procedures, as
well as explanations of associated tables and software.

     Specific strategies in Chapter 19 apply the concept of retesting. Generally speaking, almost any
prediction limit procedure in detection monitoring should be combined with an appropriate  retesting
strategy. The reason is that when testing  a large  number of compliance point  samples,  it is almost
guaranteed that one or more measurements will exceed an upper prediction limit. Resampling of those
wells where an exceedance has  occurred can either verify the initial evidence of a release or disconfirm
it, while avoiding unnecessary false positives.

     Chapter  6 introduced a  number of key terms  used in  the Unified Guidance, especially for
prediction limit and control chart tests.   The guidance applies the  term comparison to individual future
measurements or sample statistics evaluated against a prediction limit (or control chart limit), and the
term test to represent a series of future data comparisons that ultimately result in a statistical decision. A
\-of-m retesting procedure (described below), for instance, might involve comparison of up to m distinct
sample measurements against  the  prediction limit. Each of  these individual samples involves  a
comparison, but only after all the necessary individual comparisons have been made is the test complete.
This  distinction  becomes  particularly  important  when  properly  determining  SWFPRs, a subject
discussed both in Chapter 6 and Chapter 19.

     One or more future observations are collected for purposes of testing compliance well data, as
distinct from the background sample from which the prediction limit is constructed. Background data
can be obtained from upgradient wells or in combination with historical, uncontaminated compliance
well data. In intrawell testing, data from an individual compliance well constitute both the background
and future  samples.  The two data sets need to be  distinct and may not overlap, even if the historical
                                              18-2                                   March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance

background data is periodically updated with previously evaluated future samples. The key idea is that at
any given point in time, background and future data sets are clearly distinguished.

     Formally, prediction limits are constructed to contain one or more future observations or sample
statistics generated from the background population with a  specified probability  equal to (1-a).  The
probability (1-a) is known as the confidence level of the limit. It represents the chance — over repeated
applications of the limit to many similar  data sets — that the prediction limit will contain future
observations or statistics drawn from its background population.

     A sample of n background measurements is used to construct the prediction limit. Under the null
hypothesis  that the compliance point population  is identical to background, a set  of m independent
compliance point observations or a statistic like the mean based on those observations (i.e., the future
data) is then compared against the prediction limit. For the prediction limit to serve as a valid statistical
test, the future observations are initially presumed to follow the same distribution as background.

     Only  background values are used to construct the prediction limit. But the probability that the
limit contains all m  future observations or sample  statistics derived from those future data does not
depend solely on the observed background.  It is also based  on the number of future measurements or
sample statistics  used in the comparison  and how  the  individual  comparisons are  conducted.  To
underscore this point,  consider  the  general  equation for  a prediction  limit based on normal or
transformably normal populations, given by


                                          PL = x + xs                                     [18.1]

where x is the sample mean in background, s is the background standard deviation, and K is a multiplier
depending  on the type of prediction limit under construction. The simplest type of prediction limit test
compares a specific number of individual future observations to the limit (PL). For example, do all three
compliance measurements collected during a 6-month period fall within the prediction interval?  The
multiplier K and hence  the prediction limit itself, changes  depending on whether  one, two or three
compliance observations will be compared  against PL. More generally,  the K-multiplier is selected to
account not only for the  number of future comparisons, but also for the rules of the comparison strategy
and the number of simultaneous tests to be conducted (e.g.., the number of monitoring constituents times
the number of compliance wells).

     In the simplest case of a successive comparison  of m individual future measurements against PL,
the test is labeled as an m-of-m prediction limit. All m of the future observations need to fall within the
prediction interval  for the test to 'pass'  — that is, be no greater than PL. If any one or more of the future
values exceed the  PL, the test fails and the well is  deemed to have a statistically significant increase
[SSI] or constitute an exceedance.

     The ^-multiplier appropriate for an m-of-m prediction limit test  is different  from the multipliers
that would be computed for other kinds of comparison rules. Another simple type is  a  comparison of a
single future mean of order p. Here,/? future measurements are collected and averaged before comparing
against PL. If the order-/? mean is no greater than PL, the test passes; otherwise, it fails.  A test following
this rule is labeled a 1-of-l prediction limit on a future mean. The important thing to remember is that
the K-multiplier and thus the prediction limit will differ depending on whether or not the/? future values
are first averaged or simply compared against PL one-by-one.  The choice to use one rule versus the other
                                             IJTs                                    March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance

impacts the magnitude  of the prediction limit and ultimately its expected statistical power and false
positive rate.

     Other comparison rules of substantial benefit in groundwater monitoring are l-of-m prediction
limit on future observations or  a statistic like the mean or median. This test requires at least one of m
successive observations or statistic to fall within the prediction interval in order to pass. Operationally
this means that if an initial compliance well measurement is no greater than PL, the test is complete and
no further sampling need be done. If the initial value exceeds PL, one or more of (m-l) resamples need
to be obtained.  Since these additional measurements are  collected sequentially over sufficiently long
time periods to maintain approximate statistical independence (Chapter 3), the first resample to fall
within the prediction interval also ends the test as 'inbounds' or passing, frequently obviating the need to
gather all m measurements.

     Another comparison rule of some use is known as the California strategy, first developed for the
State of California RCRA program.  The California strategy can be construed as a conditional rule: if an
initial  future observation is no greater than PL, further comparisons are not needed and the test passes.
However, if the initial observation exceeds the PL, 2-of-2 or 3-of-3 resamples all need to not exceed the
PL in order for the well to remain in compliance. A slight modification to this rule termed the modified
California approach has better  statistical power and false positive rate characteristics than the original
California strategies, and is therefore included as a potential prediction limit test.
 18.1.1       BASIC REQUIREMENTS FOR PREDICTION  LIMITS

     All prediction limits share certain basic assumptions when applied as tests of groundwater. Further,
parametric prediction limits as presented in the Unified Guidance require the sample data to be either
 normally-distributed or normalized via a transformation. The key points can be summarized as follows:

    1.  background and future sample measurements need to be identically and independently distributed
       (the i.i.d. presumption; see Chapter 3);
    2.  sample data do not exhibit temporal non-stationarity in the form of trends, autocorrelation, or
       other seasonal or cyclic variation;
    3.  for interwell  tests (e.g., upgradient-to-downgradient comparisons),  sample data do not exhibit
       non-stationary distributions in the form of significant natural spatial variability;
    4.  background data do not include statistical outliers (a form of non-identical distributions);
    5.  for parametric prediction limits, background data are normal  or can  be normalized using a
       transformation; and
    6.  a minimum of 8 background measurements is available; more for non-parametric limits or when
       accounting for multiple, simultaneous prediction limit tests.
       The first assumption implies that background data are randomly drawn from a single common
 parent population, especially if aggregated from more than one source well.  As discussed in Chapter 5,
 analysis  of variance [ANOVA] can be used to determine the appropriateness of pooling data from
                                              18-4                                   March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance

different background wells. There is also a presumption that the compliance point measurements follow
the same distribution as background in the absence of a release.

       The  second assumption is corollary to the first, and  requires that  the background data are
stationary over time (Chapter 3).  This can be evaluated  with one or more techniques  described in
Chapter 14  on temporal variability.  These account  for trends, autocorrelation,  or  other variation,
perhaps by utilizing data residuals instead of the raw measurements. If the background residuals meet
the basic points above, they can be used to construct an adjusted prediction limit. Residuals of the future
observations  would also  need to  be  computed and compared against the adjusted prediction limit to
ensure a valid and consistent test.

     The  second assumption also requires that there be only a single source of variation in the data,
when using the usual sample standard deviation (s) to  compute the  prediction limit. If there are other
sources of variation such as seasonal  patterns or temporal variation in lab analytical performance, these
should be included in the estimate of variability.  Otherwise s is likely  to be biased.  One method to
accomplish this is by use of an appropriate  ANOVA model to include temporal factors  affecting the
variability (Chapter 14). Determination of the components  of variance in more complicated models is
beyond the scope of this guidance and may require consultation with a professional statistician.

     The  third assumption requires that background  and compliance point populations be identical in
distribution, absent a release, for interwell tests. Spatial variation violates this  assumption since the well
population means (u) will be different, making it impossible to know whether an apparent upgradient-to-
downgradient difference is attributable  to  a release  or simply variations in  natural  groundwater
concentration levels. The assumption also requires that  each population share a common variance (a2).
Tests of equal variance (i.e., homoscedasticity) when using  prediction limits  may be possible either by
examining groups  of historical background and compliance point  data or by  performing periodic tests
when enough compliance point measurements have been accumulated to make  a diagnostic test possible.

     The  fourth assumption  implies that background data should be screened  for outliers using the
techniques in Chapter 12. Statistical  outliers can potentially inflate a prediction limit and severely limit
its statistical power and accuracy by over-inflating both the sample background mean (x ) and especially
the background standard deviation (s). The Unified Guidance discourages automated removal of outliers
from background samples, but all possible outliers should be examined to determine whether a cause can
be identified (see discussion in Chapter 6). In some cases, an apparent outlier may represent  a valid
portion of the underlying background population that has not yet been sampled or observed. It also could
represent evidence that conditions  in background have changed or are changing.

     The  fifth assumption of normality for parametric prediction limits can be  evaluated using the
diagnostic  techniques described  in  Part II of the guidance. If  skewed  background  data can be
normalized via a transformation (e.g., the natural logarithm), the prediction limit should be constructed
on the transformed background values. The resulting  limit should either be: 1) back-transformed to the
concentration domain (e.g.,  by exponentiation) when  comparing future individual  compliance
observations; or 2) left in the transformed scale when compared to  future mean compliance data also
based on the same transformation.   In the latter case, use of a logarithmic transformation results in
evaluating population medians or geometric means and not the arithmetic means.
                                              18-5                                    March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance

     When normality  cannot be justified,  a  non-parametric  prediction limit  should  be considered
instead. A non-parametric limit assumes only that all the data  come from the same, usually unknown,
continuous population. Non-parametric  prediction limits generally require a much larger number of
background observations in order to provide the same level of confidence (1-a) as a comparable
parametric limit. Consequently, the Unified Guidance recommends that a parametric model be fit to the
data if at all possible.

     The last assumption concerns sufficient background sample sizes.  A prediction interval can be
computed with  as  few  as  three  observations  from background.  However, this  can result  in  an
unacceptably large upper prediction limit and a test with very limited statistical power. A sample size of
eight  or more is generally needed to derive an adequate parametric  prediction limit,  especially if a
retesting strategy is  not employed. The exact requirements depend on the number of simultaneous tests
(i.e.., number of wells times number of constituents per well) to  be made against the prediction limit and
the type of retesting strategy adopted (see Chapter 19 for more discussion of retesting strategies).

     If a minimum  schedule of quarterly sampling is being followed and there is only one background
well, at least two years of data will  be  needed before constructing the prediction limit.1  If data from
multiple background wells screened in comparable hydrologic  conditions can reasonably be combined
(see Chapter 5), pooling background data to increase background sample sizes is encouraged.
18.1.2       PREDICTION LIMITS WITH CENSORED DATA

     When  a  sample contains a substantial fraction of non-detects or left-censored measurements,  it
may be impossible to even approximately normalize the data  A sample data set may originate from a
normal or transformable-to-normal population, but the uncertainty surrounding both the censored values
and the consequent shape of the lower tail of the distribution  prevents a clear identification.  If the
apparent underlying distribution is not normal or transformable to normality, a non-parametric prediction
limit (Section  18.3) should be used.

     Given  that non-parametric prediction limits  typically  have much steeper  background data
requirements than their parametric counterparts, one remedy is  to attempt a fit to normality by using
censored  probability  plots (Chapter  15)  in  conjunction  with either the Kaplan-Meier  or robust
regression on  order statistics  [ROS]  techniques  (Chapter  15)  for  left-censored  data.  Censored
observations prevent a full and complete ordering of the sample, making it difficult to assess normality
with standard probability plots (Chapter 9). Censored probability plots, on the other hand, only graph
the detected values, but do  so based on a partial ordering and ranking of the sample. Data that appear
distinctly non-normal on a standard probability plot (where non-detects are perhaps replaced by half their
reporting  limits [RLs] to allow plotting)  can sometimes  appear reasonably  normal on a censored
probability plot. Transformations can also be applied and the censored  probability plot  reconstructed to
see if the data can be normalized in that fashion.
1  The Unified Guidance does not recommend that only one background well be used in any kind of interwell or upgradient-
  to-downgradient comparison. Multiple background wells are always preferred so that tests for spatial variability may be
  made and the exact nature of background better understood.

                                             18-6                                    March 2009

-------
Chapter 18. Prediction Limit Primer                                      Unified Guidance

     If the censored probability plot is close to linear and the sample approximately normalized,  an
estimated mean and standard deviation should be computed. These estimates will not be the same if each
non-detect were replaced by  half its RL, and the sample mean calculated from the resulting imputed
sample. To properly account for the censoring, the estimated mean (denoted as //) and the estimated
standard deviation (
-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance

compliance data do not come from the same distribution as background, but instead are elevated above
background.2

     With intrawell comparisons,  a prediction limit can be computed on historical data or intrawell
background to contain a specified number (m) of future (i.e., more recent) observations from the same
well.  If any  of the future values exceeds the upper prediction limit,  there is evidence of recent
contamination at the well.

       REQUIREMENTS AND ASSUMPTIONS

     As noted in Section 18.1, the prediction limit test on m future values is designated as an m-of-m
test. Each of the m individual future observations need to be compared to the prediction limit [PL]. All
should be no greater than PL for the test to pass. The number of future observations to be collected (m)
need to be specified in advance in order to correctly compute the K-multiplier from equation [18.1].
Consequently, if compliance  data  are collected on a regular schedule, the  prediction interval can be
constructed to cover a specified time period of future sampling. Usually this period will coincide with
the time between statistical evaluations specified in the site permit (e.g., on a semi-annual or annual
basis). Keep in mind also that m denotes the number of consecutive sampling events being compared to
the prediction limit at a given well for a given constituent.

     As discussed in more detail in Chapter 6, a new prediction limit should be constructed prior to
each statistical evaluation for interwell tests, when additional background data have been collected along
with the new compliance point measurements. Unless there is evidence of characteristic changes within
background groundwater quality (e.g., as demonstrated by observable trends in background), background
data  should  be  amassed or accumulated  over time. Earlier background measurements need not be
discarded, both to maintain an adequate background sample size and also because a larger span of
sampling results will provide a better statistical description of the underlying background population.
The revised prediction limit will then reflect a larger background sample size, n, but possibly the same
number, m, of future values to be predicted at the next statistical evaluation.

     For intrawell tests, the prediction limits should be revised only after intrawell background has been
updated (Chapter 5). Such updating may not coincide with the regular schedule of statistical evaluations
if done, for instance, every two years or so. In that case, the same intrawell prediction limit might be
used for multiple evaluations before being revised.

       PROCEDURE

Step 1.   Calculate the  sample  mean x ,  and standard  deviation s,  from  the set of  n background
         measurements.

Step 2.   Specify the number of individual future observations  (m)  from the  compliance well  to be
         included in the prediction interval for an m-of-m test. For an upper prediction limit with an
         overall (1-oc) confidence test level for the m comparisons, use the equation:
2 In the context of the Unified Guidance,  m represents the number of consecutive samples being compared in the prediction
  limit test for a given well and constituent.

                                             18-8                                    March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance
                                    PL = x+t.  ,   ,jl + -                               [18.3]
                                             \-a/m,n-\ A                                    L    J

         It is assumed that exactly m consecutive sample values from the compliance point will be
         compared against the upper PL. Note that the quantile from a Student's ^-distribution used in
         equation [18.3] has  two parameters: the degrees of freedom («-l) and a joint comparison
         confidence level (l-a/m). Most Student's ^-quantiles can be found directly or approximated
         through interpolation by looking in Table 16-1 of Appendix D.

         Note: equation [18.3] assumes the prediction limit is applied to only one constituent at a single
         well.  If multiple  tests need to  be performed (e.g., on  multiple  wells and/or multiple
         constituents), the prediction limit takes the form:

                                          PL = x + Ks                                     [18.4]

         where the K-multiplier is determined using one of the strategies described in Chapter 19.

         If a log transformation is applied to the data to bring about  approximate normality, the upper
         PL  should be constructed using the log-mean (j7) and log-standard deviation (Sy), using the
         equation:
                                                                                         [18.5]
         If multiple tests must be conducted and a log transformation has been applied to the data, the
         upper PL will have the form:

                                                                                         [18.6]

         Note: other  transformations besides the natural logarithm are handled in a similar manner;
         compute the prediction limit on the transformed  data, then  back-transform the limit to the
         original concentration scale prior to comparison with any future observations.

Step 3.   Once the  prediction limit (PL) has been calculated, compare each of m  compliance point
         future values against PL.  If all of these measurements are no greater than PL, the test passes
         and the well is deemed to be in compliance. If, however, any compliance point  concentration
         exceeds PL,  there is statistically significant evidence of an increase over background.

       ^EXAMPLE 18-1

     The data in the table below represent quarterly arsenic  concentrations measured in a  single well at
a solid waste  landfill.  Calculate an  intrawell upper prediction limit for  4  future samples  with  95%
confidence and determine whether there is  evidence at the annual  statistical evaluation  of  a possible
release during Year 4 of monitoring.
                                             18-9                                    March 2009

-------
Chapter 18. Prediction Limit Primer
                         Unified Guidance
               Intrawell Background
       Sampling Period	Arsenic (ppb)
          Compliance Data
Sampling Period	Arsenic (ppb)
Year 1



Year 2



YearS






12.6 Year 4
30.8
52.0
28.1
33.3
44.0
3.0
12.8
58.1
12.6
17.6
25.3
n = 12
Mean = 27.52
SD = 17.10
48.0
30.3
42.5
15.0











       SOLUTION
Step 1.   First check the sample data for the key points identified in Section 18.1.1. As an example, a
         Shapiro-Wilk test on  the background  data gives a test statistic of SW = 0.947. The critical
         point at the a = .05  level for the Shapiro-Wilk test on n = 12 observations is 0.859. Since the
         test statistic exceeds the  critical point, there is insufficient evidence to reject an assumption of
         normality.

Step 2.   Compute the  prediction interval  using the  raw background data. The sample  mean and
         standard deviation of the 12 background samples are 27.52 ppb and  17.10 ppb, respectively.

Step 3.   A single future year of compliance data then is compared to the prediction limit, leading to a
         test of m = 4 individual  measurements. Setting the overall confidence level to (1-a)  = 95%,
         the probability used  to determine an  appropriate Student's ^-quantile needs to be set to
         (l -ex/m)  =  1-.05/4 = .9875.  The ^-distribution with probability  .9875  and  (n-1) = 11
         degrees of freedom in Table 16-1 of Appendix D results in a ^-quantile  of 2.593. Using
         equation [18.3], the upper prediction limit can be computed as:


           PZ = 27.52+ r9875n(l7.1o)Jl + —= 27.52+ 2.593(l7.10yL0833 =73.67 ppb


Step 4.   Compare the  upper PL to each compliance  measurement in  Year 4. None of the  four
         observations exceeds 73.67 ppb. Consequently, there is no statistically significant evidence of
         arsenic contamination  during that year. -^
                                             18-10
                                 March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance



18.2.2       PREDICTION LIMIT FOR A FUTURE MEAN

       BACKGROUND AND PURPOSE

     Although prediction limits are often constructed as bounds on extreme individual measurements,
they can also be formulated to predict an acceptable range of concentrations for the mean  of p future
values. The comparison rule for the test is then different: instead of requiring all of a set of m individual
values to fall within the prediction interval for the test to pass, only the average of the (p) future values
should not exceed the prediction limit.

     In this setting, the prediction limit  for a future mean is  more nearly akin to a Mest or parametric
ANOVA, since the mean of the compliance point well is compared to a limit based on the background
mean. The principal differences in using a prediction limit as opposed to those tests are: first that the
variability of the compliance point population is assumed to be identical to that in background. With a t-
test or ANOVA,  each distinct well group contributes to the overall  estimate of variability, not merely the
background  values.  Secondly,  t-tests and especially ANOVA are typically utilized as interwell tests,
whereas prediction limits for a future mean can be constructed  for either interwell or intrawell testing.

     The hypothesis  being tested  when using  a prediction limit  for a future  mean in detection
monitoring is exactly the same as that posited for a prediction limit for m future values, namely, HQ\
background  population is identical  to compliance population (implying  (ic  < M-BG) vs. H\. compliance
mean is greater than background mean (i.e., (IDG > (IBG)- However, the statistical properties of the two
prediction interval formulations are somewhat different.

     For the same background sample size («), false positive rate (a),  and number of future samples
where/? = m, the power of the prediction limit for a future mean of order/? with normally-distributed
data is generally higher than for a  prediction limit of the next m individual  future observations.  This
suggests that when feasible and appropriately implemented, a prediction limit strategy based on future
means may be more environmentally protective than  a strategy based on predicting individual future
measurements. A few examples of the power differences are presented in Figures 18-1 and 18-2.
                                             18-11                                   March 2009

-------
Chapter 18.  Prediction Limit Primer
                                                     Unified Guidance
    1,0
        Figure 18-1.  Comparison of Prediction limits (BG = 8, a = .01, 1 test)

                         p=2                                           p=3
   0.8 -


 L. 0-6 -
 0)
 I
 a 0.4 -


   0,2 -


   0,0
	PL on mean
-----PL on values
——- PL w/ retests (1-of-2)
                    234
                    SD Units Over BG
                             1,0
                             0.8 -

                             0,6 -

                             0.4

                             o.z H

                             o.o
                           0,0
                                                   p = 4
                                           SD Units Over BG
                             • PL on mean
                         	PL on values
                         	PL w/retests
                                       _l,|,|,p
                                        1234
                                              SD Units Over BG
     Even when a retesting strategy is employed, such as the l-of-m schemes for prediction limits on
individual values described in Chapter 19, the statistical power at best matches that of a prediction limit
on a single future mean with no retesting, when the same numbers of background and compliance point
measurements are used.  As Figure 18-2 illustrates, for some cases the l-of-m power is comparatively
lower. Under background conditions, l-of-m strategies provide an earlier indication of uncontaminated
groundwater, since a single observation can indicate uncontaminated  groundwater. By contrast, all p =
m individual samples need to  be collected to form a mean of order/?  = m when using a prediction limit
test for a single future mean.  With a groundwater release, no such potential time savings exists. In that
case, all p or m samples need to be collected with either type of prediction limit.
                                             18-12
                                                             March 2009

-------
Chapter 18.  Prediction Limit Primer
                                                     Unified Guidance
       Figure 18-2. Comparison of Prediction limits (BG =  20, a = .05, 1 test)
     1,0
     0.8 -


   ^ 0,6-
   QJ
   a
   a 0<4_


     0,2 -


     0.0
         0
                          p=2
	-PL on mean
	PL w/ retests (1-of-2)
	• PL on values	
  234

SD Units Over B6
                                                   1.0
                             0.8-

                           u 0.6
                           *
                           Q
                           °- 0.4

                             0.2 -|

                             0.0
0
                                                                       p=3
    PL on mean
    PL w/ retests (1-of-3)
	PL on values
                                             2       3

                                           SD Units Over BG
              T
               4
     1.0

     0.8 H

     0.6

     0.4

     0.2

     0,0
                                                     PL on mean
                                                     PL w/ retests (1-of-4)
                                                     p|_ on va|ues
                                                 i
                                                 2
                                         i
                                         4
                                              SD Units Over BG

       REQUIREMENTS AND ASSUMPTIONS

     Although a prediction limit for a future mean is generally preferable in terms of statistical power
for identifying potential contamination, it is not always practical to implement.  To accommodate the
large number of statistical tests that all but the smallest sites must contend with, the Unified Guidance
recommends that  almost any prediction limit  be implemented in conjunction with a retesting strategy
(Chapter 19). Otherwise, the prediction limit formulations provided in this chapter will likely fall short
of providing an adequate balance between false negative and positive decision errors.  Retesting with a
prediction limit for a future mean will necessitate the collection of p additional measurements to form
the resampled mean, whenever the initial future mean exceeds the prediction limit.  Since all prediction
limit tests assume that both the background  and compliance data are statistically independent, there
needs to generally be enough temporal spacing between sampling events to avoid introducing significant
autocorrelation in the series of compliance point values.
                                             18-13
                                                             March 2009

-------
Chapter 18. Prediction Limit Primer                                       Unified Guidance

     If semi-annual evaluation of groundwater quality is required, and depending on data characteristics
(see Chapter 14 discussions on temporal variability), there may not be sufficient time for collecting at
least 4 independent groundwater measurements from a given well over a six-month period. This would
be the minimum needed to form an initial mean and potentially a resample mean of order 2.  To avoid
this dilemma, the guidance discusses an alternate approach in Chapter 19 for using 1-of-l  prediction
limit tests on means.

     Like the parametric prediction limit for m future values, the  prediction  limit on a future mean
assumes that the background data used to construct the limit are either normally-distributed or can be
normalized. If a transformation is used (e.g., the natural  logarithm) and the limit built on the transformed
values, the prediction limit should not be back-transformed before comparing to the compliance point
data. Rather, because of transformation bias in the mean, the compliance point data should also be
transformed, and the future mean computed from the transformed compliance measurements. Then the
mean of the transformed values (e.g.,  log-mean)  should be  compared to the prediction limit in the
transformed  domain. As previously mentioned, the prediction limit in the logarithmic  domain is not a
test of the arithmetic mean, but rather of the geometric  mean or median (also see Chapter 16). In most
situations, a  decision that the lognormal median at the  compliance point exceeds background will also
imply that the lognormal arithmetic mean exceeds background.

       PROCEDURE

Step 1.   Calculate  the sample mean, x , and the standard deviation, s, from the set of n background
         measurements.

Step 2.   Specify the order (p) of the mean to be predicted (i.e., the number of individual compliance
         observations to be averaged). If the background data are approximately normal and an upper
         prediction limit with confidence level (1-cc) is desired, use the equation:


                                                                                         [18.7]


         where it is assumed that an average of p consecutive sample values from the compliance point
         will be  compared against PL. Note that the Student's ^-quantile used  in the equation has two
         parameters: the degrees of  freedom  (n-\)  and the cumulative  probability  (l-cc).  Most
         Student's  ^-quantile values can  be found directly or approximated through interpolation by
         using  Table 16-1 in Appendix D

         Note  also  that equation [18.7] assumes that  the  prediction limit is  applied  to  only  one
         constituent at a single well. If multiple tests are to be conducted and  a retesting procedure is
         employed, the prediction limit will take the form of equation [18.4] where the K-multiplier is
         determined using the tables described in Chapter 19.

Step 3.   If  a log transformation is applied to normalize the background sample, the upper PL on the
         log-scale should be constructed using the log-mean (j7) and log-standard deviation (Sy), using
         equation [18.8]:
                                             18-14                                   March 2009

-------
Chapter 18.  Prediction Limit Primer
Unified Guidance
                                                                                         [18.8]
         Note that unlike the lognormal prediction limit for future values, the limit in equation [18.8] is
         not exponentiated back to the concentration domain. Also, equation [18.8] only applies to a
         single test (i.e., one constituent at a  single well).  If multiple tests are to be performed, the
         prediction limit will have the form:
                                         PL = y + KS
                                              •/     v
            [18.9]
         where the K-multiplier is again determined from the tables described in Chapter 19.

         Other transformations are handled similarly: construct the prediction limit on the transformed
         background, but do not back-transform the limit.

Step 4.   Once the limit has been computed, compare the compliance point mean against the prediction
         limit.  If the compliance point mean is below the upper  PL, the test passes. If the mean
         exceeds the PL, there is statistically significant evidence of an increase over background.

       ^EXAMPLE 18-2

     The table below contains chrysene concentration data found in water samples obtained from two
background wells (Wells 1 and 2) and a compliance well (Well  3). Compute the upper prediction limit
for a future mean of order 4 with 99% confidence and determine whether there is evidence of possible
chrysene contamination.

Month
1
2
3
4
Mean
SD
Log-mean
Log-SD

Well 1
6.9
27.3
10.8
8.9
13.47
9.35
2.451
0.599
Chrysene (ppb)
Background
Well 2
15.1
7.2
48.4
7.8
19.62
19.52
2.656
.881

Joint




16.55
14.54
2.533
.706
Compliance
Well 3
68.0
48.9
30.1
38.1
46.28
16.40
3.789
0.349
SOLUTION
Step 1.   Before constructing the prediction limit, check the key assumptions. Assuming there is no
         substantial natural spatial variability and it is appropriate to combine the background wells
         into a single data pool, the algorithm for  a parametric prediction limit presumes that the
         background data jointly originate from a single normal population. Running the Shapiro-Wilk
         test on the pooled set of eight background measurements gives SW = 0.7289 on the original
                                             18-15
        March 2009

-------
Chapter 18. Prediction Limit Primer                                       Unified Guidance

         scale and SW = 0.8544 after log-transforming the data. Since the critical point for the test at
         the a = .10 level of significance is sw.\o,s = 0.851 (from Table  10-3 of Appendix D), the
         results suggest that the data should be fit to a lognormal model. The log-transformed statistics
         for the joint background and compliance well are also found in the above table.

Step 2.   Construct the prediction limit on the pooled and logged background observations. Then n = 8,
         the log-mean is 2.533, and the log-standard deviation is 0.706. Since there are 4 observations
         in the compliance well, take/? = 4 as the order of the mean to be predicted. Then setting (1-a)
         = .99, the Student's ^-quantile with (n-\) = 7 degrees of freedom and cumulative probability of
         .99 is found from Table 16-1 in Appendix D to  be 2.998. Using equation [18.8], the upper
         prediction limit on the log-scale is computed as:


                        PL = 2.533 + (2.998)(0.706)J- + - = 3.83 log(ppb)


Step 3.   Compare the log-mean of the chrysene measurements at Well 3 against the upper prediction
         limit.  Since it is less than the limit, there is insufficient evidence of chrysene contamination at
         this well at the a = 0.01 significance level. ~4
18.3 NON-PARAMETRIC PREDICTION  LIMITS

     Two basic remedies are available when a data set cannot be even approximately normalized, often
due to the presence of a significant fraction  of non-detects. If the sample includes left-censored data
(e.g., non-detects), a fit to normality can be attempted using censored probability plots (Chapter 15) in
conjunction  with  either the Kaplan-Meier or Robust Regression on Order Statistics [Robust ROS]
techniques (Chapter 15). If a reasonable normality fit can be found, a parametric prediction limit can be
applied.  Otherwise, a non-parametric prediction limit can be considered. A non-parametric upper
prediction limit is constructed by setting the  limit as  a large order statistic selected from background
(e.g., the maximum or second-largest background value).

     As with their  parametric  counterparts,  non-parametric  prediction  limits have  an  associated
confidence level (1-a) which indicates the probability that the prediction interval [0, PL] will accurately
contain all m of a set of m future values over repeated application on many similar data sets.  Unlike
parametric limits,  the confidence level for non-parametric limits is not adjustable. Despite being easily
constructed for a  fixed background sample size and the number of comparisons, the confidence level
associated with the any  maximal value used  as the prediction limit is also fixed. To increase the
confidence level,  the primary  choices are to decrease the  number of future values to be predicted, or
increase the  number of background observations.

     If existing background can be supplemented with data collected from other background wells (e.g.,
in interwell  testing), a non-parametric test confidence level can be increased.  Larger samples also
provide a better characterization of site spatial variability. Unfortunately, it may always not be possible
to supplement background. In these cases, another option to achieve a desired confidence level  and
                                             18-16                                   March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance

correspondingly control the false  positive rate is to incorporate a retesting strategy  as outlined in
Chapter 19

     Although non-parametric prediction limits  do not require a presumption  of normality, other
assumptions apply equally to  both parametric and non-parametric limits.  Checks should be made of
statistical independence, identical distributions (under the null hypothesis), and stationarity over time
and space as discussed in Chapter  3 and Part II of the guidance. One particular caution for non-
parametric limits is that background should ideally be screened ahead of time for possible outliers, since
the upper prediction limit may be set to the background maximum or second highest observed value.
Unfortunately, this often cannot be accomplished  with  a formal statistical  test. Outlier tests are rather
sensitive to the underlying distribution of the data. If this distribution cannot be adequately determined
due to the presence of non-detects, an outlier test is not likely to give reliable results.

     Instead of a formal test,  it may be possible to screen for outliers using box  plots (Chapter  12).
Even with  non-detects, the box plot 'whiskers' delineating the concentration range associated with
possible  outliers are computed from the  sample  lower and upper quartiles (i.e., the  25th  and 75th
percentiles), which may  or may not be  impacted by data  censoring, or perhaps mildly  so when
computing the lower quartile. For large fractions of non-detects, the best that can usually be done is to
identify a suspected outlier through close examination of laboratory results and chain-of-custody reports.

     One of two steps can be taken in the event a possible outlier is flagged. If an error has occurred, it
should be corrected before constructing the prediction limit. If an error is merely suspected but cannot
be proven, the prediction limit can be constructed  as another order statistic from background instead of
the maximum (e.g., the second largest value). This  will prevent the suspected outlier from being adopted
as the upper prediction limit without ignoring the possibility that it may be a real measurement.
18.3.1       PREDICTION LIMIT FOR M FUTURE VALUES

       BACKGROUND AND PURPOSE

     Given n background measurements and a desired confidence level  (1-a),  a non-parametric
prediction limit test for m future values is an m-of-m comparison rule. All m future samples need to not
exceed the upper prediction limit for the test to pass.  Thus the procedure is an exact parallel  to the
parametric prediction limit for future values. Because the method is non-parametric, no distributional
model needs to be fit to the background measurements. It is assumed that the compliance point data
follow the same distribution  as background under the null hypothesis — even if this distribution is
unknown.  Although no distributional model is assumed,  order statistics of any random sample follow
certain probability laws which allow the statistical properties of the non-parametric prediction limit to be
determined.

     Once an order statistic  of the  sample  data (e.g., the maximum value) is selected as the upper
prediction limit, Guttman (1970) has shown that the statistical coverage of the interval — that  is, the
fraction of the  background population  actually  contained  within the  prediction interval — when
constructed repeatedly over many data sets, has a beta probability density with cumulative  distribution
equal to

                                             18-17                                   March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance
where n = sample size,y = (rank of prediction limit value), and F(W) = (n-\)\ = (w-l)x(w-2)...x2xl
denotes the gamma function. If the maximum is selected as the prediction limit, its rank is equal to n and
507 = n. If the second largest value is chosen as the limit, its rank would be equal to (w-1) and soy = (n-
1).  The confidence probability for predicting that one future observation (i.e., m = 1) from a compliance
well does not exceed the prediction limit is equal to the expected or average coverage of the non-
parametric prediction limit.

     Because of these properties, the confidence probability for a prediction limit on one future
measurement can be shown to equal (1-oc) =jl(n+\). If the background maximum is taken as the upper
prediction limit, the  confidence level thus becomes  n/(n+l).  Gibbons (1991a) has  shown that the
probability of having m future samples all not exceed such a limit is (1-a) = nl(n+m). More generally,
the same probability when they'th order statistic is taken as the upper prediction limit becomes (Davis
and McNichols, 1999):

                                            .             ..
     Table 18-1 in Appendix D lists these confidence levels for various choices of j, n, and m. The
false positive rate  (a) associated with a given prediction limit can be  computed as one minus the
confidence level.  As  this table illustrates, the penalty for not knowing the form of the underlying
distribution can be severe.  If  a non-parametric  prediction  limit is to  be used, more background
observations are needed compared to the parametric setting in order to construct a prediction interval
with sufficiently high confidence. As an example, to predict m = 2 future samples with 95% confidence,
at least 38 background samples are needed. Parametric prediction intervals do not  require as many
background measurements precisely because the form  of the  underlying distribution is  assumed to be
known.

     It is possible to  create  an  approximate non-parametric limit with background data  containing all
non-detects, by using  the RL (often a quantitation limit) as the PL. A quantified value  above the PL
would  constitute an exceedance.  A superior procedure is recommended in this  guidance, using the
Double Quantification Rule described in Chapter 6.

       PROCEDURE

Step 1.   Sort the  background  data into ascending order and set the prediction limit equal to the
         maximum,  the second-largest  observed value or another large background  order statistic.
         Then use Table 18-1 of Appendix D to determine the confidence level (1-oc) associated with
         predicting the next m future samples.

Step 2.   Compare each of the  m compliance point measurements to the upper prediction limit [PL].
         Identify significant evidence of possible contamination at the compliance well if one or more
         measurements exceed the PL.

                                            18-18                                   March 2009

-------
Chapter 18.  Prediction Limit Primer
Unified Guidance
Step 3.   Because the risk of false positive decision errors is greatly increased if the confidence level
         drops substantially below a target rate of at least 90% to 95%, the actual confidence level (as
         identified by equation [18.11]) needs to be routinely reported and noted whenever it is below
         the target level.

         Note that equation [18.11] assumes the prediction limit is applied to only one constituent at a
         single well. If multiple tests must be conducted and  a  retesting procedure is employed,  the
         confidence level of the prediction limit must be determined using the tables described in
         Chapter 19

       ^EXAMPLE  18-3

     Use the following trichloroethylene data to compute a non-parametric upper prediction limit for the
next m = 4  monthly measurements from a downgradient well  and determine the level  of confidence
associated with the prediction limit.

Month
1
2
3
4
5
6
Trichloroethylene Concentrations (ppb)
BW-1
<5
<5
8
<5
9
10
Background Wells
BW-2
7
6.5
<5
6
12
<5
BW-3
<5
<5
10.5
<5
<5
9
Compliance
CW-4


7.5
<5
8
14
SOLUTION
Step 1.   Determine  the  background maximum  and use this value to estimate the non-parametric
         prediction  limit.  In  this  case, the maximum  value of the n  =  18 pooled background
         observations is 12 ppb. Set PL = 12 ppb.

Step 2.   Compare each of the downgradient measurements against the prediction limit. Since the value
         of 14 ppb for Month 6 exceeds PL, conclude that there is statistically significant evidence of
         an increase over background at CW-4.

Step 3.   Compute the confidence level  and false positive rate associated with the prediction limit.
         Since four future samples are being predicted and n = 18, the confidence level equals nl(n + m)
         = 18/22 = 82%. Consequently, the Type I error or false positive rate is at most (1 - 0.82) =
         18% and the test is significant at the a = 0.18 level. This means there is nearly a one in five
         chance that the test has been falsely triggered. Only additional background data and/or use of a
         retesting strategy would lower the false positive rate. -4
                                             18-19
        March 2009

-------
Chapter 18.  Prediction Limit Primer                                       Unified Guidance



18.3.2       PREDICTION LIMIT FOR A FUTURE MEDIAN

       BACKGROUND AND PURPOSE

     A prediction limit for a future median is a non-parametric alternative to a parametric prediction
limit for a future mean (Section  18.2.2)  when  the  sample cannot be  normalized. In groundwater
monitoring, the most practical application for this kind of limit is for medians of order 3 (i.e., the median
of three consecutive measurement values), although the same procedure could theoretically be employed
for medians of any odd order (e.g.., 5, 7, etc.). The comparison rule in this case is that the test passes only
if the median of a set of 3 compliance point measurements does not exceed the upper prediction limit.
Note that this is also the same as a 2-of-3 test, whereby the well is deemed in compliance if at least 2 of
3  consecutive observations  fall  within  the  prediction  interval. Therefore, only  2 independent
observations will  generally be  needed  to  complete  the test  at uncontaminated wells.   The third
measurement will be irrelevant if the first two pass and so will not need to be collected.

     Given n  background measurements  and a desired confidence level  (1-oc),  a non-parametric
prediction limit for a future median involves a confidence probability that the median of the next/? future
observations will not exceed the limit. As noted in Section 18.3.1, order statistics of any random sample
follow certain probability laws. In particular, the statistical coverage (C) of a prediction limit estimated
by they'th order statistic (that is, they'th largest value) in background will follow a beta distribution with
parameters j and (n+\-f). Following the notation  of Davis  and McNichols (1987), the  conditional
probability that the median of 3 independent future values will not exceed the non-parametric prediction
limit can be shown to equal

                          Pr -{Future median inbounds \X..n J= 3C2 - 2C3                   [18.12]

where Xj-n  denotes that the prediction  limit equals the y'th largest order statistic in a sample  of n
observations and a  conditional  probability denotes the chance that  an event will occur given the
observance of another event (in  this case, after having observed ^j:n). The (unconditional) confidence
probability (1-00 can then be derived by taking the expected value of equation [18.12] with respect to
the random variable C. Using standard properties of the beta distribution, this probability becomes:
                                          kn -2j + 5Y; +
                                                '   A'                                 [18.13]
     Thus the confidence level associated with a prediction limit for a future median of order 3 depends
simply on the sample size of background (n) and the order statistic selected as the upper prediction limit
(/). Table 18-2 in Appendix D provides values of the confidence level for various n and choices of the
order statistic. Like the non-parametric prediction limit for m future values, ease of construction comes
with a price.  More background measurements are  required to achieve the same levels of confidence
attainable via a parametric prediction limit for a future mean. For instance, to achieve 99% confidence in
predicting  a median of order 3 in a single  test, at least 22 background observations are needed if the
maximum is  selected as the upper prediction limit, and at least 40 background observations are needed if
the prediction limit is set to the second largest measurement. Parametric prediction intervals  do not
                                             18-20                                   March 2009

-------
Chapter 18. Prediction Limit Primer                                       Unified Guidance

require  as many background samples precisely because  the  form of the underlying  distribution is
assumed to be known.

       REQUIREMENTS AND ASSUMPTIONS

     Once an order statistic (of rank f) is selected as the upper prediction limit, the confidence level is
fixed by the number of background samples (n). The confidence level can only be increased by enlarging
background. However, equation  [18.13] is only applicable for the case of predicting a future median of a
single constituent at a single well. To account for multiple tests and to incorporate a retesting strategy
(both of which are usually needed), the specific strategies  and  tables of confidence levels presented in
Chapter 19 should be consulted.

       PROCEDURE

Step 1.   Sort the background data into ascending order and set the upper prediction limit [PL] equal to
         one of the following:  the background maximum, the second largest value, or another large
         order statistic in background.  If the largest background measurement is a non-detect, set an
         approximate upper prediction limit as the RL most appropriate to the data (usually the lowest
         achievable quantitation limit [QL]).

Step 2.   Compute the median of the next three consecutive compliance point measurements. Compare
         this  statistic to  the  upper  prediction   limit.  Identify significant evidence  of  possible
         contamination at the compliance well if the median exceeds PL.  If PL equals the RL, identify
         an exceedance, if the median is quantified above the reporting limit.

Step 3.   Based on  the background sample size («), use Table 18-2 of Appendix D to determine the
         confidence level  (1-oc) associated  with  predicting the median  of the  next p = 3  future
         measurements. Because the risk of false positive errors is greatly increased if the confidence
         level drops much below a targeted rate of at least 90% to 95%, the actual confidence level (as
         identified  in equation [18.13]) should be routinely reported and noted whenever it is below the
         target level.

         Note that  equation [18.13] assumes the prediction limit is applied to only one constituent at a
         single well. If multiple tests are  conducted  and a retesting procedure  is employed,  the
         confidence level of the prediction limit needs to be determined using the tables described in
         Chapter 19

       ^EXAMPLE 18-4

     Use the following xylene background data to establish a non-parametric upper prediction limit for
a future median of order 3. Then determine if the compliance well shows evidence of excessive xylene
contamination.
                                            18-21                                   March 2009

-------
Chapter 18. Prediction Limit Primer
Unified Guidance
Month
1
2
3
4
5
6
7
8
Well 1
<5
<5
7.5
<5
<5
<5
6.4
6.0
Xylene Concentrations (ppb)
Background
Well 2 Well 3
9.2
<5
<5
6.1
8.0
5.9
<5
<5
<5
5.4
6.7
<5
<5
<5
<5
<5
Compliance
Well 4





<5
7.8
10.4
SOLUTION
Step 1.   The maximum value in the set of pooled background measurements is 9.2. Assign this value
         as the non-parametric upper prediction limit, PL = 9.2.

Step 2.   Compute the median of the  three compliance measurements. This statistic  equals 7.8 ppb.
         Since the median does not exceed PL, there is insufficient evidence of xylene contamination at
         Well 4, despite the fact that the maximum at Well 4 is larger than the maximum observed in
         background.

Step 3.   Compute the confidence level and false positive rate associated with this prediction limit.
         Given that n = 24 and the order statistic selected is the maximum (i.e.J = n\ use Table 18-2
         in Appendix D to determine that the confidence level for predicting a future median of order 3
         equals 99.1% and therefore the Type I error or false positive rate is at most 0.9%.  -4
                                            18-22
        March 2009

-------
Chapter 19.  Prediction Limits with Retesting                             Unified Guidance

    CHAPTER 19.   PREDICTION  LIMIT STRATEGIES WITH
                                   RETESTING
        19.1  RETESTING STRATEGIES	19-1
        19.2  COMPUTING SITE-WIDE FALSE POSITIVE RATES [SWFPR]	19-4
          19.2.1   Basic Subdivision Principle	19-7
        19.3  PARAMETRIC PREDICTION LIMITS WITH RETESTING	19-11
          19.3.1   Testing Individual Future Values	19-15
          19.3.2   Testing Future Means	19-20
        19.4  NON-PARAMETRIC PREDICTION LIMITS WITH RETESTING	19-26
          19.4.1   Testing Individual Future Values	19-30
          19.4.2   Testing Future Medians	19-31
     This chapter is a core part of the recommended statistical approach to detection monitoring. Even
the smallest of facilities will perform enough statistical tests on an annual basis to justify use of a
retesting strategy. Such  strategies are described in detail in this chapter in conjunction with prediction
limits.  First, the Unified Guidance considers the concept and computation  of site-wide false positive
rates [SWFPR].  Then different retesting strategies useful for groundwater monitoring are presented,
including:

    »»»  Parametric prediction limits with retesting (Section 19.3), and
    »«»  Non-parametric prediction limits with retesting (Section 19.4)
 19.1 RETESTING STRATEGIES

     Retesting is a statistical strategy designed to efficiently solve the problem of multiple comparisons
 (i.e., multiple, simultaneous statistical tests). An introduction to multiple comparisons is presented in
 Chapter 6. At first glance, formal retesting seems little different than a repackaged form of verification
 resampling, a practical technique used  for years to double-check  or  verify the results of initial
 groundwater sampling. Indeed,  all retesting schemes  are predicated on the idea that when the initial
 groundwater  results  indicate the presence of  potentially contaminated groundwater,  one or more
 additional groundwater samples should be collected and tested to determine whether or not the first
 results were accurate.

     The difference between  formal retesting  schemes and  verification resampling  found  in  the
 regulations is that the former explicitly incorporates the resample(s) into the calculation of the statistical
properties of the overall test. A statistical "test" then needs to be redefined to include not only the
 statistical manipulation of the  initial groundwater sampling  results, but also that for any further
 resamples. Both the initial samples and the resamples are integral components of any retesting method.

     The principal advantage of retesting is that very large monitoring networks can be  statistically
 tested without necessarily sacrificing either an acceptable false positive rate or adequately high effective
power. Data requirements for a typical retesting scheme are often less onerous than those required for an
 analysis of variance (ANOVA). Instead of having to  sample each well perhaps four times during  any
                                              19^1                                    March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

given evaluation period,  many of the retesting strategies discussed below involve a minimum of one
new sample at each compliance well.   Resamples are collected only at wells where the initial results
exceed a limit, and no explicit post-hoc testing of individual wells is necessary as with ANOVA in order
to identify a contaminated well.

     Since a statistical test utilizing retesting is not complete until all necessary resamples have  been
evaluated, it is important to outline  the formal decision rules or scheme associated with each retesting
strategy. Retesting schemes presented in the Unified Guidance fall into two types:  l-of-m  and the
modified California approach. The  l-of-m approach was initially  suggested by Davis  and McNichols
(1987) as  part of a broader method termed "p-of-m." The l-of-m  scheme assumes that as many  as m
samples might be collected for a particular constituent at a given well, including the initial groundwater
sample and up to (m-l) resamples.

     l-of-m  schemes  are  particularly attractive  as retesting strategies. If  the initial  groundwater
observation is in-bounds, the test is complete and no resamples need to be collected. Only when the first
value exceeds the background prediction limit, does additional sampling come into play. For practical
reasons, only l-of-m schemes with m no greater than 4 are considered in the Unified Guidance. A l-of-4
retesting plan implies that up to  4 groundwater measurements  may  have to be collected  at  each
compliance well, including the initial observation and 3  possible resamples. For the test to be valid, all
of these sample measurements need to be statistically independent. This generally requires that sufficient
time elapses between resample collection so that the assumption of statistical  independence or lack of
autocorrelation is reasonable (see the discussion in  Chapter 14). Because many groundwater evaluations
are  conducted on a semi-annual basis,  three will generally be a practical upper bound on the number of
independent resamples that might be collected. Thus the l-of-2,  l-of-3, and l-of-4 retesting schemes are
included below.

     The  second type of retesting scheme is known as the modified California approach. The decision
rules for this test are slightly different from the l-of-m schemes, although the test passes as before if the
initial groundwater measurement  is inbounds.  If it exceeds the background limit,  two  of the three
resample need to be inbounds for the test to pass. The modified California strategy  thus requires  a
majority of the resamples to be inbounds for a compliance well test to be deemed 'in bounds'. A l-of-4
scheme could have both the initial value and the first two resamples be out-of-bounds, yet pass the test
with an inbounds result from the third resample.  Although the modified  California test appears to be
more stringent, the prediction limit for a l-of-4 test under the same input  conditions will be lower and
hence be more likely to trigger single comparison exceedances.   With the prediction  limits  correctly
defined, both will have identical false positive errors for any specific monitoring design.  The guidance
also provides the same four non-parametric versions of the l-of-m and modified California tests for
future values.

     A useful variation on the l-of-m  retesting  scheme for individual  measurements is the  l-of-m
strategy for means or medians. Instead of testing a series of individual values, a series  of means  or
medians of order p is tested.  The order of the mean  or median refers  to the number of individual
measurements used to compute the statistic. For example, l-of-2 retesting with means of order 2 requires
that a pair of initial observations be averaged and  the resulting mean compared against the background
limit. If that initial mean is out-of-bounds, a second pair of observations (i.e., two resamples) would be
                                              19-2                                    March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

collected and averaged to form the resample mean. The test would fail only if both the initial mean and
the resample mean exceeded the background limit.

     Retesting schemes for means or medians have steeper data requirements than retesting strategies
for individual measurements and may  not  be  practical at many sites. Nevertheless, the statistical
properties (e.g., power and false positive rate)  associated with the testing of means and medians are
superior to comparable  plans on  individual  observations. The  Unified Guidance provides five mean
retesting plans: 1-of-l, l-of-2, or l-of-3 for means of order 2; and 1-of-l and l-of-2 for means of order
3.  The guidance also provides 1-of-l and l-of-2 tests of medians of size 3 as non-parametric options.

     These plans  were  chosen to limit the maximum possible number of distinct and independent
sampling measurements per compliance well during a single evaluation period to six. In fact,  the data
requirements  vary substantially by scheme. With means of order 2, the 1-of-l plan requires a maximum
of two new sample measurements; the l-of-2 plan requires as many as four; while only the l-of-3 plan
might need a  total of six. For means of order 3, the 1-of-l plan requires three new measurements to form
the single mean; the  l-of-2 plan might require up to six.  But for higher order \-of-m mean or median
tests, only the initial  samples may be  needed to identify a 'passing' test outcome under most background
conditions.

     The  three 1-of-l mean and  median plans  provided in the guidance are technically not retesting
schemes. The decision rule for these plans merely requires a  comparison of a single mean or median
against the background limit. If the initial mean or median comparison is inbounds, the test passes. If
not, the test fails. The fact that each average is computed from multiple individual measurements implies
that an implicit retest or verification resampling is built into these strategies. The statistical properties of
the 1-of-l plans can  often be better than comparable \-of-m schemes for individual values, with fairly
similar sampling requirements.

     The  Unified Guidance provides 1-of-l and l-of-2 non-parametric prediction limit tests for future
medians of order 3. By  'median of order 3', it means that the median  or 'middle value' of a set of three
consecutive sampling events. In the l-of-2 case, the test passes if either the initial median is inbounds or,
if not, when the resample median is inbounds. The 1-of-l scheme does not involve any resampling, but
does require  at least two distinct sampling measurements to  determine whether the initial median is
inbounds.1

     As discussed in Chapter 6, proper design of a groundwater detection monitoring program will
generally require an initial choice of a retesting scheme before future or compliance sampling data have
been collected. As a practical matter, sample collection should be spaced far enough apart in time to
ensure that any potentially needed resamples are statistically independent. Thus, the maximum number
of resamples  need to be  known in advance in order to structure a feasible sampling plan for a particular
retesting strategy.  Each retesting scheme also involves a different set of decision rules for evaluating the
status of  any given  compliance well. The rules will determine how the background  limit will  be
computed.  Given the same background sample and  group of compliance wells,  different retesting
schemes lead  to different background  limits on the same data.
1 As noted in Chapter 18, the 1-of-l retesting scheme for medians of order 3 is equivalent as a decision rule to a 2-of-3
  scheme for individual measurements.

                                              ilTs                                    March 2009

-------
Chapter 19.  Prediction Limits with Retesting                               Unified Guidance

     If parametric prediction limits are used, the general formula for the limit introduced in Chapter 18
isx + KS .   The ^-multiplier and thus the prediction limit will vary depending on which  l-of-m or
modified California plan is chosen. The ^-multipliers also depend on the monitoring evaluation schedule
in place  at the facility. In typical  applications,  it is  expected that  the  background  sample used in
statistical  evaluations from any given year will either be static or  substantially overlap  from  one
evaluation  to the next.  The same background observations are likely to be utilized or will substantially
overlap if newer background data are added to the existing  pool.  Since at least a  subset  of the
background measurements will be commonly employed in all the evaluations, there will be a statistical
dependence exhibited between distinct evaluations (see Section 19.2 below). The number of evaluations
per year against a  common  background will affect  the correct identification of prediction  limits.
Consequently, the evaluation  schedule (i.e., annual, semi-annual, quarterly)  also needs to be known or
specified in advance.2
19.2 COMPUTING SITE-WIDE FALSE  POSITIVE RATES [SWFPR]

     As discussed in Chapter 6, the fundamental  purpose of detection monitoring is to accurately
identify a significant change in groundwater relative  to background conditions. To meet this objective,
statistical monitoring programs should be designed with the twin goals of ensuring adequate statistical
power to flag well-constituent pairs elevated above background levels and limiting the risk of falsely
flagging uncontaminated wells across an entire facility. The latter is accomplished by addressing the site-
wide false positive rate [SWFPR]. Both goals contribute to accurate evaluation of groundwater and to
the validity of statistical groundwater monitoring programs.

     Retesting significantly aids this process of meeting both criteria. However, it  can be much easier
to design and implement an appropriate retesting scheme if one understands how the SWFPR is derived.
The SWFPR is based on the assumptions that no contamination is actually present at on-site monitoring
wells, and that each well-constituent pair in the network behaves independently of other constituents and
wells from a statistical standpoint.  If Q denotes the probability that a particular well-constituent pair will
be falsely declared  an exceedance (a false positive  error), the probability  of at least one such  false
positive error among r independent tests is given by:

                                                                                            [19.1]

(l-<2) equals the chance that the test will  correctly identify  the well-constituent pair as 'inbounds.' The
value of Q itself will depend on the type of retesting scheme being used.
  The Unified Guidance distinguishes between the statistical evaluation (or testing) schedule and the sampling schedule.
  Regularly scheduled sampling events might occur quarterly, even though a statistical evaluation of the data only occurs
  semi-annually or annually. Further, resamples do not constitute regular sampling events, since they are only collected at
  wells with initial exceedances, but they are associated with the data for a particular evaluation. By separately identifying
  the evaluation schedule, there is 1) less confusion about the role of resamples in the testing process, and 2) opportunity to
  design monitoring programs, so as to allow for multiple individual observations to be collected prior to each evaluation.

                                               19-4                                      March 2009

-------
Chapter 19.  Prediction Limits with Retesting                               Unified Guidance

     Consider a l-of-3  retesting  plan for future observations.   A  false positive  at a  given well-
constituent pair will be registered only if all three observations — the initial groundwater measurement
and two resamples — exceed the background prediction or control limit. If oo represents the probability
that one of these observations exceeds the background limit, Q can be calculated as  Q)XQ)XQ) (since
the initial measurement and resamples are statistically independent) and the SWFPR as:

                                                                                          [19.2]

     By setting the target site-wide a equal to 0.10 and solving for oo, one could potentially  compute the
individual comparison false positive rate (aCOmp = co) associated with any single comparison against the
background limit.   This would identify the individual per-comparison confidence level (1 - Ocomp)
necessary to compute the background limit in the first place.3 If the background limit is computed as a
prediction limit for the next single future measurement (i.e., m = 1 in a l-of-m scheme), then oo equals
the probability that a single new observation (independent of background) exceeds the prediction limit,
and (l-oo) equals the confidence level of that prediction limit.  Further, since oo  can  be obtained from
equation [19.2] as:
                                       fi> = ^l-(l-ff)                                    [19.3]

the upper prediction limit for a site involving 500 tests (for instance, 50 wells and 10 constituents per
well) and 20 background  samples could be  computed using an individual, per-comparison confidence
level of
                          1 - (O = 1 - jjl - (l - . 10 j/5°° = 1 - .0595 = 94.0%

leading to a final prediction limit of
where x and s are the background sample mean and standard deviation.

     Unfortunately,  certain  statistical  dependencies  render the foregoing  calculations somewhat
inaccurate. Whether or not a resample exceeds the background limit for any constituent depends partly
on whether the initial  observation for that  test also eclipsed the limit.  This  is because the  same
background data are used in the  comparison  of both the  initial groundwater measurement and the
resamples. This creates a statistical dependence between the comparisons,  even when the compliance
point observations themselves are statistically independent. If the background data sample mean happens
to be low relative to the true population mean, the background limit will tend to be low.  Each of the
compliance point observations  (whether the first measurement or subsequent resamples) will have a
3 Note that acomp does not represent the false positive rate for the complete 1 -of-3 test, but is being treated for the sake of
  argument as a one of a series of 3 individual and independent tests.
                                              ilTs                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                               Unified Guidance

greater than expected chance of exceeding it. Likewise, if the background sample mean is substantially
higher than the population mean, the background limit will tend to be high, resulting in a lower-than-
expected chance of exceedance for each of the compliance measurements.

     A similar dependence occurs for each well-constituent pair tested against a single background
across evaluation periods (see discussions in Chapter 5 and Section 19.1). A further dependence occurs
when well-constituent  pairs  from many  compliance wells are compared  to  a  common  interwell
background.  The tests during each statistical  evaluation again share a common (or nearly common)
background, thus impacting the individual  test false positive rate (atest) and the SWFPR (a) in turn.
Three common evaluation strategies are considered in the Unified Guidance: quarterly, semi-annual,  and
annual.  The SWFPR is computed on a cumulative, annual basis, with the assumption that background
and the associated background limit will not be updated  or recomputed (especially for intrawell tests)
more often than every one to two years.4

     These dependencies between successive comparisons and tests against the background limit during
retesting means that the derivation above will generally not result in a background limit with the targeted
annual  SWFPR of  10%.  The actual false  positive rate (a) will be somewhat  higher  and can be
substantially higher if the background sample size (n) is small to moderate (say less than 50 samples). In
part, this is because the correlation between successive comparisons against a common background limit
is on the order of !/(!+«).  That is, the smaller the background size, the greater the correlation between
the resamples and test comparisons.  The impact on the  SWFPR is  also greater if this dependence is
ignored.

     Fortunately, as Gibbons (1994) has noted, the solution suggested in the previous example will be
approximately valid for large background data sets  (say n > 50), since then the  correlation between
successive  resamples  and/or  tests is minimal.  In  fact, an  approximate  solution  for the  modified
California  and more general  l-of-m retesting schemes  can  also be derived. In  the  case of l-of-m
schemes, the probability Q of a false positive (for m = 1 to  4) is Q)m, leading to a SWFPR of:

                                                                                           [19.4]

     Solving for co in equation [19.4] leads to an approximate individual comparison false positive  rate
     p = co) of:
                                                                                           [19.5]

     For the modified California plan, a false positive for a given well-constituent pair during a single
evaluation will be registered only if both the initial measurement and at least two of three resamples are
  Even with these assumptions, not all the statistical dependence will be accounted for at every site or for all constituents.
  Even when background is updated with new measurements, some of the already existing background values are likely to be
  used in re-computing the background limit. Some well-constituent pairs may be correlated, contradicting the assumption of
  independence between tests at the same well or for the same constituent at different wells. The Unified Guidance also does
  not presume to compute the SWFPR for other multi-year periods or for the life of the facility.

                                               19-6                                    March 2009

-------
Chapter 19. Prediction Limits with Retesting                              Unified Guidance

out-of-bounds (i.e., exceed the background limit). Consequently, the probability Q of a false positive for
that pair may be expressed as:

                                                                                        [19.6]

As before, oo represents the probability of any single observation exceeding the background limit. Both
the initial and any resample comparisons against the limit are assumed to be statistically independent.
Given Q, the approximate overall false positive rate then becomes:
                               a = SWFPR = \-l-G? 4 -3(0                            [19.7]

Since oo will always be small in practice, one can usually ignore the term oo4 when expanding the right-
hand side of equation [19.7]. Then the approximate SWFPR becomes:
                                                                                        [19.8]
                                             i_       _i

Leading to a solution for oo:


                                     «^l-(l-tfj/r^-                                [19.9]


which can again be used to construct a background limit for a single new observation.

     As an  example, if the target SWFPR is 10% and one must test r = 200 comparisons using the
modified California plan, oo would become:
                               (O » >/l - .901/20°I - = .0508 = 5.1%
                                             V4

If the background limit is a prediction limit for the next future value, a confidence level of approximately
94.9% would be needed to achieve the desired overall false positive rate of 10%. This assumes that the
background  sample size  is sufficiently large (say n  > 50) to make the correlation between retests
negligible.  In similar fashion, the respective single comparison error rates for the l-of-2 through l-of-4
tests of future observations in this example would respectively be: co = .0229, .0808,  and .1515.
19.2.1       BASIC SUBDIVISION PRINCIPLE

     The previous section highlighted certain dependencies in statistical tests due to comparisons of one
or more samples  or sample sets against a common background.  In the sitewide design of a facility
detection monitoring system,  the overall  target design SWFPR is proportionately  divided  among all
relevant tests conducted in an annual period.  Depending on the type of testing (e.g., interwell versus
                                             19-7                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                               Unified Guidance

intrawell,  or a parametric versus non-parametric), the target error rates for a portion of the total set of
potential tests may need to be calculated.

     Identifying  false positive  target rates is important when considering non-parametric prediction
limit tests.  The cumulative target error rate for a group of annual tests against a single constituent is
needed to compare with the achievable levels in Tables 19-19 through  19-24 in Appendix D.  The
latter achievable rates take into account the dependencies previously discussed,  /c-multiple Tables 19-1
to 19-18 in Appendix D for parametric prediction  limit tests have already made use of target false
positive rate calculations which are generally not needed for identifying the appropriate multipliers.  The
various dependencies against a common background  are accounted for in the /c-multiple tables to meet
the nominal target rates.  R-script software  for certain parametric prediction limit tests discussed in a
following section and in Appendix C also makes use of a target per-test false positive error rate as input.

     In assigning target rates, the Unified Guidance uses  a basic subdivision principle which makes
certain assumptions. First and foremost, it is assumed that the total suite of tests can be subdivided into
mutually exclusive, independent  tests.   Each relevant annual statistical test is assigned the same single
test error rate (atest).  Using the properties of the Binomial distribution, the target single test error rate can
be obtained using equation [19.10] for r total annual tests.  The total number of annual tests r is  the
product of the number of compliance wells (w), the number of valid constituents (c), and the number of
evaluations per year («#) or r = w x c x «#, with a = SWFPR:

                                         atest=l-(l-a}l!r                                  [19.10]

     Then a cumulative  false positive rate can be assessed for any appropriate subset of tests. This
principle would apply, for instance, if there is more than one regulated unit at a site and each regulated
unit can be treated independently. A consistent portion of the overall targeted false positive rate a would
be assigned to each regulated  unit (aunit), using  a rearrangement of equation [19.10].  If a facility with
three units B, C, and D had 120 total annual tests (b + c + d= 120 = r), the cumulative target error rate
for Unit B would be:  aUnitg =1 - (\-atest)band  similarly for Units C and  D.   These three cumulative
error rates will approximately (but  not exactly) sum to a total sitewide  value close to the  SWFPR.
However, as joint independent  tests taken together,  the annual SWFPR is in fact exactly  10%.  The
Bonferroni  assumption makes  use  of the approximately  linearity  of such  error rates  for SWFPR
calculations (discussed below).

     The ways in which the overall SWFPR might be partitioned will vary with each site, considering
units, types  of tests, number of wells, constituents and evaluations per year. If unit-specific cumulative
false positive rates were established, the  group of tests associated with  each monitoring constituent
within each unit  might be separately considered.  Each group might potentially be further subdivided
into intrawell versus interwell tests, or prediction limits versus control charts, etc., assuming a mixture of
statistical  methods is employed. By using the  subdivision principle in a  consistent way, the targeted
SWFPR can be accurately maintained.
5 The Unified Guidance does not presume that every statistical test is in fact independent.  Tests or groups are treated as if
  independent,  however, to  allow  the computation of nominal target false positive rates and/or to be consistent with
  regulatory constraints (e.g., all constituents must be tested separately).

                                               19-8                                     March 2009

-------
Chapter 19.  Prediction Limits with Retesting                               Unified Guidance

     One important use when calculating SWFPR rates  is to account for multiple constituents.   In
particular, non-parametric test theory is applied to only  a single constituent at  a time.  Since each
constituent has its own set of background data and presuming the constituents behave independently of
one another, the dependence caused by using a common background pertains only to those comparisons
made against the  background for that constituent. To clarify this concept, suppose a total set of r tests
consists of c separate chemicals each monitored at w wells annually (i.e.., r = ex wx ns and HE = 1). For
each constituent,  the dependence caused by a common background only applies to the w comparisons
(one for each well) made for that monitoring parameter. This means that the overall target a= SWFPR
needs to be apportioned into a fraction for each constituent, called the per-constituent false positive rate
or ac. This can be done using the Binomial formula based on the single test error rate for w wells as:
ac = 1 - (l-atest }W'"E or by partitioning the overall a to each constituent c:

                                   ac=l-(\-a)ltc

       The two calculations are equivalent under these conditions, with the latter equation somewhat
easier to use.

     A similar  situation  occurs at sites requiring a combination  of interwell  and  intrawell  tests.
Computation of the SWFPR can be appropriately handled using the basic subdivision principle. For
interwell tests, measurements collected at each  compliance  well  are  compared against a common
interwell background, creating a degree of statistical dependence not only between successive individual
test comparisons  (i.e., initial sample and any resamples) at  a given well, but also between tests at
different compliance wells. With intrawell tests, each  well supplies its  own background.  This implies
that the component of between-well test dependence is eliminated, changing the way K-multipliers for
intrawell background limits with retesting are computed.

     For a given set of r well-constituent pairs, / tests to be  conducted on an interwell basis, and the
remaining (r - /) tests conducted as intrawell, two cumulative false positive rates need to be computed.
The single test false positive error rate atest approach can be used: amier = 1 - (l - atest)' for the subset of /
interwell tests, and  amira =1 - (l-cctest)r~' for the subset  of  r   I intrawell tests,  in order to correctly
maintain  the  SWFPR  equal   to  a.  A   somewhat more   direct   approach   can  also  be
used: amter = 1 - (l-a)1/r for the  interwell tests and  amtra  = 1 - (l-df)(r"')/r for the intrawell tests.   The
two sets of equations are consistent.

     In general, the subdivision principle works as follows.  If a group of r tests with targeted false
positive rate, a, is divided into s distinct and mutually exclusive independent subsets, the false positive
rate for each subset (asub) can be computed as:
                                              19-9                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

     The basic subdivision principle does not guarantee that the resulting detection monitoring program
will have sufficient effective power to match the EPA reference power curve (ERPC). The foregoing
calculations merely point to the correct overall false positive rate.

     As discussed in Section 6.2.2 of Chapter 6, a simpler approach would be to partition the overall
SWFPR among a facility's annual number of tests, and can make use of the Bonferroni approximation.
With low false positive rates characteristic  of detection monitoring design, the  total  SWFPR can be
divided by the number of annual tests for any of the various combinations of constituents, separate units,
or interwell versus intrawell tests.  The Bonferroni approach results in slightly different false positive
values than by directly using the Binomial formula, as described above.

     As an overall example, assume a facility with w = 20 wells monitored twice per year  (HE = 2)  for c
= 8 constituents   Further, assume that 5 of the constituents can be monitored interwell and 3 need to be
handled as intrawell comparisons.  Non-parametric prediction limits will be considered for  all  tests.
Calculate the target cumulative false positive error rates for interwell and intrawell comparisons, with
the SWFPR = a = .1.
     This site  has  a total  of r  = wxcxns = 20x8x2 = 320 tests per year.   For the  five interwell
constituents, there are 20x2x5 = 200 tests, with 20x2x3 = 120  intrawell tests. Each of the 5 interwell
constituents will have 20x2 = 40 tests against a common background, while 2 semi-annual sample tests
will be made against each of the 20x3 = 60 intrawell backgrounds.

     From  equation  [19.10],  the  single   test false  positive  error   rate  is:cctest =1- (l-cc)
= 1 - (l- .l)1/32°  = .0003292  . Each set of interwell constituent tests will have a cumulative false positive
error rate ac  for the 40 annual  tests as:  ac = 1 - (l - or)1/c = l-(l-.l)1/8  =.01308.   Note that all 8
constituents are used in the equation, since the same false positive error rate is uniformly applied to all
distinct subgroup tests.    The  result  can be obtained using the  single  test error rate  equation:
ac=\- (l-atest)w'"E  = 1 - (1-.0003292)40 = .01308 . This target value would be used to compare with
achievable non-parametric test error rates for the same input conditions. The cumulative interwell error
rate for all five constituents  can be calculated as: amier = 1 - (l - ac )c = 1 - (l - .01308)5 = .06371 .

     For  the intrawell tests,  the simplest  approach uses the single test error rate for two tests:
a2-mtra = ! - ^-ateStT"E = 1 - (1-.0003292)12 = .0006583 . This would be  the cumulative error rate to
consider with non-parametric intrawell tests.   The overall intrawell cumulative error rate for the sixty
tests would then be: a60_mira = 1 - (l-tf2_mtrj"c = 1 - (1-.0006583)60 = .03873 .

     If the two overall interwell and intrawell cumulative  error rates were added, the sum  is .1024, quite
close to the nominal  10% SWFPR.   It is exactly that value when considered jointly.  By comparison the
single test error rate using the Bonferroni approximation  would be .1/320 = .0003125, while the exact
Binomial value is .0003292. The estimated interwell cumulative error for a single constituent would be
40 times the  single test value or .0125 (versus the calculated .01308). For many non-parametric test
considerations, these differences are relatively minor.
                                             19-10                                   March 2009

-------
Chapter 19. Prediction Limits with Retesting                              Unified Guidance



19.3 PARAMETRIC PREDICTION  LIMITS  WITH  RETESTING

       BACKGROUND AND  PURPOSE

     Upper prediction limits for m future observations and for future means were described in Chapter
18. Applied to a network of statistical  comparisons in detection monitoring, these procedures can be
considered an extension  to Dunnett's multiple comparison with control [MCC]  procedure (Dunnett,
1955). These procedures explicitly incorporate retesting that is applicable to a wider variety  of cases
than addressed by Dunnett.

     Retesting can be incorporated with either interwell or intrawell prediction limits. Depending on
which approach is adopted, there  is a distinct difference in the K-multipliers of the general  prediction
limit formula. In an interwell retesting strategy, there are at least two forms of statistical dependence that
impact the SWFPR.  One is that  each initial measurement or resample at a given compliance well is
compared against the same  background. A second is the dependence among  compliance  wells  and
number of annual evaluations, all  of which are compared against a common upgradient background. In
intrawell  retesting, this second form of dependence is either essentially eliminated if there is only one
annual statistical evaluation  or else substantially reduced in the event of multiple evaluations.6  The
remaining dependence is among successive resamples at each well.

     To  account for the basic  differences between interwell  and  intrawell prediction limit tests,  an
extensive series of tables is provided in Appendix D listing a wide combination of background sample
sizes, numbers of wells,  numbers of constituents, and distinctions between interwell and intrawell tests.
In conjunction with an evaluation  schedule (i.e., annual, semi-annual, or quarterly), these tables can be
used to design and implement  specific parametric retesting strategies in this chapter. All of the K-
multiplier tables for parametric prediction limits are structured to meet an annual  SWFPR of  10% per
year and  to accommodate groundwater networks ranging in size from one to  8,000 total statistical tests
per year.  The Unified  Guidance tables are more  extensive than similar tables in Gibbons  (1994b).
Further, each table is designed to indicate the effective power of the ^-multiplier entries.

     If a particular network  configuration is not directly covered in the Appendix D  tables, two basic
options are available. First, bilinear interpolation can be used to derive an approximate ^-multiplier (see
below for guidance on table interpolation). Second, the free-of-charge, open source,  and widely available
R statistical  programming package (www.r-project.org) can  be  employed to compute  an exact K-
multiplier. Further instructions and the two template codes used to compute the Unified  Guidance K-
multiplier tables are provided in Appendix C.  After installing the R package, these template codes can
be run by supplying specific  parameters for the network of interest (e.g., number of wells,  constituents,
background sample size, etc.).  Some familiarity with properly installing a program like R  is helpful.
Appendix C explains how to execute a pre-batched  set of commands. No other technical programming
experience is needed.
6 If multiple evaluations occur each year, new compliance samples each evaluation period are tested against the common
  intrawell background.

                                             19-11                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

       REQUIREMENTS AND ASSUMPTIONS

     The  basic  assumptions  of parametric prediction  limits were described in Chapter 18. These
include data that are normal or can be normalized (via a transformation), lack of outliers, homogeneity of
variance between the background and compliance point  populations, absence of trends  over time,
stationarity, and statistical independence of the observations.

     The Unified Guidance provides separate /c-tables for interwell and intrawell limits.  One of these
approaches should be justified before computing prediction limits. To use interwell prediction limits,
there should be no significant natural spatial variation among the mean concentrations at different well
locations. Otherwise, a prediction limit test could give meaningless results, since average downgradient
levels might naturally be higher than background even in the absence of a contaminant release. The
assumption of spatial variability should therefore be checked using the methods in Chapter 13.

     While intrawell testing eliminates the problem of natural spatial variability, intrawell background
often is developed using the first n samples from each compliance point well. Since historical data from
compliance  wells need  to be  utilized  to   do  this, these  groundwater measurements  should  be
uncontaminated. The number  of intrawell background samples available may also be rather limited,  n
will tend to be  initially small prior to any updating of background.  Such constraints will limit the
intrawell retesting schemes that can both minimize the SWFPR yet maintain effective power similar to
the ERPCs.

      One possible way to overcome this limitation is to estimate a pooled standard deviation across
many wells along the lines suggested by Davis  (1998). Such a calculation is no more difficult than a one-
way ANOVA  (Chapter 13) for identifying on-site spatial variability. The mean squared error [MSB]
component of the F-statistic in ANOVA gives an estimate of the average per-well variability.  To the
extent that mean levels vary by well  location but the population standard deviation does not, a one-way
ANOVA can be run on a collection of wells (both background and compliance) to estimate the average
within-well variance, and hence, the common  intrawell standard deviation (see Chapter 13 for further
details and examples).

     Instead of a standard deviation  estimate based solely on intrawell background at a single well with
its attendant limits in size and degrees of freedom, the mean concentration level can be estimated on a
well-specific basis, while the  standard deviation  is estimated utilizing a collection of wells leading to
much larger degrees of freedom. Although the  intrawell background size for a given well might be small
(e.g., n = 4 or 8), the ^-multiplier used to construct the prediction limit is based on both the effective
sample size (i.e., degrees of freedom plus one) and the intrawell sample size (n).

     The pooled standard deviation for intrawell comparisons can be utilized if the population standard
deviation is approximately constant across wells. Many data sets may not appear so initially; however,
any transformation to normality must first be taken into account. The standard deviation is only assumed
to be constant  on the transformed scale. Furthermore, once any transformation is applied, the collection
of wells should explicitly be tested for homogeneity of variance using the tools in Chapter 11. Only if
the assumption of equal variances across wells seems reasonable should the pooled standard deviation
estimate be used.
                                             19-12                                  March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

     With  little or no spatial variability among well locations, an interwell test might be considered.
However, the  sample standard deviation (s) computed from background may not  adequately estimate
true background variability.  This can happen when there is a temporal component to the variability
affecting all wells at a site or regulated unit in parallel fashion, or when there is a significant degree of
autocorrelation between successive samples.

     A random, temporal  component to  the  variability can result from changes to the laboratory
analytical method or field sampling methodology, periodic re-calibration of lab instruments, or other
sample handling or preparation artifacts that tend to impact all observations collected during a given
sampling event. Such a temporal component can sometimes be identified through the use of parallel time
series plots (Section 14.2.1) or through  a  one-way ANOVA using  time-of-sampling as the factor
(Section 14.2.2). Results of the  ANOVA  can  be used to derive a better estimate of the background
population  standard deviation (a), along with adjusted degrees of freedom for use in constructing the
upper prediction limit (see Chapter 14 for further details and an example).

     When autocorrelation is present, methods to adjust the standard deviation estimate and degrees of
freedom  entail possibly modeling the autocorrelation function. This issue is beyond the scope of the
Unified Guidance and consultation with a professional statistician is recommended. The most practical
way to avoid  significant autocorrelation between samples is  to allow enough time to lapse between
sampling events. Precisely how much time will vary from site to site, but Gibbons (1994a) and others
(for instance,  American Society for  Testing and Materials, 2005)  recommend that the  frequency of
sampling be no more frequent than quarterly. Alternatively, a pilot study can be run on two or three wells
with the  sample autocorrelation function estimated from the results (Sections 14.3.1 and 14.2.3). The
minimum lag (i.e., time) between sampling events at  which the autocorrelation is effectively zero can be
used as an appropriate sampling interval.

       APPENDIX TABLES FOR PARAMETRIC RETESTING PLANS

     The Unified Guidance provides tables of ^-multipliers for both interwell and intrawell prediction
limits with  retesting. It also provides separate tables for predicting individual future values versus future
means. Four distinct retesting schemes  are presented in the  case of  prediction limits for individual
values:  l-of-2, l-of-3,  l-of-4, and the modified California plan schemes.  Five distinct schemes are
presented for the case of future means: 1-of-l, l-of-2, and l-of-3 for means of order 2, and 1-of-l and 1-
of-2 for means of order 3.

     Both  the Appendix D  interwell retesting tables (Tables 19-1 through 19-9) and  the  intrawell
retesting tables (Tables 19-10 through 19-18) are similarly structured.  Separate sub-tables are provided
for a  range of possible  monitoring constituents  (c = 1 to 40) and for each of the retesting schemes
mentioned  above. Each  table is divided into three parallel sections, one  section  applicable to annual
statistical evaluations, one to semi-annual  evaluations, and one to quarterly evaluations. Within each
section, ^-multipliers are listed for all combinations of background sample size (from n = 4 to 150) and
number of wells (from w = 1 to 200).  These ^-multipliers are computed to meet a target annual SWFPR
of 10%, as discussed in Chapter 6.

     The Appendix tables  also list those ^-multipliers which achieve  adequate  effective power
compared to the ERPCs. The ^-multipliers are bolded when the effective power consistently exceeds the
appropriate  ERPC for mean level  increases  above background  of 3 or  more  standard deviations
                                             19-13                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                               Unified Guidance

(designated as 'good' power). The  multipliers are italicized and shaded when the effective power is
somewhat less, but still consistently exceeds the ERPC at mean level increases of 4 or more standard
deviations above background (designated as 'acceptable' power). Non-bolded, non-italicized entries
achieve the target SWFPR, but have low power.

     To use the tables,  certain key  statistical parameters should be known or identified. These include
whether the prediction  limit tests are interwell or intrawell,  the evaluation schedule (annual,  semi-
annual, or quarterly), the number of constituents (c), the  size of the background sample  (n),  and the
number of compliance  wells to be tested (w).  In the interwell  case, it is presumed that there  are n
(upgradient) background measurements for each constituent (c).  The listed ^-multiplier would then be
applied to each of c prediction limits, one for each monitoring constituent.  The intrawell case presumes
that there are n well-specific background measurements designated at each well-constituent pair, thus
giving w  x c separate sets of intrawell background. Here, the K-multiplier would be applied to each of w
x c distinct prediction limits.

     In situations where a mixture of test types is needed (e.g., intrawell testing for some constituents,
interwell  for others), the Unified Guidance tables can still be employed. The ^-multipliers are computed
to apportion an equal share of the overall cumulative SWFPR to each of the w x c tests that need to be
run during a given statistical evaluation. Because of this fact, if r of the constituents are analyzed using
interwell  tests, but (c - r) of the constituents are handled using intrawell limits, correct prediction limits
can be developed by first selecting an interwell K-multiplier based  on all  c constituents, and  then
selecting an intrawell K-multiplier also  based on c constituents. This will ensure that the target SWFPR
is met, although each multiplier is respectively applied only to a subset of the monitoring list.

     Some background  samples might be of different sizes, either for different constituents or at distinct
wells (e.g., when using  intrawell background). Again the Unified Guidance tables  can be inspected to
select a different K-multiplier for each  distinct n.  However, each multiplier should be chosen as  if the
background sample sizes were equal for all w x c tests. Thus, while a multiplier based on n\ background
observations is applied  only to those  tests  involving  that  sample size, it should be  selected from the
Appendix D tables as if it will be applied to all the tests.

     For network configurations not listed  in Tables  19-1 to 19-18 in Appendix D, an appropriate K-
multiplier can be estimated using bilinear interpolation. Such interpolation will be fairly accurate as long
as adjacent table entries are used, representing the closest values to the desired combination of number
of wells (w) and background sample size (n).

     In general, to calculate a KW* „*, where w* and n* are the desired input points  that lie  between the
closest table entries as: w\ < w* < w^, and n\  < n* < «2, first calculate the fractional terms:

                        _(w*-Wl]          _(n*-nl)
                     Jw - 7	r   ana   Jn ~ 7	v
                          (w2-wl)             \n2-ni)

     The interpolated K-multiplier can then be computed as:
                                             19-14                                    March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

      For example, suppose a ^-multiplier is needed for a l-of-3 interwell prediction limit test for
individual values using an annual evaluation schedule. Assume the monitoring network consists of c = 5
constituents monitored at w = 28 compliance wells, using n = 17 upgradient background measurements
on which to base the prediction limit. From Table 19-2 in Appendix D, the closest table entries, «>_„ to
the desired combination are K2o,i& = 1.59, Kso,i6 = 1-70, ^20,20 = 1.52, and Kso,20 = 1.62. The interpolated
value, K25,is, can then be found using the equations in [19.12]:

              _ (28-20) _           _ (17-16) _
           Jw  (30-20)          J"    (20-16)


           Ar2518 =(l-.8)(l-.25)-Ar2016 +.8(l-.25)-AT3016 + (l-.8)-.25-r2020+.8-.25-K-3020
                = .15-1.59+ .60-1.70+ .05-1.52+ .20-1.62   =1.659

     Important considerations in designing a reasonable retesting scheme for detection monitoring are
discussed in Chapter 6. Given a background sample and a particular network configuration and size,
parametric  l-of-m plans tend to increase in statistical power as the order of m increases. All of the
schemes  have greater power with  larger background  sample sizes (n). Furthermore, plans involving
prediction limits for future means tend to be more powerful than similar plans using prediction limits for
individual observations. So if the ^-multiplier for a particular plan is not bolded or italicized, another
plan can  be sought to achieve sufficient effective power using more resamples or perhaps changing to a
mean prediction limit. Alternatively, the background sample size might need to be augmented if feasible,
prior to implementing the retesting procedure.
19.3.1      TESTING INDIVIDUAL FUTURE VALUES

     The advantages to using a prediction limit for future individual values include: 1) the ability to
explicitly control the SWFPR across a series of well-constituent pairs; and 2) greater flexibility than that
provided by prediction limits for future means (Section  19.3.2) to handle temporal autocorrelation. In
those cases when the sampling frequency needs to be reduced to maximize statistical independence of
the observations, the method can be applied to evaluations of a single new measurement (plus possible
resamples) at each compliance point well.

     To properly implement  a prediction limit strategy for future values with retesting, it needs to be
feasible to collect 2 to 4 independent measurements at each compliance well during a given evaluation
period.  All initial and any resamples are assumed to be statistically independent and thus should exhibit
no autocorrelation.

     If statistical evaluations are done annually, it may be possible to collect data on a quarterly basis
and meet the minimal sampling requirements of any of the resampling schemes discussed in the Unified
Guidance. However, more  frequent evaluations (say semi-annual or quarterly)  will require that new
samples  be  collected  perhaps  monthly  or every  six  weeks. In  these cases,  explicit tests  for
autocorrelation may need to be conducted before adopting a  l-of-m retesting scheme with m > 2 or a

                                             19-15                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

modified California plan. If significant autocorrelation is identified, the sampling frequency may need to
be reduced and/or an alternate strategy utilizing fewer resamples may need to be adopted instead.

       PROCEDURE

Step 1.   Identify the overall targeted annual false positive rate (SWFPR = a =  0.10).  Determine the
         number of wells (w) to be monitored and the number of constituents (c) to be sampled at each
         well. Also determine whether the evaluation schedule at the unit or facility is annual,  semi-
         annual or quarterly.

Step 2.   Decide on the number  of observations (m) to be  predicted. To  incorporate retesting,  a
         maximum of two independent measurements should be collected from every compliance well
         during  each  evaluation period  to use  a  l-of-2  retesting scheme, three  independent
         measurements if a l-of-3 plan is desired, and four independent measurements if either a l-of-4
         plan or a modified California plan is employed.

Step 3.   For interwell prediction limits given a background sample  of n measurements,  compute the
         background sample mean (x ) and standard deviation (s) for each constituent. Then, based on
         the evaluation schedule (annual, semi-annual or quarterly), c, n, w,  and the  specific retesting
         scheme chosen, use  Tables 19-1  to  19-4 in Appendix D to determine  a  ^-multiplier
         possessing acceptable  statistical power. Interpolate within the  tables  to  find the closest
         multiplier if an exact value is not available.

         For intrawell prediction limits,  designate n early measurements as intrawell background for
         each well-constituent  pair; compute  the  intrawell  background mean (x) and standard
         deviation (s) for each case. Given the evaluation schedule,  c, n, w,  and the chosen retesting
         scheme, use Tables 19-10  to 19-13 in Appendix D to determine an acceptably powerful K-
         multiplier. Note: if the  intrawell  background  sample size varies  by  well, a  series  of K-
         multipliers should be computed, one for each distinct n.

         For each ^-multiplier, calculate the upper prediction limit with (1- a) confidence as:

                                        PLl_a = x+Ks                                  [19.13]

         If data were transformed prior to constructing the prediction interval, back-transform the
         prediction limit before making  comparisons against the  compliance point data. Unlike  a
         prediction limit for future means, the formula for predicting m future values does not involve
         any transformation bias if the comparison is made in the original measurement domain.

Step 4.   Collect an initial measurement from each well-constituent  pair being tested. Compare each
         value against either 1) the upper  prediction limit based on upgradient background in the
         interwell  case  or  2)  the intrawell prediction  limit specific to that  well-constituent  pair.
         Depending on  the  retesting scheme chosen, if any initial compliance point concentration
         exceeds the limit, collect 1  to 3 additional resamples at that well.  If feasible, analyze only for
         those constituents which exhibited  initial  exceedances.  Compare these values sequentially
         against the upper prediction limit. If the test 'passes' prior  to collection of all the scheduled
         resamples, the remaining resamples do not need to be gathered or compared against PL.

                                             19-16                                   March 2009

-------
Chapter 19. Prediction Limits with Retesting
Unified Guidance
Step 5.   Decide that the test at a given well passes (i.e., the well is in-compliance) if any one or more
         of the resamples does not exceed PL when using a l-of-m scheme or when at least 2 resamples
         do not exceed PL when  using the modified California scheme. Identify the well as failing
         when either (1) all resamples using a  \-of-m plan also exceed the prediction limit, or (2) at
         least two of three resamples using a modified California plan exceed PL.

       ^EXAMPLE 19-1

     A large hazardous waste facility with 50 compliance wells is to monitor 10 naturally-occurring
inorganic parameters in addition to 30 non-naturally occurring volatile organic compounds that have
never been detected on-site. Groundwater evaluations are performed  on a semi-annual  basis.  If the
regulating authority will allow up to two resamples per exceedence of the background  concentration
limit, construct an interwell prediction limit with adequate statistical power and false  positive rate
control on the following pooled set (n = 25) of background sulfate measurements.
BG Well
GW-01



GW-04



GW-08







GW-09








Sampling Date
07-08-99
09-12-99
10-16-99
11-02-99
07-09-99
09-14-99
10-12-99
11-15-99
10-12-97
11-16-97
01-28-98
04-20-99
06-04-02
09-16-02
12-02-02
03-24-03
10-16-97
01-28-98
04-12-98
07-12-98
01-30-00
04-24-00
10-24-00
12-01-02
03-24-03
Sulfate
(mg/l)
63
51
60
86
104
102
84
72
31
84
65
41
51.8
57.5
66.8
87.1
59
85
75
99
75.8
82.5
85.5
188
150
Log (Sulfate)
log(mg/l)
4.143
3.932
4.094
4.454
4.644
4.625
4.431
4.277
3.434
4.431
4.174
3.714
3.947
4.052
4.202
4.467
4.078
4.443
4.317
4.595
4.328
4.413
4.449
5.236
5.011
       SOLUTION
Step 1.   Assume for purposes of the example that there are no significant spatial differences among the
         well locations, either  upgradient or downgradient. A check  of normality  of the  pooled
         background  sulfate measurements indicates that  the  interwell prediction limit should  be
         constructed on the logged sulfate measurements rather than the raw concentrations.
                                             19-17
        March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

Step 2.   Groundwater evaluations must be conducted semi-annually (S). By excluding never-detected
         organic chemicals from the SWFPR calculation, the number of constituents that are to be
         considered is c = 10 at each of w = 50 wells.

Step 3.   Since a maximum  of two resamples  will be allowed  during any given evaluation period,
         neither the l-of-4 nor the modified California retesting plan are an option. Consequently, only
         a l-of-2 or l-of-3 retesting strategy is appropriate. With n = 25 background measurements,
         Tables  19-1  and 19-2 in Appendix  D should be examined for a semi-annual evaluation
         schedule to determine ^-multipliers with adequate power. The multiplier  of K= 2.75 for a 1-
         of-2 plan has 'acceptable' power compared to the semi-annual ERPC, but the multiplier of K=
         2.00 for a l-of-3 plan  has 'good'  power. Use  the  latter value  to construct  the  interwell
         prediction limit.

Step 4.   The sample log-mean and log-standard deviation of the sulfate background measurements are
         y = 4.32 and sy = 0.376, respectively.  Use these  values and the K-multiplier to compute the
         prediction limit on the log-scale as

                            PL = y + Ks  =4.32 + 2.00x0.376 = 5.072
                                      y

         Then exponentiate the limit to back-transform it to the original measurement domain, for a
         final sulfate prediction limit of PL = e5'072 = 159.5 mg/1.

Step 5.   Compare the final prediction limit against  one new sulfate measurement from each of the 50
         compliance  point wells. For any exceedence, compare the first of two resamples to  the
         prediction limit.  If the limit is still  exceeded,  test the  second resample.  If all  three
         measurements (initial plus two resamples)  are above the prediction limit at any  specific well,
         declare that a statistically  significant exceedence  for sulfate has been identified. If, however,
         neither of the resamples exceeds the limit, judge the evidence to be insufficient to declare the
         well to be out-of-compliance. A

       ^EXAMPLE 19-2

     Due to significant natural spatial variability, an intrawell testing scheme needs to be adopted at a
solid waste landfill that monitors for 5 inorganic constituents at each of 10 compliance wells. If only one
year's  worth  of quarterly  sampling data is available at each well, but  no recent contamination is
suspected, develop an appropriate modified California intrawell retesting plan for the following chloride
measurements.  Assume that one statistical evaluation must be conducted each year.
                                             19-18                                  March 2009

-------
Chapter 19.  Prediction Limits with Retesting
Unified Guidance
Well ID
GW-09



GW-12



GW-13



GW-14



GW-15



Chloride
(mg/l)
22
18.4
39.9
33.7
78
70
61
65.8
75.1
65.6
67
55.3
59.2
57.1
41.1
47.7
35
56.8
69.8
41.3
Well Mean ± SD
(mg/l)
28.5 ± 10.021



68.7 ± 7.208



65.75 ± 8.128



51.28 ± 8.427



50.72 ± 15.672



Well ID Chloride
(mg/l)
GW-16 31
34.6
60.1
48.7
GW-24 23.4
36.4
31.1
45
GW-25 33.5
30.2
23.1
38.7
GW-26 79.8
61.3
57.8
44.8
GW-28 37.7
26.6
45.7
42
Well Mean ± SD
(mg/l)
43.6 ± 13.392



33.98 ± 9.083



31.38 ± 6.533



60.92 ± 14.447



38.0 ± 8.273



       SOLUTION
Step 1.   With c = 5 constituents, w = 10 wells, one annual evaluation, and an intrawell background size
         for each well of only n = 4, Table 19-13 in Appendix D can be examined to locate a possible
         r-multiplier,  leading to an interpolated K= 4.33.  Although this multiplier will adequately
         control the annual SWFPR to 10% or less, it yields low power for identifying contamination.
         As an alternative, try computing a pooled standard  deviation across the compliance wells for
         chloride.

Step 2.   Side-by-side box plots (Section 11.1) of the chloride values exhibit no obvious differences in
         spread or variation. The F-statistic for Levene's test  (Section 11.2) is also non-significant (F =
         1.0673) at the a = 5% level, suggesting that the variances are not unequal  and that a pooled
         standard deviation can be appropriately formed.

Step 3.   Conduct a one-way  ANOVA on all chloride measurements from the  10 compliance wells,
         using Wells as the main factor (Section  13.2.2). The ANOVA table is presented below.
Source of Variation
Between Wells
Error (within wells)
Total
Sums of Squares
7585.25
3350.37
10935.62
Degrees of
Freedom
9
30
39
Mean
Squares
842.81
111.68
F-Statistic
7.55
                                             19-19
        March 2009

-------
Chapter 19.  Prediction Limits with Retesting                               Unified Guidance

Step 4.   Compute the square root of the Error Mean Squares (also called the root mean squared error
         or RMSE) component in the ANOVA table to derive an estimate of the pooled  intrawell
         standard deviation  of sp = 10.568. This estimate  of the average intrawell variation has 30
         degrees of freedom [df\, computed by multiplying (4-1) = 3 degrees of freedom per well times
         the number of wells, or df= 3x10 = 30.

Step 5.   The Appendix D tables are not used to derive ^-multipliers when a pooled standard deviation
         estimate is used for intrawell prediction limits.  R script listed in Appendix C is used (see
         Section  13.3).  For a  modified California retesting strategy with n = 4 and df'= 30, the K-
         multiplier becomes K=  1.98.7 This value not only controls the SWFPR but  also has good
         statistical power. So use this multiplier  along with the pooled intrawell  standard deviation to
         compute an intrawell prediction  limit for each  compliance well. As an example, since the
         mean for chloride at well GW-09 is 28.5, the intrawell prediction limit would be:

                              PL = 28.5 + 1.98 x 10.568 = 49.4 mg/1

         Prediction limits for the other compliance wells would be computed similarly. ~4
19.3.2       TESTING FUTURE MEANS

       BACKGROUND AND REQUIREMENTS

     The background, requirements, and assumptions for a prediction limit on future means of order p
are essentially identical to those for prediction limits for future values (Section 19.3).  For a comparable
level of sampling effort, predicting a future mean offers increased effective power compared to a strategy
that uses prediction limits for individual future values.  To properly implement a prediction limit strategy
for future means with retesting, it must be feasible to collect 2 to 6 independent measurements at each
compliance well during a given evaluation period. All initial and resample measurements are assumed
to be statistically independent.

     To  include explicit retesting, it  should be feasible to collect  either 2p  or 3p independent
measurements per well  during each evaluation. The initial p observations are used to form the initial
mean, while  the remaining  values are used  to form either one or two resample means. If statistical
evaluations are  done annually, it may be possible  to  collect quarterly data and  meet the  minimal
sampling requirements  for p = 2  and a l-of-2 retesting scheme. For more frequent semi-annual  or
quarterly evaluations, a  larger order p or a retesting scheme entailing two resample means will require
that new samples be collected perhaps monthly or every six weeks. An explicit test for autocorrelation
should be made before  adopting the  strategy presented here.  If significant autocorrelation exists, the
frequency of  sampling may need to be reduced and alternate prediction limit strategies considered such
as a 1-of-l plan for a future mean (see Section 19.1) or individual future values (Section 19.3.1).
7  The EPA Region 8 approximation equation described in Chapter 13, Section 13.3 provides a K-multiple estimate of 1.99
  for individual wells at n = 4. The annual K-factor for w = 10 and c =5 and n = 31 in Table 19-13 of Appendix D is
  interpolated as K = 1.508.  Using the appropriated, b & c coefficients from Chapter 13, Note 2 for the modified California
  plan, results are quite close to that generated from R-script.

                                             19-20                                    March  2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

     An important difference between testing means versus individual values is that in some cases it
may not be necessary to implement a retest at all. As noted above, for  the same degree  of sampling
effort, a prediction limit for a mean of two or more observations can provide greater effective power than
a prediction limit for the same number of individual values, even if a resampled mean is not collected. In
other words, when a l-of-2 retesting plan for individual  observations is compared to a 1-of-l plan for
means  of order 2,  the  1-of-l mean-based scheme generally has  greater power  for identifying real
concentration increases if background samples sizes are n > 10 (compare K-multiple power ratings at
higher n, c, and w in Tables 19-1 and 19-5 of Appendix D) A similar comparison holds between a 1-of-
3 retesting plan for individual observations and a 1-of-l plan for a mean of order 3 (Table 19-2 versus
Table 19-8 in Appendix D)

     Even more powerful prediction limits for  future means are  possible when  explicit retesting is
added to the procedure. However, the minimum sampling increases substantially. With a l-of-2 retesting
plan for means of order 2, as many as four independent groundwater measurements needs to be collected
and analyzed per evaluation period. With a l-of-3 plan for means of order 2 or a l-of-2 plan  for means
of order 3, the  sampling increases to as  many as six independent observations per period.   The latter
plans may only be feasible for a single annual evaluation.

     A problem common to all future mean prediction limits arises if the data have to be normalized via
a transformation. In this case, all  comparisons need to be made on the transformed data in order to avoid
a transformation bias.  As a consequence, the procedure is not a direct test of the background and
compliance point arithmetic means. The test is still valid  as a measure of significant mean differences in
the transformed domain (e.g.,  a test of geometric mean differences for logarithmic data).  To the extent
that the populations being compared  share a common variance in the transformed domain, it may also
indicate that a significant difference on the transformed scale also corresponds to a significant difference
in the arithmetic means of the original populations.

     A final potential drawback is that although a \-of-m plan for future observations and a 1-of-l  plan
for means of  order p = m seem  to require the same  total  sampling  effort,  a prediction limit for
observations can actually entail  less sampling. For a future mean test of order p = m, m  individual
measurements  will always need  to be collected  and  analyzed. With  a prediction  limit for  individual
observations, the first sample is analyzed and compared to the limit. If it passes (i.e., does not exceed the
limit) there is  no need to test the second or subsequent observations. Any subsequent resample that
passes, also indicates that no further resample comparisons are needed for that test.

     Under typical conditions at a site where most or all  tested well-constituent  pairs are likely to be at
background conditions, there is a substantial savings in the number of samples  for future observations
versus means of the same size. It can also be noted that the same principle is true for a l-of-2 test of a
mean of order 2. Under background conditions, the two initial mean samples may be all that is required.
When groundwater is contaminated, both the l-of-m retesting plan for observations and the 1-of-l  plan
for a mean of order p = m require exactly the  same amount of sampling and analysis to  identify a
significant exceedance.
                                             19-21                                   March 2009

-------
Chapter 19. Prediction Limits with Retesting                              Unified Guidance

       PROCEDURE

Step 1.   Identify the number of wells (w) to be monitored and the number of constituents (c) to be
         sampled at each well. Also identify the evaluation schedule as annual (A), semi-annual (S), or
         quarterly (Q).

Step 2.   Decide on the order (/?) of the future mean to be predicted. To incorporate retesting,  it needs to
         be possible to collect 2/? independent samples during each evaluation period to use a l-of-2
         retesting scheme, or 3/? independent samples if a l-of-3 plan is desired.

Step 3.   If an interwell  prediction limit  is needed,  use the common sample  of n (upgradient)
         background  measurements to compute the background  sample mean  (x)  and standard
         deviation (s). Given the n background measurements, w,  c, p, and the evaluation schedule
         (annual, semi-annual or quarterly), use Tables 19-5 to 19-9 in Appendix D to determine a K-
         multiplier possessing acceptable statistical power.  Calculate  the upper prediction limit on
         background as:

                                         PL = x + Ks                                   [19.14]

         If intrawell prediction limits are needed, designate n early measurements at each compliance
         well as intrawell background.  Compute  the background  sample  mean  (x)  and standard
         deviation (s) for each well. Then, based on n, w, c, p, and the number of evaluations per year,
         use Tables 19-14 to 19-18 in Appendix D to determine an adequately powerful ^-multiplier.
         Compute an intrawell prediction limit for each compliance well using equation [19.14]. Note:
         if the intrawell background sample sizes vary by well, a series of ^-multipliers will  need to be
         identified in these Appendix D tables, one for each distinct n.

         If the background data were transformed prior to  constructing the prediction  limit, also
         transform any compliance point data before making comparisons against the prediction limit.
         In particular,  compute the comparison mean of order p using the transformed values,  rather
         than transforming the sample mean of the raw concentrations.

Step 6.   Collect/? initial measurements from each compliance well.  Compute the mean of order/? for
         each well, first transforming the  data if necessary  using the same function  applied to
         background. Then compare each mean against  the upper prediction  limit.  If retesting is
         desired, for any compliance point mean that exceeds the limit, collect either/? or 2p additional
         resamples at that well, depending on the retesting scheme chosen. Form either one or two
         resample means of order p from these  data;  compare these means  sequentially to the  upper
         prediction limit.

Step 7.   Identify the well as potentially contaminated when either 1) the initial mean of order/? exceeds
         the limit in a 1-of-l plan, or 2) the initial mean and all resample means using a l-of-2 or 1-of-
         3 plan also exceed the prediction limit. Deem the well to be  in-compliance if either  1) the
         initial mean does not exceed the prediction limit, or 2) any of the resample means do not
         exceed the limit.
                                            19-22                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                             Unified Guidance

       ^EXAMPLE 19-3

     Suppose a large facility with minimal natural  spatial  variation is to monitor for 20  separate
naturally-occurring  inorganic  chemicals  along  with a  number of  other  never  detected  organic
constituents. If 100 compliance wells are to be  tested every six months and 25 background sample
measurements are  available, which resampling plans can control the SWFPR,  providing acceptable
statistical power? Assume that the data for  each inorganic compound can be normalized and that the
temporal autocorrelation between successive samples at the same well is minimal, provided that no more
than four samples are collected during any semi-annual period.

       SOLUTION
Step 1.   The frequency  of statistical  evaluations  is  semi-annual  (S).  Excluding  never-detected
         compounds from the SWFPR calculation leaves c = 20 constituents that need to be explicitly
         tested  at each  of  w =  100 wells. For each of these  constituents, since the  data can  be
         normalized, assume that an  interwell  prediction limit  can be constructed  using  n  =  25
         background measurements.

Step 2.   Determine  r-multipliers and power ratings for seven possible prediction limit retesting plans
         excluding the  l-of-3 mean order  2 and the l-of-2  mean order 3 tests. Use the sub-tables
         identified as "20 COCs, Semi-Annual" for n =25 and w = 100 in interwell Tables  19-1 through
         10-9 in Appendix D, to obtain the  following:
Prediction Limit Plan
l-of-2, observations
l-of-3, observations
l-of-4, observations
Mod. California, observations
1-of-l, mean order 2
l-of-2, mean order 2
1-of-l, mean order 3
K-Multiplier
3.13
2.31
1.81
2.54
3.56
2.29
2.95
Power
Low
Good
Good
Good
Acceptable
Good
Good
Total Samples
2
3
4
4
2
4
3
Step 3.   Compare the various plans in terms of statistical power and typical sampling effort. The only
         plan with low power is the l-of-2 scheme for observations.  The 1-of-l mean order 2 has
         acceptable power.  The other plans all have good power (i.e.., ones consistently meeting or
         bettering  the ERPC for mean-level increases  above background of 3 or more standard
         deviations), but potentially require either 2 or 3 resamples.

         Restricting attention to those with good power, the least potential sampling  effort is required
         by the 1-of-l plan for a mean of order 3  or a l-of-3 plan for observations.  These two plans
         would requires less total sampling than the l-of-4 plan for observations, the l-of-2 mean order
         2 plan and the  same or less sampling than the modified California plan for observations in
         identifying a contaminant release.

         If groundwater is not contaminated, the l-of-m plans for observations require a minimum of 1
         measurement to demonstrate that the  well  is  in-bounds (i.e., when the initial measurement
         does not exceed the background limit) as  does the modified California plan. The l-of-2 plan
         for a mean of order 2 requires a minimum of 2 measurements, and the  1-of-l plan for a mean
         of order 3 requires a minimum of 3 measurements. On balance, the l-of-3 plan for individual
         observations or the  l-of-2  plan  for a mean of order 2  may  provide the best  compromise
                                             19-23                                  March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

         between minimizing  sampling  effort and  offering  a  higher  probability of identifying
         contaminated groundwater. -^

       ^EXAMPLE  19-4

     Use the chloride data of Example 19-2 to compute and contrast prediction limits for a future mean
of order 2, with and  without  explicit retesting.  Assume as before that 10 wells are monitored for 5
inorganic constituents, and evaluated on an annual basis.

       SOLUTION
Step 1.   The chloride data in Example 19-2 showed significant spatial variability, suggesting the use
         of intrawell prediction limits.  Furthermore, a  one-way ANOVA evaluation of the w = 10
         compliance  wells indicated that a pooled standard deviation estimate of sp = 10.568 with 30
         degrees of  freedom could be used to  build  intrawell  prediction limits, instead  of using
         individual variance estimates from each compliance well.

Step 2.   With c = 5 constituents,  w =  10 wells  to be  monitored, one annual  evaluation (A),  and a
         pooled degrees of freedom of df= 30, the R script in  Appendix C can be repeatedly run to
         determine ^-multipliers for each retesting scheme for prediction limits  on means of order 2.
         Since the sample size for each of the 10 wells is the same n = 4, the following multiples were
         generated from the R-script for the 1-of-l to l-of-3 tests of mean order 2: K = 2.68, 1.88 and
                          o 	
         1.51, respectively.  The prediction limits can then be constructed using equation [19.15], as
         shown for the first five compliance wells in the table below.

                                          PL  = x+fcsp                                   [19.15]

Step 3.   While the power of each retesting plan is rated 'good' compared to  the  annual-evaluation
         ERPC,  the  prediction limits  are  obviously higher when less (or no) explicit retesting is
         conducted. Depending on conditions at the site,  the range of approximately 13 mg/1 of chloride
         in the well-specific prediction limits may or may not be important in deciding which strategy
         to use. The  1-of-l plan for a mean of order 2 requires fewer total samples than the other plans.
         In some situations, the higher initial limits may be outweighed by the savings in sampling cost.

         On the other hand, the ERPC provides a  minimal standard for assessing statistical power.
         There can be a range of power curves even among plans all rated as 'good' seen in Figure 19-
         1 below, where the full effective power curves for these three strategies  are presented. Clearly,
         the l-of-2 and  l-of-3 plans for means of order 2  have visibly higher power than the 1-of-l
         retesting scheme. If site conditions permit, it may be beneficial to incorporate the l-of-2 plan
         as a reasonable compromise  between the gain in statistical power versus  the increase in
         sampling requirements (for contaminated wells). -^
8  Using the Region 8 approximation equation in Chapter 13, the corresponding K-multiples were 2.69,  1.89 and 1.52,
  respectively, based on tabular values at n = 31 of 2.258, 1.364 & .946 and using the appropriate A, b & c  coefficients for
  each test. Results are very comparable to the R-script values.

                                             19-24                                    March 2009

-------
Chapter 19. Prediction Limits with Retesting
Unified Guidance

Well ID

GW-09


GW-12


GW-13


GW-14


GW-15


Retesting
Plan
1-of-l
l-of-2
l-of-3
1-of-l
l-of-2
l-of-3
1-of-l
l-of-2
l-of-3
1-of-l
l-of-2
l-of-3
1-of-l
l-of-2
l-of-3
K Multiplier

2.68
1.88
1.51
2.68
1.88
1.51
2.68
1.88
1.51
2.68
1.88
1.51
2.68
1.88
1.51
Power Rating

Good
Good
Good
Good
Good
Good
Good
Good
Good
Good
Good
Good
Good
Good
Good
Well Mean
(mg/l)
28.50
28.50
28.50
68.70
68.70
68.70
65.75
65.75
65.75
51.28
51.28
51.28
50.72
50.72
50.72
Prediction
Limit
56.82
48.37
44.46
97.02
88.57
84.66
94.07
85.62
81.71
79.60
71.15
67.24
79.04
70.59
66.68
  Figure 19-1. Comparison of Power Curves for 1-of-m Plans for Mean of Order 2
                        1.00-
                         .90-


                         .80-


                         .70


                         .60 H

                       S
                       I  .50-
                         .40-


                         .30-


                         .20


                         .10-


                         .00
                                                  «••  0 - .
                                        SDs Above BG
                                        19-25
       March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

19.4 NON-PARAMETRIC PREDICTION LIMITS WITH RETESTING

       BACKGROUND AND PURPOSE

     When parametric prediction limits are not appropriate, either due to a large fraction of non-detects
or data that cannot be normalized, retesting can be conducted using non-parametric prediction limits.
The  Unified  Guidance discusses  retesting schemes for both  individual  future values and for future
medians  (in  parallel  to the parametric options discussed in Section 19.3). Tests on individual
observations include the three \-of-m plans and modified California plan approaches. Tests on future
medians include the 1-of-l and  l-of-2 plans for medians of order 3. The basic strategy is to establish a
non-parametric prediction limit for each monitoring constituent based on background measurements so
that it accounts for the number of well-constituent tests in the overall network. Instead of determining a
^•-multiplier, a non-parametric limit is computed as an order statistic from the background  sample.  The
term order statistic refers to one of the values in a sorted (or ordered) data set.

     In order to maintain adequate  statistical power while minimizing the overall false positive rate,
retesting will almost always be needed as part of the detection monitoring system design. As in the
parametric case, a specific number  of additional, independent resamples will  potentially need to be
collected for each compliance well test.  The initial and subsequent resamples are then compared against
the non-parametric prediction limit.

     The largest  or second-largest value in background is often  selected  as  a non-parametric limit,
representing the nth or (w-l)th order statistics.  With higher level l-of-m tests of observations, an even
lower order  statistic may  be more appropriate in achieving an optimal  balance between the  desired
SWFPR and adequate statistical power.  This can be particularly true if the background sample size is
large, but depends on the overall network design requirements.  Although the Unified Guidance provides
tables of non-parametric limits only for the largest and second-largest order statistics, EPA Region 8 has
released software  written in Visual Basic® labeled the Optimal Rank Values Calculator that computes
the optimal choice of order statistic for l-of-m retesting plans for m = 1 to 4.  The program also provides
approximate statistical power estimates based on  user inputs of a target cumulative false positive rate,
background  sample size,  and number of simultaneous tests  to  be conducted.  The software  and
explanatory narrative will be provided on the EPA website.9

       REQUIREMENTS AND  ASSUMPTIONS

     When more independent data  are added to the testing procedure, retesting with non-parametric
prediction limits leads to more powerful and more accurate assessments of  possible contamination. As
with parametric retesting schemes,  a balance must be struck between 1) quick identification  and
confirmation of contaminated groundwater and 2) statistical independence of successive resamples.  All
retesting strategies depend on the assumption of statistical independence between successive resamples.
This trade-off is typically resolved by allowing enough time between resamples to allow both the well to
  The calculator, an accompanying narrative, fact sheet and this guidance will be located  on the EPA website:
  http ://www. epa. gov/hazard/correctiveaction/resources/guidance/sitechar/gwstats/index.htm.  If the calculator cannot be
  accessed, contact Mike Gansecki for assistance (email: gansecki.mike(g),epa.gov: or phone: 303- 312-6150.)
                                             19-26                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

recharge and additional groundwater to flow past the well screen, and by limiting the number of possible
resamplesto2 or 3.

     Non-parametric retesting   schemes  offer  somewhat  less   flexibility  than  their  parametric
counterparts. As with other non-parametric statistical intervals, the same SWFPR control afforded by a
parametric interval based on a small n cannot usually be attained in a non-parametric interval;  larger
sample sizes are almost always necessary, ^-multipliers for parametric prediction limits are continuous
statistical  parameters that can be adjusted to match a desired false positive rate for  even the smallest
sample sizes. By contrast, the bounds of non-parametric intervals are restricted to values in the observed
background sample.  For a given sample size and number of tests to be run, any  order statistic selected
from background as the non-parametric prediction limit results in a discrete probability of false positive
error. Altering the  prediction limit by  selecting a different order statistic changes the  false positive rate
only in discrete probability steps, providing a less efficient means of controlling the SWFPR.

     The  non-parametric prediction limit tests provided in the Unified Guidance do  not  require the
underlying distribution to be normal.   One potentially attractive application is for background data sets
containing higher percentages of non-detects which cannot be normalized.  For  some constituent data
sets, it may be possible to pool data from several upgradient and historical compliance wells to generate
much larger total background sizes.  A non-parametric Kruskal-Wallis test of medians can establish that
these data are appropriate for pooling.

     Since larger background sample  sizes are needed because no distributional model is posited, the
non-parametric testing  schemes are   most applicable to interwell  comparisons.   Small intrawell
background sample sizes make it difficult for any of the non-parametric test options to be applied which
can meet  the  SWFPR cumulative  false positive design objective. Unlike  parametric intrawell tests,
effective sample sizes cannot be expanded by estimating a common pooled standard  deviation across a
number of wells. This conclusion is generally true no matter what order statistic is  used to estimate the
non-parametric prediction limit.   But there are other considerations which might  allow intrawell testing
using non-parametric alternatives. For a given sample size, target false positive,  a fixed maximum and
number of total tests, the higher \-of-m tests of future observations will  have lower achievable false
positive errors, with the l-of-4  test the lowest.  If the background sample size is  increased through
periodic additions, this false positive will  continue to drop.    The power of these  tests using the
maximum with small sample  sizes is almost always greater than  the EPA reference levels.  A temporary
strategy might be to utilize the highest order \-of-m test for intrawell purposes until larger sample sizes
are available.  However, the target cumulative false positive rate may not initially be met. With  larger
sample sizes, it may also be possible to decrease the m of the test and  still  achieve  the target false
positive rate.

     Even interwell  comparisons between upgradient and downgradient wells are acceptable only if the
degree of  spatial variability is insignificant. Fortunately, spatial  variability may be less of a problem in
those cases where a non-parametric retesting scheme might be implemented, i.e.., when the detection rate
of the chemical being monitored is  fairly low. High constituent non-detect rates  tend to result in more
uniform   spatial   distribution  across  site  wells,  allowing  for  similar median  concentrations.
                                              19-27                                    March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance
       APPENDIX TABLES FOR N ON-PARAMETRIC PREDICTION LIMITS

     To design appropriate non-parametric prediction  limits with  retesting, the Unified  Guidance
provides separate tables for predicting individual future values  versus future medians. Four  distinct
retesting schemes are presented in the case of prediction limits for individual values: l-of-2, l-of-3, 1-of-
4, and modified California  plan schemes. Two distinct  schemes are presented for the case of future
medians: 1-of-l and l-of-2 for medians of order 3.

     Unlike  the  tables for parametric prediction  limits discussed in Section  19.3, non-parametric
prediction limits do not involve ^-multipliers. Instead, the entries in Tables 19-19 to 19-24 of Appendix
D consist of pet'-constituent significance levels. These levels represent the achievable false positive rate
(oiconst) associated with each tested  constituent for a given retesting scheme, choice of non-parametric
prediction limit, and network configuration (i.e., number of wells [w] and background sample size [w]).10
The non-parametric prediction limit can be estimated via any order statistic from the background sample.
However, the most practical limits  are usually  either the maximum observed background value or the
second-highest value. Consequently, the Unified Guidance provides tables for these two options.

     Each table for the six specific non-parametric tests contains two sub-tables.  One uses a limit based
on the background maximum and the other the second-highest background value. All the tables are
otherwise similarly structured. Within each table and sub-tables, per-constituent significance levels are
given for all combinations of background sample size (n = 4 to 200)  and number of wells (w  = 1 to 200).
These significance levels can be used to meet a target annual SWFPR of 10%, discussed in Chapter 6.

     Correct use of these tables involves a few important considerations. First, if an interwell prediction
limit  is  desired,  the  target per-constituent false positive  rate  (aCOnst)  needs to be  computed.  Any
prediction limit strategy selected should have a table entry no greater  than aCOnst in order to ensure that
the annual SWFPR is no greater than 10%. To compute this target rate, use the formula:

                                        a   = l-(i-orfc                                 [19.16]
                                         const       \     /                                   L     J

where c equals the number of monitoring constituents and a is the SWFPR = 0.10.

     Unlike the tables for parametric prediction limits, separate tables are not provided for each of the
three  most common evaluation  schedules (i.e., annual,  semi-annual, and quarterly).   The number of
'wells' in each non-parametric table must be regarded as the actual number of compliance wells (w) times
the number of annual statistical evaluations (WE = 1, 2, or 4).  For using these tables,  let w* = w x «E.
This adjustment is necessary because  on  each  evaluation,  w wells should be  compared  against  a
prediction limit computed from a common interwell background. A  site with w* wells tested annually is
statistically equivalent to a site having w distinct well locations tested n£ times per year (w x n£ tests).
10 Per-constituent rates instead of network-wide false positive rates  are given in these tables and  those  of Davis  and
  McNichols (1994; 1999) for computational reasons. Although the mathematical algorithm is exact, it is difficult to compute
  with accuracy for a large number of tests (r). Hence the decomposition of r into constituents (c)  times  wells (w). By
  calculating the per-constituent false positive rate, only the number of wells (w) need be varied.

                                              19-28                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                               Unified Guidance

     Once w* is computed in this way, the table entry corresponding to w* and n represents the
achievable annual false positive rate per constituent. As noted, this rate should not exceed the target rate
(oiconst) in order to meet the overall  SWFPR.  If aCOnst is exceeded for a given choice of retesting scheme
and choice of non-parametric prediction limit, a different limit or scheme should be considered. In
general, selecting a l-of-m retesting scheme with larger m will  lead to a lower  achieved false positive
rate. Also,  per-constituent significance levels for the modified California approach are generally larger
than those for the l-of-m plans.


     If intrawell prediction limits  are needed, a somewhat different method needs to be employed to
correctly use the per-constituent significance levels in Tables 19-19 through 19-24 of Appendix D. In
this case, a target per well-constituent pair false positive rate (awc) needs to be first computed using the
equation:

                                       aw.c=l-(l-afw'c)                                [19.17]

where  a is the  SWFPR, w  equals the actual number  of compliance wells and c is  the  number of
monitoring constituents.  Then the placeholder w* for the non-parametric tables is to be equated with
the number of annual  statistical evaluations (w* = n^ =  1, 2, or  4). w* represents the number of times
per year that the common intrawell background  at  any given well-constituent  pair will be compared
against new compliance  measurements from that well.  The table  entry corresponding to w* and the
intrawell background  sample size  n may be regarded  as the achievable false  positive rate per well-
constituent pair. This rate should not exceed the target rate, aw.c, if the overall SWFPR is to be met.

     The same approach presented in Section 19.3 is used if a mixture of test methods is needed (e.g.,
parametric  prediction limits for some constituents, and non-parametric limits for  other constituents). By
construction, the target SWFPR is evenly proportioned across the list of monitored constituents. As long
as the  significance level per constituent (interwell case) or per well-constituent  pair (intrawell case) is
computed using all c constituents and not just those for which a non-parametric prediction limit test will
be applied, the SWFPR will not exceed a = 0.10 on an annual basis.

     Tables 19-19 through 19-24 in Appendix D provide the same bold, italicized or plain text used to
identify 'good', 'acceptable' and 'low' power  ratings following the ERPC  3  and 4 standard deviation
reference criteria as in the parametric prediction limit tables.

     As final technical  notes about these tables, the  significance levels listed as table  entries are
presented using a short-hand notation in order to compactly present a wide range of false positive rates.
In this  notation, the first four non-zero digits of the significance level are given, followed if necessary, by
the symbol -d.  The value d represents the number of leading zeros to the right of the decimal point. This
is equivalent to taking the non-zero portion  of the  entry and multiplying it by ICT^ to get the actual
significance level.  As  an  example,  if the  entry  is   .4251-4, the equivalent significance level  is
.00004251. Entries without the -d symbol  are the  actual  fractional significance  levels where no
adjustment is needed.

     For network configurations (number of wells [w]  and  background sample size [n]) not listed in
Tables 19-19 through 19-24 in Appendix D, bilinear  interpolation can be used to approximate the
significance level associated with the desired configuration.  As discussed in Section 19.3, interpolation
                                             19-29                                    March 2009

-------
Chapter 19. Prediction Limits with Retesting                              Unified Guidance

should be restricted to the closest four adjacent table entries.  The shorthand significance level notations
in the tables should first be converted to actual fractions before interpolating.
19.4.1      TESTING INDIVIDUAL FUTURE VALUES

       BACKGROUND AND  REQUIREMENTS

     The Unified Guidance recommends two variations of non-parametric prediction limits for use in
groundwater  detection monitoring.  The first  is the prediction  limit for individual  future values,
introduced in Section 18.3.1.  The other is  the prediction limit for future medians, detailed in Section
18.3.2. Basic requirements for non-parametric prediction limits are outlined in those sections.

     The main advantage to  a  prediction limit for future values  is its overall flexibility and ease of
implementation. Fewer data from each compliance well are needed to implement the test compared to a
prediction limit for a future median.  Only an initial observation  from each compliance point may be
needed to identify a well-constituent pair 'in-bounds'; initial exceedances can be followed by up to a
maximum of three additional individual resamples. Once the non-parametric upper prediction limit has
been selected from background  as a large order statistic (often the maximum or second-largest value),
each compliance point measurement is compared directly against this upper limit.

     The user should decide which  retesting scheme to use and how many resamples per well are
feasible, given  that the  measurements from any well during a given evaluation period need  to  be
statistically independent. Tables 19-19 through  19-22 in Appendix D can be employed to compare the
achievable false positive rates  of different schemes and to determine whether they  exhibit adequate
effective power. The user can also explore EPA Region VIII's Optimal Rank  Values Calculator software
to consider order statistics other than the maximum or second-largest.

       PROCEDURE

Step 1.   For an interwell test,  use the number of monitoring constituents (c) in equation  [19.16] to
         determine the  target per-constituent false positive rate (aCOnst). Also multiply  the number of
         yearly statistical evaluations (WE)  by the actual number of compliance wells  (w) to determine
         the look-up table entry, w *.  Then depending on the background sample size n and w, choose a
         type of non-parametric prediction limit (i.e.., maximum or 2nd highest value in background)
         and a retesting scheme for individual  observations using Tables 19-19 through 19-22 in
         Appendix D.  The final plan should have an achieved significance level no greater than Oconst
         and also should be labeled with 'acceptable' or 'good' power in the Appendix tables.

Step 2.   For an intrawell test, use the number of constituents (c) and the actual number of compliance
         wells (w) in equation [19.17] to compute the target significance level per well-constituent pair
         (aw_c). Set w * in the look-up table equal to the  number of yearly evaluations, nE. Based on w *
         = WE and the intrawell  background sample size n, choose a non-parametric prediction limit and
         retesting scheme so that the achieved well-constituent pair significance level  (i.e.., the selected
         table  entry) does not exceed  the target significance level, aw.c, and  also is labeled with
         'acceptable' or 'good'  statistical power.

                                            19-30                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

Step 3.   Sort the background data into ascending order and set the upper prediction limit equal to an
         appropriate  order statistic of the data (e.g.,  the maximum  or  the second-largest observed
         value). If all constituent measurements in a background sample are non-detect, use the Double
         Quantification rule in Chapter 6.  The constituent  should not be included in calculations for
         identifying the target false positive.

Step 4.   Collect one initial measurement per compliance well. Then compare each initial measurement
         against the  upper prediction limit.  Depending  on the retesting scheme  chosen, for  any
         compliance point value that exceeds the limit, collect one to  three additional resamples from
         that well.  Again compare the resamples against the upper prediction limit.

Step 5.   Identify any well with an initial exceedance as potentially contaminated when either (1) all
         resamples using a l-of-2, l-of-3, or l-of-4 plan also exceed the prediction limit, or (2) at least
         two resamples  exceed the limit using a modified  California retesting  scheme. Conversely,
         declare a well to have 'passed' the test if either 1) the initial measurement does not exceed the
         prediction limit, 2) any resamples from a \-of-m scheme do not exceed the limit, or 3) at least
         2 of 3 resamples from a modified California approach do not exceed the limit.
19.4.2      TESTING  FUTURE MEDIANS

       BACKGROUND AND REQUIREMENTS

     Prediction limits for  a future median based on either a single or with one repeat (lof-1 or l-of-2
tests) are two non-parametric procedures recommended as retesting methods in the Unified Guidance.
Compared to a prediction  limit for future individual values, the prediction of a median (Chapter 18)
often requires more data  to be collected from each compliance  well  particularly if resampling is
included. Slightly greater statistical manipulation is also needed once the data are in hand. For the 1-of-l
test,  the initial median to be predicted requires at least two initial observations from each compliance
point, and any resample medians will require additional sets of up to three measurements, all of which
needs to be statistically independent.

     Given equal amounts of data and the same input conditions, a prediction limit for a future median
tends to be more statistically powerful than a prediction limit for individual values. This is true whether
one uses a fixed order statistic or selects across a range of order statistics to form the prediction limit.
Because of this and the fact that both spatial variability and autocorrelation may be less of a problem (or
at least less easily assessed) when the detection rate is low and a non-parametric strategy is needed, the
Unified Guidance includes Appendix D  tables for both a 1-of-l scheme and a l-of-2 scheme to predict
medians of order  3. The l-of-2 median test will have a lower achievable false positive rate than the 1-
of-1  version, with all other conditions equal.

     Depending on the number of annual evaluations and the test configuration, care needs to be taken
that potentially needed samples are far enough apart in time.  The series of observations from any well is
assumed to be uncorrelated.  If autocorrelation is a problem, a prediction limit for future values (Section
19.4.1) should be considered in which the per-well sampling requirements with  explicit retesting are
more modest.
                                             19-31                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                              Unified Guidance

       PROCEDURE

Step 1.   For an interwell test, use the number of monitoring constituents (c) in equation  [19.16] to
         determine the target per-constituent false positive rate (aCOnst). Also multiply the number of
         yearly statistical evaluations (WE) by the actual number of compliance wells (w) to determine
         the look-up table margin value, w *. Then, depending on the background sample size n and w *,
         choose a type  of non-parametric prediction  limit  (i.e., maximum or 2nd highest value in
         background) and a retesting scheme for future medians using Tables 19-23  to 19-24 in
         Appendix D. The final plan should have an achieved significance level no  greater than ocCOnst,
         and also should be labeled with 'acceptable' or 'good' power in the Appendix tables.

Step 2.   For an intrawell test, use the number of constituents (c) and the actual number of compliance
         wells (w) in equation [19.17] to compute the target significance level per well-constituent pair
         (aw_c). Set w* in the look-up table margin equal to the number of yearly evaluations, WE- Based
         on w * = nE and the intrawell background  sample size («), choose a non-parametric prediction
         limit  and retesting scheme for future medians so that  the  achieved well-constituent  pair
         significance level (i.e.., the selected table entry) does not exceed the target  significance level,
         aw_c, and also is labeled with 'acceptable' or 'good' statistical power.

Step 3.   Sort background into ascending order and set the upper prediction limit equal  to  a large
         background order statistic (e.g., the  maximum or  second largest value).  If all constituent
         measurements in a background sample are non-detect, use the Double Quantification rule in
         Chapter 6.  The constituent should not be included in calculations identifying the target false
         positive rate.

Step 4.   Collect  two initial  measurements  per compliance  well. If both do not  exceed  the upper
         prediction limit, the test passes since the median of order 3  will also not exceed the limit.
         There is no need to collect the third  initial observation or any resamples. If both exceed the
         prediction limit, the median will also exceed the limit. There is no need to collect the third
         initial measurement. If using a 1-of-l plan,  move  to  Step 5.  Otherwise, collect up to three
         resamples in order to assess the resample median.

         If one initial measurement is above  and  one below the limit, collect a  third observation to
         determine the position of the median relative to the prediction  limit. In all cases,  if two or
         more of the compliance point  observations are non-detect, set the median  equal to the
         quantification level (QL).

Step 5.   Compare the median value for each compliance well against the upper prediction limit. If a 1-
         of-2 retesting scheme is selected and any compliance point median exceeds the limit, collect
         up to three additional resamples from that well. Compute the resample median and compare
         this value to the upper prediction limit.

         Identify a compliance well as potentially contaminated when either the initial median exceeds
         the upper prediction limit for a 1-of-l plan, or both the initial median and the resample median
         exceed the prediction limit in a l-of-2 plan. Conversely, declare a well to have passed the test
         if the initial median does not exceed the prediction limit, or the resample median in a l-of-2
         scheme does not exceed it.
                                             19-32                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting                             Unified Guidance

       ^EXAMPLE 19-5

     The  following trace mercury data have been  collected  in the past year from a site with four
background wells and 10 compliance wells (two of which are shown below). The facility must monitor
for five constituents, including mercury. Assuming that the percentage of non-detects in background is
too high to make a parametric analysis appropriate  or feasible,  compare interwell non-parametric
prediction limits for both observations and medians  at the  annual statistical evaluation, and determine
whether either compliance well indicates significant evidence of mercury contamination.  Further
assume that the sequentially reported compliance well data  below are obtained as needed for the
different test comparisons.

Event
1
2
3
4
5
6
Mercury Concentrations
BG-1 BG-2 BG-3
.21 <.2 <.2
<.2 <.2 .23
<.2 <.2 <.2
<.2 .21 .23
<.2 <.2 .24

BG-4
<.2
.25
.28
<.2
<.2

(ppb)
CW-1
.22
.20
<.2
.25
.24
<.2

CW-2
.36
.41
.28
.45
.43
.54
       SOLUTION
Step 1.   Using a target SWFPR of 10%, compute the target per-constituent false positive rate, noting
         that   the   monitoring   list   consists   of   five   parameters.   This   implies    that
         (Xconst =1 - (l-.l)  =.021 using equation [19.16]. Since the detection rate in background is
         only 35%,  it is reasonable to consider non-parametric prediction limits with retesting.  The
         background sample size of n = 20 is to be used to construct an interwell prediction limit for all
         w =  10 compliance wells. Since  there is only one annual evaluation (WE = 1), the look-up table
         margin value of w* equals w x «E = 10.

Step 2.   Determine  potentially applicable retesting  plans. First  consider non-parametric prediction
         limits for individual observations with n = 20 and w = 10.  Consulting Tables 19-19 through
         19-22 in Appendix D, only the l-of-3, l-of-4, and modified California plans meet (i.e.., do not
         exceed) the target false positive  rate of 2.1%. To use the  l-of-3 or modified California plans,
         the prediction limit needs to be  set to the maximum background measurement. In the l-of-4
         plan, the prediction  limit  can  be set to either  the maximum or  second-highest value in
         background using the Appendix D tables. A final  l-of-4 plan determined with the Optimal
         Rank Values Calculator allows the use of the 3rd  highest value. All of these plans boast good
         power compared to the annual ERPC. Both the l-of-4 and  modified California schemes may
         require as many as 3 separate and independent resamples in addition to the initial observation.

         Consider tests for future medians of order 3 in Tables 19-23 and 19-24 in Appendix D. Only
         the l-of-2 plan using the  maximum background value as the prediction limit meets the Oconst
         target.  It also  has good power,  but requires 3 initial measurements and up to 3 additional
         individual resamples.


                                             19-33                                   March 2009

-------
Chapter 19.  Prediction Limits with Retesting
Unified Guidance
Step 3.   Sort the combined background data and compute the possible prediction limits as PL(n) = .28
         ppb, PL(n-i) = .25 ppb, and PZ(n-2) = .24 ppb, respectively representing the maximum, second-
         largest, and third-largest background values.

Step 4.   Determine the test outcomes at  each compliance well  using the various  retesting plans, as
         shown in the table below. For the prediction limits on individual observations, the first sample
         collected during  Event  1 is  used as  the initial  screen to determine if  any  resampling is
         necessary. The first 3 measurements at each compliance  well are used to form the initial
         comparison. The median at CW-1 is .20 ppb, while that at CW-2 is .41 ppb.
Compliance
Well
CW-1





CW-2





Retesting
Plan
l-of-3
l-of-4, Max
l-of-4, 2nd
l-of-4, 3rd
Mod-Cal
l-of-2, Med
l-of-3
l-of-4, Max
l-of-4, 2nd
l-of-4, 3rd
Mod-Cal
l-of-2, Med
Achieved a
.0055
.0009
.0046
.0135
.0140
.0060
.0055
.0009
.0046
.0135
.0140
.0060
# Initial
Samples
1
1
1
1
1
3
1
1
1
1
1
3
Resamples
Required
0
0
0
0
0
0
2
2
3
3
3
3
BG
Limit
.28
.28
.25
.24
.28
.28
.28
.28
.25
.24
.28
.28
Result
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Fail
Fail
Fail
Fail
         All of the acceptable plans indicate that CW-1 is not statistically different from background,
         although more initial sampling is required for the l-of-2 retesting plan with medians. For CW-
         2, the results  are more problematic. The l-of-3 and l-of-4 plans based on the maximum
         background value allow the well to pass, while the other four plans indicate a significant
         difference from background. The least  degree of sampling is required by the l-of-3 plan; at
         some facilities, greater  sampling efforts may not be feasible.  When a well is likely to be
         contaminated,  the number of samples required to actually make a decision about the well is
         similar across the plans with the exception of the l-of-2 prediction limit on a median.

         A further consideration is that  although the power  of each plan exceeds the annual ERPC
         when additional resampling is  possible, it is helpful to compare the full  power curves  of
         multiple plans to determine whether a particular plan offers greater power than the rest. Figure
         19-2 displays  an overlay of the six power curves associated with the retesting plans in this
         example.  For  these  inputs,  the l-of-2  retesting plan for a median of order 3 using the
         background maximum and the  l-of-4 plan on individual observations using the 3rd highest
         background value achieve the best overall power (shown as a single curve on Figure 19-2).
                                             19-34
        March 2009

-------
Chapter 19.  Prediction Limits with Retesting
                                       Unified Guidance
                    Figure 19-2. Comparison of Full Power Curves
           .00
                                      -••-• ©
                                                                                1-of-4,
                                                                                1-of-4, 2nd
                                                                                Mod-Cal
                                                                                1-of-2» Med
                                                                         	& l-of-4, 3rd
               cP   #
                       V   V
v   v
 SDs Above BG
                          4>  ^
         As seen in Figure 19-2, the two plans that pass the second compliance well have visibly lower
         power — especially in the range of 2 to 3.5 standard deviations above background — than the
         four plans that failed CW-2. In such a situation, the user needs to carefully balance the risks
         and benefits of each acceptable resampling plan. In some cases, the cost of greater amounts of
         resampling may be outweighed by the added sensitivity of the test to evidence of groundwater
         contamination. -4
                                          19-35
                                              March 2009

-------
Chapter 19. Prediction Limits with Retesting                          Unified Guidance
                     This page intentionally left blank
                                        19-36                               March 2009

-------
Chapter 20. Control Charts	Unified Guidance

CHAPTER 20.   MULTIPLE COMPARISONS  USING CONTROL
                                     CHARTS

       20.1   INTRODUCTION TO CONTROL CHARTS	20-1
       20.2   BASIC PROCEDURE	20-2
       20.3   CONTROL CHART REQUIREMENTS AND ASSUMPTIONS	20-6
         20.3.1  Statistical Independence and Stationarity	20-6
         20.3.2  Sample Sizes and Updating Background	20-8
         20.3.3  Normality andNon-Detect Data	20-9
       20.4   CONTROL CHART PERFORMANCE CRITERIA	20-11
         20.4.1  Control Charts with Multiple Comparisons	20-12
         20.4.2  Retesting in Control Charts	20-14
     This chapter describes  control  charts,  a second  recommended  core  strategy for detection
monitoring. Control charts  are a useful and powerful alternative to prediction limits. The Unified
Guidance is the first EPA document to discuss retesting and simultaneous testing of multiple wells
and/or constituents as they relate to control charts. Research of these topics is still ongoing.
20.1 INTRODUCTION TO CONTROL CHARTS

     Control charts are a viable alternative to parametric prediction limits for testing groundwater in
detection monitoring. They are similar to prediction limits for future observations in that a control chart
limit is estimated from background and then compared to a sequence of compliance point measurements.
If any of these values exceeds the control limit, there is  initial evidence that the  compliance point
concentrations exceed background.

     Control charts can be constructed as either interwell or intrawell tests. The main difference is how
background is defined and what measurements are utilized to build the control limit. Interwell control
charts establish the control limit  from designated upgradient and potentially other background wells.
Intrawell control charts, on the other  hand, employ historical measurements from a  compliance point
well as background.  Intrawell tests can only be appropriately applied if the historical compliance well
background is uncontaminated.

     An advantage of control charts over prediction limits is that a control chart graphs the compliance
data over time.  Certain varieties can also evaluate gradual increases above background over the period
of monitoring.   Trends  and changes in concentration  levels  can be easily  seen  since the sample
observations are consecutively plotted  on the chart.  This provides the analyst an historical overview of
the pattern of measurement levels.  Prediction limits are typically constructed to allow  only point-in-time
comparisons between the most recent  compliance data and  background, making long-term trends more
difficult to identify.
1  Long-term results from repeated application of a prediction limit can be plotted over time, creating a graph similar in nature
  to a control chart. But this has been infrequently done in practice.

                                            20-1                                  March 2009

-------
Chapter 20. Control Charts	Unified Guidance

     As a well-established statistical methodology, there are many kinds of control charts. Historically,
control charts have been put to great use in quality engineering and manufacturing, but have more
recently been adapted for use in groundwater monitoring.   The specific control chart recommended in
the Unified Guidance is known as a combined Shewhart-CUSUM control chart (Lucas, 1982). It is a
'combined' chart because it simultaneously utilizes two separate control chart evaluation procedures.
The Shewhart portion is almost identical to a prediction limit in that  compliance  measurements are
individually compared against a background limit. The cumulative sum [CUSUM] portion sequentially
analyzes  each new measurement with prior compliance data.  Both portions are used to  assess the
similarity of compliance data to background in detection monitoring.

     The  Shewhart-CUSUM control chart  works as follows. Appropriate background data  are first
collected from the specific compliance well for intrawell  comparisons or from separate background
wells for interwell tests.   The  baseline parameters for the chart, estimates of the mean  and  standard
deviation, are obtained  from these background  data.  These baseline  measurements characterize the
expected background concentrations at compliance wells.

     As future compliance  observations are collected, the baseline parameters are used to standardize
the newly gathered data. After these measurements  are standardized  and plotted,  a control chart is
declared out-of-control if future concentrations exceed the baseline control limit. This is indicated on the
control chart when either the Shewhart or CUSUM plot traces begins to exceed a control limit. The limit
is based on the rationale that if the well remains uncontaminated as it was during the baseline period,
new standardized observations  should not deviate substantially from the baseline mean. If a release
occurs, the standardized values  will deviate significantly from baseline and tend to exceed the control
limit. The historical baseline parameters then no longer accurately represent current well concentration
levels.

     Combined  Shewhart-CUSUM control  charts initially  featured two control limits,  one for testing
the Shewhart portion of the chart, one for testing the CUSUM portion of the chart.  Later research on
control charts (Davis,  1999; Gibbons, 1999)  indicated that  having separate control  limits for the
Shewhart and CUSUM procedures is generally not important.  Both control chart traces can instead be
compared to a single control limit. This modification not only makes the control chart method slightly
easier to apply, but also aids in measuring the statistical performance  of control charts over a variety of
monitoring networks.
20.2 BASIC PROCEDURE

     The  basic  procedure  for  constructing a control  chart is presented  below. Requirements  and
assumptions for control charts are discussed in later sections:

Step 1.   Given n background measurements (XB), estimate the baseline parameters by computing the
         sample mean (XB ) and standard deviation (53).

Step 2.   For  a  compliance  point measurement (*;)  collected on sampling  event  7}, compute the
         standardized concentration Z\.

                                             20-2                                   March 2009

-------
Chapter 20. Control Charts                                                 Unified Guidance
Step 3.   For each sampling event 71;, use the standardized concentrations from Step 2 to compute the
         standardized CUSUM Si. Set So = 0 when computing the first CUSUM Si.
                                   S, . =max[0, (Z,. -*)+£,._!]                              [20.2]

         The notation max[^4, B] in equation [20.2] refers to picking the maximum of quantities A and
         B. Furthermore,  the  parameter k designates half the displacement or shift  in  standard
         deviations that should be quickly detected on a control chart. Often k is set equal to 1, meaning
         that the control chart will be designed to rapidly detect upward concentration shifts of at least
         two standard deviations. Since Z; is standardized by the estimated baseline standard deviation,
         an increase  of r units in Z; corresponds to an increase of r standard deviations above  the
         baseline mean in the domain of concentrations *;.

Step 4.   To plot the control  chart in concentration units, compute the  non-standardized CUSUMs  S°
         with the equation:

                                        S; = xB+St-sB                                   [20.3]

Step 5.   Calculate the non-standardized control  limit used  to assess  compliance  of  both future
         measurements (x;) and non-standardized CUSUMs (£/;). Traditionally, two parameters were
         used to compute standardized limits: the decision internal value (h) and the Shewhart Control
         Limit (SCL). The Unified Guidance instead recommends only one standardized control limit
         (K). Compute the non-standardized control limit (/zc) as:

                                         hc = xB + h-sB                                    [20.4]

Step 6.   Construct the control  chart by plotting both the compliance  measurements (x;) and the non-
         standardized CUSUMs (S°.)  on the ^-axis against the  sampling events 7] along the x-axis.
         Also draw a horizontal line at the concentration value equal to the control limit, hc.

Step 7.   Moving forward in  time from the first plotted sampling event T\, declare the  control chart to
         be potentially out-of-control if either of two situations occurs: 1) the trace of non-standardized
         concentrations exceeds hc; or 2) the CUSUMs become too large, exceeding hc.

         The first case  signifies a rapid increase in concentration level among the most recent sample
         data.  The second can represent either a sudden rise in concentration levels or a gradual
         increase over time. A gradual  increase or trend is particularly indicated if the CUSUM exceeds
         the control limit but the compliance concentrations do not. The reason for this is that several
         consecutive, small,  increases in x; will not trigger the  control limit,  but may cause a large
         enough increase in the CUSUM. As such, a control chart can indicate the  onset  of either
         sudden or gradual contamination at the compliance point.
                                             20-3                                    March 2009

-------
Chapter 20. Control Charts
                                                                           Unified Guidance
       ^EXAMPLE 20-1

     For background nickel data collected during 8 months in 1995 shown below, construct an intrawell
control chart and compare it with the first 8 months of the compliance period (1996):
                                        Nickel Concentration (ppb)
Month
1
2
3
4
5
6
7
8
Baseline Period
(1995)
32.8
15.2
13.5
39.6
37.1
10.4
31.9
20.6
Compliance Period
(1996)
19.0
34.5
17.8
23.6
34.8
28.8
23.7
81.8
Step 1
       SOLUTION
         As discussed in Section 20.3.3, control charts are a parametric procedure requiring normal or
         normalized data. Test the n = 8 baseline measurements for normality. A probability plot of
         these data provided in Figure 20-1  exhibits  a mostly  linear trend.  The  Shapiro-Wilk test
         statistic computed for these data isW= 0.896.  Compared to the a = .10 level critical point of
         w.io,8 = 0.851  (Table 10-3 of Appendix  D), the Shapiro-Wilk test indicates that the baseline
         data  are  approximately  normal.   Construct  the control  chart using the original nickel
         measurements.

                  Figure 20-1. Probability Plot of Baseline Nickel Data
                S  o
                IM
                   -1
                   -2
                             10          20          30           40

                                   Nickel Concentration (ppb)
                                            20-4
                                                                                  March 2009

-------
Chapter 20. Control Charts	Unified Guidance

Step 2.   Use the 1995 baseline nickel data to compute the sample mean and standard deviation:  XB  =
         25.14 ppb and SB= 11.518 ppb. Then compute the standardized concentration Z; for each 1996
         compliance period sampling event using equation [20.1]. These values are listed in the fourth
         column of the table below.
Month
1
2
3
4
5
6
7
8
Tj
1
2
3
4
5
6
7
8
Nickel (ppb)
19.0
34.5
17.8
23.6
34.8
28.8
43.7
81.8
Zj
-0.53
0.81
-0.64
-0.13
0.84
0.32
1.61
4.92
Z-k
-1.53
-0.19
-1.64
-1.13
-0.16
-0.68
0.61
3.92
s.
0.00
0.00
0.00
0.00
0.00
0.00
0.61
4.53
Sic
25.14
25.14
25.14
25.14
25.14
25.14
32.16
77.31
Step 3.   Compute the standardized CUSUMs as follows. First let the shift displacement parameter k =
         1 and set So = 0. After subtracting k from each Z;, calculate the CUSUM using equation [20.2]
         .  Note that none of the CUSUMs are positive until the first occurrence of a positive quantity
         (Z; - K).  As shown in the sixth column above, the standardized CUSUMs for the 6th, 7th and
         8th events are calculated as:
                              S6 = max[o,(o.32 -1)+ o] = 0

                              S7=max[o,(l.61-l)+o] = 0.61

                              Ss = max[o,(4.92 -1)+ 0.6l] = 4.53
Step 4.   Calculate the non-standardized CUSUMs (Sc.} using the  individual Z;, baseline mean and
         standard deviation parameters in equation [20.3]. These values are listed in the last column of
         the table above. For the 8th sampling event, this calculation gives:

                               Sc%= 25.14 + 11.518(4.53)= 77.31

Step 5.   Compute the non-standardized  control limit using  equation [20.4]. For  purposes  of this
         example, set h = 5; the non-standardized limit becomes:

                               hc = 25.14 + 11.518(5)= 82.73 ppb

Step 6.   Using the compliance period nickel concentrations and the non-standardized CUSUMs, plot
         the control chart as in Figure 20-2. The combined chart indicates there is insufficient evidence
         of groundwater contamination in 1996 because neither the  nickel concentrations nor the
         CUSUM statistics exceed the control  limit for the  months examined.  However, both traces
         nearly exceed /zc, and conceivably might do so in future sampling events if the apparent trend
         continues. If that were to happen, retesting can be performed to better determine whether the
                                             20-5                                   March 2009

-------
Chapter 20.  Control Charts
                                                            Unified Guidance
         increase was one or a series of chance fluctuations or an actual mean-level change in nickel
         concentrations. -^

       Figure 20-2. Shewhart-CUSUM Control Chart for Nickel Measurements

           100

                                                                  h-Bn ±
        JH
        Q.
        CL
            75 -
         o  50 H
         CD
        Jii
         CJ
            25 -
                                     \
                                     4
                              \
                              5
\
6
                                                           -CUSUM
                                                           "Ntkel
                                 Sam plhg Event
20.3 CONTROL CHART REQUIREMENTS AND ASSUMPTIONS

     As with other statistical methods, control charts are based on certain assumptions about the sample
data. There are also some minimum requirements for constructing them.  None of the assumptions or
requirements are unique to control charts, although there are some special issues.
20.3.1
STATISTICAL INDEPENDENCE AND STATIONARITY
     The methodology for control charts assumes that the sample data are statistically independent. A
control chart can give misleading results if consecutive sample measurements are serially correlated (i.e.,
autocorrelated). For this reason, it is important to  design a sampling plan so that distinct volumes of
groundwater are analyzed at each sampling event (Section 14.3.1). Duplicate laboratory analyses (i.e.,
aliquot or field splits) should also not be treated as independent observations when constructing a control
chart. Gibbons (1999) recommends that control chart observations be collected no more frequently than
quarterly. Since  physical independence does generally not guarantee  statistical  independence (Section
14.1), a test of autocorrelation using the sample autocorrelation function or rank von Neumann ratio tests
(Section  14.2)  should be performed to determine whether the current  sampling  interval affords
uncorrelated measurements.

     If the background data exhibit a clear seasonal cyclical pattern, the values should be deseasonalized
before computing the control  chart baseline parameters.  For a seasonal pattern  at a single  well,  the
                                           20-6
                                                                   March 2009

-------
Chapter 20.  Control Charts	Unified Guidance

method of Section  14.3.3.1 can be used to  create adjusted measurements having a stable mean.  At
several or a group of wells indicating a common seasonal pattern, the adjusted values can be computed
using a one-way analysis of variance [ANOVA] for temporal effects (Section 14.3.3.2). When baseline
data are  deseasonalized,  it  is  essential  that  newly  collected compliance  measurements also be
deseasonalized in the same manner.  It is presumed that the same pattern or physical cause will impact
future data in the same manner as for the baseline measurements.

     To deseasonalize  compliance point  measurements, simply use the  seasonal and  grand  means
estimated from background in  computing the adjusted compliance  point values.  If the  control chart
remains in control  following deseasonalizing, the  existing background can be updated with the newer
measurements.  However, the revised background set should be checked again for seasonality and the
seasonal and grand means re-computed,  in order to  more accurately adjust future measurements.

     Control charts also assume that the background mean is stationary over time. This means there
should be no apparent upward or downward trend in the background measurements. A  trend imparts
greater-than-expected variation to the background data, increasing the baseline standard deviation and
ultimately the control limit. The net result is  a control chart that has less power to identify groundwater
contamination. Tests for trend described in  Chapter 17 can  be used to check the assumption of no
background trends. Should an upward or downward trend be verified, the  background data  should not be
de-trended. While  it is possible to construct and use a control chart with  de-trended background and
future  data, the  assumption that the trend will continue indefinitely is very  problematic.   The trend
should first be investigated to ensure that background has been properly designated.  Other monitoring
wells should be  checked to see if the same trend  is occurring, indicating either evidence of an earlier
release or possibly a sitewide change in  the aquifer. In any case, a switch should be made to a trend test
rather than a control chart.

     As noted, control charts can be employed as  either interwell or intrawell tests. However, interwell
control  charts require a spatially stationary mean  across the monitoring network. If spatial variability
exists  among background wells for certain constituents,  interwell control  charts will  be no  more
interpretable than prediction limits.  A related problem can plague intrawell control charts if there is prior
spatial variability (i.e.,  some compliance wells are already contaminated prior to selection of intrawell
backgrounds). Historical observations  should be  used  as baseline data in intrawell tests only if the
compliance wells are known to be unaffected by a release from the monitored unit. Otherwise, the
control  limit based on the greater-than-expected  background values may  be  set too high to identify
current contamination.
                                             20-7                                    March 2009

-------
Chapter 20. Control Charts                                                 Unified Guidance
20.3.2       SAMPLE SIZES AND UPDATING  BACKGROUND

     Both background mean and standard deviation estimates are needed to construct a control chart
limit. The Unified Guidance recommends at least n = 8 measurements for the defining the baseline,
particularly to ensure an accurate standard deviation estimate. Baseline observations are traditionally not
plotted on the chart, although it may be visually helpful to include background values on the plot using a
distinct symbol (e.g., hollow instead of filled symbol).

     Whether baseline observations are obtained from upgradient background wells for interwell testing
or from individual compliance well historical  data for intrawell use, these data are only small  random
samples used to estimate the true background population characteristics.  Any particular sample  set may
not be adequately representative.  Because of this likelihood, the background sample size requirements
suggested above for constructing a control chart should be regarded as a minimum. More background
observations  should preferably be added to  the  initial set to improve  the characterization of the
background distribution.

     For interwell control charts, periodic updating of background (Chapter 5) poses no difficulty. New
observations should be collected at background wells on each sampling event. Then, every 1-2 years, the
newly collected background should be added to the existing  background pool after testing/checking for
statistical similarity. The revised background can be used to re-compute the baseline parameters and, in
turn, the control limit.

     Updating background for intrawell control charts  depends on the control  chart remaining 'in-
control' for several consecutive sampling events. As long as a confirmed exceedance does not occur, the
in-control compliance measurements collected  since the last background update can be tested against the
existing background for statistical similarity using  a Student's t- or Wilcoxon rank-sum test (Section
5.3).  ASTM Standard D6312-98 (1999) recommends  testing the newly revised background set for
trends, using  trend tests including those in Chapter 17.  The ASTM methodology is intended to avoid
incorporating  a  subtle trend  into the  control chart  background, which  influences the re-computed
baseline parameters  and weakens the statistical  power  of  the control chart to identify contaminant
releases.

     If the comparison of recent in-control  measurements against existing background indicates a
statistically significant difference, it may reflect changes in natural groundwater conditions unrelated to
contamination events. In these circumstances, it is possible to update background by creating a 'moving
window.' The background  sample size  n remains  fixed, with only the most recent n  measurements
included as background for computing baseline parameters.  Earlier sampling events are  excluded. The
overriding goal is to ensure that background reflects the most current and representative groundwater
conditions (Chapter 3).

     Despite the apparent benefits, the statistical performance of control charts is  only partially known
when background is periodically updated.  Davis (1999) has performed the most extensive simulations
of this question.  He suggests that substantially  different simulation results occur with the CUSUM
portion when background is periodically updated (especially early on) and combined with either a small
maximum run length or a 'warm-up' period or both (see Section 20.4.1).

                                             20-8                                   March 2009

-------
Chapter 20. Control Charts	Unified Guidance

     Two  other issues affect both  control  charts and  prediction limits when  updating  intrawell
background. First, if background is periodically augmented by adding new measurements (either from
upgradient  background wells or  from  recent  in-control  compliance  measurements), the  overall
background sample  size is increased.  This in turn should cause the prediction or control chart limit to
decrease.

      For instance,  prediction limit tables in Chapter 19 demonstrate that as the background  sample
size increases, lower prediction limit ^multipliers are appropriate. The expanded background sample is
used to re-compute  the prediction limit, provided that the measurements added to  background do not
indicate an adverse change  in groundwater quality. New compliance measurements are then tested
against the revised prediction limit. But the same cannot be done with control charts unless the CUSUM
is reset to zero. The reason  is that the CUSUM will have already been affected by those compliance
measurements now being added to  intrawell  background.   An independent  comparison between
compliance point values and background  is  thus precluded. Consequently,  the  Unified Guidance
recommends that the CUSUM portion of the  control  chart be reset after each periodic update of intrawell
background.2

     The second issue is how to update intrawell background when an initial measurement has exceeded
the control  or prediction limit, but one or more resamples disconfirm the exceedance. Routine detection
monitoring continues in this  situation.  No confirmed exceedance is registered for a prediction limit test
and the control  chart remains in-control.  Should the initial exceedance be included or excluded when
later updating intrawell background?

     The Unified Guidance recommends a strategy  parallel to the handling of outliers (Chapter 12). If
the exceedance can be shown to be a measurement in error or a confirmed outlier, it  should be excluded
from the revised background. Otherwise,  any disconfirmed exceedances (including  any resamples  that
exceed the background limit but are disconfirmed by other resamples) should probably be included when
updating the background. The reason is that background limits designed to incorporate retesting are
computed as  low as possible to ensure adequate statistical power.  The trade-off  is that compliance
measurements legitimately similar  to  background but drawn from  the upper tail of the distribution,
sometimes  exceed  the limit and have  to  be disconfirmed  with a  resample. Any exceedance  not
documented as  an error or  outlier is most  likely representative of some portion  of the background
population that previously had gone unsampled or unobserved.
20.3.3       NORMALITY AND NON-DETECT DATA

     The  combined Shewhart-CUSUM control  chart is a parametric procedure.  This  implies that
background  used to estimate the baseline parameters should either be  normal or normalized via a
transformation. Normality can be tested on either the raw measurement or transformed scale using one of
the goodness-of-fit  techniques described in Chapter 10. If the hypothesis  of normality is accepted,
  The same 'overlapping' dependence between the CUSUM and revised background will also be true when background is
  updated using a 'moving window' approach. The CUSUM should therefore be reset in these cases too. However, since the
  background sample size is kept fixed, the standardized control limit (h) will not decrease as it does when background is
  augmented.
                                             20-9                                   March 2009

-------
Chapter 20.  Control Charts	Unified Guidance

construct the control chart on the raw measurements. If it is rejected, try a transformation and retest the
transformed data for normality. If the transformation  works to normalize background, construct the
control  chart on the transformed measurements,  being sure to use the same transformation on  both
background and the compliance values to be plotted.

     Unlike prediction limits, no non-parametric version of the combined Shewhart-CUSUM control
chart exists. If the background sample cannot be normalized perhaps  due to a large fraction of  non-
detects, a non-parametric prediction limit should be considered (Section 19.4).  Control charts will be
most appropriate for those constituents with a reasonably high detection  frequency. These include many
inorganic constituents (e.g., certain trace elements, indicators and geochemical monitoring parameters)
that occur naturally in groundwater, or for other persistently detected, site-specific organic chemicals.

     If no  more than 10-15% of the data  are non-detect, it may be possible to normalize  the data via
simple substitution (Section 15.2)  of half the reporting limit [RL] for each background non-detect.  A
normalizing transformation can sometimes be found using a censored probability plot (Chapter 15) for
background data containing a substantial  fraction of non-detects up to 50%.  A censored estimation
technique such as Kaplan-Meier or Robust Regression on Order Statistics [Robust ROS] (Chapter 15)
can then be used to compute estimates of the baseline mean  (JUB) and standard deviation (<7B) that
account for the left-censored measurements. These adjusted estimates  should replace the  background
sample  mean  (XB)  and standard deviation (SB) in the control chart equations of  Section 20.2.  The
Unified Guidance  differs  somewhat from the recommended approach in  ASTM Standard D6312-98
(ASTM, 1999), which is to set all non-detects identically to zero.

     No matter how background non-detects are treated, control charts require an additional step for
future observations that isn't  needed with  prediction limits. Each new  compliance point measurement
statistic must be added to the  CUSUM associated with previous sampling events. If the new observation
is  a  non-detect,  some value  (typically a  fraction of the RL)  needs to be imputed for the censored
measurement in order to update the CUSUM. The Unified Guidance recommends  that half the RL be
substituted  for these measurements.3
3 If an intrawell control chart is constructed and it remains 'in-control' until the next background update, any non-detects
  observed in the meantime should be treated as left-censored measurements for purposes of updating the baseline mean and
  standard deviation estimates. In other words, the simple substitution of RL/2 should only apply temporarily to compute an
  updated CUSUM.

                                             20-10                                    March 2009

-------
Chapter 20. Control Charts                                                 Unified Guidance
20.4 CONTROL CHART  PERFORMANCE CRITERIA

     A significant difference exists between  control charts and prediction limits in setting statistical
performance criteria. Standard equations described in previous chapters allow the user to generate an
exact confidence level (1-a) for prediction limits. Obtaining similar confidence levels for the Shewhart-
CUSUM control charts needs to be done experimentally through varying the two background control
chart limits (K) and the displacement parameter (k\ as well as the retesting options.  The control chart
parameter limits in the two previous EPA RCRA statistical guidance documents were based on work by
Lucas  (1982), Hockman  & Lucas (1987), and  Starks (1988).   Monte Carlo  simulations for various
combinations  of control chart  parameters  (without  retests)  were  used to  develop  the  overall
recommendations in their papers.

     The specific parameter choices were not fixed, but appeared to work best in simulations at a single
well. Starks (1988) recommended setting h = 5 and k = 1 for standardized measurements, especially in
the early stages of monitoring. He further suggested that after 12 consecutive in-control measurements,
the baseline mean and standard deviation be updated to include more recent sampling measurements.
The values of k and SCL (the separate Shewhart control limit) could then be reduced to k = 0.75 and SCL
= 4.0.  In effect, this tightens the control chart limits to reflect that additional data are available to better
characterize the baseline population.

     More recent  research (notably Gibbons,  1999)  has  demonstrated  that control  charts from the
quality control literature do not account for several important characteristics of groundwater monitoring
networks. The most important is  the problem of multiple comparisons (i.e.., the need to simultaneously
conduct testing of many  well-constituent pairs  during an evaluation period described in Chapter 6).
Control chart performance is typically assessed on an individual well basis, rather than over a network of
simultaneous tests. The recommended control limits have no obvious connection to the expected false
positive rate (a),  nor is the traditional control  limit adjustable like the r-factor in prediction limits.
There is a need to account for differences in background sample sizes, a desired false positive rate, and
the number of monitoring network tests in similar fashion to prediction limits. Moreover, early research
and guidance did not address the issue of retesting in control  charts. Retesting provides substantial
improvements in prediction limit performance, and its potential needs to be evaluated  for control charts.

     It is standard practice to discuss the performance of prediction limits in terms of statistical power
and false positive rates.  However, statistical performance of control charts is usually measured via the
average run length [ARL].  The ARL is the average number of sampling events before the control limit
is  first exceeded, identifying  an  'out-of-control' process.  Ideally,  the ARL should  be large when the
mean concentration of the tested constituent is at or near the baseline average, but increasingly smaller as
the true mean is gradually shifted  above baseline.

     Put in standard statistical terms, the control chart should not easily or quickly signal false evidence
of a release when a release has not occurred. To have a low false positive rate when  the null hypothesis
of no contamination is true, the chart should stay 'in-control' for a long time indicated by a large ARL.
The statistical power for detecting a release when it occurs should be as high as possible. A short ARL
will indicate that a control chart is quickly determined to be out-of-control.

                                             20-11                                  March 2009

-------
Chapter 20. Control Charts	Unified Guidance

     False positive rates (a) for CUSUM control charts cannot be equated precisely with ARLs.  But it
has been found that the ARLs closely follow a geometric distribution pattern with a mean equal to (I/a).
Thus, a control  chart with an ARL of 100 would have an associated false positive  rate of roughly 1%.
The relationship is  not  exact, especially  for combined Shewhart-CUSUM control  charts.  It is  also
affected by the randomness in the background data used to establish the control chart baseline.

     Thus, the Unified Guidance offers a new framework for measuring  control chart statistical
performance.  It is suggested  that measuring false  positive rates in control charts  be conducted by
establishing a time frame or run length of interest, specifically, a period of one year. A false positive is
counted if the chart has a confirmed exceedance sometime during the year, under the assumption of no
contaminant release. Statistical power is similarly evaluated for a fixed time interval (e.g., one year) by
measuring the proportion of run lengths with confirmed exceedances during that interval. In this way,
both the false positive rate and power are tied to a specific one-year time frame.

     This  framework is consistent  with the  guidance  recommendations  that  prediction  limit
performance be measured  according  to  an annual,  cumulative 10%  site-wide  false positive  rate
[SWFPR]  and that cumulative, annual effective power be comparable to the EPA reference power curves
[ERPC]. The suggested framework for control charts allows a direct comparison with prediction limits
when designing alternate statistical approaches.
20.4.1      CONTROL CHARTS WITH MULTIPLE COMPARISONS

     Until recently, control charts were not designed to address the SWFPR when testing multiple well-
constituent pairs.  Furthermore, it was not clear to a user how to adjust for multiple tests using fixed
control limits (SCL, k and h). Because of these problems, Gibbons (1999) performed a series of Monte
Carlo simulations  to  gauge  intrawell control chart performance for up to 500  simultaneous tests.
Gibbons also examined the outcomes when the single Shewhart and CUSUM decision limit was allowed
to vary between h = (4.5, 5.0, 5.5, and 6.0}.  He found that control charts could be designed with both
high power and a low SWFPR, as long as retesting was incorporated into the methodology.

     Additional Monte Carlo simulation work was performed by Davis (1999). He found that control
charts perform similarly to prediction limits when both use retests.  But he also noted  that certain
favorable outcomes in Gibbons (1999) were the result of combining frequent updating of background
and a 'warm-up' period for the chart. In the latter period, any control limit exceedances were ignored.
The simulations were based on small maximum run lengths.

     Other researchers have noted (for instance,  Lucefio and Puig-Pey,  2000) that  the  run  length
distribution of CUSUM control charts is often close to geometric.  This implies that even when the ARL
is large, there can be significant probability of an early failure. The difficulty in a real-life setting is that
one will not know whether an early exceedance of the control limit is due to contaminated groundwater
or simply a false positive exceedance for an otherwise  in-control chart. This guidance recommends
against the use of  'warm-up' periods when implementing or assessing the performance of Shewhart-
CUSUM control charts.
                                            20-12                                  March 2009

-------
Chapter 20.  Control Charts _ Unified Guidance

     Gibbons (1999) provides results for a number of control chart limit options, but does not determine
limits which can provide exact false positive rate control.  A number of potential commonly applied
retesting strategies are also not evaluated.  In contrast, both Gibbons (1994) and the Unified Guidance
(Chapter 19) do provide such control for prediction limits using a wider array of retesting strategies.

     Facilities may need  to  conduct theirown  specific Monte  Carlo simulations  if  the published
literature options cannot be applied at their site.  Simulations might be needed for either intrawell  or
interwell control charts or both.   Overall  methodologies for Monte Carlo simulations are provided
below.  The first step for either type  test is a simulation of the cumulative annual false positive rate.
Then a second simulation measures the cumulative, annual statistical power.

     To perform an intrawell simulation, repeat the following steps for a large number  of simulations
(e.g.,NSim= 10,000):

  1.    Determine the total  number of well-constituent pairs for which statistical testing  is required,  as
       well as the number  of pairs at  which intrawell control charts will be constructed. Use the basic
       subdivision  principle (Section  19.2.1) to determine the per-test  false positive rate (atest)
       associated with each control chart that meets the target SWFPR.

  2.    Determine the intrawell background sample size (n). Generate n standard normal measurements.
       Then form baseline estimates by computing the sample mean (XB ) and standard deviation (SB).

  3.    Pick a set of possible standardized control limits (h). Choose a maximum run length (M), based
       on the number of sampling events conducted each year (e.g.,M= 4 for quarterly sampling).

  4.    For each potential control  limit (//), compute the non-standardized control limit using equation
       [20.4]. Then simulate the behavior of the control chart from sampling event 1 to sampling event
       M by generating standard normal compliance measurements for each event.  Generate enough
       random measurements to  account  for  resamples  potentially  needed with a selected  retesting
       strategy.

  5.    Test the initial measurement associated with each sampling event against the non-standardized
       control limit. Also  form the CUSUM for events  1  to M using equations [20.2] and [20.3].
       Compare the non-standardized CUSUM against the control limit.

  6.    If either the initial measurement or the CUSUM exceeds hc, use the resample(s) for that sampling
       event to perform a retest (see below). If the retest confirms the initial exceedance, record a false
       positive for that particular simulation (out ofWsim).

  7.    After all Nsjm runs have been conducted, compute the observed false positive rate (an) associated
       with each possible standardized control limit (h) by dividing Ns[m into the number of observed
       false positives. Set the final control limit equal to that value of h for which an is closest to
     The simulation for an interwell control chart is similar to the intrawell case,  with a few key
differences. First, instead of a per-test false positive rate, the basic subdivision principle must be used to
compute a per -constituent false  positive rate (aCOnst).  The reason  is that the  same  background
measurements for a given  constituent are used to test each of the compliance wells in the network.

                                             20-13                                    March 2009

-------
Chapter 20. Control Charts	Unified Guidance

Secondly, when generating standard normal compliance point measurements in Step 4 of the intrawell
simulation, a set of such random observations needs to be generated for each of the w wells  in the
network. The behavior of w control charts must be simulated using a common set of background data
and single control limit for each one.

     Once a control limit meeting the target SWFPR has been established, a second Monte Carlo
simulation is run to determine the statistical power of the control chart. Since effective power is defined
as the ability to flag a single contaminated well-constituent pair, the basic steps are the same for either
interwell or intrawell control charts. Repeat the following over a large number of simulations (A/sim).

  1.    Determine the  background  sample size (n). Generate n standard normal measurements. From
       these, form baseline estimates by computing the sample mean (XB ) and standard deviation (SB).

  2.    Using the standardized control limit (h) chosen in the first Monte Carlo simulation, compute a
       non-standardized control limit using equation [20.4]. Then simulate  the behavior of the control
       chart from  sampling  event 1 to  sampling event  M by generating sets of normal N(A,1)
       compliance measurements for each event, where A varies from  1 to 5 by unit steps.  Generate
       enough random measurements in  each set to account for resamples potentially needed with a
       selected retesting strategy.

  3.    For  each   set   of  successively  higher-valued  compliance  measurements,  test  the   initial
       measurement associated  with each sampling event  against the non-standardized control limit.
       Also form the CUSUM for  events 1 to Musing  equations [20.2] and [20.3]. Compare the non-
       standardized CUSUM against the control limit.

  4.    If either the initial measurement or the CUSUM exceeds hc, use the resample(s) for that sampling
       event to perform a retest (see below). If the  retest confirms the initial exceedance, record a true
       detection for that particular mean-level A and simulation (out of TVsim).

  5.    After all Nsjm runs have been conducted, compute the observed power (1-0) associated with each
       true mean level  (A) by dividing Nsim into the number of observed detections.  The simulated
       effective power curve for standardized control limit (h) is a plot of (1-0) versus A for A = 1 to 5.

     If the  standardized control limit identified during Monte Carlo simulation has effective  power
comparable to the appropriate ERPC (matching the site-specific sampling frequency to one of the three
curves in  Chapter 6:  quarterly,  semi-annual, or annual), h can be used to form site-specific control
limits. For interwell limits, compute the (upgradient) background mean and  standard deviation for each
monitoring constituent and use equation [20.4] to form the final, non-standardized control limits. For
intrawell limits, use the same equation only with intrawell background at each well-constituent pair.
20.4.2       RETESTING IN CONTROL CHARTS

     Control chart and prediction limit tests are only practical for most monitoring networks if retesting
is part of the procedure, demonstrated both by Gibbons  (1999) and Davis (1999).  A key issue is to
decide how control chart retesting should be conducted. Practical  retesting strategies for prediction

                                            20-14                                   March 2009

-------
Chapter 20. Control Charts	Unified Guidance

limits on future observations are described in Section 19.1, including both \-of-m (for m = 2, 3, 4) and
modified California plans.

     ASTM  Standard D6312-98  (1999)  recommends  a l-of-2  retesting  strategy: whenever  an
exceedance of the control limit occurs on a  given sampling event, the next quarterly sampling event is
used as the  resample. Furthermore, if the exceedance is not  confirmed by the resample, the ASTM
standard recommends that the initial exceedance be replaced in the CUSUM by the follow-up sampling
event, thus implicitly assuming that the initial observation was an error.

     Gibbons (1999) considers the performance of other retesting plans, including l-of-2, l-of-3, and
the original Cal-3 plan (see Section 19.1 and Appendix B).  For each plan, resampling is triggered when
the most recent observation  either by itself  exceeds  or causes the CUSUM to exceed  the limit. Then,
each resample (if  more than one) is compared  against h.   The initial exceedance  measurement is
removed from the CUSUM computation, replaced by the resample, and then re-compared to the control
limit. A statistically significant  increase [SSI] is  declared only if the resample  verifies  the  initial
exceedance (or both resamples for a l-of-3 plan).

     Gibbon's study and  ASTM Standard  D6312-98  raises an important concern  as  to  the  most
statistically powerful  treatment of the CUSUM when  an initial exceedance is not confirmed by retesting.
A second concern addresses when resamples  should be collected.

     The Unified Guidance suggests two practical possibilities to address the first concern.  The initial
exceedance can be removed  from the CUSUM altogether, re-setting the  CUSUM to its value from the
previous sampling  event.  As noted  above, this is essentially assuming the first sampling event was in
error.  Another  option is to replace the initial exceedance by the first resample which disconfirms the
exceedance, and then re-compute the CUSUM with that resample.

     In either  strategy, the  effects  on statistical  power and  accuracy  should be simulated when
constructing site-specific control  limits as in the procedure outlined  above. Both the false positive rate
and power depend on a faithful simulation  of all aspects of the control  chart testing procedure.  This
includes background  sample  size, the number of well-constituent pairs evaluated, the retesting strategy
and how the CUSUM is adjusted for resampling.

     The second issue concerns when resamples should be collected.  The Unified Guidance does not
recommend using the next scheduled  sampling event as a resample. If  the exceedance were due to a
laboratory analytical  error or calculation mistake, a more  quickly retrieved resample  can resolve the
discrepancy without waiting until  the next quarterly or semi-annual monitoring event.

     Where multiple resamples  are used (a l-of-3  plan, for  instance),  one would have to wait two
additional sampling rounds simply to collect the resamples. These in  turn could not be plotted on the
control chart as regular sampling events without intermingling the roles of resamples  and non-resamples,
thereby complicating the interpretation and  assessment  of control chart performance. The common
guidance recommendation is to  identify an intermediate period or periods for resampling between
regularly scheduled evaluations for both control  charts and prediction limits.
                                             20-15                                   March 2009

-------
Chapter 20.  Control Charts                                          Unified Guidance
                     This page intentionally left blank
                                       20-16                               March 2009

-------
PART IV. COMPLIANCE & CORRECTIVE ACTION TESTS	Unified Guidance
     PART  IV:   COMPLIANCE/ASSESSMENT
         AND  CORRECTIVE  ACTION TESTS
    This  last  part  of  the  Unified  Guidance  addresses  statistical  methods  useful  in
compliance/assessment and corrective action monitoring, where single-sample testing is required against
a fixed groundwater protection standard [GWPS]. These standards include not only health- or risk-based
limits,  but also those derived from background as a fixed standard.  The full subject of background
GWPS testing is treated in Section 7.5, but any of the procedures in the following chapters might be
applied to single-sample background tests.

    The primary tool  for both stages of monitoring is the confidence interval. Several varieties of
confidence intervals are presented  in Chapter 21,  including confidence  intervals around  means,
medians, and upper percentiles for stationary populations, and confidence bands around a trend for cases
where groundwater concentrations are actively changing.

    Strategies to implement confidence interval tests are discussed in Chapter 22. In particular, the
focus is on designing tests with reasonable statistical performance in terms of power and per-test false
positive rates.

    Chapter 7  of Part I provides  a discussion of the overall compliance/assessment and corrective
action  monitoring network design.  Program  elements such as the appropriate hypothesis structure,
selecting the appropriate  parameter for comparison  to a  fixed limit GWPS, sampling frequency,
statistical power,  and confidence levels are covered. These final two chapters present the tests in greater
detail.
                                                                       March 2009

-------
PART IV.  COMPLIANCE & CORRECTIVE ACTION TESTS                 Unified Guidance
                    This page intentionally left blank
                                                                      March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance

             CHAPTER 21.   CONFIDENCE  INTERVALS
       21.1   PARAMETRIC CONFIDENCE INTERVALS	21-1
         21.1.1  Confidence Interval Around Normal Mean	21-3
         21.1.2  Confidence interval Around Lognormal Geometric Mean	21-5
         21.1.3  Confidence Interval Around Lognormal Arithmetic Mean	21-8
         21.1.4  Confidence Interval Around Upper Percentile	21-11
       21.2   NON-PARAMETRIC CONFIDENCE INTERVALS	21-14
       21.3   CONFIDENCE INTERVALS AROUND TREND LINES	21-23
         21.3.1  Parametric Confidence Band Around Linear Regression	21-23
         21.3.2  Non-Parametric Confidence Band Around The il-Sen Line	21-30
     Confidence intervals are the recommended general statistical strategy in compliance/assessment or
corrective action monitoring.  Groundwater monitoring data must typically be compared to a fixed
numerical limit set as a GWPS. In compliance/assessment, the comparison is made to determine whether
groundwater concentrations have increased above the compliance standard. In corrective action, the test
determines whether  concentrations have decreased below a  clean-up  criterion or compliance level. In
compliance/assessment monitoring, the lower confidence limit [LCL] is of primary interest, while the
upper confidence limit  [UCL] is most important in corrective action.  For single-sample background
GWPS testing, the hypothesis structures are the same as for fixed-limit health-based standards. Where a
GWPS is based on two- or multiple sample testing, a  somewhat different hypothesis structure is used
(Section 7.5) and detection monitoring test procedures in Part III are applicable.

     General strategies for using confidence intervals in compliance/assessment or corrective action
monitoring are presented in Chapter 7,  including discussion of how regulatory standards should be
matched to particular statistical parameters (e.g., mean or upper percentile). More specific strategies and
examples are detailed in Chapter 22. In  this  chapter, basic  algorithms  and equations for each type of
confidence interval are described, along with an example of the calculations involved.
21.1 PARAMETRIC CONFIDENCE INTERVALS

     Confidence intervals are designed to estimate statistical characteristics of some  parameter of a
sampled population. Parametric confidence intervals  do this for  known  distributional models, e.g.,
normal, lognormal, gamma, Weibull, etc. Given a statistical parameter of interest such as the population
mean (u),  the lower and upper limits  of a confidence interval define the most probable concentration
range in which the true parameter ought to lie.

     Like any estimate, the  true parameter may not be located within the confidence interval. The
frequency  with which this error tends to occur (based on repeated confidence intervals on different
samples of the same sample size and from the same population) is denoted a, while its complement (1-
a) is known as the confidence level. The confidence level represents the percentage of cases where a
confidence interval constructed according to a  fixed algorithm or equation will  actually  contain  its
                                            21-1                                  March 2009

-------
Chapter 21.  Confidence Intervals	Unified Guidance

intended target, e.g., the population mean.  Section 7.2 discusses the difference between one- and two-
sided confidence intervals and how the a error is assigned.

     A point worth clarifying is the  distinction between a as the  complement of the confidence level
when constructing a confidence  interval  and the significance level (a) used in hypothesis testing.
Confidence intervals are often used strictly for estimation of population quantities. In that case, no test is
performed, so a does not represent a false positive rate.  Rather, it is simply the fraction of similar
intervals that do not contain their intended target.

     The  Unified Guidance focuses  on confidence interval limits compared to a fixed standard as a
formal test procedure. In this  case, the  complement (a) of the confidence  level used to generate the
confidence interval is equivalent to the  significance level (a)  of the test.  This assumes that the true
population parameter under the null hypothesis is no greater than the standard in compliance/assessment
monitoring or not less than the  standard in corrective action.1

     The  parametric confidence  intervals presented  in the Unified Guidance share some common
statistical  assumptions.  The most basic is that measurements used  to construct a confidence interval be
independent and identically distributed [i.i.d.]. Meeting this assumption requires that there be no outliers
(Chapter  12), a stationary mean and variance over the period during which observations are collected
(Chapters 3 and 14), and no autocorrelation between successive sampling events  (Chapter 14). In
particular,  sampling events  should  be  spaced  far  enough  apart  so that  approximate  statistical
independence  can be assumed (at many sites, observations should not be sampled more often than
quarterly).  Sample data should also be examined for  trends. The  mean  is  not stationary  under a
significant trend, as assumed in applying the other methods of this section. An apparent trend may need
to be handled by computing a confidence band around the trend line (Section 21.3).

     Another  common assumption is that the sample data are either normal in distribution or can be
normalized via a transformation (Chapter 10). Normality can be difficult to check if the sample contains
a significant number of left-censored measurements (i.e., non-detects). The  basic options for censored
samples are presented in Chapter 15. If the  non-detect percentage is no more than 10-15%, it may be
possible to assess normality by first substituting one-half of the reporting limit [RL] for each non-detect.
For higher non-detect percentages up  to 50%, the Unified Guidance recommends computing a censored
probability plot using either the Kaplan-Meier or Robust Regression on Order Statistics [Robust ROS]
techniques (both in Chapter 15).

     If a censored probability plot suggests that the sample (or some transformation of the sample) is
normal, either Kaplan-Meier or Robust ROS  can be used to construct estimates of the mean (//) and
standard deviation (GWPS.

                                             21^2                                    March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance

21.1.1      CONFIDENCE INTERVAL AROUND NORMAL MEAN

       BACKGROUND AND PURPOSE

     When compliance point data is to be compared to a fixed standard (e.g., a maximum concentration
limit  [MCL])  and the  standard in question is  interpreted  to  represent an  average  or  true  mean
concentration, a confidence interval around the mean is the method of statistical choice. A confidence
interval around the mean is designed to estimate the true average of the underlying population, while at
the same time accounting for variability in the sample data.

       REQUIREMENTS AND ASSUMPTIONS

     Confidence intervals around the mean of a normal distribution should only be constructed if the
data  are approximately normal or at least are reasonably  symmetric  (i.e., the skewness coefficient is
close to zero). An inaccurate confidence interval is likely to result if the sample data are highly non-
normal, particularly for right-skewed  distributions. If the observations are better fit by a lognormal
distribution,  special equations or methods need to be used to construct an accurate confidence interval on
the arithmetic mean with  a specified  level  of confidence (Section 21.1.3).   Therefore, checking for
normality is an important first step.

     A confidence interval should  not be constructed with less than 4 measurements per compliance
well, and preferably 8 or more. The equation for  a normal-based confidence interval around the mean
involves estimating the population standard deviation  via the sample standard deviation (s).  This
estimate can often be imprecise using  a small sample size (e.g., n < 4).  The equation also involves a
Student's ^-quantile based on n-\ degrees of freedom  [df\, where n equals the sample size.   The t-
quantile is large for small n, leading to a much wider confidence interval than would occur with a larger
sample size.  For a 99% confidence level, the appropriate  ^-quantile would be t = 31.82 for n = 2, t =
4.54  for n = 4, and t = 3.00 for n = 8.

     This last consideration  is important since statistically significant evidence of a violation during
compliance/assessment or success during corrective action  is indicated only when the entire confidence
interval is to one side of the standard (i.e., it does not straddle the fixed standard; see Chapter 7). For a
small sample size, the confidence interval may be so wide that a statistical difference is unlikely to be
identified. This can happen  even if the true mean groundwater concentration is  different from  the
compliance or clean-up standard, due to the statistical uncertainty associated with the  small number of
observations. More specific recommendations on appropriate sample sizes are presented in Chapter 22,
where the statistical power of the confidence interval tests is explored.

       PROCEDURE

Step  1.   Check the basic statistical assumptions of the sample as discussed above.  Assuming a normal
         distributional model is acceptable, calculate the sample mean (x ) and standard deviation (s).

Step  2.   Given a sample of size n and the desired level of confidence (1-oc), for each compliance well
         calculate either the lower confidence limit (for compliance/assessment monitoring) with the
         equation:
                                             21-3                                    March 2009

-------
Chapter 21.  Confidence Intervals _ Unified Guidance
                                                                                         [2L1]

         or the upper confidence limit (for corrective action) with the equation:
         where ^-a,»-i is obtained from a Student's Stable with (w-1) degrees of freedom (Table 16-1
         in Appendix D).  To construct a two-sided interval with overall confidence level equal to (1-
         a), substitute a/2 for a in the above equations.

Step 3.   Compare the limit calculated in Step 2 to the fixed compliance or clean-up standard (e.g., the
         MCL or alternate concentration limit [ACL]. For compliance/assessment monitoring, the LCL
         in equation [21.1] should be used  to compute the  test. For  corrective action, the UCL  in
         equation [21.2] should be used instead.

       ^EXAMPLE 21-1

     The table below lists concentrations  of the pesticide Aldicarb in three compliance  wells. For
illustrative purposes, the health-based standard in compliance monitoring for Aldicarb has been set to 7
ppb. Determine at the a = 5% significance  level  whether or not any of the wells should be flagged  as
being out of compliance.
Sampling Date
January
February
March
April
Mean
SD
Skewness (yj
Shapiro-Wilk (W)
Well 1
19.9
29.6
18.7
24.2
23.10
4.93
0.506
0.923
Aldicarb Concentration (ppb)
Well 2
23.7
21.9
26.9
26.1
24.65
2.28
-0.234
0.943
Well 3
5.6
3.3
2.3
6.9
4.52
2.10
0.074
0.950
       SOLUTION
Step 1.   First test the data for non-normality and/or significant skewness. Based on four samples per
         well, the skewness coefficients and Shapiro-Wilk statistics have been computed and are listed
         above. None of the skewness coefficients are significantly different from zero. In addition, the
         a = .10 critical point for the Shapiro-Wilk test with n = 4 (as presented in Chapter 10) is
         0.792, less  than each of the Shapiro-Wilk statistics; consequently, there is  no significant
         evidence of non-normality. Construct a normal-based confidence interval around the mean.

Step 2.   Calculate the sample mean and standard deviation  of the Aldicarb concentrations  for  each
         compliance well. These statistics are listed above.


                                             21-4                                    March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance

Step 3.   Since a = 0.05, the confidence level must be set to (1-a) = 0.95. Obtain the upper 95th
         percentile  of the  ^-distribution with (n-Y) =  3  degrees of freedom from Table 16-1 in
         Appendix D, namely £95,3 = 2.353. Then calculate the lower confidence limit [LCL] for each
         well's mean concentration, using equation [21.1]:

                      Well 1: ZCZ95 = 23.10-(2.353 x4.93y>/4 = 17.30 ppb

                      Well 2: LCL95 = 24.65 - (2.353 x 2.28)/>/4 = 21.97 ppb

                      Well 3: LCL95 = 4.52 - (2.353 x 2. lo)/>/4 = 2.05 ppb

Step 4.   Compare each LCL to the compliance standard of 7 ppb. The LCLs for Well 1 and Well 2 lie
         above 7 ppb, indicating that the mean concentration of Aldicarb in both of these wells
         significantly exceeds the compliance standard. However, the LCL for Well 3 is below 7 ppb.
         providing insufficient evidence at the  a = 0.05  level  that the  mean in  Well  3 is out of
         compliance.  -^
21.1.2      CONFIDENCE INTERVAL AROUND LOGNORMAL GEOMETRIC MEAN

       PURPOSE AND BACKGROUND

     For  many  groundwater  monitoring  constituents, neither  the assumption  of normality nor
approximate symmetry holds for the  original concentration data. Often the underlying population  is
heavily right-skewed, characterized by a majority of lower level concentrations combined with a long
right-hand tail of infrequent but  extreme  values.   A  model such  as the lognormal distribution  is
commonly used to analyze such data.

     The lognormal is traditionally designated by the notation A((i,  a) (Aitchison and Brown, 1976),
where (I and a denote parameters controlling the location and  scale of the population.  Typically
designated as N((i, a), a normal distribution also has parameters (I and a which denote the true mean and
standard deviation.  These two parameters play different roles in lognormal  distributions.   The key
distinction is between the arithmetic domain  (or the  original measurement scale of the data) and the
logarithmic domain. The latter denotes the mathematical space following a logarithmic transformation.
Transformed lognormal data are normally-distributed in the logarithmic domain.  In this new domain, (I
represents the true mean of the log-transformed measurements— that is, the log-mean.  Likewise,  a
represents the true standard deviation of the log-transformed values or the log-standard deviation.

     A  common misperception is to  assume  that a standard equation for a normal-based confidence
interval  can be applied to log-transformed data, with the interval endpoints then back-transformed (i.e.,
exponentiated) to the arithmetic domain to get a confidence interval around the lognormal arithmetic
mean. Invariably, such an interval will underestimate the true mean.  The Student t- confidence interval
applies to a geometric mean of the lognormal population when back-transformed, rather than the higher-
valued arithmetic  mean. The reason is that the sample log-mean gives  an estimate of the lognormal
parameter (i. When this estimate is back-transformed  to the arithmetic domain, one has an estimate of
                                            21-5                                   March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance

exp((i) — the lognormal geometric mean — not an estimate of the lognormal arithmetic mean, which is
expressed as exp((i + .5a2).

     Although a confidence interval around the lognormal geometric mean is not an accurate estimate of
the arithmetic mean, there are instances where such an interval may be helpful. While many GWPSs are
interpreted  to represent long-term arithmetic averages, some (as  detailed in Chapter 7) can better
represent medians or percentiles of the underlying distribution. Because the lognormal geometric mean
is  equivalent to the  median, a geometric mean  may in some cases be a better statistical parameter of
comparison  than  the  lognormal  arithmetic mean. Furthermore, when  the lognormal  coefficient  of
variation is large, the  arithmetic mean is  substantially  larger than the geometric mean,  mostly due to
infrequent but extreme individual measurements. The bulk of individual observations are located much
closer to the geometric mean. It may be that a comparison of the GWPS to the geometric mean rather
than to the arithmetic mean will provide a more reasonable test of long-term concentration levels.

     Special equations or computational methods are used to construct an accurate confidence interval
with a specified level of confidence (Section 21.1.3) when an estimate of the arithmetic mean is needed
and the observations are approximately normal.  There is another factor to consider when estimating an
upper confidence limit on the lognormal arithmetic mean using Land's procedure (described in Section
21.1.3) or other possible procedures  (see for instance Singh et al.,  1997).  When used with highly
variable data, it can lead to severely-biased, high estimates of the confidence limit.  This can make it
very difficult to evaluate the success of corrective action measures.

     In these cases, precise parametric estimation  of the arithmetic mean may have to be foregone in
favor of an alternate statistical procedure.  One such alternative is a non-parametric confidence interval
around the median (Section 21.2). Another alternative when the sample is approximately lognormal is
an estimate  around  the geometric mean which  is equivalent to the population median. A third more
computationally intensive option is a  bootstrap confidence  interval  around the lognormal arithmetic
mean (see discussion in Section 21.1.3). Unlike  the first two options, this last alternative allows a direct
estimate of the arithmetic mean.

       REQUIREMENTS AND ASSUMPTIONS

     Confidence  intervals around the geometric mean of  a lognormal distribution should only be
constructed if the log-transformed data are approximately normal or at least reasonably symmetric (i.e.,
the skewness coefficient in the logarithmic domain is close to zero). The methods of Chapter 10 can be
used to test normality of the log-transformed values. If the log-transformed sample contains non-detects,
normality on the log-scale  should be assessed using a censored probability plot.  Adjusted estimates of
the mean and standard deviation on the log-scale can then be substituted for the log-mean (J7) and log-
standard deviation (sy) in the equations below.   Like a normal  arithmetic mean, a confidence interval
around the lognormal geometric mean should not be constructed without a minimum of 4 measurements
per compliance well, and preferably with 8  or more.

       PROCEDURE

Step 1.   Take  the logarithm of each measurement,  denoted as y\, and check the n log-transformed
         values for normality. If the log-transformed measurements are approximately normal, calculate


                                             21-6                                    March 2009

-------
Chapter 21. Confidence Intervals _ Unified Guidance

         the log-mean (J) and log-standard deviation (sy). If the normal model is rejected, consider
         instead a non-parametric confidence interval (Section 21.2).

Step 2.   Given  the  desired   level  of   confidence   (1-00,   calculate   either  the  LCL   (for
         compliance/assessment monitoring) with the equation:

                                              (           s ^
                                                                                         [21.3]
         or the UCL (for corrective action) with the equation:

                                              (
                                                                                         [21.4]
         where ^-a,»-i is obtained from a Student's Stable with (n-\) degrees of freedom (Table 16-1
         in Appendix D).  In order to construct a two-sided interval with the overall confidence level
         equal to (1-a), substitute a/2 for a in the above equations.

Step 3.   Compare the limits calculated in Step 2 to the fixed compliance or clean-up standard (e.g., the
         MCL or ACL). For compliance/assessment, use the LCL in equation [21.3]. For corrective
         action, use the UCL in equation [21.4].

         Note in either case that the regulatory authority will have to approve the use of the geometric
         mean as a reasonable basis of comparison against the  compliance standard.  In some cases,
         there may be few other  statistical options. However, stakeholders should understand that the
         geometric and  arithmetic  means  estimate  two distinct  statistical  characteristics  of the
         underlying lognormal population.

       ^EXAMPLE 21-2

     Suppose the following 8 sample measurements of benzene (ppb) have been collected at a landfill
that previously handled smelter waste and is now undergoing remediation  efforts. Determine whether or
not there is statistically significant evidence at the a = 0.05 significance level that the true geometric
mean benzene concentration has fallen below the permitted MCL of 5 ppb.
Sample Month
1
2
3
4
5
6
7
8
Benzene (ppb)
0.5
0.5
1.6
1.8
1.1
16.1
1.6
<0.5
Log Benzene
log(ppb)
-0.693
-0.693
0.470
0.588
0.095
2.779
0.470
-1.386

                                             21-7                                   March 2009

-------
Chapter 21. Confidence Intervals                                         Unified Guidance
       SOLUTION
Step 1.   To estimate an upper confidence bound on the geometric mean benzene concentration with
         95% confidence, first test the skewness and normality of the data set. Since the one non-detect
         concentration is unknown but presumably between 0 ppb and the RL of 0.5 ppb, a reasonable
         compromise is to impute this value at 0.25 ppb, half the RL. The skewness is computed as yi =
         2.21, a value too high to suggest the data are normal.  In addition, a Shapiro-Wilk test statistic
         on the raw measurements works out to SW = 0.521, failing an assumption of normality at far
         below a significance level of a = 0.01.

         On the other hand,  transforming the data  via natural logarithms  gives a smaller skewness
         coefficient of yi = 0.90 and a Shapiro-Wilk statistic of W= 0.896. Because these values are
         consistent with normality on the log-scale (the critical point for the  Shapiro-Wilk test with n =
         8 and a = 0.10 is 0.818), the data set should be treated as lognormal for estimation purposes.
         As a consequence, equation [21.4] can be used to construct a one-sided UCL on the geometric
         mean.

Step 2.   Compute the sample log-mean  and log-standard deviation. This gives  y = 0.2037 log(ppb)
         and sy = 1.2575 log(ppb).

Step 3.   Apply the log-mean and log-standard deviation into equation [21.4] for a UCL with a = .05, n
         = 8, and 7 degrees of freedom. This gives an estimated limit of:

                           (        s^
                UCL95 = exp y + t95 7-f=\= exp(2037 + 1.895 x .4446)= 2.847 ppb
Step 4.   Compare the UCL to the MCL of 5 ppb. Since the limit is less than the fixed standard, there is
         statistically significant  evidence that the benzene geometric mean,  and consequently, the
         median benzene concentration, is  less than 5 ppb. However, this calculation does not show
         that  the benzene arithmetic  mean is  less  than the MCL. Extreme individual  benzene
         measurements could show up with enough regularity to cause the arithmetic mean to be higher
         than 5 ppb. A
21.1.3      CONFIDENCE INTERVAL AROUND LOGNORMAL ARITHMETIC MEAN

       PURPOSE AND BACKGROUND

     Estimation  of a lognormal arithmetic mean is not completely straightforward. As  discussed in
Section 21.1.2, applying standard equations for normal-based confidence limits around the mean to log-
transformed measurements and then exponentiating the limits, results  in confidence intervals that are
invariably underestimate the arithmetic mean.

     Inferences on arithmetic means for certain kinds of skewed populations can be made either exactly
or approximately through the  use of special techniques. In particular, if a confidence interval on the
arithmetic mean is desired, Land (1971; 1975) developed an exact technique along with extensive tables
                                            2T8March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance

for implementing  it when the underlying population  is  lognormal.  Land also developed  a more
complicated approximate technique (for a full description and  examples see EPA,  1997) when the
population  can  be transformed to  normality via any other increasing, 1-1, and  twice  differentiable
transformation (e.g.., square, square root, cube root, etc.).

     Although the core of Land's procedure is a correction for the so-called 'transformation bias' that
occurs when making back-transforming estimates from the logarithmic domain to the raw concentration
domain, it  can  produce unacceptable results, particularly with UCLs.  The Unified Guidance advises
caution  when applying Land's procedure,  particularly when the lognormal  population has  a  high
coefficient of variation. In those cases, the user may want to consider alternate techniques, such as those
discussed in Singh, et al (1997 and 1999). One option is to use EPA's free-of-charge Pro-UCL software
Version 4.0 (www.epa.gov/esd/tsc/software.htm).  It computes a variety of upper confidence limits,
including a bootstrap confidence interval around the arithmetic mean. This technique can be applied to
lognormal data to get a direct, non-parametric UCL that tends to be less  biased and to give less extreme
results than Land's procedure.

     For cases  or sample sizes  not covered by Tables 21-1 through 21-8 in Appendix D when using
Land's procedure, Gibbons and  Coleman (2001) describe a method of approximating the necessary H-
factors. The same authors review other alternate parametric methods for computing UCLs.

       REQUIREMENTS AND ASSUMPTIONS

     Confidence intervals around the arithmetic mean of a lognormal distribution should be constructed
only if the data pass a test of approximate normality on the  log-scale. While  many groundwater and
water quality populations tend to follow the lognormal  distribution, the data should first be tested for
normality on the original concentration scale. If such a test fails, the sample can be  log-transformed and
re-tested. If the  log-transformed sample contains  non-detects, normality on the log-scale should be
assessed using a censored probability plot  (Chapter 15).  If a lognormal model  is tenable, adjusted
estimates of the mean and standard deviation on the log-scale can be substituted for the log-mean (J7)
and log-standard deviation (sy) in the equations below.

     As with normal-based confidence intervals, the confidence interval here should not be constructed
with fewer than 4 measurements per compliance well, and preferably with 8 or more. The reasons are
similar: the equation for a lognormal-based confidence interval around the arithmetic mean depends on
the sample log-standard deviation (sy), used  as an estimate  of the underlying  log-scale  population
standard deviation. This estimate can be quite imprecise when fewer than 4 to 8 observations  are used.
A special factor (H) was developed by Land to account for variability  in a skewed population. These
factors are larger for smaller samples sizes, and need to be exponentiated to estimate the final confidence
limits (see below). Consequently there is a  significant penalty associated with estimating the arithmetic
mean using a small sample size, occasionally seen in remarkably wide confidence limits. The effect is
especially noticeable when computing an UCL for corrective action monitoring.

       PROCEDURE

Step 1.   Test the log-tranformed  sample for normality. If the lognormal model provides a reasonable
         fit, denote the log-transformed measurements by^i and move to Step 2.
                                             21-9                                   March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance

Step 2.   Compute the sample log-mean (J) and log-standard deviation (sy).

Step 3.   Obtain the correct bias-correction factor(s) (//«) from Land's (1975) tables  (Tables  21-1
         through  21-8 in Appendix D), where the correct factor depends  on the sample size («), the
         sample log-standard deviation (sy), and the desired confidence level (1-oc).

Step 4.   Plug these factors into one of the equations given below for the LCL or UCL (depending on
         whether the comparison applies to compliance/assessment monitoring or to corrective action).
         Note that to construct a two-sided interval with an overall confidence level  of (1-00, the
         equations should be applied by substituting a/2 for a.

                                                            ^
                                                                                       [21.5]
                                       = exp  y + .5s2 +^=     I                           [21.6]
Step 5.   Compare the confidence limit computed in Step 4 to the fixed compliance or clean-up
         standard. In compliance/assessment monitoring, use the LCL of equation [21.5]. In corrective
         action, use equation [21.6] for the UCL.

       ^EXAMPLE 21-3

     Determine whether the  benzene concentrations  of  Example 21-2  indicate that  the benzene
arithmetic mean is below the permitted MCL of 5 ppb at the a = 0.05 significance level.

       SOLUTION
Step 1.   From Example 21-2, the benzene data were found to fail a test of normality, but passed a test
         of lognormality (i.e., they were approximately normal on the log-scale). As a consequence,
         Land's equation  in [21.6]  should  be  used to construct a one-sided UCL on the arithmetic
         mean.

Step 2.   Compute the log-mean and log-standard deviation from the log-scale data. This gives  y  =
         0.2037 log(ppb) and sy = 1.2575 log(ppb).

Step 3.   Using Table 21-6 in Appendix D, pick the appropriate //-factor for estimating confidence
         limits around a lognormal arithmetic mean, noting that to achieve 95% confidence for a one-
         sided UCL, one must use (1-a) = 0.95. With a sample size of n = 8 and a standard deviation
         on the log-scale of 1.2575 log(ppb), H.95 =  4.069.

Step 4.   Plug these values along with the log-mean of 0.2037 log(ppb) into equation  [21.6] for the
         UCL. This leads to a 95% one-sided confidence limit equal to:
                                            21-10                                  March 2009

-------
Chapter 21. Confidence Intervals _ Unified Guidance
           (         ,      x  (l.2575Y4.069Yl
UCL95=exp\ .2037 + .5(1. 5813)+-^ - ft - ^1=
                                                                   18.7 ppb
Step 5.   Compare the UCL against  the MCL of 5 ppb. Since the UCL is  greater than the MCL,
         evidence is  not sufficient at the 5% significance  level to conclude that the true benzene
         arithmetic mean concentration is now below the MCL. This conclusion holds despite the fact
         that  all but one  of the benzene measurements is less  than  than 5 ppb.  In lognormal
         populations, it is not uncommon to see one or two seemingly extreme measurements coupled
         with a majority of much lower concentrations.  Since these  extreme measurements  help
         determine the location of the arithmetic mean, it is  not unreasonable to expect that the true
         mean might be larger than 5 ppb.

         The  contrast in this result to Example 21-2 is noteworthy. In that case,  the UCL on the
         geometric mean was only 2.85 ppb. The  estimated lognormal coefficient of variation  with
         these data (Chapters 3 and 10) is CV= 1.965, somewhat on the high side. It is no surprise that
         results for the arithmetic and geometric means on the same sample are rather different. Neither
         estimator is  necessarily invalid, but a decision needs to be made as to whether the MCL for
         benzene in this  setting  should be better compared  to an arithmetic mean or to a geometric
         mean/ median for lognormal distributions. ~4
21.1.4      CONFIDENCE INTERVAL AROUND UPPER PERCENTILE

       BACKGROUND AND PURPOSE

     Although most MCLs and ACLs appear to represent arithmetic or long-term averages (Chapter 7),
they can also be interpreted as standards not to be exceeded with any regularity. Other fixed standards
like nitrate/nitrite attempt to limit short-term risks and thus represent upper percentiles instead of means.
In these cases, the appropriate confidence interval is one built around a specific upper percentile.

     The  particular upper percentile  chosen will  depend on what the fixed  compliance standard
represents or is  intended  to represent. If the standard is a concentration that  represents  the  90th
percentile, the  confidence interval  should be built around the upper 90th percentile. If the standard is
meant to be a maximum,  'not to be  exceeded,' concentration, a slightly different strategy should be used.
Since there is no maximum value associated with continuous distributions like normal and lognormal, it
is not possible to construct a confidence interval around the population maximum.  Instead, one must
settle for a confidence interval around a sufficiently high percentile,  one that will exceed nearly all of the
population measurements.  Possible choices  are the  upper 90th to 95th percentile. By  estimating the
location of these percentiles, one needs to determine whether a sufficiently small fraction (e.g., at most 1
in 10 or 1 in 20) of the possible measurements will ever exceed the standard. For even greater protection
against exceedances, the upper 99th percentile could be selected,  implying  that at most 1 in  100
measurements  would ever exceed  the standard.  But as noted in Chapter 7, selection of very  high
percentiles using non-parametric tests can make it extremely difficult to demonstrate corrective action
success.
                                            21-11                                  March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance

       REQUIREMENTS AND ASSUMPTIONS

     The equations for constructing parametric confidence intervals around an upper percentile assume
that the data are normally distributed, at least approximately. If the data  can  be normalized via a
transformation, the observations should first be transformed before computing the confidence interval.
Unlike confidence intervals around an arithmetic mean for transformed data, no  special equations are
required to construct similar intervals around an upper percentile.  The same equations used for normal
data can be applied to data in the transformed domain. The only additional step is that the confidence
interval limits must be back-transformed prior to comparing them against the fixed  standard.

     The confidence interval presented here should not be constructed with fewer than 4 measurements
per compliance well, and preferably  with 8  or more. Too  small a sample size leads to imprecise
estimates of the sample standard deviation (s). Another reason is that the confidence interval equation
involves a special multiplier T, which depends on both the desired confidence level  (1-a) and the sample
size (n). When n is quite small, the T multiplier is much greater.  This  leads to a much wider confidence
interval than that  obtained with a larger n,  and therefore much greater statistical uncertainty.  For
example, at a confidence level of 95%, the appropriate T  multiplier for an upper one-sided limit on the
95th percentile is T = 52.559 when n = 2,r = 6.602 when n = 4, and r  = 3.640 when n = 8.

     When determining the r factor(s) needed for a confidence interval around an upper percentile, it
should be noted that unlike the symmetric Student's ^-distribution, separate T factors need to be
determined for the LCL and UCL. Since an upper percentile like the 95th is generally larger than the
population mean, the  equations for both the lower (i.e.., LCL) and upper (i.e., UCL) limits involve
adding a multiple of the standard deviation to the sample mean. The only difference is that a smaller
multiple TLCL is used  for the LCL, while a larger TUCL is used for the upper confidence limit. For certain
choices of w, P and 1-a, the multiple TLCL can even be negative.

       PROCEDURE

Step 1.   Test the raw data for normality.  If approximately normal, construct the interval on the original
         measurements. If the data can be normalized via a  transformation, construct the interval on the
         transformed values.

Step 2.   For a normal sample,  compute  the sample mean  (x) and standard deviation (s). If the data
         have been transformed,  compute  the mean  and  standard deviation  of the  transformed
         measurements.

Step 3.   Given the percentile (P) to be estimated, sample size («),  and the desired confidence level
         (1-a),  use Tables 21-9 and 21-10 in Appendix D to determine the  rfactor(s) needed to
         construct the appropriate one-sided or two-sided interval. A one-sided LCL is then computed
         with the equation:

                                                                                        [21.7]

         where T(P;  n, a) is the lower a factor for the Pth  percentile given n sample measurements. A
         one-sided UCL is given similarly by the equation:
                                            21-12                                   March 2009

-------
Chapter 21. Confidence Intervals _ Unified Guidance

                                                                                       [21.8]

         Finally, a two-sided confidence interval is computed by the pair of equations for the LCL and
         UCL:

                                 LCLl_a/2=x+s-T(P;n,a/2)                            [21.9]

                                UCLl_a/2=x+s-T(P;n,l-a/2)                         [21.10]

Step 4.   If the data have been transformed, the equations of Step 3  would be used but with two
         changes:  1) the mean and standard deviation of the transformed values are substituted for
         x and s;  and  2) the resulting limits back-transformed to get  final confidence limits in  the
         concentration  domain. If a logarithmic transformation has been employed, the log-mean and
         log-standard deviation would be substituted for the sample mean and standard deviation. The
         resulting limit(s) must be exponentiated to get the final confidence limits, as in the equations
         below:
                                                                                      [21.11]

                                     = exp ;7 + VrP; w,l-or                          [21.12]
Step 5.   Compare the confidence limit(s) computed in Step 3 (or Step 4) versus the fixed compliance
         or clean-up standard. In compliance/assessment, use the LCL of equation [21.7]. In corrective
         action, use  equation [21.8] for the UCL.

         Note that although the above equations differentiate between the a-error used with  the LCL
         and 1-a for the UCL, Tables 21-9 and 21-10 in Appendix D are constructed identically.  The
         a-error is represented by its confidence complement 1-a in Table 21-10 of Appendix D.

       ^EXAMPLE 21-4

     Assume that a facility permit has established an ACL of 30 ppb that should not be exceeded more
than 5% of the time. Use the  Aldicarb  concentrations and diagnostic statistical  information from
Example 21-1 to evaluate data from the three compliance wells.  Determine whether any of the wells
should be flagged as being out of compliance.

     SOLUTION
Step 1.   From Example  21-1,  all of the wells pass a normality  test. Use the sample mean and
         standard deviation for each compliance well, from the tabular information in Example 21-1.

Step 2.   Select the correct r factor  from Table 21-10  of Appendix D to construct a 99% LCL  on the
         upper 95th  percentile. The upper 95th percentile is needed because the permitted ACL cannot
         be exceeded more than 5% of the time, implying that 95%  of all the Aldicarb measurements
         should fall below the fixed standard. With n = 4 observations per well, this leads to r (P; «, a)
         = r(.95;4, .01) = 0.443.

                                            21-13                                   March 2009

-------
Chapter 21.  Confidence Intervals	Unified Guidance

Step 3.   Compute the LCL for each well as follows using equation [21.7]:

                           Welll: LCL99 = 23.10 + (0.443)(4.93) = 25.2%ppb
                           Welll :LCL99 = 24.65 + (0.443)(2.28) = 25.66ppb
                           Well3 :LCL99 = 4.52 + (0.443)(2.10) = 5A5ppb

Step 4.   Compare each LCL against the ACL of 30 ppm.  Since each well LCL is less than the ACL,
         there is  insufficient statistical evidence that the upper 95th percentile of the  Aldicarb
         distribution exceeds the fixed standard. Consequently, there is no conclusive evidence that
         more than 5% of the Aldicarb concentrations will exceed the ACL.

         If the  site were in corrective  action instead of compliance/assessment monitoring, UCLs
         around the 95th percentile would be needed instead of LCLs.  In  that case, with n  = 4
         observations per well, r(P; n, 1-oc) = r(.95; 4, .99) = 9.083 from Table 21-9 of Appendix D.
         Then, the respective well UCLs would be:

                           Welll: UCL99 = 23.10 + (9.083)(4.93) = 61 .^ ppb
                           Well2:UCL99 = 24.65 + (9.083)(2.28) = 45.36 ppb
                           Welft :UCL99 = 4.52 + (9.083)(2.10) = 23.59ppb

         In this case, two of the three wells would not meet the corrective action limit of 30 ppb. -^
 21.2 NON-PARAMETRIC CONFIDENCE INTERVALS

       BACKGROUND AND PURPOSE

     A non-parametric confidence interval  should be considered when a sample is non-normal and
 cannot be normalized, perhaps due to a significant fraction of non-detects.  Non-parametric confidence
 interval endpoints are generally chosen as order statistics of the sample data. The specific order statistics
 selected will depend on the sample size (n), the desired confidence level  (1-a), and the population
 characteristic being estimated.

     Since the data are not assumed to follow a particular distribution, it is generally not possible to
 construct a confidence interval around the population mean.  One fairly rare exception would be if it
 were already known that the distribution is  symmetric (where the mean is also the median). Sample
 order statistics represent, by definition, concentration  levels exceeded by a certain number and hence a
fraction of the sample values. They are excellent estimators of the percentiles of a distribution, but not
 of quantities like the arithmetic mean.  The latter entails summing the data values and averaging the
 result.  In positively-skewed populations, not only is the arithmetic mean greater than the median, it also
 may not correspond to any particular percentile.

     Non-parametric confidence intervals can be developed either around a measure of the center of the
 population (i.e., the  population median or 50th percentile) or around an upper or lower percentile (e.g.,
 the upper 90th). The choice of percentile affects which order statistics are selected as interval endpoints.


                                            21-14                                  March 2009

-------
Chapter 21.  Confidence Intervals _ Unified Guidance

The sample median is generally estimated using a smaller order statistic than that used for an upper 95th
percentile.

     Despite the distinction between non-parametric confidence intervals around the median and similar
intervals around an upper or lower percentile, the mathematical algorithm used to construct both types is
essentially  identical. Given an unknown P x 100th percentile of interest (where P is between  0 and  1)
and a sample of n  concentration measurements, the probability that any randomly selected measurement
will be less than the P x 100th percentile is simply P. Then the probability that the measurement will
exceed the P x  100th percentile is (I-/3).  Hence the number of sample values falling below the P x
100th percentile out of a set  of n should follow a binomial distribution with parameters n and success
probability P, where 'success' is defined as the event that a sample measurement is below the P x 100th
percentile.

     Because of this connection, the binomial distribution can be used to determine the probability that
the interval formed by a given pair of order statistics will contain the percentile  of interest. This kind  of
probability  calculation makes repeated  use of the cumulative binomial  distribution, often  denoted
Bin(x',nj)).   It represents the probability of x or fewer successes occurring in n trials with success
probability/?. The  computational equation for this expression2 can be written as:


                                                                                         [21.13]
      To make statistical inferences about the P x 100th percentile, P (expressed as a fraction) would be
substituted for/? in equation [21.13]. It can  be seen why the  same basic algorithm applies both to
confidence intervals around the median and around upper percentiles like the 95th. If an interval around
the median is desired, one would set P = 0.50.  For an interval needed around the upper 95th percentile,
one would set/1 = 0.95 and perform similar calculations.

      When constructing non-parametric confidence intervals, the type of confidence interval needs to be
matched against the kind of fixed standard to which it will be compared. Since the arithmetic mean
cannot be estimated directly, a confidence interval  around the median should be used for those cases
where the compliance standard represents an average. Some fixed standards can,  of course, be directly
interpreted as median concentration levels, but even for those standards representing arithmetic averages,
the confidence interval  on the median will  give the 'next best' comparison when a non-parametric
method is used.

      The interpretation of a confidence interval on the  median  is similar to  that of a  parametric
confidence interval around the mean. In compliance/assessment monitoring, if the LCL with confidence
level  (1-a)  exceeds the compliance  standard, there is statistically significant evidence that the  true
                            n}
   The mathematical expression      refers to the combination of n events taken / at a time.  It can be calculated as:
                           (?)
  n!/(i!x[n-i]!), where n! = {n x (n-1) x... x 2 x 1}.  By convention, 0!=\.
                                              21-15                                   March 2009

-------
Chapter 21.  Confidence Intervals	Unified Guidance

population median is higher than the standard. In corrective action monitoring, if the UCL is below the
clean-up standard, one can conclude that the true population median is less than the standard with a-
level significance.

       REQUIREMENTS AND ASSUMPTIONS

     Because a non-parametric confidence interval does not assume a specific distributional form for
the underlying population, there is no need to fit a probability model to the data.  If a significant portion
of the data are non-detect, it may be impossible to adequately fit such a model.. The non-parametric
confidence interval method only requires the ability to rank the sample data values and pick out selected
order statistics as the interval endpoints. Unfortunately, this ease of construction comes with a price. As
opposed to parametric intervals, non-parametric confidence intervals tend to be wider and  generally
require larger sample sizes to achieve comparably high confidence levels.  To compute the LCL around
the median with 99%  confidence, at least 7 compliance point measurements are needed in the non-
parametric case.  Therefore, sample  data  should be fit to a specific probability  distribution whenever
possible.

     The general method  for constructing non-parametric confidence intervals involves an iterative
testing procedure, where potential endpoints are selected from the sorted data  values (i.e.,  order
statistics) and then tested to determine what confidence level is associated with those endpoints. If the
initial choice of order statistics gives an interval with insufficient confidence, the interval needs to be
widened and tested again.  Clearly, the greatest confidence will be  associated with an interval defined by
the minimum and maximum observed sample values. But if the sample size n is  small, even the largest
possible confidence level may be less than the desired target confidence (e.g., (1-a) = 0.99).  As such,
the actual or achieved confidence level  needs to be listed when  reporting results of a non-parametric
confidence interval test.

     It  may be especially  difficult to achieve target confidence  levels  around upper percentiles even
when the sample size is fairly large.  An instructive example is when estimating an upper 95th percentile
with a  sample size of n = 20. In that case,  the  highest possible two-sided confidence  level is
approximately 64%, achieved when the minimum and maximum data values are taken as the interval
endpoints. The confidence level is substantially less than the usual targets of 90% or more,  and has very
limited value as a decision basis.

     The width of a confidence interval (which expands as the level  of confidence increases) should be
balanced against the desire to construct an interval narrow enough to provide useful  information about
the probable  location of the underlying  population characteristic (e.g., the P =  95th percentile in the
above example).  A reasonable  goal is to construct the shortest interval possible that still approaches the
highest confidence level.  In the example,  a confidence level of almost 63% could be achieved by setting
the 17th and 20th ordered  sample values as the confidence interval endpoints. The 20th ordered value is
obviously the maximum observation and cannot be changed. However, if any ranked value less than the
17th is  taken as the lower endpoint, the confidence level will increase only slightly, but the overall
interval will be unnecessarily widened.

     An iterative process is used to construct non-parametric confidence limits.  It is recommended that
a stopping rule be used to decide when the improvement in the confidence level  brought  about  by
picking more extreme order statistics is outweighed by the loss of information from making the interval

                                             21-16                                   March 2009

-------
Chapter 21. Confidence Intervals _ Unified Guidance

too wide. A reasonable stopping rule might be to end the iterative computations if the confidence level
changes by less than 1 or 2 percent when a new set of candidate ranks is selected.

     Repeated calculation of cumulative binomial distribution probabilities Bin(x;n,p) are quite tedious
when performed manually. One can make use of either an extensive table of binomial probabilities or a
software package that computes them. Almost all commercial statistical packages will compute binomial
probabilities. For  small sample sizes  up to n < 20, Table 21-11 in Appendix D provides achievable
confidence levels  for various  choices of the  sample order statistic endpoints such as the median and
common upper percentiles.

     Tied values  do not affect the procedure for constructing non-parametric confidence intervals. All
tied values (including any non-detects  treated as ties) should be regarded as distinct measurements.
Because of this, ties can be arbitrarily broken when ranking the data. For example,  a list of 6 values
including 3 non-detects would be ordered as [<5, <5, <5,  8, 12, 20] and given the set of ranks [1, 2, 3, 4,
5, 6]. Note that it  is possible for the LCL to be set equal to the RL used for non-detects.

       PROCEDURE  FOR A CONFIDENCE INTERVAL AROUND THE MEDIAN

Step 1 .   Given a sample of size w, order the measurements from least to greatest.  Denote the ordered
         values by *(i), X(2),..., X(«),  where x^ is the rth  concentration value in the ordered list and
         numbers 1 through n represent the data ranks.

Step 2.   Given P = .50, pick candidate interval endpoints by choosing ordered data values with  ranks
         as close to and as symmetrical as possible around the product of (n+l) x 0.50. If this last
         quantity is a fraction (an even-numbered sample size), the ranks immediately above and below
         it can be selected as candidate endpoints. If the  product (n+\) x 0.50 is an integer (an odd-
         numbered sample size), add 1 and subtract 1 to get the upper and lower candidate endpoints.
                                                                                            *
         Once  the candidate  endpoints have been  selected, denote the ranks of these endpoints by L
         and U.

Step 3.   For a two-sided confidence  interval, compute the confidence level  associated with the
         tentative endpoints  L*  and  if  by taking  the  difference  in the  cumulative binomial
         probabilities given by the equation:

                                                              r/*-i f  \ f  \n
                   l-a=Bm(u*-l; n,.5o)-Bin(i: -1; «,.5o)= V I  " \\ -I               [21.14]
         For a one-sided LCL, compute the confidence level associated with endpoint L  using the
         equation:
                                                                                       [21.15]
         For a one-sided UCL, compute the confidence level associated with endpoint U  using the
         equation:

                                            21-17                                  March 2009

-------
Chapter 21. Confidence Intervals _ Unified Guidance
                                                     r/*_i / N / \

                            \-a=Bin(u* - 1; W,.50)=
         To minimize the amount of direct computation needed, these equations have been used to
         compute  selected cases  over a range  of sample sizes for the median in Table  21-11 of
         Appendix D

Step 4.   If the  candidate endpoint(s)  do not achieve  the  desired confidence level, compute new
         candidate endpoints (L*-l) and (£/*+!) and re-calculate the achieved confidence level. Repeat
         this process until the target confidence level is achieved. If one candidate endpoint  already
         equals the data minimum or maximum, only change the rank of the other endpoint.  If neither
         endpoint rank can be changed, set either:  1) the minimum concentration value as a one-sided
         LCL; 2) the maximum concentration value as a one-sided UCL; or 3) the interval spanned by
         the range of the sample as a two-sided confidence  interval around the median. In each case,
         report the achieved confidence level associated with the chosen confidence limit(s).

Step 5.   Compare the confidence  limit(s) computed in Step  4 versus the fixed compliance or clean-up
         standard.  In compliance/assessment monitoring, use the LCL derived as the order statistic with
         rank L* . In corrective action monitoring, use the UCL derived as the order statistic with rank
         U*.

       ^EXAMPLE 21-5

     Use the following four years of well beryllium concentrations, collected quarterly for a total of n =
16 measurements, to compute a non-parametric LCL on the median concentration with (l-oc) = 99%
confidence.
                            SAMPLE DATA                ORDERED DATA
Date
2002, 1st Q
2002, 2nd Q
2002, 3rd Q
2002, 4th Q
2003, 1st Q
2003, 2nd Q
2003, 3rd Q
2003, 4th Q
2004, 1st Q
2004, 2nd Q
2004, 3rd Q
2004, 4th Q
2005, 1st Q
2005, 2nd Q
2005, 3rd Q
2005, 4th Q
Beryllium (ppb)
3.17
2.32
7.37
4.44
9.50
21.36
5.15
15.70
5.58
3.39
8.44
10.25
3.65
6.15
6.94
3.74
Be
2.32
3.17
3.39
3.65
3.74
4.44
5.15
5.58
6.15
6.94
7.37
8.44
9.50
10.25
15.70
21.36
Rank
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
       SOLUTION
Step 1.   Order the 16 measurements from least to greatest and determine the rank associated with each
         value (listed above in the last two columns). The smallest observation, 2.32 ppb, receives the
         smallest rank, while the largest value, 21.36 ppb, receives a rank of 16.

                                            21-18                                  March 2009

-------
Chapter 21. Confidence Intervals
                                                        Unified Guidance
Step 2.   Since a confidence interval on the median must be constructed, the desired percentile is the
         50th (i.e., P = 0.50). Therefore the quantity (w+1) xP = 17 x 0.50 = 8.5. The data ranks closest
         to this value are L* = 8 and if = 9, so these are used as initial candidate endpoints.

Step 3.   Using the cumulative binomial distribution, and recognizing that only a lower confidence limit
         is needed, use equation [21.15] to calculate the actual confidence level associated with the
         order statistic x^y.


             \-a=\-Bin(L* -I; n,p)=l-5»i(7; 16,.50)=l-£i 16 I (50J6 = 0.4018
                                                           X=0\XJ

         Since the achieved confidence level is much less than 99%, subtract 1 from L* and recompute
         the  confidence level. Repeat this process until the confidence level is at least 99%. Since the
         achieved confidence when L* = 4 is equal to .9894 or approximately 99%, the LCL should be
         selected as X(4) (i.e., the 4th order statistic in the data set, also equal to the fourth smallest
         measurement), which equals 3.65  ppm. With statistical confidence of 98.94%,  one can assert
         that the true median beryllium concentration in the underlying population is no less than  3.65
         ppm.

Step 4.   In this example,  a lognormal model  could also have  been fit to the  sample.  Indeed the
         probability plot in Figure 21-2 below indicates good agreement with a lognormal fit, enabling
         a comparison between the non-parametric LCL with that derived from assuming a parametric
         model for the same data.
                Figure 21-2.  Probability Plot on Logged Beryllium Data

                   2
               CD
               O
               o
               to
               I
0
                      0
                 1             2            3
                   Log Beryllium log(ppm)
                                            21-19
                                                                March 2009

-------
Chapter 21. Confidence Intervals _ Unified Guidance

Step 5.   Since the non-parametric LCL was constructed around the population median, the fairest
         comparison is to construct a lognormal-based confidence interval around the median and not
         the arithmetic  mean. As  discussed in Section 21.1.2, this is equivalent to constructing a
         confidence interval around the lognormal geometric mean.  This can be built via a normal-
         based confidence interval  around the mean using the log-transformed measurements and then
         exponentiating the interval limits. Thus, using equation [21.3]  with the  log-mean and log-
         standard deviation given by y  = 1.8098 log(ppm) and sy = 0.60202 log(ppm) respectively, one
         can compute the 99% LCL as:


                 =exp| J-r99n ^ I  =exPri.8098-(2.602X60202)/Vl6l = 4.13 ppm
                             '             L                           J
         The non-parametric LCL around the median is  slightly lower than the limit computed by
         assuming  an underlying lognormal  distribution.  Given the  apparent  lognormal  fit,  the
         parametric LCL is probably a slightly better estimate, but the non-parametric method performs
         well nonetheless.

         The chief virtue of using a parametric confidence interval is the ability to generate estimates at
         any confidence level  even  with small sample sizes.  On the other  hand,  if the data are
         lognormally-distributed,  a confidence interval on the arithmetic mean may be preferred for
         comparisons to a fixed standard, depending on the type of standard. The advantage of a non-
         parametric interval around the median is its greater flexibility to define confidence intervals on
         non-normal data sets.  -^
       PROCEDURE FOR A CONFIDENCE INTERVAL AROUND A PERCENTILE

Step 1.   Given a sample of size n, order the measurements from least to greatest. Denote the ordered
         values by *(i),  *(2), ..., *(„), where x^ is the rth  concentration value in the ordered list and
         numbers 1 through n represent the data ranks.

Step 2.   Given the desired percentile P, pick candidate interval endpoints by choosing  ordered data
         values with ranks as close to and as symmetrical as possible around the product (w+1) x P,
         where n is the  sample size and P is expressed as a fraction. If this last quantity is a fraction
         (even-numbered sample size), the ranks immediately above and below it can be selected as
         candidate endpoints (unless the fraction  is larger than n, in which case the maximum rank n
         would be chosen as the upper endpoint).  If the product (w+1) x P is an integer (odd-numbered
         sample size), add 1 and subtract 1 to get the upper and lower candidate endpoints.  Once the
         candidate endpoints have been selected, denote these by L  and u.

Step 3.   For a two-sided  confidence interval,  compute  the confidence level  associated  with  the
         tentative endpoints  L  and  u   by taking the  difference  in the  cumulative  binomial
         probabilities given by the equation:
                                            21-20                                  March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance


                  l-a= Bin(iJ* -1; n,p)-Bin(z* - 1; n,P)= £ I  " \PX(l-Pj~*           [21.17]
                                                                                   ^k
         For a one-sided LCL, compute the confidence level associated with the endpoint L  using the
         equation:


                                                       n                               [21.18]
         For a one-sided UCL, compute the confidence level associated with the endpoint if using the
         equation:

                                                  u*-\( A
                          l-a=Bin(u*-l; n,p)= £ I " \PX (l - P J"                    [21.19]
         To minimize the amount of direct computation, these equations have been used to compute
         selected cases over a range of sample  sizes and for certain  percentiles in Table 21-11 of
         Appendix D

Step 4.   If the candidate endpoint(s) do not achieve the desired or target confidence level, compute new
         candidate  endpoints, (£*-!) and (C/*+l),  and re-calculate the achieved confidence level.
         Repeat this process until the target confidence level is achieved.  If one candidate endpoint
         already equals the data minimum or maximum,  only change the rank of the other endpoint. If
         neither endpoint rank can be changed, set either: 1) the minimum concentration value as a one-
         sided LCL;  2) the maximum concentration value as a one-sided UCL; or 3)  the interval
         spanned by the range of the sample data as a two-sided confidence interval around the Pth
         percentile. In  each case, report the achieved confidence  level associated with the chosen
         confidence limit(s).

Step 5.   Compare the confidence limit(s) computed in Step 4 versus the fixed compliance or clean-up
         standard. In compliance/assessment monitoring,  use the LCL derived as the order statistic with
         rank L*. In corrective action monitoring, use the UCL derived as the order statistic with rank
         U*.

       ^EXAMPLE 21-6

     Use the following 12 measurements of nitrate at a well used for drinking water to determine with
95% confidence whether or not the infant-based, acute risk standard  of 10 mg/L has  been violated.
Assume that the risk standard represents an upper 95th percentile limit on nitrate concentrations.
                                            21-21                                  March 2009

-------
Chapter 21. Confidence Intervals                                         Unified Guidance
Sampling Date
7/28/99
9/3/99
11/24/99
5/3/00
7/14/00
10/31/00
12/14/00
3/27/01
6/13/01
9/16/01
11/26/01
3/2/02
Nitrate (mg/L)
<5.0
12.3
<5.0
<5.0
8.1
<5.0
11.0
35.1
<5.0
<5.0
9.3
10.3
Rank
(1)
(11)
(2)
(3)
(7)
(4)
(10)
(12)
(5)
(6)
(8)
(9)
SOLUTION
Step 1.   Half of the sample concentrations are non-detects,  making a test  of normality extremely
         difficult. One could attempt to fit these data via the Kaplan-Meier or Robust ROS adjustments
         (see  Chapter 15), but here a  non-parametric confidence interval  around  the upper 95th
         percentile will be constructed.

Step 2.   Order the data values from least to greatest and assign ranks as in the last column of the table
         above. Note that the apparent ties among the non-detects have been arbitrarily broken in order
         to give a unique rank to each measurement.

Step 3.   Using Table 21-11 in Appendix D for n = 12, there  is approximately 88% confidence
                              ?k
         associated with using L  = 11 as the rank of the lower confidence bound and approximately
         98% confidence associated with using L =10. Since the target confidence level is 95%, it can
         only be achieved by using a rank of 10 or less. Thus the non-parametric LCL needs to be set to
         the  10th smallest observation or X(io). Scanning the list of nitrate measurements, the LCL =
         ll.Oppm.

Step 4.   Since the order statistic x(i0)  achieves a confidence level  of 98%, one can conclude that the
         true  upper 95th percentile  nitrate concentration is  no  smaller than  11.0 ppm  with 98%
         confidence. Even by this more stringent confidence level, the acute risk standard for nitrate is
         violated and  there is statistically  significant evidence that at  least 1  of every 20 nitrate
         measurements from the well will exceed 10 mg/L.

Step 5.   If the well was being remediated under corrective action monitorign, the fixed standard would
         be compared against a one-way UCL around the upper 95th percentile. In that case, for n = 12,
         Table 21-11 of Appendix D indicates that the maximum  observed value of 35.1 mg/L taken
         as the UCL achieves a confidence  level of only 46%. 95% confidence could not be achieved
         unless at least 59 sample measurements were available and the UCL was set to the maximum
         of those  values.   The  remedial  action would  be  considered  successful  only if all  59
         measurements were below the fixed standard of 10 mg/L. ~4
                                            21-22                                  March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance

21.3 CONFIDENCE INTERVALS AROUND TREND LINES

     It was assumed that the underlying population is stable, (i.e., characteristics like the mean, median,
or upper percentiles are stationary over  the period of sampling)  for the confidence intervals so far
presented in this chapter. In some cases, however, the concentration data will exhibit a trend. Examples
might include successful remediation efforts that serve to gradually drive down a well's concentration
levels, or interception of an intensifying plume of contaminated groundwater.3

     The problem with ignoring a discernible trend when building a confidence interval is that the
interval will incorporate not only the natural variability in the underlying population, but also additional
variation induced by the trend itself. The net result is a confidence interval that can be much wider than
expected for a given confidence level and sample size (n).  A wider confidence interval makes it more
difficult  to  demonstrate  an  exceedance or  return to  compliance  versus  a  fixed  standard  in
compliance/assessment or corrective action monitoring. The confidence interval will have less statistical
power to identify compliance violations, or to judge the success of remedial efforts.

     When a linear trend is present, it is possible to  construct an appropriate confidence interval built
around the estimated trend. A continuous series of confidence intervals is estimated at each point along
the trend, termed a simultaneous confidence band. An  upper or lower confidence band will tend to
follow the estimated trend line whether the trend  is increasing or decreasing. It is computed once the
trend line has been estimated.

     Construction of a confidence interval around a trend line presumes that a trend actually exists. The
algorithms presented in this section assume that a trend is readily discernible on a time series plot of the
measurements and that it is essentially linear. Otherwise, the results may be less than credible.
21.3.1      PARAMETRIC CONFIDENCE BAND AROUND LINEAR REGRESSION

       BACKGROUND AND PURPOSE

     A standard method for estimating a linear trend is linear regression, introduced in Chapter 17. In
this section, equations  for constructing a linear regression are extended to form a confidence band
around the trend. Although  a parametric technique, there is  no requirement  that the concentration
measurements be normal or transformable to normality.   Instead, the residual  concentrations after
subtracting out the estimated trend line should be roughly normal in distribution or at least symmetric.

     By  way of interpretation, each  point along the trend line is  an estimate  of  the true mean
concentration at that point in time. As the underlying population mean either increases or decreases, the
confidence band similarly increases or decreases to reflect this change.

     Although  the equations presented below can be used to simultaneously construct a confidence
interval around  each  point on the trend line, in practice, the user will want to compute a confidence
3 This might occur if the well screen first intercepts the leading edge of the plume, followed by the more heavily contaminated
  core.

                                            21-23                                  March 2009

-------
Chapter 21.  Confidence Intervals	Unified Guidance

interval for a  few or several of the most recent sampling events. Because the individual confidence
intervals comprising the simultaneous confidence band have a joint confidence level of (1-a), no matter
how many  confidence intervals are constructed, the overall false positive rate associated with the entire
set of tests  against the fixed standard will be no greater than a pre-specified a.

       REQUIREMENTS AND ASSUMPTIONS

     To accurately estimate a confidence band, the sample variance should be stationary or constant as a
function of time.  Although the mean level may be increasing or decreasing with time, the level of
variation about the mean should be essentially the same.

     Once a linear regression is fitted to the data, the residuals around the trend line should be tested for
normality  and apparent skewness. Inferences concerning a linear regression are generally appropriate
when two  conditions hold: 1) the residuals from the regression are  approximately normal or at least
reasonably symmetric in distribution; and 2) a plot of residuals versus concentrations indicates a scatter
cloud of essentially uniform  vertical  thickness  or width.  That  is, the  scatter cloud does not tend to
increase in width with the level of concentration or exhibit any kind of regular pattern other than looking
like a random scatter of points.

     If one or both  of these conditions is seriously violated, it may indicate that the basic trend is either
non-linear, or the size of the variance is not independent of the  mean level.  If the variance is roughly
proportional to mean concentrations, one possible remedy is to try a transformation  of the measurements
and re-estimate the linear regression. This will change the interpretation of the estimated regression from
a linear trend of the form y = a + bt, where y and t represent concentration and time respectively, to a
non-linear  pattern.  As  an  example, if the concentration  data  are  transformed  via  logarithms, the
regression  equation will have the form logy = a + bt.  On  the original concentration  scale, the  trend
function will then have the form y = expfa + bt\

     When the regression  data are transformed in this way, the  estimated trend  in the concentration
domain (after back-transforming) no longer represents the original mean. The transformation induces a
bias in  the confidence intervals  comprising the confidence band when converted  back to the original
scale as in the case of samples with no trend. If a log transformation  is used, for instance, the back-
transformed confidence band  around the trend line represents confidence intervals  around the original-
scale geometric means and not the arithmetic means. If a comparison of an estimated geometric mean or
similar  quantity  to  the fixed  standard makes sense,  computing a trend line  on the transformed  data
should  be  acceptable.  However, if  a  confidence interval around  an arithmetic mean is  required,
consultation with a professional statistician may be necessary.

     The technique presented here produces a confidence interval around the mean  as a function of time
and not an upper percentile.  Thus, we recommend that the use of this method be restricted to  cases
where the fixed standard represents a mean concentration and not an explicit upper percentile or a 'not-
to-exceed' limit.

     At least 8 to 10 measurements should be available when computing  a confidence band around a
linear regression. There must be  enough data to not only estimate the trend function but also to compute
the variance around the trend line. In the simplest case when no trend  is present, there are (w-1) degrees

                                             21-24                                   March 2009

-------
Chapter 21.  Confidence Intervals	Unified Guidance

of freedom [df] in a sample of size n with which to estimate the population variance. With a linear trend,
however, the available degrees of freedom df'is reduced to (n-2). For moderate to large samples, loss of
one or two degrees of freedom makes little difference. But for the smallest samples, the impact on the
resulting confidence limits can be substantial.

     One last assumption is that there should be few if any non-detects when computing the regression
line and its associated confidence band. As a matter of common sense, a readily discernible trend in a
data set (either increasing or decreasing) should be based on quantified  measurements. Changes in
detection  and/or RLs over time can appear as a declining trend, but may actually be an artifact of
improved  analytical methods.   Such artifacts  of plotting and data reporting should  generally not be
considered real trends.

       PROCEDURE

Step 1.    Construct a time series plot of the measurements. If a discernible trend is  evident, compute a
          linear regression of concentration against sampling date (time),  letting X[ denote the rth
          concentration value and  t\ denote the rth  sampling date. Estimate  the linear slope with the
          equation:

                                                     -l)-^2                             [21.20]


          This estimate leads to the regression equation, given by:

                                         x = x + b-(t-t)                                  [21.21]

          where  t denotes the mean sampling date, s2t is the variance of the sampling dates, x is the
          mean concentration level, and x represents the estimated mean concentration at time t.

Step 2.    Compute the regression residual at each sampling event with the equation:

                                          rt=xt-xt                                    [21.22]

          Check the set of residuals for lack of normality and significant skewness using the techniques
          in Chapter 10. Also,  plot the residuals against the estimated regression values (x.) to check
          for non-uniform vertical thickness in the  scatter cloud. If the  residuals are non-normal  and
          substantially skewed and/or the scatter cloud appears to have a definite pattern (e.g., funnel-
          shaped; 'U' -shaped; or, residuals mostly positive on one end of the graph and mostly negative
          on the other end, instead of randomly scattered around the horizontal line r  = 0), repeat Steps 1
          and 2 after first transforming the concentration data.

Step 3.    Calculate the estimated variance around  the regression line (also known as the mean squared
          error [MSB]) with the equation:


                                         s2=—!—Yr2                                  [21.23]
                                         e      r\ ^^  i                                   L     J

                                             21-25                                   March 2009

-------
Chapter 21. Confidence Intervals
                                                   Unified Guidance
Step 4.   Given confidence level (1-a) and a point in time (to) at which a confidence interval around the
         trend line is desired, compute the lower and  upper confidence limits  with  the respective
         equations:
       — x  —  2v  • F
       — A.Q  ,\^e  -M-2a,2, n-2
                                                       n   (n-l}-s2t
                                                                                       [21.24]
UCL,_a = x0 +
\lSe-Fl-2a2n-2
— +
                                                       n   (n-l)-sf
                                                                                       [21.25]
         where  x0  is the estimated mean concentration at time t0 from the regression using equation
         [21.21], and F\^a, i, n-2 is the upper (l-2a)th percentage point from an F-distribution with 2
         and (n-2) degrees of freedom. Values for F can be found in Table 17-1 of Appendix D.

Step 5.   Depending on whether the regulated unit is in  compliance/assessment or corrective action
         monitoring, compare the appropriate confidence limit against the GWPS. Multiple confidence
         limits can be computed at a single compliance point well without increasing the significance
         level (a) of the comparison. It is  possible to estimate at what point in time (if ever)  the
         confidence limit first lies completely to one side of the fixed comparison standard,  without
         risking an unacceptable false positive rate increase for that well.

       ^EXAMPLE 21-7

     Trichloroethylene [TCE] concentrations are being monitored at a site undergoing remediation. If
the GWPS for TCE has been set at 20 ppb, test the following 10 measurements collected at a compliance
point well over the last two and a half years to determine if the clean-up goal has been reached at the a =
0.05 level of significance.
Month Sampled
2
4
8
11
13
16
20
23
26
30
TCE Concentration
(ppb)
54.2
44.3
45.4
38.3
27.1
30.2
28.3
17.6
14.7
4.1
Regression Estimates
51.735
48.530
42.119
37.311
34.106
29.298
22.888
18.080
13.272
6.861
Residuals
2.465
-4.230
3.281
0.989
-7.006
0.902
5.412
-0.480
1.428
-2.761
SOLUTION
Step 1.   Construct a time series plot of the TCE measurements as in the graph below (see Figure 21-
         3). A general downward, linear trend is evident. Then compute the estimated regression line
                                            21-26
                                                           March 2009

-------
Chapter 21. Confidence Intervals
                                                                 Unified Guidance
         using equations [21.20] and [21.21], first determining that the mean time value is  t =15.3,
         the variance of time values is s2t = 88.2333 , and the mean TCE measurement is x = 30.42ppb:

     b = [(2-15.3)- 54.2 + (4-15.3)- 44.3 + ... +(30-15.3)- 4.l]/(9x88.2333) = -1.603ppp/month

                                  y = 30.42-1.603 -(f-15.3)

      Figure 21-3. Time  Series Plot and  Regression Line of TCE Measurements
              o
              c
              Q
              LJ
              LU
              O
         60 -
         55 -
         50 -
         45 -
         40 -
         35 -
         30 -
         25 -
         20 -
         15 -
         10 -
           5 -
           0
                       0
                             10
15
20
25
30
                                           Month Sampled
Step 2.
Compute the regression residuals using equation [21.22] (listed in the table above). Note that
the residuals are found by first computing the regression line estimate for each sampled month
(i.e., t = 2, 4, 8, etc.) and then subtracting these estimates from the actual TCE concentrations.
A probability plot of the regression residuals appears reasonably linear (Figure 21-4) and the
Shapiro-Wilk statistic computed from these data yields SW= 0.962, well above the a = 0.05
critical point for n =  10 of sw.os.io = 0.842. Thus, normality of the residuals cannot be rejected.

In addition, a plot of the residuals versus the regression line estimates (Figure 21-5) exhibits
no unusual pattern,  merely random variation about the residual  mean  of zero. Therefore,
proceed to compute a confidence interval around the trend line.
                                           21-27
                                                                         March 2009

-------
Chapter 21. Confidence Intervals
          Unified Guidance
                   Figure 21-4. Probability Plot of TCE Residuals
                1 -
            CD

            S   o
            C.O
            I
               -1  -
               -2
                  -10
 \      '      I      '
-5            0

    Residual TCE (ppb)
\
5
10
      Figure 21-5. Scatterplot of TCE Residuals vs.  Regression Line Estimates

                10
            CL
            CL
            CO
            en
                 5 -
                 o -
                -5 -
               -10
                      ^    I   I    I   I    I   I    I    I   I    I   I
                    0   5   10  15  20 25  30 35  40  45 50  55 60  65

                               Estimated TCE Cone (ppb)
                                      21-28
                 March 2009

-------
Chapter 21.  Confidence Intervals
                                                                 Unified Guidance
Step 3.   Compute the variance around the estimated trend line using equation [21.23]:

                      s] = l.[(2.465)2+(-4.230)2 +...+(-2.76l)2]= 15.60
Step 4.
Step 5.
Since the comparison to the GWPS of 20 ppb is to be made at the a = 0.05 significance level,
the confidence  limit is  (1-oc)  = 95% confidence. Since the remediation  effort  aims to
demonstrate that the true mean TCE level has dropped below 20 ppb, a one-way UCL needs to
be determined using equation [21.25].  A logical point  along the trend to examine is the last
sampling event at to  = 30. Using the estimated regression value at to = 30, and the  fact that
^.90,2,8 = 3.1131, the UCL on the mean TCE concentration at this point becomes:
               UCLgs =6.861+  |2xl5.60x3.1131x
                                         1  , (30-15.3?
                                         10   8x88.2333
= 13.14 ppb
         Since this upper limit is less than the GWPS for TCE, conclude that the remediation goal has
         been achieved by to = 30. In fact, other times can also be tested using the same equation. At
         the next to last sampling event (to = 26), the UCL is:
              UCLn =13.272+  2x15.60x3.1131X
                  95
                                          1   (26-15.3J
                                         10  8x88.2333
 = 18.32 ppb
which also meets the remediation target at the a = 0.05 level of significance.

If the linear trend is ignored, a onew-way UCL of the mean might have been used.  The overall
TCE sample mean x = 30.42, the TCE standard deviation s = 15.508,  and the upper 95th
percentage point of the ^-distribution with 9 degrees of freedom is  ^.95,9 = 1.8331.  Using
equation [21.2] with the same data yields the following:
                       UCL95 = 30.42 + (l.833 l)(l5.508yVTo = 39.41 ppb
         Had the linear trend been ignored when computing the UCL, the remediation target would not
         have been achieved.  The downward trend induces the largest part of the variation observed
         over the two and a half years of sampling and needs to be taken into account. -^
21.3.2      NON-PARAMETRIC CONFIDENCE BAND AROUND THEIL-SEN  LINE

       BACKGROUND AND PURPOSE

     The Theil-Sen trend line is introduced in Section 17.3.3 as a non-parametric alternative to linear
regression. Whether due to the presence of non-detects or trend residuals that cannot be normalized, the
                                           21-29
                                                                        March 2009

-------
Chapter 21. Confidence Intervals	Unified Guidance

Theil-Sen method can usually construct a trend estimate without some of the assumptions needed by
linear regression.

     The Theil-Sen trend line is non-parametric because it combines the median pairwise slope (Section
17.3.3) with the median concentration value and the median sample date to construct the trend. Because
of this construction, the Theil-Sen line estimates the change in median concentration over time and not
the mean as in linear regression.

     There are no simple formulas to construct a confidence band around the Theil-Sen line. However, a
more computationally-intensive technique — bootstrapping — can be employed instead. The conceptual
algorithm is fairly simple.  First consider the set of n pairs of measurements used to construct the Theil-
Sen trend.  Each pair consists of a sample date (Yi) and the concentration value measured on that date (x;)
as a statistical sample. Next, repeatedly draw  samples of size n with replacement from the original
sample of pairs. These artificially constructed samples are known as bootstrap samples. At least 500 to
2,000 bootstrap samples are generated in order to improve the accuracy of the final confidence band.
Note that a bootstrap sample is not precisely the same as the  original because pairs are sampled with
replacement.  This means that  a given pair might show up multiple times in any particular bootstrap
sample.

     For each bootstrap sample, use the  Theil-Sen  algorithm to construct  an  associated  trend line
(Section 17.3.3). Each of these trend  lines is known as a bootstrap replicate. Finally, determine the
distribution of the bootstrap replicates and select certain percentiles of this distribution to form lower
and upper confidence limits. These limits can be constructed to represent a non-parametric simultaneous
confidence band around the Theil-Sen trend line with (1-a) confidence.

       REQUIREMENTS AND ASSUMPTIONS

     The key requirements for constructing a confidence band around a Theil-Sen trend are the same as
for the Theil-Sen procedure itself (Section 17.3.3).  As a non-parametric procedure, the trend residuals
do not have to be normal or have equal variance across the data range.  But the residuals are assumed to
be statistically independent.  Approximate checks of this  assumption can be made using the techniques
of Chapter 14, after removing the estimated Theil-Sen trend and as long as there aren't too many non-
detects.  It is  also important to have at least 8-10 observations from which to construct the bootstrap
samples.

     Non-detects can be accommodated by the Theil-Sen method as long as the detection frequency is at
least 50%, and the censored values occur in the lower part of the observed concentration range. Then the
median  concentration value and the median pairwise slope used to compute the Theil-Sen trend will be
based on clearly quantified values.

     Since there are no simple mathematical equations which can construct the  Theil-Sen confidence
band, a  computer software program is essential for performing the calculations. Perhaps the best current
solution is to use the open-source, free-of-charge, statistical computing package R (www.r-project.org).
A template program (or script) written in R  to compute a  Theil-Sen confidence band is listed in
Appendix  C. This script can be adapted to any  site-specific data set  and used as many times as
necessary, once the R computing environment has been installed.
                                             21-30                                   March 2009

-------
Chapter 21. Confidence Intervals _ Unified Guidance

       PROCEDURE

Step 1.   Given the original sample of n measurements, form a sample of n pairs (ft, xj), where each pair
         consists of a sample date (ft) and the concentration measurement from that date (*;).

Step 2.   Form B bootstrap samples by repeatedly sampling n pairs at random with replacement from
         the original sample of pairs in Step 1. Typically, set B > 500.

Step 3.   For each bootstrap  sample, construct a Theil-Sen trend line using the  algorithm in Section
         17.3.3. Denote each of these B trend lines as a bootstrap replicate.

Step 4.   Determine a series of equally  spaced  time  points  (ft)  along the range  of sampling dates
         represented in the original sample, j = 1 to m. At each time point, use the Theil-Sen trend line
         associated with each bootstrap replicate to compute an estimated  concentration ( x^ ). There
         will be B  such  estimates at each  of the m equally-spaced  time points when this step is
         complete.

Step 5.   Given a confidence level (1-a) to construct a two-sided confidence band, determine the lower
         (a/2)th and the upper (l-a/2)th percentiles, denoted  x^2*  and  Xj~a^ from the distribution of
         estimated concentrations at each time point (ft).  The collection of these lower and upper
         percentiles along the range of sampling dates (ft, j = 1 to m) forms the bootstrapped confidence
         band. To construct  a lower confidence band, follow the same strategy.  But determine the
         lower ath percentile x^   from the distribution of estimated concentrations at each time point
         (ft). For an upper confidence band, compute the  upper (l-a)th percentile, x'1"^  at each time
         point (ft).

Step 6.   Depending on whether the regulated unit is in  compliance/assessment or corrective action
         monitoring,  compare the appropriate confidence band  against the GWPS. Estimate at what
         point in  time  (if ever) the confidence band first sits  completely to one side  of the fixed
         comparison standard.

       ^EXAMPLE 21-8

     In Example 17-7,  a Theil-Sen trend line was  estimated for the following sodium measurements.
Note that the sample dates are recorded as the year  of collection  (2-digit format), plus  a fractional part
indicating when during the year the sample was collected.  Construct a two-sided 95% confidence band
around the trend line.
Sample Date
(yr)
89.6
90.1
90.8
91.1
92.1
93.1
94.1
95.6
96.1
96.3
Sodium Cone.
(ppm)
56
53
51
55
52
60
62
59
61
63
                                            21-31                                   March 2009

-------
Chapter 21. Confidence Intervals                                         Unified Guidance
       SOLUTION
Step 1.   Designate the n = 10 (sample date, concentration) pairs as the original sample for purposes of
         bootstrapping. Set the number of bootstrap samples to NB = 500.

Step 2.   Sample at random and with replacement NE = 500 times from the original sample to form the
         bootstrap samples. Compute a bootstrap replicate Theil-Sen trend line for each bootstrap
         sample. This gives 500 distinct linear trend lines.

Step 3.   Divide the observed range of sampling dates from  89.6 to 96.3 into m = 101 equally-spaced
         time points, /j (note: choice of m is arbitrary, depending on how often along the time range an
         estimate  of the  confidence band  is needed).  At each time point, compute the  Theil-Sen
         concentration estimate using each bootstrap replicate trend. This leads to 500 estimates of the
         form:
                                       =x
QB-(t,-7B)
         where XB is the median concentration of the Bth bootstrap sample, QB is the Theil-Sen slope
         of the 5th bootstrap sample, and t Bis the median sampling date of the 5th bootstrap sample.

Step 4.   Given a two-way confidence level of 95%, compute the lower a/2 = 0.05/2 = 0.025 and upper
         (l-a/2) = (1-0.05/2) = 0.975 sample percentiles (Chapter 3) for the set of 500 concentration
         estimates associated with each time point (/j). This entails sorting each set and finding the
         value closest to rank («+l) x p, where p = desired  percentile. In a list of n = 500, find the
         sorted values  closest to the ranks 501 x 0.025 = 12.525  for the lower percentile and 501 x
         0.975 = 488.475 for the upper percentile. Collectively, the lower and upper percentiles plotted
         by the time points give an approximation to the 95% two-sided confidence band.

Step 5.   Plot the lower and upper confidence bands as well as the original Theil-Sen trend line and the
         raw sodium measurements, as in Figure  21-6.  The fact that the trend is increasing over time is
         confirmed by the rising confidence band. ~4
                                            21-32                                  March 2009

-------
Chapter 21.  Confidence Intervals
Unified Guidance
      Figure 21-6. 95% Theil-Sen Confidence Band on Sodium Measurements
                          I — Theil-Sen Trend
                          I • • 95% Cert Band
                          90    01    92     93     94    98    96
                                       21-33
       March 2009

-------
Chapter 21.  Confidence Intervals                                    Unified Guidance
                     This page intentionally left blank
                                       21-34                               March 2009

-------
Chapter 22. Compliance & Corrective Action Tests	Unified Guidance

        CHAPTER  22.  COMPLIANCE/ASSESSMENT  AND
                      CORRECTIVE ACTION TESTS
       22.1   CONFIDENCE INTERVAL TESTS FOR MEANS	22-1
         22.1.1  Pre-SpeciJying Power In Compliance/Assessment	22-2
         22.1.2  Pre-SpeciJying False Positive Rates in Corrective Action	22-9
       22.2   CONFIDENCE INTERVAL TESTS FOR UPPER PERCENTILES	22-18
         22.2.1  Upper Percentile Tests in Compliance/Assessment	22-19
         22.2.2  Upper Percentile Tests in Corrective Action	22-20
     Chapter 7 lays out general strategies for statistical testing in compliance/assessment and corrective
action monitoring via the use of confidence intervals. Procedures for constructing confidence intervals
are described in Chapter  21.  This  chapter discusses potential  methods for developing confidence
interval tests so that adequate statistical power is maintained in compliance/assessment monitoring and
false positive rates are minimized in corrective action monitoring.
22.1 CONFIDENCE  INTERVAL TESTS FOR  MEANS

     As  discussed in  Chapter 7, EPA's  primary concern in compliance/assessment and  corrective
action  monitoring  is the identification and  remediation of contaminated groundwater.  The  basic
statistical hypotheses are reversed in these two phases of monitoring as described in Chapter 21 and
earlier. The lower confidence limit [LCL] is of most interest in compliance/assessment, while the upper
confidence limit [UCL] is used in corrective action. Statistical power is also of greater concern to the
regulatory agency  in compliance/assessment— representing the probability that contamination above a
fixed standard will be identified. A sufficiently conservative false positive rate during corrective action
is important from a regulatory standpoint, since a false positive implies that contaminated groundwater
has been falsely declared to meet a compliance standard.  The reverse of these risks is generally true for
a regulated entity.

     To ensure that contaminated groundwater is treated in ways that are statistically sound, the two
specific strategies which follow separately address compliance/assessment monitoring and formal testing
in  corrective action.  The latter occurs after the completion of remedial activities  or when potential
compliance can be anticipated. Each strategy  is designed to allow stakeholders on both sides  of the
regulator/regulated  divide to understand the expected  statistical performance of a given confidence
interval test.

     The two strategies which follow are based on the behavior of the normal mean confidence interval.
They  especially assume  that the  monitoring data  are stationary over the period of record.   Other
important assumptions were discussed in Chapter 21.  In the discussion which follows, consideration is
given  to data that is normal following a logarithmic transformation and the possible tests which can be
applied.
                                            22-1                                   March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests                        Unified Guidance
22.1.1       PRE-SPECIFYING  POWER IN COMPLIANCE/ASSESSMENT

     In most statistical literature including Gibbons & Coleman (2001) comparing a confidence interval
against a fixed standard, a low false positive error rate (a) is chosen or recommended without respect to
the power  of the test. However, the power to detect increases above a fixed standard using a lower
confidence limit around the mean can be negligible when contaminant variability is high and the sample
size is small (Chapter 7). To remedy this problem, the Unified Guidance suggests an alternate strategy.
That is, instead of pre-specifying the false positive rate a prior to computing confidence interval limits, a
desired level of power (1-P) should be set as an initial target.

     Ideally one would  like to simultaneously minimize a and maximize power by also minimizing P
(i.e., the false negative rate). However, this is generally  impossible given a fixed sample size (Chapter
3), since there is a trade-off between power and the false positive rate. Especially for small sample sizes,
fixing  a low a often leads to  less than desirable power. Conversely, pre-specifying a  high power
necessitates a higher than typical false positive rate.  Larger sample sizes are needed if both power and a
are pre-specified. High variability at a fixed sample size both lowers power and/or increases the need for
a larger false positive error rate.

     A number of considerations are relevant when constructing mean  confidence limits to achieve
adequate statistical power. In most Agency risk assessment evaluations, chronic risk levels are generally
proportional to  the average concentration. Development of MCLs  followed  similar proportional  risk
methodologies. Fixed health-based limits which can serve as groundwater protection standards [GWPS]
also cover  an enormous  concentration range when both carcinogenic and non-carcinogenic constituents
are included.

     Another relevant factor pertains to those situations where the true mean concentrations lie quite
close to either side of a compliance standard.  The difference between complying and not complying
with the GWPS in terms of the true mean  concentration level may be  so small as to make  a clear
determination of compliance very difficult (Figure 22-1).  Only sufficiently large differences relative to
a standard are likely to be determined with a high level of certainty (i.e., statistical power).
                                             22-2                                   March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests
Unified Guidance
     Figure 22-1.  True Means Too Close to Standard to Clearly Identify Violation
                                                              True Mean Under H.
                      GWPS
                     True Mean Under H
                                 0
                             Confidence Intervals too wide to determine compliance of true mean
     With the wide range of GWPS in  place  and recognizing that risk factors are proportional or
multiplicative rather than additive (e.g., a  1CT6 cancer risk), it would be appropriate to use a consistent
measure of increased risk that is independent of the actual GWPS concentration level. While ultimately
the decision of the regulatory authority, the Unified Guidance suggests a proportional increase (i.e., a
ratio) above the GWPS, which is identified at some predetermined level of statistical power to judge the
appropriateness of any specific mean confidence interval test.

     For compliance/assessment  monitoring purposes, increases in the true concentration mean of 1.5
and 2 times a fixed standard are  evaluated at a range of confidence levels. While this is not quite the
same as evaluating an absolute mean increase for a given constituent, the use of a risk ratio (R) does in
fact define a specific increase in  concentration level. For example,  a risk ratio  of 1.5 would identify a
critical increase above the 15 |ig/l MCL standard for lead of 22.5 - 15 = 7.5 |ig/l, while for chromium
with an MCL = 100 ug/1, the  absolute increase would be 50 ug/1.  Each represents a 50% increase in risk
relative to the GWPS.

     Two approaches for assessing statistical power in compliance/assessment monitoring are provided
using these critical risk ratios, based on different assumptions regarding  sample variability. In the first
approach, a constant population variance is assumed, equal to the standard  (i.e.., GWPS) being tested.
Under the null hypothesis that the true population mean is no greater than the GWPS, this  assumption
corresponds to having a coefficient of variation [CV\ of 1 when the true mean equals the standard.
Although observed sample variability is ignored, this case can be considered a relatively conservative
approach.
                                              22-3
        March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests _ Unified Guidance

     Assuming CV = 1, the relationship between the risk ratio (R), statistical power (I-/?), sample size
(«), and the false positive rate (a) can be obtained using the following equation:

                                                                                         [22.1]


where  tl_an_l  is the (l-a)th Student's ^-quantile  with (n-l)  degrees of freedom  and  Grn_j(» A)
represents the cumulative non-central ^-distribution  with (n-l) degrees of freedom and non-centrality
parameter A. By fixing a desired  or target  power level, equation [22.1]  can be  used to  choose the
necessary a based on the available  sample size n.   Alternatively, the equation can be used to determine
the sample size (n) needed to allow  for a pre-determined choice of a.

     Numerical tabulations of equation [22.1] are found in Tables 22-1 and 22-2 in  Appendix D. These
tables cover a practical range of n = 3 to 40 and a = .001 to .20, and offer combinations of the minimum
false positive rate (a) and sample  size  (n) for several fixed levels of power.  These can  be  used to
construct lower confidence limits having a pre-specified  level of power. It is important to note that the
listed combinations  are the smallest a-values resulting in  the targeted power. For a fixed w, use of an de-
value larger than that listed in the tables will provide even greater power than the target. Similarly, for
given a, use of a larger sample size than that listed in the  tables will also result in greater power than the
target.

     Minimum parameter values are presented in Tables 22-1 and 22-2 of Appendix D to document
how the desired power level can be achieved with as  few  observations and as small a false positive error
rate as possible.  It is also true that  an assumption of CV = 1 should be somewhat conservative at many
sites. Actual power will be higher than that listed in these tables if the coefficient of variation is smaller.
Not every power level is achievable in every combination of n and a, so some of the entries in these two
tables are left blank.

     The second approach requires an estimate of the population coefficient of variation. In this case,
the required (but approximate)  false positive rate of the test can be directly obtained from equation
[22.2], where R is the desired risk ratio, n is the sample size, CV is the estimated sample coefficient of
variation, ^_  ^ is  the (l-(3)th Student's ^-quantile with (n-l) degrees of freedom,  and FT n_l(*)  is the
cumulative (central) Student's ^-distribution function:


                                                                                         [22.2]
     Equation [22.2] was evaluated for sample sizes varying from n = 4 to 12 and for CFs ranging from
0. 1 to 3 .0 at two target combinations of power and risk ratio — R = 1.5 at 50% power and R = 2 at 80%
power. Results of these calculations are provided in Table 22-3  of Appendix D. Similar to the critical
power targets recommended by the Unified Guidance in detection monitoring (i.e., 55-60% power at 3 a
above background, and 80-85% power at 4o over background), two high power targets at proportionally
increasing risk ratios were also chosen for this setting.
                                             22-4                                    March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests	Unified Guidance

     Table 22-3 in Appendix D provides the approximate minimum false positive rate (a) necessary to
achieve each power target in  a single confidence interval test. The  shaded and italicized entries in the
table represent those cases where the minimum a is below the RCRA regulatory limitation of a = .01
from §264.97(i)(2) for an individual test false positive error rate. For these situations, the user would
need to set a = 0.01, which in  turn would provide even greater statistical power than the target.

     For higher estimated CFs, many of the entries in this table exceed a = .5 (bolded entries). These
cases illustrate the difficulty of simultaneously  attaining the  recommended  level  of power  while
controlling the false positive rate, especially for small sample sizes  and highly variable data.  Setting a
lower a, results in insufficient statistical power. On the other hand, setting a > .5 amounts to a simple
comparison of the sample mean against the fixed standard, with essentially no adjustment for sample
variability or uncertainty. Similar to the first approach,  a maximum  false positive rate of a  = .2 is a
reasonable upper bound which implies at most a l-in-5 chance of an error.

     Generally speaking, setting 80% power at a risk ratio of R = 2 in Table 22-3 of Appendix D is
more constraining (requiring higher a's) than 50%  power at a risk ratio of R = 1.5, although the effect
can be reversed for low CVs  and sample sizes. To  meet both targets simultaneously for a given «, the
larger of the corresponding significance levels (a) should be selected. Guidance users may choose either
of the  two approaches described above. Other ratio and power options not covered in Tables  22-1
through 22-3 of Appendix D can be handled by  direct computation using either equation  [22.1]  or
equation [22.2]. The first method makes an a priori assumption about the CV. The second method is
approximate, depending on a sample CV estimate which might be erratic at small sample sizes and larger
true population CFs especially if the compliance data are non-normal.

     Both approaches are directly applicable to the  normal mean LCL test in Section 21.1.1. While the
CV can  be  directly  estimated  using  sjx on  the  original concentration data,  this  statistic will
underestimate the likely variability  when data are lognormal.  In  that case, the logarithmic CV estimate
in Chapter 10, Section 10.4  should be used. If the data best fit a lognormal distribution, a number of
considerations follow:

        *»*  It is possible to misapply the normal mean  confidence interval test using the original
            concentration data, even when the data stem from a lognormal  distribution.  The mean is
            relatively robust  with respect to departures from normality as long as the CV variability is
            not too great.  If the predetermined false positive error a is  selected based on the normal
            power criteria above, the resulting LCL test will be  at least as powerful as the normal test.
            The actual false positive error rate will  also differ.

        »»»  If a geometric mean test in Section 21.1.2 is used, the LCL  should be computed from the
            logarithmically transformed data. Tables 22-1 to 22-3 in Appendix D are based on normal
            distribution  assumptions and the  error rates  are very conservative with respect to the
            achievable power.   As an example, given a data set  from  a lognormal distribution with n =
            10, and an estimated CV = .8, an alpha value of .151 can be  identified from Table 22-3 in
            Appendix D.  The  actual power to detect a doubling above a GWPS at 80% confidence
                                             22-5                                    March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests
Unified Guidance
            would result in a power level of 94.5%.  The false positive needed to detect a geometric
            mean doubling for this example to meet the above criteria would be a = .026.1

         »»»  The Land lower confidence interval test from Section 21.1.3 can also be used.  But since
            there  are limited a-choices in the tables,  the guidance option is to select a fixed limit of .01,
            .05, or . 1.  If data are truly lognormal, the power of this test is at least as great as would be
            predicted by equations [22.1] and [22.2]. Otherwise,  professional statistical assistance may
            be necessary.

       Since compliance data  will often be  pooled over time to increase  the eventual sample size
(Chapter 7), the two approaches can be combined by determining the false positive rate for a given risk
ratio and power level during  the first year  with Tables 22-1  and  22-2 of Appendix D. Tests in
subsequent years might use the  second power approach when a better CV estimate (using more data) can
be derived. Overall,  each approach should provide a reasonable manner of adjusting the individual test
false positive rate (a) to ensure adequate power to detect real contaminant increases. As a general guide,
the Unified  Guidance suggest formulating power in terms of risk ratios no  higher than R = 2. There
should  be  at  least  70-80%   statistical  power for  detecting increases of that magnitude  during
compliance/assessment monitoring.

       ^EXAMPLE 22-1

     Compliance monitoring recently began at a solid waste landfill.  Measurements for vinyl chloride
during detection monitoring are listed below for two compliance wells. If a value of 5 ppb vinyl chloride
is used as the  GWPS and a confidence interval test must have 80% power for detecting an increase in
mean  vinyl chloride levels  of twice the GWPS,  how  should the  confidence  interval  bounds be
constructed  and what do they indicate?  Assume that compliance  monitoring  began with Year 2 of the
sampling record and that  annual groundwater evaluations are required.

                               Vinyl Chloride Concentrations (ppb)
Sample
Ql, Yrl
Q2, Yrl
Q3, Yrl
Q4, Yrl
Ql, Yr2
Q2, Yr2
Q3, Yr2
Q4, Yr2
GW-1
6.3
9.5
8.1
11.9
7.3
11.2
6.0
7.5
GW-2
5.9
3.0
8.8
12.0
11.2
8.6
12.6
7.2
Sample
Ql, Yr3
Q2, Yr3
Q3, Yr3
Q4, Yr3
Ql, Yr4
Q2, Yr4
Q3, Yr4
Q4, Yr4
GW-1
8.4
6.4
8.9
4.9
9.6
9.7
8.7
8.7
GW-2
13.8
5.6
11.0
9.8
6.3
10.4
7.5
9.7
       SOLUTION
Step 1.   Assume for purposes of this example that the vinyl chloride data are approximately normal. In
         practice, this should be explicitly checked. Also evaluate potential trends in the vinyl chloride
  For users with access to statistical software containing the cumulative non-central t-distribution, the inverse non-central t
  CDF can be used to identify the appropriate false positive level.  For sample size df =n -1= 9, and a non-central t parameter
  = 8 = *
-------
Chapter 22. Compliance & Corrective Action Tests
                                                                           Unified Guidance
         measurements  over time,  as  in the time series plot of Figure  22-2.  Despite  apparent
         fluctuations, no obvious trend is observed. So treat these data as if the population has a stable
         mean at least for the time frame indicated in the sampling record.

                      Figure 22-2.  Vinyl Chloride Time Series Plot
j*-%.

ie Cp|
n o o
, , . . 1 , . . , 1
'£
3
o
5
J> n
1
• GW-1
-1- GW-2
	 MCL-5 ppb
+
+
+ • +
• + • • +
+ + . . .
• » +
* :
.........................................B.. .............
+
, , , i , , , i , , , i , , ,
2345
                                        Sampling Event (yr.qtr)
Step 2.
Step3.
         Given that compliance monitoring began in Year 2, use the four measurements available from
         each well to construct lower confidence limits. Since 80% power is desired for detecting vinyl
         chloride increases of two times the 5 ppb GWPS, Table 22-2 in Appendix D indicates that for
         n = 4, a false positive rate of a = 0.163 must be used to guarantee the  desired power. This
         corresponds to a Student's ^-quantile of tl_an_l = tg3J3 = 1.1714 . Then using the sample means
         and standard  deviations of the Year 2 vinyl chloride measurements, the  lower  confidence
         limits can be computed as:
                  LCLGW_2 =x-f1_aB_1-^ = 9.9-1.1714^.4468/>/4)=8.5ppb
         Since both lower confidence limits exceed the GWPS, there is statistically significant evidence
         of an increase in vinyl chloride at these wells above the compliance limit. Such a conclusion
         also  seems reasonable from Figure 22-2. However, the chance is better than 15% (i.e., a =
                                            22-7
                                                                                  March 2009

-------
Chapter 22. Compliance & Corrective Action Tests	Unified Guidance

         16.3%) that the  apparent exceedance is merely a statistical artifact. If power criteria  are
         ignored and a fixed minimum rate of a = .01 is used, the lower confidence limits would be:


                   LCLrw  = x-t   ,-^= = 8.0-4.541(2.2346/74")= 2.9 ppb
                       GW-l       a,n-\ ^            \     I    J^    ff


                         v_2=x- tan_l -^= = 9.9- 4.541 ^.4468/V4 )= 4.3 ppb
         Neither limit now exceeds the GWPS, so the vinyl chloride concentrations would be judged in
         compliance with this test, illustrating the lack of power in lowering the false positive rate (a).

Step 4.   To increase the confidence level (i.e., by lowering a) of the tests at the end of the first year of
         compliance monitoring (i.e., Year 2 in the preceding table of vinyl chloride values) without
         losing statistical power, combine the measurements from Years 1 and 2, where Year 1 samples
         represent the final measurements from  detection monitoring prior to the start of compliance
         monitoring. In this case,  n =  8, and the minimum  false positive  rate from Table 22-2  of
         Appendix D can be lowered to a = .046 or approximately 4.5%. Then the re-computed lower
         confidence limits LCLow_l =7.0 ppb andLCLGW_2 =6.4 ppb again  both exceed the GWPS,
         indicating significant evidence  of a compliance violation.

Step 5.   If the strategy presented in Step 4 of combining measurements from detection monitoring and
         compliance monitoring is considered untenable, additional confirmation of the results  can be
         made at the end of Year 3 by combining the first two years of compliance monitoring samples
         and ignoring the measurements from Year 1. Again  with n = 8,  the minimum false positive
         rate guaranteeing at least 80% power will be a = .046. The lower confidence limits are then:
Step 6.   An even lower false positive rate can be achieved after the first three years of compliance
         monitoring. Pooling these  measurements gives n = 12. Then Table 22-2 in Appendix D
         identifies a minimum false  positive rate of a = .013 or less than 1.5%. In this case, the lower
         confidence limits  LCLGW_l=6.'&ppb  and  LCLGW_2 = 7.6 ppb again  exceed the  GWPS,
         confirming the previous vinyl chloride exceedances  from  either Year 2, Years  1 and 2
         combined, or Years 2 and 3 combined. Furthermore, not only is the false positive rate quite
         low, but the power of the test still meets the pre-specified target. ^
                                            22-8                                   March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests	Unified Guidance

22.1.2       PRE-SPECIFYING FALSE POSITIVE RATES IN CORRECTIVE ACTION

     As  noted earlier,  the primary regulatory  concern in formal corrective action testing is false
declarations of remedial success. If groundwater is truly contaminated above a regulatory standard yet a
statistical test result indicates the concentrations are no longer so elevated, then on-going contamination
has been missed and the remedial process should not be exited. Statistically, this idea translates into a
desire to minimize the corrective action false positive rate  (a). False positives in corrective action are
precisely those decisions where  the true concentration mean  is  falsely  identified  to be below  the
regulatory standard, when in fact it still exceeds the standard.

     Constructing confidence interval tests by fixing a low target false positive rate is straightforward.
All of the confidence interval tests presented in Chapter 21 can be calibrated for choice of a. What is
not straightforward is how best to incorporate  statistical  power  in  corrective  action. As with any
confidence interval test, selecting a low a when the sample size is small typically results in a confidence
limit with low power. Power under corrective action monitoring represents the probability that the upper
confidence limit [UCL]  will fall below the fixed standard when in fact the true population mean  is also
less than the standard. Facilities undergoing remediation clearly have  an interest in demonstrating the
success of those clean-up efforts. They therefore may want to maximize the  power of the confidence
interval tests during corrective action, under the constraint that a must be kept low.

     What  statistical power criteria might  a  facility reasonably define in corrective action testing?
Because of the orders of magnitude range found among various GWPS, a risk ratio approach similar to
what is suggested in Section 22.1.1; only  in this case, the target ratios (R) are less than one. While a true
mean at a level of R = 0.9 times a given standard might be declared in compliance very infrequently, one
at R = 0.5  times or R = 0.25 times the standard should meet the compliance requirements much more
often. By using a consistent risk ratio across a variety of constituents,  absolute decreases in the mean
concentration are consistent with an assumed level of risk.

     Unlike the risk ratio method detailed for compliance/assessment monitoring, where power was pre-
specified but a combination of the false positive rate (a) and  sample  size (n) might be varied to meet that
power level, in corrective action both power and a are likely to be pre-specified (power by the facility
and a by the regulatory authority). The remaining component is how large a  sample  size is needed to
attain the desired level of power, given a pre-specified false positive  rate (a).

     The  normal distribution  can be used to  estimate sample size requirements for  such  risk  ratios,
given a specific false positive rate (a) and desired level of power (l-(3). There is likely to be uncertainty,
however,  in the  degree of sample variation,  as expressed  by the CV. Since the  constituents in a
contaminated  aquifer may be  modified  by remedial actions, it can  be  difficult to estimate  future
variability (and the CV) from pre-treatment data. In some situations, a  decrease in the mean over time
might be paralleled by  a decrease in total variation. If proportional, the CV would  remain relatively
constant. However, the CV could decrease  or increase depending on aquifer conditions,  constituent
behavior, etc.  The best that can be  recommended is to develop  an estimate of the  expected future CV
under conditions of aquifer stability.

     As with compliance/assessment testing, future year estimates  of the CV could be developed from
the accumulated previous years' data. Sample sizes necessary to meet specific power  targets (l-(3) can


                                              2^9                                     March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests _ Unified Guidance

then be generated from the following approximate equation, where R = fractional risk ratio (less than
1.0), (1-oc) is the desired confidence level, and CV  = estimated coefficient of variation:
V/(l-R)}2                          [22.3]
                                                      C

Since n appears on both sides of equation [22.3], it has to be solved iteratively for trial-and-error choices
of w, making it difficult to calculate without a proper computing environment. Tables 22-4 to 22-6 in
Appendix D provide requisite sample sizes (n) based on equation [22.3] for three specific risk ratios (R
= .75, .50, and .25) over a variety of inputs of a,  (3, and CV.

     These tables can be consulted when designing a remedial program, especially when determining a
sampling frequency adequate for generating the  minimally needed sample size over a specific period of
time. For example, to detect a drop in the true mean down to 0.75 x GWPS (i.e., R = 0.75) with 80%
power when CV ' = 0.6,  Table 22-4 in Appendix D indicates that a minimum of n = 16 observations are
needed to have a false positive rate (a) no greater than 10%. Demonstrating such a reduction over the
next two years might then require the collection of 8 measurements per year (or two per quarter) from
the compliance well involved. 2

     While  Tables  22-4 to  22-6 in Appendix D  identify the sampling requirements needed to
simultaneously meet pre-specified targets for power (1-0) and the false positive rate (a), they come with
some  limitations. First, many of the minimum sample  sizes are prohibitively large  when sample
variation as  measured by the CV is substantial. Proving the success of any remedial program will be
difficult when the compliance data exhibit significant relative variability. Less sampling is required to
demonstrate a more substantial concentration drop below the compliance standard than to demonstrate a
slight decrease (e.g., compare the sample  sizes forR = 0.75 to R = 0.25). This fact mirrors the statistical
truth in both detection  and compliance/assessment monitoring that highly contaminated wells are more
easily identified (and require fewer observations  to do so) than are only mildly contaminated wells.

     Another limitation of equation [22.3]  is that it assumes all  n measurements are statistically
independent. This assumption puts practical limits on the amount of sampling at a compliance well that
can reasonably be achieved over a specific time period. Samples  obtained  too frequently may  be
autocorr elated and thus violate statistical independence. Minimum sample sizes do  not apply to data
exhibiting an  obvious trend,  and  are appropriate  only when the aquifer is  in a relatively  steady-state.
Alternate methods to construct confidence bands around trends are presented in  Chapter 21. However,
equation  [22.3] cannot  be used to plan sample sizes in  this setting. Finally, Tables 22-4 to 22-6 in
Appendix D are based on an assumption of normally-distributed data. Although non-normal  data sets
might be approximated to some degree by the  range of  CVs considered, more sophisticated methods
might be needed to compute sample size requirements for such data. This might entail consultation with
a professional statistician.
 A slightly more approximate direct calculation using the standard normal distribution instead of Student t-values will also

  provide the needed sample size estimate as: n = \R- (zl_a + z^a)- CV \\.—R)\ .   The recommended sample size in
  the example above is rounded to n = 15 using the z-normal equation.   The estimate can be improved and made more
  conservative by adding an additional sample.
                                             22-10                                    March 2009

-------
Chapter 22. Compliance & Corrective Action Tests
Unified Guidance
       ^EXAMPLE 22-2

     Suppose  elevated levels of specific conductance (jimho) shown in the  table below  must be
remediated at a hazardous waste facility. If the clean-up standard has been set atL= 1000 (imho, at what
point should remediation efforts be declared a success for the two  compliance  well data in the table
below? Assume that the risk of false positive error needs to be no greater than a =  0.05 at either well.
Well ID
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12
GW-12



Date
10-16-87
01-28-88
04-13-88
06-15-88
10-12-88
12-20-88
04-19-89
10-12-89
04-25-90
07-19-90
10-23-90
02-13-91
06-27-91
09-10-91
12-06-91
03-18-92
06-03-92
09-16-92
12-02-92
03-24-93



Spec. Cond.
2100
2550
2360
2405
2560
1163
1880
1650
2410
862
1114
1346
909
888
749
515
180
526
610
570



Well ID
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
GW-13
Date
10-16-87
01-27-88
04-13-88
07-12-88
10-12-88
12-19-88
01-31-89
04-19-89
07-10-89
10-10-89
01-29-90
04-25-90
07-23-90
10-24-90
02-13-91
06-27-91
09-12-91
12-04-91
03-20-92
06-04-92
09-17-92
12-01-92
03-24-93
Spec. Cond.
2200
1463
935
809
469
465
374
499
503
590
403
527
513
451
622
495
420
634
526
472
442
530
625
       SOLUTION
Step 1.   First consider the data in well GW-12. A time series plot of the most recent 20 specific
         conductance values is shown in Figure 22-3.  This plot indicates a fairly linear downward
         trend, suggesting that a trend line should be fit to the data,  along with an upper confidence
         bound around the trend.
                                            22-11
        March 2009

-------
Chapter 22. Compliance & Corrective Action Tests
                                                 Unified Guidance
  Figure 22-3. Time Series Plot of Specific Conductance Measurements at GW-12
                3000
             •^ 2500  -

             ^ 2000  -
              U
              i 1500  -
              o
             0 1000  -
             'u
             £  500  -
                     1986
               1988
n     I
    1990
 Sampling Date
              1992
1994
        Figure 22-4. Regression of Specific Conductance vs. Sampling Date
                3000
             D
             O
2500 -

2000 -

1500 -

1000 -

 500 -
                    1986
               1988
   1990
Sampling Date
               1992
1994
                                      22-12
                                                        March 2009

-------
Chapter 22. Compliance & Corrective Action Tests
                          Unified Guidance
Step 2.   Fit  a  regression  line of specific conductance versus  sampling date using the formulas in
         Section 21.3. The equation of estimated trend line shown on Figure 22-4 is:

                                  j> = 790360- 396.36 -t

Step 3.   Examine the trend residuals. A probability plot of the residuals is given  in Figure 22-5. Since
         this plot is reasonably linear  and the Shapiro-Wilk test statistic for these residuals (SW =
         .9622) is much larger than the  1% critical point for n = 20 (sw.oi, 20 = 0.868), there is no reason
         to reject the assumption of normality.
      Figure 22-5.  Probability Plot of Specific Conductance Residuals at GW-12

                    2.50
                    1.25 -

                 £
                 I  0'00 ~
                 N

                   -1.25 -|


                   -2.50
                                      T
                        -1000
I
0
                         T
-500          0         500

   Specific Conductance Residuals
1000
         Also plot the residuals against sampling date (Figure 22-6). As no unusual pattern is evident
         on this  scatter plot (e.g., trend,  funnel-shape,  etc.)  and the variability of the residuals is
         reasonably constant across the range of sampling  dates, the key  assumptions of the linear
         regression appear to be satisfied.

Step 4.   Since the false positive error rate  must be no greater than 5%, use a = .05 when constructing
         an upper confidence band around the regression line. Using the formulas in Section  21.3 at
         each observed sampling date, both a 95% upper  confidence band and a 95% lower confidence
         band are computed and shown in Figure 22-7.  Only the upper  confidence band is needed to
         measure the  success of the remedial effort.  Note that the formula uses an F-confidence level
         of 1-a or .95 for a  one-sided confidence interval. The lower 95%  confidence band is shown
         for illustrative purposes and  the confidence level between the upper  and lower bands is
         actually 90%.
                                            22-13
                                 March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests
                                                 Unified Guidance
       Figure 22-6. Plot of Specific Conductance Residuals vs. Sampling Date
               1000
           a   500 -
           H
            •I
            a
            a

           D
            O
            u
            PL,

           BO
                  0 -
               -500 -
              -1000
                    1986
              1988       1990
                     1992
          1994
                                        Sampling Date
        Figure 22-7. 95% Confidence Bounds Around Trend  Line at GW-12
               a.
               °ssj"

               Bi




               1
               D

               D
               u
               mi
               PL,

               KJ
3000




2500  -




2000  -_





1500  -_




1000  -_




 500  -
                     0
Regression Line


95 %Conf Limits


Clean-Up Std
                      1986
                  I


                1988
            1990
                                         Sampling Date
1992
1994
                                       22-14
                                                        March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests
                                        Unified Guidance
Step 5.   Determine the first time point at which the remediation effort should be judged successful. In
         Figure 22-7, the upper confidence band drops below the clean-up standard of L = 1000 (imho
         in the second quarter of 1992, so well GW-12 could be declared in compliance at this point.

Step 6.   Now consider compliance well GW-13. A  time series plot  of the specific conductance
         measurements in this case (Figure 22-8) shows an initially steep drop in conductance level,
         followed by a more or less stable mean for the rest of the sampling record. The best strategy in
         this situation is to remove the four earliest measurements and then compute  an upper
         confidence limit on the remaining values.
  Figure 22-8. Time Series Plot of Specific Conductance Measurements at GW-13
               2500
           o
               2000  -
               1500  -
               1000  H
            B,   500  H
           63
                   0
                     1986
   I      '       I       ^
1988         1990

           Sampling Date
1992
1994
Step 7.   Before computing an upper confidence limit, test normality of the data. If the entire sampling
         record is included, the Shapiro-Wilk test statistic is only .5804, substantially below the 1%
         critical point with n = 23 of sw.oi,23 = 0.881, indicating a non-normal pattern. Certainly, a
         transformation  of the data could be attempted.  But simply  removing the first four values
         (representing the steep drop in conductance levels) gives a  Shapiro-Wilk statistic equal to
         .9536, passing  the  normality test easily. Further confirmation is  found by comparing the
         probability plots in Figures 22-9 and  22-10. In  the first plot, all the data from GW-13 are
         included, while in the second the first four values have been removed.
                                           22-15
                                               March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests
                                           Unified Guidance
Step 8.   Another instructive comparison is to compute the upper confidence limits on the same data
         with and without the first four values.  Consider the initial 8 conductance measurements,
         representing the first two years of quarterly data under corrective action. If all 8 values are
         used to compute the upper 95% confidence bound (taking 95% so that a =  .05) and the
         formula for a confidence interval around a normal mean from Section 21.1 is applied, the limit
         becomes:

               UCL  = x + t__ -4= = 901.75 + 1.8946 x 635'Z.126 = 1327.6 //mho
                           l_an_l
                                                          .
                                                       V8
         While this limit exceeds the clean-up standard of L = 1000 umho, the same limit excluding the
         first four measurements is easily below the compliance standard:

                                               54 0085
                             = 451.75 + 2.3534x—'-FT— = 515.3 //mho
        Figure 22-9. Probability Plot at GW-13 Using Entire Sampling Record
             B
             a
             u
             S3
                 2 -
                 1 -
             N   0 -
                -2
                    0
500
1000
1500
   I
2000
2500
                                   Specific Conductance (jj,mho)
                                          22-16
                                                   March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests
                                                                 Unified Guidance
     Figure 22-10. Probability Plot at GW-13  Excluding  First Four Measurements
              B
              D
              o
              m
              N
                  1  -
         0 -
                 -1  -
                 -2
                    300
                        400
500
600
700
                                     Specific Conductance (imho)
Step 9.
Based on the calculation in Step 8, the clean-up standard is certainly met by early 1990 at
GW-13. However, it is also instructive to examine the confidence bounds on larger sets of data
from the stable portion of the sampling record. For instance, if the initial 4 measurements are
excluded and then the next 8 values are used, the upper 95% confidence bound is:
               UCLM = 478.75 + 1. 8946 x
                                                       = 524.5 //mho
         If all of the last 19 specific conductance values are used, a similar 95% confidence bound
         becomes:

                       UCL95 =503.158+1.7341x74J^.6° = 532.7//mho


Step 10.  Both of the limits in Step 9 easily meet the clean-up standard of L = 1000 (imho. However, the
         amount of data used in the latter case is more than double than that of the former, which can
         impact the relative statistical power  of the upper confidence limit  for detecting decreases
         below the fixed standard.  Given that  the specific conductance seems to level off at close to
         500 (imho, or one-half the clean-up standard, and given that the CFis approximately equal to
         .15,  Table 22-5 in Appendix D  (looking under  CV =  0.2) indicates that  at  least  6
         measurements are needed to have a 95% chance of detecting a drop in conductance level to
         half the standard. So in this example, both UCLs are sufficiently powerful for detecting such a
         decrease. -^
                                           22-17
                                                                         March 2009

-------
Chapter 22. Compliance & Corrective Action Tests _ Unified Guidance

22.2 CONFIDENCE INTERVAL TESTS FOR UPPER PERCENTILES

     For fixed standards which represent an upper percentile or maximum, the proper comparison in
compliance/assessment monitoring utilizes the lower confidence limit around an upper percentile tested
against the GWPS.  In formal corrective action testing, the appropriate comparison employs an upper
confidence limit around an upper percentile. Parametric and non-parametric confidence intervals around
percentiles are presented in Chapter 21.

     While the basic comparison is similar to confidence intervals around a mean, two points should be
noted. First, any numerical  standard identified as a maximum concentration 'not to be exceeded' needs to
be treated statistically as an upper percentile. The reason is that while every observed data set has a finite
maximum, there is no way to estimate  the confidence bounds  around the maximum of a continuous
population. The true 'maximum' is always positive infinity, illustrating a point of breakdown between
mathematical models and physical reality. Nonetheless, confidence limits around an upper 90th to 99th
percentile can be used as  a  close approximation to a maximum or some  limit likely to only be
infrequently exceeded.

     Secondly, computing statistical power for an interval around an upper percentile is similar to but
not quite the same as, statistical power for an interval around the mean.  Statistical  power for a
compliance/assessment test of the upper 90th percentile is derived by considering whether more than
10% of all the population measurements exceed the GWPS.  If so, the 90th percentile must also exceed
the standard.  In  corrective action testing, the  equivalent question is whether less than 10%  of the
measurements exceed the GWPS.   In that case, the true 90th percentile must also be  less than the
standard.

     Statistically, each observation is  set equal  to  0  or  1  depending  on whether the  measured
concentration is less than  or greater than the fixed standard. Then the percentage  of measurements
exceeding the GWPS is given by the average of the set of zeros and ones. In other words, the problem is
similar to estimating an arithmetic mean.

     The similarity ends,  however,  when it  comes  to setting power  targets.   For  mean-based
evaluations, power at true mean concentration levels is equivalent to a fixed multiple or fraction of the
GWPS (e.g., 1.5 or 2 times the standard;  0.25 or 0.5 times the standard). But for upper percentile power,
the alternative hypothesis is defined in terms of the actual percentage of measurements either exceeding
the standard in compliance/assessment  monitoring (e.g., 20% or 30% instead of the null hypothesis
value of 10%) or exceeding the clean-up level in corrective action monitoring (e.g., 2% or 5% instead of
10%). In both hypothesis  frameworks, the actual fraction  of measurements above the standard can be
denoted by p. Furthermore, the power formulas rely on a normal approximation  to  the binomial
distribution.   If p is the probability that an individual observation exceeds the GWPS, and po is the
percentage of values exceeding the GWPS when the (l-po)th upper percentile  concentration equals the
standard, the quantity:
                                                                                       P2.4]
                                            22-18                                  March 2009

-------
Chapter 22. Compliance & Corrective Action Tests	Unified Guidance

has an  approximately standard normal  distribution  under either  the  compliance/assessment null
hypothesis/? po.

     Under the  compliance/assessment alternative hypothesis  (HA), the true fraction exceeding the
standard is greater than the null value (p =p\ >po).  With the corrective action alternative hypothesis, the
true fraction is less than the null value (p  = p\ 
-------
Chapter 22.  Compliance & Corrective Action Tests	Unified Guidance

     Values of n for various choices of power level (l-(3), Type I error rate (a), and upper percentile (1-
po) are tabulated in Table 22-7 in Appendix D. These can be used to maintain a specific level of power
when employing a confidence interval around an upper percentile in compliance/assessment monitoring.
The percentiles covered in this table include the 90th, 95th, 98th, and 99th. Levels  of statistical power (1-
(3) provided include .50, .60, .70, .80, .90, .95, and .99, while the false positive rate (a) ranges from .20
down to .01.  Specific cases not covered by Table 22-7 in Appendix D can be  computed directly with
equation [22.5].

       ^EXAMPLE 22-3

     Suppose a compliance limit  for the pressure under  which chlorine gas  is stored  in a moving
container  (for instance, a rail  car) is  designed to protect  against acute, short-term exposures due to
ruptures or leaks in the container.  If the compliance limit represents an upper 90th percentile of the
possible range of pressures that might be used to seal a  series of such containers, how many containers
should be sampled/tested to ensure that if in fact 30% or more of the container  pressures exceed the
limit, violation of the standard will be  identified with 90% probability and exhibit only a 5% chance of
false positive error?

       SOLUTION
Step 1.    Since the compliance limit on chlorine gas pressure represents the 90th percentile, at most
          10% of the container pressures should exceed  this limit under normal operations. In statistical
          notation, po = 0.10 and (l-po) = 0.90.  If there is a problem with the process used to seal the
          containers and  30% of the pressures instead exceed the  limit, this amounts to considering a
          multiple of k= 3 times the nominal exceedance amount.

Step 2.    Since a violation of the pressure standard by  at least 3po or 30% needs  to be identified with
          90% probability,  the target power is  (l-(3) =  0.90. Also, the chance of constructing a lower
          confidence limit on the true 90th percentile gas pressure that falsely identifies an exceedance
          of the standard must be kept to a = .05.

Step 3.    Looking in Table 22-7 in Appendix D under the 90th percentile and  k = 3, the necessary
          minimum sample size is n = 30. Thus, 30 similarly-sealed containers should be tested for gas
          pressure so that a confidence interval around  the 90th percentile can be constructed on these
          30 measurements using either the parametric or non-parametric formulas in Chapter 21. ~4
22.2.2       UPPER PERCENTILE TESTS IN CORRECTIVE ACTION

     Equation  [22.5] can also be used in  formal  corrective action testing. In this  setting, an upper
confidence limit [UCL] around an upper percentile  is of interest and the false positive rate (a) needs to
be minimized to ensure a low probability of falsely or prematurely declaring remedial success. In
practice,  a should be pre-specified to a low value. Then, different values for power (l-(3) can be input
into equation [22.5] until the resulting minimum sample size (n) either matches the available amount of
sampling data or is feasible to collect in future sampling.

     Once the  minimum sample  size is computed and these n measurements are used to construct a
UCL on the upper percentile (1-po) of interest (e.g., the 95th), there will be a (l-(3) x  100% chance that
                                             22-20                                  March 2009

-------
Chapter 22.  Compliance & Corrective Action Tests	Unified Guidance

the UCL  will be less than the clean-up standard when in fact  no more than kpo x  100%  of the
measurements actually exceed the standard. For instance, if k = 1/2, (l-(3) will be the power of the test
when in fact half as many of the measurements exceed the standard as are nominally allowed.

     Equation [22.5] also implies that the  UCL will falsely drop below the clean-up standard  with
probability a. That is, when the true percentage of measurements exceeding the standard is actually po or
greater — indicating that  the clean-up standard has not been met — the test will still declare the
remedial effort successful a x 100% of the time.

     Values of n for various choices of power level (l-(3), Type I error rate (a), and upper percentile (1-
PO) are tabulated in Table 22-8 in Appendix D. This table can be used to determine or adjust the feasible
power level based on a pre-specified a when  employing a confidence interval around an upper percentile
in corrective action. Note that the minimum  sample sizes in  Table 22-8 of Appendix  D are generally
quite large, especially for small error rates (a). Because of the regulatory interest in minimizing the risk
of prematurely exiting remediation, statistical comparisons in corrective action are likely to initially  have
fairly low power. As the clean-up process  continues,  enough additional data can be  accumulated to
adequately raise the odds of declaring the remediation a success when in fact it is.

       ^EXAMPLE 22-4

     Suppose excessive nitrate levels must be remediated in a rural drinking water supply. If the clean-
up  standard for  infant nitrate  exposure  represents an upper 95th  percentile  of the concentration
distribution, what sample size (n) should be  selected to ensure that if true nitrate levels drop below the
clean-up standard, the remediation effort will be judged successful with at least 80% probability?

SOLUTION
Step 1.  Examining Table 22-8 in Appendix D under the 95th percentile and power = (l-(3) = .80, a
         choice of n cannot be  made until two other statistical parameters  are fixed: the false positive
         rate  (a) and the relative fraction  of exceedances  (p).  The false positive rate  governs the
         likelihood that the upper confidence limit on nitrate will be below the clean-up standard,  even
         though more than 5%  of all nitrate measurements are above the compliance standard (so that
         the true 95th percentile for nitrate still exceeds the clean-up criterion). The relative fraction of
         exceedances (p)  sets the true percentage of individual nitrate concentrations  that exceed the
         clean-up standard under the alternative hypothesis (//A); that is, what fraction of nitrate values
         are exceedances when the clean-up standard is truly met.

         Unfortunately, no matter what choices of a and p are selected in Table 22-8 of Appendix D,
         the smallest required sample size is n = 55, when a = .20 and/? = .25. Even if it is practical
         and affordable to test 55 samples of groundwater for nitrate, the chance of falsely declaring the
         remediation effort a success will still be 20%. To cut that  probability in half to a = .10, n  = 99
         samples needs to be tested.
                                             22-21                                    March 2009

-------
Chapter 22. Compliance & Corrective Action Tests	Unified Guidance

Step 2.   To lessen the required sampling effort,  consider the alternatives. Lower sample  sizes are
         needed  if the percentile of interest  is less extreme,  for instance if the clean-up standard
         represents a 90th percentile instead of the 95th. In this case, only n = 48 samples are needed
         for 80% power and  a 10% false positive rate with p = .25. Of course, more frequent
         exceedances  of the compliance limit are then allowed (i.e.,  10% versus 5% of the  largest
         nitrate concentrations).

         Another less desirable option is to raise the a level of the test. This raises the risk of falsely
         declaring the remediation effort to be a success. One  could also lower/?. At p = .25  for the
         95th percentile, 80% power is guaranteed only when the true nitrate exceedance frequency is
         one-fourth the maximum allowable rate— i.e., when the true rate of exceedances is .25 x 5% =
         1.25%.  Exceedance rates greater than this will be associated with less than  80% power. But
         while lowering/? and keeping other parameters constant will indeed decrease n, it also has the
         effect of requiring a very low actual exceedance rate before the power of the test will be
         sufficiently high. Aip = .10 for the 95th percentile, for instance, the true exceedance rate then
         needs to be only . 10 x 5% = 0.5% to maintain the same level of power.

         The final option is to lower the desired power. Power in this setting is the probability that the
         UCL  on nitrate will be below the  clean-up  standard, when the  groundwater is  no longer
         contaminated above the standard. When the true nitrate levels are sufficiently low to meet the
         compliance standard, demonstrating this fact will only occur with high probability (i.e., high
         power)  when the sample size is fairly large. By taking a greater chance that the status of the
         remediation will be declared inconclusive  (i.e., when the UCL still exceeds the clean-up
         standard even though the true nitrate levels have dropped), power could be lowered to 70% or
         60% for instance, with a corresponding reduction in the required n. To illustrate, if the power
         is set at 60% instead of 80% for the 95th percentile and the false positive rate is set at a = .10,
         the required sample size would drop from n = 99 to n = 68.

Step 3.   In many groundwater contexts, the minimum sample sizes of Table 22-8 in Appendix D may
         seem excessive.  Certainly, the sampling requirements associated with upper percentile clean-
         up standards  are  substantially greater than those needed to test  mean-based  standards.
         However,  remediation efforts often last several years, so it may be possible to accumulate
         larger amounts of data for statistical use than  is possible  in, say, detection or compliance
         monitoring.  In any event, it is important  to recognize how the type of standard and the
         statistical parameters  associated  with a confidence interval test impact the amount of data
         necessary to  run the comparison. Each parameter should be assessed and interpreted in the
         planning stages of an analysis, so that the pros and cons of each choice can be weighed.

Step 4.   Once a sample  size has been selected and the data  collected, either a parametric or  non-
         parametric  upper confidence limit should be constructed  on the nitrate measurements and
         compared to the clean-up standard. -^
                                             22-22                                   March 2009

-------
APPENDICES	Unified Guidance
        STATISTICAL ANALYSIS OF
  GROUNDWATER MONITORING DATA AT
            RCRA FACILITIES

           UNIFIED GUIDANCE

              APPENDICES
OFFICE OF RESOURCE CONSERVATION AND RECOVERY

PROGRAM IMPLEMENTATION AND INFORMATION DIVISION

U.S. ENVIRONMENTAL PROTECTION AGENCY
MARCH 2009
                                     March 2009

-------
APPENDICES                                                    Unified Guidance
                    This page intentionally left blank
                                                                      March 2009

-------
APPENDICES	Unified Guidance





                APPENDICES--  TABLE OF  CONTENTS




    APPENDIX A.  REFERENCES, GLOSSARY & INDEX



          A.I REFERENCES	A-2



          A.2 GLOSSARY	A-8



          A.3 INDEX	A-12





    APPENDIX B.  HISTORICAL NOTES




          B.I Past Guidance for Checking normality	B-2



          B.2 The CABF Procedure	B-4



          B.3 Past Guidance For Non-Detects	B-5



          B.4 Trend Tests	B-6



          B.5 Prediction Limits and Retesting	B-7



                B.5.1  Retesting Schemes	B-7



                B.5.2  Tolerance Screens	B-9



                B.5.3  Non-Parametric Retesting Schemes	B-ll





     APPENDIX C. TECHNICAL APPENDIX




          C.I Special Study: Normal vs.  Lognormal Prediction Limits	C-2



                C.I.I  Results For Normal Data	C-2



                C.I.2 Results for Lognormal Data	C-4



          C.2 Calculating Statistical Power	C-10



                C.2.1Statistical Power of Welch's T-Test	C-10



                C.2.2 Power of Prediction limits  for future mean vs. ObservationsC-12



                C.2.3 Computing Power  with Lognormal Data	C-13



          C.3 R Scripts	C-16



                C.3.1  Parametric Intrawell Prediction Limit Multipliers	C-16



                C.3.2  Theil-Sen Confidence Band	C-20
                                                                         March 2009

-------
APPENDICES	Unified Guidance





      APPENDIX D.   STATISTICAL TABLES




             TABLE 10-1 Percentiles of the Standard Normal Distribution	D-l



             TABLE 10-2 Coefficients for Shapiro-Wilk Test of Normality	D-3



             TABLE 10-3 a-Level Critical Points for Shapiro-Wilk Test, n< =50	D-5



             TABLE 10-4«-Level Critical Points for Shapiro-Wilk Test, n from 50 to 100	D-6



             TABLE 10-5 a- Critical Points for Correlation Coefficient Test	D-7



             TABLE 10-6 Shapiro-Wilk Multiple Group Test, (G) Values for n from 7 to 50	D-8



             TABLE 10-7 Shapiro-Wilk Multiple Group Test, (G) Values for n from 3 to 6	D-9







             TABLE 12-1 a-Level Critical Points for Dixon's Outlier Test, n from 3 to 25	D-10



             TABLE 12-2 a-Level Critical Points for Rosner's OutiierTest	D-ll



             TABLE 14-1 Approximate a-Level Critical Points for Rank vonNeumann Ratio Test	D-13



             TABLE 15-1 Percentiles of the Student's t-Distribution	D-15







             TABLE 17-1 Percentiles of the ^Distribution for (1-a) = .8, .9, .95, .98 & .99	D-17



             TABLE 17-2 Percentiles of the Chi-Square Distribution for df from 1 to 100	D-23



             TABLE17-3 Upper Tolerance Limit Factors with /Coverage for n from 4 to 100	D-24



             TABLE 17-4 Minimum Coverage of Non-Parametric Upper Tolerance Limits	D-25



             TABLE 17-5 Significance Levels (a) for Mann-Kendall Trend Tests for n from 4-100	D-27



             TABLE 18-1 Confidence Levels of Non-Parametric Prediction Limits for n from 4-60	D-28



             TABLE 18-2 Confidence for Non-Parametric Prediction Limits on Future Median	D-31
                                                                                            March 2009

-------
APPENDICES	Unified Guidance





      APPENDIX  D.  STATISTICAL TABLES (CONT'D)




             TABLE 19-1  /(--Multipliers for l-of-2 Interwell Prediction Limits on Observations	D-34



             TABLE 19-2  K-Multipliers for l-of-3 Interwell Prediction Limits on Observations	D-43



             TABLE 19-3  /c-Multipliers for l-of-4 Interwell Prediction Limits on Observations	D-52



             TABLE 19-4  /(--Multipliers for Mod. Cal. Interwell Prediction Limits on Observations	D-61



             TABLE 19-5  K-Multipliers for 1-of-l Interwell Prediction Limits on Means of Order 2	D-72



             TABLE 19-6  K-Multipliers for l-of-2 Interwell Prediction Limits on Means of Order 2	D-81



             TABLE 19-7  /(--Multipliers for l-of-3 Interwell Prediction Limits on Means of Order 2	D-90



             TABLE 19-8  K-Multipliers for 1-of-l Interwell Prediction Limits on Means of Order 3	D-99



             TABLE 19-9  K-Multipliers for l-of-2 Interwell Prediction Limitson MeansofOrderS	D-108








             TABLE 19-10 K-Multipliers for l-of-2 Intrawell Prediction Limits on Observations	D-l 18



             TABLE 19-11 K-Multipliers for l-of-3 Intrawell Prediction Limits on Observations	D-127



             TABLE 19-12 /(--Multipliers for l-of-4 Intrawell Prediction Limits on Observations	D-136



             TABLE 19-13 K-Multipliers for Mod. Cal. Intrawell Prediction Limits on Observations	D-145



             TABLE 19-14 /(--Multipliers for 1-of-l Intrawell Prediction Limits on Mean Order 2	D-156



             TABLE 19-15 K-Multipliers for l-of-2 Intrawell Prediction Limits on  Mean Order 2	  D-165



             TABLE 19-16 K-Multipliers for l-of-3 Intrawell Prediction Limits on Mean Order 2	D-174



             TABLE 19-17 /(--Multipliers for 1-of-l Intrawell Prediction Limits on  Mean Order 3	  D-183



             TABLE 19-18 K-Multipliers for l-of-2 Intrawell Prediction Limits on  Mean Order 3	  D-192








             TABLE 19-19 Per-Constituent Significance Levels for Non-Parametric l-of-2 Plan	D-202



             TABLE 19-20 Per-Constituent Significance Levels for Non-Parametric l-of-3 Plan	D-206



             TABLE 19-21 Per-Constituent Significance Levels for Non-Parametric l-of-4 Plan	D-210



             TABLE 19-22 Per-Constituent Significance Levels for Non-Parametric Mod. Cal. Plan	D-214



             TABLE 19-23 Per-Constituent Significance Levels for Non-Param. 1-of-l Median Plan	D-219



             TABLE 19-24 Per-Constituent Significance Levels for Non-Param. l-of-2 Median Plan	D-223




                                                                                              March 2009

-------
APPENDICES	Unified Guidance





     APPENDIX D.  STATISTICAL TABLES (CONT'D)




            TABLE 21-1  Land (H) Factors for 1%LCL on a Lognormal Arithmetic Mean	D-228



            TABLE 21-2  Land (H) Factors for 2.5% LCL on a Lognormal Arithmetic Mean	D-230



            TABLE 21-3  Land (H) Factors for 5% LCL on a Lognormal Arithmetic Mean	D-232



            TABLE 21-4  Land (H) Factors for 10% LCL on a Lognormal Arithmetic Mean	D-234



            TABLE 21-5  Land (H) Factors for 90% UCL on a Lognormal Arithmetic Mean	D-236



            TABLE 21-6  Land (H) Factors for 95% UCL on a Lognormal Arithmetic Mean	D-238



            TABLE 21-7  Land (H) Factors for 97.5%UCL on a Lognormal Arithmetic Mean	D-240



            TABLE 21-8  Land (H) Factors for 99% UCL on a Lognormal Arithmetic Mean	D-242



            TABLE 21-9  Factors (r) for Parametric Upper Confidence Bounds on Percentiles (P)	D-245



            TABLE 21-10  Factors (r) for Parametric Lower Confidence Bounds on Percentiles (P)	D-247



            TABLE 21-11 One-sided Non-ParametricConf. Bnds.on Median, 95th &99"1 Percentiles	D-249







            TABLE 22-1  Combs, of r? and « Achieving Powerto Detect Increases of l.SxGWPS	D-256



            TABLE 22-2  Combs, of n and a Achieving Power to Detect Increases of 2.0xGWPS	D-257



            TABLE 22-3 Minimum Individual Test a Meeting Power criteria, given r? and CV	D-258



            TABLE 22-4 Minimum n to Detect Increases of .75xGWPS, given CV,  1-/3, and a	D-259



            TABLE 22-5 Minimum n to Detect Increases of .SxGWPS, given O/,l-/3, and a	D-261



            TABLE 22-6 Minimum n to Detect Increases of .25xGWPS, given CV,  1-/3, and a	D-263



            TABLE 22-7  Minimum n to Detect kpo Incr. over Percentile l-p0, with 1-/3 and a,k>l	D-265



            TABLE 22-8  Minimum n to Detect kpo Incr. over Percentile l-p0, with 1-/3 and a, k < 1	D-267
                                                                                      March 2009

-------
Appendix A—References, Glossary & Index	Unified Guidance



     APPENDIX A.  REFERENCES, GLOSSARY & INDEX





        A.I REFERENCES	A-2


        A.2 GLOSSARY	A-8


        A.3 INDEX	A-12
                                A-l
                                                           March 2009

-------
Appendix A—References, Glossary & Index	Unified Guidance

A.I  REFERENCES

Aitchison J (1955). On the distribution of a positive random variable having a discrete probability mass
     at the origin. Journal of American Statistical Association, 50(272), 901-908.
Aitchison  J  & Brown  JAC  (1976). The Lognormal Distribution.  Cambridge,  England: Cambridge
     University Press.
American Society for Testing and Materials  (2004). D7048-04: Standard Guide for Applying Statistical
     Methods for Assessment and  Corrective Action Environmental Monitoring Programs. West
     Conshohocken, PA: ASTM International.
 American Society for Testing and Materials (2005). D6312-98[2005]: Standard Guide for Developing
     Appropriate Statistical  Approaches for  Ground Water Detection Monitoring Programs. West
     Conshohocken, PA: ASTM International.
Barari A & Hedges LS (1985). Movement of water in glacial till. Proceedings of the 17th International
     Congress of the International Association ofHydrogeologists, 129-134.
Barnett V & Lewis T (1994). Outliers in Statistical Data (3rd Edition). New York: John Wiley & Sons.
Bartels R (1982). The rank version of von Neumann's ratio test for randomness. J. Amer. Stat. Assn, 77,
     40-46.
Best DI &  Rayner  CW (1987).  Welch's approximate  solution for  the Behren's-Fisher  problem.
     Technometrics, 29, 205-210.
Bhaumik DK &  Gibbons RD (2006). One-sided approximate prediction intervals for at  least p of m
     observation from a gamma population at each of r locations. Technometrics,  48, 112-119.
Box GEP & Cox DR  (1964). An analysis of transformations (with  discussion). Journal of Royal
     Statistical Society Series B, 26, 211-252.
Brown KW  & Andersen  DC (1981). Effects of Organic Solvents on the Permeability of Clay Soils.
     Cincinnati, OH:  USEPA. EPA 600/2-83-016, Pub. No. 83179978.
Cameron K (1996). RCRA leapfrog: how statistics shape and in turn are shaped by regulatory mandates.
     Remediation, 7,  15-25.
Cameron K  (2008). Weibull prediction limits with retesting. Proceedings of the  2007 Joint Statistical
     Meetings, (in press)
Chatfield C (2004).  The  Analysis  of Time Series: An Introduction (6th Edition). Boca Raton, FL:
     Chapman and Hall.
Cohen AC Jr (1959). Simplified estimators for the normal distribution when samples are single censored
     or truncated. Technometrics, 1, 217-237.
Cohen AC Jr (1963). Progressively censored samples in life testing. Technometrics, 5(3), 327-339.
Cook RD & Weisberg S  (1999). Applied Regression Including Computing and  Graphics. New York:
     John Wiley  & Sons.
Cox DR & Hinkley DV  (1974). Theoretical Statistics. London: Chapman & Hall.
                                            A-2
                                                                                  March 2009

-------
Appendix A— References, Glossary & Index _ Unified Guidance
Davis CB (1994). Environmental regulatory statistics. In GP Patil & CR Rao (Eds.) Handbook of
     Statistics, Volume 12: Environmental Statistics, Chapter 26. New York: Elsevier Science B.V.
Davis CB (1998). Ground-Water Statistics  and Regulations:  Principles,  Progress, and Problems.
     Henderson, NV: Environmetrics and Statistics Ltd.
Davis CB (1999). EnviroStat Technical  Report 99-1: Comparisons of Control Chart and Prediction
     Limit Procedures Recommended for  Groundwater  Detection  Monitoring. Henderson, NV:
     Environmetrics and Statistics Ltd.
Davis CB & McNichols RB (1999). Simultaneous nonparametric prediction limits. Technometrics, 41,
Davis CB & McNichols RJ (1987). One-sided intervals for at least p of m observations from a normal
     population on each of r future occasions. Technometrics, 29(3), 359-370.
Davis CB & McNichols RJ (1994).  Ground water monitoring statistics update: part I: progress since
     1988. Ground Water Monitoring and Remediation, 14(4),  148-158; and: part II: nonparametric
     prediction limits. 14(4), 159-175.
Davison AC & Hinkley DV (1997). Bootstrap Methods and Their Application. Cambridge: Cambridge
     Univ. Press.
Draper NR &  Smith H (1998). Applied Regression Analysis (3rd Edition) . New York: John Wiley &
     Sons.
Dunnett CW (1955). A multiple comparisons procedure for comparing several treatments with a control.
     J. Amer. Stat. Assn, 50, 1096-1121.
Efiron B (1979). Bootstrap methods: another look at the jackknife. Annals of Statistics, 7, 1-26.
Evans M, Hastings N, & Peacock B (1993). Statistical Distributions (2nd Edition). New York: John
     Wiley &  Sons.
Filliben JJ (1975). The probability plot  correlation coefficient test for normality. Technometrics, 17,
     111-117.
Freeze RA & Cherry JA (1979). Ground Water. Englewood Cliffs, NJ: Prentice Hall, Inc.
Gan FF & Koehler KJ (1990).  Goodness-of-fit tests based on p-p probability plots. Technometrics,
     32(3), 289-303.
Gayen AK (1949). The distribution  of "Student's" t in random samples of any  size drawn from non-
     normal universes. Biometriha, 36, 353-369.
Gehan EA (1965). A generalized Wilcoxon test for comparing  arbitrarily singly-censored  samples.
     Biometrika, 52, 203-223.
Gibbons RD (1987). Statistical models for the analysis of volatile organic compounds in waste disposal
     sites. Ground Water, 25(5), 572-580.
Gibbons  RD (1990).  A general statistical procedure for ground-water detection monitoring  at waste
     disposal facilities. Ground Water, 28(2), 235-243.
Gibbons  RD  (199 la).  Some additional  nonparametric  prediction limits for ground-water detection
     monitoring  at waste disposal facilities. Ground Water, 29(5), 729-736.
                                                                                   March 2009

-------
Appendix A—References, Glossary & Index	Unified Guidance

Gibbons RD (1991b). Statistical tolerance limits for ground-water monitoring. Ground Water, 29, 563-
     570.
Gibbons RD (1994a). The folly of Subtitle D statistics: when greenfield sites fail. Paper presented at the
     Proceedings of Waste Technology '94, Charleston, SC.
Gibbons RD (1994b). Statistical Methods for Groundwater Monitoring. New York: John Wiley & Sons.
Gibbons RD (1999). Use of combined Shewhart-CUSUM control charts for ground-water monitoring
     applications. Ground Water, 37(5), 682-691.
Gibbons  RD  &  Coleman  DE  (2001).  Statistical Methods for  Detection and Quantification of
     Environmental Contamination. New York: John Wiley & Sons.
Gilbert RO  (1987). Statistical Methods for Environmental  Pollution Monitoring.  New  York: Van
     Nostrand Reinhold.
Gilliom RJ & Helsel DR (1986). Estimation of distributional parameters for censored trace level water
     quality data: I. Estimation techniques. Water Resources Research, 22, 135-146.
Guttman  I (1970). Statistical Tolerance Regions:  Classical and Bayesian. Darien, CT: Hafner
     Publishing.
Hahn GJ  & Meeker WQ (1991). Statistical Intervals: A Guide for Practitioners.  New York: John Wiley
     & Sons.
Heath RC (1983). Basic Ground-Water Hydrology. US Geological Survey, Water Supply Paper 2220.
Helsel DR (2005). Nondetects and Data Analysis. New York: John Wiley & Sons.
Helsel DR & Gilliom JR (1986). Estimation of distributional parameters for censored trace level water
     quality data: II. verification and applications,  Water Resources Research, 22, 147-155.
Helsel DR & Cohn TA (1988). Estimation of descriptive statistics for multiply censored water quality
     data. Water Resources Research, 24, 1997-2004.
Helsel DR & Hirsch RM (2002). Chapter A3:  Statistical Methods in Water Resources:  Techniques of
     Water-Resources Investigations of the United States Geological Survey: Book  4,  Hydrologic
     Analysis and Interpretation. United States Geological Survey.
Hem JD (1989). United States Geological Survey Water-Supply Paper 2254: Study and Interpretation of
     the Chemical Characteristics of Natural Waters. Reston, VA: USGS.
Hintze X (2001). User's Guide, NCSS Statistical System for Windows. Kaysville, UT: Number Cruncher
     Statistical Systems.
Hockman KK & Lucas JM (1987). Variability reduction through sub-vessel  CUSUM control. Journal of
     Quality Technology, 19, 113-121.
Hollander M & Wolfe DA (1999). Nonparametric Statistical Methods (2nd Edition). New York: John
     Wiley &  Sons.
Kaplan EL & Meier P (1958). Non-parametric estimation from incomplete observations. J. Amer. Stat.
     Assn,53,457-481.
Knusel L (1994). The prediction problem as the dual form of the two-sample problem with  applications
     to the Poisson and the binomial distribution. The American Statistician, 48(3), 214-219.
                                             A^4
                                                                                    March 2009

-------
Appendix A—References, Glossary & Index	Unified Guidance
Land CE (1971). Confidence intervals for linear functions of the normal mean and variance. Annals of
     Mathematical Statistics, 42, 1187-1205.
Land CE (1975). Tables of confidence limits for linear functions of the normal mean and variance, In
     Selected Tables  in Mathematical Statistics,  v. 111. (pp.  385-419).  Providence, RI: American
     Mathematical Society.
Lehmann EL (1975). Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day.
Lucas  JM (1982). Combined  Shewhart-CUSUM  quality control  schemes. Journal of Quality
     Technology, 14, 51-59.
Luceno A & Puig-Pey J (2000). Evaluation of the run-length probability distribution for CUSUM charts:
     assessing chart performance. Technometrics, 42(4), 411-416.
Madansky A (1988). Prescriptions for Working Statisticians. New York: Springer-Verlag.
McNichols RJ & Davis CB (1988).  Statistical issues and problems in ground water detection monitoring
     at hazardous waste facilities. Ground Water Monitoring Review, 8, 135-150.
Miller RJ (1981). Simultaneous Statistical Inference. New York:  Springer-Verlag.
Miller RJ (1986). BeyondANOVA: Basics of Applied Statistics. New York: John Wiley & Sons.
Milliken GA &  Johnson DE (1984). Analysis of messy data:  volume  1, designed experiments. Belmont,
     CA: Lifetime Learning Publications.
Moser BK & Stevens GR (1992). Homogeneity of variance in the two-sample means test. The American
     Statistician, 46(1), 19-21.
Mull DS, Liebermann TD,  Smoot JL, & Woosley JLH (1988). Application of Dye-Tracing Techniques
    for Determining Solute Transport Characteristics of Ground Water in Karst Terranes. USEPA.
     EPA 904/6-88-001.
Odeh RE & Owen DB (1980). Tables for Normal Tolerance Limits, Sampling Plans,  and Screening.
     New York: Marcel Dekker, Inc.
Ott WR (1990). A physical explanation of the lognormality of pollutant concentrations. Journal of Air
     Waste Management Association, 40, 1378-1383.
Quinlan JF (1989).  Ground-Water Monitoring in Karst Terranes: Recommended Protocols and Implicit
     Assumptions. USEPA. EPA/600/X-89/050.
RosnerB (1975). On the detection of many outliers. Technometrics, 17, 221-227'.
Ryan TA  & Joiner BL (1990). Normal probability plots and tests for normality. Minitab Statistical
     Software: Technical Reports, November,  1-14.
Shapiro SS & Francia RS (1972). An approximate analysis of variance test for normality. J. Amer. Stat.
     Assn, 67(337), 215-216.
Shapiro SS, &  Wilk MB  (1965).  An analysis  of variance test for normality (complete  samples).
     Biometrika,52, 591-611.
Singh  AK,  Singh A,  &  Engelhardt M  (1997).  The  Lognormal Distribution  in  Environmental
     Applications. EPA/600/R-97/006, 1-20.

                                             ^5
                                                                                  March 2009

-------
Appendix A—References, Glossary & Index	Unified Guidance

Singh AK,  Singh A, & Engelhardt M (1999). Some practical aspects of sample  size and power
     computations  for  estimating the mean of positively skewed distributions in environmental
     applications. EPA/600/S-99/006, 1-37.
Starks TH (1988). Draft Report: Evaluation of Control Chart Methodologies for RCRA Waste Sites.
     Univ of Nevada, Las Vegas, NV: Environmental Research Center. CR814342-01-3.
Steel RGD (1960). A rank sum test for comparing all pairs of treatments. Technometrics, 2, 197-207.
Tarone RE & Ware J (1977). On distribution-free tests for equality of survival distributions. Biometrika,
     64, 156-160.
Todd DK (1980). Ground Water Hydrology. New York: John Wiley & Sons.
Tukey J (1977). Exploratory Data Analysis. Reading, MA: Addison-Wesley.
US  Department of the Interior [USDI] (1968),  Water Quality Criteria, Federal Water  Pollution
     Control Administration, Washington, D.C., reprinted by USEPA 1972
US Environmental Protection Agency (1986).  Resource  Conservation and Recovery  Act (RCRA)
     Ground-Water  Monitoring  Technical  Enforcement  Guidance Document. Washington,  DC:
     USEPA. OSWER-9950.1.
US Environmental Protection Agency (1989a). Statistical Analysis of Ground-Water Monitoring Data at
     RCRA Facilities: Interim Final Guidance. Washington, DC: USEPA, Office of Solid Waste.
US Environmental Protection Agency (1989b).  Risk Assessment Guidance for Superfund Volume I,
     Human Health Evaluation Manual (Part A), Washington, D.C., OSWER, EPA/540/1-89/002
US Environmental Protection Agency  (1992a).  Methods for  Evaluating the Attainment of Cleanup
     Standards: Volume 2:  Ground Water. Washington, DC: OSWER, EPA 230-R-92-014.
US Environmental Protection Agency (1992b). Statistical Analysis of Ground-Water Monitoring Data
     at RCRA Facilities: Addendum to Interim Final Guidance. Washington, DC: USEPA,  Office of
     Solid Waste.
US  Environmental Protection Agency (1992c).  Supplemental Guidance to RAGS: Calculating the
     Concentration Term, Washington, D.C., OSWER, Publication 9285.7-081
US Environmental  Protection Agency (1993). Statistical Support Document for Proposed Effluent
     Limitations Guidelines  and Standards for the Pulp, Paper, and Paperboard Point Source
     Category. Washington, DC: USEPA. EPA-821-R-93-023.
US Environmental Protection Agency (1997). Geostatistical Sampling and Evaluation Guidance for
     Soils and Solid Media (Draft). Washington, DC: USEPA, Office of Solid Waste.
US Environmental Protection Agency (1998).  Guidance for Data Quality Assessment:  Practical
     Methods for Data Analysis, EPA QA/G-9 (QA97 Version). Washington, DC: USEPA,  Office of
     Research and Development. EPA/600/R-96/084.
US Environmental  Protection Agency (2004).  Local Limits Development  Guidance,  Appendix Q
     Methods for Handling Data Below Detection Level, Office of Water Management, EPA 833-R-04-
     002B.
von Neumann J (1941). Distribution  of the ratio  of the  mean square successive difference to the
     variance. Annals of Mathematical Statistics, 12, 367-395.
                                            ^6
                                                                                 March 2009

-------
Appendix A—References, Glossary & Index	Unified Guidance

Welch BL (1937). The significance of the difference between two means when the population variances
     are unequal. Biometrika, 29, 350-362.
Wilk MB & Shapiro SS (1968). The joint assessment of normality of several independent samples.
     Technometrics, 10(4), 825-839.
                                           A-7
                                                                                March 2009

-------
Appendix A—References, Glossary & Index
                                                 Unified Guidance
A.2  GLOSSARY

    Alpha (a) level
    1-of-m Plan
    Accuracy

    ACL


    Aliquot replicates
    ANOVA


    Appendix I


    Appendix II


    Autocorrelation


    Background


    Beta (P) level
    Bias
    Box Plot


    Calibration
    CERCLA
    Confidence Interval
    Confidence Level
Decimal level of significance or false positive error of a statistical test
Retesting plan consisting of an initial sample followed by up to (m-\)
resamples;  resamples  are  collected only if initial sample exhibits  a
statistical difference
Closeness of a measured or computed value to its "true" value, where
the true value is obtained with perfect information.
Alternate Concentration Limit; a fixed standard or clean-up action level
alternative to prescribed RCRA regulatory health- or background limits
Physical splits of a single water quality sample for multiple analyses
Analysis  of Variance; a  statistical method for identifying differences
among several population means or medians
40 CFR  Part  258 chemical parameter  list for Subtitle  D  detection
monitoring programs

40 Part 258 CFR chemical  parameter list for Subtitle D compliance or
assessment monitoring programs
Correlation of values of a single variable data  set over successive time
intervals
Natural or baseline groundwater quality at a site; can be characterized by
upgradient, historical, or sometimes sidegradient water quality
Decimal value representing a false negative error rate in a statistical test
Systematic deviation between a measured (i.e., observed)  or computed
value and its true value. Bias is affected by faulty instrument calibration
and other measurement errors, systematic errors during data collection,
and sampling errors such as incomplete spatial randomization during the
design of sampling programs.

Plot  of selected descriptive statistics at a monitoring point (e.g., mean,
median, upper and lower quartiles)

Comparison of a measurement standard, instrument,  or  item with  a
standard or instrument of higher precision and lower bias to detect and
quantify inaccuracies and to report or eliminate those  inaccuracies by
adjustments. Also used to quantify instrument measurements of a given
concentration in a given sample.
Comprehensive Environmental Response, Compensation and Liability
Act  (or  Superfund);  statute  for non-active  hazardous  waste  site
management and remediation
Statistical interval designed to bound the true value of a population
parameter such as the mean or an upper percentile

Degree of confidence  associated with  a statistical estimate or  test,
denoted as (1 - a)
                                                 A-8
                                                                                           March 2009

-------
Appendix A—References, Glossary & Index
                                                 Unified Guidance
    Coverage

    Critical value
    Degrees of freedom

    Descriptive Statistics
    Effective Power

    EPA Reference Power Curves
    (ERPC)

    False Negative
    False Positive
    GWPS
    Heterogeneous
    Histogram

    Homogeneous
    Horn oscedasti city
    Hypothesis
    Independent & Identically
    Distributed (i.i.d)
    Indicator Parameters
    Interwell
    Intrawell

    Mann-Kendall Test
Fraction of a population expected to be contained within a tolerance
interval
Predetermined decision level for a test of statistical hypotheses
The  number  of ways which members of a data set or sets can  be
independently varied
Statistics used to organize and summarize sample data
In a groundwater  network of statistical tests, the power of the test
method to identify a single well contaminated by a single constituent
Recommended   standards  for  comparing   performance  of RCRA
statistical  methods in  detection monitoring;  based  on  individual
prediction limit using n = 10 background samples and a = .01
Finding of no statistically significant difference when there is, in fact, a
physical difference  in the underlying populations or between  a single
population and a fixed compliance standard; also known as beta (P) or
Type II error
Finding a statistically  significant difference when  there is, in fact,  no
physical difference  in the underlying populations or between  a single
population and a fixed compliance standard;  also known as alpha (a),
significance level, or Type I error
(Ground Water Protection Standards)  Concentration  limits set by the
regulatory  agency  as  a standard  to  be  attained  in  groundwater
monitoring.    These may be fixed health- or risk-based limits  (e.g.
MCLs) or a background level.
Non-uniform in structure or composition throughout

Graphical  representation of frequency with data values grouped into
specified numerical ranges
Uniform in structure and composition throughout
Equality of variance among sets of data
One  of two statements made about potential outcomes of a statistical
test.  The null and alternative hypothesis statements refer to the condition
of a population  parameter. The null hypothesis is favored, unless the
statistical  test demonstrates the  greater likelihood of the  alternative
hypothesis.
Groundwater measurements having the same statistical distribution and
exhibiting no statistical dependence or correlation

Chemical parameters whose presence or elevation is possibly indicative
of a facility release
Comparisons between distinct monitoring wells
Comparisons  over time  at a given monitoring well between early and
later measurements
Non-parametric test of trend
                                                 A-9
                                                                                           March 2009

-------
Appendix A—References, Glossary & Index
                                                  Unified Guidance
    MCL


    MDL



    Modified California Plan



    Non-detects (NDs)

    Non-parametric Test

    Normal distribution


    Outlier

    Parametric Test


    Percentile




    Population


    PQL or QL



    Precision



    Prediction Interval


    Prediction Limit

    Probability

    Probability Distribution



    Proportion



    Random sample
Maximum Contaminant Level; a fixed water quality standard defined
under the Safe Drinking Water Act and used in 40 CFR 258.40(e)(3)
Method Detection Limit—the minimum concentration of a substance
that can be measured and reported with 99% confidence that the analyte
concentration is greater than zero in a specific matrix.

Retesting  plan consisting  of an  initial  sample  followed  by three
resamples; if initial value exhibits a statistical difference, two of three
resamples must not exhibit a difference for the test to 'pass'
Observations below the MDL, RL, or QL

Statistical test that does not depend on knowledge of the distribution of
the sampled population
A family of symmetric continuous  probability distributions defined by
two finite parameters, the mean and variance
Value unusually discrepant from rest of a series of observations
Statistical test that depends  upon  or assumes observations from  a
particular probability distribution or distributions
The specific value of a distribution that divides the distribution such that
p percent of the distribution is equal to or below that value. If the 95th
percentile  is X, it means that 95 percent of the values in the statistical
sample are less than or equal to X.
All possible  measurements/values  over a period  of time  at  a given
location, series of locations, or over a spatial or volumetric extent
Practical Quantification  Limit—lowest concentration   level for  an
analytical method which can be reliably achieved within specified limits
of precision and accuracy under routine laboratory operating conditions

A measure of mutual agreement among individual measurements of the
same  property, usually under prescribed similar conditions, expressed
generally in terms of the sample standard deviation.
Statistical interval  constructed  from background  data on  the next
'future' sample or samples arising from the same population
Upper or lower limit of a prediction interval

Quantitative measure of uncertainty about the occurrence of a random or
uncertain event

Numerical   statistical  pattern  associated  with  a  population  of
measurements;  many  common   patterns   can be  described  using
mathematical formulas
A population proportion (p) is the  ratio of  the number of units of a
population that have the specified characteristic or attribute (M) to the
total number of units in the population (N).
Collected data which are based only on their probability of occurrence
in random fashion
                                                 A-10
                                                                                           March 2009

-------
Appendix A—References, Glossary & Index
                                                  Unified Guidance
    Ranking

    RCRA

    Reporting Limit

    Residual
    ROS

    Sample
    SDWA

    Seasonality

    Sen's Slope Estimator

    SWFPR
    Spearman's Test
    Statistical Parameter

    Statistical Power

    Statistically Significant
    Difference (or Increase)
    Time Series Plot
    Tolerance Interval

    Tolerance Limit
    Trace Value

    Random Variable

    Variance

    Verification Resampling or
    Retesting Plan
Assignment of numbers to an ordered data set indicating their relative
position, generally integer values from 1 to n for the smallest to largest
values in a sample of size n (unless specified in reverse rank order)
Resource  Conservation and Recovery Act;  statutory provisions for
active facility hazardous (Subtitle C) and non-hazardous waste (Subtitle
D) definition, storage, treatment and disposal
Reporting Limit—lowest concentration level for an analytical method
which can be reliably measured by a laboratory
Typically, the difference of a value in a data set from its mean
Regression on order statistics, either parametric or robust; techniques for
fitting non-detect data to a single distribution
Set of measurements from a population (can be as few as one)
Safe Drinking Water Act; statute under which drinking water standards
are promulgated and water treatment sites regulated
The presence of seasonal effects on ground water quality observations;
effects may be natural or man-made.
Non-parametric method to estimate the rate of change of concentration
levels over time
Site  Wide False Positive Rate;  design probability of at least one
statistically  significant  finding among  a  network of  statistical test
comparisons at a group of uncontaminated wells
Non-parametric test of trend using data ranks
A numerical characteristic  of a  statistical population  or  probability
distribution
Strength of  a test  to identify  an  actual release  of  contaminated
groundwater or difference from a compliance standard
Statistical difference exceeding a test limit large enough to  account for
data variability and chance
Graphical plot of individual concentration values over time
Statistical interval  constructed to 'cover' a specified proportion of the
underlying population of measurements
The upper or lower limit of a tolerance interval
Measured value  close  to,  but above  the  limit of detection; may lie
between the MDL and the QL
A numerical value or characteristic that can assume different values on
different sampling events or at different locations
A measure of spread or dispersion calculated as the average of squared
differences from the mean in a set of data or a population
A plan  to  collect  an additional resample  or  resamples  to  confirm  or
disconfirm an initial statistically significant finding
                                                 A-ll
                                                                                            March 2009

-------
Appendix A—References, Glossary & Index
                            Unified Guidance
A.3   INDEX

       — A 	

       Alpha (a) error  3-19
       Alternative hypothesis  3-12
       Aitchison' s method for non-detect data  15-6
       Aliquot replicate data 2-11,6-27
       ANOVA (Analysis of variance),
        diagnostic testing 6-40
        equality of variance, Levene's test 11-4
        formal testing and problems with 2-14, 6-38, 17-3
        Kruskal-Wallis formal detection test 8-29, 17-9
        parametric one-way detection test 8-28, 17-1
        pooled variance  13-8
        spatial variability test 8-16, 13-5
        temporal variability 8-17,14-6
        two-way 14-34
       Assumptions for statistical testing, general,
        i.i.d (independent and identically distributed) 3-4
        normality 3-7
        lack of statistical outliers  3-7
        stationarity 3-5
        statistical independence 3-4
       Assumptions,
         for compliance and corrective action monitoring
         (see Design,  compliance/corrective action programs)
        for detection monitoring
         (see Design,  detection monitoring programs)
       ASTM statistical guidance  1-3
       Autocorrelation function test 8-18,14-12
       Background,
        assumptions for data,
          appropriate data  5-1
          autocorrelation  5-4
          independence 5-4
          outliers 5-4
          representative data 5-1
          spatial variability 5-6
          trends
            intrawell 5-7
            use of residuals 5-7
        establishing, selecting monitoring constituents
          and sample sizes  5-3
        expanding background sample sizes  5-8
        importance of 5-1
        review of historical data 5-9
  updating,
   how to update 5-12
   presence of trends 5-14
   when to update 5-12
   with retesting data 5-13
  usedasGWPS 7-19
Beta (P) error 3-19
Beta distribution for non-parametric prediction limits
  18-18, 18-20
Binomial distribution,
  SWFPR calculations 6-7, 19-4, 19-7, 19-28
  upper percentile tests 21-14,21-17
Bonferroni approximation,
  post-hoc ANOVA contrasts 6-3,17-6
  SWFPR calculations 6-7,19-7
Boxplots,
  design  8-9, 9-5
  spatial variability screening 13-2
  outlier screening method 12-5
Censored probability plot 8-22, 15-7, 15-13
Censoring 15-1
Central limit theorem,  3-16
  applied to logarithmic data  10-7
Chi-squared distribution,
  used with Kruskal-Wallis test, 17-8, 17-12
  table D-23
Coefficient of skewness,
  method summary  8-11
  screening method for normality 10-9
Coefficient of variation, definition 3-10
  as test of normality B-l
  method summary  8-12
  screening method for normality 10-9
Cohen's method for non-detect data 8-24, 15-21
Common statistical measures, general definitions,
  coefficient of variation 3-10
  correlation coefficient  3-12
  interquartile range 3-10
  logarithmic coefficient of variation 3-11
  logarithmic mean  3-11
  logarithmic standard deviation 3-11
  median  3-10
  percentile 3-2
  quartiles 3-10
  sample mean 3-9
                                                           A-12
                                                                                                               March 2009

-------
Appendix A—References, Glossary & Index
                           Unified Guidance
        sample percentile 3-10
        sample standard deviation 3-9
       Compliance/corrective action tests,
        confidence intervals,
          around trend lines,
            non-parametric Theil-Sen 8-45,21-30
            parametric linear regression 8-44, 21-23
          design (see Design, compliance/corr.act. program)
          non-parametric 21-14
            median 8-42,21-17
            upper percentile 8-43,21-21
          parametric, 21-1
            lognormal arithmetic mean  8-40,21-8
            lognormal geometric mean  8-39,21-5
            normal mean 8-38,21-3
            upper percentile 8-41,21-11
        pre-specifying power in compliance tests 22-2
        pre-specifying power in corrective action  22-9
        upper percentiles in compliance testing  22-19
        upper percentiles in corrective action tests 22-20
       Concepts, key statistical,
        Central Limit Theorem 3-16
        continuous distribution 3-7
        detection limits 3-9
        distribution 3-2
        family of distributions  3-9
        equality of variance 3-6
        i.i.d. 3-4
        ladder of powers 3-8
        normalizing transformation  3-8
        outliers  3-7
        pairwise correlation 3-5
        percentile 3-2
        population 3-2
        population mean 3-9
        population variance 3-9
        probability distribution 3-8
        random sample 3-3
        reporting limit 3-9
        representative sample  3-2
        sample  3-2
        sample size 3-2
        sampling distribution 3-16
        seasonal variability  3-3
        spatial variability  3-5
        stationarity 3-3
        statistic   3-2
        temporal variation  3-6
        trend 3-6
       Conditional probability, California plan 18-4
       Confidence bands 7-16,21-22,21-28	
Confidence intervals,
  around trend lines,
   non-parametric Theil-Sen 8-45,21-30
   parametric linear regression 8-44, 21-23
  non-parametric 21-14
     median  8-42,21-17
     upper percentile 8-43,21-21
  parametric,  21-1
   lognormal arithmetic mean 8-40,21-8
   lognormal geometric mean 8-39,21-5
   normal mean 21-3
   upper percentile  21-11
  tests using,
   lower (LCL) 4-6, 7-3
   upper (UCL) 7-3
Confidence level 21-1
Continuity correction,
  Mann-Kendall trend test 17-31
  Wilcoxon rank-sum test 16-18
Contrasts for ANOVA detection  tests,
  parametric Bonferroni 17-3,17-6
  non-parametric 17-9,17-12
Control charts, Shewhart-CUSUM,
  in detection monitoring design  6-45
  introduction 8-38,20-1
   performance criteria, 20-11
   multiple comparisons  20-12
   retesting 20-14
   method summary 8-38
   use of Monte Carlo simulations 20-13
  requirements and assumptions 20-6
Correlation coefficient, definition 3-12
Cumulative false positive errors,
  in compliance/assessment monitoring. 7-10
  in corrective action monitoring 7-12
     D
Darcy equation,
  autocorrelation and sampling interval 6-26, 8-20,
  14-19
Design, compliance/corrective action programs,
  assumptions, impact of,
   left-censored or non-detect data  7-18
   lognormal and other normalized data 7-19
   non-normal data 7-19
   sample variability 7-17
  comparisons to background GWPS, 7-19
   ACL  7-21
   mean prediction limit 7-20
  elements of 7-2
                                                           A-13
                                                                                                              March 2009

-------
Appendix A—References, Glossary & Index
                           Unified Guidance
        ground-water protection standards [GWPS], 7-6
          ACL  7-7,7-10,7-21
          central tendency vs. upper percentiles 7-6
          MCLs 7-6
          representative parameters for 7-6
          problems of interpretation 7-8
        hypothesis testing structures, 7-2
          lower confidence limits for compliance 7-4
          upper confidence limits for corr. action 7-5
          use of one way confidence intervals 7-2
        introduction 7-1
        recommended strategies,  7-13
          confidence interval type 7-13
          sequential data pooling  7-13
        statistical program design,
          cumulative false positive in compliance mon. 7-10
          cumulative false positive in corr. act. mon. 7-12
          false positives and power in compliance mon.  7-8
          false positives and power in corr. act mon. 7-11
        shifts and trends, accounting for,
          confidence bands 7-16
          moving window 7-15
       Design, detection monitoring program,
        assumptions in,
          independence, statistical, 6-24
            aliquot replicate data  6-27
            Darcy's equation and  autocorrelation 6-26
            data mixtures 6-27
            i.i.d. 6-25
            random data, importance of 6-24
            temporal correlation corrections 6-28
          interwell versus intrawell tests,  6-28
            background-downgradient  assumptions for
            interwell testing  6-28
            tradeoffs in design 6-31
          non-detect data, 6-36
            MDL 6-36
            reporting limits 6-37
            use of techniques 6-36
          outliers, 6-34
            automated screening 6-35
            recommendations 6-35
        elements of detection monitoring design,
          effect sizes and data based power curves,
            data-based power curves 6-19
            effect sizes 6-18
          multiple comparisons problem,  6-2
            recommended guidance criteria 6-4
          power, recommendations for,
            effective power 6-13
            EPA reference power  curves [ERPC] 6-16	
     generating the ERPCs 6-14
     introduction to power curves 6-13
     non-central t-distribution 6-14
   sites using more than one statistical method 6-21
   site-wide false positive rate [SWFPR],  6-16
     development and rationale  6-7
     double quantification rule 6-11
     number of tests and constituents 6-9
  introduction to 6-2
  site design examples 6-46
  tests for detection monitoring design,
   ANOVA,
     diagnostic testing  6-40
     formal testing  and problems with 6-38
   control charts 6-45
   intervals, statistical,
     general  6-42
     prediction limits  6-43
     tolerance limits 6-44
   trend tests, 6-41
     use of residuals 6-41
   two-sample tests,
     diagnostic use 6-38
     t-tests and non-parametric options 6-37
Detection monitoring tests, formal
  ANOVA,
   Kruskal-Wallis ANOVA  8-29,17-9
   one-way parametric ANOVA 8-28,17-1
  control charts, Shewhart-CUSUM,
   introduction  8-38,20-1
   performance criteria, 20-11
     multiple comparisons 20-12
     retesting 20-14
     method  summary  8-38
     use of Monte Carlo simulations 20-13
   design (see Design, detection monitoring program)
   requirements and assumptions 20-6
  prediction limits, single tests,
   introduction  18-1
   non-parametric future median 8-36, 18-20
   non-parametric future values 8-35, 18-17
   parametric future mean 8-34,18-11
   parametric future values 8-33, 18-7
  prediction limits, using repeat testing,
   basic subdivision principle 19-7
   computing sitewide false positive 19-4
   non-parametric K-tables usage 19-27
   non-parametric tests general  19-26
   non-parametric future medians 8-36, 19-31
   non-parametric future values 8-35, 19-30
   parametric tests general  19-11	
                                                           A-14
                                                                                                               March 2009

-------
Appendix A—References,  Glossary & Index
                            Unified Guidance
          parametric K-tables usage 19-13
          parametric future means 8-34,19-20
          parametric future values 8-33,19-15
          R-script for parametric tests C-16
          strategies 19-1
        tolerance limits,
          general 8-30, 17-14
          non-parametric tolerance limits 8-30, 17-18
          parametric tolerance limits 8-30, 17-15
        trend tests,
          general 17-21
          linear regression trend test 8-31,  17-23
          Mann-Kendall trend test 8-32,17-30
          R-script for Theil-Sen confidence band C-20
          Theil-Sen trend line 8-32,17-34
        two-sample tests,
          pooled variance t-test 8-25,16-1
          Tarone-Ware test  8-27, 16-20
          Welch's test 8-25,16-7
          Wilcoxon rank-sum test 8-26,16-14
       Dixon's test for outliers 8-15,12-8
       Double quantification rule 6-11
       Effect sizes and data based power curves,
          data-based power curves 6-19
          effect sizes 6-18
       Equality of variance, screening methods,
        box plots 11-2
        mean-standard deviation scatter plot 8-15, 11-8
       Equality of variance, test, Levene's  8-14, 11-4
       Errors in hypothesis testing (see Hypothesis testing)
       Estimate, interval (see Confidence intervals)
       Exploratory tools, summaries and design,
        box plots 8-9,9-6
        histograms  8-10, 9-7
        probability  plots  8-11,9-15
        scatter plots 8-10,9-12
        time series plots 8-9, 9-1
       Factors in ANOVA tests 6-40,14-6
       Family of probability distributions 3-9
       F-distribution 3-17, tables D-17
       Filliben's   test   (see   Probability   plot  correlation
       coefficient)
       Fitting distributions (see Normality  screening methods
       and tests)
       F-tests, for ANOVA 11-6, 13-7, 13-11, 14-9, 17-5
	  Q 	

Gamma distribution 3-8,10-1
Geometric mean, 10-3
  compliance monitoring test for  21-5
Goodness-of-fit tests (see Normality screening methods
and tests)
Groundwater monitoring and tests,
  context for 4-1
  statistical programs 4-3
   compliance or assessment monitoring 4-4
   confidence limits 4-6
   corrective action monitoring  4-5
   detection monitoring 4-3
   regulatory options 4-5
  statistical significance factors 4-5
   analytical 4-10
   data errors 4-11
   geochemical 4-9
   hydrological 4-9
   statistical 4-8
   well system design 4-8
Groundwater protection standards [GWPS], 7-6
   ACL 7-7,7-10,7-21
   background used as 7-19
   central tendency vs. upper percentiles 7-6
   MCLs 7-6
   regulatory options 2-12
   representative parameters for  7-6
   problems of interpretation 7-8

—  H 	

Histogram design 8-10, 9-7
Homoscedasticity (see Equality of variance)
Hypothesis testing framework, general,
  alternative hypothesis 3-12
  false negative errors (Type II) 3-12, 3-18,3-22
  false positive errors (Type I)  3-12, 3-15, 3-22
  hypothesis testing 3-12
  null hypothesis  3-12
  power 3-18
  simple versus compound hypotheses 7-11
  truth table 3-19
Hypothesis testing,
  in compliance/corrective action monitoring., 7-2
   confidence interval type 7-13
   false positives and power in compliance mon. 7-8
   false positives and power in corr. act mon.  7-11
   lower confidence limits for compliance  7-4
                                                            A-15
                                                                                                                March 2009

-------
Appendix A—References, Glossary & Index
                           Unified Guidance
          one-way versus two-way errors 7-5
          sequential data pooling  7-13
          upper confidence limits for corr. action 7-5
          use of one way confidence intervals 7-2
        in detection monitoring 2-9, 3-12, 4-3
       I.i.d (independent and identically distributed) 3-4, 6-25
       Independence, statistical  3-4
        in detection monitoring design  6-24
        versus physical independence  14-2
       Interquartile range, definition 3-10
       Intervals, statistical,  6-42
        confidence 6-42,21-1
        prediction  6-43, 18-1
        tolerance 6-44, 17-14
        control chart  6-42,20-1
       Interwell versus intrawell tests,
        in background data  5-6
        in detection monitoring design  6-28
        with prediction limits 19-9, 19-11,19-27, 19-28

       	j  	
       — K 	

       Kaplan-Meier method for non-detect data  8-23,15-7
       Kruskal-Wallis test,
        for determining spatial variability 13-6,  17-9
        one-way detection monitoring test 8-29, 17-9
       Ladder of powers transformations 10-4
       Level a test or level of significance 3-15
       Levene's test for equality of variance 8-14, 11-4
       Linear combination of variables 3-16, 10-6
       Linear regression 8-31, 14-36, 17-23,21-23
       Lognormal data,
        comparison to normal default study 10-7, C-2
        in compliance monitoring design  7-19
        problems with Land UCL 21-9
        t-tests
          two sample 16-10
          versus fixed GWPS 21-3.21-5
       Logarithmic distribution measures, definitions,
        coefficient of variation  3-11
        sample mean 3-11
        sample standard deviation 3-11	
—  M 	

Mann-Kendall trend test 8-32,17-30
MCLs in compliance monitoring 7-6
Mean 3-9
Mean-standard deviation scatter plot  8-15, 11-8
Median, definition  3-10
Method Detection Limit 6-11
Mixture distributions with non-detect data 15-6
Monte Carlo simulations,
  control charts 20-13
  detection monitoring test comparisons  6-23
  normal vs. lognormal  default assumptions 10-7, C-2
  power with lognormal data C-13
Moving window strategy for compliance monit.  7-15
Multiple comparisons problem, 6-2
Multiple non-detect data limits  15-1
     N
Non-centrality parameter, 6-14, 6-21, C-10,
  using R-script  13-10, C-16
Non-detect data
  general considerations 6-36, 15-1
  in compliance/corr.action design 7-18
  in detection monitoring design  6-36
  methods for imputing values
   Cohen's method 8-24, 15-21
   Kaplan-Meier 8-23, 15-7
   mixture distributions 15-6
   parametric ROS 8-24,15-23
   robust ROS  8-23,15-13
   simple substitution  8-21, 15-3, B-5
   test of proportions  B-5
Normal distribution,
  approximation to binomial  22-18
  importance of  10-5
  standard (see Standard normal distribution)
Normality screening methods,
  coefficient of variation 8-12, 10-9, B-l
  coefficient of skewness 8-11, 10-9
Normality, tests of,
  probability plot correlation coefficient 8-13,
   10-16, 10-23
  Shapiro-Wilk n< 50  8-12, 10-13, 10-22
  Shapiro-Francia n > 50 8-12, 10-14
  Shapiro-Wilk group test 8-13,10-19
Null hypothesis 3-12
     O
                                                           A-16
                                                                                                              March 2009

-------
Appendix A—References, Glossary & Index
                           Unified Guidance
       One-tailed versus two-tailed test  7-5
       One-way ANOVA (see ANOVA)
       Optimal rank values calculator 19-26, 19-33
       Outlier screening methods,
        probability plot 12-1
        box plots 12-5
       Outlier tests,
        Dixon's  8-15, 12-8
        Rosner's  8-16, 12-12
       Outliers, swamping 12-11

       	p 	

       Parameter,
        definition, statistical 3-9
        non-centrality, 6-14,6-21,0-10
       Parametric ROS for non-detect data 8-24, 15-23
       Partial ordering of censored data  10-2
       Percentile  3-2
       Plots, design and example,
        box 8-9, 9-6
        histograms 8-10,  9-7
        probability 8-11,9-15
        scatter 8-10, 9-12
        time series 8-9, 9-1
       Pooled variance using ANOVA 13-8
       Population  3-2
       Power,
        detailed calculations for,
          ERPC  6-13
          prediction limits C-12
          upper percentiles 22-18
          using lognormal data C-13
          Welch'st-test C-10
        in compliance monitoring 7-8,  22-1
        in corrective action monitoring  7-11, 22-8
        in detection monitoring,
          effective power 6-13
          EPA reference power curves [ERPC] 6-16
          generating the ERPCs 6-14
          introduction to power curves  6-13
          non-central t-distribution 6-14
        Monte Carlo simulations for control charts 20-13
       Prediction limits, single tests,
        introduction 18-1
        non-parametric future median 8-36,18-20
        non-parametric future values 8-35,  18-17
        parametric future mean  8-34,18-11
        parametric future values 8-33,  18-7
       Prediction limits, using repeat testing,	
  basic subdivision principle  1 9-7
  computing sitewide false positive 1 9-4
  non-parametric Appendix K-tables usage 1 9-27
  non-parametric tests general 19-26
  non-parametric future medians 8-36, 19-31
  non-parametric future values 8-33, 19-30
  parametric tests general 19-11
  parametric Appendix K-tables usage 19-13
  parametric future means 8-34,19-20
  parametric future values 8-35,19-15
  R-script for parametric tests C-16
  strategies  19-1
Probability  3-7
Probability distribution  3-2
Probability plot correlation coefficient,
  method summary 8-13
  test of normality  10-16
Probability plots,
  design  9-15
  outlier screening method 12-1
     Q
Quartile, definition 3-10
Random data, importance of 3-3, 6-24
Rank von Neumann ratio test 8-19,14-18
Ranking of data 10-13 to 10-23
  partial ranking  15-7,16-16,18-6
RCRA regulatory discussions,
  Addendum 1992 guidance 2-4
  general guidance recommendations,
   compliance/assess, and corrective action 2-15
   detection monitoring methods  2-14
   interim status monitoring  2-13
  groundwater protection standards 2-12
  hypothesis tests  2-9
  interim final guidance [IFG]  1989 2-4
  performance standards 2-6
  recent regulatory modifications (2006) 2-5
  sampling frequency requirements 2-10
  statistical methods 2-6
  summary 2-1
Regression (see Linear regression)
Relative frequency distribution, histograms  9-11
Rejection region,
  lower-tailed test 7-5
  upper-tailed test 7-5
Representative sample 3-2 _
                                                           A-17
                                                                                                               March 2009

-------
Appendix A—References,  Glossary & Index
                            Unified Guidance
       Residual, regression 17-25
       Residual analysis  17-28
       Robust ROS method for non-detect data 8-23, 15-13
       Root mean square error with pooled variance 13-8
       Rosner's test for outliers  8-16, 12-12
       R- script,
        intrawell pooled variance 13-10, C-16
        modified California plan  C-l 8
        parametric prediction limits C-16
        Theil-Sen confidence band 21-30, C-20
       Sample correlation coefficient, definition  3-10
       Sample mean, definition 3-9
       Sample percentile, definition 3-10
       Sample standard deviation, definition 3-9
       Sample size recommendations,
        background data 5-2, 5-7
        compliance and corrective action tests,
          sequential pooling  22-6
          corrective action using power criteria  22-9, 22-21
       Sample variability in compliance mon. design 7-17
       Sampling distribution 3-16
       Scatter plots,
        design 9-12
        mean-standard deviation for equality of
        variance screen  11-8
       Seasonal Mann-Kendall test for trend 8-21, 14-37
       Sequential data pooling in compliance mon. 7-13
       Shapiro-Wilk test of normality  8-12, 10-13
       Shapiro-Francia test of normality  10-15
       Shapiro-Wilk group test normality 8-13, 10-19
       Shewhart-CUSUM (see Control charts)
       Simple substitution for non-detect data  8-21, 15-3
       Simultaneous confidence intervals 17-6, 17-12
       Site-wide false positive rate [SWFPR],
        calculations for  6-7, 19-4, 19-7
        development and rationale 6-7
        double quantification rule 6-11
        number of tests and constituents 6-9
        recommended criteria
        subdivision principle 6-7, 1 9-7
       Skewness coefficient 8-11,10-9
       Spatial variability,
        introduction 13-1
        screening methods,
          box plots 13-2
        pooled variance using ANOVA  13-8
        tests,
          one-way parametric ANOVA  8-16,13-5 _
   Kruskal-Wallis (non-parametric) 13-6, 17-9
  use of R-script for pooled variance 13-10, 13-12
Standard deviation, definition 3-9
Standard normal distribution,
  approximations for,
   Mann-Kendall trend test 17-31
   Shapiro-Wilk group test 10-20
  in probability plots 8-11, 9-15, 12-1
  R-script calculations C-16
  table D-l
  used in ERPC estimation 6-5, 6-9
  Z-transform used in tests,
   control charts  20-3
   contrasts for Kruskal-Wallis 17-12
   probability plot correlation coefficient 10-16
   parametric ROS 15-23
   Wilcoxon rank-sum 16-18
Stationarity, 3-5
Statistic 3-2
Statistical Significance 4-5
Subdivision principle for SWFPR calculations 6-7, 19-7
SWFPR (see Site-wide false positive rate)
Symmetric distribution 10-9
Tarone-Ware test detection monitoring 8-27, 16-20
Transformations, distributional,
 importance of normalizing 10-5
 ladder of powers 10-4
 logarithmic 10-6
 other distributions 10-1
t-tests,
  pooled variance t-test for detection monitoring
   8-25, 16-1
  updating background data with  5-12
  Welch's test for detection monitoring  8-25, 16-7
t-confidence intervals for means 7-4,21-3, 21-5
t-distribution table D-l5
Temporal dependence,  general 14-1
Temporal variability,
  corrections for, 6-28
   stationary mean seasonal pattern 8-20, 14-28
   sampling interval with Darcy's eq.  8-20, 14-19
   sampling frequency adjustment 14-18
   temporal effect across wells  14-33
   temporal effect using ANOVA 8-17, 14-35
   temporal effect linear trend 14-36
   trends using seasonal Mann-Kendall test
   8-21, 14-37
  screening methods,	
                                                           A-18
                                                                                                               March 2009

-------
Appendix A—References,  Glossary & Index
                           Unified Guidance
          time series parallel plots 14-3
        tests,
          ANOVA one-way  8-14, 14-6
          autocorrelation function 8-18,14-12
          rank von Neumann ratio test  8-19, 14-18
       Tests of hypotheses (see Hypothesis testing..)
       Test of proportions for non-detect data B-5
       Theil-Sen trend line  8-32,17-34
       Time series plots,
        design 8-9, 9-1
        temporal variability screening  14-3
       Tolerance limits, detection monitoring tests,
          general 8-30, 17-14
          non-parametric tolerance limits 8-30, 17-18
          parametric tolerance limits 8-30, 17-15
       Transformations to normality 10-3
       Trends,
        accounting for in compliance/corr. act. mon.,
          confidence bands 7-16
          moving window 7-15
       Truth table for hypothesis testing 3-16
       Tukey hinges for box plots 3-10
       Two-factor ANOVA 6-40,14-6
       Two-tailed test errors, 7-5
       Type I error (see Hypothesis testing....)
Type II error (see Hypothesis testing....)
Variance stabilizing transformation 11-1
Variation, coefficient of (see Coefficient of variation)
    W
Weibull distribution  3-8,10-1
Welch's test for detection monitoring  8-25, 16-7
Wilcoxon rank-sum test,
  for detection monitoring 8-26,16-14
  updating background data with 5-12
Z-distribution (see Standard normal distribution)
                                                          A-19
                                                                                                            March 2009

-------
Appendix A—References, Glossary & Index                            Unified Guidance
                      This page intentionally left blank
                                       A-20
                                                                         March 2009

-------
Appendix B.  Historical Notes	Unified Guidance

                APPENDIX  B.  HISTORICAL  NOTES
       B.I  PAST GUIDANCE FOR CHECKING NORMALITY	B-2
       B.2  THE CABF PROCEDURE	B-4
       B.3  PAST GUIDANCE FOR NON-DETECTS	B-5
       B.4  TREND TESTS	B-6
       B.5  PREDICTION LIMITS AND RETESTING	B-7
         B.5.1   Retesting Schemes	B-7
         B.5.2   Tolerance Screens	B-9
         B.5.3   Non-Parametric Retesting Schemes	B-10
                                          B-l                                 March 2009

-------
Appendix B. Historical Notes	Unified Guidance

B.I  PAST GUIDANCE  FOR CHECKING NORMALITY

     The 1989 Interim Final Guidance [IFG] outlined three different methods for checking normality:
the coefficient of variation [CV]  test, probability plots, and the chi-square test. Of these three,  only
probability plots are recommended within the Unified Guidance. The coefficient-of-variation and chi-
square tests each have potential problems or are inferior to alternate methods. These alternatives include
the coefficient of skewness, the Shapiro-Wilk or Shapiro-Francia tests, and Filliben's probability plot
correlation coefficient.

     The coefficient of variation [CV] test in the original 1982 RCRA Part 264 groundwater monitoring
regulations was recommended within the  IFG because it is easy to calculate and amenable to small
sample sizes. To ensure that a normal model with a significant fraction of negative concentration values
was not  fit to positive data, the  IFG recommended that a  sample CV be less than one to indicate
'normality.' The test was inexact since the distribution of sample CV's from a truly normal population
itself is a function of both sample size and the true coefficient of variation. Truly normal populations of
positive-valued data are likely to have a  CV of 0.3 or  lower, although individual sample CV's will
occasionally  exceed one, depending  on the  sample size. It was  also possible to incorrectly reject
normality using this criterion even when the population was really normal.

     While the  coefficient  of variation indirectly offers an  estimate  of  skewness  and  hence
normality/non-normality,  there are better formal tests to accomplish both goals. The Unified Guidance
recommends estimating skewness of a data set using the coefficient of skewness (Section 10.4), along
with other tests of normality in  Chapter 10.  Nevertheless, the coefficient of variation provides a
measure  of intrinsic variability in positive-valued data sets.  Although approximate, the coefficient of
variation can indicate the relative variability of certain data,  especially with small sample sizes and in
the absence of other formal tests.

     The CV is also a valid measure of the multiplicative relationship between the  mean and the
standard   deviation  for  positively-valued random  variables. The estimator   CV = s/x  reasonably
approximates  the  true  CV for  non-negative  normal  populations.  In  lognormal populations,  the
coefficient of variation  can also  be used  in  evaluations  of  statistical power.  For the lognormal
distribution, the population coefficient of variation works out to be:
where  ay is the population log-standard deviation. Because of this,  instead of a ratio between the
standard deviation and the mean, the lognormal coefficient of variation is usually estimated by
where  s  is the log-standard deviation. This last estimate is usually more accurate than the simple ratio
of standard deviation-to-mean,  especially when  the underlying population coefficient of variation is
high. However, neither coefficient of variation estimator is a satisfactory test as to whether a data set is
truly normal or lognormal.

                                              B-2                                    March 2009

-------
Appendix B.  Historical Notes
Unified Guidance
     The chi-square test was also recommended within the IFG. Though an acceptable goodness-of-fit
test, it is not considered the most sensitive or powerful test of normality (Can and Koehler, 1990). The
downside to the chi-square test can be explained by considering the behavior of parametric tests based
on the normal distribution. Most tests, like the t-test or parametric prediction limits, which assume that
the underlying data are normal, give fairly robust results when the normality assumption fails over the
middle ranges of the data distribution. That is, if the extreme tails are approximately normal in shape
even if the middle part of the density is not, these parametric tests will still tend to produce valid results.
However, if the extreme tails are non-normal  in shape (e.g., highly skewed), normal-based tests can lead
to false  conclusions, meaning that either a data transformation or a non-parametric technique should be
used instead.

     The chi-square test entails a division of the sample data into 'bins' or  'cells' representing distinct,
non-overlapping ranges  of the data (Figure B-l). In each bin, an expected value is computed based on
the number of data points that would be found if the normal distribution provided an appropriate  model.
The  squared  difference between the expected number and observed number is then  computed  and
summed over all the bins to calculate the chi-square test statistic.

             Figure B-l.  How the Chi-Square Goodness-of-Fit Test Works
                          0.6 -

                          0.5 -

                          0.4 -

                          0.3 -

                          0.2 -

                          0.1 -

                          0.0 -
     If the chi-square test indicates that the data are not normal, it may not be clear what ranges of the
data most violate the normality assumption. Departures from normality in  the middle bins are given
nearly  the same weight as departures in bins representing the extreme tails, and all the departures are
summed together to form the test statistic. As such, the chi-square test is not as powerful for detecting
departures from normality in the extreme tails of the distribution, the areas most crucial to the validity of
parametric tests like the ^-test or ANOVA (Miller,  1986). This implies that if there are departures in the
tails, but the middle portion of the data distribution is approximately normal,  the chi-square test may not
register as statistically significant even when better tests of normality would.
                                              B-3
        March 2009

-------
Appendix B.  Historical Notes	Unified Guidance

     The IFG  also suggested that the original data should be presumed to be normal prior to testing the
distributional assumption. If a statistical test rejected the model of normality, the data could be checked
instead for lognormality by evaluating their natural logarithms. The 1992 Addendum to Interim Final
Guidance  [Addendum]  noted  that many data sets in environmental monitoring are better  fit  by a
lognormal than by a normal distributional model. Primarily on that basis, it was recommended that the
lognormal  distribution replace the  normal as the default model  for groundwater analysis, especially
since for small data sets, the available tests of normality have limited statistical power to reject the null
hypothesis of normality,  even  if the  data arise from  a lognormal distribution. The Unified Guidance
brings this argument around almost full circle by arguing that the normal model is a slightly better
default for small samples, but that distributional testing is recommended in any case in  order to establish
the most appropriate model (Section 10.3).

B.2 THE CABF PROCEDURE

     Facilities operating under a RCRA permit specifying Cochran's Approximation to the Behrens-
Fisher Student's ^-test [CABF]  may change this method to a more appropriate procedure at the time of
State or Regional permit review and  update. Owners and operators may also apply for  a permit
modification under §270.41 (a) (3). This change is considered a Class 1 permit modification, which must
be made with prior approval from the Director.1  Depending on the nature of the permit conditions, it
may also be appropriate, on a facility-specific basis,  for an oversight agency to approve a change of
method without a formal permit modification.

     Under appropriate  circumstances,  an owner or operator may wish to continue using a t-test type
procedure.  However, instead of the CABF method, it is recommended that either a pooled  variance
Student's f-test or  a variant of this test due to Welch (1937) be employed (Chapter 16). Not only is
Welch's test a more standard type of f-test than the CABF procedure, but research has shown it to be
equivalent or preferable to other varieties of the f-test (Moser and Stevens,  1992).

     Circumstances appropriate for the use of a t-test procedure might include facilities with very few
monitoring wells (e.g., three or less)  and that monitor for a very  limited number of constituents  (e.g.,
one or two). As long as no more than 5 to 10 statistical comparisons are being made each year, running a
^-test at the 0.01 level of significance in each case should result in at most a 10% annual probability of
any comparison registering as a false positive when there is no actual contamination.

     One  of the problems with the CABF procedure in practice was the use of aliquot replicate samples
to bolster the total  sample size  (Section 2.2.4). Both the pooled variance ^-test and Welch's t-test make
the assumption that the  sample  observations are statistically independent. Though  aliquot  replicate
sampling increases the number of  available measurements, aliquot replicate samples mostly provide
information about analytical variability and accuracy, and tend to be highly correlated. Since the goal of
a RCRA groundwater statistical program is to provide data about hydro-geochemical variability in the
(uppermost) aquifer below the facility, aliquot replicate  sampling (like the CABF  procedure itself)
should be  avoided unless a more sophisticated components of variance model is used to account for the
separate effects of analytical variability and natural groundwater variance.
  See 53 FR 37912, September 28, 1988 for more details about the permit modification process.

                                              B-4                                    March 2009

-------
Appendix B.  Historical Notes	Unified Guidance

B.3  PAST GUIDANCE FOR NON-DETECTS

     Guidance for handling non-detect  measurements was first offered in  the  1989 Interim Final
Guidance [IFG]. There the  basic recommendations included the following:  1) if  less than 15% of all
samples are non-detect,  replace each non-detect by half its detection or quantitation limit [QL]  and
proceed with a parametric analysis, such as ANOVA,  tolerance limits, or prediction limits; 2) if the
percentage of non-detects is between 15 and 50, either use Cohen's adjustment to the sample mean and
variance in order to proceed with a parametric analysis,  or employ a non-parametric procedure by using
the ranks  of the observations and treating all non-detects as tied values; 3) if the percentage of non-
detects is greater than 50, use the test of proportions.

     In the 1992 Addendum to Interim Final Guidance [Addendum],  the recommendation for data sets
with small fractions of non-detects (i.e., < 15%) was left unchanged; however, for cases with moderate
detection  rates (i.e.,  non-detects comprising  15% to  50%  of the  data),  Cohen's  adjustment was
supplemented  by Aitchison's method  for data sets in  which non-detects could  be  regarded as zero
concentrations. In addition,  the test of proportions was  deleted from the Addendum. Instead, for large
fractions  of non-detects, three options  were suggested:  1) for two sample comparisons, the Wilcoxon
rank-sum  test  was recommended over the  test of proportions; 2) for moderately large background
samples,  the  Addendum recommended  non-parametric prediction and  tolerance limits;  and 3) for
extremely low detection  rates (e.g., > 90% non-detects)  and small background samples, the Addendum
recommended the use of Poisson prediction and tolerance limits.

     The  test  of proportions was not recommended in  the Addendum, even  for detection rates under
50%, for the following reason. Although acceptable as a statistical procedure, the test of proportions
does not account for potentially different  magnitudes  among the concentrations of detected  values.
Rather,  each sample  is treated essentially as a  '0'  or T  depending on whether  the measured
concentration is below or above the QL. The test of proportions ignores information about concentration
magnitudes, and hence is often less powerful than a non-parametric rank-based test like the Wilcoxon
rank-sum, even after adjusting for a large fraction of tied observations  (e.g., non-detects). In part, this is
because the ranks of a data set preserve additional information about the relative magnitudes  of the
concentration values, information which is lost when all  observations are scored as O's and 1 's.

     Furthermore,  small-scale Monte Carlo simulations  comparing  the test of proportions to the
Wilcoxon rank-sum test showed that for small to moderately large proportions of non-detects (say 0% to
60%), the  Wilcoxon rank-sum procedure  adjusted for ties was  more  powerful in identifying  real
concentration differences than the test of proportions.  When the percentage  of non-detects was quite
high (at least 70% to 75%), the test of proportions was  occasionally more powerful than the Wilcoxon
for extremely small group sample sizes  (e.g., no  more  than 4 to 6 measurements per group), but the
results of the two tests usually led to the same  conclusion. Consequently, the  Wilcoxon rank-sum test
was recommended in all cases where non-detects constituted more than 15 percent of the samples.

     The revised Unified Guidance also places  less emphasis on Cohen's method. The reason is that it
could only accommodate a single censoring limit (e.g.,  reporting limit  [RL]) in its original formulation
and  assumed  that all  quantified  values were necessarily greater  than this limit. Because  many
environmental data sets include multiple reporting and/or detection limits and an intermixing of detects
and non-detects, two  other methods are now recommended that are designed to handle more complex
data configurations (Chapter 15).   Cohen's  and the parametric ROS method may have  limited

                                             B-5                                   March 2009

-------
Appendix B. Historical Notes	Unified Guidance

applicability  when  both  detect and non-detect data are  expected  to stem  from a single parametric
distribution and a single censoring limit can be used.

B.4  TREND  TESTS

      The Unified Guidance recommends trend testing as an alternative to prediction limits or control
charts when those methods are not suitable. To understand the basis for this recommendation, it may
help to  consider how intrawell comparisons initially supplemented, and then came in many cases to
supplant, interwell comparisons.

      In the 1989 IFG and the  1992 Addendum, the  recommended statistical methods closely followed
the 1988  and 1991  Final Rules published in the  Federal Register. Although these methods  replaced
historical  use of the  CABF Student's ^-test,  there  was  still an emphasis  on interwell comparisons
between background and downgradient wells through the use of t-tests and ANOVA. Indeed, where
justified, interwell comparisons  provide undeniable conceptual  advantages  over  other kinds of tests.
When (upgradient)  background measurements  can  be  used   to  establish a  reasonable  baseline
concentration level,  such  data offer   invaluable  information  about  site-specific   conditions  at
uncontaminated locations and the level of variability one should expect to encounter in the absence of
events that precipitate groundwater contamination.

      Unfortunately, ANOVA and t-tests  all involve a comparison of population means under the key
assumption that the  populations have not changed over time. The underlying distributions in each group
or well  are assumed to be stable over the  period of monitoring, so that concentration measurements
fluctuate randomly around a constant mean level. Stability, of course, is not guaranteed. Several factors
can impact the statistical  characteristics of the  underlying aquifer at either upgradient or downgradient
wells, including natural  fluctuations in  aquifer parameters, migration  of contaminants from off-site
sources, changes in the mixture of deposited waste and its geochemical interaction with the subsurface
environment, and alterations in geochemistry from 'percolation' effects due to past  waste  disposal
practices or land usage.

      EPA's hope in the 1989 IFG was that ANOVA-type comparisons would be done quickly enough
(e.g.,  every six months)  that the underlying populations could be considered essentially static during
each testing period.  At some sites, this may be a reasonable assumption. However,  in practice, sampling
is now done  on  a quarterly, semi-annual, or annual basis. In order to gather the  four to five samples
needed— at  a minimum  — to  run a ^-test or ANOVA, at least  one to four years  of  sampling is
necessary. Over this length of time, the statistical characteristics of groundwater may or may not change.

      Furthermore,  interwell comparisons between upgradient and downgradient well locations are not
always appropriate,  either due  to natural spatial variability, screening of background and downgradient
wells in different hydrostratigraphic positions, effects of groundwater mounding, etc. In such cases, the
appropriate statistical approach is to use  an intrawell test at each compliance location.  Intrawell tests
involve  a comparison only of data collected at that specific well location,  thus  eliminating  spurious
differences that  might arise due to natural spatial variability  or  other background-to-downgradient
differences not attributable to the presence of contaminated groundwater.

      Two basic  intrawell techniques are described in the Unified Guidance:  intrawell prediction  limits
and  control  charts.  Both  designate some  portion of the  historical sampling  record as intrawell
'background' for that well. Ideally, this intrawell background should consist of measurements known to
                                              B-6                                   March 2009

-------
Appendix B.  Historical Notes	Unified Guidance

be uncontaminated. Furthermore, both methods assume (unless special adjustments are made) that the
intrawell background represents a random sample from a stable population, just as with the  t-test and
ANOVA. If the population mean and/or standard deviation change while intrawell background is being
compiled, results of either prediction limit or control chart tests against more recent data from the well
can be severely biased or altogether inaccurate.

     For  these  reasons,  neither  prediction  limits  nor control charts  are  appropriate  for  every
circumstance where an intrawell  test is warranted. The Unified Guidance recommends trend  testing as
an alternative  to prediction limits or control charts when those methods are not suitable as intrawell
techniques (Chapter 17). Tests for trend are specifically designed to identify groundwater populations
whose mean concentration levels are  not stable over time,  but rather are significantly increasing (or
decreasing).

B.5  PREDICTION LIMITS AND RETESTING

B.5.1 RETESTING  SCHEMES

     Since roughly  1987, several  different  retesting  schemes  have been suggested  in  regulatory
documents or published in scientific literature. Classification of these schemes shows that they fall into
three basic types: l-of-m, California, and tolerance screens. The  l-of-m approach was initially suggested
by Davis and  McNichols  (1987) as part of a broader method termed 'p-of-m.' Essentially the p-of-m
approach assumed  that as many as m  observations would be collected for a particular constituent at a
given well, including  the initial groundwater measurement and up to (m-l) resamples.  As long as at
least p of these observations were below a predetermined upper prediction limit, the constituent would
'pass' the test at that well, allowing detection monitoring to continue.

     Davis and McNichols  determined how to calculate the necessary prediction  limits so that the
overall  false  positive  rate would  remain below a fixed value (say 5%, as targeted in  the  1992
Addendum), even when the same testing procedure was applied over many different testing periods (r in
their terminology). By applying the same  technique to r different well-constituent pairs (and assuming
mutual  statistical independence among constituents and  compliance wells) instead  of to r different
testing or evaluation periods, one then has a retesting scheme that can be applied at a large variety of
monitoring networks  while  ensuring  that the site-wide false positive rate  [SWFPR] is  kept to  a
minimum.

     In practice, though the p-of-m strategy provides a great deal of flexibility in designing a retesting
scheme, only those schemes known as l-of-m are typically useful in the current regulatory context of
groundwater monitoring. Consider, for example, a 2-of-3 strategy. By definition, if at least two of three
groundwater samples are below the upper prediction limit (i.e.,  are 'in-bounds'), the constituent passes
and is not flagged  as exceeding background. Since at least two samples must be  'in-bounds,' it is not
enough  to collect one initial groundwater measurement and show that it is below the prediction limit. At
least one additional resample must always be collected and measured,  l-of-m  strategies, by contrast,
only require a  single groundwater observation to pass. If the initial measurement is below the prediction
limit,  the constituent passes the overall test and no resamples need be collected.

     The second retesting scheme, known as California-style plans, was suggested partly in response to
perceived problems with  the  l-of-m  plans. California regulators noted, for  instance,  that a  l-of-3
retesting scheme would allow a constituent in a given well to pass even if both the initial groundwater
                                              B-7                                    March 2009

-------
Appendix B.  Historical Notes	Unified Guidance

measurement and one of the two retests exceeded the predetermined prediction limit. The only way for
that well-constituent pair to fail would be if all three measurements — the initial and the two resamples
— exceeded the prediction limit. To many regulators (and not just those in  California)  the  l-of-m
scheme appeared to practically guarantee that contaminated wells would go unidentified, 'passing' the
test each time and undermining protection of human health and the environment.

     In 1991, California received explicit approval from EPA  to use an alternate retesting scheme
constructed as follows. For each well-constituent pair, collect an initial groundwater observation. If this
initial measurement is in-bounds (i.e., below the prediction limit), the test for that pair passes and no
resamples need be collected. If the  initial  measurement exceeds  the prediction limit, two  or possibly
three  resamples must be collected  and each must be in-bounds for  the test to  pass. If  any of the
resamples exceeds the prediction limit (i.e., is 'out-of-bounds'), the test fails and possible groundwater
contamination is indicated.

     The  California strategy was seen as a more environmentally 'conservative'  approach to retesting.
An initially high  groundwater measurement would only be deemed 'spurious' if all the  subsequent
resamples were below the target prediction limit, providing at least double reconfirmation that the well
was 'clean' for that constituent. Unfortunately, the  more stringent requirements of the California plans
came  with unexpected  consequences. A  California retesting plan typically requires a larger target
prediction limit (or 'trigger level') than a l-of-m plan with a comparable number of resamples, in order
to achieve the same overall  SWFPR. Since a larger trigger level corresponds to a less  statistically
powerful test, a given California plan may or may not have adequate effective power even if a similar 1-
of-m plan  does.

     The  net result is that l-of-m retesting schemes often provide greater statistical power for detecting
real groundwater contamination, particularly in large networks, even though not every resample need be
below the prediction limit. If the trigger level is low enough, at least one of the resamples may exceed
the prediction  limit even when there is no contamination. So these cases should not automatically be
classified  as verified contamination.  Conversely, a lower prediction limit increase the odds (i.e., power)
that truly contaminated groundwater will be  identified, since  both the initial observation and any
resamples will be more likely to exceed a lower trigger level than one set to a higher benchmark.

B.5.2 TOLERANCE SCREENS

     A final type of retesting scheme might be termed the tolerance screen approach. First suggested by
Gibbons (1991b), this approach was  modified and recommended by EPA in the 1992 Addendum, but —
for reasons discussed below — is not recommended  within the Unified Guidance. In contrast to the 1-of-
m and California-style plans, which make use of repeated prediction limits as the trigger levels, the
tolerance screen involves a two-stage testing procedure as follows. An initial groundwater measurement
is collected from  each well in the network and compared to an upper tolerance  limit with specified
coverage and confidence levels. If any measurement exceeds the tolerance limit, one or more resamples
are collected from that well and these measurements are compared against an upper: prediction limit.

     Other than the use of a tolerance limit instead of a prediction limit as the 'screen' for the initial
groundwater measurement, the rules for passing the test are the same as  a modified California approach
described  in Section  19.1. Either the first observation  must be below the  tolerance limit (i.e., 'in-
bounds') or q-of-(m-l)  resamples must  be below the prediction  limit.  If both of these conditions are
violated, possible groundwater contamination is indicated.
                                              B-8                                    March 2009

-------
Appendix B.  Historical Notes _ Unified Guidance

     The  use  of two  separate trigger levels (i.e., tolerance limit and prediction limit) for the initial
observation versus the resamples may seem an  unnecessary complication in developing a retesting
procedure. However, there are two advantages to this approach. For one, the  tolerance and prediction
limits are computed on the same background data and both these calculations are done prior to any data
comparisons. Secondly, by allowing two different trigger levels, greater flexibility is gained in designing
— for a given sized network of comparisons — a retesting scheme that meets a target SWFPR.

     Gibbons'  (1991b) original tolerance screen approach advocated constructing a 95% confidence
tolerance limit with a degree of coverage that would vary depending on the network size. For 100 tests,
Gibbons reasoned that a tolerance limit with 95% coverage would result in as many as 5 exceedances of
the initial  trigger just by chance (i.e., even when no contamination was present). Any such exceedance
would then require that a resample be collected at that well and compared to a prediction limit with 95%
confidence for the next 5 future samples (m = 5), in order to maintain an overall 5% SWFPR. The same
type  of false positive rate control could be achieved by setting the  degree of coverage to 99%, so that
only 1 exceedance would be expected in 100 tests against the tolerance limit. In this case, the prediction
limit would be computed with 95% confidence but m  = 1 instead. In all cases, the number of future
measurements  (m) being predicted  would equal the number of measurements possibly expected to
exceed the tolerance limit just by chance.

     To offer  even greater flexibility, EPA recommended a modification to Gibbons' tolerance screen
within the 1992 Addendum. To understand why a modified version was adopted, note that the formula
for an upper prediction limit on the next m future samples with fl - aj confidence may be expressed as
follows:
Careful examination of this formula shows that the effect of changing the number of future samples m
for a given confidence level is equivalent to changing the confidence level associated with a prediction
limit for a single future observation (m= 1).

     Because of this, EPA suggested three alterations to Gibbons' original scheme: 1) instead of fixing
the level of confidence and varying the number of future samples m,fixm= 1 and allow the confidence
level of the prediction limit to vary; 2) allow more than one resample per comparison up to a practical
maximum of three; and 3) use a tolerance limit with average coverage instead of minimum coverage.
While  Gibbons offered power comparisons with his scheme against either a single tolerance limit or a
single  prediction  limit, the  Addendum  offered recommended choices of degrees  of coverage and
confidence levels that  would simultaneously limit  the SWFPR  to  approximately  5%  and generate
effective power at least as high as the EPA reference power curve.

     Unfortunately, as Davis and McNichols (1994) noted, the Monte Carlo simulations used in the
Addendum to generate recommended retesting plans based on the tolerance screen approach were partly
flawed. Two criticisms were particularly  relevant. First, Davis and McNichols noted that the Appendix
to the Addendum  spoke of networks in terms of number of wells rather than the number of tests. Since
the total number  of tests is  a product of the number of wells and the number of constituents being
                                             B-9                                    March 2009

-------
Appendix B.  Historical Notes	Unified Guidance

monitored in each well, they suggested that the Addendum recommendations for retesting plans might
elevate the SWFPR above 5% (the recommended per-evaluation rate in 1992).

     The reason is that if a particular plan in the Addendum was only applicable to a single constituent
(albeit across a large number of wells), a similar but separate plan would be needed for each constituent.
This in turn would imply that  the target  overall false positive  rate of 5% would  only apply per
constituent,  meaning that  tests for many constituents in the  same network  would lead to  a  sharply
elevated SWFPR. Of course, the  text of the Addendum clearly spoke of tests as a combination of wells
and constituents. Still, Davis and McNichols were correct to note that some may have misunderstood the
contextual meaning of the phrase 'wells' in the  Appendix and  also  in the table on non-parametric
retesting  strategies, which was  naively used as  a simple shorthand for  the more awkward 'well-
constituent pairs.'

     A second criticism related to the algorithm used to simulate  the effective power of the tolerance
screen plans. Davis and McNichols correctly observed that while effective power was defined in terms
of a single well contaminated by a single constituent, the power  curves illustrated in the Addendum
Appendix mistakenly added those cases where the contaminated well failed the overall testing procedure
to those where uncontaminated wells failed the procedure  (i.e., instances of false positives). The net
effect was to  slightly  raise the  stated power above the actual power,  especially  at lower standard
deviation shifts in the mean level above background (e.g., 0 < A < 2). As a result of this criticism, all
calculations in the  Unified Guidance with respect  to  retesting  plans have been divided into  two
components: 1) computation of the SWFPR based on the total number of tests, taken as a product of
wells times constituents, and 2) computation of the effective power based on a single contaminated well-
constituent pair.

     A third criticism can now  be added to those offered by Davis and  McNichols. Given  a fixed
background  sample, one drawback to  both  l-of-m and California-style plans is that they have limited
flexibility when it comes to controlling the  SWFPR below a target level (e.g., 10%) over a variety of
network sizes. In some cases, sufficient  false positive  rate control  and  adequate power can only be
achieved by switching, say, from a l-of-2 plan to a l-of-3 plan, or from  a  California plan to a l-of-m
scheme, or by  increasing the background sample size. The problem is that using the same trigger level
— here a prediction limit — to test both the  initial measurement and any resamples restricts the number
of simultaneous tests  that can be accommodated. The  EPA tolerance screen approach  uses different
trigger levels at each stage, allowing greater manipulation of the statistical parameters used to construct
the tolerance and prediction limits and ultimately more  flexibility in designing a retesting scheme that
can meet a target SWFPR for a fixed background size over a wide variety of networks.

     Despite this advantage,  new research  done in preparing the  Unified Guidance indicates that the
effective  power of any tolerance screen retesting procedure is always less than a comparable  scheme
based on a single repeated trigger value. The gain in flexibility in controlling false positive rates is real,
but the most powerful retesting procedures will be of the l-of-m or modified  California-style varieties.
Because of  this loss in effective power, the Unified Guidance recommends  an appropriate  l-of-m or
modified California-style plan (Chapter 19).

B.5.3 NON-PARAMETRIC RETESTING SCHEMES

     In the Addendum,  two basic  approaches  to  non-parametric  retesting were described,  each
suggested by Gibbons (1990;  1991a). Both of these strategies  defined the upper prediction limit as the
                                             B-10                                   March 2009

-------
Appendix B.  Historical Notes	Unified Guidance

maximum observed background value. Once in hand, one new observation was collected from each
downgradient  well  and compared against the non-parametric prediction limit. Measurements  that
exceeded  the  prediction  limit were then retested. In his  1990 article,  Gibbons presented tables of
approximate network-wide significance levels for the case  of l-of-m  retesting plans. Gibbons'  1991
article detailed the more stringent non-parametric California plans, giving exact false positive rates, but
only in the case where the prediction limit was defined as the maximum of background.

     Both of  these efforts were superseded by Davis and  McNichols (1994),  who give  exact false
positive rates are given for both l-of-m and strict California retesting strategies. In addition, Davis and
McNichols compute these false positive rates when the non-parametric prediction limit is taken as either
the maximum  background value or the second-largest background concentration. The latter calculation
is helpful  in two ways. First, if a particular background concentration is unusually high and possibly an
outlier,  one could  choose to fix the non-parametric prediction  limit as the  second-highest (and
presumably  more  representative)  background  concentration. The statistical characteristics of the
retesting scheme would still be assured without having to 'throw out' the suspected background outlier.
Secondly, the  statistical power  of prediction limits based on the second-largest  background value  is
greater than for those prediction limits based on the maximum. For large background samples (n), use of
this alternate prediction limit may be  the only option at some sites to achieve both the targeted false
positive rate and sufficient effective power.

     While  the tables in the Davis and McNichols article are extremely useful, they do not include
results for the  modified California retesting scheme with m =  4, in which either the initial measurement
or two of three resamples must be in-bounds. To complete the tables needed for the Unified Guidance, a
variation of the Davis and McNichols algorithm was initially used to calculate the significance levels of
the modified California retesting scheme with m  =  4. Since  that time, Davis and McNichols (1999)
published an  exact algorithm not only for  the l-of-m  and strict  California  plans, but also for the
modified California plan first suggested in an earlier draft  of the  Unified Guidance. Following their
algorithm with some  minor computational  adjustments,  the  Unified  Guidance  tables  have  been
recomputed, covering first the non-parametric l-of-m plans and  then the non-parametric modified
California plan (Chapter  19). In each case, results are provided for non-parametric prediction limits
taken either as the maximum value in background or as the second-largest concentration.

     To measure the statistical power of these non-parametric retesting strategies, Davis and McNichols
estimated power using Monte Carlo simulation with  normally-distributed random variates. They then
offered  a new measure of power labeled the Modified Addendum Criterion or MAC, which  rated
schemes against an EPA reference power curve with n = 8  background samples. Recognizing that a
particular power curve might only exceed the EPA reference power curve at large mean concentration
shifts (A)  above background, the MAC evaluated at what percentage power a  proposed scheme did in
fact begin to exceed the EPA reference power curve (e.g., starting at 30% power, or 50% power, etc.).

     In the  Unified Guidance, effective power of non-parametric  retesting schemes is measured in a
similar,  though not  identical, manner. One  difference is that the recommended EPA reference power
curves are based  on  10  rather than  8 background  samples. With n  =  8, there is less than a 50%
probability of identifying a mean concentration increase above background of 3 standard deviations and
less than an 80%  chance of identifying an increase of 4 standard deviations. Another mostly semantic
difference is that the schemes in the Unified Guidance are evaluated on whether or not they exceed the
EPA reference power curve for concentrations exceeding background  by a given number of standard

                                             B-ll                                    March 2009

-------
Appendix B.  Historical Notes	Unified Guidance

deviation units (e.g., 3 or 4 standard deviations),  instead of at a particular power percentage (e.g., 30%,
70%, etc.).

     To actually compute effective power,  Monte Carlo simulations were not utilized in the Unified
Guidance. Rather, since the underlying data were assumed to be normal, a simple modification to the
numerical integration algorithms presented  in Davis and McNichols (1987)  was used to compute the
power directly. Of course, if the data are normal  in the first place, a parametric retesting scheme would
be more appropriate. Non-parametric strategies should only be considered when the data appear to be
distinctly non-normal or exhibit too many non-detects to judge normality.  Nevertheless, since the true
underlying distribution  is unknown, the usual method of attack is to measure the statistical power that
results when the underlying distribution is taken to be normal.
                                             B-12                                   March 2009

-------
Appendix C. Technical Appendix	Unified Guidance

                APPENDIX C. TECHNICAL APPENDIX
       C.I   SPECIAL STUDY: NORMAL vs. LOGNORMAL PREDICTION LIMITS	2
         C.I.I   Results For Normal Data	2
         C.I.2   Results for Lognormal Data	4
       C.2   CALCULATING STATISTICAL POWER	10
         C.2.1   Statistical Power of Welch's T-Test	10
         C.2.2   Power of Prediction limits for future mean vs. Observations	12
         C.2.3   Computing Power with Lognormal Data	13
       C.3   R SCRIPTS	16
         C.3.1   Parametric Intrawell Prediction Limit Multipliers	17
         C.3.2   Theil-Sen Confidence Band	20
                                            C-l                                   March 2009

-------
Appendix C.  Technical Appendix	Unified Guidance

C.I  SPECIAL  STUDY:  NORMAL VS. LOGNORMAL PREDICTION LIMITS

     Section 10.3  outlines the strategy for distributional testing in the Unified Guidance. Among these
recommendations is that the  normal distribution should be treated as a default model until specific
testing indicates otherwise. To establish this recommendation,  a special study was conducted for the
Unified Guidance to answer two key questions: 1) what are the consequences of incorrectly applying
statistical  techniques  based  on one  distributional  assumption  (normal or  lognormal),  when  the
underlying distribution is, in fact, the other? and 2) what is the impact on statistical power and accuracy
of assuming the wrong underlying distribution? These questions were tested for prediction limit tests in
detection monitoring (and, by extension, for control charts).

     The  general effects of violating test assumptions  can be measured in terms of false positive and
negative error rates (and therefore power). A series of  Monte Carlo simulations was generated for the
Unified Guidance  to  evaluate the impacts on prediction limit false positive  error rates and statistical
power of using normal and lognormal distributions when applied  either correctly or incorrectly to the
underlying 'true' distributions. For varying inputs of background sample size,  population coefficients of
variation and confidence levels, sample data sets were  generated and prediction limits computed for a
single future observation using either a normal prediction limit [NorPL] or a lognormal prediction limit
[LgnPL], as given in the  equations below,  x and sx are the mean and standard deviation respectively of
the original measurements, while y and s  represent the log-mean and log-standard deviation:


                                 NorPL  = x + t ,,  s Jl + —
                                       I-a       n-l,l-a x\l
                                     -a = exP
     To evaluate prediction limit performance, for each choice of inputs and statistical parameters, one
million (N = 1,000,000) simulated normal background data sets and one million lognormal background
data sets were generated and tested against each limit. When the  underlying distribution was normal, a
fixed unit  standard deviation was coupled with a series of increasing mean levels to vary the population
coefficient of variation. Then, to measure power in each case, new measurements were generated  from
similar normal  models  with  mean  levels incremented  by  k  standard  deviation  units  above the
background mean, for k ranging from 0 to 5. A parallel evaluation was also conducted when a retest was
added to the procedure. In this case, the prediction limits were constructed using the K multiples for a 1-
of-2 retesting scheme as described in Chapter 19. A summary of these results is given in Figure C-l.

C.I.I RESULTS FOR NORMAL DATA

     If the underlying population is truly normal, treating the sample data as lognormal in constructing
a prediction limit can have significant consequences. Figure C-l presents  key results, either averaged
over all the statistical input parameters or broken down by sample size, confidence level, and coefficient
of variation. These statistics include the average  ratio between the normal prediction limit [NorPL] and
the lognormal prediction limit [LgnPL], the average difference between the nominal (i.e., expected)  false
positive rate (a) of the test and the observed false positive  rate, and the average percentage of cases

                                             C^2                                    March  2009

-------
Appendix C.  Technical Appendix	Unified Guidance

where two statistical power targets were met, those being 50% power at 3 standard deviations above the
background mean and 80% power at 4 standard deviations above the background mean.

     With  no retesting and  truly normal data, the  lognormal prediction limits were in every case
considerably longer and thus less powerful than the normal  prediction  limits.  The discrepancies in
performance were smallest for larger sample sizes, lower confidence levels, and smaller coefficients of
variation. However, in only one of the category breakdowns (1-a =  0.995) did the normal prediction
limits fail to meet both power targets at least half the time, while the  lognormal limits jointly met both
power targets  less than half the time in all  cases except one  (1-a = 0.90). As to  false positives, the
lognormal limits consistently exhibited  less than the expected (nominal) false positive rate. The normal
prediction limits tended to have slightly higher than nominal error rates.

     When retesting was  added to the  procedure, the performance of both limits improved. The false
positive rates  of both were  closer to  the nominal rates, though the normal prediction limits were
relatively closer to the expected rates. While  power improved across the board compared to not using a
retest, the normal limits were on average about  13% shorter than the lognormal limits, leading again to a
measurable loss of statistical power for the lognormal prediction limits. Particularly noticeable was the
significant  difference  in power at higher confidence levels, the very kinds of confidence levels needed
when designing retesting strategies for multiple tests against a prediction limit.

     On balance, misapplication of logarithmic prediction limits to normal data consistently resulted in
lower power  (often considerably) and  false  positive rates that were lower than expected, unless the
population  coefficient of variation was quite small,  the background sample size was larger, and the
confidence level more moderate. Since a  lognormal prediction limit will  be applied only  if the
underlying population is thought or assumed to be lognormal,  it is helpful to  gauge how these factors
work in practice. On one hand, the higher confidence levels and consequently lower a values needed for
retesting strategies with simultaneous tests (Chapter 19) would argue  against presuming the underlying
data to be  lognormal  without specific goodness-of-fit testing.  In  other words, if the data are actually
normal but the lognormal prediction limit is misapplied, a high price in statistical power may be paid.

     In terms  of sample size, the greatest penalties from misapplying  lognormal prediction limits occur
for smaller background sizes.  Since goodness-of-fit tests are least able to distinguish between normal
and lognormal data with small samples, small background samples be not  be presumed to be lognormal
as a default unless other site-specific evidence suggests otherwise.  For larger sample sizes, goodness-of-
fit tests have much better discriminatory power, enabling a better indication of which model to use.

     With  regard to  coefficient of variation  [CV], the guidelines are less clear cut.  Given  that
groundwater data are generally positive in value, truly normal populations are likely to have population
coefficients of variation of 0.3 or lower.  Larger coefficients of variation would  result in a significant
fraction of negative measurements. In addition,  the probability of observing a large sample coefficient of
variation from a normal population with population coefficient of variation of 0.3 or less is rather small.

     However, the measurement and censoring of small concentration values complicates the picture.
Such values are measured below a reporting  limit  [RL]  and are generally listed as 'less thans.' A
measurement process  that is normal with high  coefficient of variation and mean close to the RL can
generate a mixture of left-censored and detected values with  fairly high coefficient of variation yet not
be lognormal.  In fact, the cases in Figure C-l with  higher  coefficients of variation were analyzed in
essentially  this fashion, with negative values  imputed to a small,  positive  reporting  limit  prior to
                                              C-3                                     March 2009

-------
Appendix C.  Technical Appendix	Unified Guidance

calculation of the prediction  limits.  The results  indicate a substantial loss of performance  when
lognormal  limits  are misapplied to these left-censored normal data sets, with or without retesting.
Therefore, the observed CV should not be used as the sole criterion of whether to presume an underlying
normal  or  lognormal  data model.  Rather, if large fractions  of censored data  are present,  censored
probability plots (Chapter 15) should be constructed to aid in choosing an appropriate distribution.

C.I.2 RESULTS FOR  LOGNORMAL DATA

     Do normal-based prediction limits suffer in a  similar comparison when the  underlying population
it really lognormal? The  results from applying  normal and lognormal prediction limits to underlying
lognormal data are presented in Figure C-2. There, the summaries are similar to Figure C-l with one
important  exception.  As explained in  Chapter  10  and Appendix Section  C.2,  the lognormal
distribution is not an additive model.  Because of this fact, the  distributional alternatives  used  in
assessing  the  statistical  power of  a lognormal-based  prediction  limit  usually involve  setting the
alternative  mean to a multiple of the background mean while keeping a constant lognormal coefficient
of variation.

     The net effect is that the power of lognormal-based tests depends greatly on the actual level  of the
coefficient of variation. This is different from normal-based power analyses, where  the coefficient of
variation only plays a role in terms of the degree of censoring in the data (thus affecting power through
the handling of left-censored values, i.e., non-detects). Because the achievable power varies over such a
large range — depending on the level of skewness of the specific lognormal distribution — reference
statistical power for lognormal models must be tied  to the observed background coefficient of variation.
However, since a performance comparison across coefficient of variation levels was needed for the
results of Figure  C-2, a single benchmark was used to assess the comparative power of the normal and
lognormal prediction limits. While imperfect for practical use, this benchmark was set at 25% power for
alternatives of three times the background mean and 50% power at five times the background mean.

     For an underlying lognormal model with no  retesting, Figure C-2 indicates that while the false
positive rates of lognormal-based prediction limits are essentially as advertised (i.e., a 95% confidence
prediction limit has close  to the nominal  5% false positive rate), the false positive rates of normal-based
limits are higher than expected, often substantially, especially for higher confidence  levels and higher
coefficients of variation.  The  most significant  drawback to misapplying normal prediction limits  to
lognormal  data would then be an excessive  site-wide  false positive rate from using such limits  on
multiple well-constituent  pairs.

     However, the situation changes dramatically with the addition  of even a single retest. In this case,
the lognormal prediction  limits are still more  accurate than the normal limits, in terms of having false
positive rates closer to the  nominal targets.  Nevertheless,  with the  added retest, the achieved false
positive rates for  the normal limits tend to be less  than  the expected rates, especially  for moderate to
larger sample  sizes.  In addition,  except for  very  skewed lognormal distributions, the power of the
normal limits is comparable or greater than the power of the lognormal limits.
                                              C-4                                    March 2009

-------
  Appendix C. Technical Appendix
Unified Guidance
    Figure C-l. Accuracy and Power of Normal vs. Lognormal  Prediction Limits When
                                 Underlying Data Are Normal

No Retesting, 1-of-l Scheme
Category
ALL
N 4
8
12
16
(1-oc) 0.900
0.950
0.990
0.995
CV 0.125
0.250
0.333
0.500
0.667
0.752
1.000

Assumed
Model
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
Length
Ratio
0.6599
0.5729
0.6713
0.6935
0.7020
0.8437
0.7436
0.5583
0.4940
0.9504
0.8401
0.7439
0.5794
0.5112
0.4989
0.4954

a-Error
0.00611
-0.01965
0.00643
-0.01590
0.00607
-0.01962
0.00599
-0.02107
0.00596
-0.02201
0.00898
-0.03843
0.00857
-0.02884
0.00416
-0.00749
0.00274
-0.00385
-0.00005
-0.00740
-0.00009
-0.01486
0.00015
-0.01959
0.00266
-0.02614
0.00830
-0.02611
0.01160
-0.02458
0.02021
-0.01887
Power-
50%
0.759
0.348
0.500
0.286
0.679
0.357
0.857
0.357
1.000
0.393
1.000
1.000
1.000
0.393
0.643
0.000
0.393
0.000
0.688
0.500
0.688
0.438
0.688
0.438
0.750
0.250
0.813
0.250
0.813
0.250
0.875
0.313
Power-80%
0.741
0.277
0.500
0.107
0.607
0.321
0.857
0.321
1.000
0.357
1.000
0.857
1.000
0.250
0.607
0.000
0.357
0.000
0.688
0.438
0.688
0.438
0.688
0.313
0.688
0.188
0.813
0.188
0.813
0.188
0.813
0.188
Power-
Both
0.741
0.277
0.500
0.107
0.607
0.321
0.857
0.321
1.000
0.357
1.000
0.857
1.000
0.250
0.607
0.000
0.357
0.000
0.688
0.438
0.688
0.438
0.688
0.313
0.688
0.188
0.813
0.188
0.813
0.188
0.813
0.188
   Legend.  Category: N = Sample size; (1-a) = Nominal confidence level; CV = Coefficient of variation of
            underlying normal distribution. For each case, results for all simulations with that characteristic
            were averaged to derive that line of the figure.
            Assumed Model: Whether normal or lognormal formulas were used to compute the prediction
            limits.
            Length Ratio: Ratio of the normal prediction limit to the lognormal prediction limit.
            cc-error: Achieved false positive rate minus nominal false positive rate.
            Power-50%: Fraction of simulations in which 50%  power target at 3  standard deviations above
            background was met by the prediction limit.
            Power-80%: Fraction of simulations in which 80%  power target at 4  standard deviations above
            background was met by the prediction limit.
            Power-Both: Fraction of simulations in which both the 50% and 80% power targets were met.
                                               C-5
        March 2009

-------
   Appendix C.  Technical Appendix
Unified Guidance
Retesting, l-of-2 Scheme
Category
ALL
n 4
8
12
16
(1-oc) 0.900
0.950
0.990
0.995
CV 0.125
0.250
0.333
0.500
0.667
0.752
1.000

Assumed
Model
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
Length
Ratio
0.8712
0.7870
0.8815
0.9034
0.9129
1.0370
0.9543
0.7807
0.7129
0.9867
0.9486
0.9065
0.8289
0.8051
0.8048
0.8178

a-Error
0.00134
0.00052
0.00254
-0.00302
0.00130
0.00070
0.00066
0.00157
0.00087
0.00283
-0.00134
0.01496
0.00235
-0.00485
0.00252
-0.00510
0.00185
-0.00295
0.00009
0.00023
0.00003
-0.00048
0.00016
-0.00164
0.00095
-0.00190
0.00242
0.00132
0.00270
0.00241
0.00306
0.00371
Power-
50%
0.911
0.670
0.643
0.500
1.000
0.643
1.000
0.714
1.000
0.821
1.000
1.000
1.000
1.000
0.893
0.500
0.750
0.179
0.875
0.875
0.875
0.813
0.875
0.688
0.938
0.563
0.938
0.563
0.938
0.563
0.938
0.625
Power-80%
0.884
0.625
0.536
0.500
1.000
0.607
1.000
0.679
1.000
0.714
1.000
1.000
1.000
1.000
0.786
0.321
0.750
0.179
0.875
0.875
0.875
0.813
0.875
0.625
0.875
0.500
0.875
0.500
0.875
0.500
0.938
0.563
Power-
Both
0.884
0.625
0.536
0.500
1.000
0.607
1.000
0.679
1.000
0.714
1.000
1.000
1.000
1.000
0.786
0.321
0.750
0.179
0.875
0.875
0.875
0.813
0.875
0.625
0.875
0.500
0.875
0.500
0.875
0.500
0.938
0.563
   Legend.   Category: N = Sample size; (1-a) = Nominal confidence level; CV = Coefficient of variation of
             underlying normal distribution. For each case, results for all simulations with that characteristic
             were averaged to derive that line of the figure.
             Assumed Model: Whether normal or lognormal formulas were used to compute the prediction
             limits.
             Length Ratio: Ratio of the normal prediction limit to the lognormal prediction limit.
             a-error: Achieved false positive rate minus nominal false positive rate.
             Power-50%: Fraction of simulations in which 50% power target at 3 standard deviations above
             background was met by the prediction limit.
             Power-80%: Fraction of simulations in which 80% power target at 4 standard deviations above
             background was met by  the prediction limit.
             Power-Both: Fraction of simulations in which both the 50% and 80% power targets were met.
                                                 C-6
        March 2009

-------
Appendix C.  Technical Appendix
Unified Guidance
  Figure C-2.  Accuracy and Power of Normal vs. Lognormal  Prediction Limits When
                             Underlying Data Are Lognormal
  No Retesting,  1-of-l Scheme
Category
ALL

n







(1-a)







CV





















4

8

12

16

0.900

0.950

0.990

0.995

0.125

0.250

0.500

0.750

1.000

1.250

1.500

2.000

2.500

3.000

Assumed
Model
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
Length
Ratio
0.6082

0.4444

0.6006

0.6723

0.7153

0.8985

0.7108

0.4472

0.3762

0.9549

0.8749

0.7282

0.6278

0.5629

0.5204

0.4915

0.4566

0.4378

0.4266

ex-Error
0.03557
-0.00001
0.04682
0.00000
0.03782
-0.00002
0.03102
0.00001
0.02660
-0.00003
0.02509
-0.00001
0.04175
-0.00002
0.04037
0.00000
0.03505
0.00000
0.00733
-0.00003
0.01419
-0.00004
0.02535
-0.00004
0.03315
0.00006
0.03829
-0.00008
0.04216
0.00009
0.04479
-0.00002
0.04815
0.00003
0.05037
-0.00004
0.05189
-0.00002
Power-25%
0.806
0.500
0.775
0.400
0.825
0.500
0.825
0.550
0.800
0.550
1.000
0.950
0.975
0.525
0.700
0.275
0.550
0.250
1.000
1.000
1.000
1.000
1.000
0.813
1.000
0.500
0.938
0.438
0.938
0.375
0.750
0.250
0.500
0.250
0.500
0.188
0.438
0.188
Power-50%
0.581
0.412
0.550
0.325
0.600
0.425
0.600
0.450
0.575
0.450
0.750
0.650
0.675
0.475
0.475
0.275
0.425
0.250
1.000
1.000
1.000
1.000
1.000
0.813
0.938
0.500
0.813
0.438
0.500
0.188
0.438
0.188
0.125
0.000
0.000
0.000
0.000
0.000
Power-Both
0.581
0.412
0.550
0.325
0.600
0.425
0.600
0.450
0.575
0.450
0.750
0.650
0.675
0.475
0.475
0.275
0.425
0.250
1.000
1.000
1.000
1.000
1.000
0.813
0.938
0.500
0.813
0.438
0.500
0.188
0.438
0.188
0.125
0.000
0.000
0.000
0.000
0.000
Legend.   Category: N = Sample size; (1-a) = Nominal confidence level; CV = Coefficient of variation of
          underlying lognormal distribution. For each case, results for all simulations with that
          characteristic were averaged to derive that line of the figure.
          Assumed Model: Whether normal or lognormal formulas were used to compute the prediction
          limits.
          Length Ratio: Ratio of the normal prediction limit to the lognormal prediction limit.
          cc-error: Achieved false positive rate minus nominal false positive rate.
          Power-25%: Fraction of simulations in which 25% power target at 3 times the background
          mean was met by the prediction limit.
          Power-50%: Fraction of simulations in which 50% power target at 5 times the background
          mean was met by the prediction limit.
          Power-Both: Fraction of simulations where both 25% and 50% power targets were met.
                                             C-7
        March 2009

-------
Appendix C. Technical Appendix
Unified Guidance
 Retesting, l-of-2 Scheme
Category
ALL

n







(1-a)







CV





















4

8

12

16

0.900

0.950

0.990

0.995

0.125

0.250

0.500

0.750

1.000

1.250

1.500

2.000

2.500

3.000

Assumed
Model
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
normal
lognormal
Length
Ratio
0.9920

0.7334

0.9830

1.0931

1.1586

1.3532

1.1500

0.7890

0.6759

0.9889

0.9684

0.9332

0.9199

0.9253

0.9428

0.9673

1.0266

1.0913

1.1566

ct-Error
-0.00405
0.00009
0.00804
0.00022
-0.00354
0.00016
-0.00897
-0.00020
-0.01175
0.00018
-0.02895
0.00027
-0.00478
0.00009
0.00892
0.00000
0.00860
0.00000
-0.00035
0.00011
-0.00099
0.00008
-0.00214
0.00008
-0.00317
0.00018
-0.00416
0.00008
-0.00481
0.00014
-0.00527
0.00011
-0.00606
0.00004
-0.00662
0.00003
-0.00696
0.00004
Power-25%
0.587
0.600
0.625
0.550
0.600
0.600
0.575
0.625
0.550
0.625
0.800
1.000
0.700
0.750
0.475
0.375
0.375
0.275
1.000
1.000
1.000
1.000
1.000
0.938
0.938
0.688
0.688
0.500
0.500
0.500
0.438
0.500
0.188
0.375
0.063
0.250
0.063
0.250
Power-50%
0.544
0.544
0.525
0.425
0.550
0.550
0.550
0.600
0.550
0.600
0.700
0.850
0.600
0.625
0.475
0.375
0.400
0.325
1.000
1.000
1.000
1.000
1.000
0.938
1.000
0.813
0.688
0.500
0.500
0.438
0.250
0.375
0.000
0.188
0.000
0.188
0.000
0.000
Power-Both
0.537
0.531
0.500
0.425
0.550
0.550
0.550
0.575
0.550
0.575
0.700
0.850
0.600
0.625
0.475
0.375
0.375
0.275
1.000
1.000
1.000
1.000
1.000
0.938
0.938
0.688
0.688
0.500
0.500
0.438
0.250
0.375
0.000
0.188
0.000
0.188
0.000
0.000
Legend.   Category: N = Sample size; (1-a) = Nominal confidence level; CV = Coefficient of variation of
          underlying lognormal distribution. For each case, results for all simulations with that
          characteristic were averaged to derive that line of the figure.
          Assumed Model: Whether normal or lognormal formulas were used to compute the prediction
          limits.
          Length Ratio: Ratio of the normal prediction limit to the lognormal prediction limit.
          cc-error: Achieved false positive rate minus nominal  false positive rate.
          Power-25%: Fraction of simulations in which 25% power target at 3  times the background
          mean was met by the prediction limit.
          Power-50%: Fraction of simulations in which 50% power target at 5  times the background
          mean was met by the prediction limit.
          Power-Both: Fraction of simulations in which both the 25% and 50% power targets were met.
                                               C-8
        March 2009

-------
Appendix C.  Technical Appendix	Unified Guidance

     On balance,  adding a retest to the testing  procedure  significantly minimizes  the  penalty  of
misapplying normal prediction limits to lognormal data, as long one uses a sample size of at least 8 and
the coefficient of variation is  not too large.  Consequently, for most situations,  there is less penalty
associated with making a default assumption of normality than in making a default assumption  of
lognormality. With highly skewed data,  say with large  coefficients of variation of 1.5 or more,
goodness-of-fit tests tend to better discriminate between the normal and lognormal models. Again such
diagnostic  testing  should be done explicitly, rather than simply  assuming the data are normal  or
lognormal.

     The most problematic cases occur for very small background sample sizes, where a misapplication
of prediction limits in either direction can result in poorer statistical performance, even with retesting. In
some situations, testing may have to  done on an interim or ad-hoc basis  until more data is collected.
Still, the Unified Guidance does not recommend an automatic default assumption of lognormality.
                                              C-9                                    March 2009

-------
Appendix C.  Technical Appendix	Unified Guidance

C.2  CALCULATING STATISTICAL POWER

       C.2.1  STATISTICAL POWER OF  WELCH'S T-TEST

     The statistical power of any test represents the probability that the alternative hypothesis, HA, will
be accepted, given that the  null hypothesis, HO, is actually false. In groundwater monitoring, power
usually represents the probability  that the  compliance  point concentrations will be  identified  as
significantly higher than background, when in fact they are higher. Of course, statistical power is not a
single  number, but rather a function  of the  increase in the compliance  population mean above the
background average. This fact makes the  exact power of many tests difficult to calculate, especially
since many test statistics have a complicated distributional behavior under the alternative hypothesis.

     The critical points or percentage points of any test are computed under the assumption that the null
hypothesis  is true. In the case of Welch's t-test, the ^-statistic approximately follows a Student's t-
distribution under HO. That is not the case,  however, when the alternative hypothesis is true; then the t-
statistic follows what is known as  a non-central  ^-distribution with  non-centrality  parameter  5.
Essentially, the non-centrality parameter 5 governs the average or expected  value of the r-statistic.

     When the null hypothesis is true, so that the two population means are equal, the ^-statistic should
tend to be close to zero. The  distribution of the ^-statistic is in fact centered at zero in this case, meaning
that the usual Student's ^-distribution can be regarded as a non-central ^-distribution with non-centrality
parameter equal to zero.

     When HA is true instead, and the compliance point population mean is larger than the background
mean, Welch's r-statistic will tend to be positive rather than centered at zero. The actual  center of the
distribution will depend  on  precisely how much larger the compliance  point mean is  compared to
background. However, if ax  represents the standard deviation of the first population and  ay represents
the standard deviation of the  second population, it can be shown that the two-sample Welch's ^-statistic
approximately follows a non-central ^-distribution with degrees of freedom equal to
                           df =
                                 n    n
                                                             
-------
Appendix C.  Technical Appendix	Unified Guidance

power is essentially impossible. Instead, an approximate power can be computed by substituting the
sample  variances  for  their  population  counterparts into equations  [C.I]  and  [C.2].  By letting
/ = o-2ylu2x, the non-centrality parameter becomes
                                                                                          [C.3]
where  k represents the  increase in standard deviation units above the background mean. The non-
centrality parameter can be approximated by substituting / = s2js2x for/in [C.3].

     Using this formulation, the approximate statistical power of Welch's f-test can be computed by
repeatedly increasing k (e.g., in half units starting with 0.5) and determining the probability of exceeding
the original critical point, tcp, under the non-central ^-distribution. A concise summary of the non-central
t-distribution  can  be found  in  Evans, Hastings, and  Peacock  (1993). Percentage points of  this
distribution can be computed in selected standard statistical packages, including the free, open-source
statistical software R (www.r-project.org).

^EXAMPLE C-l

     Determine the approximate power of the ^-test on benzene data used in Example 16-1.

SOLUTION
Step 1.   Since Welch's ^-test was  run on the logged benzene measurements, power should also be
         computed using the logged values. In that case, the degrees of freedom was approximated at
         df = 11 and the critical point at a = .05 was found to  be tcp  = 1.796 from  the Student's t-
         distribution in  Table 16-1.

Step 2.   Determine the non-centrality parameter 5 from equation [C.3], substituting / = s2y/s2x for/
         Since nx = ny  = 8, the sample downgradient log-standard deviation  is sy =  1.9849, and the
         sample  background  log-standard deviation is  sx = 1.0826. Plugging these values into /
         gives/ = (l.9849)2/(l-0826)2 = 3.362. The approximate non-centrality parameter becomes
                                             8x8       4.354)
                                                         v     >
                                        V8 + 3.362x8

         where k represents the increase above the benzene background log-scale mean in log-standard
         deviation units.

Step 3.   Systematically  increase k from  0.5  to 5  in steps of  0.5 to determine the non-centrality
         parameter 5 at each point to be computed on the power curve (presented in the table below) .
         Then determine each power value by calculating from the non-central ^-distribution, with non-
         centrality parameter 5 and df= 1 1, the probability of exceeding the original critical point of tcp
         = 1.796.
                                             C-ll                                   March 2009

-------
Appendix C.  Technical Appendix                                          Unified Guidance
k
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5
0.677
1.354
2.031
2.708
3.385
4.062
4.739
5.416
6.093
6.770
power
0.1565
0.3541
0.6022
0.8135
0.9360
0.9843
0.9973
0.9997
1.0000
1.0000
Step 4.   Interpret the power results. The table in Step 3 shows an approximate probability of 81% for
         detecting a two log-standard deviation increase above the background mean benzene level. If
         the data had been  analyzed in the  original units, a two standard deviation increase would
         translate into almost 11  ppb (using the sample background standard deviation of 5.31 ppb
         from Example 16-1 as an estimate of the true standard deviation).

         However,  in the logarithmic domain, the interpretation is a bit different. As discussed in
         Section C.2.3, adding ko to the  log-scale mean is equivalent to multiplying the arithmetic
         mean by  exp(£a). Therefore,  a two  log-standard  deviation  increase  in the log-scale
         background  mean is roughly equivalent to multiplying the original background mean by a
         factor of exp(2 x 1.0826) = 8.7, taking the sample log-scale background standard deviation of
         1.0826 as an estimate for the true log-scale standard deviation.

         Consequently,  if the true background mean for benzene is close to the sample value of 3 ppb,
         the test will have more than 80% power for detecting a downgradient benzene mean of at least
         3 x 8.7 ~ 26 ppb or larger. -4
C.2.2 POWER OF PREDICTION LIMITS FOR FUTURE MEAN VS. OBSERVATIONS

     The  Unified  Guidance discusses  two basic kinds of parametric prediction limits:  those  for
individual future observations and those for future means. Analytical expressions for the statistical
power of each can be written and compared using the same sample size (n) , the same false positive rate
(a) , and the same number of future measurements (p = m).

     The power of a prediction limit for a future mean of order/? (that is, a mean ofp individual future
values) with normally- distributed data can be expressed in the equation
                           l-/?=Pr T  ,1 S=. - + -\>t.    .                         [C.4]
                              ^       n-l                   l-a,n-l                         L    J
where (1-|3) is a notation for power, A is the true difference (in standard deviation units) between the
background and compliance point population means, and
                                            C-12                                   March 2009

-------
Appendix C.  Technical Appendix                                          Unified Guidance
                                                                                         [C.5]
                                      n-1
denotes a random variable distributed according to the non-central  ^-distribution with non-centrality
parameter 5 and (w-1) degrees of freedom.

     By contrast, the power of a prediction limit for/? individual future values can be derived using the
formulation in Davis and McNichols (1987), leading to the expression

                                                                 '     "                  [C.6]


where  in  this case  O^fw) denotes the inverse standard normal  transformation. The non-central t-
distribution is required in each case, with further integration of the non-central t cumulative distribution
function [CDF] needed for the case ofp individual future measurements. These formulas are utilized in
Chapter 18 to provide graphical power comparisons between prediction limits for future means versus
prediction limits for individual values.

C.2.3 COMPUTING POWER WITH  LOGNORMAL DATA

     The special Monte  Carlo  study presented in  Section C.I involved a computation of statistical
power when the underlying data are lognormal in distribution rather than normal.  In the case of normal
data, effective power is computed by adding an upward  'shift' in the  mean of the baseline distribution,
in order to simulate an increasing compliance point concentration. Adding such a shift does not increase
the variance (a2)  of the shifted distribution, only the mean (|j,).

     With lognormal data, both the mean and variance depend  on the two distributional parameters, (j,
and a.  Adding a  shift to the log-mean (j, on the log-scale thus increases both the variance and the mean
in the concentration domain, confusing the usual interpretation of power as the ability to detect upward
changes in the mean level when all other factors (including the variance) are held constant.

     In fact,  if computations  are conducted on the log-scale and a shift (A) is added to the log-mean
parameter (|j,), the  effect is to multiply the lognormal mean in the arithmetic domain by  a factor of
exp(A\ To see this, note that  the lognormal mean is written as
                                                                                         [C.7]

An additive shift to the log-mean results in a change to the (arithmetic) lognormal mean of

                   MA = exp// + A + .5cr2=exp(A)exp// + .5cr2=exp(A)M               [C.8]
     To compute statistical power, one must assess test performance both under background conditions
and under increasing levels of contamination. But the power that can be expected with lognormal data
varies depending on the lognormal coefficient of variation [CV] . For a fixed coefficient of variation, as
lognormal concentrations increase, the lognormal standard deviation increases proportionally to the
                                             CMS                                   March 2009

-------
Appendix C.  Technical Appendix	Unified Guidance

lognormal mean. Because of this — and in contrast to the case of normal data — a different lognormal
power curve could be associated with each unique value of the CV.

     To sidestep this problem, the Unified Guidance assumes that if background is lognormal, the same
coefficient of variation [CV\ will apply to both the background and compliance point populations. This
assumption has two important  consequences: 1) compliance point data with mean  levels higher than
background will tend to also be more variable than the background measurements, a  common empirical
truth in environmental data sets;  and 2)  on the log-scale, the log-variance parameter  (a2) will be the
same  in both  populations. The reason that this  second consequence holds is  that the log-standard
deviation parameter is solely a function  of the  coefficient of variation, as expressed in the following
equation:
                                                                                         [C.9]
Thus if the CFis held constant, to will the log-standard deviation parameter (o).
     The upshot of the second consequence is that all power computations for lognormal data can be
done in the log-domain, using the fact that the transformed data will be normally distributed and that the
background and compliance  point populations will have a common standard deviation. Consequently,
the computational framework for simulating statistical power of lognormal data is almost precisely the
same as the framework for the normal case.

     In particular, the power  curve for a  given  test  can always be  generated —  without loss of
generality — by assuming that the background data follow  (perhaps in the log-domain)  a  standard
normal distribution, and that the compliance point  data follow (again in the  log-domain for lognormal
populations) a normal distribution with unit variance and shifted mean equal to ko = k, since a is
assumed for computational purposes equal to 1 . Then the multiplier k is typically allowed to range from
0 to 5, as this adequately sketches out the normal power curve in most situations.

     The only aspect of the lognormal case that differs from the normal is the scaling of the horizontal
axis of the power curve.  In the log-domain, the curve  documents power at increasing  multiples  of k log-
standard deviations (a) above the background log-mean (|j,) . To interpret these values in terms  of the
original concentrations, the background mean has to be reconstructed using the formula
while the compliance point (arithmetic) mean corresponding to the ko log-scale increase becomes

                           Mcw = exp ([i + ker + O.Scr2 )= MBG exp(far)                    [C.

or equivalently, a multiple of exp(£a) times the mean background level.
                                             C-14                                   March 2009

-------
Appendix C.  Technical Appendix	Unified Guidance

^EXAMPLE  C-2

     Suppose  that background data are fit best by a lognormal distribution with CV = 0.5. What steps
must be taken to  simulate the statistical performance of a lognormal prediction limit on observations
with a single verification resample?

SOLUTION
Step 1.   Compute the background log-standard deviation parameter as:
                           a =  lol + CF2=   lol + .25= 0.4724
         taking multiplier k = 0 to represent the background population.

Step 2.   Generate simulated random values from a standard normal distribution with zero mean and
         unit  standard  deviation.  These values  represent simulated and standardized  log-domain
         background measurements.

Step 3.   Compute the background prediction limit for lognormal data with a single resample using the
         formula:

                                                                                        [C.I 2]

         where y  and sy are respectively the log-mean and log-standard deviation, and K is taken from
         the Chapter 19 tables in the Appendix,  depending on  the background sample size, the
         number of tests to be run, and the type of l-of-2 retesting  plan (interwell or intrawell). Note
         that the simulated background values do not need to be exponentiated prior to computing the
         background prediction limit, due to the construction of formula [C.I2].

Step 4.   For any specific k in the range from 0 to 5  (with increasing steps of 0.5), set the compliance
         point log-mean equal to HA = k. Use this result to generate two normal measurements with
         shifted mean (j,A and unit standard deviation.  The two simulated values represent an original
         sample and a possible resample from the contaminated compliance point. Exponentiate these
         two  values  to get  simulated lognormal measurements from  the  desired  (alternative)
         distribution.

Step 5.   Compare the simulated values against the  background prediction limit. If both exceed the
         limit, increment the count of cases associated with k in which a difference from background
         has been identified. If only one or none exceeds the limit, do not increment the count.

Step 6.   Repeat Steps 2 through 5 a large number of iterations (say  10,000 or more) and determine the
         fraction of cases for given k at which an exceedance of background is found. This fraction
         represents the  estimated power of the lognormal prediction limit in the log-domain of a ko
         increase above the  background log-mean.  Equivalently, with a population  CV = 0.5, this
         represents a compliance point mean level of exp(#cr)= exp(& x .4724") times the (arithmetic)
         background mean. Repeat this entire process  for each k in  the range of 0 to 5 to  estimate the
         full lognormal  power curve for that prediction  limit. -4

                                            CMS                                   March 2009

-------
Appendix C. Technical Appendix	Unified Guidance

C.3  R SCRIPTS

     Certain calculations in the Unified Guidance cannot  easily be  done either by hand, with a
spreadsheet,  or even within many common statistical  packages.  In some cases, proprietary software
tailored to groundwater statistics can be consulted. Barring that, an alternate solution is to download and
install the free-of-charge, open source, statistical analysis and programming environment R software. It
can be utilized to perform or program almost any kind of statistical test or calculation. However, with its
power and flexibility comes a somewhat steeper learning curve for new programming language.

     One of R's advantages is the ability to run 'scripts,'  short pre-written programs that can be  run
repeatedly to perform specific  statistical calculations.  Scripts can be easily tailored to data- or site-
specific configurations  using a  simple  text  editor.  Because  users  of the Unified  Guidance may
occasionally  need calculations not covered in the Appendix tables or which are unavailable in standard
statistical  software, a small number  of R scripts are listed below. These scripts can be  modified as
necessary and then run in R, once the R environment is installed on a personal computer. They  are
provided as a courtesy to users  of the Unified Guidance and are provided without any guarantees or
implied warranties.

     The  scripts provided in the Unified Guidance below cover two specific topics: 1) calculation of
parametric intrawell prediction  limit K-multipliers  used with retesting, especially in  cases where a
pooled standard deviation  estimate might  be used in place of the  usual  sample standard deviation
(Section  13.3); and 2) computation of a bootstrapped non-parametric confidence band around a Theil-
Sen trend line (Section 21.3.2).

       It is first necessary to install  the R-software. As of this date, the latest version is  2.7.2.  The
program can  be downloaded from the website: http://cran.r-project.org.  Versions are available for most
current Windows operating systems,  as well as other types.   Once the program has been  downloaded
(approximately 30 mb), it can be accessed through a self-installed desktop icon.

       The R-scripts should first be transferred to a working directory; copies are provided with  the
distribution CD. If copied directly from the guidance Acrobat pdf using a text editor such as Notepad, it
will be necessary to copy each page of the script separately and combine (avoiding unnecessary margin,
header and footer information).  Each file should be named and saved with the extension changed to a
xxx. r format.   It may be necessary to add a number of additional comment codes (#) at the beginning of
the scripts using the text editor, so that each line of narrative text is first identfied by a  comment code.
To run the scripts:

     1) Open the R-software from the desktop icon; you will be in the R-console window;

     2) Click File on the toolbar, select Change_dir and hit [Enter]; set the working directory to  the
one with your scripts; hit [Enter];

     3) Then click File and select Open Script [Enter]; Click on the desired R-script file and hit Enter;

      4) In the R-console window; change script inputs as desired; Click Edit on the toolbar and select
Run All.
                                              C-16                                   March 2009

-------
Appendix C. Technical Appendix	Unified Guidance


     5) The program will run behind the console window.  Outputs can be read by minimizing the R-
editor.  Using the side scrollbar, check the R-script text run to determine if any errors occurred.   As
noted above,  it may be necessary to add the comment code (#) where line length has been exceeded.  To
run additional inputs within a script, simply modify the inputs in the R-console window and then follow
steps 4) and 5).  To run other scripts, minimize R, select the new script, adjust as appropriate and follow
steps 3) to 5).

     6) If an effect size power level is desired for the two prediction limit scripts, change one of the two
values in parentheses on the line del = c(3,4) and run again.

C.3.1 PARAMETRIC INTRAWELL  PREDICTION LIMIT MULTIPLIERS

1-of-m Retesting Plans

# R Script for 1-of-m retesting plans
# Compute multiplier for intrawell prediction limit using either regular or pooled SD estimate
# and 1-of-m  retesting for either observations or means of order p
# Solve for kappa given an SWFPR adjusted for nbr of constituents  and wells;
# then rate by effective power
# ne = number of yearly evaluations
# Note: ne=4 (quarterly eval), ne=2 (semi-annual), ne=l (annual)
# n = intrawell BG sample size; w = # wells; coc = # constituents
# df =  degrees of freedom associated with variance estimate  of prediction limit formula
# Note: if the usual std deviation for a single well is used,  set df = (n-1);
#       if using a pooled SD estimate across w equal sized wells, set df= w*(n-l) or
#       df = (sum of well n's) -  w, if  w pooled wells are  of different sizes
# alph  = per-test false positive rate
# m =  type of 1-of-m retesting scheme (usually m= 1,2,3,or  4)
# ord = order of the mean to be predicted (for tests on observations, set ord=l)
# swfpr is the targeted network-wide false positive rate, by default  set to 10%
# Rate power at 3 and 4 SD  units above BG;
# use ERPC power values as the reference power
# user supplied values of n, w, coc, df, evaluation frequency,  m, and ord
n= 4
w=  10
coc= 5
df= w*(n-l)
ne=  1
m= 3
ord= 2

swfpr= .1
alph= 1 - (l-swfpr)'Xl/(coc*w))
ref= c()
if (ne= = l) ref= c(.54,.81)
if (ne==2) ref= c(.59,.85)
if (ne==4) ref= c(.60,.86)

# default tolerance values for convergence
tol=  .000001
to!2= .0001

# default lower and upper limits on range for desired multiplier
11= 0
ul= 15

# recursive function to compute correct multiplier within limits (lo,hi)

                                              C-17                                     March 2009

-------
Appendix C.  Technical Appendix	Unified Guidance

kfind= function(lo,hi,n,alph,ne,tol) {
       if (abs(hi-lo)

del= c(3,4)
pow= c()
powrate=  c()

kap= kfind(ll,ul,n,alph,ne,tol)

for (jj in l:length(del))  {
       dc= delFjj]
       tt= sqrt(n)*kap
       nc= function(x) {sqrt(n)*(qnorm(x)/sqrt(ord) + del[jj])>
       h= function(x) {
               if (ne= = l) {
                      m*((l-x)/Xm-l))*pt(tt,df,nc(x))
                      >
               else {
                      ne*m*((l
               >
       pow[jj]= 1 - integrate(h,0,l,stop.on.error=F)$value
       >

if ((pow[l] >= ref[l]) && (pow[2] >= ref[2]))  powrate= 'GOOD'
if ((pow[l] < ref[l]) && (pow[2] >= ref[2])) powrate=  'ACCEPTABLE'
if ((pow[l] >= ref[l]) && (pow[2] < ref[2])) powrate=  'ACCEPTABLE'
if ((pow[l] < ref[l]) && (pow[2] < ref[2])) powrate= 'LOW

print(paste('intrawell lofm'),quote=F)
print(paste('n,w,coc,ne= ',n,w,coc,ne),quote=F)
print(paste('m,ord=',m,ord),quote=F)
print(paste('ref power from ERPC at 3 and 4 SDs'),quote=F)
print(ref)
print(paste('kappa=',round(kap,2)),quote=F)
print(paste('calculated power at 3 and 4 SDs'),quote=F)
print(round(pow,3))
print(paste('power rating=',powrate),quote=F)
                                                 C-18                                      March 2009

-------
Appendix C.  Technical Appendix	Unified Guidance


Modified California Retesting Plans

# R Script for modified California plan
# Compute multiplier for intrawell prediction limit using regular
# or pooled SD estimate and modified Calif retesting for observations
# Solve for kappa given  an SWFPR adjusted for number of constituents and wells;
# then rate by effective power
# ne = number of yearly evaluations
# Note: ne=4 (quarterly eval), ne=2 (semi-annual), ne=l (annual)
# n = intrawell  BG sample size; w = # wells; coc = # constituents
# df = degrees  of freedom associated with variance estimate of prediction limit formula
# Note: if the usual std deviation is used, set df = (n-1);
#       if using a pooled SD estimate across w wells, set df= w*(n-l)
# alph = per-test false positive rate
# swfpr is the targeted network-wide false positive rate, by default set to 10%
# Rate power at 3 and 4 SD units above BG;  use ERPC power values as the reference power
# user supplied values of n, w, coc, df, and evaluation frequency
n= 4
w= 10
coc= 5
df= w*(n-l)
ne=  1

swfpr= .1
alph= 1 - (l-swfpr)^(l/(coc*w))

ref= c()

if (ne= = l) ref= c(.54,.81)
if (ne==2) ref= c(.59,.85)
if (ne==4) ref= c(.60,.86)

# default tolerance values  for convergence
tol=  .000001
to!2= .0001

# default lower and upper  limits on range for desired multiplier
11= 0
ul= 15

# recursive function to compute correct multiplier within limits (lo,hi)
kfind= function(lo,hi,n,alph,ne,tol) {
       if (abs(hi-lo)
-------
Appendix C. Technical Appendix	Unified Guidance

           {
              stop('bad limits') }
       >

del= c(3,4)
pow= c()
powrate= c()

kap= kfind(ll,ul,n,alph,ne,tol)

for (jj in l:length(del)) {
       dc= del[jj]
       tt= sqrt(n)*kap
       nc= function(x) {sqrt(n)*(qnorm(x) + del[jj])>
       h= function(x) {
              if (ne= = l) {
                     (1 + 6*x - 15*x^2 + 8*x^3)*pt(tt,df,nc(x))
                     >
                  {
                     ne*(x*(l + 3*x - 5*x^2 + 2*x/^3))/Xne-l)*(l + &*x - 15*x^2 +
8*x^3)*pt(tt,df,nc(x))
                     >
              >
       pow[jj]=  1 - integrate(h,Q,l,stop.on.error=F)$value
       >

if ((pow[l] >= ref[l]) && (pow[2] >= ref[2])) powrate= 'GOOD'
if ((pow[l] < ref[l]) && (pow[2] >= ref[2])) powrate= 'ACCEPTABLE'
if ((pow[l] >= ref[l]) && (pow[2] < ref[2])) powrate= 'ACCEPTABLE'
if ((pow[l] < ref[l]) && (pow[2] < ref[2])) powrate= 'LOW

print(paste('intrawell  modCal'),quote=F)
print(paste('n,w,coc,ne= ',n,w,coc,ne),quote=F)
print(paste('ref power from ERPC at 3 and 4 SDs'),quote=F)
print(ref)
print(paste('kappa=',round(kap,2)),quote=F)
print(paste('calculated power at 3 and 4 SDs'),quote=F)
print(round(pow,3))
print(paste('power rating=',powrate),quote=F)

C.3.2THEIL-SEN CONFIDENCE BAND

# R script for Theil-Sen Confidence band
# Compute bootstrapped confidence band around Theil-Sen trend line
# user inputs:  list of x-values, list of y-values, desired confidence level
# Note: replace numbers in parentheses below with specific x and y values
#       corresponding to data-specific ordered pairs
# x-values should be numeric values representing sampling dates or events
# y-values should be concentration values  corresponding to these dates or events
# Script produces a plot of the Theil-Sen trend line, the confidence  band around the trend,
# and an overlay of the actual data values

x= c(89.6,90.1,90.8,91.1,92.1,93.1,94.1,95.6,96.1,96.3)
y= c(56,53,51,55,52,60,62,59,61,63)
conf = .90

elimna= function(m){
#
# remove any  rows of data having missing values
m= as.matrix(m)

                                               C^20                                     March 2009

-------
Appendix C. Technical Appendix _ Unified Guidance

ikeep= c(l:nrow(m))
for(i in l:nrow(m)) if (sum(is.na(m[i, ])> = !)) ikeep[i]= 0
elimna= m[ikeep[ikeep> = l],]
elimna
>

theilsen2= function(x,y){
#
# Compute the Theil-Sen regression estimator
# Do not compute residuals in this version
# Assumes missing pairs already removed
#
ord= order(x)
xs= x[ord]
ys= y[ord]
vecl= outer(ys,ys,"-")
vec2= outer(xs,xs,"-")
vl= vecl[vec2>0]
v2= vec2[vec2>0]
slope= median(vl/v2)
coef= 0
coef[l]= median(y)-slope*median(x)
coef[2]= slope
list(coef=coef)
>

nb=
temp= matrix(c(x,y),ncol=2)
temp= elimna(temp)                       #remove any pairs with missing values
x= temp[,l]
y= temp[,2]
n= length(x)
ord= order(x)
cut= min(x) + (G:lGG)*(max(x)-min(x))/100  #compute 101 cut pts
tO= theilsen2(x,y)                          #compute trend line on original data
tmp= matrix(nrow=nb,ncol =
for (i in l:nb) {
       idx= sample(ord,n,rep=T)
       xboot= x[idx]
       yboot= y[idx]
       tboot= theilsen2(xboot,yboot)
       tmp[i,]= tboot$coef[l] + cut*tboot$coef[2]
       >

lb= 0; ub= 0
for(i in 1:101){
       lb[i]= quantile(tmp[,i],c((l-conf)/2))
       ub[i]= quantile(tmp[,i],c((l+conf)/2))
       >
tband= Iist(xcut=cut,lo=lb,hi=ub,ths0=t0)
yt= tband$thsO$coef[l] + tband$thsO$coef[2]*tband$xcut
plot(yt~tband$xcut,type=T,xlim=range(x),ylim=c(min(tband$lo),max(tband$hi)),xlab='Date',ylab='Conc')
points(x,y,pch = 16)
lines(tband$hi~tband$xcut,type=T,lty=2)
lines(tband$lo~tband$xcut,type=T,lty=2)
                                               C-21                                     March 2009

-------
Appendix C. Technical Appendix                                     Unified Guidance
                    This page intentionally left blank
                                       C-22                              March 2009

-------
Appendix D. Statistical Tables	Unified Guidance
            APPENDIX D:  STATISTICAL TABLES
                                                         March 2009

-------
Appendix D. Statistical Tables	Unified Guidance

                                               D  STATISTICAL TABLES

D.I   TABLES FROM CHAPTERS  10 THROUGH  18
            TABLE 10-1 Percentiles of the Standard Normal Distribution	D-l
            TABLE 10-2 Coefficients for Shapiro-Wilk Test of Normality	D-3
            TABLE 10-3 a-Level Critical Points for Shapiro-Wilk Test, n< =50	D-5
            TABLE 10-4«4£vel Critical Points for Shapiro-Wilk Test, n from 50 to 100	D-6
            TABLE 10-5 a- Critical Points for Correlation Coefficient Test	D-7
            TABLE 10-6 Shapiro-Wilk Multiple Group Test, (G) Values for nfrom 7 to 50	D-8
            TABLE 10-7 Shapiro-Wilk Multiple Group Test, (G) Values for n from 3 to 6	D-9
            TABLE 12-1 a-Level Critical Points for Dixon's Outlier Test, n from 3 to 25	D-10
            TABLE 12-2 c^Level Critical Points for Rosner's OutiierTest	D-ll
            TABLE 14-1 Approximate a-Level Critical Points for Rank vonNeumann Ratio Test	D-13
            TABLE 15-1 Percentiles of the Student's  t-Distribution	D-15
            TABLE 17-1 Percentiles of the F-Distribution for (1-a) = .8, .9, .95, .98 & .99	D-17
            TABLE 17-2 Percentiles of the Chi-Square Distribution for df from 1 to 100	D-23
            TABLE17-3 Upper Tolerance Limit Factors with y Coverage for n from 4 to 100	D-24
            TABLE 17-4  Minimum Coverage of Non-Parametric Upper Tolerance Limits	D-25
            TABLE 17-5 Significance  Levels («) for Mann-Kendall Trend Tests for nfrom 4-100	D-27
            TABLE 18-1  Confidence  Levels of Non-Parametric Prediction Limits for nfrom 4-60	D-28
            TABLE 18-2  Confidence for Non-Parametric Prediction Limits on Future Median	D-31
                                                                                                                         March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
Table 10-1. Percentiles
p
0.50
0.51
0.52
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.60
0.61
0.62
0.63
0.64
0.65
0.66
0.67
0.68
0.69
0.70
0.71
0.72
0.73
0.74
0.75
0.000
0.0000
0.0251
0.0502
0.0753
0.1004
0.1257
0.1510
0.1764
0.2019
0.2275
0.2533
0.2793
0.3055
0.3319
0.3585
0.3853
0.4125
0.4399
0.4677
0.4959
0.5244
0.5534
0.5828
0.6128
0.6433
0.6745
0.001
0.0025
0.0276
0.0527
0.0778
0.1030
0.1282
0.1535
0.1789
0.2045
0.2301
0.2559
0.2819
0.3081
0.3345
0.3611
0.3880
0.4152
0.4427
0.4705
0.4987
0.5273
0.5563
0.5858
0.6158
0.6464
0.6776
0.002
0.0050
0.0301
0.0552
0.0803
0.1055
0.1307
0.1560
0.1815
0.2070
0.2327
0.2585
0.2845
0.3107
0.3372
0.3638
0.3907
0.4179
0.4454
0.4733
0.5015
0.5302
0.5592
0.5888
0.6189
0.6495
0.6808
0.003
0.0075
0.0326
0.0577
0.0828
0.1080
0.1332
0.1586
0.1840
0.2096
0.2353
0.2611
0.2871
0.3134
0.3398
0.3665
0.3934
0.4207
0.4482
0.4761
0.5044
0.5330
0.5622
0.5918
0.6219
0.6526
0.6840
of Standard Normal Distribution
0.004
0.0100
0.0351
0.0602
0.0853
0.1105
0.1358
0.1611
0.1866
0.2121
0.2378
0.2637
0.2898
0.3160
0.3425
0.3692
0.3961
0.4234
0.4510
0.4789
0.5072
0.5359
0.5651
0.5948
0.6250
0.6557
0.6871
0.005
0.0125
0.0376
0.0627
0.0878
0.1130
0.1383
0.1637
0.1891
0.2147
0.2404
0.2663
0.2924
0.3186
0.3451
0.3719
0.3989
0.4261
0.4538
0.4817
0.5101
0.5388
0.5681
0.5978
0.6280
0.6588
0.6903
0.006
0.0150
0.0401
0.0652
0.0904
0.1156
0.1408
0.1662
0.1917
0.2173
0.2430
0.2689
0.2950
0.3213
0.3478
0.3745
0.4016
0.4289
0.4565
0.4845
0.5129
0.5417
0.5710
0.6008
0.6311
0.6620
0.6935
0.007
0.0175
0.0426
0.0677
0.0929
0.1181
0.1434
0.1687
0.1942
0.2198
0.2456
0.2715
0.2976
0.3239
0.3505
0.3772
0.4043
0.4316
0.4593
0.4874
0.5158
0.5446
0.5740
0.6038
0.6341
0.6651
0.6967
0.008
0.0201
0.0451
0.0702
0.0954
0.1206
0.1459
0.1713
0.1968
0.2224
0.2482
0.2741
0.3002
0.3266
0.3531
0.3799
0.4070
0.4344
0.4621
0.4902
0.5187
0.5476
0.5769
0.6068
0.6372
0.6682
0.6999
0.009
0.0226
0.0476
0.0728
0.0979
0.1231
0.1484
0.1738
0.1993
0.2250
0.2508
0.2767
0.3029
0.3292
0.3558
0.3826
0.4097
0.4372
0.4649
0.4930
0.5215
0.5505
0.5799
0.6098
0.6403
0.6713
0.7031

                                                      D-l
                                                                                                       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
Table 10-1. Percentiles
p
0.76
0.77
0.78
0.79
0.80
0.81
0.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
0.000
0.7063
0.7388
0.7722
0.8064
0.8416
0.8779
0.9154
0.9542
0.9945
1.0364
1.0803
1.1264
1.1750
1.2265
1.2816
1.3408
1.4051
1.4758
1.5548
1.6449
1.7507
1.8808
2.0537
2.3263
0.001
0.7095
0.7421
0.7756
0.8099
0.8452
0.8816
0.9192
0.9581
0.9986
1.0407
1.0848
1.1311
1.1800
1.2319
1.2873
1.3469
1.4118
1.4833
1.5632
1.6546
1.7624
1.8957
2.0749
2.3656
0.002
0.7128
0.7454
0.7790
0.8134
0.8488
0.8853
0.9230
0.9621
1.0027
1.0450
1.0893
1.1359
1.1850
1.2372
1.2930
1.3532
1.4187
1.4909
1.5718
1.6646
1.7744
1.9110
2.0969
2.4089
0.003
0.7160
0.7488
0.7824
0.8169
0.8524
0.8890
0.9269
0.9661
1.0069
1.0494
1.0939
1.1407
1.1901
1.2426
1.2988
1.3595
1.4255
1.4985
1.5805
1.6747
1.7866
1.9268
2.1201
2.4573
of Standard Normal Distribution
0.004
0.7192
0.7521
0.7858
0.8204
0.8560
0.8927
0.9307
0.9701
1.0110
1.0537
1.0985
1.1455
1.1952
1.2481
1.3047
1.3658
1.4325
1.5063
1.5893
1.6849
1.7991
1.9431
2.1444
2.5121
0.005
0.7225
0.7554
0.7892
0.8239
0.8596
0.8965
0.9346
0.9741
1.0152
1.0581
1.1031
1.1503
1.2004
1.2536
1.3106
1.3722
1.4395
1.5141
1.5982
1.6954
1.8119
1.9600
2.1701
2.5758
0.006
0.7257
0.7588
0.7926
0.8274
0.8633
0.9002
0.9385
0.9782
1.0194
1.0625
1.1077
1.1552
1.2055
1.2591
1.3165
1.3787
1.4466
1.5220
1.6072
1.7060
1.8250
1.9774
2.1973
2.6521
0.007
0.7290
0.7621
0.7961
0.8310
0.8669
0.9040
0.9424
0.9822
1.0237
1.0669
1.1123
1.1601
1.2107
1.2646
1.3225
1.3852
1.4538
1.5301
1.6164
1.7169
1.8384
1.9954
2.2262
2.7478
0.008
0.7323
0.7655
0.7995
0.8345
0.8705
0.9078
0.9463
0.9863
1.0279
1.0714
1.1170
1.1650
1.2160
1.2702
1.3285
1.3917
1.4611
1.5382
1.6258
1.7279
1.8522
2.0141
2.2571
2.8782
0.009
0.7356
0.7688
0.8030
0.8381
0.8742
0.9116
0.9502
0.9904
1.0322
1.0758
1.1217
1.1700
1.2212
1.2759
1.3346
1.3984
1.4684
1.5464
1.6352
1.7392
1.8663
2.0335
2.2904
3.0902

                                                      D-2
                                                                                                       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Table 10-2.
i/n
1
2
3
4
5
i/n
1
2
3
4
5
6
7
8
9
10
i/n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
i/n
1
2
3
4
5
6
7
8
9
10
2
0.7071
	
	
	
	
11
0.5601
.3315
.2260
.1429
.0695
0.0000
	
	
	
	
21
0.4643
.3185
.2578
.2119
.1736
0.1399
.1092
.0804
.0530
.0263
0.0000
	
	
	
	
31
0.4220
.2921
.2475
.2145
.1874
0.1641
.1433
.1243
.1066
.0899
Coefficients [an-i+i] for
3
0.7071
.0000
	
	
	
12
0.5475
.3325
.2347
.1586
.0922
0.0303
	
	
	
	
22
0.4590
.3156
.2571
.2131
.1764
0.1443
.1150
.0878
.0618
.0368
0.0122
	
	
	
	
32
0.4188
.2898
.2463
.2141
.1878
0.1651
.1449
.1265
.1093
.0931
4
0.6872
.1677
	
	
	
13
0.5359
.3325
.2412
.1707
.1099
0.0539
.0000
	
	
	
23
0.4542
.3126
.2563
.2139
.1787
0.1480
.1201
.0941
.0696
.0459
0.0228
.0000
	
	
	
33
0.4156
.2876
.2451
.2137
.1880
0.1660
.1463
.1284
.1118
.0961
5
0.6646
.2413
.0000
	
	
14
0.5251
.3318
.2460
.1802
.1240
0.0727
.0240
	
	
	
24
0.4493
.3098
.2554
.2145
.1807
0.1512
.1245
.0997
.0764
.0539
0.0321
.0107
	
	
	
34
0.4127
.2854
.2439
.2132
.1882
0.1667
.1475
.1301
.1140
.0988
Unified Guidance
Shapiro-Wilk Test of Normality
6
0.6431
.2806
.0875
	
	
15
0.5150
.3306
.2495
.1878
.1353
0.0880
.0433
.0000
	
	
25
0.4450
.3069
.2543
.2148
.1822
0.1539
.1283
.1046
.0823
.0610
0.0403
.0200
.0000
	
	
35
0.4096
.2834
.2427
.2127
.1883
0.1673
.1487
.1317
.1160
.1013
7
0.6233
.3031
.1401
.0000
	
16
0.5056
.3290
.2521
.1939
.1447
0.1005
.0593
.0196
	
	
26
0.4407
.3043
.2533
.2151
.1836
0.1563
.1316
.1089
.0876
.0672
0.0476
.0284
.0094
	
	
36
0.4068
.2813
.2415
.2121
.1883
0.1678
.1496
.1331
.1179
.1036
8
0.6052
.3164
.1743
.0561
	
17
0.4968
.3273
.2540
.1988
.1524
0.1109
.0725
.0359
.0000
	
27
0.4366
.3018
.2522
.2152
.1848
0.1584
.1346
.1128
.0923
.0728
0.0540
.0358
.0178
.0000
	
37
0.4040
.2794
.2403
.2116
.1883
0.1683
.1503
.1344
.1196
.1056
9
0.5888
.3244
.1976
.0947
.0000
18
0.4886
.3253
.2553
.2027
.1587
0.1197
.0837
.0496
.0163
	
28
0.4328
.2992
.2510
.2151
.1857
0.1601
.1372
.1162
.0965
.0778
0.0598
.0424
.0253
.0084
	
38
0.4015
.2774
.2391
.2110
.1881
0.1686
.1513
.1356
.1211
.1075
, n= 2(1)50
10
0.5739
.3291
.2141
.1224
.0399
19
0.4808
.3232
.2561
.2059
.1641
0.1271
.0932
.0612
.0303
.0000
29
0.4291
.2968
.2499
.2150
.1864
0.1616
.1395
.1192
.1002
.0822
0.0650
.0483
.0320
.0159
.0000
39
0.3989
.2755
.2380
.2104
.1880
0.1689
.1520
.1366
.1225
.1092






20
0.4734
.3211
.2565
.2085
.1686
0.1334
.1013
.0711
.0422
.0140
30
0.4254
.2944
.2487
.2148
.1870
0.1630
.1415
.1219
.1036
.0862
0.0697
.0537
.0381
.0227
.0076
40
0.3964
.2737
.2368
.2098
.1878
0.1691
.1526
.1376
.1237
.1108
Source: Madansky (1988)
Footnote. The notation n = 2(1)50 is shorthand for nfrom 2 to 50 in unit steps
                                           D-3
March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Table 10-2.
i/n
11
12
13
14
15
16
17
18
19
20
i/n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
31
0.0739
.0585
.0435
.0289
.0144
0.0000
	
	
	
	
41
0.3940
.2719
.2357
.2091
.1876
0.1693
.1531
.1384
.1249
.1123
0.1004
.0891
.0782
.0677
.0575
0.0476
.0379
.0283
.0188
.0094
0.0000
	
	
	
	
Coefficients [an-i+i] for
32
0.0777
.0629
.0485
.0344
.0206
0.0068
	
	
	
	
42
0.3917
.2701
.2345
.2085
.1874
0.1694
.1535
.1392
.1259
.1136
0.1020
.0909
.0804
.0701
.0602
0.0506
.0411
.0318
.0227
.0136
0.0045
	
	
	
	
33
0.0812
.0669
.0530
.0395
.0262
0.0131
.0000
	
	
	
43
0.3894
.2684
.2334
.2078
.1871
0.1695
.1539
.1398
.1269
.1149
0.1035
.0927
.0824
.0724
.0628
0.0534
.0442
.0352
.0263
.0175
0.0087
.0000
	
	
	
34
0.0844
.0706
.0572
.0441
.0314
0.0187
.0062
	
	
	
44
0.3872
.2667
.2323
.2072
.1868
0.1695
.1542
.1405
.1278
.1160
0.1049
.0943
.0842
.0745
.0651
0.0560
.0471
.0383
.0296
.0211
0.0126
.0042
	
	
	
Unified Guidance
Shapiro-Wilk Test of Normality
35
0.0873
.0739
.0610
.0484
.0361
0.0239
.0119
.0000
	
	
45
0.3850
.2651
.2313
.2065
.1865
0.1695
.1545
.1410
.1286
.1170
0.1062
.0959
.0860
.0775
.0673
0.0584
.0497
.0412
.0328
.0245
0.0163
.0081
.0000
	
	
36
0.0900
.0770
.0645
.0523
.0404
0.0287
.0172
.0057
	
	
46
0.3830
.2635
.2302
.2058
.1862
0.1695
.1548
.1415
.1293
.1180
0.1073
.0972
.0876
.0785
.0694
0.0607
.0522
.0439
.0357
.0277
0.0197
.0118
.0039
	
	
37
0.0924
.0798
.0677
.0559
.0444
0.0331
.0220
.0110
.0000
	
47
0.3808
.2620
.2291
.2052
.1859
0.1695
.1550
.1420
.1300
.1189
0.1085
.0986
.0892
.0801
.0713
0.0628
.0546
.0465
.0385
.0307
0.0229
.0153
.0076
.0000
	
38
0.0947
.0824
.0706
.0592
.0481
0.0372
.0264
.0158
.0053
	
48
0.3789
.2604
.2281
.2045
.1855
0.1693
.1551
.1423
.1306
.1197
0.1095
.0998
.0906
.0817
.0731
0.0648
.0568
.0489
.0411
.0335
0.0259
.0185
.0111
.0037
	
, n= 2(1)50
39
0.0967
.0848
.0733
.0622
.0515
0.0409
.0305
.0203
.0101
.0000
49
0.3770
.2589
.2271
.2038
.1851
0.1692
.1553
.1427
.1312
.1205
0.1105
.1010
.0919
.0832
.0748
0.0667
.0588
.0511
.0436
.0361
0.0288
.0215
.0143
.0071
.0000
40
0.0986
.0870
.0759
.0651
.0546
0.0444
.0343
.0244
.0146
.0049
50
0.3751
.2574
.2260
.2032
.1847
0.1691
.1554
.1430
.1317
.1212
0.1113
.1020
.0932
.0846
.0764
0.0685
.0608
.0532
.0459
.0386
0.0314
.0244
.0174
.0104
.0035
Source: Madansky (1988)
Footnote. The notation n = 2(1)50 is shorthand for nfrom 2 to 50 in unit steps
                                           D-4
March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
        Table 10-3. a-Level Critical Points for Shapiro-Wilk Test, n = 3(1)50
n\a
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
0.01
0.753
0.687
0.686
0.713
0.730
0.749
0.764
0.781
0.792
0.805
0.814
0.825
0.835
0.844
0.851
0.858
0.863
0.868
0.873
0.878
0.881
0.884
0.888
0.891
0.894
0.896
0.898
0.900
0.902
0.904
0.906
0.908
0.910
0.912
0.914
0.916
0.917
0.919
0.920
0.922
0.923
0.924
0.926
0.927
0.928
0.929
0.929
0.930
0.05
0.767
0.748
0.762
0.788
0.803
0.818
0.829
0.842
0.850
0.859
0.866
0.874
0.881
0.887
0.892
0.897
0.901
0.905
0.908
0.911
0.914
0.916
0.918
0.920
0.923
0.924
0.926
0.927
0.929
0.930
0.931
0.933
0.934
0.935
0.936
0.938
0.939
0.940
0.941
0.942
0.943
0.944
0.945
0.945
0.946
0.947
0.947
0.947
0.10
0.789
0.792
0.806
0.826
0.838
0.851
0.859
0.869
0.876
0.883
0.889
0.895
0.901
0.906
0.910
0.914
0.917
0.920
0.923
0.926
0.928
0.930
0.931
0.933
0.935
0.936
0.937
0.939
0.940
0.941
0.942
0.943
0.944
0.945
0.946
0.947
0.948
0.949
0.950
0.951
0.951
0.952
0.953
0.953
0.954
0.954
0.955
0.955
Source: Madansky (1988)
Footnote. The notation n = 3(1)50 is shorthand for nfrom 3 to 50 in unit steps
                                        D-5
       March 2009

-------
Appendix D.  Chapters 10 to 18 Tables
Unified Guidance
      Table 10-4. a-Level Critical Points for Shapiro-Francfa Test, n = 50(1)99
n\a
50
51
53
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
85
87
89
91
93
95
97
99
0.01
0.935
0.935
0.938
0.940
0.944
0.945
0.947
0.947
0.948
0.950
0.951
0.953
0.956
0.956
0.957
0.957
0.958
0.960
0.961
0.961
0.961
0.962
0.963
0.965
0.965
0.967
0.05
0.953
0.954
0.957
0.958
0.961
0.962
0.963
0.964
0.965
0.966
0.966
0.967
0.968
0.969
0.969
0.970
0.970
0.971
0.972
0.972
0.972
0.973
0.973
0.974
0.975
0.976
0.10
0.963
0.964
0.964
0.965
0.966
0.967
0.968
0.970
0.971
0.971
0.972
0.972
0.973
0.973
0.974
0.975
0.975
0.976
0.977
0.977
0.977
0.978
0.979
0.979
0.979
0.980
Source: Shapiro & Francfa (1972)
Footnote. The notation n = 50(1)99 is shorthand for nfrom 50 to 99 in unit steps
                                         D-6
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
  Table 10-5. a-Critical Pts., Prob. Plot Correlation Coeff. Test, n = 3(1)50(5)100
n\a
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
55
60
65
70
75
80
85
90
95
100
0.01
0.869
0.822
0.822
0.835
0.847
0.859
0.868
0.876
0.883
0.889
0.895
0.901
0.907
0.912
0.912
0.919
0.923
0.925
0.928
0.930
0.933
0.936
0.937
0.939
0.941
0.943
0.945
0.947
0.948
0.949
0.950
0.951
0.952
0.953
0.955
0.956
0.957
0.958
0.958
0.959
0.959
0.960
0.961
0.962
0.963
0.963
0.964
0.965
0.967
0.970
0.972
0.974
0.975
0.976
0.977
0.978
0.979
0.981
0.025
0.872
0.845
0.855
0.868
0.876
0.886
0.893
0.900
0.906
0.912
0.917
0.921
0.925
0.928
0.931
0.934
0.937
0.939
0.942
0.944
0.947
0.949
0.950
0.952
0.953
0.955
0.956
0.957
0.958
0.959
0.960
0.960
0.961
0.962
0.962
0.964
0.965
0.966
0.967
0.967
0.967
0.968
0.969
0.969
0.970
0.970
0.971
0.972
0.974
0.976
0.977
0.978
0.979
0.980
0.981
0.982
0.983
0.984
0.05
0.879
0.868
0.879
0.890
0.899
0.905
0.912
0.917
0.922
0.926
0.931
0.934
0.937
0.940
0.942
0.945
0.947
0.950
0.952
0.954
0.955
0.957
0.958
0.959
0.960
0.962
0.962
0.964
0.965
0.966
0.967
0.967
0.968
0.968
0.969
0.970
0.971
0.972
0.973
0.973
0.973
0.974
0.974
0.974
0.975
0.975
0.977
0.978
0.980
0.981
0.982
0.983
0.984
0.985
0.985
0.985
0.986
0.987
0.10
0.891
0.894
0.902
0.911
0.916
0.924
0.929
0.934
0.938
0.941
0.944
0.947
0.950
0.952
0.954
0.956
0.958
0.960
0.961
0.962
0.964
0.965
0.966
0.967
0.968
0.969
0.969
0.970
0.971
0.972
0.973
0.973
0.974
0.974
0.975
0.975
0.976
0.977
0.977
0.978
0.978
0.978
0.978
0.979
0.979
0.980
0.980
0.981
0.982
0.983
0.984
0.985
0.986
0.987
0.987
0.988
0.989
0.989
Source: Filliben (1975)
                                       D-7
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
    Table 10-6. Shapiro-Wilk Multiple Group Test: Values to Compute G\ for n =

                                      7(1)50
n
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Y
-2.356
-2.696
-2.968
-3.262
-3.485
-3.731
-3.936
-4.155
-4.373
-4.567
-4.713
-4.885
-5.018
-5.153
-5.291
-5.413
-5.508
-5.605
-5.704
-5.803
-5.905
-5.988
-6.074
-6.150
8
1.245
1.333
1.400
1.471
1.515
1.571
1.613
1.655
1.695
1.724
1.739
1.770
1.786
1.802
1.818
1.835
1.848
1.862
1.876
1.890
1.905
1.919
1.934
1.949
B
.4533
.4186
.3900
.3660
.3451
.3270
.3111
.2969
.2842
.2727
.2622
.2528
.2440
.2359
.2264
.2207
.2157
.2106
.2063
.2020
.1980
.1943
.1907
.1872
n
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50




Y
-6.248
-6.324
-6.402
-6.480
-6.559
-6.640
-6.721
-6.803
-6.887
-6.961
-7.035
-7.111
-7.188
-7.266
-7.345
-7.414
-7.484
-7.555
-7.615
-7.677




S
1.965
1.976
1.988
2.000
2.012
2.024
2.037
2.049
2.062
2.075
2.088
2.101
2.114
2.128
2.141
2.155
2.169
2.183
2.198
2.212




E
.1840
.1811
.1781
.1755
.1727
.1702
.1677
.1656
.1633
.1612
.1591
.1572
.1552
.1534
.1516
.1499
.1482
.1466
.1451
.1436




Source: Gibbons (1994)
Footnote. The notation n = 7(1)50 is shorthand for nfrom 7 to 50 in unit steps
                                        D-8
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
     Table 10-7. Shapiro-Wilk Multiple Group Test: Values of G\ for n = 3(1)6

u
-7.0
-5.4
-5.0
-4.6
-4.2
-3.8
-3.4
-3.0
-2.6
-2.2
-1.8
-1.4
-1.0
-0.6
-0.2
0.0
0.2
0.4
0.6
1.0
1.4
1.6
1.8
2.2
2.6
3.0
3.4
3.8
4.2
4.6
5.0
5.4
5.8
6.2
6.6
7.0
7.4
7.8
8.2
8.6
n
(6 =
W
.7502
.7511
.7517
.7525
.7537
.7555
.7581
.7619
.7673
.7749
.7855
.7995
.8172
.8386
.8625
.8750
.8875
.8997
.9114
.9328
.9505
.9580
.9645
.9751
.9827
.9881
.9919
.9945
.9963
.9975
.9983
.9989
.99925
.99949
.99966
.99977
.99985
.99990
.99993
.99995
= 3
.7500)
G;
-3.291
-2.810
-2.678
-2.543
-2.400
-2.254
-2.099
-1.937
-1.767
-1.589
-1.404
-1.210
-1.010
-0.805
-0.599
-0.496
-0.395
-0.294
-0.195
-0.003
0.181
0.268
0.354
0.516
0.669
0.812
0.947
1.074
1.195
1.309
1.418
1.522
1.621
1.717
1.809
1.899
1.985
2.068
2.149
2.226
n = 4
(s = .6297)
W GI





.6378
.6417
.6473
.6553
.6666
.6822
.7030
.7293
.7609
.7964
.8149
.8333
.8514
.8688
.9004
.9267
.9378
.9475
.9631
.9744
.9824
.9880
.9919
.9945
.9963
.9975
.9983
.9989
.9993
.9995
.9997
.9998
.9998
.9999
.9999





-3.497
-3.270
-3.043
-2.839
-2.642
-2.441
-2.222
-1.964
-1.664
-1.309
-1.122
-0.944
-0.766
-0.573
-0.192
0.148
0.298
0.451
0.739
0.998
1.202
1.426
1.660
1.847
2.028
2.193
2.341
2.483
2.628
2.754
2.869
2.971
3.084
3.224
3.359
n
W







.5733
.5831
.5968
.6156
.6407
.6726
.7108
.7537
.7761
.7984
.8203
.8413
.8795
.9114
.9248
.9365
.9553
.9690
.9788
.9855
.9902
.9934
.9955
.9970
.9980
.9986







= 5
.5521)
GI







-4.013
-3.698
-3.383
-3.113
-2.874
-2.558
-2.181
-1.815
-1.635
-1.418
-1.200
-0.970
-0.513
-0.057
0.174
0.374
0.745
1.087
1.403
1.673
1.907
2.136
2.455
2.850
3.245
3.640







n = 6
(s = .4963)
W GI












.6318
.6748
.7230
.7482
.7733
.7979
.8215
.8645
.9004
.9154
.9285
.9498
.9652
.9761
.9837
.9890
.9926
.9950
.9966
.9977




















-3.719
-2.878
-2.273
-2.068
-1.858
-1.614
-1.383
-0.842
-0.349
-0.075
0.182
0.653
1.045
1.440
1.838
2.170
2.512
2.748
3.090
3.540








Source: Wilk & Shapiro (1968)
Footnote. The notation n = 3(1)6 is shorthand for n from 3 to 6 in unit steps
                                        D-9
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
       Table 12-1. a-Level Critical Points for Dixon's Outlier Test, n = 3(1)25
n\a
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
.01
0.988
0.889
0.780
0.698
0.637
0.683
0.635
0.597
0.679
0.642
0.615
0.641
0.616
0.595
0.577
0.561
0.547
0.535
0.524
0.514
0.505
0.497
0.489
.05
0.941
0.765
0.642
0.560
0.507
0.554
0.512
0.477
0.576
0.546
0.521
0.546
0.525
0.507
0.490
0.475
0.462
0.450
0.440
0.430
0.421
0.413
0.406
.10
0.886
0.679
0.557
0.482
0.434
0.479
0.441
0.409
0.517
0.490
0.467
0.492
0.472
0.454
0.438
0.424
0.412
0.401
0.391
0.382
0.374
0.367
0.360
Source: USEPA (1998)
Footnote. The  notation n = 3(1)25 is shorthand for nfrom 3 to 25 in unit steps
                                        D-10
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
             Table 12-2. a-Level Critical Points for Rosner's Outlier Test
n\a
20


30


40


50


60


80


100


.05
2.83
2.52

3.05
2.67

3.17
2.77

3.27
2.85

3.34
2.90

3.45
2.97

3.52
3.03

k= 2
.01
3.09
2.76

3.35
2.92

3.52
2.98

3.61
3.08

3.70
3.17

3.80
3.23

3.87
3.28

.05
2.88
2.60
2.45
3.12
2.73
2.56
3.22
2.81
2.62
3.34
2.89
2.68
3.42
2.95
2.73
3.49
3.03
2.81
3.60
3.10
2.86
k = 3
.01
3.13
2.83
2.68
3.41
3.01
2.75
3.58
3.03
2.82
3.68
3.15
2.89
3.75
3.20
2.95
3.85
3.27
3.01
3.97
3.34
3.06
Source: Barnett & Lewis (1994)
Footnote, k = number of suspected outliers. Since k critical points are needed for each test, there
are 2 values under each k = 2 entry, 3 under each k = 3 entry, etc.
                                          D-ll
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
        Table 12-2. a-Level Critical Points for Rosner's Outlier Test (cont.)
n\a
20




30




40




50




60




80




100




.05
2.95
2.63
2.49
2.39

3.16
2.77
2.59
2.49

3.32
2.86
2.67
2.55

3.40
2.93
2.72
2.59

3.48
2.98
2.77
2.63

3.57
3.05
2.84
2.69

3.64
3.13
2.89
2.74

k = 4
.01
3.20
2.83
2.68
2.58

3.48
3.02
2.79
2.70

3.64
3.10
2.87
2.74

3.74
3.18
2.92
2.78

3.82
3.20
2.97
2.82

3.91
3.31
3.04
2.87

3.96
3.34
3.06
2.90

.05
2.97
2.65
2.51
2.42
2.37
3.19
2.78
2.60
2.51
2.45
3.31
2.88
2.69
2.55
2.47
3.45
2.96
2.74
2.61
2.52
3.51
3.01
2.77
2.65
2.56
3.61
3.11
2.86
2.72
2.62
3.70
3.16
2.91
2.77
2.67
k= 5
.01
3.18
2.89
2.69
2.61
2.57
3.48
3.03
2.80
2.74
2.62
3.63
3.13
2.89
2.74
2.65
3.77
3.21
2.94
2.79
2.70
3.81
3.24
2.96
2.83
2.72
3.93
3.36
3.08
2.89
2.76
4.01
3.42
3.10
2.93
2.84
                                       D-12
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
 Table 14-1. Approximate a-Level Critical Points for Rank von Neumann Ratio Test

                            for n = 4(1)30(2)50(5)100
n\a
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
32
34
36
38
40
42
44
46
48
50
55
60
65
70
75
80
85
90
95
100
.005

	
0.29
0.50
0.55
0.57
0.62
0.67
0.71
0.74
0.78
0.81
0.84
0.87
0.89
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.05
1.07
1.08
1.10
1.11
1.13
1.16
1.18
1.20
1.22
1.24
1.25
1.27
1.28
1.29
1.33
1.35
1.38
1.40
1.42
1.44
1.45
1.47
1.48
1.49
.01

	
0.46
0.54
0.62
0.67
0.72
0.77
0.81
0.84
0.87
0.90
0.93
0.96
0.98
1.01
1.03
1.05
1.07
1.09
1.10
1.12
1.13
1.15
1.16
1.18
1.19
1.21
1.23
1.25
1.27
1.29
1.30
1.32
1.33
1.35
1.36
1.39
1.41
1.43
1.45
1.47
1.49
1.50
1.52
1.53
1.54
.025

0.40
0.63
0.64
0.76
0.82
0.89
0.93
0.96
1.00
1.03
1.05
1.08
1.10
1.13
1.15
1.17
1.18
1.20
1.22
1.23
1.25
1.26
1.27
1.28
1.30
1.31
1.33
1.35
1.36
1.38
1.39
1.41
1.42
1.43
1.45
1.46
1.48
1.50
1.52
1.54
1.55
1.57
1.58
1.59
1.60
1.61
.05

0.70
0.80
0.86
0.93
0.98
1.04
1.08
1.11
1.14
1.17
1.19
1.21
1.24
1.26
1.27
1.29
1.31
1.32
1.33
1.35
1.36
1.37
1.38
1.39
1.40
1.41
1.43
1.45
1.46
1.48
1.49
1.50
1.51
1.52
1.53
1.54
1.56
1.58
1.60
1.61
1.62
1.64
1.65
1.66
1.66
1.67
.10
0.60
	
0.97
1.11
1.14
1.18
1.23
1.26
1.29
1.32
1.34
1.36
1.38
1.40
1.41
1.43
1.44
1.45
1.46
1.48
1.49
1.50
1.51
1.51
1.52
1.53
1.54
1.55
1.57
1.58
1.59
1.60
1.61
1.62
1.63
1.63
1.64
1.66
1.67
1.68
1.70
1.71
1.71
1.72
1.73
1.74
1.74
Sources: Bartels (1982), Madansky (1988)
Footnote. The notation n = 4(1)30(2)50(5)100 is shorthand for n from 4 to 30 in unit steps, then
from 30 to 50 by 2's, then from 50 to 100 by 5's
                                       D-13
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables                               Unified Guidance
                     This page intentionally left blank
                                       D-14                              March 2009

-------
Appendix D.  Chapters 10 to 18 Tables
                                                                                           Unified Guidance
                                Table 16-1. Percentiles of Student's t-Distribution
 df\P
.75
        .80
.85
.90
.95
.96
.97
.975
.98
.9833
.9875
.99
.995
.999
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1.000
0.816
0.765
0.741
0.727
0.718
0.711
0.706
0.703
0.700
0.697
0.695
0.694
0.692
0.691
0.690
0.689
0.688
0.688
0.687
0.686
0.686
0.685
0.685
0.684
0.684
0.684
0.683
0.683
0.683
1.376
1.061
0.978
0.941
0.920
0.906
0.896
0.889
0.883
0.879
0.876
0.873
0.870
0.868
0.866
0.865
0.863
0.862
0.861
0.860
0.859
0.858
0.858
0.857
0.856
0.856
0.855
0.855
0.854
0.854
1.963
1.386
1.250
1.190
1.156
1.134
1.119
1.108
1.100
1.093
1.088
1.083
1.079
1.076
1.074
1.071
1.069
1.067
1.066
1.064
1.063
1.061
1.060
1.059
1.058
1.058
1.057
1.056
1.055
1.055
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
7.916
3.320
2.605
2.333
2.191
2.104
2.046
2.004
1.973
1.948
1.928
1.912
1.899
1.887
1.878
1.869
1.862
1.855
1.850
1.844
1.840
1.835
1.832
1.828
1.825
1.822
1.819
1.817
1.814
1.812
10.579
3.896
2.951
2.601
2.422
2.313
2.241
2.189
2.150
2.120
2.096
2.076
2.060
2.046
2.034
2.024
2.015
2.007
2.000
1.994
1.988
1.983
1.978
1.974
1.970
1.967
1.963
1.960
1.957
1.955
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
15.895
4.849
3.482
2.999
2.757
2.612
2.517
2.449
2.398
2.359
2.328
2.303
2.282
2.264
2.249
2.235
2.224
2.214
2.205
2.197
2.189
2.183
2.177
2.172
2.167
2.162
2.158
2.154
2.150
2.147
19.043
5.334
3.738
3.184
2.910
2.748
2.640
2.565
2.508
2.465
2.430
2.402
2.379
2.359
2.342
2.327
2.315
2.303
2.293
2.285
2.277
2.269
2.263
2.257
2.251
2.246
2.242
2.237
2.233
2.230
25.452
6.205
4.177
3.495
3.163
2.969
2.841
2.752
2.685
2.634
2.593
2.560
2.533
2.510
2.490
2.473
2.458
2.445
2.433
2.423
2.414
2.405
2.398
2.391
2.385
2.379
2.373
2.368
2.364
2.360
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
318.309
22.327
10.215
7.173
5.893
5.208
4.785
4.501
4.297
4.144
4.025
3.930
3.852
3.787
3.733
3.686
3.646
3.610
3.579
3.552
3.527
3.505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.385
                                                        D-15
                                                                                                          March 2009

-------
Appendix D.  Chapters 10 to 18 Tables
                                                                                           Unified Guidance
                             Table 16-1. Percentiles of Student's t-Distribution (cont.)
 df\P
.75
        .80
.85
.90
.95
.96
.97
.975
.98
.9833
.9875
.99
.995
.999
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
70
80
90
100
110
120
0.682
0.682
0.682
0.682
0.682
0.681
0.681
0.681
0.681
0.681
0.681
0.680
0.680
0.680
0.680
0.680
0.680
0.680
0.680
0.679
0.679
0.679
0.679
0.679
0.679
0.679
0.679
0.679
0.679
0.679
0.678
0.678
0.677
0.677
0.677
0.677
0.853
0.853
0.853
0.852
0.852
0.852
0.851
0.851
0.851
0.851
0.850
0.850
0.850
0.850
0.850
0.850
0.849
0.849
0.849
0.849
0.849
0.849
0.848
0.848
0.848
0.848
0.848
0.848
0.848
0.848
0.847
0.846
0.846
0.845
0.845
0.845
1.054
1.054
1.053
1.052
1.052
1.052
1.051
1.051
1.050
1.050
1.050
1.049
1.049
1.049
1.049
1.048
1.048
1.048
1.048
1.047
1.047
1.047
1.047
1.046
1.046
1.046
1.046
1.046
1.046
1.045
1.044
1.043
1.042
1.042
1.041
1.041
1.309
1.309
1.308
1.307
1.306
1.306
1.305
1.304
1.304
1.303
1.303
1.302
1.302
1.301
1.301
1.300
1.300
1.299
1.299
1.299
1.298
1.298
1.298
1.297
1.297
1.297
1.297
1.296
1.296
1.296
1.294
1.292
1.291
1.290
1.289
1.289
1.696
1.694
1.692
1.691
1.690
1.688
1.687
1.686
1.685
1.684
1.683
1.682
1.681
1.680
1.679
1.679
1.678
1.677
1.677
1.676
1.675
1.675
1.674
1.674
1.673
1.673
1.672
1.672
1.671
1.671
1.667
1.664
1.662
1.660
1.659
1.658
1.810
1.808
1.806
1.805
1.803
1.802
1.800
1.799
1.798
1.796
1.795
1.794
1.793
1.792
1.791
1.790
1.789
1.789
1.788
1.787
1.786
1.786
1.785
1.784
1.784
1.783
1.782
1.782
1.781
1.781
1.776
1.773
1.771
1.769
1.767
1.766
1.952
1.950
1.948
1.946
1.944
1.942
1.940
1.939
1.937
1.936
1.934
1.933
1.932
1.931
1.929
1.928
1.927
1.926
1.925
1.924
1.924
1.923
1.922
1.921
1.920
1.920
1.919
1.918
1.918
1.917
1.912
1.908
1.905
1.902
1.900
1.899
2.040
2.037
2.035
2.032
2.030
2.028
2.026
2.024
2.023
2.021
2.020
2.018
2.017
2.015
2.014
2.013
2.012
2.011
2.010
2.009
2.008
2.007
2.006
2.005
2.004
2.003
2.002
2.002
2.001
2.000
1.994
1.990
1.987
1.984
1.982
1.980
2.144
2.141
2.138
2.136
2.133
2.131
2.129
2.127
2.125
2.123
2.121
2.120
2.118
2.116
2.115
2.114
2.112
2.111
2.110
2.109
2.108
2.107
2.106
2.105
2.104
2.103
2.102
2.101
2.100
2.099
2.093
2.088
2.084
2.081
2.078
2.076
2.226
2.223
2.220
2.217
2.215
2.212
2.210
2.207
2.205
2.203
2.201
2.199
2.198
2.196
2.195
2.193
2.192
2.190
2.189
2.188
2.186
2.185
2.184
2.183
2.182
2.181
2.180
2.179
2.178
2.177
2.170
2.165
2.160
2.157
2.154
2.152
2.356
2.352
2.348
2.345
2.342
2.339
2.336
2.334
2.331
2.329
2.327
2.325
2.323
2.321
2.319
2.317
2.315
2.314
2.312
2.311
2.310
2.308
2.307
2.306
2.304
2.303
2.302
2.301
2.300
2.299
2.291
2.284
2.280
2.276
2.272
2.270
2.453
2.449
2.445
2.441
2.438
2.434
2.431
2.429
2.426
2.423
2.421
2.418
2.416
2.414
2.412
2.410
2.408
2.407
2.405
2.403
2.402
2.400
2.399
2.397
2.396
2.395
2.394
2.392
2.391
2.390
2.381
2.374
2.368
2.364
2.361
2.358
2.744
2.738
2.733
2.728
2.724
2.719
2.715
2.712
2.708
2.704
2.701
2.698
2.695
2.692
2.690
2.687
2.685
2.682
2.680
2.678
2.676
2.674
2.672
2.670
2.668
2.667
2.665
2.663
2.662
2.660
2.648
2.639
2.632
2.626
2.621
2.617
3.375
3.365
3.356
3.348
3.340
3.333
3.326
3.319
3.313
3.307
3.301
3.296
3.291
3.286
3.281
3.277
3.273
3.269
3.265
3.261
3.258
3.255
3.251
3.248
3.245
3.242
3.239
3.237
3.234
3.232
3.211
3.195
3.183
3.174
3.166
3.160
                                                        D-16
                                                                                                          March 2009

-------
Appendix D. Chapters 10 to 18 Tables
                         Unified Guidance
                            Table 17-1. Percentiles of F-Distribution for (1-a)  = .80
      V2\Vl
10
      11
12
13
                         14
                               15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
55
60
70
80
90
100
110
120
9.47
3.56
2.68
2.35
2.18
2.07
2.00
1.95
1.91
1.88
1.86
1.84
1.82
1.81
1.80
1.79
1.78
1.77
1.76
1.76
1.75
1.75
1.74
1.74
1.73
1.73
1.73
1.72
1.72
1.72
1.71
1.70
1.69
1.69
1.68
1.68
1.67
1.67
1.67
1.66
1.66
1.66
12.00
4.00
2.89
2.47
2.26
2.13
2.04
1.98
1.93
1.90
1.87
1.85
1.83
1.81
1.80
1.78
1.77
1.76
1.75
1.75
1.74
1.73
1.73
1.72
1.72
1.71
1.71
1.71
1.70
1.70
1.69
1.68
1.67
1.66
1.66
1.65
1.65
1.64
1.64
1.64
1.63
1.63
13.06
4.16
2.94
2.48
2.25
2.11
2.02
1.95
1.90
1.86
1.83
1.80
1.78
1.76
1.75
1.74
1.72
1.71
1.70
1.70
1.69
1.68
1.68
1.67
1.66
1.66
1.66
1.65
1.65
1.64
1.63
1.62
1.61
1.60
1.60
1.60
1.59
1.58
1.58
1.58
1.57
1.57
13.64
4.24
2.96
2.48
2.24
2.09
1.99
1.92
1.87
1.83
1.80
1.77
1.75
1.73
1.71
1.70
1.68
1.67
1.66
1.65
1.65
1.64
1.63
1.63
1.62
1.62
1.61
1.61
1.60
1.60
1.58
1.57
1.57
1.56
1.55
1.55
1.54
1.53
1.53
1.53
1.52
1.52
14.01
4.28
2.97
2.48
2.23
2.08
1.97
1.90
1.85
1.80
1.77
1.74
1.72
1.70
1.68
1.67
1.65
1.64
1.63
1.62
1.61
1.61
1.60
1.59
1.59
1.58
1.58
1.57
1.57
1.57
1.55
1.54
1.53
1.52
1.52
1.51
1.50
1.50
1.49
1.49
1.49
1.48
14.26
4.32
2.97
2.47
2.22
2.06
1.96
1.88
1.83
1.78
1.75
1.72
1.69
1.67
1.66
1.64
1.63
1.62
1.61
1.60
1.59
1.58
1.57
1.57
1.56
1.56
1.55
1.55
1.54
1.54
1.52
1.51
1.50
1.49
1.49
1.48
1.47
1.47
1.46
1.46
1.46
1.45
14.44
4.34
2.97
2.47
2.21
2.05
1.94
1.87
1.81
1.77
1.73
1.70
1.68
1.65
1.64
1.62
1.61
1.60
1.58
1.58
1.57
1.56
1.55
1.55
1.54
1.53
1.53
1.52
1.52
1.52
1.50
1.49
1.48
1.47
1.46
1.46
1.45
1.44
1.44
1.43
1.43
1.43
14.58
4.36
2.98
2.47
2.20
2.04
1.93
1.86
1.80
1.75
1.72
1.69
1.66
1.64
1.62
1.61
1.59
1.58
1.57
1.56
1.55
1.54
1.53
1.53
1.52
1.52
1.51
1.51
1.50
1.50
1.48
1.47
1.46
1.45
1.44
1.44
1.43
1.42
1.42
1.41
1.41
1.41
14.68
4.37
2.98
2.46
2.20
2.03
1.93
1.85
1.79
1.74
1.70
1.67
1.65
1.63
1.61
1.59
1.58
1.56
1.55
1.54
1.53
1.53
1.52
1.51
1.51
1.50
1.49
1.49
1.49
1.48
1.46
1.45
1.44
1.43
1.43
1.42
1.41
1.41
1.40
1.40
1.39
1.39
14.77
4.38
2.98
2.46
2.19
2.03
1.92
1.84
1.78
1.73
1.69
1.66
1.64
1.62
1.60
1.58
1.57
1.55
1.54
1.53
1.52
1.51
1.51
1.50
1.49
1.49
1.48
1.48
1.47
1.47
1.45
1.44
1.43
1.42
1.41
1.41
1.40
1.39
1.38
1.38
1.38
1.37
14.84
4.39
2.98
2.46
2.19
2.02
1.91
1.83
1.77
1.72
1.69
1.65
1.63
1.61
1.59
1.57
1.56
1.54
1.53
1.52
1.51
1.50
1.50
1.49
1.48
1.48
1.47
1.47
1.46
1.46
1.44
1.42
1.41
1.41
1.40
1.39
1.38
1.38
1.37
1.37
1.36
1.36
14.90
4.40
2.98
2.46
2.18
2.02
1.91
1.83
1.76
1.72
1.68
1.65
1.62
1.60
1.58
1.56
1.55
1.53
1.52
1.51
1.50
1.49
1.49
1.48
1.47
1.47
1.46
1.46
1.45
1.45
1.43
1.41
1.40
1.39
1.39
1.38
1.37
1.37
1.36
1.36
1.35
1.35
14.95
4.40
2.98
2.45
2.18
2.01
1.90
1.82
1.76
1.71
1.67
1.64
1.61
1.59
1.57
1.55
1.54
1.53
1.51
1.50
1.49
1.49
1.48
1.47
1.46
1.46
1.45
1.45
1.44
1.44
1.42
1.40
1.39
1.38
1.38
1.37
1.36
1.36
1.35
1.35
1.34
1.34
15.00
4.41
2.98
2.45
2.18
2.01
1.90
1.82
1.75
1.70
1.67
1.63
1.61
1.58
1.56
1.55
1.53
1.52
1.51
1.50
1.49
1.48
1.47
1.46
1.46
1.45
1.44
1.44
1.43
1.43
1.41
1.40
1.38
1.38
1.37
1.36
1.35
1.35
1.34
1.34
1.33
1.33
15.04
4.42
2.98
2.45
2.18
2.01
1.89
1.81
1.75
1.70
1.66
1.63
1.60
1.58
1.56
1.54
1.53
1.51
1.50
1.49
1.48
1.47
1.46
1.46
1.45
1.44
1.44
1.43
1.43
1.42
1.40
1.39
1.38
1.37
1.36
1.35
1.35
1.34
1.33
1.33
1.32
1.32
                                                      D-17
                                                                                                       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
                            Table 17-1. Percentiles of F-Distribution for (1-a) = .90
V2\Vi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
55
60
70
80
90
100
110
120
1
39.86
8.53
5.54
4.54
4.06
3.78
3.59
3.46
3.36
3.29
3.23
3.18
3.14
3.10
3.07
3.05
3.03
3.01
2.99
2.97
2.96
2.95
2.94
2.93
2.92
2.91
2.90
2.89
2.89
2.88
2.85
2.84
2.82
2.81
2.80
2.79
2.78
2.77
2.76
2.76
2.75
2.75
2
49.50
9.00
5.46
4.32
3.78
3.46
3.26
3.11
3.01
2.92
2.86
2.81
2.76
2.73
2.70
2.67
2.64
2.62
2.61
2.59
2.57
2.56
2.55
2.54
2.53
2.52
2.51
2.50
2.50
2.49
2.46
2.44
2.42
2.41
2.40
2.39
2.38
2.37
2.36
2.36
2.35
2.35
3
53.59
9.16
5.39
4.19
3.62
3.29
3.07
2.92
2.81
2.73
2.66
2.61
2.56
2.52
2.49
2.46
2.44
2.42
2.40
2.38
2.36
2.35
2.34
2.33
2.32
2.31
2.30
2.29
2.28
2.28
2.25
2.23
2.21
2.20
2.19
2.18
2.16
2.15
2.15
2.14
2.13
2.13
4
55.83
9.24
5.34
4.11
3.52
3.18
2.96
2.81
2.69
2.61
2.54
2.48
2.43
2.39
2.36
2.33
2.31
2.29
2.27
2.25
2.23
2.22
2.21
2.19
2.18
2.17
2.17
2.16
2.15
2.14
2.11
2.09
2.07
2.06
2.05
2.04
2.03
2.02
2.01
2.00
2.00
1.99
5
57.24
9.29
5.31
4.05
3.45
3.11
2.88
2.73
2.61
2.52
2.45
2.39
2.35
2.31
2.27
2.24
2.22
2.20
2.18
2.16
2.14
2.13
2.11
2.10
2.09
2.08
2.07
2.06
2.06
2.05
2.02
2.00
1.98
1.97
1.95
1.95
1.93
1.92
1.91
1.91
1.90
1.90
6
58.20
9.33
5.28
4.01
3.40
3.05
2.83
2.67
2.55
2.46
2.39
2.33
2.28
2.24
2.21
2.18
2.15
2.13
2.11
2.09
2.08
2.06
2.05
2.04
2.02
2.01
2.00
2.00
1.99
1.98
1.95
1.93
1.91
1.90
1.88
1.87
1.86
1.85
1.84
1.83
1.83
1.82
7
58.91
9.35
5.27
3.98
3.37
3.01
2.78
2.62
2.51
2.41
2.34
2.28
2.23
2.19
2.16
2.13
2.10
2.08
2.06
2.04
2.02
2.01
1.99
1.98
1.97
1.96
1.95
1.94
1.93
1.93
1.90
1.87
1.85
1.84
1.83
1.82
1.80
1.79
1.78
1.78
1.77
1.77
8
59.44
9.37
5.25
3.95
3.34
2.98
2.75
2.59
2.47
2.38
2.30
2.24
2.20
2.15
2.12
2.09
2.06
2.04
2.02
2.00
1.98
1.97
1.95
1.94
1.93
1.92
1.91
1.90
1.89
1.88
1.85
1.83
1.81
1.80
1.78
1.77
1.76
1.75
1.74
1.73
1.73
1.72
9
59.86
9.38
5.24
3.94
3.32
2.96
2.72
2.56
2.44
2.35
2.27
2.21
2.16
2.12
2.09
2.06
2.03
2.00
1.98
1.96
1.95
1.93
1.92
1.91
1.89
1.88
1.87
1.87
1.86
1.85
1.82
1.79
1.77
1.76
1.75
1.74
1.72
1.71
1.70
1.69
1.69
1.68
10
60.19
9.39
5.23
3.92
3.30
2.94
2.70
2.54
2.42
2.32
2.25
2.19
2.14
2.10
2.06
2.03
2.00
1.98
1.96
1.94
1.92
1.90
1.89
1.88
1.87
1.86
1.85
1.84
1.83
1.82
1.79
1.76
1.74
1.73
1.72
1.71
1.69
1.68
1.67
1.66
1.66
1.65
11
60.47
9.40
5.22
3.91
3.28
2.92
2.68
2.52
2.40
2.30
2.23
2.17
2.12
2.07
2.04
2.01
1.98
1.95
1.93
1.91
1.90
1.88
1.87
1.85
1.84
1.83
1.82
1.81
1.80
1.79
1.76
1.74
1.72
1.70
1.69
1.68
1.66
1.65
1.64
1.64
1.63
1.63
12
60.71
9.41
5.22
3.90
3.27
2.90
2.67
2.50
2.38
2.28
2.21
2.15
2.10
2.05
2.02
1.99
1.96
1.93
1.91
1.89
1.87
1.86
1.84
1.83
1.82
1.81
1.80
1.79
1.78
1.77
1.74
1.71
1.70
1.68
1.67
1.66
1.64
1.63
1.62
1.61
1.61
1.60
13
60.90
9.41
5.21
3.89
3.26
2.89
2.65
2.49
2.36
2.27
2.19
2.13
2.08
2.04
2.00
1.97
1.94
1.92
1.89
1.87
1.86
1.84
1.83
1.81
1.80
1.79
1.78
1.77
1.76
1.75
1.72
1.70
1.68
1.66
1.65
1.64
1.62
1.61
1.60
1.59
1.59
1.58
14
61.07
9.42
5.20
3.88
3.25
2.88
2.64
2.48
2.35
2.26
2.18
2.12
2.07
2.02
1.99
1.95
1.93
1.90
1.88
1.86
1.84
1.83
1.81
1.80
1.79
1.77
1.76
1.75
1.75
1.74
1.70
1.68
1.66
1.64
1.63
1.62
1.60
1.59
1.58
1.57
1.57
1.56
15
61.22
9.42
5.20
3.87
3.24
2.87
2.63
2.46
2.34
2.24
2.17
2.10
2.05
2.01
1.97
1.94
1.91
1.89
1.86
1.84
1.83
1.81
1.80
1.78
1.77
1.76
1.75
1.74
1.73
1.72
1.69
1.66
1.64
1.63
1.61
1.60
1.59
1.57
1.56
1.56
1.55
1.55

                                                     D-18
                                                                                                      March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
Table 17-1.
V2\Vi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
55
60
70
80
90
100
110
120
1
161.45
18.51
10.13
7.71
6.61
5.99
5.59
5.32
5.12
4.96
4.84
4.75
4.67
4.60
4.54
4.49
4.45
4.41
4.38
4.35
4.32
4.30
4.28
4.26
4.24
4.23
4.21
4.20
4.18
4.17
4.12
4.08
4.06
4.03
4.02
4.00
3.98
3.96
3.95
3.94
3.93
3.92
2
199.50
19.00
9.55
6.94
5.79
5.14
4.74
4.46
4.26
4.10
3.98
3.89
3.81
3.74
3.68
3.63
3.59
3.55
3.52
3.49
3.47
3.44
3.42
3.40
3.39
3.37
3.35
3.34
3.33
3.32
3.27
3.23
3.20
3.18
3.16
3.15
3.13
3.11
3.10
3.09
3.08
3.07
3
215.71
19.16
9.28
6.59
5.41
4.76
4.35
4.07
3.86
3.71
3.59
3.49
3.41
3.34
3.29
3.24
3.20
3.16
3.13
3.10
3.07
3.05
3.03
3.01
2.99
2.98
2.96
2.95
2.93
2.92
2.87
2.84
2.81
2.79
2.77
2.76
2.74
2.72
2.71
2.70
2.69
2.68
4
224.58
19.25
9.12
6.39
5.19
4.53
4.12
3.84
3.63
3.48
3.36
3.26
3.18
3.11
3.06
3.01
2.96
2.93
2.90
2.87
2.84
2.82
2.80
2.78
2.76
2.74
2.73
2.71
2.70
2.69
2.64
2.61
2.58
2.56
2.54
2.53
2.50
2.49
2.47
2.46
2.45
2.45
5
230.16
19.30
9.01
6.26
5.05
4.39
3.97
3.69
3.48
3.33
3.20
3.11
3.03
2.96
2.90
2.85
2.81
2.77
2.74
2.71
2.68
2.66
2.64
2.62
2.60
2.59
2.57
2.56
2.55
2.53
2.49
2.45
2.42
2.40
2.38
2.37
2.35
2.33
2.32
2.31
2.30
2.29
Percent! les of F- Distribution for (1-a)
6
233.99
19.33
8.94
6.16
4.95
4.28
3.87
3.58
3.37
3.22
3.09
3.00
2.92
2.85
2.79
2.74
2.70
2.66
2.63
2.60
2.57
2.55
2.53
2.51
2.49
2.47
2.46
2.45
2.43
2.42
2.37
2.34
2.31
2.29
2.27
2.25
2.23
2.21
2.20
2.19
2.18
2.18
7
236.77
19.35
8.89
6.09
4.88
4.21
3.79
3.50
3.29
3.14
3.01
2.91
2.83
2.76
2.71
2.66
2.61
2.58
2.54
2.51
2.49
2.46
2.44
2.42
2.40
2.39
2.37
2.36
2.35
2.33
2.29
2.25
2.22
2.20
2.18
2.17
2.14
2.13
2.11
2.10
2.09
2.09
8
238.88
19.37
8.85
6.04
4.82
4.15
3.73
3.44
3.23
3.07
2.95
2.85
2.77
2.70
2.64
2.59
2.55
2.51
2.48
2.45
2.42
2.40
2.37
2.36
2.34
2.32
2.31
2.29
2.28
2.27
2.22
2.18
2.15
2.13
2.11
2.10
2.07
2.06
2.04
2.03
2.02
2.02
9
240.54
19.38
8.81
6.00
4.77
4.10
3.68
3.39
3.18
3.02
2.90
2.80
2.71
2.65
2.59
2.54
2.49
2.46
2.42
2.39
2.37
2.34
2.32
2.30
2.28
2.27
2.25
2.24
2.22
2.21
2.16
2.12
2.10
2.07
2.06
2.04
2.02
2.00
1.99
1.97
1.97
1.96
10
241.88
19.40
8.79
5.96
4.74
4.06
3.64
3.35
3.14
2.98
2.85
2.75
2.67
2.60
2.54
2.49
2.45
2.41
2.38
2.35
2.32
2.30
2.27
2.25
2.24
2.22
2.20
2.19
2.18
2.16
2.11
2.08
2.05
2.03
2.01
1.99
1.97
1.95
1.94
1.93
1.92
1.91
= .95
11
242.98
19.40
8.76
5.94
4.70
4.03
3.60
3.31
3.10
2.94
2.82
2.72
2.63
2.57
2.51
2.46
2.41
2.37
2.34
2.31
2.28
2.26
2.24
2.22
2.20
2.18
2.17
2.15
2.14
2.13
2.07
2.04
2.01
1.99
1.97
1.95
1.93
1.91
1.90
1.89
1.88
1.87

12
243.91
19.41
8.74
5.91
4.68
4.00
3.57
3.28
3.07
2.91
2.79
2.69
2.60
2.53
2.48
2.42
2.38
2.34
2.31
2.28
2.25
2.23
2.20
2.18
2.16
2.15
2.13
2.12
2.10
2.09
2.04
2.00
1.97
1.95
1.93
1.92
1.89
1.88
1.86
1.85
1.84
1.83

13
244.69
19.42
8.73
5.89
4.66
3.98
3.55
3.26
3.05
2.89
2.76
2.66
2.58
2.51
2.45
2.40
2.35
2.31
2.28
2.25
2.22
2.20
2.18
2.15
2.14
2.12
2.10
2.09
2.08
2.06
2.01
1.97
1.94
1.92
1.90
1.89
1.86
1.84
1.83
1.82
1.81
1.80

14
245.36
19.42
8.71
5.87
4.64
3.96
3.53
3.24
3.03
2.86
2.74
2.64
2.55
2.48
2.42
2.37
2.33
2.29
2.26
2.22
2.20
2.17
2.15
2.13
2.11
2.09
2.08
2.06
2.05
2.04
1.99
1.95
1.92
1.89
1.88
1.86
1.84
1.82
1.80
1.79
1.78
1.78

15
245.95
19.43
8.70
5.86
4.62
3.94
3.51
3.22
3.01
2.85
2.72
2.62
2.53
2.46
2.40
2.35
2.31
2.27
2.23
2.20
2.18
2.15
2.13
2.11
2.09
2.07
2.06
2.04
2.03
2.01
1.96
1.92
1.89
1.87
1.85
1.84
1.81
1.79
1.78
1.77
1.76
1.75

                                                      D-19
                                                                                                       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
Table 17-1.
V2\Vi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
55
60
70
80
90
100
110
120
1
1012.55
48.51
20.62
14.04
11.32
9.88
8.99
8.39
7.96
7.64
7.39
7.19
7.02
6.89
6.77
6.67
6.59
6.51
6.45
6.39
6.34
6.29
6.25
6.21
6.18
6.14
6.11
6.09
6.06
6.04
5.94
5.87
5.82
5.78
5.74
5.71
5.67
5.64
5.61
5.59
5.57
5.56
2
1249.50
49.00
18.86
12.14
9.45
8.05
7.20
6.64
6.23
5.93
5.70
5.52
5.37
5.24
5.14
5.05
4.97
4.90
4.84
4.79
4.74
4.70
4.66
4.63
4.59
4.56
4.54
4.51
4.49
4.47
4.38
4.32
4.27
4.23
4.20
4.18
4.14
4.11
4.09
4.07
4.05
4.04
3
1350.50
49.17
18.11
11.34
8.67
7.29
6.45
5.90
5.51
5.22
4.99
4.81
4.67
4.55
4.45
4.36
4.29
4.22
4.16
4.11
4.07
4.03
3.99
3.96
3.93
3.90
3.87
3.85
3.83
3.81
3.73
3.67
3.62
3.59
3.56
3.53
3.49
3.47
3.45
3.43
3.41
3.40
4
1405.83
49.25
17.69
10.90
8.23
6.86
6.03
5.49
5.10
4.82
4.59
4.42
4.28
4.16
4.06
3.97
3.90
3.84
3.78
3.73
3.69
3.65
3.61
3.58
3.55
3.52
3.50
3.47
3.45
3.43
3.35
3.30
3.25
3.22
3.19
3.16
3.13
3.10
3.08
3.06
3.05
3.04
5
1440.61
49.30
17.43
10.62
7.95
6.58
5.76
5.22
4.84
4.55
4.34
4.16
4.02
3.90
3.81
3.72
3.65
3.59
3.53
3.48
3.44
3.40
3.36
3.33
3.30
3.28
3.25
3.23
3.21
3.19
3.11
3.05
3.01
2.97
2.94
2.92
2.88
2.86
2.84
2.82
2.81
2.80
Percent! les of F- Distribution for (1-a)
6
1464.45
49.33
17.25
10.42
7.76
6.39
5.58
5.04
4.65
4.37
4.15
3.98
3.84
3.72
3.63
3.54
3.47
3.41
3.35
3.30
3.26
3.22
3.19
3.15
3.13
3.10
3.07
3.05
3.03
3.01
2.93
2.88
2.83
2.80
2.77
2.75
2.71
2.68
2.66
2.65
2.63
2.62
7
1481.80
49.36
17.11
10.27
7.61
6.25
5.44
4.90
4.52
4.23
4.02
3.85
3.71
3.59
3.49
3.41
3.34
3.27
3.22
3.17
3.13
3.09
3.05
3.02
2.99
2.97
2.94
2.92
2.90
2.88
2.80
2.74
2.70
2.67
2.64
2.62
2.58
2.55
2.53
2.52
2.50
2.49
8
1494.99
49.37
17.01
10.16
7.50
6.14
5.33
4.79
4.41
4.13
3.91
3.74
3.60
3.48
3.39
3.30
3.23
3.17
3.12
3.07
3.02
2.99
2.95
2.92
2.89
2.86
2.84
2.82
2.80
2.78
2.70
2.64
2.60
2.56
2.54
2.51
2.48
2.45
2.43
2.41
2.40
2.39
9
1505.34
49.39
16.93
10.07
7.42
6.05
5.24
4.70
4.33
4.04
3.83
3.66
3.52
3.40
3.30
3.22
3.15
3.09
3.03
2.98
2.94
2.90
2.87
2.83
2.81
2.78
2.76
2.73
2.71
2.69
2.61
2.56
2.51
2.48
2.45
2.43
2.39
2.37
2.35
2.33
2.32
2.30
10
1513.69
49.40
16.86
10.00
7.34
5.98
5.17
4.63
4.26
3.97
3.76
3.59
3.45
3.33
3.23
3.15
3.08
3.02
2.96
2.91
2.87
2.83
2.80
2.77
2.74
2.71
2.69
2.66
2.64
2.62
2.55
2.49
2.44
2.41
2.38
2.36
2.32
2.30
2.28
2.26
2.25
2.23
= .98
11
1520.56
49.41
16.81
9.94
7.28
5.93
5.11
4.58
4.20
3.92
3.70
3.53
3.39
3.27
3.18
3.09
3.02
2.96
2.91
2.86
2.81
2.77
2.74
2.71
2.68
2.65
2.63
2.61
2.58
2.57
2.49
2.43
2.39
2.35
2.32
2.30
2.26
2.24
2.22
2.20
2.19
2.18

12
1526.31
49.42
16.76
9.89
7.23
5.88
5.06
4.53
4.15
3.87
3.65
3.48
3.34
3.23
3.13
3.05
2.97
2.91
2.86
2.81
2.76
2.73
2.69
2.66
2.63
2.60
2.58
2.56
2.54
2.52
2.44
2.38
2.34
2.30
2.27
2.25
2.21
2.19
2.17
2.15
2.14
2.12

13
1531.20
49.42
16.72
9.85
7.19
5.83
5.02
4.49
4.11
3.83
3.61
3.44
3.30
3.18
3.09
3.00
2.93
2.87
2.81
2.77
2.72
2.68
2.65
2.62
2.59
2.56
2.54
2.51
2.49
2.47
2.39
2.34
2.29
2.26
2.23
2.21
2.17
2.14
2.12
2.10
2.09
2.08

14
1535.40
49.43
16.69
9.81
7.16
5.80
4.98
4.45
4.07
3.79
3.57
3.40
3.26
3.15
3.05
2.97
2.89
2.83
2.78
2.73
2.68
2.65
2.61
2.58
2.55
2.52
2.50
2.48
2.45
2.44
2.36
2.30
2.25
2.22
2.19
2.17
2.13
2.10
2.08
2.07
2.05
2.04

15
1539.05
49.43
16.66
9.78
7.12
5.76
4.95
4.42
4.04
3.76
3.54
3.37
3.23
3.11
3.02
2.93
2.86
2.80
2.74
2.70
2.65
2.61
2.58
2.55
2.52
2.49
2.46
2.44
2.42
2.40
2.32
2.26
2.22
2.18
2.16
2.13
2.10
2.07
2.05
2.03
2.02
2.01

                                                      D-20
                                                                                                       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
Table 17-1.
V2\Vi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
55
60
70
80
90
100
110
120
1
4052.18
98.50
34.12
21.20
16.26
13.75
12.25
11.26
10.56
10.04
9.65
9.33
9.07
8.86
8.68
8.53
8.40
8.29
8.18
8.10
8.02
7.95
7.88
7.82
7.77
7.72
7.68
7.64
7.60
7.56
7.42
7.31
7.23
7.17
7.12
7.08
7.01
6.96
6.93
6.90
6.87
6.85
2
4999.50
99.00
30.82
18.00
13.27
10.92
9.55
8.65
8.02
7.56
7.21
6.93
6.70
6.51
6.36
6.23
6.11
6.01
5.93
5.85
5.78
5.72
5.66
5.61
5.57
5.53
5.49
5.45
5.42
5.39
5.27
5.18
5.11
5.06
5.01
4.98
4.92
4.88
4.85
4.82
4.80
4.79
3
5403.35
99.17
29.46
16.69
12.06
9.78
8.45
7.59
6.99
6.55
6.22
5.95
5.74
5.56
5.42
5.29
5.18
5.09
5.01
4.94
4.87
4.82
4.76
4.72
4.68
4.64
4.60
4.57
4.54
4.51
4.40
4.31
4.25
4.20
4.16
4.13
4.07
4.04
4.01
3.98
3.96
3.95
4
5624.58
99.25
28.71
15.98
11.39
9.15
7.85
7.01
6.42
5.99
5.67
5.41
5.21
5.04
4.89
4.77
4.67
4.58
4.50
4.43
4.37
4.31
4.26
4.22
4.18
4.14
4.11
4.07
4.04
4.02
3.91
3.83
3.77
3.72
3.68
3.65
3.60
3.56
3.53
3.51
3.49
3.48
5
5763.65
99.30
28.24
15.52
10.97
8.75
7.46
6.63
6.06
5.64
5.32
5.06
4.86
4.69
4.56
4.44
4.34
4.25
4.17
4.10
4.04
3.99
3.94
3.90
3.85
3.82
3.78
3.75
3.73
3.70
3.59
3.51
3.45
3.41
3.37
3.34
3.29
3.26
3.23
3.21
3.19
3.17
Percent! les of F- Distribution for (1-a)
6
5858.99
99.33
27.91
15.21
10.67
8.47
7.19
6.37
5.80
5.39
5.07
4.82
4.62
4.46
4.32
4.20
4.10
4.01
3.94
3.87
3.81
3.76
3.71
3.67
3.63
3.59
3.56
3.53
3.50
3.47
3.37
3.29
3.23
3.19
3.15
3.12
3.07
3.04
3.01
2.99
2.97
2.96
7
5928.36
99.36
27.67
14.98
10.46
8.26
6.99
6.18
5.61
5.20
4.89
4.64
4.44
4.28
4.14
4.03
3.93
3.84
3.77
3.70
3.64
3.59
3.54
3.50
3.46
3.42
3.39
3.36
3.33
3.30
3.20
3.12
3.07
3.02
2.98
2.95
2.91
2.87
2.84
2.82
2.81
2.79
8
5981.07
99.37
27.49
14.80
10.29
8.10
6.84
6.03
5.47
5.06
4.74
4.50
4.30
4.14
4.00
3.89
3.79
3.71
3.63
3.56
3.51
3.45
3.41
3.36
3.32
3.29
3.26
3.23
3.20
3.17
3.07
2.99
2.94
2.89
2.85
2.82
2.78
2.74
2.72
2.69
2.68
2.66
9
6022.47
99.39
27.35
14.66
10.16
7.98
6.72
5.91
5.35
4.94
4.63
4.39
4.19
4.03
3.89
3.78
3.68
3.60
3.52
3.46
3.40
3.35
3.30
3.26
3.22
3.18
3.15
3.12
3.09
3.07
2.96
2.89
2.83
2.78
2.75
2.72
2.67
2.64
2.61
2.59
2.57
2.56
10
6055.85
99.40
27.23
14.55
10.05
7.87
6.62
5.81
5.26
4.85
4.54
4.30
4.10
3.94
3.80
3.69
3.59
3.51
3.43
3.37
3.31
3.26
3.21
3.17
3.13
3.09
3.06
3.03
3.00
2.98
2.88
2.80
2.74
2.70
2.66
2.63
2.59
2.55
2.52
2.50
2.49
2.47
= .99
11
6083.32
99.41
27.13
14.45
9.96
7.79
6.54
5.73
5.18
4.77
4.46
4.22
4.02
3.86
3.73
3.62
3.52
3.43
3.36
3.29
3.24
3.18
3.14
3.09
3.06
3.02
2.99
2.96
2.93
2.91
2.80
2.73
2.67
2.63
2.59
2.56
2.51
2.48
2.45
2.43
2.41
2.40

12
6106.32
99.42
27.05
14.37
9.89
7.72
6.47
5.67
5.11
4.71
4.40
4.16
3.96
3.80
3.67
3.55
3.46
3.37
3.30
3.23
3.17
3.12
3.07
3.03
2.99
2.96
2.93
2.90
2.87
2.84
2.74
2.66
2.61
2.56
2.53
2.50
2.45
2.42
2.39
2.37
2.35
2.34

13
6125.86
99.42
26.98
14.31
9.82
7.66
6.41
5.61
5.05
4.65
4.34
4.10
3.91
3.75
3.61
3.50
3.40
3.32
3.24
3.18
3.12
3.07
3.02
2.98
2.94
2.90
2.87
2.84
2.81
2.79
2.69
2.61
2.55
2.51
2.47
2.44
2.40
2.36
2.33
2.31
2.30
2.28

14
6142.67
99.43
26.92
14.25
9.77
7.60
6.36
5.56
5.01
4.60
4.29
4.05
3.86
3.70
3.56
3.45
3.35
3.27
3.19
3.13
3.07
3.02
2.97
2.93
2.89
2.86
2.82
2.79
2.77
2.74
2.64
2.56
2.51
2.46
2.42
2.39
2.35
2.31
2.29
2.27
2.25
2.23

15
6157.28
99.43
26.87
14.20
9.72
7.56
6.31
5.52
4.96
4.56
4.25
4.01
3.82
3.66
3.52
3.41
3.31
3.23
3.15
3.09
3.03
2.98
2.93
2.89
2.85
2.81
2.78
2.75
2.73
2.70
2.60
2.52
2.46
2.42
2.38
2.35
2.31
2.27
2.24
2.22
2.21
2.19

                                                      D-21
                                                                                                       March 2009

-------
Appendix D. Chapters 10 to 18 Tables                                                            Unified Guidance
                                   This page intentionally left blank
                                                     D-22
                                                                                                     March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
     Table 17-2. Percentiles of Chi-Square Distribution for df = 1(1)30(5)100
df \(l-a)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
0.90
2.706
4.605
6.251
7.779
9.236
10.645
12.017
13.362
14.684
15.987
17.275
18.549
19.812
21.064
22.307
23.542
24.769
25.989
27.204
28.412
29.615
30.813
32.007
33.196
34.382
35.563
36.741
37.916
39.087
40.256
46.059
51.805
57.505
63.167
68.796
74.397
79.973
85.527
91.061
96.578
102.079
107.565
113.038
118.498
0.95
3.841
5.991
7.815
9.488
11.070
12.592
14.067
15.507
16.919
18.307
19.675
21.026
22.362
23.685
24.996
26.296
27.587
28.869
30.144
31.410
32.671
33.924
35.172
36.415
37.652
38.885
40.113
41.337
42.557
43.773
49.802
55.758
61.656
67.505
73.311
79.082
84.821
90.531
96.217
101.879
107.522
113.145
118.752
124.342
0.975
5.024
7.378
9.348
11.143
12.833
14.449
16.013
17.535
19.023
20.483
21.920
23.337
24.736
26.119
27.488
28.845
30.191
31.526
32.852
34.170
35.479
36.781
38.076
39.364
40.646
41.923
43.195
44.461
45.722
46.979
53.203
59.342
65.410
71.420
77.380
83.298
89.177
95.023
100.839
106.629
112.393
118.136
123.858
129.561
0.98
5.412
7.824
9.837
11.668
13.388
15.033
16.622
18.168
19.679
21.161
22.618
24.054
25.472
26.873
28.259
29.633
30.995
32.346
33.687
35.020
36.343
37.659
38.968
40.270
41.566
42.856
44.140
45.419
46.693
47.962
54.244
60.436
66.555
72.613
78.619
84.580
90.501
96.388
102.243
108.069
113.871
119.648
125.405
131.142
0.99
6.635
9.210
11.345
13.277
15.086
16.812
18.475
20.090
21.666
23.209
24.725
26.217
27.688
29.141
30.578
32.000
33.409
34.805
36.191
37.566
38.932
40.289
41.638
42.980
44.314
45.642
46.963
48.278
49.588
50.892
57.342
63.691
69.957
76.154
82.292
88.379
94.422
100.425
106.393
112.329
118.236
124.116
129.973
135.807
Footnote. The notation df = 1 (1)30(5)100 is a shorthand for c/ffrom 1 to 30 by unit steps, then
from 35 to 100 by 5's
                                        D-23
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
 Table 17-3. Upper Tolerance Limit Factors With y Coverage for n = 4(1)30(5)100
95%
n\v
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
0.90
4.162
3.407
3.006
2.755
2.582
2.454
2.355
2.275
2.210
2.155
2.109
2.068
2.033
2.002
1.974
1.949
1.926
1.905
1.886
1.869
1.853
1.838
1.824
1.811
1.799
1.788
1.777
1.732
1.697
1.669
1.646
1.626
1.609
1.594
1.581
1.570
1.559
1.550
1.542
1.534
1.527
Confidence
0.95
5.144
4.203
3.708
3.399
3.187
3.031
2.911
2.815
2.736
2.671
2.614
2.566
2.524
2.486
2.453
2.423
2.396
2.371
2.349
2.328
2.309
2.292
2.275
2.260
2.246
2.232
2.220
2.167
2.125
2.092
2.065
2.042
2.022
2.005
1.990
1.976
1.964
1.954
1.944
1.935
1.927
99%
0.99
7.042
5.741
5.062
4.642
4.354
4.143
3.981
3.852
3.747
3.659
3.585
3.520
3.464
3.414
3.370
3.331
3.295
3.263
3.233
3.206
3.181
3.158
3.136
3.116
3.098
3.080
3.064
2.995
2.941
2.898
2.862
2.833
2.807
2.785
2.765
2.748
2.733
2.719
2.706
2.695
2.684
0.90
7.380
5.362
4.411
3.859
3.497
3.240
3.048
2.898
2.777
2.677
2.593
2.521
2.459
2.405
2.357
2.314
2.276
2.241
2.209
2.180
2.154
2.129
2.106
2.085
2.065
2.047
2.030
1.957
1.902
1.857
1.821
1.790
1.764
1.741
1.722
1.704
1.688
1.674
1.661
1.650
1.639
Confidence
0.95
9.083
6.578
5.406
4.728
4.285
3.972
3.738
3.556
3.410
3.290
3.189
3.102
3.028
2.963
2.905
2.854
2.808
2.766
2.729
2.694
2.662
2.633
2.606
2.581
2.558
2.536
2.515
2.430
2.364
2.312
2.269
2.233
2.202
2.176
2.153
2.132
2.114
2.097
2.082
2.069
2.056

0.99
12.387
8.939
7.335
6.412
5.812
5.389
5.074
4.829
4.633
4.472
4.337
4.222
4.123
4.037
3.960
3.892
3.832
3.777
3.727
3.681
3.640
3.601
3.566
3.533
3.502
3.473
3.447
3.334
3.249
3.180
3.125
3.078
3.038
3.004
2.974
2.947
2.924
2.902
2.883
2.866
2.850
Source of algorithm used to compute table: Odeh & Owen (1980)
Footnote. The notation n = 4(1)30(5)100 is a shorthand for n from 4 to 30 by unit steps, then
from 35 to 100 by 5's
                                        D-24
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
 Table 17-4. Minimum Coverage of Non-Parametric Upper Tolerance Limit for n =
                                  4(1)30(5)100
n\(l-cO
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
0.95
0.473
0.549
0.607
0.652
0.688
0.717
0.741
0.762
0.779
0.794
0.807
0.819
0.829
0.838
0.847
0.854
0.861
0.867
0.873
0.878
0.883
0.887
0.891
0.895
0.899
0.902
0.905
0.918
0.928
0.936
0.942
0.947
0.951
0.955
0.958
0.961
0.963
0.965
0.967
0.969
0.970
Maximum
0.99
0.316
0.398
0.464
0.518
0.562
0.599
0.631
0.658
0.681
0.702
0.720
0.736
0.750
0.763
0.774
0.785
0.794
0.803
0.811
0.819
0.825
0.832
0.838
0.843
0.848
0.853
0.858
0.877
0.891
0.903
0.912
0.920
0.926
0.932
0.936
0.940
0.944
0.947
0.950
0.953
0.955
2nd
0.95
0.248
0.342
0.418
0.479
0.529
0.570
0.605
0.635
0.661
0.683
0.703
0.720
0.736
0.749
0.762
0.773
0.783
0.793
0.801
0.809
0.817
0.823
0.830
0.836
0.841
0.846
0.851
0.871
0.886
0.898
0.908
0.916
0.923
0.929
0.934
0.938
0.942
0.945
0.948
0.951
0.953
Largest
0.99
0.140
0.222
0.294
0.356
0.410
0.455
0.495
0.530
0.560
0.587
0.610
0.632
0.651
0.668
0.683
0.698
0.711
0.723
0.734
0.744
0.753
0.762
0.770
0.778
0.785
0.792
0.798
0.824
0.845
0.861
0.874
0.885
0.894
0.902
0.908
0.914
0.919
0.924
0.928
0.932
0.935
Footnotes. Maximum, 2nd Largest refer to Largest and next largest sample values
The notation n = 4(1)30(5)100 is a shorthand for n from 4 to 30 by unit steps, then from 35 to
100 by 5's
                                       D-25
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables                               Unified Guidance
                    This page intentionally left blank
                                       D-26                              March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
                 Table 17-5. Significance Levels (a) for Mann-Kendall Trend Test for n = 4(1)10
n = 4
S a
0 0.6250
2 0.3750
4 0.1670
6 0.0420



















n = 5
S a
0 0.5920
2 0.4080
4 0.2420
6 0.1170
8 0.0420
10 0.0083

















n = 6
S a
1 0.5000
3 0.3600
5 0.2350
7 0.1360
9 0.0680
11 0.0280
13 0.0083
15 0.0014















n = 7
S a
1 0.5000
3 0.3860
5 0.2810
7 0.1910
9 0.1190
11 0.0680
13 0.0350
15 0.0150
17 0.0054
19 0.0014
21 0.0002












n = 8
S a
0 0.5480
2 0.4520
4 0.3600
6 0.2740
8 0.1990
10 0.1380
12 0.0890
14 0.0540
16 0.0310
18 0.0160
20 0.0071
22 0.0028
24 0.0009
26 0.0002
28 0.0000








n = 9
S a
0 0.5400
2 0.4600
4 0.3810
6 0.3060
8 0.2380
10 0.1790
12 0.1300
14 0.0900
16 0.0600
18 0.0380
20 0.0220
22 0.0120
24 0.0063
26 0.0029
28 0.0012
30 0.0004
32 0.0001
34 0.0000
36 0.0000




n = 10
S a
1 0.5000
3 0.4310
5 0.3640
7 0.3000
9 0.2420
11 0.1900
13 0.1460
15 0.1080
17 0.0780
19 0.0540
21 0.0360
23 0.0230
25 0.0140
27 0.0083
29 0.0046
31 0.0023
33 0.0011
35 0.0005
37 0.0002
39 0.0001
41 0.0000
43 0.0000
45 0.0000
Source: Gilbert (1987)
Footnote: Notation n = 4(1)10 is shorthand for n from 4 to 10 by unit steps
                                                      D-27
                                                                                                       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
Table 18-1. Confidence Levels of Non-Parametric Prediction Limits for Next m Values (PL = j'th Order Statistic)
for n = 4(1)60
n m = 1
4 0.800
5 0.833
6 0.857
7 0.875
8 0.889
9 0.900
10 0.909
11 0.917
12 0.923
13 0.929
14 0.933
15 0.938
16 0.941
17 0.944
18 0.947
19 0.950
20 0.952
21 0.955
22 0.957
23 0.958
24 0.960
25 0.962
26 0.963
27 0.964
28 0.966
29 0.967
j =
m = 2
0.667
0.714
0.750
0.778
0.800
0.818
0.833
0.846
0.857
0.867
0.875
0.882
0.889
0.895
0.900
0.905
0.909
0.913
0.917
0.920
0.923
0.926
0.929
0.931
0.933
0.935
n
m = 3
0.571
0.625
0.667
0.700
0.727
0.750
0.769
0.786
0.800
0.812
0.824
0.833
0.842
0.850
0.857
0.864
0.870
0.875
0.880
0.885
0.889
0.893
0.897
0.900
0.903
0.906
m = 4
0.500
0.556
0.600
0.636
0.667
0.692
0.714
0.733
0.750
0.765
0.778
0.789
0.800
0.810
0.818
0.826
0.833
0.840
0.846
0.852
0.857
0.862
0.867
0.871
0.875
0.879
m = 1
0.600
0.667
0.714
0.750
0.778
0.800
0.818
0.833
0.846
0.857
0.867
0.875
0.882
0.889
0.895
0.900
0.905
0.909
0.913
0.917
0.920
0.923
0.926
0.929
0.931
0.933
j =
m = 2
0.400
0.476
0.536
0.583
0.622
0.655
0.682
0.705
0.725
0.743
0.758
0.772
0.784
0.795
0.805
0.814
0.823
0.830
0.837
0.843
0.849
0.855
0.860
0.865
0.869
0.873
n-1
m = 3
0.286
0.357
0.417
0.467
0.509
0.545
0.577
0.604
0.629
0.650
0.669
0.686
0.702
0.716
0.729
0.740
0.751
0.761
0.770
0.778
0.786
0.794
0.800
0.807
0.813
0.819
m = 4
0.214
0.278
0.333
0.382
0.424
0.462
0.495
0.524
0.550
0.574
0.595
0.614
0.632
0.648
0.662
0.676
0.688
0.700
0.711
0.721
0.730
0.739
0.747
0.755
0.762
0.769
m = 1
0.400
0.500
0.571
0.625
0.667
0.700
0.727
0.750
0.769
0.786
0.800
0.812
0.824
0.833
0.842
0.850
0.857
0.864
0.870
0.875
0.880
0.885
0.889
0.893
0.897
0.900
j =
m = 2
0.200
0.286
0.357
0.417
0.467
0.509
0.545
0.577
0.604
0.629
0.650
0.669
0.686
0.702
0.716
0.729
0.740
0.751
0.761
0.770
0.778
0.786
0.794
0.800
0.807
0.813
n-2
m = 3
0.114
0.179
0.238
0.292
0.339
0.382
0.420
0.453
0.484
0.511
0.535
0.558
0.578
0.596
0.614
0.629
0.644
0.657
0.670
0.681
0.692
0.702
0.712
0.720
0.729
0.737
m = 4
0.071
0.119
0.167
0.212
0.255
0.294
0.330
0.363
0.393
0.421
0.446
0.470
0.491
0.511
0.530
0.547
0.563
0.578
0.592
0.605
0.618
0.629
0.640
0.651
0.660
0.670
Footnotes: Notation n = 4(1)60 is shorthand for n from 4 to 60 by unit steps
PL = Prediction Limit
                                                      D-28
                                                                                                       March 2009

-------
Appendix D. Chapters 10 to 18 Tables
Unified Guidance
         Table 18-1. Confidence Levels of Non-Parametric Prediction Limits for Next m Values (PL = j'th Order
                                          Statistic) for n = 4(1)60
n m = 1
30 0.968
31 0.969
32 0.970
33 0.971
34 0.971
35 0.972
36 0.973
37 0.974
38 0.974
39 0.975
40 0.976
41 0.976
42 0.977
43 0.977
44 0.978
45 0.978
46 0.979
47 0.979
48 0.980
49 0.980
50 0.980
51 0.981
52 0.981
53 0.981
54 0.982
55 0.982
56 0.982
57 0.983
58 0.983
59 0.983
60 0.984
j =
m = 2
0.938
0.939
0.941
0.943
0.944
0.946
0.947
0.949
0.950
0.951
0.952
0.953
0.955
0.956
0.957
0.957
0.958
0.959
0.960
0.961
0.962
0.962
0.963
0.964
0.964
0.965
0.966
0.966
0.967
0.967
0.968
n
m = 3
0.909
0.912
0.914
0.917
0.919
0.921
0.923
0.925
0.927
0.929
0.930
0.932
0.933
0.935
0.936
0.938
0.939
0.940
0.941
0.942
0.943
0.944
0.945
0.946
0.947
0.948
0.949
0.950
0.951
0.952
0.952
m = 4
0.882
0.886
0.889
0.892
0.895
0.897
0.900
0.902
0.905
0.907
0.909
0.911
0.913
0.915
0.917
0.918
0.920
0.922
0.923
0.925
0.926
0.927
0.929
0.930
0.931
0.932
0.933
0.934
0.935
0.937
0.938
m = 1
0.935
0.938
0.939
0.941
0.943
0.944
0.946
0.947
0.949
0.950
0.951
0.952
0.953
0.955
0.956
0.957
0.957
0.958
0.959
0.960
0.961
0.962
0.962
0.963
0.964
0.964
0.965
0.966
0.966
0.967
0.967
j =
m = 2
0.877
0.881
0.884
0.887
0.890
0.893
0.896
0.899
0.901
0.904
0.906
0.908
0.910
0.912
0.914
0.916
0.918
0.919
0.921
0.922
0.924
0.925
0.927
0.928
0.929
0.930
0.932
0.933
0.934
0.935
0.936
n-1
m = 3
0.824
0.829
0.834
0.838
0.842
0.846
0.850
0.854
0.857
0.861
0.864
0.867
0.870
0.872
0.875
0.878
0.880
0.882
0.885
0.887
0.889
0.891
0.893
0.895
0.897
0.898
0.900
0.902
0.903
0.905
0.906
m = 4
0.775
0.782
0.787
0.793
0.798
0.803
0.808
0.812
0.816
0.821
0.825
0.828
0.832
0.835
0.839
0.842
0.845
0.848
0.851
0.853
0.856
0.859
0.861
0.863
0.866
0.868
0.870
0.872
0.874
0.876
0.878
m = 1
0.903
0.906
0.909
0.912
0.914
0.917
0.919
0.921
0.923
0.925
0.927
0.929
0.930
0.932
0.933
0.935
0.936
0.938
0.939
0.940
0.941
0.942
0.943
0.944
0.945
0.946
0.947
0.948
0.949
0.950
0.951
j =
m = 2
0.819
0.824
0.829
0.834
0.838
0.842
0.846
0.850
0.854
0.857
0.861
0.864
0.867
0.870
0.872
0.875
0.878
0.880
0.882
0.885
0.887
0.889
0.891
0.893
0.895
0.897
0.898
0.900
0.902
0.903
0.905
n-2
m = 3
0.744
0.751
0.758
0.764
0.770
0.776
0.781
0.786
0.791
0.796
0.801
0.805
0.809
0.813
0.817
0.820
0.824
0.827
0.831
0.834
0.837
0.840
0.842
0.845
0.848
0.850
0.853
0.855
0.857
0.860
0.862
m = 4
0.678
0.687
0.695
0.702
0.709
0.716
0.723
0.729
0.735
0.741
0.746
0.751
0.756
0.761
0.766
0.770
0.774
0.779
0.783
0.786
0.790
0.794
0.797
0.801
0.804
0.807
0.810
0.813
0.816
0.819
0.821
                                                    D-29
                                                                                                   March 2009

-------
Appendix D. Chapters 10 to 18 Tables                                                            Unified Guidance
                                   This page intentionally left blank
                                                     D-30
                                                                                                     March 2009

-------
Appendix D.  Chapters 10 to 18 Tables
Unified Guidance
   Table 18-2. Confidence Levels for Non-Parametric Prediction Limit on Future
            Median of Order 3 (PL = jth Order Statistic) for n = 4(1)60
n
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
j = n
0.857
0.893
0.917
0.933
0.945
0.955
0.962
0.967
0.971
0.975
0.978
0.980
0.982
0.984
0.986
0.987
0.988
0.989
0.990
0.991
0.991
0.992
0.993
0.993
0.994
0.994
0.994
0.995
0.995
0.995
0.995
0.996
0.996
0.996
0.996
0.997
0.997
0.997
0.997
0.997
0.997
0.997
0.997
0.998
0.998
0.998
0.998
0.998
0.998
0.998
0.998
0.998
0.998
0.998
0.998
0.998
0.998
Footnotes: Notation n = 4(1)60 is shorthand
j = n-l
0.629
0.714
0.774
0.817
0.848
0.873
0.892
0.907
0.919
0.929
0.937
0.944
0.949
0.954
0.959
0.962
0.966
0.968
0.971
0.973
0.975
0.977
0.978
0.980
0.981
0.982
0.983
0.984
0.985
0.986
0.987
0.987
0.988
0.989
0.989
0.990
0.990
0.991
0.991
0.991
0.992
0.992
0.992
0.993
0.993
0.993
0.994
0.994
0.994
0.994
0.994
0.995
0.995
0.995
0.995
0.995
0.995
for n from 4 to
j = n-2
0.371
0.500
0.595
0.667
0.721
0.764
0.797
0.824
0.846
0.864
0.879
0.892
0.903
0.912
0.920
0.927
0.933
0.939
0.943
0.948
0.951
0.955
0.958
0.961
0.963
0.965
0.967
0.969
0.971
0.973
0.974
0.975
0.977
0.978
0.979
0.980
0.981
0.982
0.982
0.983
0.984
0.985
0.985
0.986
0.986
0.987
0.987
0.988
0.988
0.989
0.989
0.989
0.990
0.990
0.990
0.991
0.991
60 by unit steps;PL = Prediction Limit
                                      D-31
       March 2009

-------
Appendix D. Chapters 10 to 18 Tables                               Unified Guidance
                    This page intentionally left blank
                                       D-32                              March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations	Unified Guidance
                                         D  STATISTICAL  TABLES

D.2 TABLES  FROM CHAPTER 19:  INTERWELL PREDICTION  LIMITS  FOR FUTURE VALUES

    TABLE 19-1  K-Multipliers for 1 -of-2 Interwell Prediction Limits on Observations	D-34
    TABLE 19-2 /c-Multipliers for 1 -of-3 Interwell Prediction Limits on Observations	D-43
    TABLE 19-3 /c-Multipliers for 1 -of-4 Interwell Prediction Limits on Observations	D-52
    TABLE 19-4 K-Multipliers for Mod. Cal. Interwell Prediction Limits on Observations	D-61
                                                       D-33                                              March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                     Unified Guidance
        Table 19-1. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Observations (1 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.78 0.67 0.61 (
1.21 1.03 0.95 (
1.47 1.23 1.13
1.65 1.37 1.26
1.79 1.48 1.36
2.09 1.71 1.56
2.34 1.90 1.73
2.52 2.03 1.85
2.65 2.14 1.94
2.89 2.32 2.10
3.06 2.45 2.21 ;
3.19 2.54 2.29 ;
3.29 2.62 2.36 ;
3.41 2.71 2,44 ;
3.57 2.83 2.55 ;
3.69 2.92 2.63
3.79 3.00 2.69
3.87 3.06 2.75
3.93 3.11 2.79
).59 0.57 0.54 (
).90 0.88 0.84 (
.08 1.05 1.01 (
.20 1.16 1.12
.29 1.25 1.20
.48 1.43 1.37
.64 1.58 1.51
.75 1.68 1.61
.83 1.76 1.68
.98 1.90 1.81
2.08 2.00 1.90
2.16 2.08 1.97
1.22 2.13 2.03
2.30 2.21 2.10 ;
2.40 2.30 2.18 ;
2,47 2.37 2.25 ;
2:53 2,42 2.30 ;
2.58 2,47 2.34 ;
2.62 .2,51 2.38 ;
).53 (
).82 (
).98 (
.09
.17
.34
.47
.56
.64
.76
.85
.91
.97
2.03
Ml ;
2.18 ;
1.23 ;
2.27 ;
2.30 ;
).52 0.51 (
).81 0.80 (
).96 0.95 (
.07 1.06
.15 1.13
.31 1.29
.44 1.42
.53 1.51
.60 1.57
.72 1.69
.80 1.78
.87 1.84
.92 1.89
.98 1.95
2.06 2.02 ;
1.12 2.08 ;
2.17 2.13 ;
2.21 2.17 ;
2.24 2.20 ;
).51 (
).79 (
).94 (
.05
.12
.28
.40
.49
.56
.67
.75
.82
.86
.92
2.00
2.06 ;
>.io ;
M4 ;
M7 ;
).50 0.50 (
).79 0.78 (
).94 0.93 (
.04 1.03
.11 1.11
.27 1.26
.39 1.39
.48 1.47
.54 1.53
.66 1.65
.74 1.73
.80 1.79
.85 1.83
.91 1.89
.98 1.96
2.04 2.02 ;
2.08 2.07 ;
M2 2.10 ;
M5 2.13 ;
).50 0.50 0.49 (
).78 0.77 0.77 (
).93 0.92 0.92 (
.03 1.02 1.02
.10 1.10 1.09
.25 1.25 1.24
.38 1.37 1.36
.46 1.45 1.44
.53 1.52 1.51
.64 1.63 1.62
.72 1.70 1.69
.78 1.76 1.75
.82 1.81 1.80
.88 1.86 1.85
.95 1.94 1.92
2.01 1.99 1.98
2.05 2.03 2.02 ;
2.09 2.07 2.05 ;
M2 2.10 2.09 ;
).49 0.49 (
).77 0.77 (
).92 0.91 (
.01 1.01
.09 1.08
.24 1.23
.36 1.35
.44 1.43
.50 1.50
.61 1.60
.69 1.68
.74 1.74
.79 1.78
.84 1.84
.91 1.91
.97 1.96
2.01 2.00
2.04 2.04 ;
2.07 2.07 ;
).49 0.49 (
).76 0.76 (
).91 0.91 (
.01 1.00
.08 1.08
.23 1.22
.35 1.34
.43 1.42
.49 1.49
.60 1.59
.67 1.67
.73 1.72
.78 1.77
.83 1.82
.90 1.89
.95 1.94
.99 1.98
2.03 2.02 ;
2.06 2.05 ;
).49
).76
).91
.00
.07
.22
.34
.42
.48
.59
.66
.72
.76
.81
.88
.93
.97
2.01
2.04
     Table 19-1. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Observations (1 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4 6 8 10 12 16
1.21 1.03 0.95 0.90 0.88 0.84 (
1.65 1.37 1.26 .20 1.16 1.12
1.91 1.57 1.44 .37 1.32 1.27
2.09 1.71 1.56 .48 1.43 1.37
2.23 1.82 1.65 .57 1.51 1.45
2.52 2.03 1.85 .75 1.68 1.61
276 2.22 2.01 .90 1.83 1.74
2.93 2.35 2.12 2.00 1.93 1.83
3.06 2.45 2.21 2.08 2.00 1.90
329 2.62 2.36 2.22 2.13 2.03
345 2.74 2.47 2.32 2.23 2.12 ;
3.57 2.83 2.55 2.40 2.30 2.18 ;
3.67 2.91 2.61 2.46 2.36 2.24 ;
3.79 3.00 2.69 2.53 2.42 2.30 ;
3.93 3.11 2,79 2.62 2.51 2.38 ;
4.05 3.19 2,87 2.69 2.58 2.44 ;
4.14 3.26 2.93 2.75 2.63 2.49 ;
4.21 3.32 2.98 2,79 2.68 2.54 ;
4.28 3.37 3.02 2.84 2.72 2.57 ;

20
).82 (
.09
.24
.34
.41
.56
.69
.78
.85
.97
2.05 ;
Ml ;
M6 ;
2.23 ;
2.30 ;
2.36 ;
2.4i ;
2.45 ;
1.49 ;

25 30
).81 0.80 (
.07 1.06
.21 1.19
.31 1.29
.38 1.36
.53 1.51
.65 1.63
.74 1.71
.80 1.78
.92 1.89
2.00 1.97
2.06 2.02 ;
Ml 2.07 ;
M7 2.13 ;
1.24 2.20 ;
2.30 2.26 ;
2.35 2.30 ;
2.39 2.34 ;
1.42 2.37 ;

35
).79 (
.05
.18
.28
.35
.49
.61
.69
.75
.86
.94
2.00
2.05 ;
MO ;
M7 ;
2.23 ;
2.27 ;
2.3i ;
2.34 ;

40 45
).79 0.78 (
.04 1.03
.18 1.17
.27 1.26
.34 1.33
.48 1.47
.60 1.59
.68 1.67
.74 1.73
.85 1.83
.92 1.91
.98 1.96
2.03 2.01 ;
2.08 2.07 ;
2.15 2.13 ;
2.21 2.19 ;
2.25 2.23 ;
2.28 2.27 ;
2.32 2.30 ;

50 60 70
).78 0.77 0.77 (
.03 1.02 1.02
.16 1.16 1.15
.25 1.25 1.24
.32 1.31 1.31
.46 1.45 1.44
.58 1.57 1.56
.66 1.64 1.63
.72 1.70 1.69
.82 1.81 1.80
.90 1.88 1.87
.95 1.94 1.92
2.00 1.98 1.97
2.05 2.03 2.02 ;
2.12 2.10 2.09 ;
2.17 2.15 2.14 ;
2.22 2.19 2.18 ;
2.25 2.23 2.21 ;
2.28 2.26 2.24 ;

80 90
).77 0.77 (
.01 1.01
.15 1.14
.24 1.23
.30 1.30
.44 1.43
.55 1.55
.63 1.62
.69 1.68
.79 1.78
.86 1.85
.91 1.91
.96 1.95
2.01 2.00
2.07 2.07 ;
2.12 2.12 ;
2.17 2.16 ;
2.20 2.19 ;
2.23 2.22 ;

100 125
).76 0.76 (
.01 1.00
.14 1.14
.23 1.22
.30 1.29
.43 1.42
.54 1.53
.62 1.61
.67 1.67
.78 1.77
.85 1.84
.90 1.89
.94 1.93
.99 1.98
2.06 2.05 ;
2.11 2.10 ;
2.15 2.14 ;
2.18 2.17 ;
2.21 2.20 ;

150
).76
.00
.13
.22
.29
.42
.53
.60
.66
.76
.83
.88
.92
.97
2.04
2.09
2.13
2.16
2.19

                                                    D-34
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
Unified Guidance
       Table 19-1.  K-Multipliers for 1-of-2  Interwell Prediction Limits on Observations (1 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
1.65 1.37 1.26 .20 1.16 1.12
2.09 1.71 1.56 .48 1.43 1.37
2.34 1.90 1.73 .64 1.58 1.51
2.52 2.03 1.85 .75 1.68 1.61
2.65 2.14 1.94 .83 1.76 1.68
2.93 2.35 2.12 2.00 1.93 1.83
316 2.52 2.28 2.15 2.06 1.96
3.33 2.65 2.39 2.24 2.16 2.05
345 2.74 2.47 2.32 2.23 2.12 ;
3.67 2.91 2.61 2.46 2.36 2.24 ;
3.82 3.02 2.71 2.55 2.44 2.32 ;
3.93 3.11 2.79 2.62 2.51 2.38 ;
4.03 3.18 2.85 2.68 2.57 2.43 ;
4.14 3.26 2.93 2.75 2.63 2.49 ;
4.28 3.37 3.02 2.84 2.72 2.57 ;
4.39 3.45 3.10 2.90 2.78 2.63 ;
4.47 3.52 3,15 2.96 2.83 2.68 ;
4.54 3.57 3,20 3.00 2.87 2.72 ;
4.61 3.62 3.24 3.04 2.91 2.75 ;
20
.09
.34
.47
.56
.64
.78
.90
.99
2.05 ;
2.16 ;
1.24 ;
2.30 ;
2.35 ;
2.4i ;
1.49 ;
2.54 ;
2.59 ;
2.63 ;
2.66 ;
25 30
.07 1.06
.31 1.29
.44 1.42
.53 1.51
.60 1.57
.74 1.71
.86 1.83
.94 1.90
2.00 1.97
Ml 2.07 ;
2.19 2.15 ;
1.24 2.20 ;
1.29 2.25 ;
2.35 2.30 ;
1.42 2.37 ;
2.47 2.43 ;
2.52 2.47 ;
2.56 2.51 ;
2.59 2.54 ;
35
.05
.28
.40
.49
.56
.69
.80
.88
.94
2.05 ;
2.12 ;
2.17 ;
2.22 ;
2.27 ;
2.34 ;
2.39 ;
2.44 ;
2.47 ;
2.50 ;
40 45
.04 1.03
.27 1.26
.39 1.39
.48 1.47
.54 1.53
.68 1.67
.79 1.78
.86 1.85
.92 1.91
2.03 2.01 ;
2.10 2.08 ;
2.15 2.13 ;
2.20 2.18 ;
2.25 2.23 ;
2.32 2.30 ;
2.37 2.35 ;
2.41 2.39 ;
2.44 2.42 ;
2.47 2.45 ;
50 60 70
.03 1.02 1.02
.25 1.25 1.24
.38 1.37 1.36
.46 1.45 1.44
.53 1.52 1.51
.66 1.64 1.63
.77 1.75 1.74
.84 1.82 1.81
.90 1.88 1.87
2.00 1.98 1.97
2.07 2.05 2.03 ;
2.12 2.10 2.09 ;
2.16 2.14 2.13 ;
2.22 2.19 2.18 ;
2.28 2.26 2.24 ;
2.33 2.31 2.29 ;
2.37 2.35 2.33 ;
2.41 2.38 2.36 ;
2.44 2.41 2.39 ;
80 90
.01 1.01
.24 1.23
.36 1.35
.44 1.43
.50 1.50
.63 1.62
.73 1.73
.80 1.80
.86 1.85
.96 1.95
2.02 2.02 ;
2.07 2.07 ;
2.12 2.11 ;
2.17 2.16 ;
2.23 2.22 ;
2.28 2.27 ;
2.32 2.31 ;
2.35 2.34 ;
2.38 2.37 ;
100 125
.01 1.00
.23 1.22
.35 1.34
.43 1.42
.49 1.49
.62 1.61
.72 1.71
.79 1.78
.85 1.84
.94 1.93
2.01 2.00
2.06 2.05 ;
2.10 2.09 ;
2.15 2.14 ;
2.21 2.20 ;
2.26 2.24 ;
2.30 2.28 ;
2.33 2.31 ;
2.36 2.34 ;
150
.00
.22
.34
.42
.48
.60
.71
.78
.83
.92
.99
2.04
2.08
2.13
2.19
2.23
2.27
2.30
2.33
        Table 19-1. K-Multipliers for 1-of-2 Interwell Prediction Limits on Observations (2 COCs, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
1.27 1.05 0.97 0.92 0.89 0.85 (
1.76 1.42 1.29 .22 1.18 1.13
2.05 1.63 1.48 .39 1.34 1.28
2.27 1.78 1.61 .51 1.45 1.39
2.43 1.90 1.71 .60 1.54 1.47
279 2.15 1.91 .79 1.72 1.63
3.10 2.36 2.09 .95 1.87 1.77
3.32 1,5? 2.22 2.07 1.97 1.86
3.48 2.62 2.31 2.15 2.05 1.94
3.78 2.83 2,48 . 2.31 2.20 2.07 ;
3.99 2.97 2.60 2.41 2.30 2.16 ;
4.15 3.08 2.70 "z.'sa" 2.37 2.23 ;
4.28 3.17 2.77 2,58 2,44 2.29 ;
4.43 3.27 2.86 2.65 2,51 2.36 ;
4.63 3.41 2.97 2.75 2.61 2,44 \
4.78 3.51 3.06 2.83 2.68 2,51
4.90 3.60 3.13 2.89 2.74 2,56 .
5.00 3.67 3.19 2.94 2.79 2,61 .
5.08 3.73 3.24 2.99 2.83 2.65 .,
20
).83 (
.10
.25
.35
.42
.58
.71
.80
.87
2.00
2.08 ;
2.15 ;
2.20 ;
2.27 ;
2.35 ;
2.4i- ;
2.46 ;
2.51 .
2,54 •' ,
25 30
).81 0.80 (
.08 1.06
.22 1.20
.32 1.30
.39 1.37
.54 1.52
.67 1.64
.76 1.72
.82 1.79
.94 1.90
2.02 1.98
2.09 2.04 ;
2.14 2.09 ;
2.20 2.15 ;
2.28 2.23 ;
2.34 2.28 ;
2.38 2.33 ;
2.42". 2.37 ;
2.48 2.4Q :
35
).79 (
.05
.19
.28
.35
.50
.62
.70
.77
.88
.96
2.01
2.06 ;
2.12 ;
2.19 ;
2.25 ;
2.29 ;
2.33 ;
2.37 ;
40 45
).79 0.78 (
.04 1.04
.18 1.17
.27 1.26
.34 1.33
.49 1.48
.60 1.59
.69 1.67
.75 1.73
.86 1.84
.93 1.92
.99 1.98
2.04 2.02 ;
2.10 2.08 ;
2.17 2.15 ;
2.22 2.20 ;
2.27 2.25 ;
2.30 2.28 ;
2.34 2.31 ;
50 60 70
).78 0.78 0.77 (
.03 1.02 1.02
.17 1.16 1.15
.26 1.25 1.24
.33 1.32 1.31
.47 1.46 1.45
.58 1.57 1.56
.66 1.65 1.64
.72 1.71 1.70
.83 1.81 1.80
.91 1.89 1.87
.96 1.94 1.93
2.01 1.99 1.97
2.06 2.04 2.03 ;
2.13 2.11 2.09 ;
2.19 2.16 2.14 ;
2.23 2.20 2.19 ;
2.27 2.24 2.22 ;
2.30 2.27 2.25 ;
80 90
).77 0.77 (
.02 1.01
.15 1.14
.24 1.23
.30 1.30
.44 1.44
.55 1.55
.63 1.62
.69 1.68
.79 1.79
.86 1.86
.92 1.91
.96 1.95
2.01 2.01 ;
2.08 2.07 ;
2.13 2.12 ;
2.17 2.16 ;
2.21 2.20 ;
2.24 2.23 ;
100 125
).77 0.76 (
.01 1.01
.14 1.14
.23 1.23
.30 1.29
.43 1.43
.54 1.54
.62 1.61
.68 1.67
.78 1.77
.85 1.84
.90 1.89
.95 1.93
2.00 1.99
2.06 2.05 ;
2.11 2.10 ;
2.15 2.14 ;
2.19 2.17 ;
2.22 2.20 ;
150
).76
.00
.13
.22
.29
.42
.53
.61
.66
.76
.83
.88
.93
.98
2.04
2.09
2.13
2.16
2.19
                                                    D-35
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Observations
Unified Guidance
     Table 19-1. K-Multipliers for 1-of-2 Interwell Prediction Limits on Observations (2 COCs, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
1.76 1.42 1.29 .22 1.18 1.13
2.27 1.78 1.61 .51 1.45 1.39
2.57 2.00 1.79 .68 1.61 1.53
279 2.15 1.91 .79 1.72 1.63
2.96 2.27 2.01 .88 1.80 1.71
332 2.51 2.22 2.07 1.97 1.86
362 2.71 2.39 2.22 2.12 2.00
3.83 2.86 2.51 2.33 2.22 2.09 ;
3.99 2.97 2.60 2.41 2.30 2.16 ;
4.28 3.17 2.77 2.56 2.44 2.29 ;
4.48 3.30 2,89 2.67 2.53 2.38 ;
4.63 3.41 2.97 2.75 2.61 2.44 ;
4.75 3.49 3.05 2,81 2.67 2.50 ;
4.90 3.60 3.13 2,89. 2.74 2.56 ;
5.08 3.73 3.24 2.99 2,83 2.65 ;
5.23 3.82 3.33 3.06 2.9O 2.71 ;
5.34 3.90 3.39 3.13 2,96 2,76 \
5.43 3.97 3.45 3.18 3.01 2,81 :
5.52 4.03 3.50 3.22 3.05 2,85 '4
20
.10
.35
.48
.58
.65
.80
.93
1.02
1.08 :
1.20 :
1.29 ;
2.35 ;
2.40 ;
>.46 :
1.54 ;
2.60 ;
2.65 ;
2.69 ;
2.73 ;
25 30
.08 1.06
.32 1.30
.45 1.43
.54 1.52
.61 1.58
.76 1.72
.88 1.84
.96 1.92
1.02 1.98
1.14 2.09 ;
1.21 2.17 ;
2.28 2.23 ;
2.32 2.27 ;
2.38 2.33 ;
2.46 2.40 ;
2.52 2.46 ;
2.56 2.50 ;
2.60 2.54 ;
2.64 2.57 ;
35
.05
.28
.41
.50
.57
.70
.82
.89
.96
2.06 ;
2.14 ;
2.19 ;
2.24 ;
2.29 ;
2.37 ;
2.42 ;
2.46 ;
2.50 ;
2.53 ;
40 45
.04 1.04
.27 1.26
.40 1.39
.49 1.48
.55 1.54
.69 1.67
.80 1.78
.88 1.86
.93 1.92
2.04 2.02 ;
2.11 2.09 ;
2.17 2.15 ;
2.21 2.19 ;
2.27 2.25 ;
2.34 2.31 ;
2.39 2.37 ;
2.43 2.41 ;
2.47 2.44 ;
2.50 2.47 ;
50 60 70
.03 1.02 1.02
.26 1.25 1.24
.38 1.37 1.36
.47 1.46 1.45
.53 1.52 1.51
.66 1.65 1.64
.77 1.76 1.74
.85 1.83 1.82
.91 1.89 1.87
2.01 1.99 1.97
2.08 2.06 2.04 ;
2.13 2.11 2.09 ;
2.18 2.15 2.14 ;
2.23 2.20 2.19 ;
2.30 2.27 2.25 ;
2.35 2.32 2.30 ;
2.39 2.36 2.34 ;
2.42 2.39 2.37 ;
2.45 2.42 2.40 ;
80 90
.02 1.01
.24 1.23
.36 1.35
.44 1.44
.50 1.50
.63 1.62
.74 1.73
.81 1.80
.86 1.86
.96 1.95
2.03 2.02 ;
2.08 2.07 ;
2.12 2.11 ;
2.17 2.16 ;
2.24 2.23 ;
2.29 2.27 ;
2.32 2.31 ;
2.36 2.35 ;
2.39 2.37 ;
100 125
.01 1.01
.23 1.23
.35 1.34
.43 1.43
.49 1.49
.62 1.61
.72 1.71
.80 1.79
.85 1.84
.95 1.93
2.01 2.00
2.06 2.05 ;
2.10 2.09 ;
2.15 2.14 ;
2.22 2.20 ;
2.26 2.25 ;
2.30 2.29 ;
2.34 2.32 ;
2.36 2.35 ;
150
.00
.22
.34
.42
.48
.61
.71
.78
.83
.93
.99
2.04
2.08
2.13
2.19
2.24
2.27
2.31
2.33
      Table 19-1. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Observations (2 COCs, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
2.27 1.79 1.61 1.51 1.45 1.39 .35
2.79 2.15 1.91 1.79 1.72 1.63 .58
3.10 2.36 2.09 1.95 1.87 1.77 .71
332 2.51 2.22 2.07 1.97 1.86 .80
348 2.62 2.31 2.15 2.05 1.94 .87
383 2.86 2.51 2.33 2.22 2.09 2.02
412 3.06 2.68 2.48 2.36 2.22 2.14
4.32 3.20 2.80 2.59 2.46 2.31 2.22
4.48 3.30 2.89 2.67 2.53 2.38 2.29
4.75 3.49 3.05 2.81 2.67 2.50 2.40
4.94 3.63 3,16 2.91 2.76 2.58 2.48
5.08 3.72 3.24 2.99 2.83 2.65 2.54
5.20 3.81 3.31 3.05 2.89 2.70 2.59
5.34 3.91 3.39 3,13 2.96 2.76 2.65
5.52 4.03 3.50 3.22 3.05 2.85 2.73
5.65 4.12 3.58 3.29 3.12 2.91 2.79
5.76 4.20 3.64 3.35 3,17 2.96 2.83
5.85 4.26 3.70 3.40 3.22 3.00 2.87
5.93 4.32 3.74 3.45 3,26 3.04 2.91
25
.32
.54
.67
.75
.82
.96
2.07
2.15
2.22
2.32
2.40
2.46
2.51
2.56
2.64
2.69
2.74
2.77
2.81
30
1.30
1.52
1.64
1.72
1.79
1.92
2.03
2.11
2.17
2.27
2.35
2.40
2.45
2.50
2.57
2.63
2.67
2.71
2.74
35
.28
.50
.62
.70
.77
.89
2.00
2.08 ;
2.14 ;
2.24 ;
2.31 ;
2.37 ;
2.4i ;
2.46 ;
2.53 ;
2.58 ;
2.63 ;
2.66 ;
2.69 ;
40
.27
.49
.60
.69
.75
.88
.98
2.06
2.11
2.21
2.28
2.34
2.38
2.43
2.50
2.55
2.59
2.63
2.66
45
1.26
1.48
1.59
1.67
1.73
1.86
1.97
2.04 ;
2.09 ;
2.19 ;
2.26 ;
2.3i ;
2.36 ;
2.4i ;
2.47 ;
2.52 ;
2.56 ;
2.60 ;
2.63 ;
50
.26
.47
.58
.66
.72
.85
.95
2.02
2.08
2.18
2.24
2.30
2.34
2.39
2.45
2.50
2.54
2.58
2.61
60
1.25
1.46
1.57
1.65
1.71
1.83
1.93
2.00
2.06
2.15
2.22
2.27
2.31
2.36
2.42
2.47
2.51
2.54
2.57
70 80
1.24 .24
1.45 .44
1.56 .55
1.64 .63
1.70 .69
1.82 .81
1.92 .91
1.99 .98
2.04 2.03
2.14 2.12
2.20 2.19
2.25 2.24
2.29 2.28
2.34 2.32
2.40 2.39
2.45 2.43
2.49 2.47
2.52 2.50
2.55 2.53
90 100
1.23 .23
1.44 .43
1.55 .54
1.62 .62
1.68 .68
1.80 .80
1.90 .89
1.97 .96
2.02 2.01
2.11 2.10
2.18 2.17
2.23 2.22
2.26 2.26
2.31 2.30
2.37 2.36
2.42 2.41
2.46 2.45
2.49 2.48
2.52 2.50
125 150
1.23 .22
1.43 .42
1.54 .53
1.61 .61
1.67 .66
1.79 .78
1.88 .87
1.95 .94
2.00 .99
2.09 2.08
2.15 2.14
2.20 2.19
2.24 2.23
2.29 2.28
2.35 2.33
2.39 2.38
2.43 2.41
2.46 2.45
2.48 2.47
                                                    D-36
                                                                                                  March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Observations
Unified Guidance
        Table 19-1. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Observations (5 COCs, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
2.02 1.58 1.42 1.33 1.28 1.22 .18
2.62 1.97 1.74 1.63 1.56 1.48 .43
300 2.20 1.93 1.80 1.72 1.62 .57
327 2.37 2.07 1.92 1.83 1.72 .66
3.49 1,5? 2.18 2.01 1.91 1.80 .74
3.96 2.79 2.40 2.21 2.09 1.96 .89
4.37 3.03 2.59 2.38 2.25 2.10 2.02
4.66 3.20 2.73 2,49 2.35 2.19 2.11 ;
4.88 3.34 2.84 2.59 '2, '44" 2.27 2.17 ;
5.28 3.58 3.03 2.75 2,59 2.4O 2.30 ;
5.56 3.75 3.16 2.87 2.69 2,49 2.38 ;
5.77 3.88 3.26 2.96 2.77 2,57 2,45 \
5.94 3.98 3.35 3.03 2.84 2.62 2.5O .
6.15 4.11 3.45 3.12 2.92 2.69 2,57 ,
6.42 4.27 3.58 3.23 3.02 2.78 2.65
6.62 4.40 3.68 3.32 3.10 2.85 2.71
6.79 4.50 3.75 3.39 3.16 2.91 2.77
6.92 4.58 3.82 3.45 3.22 2.96 2.81
7.04 4.65 3.88 3.49 3.26 3.00 2.85
25 30
.16 1.14
.40 1.37
.53 1.50
.62 1.59
.69 1.66
.83 1.80
.95 1.91
2.04 1.99
2.10 2.06 ;
1.22 2.17 ;
2.30 2.24 ;
2.36 2.30 ;
2.41 2.35 ;
2,47 2,41 :
2:55\ 2,48 *
2,61 2,54 i
2.66 2.59 <
2.70 2,63 .
2.13 2.66 i
35
.13
.36
.48
.57
.64
.77
.89
.96
1.02 ;
2.13 ;
2.21 ;
i.26 :
i.3i :
1.37 :
2.44 .
2,43 .
2,54 ,
2,58 .
2.61
40 45
.12 1.11
.35 1.34
.47 1.46
.56 1.54
.62 1.61
.75 1.74
.86 1.85
.94 1.92
2.00 1.98
Ml 2.09 ;
2.18 2.16 ;
2.23 2.21 ;
2.28 2.26 ;
1.34 2.31 ;
2.40 2.38 ;
2,46 ' '2','43'"'-'.
2.5O 2:41.' .
2,54 2,51 ,
2.57 2.54 .j
50 60 70
.11 1.10 1.09
.33 1.32 1.31
.45 1.44 1.43
.54 1.52 1.51
.60 1.59 1.57
.73 1.71 1.70
.84 1.82 1.80
.91 1.89 1.88
.97 1.95 1.93
1.07 2.05 2.03 ;
2.14 2.12 2.10 ;
2.19 2.17 2.15 ;
2.24 2.21 2.19 ;
1.29 2.26 2.24 ;
2.36 2.33 2.31 ;
2.41 "'"2.38" 2.36 ;
2.45 2,42 . '2.'4'6'.""<
2,49 2,45 . 2,43 .
2.52 2.48 2.46 .
80 90
.09 1.09
.31 1.30
.42 1.42
.51 1.50
.57 1.56
.69 1.69
.80 1.79
.87 1.86
.92 1.91
1.02 2.01 ;
2.09 2.08 ;
2.14 2.13 ;
2.18 2.17 ;
2.23 2.22 ;
1.29 2.28 ;
2.34 2.33 ;
2.38 2.37 :
2,41 2.40 .
2.44 2,43 ,
100 125
.08 1.08
.30 1.29
.42 1.41
.50 1.49
.56 1.55
.68 1.67
.78 1.77
.85 1.84
.91 1.89
2.00 1.99
1.07 2.05 ;
2.12 2.10 ;
2.16 2.14 ;
1.21 2.19 ;
1.27 2.25 ;
2.32 2.30 ;
2.35 2.34 ;
2,39 2.37 :
2,41 2.39 <
150
.08
.29
.40
.48
.54
.66
.76
.83
.89
.98
2.04
2.09
2.13
2.18
2.24
2.29
2.32
2.35
2.38
     Table 19-1. K-Multipliers for 1-of-2 Interwell Prediction Limits on Observations (5 COCs, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.62
3.27
3.67
3.96
4.18
4.66
5.06
5.34
5.56
5.94
6.21
6.42
6.59
6.79
7.04
7.23
7.38
7.52
7.62
6 8 10 12 16 20 25 30 35
1.97 1.74 1.63 1.56 1.48 .43 .40 1.37 .36
2.37 2.07 1.92 1.83 1.72 .66 .62 1.59 .57
2.62 2.26 2.09 1.98 1.86 .80 .75 1.71 .69
2.79 2.40 2.21 2.09 1.96 .89 .83 1.80 .77
2.92 2.51 2.30 2.18 2.04 .96 .90 1.86 .83
3.20 2.73 2.49 2.35 2.19 2.11 2.04 1.99 .96
3.45 2.92 2.66 2.51 2.33 2.23 2.15 2.11 2.07
3.62 3.06 ""2.'78" 2.61 2.42 2.32 2.24 2.19 2.15
3.75 3.16 2,87 2.69 2.49 2.38 2.30 2.24 2.21
3.98 3.35 3.03 2,84 2.62 2.50 2.41 2.35 2.31
4.15 3.48 3.14 2,94 2.71 2.59 2.49 2.43 2.38
4.27 3.58 3.23 3.02 2. 78 2.65 2.55 2.48 2.44
4.37 3.66 3.30 3.09 2,84 2.70 2.60 2.53 2.48
4.50 3.75 3.39 3.16 2.91 2.77 2.66 2.59 2.54
4.65 3.88 3.49 3.26 3.OO -'2.85, 2.73 2.66 2.61
4.77 3.98 3.58 3.34 3.06 2.91- 2,79 2.72 2.66
4.87 4.05 3.65 3.40 3.12 .-2.96 2.84 2.76 2.70
4.95 4.12 3.70 3.45 3.17 3,00. 2,88 2.8O 2.74
5.02 4.17 3.75 3.50 3.20 3,04 2.91 2,83 2,77
40
.35
.56
.67
.75
.82
.94
2.05
2.12
2.18
2.28
2.35
2.40
2.45
2.50
2.57
2.62
2.66
2.70
2.73
45
1.34
1.54
1.66
1.74
1.80
1.92
2.03
2.10
2.16
2.26
2.33
2.38
2.42
2.47
2.54
2.59
2.63
2.67
2.70
50
.33
.54
.65
.73
.79
.91
2.01
2.09
2.14
2.24
2.31
2.36
2.40
2.45
2.52
2.57
2.61
2.64
2.67
60
1.32
1.52
1.63
1.71
1.77
1.89
1.99
2.06
2.12
2.21
2.28
2.33
2.37
2.42
2.48
2.53
2.57
2.60
2.63
70
1.31
1.51
1.62
1.70
1.76
1.88
1.98
2.05 ;
2.10 ;
2.19 ;
2.26 ;
2.3i ;
2.35 ;
2.40 ;
2.46 ;
2.5i ;
2.54 ;
2.58 ;
2.60 ;
80
.31
.51
.62
.69
.75
.87
.97
2.03
2.09
2.18
2.24
2.29
2.33
2.38
2.44
2.49
2.52
2.56
2.58
90
1.30
1.50
1.61
1.69
1.74
1.86
1.96
2.02 ;
2.08 ;
2.17 ;
2.23 ;
2.28 ;
2.32 ;
2.37 ;
2.43 ;
2.47 ;
2.5i ;
2.54 ;
2.57 ;
100
.30
.50
.60
.68
.74
.85
.95
2.02
2.07
2.16
2.22
2.27
2.31
2.35
2.41
2.46
2.50
2.53
2.55
125 150
1.29 .29
1.49 .48
1.60 .59
1.67 .66
1.73 .72
1.84 .83
1.94 .93
2.00 .99
2.05 2.04
2.14 2.13
2.20 2.19
2.25 2.24
2.29 2.28
2.34 2.32
2.39 2.38
2.44 2.43
2.48 2.46
2.51 2.49
2.53 2.52
                                                    D-37
                                                                                                  March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
Unified Guidance
Table 19-1. K- Multipliers for 1-of-2 Interwell Prediction Limits on Observations (5 COCs,
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.27
3.96
4.37
4.66
4.88
5.34
5.74
6.01
6.21
6.59
6.84
7.04
7.20
7.38
7.62
7.81
7.96
8.09
8.19
Table
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.69
3.43
3.88
4.24
4.51
5.09
5.60
5.95
6.23
6.73
7.08
7.35
7.58
7.83
8.17
8.43
8.63
8.80
8.94
6 8 10 12 16 20 25 30 35 40
2.37 2.07 1.92 1.83 1.72 1.66 1.62 1.59 1.57 1.56
2.79 2.40 2.21 2.09 1.96 1.89 1.83 1.80 1.77 1.75
3.03 2.59 2.38 2.25 2.10 2.02 1.95 1.91 1.89 1.86
3.20 2.73 2.49 2.35 2.19 2.11 2.04 1.99 1.96 1.94
3.34 2.84 2.59 2.44 2.27 2.17 2.10 2.06 2.02 2.00
362 3.06 2.78 2.61 2.42 2.32 2.24 2.19 2.15 2.12
3.86 3.24 2.94 2.76 2.55 2.44 2.35 2.29 2.25 2.22
4.02 3.37 3.06 2.86 2.64 2.52 2.43 2.37 2.33 2.30
4.15 3.48 3,14 2.94 2.71 2.59 2.49 2.43 2.38 2.35
4.37 3.66 3.30 3.09 2.84 2.70 2.60 2.53 2.48 2.45
4.53 3.78 3.41 3.W 2.93 2.79 2.68 2.60 2.55 2.52
4.65 3.88 3.49 3,26 3.00 2.85 2.73 2.66 2.61 2.57
4.75 3.96 3.56 3.32 3.05 2.90 2.78 2.71 2.65 2.61
4.87 4.05 3.65 3.40 3.7,2 2.96 2.84 2.76 2.70 2.66
5.02 4.17 3.75 3.50 3.2O 3.04 2.91 2.83 2.77 2.73
5.13 4.26 3.83 3.57 3.27 3.10 2.97 2.88 2.82 2.78
5.23 4.34 3.90 3.63 3,32 3,15 3.02 2.93 2.87 2.82
5.30 4.40 3.95 3.68 3,37 3,19 3.05 2.96 2.90 2.86
5.37 4.45 4.00 3.72 3.41 3,23 3.09 3.00 2.93 2.89
19-1. K-Multipliers for 1-of-2 Interwell Prediction
6 8 10 12 16 20 25 30 35 40
1.99 1.76 1.64 1.56 1.48 .43 .40 1.37 .36 .35
2.42 2.09 1.93 1.84 1.73 .67 .62 1.59 .57 .56
2.68 2.29 2.11 2.00 1.87 .80 .75 1.71 .69 .68
2.86 2,44 2.23 2.11 1.97 .89 .84 1.80 .77 .76
3.01 2.55 2.33 2.20 2.05 .97 .91 1.86 .84 .82
3.33 2.79 2,53 2.38 2.21 2.11 2.04 2.00 .97 .94
3.60 3.00 2.71 2,54 2.34 2.24 2.16 2.11 2.08 2.05
3.80 3.14 2.83 2.65 "2.44 " 2.33 2.25 2.19 2.15 2.12
3.96 3.26 2.93 2.73 2,51 2,40 2.31 2.25 2.21 2.18
4.22 3.47 3.10 2.89 2.65 2,52 2,42 2.36 2.32 2.28
4.42 3.61 3.23 3.00 2.74 2,61 2,50 2,44 2,39 2.35
4.56 3.72 3.32 3.08 2.82 2.67 2.58 2,49 2,45 2^41
4.69 3.81 3.40 3.15 2.88 2.73 2,61 2,54. 2,49 2,45
4.83 3.93 3.49 3.23 2.95 2.79 2.67 2,60 2,55 2,51
5.02 4.07 3.61 3.34 3.04 2.88 2.75 2.67 2,62 2,58
5.16 4.17 3.70 3.42 3.11 2.94 2.81 2.73 2.67 2,63
5.27 4.27 3.78 3.49 3.17 2.99 2.86 2.77 2.72 2,67
5.37 4.34 3.84 3.55 3.22 3.04 2.90 2.81 2.75 2.71
5.46 4.40 3.89 3.60 3.26 3.08 2.94 2.85 2.79 2.74
45
1.54
1.74
1.85
1.92
1.98
2.10
2.20
2.27
2.33
2.42
2.49
2.54
2.58
2.63
2.70
2.75
2.79
2.82
2.85
Limits
45
1.34
1.55
1.66
1.74
1.80
1.93
2.03
2.11
2.16
2.26
2.33
: ' ' "2.3Q '
2,43
2,48:
2.55
2,60
2,64
2,67
2.70
50
.54
.73
.84
.91
.97
2.09
2.19
2.25
2.31
2.40
2.47
2.52
2.56
2.61
2.67
2.72
2.76
2.79
2.82
60
1.52
1.71
1.82
1.89
1.95
2.06
2.16
2.23
2.28
2.37
2.43
2.48
2.52
2.57
2.63
2.68
2.72
2.75
2.78
70
1.51
1.70
1.80
1.88
1.93
2.05
2.14
2.21
2.26
2.35
2.41
2.46
2.50
2.54
2.60
2.65
2.69
2.72
2.75
80
.51
.69
.80
.87
.92
2.03
2.13
2.19
2.24
2.33
2.39
2.44
2.48
2.52
2.58
2.63
2.67
2.70
2.72
on Observations (10
50
.33
.54
.65
.73
.79
.91
2.02
2.09
2.14
2.24
2.31
2.36
2,40
2,46
2,52
2,57
2.61
,2.65
2.68
60
1.32
1.52
1.64
1.71
1.77
1.89
1.99
2.06
2.12
2.21
2.28
2.33
2.37
2,42
2.49
2,54
2,57
2.61
2,84
70
1.31
1.51
1.63
1.70
1.76
1.88
1.98
2.05
2.10
2.19
2.26
2.31
2.35
2,40
2,46.
2,51
2,55
2.58
2.61
80
.31
.51
.62
.69
.75
.87
.97
2.03
2.09
2.18
2.24
2.29
2.33
2,38
2.44
2,49
2,53
2.56
2.59
90
1.50
1.69
1.79
1.86
1.91
2.02
2.12
2.18
2.23
2.32
2.38
2.43
2.46
2.51
2.57
2.61
2.65
2.68
2.70
Quarterly)
100
.50
.68
.78
.85
.91
2.02
2.11
2.17
2.22
2.31
2.37
2.41
2.45
2.50
2.55
2.60
2.63
2.66
2.69
125
1.49
1.67
1.77
1.84
1.89
2.00
2.09
2.16
2.20
2.29
2.35
2.39
2.43
2.48
2.53
2.58
2.61
2.64
2.67
150
.48
.66
.76
.83
.89
.99
2.08
2.15
2.19
2.28
2.34
2.38
2.42
2.46
2.52
2.56
2.59
2.62
2.65
COCs, Annual)
90
1.30
1.50
1.61
1.69
1.74
1.86
1.96
2.02
2.08
2.17
2.23
2.28
2.32
2.37
'2.43"
2,47
2,51
2.54
2.57
100
.30
.50
.60
.68
.74
.85
.95
2.02
2.07
2.16
2.22
2.27
2.31
2.36
'"2.42"
2,46
.2.50
2,53
2,56
125
1.29
1.49
1.60
1.67
1.73
1.84
1.94
2.00
2.05
2.14
2.20
2.25
2.29
2.34
2, 40
2.44
2,48
2,51
2,53
150
.29
.48
.59
.66
.72
.83
.93
.99
2.04
2.13
2.19
2.24
2.28
2.32
"•'2.38 '
2,43
2,46
2.49
2.52 .

                                                      D-38
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
Unified Guidance
Table 19-1. K-Multipliers for 1-of-
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.43
4.24
4.72
5.09
5.37
5.95
6.46
6.82
7.08
7.58
7.92
8.17
8.37
8.63
8.94
9.19
9.39
9.56
9.70
Table
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.24
5.09
5.60
5.94
6.22
6.82
7.30
7.64
7.92
8.37
8.71
8.94
9.17
9.39
9.68
9.90
10.13
10.24
10.41
6
2.42
2.86
3.13
3.33
3.48
3.80
4.08
4.27
4.42
4.69
4.88
5.02
5.13
5.27
5.46
5.60
5.70
5.80
5.88
19-1.
6
2.86
3.33
3.60
3.80
3.96
4.27
4.55
4.72
4.88
5.14
5.31
5.46
5.57
5.71
5.88
6.01
6.11
6.19
6.28
8 10
2.09 1.93
2.44 2.23
2.65 2.40
2,79 2.53
2.90 2.63
3.14 "2,83"
3.35 3.01
3.49 3.13
3.61 3.23
3.81 3.40
3.96 3.52
4.07 3.61
4.15 3.69
4.27 3.78
4.40 3.89
4.51 3.98
4.59 4.06
4.66 4.12
4.73 4.17
12 16
1.84 1.73
2.11 1.97
2.27 2.11
2.38 2.21
2.46 2.28
2.65 2.44
2.8O 2.57
2,91 2.67
3.00 2.74
3.15 2,88
3.26 .2,97
3.34 3.04
3.41 3.10
3.49 3.17
3.60 3.26
3.68 3.33
3.74 3.39
3.80 3.43
3.85 3.48
2 Interwell Prediction Limits on Observations (10 COCs, Semi-Annual)
20 25
1.67 1.62
1.89 1.84
2.02 1.96
2.11 2.04
2.18 2.11
2.33 2.25
2.45 2.36
2.54 2.44
2.61 2.50
2.73 2.61
2,81 2.69
2.88: 2,75
2,93 2.8O
2,99 2,86 ,
3.08 2,94
3.14 3.OO
3.19 3.O4
3.24 . 3.68
3.27 3.12
30 35
1.59 1.57
1.80 1.77
1.92 1.89
2.00 1.97
2.06 2.03
2.19 2.15
2.30 2.26
2.38 2.33
2.44 2.39
2.54 2.49
2.62 2.56
2.67 2.62
2.72 2.66
2.77 2.72
2.85 2. 79
2.9Q 2.84
.2.95 2.88
2.99 . 2.92
3.Q2 2.95
40 45
1.56 1.55
1.76 1.74
1.87 1.85
1.94 1.93
2.00 1.99
2.12 2.11
2.23 2.21
2.30 2.28
2.35 2.33
2.45 2.43
2.52 2.49
2.58 2.55
2.62 2.59
2.67 2.64
2.74 2.70
'2.'79""-2,76
2.83 2.79
2.87 2.83
2.90 2.86
K- Multipliers for 1-of-2 Interwell Prediction Limits
8 10
2.44 2.23
2.79 2.53
2.99 2.71
3,15 2.83
3.26 2.93
3.50 3, 13
3.70 3.30
3.84 3.42
3.96 3.52
4.15 3.69
4.29 3.80
4.41 3.90
4.49 3.97
4.59 4.05
4.72 4.17
4.83 4.26
4.92 4.33
4.97 4.39
5.03 4.44
12 16
2.11 1.97
2.38 2.21
2.54 2.34
2.65 2.44
2.73 2.52
2.91 2.67
3.06 2.80
~"3~.i8" 2.90
3,26 2.97
3.41 3.10
3.52 3,19
3.60 3.26
3.66 3,32
3.74 3,39
3.84 3.47
3.93 3.54
3.98 3.60
4.04 3.64
4.08 3.69
20 25
1.89 1.84
2.11 2.04
2.24 2.16
2.33 2.24
2.40 2.31
2.54 2.44
2.66 2.55
2.74 2.63
2.81 2.69
2.93 2.80
3.01 2.88
3.08 2.94
3,13 2.99
'3:19 3.04
3,28 3. 12
3,33 3,18
3,39 3.22
3.43 3.26
3.47 3.3Q
30 35
1.80 1.77
2.00 1.97
2.11 2.08
2.19 2.15
2.25 2.21
2.38 2.33
2.48 2.44
2.56 2.51
2.62 2.56
2.72 2.66
2.79 2.73
2.85 2.79
2.89 2.83
2.95 2.88
3.02 2.95
3.07 3.00
3.-12- 3.04
3:18 3.08
3,78 3.11
40 45
1.76 1.74
1.94 1.93
2.05 2.03
2.12 2.11
2.19 2.16
2.30 2.28
2.40 2.37
2.47 2.44
2.52 2.49
2.62 2.59
2.69 2.65
2.74 2.71
2.78 2.74
2.83 2.79
2.90 2.86
2.95 2.91
2.99 2.95
3.02 2.98
3.05 3.01
50 60
1.54 1.52
1.73 1.71
1.84 1.82
1.91 1.89
1.97 1.95
2.09 2.06
2.19 2.16
2.26 2.23
2.31 2.28
2.40 2.37
2.47 2.44
2.52 2.49
2.56 2.52
2.61 2.57
2.68 2.64
2.73 2.68
"2,77" 2.72
2, SO 2,76
2,83 2,78
70 80
1.51 .51
1.70 .69
1.81 .80
1.88 .87
1.93 .92
2.05 2.03
2.14 2.13
2.21 2.19
2.26 2.24
2.35 2.33
2.41 2.39
2.46 2.44
2.50 2.48
2.55 2.53
2.61 2.59
2.65 2.63
2.69 2.67
2.72 2.70
2,75 2.73
on Observations (10
50 60
1.73 1.71
1.91 1.89
2.02 1.99
2.09 2.06
2.14 2.12
2.26 2.23
2.35 2.32
2.42 2.39
2.47 2.44
2.56 2.52
2.63 2.59
2.68 2.63
2.72 2.67
2.77 2.72
2.83 2.78
2.88 2.83
2.91 2.86
2.95 2.90
2.97 2.92
70 80
1.70 1.69
1.88 1.87
1.98 1.97
2.05 2.04
2.10 2.09
2.21 2.19
2.30 2.28
2.36 2.35
2.41 2.39
2.50 2.48
2.56 2.54
2.61 2.59
2.65 2.62
2.69 2.67
2.75 2.73
2.79 2.77
2.83 2.81
2.86 2.84
2.89 2.86
90 100
1.50 .50
1.69 .68
1.79 .78
1.86 .85
1.91 .91
2.02 2.02
2.12 2.11
2.18 2.17
2.23 2.22
2.32 2.31
2.38 2.37
2.43 2.42
2.46 2.45
2.51 2.50
2.57 2.56
2.62 2.60
2.65 2.64
2.68 2.67
2.71 2.69
125 150
1.49 .48
1.67 .66
1.77 .76
1.84 .83
1.89 .89
2.00 .99
2.09 2.08
2.16 2.15
2.20 2.19
2.29 2.28
2.35 2.34
2.40 2.38
2.43 2.42
2.48 2.46
2.53 2.52
2.58 2.56
2.61 2.60
2.64 2.63
2.67 2.65
COCs, Quarterly)
90 100
1.69 1.68
1.86 1.85
1.96 1.95
2.03 2.02
2.08 2.07
2.18 2.17
2.27 2.26
2.33 2.32
2.38 2.37
2.46 2.45
2.52 2.51
2.57 2.56
2.61 2.59
2.65 2.64
2.71 2.69
2.75 2.74
2.79 2.77
2.82 2.80
2.84 2.83
125 150
1.67 1.66
1.84 1.83
1.94 1.93
2.00 1.99
2.05 2.04
2.16 2.15
2.24 2.23
2.31 2.29
2.35 2.34
2.43 2.42
2.49 2.48
2.53 2.52
2.57 2.55
2.61 2.60
2.67 2.65
2.71 2.69
2.74 2.73
2.77 2.76
2.80 2.78

                                                      D-39
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
Unified Guidance
Table 19-1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.53
4.41
5.00
5.43
5.77
6.50
7.13
7.58
7.92
8.54
9.00
9.34
9.62
9.96
10.36
6
2.44
2.91
3.20
3.42
3.57
3.93
4.24
4.46
4.63
4.95
5.17
5.34
5.48
5.65
5.85
10.70 6.02
10.92
11.15
11.38
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.41
5.43
6.05
6.50
6.84
7.58
8.20
8.66
9.00
9.62
10.02
10.36
10.64
10.92
11.38
11.66
11.88
12.11
12.28
6.16
6.28
6.36
. K-Multipliers for 1-of-2 Interwell Prediction Limits on Observations (20
8
2.11
2,46
2.67
2.82
2.95
3.20
3.43
3.59
3.71
3.94
4.10
4.22
4.32
4.44
4.59
4.72
4.80
4.89
4.96
10 12 16 20 25 30 35 40 45 50
1.94 1.85 1.73 1.67 1.63 1.59 1.58 1.56 1.55 1.54
2.24 2.11 1.97 1.89 1.84 1.80 1.77 1.76 1.74 1.73
2.43 2.28 2.11 2.03 1.96 1.92 1.89 1.87 1.85 1.84
2,55 2.39 2.21 2.12 2.05 2.00 1.97 1.94 1.93 1.92
2.65 2.48 2.29 2.19 2.11 2.06 2.03 2.00 1.99 1.97
2.86 2.67 '2.45' 2.33 2.25 2.19 2.16 2.13 2.11 2.09
3.05 2.83 2,59, 2.46 2.36 2.31 2.26 2.23 2.21 2.19
3.18 2.94 2.69 -2,55 2,45 2.38 2.33 2.31 2.28 2.26
3.28 3.03 2.76 2.62 2,57 "2,44".'"2;39" • 2.36 2.33 2.31
3.47 3.20 2.90 2.74 2,62 2,55 2. SO, 2.46 2.43 2.41
3.59 3.31 2.99 2.83 2.70 2,62 2,57 2,53 2, SO 2,48
3.70 3.40 3.07 2.89 2.76 2.68 2.62 2.58 2,55 2,52
3.78 3.47 3.13 2.95 2.81 2.73 2,67 2,62 2,50 2,57
3.88 3.56 3.20 3.01 2.87 2.78 2.72 2.68 2,64 2.62
4.01 3.67 3.30 3.10 2.95 2.86 2.79 2.74 2.71 2,68
4.11 3.76 3.37 3.17 3.01 2.91 2.84 2.80 2.76 2.73
4.19 3.83 3.43 3.22 3.06 2.96 2.89 2.84 2.80 2.77
4.25 3.89 3.48 3.26 3.10 3.00 2.92 2.87 2.84 2.80
4.31 3.94 3.53 3.30 3.13 3.03 2.96 2.90 2.86 2.83
60
.52
.71
.82
.89
.95
2.06
2.16
2.23
2.28
2.37
2,44
2,49
. 2,53 •
'2.57-
2,64
2,69
2.72
2.76
2.79
70
1.51
1.70
1.81
1.88
1.93
2.05
2.14
2.21
2.26
2.35
2,41
2.46
2. SO
2,55
2,67
2,66
2.69
2.72
2.75
80
.51
.69
.80
.87
.92
2.04
2.13
2.20
2.25
2.33
2.40.
2.44
2.48
• 2.53
2,59
2.63
2,67
2.70
2.73
COCs, Annual)
90
1.50
1.69
1.79
1.86
1.92
2.03
2.12
2.18
2.23
2.32
2,38
2,43
2,47
2,57
' 2,57
2,62
2,65
2.68
2.71
9-1. K-Multipliers for 1-of-2 Interwell Prediction Limits on Observations (20 COCs,
6
2.91
3.42
3.71
3.93
4.10
4.46
4.78
5.00
5.17
5.48
5.70
5.85
5.99
6.16
6.36
6.53
6.65
6.76
6.84
8
2.46
2,82
3.05
3.20
3.32
3.59
3.81
3.97
4.10
4.32
4.48
4.59
4.69
4.80
4.96
5.09
5.17
5.26
5.31
10 12 16 20 25 30 35 40 45 50
2.24 2.11 1.97 1.89 1.84 1.80 1.77 1.76 1.74 1.73
2.55 2.39 2.21 2.12 2.05 2.00 1.97 1.94 1.93 1.92
2.73 2.55 2.35 2.25 2.16 2.11 2.08 2.05 2.03 2.02
'2.86" 2.67 2.45 2.33 2.25 2.19 2.16 2.13 2.11 2.09
2.96 2,76 2.52 2.40 2.31 2.26 2.21 2.19 2.16 2.14
3.18 2,94 2.69 2.55 2.45 2.38 2.33 2.31 2.28 2.26
3.36 3.11 2.82 2.67 2.56 2.49 2.44 2.40 2.38 2.35
3.49 3.22 2,92 2,76 2.64 2.56 2.51 2.48 2.44 2.42
3.59 3.31 .2,99 2,83 2.70 2.62 2.57 2.53 2.50 2.48
3.78 3.47 3.13 2,95 2.81 2.73 2.67 2.62 2.59 2.57
3.91 3.59 3.23 3.O3 2.89 2,80 2.74 2.69 2.66 2.63
4.01 3.67 3.30 3.10 2,95 2,86 '2.79" 2.74 2.71 2.68
4.09 3.74 3.36 3.16 3:00 2,90 2,84 2,79 2,75. 2.72
4.19 3.83 3.43 3.22 3,06 2,96 2,89 2,84 2,80 2,77
4.31 3.94 3.53 3.30 3.13 3.03 2,96 2,90 2,86 '2.83.
4.41 4.03 3.60 3.37 3.20 3,08 3,07 2,96 2.91 2,88
4.49 4.10 3.66 3.42 3.24 3.13 3,05 3,00 2,95 2,92
4.56 4.15 3.71 3.47 3.28 3.17 3,09 3,03 2.99 2.95
4.61 4.20 3.75 3.50 3.32 3.20 3.12 3,06 3,02 2,98
60
1.71
1.89
1.99
2.06
2.12
2.23
2.32
2.39
2.44
2.53
2.59
2.64
2.68
2.72
2,79
. 2,83
2,87
2,90
2,93
70
1.70
1.88
1.98
2.05
2.10
2.21
2.30
2.37
2.41
2.50
2.56
2.61
2.65
2.69
2,75
2, BO
2,83
2,86
2,89
80
1.69
1.87
1.97
2.04
2.09
2.20
2.28
2.35
2.40
2.48
2.54
2.59
2.62
2.67
2.73
2,77
2,87
2,84
2,86
90
1.69
1.86
1.96
2.03
2.08
2.18
2.27
2.33
2.38
2.47
2.52
2.57
2.61
2.65
2.71
2,75
2,79
2,82
2,84
100
.50
.68
.78
.85
.91
2.02
2.11
2.17
2.22
2.31
2.37
2.42
2,45
2,50
2,56
2,60
2,64
2,67
2,69
125
.49
.67
.77
.84
.89
2.00
2.09
2.16
2.21
2.29
2.35
2,40
2, 43
2,48
2,54
2,58
2,67
2,64
2,67
150
1.48
1.66
1.76
1.83
1.89
1.99
2.09
2.15
2.19
2.28
2.34
2.38
'•2.42
2.46
2.-S2
2.56
2,60
2,62
•2.85
Semi-Annual)
100
1.68
1.85
1.95
2.02
2.07
2.17
2.26
2.32
2.37
2.45
2.51
2.56
2.60
2.64
2.69
2.74
2,77
2,80
2,83
125
1.67
1.84
1.94
2.00
2.05
2.16
2.25
2.31
2.35
2.43
2.49
2.54
2.57
2.61
2.67
2.71
2.74
' '2J7"
2,80
150
1.66
1.83
1.93
1.99
2.04
2.15
2.23
2.29
2.34
2.42
2.48
2.52
2.55
2.60
2.65
2.69
2.73
'"2.76"
2,78

                                                      D-40
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
Unified Guidance
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.43
6.50
7.13
7.58
7.92
8.66
9.22
9.68
10.02
10.58
11.04
11.38
11.60
11.83
12.28
12.51
12.73
12.96
13.19
Table
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.56
5.69
6.34
6.91
7.38
8.31
9.06
9.62
10.00
10.75
11.50
11.88
12.25
12.62
13.00
13.38
13.75
14.12
14.31
9-1.
6
3.42
3.93
4.24
4.46
4.63
5.00
5.31
5.54
5.71
5.99
6.22
6.36
6.50
6.67
6.84
7.01
7.13
7.24
7.30
19-1
6
2.92
3.48
3.81
4.00
4.19
4.61
4.94
5.22
5.41
5.78
6.06
6.25
6.34
6.53
6.81
7.00
7.19
7.28
7.38
K- Multipliers for 1-of-2 Interwell Prediction Limits on Observations
8
2.82
3.2O
3.42
3.59
3.70
3.98
4.18
4.35
4.46
4.69
4.86
4.97
5.06
5.17
5.31
5.43
5.54
5.60
5.65
10 12 16 20 25 30 35 40 45 50
2.55 2.40 2.21 2.11 2.04 2.00 1.97 1.94 1.93 1.92
2.86 2.67 2.45 2.34 2.25 2.19 2.16 2.13 2.11 2.09
3.05 2.82 2.60 2.45 2.37 2.30 2.26 2.23 2.21 2.19
3,18 2.95 2.68 2.55 2.45 2.38 2.33 2.31 2.28 2.26
3,28 3.03 2.77 2.62 2.51 2.44 2.40 2.35 2.33 2.31
3.50 -3,22 2.92 2.76 2.64 2.57 2.51 2.48 2.45 2.43
3.67 3.39 3.06 2.88 2.75 2.67 2.61 2.57 2.54 2.51
3.81 3.50 3, IS 2.96 2.83 2.74 2.68 2.64 2.61 2.58
3.90 3.59 3.23 3.03 2.89 2.80 2.74 2.69 2.66 2.63
4.10 3.74 3,36 3,75 3.00 2.91 2.84 2.79 2.75 2.72
4.21 3.86 3.46 3,23 3.08 2.98 2.91 2.85 2.82 2.78
4.32 3.94 3.53 3,30 3,73 3.03 2.96 2.91 2.86 2.83
4.39 4.01 3.59 3,36 3, W 3.08 3.00 2.95 2.91 2.87
4.49 4.10 3.66 3,42 ' 3,25 3,73 3.06 3.00 2.95 2.92
4.61 4.21 3.76 3.50 3,32 3,20 3,72 3.06 3.02 2.98
4.72 4.29 3.83 3.57 3,37 3:26 3.18 3.11 3.06 3.03
4.78 4.35 3.88 3.62 3,42 3.3O 3,22 3,75 3.11 3.06
4.86 4.41 3.93 3.66 3. '46 3,34 3,25 3,79 3,73 3.10
4.90 4.46 3.97 3.70 3.50 3,37 3,28 3,22 3,76 3, f 3
60
1.89
2.06
2.16
2.23
2.28
2.39
2.48
2.54
2.59
2.68
2.74
2.79
2.82
2.87
2.93
2.97
3.01
3.04
3.07
70
1.88
2.04
2.14
2.21
2.26
2.36
2.45
2.52
2.56
2.65
2.71
2.75
2.79
2.84
2.89
2.94
2.97
3.00
3.03
(20 COCs,
80
1.87
2.04
2.13
2.19
2.24
2.35
2.43
2.50
2.54
2.62
2.68
2.73
2.77
2.81
2.86
2.91
2.94
2.97
3.00
. K-Multipliers for 1-of-2 Interwell Prediction Limits on Observations (40
8
2.48
2.85
3.06
3.25
3.37
3.62
3.91
4.05
4.19
4.42
4.61
4.75
4.84
4.98
5.17
5.31
5.41
5.50
5.59
10 12 16 20 25 30 35 40 45 50
2.24 2.12 1.98 1.89 1.84 1.80 1.77 1.75 1.75 1.73
2.57 2,47 2.22 2.12 2.05 2.01 1.97 1.95 1.93 1.91
2.73 2,57 2.36 2.24 2.17 2.11 2.08 2.05 2.03 2.02
2.88 2.69 245 2.34 2.24 2.20 2.16 2.12 2.10 2.09
2.99 2.78 2,52 2,47 2.31 2.25 2.22 2.18 2.16 2.15
3.20 2.97 2.69 2,55 2.45 2.38 2.34 2.30 2.28 2.27
3.39 3.13 2.83 2.69 2,57. '2.49 '"•"2.44 ""-2,41 2,38 2.36
3.53 3.25 2.93 2.76 '2.64 2.57 2.51 2.48 2,45 '2.43'"
3.65 3.34 3.02 2.83 2.71 2,62 2,57 2,52 2,50 2,48
3.86 3.51 3.16 2.96 2.82 2.73 2,66 2,63 2,59, 2,57
4.00 3.62 3.25 3.04 2.90 2.80 2.73 2.69 2,66 2,63.
4.09 3.72 3.32 3.11 2.96 2.86 2.79 2.75 2.71 2.69
4.19 3.81 3.39 3.17 3.02 2.91 2.84 2.79 2.76 2.72
4.28 3.91 3.46 3.24 3.06 2.97 2.90 2.84 2.80 2.77
4.42 4.00 3.55 3.32 3.16 3.04 2.97 2.91 2.86 2.83
4.54 4.09 3.65 3.39 3.20 3.10 3.02 2.96 2.92 2.89
4.61 4.19 3.70 3.44 3.25 3.13 3.06 3.00 2.96 2.92
4.70 4.23 3.74 3.48 3.30 3.18 3.10 3.04 2.99 2.96
4.75 4.30 3.79 3.53 3.34 3.21 3.13 3.06 3.02 2.98
60
1.71
1.89
2.00
2.07
2.12
2.23
2.32
""2, '38"
2,44
2,52
2,59
2,64
2,68
2.72
2.78
2.83
2.88
2.90
2.93
70
1.70
1.88
1.98
2.05
2.10
2.21
2.30
2.36
2,42
. 2.5O
2,56
2,67
2,65
2.70
2.76
2.80
2.84
2.86
2.90
80
1.69
1.87
1.96
2.03
2.09
2.20
2.29
2.35
2.39
2,48
2,55
2,59
2,63
2.68
2.73
2.77
2.80
2.84
2.86
90
1.86
2.03
2.11
2.19
2.23
2.33
2.42
2.48
2.52
2.61
2.67
2.71
2.74
2.79
2.84
2.89
2.92
2.95
2.98
COCs
90
1.68
1.87
1.96
2.03
2.08
2.18
2.28
2.34
2,38
2.46
2.52 '
2,57 '
2,61
2.65
2.71
2.76
2.79
2.82
2.84
Quarterly)
100
1.85
2.02
2.11
2.17
2.22
2.33
2.41
2.47
2.51
2.60
2.65
2.69
2.73
2.77
2.83
2.87
2.90
2.93
2.96
125
1.85
2.00
2.09
2.16
2.21
2.31
2.39
2.45
2.49
2.57
2.62
2.67
2.70
2.74
2.80
2.84
2.87
2.90
2.92
150
1.83
1.99
2.09
2.14
2.19
2.29
2.38
2.43
2.48
2.55
2.61
2.65
2.69
2.72
2.78
2.82
2.85
2.88
2.90
, Annual)
100
1.68
1.86
1.95
2.02
2.07
2.17
2.27
2.32
2.37
2,45
2,57
2,56
2,59
2,64 .
2.7O
2.73
2.77
2.80
2.82
125
1.67
1.84
1.94
2.01
2.05
2.16
2.24
2.30
2.35
2,43
2,49
2,54
2,57
' ,2.62
2,66
2.71
2.75
2.77
2.79
150
1.67
1.83
1.93
2.00
2.04
2.15
2.23
2.29
2.34
2,42
2,48
2,52
2.56
. 2,59
2,65
2,69
2.72
2.76
2.78

                                                      D-41
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
Unified Guidance
Table 19-1. K-Multipliers for 1-of-2 Interwell Prediction Limits on Observations (40 COCs, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20 25 30
5.69 3.48 2,85 2.57 2.41 2.22 2.12 2.05 2.01
6.91 4.00 3.25 2,88 2.69 2.45 2.34 2.24 2.20
7.66 4.38 3.48 3.06 2,85 2.59 2.46 2.36 2.31
8.31 4.61 3.62 3.20 2,97 2.69 2.55 2.45 2.38
8.69 4.80 3.77 3.32 3.06 2,78 2.62 2.51 2.44
9.62 5.22 4.05 3.53 3.25 2,93 .'2.78' 2.64 2.57
10.38 5.59 4.33 3.74 3.41 3.06 2.9O 2,78 2.68
10.94 5.83 4.47 3.88 3.53 3.18 2.98 2,83 2.75
11.50 6.06 4.61 4.00 3.62 3.25 3.O4 2,90 2.8O
12.25 6.34 4.84 4.19 3.81 3.39 3.17 3,02 .2,91
12.62 6.62 5.03 4.33 3.91 3.48 3.25 3.09 2.98
13.00 6.81 5.17 4.42 4.00 3.55 3.32 3.16 3,04
13.38 7.00 5.27 4.52 4.09 3.62 3.38 3.20 3,09
13.75 7.19 5.41 4.61 4.19 3.70 3.44 3.25 3.13
14.31 7.38 5.59 4.75 4.30 3.79 3.53 3.34 3.21
14.88 7.56 5.69 4.84 4.38 3.88 3.60 3.39 3.27
15.06 7.75 5.78 4.94 4.47 3.93 3.65 3.44 3.32
15.25 7.84 5.88 5.03 4.52 3.98 3.70 3.48 3.34
15.62 7.94 5.97 5.08 4.56 4.02 3.74 3.52 3.39
35
1.97
2.16
2.27
2.34
2.39
2.51
2.62
2.69
2.73
2,84
2.91
2.97
3,00
3,06
3.13
3.18
3.23
3.26
3.30
40 45
1.95 1.93
2.12 2.10
2.23 2.21
2.30 2.28
2.36 2.34
2.48 2.45
2.57 2.55
2.64 2.61
2.69 2.66
2, 79 2, 76
2.85 2,82
2,91 2.86
2.95 2.91
3.OO 2,96
3.O6 . 3.O2
3.12 3.O6
3.16 3,11
3.19 3.14
3.23 3.17
Table 19-1. K- Multipliers for 1-of-2 Interwell Prediction Limits
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20 25 30
6.91 4.00 3.25 2.88 2.69 2.45 2.34 2.24 2.20
8.31 4.61 3.62 3.2O 2.97 2.69 2.55 2.45 2.38
9.06 4.94 3.91 3.39 3'. 13" 2.83 2.69 2.57 2.49
9.62 5.22 4.05 3.53 '3,25 2.93 2.76 2.64 2.57
10.00 5.41 4.19 3.65 3.34 3.02 2.83 2.71 2.62
10.94 5.83 4.47 3.88 3.53 3.18 2.98 2.83 2.75
11.69 6.16 4.75 4.07 3.72 3,32 . 3.11 2.95 2.85
12.25 6.44 4.89 4.21 3.84 3.41 3.18"" 3.03 2.92
12.62 6.62 5.03 4.33 3.91 3.48 3:25' 3.09 2.98
13.38 7.00 5.27 4.52 4.09 3.62 3,38 3.2O 3.09
13.94 7.19 5.45 4.66 4.21 3.72 3.46 3,27. 3J6"
14.31 7.38 5.59 4.75 4.30 3.79 3.53 3,34 3,21
14.69 7.56 5.69 4.84 4.38 3.86 3.58 3,39 3,25
15.06 7.75 5.78 4.94 4.47 3.93 3.65 3,44 . 3,32
15.62 7.94 5.97 5.08 4.56 4.02 3.74 3.52 3.39
16.00 8.12 6.11 5.17 4.66 4.09 3.81 3.58 3.44
16.19 8.31 6.20 5.27 4.75 4.16 3.86 3.62 3,48
16.38 8.41 6.25 5.34 4.80 4.21 3.91 3.67 3.52
16.75 8.50 6.34 5.41 4.84 4.26 3.93 3.70 3.55
35
2.16
2.34
2.44
2.51
2.57
2.69
2.78
2.85
2.91
3.00
3.07
3,13
3,18
3., 23
3.30
3,34
3.39
3,41
3,45-
40 45
2.12 2.10
2.30 2.28
2.41 2.38
2.48 2.45
2.52 2.50
2.64 2.61
2.73 2.70
2.80 2.77
2.85 2.82
2.95 2.91
3.02 2.97
3.06 3.02
3.11 3.06
3.18 3.11
3,23 3.' 17
•3.27 3,21
3,31 3,26
3.34 3.3O
3,38 3.32
50
1.91
2.09
2.20
2.27
2.31
2.43
2.51
2.58
2.63
2.72
2,78
. 2, 83
2,88
. 2,92
2,38
3,03
3..07
3,10
3,13
60
1.89
2.07
2.16
2.23
2.29
2.38
2.48
2.55
2.59
2.68
2.73
'i'.'jB
2,83
2.88
2.93
2.98
3.02
3.04
3.07
70
1.88
2.05
2.15
2.21
2.27
2.36
2.45
2.51
2.56
2.65
2.71
"•z'.'ie"
2.79
2,84 ,
2,90
2,93
2,97
3.OO
3.03.
on Observations
50
2.09
2.27
2.36
2.43
2.48
2.58
2.68
2.73
2.78
2.88
2.93
2.98
3.03
3.07
""3,'i'3"
3,18
3.21
'3,25
.' 3,27
60
2.07
2.23
2.32
2.38
2.44
2.55
2.63
2.69
2.73
2.83
2.89
2.93
2.97
3.02
3.07
3,12
3.16
3,18
•3,21'
70
2.05
2.21
2.30
2.36
2.42
2.51
2.61
2.66
2.71
2.79
2.85
2.90
2.92
2.97
3.03
3.07
3.11
"3,'l3
3.17
80
1.87
2.03
2.12
2.20
2.24
2.35
2.43
2.50
2.55
2.63
2.69
2.73
2.77
2.8O
2.86
2,91
2,95
2,97
3.0O
(40
80
2.03
2.20
2.29
2.35
2.39
2.50
2.58
2.64
2.69
2.77
2.83
2.86
2.90
2.95
3.00
3.04
3.07
3.11
3,13
90
1.87
2.03
2.12
2.18
2.23
2.34
2.42
2.48
2.52
2.61
2.66
2.71
2.75
2.79
, 2.84
2.89
2,92
2.95.
2,98
COCs,
90
2.03
2.18
2.28
2.34
2.38
2.48
2.56
2.62
2.66
2.75
2.80
2.84
2.88
2.92
2.98
3.02
3.05
3.07
3.11
100
1.86
2.02
2.11
2.17
2.22
2.32
2.41
2.46
2.51
2.59
2.65
2.70
2.73
2,77
2,82
2.88
2.9O
2,93
2.96
125
1.84
2.01
2.09
2.16
2.21
2.30
2.38
2.44
2.49
2.57
2.63
2.66
2.70
2.75
2.79
2,84
2.87
2.9O.
2,92
150
1.83
2.00
2.08
2.15
2.20
2.29
2.37
2.43
2.48
2.56
2.61
2.65
2.69
2.72
2.78
2.82
2,85
2,88
2.9O
Quarterly)
100
2.02
2.17
2.27
2.32
2.37
2.46
2.55
2.61
2.65
2.73
2.78
2.82
2.86
2.90
2.96
3.00
3.03
3.06
3.09
125
2.01
2.16
2.24
2.30
2.35
2.44
2.50
2.58
2.63
2.70
2.76
2.79
2.83
2.87
2.92
2.97
2.99
3.03
3.05
150
2.00
2.15
2.23
2.29
2.34
2.43
2.51
2.57
2.61
2.69
2.73
2.78
2.82
2.85
2.90
2.95
2.97
3.00
3.03

                                                      D-42
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                     Unified Guidance
        Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (1 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
0.33 (
0.67 (
0.87 (
.01 (
.12 (
.34 ]
.54
.68
.78
.97
2.10
2.21
2.29
2.39
2.51 :
2.61 :
2.68 :
2.75 ",
2.80 ,
6 8
).25 0.21 (
).54 0.49 (
).71 0.64 (
).82 0.75 (
).91 0.83 (
L.09 0.99 (
.25 1.13
.36 1.22
.44 1.30
.58 1.43
.69 1.52
.76 1.59
.83 1.64
.90 1.71
2.00 1.80
1.07 1.86
2.13 1.91
?,18 1.96
2.22 1.99
10 12 16
).18 0.17 0.15 (
).45 0.43 0.41 (
).60 0.58 0.54 (
).70 0.67 0.64 (
).78 0.75 0.71 (
).93 0.90 0.85 (
.06 1.02 0.97 (
.15 1.11 1.05
.22 1.17 1.11
.34 1.29 1.22
.43 1.37 1.30
.49 1.43 1.35
.54 1.48 1.40
.60 1.54 1.46
.68 1.61 1.53
.74 1.67 1.58
.79 1.72 1.62
.83 1.76 1.66
.87 1.79 1.69
20
).14 (
).39 (
).53 (
).62 (
).69 (
).82 (
).94 (
.02 (
.08
.18
.25
.31
.35
.41
.48
.53
.57
.60
.63
25 30
).13 0.12 (
).38 0.37 (
).51 0.50 (
).60 0.59 (
).67 0.66 (
).80 0.79 (
).91 0.90 (
).99 0.97 (
.05 1.03
.15 1.13
.22 1.20
.27 1.25
.32 1.29
.37 1.34
.44 1.41
.49 1.46
.53 1.50
.56 1.53
.59 1.56
35
).12 (
).36 (
).49 (
).58 (
).65 (
).78 (
).89 (
).96 (
.02
.12
.18
.23
.28
.33
.39
.44
.48
.51
.54
40 45
).ll 0.11 (
).36 0.36 (
).49 0.48 (
).58 0.57 (
).64 0.64 (
).77 0.77 (
).88 0.87 (
).95 0.94 (
.01 1.00 (
.10 1.10
.17 1.16
.22 1.21
.26 1.25
.31 1.30
.37 1.36
.42 1.41
.46 1.45
.49 1.48
.52 1.51
50 60 70
).ll 0.11 0.10 (
).35 0.35 0.35 (
).48 0.48 0.47 (
).57 0.56 0.56 (
).63 0.63 0.62 (
).76 0.75 0.75 (
).87 0.86 0.85 (
).94 0.93 0.92 (
).99 0.98 0.98 (
.09 1.08 1.07
.16 1.14 1.14
.20 1.19 1.18
.24 1.23 1.22
.29 1.28 1.27
.35 1.34 1.33
.40 1.39 1.38
.44 1.42 1.41
.47 1.45 1.44
.50 1.48 1.47
80 90
).10 0.10 (
).34 0.34 (
).47 0.47 (
).56 0.55 (
).62 0.62 (
).75 0.74 (
).85 0.85 (
).92 0.92 (
).97 0.97 (
.07 1.06
.13 1.13
.18 1.17
.22 1.21
.26 1.26
.32 1.32
.37 1.36
.40 1.40
.43 1.43
.46 1.45
100 125
).10 0.10 (
).34 0.34 (
).47 0.46 (
).55 0.55 (
).61 0.61 (
).74 0.74 (
).84 0.84 (
).91 0.91 (
).97 0.96 (
.06 1.05
.12 1.12
.17 1.16
.21 1.20
.25 1.25
.31 1.30
.36 1.35
.39 1.38
.42 1.41
.45 1.44
150
).10
).34
).46
).55
).61
).73
).84
).91
).96
.05
.11
.16
.20
.24
.30
.34
.38
.41
.43
     Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.67 0.54 0.49 (
.01 0.82 0.75 (
.21 0.98 0.89 (
.34 1.09 0.99 (
.45 1.18 1.07
.68 1.36 1.22
.87 .50 1.36
2.00 .61 1.45
2.10 .69 1.52
2.29 .83 1.64
2.41 .92 1.73
2.51 2.00 1.80
2.59 2.06 1.85
2.68 2.13 1.91
2.80 2.22 1.99
2.90 2.29 2.06
2.97 2.35 2.11
3.03 2.40 2.15 ;
3.08 2.44 2.18 ;
).45 0.43 0.41 (
).70 0.67 0.64 (
).84 0.80 0.76 (
).93 0.90 0.85 (
.00 0.96 0.92 (
.15 1.11 1.05
.28 1.22 1.16
.36 1.31 1.24
.43 1.37 1.30
.54 1.48 1.40
.62 1.55 1.47
.68 1.61 1.53
.73 1.66 1.57
.79 1.72 1.62
.87 1.79 1.69
.93 1.84 1.74
.97 1.89 1.79
2.01 1.93 1.82
2.05 1.96 1.85
).39 (
).62 (
).74 (
).82 (
).89 (
.02 (
.12
.20
.25
.35
.42
.48
.52
.57
.63
.68
.72
.76
.79
).38 0.37 (
).60 0.59 (
).72 0.71 (
).80 0.79 (
).87 0.85 (
).99 0.97 (
.09 1.08
.17 1.15
.22 1.20
.32 1.29
.38 1.36
.44 1.41
.48 1.45
.53 1.50
.59 1.56
.64 1.60
.67 1.64
.71 1.67
.73 1.70
).36 (
).58 (
).70 (
).78 (
).84 (
).96 (
.06
.13
.18
.28
.34
.39
.43
.48
.54
.58
.62
.65
.68
).36 0.36 (
).58 0.57 (
).69 0.69 (
).77 0.77 (
).83 0.83 (
).95 0.94 (
.05 1.04
.12 1.11
.17 1.16
.26 1.25
.33 1.32
.37 1.36
.41 1.40
.46 1.45
.52 1.51
.56 1.55
.60 1.59
.63 1.62
.66 1.64
).35 0.35 0.35 (
).57 0.56 0.56 (
).68 0.68 0.67 (
).76 0.75 0.75 (
).82 0.81 0.81 (
).94 0.93 0.92 (
.04 1.03 1.02
.10 1.09 1.09
.16 1.14 1.14
.24 1.23 1.22
.31 1.29 1.28
.35 1.34 1.33
.39 1.38 1.37
.44 1.42 1.41
.50 1.48 1.47
.54 1.52 1.51
.57 1.56 1.55
.60 1.59 1.57
.63 1.61 1.60
).34 0.34 (
).56 0.55 (
).67 0.67 (
).75 0.74 (
).80 0.80 (
).92 0.92 (
.02 1.01
.08 1.08
.13 1.13
.22 1.21
.28 1.27
.32 1.32
.36 1.35
.40 1.40
.46 1.45
.50 1.50
.54 1.53
.57 1.56
.59 1.58
).34 0.34 (
).55 0.55 (
).66 0.66 (
).74 0.74 (
).80 0.79 (
).91 0.91 (
.01 1.00
.07 1.07
.12 1.12
.21 1.20
.27 1.26
.31 1.30
.35 1.34
.39 1.38
.45 1.44
.49 1.48
.52 1.51
.55 1.54
.58 1.57
).34
).55
).66
).73
).79
).91
.00
.06
.11
.20
.25
.30
.33
.38
.43
.47
.51
.53
.56
                                                    D-43
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
       Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.01 0.82 0.75 (
.34 1.09 0.99 (
.54 1.25 1.13
.68 1.36 1.22
.78 1.44 1.30
2.00 1.61 1.45
2.19 1.75 1.57
2.32 1.85 1.66
2.41 1.92 1.73
2.59 2.06 1.85
2.71 2.15 1.93
2.80 2.22 1.99
2.88 2.28 2.04
2.97 2.35 2.11
3.08 2.44 2.18 ;
3.17 2.50 2.24 ;
3.24 2.56 2.29 ;
3.30 2.60 2.33 ;
3.35 2.64 2.37 ;
).70 0.67 0.64 (
).93 0.90 0.85 (
.06 1.02 0.97 (
.15 1.11 1.05
.22 1.17 1.11
.36 1.31 1.24
.48 1.42 1.34
.56 1.50 1.42
.62 1.55 1.47
.73 1.66 1.57
.81 1.73 1.64
.87 1.79 1.69
.92 1.83 1.73
.97 1.89 1.79
>.05 1.96 1.85
MO 2.01 1.90
>.15 2.05 1.94
>.18 2.09 1.97
1.22 2.12 2.00
).62 (
).82 (
).94 (
.02 (
.08
.20
.30
.37
.42
.52
.58
.63
.67
.72
.79
.83
.87
.90
.93
).60 0.59 (
).80 0.79 (
).91 0.90 (
).99 0.97 (
.05 1.03
.17 1.15
.26 1.24
.33 1.31
.38 1.36
.48 1.45
.54 1.51
.59 1.56
.63 1.60
.67 1.64
.73 1.70
.78 1.74
.82 1.78
.85 1.81
.87 1.84
).58 (
).78 (
).89 (
).96 (
.02
.13
.23
.29
.34
.43
.49
.54
.57
.62
.68
.72
.75
.78
.81
).58 0.57 (
).77 0.77 (
).88 0.87 (
).95 0.94 (
.01 1.00 (
.12 1.11
.21 1.20
.28 1.27
.33 1.32
.41 1.40
.47 1.46
.52 1.51
.56 1.54
.60 1.59
.66 1.64
.70 1.68
.73 1.72
.76 1.75
.79 1.77
).57 0.56 0.56 (
).76 0.75 0.75 (
).87 0.86 0.85 (
).94 0.93 0.92 (
).99 0.98 0.98 (
.10 1.09 1.09
.20 1.18 1.18
.26 1.25 1.24
.31 1.29 1.28
.39 1.38 1.37
.45 1.44 1.42
.50 1.48 1.47
.53 1.52 1.50
.57 1.56 1.55
.63 1.61 1.60
.67 1.65 1.64
.71 1.69 1.67
.73 1.72 1.70
.76 1.74 1.73
).56 0.55 (
).75 0.74 (
).85 0.85 (
).92 0.92 (
).97 0.97 (
.08 1.08
.17 1.17
.23 1.23
.28 1.27
.36 1.35
.42 1.41
.46 1.45
.49 1.49
.54 1.53
.59 1.58
.63 1.62
.66 1.66
.69 1.68
.71 1.71
).55 0.55 (
).74 0.74 (
).84 0.84 (
).91 0.91 (
).97 0.96 (
.07 1.07
.16 1.15
.22 1.21
.27 1.26
.35 1.34
.41 1.40
.45 1.44
.48 1.47
.52 1.51
.58 1.57
.62 1.61
.65 1.64
.68 1.66
.70 1.69
).55
).73
).84
).91
).96
.06
.15
.21
.25
.33
.39
.43
.47
.51
.56
.60
.63
.66
.68
        Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (2 COCs, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.71 0.57 0.50 (
.08 0.86 0.77 (
.30 1.03 0.92 (
.47 1.15 1.02 (
.59 1.24 1.11
.86 1.44 1.27
2.09 1.60 1.42
2.26 1.72 1.52
2.39 1.81 1.59
2.62 1.97 1.73
2.78 2.08 1.83
2.91 2.17 1.90
3.01 2,24 1.96
3.13 2.33 2.03
3.29 2.44 2.13
3.41 2.52 2.2O :
3.50 2.59 2,25 :
3.58 2.64 2,30 ;
3.65 2.69 2.34 ,
).46 0.44 0.41 (
).72 0.69 0.65 (
).86 0.82 0.77 (
).96 0.91 0.86 (
.03 0.99 0.93 (
.19 1.13 1.07
.32 1.25 1.18
.41 1.34 1.26
.48 1.41 1.32
.60 1.52 1.43
.69 1.61 1.50
.76 1.67 1.56
.81 1.72 1.61
.88 1.78 1.67
.96 1.86 1.74
>.03 1.92 1.79
>.08 1.97 1.84
1.12 2.01 1.87
2. 16 2.04 1.91
).39 (
).62 (
).75 (
).83 (
).90 (
.03
.14
.21
.27
.38
.45
.50
.55
.60
.67
.72
.76
.80
.83
).38 0.37 (
).61 0.59 (
).73 0.71 (
).81 0.79 (
).87 0.86 (
.00 0.98 (
.11 1.08
.18 1.16
.23 1.21
.33 1.31
.40 1.37
.45 1.42
.50 1.46
.55 1.51
.61 1.58
.66 1.62
.70 1.66
.74 1.69
.76 1.72
).37 (
).59 (
).70 (
).78 (
).84 (
).97 (
.07
.14
.19
.29
.35
.40
.44
.49
.55
.60
.63
.67
.69
).36 0.36 (
).58 0.57 (
).70 0.69 (
).78 0.77 (
).84 0.83 (
).96 0.95 (
.06 1.05
.13 1.12
.18 1.17
.27 1.26
.34 1.32
.38 1.37
.42 1.41
.47 1.46
.53 1.52
.58 1.56
.61 1.60
.64 1.63
.67 1.65
).35 0.35 0.35 (
).57 0.56 0.56 (
).69 0.68 0.67 (
).76 0.76 0.75 (
).82 0.82 0.81 (
).94 0.93 0.93 (
.04 1.03 1.02
.11 1.10 1.09
.16 1.15 1.14
.25 1.24 1.23
.31 1.30 1.29
.36 1.35 1.34
.40 1.38 1.37
.45 1.43 1.42
.50 1.49 1.47
.55 1.53 1.52
.59 1.57 1.55
.62 1.60 1.58
.64 1.62 1.61
).34 0.34 (
).56 0.55 (
).67 0.67 (
).75 0.74 (
).81 0.80 (
).92 0.92 (
.02 1.01
.08 1.08
.13 1.13
.22 1.22
.28 1.28
.33 1.32
.36 1.36
.41 1.40
.46 1.46
.51 1.50
.54 1.53
.57 1.56
.60 1.59
).34 0.34 (
).55 0.55 (
).67 0.66 (
).74 0.74 (
).80 0.79 (
).92 0.91 (
.01 1.00
.08 1.07
.12 1.12
.21 1.20
.27 1.26
.32 1.31
.35 1.34
.40 1.39
.45 1.44
.49 1.48
.53 1.52
.56 1.54
.58 1.57
).34
).55
).66
).74
).79
).91
.00
.06
.11
.20
.26
.30
.34
.38
.43
.48
.51
.54
.56
                                                     D-44
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
     Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (2 COCs, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.08 0.86 0.77 (
.47 1.15 1.02 (
.70 1.32 1.17
.86 1.44 1.27
.99 1.53 1.35
2.26 1.72 1.52
2.49 1.88 1.65
2.66 2.00 1.75
2.78 2.08 1.83
3.01 2.24 1.96
3.17 2.35 2.05
3.29 2.44 2.13
3.38 ,2,50 2.18 ;
3.50 2.59 2.25 ;
3.65 2.69 2.34 ;
3.77 2.77 2.41 ;
3.86 2.84 2,47 '<
3.94 2.89 2,57 ;
4.00 2.94 2,55 ;
).72 0.69 0.65 (
).96 0.91 0.86 (
.09 1.04 0.98 (
.19 1.13 1.07
.26 1.20 1.13
.41 1.34 1.26
.53 1.46 1.37
.62 1.54 1.45
.69 1.61 1.50
.81 1.72 1.61
.90 1.80 1.68
.96 1.86 1.74
2.01 1.91 1.78
2.08 1.97 1.84
2.16 2.04 1.91
1.22 2.10 1.96
2.27 2.15 2.00
2.31 2.19 2.04
2.35 2.22 2.07
).62 (
).83 (
).95 (
.03
.09
.21
.32
.39
.45
.55
.61
.67
.71
.76
.83
.88
.92
.95
.98
).61 0.59 (
).81 0.79 (
).92 0.91 (
.00 0.98 (
.06 1.04
.18 1.16
.28 1.25
.35 1.32
.40 1.37
.50 1.46
.56 1.53
.61 1.58
.65 1.61
.70 1.66
.76 1.72
.81 1.77
.85 1.81
.88 1.84
.91 1.86
).59 (
).78 (
).89 (
).97 (
.02
.14
.23
.30
.35
.44
.50
.55
.59
.63
.69
.74
.77
.81
.83
).58 0.57 (
).78 0.77 (
).88 0.88 (
).96 0.95 (
.01 1.00
.13 1.12
.22 1.21
.29 1.27
.34 1.32
.42 1.41
.48 1.47
.53 1.52
.57 1.55
.61 1.60
.67 1.65
.72 1.70
.75 1.73
.78 1.76
.81 1.79
).57 0.56 0.56 (
).76 0.76 0.75 (
).87 0.86 0.86 (
).94 0.93 0.93 (
.00 0.99 0.98 (
.11 1.10 1.09
.20 1.19 1.18
.27 1.25 1.24
.31 1.30 1.29
.40 1.38 1.37
.46 1.44 1.43
.50 1.49 1.47
.54 1.52 1.51
.59 1.57 1.55
.64 1.62 1.61
.68 1.66 1.65
.72 1.70 1.68
.75 1.73 1.71
.77 1.75 1.73
).56 0.55 (
).75 0.74 (
).85 0.85 (
).92 0.92 (
).98 0.97 (
.08 1.08
.17 1.17
.23 1.23
.28 1.28
.36 1.36
.42 1.41
.46 1.46
.50 1.49
.54 1.53
.60 1.59
.64 1.63
.67 1.66
.70 1.69
.72 1.71
).55 0.55 (
).74 0.74 (
).85 0.84 (
).92 0.91 (
).97 0.96 (
.08 1.07
.16 1.16
.22 1.22
.27 1.26
.35 1.34
.41 1.40
.45 1.44
.49 1.48
.53 1.52
.58 1.57
.62 1.61
.65 1.64
.68 1.67
.70 1.69
).55
).74
).84
).91
).96
.06
.15
.21
.26
.34
.39
.43
.47
.51
.56
.60
.63
.66
.68
      Table 19-2. K-Multipliers for 1-of-3 Interwell Prediction Limits on Observations (2 COCs, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.47 1.15 1.02 (
1.86 1.44 1.27
2.09 1.60 1.42
2.26 1.72 1.52
2.39 1.81 1.59
2.66 2.00 1.75
2.88 2.16 1.89
3.04 2.27 1.98
3.17 2.35 2.05
3.38 2.50 2.18 ;
3.54 2.61 2.27 ;
3.65 2.69 2.34 ;
3.75 2.76 2.40 ;
3.86 2,84 2.47 ;
4.00 2.94 2.55 ;
4.11 3.02 2.62 ;
4.20 3.08 2.67 ;
4.28 3.13 2.71 ;
4.34 3.17 2.75 ;
).96 0.91 0.86 (
.19 1.13 1.07
.32 1.25 1.18
.41 1.34 1.26
.48 1.41 1.32
.62 1.54 1.45
.75 1.66 1.55
.83 1.74 1.63
.90 1.80 1.68
2.01 1.91 1.78
MO 1.98 1.85
2.16 2.04 1.91
1.21 2.09 1.95
1.27 2.15 2.00
2.35 2.22 2.07
2.41 2.28 2.12 ;
1.46 2.32 2.16 ;
2.50 2.36 2.20 ;
2.53 2.39 2.23 ;
).83 (
.03
.14
.21
.27
.39
.49
.56
.61
.71
.78
.83
.87
.92
.98
2.03
2.07
2.10 ;
2.13 ;
).81 0.79 (
.00 0.98 (
.11 1.08
.18 1.16
.23 1.21
.35 1.32
.44 1.41
.51 1.48
.56 1.53
.65 1.61
.72 1.68
.76 1.72
.80 1.76
.85 1.81
.91 1.86
.96 1.91
.99 1.95
2.03 1.98
2.05 2.00
).78 (
).97 (
.07
.14
.19
.30
.39
.45
.50
.59
.65
.69
.73
.77
.83
.88
.91
.94
.97
).78 0.77 (
).96 0.95 (
.06 1.05
.13 1.12
.18 1.17
.29 1.27
.38 1.36
.44 1.42
.48 1.47
.57 1.55
.63 1.61
.67 1.65
.71 1.69
.75 1.73
.81 1.79
.85 1.83
.88 1.86
.91 1.89
.94 1.92
).76 0.76 0.75 (
).94 0.93 0.93 (
.04 1.03 1.02
.11 1.10 1.09
.16 1.15 1.14
.27 1.25 1.24
.35 1.34 1.33
.41 1.40 1.39
.46 1.44 1.43
.54 1.52 1.51
.60 1.58 1.56
.64 1.62 1.61
.68 1.66 1.64
.72 1.70 1.68
.77 1.75 1.73
.81 1.79 1.77
.85 1.82 1.81
.88 1.85 1.83
.90 1.87 1.86
).75 0.74 (
).92 0.92 (
.02 1.01
.08 1.08
.13 1.13
.23 1.23
.32 1.31
.38 1.37
.42 1.41
.50 1.49
.55 1.55
.60 1.59
.63 1.62
.67 1.66
.72 1.71
.76 1.75
.79 1.78
.82 1.81
.84 1.83
).74 0.74 (
).92 0.91 (
.01 1.00
.08 1.07
.12 1.12
.22 1.22
.31 1.30
.37 1.36
.41 1.40
.49 1.48
.54 1.53
.58 1.57
.61 1.60
.65 1.64
.70 1.69
.74 1.73
.77 1.76
.80 1.79
.82 1.81
).74
).91
.00
.06
.11
.21
.29
.35
.39
.47
.52
.56
.59
.63
.68
.72
.75
.78
.80
                                                     D-45
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
        Table 19-2. K-Multipliers for 1-of-3  Interwell Prediction Limits on Observations (5 COCs, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.27 0.98 0.87 (
1.71 1.29 1.13
1.99 1.47 1.28
2.19 1.60 1.39
2.35 1.70 1.48
2.70 1.92 1.65
3.00 2.11 1.81
3.22 2,24 1.91
3.39 2.35 2.00
3.70 2.54 2.15
3.92 2.67 2,25 ;
4.08 2.77 2,34 '*
4.22 2.86 2.40 ,
4.38 2.96 2.48
4.59 3.09 2.59 .,
4.75 3.19 2.67
4.88 3.27 2.73
4.99 3.33 2.79
5.09 3.39 2.83
).81 0.77 0.72 (
.05 1.00 0.94 (
.19 1.13 1.06
.29 1.22 1.14
.36 1.29 1.21
.52 1.43 1.34
.65 1.56 1.45
.75 1.65 1.53
.82 1.71 1.59
.95 1.83 1.70
>.05 1.92 1.77
>.12 1.98 1.83
2,18 2.04 1.88
2,25 2.10 1.94
2,34 2,18 2.01
2.41 2,25 2.07
2.46 2.3O 2.11 ;
2.51 2,34 2,15 :
2.55 2,38 2,18 '*
).70 (
).90 (
.02 (
.10
.16
.28
.39
.46
.52
.62
.69
.75
.79
.84
.91
.96
>.01
>.04
>.07
).68 0.66 (
).88 0.86 (
).99 0.97 (
.07 1.04
.12 1.10
.24 1.22
.34 1.31
.41 1.38
.47 1.43
.56 1.52
.63 1.59
.68 1.64
.72 1.68
.77 1.73
.84 1.79
.88 1.83
.92 1.87
.96 1.90
.99 1.93
).65 (
).85 (
).95 (
.03
.08
.20
.29
.36
.41
.50
.56
.61
.65
.69
.75
.80
.83
.87
.89
).65 0.64 (
).84 0.83 (
).94 0.94 (
.02 1.01
.07 1.06
.18 1.17
.28 1.26
.34 1.33
.39 1.38
.48 1.46
.54 1.52
.59 1.57
.62 1.61
.67 1.65
.73 1.71
.77 1.75
.81 1.79
.84 1.82
.86 1.84
).64 0.63 0.62 (
).83 0.82 0.81 (
).93 0.92 0.91 (
.00 0.99 0.98 (
.05 1.04 1.04
.16 1.15 1.14
.25 1.24 1.23
.32 1.30 1.29
.37 1.35 1.34
.45 1.43 1.42
.51 1.49 1.48
.56 1.54 1.52
.59 1.57 1.56
.64 1.61 1.60
.69 1.67 1.65
.73 1.71 1.69
.77 1.74 1.73
.80 1.77 1.76
.82 1.80 1.78
).62 0.62 (
).81 0.80 (
).91 0.90 (
).98 0.97 (
.03 1.02
.14 1.13
.22 1.22
.28 1.28
.33 1.32
.41 1.40
.47 1.46
.51 1.50
.55 1.54
.59 1.58
.64 1.63
.68 1.67
.71 1.70
.74 1.73
.77 1.76
).62 0.61 (
).80 0.80 (
).90 0.90 (
).97 0.96 (
.02 1.01
.13 1.12
.21 1.20
.27 1.26
.32 1.31
.40 1.39
.45 1.44
.50 1.48
.53 1.52
.57 1.56
.62 1.61
.66 1.65
.70 1.68
.72 1.71
.75 1.73
).61
).79
).89
).96
.01
.11
.20
.26
.30
.38
.44
.48
.51
.55
.60
.64
.67
.70
.72
     Table 19-2.  K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (5 COCs, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.71 1.29 1.13
2.19 1.60 1.39
2.49 1.79 1.55
2.70 1.92 1.65
2.87 2.03 1.74
3.22 2.25 1.91
3.53 2.43 2.07
3.75 ,2,57 2.17
3.92 2.67 2.25 ;
4.22 2.86 2.40 ;
4.43 2.99 2,51 \
4.59 3.09 2,59 \
4.73 3.17 2,65 *
4.88 3.27 2.73
5.09 3.39 2.83
5.24 3.49 2.91 ,
5.37 3.56 2.97 ,
5.47 3.63 3.02
5.56 3.69 3.07
.05 1.00 0.94 (
.29 1.22 1.14
.42 1.35 1.26
.52 1.43 1.34
.59 1.50 1.40
.75 1.65 1.53
.88 1.77 1.64
.97 1.85 1.71
>.05 1.92 1.77
>.18 2.04 1.88
2.27 2.12 1.95
2.34 2.19 2.01
2.39 2.24 2.06
2,46 2.30 2.11 ;
2,55 2.38 2.18 ;
2,62 2.44 2.24 ;
2,67 2,49 2.28 ;
2.72 2,53 2.32 ;
2.76 2:57' 2.35 ;
).90 (
.10
.21
.28
.34
.46
.56
.64
.69
.79
.86
.91
.95
2.01
2.07
2.12 ;
2.16 ;
>..20 ;
2.23 ;
).88 0.86 (
.06 1.04
.17 1.15
.24 1.22
.30 1.27
.41 1.38
.51 1.47
.58 1.54
.63 1.59
.72 1.68
.79 1.74
.84 1.79
.88 1.82
.92 1.87
.99 1.93
2.03 1.98
1.07 2.01
Ml 2.05 ;
2.13 2.07 ;
).85 (
.03
.13
.20
.25
.36
.45
.51
.56
.65
.71
.75
.79
.83
.89
.94
.97
2.00
2.03 ;
).84 0.83 (
.02 1.01
.11 1.10
.18 1.17
.23 1.22
.34 1.33
.43 1.42
.49 1.48
.54 1.52
.62 1.61
.68 1.66
.73 1.71
.76 1.74
.81 1.79
.86 1.84
.91 1.88
.94 1.92
.97 1.95
2.00 1.97
).83 0.82 0.81 (
.00 0.99 0.98 (
.10 1.09 1.08
.16 1.15 1.14
.21 1.20 1.19
.32 1.30 1.29
.40 1.39 1.38
.46 1.45 1.43
.51 1.49 1.48
.59 1.57 1.56
.65 1.63 1.61
.69 1.67 1.65
.73 1.70 1.69
.77 1.74 1.73
.82 1.80 1.78
.87 1.84 1.82
.90 1.87 1.85
.93 1.90 1.88
.95 1.92 1.90
).81 0.80 (
).98 0.97 (
.07 1.07
.14 1.13
.18 1.18
.28 1.28
.37 1.36
.42 1.42
.47 1.46
.55 1.54
.60 1.59
.64 1.63
.67 1.66
.71 1.70
.77 1.76
.80 1.79
.84 1.82
.86 1.85
.89 1.87
).80 0.80 (
).97 0.96 (
.06 1.06
.13 1.12
.17 1.17
.27 1.26
.35 1.34
.41 1.40
.45 1.44
.53 1.52
.58 1.57
.62 1.61
.66 1.64
.70 1.68
.75 1.73
.79 1.77
.82 1.80
.84 1.83
.86 1.85
).79
).96
.05
.11
.16
.26
.34
.39
.44
.51
.56
.60
.63
.67
.72
.76
.79
.81
.84
                                                    D-46
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
      Table 19-2. K-Multipliers for 1-of-3 Interwell Prediction Limits on Observations (5 COCs, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.19 1.60 1.39
2.70 1.92 1.65
3.00 2.11 1.81
3.22 2.25 1.91
3.39 2.35 2.00
3.75 2.57 2.17
4.05 2.75 2.32 ;
4.27 2.89 2.43 ;
4.43 2.99 2.51 ;
4.73 3.17 2.65 ;
4.93 3.30 2.75 ;
5.09 3.39 2,83 '*
5.21 3.47 2.9O \
5.37 3.56 2.97 ;
5.56 3.69 3.07 ;
5.71 3.78 3.14
5.83 3.85 3.20 ,
5.93 3.92 3.25 ',
6.02 3.97 3.30 ',
.29 1.22 1.14
.52 1.43 1.34
.65 1.56 1.45
.75 1.65 1.53
.82 1.71 1.59
.97 1.85 1.71
MO 1.97 1.82
1.20 2.06 1.90
2.27 2.12 1.95
2.39 2.24 2.06
2.48 2.32 2.13 ;
2.55 2.38 2.18 ;
2.61 2.43 2.23 ;
1.67 2.49 2.28 ;
2.76 2.57 2.35 ;
2,82 2.63 2.41 ;
2,88 2.68 2.45 ;
2,92 2.72 2.49 ;
2:96 2.76 2.52 ;
.10
.28
.39
.46
.52
.64
.74
.81
.86
.95
1.02
1.07
Ml ;
M6 ;
1.23 :
1.28 ;
2.32 ;
2.35 ;
2.38 ;
.06 1.04
.24 1.22
.34 1.31
.41 1.38
.47 1.43
.58 1.54
.67 1.63
.74 1.69
.79 1.74
.88 1.82
.94 1.89
.99 1.93
2.03 1.97
2.07 2.01
2.13 2.07 ;
2.18 2.12 ;
2.22 2.15 ;
2.25 2.18 ;
2.28 2.21 ;
.03
.20
.29
.36
.41
.51
.60
.66
.71
.79
.85
.89
.93
.97
2.03 ;
2.07 ;
2.11 ;
2.14 ;
2.16 ;
.02 1.01
.18 1.17
.28 1.26
.34 1.33
.39 1.38
.49 1.48
.58 1.56
.64 1.62
.68 1.66
.76 1.74
.82 1.80
.86 1.84
.90 1.88
.94 1.92
2.00 1.97
2.04 2.01
2.07 2.05 ;
2.10 2.07 ;
2.13 2.10 ;
.00 0.99 0.98 (
.16 1.15 1.14
.25 1.24 1.23
.32 1.30 1.29
.37 1.35 1.34
.46 1.45 1.43
.55 1.53 1.51
.60 1.58 1.57
.65 1.63 1.61
.73 1.70 1.69
.78 1.76 1.74
.82 1.80 1.78
.86 1.83 1.81
.90 1.87 1.85
.95 1.92 1.90
.99 1.96 1.94
2.03 1.99 1.97
2.05 2.02 2.00
2.08 2.04 2.02 ;
).98 0.97 (
.14 1.13
.22 1.22
.28 1.28
.33 1.32
.42 1.42
.50 1.50
.56 1.55
.60 1.59
.67 1.66
.73 1.72
.77 1.76
.80 1.79
.84 1.82
.89 1.87
.92 1.91
.95 1.94
.98 1.97
2.00 1.99
).97 0.96 (
.13 1.12
.21 1.20
.27 1.26
.32 1.31
.41 1.40
.49 1.48
.54 1.53
.58 1.57
.66 1.64
.71 1.69
.75 1.73
.78 1.76
.82 1.80
.86 1.85
.90 1.88
.93 1.91
.96 1.94
.98 1.96
).96
.11
.20
.26
.30
.39
.47
.52
.56
.63
.68
.72
.75
.79
.84
.87
.90
.93
.95
       Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (10 COCs, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.76 1.31 1.14
2.29 1.63 1.41
2.62 1.83 1.57
2.87 1.97 1.68
3.07 2.08 1.77
3.50 2.33 1.95
3.88 2.54 2.12
4.15 2.69 2,23 ;
4.36 2.81 2.32 ;
4.75 3.02 2.49
5.02 3.17 2.60 ,
5.22 3.29 2.69
5.39 3.38 2.76
5.61 3.50 2.85
5.86 3.64 2.96
6.06 3.76 3.05
6.23 3.85 3.12
6.37 3.93 3.18
6.48 3.99 3.23
.06 1.00 0.94 (
.30 1.23 1.15
.44 1.35 1.26
.53 1.45 1.35
.61 1.52 1.41
.77 1.66 1.54
.91 1.79 1.65
2.01 1.88 1.73
2.08 1.95 1.79
2,22 2.07 1.90
2.32 '2.16" 1.97
2.40 2.23 2.03
2.46 2,28 2.08
2.53 2,35 2.14 ;
2.63 2.44 2.21 '*
2.71 2.50 2.27 ..
l.ll 2.56 2,32 .
2.82 2.60 '2.38 .
2.86 2.64 2.39 .
).91 (
.10
.21
.29
.35
.47
.57
.65
.70
.80
.87
.93
.97
2.02
2.09 ;
2. 14. :
2j9 :
2.22 .
2.26 ,
).88 0.86 (
.07 1.04
.17 1.15
.25 1.22
.30 1.27
.42 1.38
.51 1.48
.58 1.54
.63 1.59
.73 1.68
.79 1.75
.84 1.79
.89 1.83
.94 1.88
2.00 1.94
2.05 1.99
2.09 2.03
2,12 2.06 ;
2.15 2.08 ;
).85 (
.03
.13
.20
.25
.36
.45
.51
.56
.65
.71
.76
.79
.84
.90
.94
.98
2.01
2.04 ;
).84 0.83 (
.02 1.01
.12 1.11
.18 1.17
.24 1.22
.34 1.33
.43 1.42
.49 1.48
.54 1.52
.63 1.61
.69 1.67
.73 1.71
.77 1.75
.81 1.79
.87 1.85
.91 1.89
.95 1.92
.98 1.95
2.00 1.98
).83 0.82 0.81 (
.00 0.99 0.98 (
.10 1.09 1.08
.16 1.15 1.14
.22 1.20 1.19
.32 1.30 1.29
.41 1.39 1.38
.47 1.45 1.43
.51 1.49 1.48
.59 1.57 1.56
.65 1.63 1.61
.69 1.67 1.65
.73 1.71 1.69
.77 1.75 1.73
.83 1.80 1.78
.87 1.84 1.82
.90 1.87 1.85
.93 1.90 1.88
.96 1.93 1.90
).81 0.80 (
).98 0.97 (
.07 1.07
.14 1.13
.18 1.18
.28 1.28
.37 1.36
.42 1.42
.47 1.46
.55 1.54
.60 1.59
.64 1.63
.68 1.67
.72 1.71
.77 1.76
.81 1.80
.84 1.83
.87 1.85
.89 1.88
).80 0.80 (
).97 0.96 (
.06 1.06
.13 1.12
.17 1.17
.27 1.26
.35 1.35
.41 1.40
.46 1.44
.53 1.52
.58 1.57
.62 1.61
.66 1.64
.70 1.68
.75 1.73
.79 1.77
.82 1.80
.84 1.83
.87 1.85
).79
).96
.05
.11
.16
.26
.34
.39
.44
.51
.56
.60
.63
.67
.72
.76
.79
.82
.84
                                                     D-47
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
    Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (10 COCs, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.29 1.63 1.41 .29 1.23 1.15
2.87 1.97 1.68 .53 1.45 1.35
3.23 2.18 1.84 .67 1.57 1.46
3.50 2.32 1.95 .77 1.66 1.54
3.70 2.44 2.04 .85 1.73 1.60
4.14 2.69 2.23 2.01 1.88 1.73
4.53 2.90 2.39 2.15 2.00 1.84
4.80 3.06 2.51 2.25 2.09 1.91
5.02 3.17 2.6O 2.32 2.16 1.97
5.39 3.38 2.76 2^6 2.28 2.08
5.66 3.53 2.88 2:56 2.37 2.16 ;
5.86 3.64 2.96 2,63 2.44 2.21 ;
6.04 3.74 3.03 2.69 2,49 2.26 ;
6.23 3.85 3.12 2.76 2,56 2.32 ;
6.48 3.99 3.23 2.86 2.64 2.39 ;
6.68 4.10 3.32 2.93 2,7? 2,45 ;
6.84 4.19 3.38 2.99 2.76 2.5O \
6.97 4.27 3.44 3.04 2.80 2,55 ;
7.07 4.34 3.49 3.08 2.84 2,57 :
.10
.29
.39
.47
.53
.65
.75
.82
.87
.97
2.04
2.09 ;
2.14 ;
2.19 ;
2.26 ;
2.3i ;
2.35 ;
2.39 ;
2.4i ;
.07 1.04
.25 1.22
.35 1.32
.42 1.38
.47 1.44
.58 1.54
.68 1.63
.74 1.70
.79 1.75
.88 1.83
.95 1.89
2.00 1.94
2.04 1.98
2.09 2.02
2.15 2.08 ;
2.20 2.13 ;
2.24 2.17 ;
2.27 2.20 ;
2.30 2.22 ;
.03
.20
.29
.36
.41
.51
.60
.66
.71
.79
.85
.90
.94
.98
2.04 ;
2.08 ;
2.12 ;
2.15 ;
2.17 ;
.02 1.01
.18 1.17
.28 1.27
.34 1.33
.39 1.38
.49 1.48
.58 1.56
.64 1.62
.69 1.67
.77 1.75
.82 1.80
.87 1.85
.90 1.88
.95 1.92
2.00 1.98
2.05 2.02 ;
2.08 2.05 ;
Ml 2.08 ;
2.13 2.10 ;
.00 0.99 0.98 (
.16 1.15 1.14
.26 1.24 1.23
.32 1.30 1.29
.37 1.35 1.34
.47 1.45 1.44
.55 1.53 1.51
.61 1.58 1.57
.65 1.63 1.61
.73 1.71 1.69
.78 1.76 1.74
.83 1.80 1.78
.86 1.83 1.81
.90 1.88 1.85
.96 1.93 1.90
2.00 1.97 1.94
2.03 2.00 1.97
2.06 2.02 2.00
2.08 2.05 2.02 ;
).98 0.97 (
.14 1.13
.22 1.22
.28 1.28
.33 1.32
.43 1.42
.50 1.50
.56 1.55
.60 1.59
.67 1.67
.73 1.72
.77 1.76
.80 1.79
.84 1.83
.89 1.88
.93 1.91
.96 1.94
.98 1.97
2.00 1.99
).97 0.96 (
.13 1.12
.21 1.20
.27 1.26
.32 1.31
.41 1.40
.49 1.48
.54 1.53
.58 1.57
.66 1.64
.71 1.69
.75 1.73
.78 1.76
.82 1.80
.87 1.85
.90 1.88
.93 1.91
.96 1.94
.98 1.96
).96
.11
.20
.26
.30
.39
.47
.52
.56
.63
.68
.72
.75
.79
.84
.87
.90
.93
.95
      Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (10 COCs, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.87 1.97 1.68 1.53 1.45 1.35
3.50 2.32 1.95 1.77 1.66 1.54
3.88 2.54 2.11 1.91 1.79 1.65
414 2.69 2.23 2.01 1.88 1.73
4.36 2,81 2.32 2.08 1.95 1.79
4.80 3.06 2.51 2.25 2.09 1.91
5.19 3.27 2.67 2.38 2.21 2.02
5.45 3.42 2,79 2.48 2.30 2.10
5.66 3.53 2.88 2.56 2.37 2.16 ;
6.04 3.74 3.03 2.69 2.49 2.26 ;
6.29 3.88 3.14 2.79 2.58 2.34 ;
6.48 3.99 3.23 2.86 2.64 2.39 ;
6.64 4.08 3.30 2,92 2.70 2.44 ;
6.84 4.19 3.38 2,99, 2.76 2.50 ;
7.07 4.34 3.49 3.08 2,84 2.57 ;
7.27 4.44 3.57 3.15 2-97 2.62 ;
7.42 4.52 3.64 3.21 2,95 2.67 ;
7.54 4.60 3.70 3.26 3.OQ 2.71 ;
7.66 4.66 3.75 3.30 3.O4 2.74 ;
.29
.47
.57
.65
.70
.82
.92
.98
2.04
2.14 ;
2.20 ;
2.26 ;
2.30 ;
2.35 ;
2.4i ;
2.47 ;
>.5i ;
2.54 ;
2.57 ;
.25 1.22
.42 1.38
.51 1.48
.58 1.54
.64 1.59
.74 1.70
.84 1.78
.90 1.85
.95 1.89
2.04 1.98
2.10 2.04
2.15 2.08 ;
2.19 2.12 ;
2.24 2.17 ;
2.30 2.22 ;
2.35 2.27 ;
2.39 2.30 ;
2.42 2.34 ;
2.44 2.36 ;
.20
.36
.45
.51
.56
.66
.75
.81
.85
.94
.99
2.04 ;
2.07 ;
2.12 ;
2.17 ;
2.22 ;
2.25 ;
2.28 ;
2.30 ;
.18 1.17
.34 1.33
.43 1.42
.49 1.48
.54 1.53
.64 1.62
.72 1.70
.78 1.76
.82 1.80
.90 1.88
.96 1.94
2.00 1.98
2.04 2.01
2.08 2.05 ;
2.13 2.10 ;
2.18 2.15 ;
2.21 2.18 ;
2.24 2.21 ;
2.26 2.23 ;
.16 1.15 1.14
.32 1.30 1.29
.41 1.39 1.38
.47 1.45 1.44
.51 1.49 1.48
.61 1.58 1.57
.69 1.66 1.65
.74 1.72 1.70
.78 1.76 1.74
.86 1.83 1.81
.92 1.89 1.87
.96 1.93 1.90
.99 1.96 1.94
2.03 2.00 1.97
2.08 2.05 2.02 ;
2.12 2.08 2.06 ;
2.15 2.12 2.09 ;
2.18 2.14 2.12 ;
2.20 2.17 2.14 ;
.14 1.13
.28 1.28
.37 1.36
.43 1.42
.47 1.46
.56 1.55
.63 1.63
.69 1.68
.73 1.72
.80 1.79
.85 1.84
.89 1.88
.92 1.91
.96 1.94
2.00 1.99
2.04 2.03 ;
2.07 2.06 ;
2.10 2.08 ;
2.12 2.10 ;
.13 1.12
.27 1.26
.35 1.35
.41 1.40
.46 1.44
.54 1.53
.62 1.60
.67 1.66
.71 1.69
.78 1.76
.83 1.81
.87 1.85
.90 1.88
.93 1.91
.98 1.96
2.02 2.00
2.05 2.02 ;
2.07 2.05 ;
2.09 2.07 ;
.11
.26
.34
.39
.44
.52
.60
.65
.68
.75
.80
.84
.87
.90
.95
.98
2.01
2.03
2.05
                                                     D-48
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
       Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (20 COCs, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.34 1.65 1.42 .30 1.23 1.15
2.99 2.00 1.69 .54 1.45 1.35
3.40 2,22 1.86 .68 1.58 1.46
3.71 2.38 1.98 .78 1.67 1.54
3.95 2.50 2.07 .87 1.74 1.61
4.48 2.77 2,27 2.03 1.89 1.73
4.96 3.01 2.44 2.18 2.02 1.85
5.29 3.18 2.57 2,28 2.11 1.92
5.57 3.32 2.67 2*36 '2,' 18'-' 1.98
6.04 3.55 2.84 2.50 2,31 2.09
6.39 3.73 2.97 2.61 2.40 2,17 :
6.64 3.87 3.07 2.69 2.47 2,23 '*
6.86 3.97 3.14 2.75 2.53 2,28 t
7.11 4.10 3.24 2.84 2.60 2,34 .,
7.46 4.27 3.36 2.94 2.69 2.42
7.70 4.40 3.46 3.02 2.76 2.48
7.91 4.51 3.54 3.08 2.82 2.53
8.09 4.59 3.60 3.13 2.87 2.57
8.24 4.67 3.66 3.18 2.91 2.60
.10
.29
.40
.47
.53
.65
.75
.82
.88
.98
2.05
>.io ;
2, 75, ;
2,20 ;
2.27 ' ,
2,32 : .
2.37 ..
2.4O .
2.44
.07 1.04
.25 1.22
.35 1.32
.42 1.38
.47 1.44
.59 1.55
.68 1.64
.75 1.70
.80 1.75
.89 1.84
.96 1.90
2.01 1.95
2.05 1.98
2.10 2.03
2. 18 2.09 ;
3.21 2.14 ;
2.25 2,18 ,
2,28 2.21 ' .
2,31 2,23* ,
.03
.20
.29
.36
.41
.52
.60
.67
.71
.80
.86
.90
.94
.98
2.04 ;
2.09 ;
2.12" :
2:15. • ,
ZJ8 . .
.02 1.01
.19 1.17
.28 1.27
.34 1.33
.39 1.38
.50 1.48
.58 1.56
.64 1.62
.69 1.67
.77 1.75
.83 1.80
.87 1.85
.91 1.88
.95 1.93
2.01 1.98
2.05 2.02 ;
2.08 2.06 ;
2.11 2.08 ;
?. 14 2.11 ;
.00 0.99 0.98 (
.16 1.15 1.14
.26 1.24 1.23
.32 1.30 1.29
.37 1.35 1.34
.47 1.45 1.44
.55 1.53 1.52
.61 1.59 1.57
.65 1.63 1.61
.73 1.71 1.69
.79 1.76 1.74
.83 1.80 1.78
.86 1.84 1.82
.91 1.88 1.86
.96 1.93 1.91
2.00 1.97 1.94
2.03 2.00 1.98
2.06 2.03 2.00
2.08 2.05 2.02 ;
).98 0.97 (
.14 1.13
.22 1.22
.28 1.28
.33 1.32
.43 1.42
.50 1.50
.56 1.55
.60 1.59
.68 1.67
.73 1.72
.77 1.76
.80 1.79
.84 1.83
.89 1.88
.93 1.91
.96 1.94
.98 1.97
2.01 1.99
).97 0.96 (
.13 1.12
.21 1.21
.27 1.26
.32 1.31
.41 1.40
.49 1.48
.54 1.53
.58 1.57
.66 1.64
.71 1.69
.75 1.73
.78 1.76
.82 1.80
.87 1.85
.90 1.88
.93 1.92
.96 1.94
.98 1.96
).96
.11
.20
.26
.30
.39
.47
.52
.56
.64
.68
.72
.75
.79
.84
.87
.90
.93
.95
    Table 19-2. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Observations (20 COCs, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.99 2.00 1.69 1.54 1.45 1.35
371 2.38 1.98 1.78 1.67 1.54
4.16 2.61 2.15 1.93 1.80 1.66
4.48 2.77 2.27 2.03 1.89 1.73
4.75 2.91 2.36 2.11 1.96 1.80
5.29 3.18 2.57 2.28 2.11 1.92
5.78 3.43 2.74 2.43 2.24 2.04
6.11 3.59 2.87 2.53 2.33 2.11
6.39 3.73 2.97 2:.61 2.40 2.17 ;
6.86 3.97 3.14 2.75 2,53 2.28 ;
7.19 4.14 3.27 2.86 2.62 2.36 ;
7.46 4.27 3.36 2.94 2,69 2.42 ;
7.66 4.38 3.44 3.00 2.75 2.47 :
7.91 4.51 3.54 3.08 2.82 2.53 '*
8.24 4.67 3.66 3.18 2.91 2.60 \
8.48 4.79 3.75 3.26 2.98 2.68 :
8.67 4.90 3.83 3.33 3.03 2.71
8.83 4.98 3.89 3.38 3.08 2.75 ,
8.98 5.06 3.95 3.42 3.12 2.78
.29
.47
.58
.65
.71
.82
.92
.99
2.05
2.15 ;
1.22 ;
2.27 ;
2.3i ;
2.37 ;
2.44 ;
2.49 :
2,53 :
2,57 • ;
2.6O ,
.25 1.22
.42 1.38
.52 1.48
.59 1.55
.64 1.59
.75 1.70
.84 1.79
.91 1.85
.96 1.90
2.05 1.98
Ml 2.04 ;
M6 2.09 ;
1.20 2.13 ;
1.25 2.18 ;
2.31 2.23 ;
2.36 2.28 ;
2.40 2.32 ;
2.43 2.35 ;
2.46 2.37 ;
.20
.36
.45
.52
.56
.67
.75
.81
.86
.94
2.00
2.04 ;
2.08 ;
M2 ;
MS ;
1.22 ;
>.26 :
1.29 ;
2.3i ;
.19 1.17
.34 1.33
.43 1.42
.50 1.48
.54 1.53
.64 1.62
.72 1.70
.78 1.76
.83 1.80
.91 1.88
.96 1.94
2.01 1.98
2.04 2.01
2.08 2.06 ;
M4 2.11 ;
MS 2.15 ;
1.21 2.18 ;
1.24 2.21 ;
2.27 2.23 ;
.16 1.15 1.14
.32 1.30 1.29
.41 1.39 1.38
.47 1.45 1.44
.51 1.49 1.48
.61 1.59 1.57
.69 1.67 1.65
.74 1.72 1.70
.79 1.76 1.74
.86 1.84 1.82
.92 1.89 1.87
.96 1.93 1.91
.99 1.96 1.94
2.03 2.00 1.98
2.08 2.05 2.02 ;
2.12 2.09 2.06 ;
2.16 2.12 2.09 ;
2.19 2.15 2.12 ;
2.21 2.17 2.14 ;
.14 1.13
.28 1.28
.37 1.36
.43 1.42
.47 1.46
.56 1.55
.64 1.63
.69 1.68
.73 1.72
.80 1.79
.85 1.84
.89 1.88
.92 1.91
.96 1.94
2.01 1.99
2.04 2.03 ;
2.07 2.06 ;
2.10 2.08 ;
2.12 2.10 ;
.13 1.12
.27 1.26
.35 1.35
.41 1.40
.46 1.44
.54 1.53
.62 1.60
.67 1.66
.71 1.69
.78 1.76
.83 1.81
.87 1.85
.90 1.88
.93 1.92
.98 1.96
2.02 2.00
2.05 2.03 ;
2.07 2.05 ;
2.09 2.07 ;
.11
.26
.34
.39
.44
.52
.60
.65
.68
.75
.80
.84
.87
.90
.95
.98
2.01
2.03
2.06
                                                    D-49
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                         Unified Guidance
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.71
4.48
4.96
5.29
5.57
6.11
6.60
6.93
7.19
7.66
7.97
8.24
8.44
8.67
8.98
9.22
9.41
9.57
9.69
Table
9-2.
6
2.38
2.77
3.01
3.18
3.32
3.59
3.84
4.00
4.14
4.38
4.54
4.67
4.78
4.90
5.06
5.18
5.28
5.37
5.44
19-2
K- Multipliers for 1-of-3 Interwell Prediction Limits on Observations (20 COCs, Quarterly)
8 10 12 16 20 25 30 35
1.98 1.78 1.67 1.54 .47 .42 1.38 .36
2.27 2.03 1.89 1.73 .65 .59 1.55 .52
2.44 2.18 2.02 1.85 .75 .68 1.64 .60
2.57 2.28 2.11 1.92 .82 .75 1.70 .67
2.67 2.36 2.18 1.98 .88 .80 1.75 .71
2,87 2.53 2.33 2.11 .99 .91 1.85 .81
305 2.68 2.46 2.22 2.09 2.00 1.94 .89
3.17 2.78 2.55 2.30 2.16 2.06 2.00 .95
3.27 2.86 2.62 2.36 2.22 2.11 2.04 2.00
40 45 50
.34 1.33
.50 1.48
.58 1.56
.64 1.62
.69 1.67
.78 1.76
.86 1.84
.92 1.89
.96 1.94
3.44 3.OO 2.75 2.47 2.31 2.20 2.13 2.08 2.04 2.01
.32
.47
.55
.61
.65
.74
.82
.88
.92
.99
3.56 3.11 2,84 2.54 2.38 2.26 2.19 2.14 2.10 2.07 2.04
3.66 3.18 '2.91 2.60 2.44 2.31 2.23 2.18 2.14 2.11 2.08
3.73 3.25 2.96 2.65 2.48 2.35 2.27 2.21 2.17 2.14 2.12
3.83 3.33 3.03 2.71 2.53 2.40 2.32 2.26 2.21 2.18 2.16
3.95 3.42 3.12 2.78. 2.60 2.46 2.37 2.31 2.27 2.23 2.21
4.03 3.50 3.19 2.84 2.65 2.51 2.42 2.36 2.31 2.28 2.25
4.11 3.56 3.24 2.89 2.70 2.55 2.45 2.39 2.34 2.31 2.28
4.17 3.61 3.29 ,2,93 2.73 2.58 2.49 2.42 2.37 2.33 2.30
4.23 3.66 3.33 2.96 2.76 2.61 2.51 2.44 2.40 2.36 2.33
60
1.30
1.45
1.53
1.59
1.63
1.72
1.79
1.85
1.89
1.96
2.01
2.05
2.08
2.12
2.17
2.21
2.24
2.26
2.29
70 80
1.29
1.44
1.52
1.57
1.61
1.70
1.77
1.83
1.87
1.94
1.99
.28
.43
.50
.56
.60
.69
.76
.81
.85
.92
.97
2.02 2.01
2.06 2.04
2.09 2.07
2.14 2.12
2.18 2.16
2.21 2.19
2.23 2.21
2.25 2.23
. K-Multipliers for 1-of-3 Interwell Prediction Limits on Observations (40
90 100
1.28 .27
1.42 .41
1.50 .49
1.55 .54
1.59 .58
1.68 .67
1.75 .74
1.80 .79
1.84 .83
1.91 .90
1.96 .94
1.99 .98
2.02 2.01
2.06 2.05
2.10 2.09
2.14 2.13
2.17 2.16
2.19 2.18
2.21 2.20
125 150
1.26
1.40
1.48
1.53
1.57
1.66
1.73
1.77
1.81
1.88
1.93
1.96
1.99
.26
.39
.47
.52
.56
.65
.72
.76
.80
.87
.91
.95
.98
2.03 2.01
2.07 2.06
2.10 2.09
2.13 2.12
2.16 2.14
2.18 2.16
COCs, Annual)
   w/n
10
12
16
20
25
30
35
40
                                         45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.05 2.03 1.70 1.55 .45 1.35 1.29
3.84 2.41 1.99 1.79 .68 1.55 1.47
4.36 2.65 2.16 1.94 .81 1.66 1.58
4.72 2.83 2,29 2.04 .90 1.74 1.65
5.05 2.97 2.39 2.12 .97 1.80 1.71
5.70 3.27 2.61 2.30 . 2.12 1.93 1.83
6.30 3.54 2.79 2.45 2.26 2.04 1.93
6.74 3.73 2.93 2.56 2.35 2.12 2.00
7.07 3.90 3.04 2.64 2.42 2,18 2.05
7.67 4.17 3.23 2.80 2.56 2.29 2.16 '*
8.11 4.36 3.37 2.91 2.65 2.37 2.22 \
8.44 4.51 3.46 3.00 2.72 2.44 2,2.8'. .
8.71 4.64 3.56 3.06 2.79 2.48 ,2,33 „
8.98 4.79 3.65 3.15 2.86 2.54 2,38 .
9.42 4.98 3.79 3.26 2.96 2.63 2.45
9.75 5.13 3.90 3.34 3.03 2.69 2.50 ,
10.02 5.24 3.98 3.41 3.09 2.74 2.55
10.24 5.35 4.06 3.47 3.14 2.78 2.59
10.41 5.43 4.12 3.52 3.19 2.81 2.62
.25 1.22 1.20
.42 1.39 1.36
.52 1.48 1.45
.59 1.55 1.52
.64 1.59 1.57
.75 1.70 1.67
.84 1.79 1.75
.91 1.85 1.81
.96 1.90 1.86
>.05 1.98 1.94
Ml 2.05 2.00
2.17 2.09 2.05 ;
2,21 2.13 2.08 ;
2.26 2,18 2,1:2 :
2.32 2.24 2.18 ,
2.37 2.29 2,23 .
2.41 2,32 2,26. .
2.44 2,35 . 2.29 ,
2.47 2.38 2.32 -
.18 1.17 1.16
.34 1.33 1.32
.43 1.42 1.41
.50 1.48 1.47
.54 1.53 1.51
.64 1.62 1.61
.72 1.70 1.69
.79 1.76 1.75
.83 1.81 1.79
.91 1.89 1.86
.96 1.94 1.92
>.01 1.98 1.96
>.05 2.02 1.99
>.09 2.06 2.03 ;
2.14 	 2~.1l" 2.09 ;
2.18 2.15 2', 13" :
2,22 2.19 2.16 ,
2.25 .2,21 2,19 ,
2.27. 2.24 2,21 .
.15 1.14
.30 1.29
.39 1.38
.45 1.44
.49 1.48
.59 1.57
.66 1.65
.72 1.70
.76 1.74
.83 1.82
.89 1.87
.93 1.91
.96 1.94
>.00 1.98
>.05 2.03 ;
>.09 2.06 ;
2. 12 2.09 ;
2.1-5 2,12 ;
2. 17 2- 14 .
.14 1.13
.29 1.28
.37 1.36
.42 1.42
.47 1.46
.56 1.55
.64 1.63
.69 1.68
.73 1.72
.80 1.79
.85 1.84
.89 1.88
.92 1.91
.96 1.95
>.01 1.99
>.05 2.03 ;
1.07 2.06 ;
MO 2.08 ;
2. 12 2.10 ;
.13
.27
.36
.41
.45
.54
.62
.67
.71
.78
.83
.87
.90
.93
.98
1.02 ;
>.05 ;
>.07 ;
>.09 ;
.12 1.12
.27 1.26
.35 1.34
.40 1.40
.44 1.44
.53 1.52
.60 1.59
.66 1.65
.69 1.68
.77 1.75
.81 1.80
.85 1.84
.88 1.87
.92 1.90
.96 1.95
>.00 1.98
>.03 2.01
>.05 2.04
1.07 2.06
                                                       D-50
                                                                                                         March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Observations
Unified Guidance
    Table 19-2. K-Multipliers  for 1-of-3  Interwell Prediction Limits on Observations (40 COCs, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20 25 30 35
3.84 2.41 1.99 1.79 1.68 1.55 1.47 .42 1.39 1.36
4.72 2.83 2.29 2.04 1.90 1.74 1.65 .59 1.55 1.52
5.29 3.09 2.48 2.19 2.03 1.85 1.75 .68 1.64 1.60
5.70 3.27 2.61 2.30 2.12 1.93 1.83 .75 1.70 1.67
6.03 3.42 2.71 2.38 2.20 1.99 1.88 .80 1.75 1.71
6.74 3.73 2.93 2.56 2.35 2.12 2.00 .91 1.85 1.81
7.34 4.01 3.12 2.71 2.48 2.23 2.10 2.00 1.94 1.90
7.78 4.20 3.26 2.83 2.58 2.31 2.17 2.07 2.00 1.96
8.11 4.36 3.37 2.91 2,65 2.37 2.22 2.11 2.05 2.00
8.71 4.64 3.56 3.06 2.79 • '2.48 2.33 2.21 2.13 2.08 ;
9.09 4.83 3.69 3.17 2.89 2.57 2.39 2.27 2.19 2.14 ;
9.42 4.98 3.79 3.26 2.96 .2,63 2.45 2.32 2.24 2.18 ;
9.70 5.10 3.88 3.32 3.02 2.67 2, SO 2.36 2.28 2.22 ;
10.02 5.24 3.98 3.41 3.09 2,74 ,2.55: 2.41 2.32 2.26 ;
10.41 5.43 4.12 3.52 3.19 2.81 2,62, 2,47 2.38 2.32 ;
10.73 5.59 4.23 3.61 3.26 2.87 2,67 ,2,52 2.43 2.36 ;
10.95 5.70 4.31 3.68 3.32 2.93 2,72 2,56 '2,46 " 2.39 ;
11.17 5.81 4.38 3.73 3.37 2.97 . 2,75 2.59 2,49 2.43 ;
11.39 5.89 4.43 3.78 3.41 3.00 2,78 ',2,62 2.52 2.45 '4
40 45 50
.34 1.33 1.32
.50 1.48 1.47
.58 1.56 1.55
.64 1.62 1.61
.69 1.67 1.65
.79 1.76 1.75
.86 1.84 1.82
.92 1.90 1.88
.96 1.94 1.92
>.05 2.02 1.99
MO 2.07 2.05 ;
>.14 2.11 2.09 ;
>.18 2.15 2.12 ;
1.22 2.19 2.16 ;
1.27 2.24 2.21 ;
>.31 2.28 2.25 ;
>.35 2.31 2.28 ;
2.37 2.34 2.31 ;
>.40 2.36 2.33 ;
60 70
.30 1.29
.45 1.44
.53 1.52
.59 1.57
.63 1.62
.72 1.70
.79 1.78
.85 1.83
.89 1.87
.96 1.94
>.01 1.99
>.05 2.03 ;
>.08 2.06 ;
1.12 2.09 ;
>.17 2.14 ;
>.21 2.18 ;
>.24 2.21 ;
1.26 2.23 ;
1.29 2.25 ;
80 90
.29 1.28
.42 1.42
.51 1.50
.56 1.55
.60 1.59
.69 1.68
.76 1.75
.81 1.80
.85 1.84
.92 1.91
.97 1.96
>.01 1.99
>.04 2.02 ;
1.07 2.06 ;
1.12 2.10 ;
>.16 2.14 ;
>.19 2.17 ;
1.21 2.19 ;
1.23 2.21 ;
100
.27
.41
.49
.54
.58
.67
.74
.79
.83
.90
.94
.98
>.01
>.05 ;
>.09 ;
>.is ;
>.ie ;
>.is ;
1.20 ;
125 150
.27 1.26
.40 1.40
.48 1.47
.53 1.52
.57 1.56
.66 1.65
.72 1.72
.78 1.77
.81 1.80
.88 1.87
.93 1.91
.96 1.95
.99 1.98
>.03 2.01
1.07 2.06
>.10 2.09
>.13 2.12
>.16 2.14
>.18 2.16
      Table 19-2. K-Multipliers for 1-of-3 Interwell Prediction Limits on Observations (40 COCs, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.72
5.70
6.30
6.74
7.07
7.78
8.38
8.77
9.09
9.70
10.08
10.41
10.68
10.95
11.39
11.66
11.94
12.10
12.27
6
2. 83
3.27
3.54
3.73
3.90
4.20
4.49
4.68
4.83
5.10
5.29
5.43
5.57
5.70
5.89
6.03
6.14
6.25
6.30
8
2.29
2.61
"Z7S"
2,93
3.04
3.26
3.45
3.58
3.69
3.88
4.01
4.12
4.20
4.31
4.43
4.54
4.61
4.69
4.75
10
2.04
2.30
2.45
2.56
2.64
2,83
2.98
3.09
3.17
3.32
3.43
3.52
3.59
3.68
3.78
3.86
3.93
3.98
4.04
12
1.90
2.12
2.26
2.35
2.42
2.58
2.72
2.8O
2,89
3.02
3.11
3.19
3.24
3.32
3.41
3.48
3.54
3.58
3.63
16
1.74
1.93
2.04
2.12
2.18
2.31
2.42
2.50
2.57
2.67
2.76
2,81
2,87
2,93
3.OO
3.O6
3.11
3.15
3.19
20
1.65
1.83
1.93
2.00
2.05
2.17
2.27
2.34
2.39
2.50
2.57
2.62
2.66
2.72
2,78"
2.84
2,88
2,91
2,95
25
.59
.75
.84
.91
.96
2.07
2.16
2.22
2.27
2.36
2.42
2.47
2.51
2.56
2.62
2.67
2.71
2.74
2,77-
30
1.55
1.70
1.79
1.85
1.90
2.00
2.09
2.15
2.19
2.28
2.33
2.38
2.42
2.46
2.52
2.57
2.60
2.63
2.66
35 40 45 50 60 70 80 90 100 125
1.52 .50 1.48 1.47 .45 1.44 .42 1.42 .41 .40
1.67 .64 1.62 1.61 .59 1.57 .56 1.55 .54 .53
1.75 .72 1.70 1.69 .66 1.65 .64 1.63 .62 .60
1.81 .79 1.76 1.75 .72 1.70 .69 1.68 .67 .66
1.86 .83 1.81 1.79 .76 1.74 .73 1.72 .71 .69
1.96 .92 1.90 1.88 .85 1.83 .81 1.80 .79 .78
2.04 2.00 1.97 1.95 .92 1.90 .88 1.87 .86 .84
2.09 2.06 2.03 2.00 .97 1.95 .93 1.92 .91 .89
2.14 2.10 2.07 2.05 2.01 1.99 .97 1.96 .94 .93
2.22 2.18 2.15 2.12 2.08 2.06 2.04 2.02 2.01 .99
2.27 2.23 2.20 2.17 2.13 2.10 2.08 2.07 2.06 2.04
2.32 2.27 2.24 2.21 2.17 2.14 2.12 2.10 2.09 2.07
2.35 2.31 2.27 2.24 2.20 2.17 2.15 2.13 2.12 2.10
2.39 2.35 2.31 2.28 2.24 2.21 2.19 2.17 2.16 2.13
2.45 2.40 2.36 2.33 2.29 2.25 2.23 2.21 2.20 2.18
2.49 2.44 2.40 2.37 2.32 2.29 2.27 2.25 2.23 2.21
2.53 2.47 2.43 2.40 2.35 2.32 2.30 2.28 2.26 2.24
2.56 2.50 2.46 2.43 2.38 2.35 2.32 2.30 2.29 2.26
2.58 2.52 2.48 2.45 2.40 2.37 2.34 2.32 2.31 2.28
150
1.40
1.52
1.59
1.65
1.68
1.77
1.83
1.88
1.91
1.98
2.02
2.06
2.08
2.12
2.16
2.19
2.22
2.24
2.26
                                                    D-51
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Observations
Unified Guidance
        Table 19-3. K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (1 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20 25 30 35
0.06 -0.01 -0.04 -0.07 -0.08 -0.10 -0.11 -0.12 -0.13 -0.13
0.36 0.25 0.21 0.18 0.16 0.13 0.12 0.11 0.10 0.09
0.52 0.40 0.34 0.31 0.29 0.26 0.24 0.23 0.22 0.21
0.64 0.50 0.44 0.40 0.37 0.34 0.32 0.31 0.30 0.29
0.73 0.58 0.51 0.47 0.44 0.41 0.39 0.37 0.36 0.35
0.92 0.73 0.65 0.60 0.57 0.53 0.51 0.49 0.48 0.47
.08 0.86 0.77 0.72 0.68 0.64 0.61 0.59 0.58 0.57
.19 0.95 0.85 0.79 0.76 0.71 0.68 0.66 0.64 0.63
.28 .02 0.91 0.85 0.81 0.76 0.73 0.71 0.69 0.68
.44 .15 1.03 0.96 0.91 0.86 0.83 0.80 0.78 0.77
.55 .23 1.10 .03 0.98 0.93 0.89 0.86 0.84 0.83
.63 .30 1.16 .09 .04 0.98 0.94 0.91 0.89 0.88
.70 .35 1.21 .13 .08 1.02 0.98 0.95 0.93 0.91
.78 .42 1.27 .19 .13 1.07 1.03 0.99 0.97 0.96
.89 .50 1.34 .26 .20 1.13 1.09 .05 1.03 1.01
.97 .56 1.40 .31 .25 1.18 1.13 .10 1.07 1.06
2.03 .61 1.44 .35 .29 1.21 1.17 .13 1.11 1.09
2.O9 .66 1.48 .39 .32 1.25 1.20 .16 1.14 1.12
2.14 .69 1.51 .42 .35 1.27 1.23 .19 1.16 1.14
40
-0.13
0.09
0.21
0.29
0.35
0.46
0.56
0.63
0.67
0.76
0.82
0.87
0.90
0.95
.00
.04
.08
.11
.13
45
-0.14
0.09
0.20
0.28
0.34
0.46
0.55
0.62
0.67
0.75
0.81
0.86
0.89
0.94
0.99
1.03
1.07
1.10
1.12
50
-0.14
0.08
0.20
0.28
0.34
0.45
0.55
0.61
0.66
0.75
0.81
0.85
0.89
0.93
0.98
1.03
1.06
1.09
1.11
60
-0.14
0.08
0.20
0.27
0.33
0.45
0.54
0.61
0.66
0.74
0.80
0.84
0.88
0.92
0.97
1.01
1.05
1.07
1.10
70
-0.14
0.08
0.19
0.27
0.33
0.44
0.54
0.60
0.65
0.73
0.79
0.83
0.87
0.91
0.96
1.01
1.04
1.06
1.09
80
-0.15
0.08
0.19
0.27
0.33
0.44
0.53
0.60
0.65
0.73
0.79
0.83
0.86
0.91
0.96
1.00
1.03
1.06
1.08
90
-0.15
0.07
0.19
0.27
0.32
0.44
0.53
0.59
0.64
0.73
0.78
0.83
0.86
0.90
0.95
0.99
1.03
1.05
1.08
100
-0.15
0.07
0.19
0.26
0.32
0.44
0.53
0.59
0.64
0.72
0.78
0.82
0.86
0.90
0.95
0.99
1.02
1.05
1.07
125
-0.15
0.07
0.18
0.26
0.32
0.43
0.52
0.59
0.63
0.72
0.77
0.82
0.85
0.89
0.94
0.98
1.01
1.04
1.06
150
-0.15
0.07
0.18
0.26
0.32
0.43
0.52
0.58
0.63
0.71
0.77
0.81
0.85
0.89
0.94
0.98
1.01
1.03
1.06
     Table 19-3. K-Multipliers  for 1-of-4  Interwell Prediction Limits on Observations (1 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

(
(
(
(















4
).36 (
).64 (
).80 (
).92 (
.01 (
.19 (
.35
.46
.55
.70
.81
.89
.95
>.03
>.14
>.21
>.28
2,33
2.31 ]
6 8
).25 0.21 (
).50 0.44 (
).64 0.56 (
).73 0.65 (
).80 0.72 (
).95 0.85 (
.08 0.97 (
.17 1.04 (
.23 1.10
.35 1.21
.44 1.29
.50 1.34
.55 1.39
.61 1.44
.69 1.51
.75 1.57
.80 1.61
.84 1.65
L.88 1.68
10 12 16
).18 0.16 0.13 (
).40 0.37 0.34 (
).52 0.49 0.46 (
).60 0.57 0.53 (
).66 0.63 0.59 (
).79 0.76 0.71 (
).90 0.86 0.81 (
).98 0.93 0.87 (
.03 0.98 0.93 (
.13 1.08 1.02 (
.20 1.15 1.08
.26 1.20 1.13
.30 1.24 1.17
.35 1.29 1.21
.42 1.35 1.27
.47 1.40 1.32
.51 1.44 1.35
.54 1.47 1.38
.57 1.50 1.41
20
).12 (
).32 (
).43 (
).51 (
).57 (
).68 (
).78 (
).84 (
).89 (
).98 (
.04
.09
.12
.17
.23
.27
.30
.33
.36
25 30
).ll 0.10 (
).31 0.30 (
).42 0.41 (
).49 0.48 (
).55 0.53 (
).66 0.64 (
).75 0.74 (
).82 0.80 (
).86 0.84 (
).95 0.93 (
.01 0.99 (
.05 1.03
.09 1.07
.13 1.11
.19 1.16
.23 1.20
.26 1.24
.29 1.26
.32 1.29
35
).09 (
).29 (
).40 (
).47 (
).52 (
).63 (
).72 (
).78 (
).83 (
).91 (
).97 (
.01
.05
.09
.14
.18
.22
.24
.27
40 45
).09 0.09 (
).29 0.28 (
).39 0.39 (
).46 0.46 (
).52 0.51 (
).63 0.62 (
).71 0.71 (
).78 0.77 (
).82 0.81 (
).90 0.89 (
).96 0.95 (
.00 0.99 (
.04 1.03
.08 1.07
.13 1.12
.17 1.16
.20 1.19
.23 1.22
.25 1.24
50 60 70 80 90 100 125 150
).08 0.08 0.08 0.08 0.07 0.07 0.07 0.07
).28 0.27 0.27 0.27 0.27 0.26 0.26 0.26
).38 0.38 0.37 0.37 0.37 0.37 0.36 0.36
).45 0.45 0.44 0.44 0.44 0.44 0.43 0.43
).51 0.50 0.50 0.49 0.49 0.49 0.48 0.48
).61 0.61 0.60 0.60 0.59 0.59 0.59 0.58
).70 0.69 0.69 0.68 0.68 0.68 0.67 0.67
).76 0.75 0.75 0.74 0.74 0.74 0.73 0.73
).81 0.80 0.79 0.79 0.78 0.78 0.77 0.77
).89 0.88 0.87 0.86 0.86 0.86 0.85 0.85
).94 0.93 0.92 0.92 0.91 0.91 0.90 0.90
).98 0.97 0.96 0.96 0.95 0.95 0.94 0.94
.02 1.01 1.00 0.99 0.99 0.98 0.98 0.97
.06 1.05 1.04 .03 1.03 .02 1.01 .01
.11 1.10 1.09 .08 1.08 .07 1.06 .06
.15 1.14 1.13 .12 1.11 .11 1.10 .09
.18 1.17 1.16 .15 1.14 .14 1.13 .12
.21 1.19 1.18 .17 1.17 .16 1.15 .15
.23 1.22 1.20 .20 1.19 .19 1.18 .17
                                                    D-52
                                                                                                  March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                       Unified Guidance
       Table 19-3. K-Multipliers for 1-of-4  Interwell Prediction Limits on Observations (1  COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
(
(

















).64 (
).92 (
.08 (
.19 (
.28 ]
.46 ]
.62
.72
.81
.95
>.06
>.14
1.20
2.28
2.37
2.45
2.51
2.56 ;
2.60 :
1.50 0.44 (
).73 0.65 (
).86 0.77 (
).95 0.85 (
L.02 0.91 (
L.17 1.04 (
.29 1.15
.37 1.23
.44 1.29
.55 1.39
.63 1.46
.69 1.51
.74 1.56
.80 1.61
.88 1.68
.94 1.73
.98 1.77
1.02 1.81
2.06 1.84
).40 0.37 0.34 (
).60 0.57 0.53 (
).72 0.68 0.64 (
).79 0.76 0.71 (
).85 0.81 0.76 (
).98 0.93 0.87 (
.08 1.03 0.97 (
.15 1.10 1.03 (
.20 1.15 1.08
.30 1.24 1.17
.36 1.30 1.23
.42 1.35 1.27
.46 1.39 1.31
.51 1.44 1.35
.57 1.50 1.41
.62 1.54 1.45
.66 1.58 1.49
.69 1.61 1.52
.72 1.64 1.54
).32 (
).51 (
).61 (
).68 (
).73 (
).84 (
).93 (
).99 (
.04
.12
.18
.23
.26
.30
.36
.40
.43
.46
.49
).31 0.30 (
).49 0.48 (
).59 0.58 (
).66 0.64 (
).71 0.69 (
).82 0.80 (
).90 0.88 (
).96 0.94 (
.01 0.99 (
.09 1.07
.14 1.12
.19 1.16
.22 1.20
.26 1.24
.32 1.29
.36 1.33
.39 1.36
.42 1.39
.44 1.41
).29 (
).47 (
).57 (
).63 (
).68 (
).78 (
).87 (
).93 (
).97 (
.05
.10
.14
.18
.22
.27
.31
.34
.36
.39
).29 0.28 (
).46 0.46 (
).56 0.55 (
).63 0.62 (
).67 0.67 (
).78 0.77 (
).86 0.85 (
).92 0.91 (
).96 0.95 (
.04 1.03
.09 1.08
.13 1.12
.16 1.15
.20 1.19
.25 1.24
.29 1.28
.32 1.31
.35 1.33
.37 1.36
).28 0.27 0.27 (
).45 0.45 0.44 (
).55 0.54 0.54 (
).61 0.61 0.60 (
).66 0.66 0.65 (
).76 0.75 0.75 (
).84 0.83 0.83 (
).90 0.89 0.88 (
).94 0.93 0.92 (
.02 1.01 1.00 (
.07 1.06 1.05
.11 1.10 1.09
.14 1.13 1.12
.18 1.17 1.16
.23 1.22 1.20
.27 1.25 1.24
.30 1.28 1.27
.32 1.31 1.30
.34 1.33 1.32
).27 0.27 (
).44 0.44 (
).53 0.53 (
).60 0.59 (
).65 0.64 (
).74 0.74 (
).82 0.82 (
).88 0.87 (
).92 0.91 (
).99 0.99 (
.04 1.04
.08 1.08
.11 1.11
.15 1.14
.20 1.19
.23 1.23
.26 1.26
.29 1.28
.31 1.30
).26 0.26 (
).44 0.43 (
).53 0.52 (
).59 0.59 (
).64 0.63 (
).74 0.73 (
).81 0.81 (
).87 0.86 (
).91 0.90 (
).98 0.98 (
.03 1.02
.07 1.06
.10 1.09
.14 1.13
.19 1.18
.22 1.21
.25 1.24
.27 1.26
.29 1.28
).26
).43
).52
).58
).63
).73
).80
).86
).90
).97
.02
.06
.09
.12
.17
.20
.23
.26
.28
Table 19-
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
0.39
0.70
0.88
.01
.12
.33
.52
.66
.76
.95
2.O9
2.19
2.27
2.37
2.50
2.60
2.68
2.75
2.81
6
0.28
0.53
0.67
0.78
0.85
1.02
1.16
1.26
1.33
1.47
1.56
1.64
1.70
1.77
1.86
1.93
1.99
'"2,'O4"
2.O8
3. K-Multipliers for 1-of-4 nterwell Prediction Limits
on
8 10 12 16 20 25 30 35 40 45 50
0.22 0.19 0.17 0.14 0.12 0.11 0.10 0.10 0.09 0.09 0
0.46 0.41 0.39 0.35 0.33 0.31 0.30 0.30 0.29 0.29 0
0.59 0.54 0.50 0.47 0.44 0.42 0.41 0.40 0.40 0.39 0
0.68 0.62 0.59 0.54 0.52 0.50 0.48 0.47 0.47 0.46 0
0.75 0.69 0.65 0.60 0.58 0.55 0.54 0.53 0.52 0.52 0
0.89 0.82 0.78 0.72 0.69 0.67 0.65 0.64 0.63 0.62 0
1.01 0.93 0.88 0.82 0.79 0.76 0.74 0.73 0.72 0.71 0
1.10 .01 0.96 0.89 0.86 0.83 0.81 0.79 0.78 0.77 0
1.16 .07 1.02 0.95 0.91 0.87 0.85 0.84 0.83 0.82 0
1.28 .18 1.12 1.04 .00 0.96 0.94 0.92 0.91 0.90 0
1.36 .26 1.19 1.11 .06 .02 1.00 0.98 0.97 0.96 0
1.43 .31 1.24 1.16 .11 .07 1.04 .02 .01 1.00 0
1.48 .36 1.29 1.20 .15 .11 1.08 .06 .04 1.03
1.54 .42 1.34 1.25 .19 .15 1.12 .10 .09 1.08
1.62 .49 1.41 1.31 .25 .21 1.18 .16 .14 1.13
1.68 .54 1.46 1.36 .30 .25 1.22 .20 .18 1.17
1.73 .59 1.50 1.40 .33 .29 1.25 .23 .21 1.20
1.77 .63 1.54 1.43 .37 .32 1.28 .26 .24 1.23
1.81 .66 1.57 1.46 .39 .34 1.31 .28 .26 1.25
.09
.28
.39
.46
.51
.62
.71
.77
.81
.89
.95
.99
.02
.07
.12
.16
.19
.22
.24
Observations (2
60
0.08
0.28
0.38
0.45
0.50
0.61
0.70
0.76
0.80
0.88
0.94
0.98
1.01
1.05
1.10
1.14
1.17
1.20
1.22
70
0.08
0.27
0.38
0.45
0.50
0.60
0.69
0.75
0.79
0.87
0.93
0.97
1.00
1.04
1.09
1.13
1.16
1.19
1.21
80
0.08
0.27
0.37
0.44
0.49
0.60
0.69
0.74
0.79
0.87
0.92
0.96
.00
.03
.08
.12
.15
.18
.20
COC,
90
0.07
0.27
0.37
0.44
0.49
0.60
0.68
0.74
0.78
0.86
0.92
0.96
0.99
1.03
1.08
1.12
1.15
1.17
1.19
Annual)
100
0.07
0.27
0.37
0.44
0.49
0.59
0.68
0.74
0.78
0.86
0.91
0.95
0.99
.02
.07
.11
.14
.17
.19
125
0.07
0.26
0.36
0.43
0.48
0.59
0.67
0.73
0.77
0.85
0.90
0.95
0.98
1.02
1.07
1.10
1.13
1.16
1.18
150
0.07
0.26
0.36
0.43
0.48
0.59
0.67
0.73
0.77
0.85
0.90
0.94
0.97
.01
.06
.10
.13
.15
.17
                                                     D-53
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
     Table 19-3. K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (2 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
(


















).70 (
.01 (
.20 (
.33 ]
.44 ]
.66 ]
.85
.98
2.09
2.27
2.40
2.50
2.59
2.68
2.81 :
2.91 :
2.99 :
3.05 :
3.11
).53 0.46 (
).78 0.68 (
).92 0.80 (
L.02 0.89 (
L.10 0.96 (
L.26 1.10
.39 1.22
.49 1.30
.56 1.36
.70 1.48
.79 1.56
.86 1.62
.92 1.67
.99 1.73
2.08 1.81
2.15 1.86
2.20 1.91
2.25 1.95
2,29 1.98
).41 0.39 0.35 (
).62 0.59 0.54 (
).74 0.70 0.65 (
).82 0.78 0.72 (
).88 0.84 0.78 (
.01 0.96 0.89 (
.12 1.06 0.99 (
.20 1.13 1.06
.26 1.19 1.11
.36 1.29 1.20
.43 1.36 1.26
.49 1.41 1.31
.53 1.45 1.35
.59 1.50 1.40
.66 1.57 1.46
.71 1.62 1.50
.76 1.66 1.54
.79 1.69 1.57
.82 1.72 1.60
).33 (
).52 (
).62 (
).69 (
).75 (
).86 (
).95 (
.01 (
.06
.15
.21
.25
.29
.33
.39
.44
.47
.50
.53
).31 0.30 (
).50 0.48 (
).60 0.58 (
).67 0.65 (
).72 0.70 (
).83 0.81 (
).91 0.89 (
).98 0.95 (
.02 1.00 (
.11 1.08
.16 1.13
.21 1.18
.24 1.21
.29 1.25
.34 1.31
.38 1.35
.42 1.38
.44 1.41
.47 1.43
).30 (
).47 (
).57 (
).64 (
).69 (
).79 (
).88 (
).94 (
).98 (
.06
.11
.16
.19
.23
.28
.32
.35
.38
.40
).29 0.29 (
).47 0.46 (
).56 0.56 (
).63 0.62 (
).68 0.67 (
).78 0.77 (
).86 0.86 (
).92 0.91 (
).97 0.96 (
.04 1.03
.10 1.09
.14 1.13
.17 1.16
.21 1.20
.26 1.25
.30 1.29
.34 1.32
.36 1.35
.38 1.37
).28 0.28 0.27 (
).46 0.45 0.45 (
).55 0.55 0.54 (
).62 0.61 0.60 (
).67 0.66 0.65 (
).77 0.76 0.75 (
).85 0.84 0.83 (
).91 0.89 0.89 (
).95 0.94 0.93 (
.02 1.01 1.00
.08 1.06 1.05
.12 1.10 1.09
.15 1.14 1.12
.19 1.17 1.16
.24 1.22 1.21
.28 1.26 1.25
.31 1.29 1.28
.33 1.32 1.30
.36 1.34 1.32
).27 0.27 (
).44 0.44 (
).54 0.53 (
).60 0.60 (
).65 0.64 (
).74 0.74 (
).82 0.82 (
).88 0.87 (
).92 0.92 (
.00 0.99 (
.05 1.04
.08 1.08
.12 1.11
.15 1.15
.20 1.19
.24 1.23
.27 1.26
.29 1.28
.31 1.31
).27 0.26 (
).44 0.43 (
).53 0.53 (
).59 0.59 (
).64 0.64 (
).74 0.73 (
).82 0.81 (
).87 0.86 (
).91 0.90 (
).99 0.98 (
.04 1.03
.07 1.07
.10 1.10
.14 1.13
.19 1.18
.22 1.21
.25 1.24
.28 1.27
.30 1.29
).26
).43
).52
).59
).63
).73
).81
).86
).90
).97
.02
.06
.09
.13
.17
.21
.24
.26
.28
       Table 19-3. K-Multipliers  for 1-of-4 Interwell Prediction Limits on Observations (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.01 0.78 0.68 (
.33 1.02 0.89 (
.52 1.16 1.01 (
.66 1.26 1.10
.76 1.33 1.16
.98 1.49 1.30
2.17 1.62 1.42
2.30 1.72 1.50
2.40 1.79 1.56
2.59 1.92 1.67
2.71 2.01 1.75
2.81 2.08 1.81
2.89 2.13 1.85
2.99 2.20 1.91
3.11 2.29 1.98
3.20 2.35 2.04
3.28 2.41 2.09
3.34 2.45 2.13
3.39 2.49 2.16
).62 0.59 0.54 (
).82 0.78 0.72 (
).93 0.88 0.82 (
.01 0.96 0.89 (
.07 1.02 0.95 (
.20 1.13 1.06
.30 1.23 1.15
.38 1.30 1.21
.43 1.36 1.26
.53 1.45 1.35
.61 1.52 1.41
.66 1.57 1.46
.70 1.61 1.49
.76 1.66 1.54
.82 1.72 1.60
.87 1.77 1.64
.92 1.81 1.68
.95 1.84 1.71
.98 1.87 1.74
).52 (
).69 (
).79 (
).86 (
).91 (
.01 (
.10
.16
.21
.29
.35
.39
.43
.47
.53
.57
.60
.63
.66
).50 0.48 (
).67 0.65 (
).76 0.74 (
).83 0.81 (
).87 0.85 (
).98 0.95 (
.06 1.03
.12 1.09
.16 1.13
.24 1.21
.30 1.27
.34 1.31
.37 1.34
.42 1.38
.47 1.43
.51 1.47
.54 1.50
.57 1.53
.59 1.55
).47 (
).64 (
).73 (
).79 (
).84 (
).94 (
.02
.07
.11
.19
.24
.28
.32
.35
.40
.44
.47
.50
.52
).47 0.46 (
).63 0.62 (
).72 0.71 (
).78 0.77 (
).83 0.82 (
).92 0.91 (
.00 0.99 (
.06 1.05
.10 1.09
.17 1.16
.23 1.21
.26 1.25
.30 1.28
.34 1.32
.38 1.37
.42 1.41
.45 1.44
.48 1.46
.50 1.48
).46 0.45 0.45 (
).62 0.61 0.60 (
).71 0.70 0.69 (
).77 0.76 0.75 (
).81 0.80 0.79 (
).91 0.89 0.89 (
).98 0.97 0.96 (
.04 1.02 1.01
.08 1.06 1.05
.15 1.14 1.12
.20 1.18 1.17
.24 1.22 1.21
.27 1.25 1.24
.31 1.29 1.28
.36 1.34 1.32
.39 1.37 1.36
.42 1.40 1.39
.45 1.43 1.41
.47 1.45 1.43
).44 0.44 (
).60 0.60 (
).69 0.68 (
).74 0.74 (
).79 0.78 (
).88 0.87 (
).95 0.95 (
.01 1.00
.05 1.04
.12 1.11
.16 1.16
.20 1.19
.23 1.22
.27 1.26
.31 1.31
.35 1.34
.38 1.37
.40 1.39
.42 1.41
).44 0.43 (
).59 0.59 (
).68 0.67 (
).74 0.73 (
).78 0.77 (
).87 0.86 (
).95 0.94 (
.00 0.99 (
.04 1.03
.10 1.10
.15 1.14
.19 1.18
.22 1.21
.25 1.24
.30 1.29
.33 1.32
.36 1.35
.38 1.37
.40 1.39
).43
).59
).67
).73
).77
).86
).93
).98
.02
.09
.14
.17
.20
.24
.28
.31
.34
.36
.38
                                                     D-54
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
        Table 19-3. K-Multipliers  for 1-of-4 Interwell Prediction Limits on Observations (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.85 0.63 0.54 (
.21 0.89 0.77 (
.43 1.04 0.90 (
.59 1.15 0.99 (
.71 1.24 1.06 (
.99 1.42 1.21
2.24 1.58 1.34
2.41 1.69 1.43
2.55 1.77 1.50
2.80 1.93 1.63
2.97 2.O4 1.72
3.11 2, 13 1.79
3.22 2.20 1.85
3.36 2.28 1.91
3.53 2.39 2.OO
3.67 2.47 2.O7
3.77 2.54 2,12
3.86 2.60 2,77
3.94 2.65 2.21
).49 0.46 0.42 (
).70 0.66 0.61 (
).82 0.77 0.72 (
).91 0.85 0.79 (
).97 0.91 0.85 (
.11 1.04 0.96 (
.22 1.15 1.06
.30 1.22 1.13
.36 1.28 1.18
.48 1.38 1.27
.56 1.46 1.34
.62 1.51 1.39
.67 1.56 1.43
.73 1.61 1.48
.80 1.68 1.54
.86 1.74 1.59
.91 1.78 1.63
.95 1.82 1.66
1.99. 1.85 1.69
).40 (
).58 (
).68 (
).75 (
).81 (
).92 (
.01 (
.07
.12
.21
.27
.32
.36
.40
.46
.51
.55
.58
.60
).38 0.37 (
).56 0.54 (
).66 0.64 (
).72 0.71 (
).78 0.76 (
).88 0.86 (
).97 0.95 (
.03 1.00 (
.08 1.05
.16 1.13
.22 1.19
.27 1.23
.30 1.27
.35 1.31
.40 1.36
.44 1.40
.48 1.44
.51 1.46
.53 1.49
).36 (
).53 (
).63 (
).69 (
).74 (
).84 (
).93 (
).99 (
.03
.11
.16
.21
.24
.28
.33
.37
.41
.43
.46
).35 0.35 (
).52 0.52 (
).62 0.61 (
).68 0.68 (
).73 0.72 (
).83 0.82 (
).91 0.90 (
).97 0.96 (
.02 1.00 (
.09 1.08
.15 1.13
.19 1.17
.22 1.21
.26 1.25
.31 1.30
.35 1.33
.38 1.37
.41 1.39
.43 1.41
).34 0.34 0.33 (
).51 0.50 0.50 (
).61 0.60 0.59 (
).67 0.66 0.65 (
).72 0.71 0.70 (
).81 0.80 0.80 (
).90 0.88 0.88 (
).95 0.94 0.93 (
).99 0.98 0.97 (
.07 1.06 1.04
.12 1.11 1.10
.16 1.15 1.13
.20 1.18 1.17
.23 1.22 1.20
.28 1.26 1.25
.32 1.30 1.29
.35 1.33 1.32
.38 1.36 1.34
.40 1.38 1.36
).33 0.33 (
).50 0.49 (
).59 0.58 (
).65 0.64 (
).70 0.69 (
).79 0.79 (
).87 0.86 (
).92 0.92 (
).96 0.96 (
.04 1.03
.09 1.08
.13 1.12
.16 1.15
.19 1.19
.24 1.23
.28 1.27
.31 1.30
.33 1.32
.35 1.34
).32 0.32 (
).49 0.49 (
).58 0.58 (
).64 0.64 (
).69 0.68 (
).78 0.78 (
).86 0.85 (
).91 0.91 (
).95 0.95 (
.03 1.02
.08 1.07
.11 1.10
.14 1.13
.18 1.17
.23 1.22
.26 1.25
.29 1.28
.32 1.30
.34 1.32
).32
).48
).57
).63
).68
).77
).85
).90
).94
.01
.06
.10
.13
.16
.21
.24
.27
.30
.32
     Table 19-3. K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.21 0.89 0.77 (
1.59 1.15 0.99 (
1.82 1.31 1.12
1.99 1.42 1.21
2.13 1.50 1.28
2.41 1.69 1.43
2.66 1.84 1.56
2.84 1.95 1.65
2.97 2.04 1.72
3.22 2.20 1.85
3.40 '"2,'3O" 1.93
3.53 2.39 2.00
3.64 2.46 2.06
3.77 2.54 2.12
3.94 2.65 2.21
4.07 2.73 2,27 '*
4.18 2.79 2,33 ;
4.27 2.85 2:37 '<
4.34 2.90 2,41 '<
).70 0.66 0.61 (
).91 0.85 0.79 (
.02 0.96 0.89 (
.11 1.04 0.96 (
.17 1.10 1.02 (
.30 1.22 1.13
.41 1.33 1.22
.49 1.40 1.29
.56 1.46 1.34
.67 1.56 1.43
.74 1.63 1.49
.80 1.68 1.54
.85 1.73 1.58
.91 1.78 1.63
.99 1.85 1.69
>.04 1.90 1.74
>.09 1.95 1.78
>.13 1.98 1.81
>.17 2.02 1.84
).58 (
).75 (
).85 (
).92 (
).97 (
.07
.16
.22
.27
.36
.42
.46
.50
.55
.60
.65
.68
.71
.74
).56 0.54 (
).72 0.71 (
).82 0.80 (
).88 0.86 (
).93 0.91 (
.03 1.00 (
.12 1.09
.18 1.14
.22 1.19
.30 1.27
.36 1.32
.40 1.36
.44 1.40
.48 1.44
.53 1.49
.58 1.53
.61 1.56
.64 1.59
.66 1.61
).53 (
).69 (
).78 (
).84 (
).89 (
).99 (
.07
.12
.16
.24
.29
.33
.37
.41
.46
.50
.53
.55
.58
).52 0.52 (
).68 0.68 (
).77 0.76 (
).83 0.82 (
).88 0.87 (
).97 0.96 (
.05 1.04
.11 1.09
.15 1.13
.22 1.21
.27 1.26
.31 1.30
.34 1.33
.38 1.37
.43 1.41
.47 1.45
.50 1.48
.53 1.51
.55 1.53
).51 0.50 0.50 (
).67 0.66 0.65 (
).76 0.75 0.74 (
).81 0.80 0.80 (
).86 0.85 0.84 (
).95 0.94 0.93 (
.03 1.01 1.00
.08 1.07 1.06
.12 1.11 1.10
.20 1.18 1.17
.25 1.23 1.21
.28 1.26 1.25
.31 1.30 1.28
.35 1.33 1.32
.40 1.38 1.36
.44 1.41 1.40
.47 1.44 1.43
.49 1.47 1.45
.51 1.49 1.47
).50 0.49 (
).65 0.64 (
).73 0.73 (
).79 0.79 (
).83 0.83 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.16 1.15
.20 1.20
.24 1.23
.27 1.26
.31 1.30
.35 1.34
.39 1.38
.41 1.41
.44 1.43
.46 1.45
).49 0.49 (
).64 0.64 (
).73 0.72 (
).78 0.78 (
).83 0.82 (
).91 0.91 (
).99 0.98 (
.04 1.03
.08 1.07
.14 1.13
.19 1.18
.23 1.22
.26 1.24
.29 1.28
.34 1.32
.37 1.36
.40 1.38
.42 1.41
.44 1.43
).48
).63
).72
).77
).81
).90
).97
.02
.06
.13
.17
.21
.24
.27
.32
.35
.38
.40
.42
                                                     D-55
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
       Table 19-3. K-Multipliers  for 1-of-4 Interwell Prediction Limits on Observations (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.59 1.15 0.99 (
1.99 1.42 1.21
2.24 1.58 1.34
2.41 1.69 1.43
2.55 1.77 1.50
2.84 1.95 1.65
3.09 2.11 1.78
3.26 2.22 1.87
3.40 2.30 1.93
3.64 2.46 2.06
3.81 2.56 2.14
3.94 "'2,65' 2.21
4.05 2.71 2.26 ;
4.18 2.79 2.33 ;
4.34 2.90 2.41 ;
4.47 2.97 2.48 ;
4.57 3.04 2.53 ;
4.66 3.09 2.57 :
4.73 3.14 2,61 '4
).91 0.85 0.79 (
.11 1.04 0.96 (
.22 1.15 1.06
.30 1.22 1.13
.36 1.28 1.18
.49 1.40 1.29
.61 1.50 1.38
.68 1.57 1.44
.74 1.63 1.49
.85 1.73 1.58
.93 1.80 1.65
.99 1.85 1.69
>.03 1.89 1.73
>.09 1.95 1.78
>.17 2.02 1.84
1.22 2.07 1.89
2.27 2.11 1.92
>.31 2.14 1.96
1.34 2.17 1.98
).75 (
).92 (
.01 (
.07
.12
.22
.31
.37
.42
.50
.56
.60
.64
.68
.74
.78
.82
.85
.87
).72 0.71 (
).88 0.86 (
).97 0.95 (
.03 1.00 (
.08 1.05
.18 1.14
.26 1.22
.31 1.28
.36 1.32
.44 1.40
.49 1.45
.53 1.49
.57 1.52
.61 1.56
.66 1.61
.70 1.65
.74 1.68
.76 1.71
.79 1.73
).69 (
).84 (
).93 (
).99 (
.03
.12
.20
.25
.29
.37
.42
.46
.49
.53
.58
.61
.64
.67
.69
).68 0.68 (
).83 0.82 (
).91 0.90 (
).97 0.96 (
.02 1.00 (
.11 1.09
.18 1.17
.23 1.22
.27 1.26
.34 1.33
.39 1.38
.43 1.41
.46 1.45
.50 1.48
.55 1.53
.59 1.57
.62 1.59
.64 1.62
.66 1.64
).67 0.66 0.65 (
).81 0.80 0.80 (
).90 0.88 0.88 (
).95 0.94 0.93 (
).99 0.98 0.97 (
.08 1.07 1.06
.16 1.14 1.13
.21 1.19 1.18
.25 1.23 1.21
.31 1.30 1.28
.36 1.34 1.33
.40 1.38 1.36
.43 1.41 1.39
.47 1.44 1.43
.51 1.49 1.47
.55 1.52 1.51
.58 1.55 1.53
.60 1.57 1.56
.62 1.60 1.58
).65 0.64 (
).79 0.79 (
).87 0.86 (
).92 0.92 (
).96 0.96 (
.05 1.04
.12 1.11
.17 1.16
.20 1.20
.27 1.26
.32 1.31
.35 1.34
.38 1.37
.41 1.41
.46 1.45
.49 1.48
.52 1.51
.54 1.53
.56 1.55
).64 0.64 (
).78 0.78 (
).86 0.85 (
).91 0.91 (
).95 0.95 (
.04 1.03
.11 1.10
.15 1.14
.19 1.18
.26 1.24
.30 1.29
.34 1.32
.36 1.35
.40 1.38
.44 1.43
.47 1.46
.50 1.49
.52 1.51
.54 1.53
).63
).77
).85
).90
).94
.02
.09
.14
.17
.24
.28
.32
.34
.38
.42
.45
.48
.50
.52
        Table 19-3. K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.24 0.91 0.78 (
1.66 1.18 1.01 (
1.92 1.34 1.14
2.11 1.46 1.23
2.27 1.55 1.31
2.61 1.75 1.46
2.91 1.92 1.60
3.13 1,05 1.70
3.30 -2, 14 1.77
3.61 2.32 1.91
3.83 2.44 ,2.0O
4.00 2.54 2;08
4.14 2.62 2.14
4.31 2.71 2.21
4.52 2.84 2.31
4.69 2.93 2.38
4.83 3.01 2.44
4.94 3.07 2.49
5.04 3.13 2.54
).71 0.67 0.61 (
).91 0.86 0.79 (
.03 0.97 0.90 (
.12 1.05 0.97 (
.18 1.11 1.02 (
.32 1.23 1.13
.44 1.34 1.23
.52 1.42 1.30
.59 1.48 1.35
.70 1.58 1.44
.79 1.66 1.51
.85 1.72 1.56
.90 1.76 1.60
1,97 1.82 1.65
2.0S 1.89 1.72
2,11 1.95 1.77
?. 16 2.OO 1.81
2.21 2,04 1.84
2.24 2.O7 1.87
).58 (
).76 (
).85 (
).92 (
).97 (
.08
.17
.23
.28
.37
.43
.47
.51
.56
.62
.66
.70
.73
.76
).56 0.54 (
).73 0.71 (
).82 0.80 (
).88 0.86 (
).93 0.91 (
.04 1.01 (
.12 1.09
.18 1.15
.23 1.19
.31 1.27
.37 1.32
.41 1.37
.44 1.40
.49 1.44
.54 1.50
.59 1.54
.62 1.57
.65 1.60
.68 1.62
).53 (
).69 (
).78 (
).84 (
).89 (
).99 (
.07
.12
.17
.24
.30
.34
.37
.41
.46
.50
.53
.56
.58
).52 0.52 (
).68 0.68 (
).77 0.76 (
).83 0.82 (
).88 0.87 (
).97 0.96 (
.05 1.04
.11 1.09
.15 1.14
.22 1.21
.28 1.26
.32 1.30
.35 1.33
.39 1.37
.44 1.42
.48 1.46
.51 1.49
.53 1.51
.55 1.53
).51 0.51 0.50 (
).67 0.66 0.65 (
).76 0.75 0.74 (
).82 0.80 0.80 (
).86 0.85 0.84 (
).95 0.94 0.93 (
.03 1.02 1.01
.08 1.07 1.06
.12 1.11 1.10
.20 1.18 1.17
.25 1.23 1.22
.29 1.27 1.25
.32 1.30 1.28
.35 1.33 1.32
.40 1.38 1.37
.44 1.42 1.40
.47 1.45 1.43
.49 1.47 1.45
.52 1.49 1.47
).50 0.49 (
).65 0.65 (
).73 0.73 (
).79 0.79 (
).83 0.83 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.16 1.15
.21 1.20
.24 1.23
.27 1.26
.31 1.30
.35 1.34
.39 1.38
.42 1.41
.44 1.43
.46 1.45
).49 0.49 (
).64 0.64 (
).73 0.72 (
).78 0.78 (
).83 0.82 (
).91 0.91 (
).99 0.98 (
.04 1.03
.08 1.07
.14 1.13
.19 1.18
.23 1.22
.26 1.25
.29 1.28
.34 1.32
.37 1.36
.40 1.39
.42 1.41
.44 1.43
).48
).63
).72
).77
).81
).90
).97
.02
.06
.13
.17
.21
.24
.27
.32
.35
.38
.40
.42
                                                     D-56
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                     Unified Guidance
     Table 19-3. K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (10 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
468
1.66 1.18 1.01 (
2.11 1.46 1.23
2.40 1.63 1.37
2.61 1.75 1.46
2.77 1.84 1.54
3.13 2.05 1.70
3.44 2.22 1.83
3.66 2,35 1.93
3.83 2.44 2.00
4.14 2.62 2.14
4.36 2.74 2.24
4.52 2.84 2,31 '*
4.66 2.92 2.37 '<
4.83 3.01 2.44 '<
5.04 3.13 2.54 ;
5.20 3.22 2.61
5.34 3.30 2.67
5.45 3.36 2.71
5.54 3.42 2.76 ,
10 12 16
).91 0.86 0.79 (
.12 1.05 0.97 (
.24 1.16 1.07
.32 1.23 1.13
.39 1.29 1.19
.52 1.42 1.30
.64 1.52 1.39
.72 1.60 1.46
.79 1.66 1.51
.90 1.76 1.60
.99 1.84 1.67
>.05 1.89 1.72
MO 1.94 1.76
M6 2.00 1.81
1.24 2.07 1.87
2,37 2.13 1.92
2,36 2.17 1.96
2,40 2.21 1.99
2.43 2.24 2.02
20
).76 (
).92 (
.01 (
.08
.13
.23
.32
.38
.43
.51
.57
.62
.66
.70
.76
.80
.84
.87
.90
25 30
).73 0.71 (
).88 0.86 (
).97 0.95 (
.04 1.01 (
.08 1.05
.18 1.15
.26 1.23
.32 1.28
.37 1.32
.44 1.40
.50 1.45
.54 1.50
.58 1.53
.62 1.57
.68 1.62
.72 1.66
.75 1.69
.78 1.72
.80 1.74
35
).69 (
).84 (
).93 (
).99 (
.03
.12
.20
.26
.30
.37
.42
.46
.49
.53
.58
.62
.65
.68
.70
40 45
).68 0.68 (
).83 0.82 (
).92 0.91 (
).97 0.96 (
.02 1.01
.11 1.09
.18 1.17
.24 1.22
.28 1.26
.35 1.33
.40 1.38
.44 1.42
.47 1.45
.51 1.49
.55 1.53
.59 1.57
.62 1.60
.65 1.62
.67 1.65
50 60 70
).67 0.66 0.65 (
).82 0.80 0.80 (
).90 0.89 0.88 (
).95 0.94 0.93 (
.00 0.98 0.97 (
.08 1.07 1.06
.16 1.14 1.13
.21 1.19 1.18
.25 1.23 1.22
.32 1.30 1.28
.37 1.34 1.33
.40 1.38 1.37
.43 1.41 1.39
.47 1.45 1.43
.52 1.49 1.47
.55 1.53 1.51
.58 1.55 1.54
.61 1.58 1.56
.63 1.60 1.58
80 90
).65 0.65 (
).79 0.79 (
).87 0.86 (
).92 0.92 (
).96 0.96 (
.05 1.04
.12 1.11
.17 1.16
.21 1.20
.27 1.26
.32 1.31
.35 1.34
.38 1.37
.42 1.41
.46 1.45
.49 1.48
.52 1.51
.54 1.53
.56 1.55
100 125
).64 0.64 (
).78 0.78 (
).86 0.85 (
).91 0.91 (
).95 0.95 (
.04 1.03
.11 1.10
.16 1.15
.19 1.18
.26 1.25
.30 1.29
.34 1.32
.37 1.35
.40 1.39
.44 1.43
.47 1.46
.50 1.49
.52 1.51
.54 1.53
150
).63
).77
).85
).90
).94
.02
.09
.14
.17
.24
.28
.32
.34
.38
.42
.45
.48
.50
.52
      Table 19-3. K-Multipliers  for 1-of-4 Interwell Prediction Limits on Observations (10 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.11 1.46 1.23
2.61 1.75 1.46
2.91 1.92 1.60
3.13 2.05 1.70
3.30 2.14 1.77
3.66 2.35 1.93
3.97 2.52 2.06
4.19 '"2,65" 2.16
4.36 2.74 2.24
4.66 2.92 2.37 ;
4.88 3.04 2.46 ;
5.04 3.13 2.54 ;
5.17 3.21 2.59 :
5.34 3.30 2,67 '*
5.54 3.42 2.78 \
5.70 3.51 2.83 ;
5.83 3.58 2.88 ;
5.94 3.64 2.93
6.03 3.70 2.97
.12 1.05 0.97 (
.32 1.23 1.13
.44 1.34 1.23
.52 1.42 1.30
.59 1.48 1.35
.72 1.60 1.46
.84 1.71 1.55
.92 1.78 1.62
.99 1.84 1.67
MO 1.94 1.76
MS 2.01 1.82
2.24 2.07 1.87
1.29 2.11 1.91
>.36 2.17 1.96
>.43 2.24 2.02
>.50 2.30 2.07
>.54 2.34 2.11
2.59 2.38 2.14 ;
2.62 2.41 2.17 ;
).92 (
.08
.17
.23
.28
.38
.47
.53
.57
.66
.71
.76
.80
.84
.90
.94
.98
>.01
>.03
).88 0.86 (
.04 1.01 (
.12 1.09
.18 1.15
.23 1.19
.32 1.28
.40 1.36
.46 1.41
.50 1.45
.58 1.53
.63 1.58
.68 1.62
.71 1.65
.75 1.69
.80 1.74
.85 1.78
.88 1.81
.91 1.84
.93 1.86
).84 (
).99 (
.07
.12
.17
.26
.33
.38
.42
.49
.54
.58
.61
.65
.70
.74
.77
.79
.82
).83 0.82 (
).97 0.96 (
.05 1.04
.11 1.09
.15 1.14
.24 1.22
.31 1.29
.36 1.34
.40 1.38
.47 1.45
.52 1.50
.55 1.53
.59 1.56
.62 1.60
.67 1.65
.71 1.68
.74 1.71
.76 1.73
.78 1.75
).82 0.80 0.80 (
).95 0.94 0.93 (
.03 1.02 1.01
.08 1.07 1.06
.12 1.11 1.10
.21 1.19 1.18
.28 1.26 1.25
.33 1.31 1.29
.37 1.34 1.33
.43 1.41 1.39
.48 1.46 1.44
.52 1.49 1.47
.55 1.52 1.50
.58 1.55 1.54
.63 1.60 1.58
.66 1.63 1.61
.69 1.66 1.64
.71 1.68 1.66
.73 1.70 1.68
).79 0.79 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.17 1.16
.24 1.23
.28 1.27
.32 1.31
.38 1.37
.43 1.42
.46 1.45
.49 1.48
.52 1.51
.56 1.55
.60 1.58
.62 1.61
.64 1.63
.66 1.65
).78 0.78 (
).91 0.91 (
).99 0.98 (
.04 1.03
.08 1.07
.16 1.15
.22 1.21
.27 1.26
.30 1.29
.37 1.35
.41 1.39
.44 1.43
.47 1.45
.50 1.49
.54 1.53
.58 1.56
.60 1.58
.62 1.60
.64 1.62
).77
).90
).97
.02
.06
.14
.20
.25
.28
.34
.39
.42
.44
.48
.52
.55
.57
.59
.61
                                                    D-57
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
        Table 19-3. K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.70 1.19 1.01 (
2.20 1.48 1.25
2.51 1.66 1.38
2.75 1.79 1.48
2.94 1.89 1.56
3.36 '"2.ii" 1.72
3.74 2.30 1.87
4.01 2.44 1.97
4.22 2.55 2.05
4.61 2.75 2.20
4.88 2.89 2.30 ,
5.10 3.00 2.39
5.27 3.09 2.45
5.49 3.20 2.53 .
5.76 3.34 2.64
5.97 3.45 2.72
6.14 3.54 2.78
6.28 3.61 2.84
6.41 3.68 2.88
).92 0.86 0.80 (
.13 1.05 0.97 (
.25 1.16 1.07
.33 1.24 1.14
.40 1.30 1.19
.54 1.43 1.30
.66 1.54 1.40
.75 1.62 1.47
.82 1.68 1.52
.94 1.78 1.61
3.03 1.86 1.68
2.O9 1.92 1.73
2,1-S "1.97 1.77
2.22 2.O3 1.82
2.30 2,n. 1.89
2.37 2.17 1.94
2.42 2.22 1,98
2.47 2.26 2.O2
2.51 2.29 2,05
).76 (
).92 (
.02 (
.08
.13
.24
.32
.39
.43
.52
.58
.63
.67
.71
.77
.82
.86
.89
.91
).73 0.71 (
).89 0.86 (
).98 0.95 (
.04 1.01 (
.08 1.05
.18 1.15
.27 1.23
.32 1.28
.37 1.33
.45 1.40
.51 1.46
.55 1.50
.58 1.53
.63 1.57
.68 1.63
.73 1.67
.76 1.70
.79 1.73
.81 1.75
).69 (
).85 (
).93 (
).99 (
.03
.13
.20
.26
.30
.37
.42
.46
.50
.54
.59
.63
.66
.68
.71
).68 0.68 (
).83 0.82 (
).92 0.91 (
).97 0.96 (
.02 1.01
.11 1.09
.18 1.17
.24 1.22
.28 1.26
.35 1.33
.40 1.38
.44 1.42
.47 1.45
.51 1.49
.56 1.54
.60 1.57
.63 1.60
.65 1.63
.67 1.65
).67 0.66 0.65 (
).82 0.81 0.80 (
).90 0.89 0.88 (
).95 0.94 0.93 (
.00 0.98 0.97 (
.08 1.07 1.06
.16 1.14 1.13
.21 1.19 1.18
.25 1.23 1.22
.32 1.30 1.28
.37 1.35 1.33
.40 1.38 1.37
.43 1.41 1.39
.47 1.45 1.43
.52 1.49 1.47
.55 1.53 1.51
.58 1.56 1.54
.61 1.58 1.56
.63 1.60 1.58
).65 0.65 (
).79 0.79 (
).87 0.86 (
).92 0.92 (
).97 0.96 (
.05 1.04
.12 1.11
.17 1.16
.21 1.20
.27 1.26
.32 1.31
.35 1.34
.38 1.37
.42 1.41
.46 1.45
.49 1.48
.52 1.51
.54 1.53
.56 1.55
).64 0.64 (
).78 0.78 (
).86 0.85 (
).91 0.91 (
).96 0.95 (
.04 1.03
.11 1.10
.16 1.15
.19 1.18
.26 1.25
.30 1.29
.34 1.32
.37 1.35
.40 1.39
.44 1.43
.48 1.46
.50 1.49
.52 1.51
.54 1.53
).63
).77
).85
).90
).94
.02
.09
.14
.17
.24
.28
.32
.34
.38
.42
.45
.48
.50
.52
     Table 19-3. K-Multipliers for 1-of-4  Interwell Prediction Limits on Observations (20 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.20 1.48 1.25
2.75 1.79 1.48
3.11 1.98 1.62
3.36 2.11 1.72
3.57 2.21 1.80
4.01 2.44 1.97
4.39 2.64 2.12
4.67 2.78 2.22
4.88 2.89 2.3O ;
5.27 3.09 2.4S '*
5.55 3.23 2.56 ;
5.76 3.34 2.64
5.93 3.43 2.70 ,
6.14 3.54 2.78
6.41 3.68 2.88
6.61 3.78 2.96
6.78 3.87 3.03
6.92 3.94 3.08
7.04 4.00 3.13
.13 1.05 0.97 (
.33 1.24 1.14
.45 1.35 1.24
.54 1.43 1.30
.61 1.49 1.36
.75 1.62 1.47
.87 1.72 1.56
.96 1.80 1.63
>.03 1.86 1.68
>.15 1.97 1.77
>.24 2.05 1.84
2.30 2.11 1.89
2.36 2.16 1.93
2'A2 2.22 1.98
2.S1 2,29 2.05
2.58 2.35 2.10
2.63 2.40 2.14 ;
2.68 2.44 2.17 ;
2.71 '2.47 2.20 ;
).92 (
.08
.17
.24
.28
.39
.47
.53
.58
.67
.73
.77
.81
.86
.91
.96
>.oo
>.03
>.06
).89 0.86 (
.04 1.01 (
.12 1.09
.18 1.15
.23 1.19
.32 1.28
.41 1.36
.46 1.42
.51 1.46
.58 1.53
.64 1.59
.68 1.63
.72 1.66
.76 1.70
.81 1.75
.86 1.79
.89 1.82
.92 1.85
.94 1.87
).85 (
).99 (
.07
.13
.17
.26
.33
.38
.42
.50
.55
.59
.62
.66
.71
.74
.77
.80
.82
).83 0.82 (
).97 0.96 (
.05 1.04
.11 1.09
.15 1.14
.24 1.22
.31 1.29
.36 1.34
.40 1.38
.47 1.45
.52 1.50
.56 1.54
.59 1.57
.63 1.60
.67 1.65
.71 1.68
.74 1.71
.76 1.74
.79 1.76
).82 0.81 0.80 (
).95 0.94 0.93 (
.03 1.02 1.01
.08 1.07 1.06
.13 1.11 1.10
.21 1.19 1.18
.28 1.26 1.25
.33 1.31 1.29
.37 1.35 1.33
.43 1.41 1.39
.48 1.46 1.44
.52 1.49 1.47
.55 1.52 1.50
.58 1.56 1.54
.63 1.60 1.58
.66 1.63 1.61
.69 1.66 1.64
.72 1.68 1.66
.74 1.70 1.68
).79 0.79 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.17 1.16
.24 1.23
.28 1.27
.32 1.31
.38 1.37
.43 1.42
.46 1.45
.49 1.48
.52 1.51
.56 1.55
.60 1.59
.62 1.61
.65 1.63
.67 1.65
).78 0.78 (
).91 0.91 (
).99 0.98 (
.04 1.03
.08 1.07
.16 1.15
.22 1.21
.27 1.26
.30 1.29
.37 1.35
.41 1.40
.44 1.43
.47 1.45
.50 1.49
.54 1.53
.58 1.56
.60 1.58
.62 1.61
.64 1.62
).77
).90
).97
.02
.06
.14
.20
.25
.28
.34
.39
.42
.44
.48
.52
.55
.57
.59
.61
                                                    D-58
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
      Table 19-3. K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (20 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.75 1.79 1.48
3.36 2.11 1.72
3.74 2.30 1.87
4.01 2.44 1.97
4.22 2.55 2.05
4.67 2.78 2.22
5.06 2.98 2.37 ;
5.34 3.12 2.47 ;
5.55 3.23 2,56 \
5.93 3.43 2,70 ;
6.20 3.57 2.81 ;
6.41 3.68 2.88 ;
6.57 3.76 2.95 '",
6.78 3.87 3.03
7.04 4.00 3.13
7.24 4.11 3.21
7.40 4.19 3.27
7.54 4.26 3.32
7.66 4.32 3.37
.33 1.24 1.14
.54 1.43 1.30
.66 1.54 1.40
.75 1.62 1.47
.82 1.68 1.52
.96 1.80 1.63
>.08 1.91 1.72
>.17 1.99 1.79
>.24 2.05 1.84
>.36 2.16 1.93
>.44 2.23 2.00
>.51 2.29 2.05
2,50 2.34 2.09
2.63 2.40 2.14 ;
2,71 2.47 2.20 ;
Z.78 2.53 2.25 ;
2.83 2.S8' 2.29 ;
2.88 2,62 2.33 ;
2.92 2,65 2.36 ;
.08
.24
.32
.39
.43
.53
.62
.68
.73
.81
.87
.91
.95
>.oo
>.06
MO
>.14 ;
>.17 ;
>.i9 ;
.04 1.01 (
.18 1.15
.27 1.23
.32 1.28
.37 1.33
.46 1.42
.54 1.49
.60 1.54
.64 1.59
.72 1.66
.77 1.71
.81 1.75
.85 1.78
.89 1.82
.94 1.87
.98 1.91
1.02 1.94
>.05 1.97
>.07 1.99
).99 (
.13
.20
.26
.30
.38
.46
.51
.55
.62
.67
.71
.74
.77
.82
.86
.89
.92
.94
).97 0.96 (
.11 1.09
.18 1.17
.24 1.22
.28 1.26
.36 1.34
.43 1.41
.48 1.46
.52 1.50
.59 1.57
.64 1.61
.67 1.65
.70 1.68
.74 1.71
.79 1.76
.82 1.79
.85 1.82
.88 1.85
.90 1.87
).95 0.94 0.93 (
.08 1.07 1.06
.16 1.14 1.13
.21 1.19 1.18
.25 1.23 1.22
.33 1.31 1.29
.40 1.38 1.36
.45 1.42 1.41
.48 1.46 1.44
.55 1.52 1.50
.59 1.57 1.55
.63 1.60 1.58
.66 1.63 1.61
.69 1.66 1.64
.74 1.70 1.68
.77 1.74 1.71
.80 1.76 1.74
.82 1.79 1.76
.84 1.81 1.78
).92 0.92 (
.05 1.04
.12 1.11
.17 1.16
.21 1.20
.28 1.27
.35 1.34
.39 1.38
.43 1.42
.49 1.48
.53 1.52
.56 1.55
.59 1.58
.62 1.61
.67 1.65
.70 1.68
.72 1.71
.74 1.73
.76 1.75
).91 0.91 (
.04 1.03
.11 1.10
.16 1.15
.19 1.18
.27 1.26
.33 1.32
.38 1.36
.41 1.40
.47 1.45
.51 1.50
.54 1.53
.57 1.55
.60 1.58
.64 1.62
.67 1.65
.70 1.68
.72 1.70
.74 1.72
).90
.02
.09
.14
.17
.25
.31
.35
.39
.44
.49
.52
.54
.57
.61
.64
.67
.69
.70
        Table 19-3. K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (40 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.25 1.49 1.25
2.85 1.81 1.49
3.25 2, 0? 1.64
3.54 2.15 1.74
3.78 2.27 1.82
4.30 2.51 2.OO
4.77 2.73 2.16
5.11 2.89 2.27
5.37 3.01 2.35
5.86 3.23 2.51
6.21 3.39 2.63
6.47 3.52 2.71
6.70 3.62 2.79
6.95 3.75 2.87
7.30 3.91 2.99
7.58 4.03 3.08
7.79 4.13 3.15
7.97 4.22 3.21
8.12 4.29 3.26
.13 1.05 0.97 (
.34 1.25 1.14
.46 1.35 1.24
.55 1.44 1.31
.62 1.50 1.36
.76 1.62 1.47
.89 1.74 1.57
1,98 1.82 1.64
2,05 1.88 1.69
2,18 1,99. 1.78
1.21 2.O7 1.85
2.34 2.13 1.90
2.40 2,19 1,95
2.48 2.25 2.00
2.57 2.33 2.O7
2.64 2.39 2,12 '
2.10 2.44 2,16 ,
2.75 2.48 2, 19 .
2.79 2.52 2.23 :.
).92 (
.08
.17
.24
.29
.39
.48
.54
.58
.67
.73
.78
.82
.87
.92
1,97
2.O1
2,04
2.O7
).89 0.86 (
.04 1.01 (
.12 1.09
.18 1.15
.23 1.19
.33 1.29
.41 1.36
.46 1.42
.51 1.46
.59 1.54
.64 1.59
.69 1.63
.72 1.66
.77 1.70
.82 1.75
.86 1.79
.90 1.83
.93 1.85
1.95 . 1.88
).85 (
).99 (
.07
.13
.17
.26
.33
.39
.43
.50
.55
.59
.62
.66
.71
.75
.78
.80
.83
).83 0.82 (
).97 0.96 (
.05 1.04
.11 1.10
.15 1.14
.24 1.22
.31 1.29
.36 1.34
.40 1.38
.47 1.45
.52 1.50
.56 1.54
.59 1.57
.63 1.60
.67 1.65
.71 1.69
.74 1.72
.77 1.74
.79 1.76
).82 0.81 0.80 (
).95 0.94 0.93 (
.03 1.02 1.01
.09 1.07 1.06
.13 1.11 1.10
.21 1.19 1.18
.28 1.26 1.25
.33 1.31 1.29
.37 1.35 1.33
.44 1.41 1.40
.48 1.46 1.44
.52 1.49 1.47
.55 1.52 1.50
.58 1.56 1.54
.63 1.60 1.58
.67 1.63 1.61
.69 1.66 1.64
.72 1.69 1.66
.74 1.71 1.68
).79 0.79 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.17 1.16
.24 1.23
.28 1.27
.32 1.31
.38 1.37
.43 1.42
.46 1.45
.49 1.48
.52 1.51
.56 1.55
.60 1.59
.62 1.61
.65 1.63
.67 1.65
).78 0.78 (
).91 0.91 (
).99 0.98 (
.04 1.03
.08 1.07
.16 1.15
.22 1.21
.27 1.26
.30 1.29
.37 1.35
.41 1.40
.44 1.43
.47 1.46
.50 1.49
.54 1.53
.58 1.56
.60 1.58
.62 1.61
.64 1.62
).77
).90
).97
.02
.06
.14
.20
.25
.28
.34
.39
.42
.44
.48
.52
.55
.57
.59
.61
                                                     D-59
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
     Table 19-3.  K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (40 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.85 1.81 1.49
3.54 2.15 1.74
3.98 2,36 . 1.89
4.30 2.51 2.00
4.56 2.63 2.08
5.11 2.89 2.27
5.59 3.11 2.42 ;
5.94 3.27 2.54 ;
6.21 3.39 2.63
6.70 3.62 2.79
7.03 3.78 2.90
7.30 3.91 2.99
7.52 4.01 3.06
7.79 4.13 3.15
8.12 4.29 3.26
8.38 4.41 3.34
8.59 4.51 3.42
8.77 4.59 3.48
8.91 4.67 3.53
.34 1.25 1.14
.55 1.44 1.31
.67 1.55 1.40
.76 1.62 1.47
.83 1.69 1.52
.98 1.82 1.64
Ml 1.93 1.73
1.20 2.01 1.80
2,27 2.07 1.85
?,40 2.19 1.95
2.5O 2,27. 2.01
1.51 2.33 2.07
2.63 2.38 2.11
2.10 2.44 2.16 ;
2.19 2.52 2.23 ;
2.86 2.58 2,28" '<
2.92 2.63 '2.32 \
2.91 2.68 2,36 ;
3.01 2.71 2.39 ;
.08
.24
.33
.39
.44
.54
.62
.68
.73
.82
.88
.92
.96
>.01
>.07
Ml
MS ;
MS ;
1.21 ;
.04 1.01 (
.18 1.15
.27 1.23
.33 1.29
.37 1.33
.46 1.42
.54 1.49
.60 1.55
.64 1.59
.72 1.66
.78 1.71
.82 1.75
.85 1.79
.90 1.83
.95 1.88
.99 1.92
>.03 1.95
>.06 1.98
>.08 2.00
).99 (
.13
.20
.26
.30
.39
.46
.51
.55
.62
.67
.71
.74
.78
.83
.86
.89
.92
.94
).97 0.96 (
.11 1.10
.19 1.17
.24 1.22
.28 1.26
.36 1.34
.43 1.41
.48 1.46
.52 1.50
.59 1.57
.64 1.61
.67 1.65
.71 1.68
.74 1.72
.79 1.76
.82 1.80
.85 1.82
.88 1.85
.90 1.87
).95 0.94 0.93 (
.09 1.07 1.06
.16 1.14 1.13
.21 1.19 1.18
.25 1.23 1.22
.33 1.31 1.29
.40 1.38 1.36
.45 1.42 1.41
.48 1.46 1.44
.55 1.52 1.50
.59 1.57 1.55
.63 1.60 1.58
.66 1.63 1.61
.69 1.66 1.64
.74 1.71 1.68
.77 1.74 1.72
.80 1.77 1.74
.82 1.79 1.76
.84 1.81 1.78
).92 0.92 (
.05 1.04
.12 1.11
.17 1.16
.21 1.20
.28 1.27
.35 1.34
.39 1.38
.43 1.42
.49 1.48
.53 1.52
.56 1.55
.59 1.58
.62 1.61
.67 1.65
.70 1.68
.72 1.71
.74 1.73
.76 1.75
).91 0.91 (
.04 1.03
.11 1.10
.16 1.15
.19 1.18
.27 1.26
.33 1.32
.38 1.36
.41 1.40
.47 1.46
.51 1.50
.54 1.53
.57 1.55
.60 1.58
.64 1.62
.67 1.65
.70 1.68
.72 1.70
.74 1.72
).90
.02
.09
.14
.17
.25
.31
.35
.39
.44
.49
.52
.54
.57
.61
.64
.67
.69
.70
      Table 19-3. K-Multipliers for 1-of-4 Interwell Prediction Limits on Observations (40 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.54 2.15 1.74 1.55 1.44 1.31
430 2.51 2.00 1.76 1.62 1.47
4.77 2.73 2.16 1.89 1.74 1.57
5.11 2.89 2.27 1.98 1.82 1.64
5.37 3.01 2.35 2.05 1.88 1.69
5.94 3.27 2.54 2.20 2.01 1.80
6.43 3.50 2.7O 2.33 2.12 1.89
6.78 3.66 2.81 2.42 2.20 1.96
7.03 3.78 2.90 2.50 2.27 2.01
7.52 4.01 3.06 2.63 2.38 2.11
7.85 4.17 3.17 2.72 2.46 2.18 ;
8.12 4.29 3.26 '2,79 2.52 2.23 ;
8.34 4.39 3.33 2.85 -2.57 2.27 ;
8.59 4.51 3.42 2.92 2.63 2.32 ;
8.91 4.67 3.53 3.01 2.71. 2.39 ;
9.16 4.79 3.61 3.08 2.77 2.44 ;
9.38 4.88 3.68 3.13 2.82 2.48 ;
9.53 4.96 3.74 3.18 2.87 2.51 ;
9.69 5.03 3.79 3.23 2.90 2.54 \
.24
.39
.48
.54
.58
.68
.77
.83
.88
.96
1.02
1.07
Ml
MS ;
>.21 ;
1.26 :
1.29 ;
2.33 ;
2.35 ;
.18 1.15
.33 1.29
.41 1.36
.46 1.42
.51 1.46
.60 1.55
.68 1.62
.73 1.67
.78 1.71
.85 1.79
.91 1.84
.95 1.88
.98 1.91
>.03 1.95
>.08 2.00
M2 2.04
M6 2.07 ;
M9 2.10 ;
>.21 2.12 ;
.13
.26
.33
.39
.43
.51
.58
.63
.67
.74
.79
.83
.86
.89
.94
.98
>.01
>.03
>.06 ;
.11 1.10
.24 1.22
.31 1.29
.36 1.34
.40 1.38
.48 1.46
.55 1.53
.60 1.58
.64 1.61
.71 1.68
.75 1.72
.79 1.76
.82 1.79
.85 1.82
.90 1.87
.94 1.90
.97 1.93
.99 1.96
>.01 1.98
.09 1.07 1.06
.21 1.19 1.18
.28 1.26 1.25
.33 1.31 1.29
.37 1.35 1.33
.45 1.42 1.41
.51 1.49 1.47
.56 1.53 1.51
.59 1.57 1.55
.66 1.63 1.61
.70 1.67 1.65
.74 1.71 1.68
.77 1.73 1.71
.80 1.77 1.74
.84 1.81 1.78
.88 1.84 1.81
.91 1.87 1.84
.93 1.89 1.86
.95 1.91 1.88
.05 1.04
.17 1.16
.24 1.23
.28 1.27
.32 1.31
.39 1.38
.46 1.45
.50 1.49
.53 1.52
.59 1.58
.63 1.62
.67 1.65
.69 1.68
.72 1.71
.76 1.75
.79 1.78
.82 1.80
.84 1.82
.86 1.84
.04 1.03
.16 1.15
.22 1.21
.27 1.26
.30 1.29
.38 1.36
.44 1.42
.48 1.46
.51 1.50
.57 1.55
.61 1.59
.64 1.62
.67 1.65
.70 1.68
.74 1.72
.77 1.75
.79 1.77
.81 1.79
.83 1.81
.02
.14
.20
.25
.28
.35
.41
.45
.49
.54
.58
.61
.64
.67
.70
.73
.76
.78
.79
                                                     D-60
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
    Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (1  COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.71 0.59 0.53 (
.07 0.88 0.81 (
.28 1.05 0.96 (
.43 1.17 1.06
.54 1.26 1.14
.78 1.44 1.30
.98 1.59 1.44
2.11 1.70 1.53
2.22 1.78 1.60
2.41 1.92 1.73
2.54 2.02 1.82
2.64 2.10 1.88
2.72 2.16 1.94
2.81 2.23 2.00
2.93 2.32 2.08
3.03 2.39 2.15 ;
3.10 2.45 2.20 ;
3.16 2.50 2.24 ;
3.21 2.54 2.27 ;
).50 0.48 0.45 (
).76 0.73 0.70 (
).90 0.87 0.83 (
.00 0.96 0.92 (
.07 1.03 0.98 (
.23 1.18 1.12
.35 1.30 1.23
.44 1.38 1.31
.51 1.44 1.37
.62 1.56 1.48
.71 1.63 1.55
.77 1.69 1.60
.82 1.74 1.65
.88 1.80 1.70
.95 1.87 1.77
>.01 1.93 1.82
>.06 1.97 1.86
MO 2.01 1.90
>.13 2.04 1.93
).44 (
).68 (
).80 (
).89 (
).95 (
.09
.20
.27
.33
.43
.50
.55
.59
.64
.71
.76
.80
.83
.86
).43 0.42 (
).66 0.65 (
).78 0.77 (
).87 0.85 (
).93 0.92 (
.06 1.04
.16 1.14
.24 1.22
.29 1.27
.39 1.36
.46 1.43
.51 1.48
.55 1.52
.60 1.57
.66 1.63
.71 1.67
.75 1.71
.78 1.74
.81 1.77
).41 (
).64 (
).76 (
).84 (
).91 (
.03
.13
.20
.25
.35
.41
.46
.50
.55
.61
.65
.69
.72
.74
).41 0.41 (
).64 0.63 (
).76 0.75 (
).84 0.83 (
).90 0.89 (
.02 1.01
.12 1.11
.19 1.18
.24 1.23
.33 1.32
.40 1.38
.44 1.43
.48 1.47
.53 1.52
.59 1.57
.63 1.62
.67 1.65
.70 1.68
.72 1.71
).40 0.40 0.40 (
).63 0.62 0.62 (
).75 0.74 0.74 (
).83 0.82 0.81 (
).89 0.88 0.87 (
.01 1.00 0.99 (
.11 1.10 1.09
.17 1.16 1.15
.22 1.21 1.20
.31 1.30 1.29
.38 1.36 1.35
.42 1.41 1.40
.46 1.45 1.43
.51 1.49 1.48
.56 1.55 1.54
.61 1.59 1.58
.64 1.62 1.61
.67 1.65 1.64
.70 1.68 1.67
).40 0.39 (
).62 0.61 (
).73 0.73 (
).81 0.81 (
).87 0.87 (
).99 0.98 (
.08 1.08
.15 1.14
.20 1.19
.28 1.28
.34 1.34
.39 1.38
.43 1.42
.47 1.46
.53 1.52
.57 1.56
.60 1.59
.63 1.62
.66 1.65
).39 0.39 (
).61 0.61 (
).73 0.72 (
).81 0.80 (
).86 0.86 (
).98 0.97 (
.08 1.07
.14 1.13
.19 1.18
.28 1.27
.33 1.33
.38 1.37
.42 1.41
.46 1.45
.51 1.50
.56 1.54
.59 1.58
.62 1.61
.64 1.63
).39
).61
).72
).80
).86
).97
.07
.13
.18
.26
.32
.37
.40
.44
.50
.54
.57
.60
.62
  Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.07 0.88 0.81 (
.43 1.17 1.06
.63 1.33 1.20
.78 1.44 1.30
.89 1.52 1.38
2.11 1.70 1.53
2.31 1.84 1.66
2.44 1.94 1.75
2.54 2.02 1.82
2.72 2.16 1.94
2.84 2.25 2.02
2.93 2.32 2.08
3.01 2.38 2.13 ;
3.10 2.45 2.20 ;
3.21 2.54 2.27 ;
3.30 2.61 2.33 ;
3.37 2.66 2.38 ;
3.43 2.70 2.42 ;
3.48 2.74 2.46 ;
).76 0.73 0.70 (
.00 0.96 0.92 (
.13 1.09 1.04
.23 1.18 1.12
.30 1.25 1.18
.44 1.38 1.31
.56 1.50 1.42
.64 1.57 1.49
.71 1.63 1.55
.82 1.74 1.65
.89 1.81 1.72
.95 1.87 1.77
>.00 1.92 1.81
>.06 1.97 1.86
>.13 2.04 1.93
>.19 2.09 1.98
2.23 2.13 2.02
2.27 2.17 2.05
>.30 2.20 2.08 ;
).68 (
).89 (
.01 (
.09
.15
.27
.37
.44
.50
.59
.66
.71
.75
.80
.86
.91
.95
.98
>.oo
).66 0.65 (
).87 0.85 (
).98 0.97 (
.06 1.04
.12 1.10
.24 1.22
.34 1.31
.40 1.38
.46 1.43
.55 1.52
.61 1.58
.66 1.63
.70 1.67
.75 1.71
.81 1.77
.85 1.81
.89 1.85
.92 1.88
.95 1.91
).64 (
).84 (
).95 (
.03
.09
.20
.30
.36
.41
.50
.56
.61
.64
.69
.74
.79
.82
.85
.88
).64 0.63 (
).84 0.83 (
).95 0.94 (
.02 1.01
.08 1.07
.19 1.18
.28 1.27
.35 1.34
.40 1.38
.48 1.47
.54 1.53
.59 1.57
.62 1.61
.67 1.65
.72 1.71
.77 1.75
.80 1.79
.83 1.81
.86 1.84
).63 0.62 0.62 (
).83 0.82 0.81 (
).93 0.93 0.92 (
.01 1.00 0.99 (
.06 1.05 1.05
.17 1.16 1.15
.26 1.25 1.24
.33 1.31 1.31
.38 1.36 1.35
.46 1.45 1.43
.52 1.50 1.49
.56 1.55 1.54
.60 1.58 1.57
.64 1.62 1.61
.70 1.68 1.67
.74 1.72 1.71
.77 1.75 1.74
.80 1.78 1.77
.83 1.80 1.79
).62 0.61 (
).81 0.81 (
).92 0.91 (
).99 0.98 (
.04 1.04
.15 1.14
.24 1.23
.30 1.29
.34 1.34
.43 1.42
.48 1.48
.53 1.52
.56 1.55
.60 1.59
.66 1.65
.70 1.69
.73 1.72
.76 1.75
.78 1.77
).61 0.61 (
).81 0.80 (
).91 0.90 (
).98 0.97 (
.03 1.03
.14 1.13
.23 1.22
.29 1.28
.33 1.33
.42 1.41
.47 1.46
.51 1.50
.55 1.54
.59 1.58
.64 1.63
.68 1.67
.71 1.70
.74 1.73
.76 1.75
).61
).80
).90
).97
.02
.13
.22
.28
.32
.40
.46
.50
.53
.57
.62
.66
.69
.72
.74
                                                     D-61
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                       Unified Guidance
   Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.43 1.17 1.06
1.78 1.44 1.30
1.98 1.59 1.44
2.11 1.70 1.53
2.22 1.78 1.60
2.44 1.94 1.75
2.62 2.08 1.87
2.74 2.18 1.96
2.84 2.25 2.02
3.01 2.38 2.13 ;
3.13 2.47 2.21 ;
3.21 2.54 2.27 ;
3.29 2.59 2.32 ;
3.37 2.66 2.38 ;
3.48 2.74 2.46 ;
3.57 2.81 2.51 ;
3.63 2.86 2.56 ;
3.69 2.90 2.60 ;
3.74 2.94 2.63 ;
.00 0.96 0.92 (
.23 1.18 1.12
.35 1.30 1.23
.44 1.38 1.31
.51 1.44 1.37
.64 1.57 1.49
.76 1.68 1.59
.83 1.76 1.66
.89 1.81 1.72
2.00 1.92 1.81
1.07 1.99 1.88
2.13 2.04 1.93
2.18 2.08 1.97
1.23 2.13 2.02
2.30 2.20 2.08 ;
2.35 2.25 2.12 ;
2.40 2.29 2.16 ;
1.43 2.32 2.19 ;
1.46 2.35 2.22 ;
).89 (
.09
.20
.27
.33
.44
.54
.61
.66
.75
.81
.86
.90
.95
2.00
2.05
2.09 ;
1.12 :
>..i4 ;
).87 0.85 (
.06 1.04
.16 1.14
.24 1.22
.29 1.27
.40 1.38
.50 1.47
.56 1.53
.61 1.58
.70 1.67
.76 1.73
.81 1.77
.84 1.81
.89 1.85
.95 1.91
.99 1.95
1.02 1.98
2.05 2.01
2.08 2.04 ;
).84 (
.03
.13
.20
.25
.36
.45
.51
.56
.64
.70
.74
.78
.82
.88
.92
.95
.98
2.01
).84 0.83 (
.02 1.01
.12 1.11
.19 1.18
.24 1.23
.35 1.34
.43 1.42
.50 1.48
.54 1.53
.62 1.61
.68 1.67
.72 1.71
.76 1.74
.80 1.79
.86 1.84
.90 1.88
.93 1.91
.96 1.94
.98 1.96
).83 0.82 0.81 (
.01 1.00 0.99 (
.11 1.10 1.09
.17 1.16 1.15
.22 1.21 1.20
.33 1.31 1.31
.41 1.40 1.39
.47 1.46 1.45
.52 1.50 1.49
.60 1.58 1.57
.65 1.64 1.62
.70 1.68 1.67
.73 1.71 1.70
.77 1.75 1.74
.83 1.80 1.79
.87 1.84 1.83
.90 1.88 1.86
.93 1.90 1.89
.95 1.93 1.91
).81 0.81 (
).99 0.98 (
.08 1.08
.15 1.14
.20 1.19
.30 1.29
.38 1.38
.44 1.43
.48 1.48
.56 1.55
.61 1.61
.66 1.65
.69 1.68
.73 1.72
.78 1.77
.82 1.81
.85 1.84
.87 1.86
.90 1.89
).81 0.80 (
).98 0.97 (
.08 1.07
.14 1.13
.19 1.18
.29 1.28
.37 1.36
.43 1.42
.47 1.46
.55 1.54
.60 1.59
.64 1.63
.67 1.66
.71 1.70
.76 1.75
.80 1.79
.83 1.82
.86 1.84
.88 1.86
).80
).97
.07
.13
.18
.28
.36
.41
.46
.53
.58
.62
.65
.69
.74
.78
.81
.83
.85
    Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.14 0.92 0.83 (
1.55 1.22 1.09
1.79 1.40 1.25
1.97 1.52 1.35
2.10 1.62 1.43
2.39 1.81 1.60
2.63 1.98 1.74
2.80 2.10 1.84
2.93 2.19 1.92
3.16 2.35 2.06
3.32 2.46 2.15
3.44 2.55 2.22 ;
3.54 2,62 2.28 ;
3.66 2.70 2.35 ;
3.81 2.81 2.44 ;
3.93 2.89 2.51 ;
4.02 2.95 2,57 '*
4.10 3.01 2,61 :
4.17 3.06 2.6S \
).78 0.74 0.71 (
.02 0.98 0.93 (
.16 1.11 1.05
.26 1.20 1.14
.34 1.27 1.20
.49 1.42 1.34
.62 1.54 1.45
.71 1.62 1.52
.78 1.69 1.58
.90 1.80 1.69
.99 1.88 1.76
2.05 1.94 1.82
2.10 1.99 1.86
1.17 2.05 1.92
2.25 2.13 1.99
2.31 2.19 2.04
2.36 2.23 2.08
2.40 2.27 2.12 ;
2.44 2.31 2.15 ;
).68 (
).90 (
.02 (
.10
.16
.29
.39
.47
.52
.62
.69
.74
.79
.84
.90
.95
.99
2.03
2.06
).66 0.65 (
).88 0.86 (
).99 0.97 (
.07 1.05
.13 1.11
.25 1.23
.35 1.32
.42 1.39
.47 1.44
.57 1.54
.64 1.60
.69 1.65
.73 1.69
.78 1.73
.84 1.79
.89 1.84
.92 1.88
.96 1.91
.98 1.94
).64 (
).85 (
).96 (
.04
.09
.21
.30
.37
.42
.51
.57
.62
.66
.71
.76
.81
.85
.88
.90
).64 0.63 (
).84 0.83 (
).95 0.94 (
.02 1.02
.08 1.07
.20 1.19
.29 1.28
.36 1.34
.41 1.39
.49 1.48
.55 1.54
.60 1.59
.64 1.62
.68 1.67
.74 1.72
.79 1.77
.82 1.80
.85 1.83
.88 1.86
).63 0.62 0.62 (
).83 0.82 0.82 (
).94 0.93 0.92 (
.01 1.00 0.99 (
.07 1.06 1.05
.18 1.17 1.16
.27 1.26 1.25
.33 1.32 1.31
.38 1.37 1.36
.47 1.45 1.44
.53 1.51 1.50
.57 1.55 1.54
.61 1.59 1.58
.65 1.63 1.62
.71 1.69 1.67
.75 1.73 1.71
.79 1.76 1.75
.82 1.79 1.78
.84 1.82 1.80
).62 0.61 (
).81 0.81 (
).92 0.91 (
).99 0.99 (
.04 1.04
.15 1.15
.24 1.24
.30 1.30
.35 1.34
.43 1.42
.49 1.48
.53 1.52
.57 1.56
.61 1.60
.66 1.65
.70 1.69
.74 1.73
.76 1.75
.79 1.78
).61 0.61 (
).81 0.80 (
).91 0.91 (
).98 0.98 (
.04 1.03
.14 1.14
.23 1.22
.29 1.28
.34 1.33
.42 1.41
.47 1.46
.52 1.51
.55 1.54
.59 1.58
.65 1.63
.69 1.67
.72 1.70
.74 1.73
.77 1.75
).61
).80
).90
).97
.02
.13
.22
.28
.32
.40
.46
.50
.53
.57
.62
.66
.70
.72
.74
                                                     D-62
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
  Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (2 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.55 1.22 1.09
1.97 1.52 1.35
2.21 1.69 1.50
2.39 1.81 1.60
2.52 1.91 1.68
2.80 2.10 1.84
3.03 2.26 1.98
3.20 2.38 2.08
3.32 2.46 2.15
3.54 2.62 2.28 ;
3.70 2.73 2.37 ;
3.81 2.81 2.44 ;
3.91 "'2.87' 2.50 ;
4.02 2.95 2.57 ;
4.17 3.06 2.65 ;
4.28 3.13 2.72 ;
4.37 3.19 2.77 ;
4.44 3.25 2.82 '*
4.51 3.29 2,85 '*
.02 0.98 0.93 (
.26 1.20 1.14
.40 1.33 1.25
.49 1.42 1.34
.56 1.49 1.40
.71 1.62 1.52
.83 1.74 1.63
.92 1.82 1.70
.99 1.88 1.76
MO 1.99 1.86
2.19 2.07 1.93
2.25 2.13 1.99
2.30 2.18 2.03
2.36 2.23 2.08
2.44 2.31 2.15 ;
2.50 2.36 2.20 ;
2.55 2.41 2.24 ;
2.59 2.45 2.28 ;
1.62 2.48 2.31 ;
).90 (
.10
.21
.29
.35
.47
.57
.64
.69
.79
.85
.90
.94
.99
2.06
Ml ;
M4 ;
MS ;
>..20 ;
).88 0.86 (
.07 1.05
.18 1.15
.25 1.23
.31 1.28
.42 1.39
.52 1.49
.58 1.55
.64 1.60
.73 1.69
.79 1.75
.84 1.79
.88 1.83
.92 1.88
.98 1.94
2.03 1.98
2.07 2.02
MO 2.05 ;
M3 2.07 ;
).85 (
.04
.14
.21
.26
.37
.46
.53
.57
.66
.72
.76
.80
.85
.90
.94
.98
2.01
2.03 ;
).84 0.83 (
.02 1.02
.13 1.12
.20 1.19
.25 1.24
.36 1.34
.45 1.43
.51 1.49
.55 1.54
.64 1.62
.70 1.68
.74 1.72
.78 1.76
.82 1.80
.88 1.86
.92 1.90
.95 1.93
.98 1.96
2.01 1.98
).83 0.82 0.82 (
.01 1.00 0.99 (
.11 1.10 1.09
.18 1.17 1.16
.23 1.22 1.21
.33 1.32 1.31
.42 1.41 1.39
.48 1.46 1.45
.53 1.51 1.50
.61 1.59 1.58
.67 1.65 1.63
.71 1.69 1.67
.74 1.72 1.71
.79 1.76 1.75
.84 1.82 1.80
.88 1.86 1.84
.91 1.89 1.87
.94 1.92 1.90
.97 1.94 1.92
).81 0.81 (
).99 0.99 (
.09 1.08
.15 1.15
.20 1.20
.30 1.30
.39 1.38
.44 1.44
.49 1.48
.57 1.56
.62 1.61
.66 1.65
.69 1.69
.74 1.73
.79 1.78
.83 1.81
.86 1.85
.88 1.87
.91 1.89
).81 0.80 (
).98 0.98 (
.08 1.07
.14 1.14
.19 1.18
.29 1.28
.37 1.37
.43 1.42
.47 1.46
.55 1.54
.61 1.59
.65 1.63
.68 1.67
.72 1.70
.77 1.75
.81 1.79
.84 1.82
.86 1.85
.89 1.87
).80
).97
.07
.13
.18
.28
.36
.42
.46
.53
.59
.62
.66
.70
.74
.78
.81
.84
.86
   Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.97 1.52 1.35
2.39 1.81 1.60
2.63 1.98 1.74
2.80 2.10 1.84
2.93 2.19 1.92
3.20 2.38 2.08
3.42 2.53 2.21 ;
3.58 2.64 2.30 ;
3.70 2.73 2.37 ;
3.91 2.87 2.50 ;
4.06 2.98 2.59 ;
4.17 3.06 2.65 ;
4.26 3.12 2.71 ;
4.37 3.19 2.77 ;
4.51 3.29 2.85 ;
4.61 3.37 2.92 ;
4.69 3.43 2.97 ;
4.77 3.48 3.01 ;
4.83 3.52 3.05 ;
.26 1.20 1.14
.49 1.42 1.34
.62 1.54 1.45
.71 1.62 1.52
.78 1.69 1.58
.92 1.82 1.70
2.04 1.93 1.81
M2 2.01 1.88
M9 2.07 1.93
2.30 2.18 2.03
2.38 2.25 2.10 ;
2.44 2.31 2.15 ;
1.49 2.35 2.19 ;
2.55 2.41 2.24 ;
1.62 2.48 2.31 ;
2.68 2.53 2.35 ;
1.73 2.58 2.39 ;
2.76 2.61 2.43 ;
2.80 2.64 2.46 ;
.10
.29
.39
.47
.52
.64
.73
.80
.85
.94
2.01
2.06
2.10 ;
2.14 ;
2.20 ;
2.25 ;
2.29 ;
2.32 ;
2.35 ;
.07 1.05
.25 1.23
.35 1.32
.42 1.39
.47 1.44
.58 1.55
.68 1.64
.74 1.70
.79 1.75
.88 1.83
.94 1.89
.98 1.94
2.02 1.97
2.07 2.02
2.13 2.07 ;
2.17 2.12 ;
2.21 2.15 ;
2.24 2.18 ;
2.26 2.20 ;
.04
.21
.30
.37
.42
.53
.61
.67
.72
.80
.86
.90
.94
.98
2.03 ;
2.08 ;
2.11 ;
2.14 ;
2.16 ;
.02 1.02
.20 1.19
.29 1.28
.36 1.34
.41 1.39
.51 1.49
.59 1.58
.65 1.64
.70 1.68
.78 1.76
.83 1.81
.88 1.86
.91 1.89
.95 1.93
2.01 1.98
2.05 2.02 ;
2.08 2.06 ;
2.11 2.08 ;
2.13 2.11 ;
.01 1.00 0.99 (
.18 1.17 1.16
.27 1.26 1.25
.33 1.32 1.31
.38 1.37 1.36
.48 1.46 1.45
.56 1.55 1.53
.62 1.60 1.59
.67 1.65 1.63
.74 1.72 1.71
.80 1.78 1.76
.84 1.82 1.80
.87 1.85 1.83
.91 1.89 1.87
.97 1.94 1.92
2.01 1.98 1.96
2.04 2.01 1.99
2.06 2.04 2.01 ;
2.09 2.06 2.04 ;
).99 0.99 (
.15 1.15
.24 1.24
.30 1.30
.35 1.34
.44 1.44
.52 1.52
.58 1.57
.62 1.61
.69 1.69
.75 1.74
.79 1.78
.82 1.81
.86 1.85
.91 1.89
.94 1.93
.97 1.96
2.00 1.99
2.02 2.01 ;
).98 0.98 (
.14 1.14
.23 1.22
.29 1.28
.34 1.33
.43 1.42
.51 1.50
.56 1.55
.61 1.59
.68 1.67
.73 1.72
.77 1.75
.80 1.78
.84 1.82
.89 1.87
.92 1.91
.95 1.93
.98 1.96
2.00 1.98
).97
.13
.22
.28
.32
.42
.49
.54
.59
.66
.71
.74
.77
.81
.86
.89
.92
.95
.97
                                                     D-63
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
    Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.80 1.36 1.20
2.30 1.69 1.47
2.62 1.88 1.63
2.84 2.02 1.74
3.02 2.13 1.83
3.39 2.36 2.01
371 2.56 2.17
3.94 2.69 2.28 ;
4.11 2.80 2.36 ;
4.43 2.99 2.51 ;
4.64 3.12 2,62 '4
4.81 3.22 2.70 ;
4.94 3.31 2.76 ;
5.10 3.40 2.84
5.31 3.53 2.95
5.46 3.63 3.02 ,
5.59 3.71 3.09
5.69 3.77 3.14
5.79 3.83 3.18
.12 1.07 1.01 (
.36 1.29 1.21
.50 1.42 1.33
.60 1.52 1.41
.68 1.59 1.48
.84 1.73 1.61
.97 1.85 1.72
1.07 1.94 1.80
2.14 2.01 1.86
1.27 2.13 1.96
1.37 2.21 2.04
2.44 2.28 2.09
2.49 2.33 2.14 ;
2.56 2.39 2.20 ;
2,65 2.47 2.27 ;
2.72 2,54 2.32 ;
2.78 2.59 2.37 ;
1.8,2 2.63 2.41 ;
2.86 2,67 2.44 ;
).97 (
.17
.28
.36
.42
.54
.64
.71
.77
.87
.94
.99
2.03
2.09 ;
2.15 ;
1.20 ;
2.24 ;
2.28 ;
2.3i ;
).94 0.93 (
.13 1.11
.24 1.22
.32 1.29
.37 1.34
.49 1.45
.58 1.55
.65 1.61
.70 1.66
.80 1.75
.86 1.81
.91 1.86
.95 1.90
2.00 1.94
2.06 2.00
Ml 2.05 ;
2.15 2.09 ;
2.18 2.12 ;
1.21 2.15 ;
).91 (
.10
.20
.27
.32
.43
.52
.58
.63
.72
.78
.82
.86
.91
.96
2.01
2.04 ;
2.07 ;
>.io ;
).90 0.90 (
.08 1.08
.18 1.17
.25 1.24
.30 1.29
.41 1.40
.50 1.49
.56 1.55
.61 1.59
.69 1.68
.75 1.73
.80 1.78
.83 1.81
.88 1.86
.93 1.91
.98 1.95
2.01 1.99
2.04 2.01
1.07 2.04 ;
).89 0.88 0.88 (
.07 1.06 1.05
.17 1.15 1.14
.23 1.22 1.21
.28 1.27 1.26
.39 1.37 1.36
.47 1.46 1.44
.53 1.51 1.50
.58 1.56 1.55
.66 1.64 1.62
.72 1.69 1.68
.76 1.74 1.72
.80 1.77 1.75
.84 1.81 1.79
.89 1.86 1.84
.93 1.90 1.88
.97 1.94 1.92
.99 1.96 1.94
1.02 1.99 1.97
).87 0.87 (
.04 1.04
.14 1.13
.20 1.20
.25 1.25
.35 1.34
.43 1.43
.49 1.48
.53 1.53
.61 1.60
.67 1.66
.71 1.70
.74 1.73
.78 1.77
.83 1.82
.87 1.86
.90 1.89
.93 1.91
.95 1.94
).87 0.86 (
.04 1.03
.13 1.12
.19 1.19
.24 1.23
.34 1.33
.42 1.41
.48 1.47
.52 1.51
.60 1.58
.65 1.64
.69 1.67
.72 1.71
.76 1.75
.81 1.79
.85 1.83
.88 1.86
.90 1.89
.93 1.91
).86
.03
.12
.18
.23
.32
.40
.46
.50
.58
.63
.67
.70
.74
.78
.82
.85
.88
.90
  Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.30 1.69 1.47 .36 1.29 1.21
2.84 2.02 1.74 .60 1.52 1.41
3.16 2.22 1.90 .74 1.64 1.53
3.39 2.36 2.01 .84 1.73 1.61
357 2.47 2.10 .91 1.80 1.67
394 2.69 2.28 2.07 1.94 1.80
4.26 2,88 2.43 2.20 2.06 1.90
4.47 3.02 2.54 2.29 2.15 1.98
4.64 3.12 2.62 2.37 2.21 2.04
4.94 3.31 2.76 2.49 2.33 2.14 ;
5.15 3.43 2,87 2.58 2.41 2.21 ;
5.31 3.53 2,95 2.65 2.47 2.27 ;
5.43 3.61 3.01 2.71 2.53 2.31 ;
5.59 3.71 3.09 2.78 2.59 2.37 ;
5.79 3.83 3.18 '"2.86" 2.67 2.44 ;
5.94 3.92 3.26 2,93 2.73 2.49 ;
6.05 4.00 3.32 2.98 2.77 2.54 ;
6.16 4.06 3.37 3.02 2.82 2.57 ;
6.24 4.11 3.41 3.06 2,85 2.60 ;
.17
.36
.47
.54
.60
.71
.81
.88
.94
2.03
MO ;
MS ;
M9 ;
>..24 ;
2.3i ;
2.36 ;
2.40 ;
2.43 ;
2.46 ;
.13 1.11
.32 1.29
.42 1.38
.49 1.45
.54 1.50
.65 1.61
.75 1.70
.81 1.76
.86 1.81
.95 1.90
2.01 1.96
2.06 2.00
MO 2.04 ;
M5 2.09 ;
2.21 2.15 ;
1.26 2.19 ;
1.29 2.23 ;
2.33 2.26 ;
2.35 2.28 ;
.10
.27
.36
.43
.48
.58
.67
.73
.78
.86
.92
.96
2.00
2.04 ;
MO ;
M4 ;
MS ;
2.21 ;
2.23 ;
.08 1.08
.25 1.24
.35 1.33
.41 1.40
.46 1.45
.56 1.55
.65 1.63
.71 1.69
.75 1.73
.83 1.81
.89 1.87
.93 1.91
.97 1.94
2.01 1.99
2.07 2.04 ;
2.11 2.08 ;
2.14 2.11 ;
2.17 2.14 ;
2.19 2.17 ;
.07 1.06 1.05
.23 1.22 1.21
.32 1.31 1.30
.39 1.37 1.36
.43 1.42 1.41
.53 1.51 1.50
.62 1.60 1.58
.67 1.65 1.64
.72 1.69 1.68
.80 1.77 1.75
.85 1.82 1.80
.89 1.86 1.84
.93 1.90 1.88
.97 1.94 1.92
2.02 1.99 1.97
2.06 2.03 2.00
2.09 2.06 2.04 ;
2.12 2.08 2.06 ;
2.14 2.11 2.08 ;
.04 1.04
.20 1.20
.29 1.28
.35 1.34
.40 1.39
.49 1.48
.57 1.56
.62 1.62
.67 1.66
.74 1.73
.79 1.78
.83 1.82
.86 1.85
.90 1.89
.95 1.94
.99 1.97
2.02 2.00
2.04 2.03 ;
2.07 2.05 ;
.04 1.03
.19 1.19
.28 1.27
.34 1.33
.38 1.37
.48 1.47
.55 1.54
.61 1.60
.65 1.64
.72 1.71
.77 1.76
.81 1.79
.84 1.82
.88 1.86
.93 1.91
.96 1.95
.99 1.97
2.02 2.00
2.04 2.02 ;
.03
.18
.27
.32
.37
.46
.53
.59
.63
.70
.75
.78
.81
.85
.90
.93
.96
.99
2.01
                                                     D-64
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
   Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.84 2.02 1.74 1.60 1.52 1.41
3.39 2.36 2.01 1.84 1.73 1.61
371 2.56 2.17 1.97 1.85 1.72
394 2.69 2.28 2.07 1.94 1.80
411 2.80 2.36 2.14 2.01 1.86
447 3.02 2.54 2.29 2.15 1.98
4.78 3.20 2.68 2.42 2.27 2.08
4.99 3.33 2.79 2.51 2.35 2.16 ;
5.15 3.43 2.87 2.58 2.41 2.21 ;
5.43 3.61 3.01 2.71 2.53 2.31 ;
5.63 3.73 3.11 2.79 2.60 2.38 ;
5.79 3.83 3,18 2.86 2.67 2.44 ;
5.91 3.90 3.25 2.92 2.71 2.48 ;
6.05 4.00 3.32 2.98 2.77 2.54 ;
6.24 4.11 3.41 3.06 2.85 2.60 ;
6.39 4.20 3.49 '"3.13" 2.91 2.66 ;
6.50 4.27 3.54 -3,18 2.96 2.70 ;
6.60 4.33 3.59 3.22., 3.00 2.73 ;
6.68 4.39 3.64 3,26 3.03 2.76 ;
.36
.54
.64
.71
.77
.88
.98
2.05
>.io ;
2.19 ;
1.26 ;
2.3i ;
2.35 ;
2.40 ;
>..46 ;
2.5i ;
2.55 ;
2.58 ;
2.6i ;
.32 1.29
.49 1.45
.58 1.55
.65 1.61
.70 1.66
.81 1.76
.90 1.85
.97 1.91
2.01 1.96
2.10 2.04 ;
2.16 2.10 ;
2.21 2.15 ;
1.25 2.18 ;
1.29 2.23 ;
2.35 2.28 ;
2.40 2.33 ;
2.44 2.36 ;
2.47 2.39 ;
2.49 2.42 ;
.27
.43
.52
.58
.63
.73
.82
.87
.92
2.00
2.06 ;
2.10 ;
2.13 ;
2.18 ;
2.23 ;
2.27 ;
2.3i ;
2.34 ;
2.36 ;
.25 1.24
.41 1.40
.50 1.49
.56 1.55
.61 1.59
.71 1.69
.79 1.77
.85 1.82
.89 1.87
.97 1.94
2.02 2.00
2.07 2.04 ;
2.10 2.07 ;
2.14 2.11 ;
2.19 2.17 ;
2.23 2.20 ;
2.27 2.24 ;
2.30 2.26 ;
2.32 2.29 ;
.23 1.22 1.21
.39 1.37 1.36
.47 1.46 1.44
.53 1.51 1.50
.58 1.56 1.55
.67 1.65 1.64
.75 1.73 1.71
.81 1.78 1.76
.85 1.82 1.80
.93 1.90 1.88
.98 1.95 1.93
2.02 1.99 1.97
2.05 2.02 2.00
2.09 2.06 2.04 ;
2.14 2.11 2.08 ;
2.18 2.15 2.12 ;
2.21 2.18 2.15 ;
2.24 2.20 2.18 ;
2.26 2.22 2.20 ;
.20 1.20
.35 1.34
.43 1.43
.49 1.48
.53 1.53
.62 1.62
.70 1.69
.75 1.74
.79 1.78
.86 1.85
.91 1.90
.95 1.94
.98 1.97
2.02 2.00
2.07 2.05 ;
2.10 2.09 ;
2.13 2.12 ;
2.16 2.14 ;
2.18 2.16 ;
.19 1.19
.34 1.33
.42 1.41
.48 1.47
.52 1.51
.61 1.60
.68 1.67
.73 1.72
.77 1.76
.84 1.82
.89 1.87
.93 1.91
.96 1.94
.99 1.97
2.04 2.02 ;
2.08 2.05 ;
2.10 2.08 ;
2.13 2.11 ;
2.15 2.13 ;
.18
.32
.40
.46
.50
.59
.66
.71
.75
.81
.86
.90
.93
.96
2.01
2.04
2.07
2.09
2.11
    Table 19-4. K-Multipliers  for Modified Calif.  Interwell Prediction Limits on Observations (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.40 1.72 1.49 .37 1.30 1.22
301 2.08 1.77 .62 1.53 1.42
3.40 2.29 1.94 .76 1.66 1.54
3.68 2.44 2.05 .86 1.75 1.62
3.89 2.57 2.15 .94 1.82 1.68
4.36 2.82 2.34 2.11 1.97 1.81
4.76 3.04 2.51 2.25 2.09 1.92
5.04 3.20 2.82 2.35 2.18 2.00
5.26 3.32 2.72 2.43 2.25 2.06
5.65 3.53 2.88 2.86 2.38 2.17 ;
5.92 3.69 3.00 2.66 2.47 2.24 ;
6.13 3.80 3.08 2.74 ' '2.'S3" 2.30 ;
6.30 3.90 3.16 2.80 2,59 2.35 ;
6.50 4.01 3.24 2.87 2,66 2.41 ;
6.76 4.15 3.36 2.97 2,74 2.48 ;
6.96 4.27 3.44 3.04 2.81 2,54 :
7.12 4.35 3.51 3.10 2.86 2.58. '4
7.25 4.43 3.57 3.15 2.90 2,62 \
7.36 4.49 3.62 3.19 2.94 2.66 \
.17
.36
.47
.55
.60
.72
.83
.90
.95
2.05
2.12 ;
2.17 ;
2.22 ;
2.27 ;
2.34 ;
2.39 ;
2.43 ;
2.47 ;
2.50 ;
.14 1.11
.32 1.29
.42 1.39
.49 1.46
.55 1.51
.66 1.62
.75 1.71
.82 1.77
.87 1.82
.96 1.91
2.03 1.97
2.08 2.01
2.12 2.05 ;
2.17 2.10 ;
2.23 2.16 ;
2.28 2.20 ;
2.32 2.24 ;
2.35 2.27 ;
2.38 2.30 ;
.10
.27
.37
.43
.48
.59
.67
.74
.78
.87
.93
.97
2.01
2.05 ;
2.11 ;
2.15 ;
2.19 ;
2.22 ;
2.24 ;
.09 1.08
.25 1.24
.35 1.34
.41 1.40
.46 1.45
.56 1.55
.65 1.63
.71 1.69
.76 1.74
.84 1.82
.90 1.87
.94 1.91
.97 1.95
2.02 1.99
2.07 2.05 ;
2.12 2.09 ;
2.15 2.12 ;
2.18 2.15 ;
2.20 2.17 ;
.07 1.06 1.05
.23 1.22 1.21
.33 1.31 1.30
.39 1.37 1.36
.44 1.42 1.41
.54 1.52 1.50
.62 1.60 1.58
.68 1.65 1.64
.72 1.70 1.68
.80 1.77 1.75
.85 1.83 1.81
.90 1.87 1.85
.93 1.90 1.88
.97 1.94 1.92
2.02 1.99 1.97
2.06 2.03 2.01
2.10 2.06 2.04 ;
2.12 2.09 2.06 ;
2.15 2.11 2.09 ;
.04 1.04
.20 1.20
.29 1.29
.35 1.35
.40 1.39
.49 1.48
.57 1.56
.63 1.62
.67 1.66
.74 1.73
.79 1.78
.83 1.82
.86 1.85
.90 1.89
.95 1.94
.99 1.98
2.02 2.01 ;
2.05 2.03 ;
2.07 2.05 ;
.04 1.03
.19 1.19
.28 1.27
.34 1.33
.39 1.38
.48 1.47
.56 1.54
.61 1.60
.65 1.64
.72 1.71
.77 1.76
.81 1.80
.84 1.83
.88 1.86
.93 1.91
.97 1.95
2.00 1.98
2.02 2.00
2.04 2.02 ;
.03
.18
.27
.32
.37
.46
.54
.59
.63
.70
.75
.78
.81
.85
.90
.93
.96
.99
2.01
                                                     D-65
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                      Unified Guidance
 Table 19-4. K-Multipliers  for Modified Calif. Interwell Prediction Limits on Observations (10 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.01 2.08 1.77 1.62 1.53 1.42
3.68 2.44 2.05 1.86 1.75 1.62
408 2.66 2.22 2.00 1.88 1.73
4.36 2.82 2.34 2.11 1.97 1.81
4.58 2.94 2.43 2.18 2.04 1.87
5.04 3.20 2.62 2.35 2.18 2.00
5.44 3.41 2.79 2.49 2.31 2.11 ;
5.71 3.57 2.91 2.59 2.40 2.18 ;
5.92 3.69 3.00 2.66 2.47 2.24 ;
6.30 3.90 3.16 2,3Q 2.59 2.35 ;
6.56 4.04 3.27 Z9O 2.67 2.42 ;
6.76 4.15 3.36 2,97 2.74 2.48 ;
6.92 4.25 3.43 3.03 2.79 2.53 ;
7.12 4.35 3.51 3.10 2,86 2.58 ;
7.36 4.49 3.62 3.19 2,94 2.66 ;
7.55 4.60 3.70 3.27 3.O1 2.71 ;
7.71 4.69 3.77 3.32 3.Q6 2.76 ;
7.83 4.76 3.82 3.37 3.10 2,80 ;
7.94 4.82 3.87 3.41 3.14 2.83. :
.36
.55
.65
.72
.78
.90
2.00
2.07
2.12 ;
1.22 :
1.29 ;
2.34 ;
2.38 ;
2.43 ;
2.50 ;
2.55 ;
2.59 ;
>.62 :
i.65 ;
.32 1.29
.49 1.46
.59 1.55
.66 1.62
.71 1.67
.82 1.77
.91 1.86
.98 1.92
2.03 1.97
1.12 2.05 ;
2.18 2.11 ;
1.23 2.16 ;
2.27 2.20 ;
2.32 2.24 ;
2.38 2.30 ;
2.42 2.34 ;
2.46 2.38 ;
2.49 2.41 ;
2.52 2.44 ;
.27
.43
.52
.59
.64
.74
.82
.88
.93
2.01
2.07 ;
2.11 ;
2.14 ;
2.19 ;
2.24 ;
2.29 ;
2.32 ;
2.35 ;
2.38 ;
.25 1.24
.41 1.40
.50 1.49
.56 1.55
.61 1.59
.71 1.69
.79 1.77
.85 1.83
.90 1.87
.97 1.95
2.03 2.00
2.07 2.05 ;
2.11 2.08 ;
2.15 2.12 ;
2.20 2.17 ;
2.24 2.21 ;
2.28 2.25 ;
2.31 2.27 ;
2.33 2.30 ;
.23 1.22 1.21
.39 1.37 1.36
.48 1.46 1.44
.54 1.52 1.50
.58 1.56 1.55
.68 1.65 1.64
.76 1.73 1.71
.81 1.78 1.77
.85 1.83 1.81
.93 1.90 1.88
.98 1.95 1.93
2.02 1.99 1.97
2.06 2.02 2.00
2.10 2.06 2.04 ;
2.15 2.11 2.09 ;
2.19 2.15 2.12 ;
2.22 2.18 2.15 ;
2.25 2.21 2.18 ;
2.27 2.23 2.20 ;
.20 1.20
.35 1.35
.43 1.43
.49 1.48
.54 1.53
.63 1.62
.70 1.69
.75 1.74
.79 1.78
.86 1.85
.91 1.90
.95 1.94
.98 1.97
2.02 2.01 ;
2.07 2.05 ;
2.10 2.09 ;
2.13 2.12 ;
2.16 2.14 ;
2.18 2.16 ;
.19 1.19
.34 1.33
.42 1.41
.48 1.47
.52 1.51
.61 1.60
.68 1.67
.73 1.72
.77 1.76
.84 1.83
.89 1.87
.93 1.91
.96 1.94
2.00 1.98
2.04 2.02 ;
2.08 2.06 ;
2.11 2.08 ;
2.13 2.11 ;
2.15 2.13 ;
.18
.32
.40
.46
.50
.59
.66
.71
.75
.81
.86
.90
.93
.96
2.01
2.04
2.07
2.09
2.11
   Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (10 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
3.68
4.36
4.76
5.04
5.26
5.71
6.09
6.36
6.56
6.92
7.17
7.36
7.52
7.71
7.94
8.12
8.27
8.38
8.49

6 8 10 12 16 20
2.44 2.05 1.86 1.75 1.62 .55
2.82 2.34 2.11 1.97 1.81 .72
3.04 2.51 2.25 2.09 1.92 .83
3.20 2.62 2.35 2.18 2.00 .90
3.32 2.72 2.43 2.25 2.06 .95
357 2.91 2.59 2.40 2.18 2.07
3.78 3.07 2.72 2.52 2.29 2.16
3.93 3,18 2.82 2.61 2.37 2.23
4.04 3.27 2.90 2.67 2.42 2.29
4.25 3.43 3.03 2.79 2.53 2.38
4.39 3.53 '"3,12" 2.88 2.60 2.45
4.49 3.62 --3.19 2.94 2.66 2.50
4.58 3.69 3,25 2.99 2.70 2.54
4.69 3.77 3.32 3.06 2.76 2.59
4.82 3.87 3.41 5,74 2.83 2.65
4.93 3.95 3.48 • 3,'2O 2.88 2.70
5.01 4.02 3.54 3.25 2.93 2.74
5.08 4.07 3.58 3,29 2.96 2.78
5.14 4.12 3.62 3,33 2.99 2.81

25
.49
.66
.75
.82
.87
.98
2.07
2.13
2.18
2.27
2.33
2.38
2.41
2.46
2.52
2.57
2.60
2.64
2.66

30
1.46
1.62
1.71
1.77
1.82
1.92
2.01
2.07
2.11
2.20
2.25
2.30
2.33
2.38
2.44
2.48
2.51
2.54
2.57

35
.43
.59
.67
.74
.78
.88
.96
2.02
2.07 ;
2.14 ;
2.20 ;
2.24 ;
2.28 ;
2.32 ;
2.38 ;
2.42 ;
2.45 ;
2.48 ;
2.50 ;

40
.41
.56
.65
.71
.76
.85
.93
.99
2.03
2.11
2.16
2.20
2.24
2.28
2.33
2.37
2.40
2.43
2.46

45 50
1.40 .39
1.55 .54
1.63 .62
1.69 .68
1.74 .72
1.83 .81
1.91 .89
1.96 .94
2.00 .98
2.08 2.06
2.13 2.11
2.17 2.15
2.21 2.18
2.25 2.22
2.30 2.27
2.34 2.31
2.37 2.34
2.39 2.37
2.42 2.39

60
1.37
1.52
1.60
1.65
1.70
1.78
1.86
1.91
1.95
2.02
2.07
2.11
2.14
2.18
2.23
2.27
2.30
2.32
2.34

70 80
1.36 .35
1.50 .49
1.58 .57
1.64 .63
1.68 .67
1.77 .75
1.84 .83
1.89 .88
1.93 .91
2.00 .98
2.05 2.03
2.09 2.07
2.12 2.10
2.15 2.13
2.20 2.18
2.24 2.22
2.27 2.24
2.29 2.27
2.31 2.29

90 100
1.35 .34
1.48 .48
1.56 .56
1.62 .61
1.66 .65
1.74 .73
1.81 .80
1.86 .85
1.90 .89
1.97 .96
2.02 2.01
2.05 2.04
2.08 2.07
2.12 2.11
2.16 2.15
2.20 2.19
2.23 2.21
2.25 2.24
2.27 2.26

125 150
1.33 .32
1.47 .46
1.54 .54
1.60 .59
1.64 .63
1.72 .71
1.79 .78
1.84 .83
1.87 .86
1.94 .93
1.99 .97
2.02 2.01
2.05 2.04
2.08 2.07
2.13 2.11
2.16 2.15
2.19 2.17
2.21 2.20
2.23 2.22

                                                    D-66
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
                                                                     Unified Guidance
    Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.13 2.11 1.79 1.63 1.53 1.42
3.89 2.50 2.08 1.88 1.76 1.62
4.36 2.74 2.25 2.02 1.89 1.74
4.71 2.91 2.38 2.13 1.98 1.82
4.98 3.05 2.48 2.21 2.05 1.88
5.56 3.34 2.89-. 2.38 2.21 2.01
6.06 3.59 2.87 2,53 2.34 2.12 ;
6.42 3.76 3.00 2,64 2.43 2.20 ;
6.69 3.90 3.10 2. 72. 2.50 2.26 ;
7.19 4.15 3.28 2.87 2,63 2.37 ;
7.53 4.32 3.40 2.97 2,73 2.45 ;
7.79 4.45 3.50 3.05 2.80 '2.51 '<
8.00 4.56 3.58 3.12 2.85 2.56-'- '<
8.25 4.69 3.67 3.20 2.92 2.62 ;
8.58 4.86 3.80 3.30 3.01 '2,70 ,.
8.83 4.98 3.89 3.38 3.08 2.7 5 .
9.02 5.09 3.97 3.44 3.14 2.80 •„
9.19 5.17 4.03 3.50 3.19 2.84 . ,
9.33 5.25 4.09 3.54 3.23 2.88
.37
.55
.65
.73
.79
.90
2.01
2.08
2.13 ;
1.23 :
i.30 ;
2.35 ;
2.40 ;
2.45 ;
2.52 ;
2,57 :
2.62 :
2.65 . .
2,68 ,
.32 1.29
.49 1.46
.59 1.55
.66 1.62
.71 1.67
.82 1.77
.92 1.86
.98 1.92
2.03 1.97
1.12 2.06 ;
2.19 2.12 ;
2.24 2.17 ;
2.28 2.20 ;
1.33 2.25 ;
2.39 2.31 ;
2.44 2.35 ;
2.48 2.39 ;
2,51 2.42 ;
2,54 2.45 ;
.27
.43
.52
.59
.64
.74
.82
.88
.93
2.01
2.07 ;
Ml ;
2.15 ;
2.19 ;
>..25 ;
1.29 :
1.33 :
i.36 :
i.38 ;
.26 1.24
.41 1.40
.50 1.49
.57 1.55
.61 1.60
.71 1.69
.80 1.77
.85 1.83
.90 1.87
.98 1.95
2.03 2.01
2.08 2.05 ;
Ml 2.08 ;
M5 2.12 ;
1.21 2.18 ;
1.25 2.22 :
1.28 2.25 ;
2.31 2.28 ;
2.34 2.30 ;
.23 1.22 1.21
.39 1.37 1.36
.48 1.46 1.44
.54 1.52 1.50
.58 1.56 1.55
.68 1.65 1.64
.76 1.73 1.71
.81 1.79 1.77
.86 1.83 1.81
.93 1.90 1.88
.99 1.95 1.93
2.03 1.99 1.97
2.06 2.03 2.00
MO 2.06 2.04 ;
M5 2.11 2.09 ;
M9 2.15 2.13 ;
1.22 2.18 2.16 ;
1.25 2.21 2.18 :
1.27 2.23 2.20 '<
.20 1.20
.35 1.35
.44 1.43
.49 1.48
.54 1.53
.63 1.62
.70 1.69
.75 1.74
.79 1.78
.86 1.85
.91 1.90
.95 1.94
.98 1.97
1.02 2.01 :
1.07 2.05 ;
Ml 2.09 ;
M4 2.12 ;
M6 2.14 ;
MS 2.17 ;
.19 1.19
.34 1.33
.42 1.41
.48 1.47
.52 1.51
.61 1.60
.68 1.67
.73 1.72
.77 1.76
.84 1.83
.89 1.87
.93 1.91
.96 1.94
2.00 1.98
2.04 2.02 ;
2.08 2.06 ;
Ml 2.09 ;
M3 2.11 ;
M5 2.13 ;
.18
.32
.40
.46
.50
.59
.66
.71
.75
.82
.86
.90
.93
.96
2.01
2.04
2.07
2.09
Ml
 Table 19-4. K-Multipliers  for Modified Calif. Interwell Prediction Limits on Observations (20 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
3.89
4.71
5.21
5.56
5.84
6.42
6.91
7.26
7.53
8.00
8.33
8.58
8.78
9.02
9.33
9.57
9.77
9.93
10.06

6
2.50
2.91
3.16
3.34
3.47
3.76
4.01
4.19
4.32
4.56
4.73
4.86
4.96
5.09
5.25
5.37
5.47
5.55
5.62

8 10
2.08 1.88
2.38 2.13
2.56 2.28
2.69 2.38
2.79 2.46
3.00 2.64
3.18 2.79
3.30 2,89
3.40 2.97
3.58 3.12
3.70 3.22
3.80 3.30
3.87 3.37
3.97 3.44
4.09 3.54
4.18 3.62
4.25 3.68
4.31 3.73
4.37 3.78

12
1.76
1.98
2.11
2.21
2.28
2.43
2.56
2.65
2.73
2,85
2,94
3.O1
3.07
3.14
3.23
3.30
3.35
3.40
3.44

16
1.62
1.82
1.93
2.01
2.07
2.20
2.31
2.39
2.45
2.56
2.64
2.70
2.74
2.8O
2.88
. 2,93
2,98
3. 02
3, OS

20
1.55
1.73
1.83
1.90
1.96
2.08
2.18
2.25
2.30
2.40
2.47
2.52
2.56
2.62
2.68
2.74
2.78
.2,81
2,85

25
.49
.66
.76
.82
.88
.98
2.08
2.14
2.19
2.28
2.34
2.39
2.43
2.48
2.54
2.59
2.63
2.66
2.69

30
1.46
1.62
1.71
1.77
1.82
1.92
2.01
2.07
2.12
2.20
2.26
2.31
2.35
2.39
2.45
2.49
2.53
2.56
2.59

35
1.43
1.59
1.68
1.74
1.79
1.88
1.97
2.03
2.07 ;
2.15 ;
2.21 ;
2.25 ;
2.29 ;
2.33 ;
2.38 ;
2.43 ;
2.46 ;
2.49 ;
2.52 ;

40 45 50 60 70 80 90 100 125
.41 1.40 1.39 .37 1.36 .35 1.35 .34 .33
.57 1.55 1.54 .52 1.50 .49 1.48 .48 .47
.65 1.63 1.62 .60 1.58 .57 1.56 .56 .54
.71 1.69 1.68 .65 1.64 .63 1.62 .61 .60
.76 1.74 1.72 .70 1.68 .67 1.66 .65 .64
.85 1.83 1.81 .79 1.77 .75 1.74 .73 .72
.93 1.91 1.89 .86 1.84 .83 1.81 .81 .79
.99 1.96 1.94 .91 1.89 .88 1.86 .85 .84
2.03 2.01 1.99 .95 1.93 .91 1.90 .89 .87
Ml 2.08 2.06 2.03 2.00 .98 1.97 .96 .94
M7 2.14 2.11 2.08 2.05 2.03 2.02 2.01 .99
1.21 2.18 2.15 2.11 2.09 2.07 2.05 2.04 2.02
1.24 2.21 2.18 2.15 2.12 2.10 2.08 2.07 2.05
2.28 2.25 2.22 2.18 2.16 2.14 2.12 2.11 2.09
2.34 2.30 2.27 2.23 2.20 2.18 2.17 2.15 2.13
2.38 2.34 2.31 2.27 2.24 2.22 2.20 2.19 2.16
2.41 2.37 2.34 2.30 2.27 2.25 2.23 2.22 2.19
2.44 2.40 2.37 2.33 2.29 2.27 2.25 2.24 2.21
2.46 2.43 2.39 2.35 2.32 2.29 2.27 2.26 2.23

150
1.32
1.46
1.54
1.59
1.63
1.71
1.78
1.83
1.86
1.93
1.97
2.01
2.04
2.07
2.11
2.15
2.17
2.20
2.22

                                                    D-67
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
Unified Guidance
   Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (20 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.72
5.57
6.06
6.42
6.69
7.27
7.73
8.09
8.32
8.79
9.08
9.35
9.52
9.76
10.08
10.28
10.49
10.63
10.75
6
2.92
3.33
3.59
3.76
3.90
4.19
4.42
4.60
4.73
4.97
5.13
5.24
5.35
5.46
5.62
5.74
5.84
5.92
5.99
8 10 12 16 20 25 30 35 40
2.38 2.13 1.98 1.82 1.73 1.66 1.62 1.59 .57
2.69 2.38 2.21 2.01 1.90 1.82 1.77 1.74 .71
2.87 2.53 2.34 2.12 2.01 1.92 1.86 1.82 .79
3.00 2.64 2.43 2.20 2.08 1.98 1.92 1.88 .85
3,10 2.72 2.50 2.26 2.13 2.03 1.97 1.93 .90
330 2.89 2.66 2.39 2.25 2.14 2.07 2.03 .99
348 3.04 2.78 2.50 2.34 2.23 2.16 2.11 2.07
3.60 3.14 2.87 2.58 2.41 2.29 2.22 2.16 2.12
3.70 3.22 2.94 2.64 2.47 2.34 2.26 2.21 2.17
3.87 3.37 3.07 2.74 2.56 2.43 2.35 2.29 2.24
3.99 3.46 3,16 2.82 2.63 2.49 2.40 2.34 2.30
4.09 3.54 3,23 2.87 2.68 2.54 2.45 2.38 2.34
4.16 3.60 ,3,28 2.92 2.73 2.58 2.48 2.42 2.37
4.25 3.68 3.35 2.98 2.78 2.63 2.53 2.46 2.41
4.37 3.78 3.44 3.05 2.85 2.69 2.59 2.52 2.46
4.45 3.85 3.50 3,11 2.90 2.73 2.63 2.56 2.50
4.53 3.91 3.56 3.16 2.94 2.77 2.67 2.59 2.54
4.58 3.96 3.60 3,20 2.97 2.80 2.70 2.62 2.57
4.64 4.01 3.64 3,23 3.00 2.83 2.72 2.64 2.59
45
1.55
1.69
1.77
1.83
1.88
1.96
2.04
2.09
2.14
2.21
2.26
2.30
2.33
2.37
2.42
2.46
2.50
2.52
2.55
50
1.54
1.68
1.76
1.81
1.85
1.94
2.02
2.07 ;
2.11 ;
2.18 ;
2.24 ;
2.27 ;
2.3i ;
2.34 ;
2.39 ;
2.43 ;
2.46 ;
2.49 ;
2.5i ;
60
.52
.66
.73
.79
.83
.91
.99
2.04
2.08
2.15
1.20
1.23
1.26
1.30
1.35
1.38
1.42
1.44
1.46
70
1.50
1.64
1.71
1.77
1.81
1.89
1.96
2.01 ;
2.05 ;
2.12 ;
2.17 ;
2.20 ;
2.23 ;
2.27 ;
2.32 ;
2.35 ;
2.38 ;
2.40 ;
2.43 ;
80
.49
.63
.70
.75
.79
.88
.95
2.00
2.03
2.10
2.15
2.18
2.21
1.25
1.29
1.33
1.35
1.38
1.40
90
1.48
1.62
1.69
1.74
1.78
1.86
1.93
1.98
2.02 ;
2.08 ;
2.13 ;
2.17 ;
2.19 ;
2.23 ;
2.27 ;
2.3i ;
2.34 ;
2.36 ;
2.38 ;
100 125
.48 .47
.61 .60
.68 .67
.73 .72
.77 .76
.85 .84
.92 .90
.97 .95
2.01 .99
1.07 2.05
1.12 2.09
2.15 2.13
2.18 2.16
1.22 2.19
1.26 2.23
1.29 2.27
1.32 2.29
1.34 2.32
1.36 2.33
150
1.46
1.59
1.66
1.71
1.75
1.83
1.89
1.94
1.97
2.04
2.08
2.11
2.14
2.17
2.22
2.25
2.27
2.30
2.32
    Table 19-4. K-Multipliers for Modified Calif. Interwell Prediction Limits on Observations (40 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20 25 30
4.02 2.53 2.09 1.88 1.76 1.63 1.55 .49 .46
4.97 2.97 2.40 2.14 1.99 1.82 1.73 .66 .62
5.56 3.24 "'2.59' 2.29 2.12 1.94 1.83 .76 .71
5.99 3.43 2.73 2.40 2.22 2.02 1.91 .83 .77
6.34 3.58 2.83 2.49 2.29 2.08 1.96 .88 .82
7.06 3.91 3.06 2.87 2.45 2.21 2.08 .99 .93
7.70 4.19 3.26 2.83 2,58" 2.32 2.18 2.08 2.01
8.14 4.40 3.40 2.94 2,68 2.40 2.26 2.14 2.08 ;
8.49 4.56 3.50 3.03 2.76 2.47 2.31 2.19 2.12 ;
9.10 4.83 3.70 3.19 2.90 2,58- 2.41 2.29 2.21 ;
9.53 5.03 3.84 3.30 2.99 2,66 2.48 2.35 2.27 ;
9.86 5.19 3.95 3.39 3.07 2.72 2.S4 2.40 2.31 ;
10.13 5.31 4.03 3.45 3.13 ,2,77 2,58 2.44 2.35 :
10.45 5.46 4.14 3.54 3.20 2.83 2.84 2.49 2.40 ;
10.86 5.65 4.27 3.65 3.30 2.91 2,7? 2,55 2.46 ;
11.17 5.80 4.38 3.74 3.37 2.97 2,76 2,60 2,50 ;
11.43 5.92 4.46 3.80 3.43 3.02 2.8O 2,64 -2,54 ;
11.63 6.02 4.53 3.86 3.48 3.06 2.84 2,67 2-57 .
11.82 6.10 4.59 3.91 3.52 3.10 2.87 2,70 2,60 ,
35
.43
.59
.68
.74
.79
.89
.97
2.03
2.07 ;
2.15 ;
2.21 ;
i.25 :
1.29 :
1.33 ;
2.39 ;
1.43 :
1.47 ;
2,50" ;
2,52 :
40
.41
.57
.65
.71
.76
.85
.94
.99
2.04 ;
Ml ;
2.17 ;
2.21 ;
>.25 :
1.29 ;
2.34 ;
2.38 ;
>.A2 ;
2.45 ;
2.47 ;
45
.40
.55
.63
.69
.74
.83
.91
.97
2.01
2.08 ;
2.14 ;
2.18 ;
1.21 :
i.25 ;
2.3i ;
2.34 ;
2.38 ;
2.4i ;
2.43 ;
50 60 70 80 90
.39 1.37 1.36 1.35 1.35
.54 1.52 1.50 1.49 1.48
.62 1.60 1.58 1.57 1.56
.68 1.65 1.64 1.63 1.62
.72 1.70 1.68 1.67 1.66
.81 1.79 1.77 1.75 1.74
.89 1.86 1.84 1.83 1.81
.95 1.91 1.89 1.88 1.86
.99 1.95 1.93 1.92 1.90
2.06 2.03 2.00 1.98 1.97
2.11 2.08 2.05 2.03 2.02 ;
2.15 2.12 2.09 2.07 2.06 ;
2.19 2.15 2.12 2.10 2.08 ;
2.23 2.19 2.16 2.14 2.12 ;
2.28 2.23 2.20 2.18 2.17 ;
2.32 2.27 2.24 2.22 2.20 ;
2.35 2.30 2.27 2.25 2.23 ;
2.37 2.33 2.30 2.27 2.25 ;
2.40 2.35 2.32 2.29 2.27 ;
100
.34
.48
.56
.61
.65
.73
.81
.85
.89
.96
2.01
2.04 ;
2.07 ;
2.11 ;
2.15 ;
2.19 ;
2.22 ;
2.24 ;
2.26 ;
125
.33
.47
.54
.60
.64
.72
.79
.84
.87
.94
.99
2.02 ;
2.05 ;
2.09 ;
2.13 ;
2.16 ;
2.19 ;
2.21 ;
2.23 ;
150
.32
.46
.54
.59
.63
.71
.78
.83
.86
.93
.97
2.01
2.04
2.07
2.11
2.15
2.17
2.20
2.22
                                                    D-68
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations
Unified Guidance
Table 19-4. K-Multipliers for Modified
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.97
5.99
6.62
7.06
7.41
8.14
8.76
9.20
9.53
10.13
10.54
10.86
11.11
11.43
11.82
12.11
12.35
12.56
12.73
Table 19-4
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.99
7.06
7.70
8.14
8.49
9.20
9.80
10.21
10.54
11.11
11.50
11.82
12.05
12.35
12.73
13.01
13.24
13.44
13.61
6
2.97
3.43
3.71
3.91
4.07
4.40
4.68
4.88
5.03
5.31
5.50
5.65
5.77
5.92
6.10
6.24
6.36
6.46
6.53
8
2.40
2.73
2,92
3.06
3.17
3.40
3.59
3.73
3.84
4.03
4.17
4.27
4.36
4.46
4.59
4.69
4.78
4.84
4.90
10
2.14
2.40
2.56
2.67
2.76
2.94
3.10
3.21
3.30
3.45
3.56
3.65
3.72
3.80
3.91
3.99
4.06
4.12
4.17
. K-Multipliers
6
3.43
3.91
4.19
4.40
4.56
4.88
5.16
5.35
5.50
5.77
5.96
6.10
6.22
6.36
6.53
6.67
6.78
6.88
6.95
8
2.73
3.06
3.26
3.40
3.50
3.73
3.93
4.06
4.17
4.36
4.49
4.59
4.68
4.78
4.90
5.00
5.08
5.15
5.21
10
2.40
2.67
2.83
2.94
3.03
3,21
3.37
3.48
3.56
3.72
3.83
3.91
3.98
4.06
4.17
4.25
4.31
4.37
4.41
12
1.99
2.22
2.35
2.45
2.53
2.68
2.82
2,92
2.99 .
3.13
3.22
3.30
3.36
3.43
3.52
3.59
3.65
3.70
3.75
16
1.82
2.02
2.13
2.21
2.27
2.40
2.52
2.60
2.66
2.77
2.85
2.91
2.96
3,02
3.1O
3.16
3.21
3.25
3.28
Calif. Interwell Prediction Limits on Observations (40 COC, Semi-Annual)
20 25 30 35 40 45 50 60 70 80
1.73 1.66 1.62 1.59 .57 1.55 1.54 .52 1.50 .49
1.91 1.83 1.77 1.74 .71 1.69 1.68 .65 1.64 .63
2.01 1.92 1.86 1.82 .80 1.77 1.76 .73 1.72 .70
2.08 1.99 1.93 1.89 .85 1.83 1.81 .79 1.77 .75
2.14 2.04 1.98 1.93 .90 1.88 1.86 .83 1.81 .79
2.26 2.14 2.08 2.03 .99 1.97 1.95 .91 1.89 .88
2.36 2.24 2.16 2.11 2.07 2.04 2.02 .99 1.96 .95
2.43 2.30 2.22 2.17 2.13 2.10 2.07 2.04 2.01 2.00
2.48 2.35 2.27 2.21 2.17 2.14 2.11 2.08 2.05 2.03
2.58 2.44 2.35 2.29 2.25 2.21 2.19 2.15 2.12 2.10
2.65 2.50 2.41 2.35 2.30 2.27 2.24 2.20 2.17 2.15
2.71 2.55 2.46 2.39 2.34 2.31 2.28 2.23 2.20 2.18
2.75 2.59 2.49 2.42 2.38 2.34 2.31 2.27 2.23 2.21
2,00 2.64 2.54 2.47 2.42 2.38 2.35 2.30 2.27 2.25
2.87' 2.70 2.60 2.52 2.47 2.43 2.40 2.35 2.32 2.29
2.92 2.75 2.64 2.57 2.51 2.47 2.44 2.39 2.35 2.33
2.97 2.79 2.68 2.60 2.54 2.50 2.47 2.42 2.38 2.36
3.0O 2,82 2.71 2.63 2.57 2.53 2.49 2.44 2.41 2.38
3:04 2,85 2.73 2.65 2.60 2.55 2.52 2.46 2.43 2.40
for Modified Calif. Interwell Prediction Limits on Observations
12
2.22
2.45
2.59
2.68
2.76
2.92
3.05
3.15
'3,22
3.36
3.45
3.52
3.58
3.65
3.75
3.82
3.87
3.92
3.96
16
2.02
2.21
2.32
2.40
2.47
2.60
2.71
2.79
2.85
2.96
3.04
3. 1O
3.15
,,3.21
3.28
3,34
3.39
3.43
3.47
20 25 30 35 40 45 50 60 70 80
1.91 1.83 1.77 1.74 1.71 1.69 1.68 .65 1.64 .63
2.08 1.99 1.93 1.89 1.85 1.83 1.81 .79 1.77 .75
2.18 2.08 2.01 1.97 1.94 1.91 1.89 .86 1.84 .83
2.26 2.14 2.08 2.03 1.99 1.97 1.95 .91 1.89 .88
2.31 2.19 2.12 2.07 2.04 2.01 1.99 .95 1.93 .92
2.43 2.30 2.22 2.17 2.13 2.10 2.07 2.04 2.01 2.00
2.53 2.39 2.31 2.25 2.20 2.17 2.15 2.11 2.08 2.06
2.60 2.45 2.36 2.30 2.26 2.22 2.20 2.16 2.13 2.11
2.65 2.50 2.41 2.35 2.30 2.27 2.24 2.20 2.17 2.15
2.75 2.59 2.49 2.42 2.38 2.34 2.31 2.27 2.23 2.21
2.82 2.65 2.55 2.48 2.43 2.39 2.36 2.31 2.28 2.26
2.87 2.70 2.60 2.52 2.47 2.43 2.40 2.35 2.32 2.29
2.92 2.74 2.63 2.56 2.50 2.46 2.43 2.38 2.35 2.32
2.97 2.79 2.68 2.60 2.54 2.50 2.47 2.42 2.38 2.36
3.04 2.85 2.73 2.65 2.60 2.55 2.52 2.46 2.43 2.40
3,09 2.90 2.78 2.70 2.64 2.59 2.55 2.50 2.46 2.43
'3.13 2.94 2.81 2.73 2.67 2.62 2.58 2.53 2.49 2.46
3,77 2.97 2.84 2.76 2.70 2.65 2.61 2.55 2.51 2.49
3.20 3.00 2.87 2.78 2.72 2.67 2.63 2.58 2.54 2.51
90
1.48
1.62
1.69
1.74
1.78
1.86
1.93
1.98
2.02
2.08
2.13
2.17
2.19
2.23
2.27
2.31
2.34
2.36
2.38
100
.48
.61
.68
.73
.77
.85
.92
.97
2.01
2.07
2.12
2.15
2.18
2.22
2.26
2.29
2.32
2.34
2.36
(40 COC,
90
1.62
1.74
1.81
1.86
1.90
1.98
2.05
2.10
2.13
2.19
2.24
2.27
2.30
2.34
2.38
2.41
2.44
2.46
2.48
100
.61
.73
.81
.85
.89
.97
2.04
2.08
2.12
2.18
2.23
2.26
2.29
2.32
2.36
2.40
2.42
2.45
2.47
125
.47
.60
.67
.72
.76
.84
.90
.95
.99
2.05
2.10
2.13
2.16
2.19
2.23
2.27
2.29
2.32
2.34
150
1.46
1.59
1.66
1.71
1.75
1.83
1.89
1.94
1.97
2.04
2.08
2.11
2.14
2.17
2.22
2.25
2.28
2.30
2.32
Quarterly)
125
.60
.72
.79
.84
.87
.95
2.02
2.06
2.10
2.16
2.20
2.23
2.26
2.29
2.34
2.37
2.39
2.42
2.43
150
1.59
1.71
1.78
1.83
1.86
1.94
2.00
2.05
2.08
2.14
2.18
2.22
2.24
2.28
2.32
2.35
2.37
2.40
2.41

                                                      D-69
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Observations                                         Unified Guidance
                                    This page intentionally left blank
                                                      D-70                                              March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means	Unified Guidance





                                          D STATISTICAL TABLES








D.2 TABLES FROM CHAPTER 19:  INTERWELL  PREDICTION  LIMITS FOR FUTURE MEANS








    TABLE 19-5 K-Multipliers for 1 -of-1 Interwell Prediction Limits on Means of Order 2	D-72



    TABLE19-6 /c-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 2	D-81



    TABLE 19-7 /c-Multipliers for 1 -of-3 Interwell Prediction Limits on Means of Order 2	D-90



    TABLE 19-8 /c-Multipliers for 1 -of-1 Interwell Prediction Limits on Means of Order 3	D-99



    TABLE 19-9 K-Multipliers for 1 -of-2 Interwell Prediction Limits on Means of Order 3	D-108
                                                        D-71                                               March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means
Unified Guidance
      Table 19-5. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 2 (1 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
1.42 1.21 1.12 .07 1.04 1.01 (
1.89 1.58 1.45 .38 1.34 1.29
2.17 1.79 1.64 .56 1.51 1.45
2.37 1.94 1.77 .68 1.62 1.56
2.52 2.05 1.87 .77 1.71 1.64
2.83 2.28 2.07 .96 1.89 1.80
3.09 2.48 2.24 2.12 2.04 1.94
3.27 2.61 2.36 2.22 2.14 2.04
3.40 2.71 2.45 2.31 2.22 2.11 ;
3.64 2.89 2.61 2.45 2.36 2.24 ;
381 3.02 2.72 2.55 2.45 2.33 ;
3.93 3.11 2.80 2.63 2.52 2.40 ;
4.03 3.19 2.87 2.69 2.58 2.45 ;
4.15 3.28 2.95 2.77 2.65 2.52 ;
4.31 3.40 3.O5 2.86 2.74 2.60 ;
4.42 3.48 3.13 2.93 2.81 2.66 ;
4.51 3.55 3.19 2.99 2.86 2.71 ;
4.59 3.61 3.24 '"3,O4" 2.91 2.76 ;
4.66 3.66 3.28 3,08 2.95 2.79 ;
20
).98 (
.26
.41
.52
.60
.76
.89
.98
2.05 ;
2.17 ;
1.26 :
1.32 :
1.37 ;
2.43 ;
2.5i ;
2.57 ;
>..62 ;
2.66 ;
2.70 ;
25 30
).97 0.96 (
.24 1.23
.39 1.37
.49 1.47
.56 1.54
.72 1.69
.85 1.82
.93 1.90
2.00 1.97
1.12 2.08 ;
1.20 2.16 ;
1.26 2.22 :
1.31 2.27 ;
2.37 2.33 ;
2.45 2.40 ;
2.51 2.46 ;
2.55 2.50 ;
2.59 2.54 ;
2.62 2.58 ;
35
).95 (
.22
.36
.45
.53
.68
.80
.88
.94
2.06 ;
2.13 ;
2.19 ;
2.24 ;
2.30 ;
2.37 ;
2.43 ;
2.47 ;
2.5i ;
2.54 ;
40 45
).94 0.94 (
.21 1.20
.35 1.34
.44 1.44
.52 1.51
.66 1.65
.78 1.77
.86 1.85
.93 1.91
2.04 2.02 ;
2.11 2.10 ;
2.17 2.16 ;
2.22 2.20 ;
2.28 2.26 ;
2.35 2.33 ;
2.40 2.38 ;
2.44 2.42 ;
2.48 2.46 ;
2.51 2.49 ;
50 60 70
).94 0.93 0.93 (
.20 1.19 1.18
.34 1.33 1.32
.43 1.42 1.41
.50 1.49 1.48
.64 1.63 1.62
.76 1.75 1.74
.84 1.83 1.82
.90 1.89 1.88
2.01 1.99 1.98
2.09 2.07 2.05 ;
2.14 2.12 2.11 ;
2.19 2.17 2.15 ;
2.24 2.22 2.21 ;
2.31 2.29 2.27 ;
2.37 2.34 2.32 ;
2.41 2.38 2.37 ;
2.44 2.42 2.40 ;
2.47 2.45 2.43 ;
80 90
).93 0.92 (
.18 1.18
.32 1.31
.41 1.40
.48 1.47
.62 1.61
.73 1.73
.81 1.80
.87 1.86
.97 1.96
2.04 2.04 ;
2.10 2.09 ;
2.14 2.13 ;
2.19 2.19 ;
2.26 2.25 ;
2.31 2.30 ;
2.35 2.34 ;
2.39 2.38 ;
2.42 2.41 ;
100 125
).92 0.92 (
.18 1.17
.31 1.31
.40 1.40
.47 1.46
.61 1.60
.72 1.71
.80 1.79
.86 1.85
.96 1.95
2.03 2.02 ;
2.08 2.07 ;
2.13 2.11 ;
2.18 2.17 ;
2.24 2.23 ;
2.29 2.28 ;
2.33 2.32 ;
2.37 2.35 ;
2.40 2.38 ;
150
).92
.17
.30
.39
.46
.60
.71
.78
.84
.94
2.01
2.06
2.11
2.16
2.22
2.27
2.31
2.34
2.37
   Table 19-5. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 2 (1 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
1.89 1.58 1.45 1.38 1.34 1.29 .26
2.37 1.94 1.77 1.68 1.62 1.56 .52
2.64 2.14 1.95 1.84 1.78 1.70 .66
2.83 2.28 2.07 1.96 1.89 1.80 .76
2.97 2.39 2.17 2.05 1.97 1.88 .83
3.27 2.61 2.36 2.22 2.14 2.04 .98
351 2.79 2.52 2.37 2.28 2.17 2.10
3.68 2.92 2.63 2.48 2.38 2.26 2.19
381 3.02 2.72 2.55 2.45 2.33 2.26
403 3.19 2.87 2.69 2.58 2.45 2.37
419 331 2.97 2.79 2.67 2.53 2.45
4.31 3.40 3.05 2.86 2.74 2.60 2.51
4.40 3.47 3.11 2.92 2.80 2.65 2.56
451 355 3.19 2.99 2.86 2.71 2.62
4.66 3.66 3.28 3.08 2.95 2.79 2.70
4.77 3.75 3,36 3.15 3.01 2.85 2.76
4.85 3.81 3.42 3.20 3.07 2.90 2.80
4.93 3.87 3.47 3.25 3.11 2.94 2.84
4.99 3.92 3.51 3,29 3.15 2.98 2.88
25
.24
.49
.62
.72
.79
.93
2.05
2.14
2.20
2.31
2.39
2.45
2.49
2.55
2.62
2.68
2.73
2.76
2.79
30
1.23
1.47
1.60
1.69
1.76
1.90
2.02
2.10
2.16
2.27
2.35
2.40
2.45
2.50
2.58
2.63
2.67
2.71
2.74
35
.22
.45
.59
.68
.74
.88
2.00
2.07 ;
2.13 ;
2.24 ;
2.31 ;
2.37 ;
2.42 ;
2.47 ;
2.54 ;
2.59 ;
2.63 ;
2.67 ;
2.70 ;
40
.21
.44
.57
.66
.73
.86
.98
2.06
2.11
2.22
2.29
2.35
2.39
2.44
2.51
2.56
2.61
2.64
2.67
45
1.20
1.44
1.56
1.65
1.72
1.85
1.96
2.04 ;
2.10 ;
2.20 ;
2.27 ;
2.33 ;
2.37 ;
2.42 ;
2.49 ;
2.54 ;
2.58 ;
2.62 ;
2.65 ;
50
.20
.43
.56
.64
.71
.84
.95
2.03
2.09
2.19
2.26
2.31
2.36
2.41
2.47
2.53
2.57
2.60
2.63
60
1.19
1.42
1.55
1.63
1.70
1.83
1.94
2.01
2.07
2.17
2.24
2.29
2.33
2.38
2.45
2.50
2.54
2.57
2.60
70 80
1.18 .18
1.41 .41
1.54 .53
1.62 .62
1.69 .68
1.82 .81
1.92 .92
2.00 .99
2.05 2.04
2.15 2.14
2.22 2.21
2.27 2.26
2.32 2.30
2.37 2.35
2.43 2.42
2.48 2.46
2.52 2.50
2.55 2.54
2.58 2.56
90 100
1.18 .18
1.40 .40
1.53 .52
1.61 .61
1.67 .67
1.80 .80
1.91 .90
1.98 .98
2.04 2.03
2.13 2.13
2.20 2.19
2.25 2.24
2.29 2.28
2.34 2.33
2.41 2.40
2.45 2.44
2.49 2.48
2.52 2.51
2.55 2.54
125 150
1.17 .17
1.40 .39
1.52 .51
1.60 .60
1.66 .66
1.79 .78
1.89 .89
1.96 .96
2.02 2.01
2.11 2.11
2.18 2.17
2.23 2.22
2.27 2.26
2.32 2.31
2.38 2.37
2.43 2.42
2.47 2.45
2.50 2.49
2.52 2.51
                                                   D-72
                                                                                                 March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means
Unified Guidance
     Table 19-5. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 2 (1  COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.37
2.83
3.09
3.27
3.40
3.68
3.91
4.07
4.19
4.40
4.55
4.66
4.75
4.85
4.99
5.09
5.18
5.24
5.30
6
1.94
2.28
2.48
2.61
2.71
2.92
3.10
3.22
3.31
3.47
3.58
3.66
3.73
3.81
3.92
4.00
4.06
4.11
4.16
8
1.77
2.07
2.24
2.36
2.45
2.63
2.78
2.89
2.97
3.11
3.21
3.28
3.34
3.42
3.51
3.58
3.63
3.68
3.72
10
1.68
1.96
2.12
2.22
2.31
2.48
2.62
2.71
2.79
2.92
3.01
3.08
3.13
3.20
3.29
3.35
3.40
3.45
3.49
12
1.62
1.89
2.04
2.14
2.22
2.38
2.51
2.60
2.67
2.80
2.88
2.95
3.00
3.07
3.15
3.21
3.26
3.30
3.34
16
1.56
1.80
1.94
2.04
2.11
2.26
2.38
2.47
2.53
2.65
2.73
2.79
2.84
2.90
2.98
3.03
3.08
3.12
3.15
20
1.52
1.76
1.89
1.98
2.05
2.19
2.31
2.39
2.45
2.56
2.64
2.70
2.75
2.80
2.88
2.93
2.98
3.01
3.04
25
1.49
1.72
1.85
1.93
2.00
2.14
2.25
2.33
2.39
2.49
2.57
2.62
2.67
2.73
2.79
2.85
2.89
2.93
2.96
30
1.47
1.69
1.82
1.90
1.97
2.10
2.21
2.29
2.35
2.45
2.52
2.58
2.62
2.67
2.74
2.79
2.83
2.87
2.90
35
.45
.68
.80
.88
.94
2.07
2.18
2.26
2.31
2.42
2.49
2.54
2.58
2.63
2.70
2.75
2.79
2.83
2.86
40
.44
.66
.78
.86
.93
2.06
2.16
2.24
2.29
2.39
2.46
2.51
2.56
2.61
2.67
2.72
2.76
2.80
2.82
45
1.44
1.65
1.77
1.85
1.91
2.04
2.15
2.22
2.27
2.37
2.44
2.49
2.53
2.58
2.65
2.70
2.74
2.77
2.80
50
.43
.64
.76
.84
.90
2.03
2.13
2.20
2.26
2.36
2.42
2.47
2.52
2.57
2.63
2.68
2.72
2.75
2.78
60
1.42
1.63
1.75
1.83
1.89
2.01
2.11
2.18
2.24
2.33
2.40
2.45
2.49
2.54
2.60
2.65
2.69
2.72
2.75
70
1.41
1.62
1.74
1.82
1.88
2.00
2.10
2.17
2.22
2.32
2.38
2.43
2.47
2.52
2.58
2.63
2.67
2.70
2.72
80
.41
.62
.73
.81
.87
.99
2.09
2.16
2.21
2.30
2.37
2.42
2.46
2.50
2.56
2.61
2.65
2.68
2.71
90
1.40
1.61
1.73
1.80
1.86
1.98
2.08
2.15
2.20
2.29
2.36
2.41
2.44
2.49
2.55
2.60
2.64
2.67
2.69
100
.40
.61
.72
.80
.86
.98
2.07
2.14
2.19
2.28
2.35
2.40
2.44
2.48
2.54
2.59
2.63
2.66
2.68
125
1.40
1.60
1.71
1.79
1.85
1.96
2.06
2.13
2.18
2.27
2.33
2.38
2.42
2.47
2.52
2.57
2.61
2.64
2.66
150
.39
.60
.71
.78
.84
.96
2.05
2.12
2.17
2.26
2.32
2.37
2.41
2.45
2.51
2.56
2.59
2.62
2.65
      Table 19-5. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 2 (2 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
2.01 1.63 1.48 1.41 1.36 1.30 .27
2.57 2.02 1.82 1.71 1.65 1.57 .53
2.90 2.25 2.01 1.89 1.81 1.72 .67
3.14 2.41 2.15 2.01 1.93 1.83 .77
3.33 2.54 2.25 2.10 2.01 1.91 .85
371 2.79 2.47 2.30 2.19 2.07 2.00
4.03 3.01 2.65 2.46 2.35 2.21 2.13
4.25 3.16 2.78 2.57 2.45 2.31 2.22
4.42 3.28 2.87 2.66 2.53 2.38 2.29
4.72 3.48 3,04 2.82 2.68 2.51 2.42
4.93 3.63 3.16 2.92 2.77 2.60 2.50
5.08 3.74 3.26 3. 01, 2.85 2.67 2.57
5.21 3.82 3.33 3,07: 2.91 2.73 2.62
5.36 3.93 3.42 3.15 . 2,99 2.79 2.68
5.55 4.06 3.53 3.25 3,O8 2.88 2.76
5.70 4.16 3.61 3.33 -.3.75 2.94 2.82
5.82 4.24 3.68 3.39 3.21 3,00 2.87
5.91 4.31 3.74 3.44 3.26 3.O4 2.92
6.00 4.37 3.79 3.49 3.30 3.O8 2.95
25
.25
.50
.64
.73
.80
.95
2.07
2.16
2.23
2.34
2.42
2.48
2.53
2.59
2.67
2.73
2.78
2.82
2.85
30
1.23
1.48
1.61
1.70
1.77
1.92
2.04
2.12
2.18
2.29
2.37
2.43
2.48
2.54
2.61
2.67
2.71
2.75
2.78
35
.22
.46
.59
.68
.75
.89
2.01
2.09 ;
2.15 ;
2.26 ;
2.34 ;
2.39 ;
2.44 ;
2.50 ;
2.57 ;
2.62 ;
2.67 ;
2.70 ;
2.74 ;
40
.21
.45
.58
.67
.74
.87
.99
2.07
2.13
1.23
1.31
1.37
2.41
2.47
2.54
2.59
2.63
2.67
2.70
45
1.20
1.44
1.57
1.66
1.72
1.86
1.97
2.05 ;
2.11 ;
2.22 ;
2.29 ;
2.34 ;
2.39 ;
2.44 ;
2.5i ;
2.56 ;
2.6i ;
2.64 ;
2.67 ;
50
.20
.43
.56
.65
.72
.85
.96
2.04
MO
1.20
1.27
1.33
1.37
2.42
2.49
2.54
2.59
2.62
2.65
60
1.19
1.42
1.55
1.64
1.70
1.83
1.94
2.02
2.08
2.18
2.25
2.30
2.34
2.40
2.46
2.51
2.55
2.59
2.62
70 80
1.19 .18
1.42 .41
1.54 .54
1.63 .62
1.69 .68
1.82 .81
1.93 .92
2.00 .99
2.06 2.05
2.16 2.15
2.23 2.22
2.28 2.27
2.32 2.31
2.38 2.36
2.44 2.43
2.49 2.47
2.53 2.51
2.56 2.55
2.59 2.58
90 100
1.18 .18
1.41 .40
1.53 .53
1.61 .61
1.68 .67
1.81 .80
1.91 .91
1.99 .98
2.04 2.03
2.14 2.13
2.21 2.20
2.26 2.25
2.30 2.29
2.35 2.34
2.41 2.40
2.46 2.45
2.50 2.49
2.53 2.52
2.56 2.55
125 150
1.17 .17
1.40 .39
1.52 .52
1.60 .60
1.67 .66
1.79 .79
1.90 .89
1.97 .96
2.02 2.01
2.12 2.11
2.18 2.17
2.23 2.22
2.27 2.26
2.32 2.31
2.39 2.37
2.43 2.42
2.47 2.46
2.50 2.49
2.53 2.52
                                                    D-73
                                                                                                  March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-5. K- Multipliers
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.57
3.14
3.47
3.71
3.88
4.25
4.56
4.77
4.93
5.21
5.40
5.55
5.67
5.82
6.00
6.13
6.24
6.34
6.42
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.14
3.71
4.03
4.25
4.42
4.77
5.06
5.25
5.40
5.67
5.86
6.00
6.10
6.24
6.42
6.55
6.65
6.74
6.82
6
2.02
2.41
2.64
2.79
2.91
3.16
3.37
3.52
3.63
3.82
3.96
4.06
4.14
4.24
4.37
4.47
4.54
4.61
4.66
9-5.
6
2.41
2.79
3.01
3.16
3.28
3.52
3.72
3.85
3.96
4.14
4.27
4.37
4.45
4.54
4.66
4.76
4.83
4.89
4.94
8
1.82
2.15
2.34
2.47
2.57
2.78
2.95
3.07
3.16
3,33
3.44
3.53
3.60
3.68
3.79
3.87
3.94
3.99
4.04
10
1.71
2.01
2.18
2.30
2.39
2.57
2.73
2.84
2.92
3.07
3.17
3.25
"&'i?"
3,39
3.49
3.56
3.62
3.67
3.72
for 1
12
1.65
1.93
2.08
2.19
2.28
2.45
2.60
2.70
2.77
2.91
3.01
3.08
3.14
3.21
3,30
3.37
3,43
3.47
3.51
K-Multipliers for
8
2.15
2.47
2.65
2.78
2.87
3.07
3.24
3.35
3.44
3.6O
3.71
3.79
3.86
3.94
4.04
4.12
4.18
4.23
4.28
10
2.01
2.30
2.46
2.57
2.66
2.84
2.99
3.09
3.17
3.32
3.41
3.49
3.55
3,62
3.72
3.79
3.84
3.89
3.93
12
1.93
2.19
2.35
2.45
2.53
2.70
2.84
2.93
3.01
3.14
3.23
3.30
3.36
3.43
3.51
3.58
3.63
3,68
3,72
-of-1 Interwell Prediction Limits on Means of Order 2 (2 COC, Semi-Annual)
16
1.57
1.83
1.97
2.07
2.15
2.31
2.44
2.53
2.60
2.73
2.81
2.88
2.93
3.00
3.08
3.14
3.19
3.24
3.27
20 25 30 35 40 45 50 60 70 80
1.53 1.50 1.48 .46 .45 1.44 .43 1.42 1.42 .41
1.77 1.73 1.70 .68 .67 1.66 .65 1.64 1.63 .62
1.91 1.86 1.83 .81 .79 1.78 .77 1.75 1.74 .73
2.00 1.95 1.92 .89 .87 1.86 .85 1.83 1.82 .81
2.08 2.02 1.98 .96 .94 1.92 .91 1.89 1.88 .87
2.22 2.16 2.12 2.09 2.07 2.05 2.04 2.02 2.00 .99
2.35 2.28 2.23 2.20 2.18 2.16 2.14 2.12 2.11 2.09
2.43 2.36 2.31 2.28 2.25 2.23 2.22 2.19 2.18 2.16
2.50 2.42 2.37 2.34 2.31 2.29 2.27 2.25 2.23 2.22
2.62 2.53 2.48 2.44 2.41 2.39 2.37 2.34 2.32 2.31
2.70 2.61 2.55 2.51 2.48 2.46 2.44 2.41 2.39 2.38
2.76 2.67 2.61 2.57 2.54 2.51 2.49 2.46 2.44 2.43
2.81 2.72 2.66 2.61 2.58 2.55 2.53 2.50 2.48 2.47
2.87 2.78 2.71 2.67 2.63 2.61 2.59 2.55 2.53 2.51
2.95 2.85 2.78 2.74 2.70 2.67 2.65 2.62 2.59 2.58
3.01 2.91 2.84 2.79 2.75 2.72 2.70 2.67 2.64 2.62
3.06 2.95 2.88 2.83 2.79 2.76 2.74 2.71 2.68 2.66
3.10 2.99 2.92 2.87 2.83 2.80 2.77 2.74 2.71 2.69
3.13 3.02 2.95 2.90 2.86 2.83 2.80 2.77 2.74 2.72
1-of-1 Interwell Prediction Limits on Means of Order 2
16
1.83
2.07
2.21
2.31
2.38
2.53
2.66
2.75
2.81
2.93
3.02
3.08
3.13
3.19
3.27
3.33
3.38
3.42
3.46
20 25 30 35 40 45 50 60 70 80
1.77 1.73 1.70 1.68 1.67 1.66 1.65 1.64 1.63 1.62
2.00 1.95 1.92 1.89 1.87 1.86 1.85 1.83 1.82 1.81
2.13 2.07 2.04 2.01 1.99 1.97 1.96 1.94 1.93 1.92
2.22 2.16 2.12 2.09 2.07 2.05 2.04 2.02 2.00 1.99
2.29 2.23 2.18 2.15 2.13 2.11 2.10 2.08 2.06 2.05
2.43 2.36 2.31 2.28 2.25 2.23 2.22 2.19 2.18 2.16
2.55 2.47 2.42 2.38 2.36 2.33 2.32 2.29 2.27 2.26
2.64 2.55 2.50 2.46 2.43 2.40 2.39 2.36 2.34 2.33
2.70 2.61 2.55 2.51 2.48 2.46 2.44 2.41 2.39 2.38
2.81 2.72 2.66 2.61 2.58 2.55 2.53 2.50 2.48 2.47
2.89 2.79 2.73 2.68 2.65 2.62 2.60 2.57 2.55 2.53
2.95 2.85 2.78 2.74 2.70 2.67 2.65 2.62 2.59 2.58
3.00 2.90 2.83 2.78 2.74 2.71 2.69 2.66 2.63 2.61
3.06 2.95 2.88 2.83 2.79 2.76 2.74 2.71 2.68 2.66
3.13 3.02 2.95 2.90 2.86 2.83 2.80 2.77 2.74 2.72
3.19 3.08 3.00 2.95 2.91 2.88 2.85 2.81 2.79 2.77
3.24 3.12 3.04 2.99 2.95 2.92 2.89 2.85 2.82 2.80
3.28 3.16 3.08 3.02 2.98 2.95 2.92 2.88 2.85 2.83
3.31 3.19 3.11 3.05 3.01 2.98 2.95 2.91 2.88 2.86
90
1.41
1.61
1.73
1.81
1.87
1.99
2.09
2.15
2.21
2.30
2.36
2.41
2.45
2.50
2.56
2.61
2.65
2.68
2.70
(2 COC
90
1.61
1.81
1.91
1.99
2.04
2.15
2.25
2.31
2.36
2.45
2.51
2.56
2.60
2.65
2.70
2.75
2.79
2.82
2.84
100
.40
.61
.72
.80
.86
.98
2.08
2.15
2.20
2.29
2.35
2.40
2.44
2.49
2.55
2.60
2.63
2.67
2.69
125
1.40
1.60
1.71
1.79
1.85
1.97
2.07
2.13
2.18
2.27
2.34
2.39
2.42
2.47
2.53
2.58
2.61
2.64
2.67
150
.39
.60
.71
.79
.84
.96
2.06
2.12
2.17
2.26
2.33
2.37
2.41
2.46
2.52
2.56
2.60
2.63
2.65
, Quarterly)
100
1.61
1.80
1.91
1.98
2.03
2.15
2.24
2.31
2.35
2.44
2.50
2.55
2.59
2.63
2.69
2.74
2.77
2.80
2.83
125
1.60
1.79
1.90
1.97
2.02
2.13
2.23
2.29
2.34
2.42
2.48
2.53
2.57
2.61
2.67
2.71
2.75
2.78
2.80
150
1.60
1.79
1.89
1.96
2.01
2.12
2.22
2.28
2.33
2.41
2.47
2.52
2.55
2.60
2.65
2.70
2.73
2.76
2.79

                                                      D-74
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table '
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.97
3.68
4.12
4.43
4.68
5.19
5.62
5.92
6.15
6.55
6.83
7.04
7.21
7.42
7.68
7.88
8.04
8.17
8.28
Table 19-
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.68
4.43
4.88
5.19
5.42
5.92
6.33
6.61
6.83
7.21
7.48
7.68
7.84
8.04
8.28
8.47
8.62
8.75
8.85
19-5
6
2.22
2.66
2.92
3.10
3.25
3.55
3.81
3.99
4.12
4.37
4.54
4.67
4.77
4.90
5.06
5.18
5.28
5.36
5.43
K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 2
8
1.97
2.32
2.52
2.67
2.78
3.O2 .
3.22
3.36
3.47
3.66
3.80
3.90
3.98
4.08
4.21
4.31
4.39
4.45
4.51
10
1.84
2.15
2.33
2.45
2.55
2.75
2.93
3. OS
3, 14.
3.31
3.43
3.52
3.59
3.68
3.79
3.88
3.95
4.00
4.05
5. K-Multipliers
6
2.66
3.10
3.37
3.55
3.69
3.99
4.23
4.41
4.54
4.77
4.94
5.06
5.16
5.28
5.43
5.55
5.64
5.72
5.79
8
2.32
2.67
2.87
3.02
3.13
3.36
3.56
3.69
3.80
3.98
4.11
4.21
4.29
4.39
4.51
4.61
4.68
4.74
4.80
10
2.15
2.45
2.63
2.75
2.85
3.05
3.22
""3,'3"4"
3.43
3.59
3.70
3.79
3.86
3.95
4.05
4.14
4.20
4.26
4.31
12
1.76
2.04
2.21
2.32
2.41
2.60
2.75
2.86
2.95
3,10
3.21
3.29
3.36
3.44
3.54
3.62
3.68
3.73
3.78
for '
12
2.04
2.32
2.48
2.60
2.68
2.86
3.02
3.13
3.21
3.3S
3.46
3.54
3.60
3.68
3.78
3.85
3.91
3.96
4.01
16
1.67
1.93
2.07
2.18
2.25
2.42
2.56
2.65
2.73
2.86
2.96
3.O3
. 3,09
3,16
3.25
3.31
3.37
3.42
3.46
20
1.62
1.86
2.00
2.09
2.17
2.32
2.45
2.54
2.61
2.73
2.81
2.88
2.93
3,00
3.O8
3.15
3.20
3,24
3.28
25
1.58
1.81
1.94
2.03
2.10
2.24
2.36
2.45
2.51
2.63
2.71
2.77
2.82
2.88
2.90
3.O2
3.O6
3.10
3,14
1-of-1 Interwell
16
1.93
2.18
2.32
2.42
2.49
2.65
2.79
2.88
2.96
3.09
3.18
3.25
3, 3O
3.37 .
3,46
3,52
3.58
3.62
3.66
20
1.86
2.09
2.23
2.32
2.39
2.54
2.66
2.75
2.81
2.93
3.02
3.08
3.13
3.20
3.28
3,34
3,39
3,43
3.47
25
1.81
2.03
2.16
2.24
2.31
2.45
2.56
2.64
2.71
2.82
2.90
2.96
3.00
3.06
3.14
3.20
3.24
3,28
3,31
30 35 40 45 50 60 70 80
1.56 1.54 1.53 1.52 .51 1.50 1.49 .48
1.78 1.76 1.74 1.73 .72 1.70 1.69 .69
1.91 1.88 1.86 1.85 .84 1.82 1.81 .80
1.99 1.96 1.94 1.93 .92 1.90 1.88 .88
2.06 2.03 2.01 1.99 .98 1.96 1.94 .93
2.20 2.16 2.14 2.12 2.10 2.08 2.06 2.05
2.31 2.27 2.25 2.22 2.21 2.18 2.17 2.15
2.39 2.35 2.32 2.30 2.28 2.25 2.23 2.22
2.45 2.41 2.38 2.35 2.34 2.31 2.29 2.27
2.56 2.51 2.48 2.45 2.43 2.40 2.38 2.37
2.64 2.59 2.55 2.53 2.50 2.47 2.45 2.43
2.70 2.65 2.61 2.58 2.56 2.52 2.50 2.48
2.74 2.69 2.65 2.62 2.60 2.56 2.54 2.52
2.80 2.75 2.71 2.67 2.65 2.61 2.59 2.57
2.87 2.82 2.77 2.74 2.72 2.68 2.65 2.63
2.93 2.87 2.83 2.79 2.77 2.73 2.70 2.68
.''2'.98" 2.92 2.87 2.84 2.81 2.77 2.74 2.72
3.01 2.95 2.91 2.87 2.84 2.80 2.77 2.75
3,05 2.98 2.94 2.90 2.87 2.83 2.80 2.78
Prediction Limits on Means of Order 2 (5
30 35 40 45 50 60 70 80
1.78 1.76 1.74 1.73 1.72 1.70 1.69 1.69
1.99 1.96 1.94 1.93 1.92 1.90 1.88 1.88
2.11 2.08 2.06 2.04 2.03 2.01 1.99 1.98
2.20 2.16 2.14 2.12 2.10 2.08 2.06 2.05
2.26 2.22 2.20 2.18 2.16 2.14 2.12 2.11
2.39 2.35 2.32 2.30 2.28 2.25 2.23 2.22
2.50 2.46 2.42 2.40 2.38 2.35 2.33 2.32
2.58 2.53 2.50 2.47 2.45 2.42 2.40 2.38
2.64 2.59 2.55 2.53 2.50 2.47 2.45 2.43
2.74 2.69 2.65 2.62 2.60 2.56 2.54 2.52
2.82 2.76 2.72 2.69 2.67 2.63 2.60 2.58
2.87 2.82 2.77 2.74 2.72 2.68 2.65 2.63
2.92 2.86 2.82 2.78 2.76 2.72 2.69 2.67
2.98 2.92 2.87 2.84 2.81 2.77 2.74 2.72
3.05 2.98 2.94 2.90 2.87 2.83 2.80 2.78
3.10 3.04 2.99 2.95 2.92 2.88 2.84 2.82
3.15 3.08 3.03 2.99 2.96 2.91 2.88 2.86
3.18 3.12 3.06 3.02 2.99 2.95 2.91 2.89
3.22 3.15 3.09 3.05 3.02 2.97 2.94 2.91
(5 COC, Annual)
90
1.48
1.68
1.79
1.87
1.93
2.04
2.14
2.21
2.26
2.35
2.42
2.47
2.51
2.55
2.61
2.66
2.70
2.73
2.76
COC,
90
1.68
1.87
1.97
2.04
2.10
2.21
2.30
2.37
2.42
2.51
2.57
2.61
2.65
2.70
2.76
2.80
2.84
2.87
2.90
100
.47
.67
.79
.86
.92
2.04
2.13
2.20
2.25
2.34
2.41
2.46
2.50
2.54
2.60
2.65
2.69
2.72
2.74
125
1.47
1.67
1.78
1.85
1.91
2.02
2.12
2.19
2.24
2.33
2.39
2.44
2.48
2.52
2.58
2.63
2.66
2.69
2.72
150
.46
.66
.77
.84
.90
2.02
2.11
2.18
2.23
2.32
2.38
2.42
2.46
2.51
2.57
2.61
2.65
2.68
2.70
Semi-Annual)
100
1.67
1.86
1.97
2.04
2.09
2.20
2.29
2.36
2.41
2.50
2.56
2.60
2.64
2.69
2.74
2.79
2.82
2.85
2.88
125
1.67
1.85
1.95
2.02
2.08
2.19
2.28
2.34
2.39
2.48
2.53
2.58
2.62
2.66
2.72
2.76
2.80
2.83
2.85
150
1.66
1.84
1.95
2.02
2.07
2.18
2.27
2.33
2.38
2.46
2.52
2.57
2.60
2.65
2.70
2.74
2.78
2.81
2.83

                                                     D-75
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-5. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.43
5.19
5.62
5.92
6.15
6.61
7.00
7.27
7.48
7.84
8.09
8.28
8.43
8.62
8.85
9.03
9.18
9.29
9.40
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.86
4.75
5.29
5.68
5.99
6.62
7.16
7.54
7.83
8.33
8.69
8.95
9.17
9.43
9.76
10.01
10.21
10.37
10.52
6 8 10 12 16 20 25 30 35 40
3.10 2.67 2.45 2.32 2.18 2.09 2.03 1.99 1.96 1.94
355 3.02 2.75 2.60 2.42 2.32 2.24 2.20 2.16 2.14
381 3.22 2.93 2.75 2.56 2.45 2.36 2.31 2.27 2.25
3.99 3.36 3.05 2.86 2.65 2.54 2.45 2.39 2.35 2.32
4.12 3.47 3.14 2.95 2.73 2.61 2.51 2.45 2.41 2.38
4.41 3.69 3.34 3.13 2.88 2.75 2.64 2.58 2.53 2.50
465 388 3.50 3.28 3.01 2.87 2.76 2.69 2.63 2.60
4.81 4.01 3,62 3.38 3.11 2.95 2.84 2.76 2.71 2.67
494 411 370 3.46 3.18 3.02 2.90 2.82 2.76 2.72
5.16 4.29 3.86 3.6O 3.30 3.13 3.00 2.92 2.86 2.82
5.32 4.42 3.97 3, JO 3.39 3.21 3.08 2.99 2.93 2.89
5.43 4.51 4.05 3.78 3.46 3.28 3.14 3.05 2.98 2.94
5.53 4.59 4.12 3.84 3.51 3.33 3.19 3.09 3.03 2.98
5.64 4.68 4.20 3.91 3,58 3.39 3.24 3.15 3.08 3.03
5.79 4.80 4.31 4.01 3,66 3.47 3.31 3.22 3.15 3.09
5.90 4.89 4.38 4.08 3.73 3.53 3.37 3.27 3.20 3.14
5.99 4.96 4.45 4.14 3, 78 3.57 3.42 3.31 3.24 3.18
6.07 5.02 4.50 4.19 3,82 3,62 3.45 3.35 3.27 3.22
6.13 5.07 4.55 4.23 3.86 3,65 3.49 3.38 3.30 3.25
45
1.93
2.12
2.22
2.30
2.35
2.47
2.57
2.64
2.69
2.78
2.85
2.90
2.94
2.99
3.05
3.10
3.14
3.17
3.20
50
1.92
2.10
2.21
2.28
2.34
2.45
2.55
2.61
2.67
2.76
2.82
2.87
2.91
2.96
3.02
3.07
3.11
3.14
3.17
60
1.90
2.08
2.18
2.25
2.31
2.42
2.51
2.58
2.63
2.72
2.78
2.83
2.87
2.91
2.97
3.02
3.06
3.09
3.12
9-5. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of
6 8 10 12 16 20 25 30 35 40
2.71 2.34 2.16 2.06 1.93 1.87 1.82 1.78 1.76 1.74
320 2.71 2.48 2.34 2.19 2.10 2.04 2.00 1.97 1.95
349 2.93 2.66 2.51 2.33 2.24 2.16 2.12 2.08 2.06
3.69 3.O9 2.80 2.62 2.43 2.33 2.25 2.20 2.17 2.14
386 321 2.90 2.72 2.51 2.40 2.32 2.26 2.23 2.20
4.20 3.46 3.11 2.91 2.68 2.55 2.46 2.40 2.35 2.32
4.49 3.68 3.29 3.O7 2.82 2.68 2.57 2.51 2.46 2.43
4.69 3.83 3.42 3.18 2.91 2.77 2.66 2.59 2.54 2.50
4.85 3.95 3.52 3.27 2.99. 2.84 2.72 2.65 2.60 2.56
5.13 4.17 3.70 3.43 3,13 2,96 2.84 2.76 2.70 2.66
5.33 4.31 3.83 3.54 3.22 3.05 2.92 2.83 2.77 2.73
5.48 4.43 3.92 3.63 3.30 3.12 2,98 2.89 2.83 2.78
5.60 4.52 4.00 3.70 3.36 3.16 3,03 2.94 2.81 2.83
5.74 4.63 4.10 3.78 3.43 3,24 3,09, 2,99 2.93 2.88
5.93 4.77 4.22 3.89 3.52 3.32 3.17 3,07 3.OO 2,95-
6.07 4.88 4.31 3.97 3.59 3.39 3.23 .3,12 3.05 3.OO
6.18 4.97 4.38 4.04 3.65 3.44 3.27 3.17, 3,10 3.04
6.28 5.04 4.45 4.10 3.70 3.48 3.32 3,21 3,13 3.08
6.36 5.10 4.50 4.15 3.74 3.52 3.35 3:24 '. 3,16 3,11
45
1.73
1.93
2.04
2.12
2.18
2.30
2.40
2.48
2.53
2.63
2.70
2.75
2.79
2.84
2.91
2,96
3. OO
3,04
3.07
50
1.72
1.92
2.03
2.10
2.16
2.28
2.38
2.45
2.51
2.60
2.67
2.72
2.76
2.81
2.88
2.93
2,97
3.00
3.O3
60
1.71
1.90
2.01
2.08
2.14
2.26
2.35
2.42
2.47
2.57
2.63
2.68
2.72
2.77
2.83
2.88
2.92
'2,95'
2,98
Order 2 (5 COC, Quarterly)
70
1.88
2.06
2.17
2.23
2.29
2.40
2.49
2.55
2.60
2.69
2.75
2.80
2.84
2.88
2.94
2.99
3.02
3.05
3.08
80
1.88
2.05
2.15
2.22
2.27
2.38
2.47
2.53
2.58
2.67
2.73
2.78
2.81
2.86
2.91
2.96
2.99
3.02
3.05
Order 2
70
1.69
1.89
1.99
2.07
2.12
2.24
2.33
2.40
2.45
2.54
2.61
2.65
2.69
2.74
2.80
2.85
2.89
2.92
2,95
80
1.69
1.88
1.98
2.05
2.11
2.22
2.32
2.38
2.43
2.52
2.59
2.63
2.67
2.72
2.78
2.82
2.86
2.89
2.92
90
1.87
2.04
2.14
2.21
2.26
2.37
2.46
2.52
2.57
2.65
2.71
2.76
2.79
2.84
2.90
2.94
2.97
3.00
3.03
100
1.86
2.04
2.13
2.20
2.25
2.36
2.45
2.51
2.56
2.64
2.70
2.74
2.78
2.82
2.88
2.92
2.96
2.99
3.01
125
1.85
2.02
2.12
2.19
2.24
2.34
2.43
2.49
2.53
2.62
2.67
2.72
2.75
2.80
2.85
2.89
2.93
2.96
2.98
150
1.84
2.02
2.11
2.18
2.23
2.33
2.42
2.48
2.52
2.60
2.66
2.70
2.74
2.78
2.83
2.87
2.91
2.94
2.96
(10 COC, Annual)
90
1.68
1.87
1.97
2.04
2.10
2.21
2.31
2.37
2.42
2.51
2.57
2.62
2.66
2.70
2.76
2.81
2.84
2.87
2.90
100
1.68
1.86
1.97
2.04
2.09
2.20
2.30
2.36
2.41
2.50
2.56
2.60
2.64
2.69
2.75
2.79
2.83
2.86
2.88
125
1.67
1.85
1.95
2.02
2.08
2.19
2.28
2.34
2.39
2.48
2.54
2.58
2.62
2.66
2.72
2.76
2.80
2.83
2.85
150
1.66
1.84
1.95
2.02
2.07
2.18
2.27
2.33
2.38
2.46
2.52
2.57
2.60
2.65
2.70
2.75
2.78
2.81
2.83

                                                     D-76
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-5. K- Multipliers for 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.75
5.68
6.23
6.62
6.92
7.54
8.06
8.42
8.69
9.17
9.50
9.76
9.96
10.21
10.52
10.76
10.95
11.10
11.24
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.68
6.62
7.16
7.54
7.83
8.42
8.91
9.25
9.50
9.96
10.28
10.52
10.71
10.95
11.24
11.47
11.65
11.80
11.93
6
8
3.20 2.71
3.69
3.99
4.20
4.36
4.69
4.98
5.18
5.33
5.60
5.78
5.93
6.04
6.18
6.36
6.49
6.60
6.69
6.77
9-5.
6
3.69
4.20
4.49
4.69
4.85
5.18
5.45
5.64
5.78
6.04
6.22
6.36
6.47
6.60
6.77
6.90
7.00
7.09
7.16
3.09
3,31
3.46
3.58
3.83
4.05
4.20
4.31
4.52
4.66
4.77
4.86
4.97
5.10
5.21
5.29
5.36
5.42
10
2.48
2.80
2.98
3.11
3.21
3.42
3.60
3.73
3.83
4.00
4.12
4.22
4.29
4.38
4.50
4.59
4.66
4.72
4.77
K-Multipliers
8
3.09
3.46
3.68
3.83
3.95
4.20
4.41
4.55
4.66
4.86
5.00
5.10
5.19
5.29
5.42
5.52
5.60
5.67
5.73
10
2.80
3.11
3.29
3.42
3.52
3.73
3.91
4.03
4.12
4.29
4.41
4.50
4.57
4.66
4.77
4.86
4.93
4.99
5.04
12
2.34
2.62
2.79
2.91
3.00
3.18
3,34
3,46
3.54
3.70
3.81
3.89
3.96
4.04
4.15
4.23
4.29
4.35
4.39
for
12
2.62
2.91
3.07
3.18
3.27
3.46
'3,6'i"
3.72
3.81
3.96
4.06
4.15
4.21
4.29
4.39
4.47
4.53
4.58
4.63
-of-1
16
2.19
2.43
2.58
2.68
2.75
2.91
3.05
3.15
3.22
3,36
3,45
'3,52
3.58
3.65
3.74
3.81
3.87
3.92
3.96
Interwell Prediction Limits on Means of Order 2 (10 COC, Semi-Annual)
20
2.10
2.33
2.46
2.55
2.62
2.77
2.89
2.98
3.05
3.16
3.25
3.32
3.37 ,
. 3,44
3,52
3.58
3.64
3.68
3.72
25
2.04
2.25
2.37
2.46
2.52
2.66
2.77
2.85
2.92
3.03
3.11
3.17
3.22
3,27.
. 3. 35
3,41
3.46
3,50 .
3.53
1-of-1 Interwell
16
2.43
2.68
2.82
2.91
2.99
3.15
3.28
3.38
3.45
3,58
3,67
3.74
3.8O
3.87
3.96
4.02
4.08
4.12
4.16
20
2.33
2.55
2.68
2.77
2.84
2.98
3.10
3.19
3.25
3.37
3.46
3.52
3.57
3,64
3.72
3,78
3.83
3.87
3.90
25
2.25
2.46
2.57
2.66
2.72
2.85
2.97
3.05
3.11
3.22
3.29
3.35
3.40
3.46
3.53
.3,59
3,63
3.67
3.71 .
30
2.00
2.20
2.32
2.40
2.46
2.59
2.70
2.77
2.83
2.94
3.01
3.07
3.11
3.17
3.24
'3.30'
3.34
3,38
3.41
35 40 45 50 60 70 80 90 100 125
1.97 1.95 1.93 1.92 1.90 1.89 1.88 1.87 1.86 1.85
2.17 2.14 2.12 2.10 2.08 2.07 2.05 2.04 2.04 2.02
2.28 2.25 2.23 2.21 2.18 2.17 2.15 2.14 2.14 2.12
2.35 2.32 2.30 2.28 2.26 2.24 2.22 2.21 2.20 2.19
2.41 2.38 2.36 2.34 2.31 2.29 2.27 2.26 2.25 2.24
2.54 2.50 2.48 2.45 2.42 2.40 2.38 2.37 2.36 2.34
2.64 2.60 2.57 2.55 2.52 2.49 2.47 2.46 2.45 2.43
2.72 2.68 2.64 2.62 2.58 2.56 2.54 2.52 2.51 2.49
2.77 2.73 2.70 2.67 2.63 2.61 2.59 2.57 2.56 2.54
2.81 2.83 2.79 2.76 2.72 2.69 2.67 2.66 2.64 2.62
2.94 2.90 2.86 2.83 2.79 2.76 2.73 2.71 2.70 2.68
3.00 2.95 2.91 2.88 2.83 2.80 2.78 2.76 2.75 2.72
3.04 2.99 2.95 2.92 2.87 2.84 2.82 2.80 2.78 2.76
3.10 3.04 3.00 2.97 2.92 2.89 2.86 2.84 2.83 2.80
3.16 3.11 3.07 3.03 2.98 2.95 2.92 2.90 2.88 2.85
3.22 3.16 3.12 3.08 3.03 2.99 2.96 2.94 2.93 2.90
3.26 3.20 3.16 3.12 3.07 3.03 3.00 2.98 2.96 2.93
3,30 3.24 3.19 3.15 3.10 3.06 3.03 3.01 2.99 2.96
3.33 3,27 3.22 3.18 3.12 3.09 3.06 3.03 3.02 2.98
150
1.84
2.02
2.11
2.18
2.23
2.33
2.42
2.48
2.52
2.60
2.66
2.70
2.74
2.78
2.83
2.88
2.91
2.94
2.96
Prediction Limits on Means of Order 2 (10 COC, Quarterly)
30
2.20
2.40
2.51
2.59
2.65
2.77
2.88
2.95
3.01
3.11
3.19
3.24
3.29
3.34
3.41
3.46
3.51
3.54
.'3.58'
35 40 45 50 60 70 80 90 100 125
2.17 2.14 2.12 2.10 2.08 2.07 2.05 2.04 2.04 2.02
2.35 2.32 2.30 2.28 2.26 2.24 2.22 2.21 2.20 2.19
2.46 2.43 2.40 2.38 2.35 2.33 2.32 2.31 2.30 2.28
2.54 2.50 2.48 2.45 2.42 2.40 2.38 2.37 2.36 2.34
2.60 2.56 2.53 2.51 2.47 2.45 2.43 2.42 2.41 2.39
2.72 2.68 2.64 2.62 2.58 2.56 2.54 2.52 2.51 2.49
2.82 2.77 2.74 2.71 2.67 2.65 2.62 2.61 2.60 2.57
2.89 2.84 2.81 2.78 2.74 2.71 2.69 2.67 2.66 2.63
2.94 2.90 2.86 2.83 2.79 2.76 2.73 2.71 2.70 2.68
3.04 2.99 2.95 2.92 2.87 2.84 2.82 2.80 2.78 2.76
3.11 3.06 3.02 2.98 2.93 2.90 2.87 2.85 2.84 2.81
3.16 3.11 3.07 3.03 2.98 2.95 2.92 2.90 2.88 2.85
3.21 3.15 3.11 3.07 3.02 2.98 2.96 2.93 2.92 2.89
3.26 3.20 3.16 3.12 3.07 3.03 3.00 2.98 2.96 2.93
3.33 3.27 3.22 3.18 3.12 3.09 3.06 3.03 3.02 2.98
3.38 3.32 3.27 3.23 3.17 3.13 3.10 3.08 3.06 3.02
3.42 3.36 3.31 3.27 3.21 3.17 3.13 3.11 3.09 3.06
3.46 3.39 3.34 3.30 3.24 3.20 3.16 3.14 3.12 3.09
3.49 3.42 3.37 3.33 3.26 3.22 3.19 3.16 3.14 3.11
150
2.02
2.18
2.27
2.33
2.38
2.48
2.56
2.62
2.66
2.74
2.79
2.83
2.87
2.91
2.96
3.00
3.03
3.06
3.09

                                                     D-77
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-5.
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.97
6.07
6.75
7.24
7.62
8.41
9.09
9.57
9.93
10.57
11.01
11.35
11.62
11.95
12.36
12.68
12.93
13.14
13.32
Table 19-5
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
6.07
7.24
7.93
8.41
8.79
9.57
10.22
10.67
11.01
11.62
12.04
12.36
12.62
12.93
13.32
13.62
13.86
14.06
14.23
6
3.25
3.79
4.12
4.35
4.54
4.92
5.26
5.49
5.67
5.99
6.22
6.39
6.53
6.69
6.91
7.07
7.20
7.31
7.40
K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 2 (20 COC, Annual)
8
2.74
3.13
3.36
3.53
3.66
3.94
4.18
4.34
4.47
4.71
4.87
5.00
5.10
5.22
5.38
5.50
5.59
5.67
5.74
10
2.49
2.82
3.O1
3,15
3.26
3.48
3.68
3.81
3.92
4.11
4.25
4.35
4.43
4.54
4.67
4.77
4.85
4.91
4.97
. K- Multipliers
6
3.79
4.35
4.69
4.92
5.11
5.49
5.82
6.04
6.22
6.53
6.74
6.91
7.04
7.20
7.40
7.56
7.68
7.79
7.88
8
3.13
3.53
3.77
3.94
4.07
4.34
4.58
4.74
4.87
5.10
5.25
5.38
5.47
5.59
5.74
5.86
5.95
6.03
6.10
10
2.82
3.15
3.34
3.48
3.59
3.81
4.01
4.14
4.25
4.43
4.57
4.67
4.75
4.85
4.97
5.07
5.15
5.21
5.27
12
2.35
2.64
2.81
2.93
3,02
3.22
3.39
3.51
3.60
3.77
3.89
3.98
4.05
4.14
4.26
4.34
4.42
4.48
4.53
for 1
12
2.64
2.93
3.10
3.22
3,31
3.51
3.68
3.80
3.89
4.05
4.17
4.26
4.33
4.42
4.53
4.61
4.68
4.74
4.79
16
2.19
2.44
2.59
2.69
2.77
2.93
3,08
3.18
3.25
3.39
3.49
3.57
3.63
3.71
3.80
3.88
3.94
3.99
4.03
-of-1
16
2.44
2.69
2.83
2.93
3.01
3.18
3.32
3,42
,3,49
3.63
3.73
3.80
3.86
3.94
4.03
4.10
4.16
4.21
4.26
20
2.10
2.33
2.47
2.56
2.63
2.78
2.91
3.0Q
3.O7
3,19,
3.28
3.35
3.40
3.47
3.56
3.62
3.68
3.72
3.76
25
2.04
2.25
2.38
2.46
2.53
2.66
2.78
2.86
2.93
3,04
3,12
3,18
3,23
3.30
3.37
3.43
3.48
3.52
3.56
Interwell
20
2.33
2.56
2.69
2.78
2.85
3.00
3.12
3.21
- 3,28
3.4O
3.49
3, 56
3.61
3.68
3.76
3.83
3.88
3.93
3.96
25
2.25
2.46
2.58
2.66
2.73
2.86
2.98
3.06
3.12
3.23
3,3 1
3,37
3,42
3.48
3.56
3.62
3.67
3.71
3.74
30
2.00
2.20
2.32
2.40
2.46
2.59
2.70
2.78
2.84
2,95
. 3.02
. 3.O8
3.13
3; 18
3,26
3.31
3.36
3.40
3.43
35
1.97
2.17
2.28
2.36
2.42
2.54
2.65
2.72
2.78
2.88
2.95
3.01
3. OS
3.11
3.18
3.23
3.27
3.31
3.34
Prediction
30
2.20
2.40
2.51
2.59
2.65
2.78
2.89
2.96
3.02
3.13
3.20
3.26
3,30
3,36
3,43.
3,49
3.53
3,57
3.6O
35
2.17
2.36
2.47
2.54
2.60
2.72
2.82
2.90
2.95
3.05
3.12
3.18
3.22
3.27
3,34
3,39
3.44
3.47
3.5O
40
1.95
2.14
2.25
2.33
2.38
2.51
2.61
2.68
2.73
2.83
2.90
2,95
3.OO
3, OS
3.12
3.17
3.21 .
3.25
3,28
Limits
40
2.14
2.33
2.43
2.51
2.56
2.68
2.78
2.85
2.90
3.00
3.07
3.12
3.16
3.21
3,28
3,33
3.37
3,40
3,43
45 50 60 70 80 90 100 125
1.93 1.92 1.90 1.89 1.88 1.87 1.86 1.85
2.12 2.11 2.08 2.07 2.05 2.05 2.04 2.02
2.23 2.21 2.19 2.17 2.15 2.14 2.14 2.12
2.30 2.28 2.26 2.24 2.22 2.21 2.20 2.19
2.36 2.34 2.31 2.29 2.28 2.26 2.25 2.24
2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34
2.58 2.55 2.52 2.49 2.47 2.46 2.45 2.43
2.65 2.62 2.58 2.56 2.54 2.52 2.51 2.49
2.70 2.67 2.63 2.61 2.59 2.57 2.56 2.54
2.80 2.77 2.72 2.70 2.67 2.66 2.64 2.62
2.86 2.83 2.79 2.76 2.73 2.72 2.70 2.68
2.89 2.88 2.84 2.80 2.78 2.76 2.75 2.72
.'2.96'. 2.92 2.88 2.84 2.82 2.80 2.78 2.76
3,01 2,97 2.92 2.89 2.86 2.84 2.83 2.80
3.07 3.04 2,98 2.95 2.92 2.90 2.88 2.85
3, 12. 3,09 3,03 2.99 '2,37" "'2,94' 2.93 2.90
3,16 .3.13 3.07 3,03 3.OO 2.98 '2, '96" 2.93
3.2O 3,16 3:10 3,06 3,03 3,01 ,2,99 2,96
3.23 3.19 3,13 3.09 3.06 3.O4 . 3,02 2,98
150
1.84
2.02
2.11
2.18
2.23
2.33
2.42
2.48
2.52
2.60
2.66
2.70
2.74
2.78
2.84
2.88
2.91
2,94
2.96
on Means of Order 2 (20 COC, Semi-Annual)
45 50 60 70 80 90 100 125
2.12 2.11 2.08 2.07 2.05 2.05 2.04 2.02
2.30 2.28 2.26 2.24 2.22 2.21 2.20 2.19
2.41 2.39 2.36 2.33 2.32 2.31 2.30 2.28
2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34
2.53 2.51 2.48 2.45 2.43 2.42 2.41 2.39
2.65 2.62 2.58 2.56 2.54 2.52 2.51 2.49
2.74 2.72 2.68 2.65 2.63 2.61 2.60 2.57
2.81 2.78 2.74 2.71 2.69 2.67 2.66 2.63
2.86 2.83 2.79 2.76 2.73 2.72 2.70 2.68
2.96 2.92 2.88 2.84 2.82 2.80 2.78 2.76
3.02 2.99 2.94 2.90 2.88 2.86 2.84 2.81
3.07 3.04 2.98 2.95 2.92 2.90 2.88 2.85
3.11 3.08 3.02 2.99 2.96 2.94 2.92 2.89
3.16 3.13 3.07 3.03 3.00 2.98 2.96 2.93
3.23 3.19 3.13 3.09 3.06 3.04 3.02 2.98
3.28 3.24 3.18 3.13 3.10 3.08 3.06 3.03
3.32. 3,27 3.21 3.17 3.14 3.11 3.09 3.06
3,35 3.31 3.24 3.20 3.17 3.14 3.12 3.09
3,38 ,3.33 • 3,27 3.23 3.19 3.17 3.15 3.11
150
2.02
2.18
2.27
2.33
2.38
2.48
2.56
2.62
2.66
2.74
2.79
2.84
2.87
2.91
2.96
3.00
3.04
3.06
3.09

                                                     D-78
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-5. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 2 (20 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
7.24
8.41
9.09
9.57
9.93
10.67
11.29
11.72
12.04
12.62
13.02
13.32
13.56
13.86
14.23
14.52
14.74
14.94
15.10
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
6.34
7.72
8.57
9.18
9.66
10.66
11.51
12.11
12.56
13.37
13.93
14.35
14.70
15.11
15.63
16.02
16.34
16.61
16.84
6
4.35
4.92
5.26
5.49
5.67
6.04
6.36
6.57
6.74
7.04
7.24
7.40
7.53
7.68
7.88
8.03
8.15
8.25
8.33
9-5.
6
3.85
4.46
4.83
5.10
5.30
5.75
6.13
6.39
6.60
6.97
7.22
7.42
7.58
7.77
8.02
8.20
8.35
8.48
8.59
8
3.53
3.94
4.18
4.34
4.47
4.74
4.97
5.13
5.25
5.47
5.63
5.74
5.84
5.95
6.10
6.21
6.30
6.37
6.44
10 12 16 20 25 30 35 40 45 50
3.15 2.93 2.69 2.56 2.46 2.40 2.36 2.33 2.30 2.28
3.48 3.22 2.93 2.78 2.66 2.59 2.54 2.51 2.48 2.46
3,68 3.39 3.08 2.91 2.78 2.70 2.65 2.61 2.58 2.55
381 3.51 3.18 3.00 2.86 2.78 2.72 2.68 2.65 2.62
3.92 3,60 3.25 3.07 2.93 2.84 2.78 2.73 2.70 2.67
414 380 3.42 3.21 3.06 2.96 2.90 2.85 2.81 2.78
4.33 3.96 3.56 3.34 3.17 3.07 3.00 2.94 2.91 2.87
4.46 4.08 3,65 3.42 3.25 3.14 3.07 3.01 2.97 2.94
4.57 4.17 3,73 3.49 3.31 3.20 3.12 3.07 3.02 2.99
4.75 4.33 3.86 3,61 3.42 3.30 3.22 3.16 3.11 3.08
4.87 4.44 3.96 3, 70 3.50 3.38 3.29 3.23 3.18 3.14
4.97 4.53 4.03 3,76 3.56 3.43 3.34 3.28 3.23 3.19
5.05 4.60 4.09 3.82 3.61 3.48 3.39 3.32 3.27 3.23
5.15 4.68 4.16 3,88 3,67 3.53 3.44 3.37 3.32 3.27
5.27 4.79 4.26 3.96 3,74 3,60 3.50 3.43 3.38 3.33
5.36 4.87 4.33 4.03 3.8O 3,66 3.56 3.48 3.43 3.38
5.44 4.94 4.38 4.08 3.85 3.7O 3.60 3.52 3.46 3.42
5.50 5.00 4.43 4.12 3,89 3,74 3,63 3.56 3.50 3.45
5.55 5.04 4.47 4.16 3.92 3.77 3.66 3,59 3.53 3.48
60
2.26
2.42
2.52
2.58
2.63
2.74
2.83
2.89
2.94
3.02
3.08
3.13
3.17
3.21
3.27
3.32
3.35
3.38
3.41
K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of
8
3.15
3.57
3.83
4.01
4.15
4.45
4.71
4.90
5.04
5.29
5.47
5.61
5.72
5.86
6.03
6.16
6.27
6.36
6.44
10 12 16 20 25 30 35 40 45 50
2.83 2.65 2.45 2.34 2.26 2.20 2.17 2.14 2.12 2.11
317 2.94 2.70 2.56 2.46 2.40 2.36 2.33 2.30 2.29
3.37 "'3,12' 2.84 2.69 2.58 2.52 2.47 2.43 2.41 2.39
3.52 3.24 2.94 2.79 2.67 2.59 2.54 2.51 2.48 2.46
3.63 3.34 3,02 2.86 2.73 2.66 2.60 2.56 2.53 2.51
3.87 3.55 3.19 3.O1 2.87 2.78 2.72 2.68 2.65 2.62
4.08 3.72 3.34 3,1.4 2,99 2.89 2.83 2.78 2.75 2.72
4.22 3.85 3.44 ,3.23 3.O7 2.97 2.90 2.85 2.81 2.78
4.34 3.95 3.52 3.30 3, 13 3.O3 2,96 2.90 2.87 2.83
4.54 4.12 3.66 3.42 3,24 3,13 3.O6 3.OO 2,96. 2.93
4.69 4.25 3.77 3.51 3.33 3,21 3.13 3.O7 3,03 ,, 2.99
4.80 4.34 3.85 3.58 3.32 3,27 3.18, 3,12 3.O8, 3,04
4.89 4.42 3.91 3.64 3.44 3.31 3,23 3.16 3,12 3,08
5.00 4.51 3.99 3.71 3.50 3.37 3,28 3,22 3,17 3,13
5.14 4.64 4.09 3.80 3.58 3.44 3.35 3,28 3,23 3:19
5.25 4.73 4.16 3.86 3.64 3.50 3.40 3.33 ,3,28 3,24
5.33 4.80 4.23 3.92 3.69 3.55 3.45 3.38 3.32 3.28
5.41 4.87 4.28 3.97 3.73 3.59 3.48 3.41 3.36 3,31
5.47 4.92 4.33 4.01 3.77 3.62 3.52 3.44 3.38 3.34
60
2.08
2.26
2.36
2.42
2.48
2.58
2.68
2.74
2.79
2.88
'•"2,94"
2.99
,3,O3
3.07
3.13
3,18
, 3,22
3,25
3,27'
70
2.24
2.40
2.49
2.56
2.61
2.71
2.80
2.86
2.90
2.99
3.04
3.09
3.13
3.17
3.23
3.27
3.30
3.33
3.36
80
2.22
2.38
2.47
2.54
2.59
2.69
2.77
2.83
2.88
2.96
3.01
3.06
3.09
3.14
3.19
3.23
3.27
3.30
3.32
Order 2
70
2.07
2.24
2.33
2.40
2.45
2.56
2.65
2.71
2.76
2.84
2.90
2.95
2,99
3.03
3.O9
3,14
3.17
3. 2O
3,23
80
2.05
2.22
2.32
2.38
2.43
2.54
2.63
2.69
2.73
2.82
2.88
2.92
2,96
3.OO
3.06
3.1O
3;14
3.17
,3.19
90
2.21
2.37
2.46
2.52
2.57
2.67
2.75
2.81
2.86
2.94
2.99
3.04
3.07
3.11
3.17
3.21
3.24
3.27
3.30
100
2.20
2.36
2.45
2.51
2.56
2.66
2.74
2.80
2.84
2.92
2.97
3.02
3.05
3.09
3.15
3.19
3.22
3.25
3.27
125
2.19
2.34
2.43
2.49
2.54
2.63
2.71
2.77
2.81
2.89
2.94
2.98
3.02
3.06
3.11
3.15
3.18
3.21
3.23
150
2.18
2.33
2.42
2.48
2.52
2.62
2.70
2.75
2.79
2.87
2.92
2.96
3.00
3.04
3.09
3.13
3.16
3.18
3.21
(40 COC, Annual)
90
2.05
2.21
2.31
2.37
2.42
2.52
2.61
2.67
2.72
2.80
2.86
2.90
2,94
2.98
3.O4
3.O8
3.11
3,14
3,17
100
2.04
2.20
2.30
2.36
2.41
2.51
2.60
2.66
2.70
2.78
2.84
2.88
2.92
2.96
3. 02
3.O6
3.O9
,3,12
3,,15
125
2.02
2.19
2.28
2.34
2.39
2.49
2.57
2.63
2.68
2.76
2.81
2.86
2.89
2.93
2,99
3.O3
3.06
3,O9
3,11
150
2.02
2.18
2.27
2.33
2.38
2.48
2.56
2.62
2.66
2.74
2.79
2.84
2.87
2.91
2,96
3.OO ,
3.O4
3.O6
3,09

                                                     D-79
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-5. K- Multipliers for 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
7.72
9.18
10.05
10.66
11.13
12.11
12.93
13.50
13.93
14.70
15.23
15.63
15.95
16.34
16.84
17.21
17.52
17.77
17.99
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
9.18
10.66
11.51
12.11
12.56
13.50
14.28
14.82
15.23
15.95
16.46
16.84
17.15
17.52
17.99
18.34
18.63
18.88
19.08
6
4.46
5.10
5.48
5.75
5.95
6.39
6.76
7.03
7.22
7.58
7.83
8.02
8.17
8.35
8.59
8.77
8.91
9.03
9.14
9-5.
6
5.10
5.75
6.13
6.39
6.60
7.03
7.39
7.64
7.83
8.17
8.41
8.59
8.73
8.91
9.14
9.31
9.44
9.56
9.66
8
3.57
4.01
4.27
4.45
4.59
4.90
5.15
5.33
5.47
5.72
5.90
6.03
6.14
6.27
6.44
6.56
6.67
6.75
6.83
10
3.17
3.52
3.72
3.87
3.99
4.22
4.43
4.58
4.69
4.89
5.03
5.14
5.23
5.33
5.47
5.57
5.66
5.73
5.79
K-Multipliers
8
4.01
4.45
4.71
4.90
5.04
5.33
5.59
5.76
5.90
6.14
6.31
6.44
6.54
6.67
6.83
6.95
7.05
7.13
7.21
10
3.52
3.87
4.08
4.22
4.34
4.58
4.78
4.92
5.03
5.23
5.36
5.47
5.55
5.66
5.79
5.89
5.97
6.04
6.10
12
2.94
3.24
3,42
3.55
3.64
3.85
4.02
4.15
4.25
4.42
4.54
4.64
4.71
4.80
4.92
5.01
5.09
5.15
5.20
for
12
3.24
3.55
'3, 72'
3.85
3.95
4.15
4.32
4.45
4.54
4.71
4.83
4.92
5.00
5.09
5.20
5.29
5.36
5.42
5.47
-of-1
16
2.70
2.94
3.09
3.19
3.27
3.44
3.59
3.69
3.77
3.91
4.01
4.09
4.15
4.23
4.33
4.40
4.46
4.51
4.56
Interwell Prediction Limits on Means of Order 2 (40 COC, Semi-Annual)
20
2.56
2.79
2.92
3.01
3.08
3.23
. 3,35
3.44
3.51
3.64
3.73
3.80
3.85
3.92
4.01
4.07
4.13
4.17
4.21
25
2.46
2.67
2.79
2.87
2.93
3.07
3.18
3.26
3.33
3.44
3,52
3,58
3.63
3.69
3.77
3.83
3.88
3.92
3.95
1-of-1 Interwell
16
2.94
3.19
3.34
3.44
3.52
3,69
3.83
3.93
4.01
4.15
4.25
4.33
4.39
4.46
4.56
4.63
4.69
4.74
4.79
20
2.79
3.01
3.14
3.23
3.30
3.44
3.57
3,66
3.73
3.85
3.94
4.01
4.06
4.13
4.21
4.28
4.33
4.38
4.42
25
2.67
2.87
2.99
3.07
3.13
3.26
3.38
3.46
3.52
3,63
3.71
'3, 77
3,82
3,88
3.95
4.01
4.06
4.10
4.14
30
2.40
2.59
2.71
2.78
2.84
2.97
3.08
3.15
3.21
3,31
3,39
.3,44
3,49
3,55
3,62
3.67
3.72
3.76
3.79
35 40 45 50 60 70 80 90 100 125
2.36 2.33 2.30 2.29 2.26 2.24 2.22 2.21 2.20 2.19
2.54 2.51 2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34
2.65 2.61 2.58 2.55 2.52 2.49 2.48 2.46 2.45 2.43
2.72 2.68 2.65 2.62 2.58 2.56 2.54 2.52 2.51 2.49
2.78 2.74 2.70 2.68 2.64 2.61 2.59 2.57 2.56 2.54
2.90 2.85 2.81 2.78 2.74 2.71 2.69 2.67 2.66 2.63
3.00 2.95 2.91 2.88 2.83 2.80 2.77 2.75 2.74 2.71
3.07 3.02 2.97 2.94 2.89 2.86 2.83 2.81 2.80 2.77
3.13 3.07 3.03 2.99 2.94 2.90 2.88 2.86 2.84 2.81
3.23 3.16 3.12 3.08 3.03 2.99 2.96 2.94 2.92 2.89
3:,30 3.23 3.18 3.14 3.09 3.05 3.02 2.99 2.98 2.94
3.35 "3,28' 3.23 3.19 3.13 3.09 3.06 3.04 3.02 2.99
3,39 ,3.32* 3.27 3.23 3.17 3.13 3.10 3.07 3.05 3.02
•3.45 3,38 3.32 3.28 3.22 3.17 3.14 3.11 3.09 3.06
3,52 3,44 3,38 3,34. 3,27 3.23 3.19 3.17 3.15 3.11
3:57 3.49 3,43 3.39 3.32 '3.27' 3.24 3.21 3.19 3.15
3,61 3,53 3,47 3,43 3,36 3,31 "3.27' 3.24 3.22 3.18
3,65 3.57 3.51 3,46 3,39 3,34 3. .30 3,27 3.25 3.21
3.68 3.6O 3,53 3:49 3,41 3.36 3,33 3.3O 3,27 3.23
150
2.18
2.33
2.42
2.48
2.52
2.62
2.70
2.75
2.79
2.87
2.92
2.96
3.00
3.04
3.09
3.13
3.16
3.18
3.21
Prediction Limits on Means of Order 2 (40 COC, Quarterly)
30
2.59
2.78
2.89
2.97
3.03
3.15
3.26
3.33
3.39
3.49
3.56
3'.62 '
3,66
3,72
3,79
3,85
3,89
3,93
3,96
35 40 45 50 60 70 80 90 100 125
2.54 2.51 2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34
2.72 2.68 2.65 2.62 2.58 2.56 2.54 2.52 2.51 2.49
2.83 2.78 2.75 2.72 2.68 2.65 2.63 2.61 2.60 2.57
2.90 2.85 2.81 2.78 2.74 2.71 2.69 2.67 2.66 2.63
2.96 2.90 2.87 2.83 2.79 2.76 2.73 2.72 2.70 2.68
3.07 3.02 2.97 2.94 2.89 2.86 2.83 2.81 2.80 2.77
3.17 3.11 3.07 3.03 2.98 2.94 2.91 2.89 2.88 2.85
3.24 3.18 3.13 3.09 3.04 3.00 2.97 2.95 2.93 2.90
3.30 3.23 3.18 3.14 3.09 3.05 3.02 2.99 2.98 2.94
3.39 3.32 3.27 3.23 3.17 3.13 3.10 3.07 3.05 3.02
3.46 3.39 3.34 3.29 3.23 3.18 3.15 3.13 3.11 3.07
3.52 3.44 3.38 3.34 3.27 3.23 3.19 3.17 3.15 3.11
3.56 3.48 3.42 3.38 3.31 3.26 3.23 3.20 3.18 3.14
3.61 3.53 3.47 3.43 3.36 3.31 3.27 3.24 3.22 3.18
3,68 3,60 3.53 3.49 3.41 3.36 3.33 3.30 3.27 3.23
3,73 3.65 3,58 3.53 3.46 3.41 3.37 3.34 3.31 3.27
3,77 3,69 3.62, 3.57 3.49 3.44 3.40 3.37 3.35 3.30
3,81 3,72 3,65 :3,6O 3.52 3.47 3.43 3.40 3.37 3.33
3,84 3,75 .3,68 3,83 3.55 3.49 3.45 3.42 3.40 3.35
150
2.33
2.48
2.56
2.62
2.66
2.75
2.83
2.87
2.92
3.00
3.05
3.09
3.12
3.16
3.21
3.25
3.28
3.30
3.33

                                                     D-80
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
      Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (1 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
(


















).69 (
.02 (
.21 (
.34 ]
.44 ]
.65 ]
.83
.95
2.05
2.21
2.33
2.41
2.48
2.57 ;
2.68 ;
2.76 :
2.82 :
2.88 :
2.92 :
).56 0.50 (
).83 0.75 (
).98 0.89 (
L.09 0.98 (
L.17 1.05 (
L.33 1.20
.47 1.32
.56 1.40
.63 1.47
.76 1.58
.85 1.66
.92 1.72
.97 1.77
2.03 1.82
1.12 1.90
2.18 1.95
2.23 2.00
2.27 2.03
2.31 2.06
).47 0.45 0.42 (
).70 0.67 0.64 (
).83 0.80 0.76 (
).92 0.88 0.84 (
).99 0.95 0.90 (
.12 1.08 1.02 (
.24 1.19 1.12
.32 1.26 1.19
.38 1.32 1.24
.48 1.42 1.34
.55 1.49 1.40
.61 1.54 1.45
.65 1.58 1.49
.71 1.63 1.54
.77 1.70 1.60
.83 1.74 1.65
.87 1.78 1.68
.90 1.82 1.71
.93 1.85 1.74
).40 (
).62 (
).73 (
).81 (
).87 (
).99 (
.08
.15
.20
.29
.35
.40
.44
.48
.54
.59
.62
.65
.68
).39 0.38 (
).60 0.59 (
).71 0.70 (
).79 0.77 (
).84 0.83 (
).96 0.94 (
.05 1.03
.12 1.10
.17 1.15
.25 1.23
.31 1.29
.36 1.33
.40 1.37
.44 1.41
.50 1.47
.54 1.51
.57 1.54
.60 1.57
.63 1.59
).37 (
).58 (
).69 (
).76 (
).82 (
).93 (
.02
.08
.13
.21
.27
.31
.35
.39
.44
.48
.52
.54
.57
).37 0.37 (
).57 0.57 (
).68 0.68 (
).75 0.75 (
).81 0.80 (
).92 0.91 (
.01 1.00 (
.07 1.06
.12 1.11
.20 1.19
.26 1.24
.30 1.29
.33 1.32
.37 1.36
.43 1.41
.47 1.45
.50 1.49
.53 1.51
.55 1.53
).36 0.36 0.36 (
).56 0.56 0.55 (
).67 0.66 0.66 (
).74 0.74 0.73 (
).80 0.79 0.78 (
).90 0.90 0.89 (
).99 0.98 0.98 (
.05 1.04 1.04
.10 1.09 1.08
.18 1.17 1.16
.24 1.22 1.21
.28 1.26 1.25
.31 1.30 1.29
.35 1.34 1.33
.40 1.39 1.38
.44 1.43 1.41
.47 1.46 1.44
.50 1.48 1.47
.52 1.51 1.49
).35 0.35 (
).55 0.55 (
).66 0.65 (
).73 0.72 (
).78 0.78 (
).88 0.88 (
).97 0.97 (
.03 1.03
.07 1.07
.15 1.15
.21 1.20
.25 1.24
.28 1.27
.32 1.31
.37 1.36
.41 1.40
.44 1.43
.46 1.45
.48 1.47
).35 0.35 (
).55 0.54 (
).65 0.65 (
).72 0.72 (
).77 0.77 (
).88 0.87 (
).96 0.96 (
.02 1.01
.07 1.06
.14 1.13
.20 1.19
.24 1.23
.27 1.26
.31 1.30
.36 1.35
.39 1.38
.42 1.41
.45 1.44
.47 1.46
).35
).54
).64
).71
).77
).87
).95
.01
.05
.13
.18
.22
.25
.29
.34
.38
.40
.43
.45
   Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.02 0.83 0.75 (
.34 1.09 0.98 (
.52 1.23 1.11
.65 1.33 1.20
.75 1.41 1.26
.95 1.56 1.40
2.12 1.69 1.52
2.24 1.78 1.60
2.33 1.85 1.66
2.48 1.97 1.77
2.59 2.05 1.84
2.68 2.12 1.90
2.74 2.17 1.94
2.82 2.23 2.00
2.92 2.31 2.06
3.00 2.37 2.12
3.06 2.41 2.16 ;
3.11 2.45 2.19 ;
3.16 2.49 2.23 ;
).70 0.67 0.64 (
).92 0.88 0.84 (
.04 1.00 0.94 (
.12 1.08 1.02 (
.19 1.14 1.08
.32 1.26 1.19
.42 1.36 1.29
.50 1.43 1.35
.55 1.49 1.40
.65 1.58 1.49
.72 1.65 1.55
.77 1.70 1.60
.82 1.74 1.64
.87 1.78 1.68
.93 1.85 1.74
.98 1.89 1.78
2.02 1.93 1.82
2.05 1.96 1.85
2.08 1.99 1.87
).62 (
).81 (
).91 (
).99 (
.04
.15
.24
.31
.35
.44
.50
.54
.58
.62
.68
.72
.75
.78
.81
).60 0.59 (
).79 0.77 (
).89 0.87 (
).96 0.94 (
.01 0.99 (
.12 1.10
.21 1.18
.27 1.24
.31 1.29
.40 1.37
.45 1.42
.50 1.47
.53 1.50
.57 1.54
.63 1.59
.67 1.63
.70 1.66
.73 1.69
.75 1.71
).58 (
).76 (
).86 (
).93 (
).98 (
.08
.17
.23
.27
.35
.40
.44
.48
.52
.57
.61
.64
.66
.69
).57 0.57 (
).75 0.75 (
).85 0.84 (
).92 0.91 (
).97 0.96 (
.07 1.06
.15 1.14
.21 1.20
.26 1.24
.33 1.32
.39 1.37
.43 1.41
.46 1.45
.50 1.49
.55 1.53
.59 1.57
.62 1.60
.64 1.63
.67 1.65
).56 0.56 0.55 (
).74 0.74 0.73 (
).84 0.83 0.83 (
).90 0.90 0.89 (
).95 0.94 0.94 (
.05 1.04 1.04
.14 1.12 1.12
.19 1.18 1.17
.24 1.22 1.21
.31 1.30 1.29
.36 1.35 1.34
.40 1.39 1.38
.44 1.42 1.41
.47 1.46 1.44
.52 1.51 1.49
.56 1.54 1.53
.59 1.57 1.56
.62 1.60 1.58
.64 1.62 1.60
).55 0.55 (
).73 0.72 (
).82 0.82 (
).88 0.88 (
).93 0.93 (
.03 1.03
.11 1.10
.16 1.16
.21 1.20
.28 1.27
.33 1.32
.37 1.36
.40 1.39
.44 1.43
.48 1.47
.52 1.51
.55 1.54
.57 1.56
.59 1.58
).55 0.54 (
).72 0.72 (
).81 0.81 (
).88 0.87 (
).93 0.92 (
.02 1.01
.10 1.09
.15 1.15
.20 1.19
.27 1.26
.32 1.31
.36 1.35
.39 1.38
.42 1.41
.47 1.46
.50 1.49
.53 1.52
.56 1.54
.58 1.56
).54
).71
).81
).87
).92
.01
.09
.14
.18
.25
.30
.34
.37
.40
.45
.48
.51
.54
.56
                                                    D-81
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                      Unified Guidance
     Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.34 1.09 0.98 (
1.65 1.33 1.20
1.83 1.47 1.32
1.95 1.56 1.40
2.05 1.63 1.47
2.24 1.78 1.60
2.40 1.90 1.71
2.51 1.99 1.78
2.59 2.05 1.84
2.74 2.17 1.94
2.85 2.25 2.01
2.92 2.31 2.06
2.99 2.36 2.11
3.06 2.41 2.16 ;
3.16 2.49 2.23 ;
3.23 2.54 2.28 ;
3.29 2.59 2.32 ;
3.34 2.63 2.35 ;
3.38 2.66 2.38 ;
).92 0.88 0.84 (
.12 1.08 1.02 (
.24 1.19 1.12
.32 1.26 1.19
.38 1.32 1.24
.50 1.43 1.35
.60 1.53 1.44
.67 1.60 1.51
.72 1.65 1.55
.82 1.74 1.64
.88 1.80 1.70
.93 1.85 1.74
.97 1.88 1.78
1.02 1.93 1.82
>.08 1.99 1.87
>.13 2.03 1.92
>.17 2.07 1.95
1.20 2.10 1.98
1.22 2.12 2.00
).81 (
).99 (
.08
.15
.20
.31
.39
.45
.50
.58
.64
.68
.71
.75
.81
.85
.88
.91
.93
).79 0.77 (
).96 0.94 (
.05 1.03
.12 1.10
.17 1.15
.27 1.24
.35 1.32
.41 1.38
.45 1.42
.53 1.50
.59 1.55
.63 1.59
.66 1.63
.70 1.66
.75 1.71
.79 1.75
.82 1.78
.85 1.81
.87 1.83
).76 (
).93 (
.02
.08
.13
.23
.31
.36
.40
.48
.53
.57
.60
.64
.69
.72
.75
.78
.80
).75 0.75 (
).92 0.91 (
.01 1.00 (
.07 1.06
.12 1.11
.21 1.20
.29 1.28
.35 1.33
.39 1.37
.46 1.45
.51 1.50
.55 1.53
.58 1.57
.62 1.60
.67 1.65
.70 1.69
.73 1.72
.76 1.74
.78 1.76
).74 0.74 0.73 (
).90 0.90 0.89 (
).99 0.98 0.98 (
.05 1.04 1.04
.10 1.09 1.08
.19 1.18 1.17
.27 1.26 1.25
.32 1.31 1.30
.36 1.35 1.34
.44 1.42 1.41
.49 1.47 1.46
.52 1.51 1.49
.55 1.54 1.52
.59 1.57 1.56
.64 1.62 1.60
.67 1.65 1.64
.70 1.68 1.67
.73 1.70 1.69
.75 1.72 1.71
).73 0.72 (
).88 0.88 (
).97 0.97 (
.03 1.03
.07 1.07
.16 1.16
.24 1.23
.29 1.28
.33 1.32
.40 1.39
.45 1.44
.48 1.47
.51 1.50
.55 1.54
.59 1.58
.63 1.62
.65 1.64
.68 1.67
.70 1.69
).72 0.72 (
).88 0.87 (
).96 0.96 (
.02 1.01
.07 1.06
.15 1.15
.23 1.22
.28 1.27
.32 1.31
.39 1.38
.43 1.42
.47 1.46
.50 1.49
.53 1.52
.58 1.56
.61 1.60
.64 1.62
.66 1.65
.68 1.67
).71
).87
).95
.01
.05
.14
.21
.26
.30
.37
.41
.45
.48
.51
.56
.59
.62
.64
.66
      Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.10 0.87 0.77 (
.47 1.14 1.02 (
.69 1.30 1.15
.85 1.42 1.25
.97 1.50 1.32
2.22 1.68 1.48
2.44 1.83 1.60
2.59 1.94 1.69
2.71 2.02 1.76
2.91 2.16 1.88
3.05 2.26 1.97
3.16 2.34 2.03
3.25 2.40 2.08
3.35 2.47 2.15
3.49 2.57 2.23 ;
3.59 2.64 2.29 ;
3.67 '"2,69" 2.34 ;
3.74 2,74 2.38 ;
3.80 2.78 2.41 ;
).72 0.69 0.65 (
).95 0.90 0.85 (
.07 1.02 0.96 (
.16 1.10 1.04
.23 1.17 1.10
.37 1.30 1.22
.48 1.41 1.32
.56 1.48 1.39
.63 1.54 1.44
.74 1.64 1.53
.81 1.72 1.60
.87 1.77 1.65
.92 1.81 1.69
.98 1.87 1.74
>.05 1.93 1.80
MO 1.99 1.85
>.15 2.03 1.89
>.18 2.06 1.92
1.22 2.09 1.94
).62 (
).82 (
).93 (
.00 (
.05
.17
.26
.33
.38
.47
.53
.58
.62
.66
.72
.77
.80
.83
.86
).60 0.59 (
).79 0.78 (
).90 0.88 (
).97 0.95 (
.02 1.00 (
.13 1.11
.22 1.20
.29 1.26
.33 1.30
.42 1.39
.48 1.44
.52 1.49
.56 1.52
.60 1.56
.66 1.62
.70 1.66
.74 1.69
.76 1.72
.79 1.74
).58 (
).77 (
).87 (
).93 (
).99 (
.09
.18
.24
.28
.36
.42
.46
.49
.54
.59
.63
.66
.69
.71
).58 0.57 (
).76 0.75 (
).86 0.85 (
).92 0.92 (
).97 0.97 (
.08 1.07
.16 1.15
.22 1.21
.27 1.25
.34 1.33
.40 1.38
.44 1.43
.47 1.46
.51 1.50
.57 1.55
.61 1.59
.64 1.62
.66 1.64
.69 1.67
).57 0.56 0.56 (
).75 0.74 0.73 (
).84 0.83 0.83 (
).91 0.90 0.89 (
).96 0.95 0.94 (
.06 1.05 1.04
.14 1.13 1.12
.20 1.19 1.18
.24 1.23 1.22
.32 1.30 1.29
.37 1.36 1.34
.41 1.40 1.38
.45 1.43 1.41
.49 1.47 1.45
.54 1.51 1.50
.57 1.55 1.54
.60 1.58 1.57
.63 1.61 1.59
.65 1.63 1.61
).55 0.55 (
).73 0.73 (
).82 0.82 (
).89 0.88 (
).94 0.93 (
.03 1.03
.11 1.11
.17 1.16
.21 1.20
.28 1.28
.33 1.33
.37 1.37
.40 1.40
.44 1.43
.49 1.48
.53 1.52
.55 1.55
.58 1.57
.60 1.59
).55 0.55 (
).72 0.72 (
).82 0.81 (
).88 0.87 (
).93 0.92 (
.02 1.02
.10 1.10
.16 1.15
.20 1.19
.27 1.26
.32 1.31
.36 1.35
.39 1.38
.43 1.41
.47 1.46
.51 1.50
.54 1.52
.56 1.55
.58 1.57
).54
).72
).81
).87
).92
.01
.09
.14
.18
.25
.30
.34
.37
.41
.45
.49
.52
.54
.56
                                                    D-82
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
   Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2  (2 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.47 1.14 1.02 (
1.85 1.42 1.25
2.07 1.57 1.38
2.22 1.68 1.48
2.34 1.76 1.55
2.59 1.94 1.69
2.80 2.08 1.82
2.94 2.18 1.90
3.05 2.26 1.97
3.25 2.40 2.08
3.38 2.49 2.17
3.49 2.57 2.23 ;
3.57 2.62 2.28 ;
3.67 2.69 2.34 ;
3.80 2.78 2.41 ;
3.90 2.85 2.47 ;
3.97 '"2,'9i" 2.52 ;
4.04 2.95 2.56 ;
4.09 2.99 2.59 ;
).95 0.90 0.85 (
.16 1.10 1.04
.28 1.22 1.14
.37 1.30 1.22
.43 1.36 1.27
.56 1.48 1.39
.68 1.59 1.48
.75 1.66 1.55
.81 1.72 1.60
.92 1.81 1.69
.99 1.88 1.75
>.05 1.93 1.80
>.09 1.98 1.84
>.15 2.03 1.89
1.22 2.09 1.94
2.27 2.14 1.99
>.31 2.18 2.03
>.35 2.22 2.06
>.38 2.24 2.08
).82 (
.00 (
.10
.17
.22
.33
.42
.48
.53
.62
.68
.72
.76
.80
.86
.90
.94
.96
.99
).79 0.78 (
).97 0.95 (
.07 1.04
.13 1.11
.18 1.16
.29 1.26
.37 1.34
.43 1.40
.48 1.44
.56 1.52
.62 1.58
.66 1.62
.69 1.65
.74 1.69
.79 1.74
.83 1.78
.86 1.82
.89 1.84
.92 1.87
).77 (
).93 (
.03
.09
.14
.24
.32
.37
.42
.49
.55
.59
.62
.66
.71
.75
.78
.81
.83
).76 0.75 (
).92 0.92 (
.01 1.01
.08 1.07
.12 1.11
.22 1.21
.30 1.29
.36 1.34
.40 1.38
.47 1.46
.53 1.51
.57 1.55
.60 1.58
.64 1.62
.69 1.67
.72 1.70
.75 1.73
.78 1.76
.80 1.78
).75 0.74 0.73 (
).91 0.90 0.89 (
.00 0.99 0.98 (
.06 1.05 1.04
.11 1.09 1.08
.20 1.19 1.18
.28 1.26 1.25
.33 1.32 1.30
.37 1.36 1.34
.45 1.43 1.41
.50 1.48 1.46
.54 1.51 1.50
.57 1.55 1.53
.60 1.58 1.57
.65 1.63 1.61
.69 1.66 1.65
.72 1.69 1.68
.74 1.72 1.70
.76 1.74 1.72
).73 0.73 (
).89 0.88 (
).97 0.97 (
.03 1.03
.08 1.07
.17 1.16
.24 1.24
.29 1.29
.33 1.33
.40 1.40
.45 1.44
.49 1.48
.52 1.51
.55 1.55
.60 1.59
.63 1.62
.66 1.65
.69 1.68
.71 1.70
).72 0.72 (
).88 0.87 (
).97 0.96 (
.02 1.02
.07 1.06
.16 1.15
.23 1.22
.28 1.27
.32 1.31
.39 1.38
.44 1.43
.47 1.46
.50 1.49
.54 1.52
.58 1.57
.62 1.60
.64 1.63
.67 1.65
.69 1.67
).72
).87
).95
.01
.06
.14
.22
.27
.30
.37
.42
.45
.48
.52
.56
.59
.62
.64
.66
     Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.85 1.42 1.25
2.22 1.68 1.48
2.44 1.83 1.60
2.59 1.94 1.69
2.71 2.02 1.76
2.94 2.18 1.90
314 2.32 2.02
3.28 2.42 2.10
3.38 2.49 2.17
3.57 2.62 2.28 ;
3.70 2.71 2.35 ;
3.80 2.78 2.41 ;
3.88 2.84 2.46 ;
3.97 2.91 2.52 ;
4.09 2.99 2.59 ;
4.19 3.06 2.65 ;
4.26 3,11 2.69 ;
4.32 3.15 2.73 ;
4.38 3.19 2.76 ;
.16 1.10 1.04
.37 1.30 1.22
.48 1.41 1.32
.56 1.48 1.39
.63 1.54 1.44
.75 1.66 1.55
.86 1.76 1.64
.93 1.83 1.70
.99 1.88 1.75
>.09 1.98 1.84
>.16 2.04 1.90
1.22 2.09 1.94
1.26 2.13 1.98
>.31 2.18 2.03
>.38 2.24 2.08
1.43 2.29 2.13 ;
>.47 2.33 2.16 ;
>.50 2.36 2.19 ;
>.53 2.39 2.22 ;
.00 (
.17
.26
.33
.38
.48
.57
.63
.68
.76
.81
.86
.89
.94
.99
>.03
>.06
>.09 ;
>.12 ;
).97 0.95 (
.13 1.11
.22 1.20
.29 1.26
.33 1.30
.43 1.40
.51 1.48
.57 1.53
.62 1.58
.69 1.65
.75 1.70
.79 1.74
.82 1.78
.86 1.82
.92 1.87
.95 1.90
.99 1.93
>.01 1.96
>.04 1.98
).93 (
.09
.18
.24
.28
.37
.45
.51
.55
.62
.67
.71
.74
.78
.83
.87
.90
.92
.94
).92 0.92 (
.08 1.07
.16 1.15
.22 1.21
.27 1.25
.36 1.34
.43 1.42
.49 1.47
.53 1.51
.60 1.58
.65 1.63
.69 1.67
.72 1.70
.75 1.73
.80 1.78
.84 1.82
.87 1.85
.89 1.87
.91 1.89
).91 0.90 0.89 (
.06 1.05 1.04
.14 1.13 1.12
.20 1.19 1.18
.24 1.23 1.22
.33 1.32 1.30
.41 1.39 1.38
.46 1.44 1.42
.50 1.48 1.46
.57 1.55 1.53
.61 1.59 1.58
.65 1.63 1.61
.68 1.66 1.64
.72 1.69 1.68
.76 1.74 1.72
.80 1.77 1.75
.83 1.80 1.78
.85 1.82 1.80
.87 1.84 1.82
).89 0.88 (
.03 1.03
.11 1.11
.17 1.16
.21 1.20
.29 1.29
.37 1.36
.41 1.41
.45 1.44
.52 1.51
.56 1.56
.60 1.59
.63 1.62
.66 1.65
.71 1.70
.74 1.73
.77 1.76
.79 1.78
.81 1.80
).88 0.87 (
.02 1.02
.10 1.10
.16 1.15
.20 1.19
.28 1.27
.35 1.34
.40 1.39
.44 1.43
.50 1.49
.55 1.53
.58 1.57
.61 1.60
.64 1.63
.69 1.67
.72 1.70
.75 1.73
.77 1.75
.79 1.77
).87
.01
.09
.14
.18
.27
.33
.38
.42
.48
.53
.56
.59
.62
.66
.69
.72
.74
.76
                                                    D-83
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
      Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.72 1.28
2.19 1.58
2.47 1.76
2.67 1.89
2.83 1.98
3.17 2.19
3.45 2.37 ;
3.65 2.49 ;
3.81 2.58 ;
4.08 2,75 ;
4.27 2.87 ;
4.42 2.96 ;
4.54 3.03 ;
4.68 3.12 ;
4.86 3.23
5.00 3.32 1
5.11 3.39
5.20 3.44 I
5.28 3.49 2
.12 1.04 0.98 (
.37 1.26 1.19
.51 1.39 1.31
.61 1.48 1.39
.69 1.54 1.45
.86 1.69 1.59
>.00 1.81 1.70
MO 1.90 1.78
M7 1.96 1.84
>.31 2.08 1.94
>.40 2.16 2.02
2.47 2.23 2.08
2.53 2.28 2.12
2.60 2.34 2.18
1.89 2.42 2.25 ;
?,76 2.48 2.31 ;
?.8£ 2.53 2.35 ;
>.86 2.57 2.39 ;
>.90 2.6Q 2.42 ;
).92 0.88 0.86 (
.11 1.06 1.03
.22 1.17 1.13
.29 1.24 1.19
.35 1.29 1.24
.47 1.40 1.35
.57 1.49 1.43
.64 1.56 1.50
.69 1.61 1.54
.78 1.69 1.63
.85 1.76 1.68
.90 1.80 1.73
.94 1.84 1.76
.99 1.89 1.81
2.06 1.95 1.86
Ml 1.99 1.91
2.15 2.03 1.94
2.18 2.06 1.97
2.21 2.09 2.00
).84 (
.01 (
.10
.16
.21
.31
.40
.46
.50
.58
.64
.68
.71
.75
.81
.85
.88
.91
.93
).83 0.82 0.81 (
).99 0.98 0.97 (
.08 1.07 1.06
.14 1.13 1.12
.19 1.18 1.16
.29 1.27 1.26
.37 1.35 1.34
.43 1.41 1.39
.47 1.45 1.43
.55 1.53 .51
.60 1.58 .56
.64 1.62 .60
.68 1.65 .63
.72 1.69 .67
.77 1.74 1.72
.81 1.78 1.75
.84 1.81 1.79
.87 1.84 1.81
.89 1.86 1.83
).80 0.79 (
).96 0.95 (
.05 1.04
.11 1.10
.15 1.14
.25 1.23
.33 1.31
.38 1.36
.42 1.40
.49 1.47
.54 1.52
.58 1.56
.61 1.59
.65 1.63
.70 1.67
.74 1.71
.77 1.74
.79 1.76
.81 1.78
).79 0.78 (
).94 0.94 (
.03 1.02
.09 1.08
.13 1.12
.22 1.21
.30 1.29
.35 1.34
.39 1.38
.46 1.45
.50 1.49
.54 1.53
.57 1.56
.61 1.59
.65 1.64
.69 1.67
.72 1.70
.74 1.73
.76 1.75
).78 0.78 0.77 (
).93 0.93 0.92 (
.02 1.01 1.01
.07 1.07 1.06
.12 1.11 1.10
.21 1.20 1.19
.28 1.27 1.26
.33 1.32 1.31
.37 1.36 1.35
.44 1.43 1.42
.48 1.48 1.46
.52 1.51 1.50
.55 1.54 1.53
.58 1.58 1.56
.63 1.62 1.60
.66 1.65 1.64
.69 1.68 1.66
.71 1.70 1.69
.73 1.72 1.71
).77
).92
.00
.06
.10
.18
.26
.31
.34
.41
.45
.49
.52
.55
.59
.63
.65
.67
.70
   Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.19 1.58 1.37
2.67 1.89 1.61
2.96 2.06 1.76
3.17 2.19 1.86
3.32 2.29 1.93
3.65 2.49 2.10
3.93 2.66 2.23 ;
4.13 2.78 2.33 ;
4.27 2,87 2.40 ;
4.54 3.03 2.53 ;
4.72 3.15 2.62 ;
4.86 3.23 2.69 ;
4.97 3.30 2.75 ;
5.11 3.39 2.82 ;
5.28 3.49 2.9O \
5.41 3.58 2,97 :
5.52 3.64 3.02 ;
5.61 3.70 3.07 ;
5.68 3.75 3.11 ;
.26 1.19 1.11
.48 1.39 1.29
.60 1.50 1.39
.69 1.59 1.47
.76 1.65 1.52
.90 1.78 1.64
2.02 1.88 1.73
MO 1.96 1.80
M6 2.02 1.85
2.28 2.12 1.94
2.36 2.20 2.01
2.42 2.25 2.06
2.47 2.30 2.10
2.53 2.35 2.15 ;
2.60 2.42 2.21 ;
2.66 2.48 2.26 ;
2.71 2.52 2.30 ;
2.75 2.56 2.33 ;
2.78 2.59 2.36 ;
.06
.24
.33
.40
.45
.56
.65
.71
.76
.84
.90
.95
.99
2.03
2.09 ;
M3 ;
M7 ;
2.20 ;
2.23 ;
.03 1.01 (
.19 1.16
.28 1.25
.35 1.31
.40 1.36
.50 1.46
.58 1.54
.64 1.59
.68 1.64
.76 1.71
.82 1.77
.86 1.81
.90 1.84
.94 1.88
2.00 1.93
2.04 1.97
2.07 2.01
MO 2.03
M2 2.06 ;
).99 (
.14
.23
.29
.33
.43
.51
.56
.60
.68
.73
.77
.80
.84
.89
.93
.96
.99
2.01
).98 0.97 (
.13 1.12
.21 1.20
.27 1.26
.32 1.30
.41 1.39
.48 1.47
.54 1.52
.58 1.56
.65 1.63
.70 1.68
.74 1.72
.77 1.75
.81 1.79
.86 1.83
.89 1.87
.92 1.90
.95 1.92
.97 1.94
).96 0.95 0.94 (
.11 1.10 1.09
.19 1.18 1.17
.25 1.23 1.22
.29 1.27 1.26
.38 1.36 1.35
.45 1.43 1.42
.50 1.48 1.47
.54 1.52 1.50
.61 1.59 1.57
.66 1.64 1.62
.70 1.67 1.65
.73 1.70 1.68
.77 1.74 1.72
.81 1.78 1.76
.85 1.82 1.80
.88 1.85 1.82
.90 1.87 1.85
.92 1.89 1.87
).94 0.93 (
.08 1.07
.16 1.15
.21 1.21
.25 1.25
.34 1.33
.41 1.40
.46 1.45
.49 1.48
.56 1.55
.60 1.59
.64 1.63
.67 1.66
.70 1.69
.75 1.73
.78 1.77
.81 1.79
.83 1.82
.85 1.84
).93 0.92 (
.07 1.06
.15 1.14
.20 1.19
.24 1.23
.32 1.31
.39 1.38
.44 1.43
.48 1.46
.54 1.53
.59 1.57
.62 1.60
.65 1.63
.68 1.66
.72 1.71
.76 1.74
.78 1.76
.81 1.79
.82 1.80
).92
.06
.13
.18
.22
.31
.37
.42
.45
.52
.56
.59
.62
.65
.70
.73
.75
.77
.79
                                                    D-84
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                      Unified Guidance
     Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.67 1.89 1.61 .48 1.39 1.29
3.17 2.19 1.86 .69 1.59 1.47
345 2.37 2.00 .81 1.70 1.57
3.65 2.49 2.10 .90 1.78 1.64
3.81 2.58 2.17 .96 1.84 1.69
413 2.78 2.33 2.10 1.96 1.80
439 2.94 2.46 2.21 2.07 1.89
4.58 3.06 2.55 2.30 2.14 1.96
4.72 3.15 2.62 2.36 2.20 2.01
4.97 3.30 2.75 2.47 2.30 2.10
5.15 3.41 2.84 2.55 2.37 2.16 ;
5.28 3.49 2.90 2.60 2.42 2.21 ;
5.39 3.56 2.96 2.65 2.47 2.25 ;
5.52 3.64 3.02 2.71 2.52 2.30 ;
5.68 3.75 3,11 2.78 2.59 2.36 ;
5.81 3.82 3.17 2.84 2.64 2.40 ;
5.91 3.89 3.22 2.88 2.68 2.44 ;
5.99 3.94 3.26 2.92 2.71 2.47 ;
6.07 3.99 3.30 2.96 2.74 2.50 ;
.24
.40
.49
.56
.61
.71
.80
.86
.90
.99
2.04
2.09 ;
2.13 ;
2.17 ;
>..23 ;
2.27 ;
2.30 ;
1.33 :
i.36 ;
.19 1.16
.35 1.31
.43 1.40
.50 1.46
.54 1.50
.64 1.59
.72 1.67
.78 1.73
.82 1.77
.90 1.84
.95 1.89
2.00 1.93
2.03 1.97
1.07 2.01
1.12 2.06 ;
2.16 2.09 ;
1.20 2.12 ;
1.22 2.15 ;
1.25 2.17 ;
.14
.29
.37
.43
.47
.56
.64
.69
.73
.80
.85
.89
.92
.96
2.01
2.04 ;
2.07 ;
>.io ;
1.12 ;
.13 1.12
.27 1.26
.35 1.34
.41 1.39
.45 1.43
.54 1.52
.61 1.59
.66 1.64
.70 1.68
.77 1.75
.82 1.80
.86 1.83
.89 1.86
.92 1.90
.97 1.94
2.01 1.98
2.04 2.01
2.06 2.03 ;
2.08 2.05 ;
.11 1.10 1.09
.25 1.23 1.22
.33 1.31 1.30
.38 1.36 1.35
.42 1.40 1.39
.50 1.48 1.47
.58 1.55 1.54
.62 1.60 1.58
.66 1.64 1.62
.73 1.70 1.68
.78 1.75 1.73
.81 1.78 1.76
.84 1.81 1.79
.88 1.85 1.82
.92 1.89 1.87
.96 1.92 1.90
.99 1.95 1.93
2.01 1.97 1.95
2.03 1.99 1.97
.08 1.07
.21 1.21
.29 1.28
.34 1.33
.38 1.37
.46 1.45
.52 1.51
.57 1.56
.60 1.59
.67 1.66
.71 1.70
.75 1.73
.77 1.76
.81 1.79
.85 1.84
.88 1.87
.91 1.89
.93 1.91
.95 1.93
.07 1.06
.20 1.19
.27 1.26
.32 1.31
.36 1.35
.44 1.43
.51 1.49
.55 1.54
.59 1.57
.65 1.63
.69 1.67
.72 1.71
.75 1.73
.78 1.76
.82 1.80
.86 1.84
.88 1.86
.90 1.88
.92 1.90
.06
.18
.26
.31
.34
.42
.48
.53
.56
.62
.66
.70
.72
.75
.79
.82
.85
.87
.89
      Table 19-6. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 2 (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.29 1.62 1.39
2.86 1.94 1.64
3.21 2.14 1.79
3.46 2.28 1.90
3.66 2.39 1.99
4.07 2.62 2.16
4.43 2.82 2.31 ;
4.68 2.96 2.42 ;
4.87 3.06 2.50 ;
5.22 3.26 2.65 '4
5.46 3.39 2,75 \
5.64 3.49 2.83 ;
5.79 3.58 2.89 ;
5.97 3.68 2.97
6.19 3.81 3.07
6.37 3.90 3.15
6.51 3.98 3.21
6.62 4.05 3.26
6.72 4.11 3.30
.27 1.20 1.11
.49 1.40 1.30
.62 1.52 1.40
.72 1.60 1.48
.79 1.67 1.53
.94 1.80 1.65
2.06 1.92 1.75
2.15 2.00 1.82
1.22 2.06 1.87
2.35 2.17 1.97
2.44 2.25 2.04
2.50 2.31 2.09
2.56 2.36 2.14 ;
2.83. 2.42 2.19 ;
2.71 2.50 2.25 ;
2.78 2.56 2.30 ;
2.83 2,60 2.35 ;
2.87 2,64 2.38 ;
>.91 2.68 2.41 ;
.07
.24
.34
.41
.46
.57
.66
.72
.77
.86
.92
.97
2.01
2.06
2.12 ;
2.16 ;
1.20 ;
2.23 ;
2.26 ;
.03 1.01 (
.20 1.17
.29 1.26
.35 1.32
.40 1.36
.50 1.46
.59 1.54
.65 1.60
.69 1.64
.78 1.72
.83 1.78
.88 1.82
.91 1.85
.96 1.89
2.01 1.95
2.06 1.99
2.09 2.02
2.12 2.05 ;
2.15 2.07 ;
).99 (
.15
.23
.29
.34
.43
.51
.57
.61
.68
.74
.78
.81
.85
.90
.94
.97
2.00
2.02
).98 0.97 (
.13 1.12
.22 1.20
.27 1.26
.32 1.30
.41 1.39
.49 1.47
.54 1.52
.58 1.56
.66 1.63
.71 1.68
.75 1.72
.78 1.75
.82 1.79
.87 1.84
.90 1.88
.93 1.91
.96 1.93
.98 1.95
).96 0.95 0.94 (
.11 1.10 1.09
.19 1.18 1.17
.25 1.23 1.22
.29 1.28 1.26
.38 1.36 1.35
.46 1.43 1.42
.51 1.48 1.47
.55 1.52 1.51
.62 1.59 1.57
.67 1.64 1.62
.70 1.68 1.66
.73 1.71 1.69
.77 1.74 1.72
.82 1.79 1.76
.85 1.82 1.80
.88 1.85 1.83
.91 1.87 1.85
.93 1.89 1.87
).94 0.93 (
.08 1.07
.16 1.15
.21 1.21
.25 1.25
.34 1.33
.41 1.40
.46 1.45
.49 1.49
.56 1.55
.61 1.60
.64 1.63
.67 1.66
.70 1.69
.75 1.74
.78 1.77
.81 1.80
.83 1.82
.85 1.84
).93 0.92 (
.07 1.06
.15 1.14
.20 1.19
.24 1.23
.32 1.31
.39 1.38
.44 1.43
.48 1.46
.54 1.53
.59 1.57
.62 1.61
.65 1.63
.68 1.67
.73 1.71
.76 1.74
.78 1.77
.81 1.79
.83 1.81
).92
.06
.13
.19
.22
.31
.37
.42
.46
.52
.56
.60
.62
.65
.70
.73
.75
.77
.79
                                                    D-85
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
   Table 19-6. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 2 (10 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.86 1.94 1.64 1.49 1.40 1.30
3.46 2.28 1.90 1.72 1.60 1.48
3.82 2.48 2.05 1.84 1.72 1.58
4.07 2.62 2.16 1.94 1.80 1.65
4.27 2.73 2.24 2.01 1.87 1.71
4.68 2.96 2.42 2.15 2.00 1.82
5.03 3.15 2.57 2.28 2.11 1.92
5.27 3.29 2.67 2.37 2.19 1.99
5.46 3.39 2.75 2.44 2.25 2.04
5.79 3.58 2.89 2.56 2.36 2.14 ;
6.02 3.71 2,99 2.65 2.44 2.20 ;
6.19 3.81 3.07 2.71 2.50 2.25 ;
6.34 3.89 3.13 2.76 2.54 2.30 ;
6.51 3.98 3.21 2.83 2.60 2.35 ;
6.72 4.11 3.30 2.91 2.68 2.41 ;
6.89 4.20 3.38 2,97 2.73 2.46 ;
7.02 4.28 3.43 3.O? .' 2.78 2.50 ;
7.13 4.34 3.48 3.O7 2.82 2.53 ;
7.23 4.40 3.53 3.10 2,85 2.56 ;
.24
.41
.50
.57
.62
.72
.81
.87
.92
2.01
2.07
2.12 ;
2.16 ;
1.20 :
i.26 ;
2.3i ;
2.34 ;
2.37 ;
2.40 ;
.20 1.17
.35 1.32
.44 1.40
.50 1.46
.55 1.51
.65 1.60
.73 1.68
.79 1.73
.83 1.78
.91 1.85
.97 1.91
2.01 1.95
2.05 1.98
2.09 2.02
2.15 2.07 ;
2.19 2.11 ;
1.22 2.14 ;
1.25 2.17 ;
2.28 2.19 ;
.15
.29
.37
.43
.48
.57
.64
.70
.74
.81
.86
.90
.93
.97
1.02
2.06 ;
2.09 ;
1.12 :
2.14 ;
.13 1.12
.27 1.26
.35 1.34
.41 1.39
.45 1.44
.54 1.52
.62 1.59
.67 1.65
.71 1.68
.78 1.75
.83 1.80
.87 1.84
.90 1.87
.93 1.91
.98 1.95
1.02 1.99
2.05 2.02
1.07 2.04 ;
2.10 2.06 ;
.11 1.10 1.09
.25 1.23 1.22
.33 1.31 1.30
.38 1.36 1.35
.42 1.40 1.39
.51 1.48 1.47
.58 1.55 1.54
.63 1.60 1.58
.67 1.64 1.62
.73 1.71 1.69
.78 1.75 1.73
.82 1.79 1.76
.85 1.82 1.79
.88 1.85 1.83
.93 1.89 1.87
.96 1.93 1.90
.99 1.96 1.93
1.02 1.98 1.95
2.04 2.00 1.97
.08 1.07
.21 1.21
.29 1.28
.34 1.33
.38 1.37
.46 1.45
.52 1.51
.57 1.56
.61 1.60
.67 1.66
.71 1.70
.75 1.74
.78 1.76
.81 1.80
.85 1.84
.88 1.87
.91 1.90
.93 1.92
.95 1.94
.07 1.06
.20 1.19
.27 1.26
.32 1.31
.36 1.35
.44 1.43
.51 1.49
.55 1.54
.59 1.57
.65 1.63
.69 1.68
.73 1.71
.75 1.73
.78 1.77
.83 1.81
.86 1.84
.88 1.86
.90 1.88
.92 1.90
.06
.19
.26
.31
.34
.42
.48
.53
.56
.62
.66
.70
.72
.75
.79
.82
.85
.87
.89
    Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (10 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.46 2.28 1.90 1.72 1.60 1.48
4.07 2.62 2.16 1.94 1.80 1.65
443 2.82 2.31 2.06 1.92 1.75
468 2.96 2.42 2.15 2.00 1.82
487 3.06 2.50 2.22 2.06 1.87
5.27 3.29 2.67 2.37 2.19 1.99
5.61 3.48 2.81 2.49 2.30 2.08
5.84 3.61 2.92 2.58 2.38 2.15 ;
6.02 3.71 2.99 2.65 2.44 2.20 ;
6.34 3.89 3,13 2.76 2.54 2.30 ;
6.56 4.01 3.23 2.85 2.62 2.36 ;
6.72 4.11 3.30 2.91 2.68 2.41 ;
6.86 4.18 3.36 2.96 2.72 2.45 ;
7.02 4.28 3.43 3.02 2.78 2.50 ;
7.23 4.40 3.53 '"^io". 2.85 2.56 ;
7.39 4.49 3.60 3, 18 2.91 2.61 ;
7.52 4.56 3.65 3,21 2.95 2.65 ;
7.62 4.62 3.70 3.25 2.99 2.68 ;
7.71 4.67 3.74 3.29 3.02 2.71 ;
.41
.57
.66
.72
.77
.87
.96
1.02
2.07
2.16 ;
1.22 ;
1.26 :
i.30 :
1.34 ;
2.40 ;
2.44 ;
2.48 ;
2.5i ;
2.54 ;
.35 1.32
.50 1.46
.59 1.54
.65 1.60
.69 1.64
.79 1.73
.87 1.81
.93 1.86
.97 1.91
2.05 1.98
2.10 2.03
2.15 2.07 ;
2.18 2.10 ;
1.22 2.14 ;
2.28 2.19 ;
1.32 2.23 :
1.35 2.26 ;
2.38 2.29 ;
2.40 2.31 ;
.29
.43
.51
.57
.61
.70
.77
.82
.86
.93
.98
1.02
2.05 ;
2.09 ;
2.14 ;
2.17 ;
1.20 ;
2.23 ;
2.25 ;
.27 1.26
.41 1.39
.49 1.47
.54 1.52
.58 1.56
.67 1.65
.74 1.72
.79 1.76
.83 1.80
.90 1.87
.94 1.92
.98 1.95
2.01 1.98
2.05 2.02
2.10 2.06 ;
2.13 2.10 ;
2.16 2.13 ;
2.18 2.15 ;
2.21 2.17 ;
.25 1.23 1.22
.38 1.36 1.35
.46 1.43 1.42
.51 1.48 1.47
.55 1.52 1.51
.63 1.60 1.58
.70 1.67 1.65
.74 1.72 1.70
.78 1.75 1.73
.85 1.82 1.79
.89 1.86 1.84
.93 1.89 1.87
.96 1.92 1.90
.99 1.96 1.93
2.04 2.00 1.97
2.07 2.03 2.00
2.10 2.06 2.03 ;
2.12 2.08 2.05 ;
2.14 2.10 2.07 ;
.21 1.21
.34 1.33
.41 1.40
.46 1.45
.49 1.49
.57 1.56
.64 1.62
.68 1.67
.71 1.70
.78 1.76
.82 1.81
.85 1.84
.88 1.86
.91 1.90
.95 1.94
.98 1.97
2.01 1.99
2.03 2.01 ;
2.05 2.03 ;
.20 1.19
.32 1.31
.39 1.38
.44 1.43
.48 1.46
.55 1.54
.62 1.60
.66 1.64
.69 1.68
.75 1.73
.79 1.77
.83 1.81
.85 1.83
.88 1.86
.92 1.90
.95 1.93
.98 1.96
2.00 1.98
2.02 1.99
.19
.31
.37
.42
.46
.53
.59
.63
.66
.72
.76
.79
.82
.85
.89
.92
.93
.96
.98
                                                    D-86
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
     Table 19-6. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 2 (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.99 1.98 1.66 1.50 1.41 1.30
3.68 2.34 1.93 1.73 1.61 1.48
412 2.56 2.09 1.87 1.73 1.59
4.43 2,71 2.20 1.96 1.82 1.66
4.68 2.84 2.29 2.04 1.88 1.72
5.20 3.10 2.48 2.19 2.02 1.83
5.65 3.32 2,65 2.33 2.14 1.93
5.96 3.48 2,76 2.42 2.23 2.01
6.20 3.60 2.85 2.50 2.29 2.06
6.63 3.82 3.01 2,63 2.41 2.16 ;
6.94 3.98 3.12 2,72 2.49 2.23 ;
7.17 4.09 3.21 2.80 2.55 2.28 ;
7.35 4.19 3.28 2.85 2.61 2.33 ;
7.58 4.31 3.37 2.92 2,87.' 2.38 ;
7.86 4.45 3.48 3.02 2,75 2.45 ;
8.08 4.56 3.56 3.09 . 2-81 2.50 ;
8.26 4.66 3.63 3.14 2.86 2.54 ;
8.40 4.73 3.68 3.19 2.90 2.58 ;
8.53 4.80 3.73 3.23 2.94 2.61 '4
.24
.41
.50
.57
.62
.73
.82
.89
.94
1.02
1.09
2.14 ;
2.18 ;
1.22 :
i.28 ;
2.33 ;
2.37 ;
2.40 ;
1.43 ;
.20 1.17
.35 1.32
.44 1.40
.51 1.46
.55 1.51
.65 1.60
.74 1.68
.80 1.74
.84 1.78
.92 1.86
.98 1.91
2.03 1.95
2.06 1.99
Ml 2.03
2.16 2.08 ;
1.20 2.12 ;
2.24 2.16 ;
1.27 2.18 ;
1.29 2.21 ;
.15
.29
.38
.43
.48
.57
.65
.70
.74
.81
.87
.91
.94
.98
2.03
2.07 ;
>.io ;
M2 ;
MS ;
.13 1.12
.28 1.26
.36 1.34
.41 1.40
.45 1.44
.54 1.52
.62 1.60
.67 1.65
.71 1.69
.78 1.76
.83 1.80
.87 1.84
.90 1.87
.94 1.91
.99 1.96
1.02 1.99
2.06 2.02 ;
2.08 2.05 ;
MO 2.07 ;
.11 1.10 1.09
.25 1.23 1.22
.33 1.31 1.30
.38 1.36 1.35
.42 1.40 1.39
.51 1.49 1.47
.58 1.56 1.54
.63 1.60 1.59
.67 1.64 1.62
.74 1.71 1.69
.78 1.75 1.73
.82 1.79 1.77
.85 1.82 1.79
.89 1.85 1.83
.93 1.90 1.87
.97 1.93 1.90
2.00 1.96 1.93
1.02 1.98 1.95
2.04 2.00 1.97
.08 1.07
.21 1.21
.29 1.28
.34 1.33
.38 1.37
.46 1.45
.53 1.52
.57 1.56
.61 1.60
.67 1.66
.72 1.70
.75 1.74
.78 1.76
.81 1.80
.85 1.84
.89 1.87
.91 1.90
.93 1.92
.95 1.94
.07 1.06
.20 1.19
.27 1.26
.32 1.31
.36 1.35
.44 1.43
.51 1.49
.55 1.54
.59 1.57
.65 1.63
.69 1.68
.73 1.71
.75 1.73
.79 1.77
.83 1.81
.86 1.84
.88 1.86
.91 1.88
.92 1.90
.06
.19
.26
.31
.34
.42
.48
.53
.56
.62
.66
.70
.72
.75
.79
.82
.85
.87
.89
   Table 19-6. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 2  (20 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.68 2.34 1.93 1.73 1.61 1.48
443 2.71 2.20 1.96 1.82 1.66
4.88 2.94 2.37 2.10 1.94 1.76
5.20 3.10 2.48 2.19 2.02 1.83
5.44 3.22 2.57 2.27 2.09 1.89
5.96 3.48 2.76 2.42 2.23 2.01
6.40 3.70 2.92. 2.56 2.34 2.11
6.70 3.86 3.04 2.65 2.43 2.18 ;
6.94 3.98 3.12 2.72 2.49 2.23 ;
7.35 4.19 3.28 2.85 2.61 2.33 ;
7.64 4.34 3.39 2.95 2.69 2.40 ;
7.86 4.45 3.48 3.02 2.75 2.45 ;
8.04 4.54 3.54 3.07 2.80 2.49 ;
8.26 4.66 3.63 3.14 2,86 2.54 ;
8.53 4.80 3.73 3.23 2.94 2.61 ;
8.74 4.91 3.81 3.30 3,00 2.66 ;
8.91 4.99 3.88 3.35 3. OS, 2.70 ;
9.05 5.07 3.93 3.40 3.09 2.74 ;
9.17 5.13 3.98 3.44 3.12 2.77 ;
.41
.57
.66
.73
.78
.89
.98
2.04
2.09
MS ;
2.24 ;
2.28 ;
>.32 :
1.37 ;
2.43 ;
2.48 ;
2.5i ;
2.55 ;
2.57 ;
.35 1.32
.51 1.46
.59 1.55
.65 1.60
.70 1.65
.80 1.74
.88 1.82
.94 1.87
.98 1.91
2.06 1.99
2.12 2.04
2.16 2.08 ;
2.20 2.12 ;
2.24 2.16 ;
2.29 2.21 ;
2.34 2.25 ;
2.37 2.28 ;
2.40 2.31 ;
2.43 2.33 ;
.29
.43
.51
.57
.61
.70
.77
.83
.87
.94
.99
2.03
2.06 ;
2.10 ;
2.15 ;
2.18 ;
2.22 ;
2.24 ;
2.26 ;
.28 1.26
.41 1.40
.49 1.47
.54 1.52
.58 1.56
.67 1.65
.74 1.72
.79 1.77
.83 1.80
.90 1.87
.95 1.92
.99 1.96
2.02 1.99
2.06 2.02 ;
2.10 2.07 ;
2.14 2.10 ;
2.17 2.13 ;
2.19 2.16 ;
2.22 2.18 ;
.25 1.23 1.22
.38 1.36 1.35
.46 1.44 1.42
.51 1.49 1.47
.55 1.52 1.51
.63 1.60 1.59
.70 1.67 1.65
.75 1.72 1.70
.78 1.75 1.73
.85 1.82 1.79
.90 1.86 1.84
.93 1.90 1.87
.96 1.92 1.90
2.00 1.96 1.93
2.04 2.00 1.97
2.08 2.04 2.01
2.10 2.06 2.03 ;
2.13 2.08 2.05 ;
2.15 2.10 2.07 ;
.21 1.21
.34 1.33
.41 1.40
.46 1.45
.50 1.49
.57 1.56
.64 1.63
.68 1.67
.72 1.70
.78 1.76
.82 1.81
.85 1.84
.88 1.86
.91 1.90
.95 1.94
.98 1.97
2.01 1.99
2.03 2.02 ;
2.05 2.03 ;
.20 1.19
.32 1.31
.39 1.38
.44 1.43
.48 1.46
.55 1.54
.62 1.60
.66 1.64
.69 1.68
.75 1.73
.80 1.78
.83 1.81
.85 1.83
.88 1.86
.92 1.90
.96 1.93
.98 1.96
2.00 1.98
2.02 1.99
.19
.31
.37
.42
.46
.53
.59
.63
.66
.72
.76
.79
.82
.85
.89
.92
.94
.96
.98
                                                    D-87
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                      Unified Guidance
    Table 19-6. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 2 (20 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4.43 2.71 2.20 1.96 1.82 1.66 1.57
5.20 3.10 2.48 2.19 2.02 1.83 1.73
5.65 3.32 2.65 2.33 2.14 1.93 1.82
5.96 3.48 2.76 2.42 2.23 2.01 1.89
6.20 3.60 2.85 2.50 2.29 2.06 1.94
6.70 3.86 3.04 2.65 2.43 2.18 2.04
7.12 4.07 3,19 2.78 2.54 2.27 2.13 ;
7.42 4.22 3.31 2.87 2.62 2.34 2.19 ;
7.64 4.34 3.39 2.95 2.69 2.40 2.24 ;
8.04 4.54 3.54 3.07 2.80 2.49 2.32 ;
8.32 4.69 3.65 3,16 2.88 2.56 2.38 ;
8.53 4.80 3.73 3,23 2.94 2.61 2.43 ;
8.70 4.89 3.80 3,29 2.99 2.65 2.47 ;
8.91 4.99 3.88 3.35 3.05 2.70 2.51 ;
9.17 5.13 3.98 3.44 3,12 2.77 2.57 ;
9.37 5.23 4.06 3.50 3,18 2.82 2.62 ;
9.53 5.32 4.12 3.56 i 3,23 2.86 2.66 ;
9.66 5.39 4.17 3.60 3,27 2.89 2.69 ;
9.78 5.45 4.22 3.64 3.3O 2.92 2.71 ;
.51 1.46
.65 1.60
.74 1.68
.80 1.74
.84 1.78
.94 1.87
2.02 1.95
1.07 2.00
2.12 2.04
1.20 2.12 ;
1.25 2.17 ;
1.29 2.21 ;
2.33 2.24 ;
1.37 2.28 ;
2.43 2.33 ;
2.47 2.37 ;
2.50 2.40 ;
2.53 2.43 ;
2.55 2.45 ;
.43
.57
.65
.70
.74
.83
.90
.95
.99
2.06 ;
Ml ;
2.15 ;
2.18 ;
1.22 :
i.26 ;
2.30 ;
2.33 ;
2.36 ;
2.38 ;
.41 1.40
.54 1.52
.62 1.60
.67 1.65
.71 1.69
.79 1.77
.86 1.84
.91 1.88
.95 1.92
2.02 1.99
2.07 2.03 ;
2.10 2.07 ;
2.13 2.10 ;
2.17 2.13 ;
2.22 2.18 ;
2.25 2.21 ;
2.28 2.24 ;
2.30 2.26 ;
2.33 2.28 ;
.38 1.36 1.35
.51 1.49 1.47
.58 1.56 1.54
.63 1.60 1.59
.67 1.64 1.62
.75 1.72 1.70
.81 1.78 1.76
.86 1.83 1.80
.90 1.86 1.84
.96 1.92 1.90
2.01 1.97 1.94
2.04 2.00 1.97
2.07 2.03 2.00
2.10 2.06 2.03 ;
2.15 2.10 2.07 ;
2.18 2.14 2.11 ;
2.21 2.16 2.13 ;
2.23 2.19 2.15 ;
2.25 2.20 2.17 ;
.34 1.33
.46 1.45
.53 1.52
.57 1.56
.61 1.60
.68 1.67
.74 1.73
.79 1.77
.82 1.81
.88 1.86
.92 1.91
.95 1.94
.98 1.96
2.01 1.99
2.05 2.03 ;
2.08 2.06 ;
2.11 2.09 ;
2.13 2.11 ;
2.15 2.13 ;
.32 1.31
.44 1.43
.51 1.49
.55 1.54
.59 1.57
.66 1.64
.72 1.70
.76 1.74
.80 1.78
.85 1.83
.89 1.87
.92 1.90
.95 1.93
.98 1.96
2.02 1.99
2.05 2.02 ;
2.07 2.05 ;
2.09 2.07 ;
2.11 2.08 ;
.31
.42
.48
.53
.56
.63
.69
.73
.76
.82
.86
.89
.91
.94
.98
2.01
2.03
2.05
2.07
      Table 19-6. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 2 (40 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.84 2.38 1.94 1.74 1.62 1.49 1.41
4.71 2.78 2.23 1.98 1.83 1.66 1.57
5.25 3.02 2.40 2.12 1.95 1.77 1.67
5.64 3.20 2.53 2.22 2.04 1.84 1.73
5.95 3.34 2,62 2.29 2.10 1.90 1.79
6.60 3.64 2.83 2.46 2.25 2.02 1.89
7.16 3.89 3.00 2,60 2.37 2.12 1.98
7.56 4.07 3.13 2, 7O 2.46 2.19 2.05
7.86 4.21 3.23 2,78 2.53 2.25 2.10
8.41 4.46 3.40 2.92 2,85. 2.35 2.19 ;
8.79 4.64 3.53 3.02 2,73 2.42 2.25 ;
9.08 4.77 3.62 3.10 2.8O 2.48 2.30 ;
9.31 4.88 3.70 3.16 2,86 2.52 2.34 ;
9.59 5.01 3.79 3.24 2.92 2.58 2.39 ;
9.95 5.18 3.91 3.34 3.01 2,65" 2.45 ;
10.23 5.31 4.00 3.41 3.07 2, 70 2.50 ;
10.45 5.42 4.08 3.47 3.13 2,74 2.54 ;
10.63 5.50 4.14 3.52 3.17 2,78 2,57" :
10.79 5.58 4.20 3.57 3.21 2,81 2.6O \
.36 1.32 1.29
.51 1.46 1.43
.59 1.55 1.51
.65 1.60 1.57
.70 1.65 1.61
.80 1.74 1.70
.88 1.82 1.78
.94 1.87 1.83
.99 1.92 1.87
2.07 1.99 1.94
2.13 2.05 1.99
2.17 2.09 2.03
2.21 2.12 2.06 ;
2.25 2.16 2.10 ;
2.31 2.22 2.15 ;
2.35 2.26 2.19 ;
2.39 2.29 2.22 ;
2.42 2.32 2.25 ;
2.44 2.34 2.27 ;
.28 1.26 1.25
.41 1.40 1.38
.49 1.47 1.46
.54 1.52 1.51
.59 1.56 1.55
.67 1.65 1.63
.74 1.72 1.70
.79 1.77 1.75
.83 1.81 1.79
.90 1.87 1.85
.95 1.92 1.90
.99 1.96 1.93
2.02 1.99 1.96
2.06 2.03 2.00
2.11 2.07 2.04 ;
2.14 2.11 2.08 ;
2.17 2.14 2.11 ;
2.20 2.16 2.13 ;
2.22 2.18 2.15 ;
.23 1.22
.36 1.35
.44 1.42
.49 1.47
.52 1.51
.60 1.59
.67 1.65
.72 1.70
.75 1.73
.82 1.80
.86 1.84
.90 1.87
.93 1.90
.96 1.93
2.00 1.98
2.04 2.01
2.06 2.03 ;
2.09 2.06 ;
2.11 2.08 ;
.21 1.21
.34 1.33
.41 1.40
.46 1.45
.50 1.49
.57 1.56
.64 1.63
.68 1.67
.72 1.70
.78 1.76
.82 1.81
.85 1.84
.88 1.87
.91 1.90
.95 1.94
.99 1.97
2.01 1.99
2.03 2.02 ;
2.05 2.03 ;
.20
.32
.39
.44
.48
.55
.62
.66
.69
.75
.80
.83
.85
.89
.93
.96
.98
2.00
2.02
.19 1.19
.31 1.31
.38 1.37
.43 1.42
.46 1.46
.54 1.53
.60 1.59
.64 1.63
.68 1.66
.74 1.72
.78 1.76
.81 1.79
.83 1.82
.86 1.85
.90 1.89
.93 1.92
.96 1.94
.98 1.96
.99 1.98
                                                    D-88
                                                                                                    March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means
                                                                    Unified Guidance
   Table 19-6. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 2 (40 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4.71 2.78 2.23 1.98 1.83 1.66 1.57
5.64 3.20 2.53 2.22 2.04 1.84 1.73
6.20 3.45 2.70 2.36 2.16 1.94 1.83
6.60 3.64 2.83 2.46 2.25 2.02 1.89
6.91 3.78 2,92 2.54 2.32 2.07 1.94
7.56 4.07 3.13 2.70 2.46 2.19 2.05
8.11 4.32 3.31 2.84 2.58 2.29 2.14 ;
8.49 4.50 3.43 2,94 2.67 2.36 2.20 ;
8.79 4.64 3.53 3.O2 2.73 2.42 2.25 ;
9.31 4.88 3.70 3.16 2,38. 2.52 2.34 ;
9.68 5.05 3.82 3.26 2.94 2.59 2.40 ;
9.95 5.18 3.91 3.34 3,07 2.65 2.45 ;
10.18 5.29 3.99 3.40 3,06 2.69 2.49 ;
10.45 5.42 4.08 3.47 3.13 2.74 2.54 ;
10.79 5.58 4.20 3.57 3.21 2.81 2.60 ;
11.05 5.70 4.28 3.64 3.27 2,87 2.65 ;
11.27 5.81 4.36 3.70 3.32 2,91-' 2.69 ;
11.44 5.89 4.42 3.75 3.37 2. 95 2.72 ;
11.60 5.96 4.47 3.79 3.41 2.98 2.75 ;
.51 1.46 1.43
.65 1.60 1.57
.74 1.68 1.65
.80 1.74 1.70
.85 1.78 1.74
.94 1.87 1.83
1.02 1.95 1.90
>.08 2.01 1.95
M3 2.05 1.99
1.21 2.12 2.06 ;
1.26 2.17 2.11 ;
>.31 2.22 2.15 ;
2.34 2.25 2.18 ;
2.39 2.29 2.22 ;
2.44 2.34 2.27 ;
2.48 2.38 2.31 ;
2.52 2.41 2.34 ;
2.55 2.44 2.37 ;
1.57 2.46 2.39 ;
.41 1.40 1.38
.54 1.52 1.51
.62 1.60 1.58
.67 1.65 1.63
.71 1.69 1.67
.79 1.77 1.75
.87 1.84 1.82
.91 1.89 1.86
.95 1.92 1.90
1.02 1.99 1.96
1.07 2.04 2.01
Ml 2.07 2.04 ;
2.14 2.10 2.07 ;
1.17 2.14 2.11 ;
1.22 2.18 2.15 ;
1.26 2.22 2.19 ;
1.29 2.25 2.21 ;
2.31 2.27 2.24 ;
2.33 2.29 2.26 ;
.36 1.35
.49 1.47
.56 1.54
.60 1.59
.64 1.62
.72 1.70
.78 1.76
.83 1.81
.86 1.84
.93 1.90
.97 1.94
2.00 1.98
2.03 2.00
2.06 2.03 ;
Ml 2.08 ;
M4 2.11 ;
M7 2.13 ;
M9 2.15 ;
1.21 2.17 ;
.34 1.33
.46 1.45
.53 1.52
.57 1.56
.61 1.60
.68 1.67
.74 1.73
.79 1.77
.82 1.81
.88 1.87
.92 1.91
.95 1.94
.98 1.96
2.01 1.99
2.05 2.03 ;
2.08 2.06 ;
Ml 2.09 ;
M3 2.11 ;
M5 2.13 ;
.32
.44
.51
.55
.59
.66
.72
.76
.80
.85
.89
.93
.95
.98
1.02
1.05 :
1.07 :
1.09 :
MI ;
.31 1.31
.43 1.42
.49 1.48
.54 1.53
.57 1.56
.64 1.63
.70 1.69
.74 1.73
.78 1.76
.83 1.82
.87 1.86
.90 1.89
.93 1.91
.96 1.94
.99 1.98
1.02 2.01
2.05 2.03
1.07 2.05
2.08 2.07
    Table 19-6. K-Multipliers  for 1-of-2  Interwell Prediction Limits on Means of Order 2 (40 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
5.64
6.60
7.16
7.56
7.86
8.49
9.02
9.39
9.68
10.18
10.53
10.79
11.01
11.27
11.60
11.85
12.05
12.22
12.37

6
3.20
3.64
3.89
4.07
4.21
4.50
4.75
4.92
5.05
5.29
5.45
5.58
5.68
5.81
5.96
6.08
6.18
6.26
6.33

8
2.53
2.83
3.00
3.13
3.23
3.43
3.60
3.73
3.82
3.99
4.11
4.20
4.27
4.36
4.47
4.56
4.63
4.68
4.74

10
2.22
2.46
2.60
2.70
2.78
2.94
3.O9
3,18
. 3,26
3.40
3.49
3.57
3.63
3.70
3.79
3.86
3.92
3.97
4.01

12
2.04
2.25
2.37
2.46
2.53
2.67
2.79
2.88
2.94
3.06
3'. 14 '
3,21
3.28
3.32
3.41
3.47
3.52
3.56
3.60

16
1.84
2.02
2.12
2.19
2.25
2.36
2.47
2.54
2.59
2.69
2.76
2.81
2.86
2.91
2.98
3.03
3..O7
3.11
3.14

20
1.73
1.89
1.98
2.05
2.10
2.20
2.29
2.35
2.40
2.49
2.55
2.60
2.64
2.69
2.75
2.79
2.83
2.86
2.89

25
.65
.80
.88
.94
.99
2.08
2.16
2.22
2.26
2.34
2.40
2.44
2.48
2.52
2.57
2.62
2.65
2.68
2.70

30
1.60
1.74
1.82
1.87
1.92
2.01
2.08
2.13
2.17
2.25
2.30
2.34
2.37
2.41
2.46
2.50
2.54
2.56
2.59

35 40 45 50 60 70 80 90 100 125
1.57 .54 1.52 1.51 .49 1.47 .46 1.45 .44 .43
1.70 .67 1.65 1.63 .60 1.59 .57 1.56 .55 .54
1.78 .74 1.72 1.70 .67 1.65 .64 1.63 .62 .60
1.83 .79 1.77 1.75 .72 1.70 .68 1.67 .66 .64
1.87 .83 1.81 1.79 .75 1.73 .72 1.70 .69 .68
1.95 .91 1.89 1.86 .83 1.81 .79 1.77 .76 .74
2.02 .98 1.95 1.93 .89 1.87 .85 1.83 .82 .79
2.08 2.03 2.00 1.97 .94 1.91 .89 1.87 .86 .84
2.11 2.07 2.04 2.01 .97 1.94 .92 1.91 .89 .87
2.18 2.14 2.10 2.07 2.03 2.00 .98 1.96 .95 .93
2.23 2.18 2.15 2.12 2.07 2.04 2.02 2.00 .99 .97
2.27 2.22 2.18 2.15 2.11 2.08 2.05 2.03 2.02 .99
2.30 2.25 2.21 2.18 2.13 2.10 2.08 2.06 2.04 2.02
2.34 2.29 2.25 2.21 2.17 2.13 2.11 2.09 2.07 2.05
2.39 2.33 2.29 2.26 2.21 2.17 2.15 2.13 2.11 2.08
2.43 2.37 2.32 2.29 2.24 2.20 2.18 2.16 2.14 2.11
2.46 2.40 2.35 2.32 2.27 2.23 2.20 2.18 2.17 2.14
2.48 2.42 2.38 2.34 2.29 2.25 2.22 2.20 2.19 2.16
2.50 2.44 2.40 2.36 2.31 2.27 2.24 2.22 2.20 2.17

150
1.42
1.53
1.59
1.63
1.66
1.73
1.79
1.83
1.86
1.91
1.95
1.98
2.00
2.03
2.07
2.09
2.12
2.14
2.15

                                                    D-89
                                                                                                  March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means
Unified Guidance
      Table 19-7. K-Multipliers for 1-of-3 Interwell Prediction Limits on Means of Order 2 (1 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
0.37 (
0.64 (
0.79 (
0.90 (
0.98 (
.15 (
.29
.38
.46
.59
.68
.75
.81
.87
.96
2.02
2.08
2.12
2.16
6
).27
).50
).62
).71
).77
).91
.02
.10
.15
.26
.33
.38
.43
.48
.55
.60
.64
.67
.70
8 10
0.22 0.19
0.43 0.39
0.54 0.50
0.62 0.57
0.68 0.63
0.80 0.75
0.90 0.84
0.97 0.90
1.03 0.95
1.12 .04
1.18 .10
1.23 .15
1.27 .18
1.32 .23
1.38 .28
1.42 .33
1.46 .36
1.49 .39
1.52 .41
12
0.17
0.36
0.47
0.54
0.60
0.71
0.80
0.86
0.91
0.99
1.05
1.09
1.13
1.17
1.22
1.26
1.30
1.32
1.35
16
0.14 (
0.33 (
0.43 (
0.50 (
0.55 (
0.66 (
0.74 (
0.80 (
0.85 (
0.93 (
0.98 (
1.02 (
1.06
1.10
1.15
1.18
1.21
1.24
1.26
20 25 30 35 40 45 50 60 70 80 90 100 125 150
).13 0.11 0.11 0.10 0.10 0.09 0.09 0.08 0.08 0.08 0.08 0.08 0.07 0.07
).31 0.30 0.29 0.28 0.27 0.27 0.26 0.26 0.25 0.25 0.25 0.25 0.24 0.24
).41 0.39 0.38 0.37 0.37 0.36 0.36 0.35 0.35 0.34 0.34 0.34 0.33 0.33
).48 0.46 0.45 0.44 0.43 0.42 0.42 0.41 0.41 0.40 0.40 0.40 0.39 0.39
).53 0.51 0.49 0.48 0.48 0.47 0.47 0.46 0.45 0.45 0.45 0.44 0.44 0.44
).63 0.61 0.59 0.58 0.57 0.56 0.56 0.55 0.54 0.54 0.54 0.53 0.53 0.53
).71 0.69 0.67 0.66 0.65 0.64 0.63 0.63 0.62 0.61 0.61 0.61 0.60 0.60
).77 0.74 0.72 0.71 0.70 0.69 0.69 0.68 0.67 0.66 0.66 0.66 0.65 0.65
).81 0.78 0.76 0.75 0.74 0.73 0.73 0.72 0.71 0.70 0.70 0.69 0.69 0.68
).89 0.86 0.84 0.82 0.81 0.80 0.79 0.78 0.78 0.77 0.76 0.76 0.75 0.75
).94 0.91 0.89 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.81 0.81 0.80 0.79
).98 0.95 0.92 0.91 0.89 0.88 0.88 0.86 0.86 0.85 0.84 0.84 0.83 0.83
.01 0.98 0.95 0.94 0.92 0.91 0.91 0.89 0.88 0.88 0.87 0.87 0.86 0.85
.05 .01 0.99 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.91 0.90 0.89 0.89
.10 .06 1.04 .02 .00 0.99 0.98 0.97 0.96 0.95 0.95 0.94 0.93 0.93
.14 .10 1.07 .05 .04 1.03 1.02 1.00 0.99 0.98 0.98 0.97 0.97 0.96
.17 .13 1.10 .08 .06 1.05 1.04 1.03 1.02 1.01 1.00 1.00 0.99 0.98
.19 .15 1.12 .10 .09 1.08 1.07 1.05 1.04 1.03 1.03 1.02 1.01 1.00
.21 .17 1.14 .12 .11 1.09 1.08 1.07 1.06 1.05 1.04 1.04 1.03 1.02
   Table 19-7. K-Multipliers for 1-of-3 Interwell Prediction Limits on Means of Order 2 (1 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

(
(

















4
).64 (
).90 (
.04 (
.15 (
.22 (
.38 ]
.52
.61
.68
.81
.89
.96
>.01
>.08
>.16
1.22
1.27
1.31
2.35
6 8
).50 0.43 (
).71 0.62 (
).83 0.73 (
).91 0.80 (
).97 0.86 (
L.10 0.97 (
.20 1.07 (
.27 1.13
.33 1.18
.43 1.27
.50 1.33
.55 1.38
.59 1.42
.64 1.46
.70 1.52
.75 1.56
.79 1.60
.82 1.62
.85 1.65
10 12 16
).39 0.36 0.33 (
).57 0.54 0.50 (
).68 0.64 0.59 (
).75 0.71 0.66 (
).80 0.76 0.71 (
).90 0.86 0.80 (
).99 0.94 0.88 (
.05 1.00 0.94 (
.10 1.05 0.98 (
.18 1.13 1.06
.24 1.18 1.11
.28 1.22 1.15
.32 1.26 1.18
.36 1.30 1.21
.41 1.35 1.26
.45 1.38 1.30
.49 1.42 1.33
.51 1.44 1.35
.54 1.46 1.37
20
).31 (
).48 (
).57 (
).63 (
).68 (
).77 (
).85 (
).90 (
).94 (
.01 (
.06
.10
.13
.17
.21
.25
.28
.30
.32
25 30
).30 0.29 (
).46 0.45 (
).55 0.53 (
).61 0.59 (
).65 0.63 (
).74 0.72 (
).82 0.80 (
).87 0.85 (
).91 0.89 (
).98 0.95 (
.03 1.00 (
.06 1.04
.09 1.06
.13 1.10
.17 1.14
.20 1.18
.23 1.20
.25 1.22
.27 1.24
35
).28 (
).44 (
).52 (
).58 (
).62 (
).71 (
).78 (
).83 (
).87 (
).94 (
).98 (
.02
.05
.08
.12
.15
.18
.20
.22
40 45
).27 0.27 (
).43 0.42 (
).51 0.51 (
).57 0.56 (
).61 0.61 (
).70 0.69 (
).77 0.76 (
).82 0.81 (
).86 0.85 (
).92 0.91 (
).97 0.96 (
.00 0.99 (
.03 1.02
.06 1.05
.11 1.09
.14 1.13
.16 1.15
.19 1.17
.20 1.19
50 60 70 80 90 100 125 150
).26 0.26 0.25 0.25 0.25 0.25 0.24 0.24
).42 0.41 0.41 0.40 0.40 0.40 0.39 0.39
).50 0.49 0.49 0.49 0.48 0.48 0.48 0.47
).56 0.55 0.54 0.54 0.54 0.53 0.53 0.53
).60 0.59 0.59 0.58 0.58 0.57 0.57 0.57
).69 0.68 0.67 0.66 0.66 0.66 0.65 0.65
).76 0.75 0.74 0.73 0.73 0.73 0.72 0.71
).80 0.79 0.79 0.78 0.77 0.77 0.76 0.76
).84 0.83 0.82 0.81 0.81 0.81 0.80 0.79
).91 0.89 0.88 0.88 0.87 0.87 0.86 0.85
).95 0.94 0.93 0.92 0.91 0.91 0.90 0.90
).98 0.97 0.96 0.95 0.95 0.94 0.93 0.93
.01 1.00 0.99 0.98 0.97 0.97 0.96 0.95
.04 1.03 1.02 .01 1.00 .00 0.99 0.98
.08 1.07 1.06 .05 1.04 .04 1.03 .02
.12 1.10 1.09 .08 1.07 .07 1.06 .05
.14 1.13 1.11 .11 1.10 .09 1.08 .08
.16 1.15 1.13 .13 1.12 .11 1.10 .10
.18 1.16 1.15 .14 1.14 .13 1.12 .11
                                                   D-90
                                                                                                 March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                      Unified Guidance
     Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2  (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
(


















).90 (
.15 (
.29 ]
.38 ]
.46 ]
.61 ]
.74
.83
.89
2.01
2.09
2.16
2.21
2.27
2.35
2.41
2.46
2.50
2.53
).71 0.62 (
).91 0.80 (
L.02 0.90 (
L.10 0.97 (
L.15 1.03 (
L.27 1.13
.37 1.22
.44 1.29
.50 1.33
.59 1.42
.65 1.47
.70 1.52
.74 1.55
.79 1.60
.85 1.65
.90 1.69
.93 1.72
.96 1.75
.99 1.78
).57 0.54 0.50 (
).75 0.71 0.66 (
).84 0.80 0.74 (
).90 0.86 0.80 (
).95 0.91 0.85 (
.05 1.00 0.94 (
.14 1.08 1.01 (
.20 1.14 1.07
.24 1.18 1.11
.32 1.26 1.18
.37 1.31 1.23
.41 1.35 1.26
.45 1.38 1.29
.49 1.42 1.33
.54 1.46 1.37
.58 1.50 1.41
.61 1.53 1.44
.63 1.56 1.46
.66 1.58 1.48
).48 (
).63 (
).71 (
).77 (
).81 (
).90 (
).97 (
.02 (
.06
.13
.18
.21
.24
.28
.32
.35
.38
.40
.42
).46 0.45 (
).61 0.59 (
).69 0.67 (
).74 0.72 (
).78 0.76 (
).87 0.85 (
).94 0.92 (
).99 0.96 (
.03 1.00 (
.09 1.06
.14 1.11
.17 1.14
.20 1.17
.23 1.20
.27 1.24
.31 1.28
.33 1.30
.36 1.32
.37 1.34
).44 (
).58 (
).66 (
).71 (
).75 (
).83 (
).90 (
).95 (
).98 (
.05
.09
.12
.15
.18
.22
.25
.28
.30
.32
).43 0.42 (
).57 0.56 (
).65 0.64 (
).70 0.69 (
).74 0.73 (
).82 0.81 (
).89 0.88 (
).93 0.92 (
).97 0.96 (
.03 1.02
.07 1.06
.11 1.09
.13 1.12
.16 1.15
.20 1.19
.24 1.22
.26 1.25
.28 1.27
.30 1.28
).42 0.41 0.41 (
).56 0.55 0.54 (
).63 0.63 0.62 (
).69 0.68 0.67 (
).73 0.72 0.71 (
).80 0.79 0.79 (
).87 0.86 0.85 (
).92 0.90 0.89 (
).95 0.94 0.93 (
.01 1.00 0.99 (
.05 1.04 1.03
.08 1.07 1.06
.11 1.09 1.08
.14 1.13 1.11
.18 1.16 1.15
.21 1.19 1.18
.24 1.22 1.21
.26 1.24 1.22
.27 1.25 1.24
).40 0.40 (
).54 0.54 (
).61 0.61 (
).66 0.66 (
).70 0.70 (
).78 0.77 (
).84 0.84 (
).89 0.88 (
).92 0.91 (
).98 0.97 (
.02 1.01
.05 1.04
.08 1.07
.11 1.10
.14 1.14
.17 1.16
.20 1.19
.21 1.21
.23 1.22
).40 0.39 (
).53 0.53 (
).61 0.60 (
).66 0.65 (
).69 0.69 (
).77 0.76 (
).83 0.83 (
).88 0.87 (
).91 0.90 (
).97 0.96 (
.01 1.00 (
.04 1.03
.06 1.05
.09 1.08
.13 1.12
.16 1.15
.18 1.17
.20 1.19
.22 1.21
).39
).53
).60
).65
).68
).76
).82
).86
).90
).95
).99
.02
.05
.08
.11
.14
.16
.18
.20
      Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2 (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
(


















).71 (
.00 (
.17 (
.30 (
.39 ]
.59 ]
.76
.88
.97
2.13
2.24
2.33
2.40
2.48
>.59 ]
2.67 ]
2.13 :
2.79 :
2.83 :
).53 0.45 (
).76 0.66 (
).89 0.77 (
).98 0.85 (
L.05 0.91 (
L.19 1.04 (
.31 1.14
.40 1.21
.46 1.27
.58 1.37
.66 1.44
.72 1.49
.77 1.53
.83 1.58
L.90 1.65
L.96 1.70
2.01 1.74
2.05 1.77
2.08 1.80
).41 0.38 0.34 (
).60 0.56 0.51 (
).70 0.66 0.61 (
).78 0.73 0.68 (
).83 0.78 0.72 (
).95 0.89 0.83 (
.04 0.98 0.91 (
.11 1.05 0.97 (
.16 1.09 1.01 (
.25 1.18 1.09
.32 1.24 1.15
.36 1.28 1.19
.40 1.32 1.22
.45 1.36 1.26
.51 1.42 1.31
.55 1.46 1.35
.59 1.50 1.38
.62 1.52 1.41
.65 1.55 1.43
).32 (
).49 (
).58 (
).64 (
).69 (
).79 (
).87 (
).92 (
).96 (
.04
.09
.13
.16
.20
.25
.29
.32
.34
.36
).30 0.29 (
).47 0.45 (
).55 0.54 (
).62 0.60 (
).66 0.64 (
).75 0.73 (
).83 0.81 (
).88 0.86 (
).92 0.90 (
.00 0.97 (
.05 1.02
.08 1.05
.11 1.08
.15 1.12
.20 1.16
.23 1.20
.26 1.23
.29 1.25
.31 1.27
).28 (
).44 (
).53 (
).59 (
).63 (
).72 (
).79 (
).84 (
).88 (
).95 (
.00 (
.03
.06
.10
.14
.17
.20
.22
.24
).28 0.27 (
).43 0.43 (
).52 0.51 (
).58 0.57 (
).62 0.61 (
).71 0.70 (
).78 0.77 (
).83 0.82 (
).87 0.86 (
).93 0.92 (
).98 0.97 (
.02 1.00 (
.04 1.03
.08 1.06
.12 1.11
.15 1.14
.18 1.17
.20 1.19
.22 1.21
).27 0.26 0.26 (
).42 0.41 0.41 (
).51 0.50 0.49 (
).56 0.55 0.55 (
).60 0.60 0.59 (
).69 0.68 0.67 (
).76 0.75 0.74 (
).81 0.80 0.79 (
).85 0.83 0.83 (
).91 0.90 0.89 (
).96 0.94 0.93 (
).99 0.98 0.97 (
.02 1.00 0.99 (
.05 1.04 1.02
.10 1.08 1.07
.13 1.11 1.10
.15 1.13 1.12
.17 1.16 1.14
.19 1.17 1.16
).25 0.25 (
).41 0.40 (
).49 0.48 (
).54 0.54 (
).58 0.58 (
).67 0.66 (
).74 0.73 (
).78 0.78 (
).82 0.81 (
).88 0.88 (
).92 0.92 (
).96 0.95 (
).98 0.98 (
.02 1.01
.06 1.05
.09 1.08
.11 1.10
.13 1.12
.15 1.14
).25 0.25 0.24
).40 0.40 0.39
).48 0.48 0.47
).54 0.53 0.53
).58 0.57 0.57
).66 0.65 0.65
).73 0.72 0.72
).77 0.77 0.76
).81 0.80 0.80
).87 0.86 0.86
).91 0.90 0.90
).95 0.94 0.93
).97 0.96 0.96
.00 0.99 0.99
.04 1.03 .03
.07 1.06 .05
.10 1.09 .08
.12 1.11 .10
.14 1.12 .12
                                                    D-91
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
   Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2  (2 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.00 (
.30 (
.47 ]
.59 ]
.68 ]
.88 ]
2.04
2.15
2.24
2.40
2.50
2.59
2.65
2.73 ;
2.83 :
2.91 :
2.97 :
3.03 :
3.07 :
).76 0.66 (
).98 0.85 (
L.10 0.96 (
L.19 1.04 (
L.26 1.09
L.40 1.21
.52 1.31
.60 1.38
.66 1.44
.77 1.53
.85 1.60
.90 1.65
.95 1.69
2.01 1.74
2.08 1.80
2.14 1.85
2.18 1.88
2.22 1.92
2.25 1.94
).60 0.56 0.51 (
).78 0.73 0.68 (
).88 0.83 0.76 (
).95 0.89 0.83 (
.00 0.94 0.87 (
.11 1.05 0.97 (
.20 1.13 1.05
.27 1.19 1.10
.32 1.24 1.15
.40 1.32 1.22
.46 1.38 1.27
.51 1.42 1.31
.54 1.45 1.34
.59 1.50 1.38
.65 1.55 1.43
.69 1.59 1.47
.72 1.62 1.50
.75 1.65 1.53
.78 1.67 1.55
).49 (
).64 (
).73 (
).79 (
).83 (
).92 (
.00 (
.05
.09
.16
.21
.25
.28
.32
.36
.40
.43
.45
.47
).47 0.45 (
).62 0.60 (
).70 0.68 (
).75 0.73 (
).80 0.77 (
).88 0.86 (
).96 0.93 (
.01 0.98 (
.05 1.02
.11 1.08
.16 1.13
.20 1.16
.23 1.19
.26 1.23
.31 1.27
.34 1.30
.37 1.33
.39 1.35
.41 1.37
).44 (
).59 (
).66 (
).72 (
).76 (
).84 (
).91 (
).96 (
.00 (
.06
.11
.14
.17
.20
.24
.27
.30
.32
.34
).43 0.43 (
).58 0.57 (
).65 0.65 (
).71 0.70 (
).75 0.74 (
).83 0.82 (
).90 0.89 (
).94 0.93 (
).98 0.97 (
.04 1.03
.09 1.07
.12 1.11
.15 1.13
.18 1.17
.22 1.21
.25 1.24
.28 1.26
.30 1.28
.32 1.30
).42 0.41 0.41 (
).56 0.55 0.55 (
).64 0.63 0.62 (
).69 0.68 0.67 (
).73 0.72 0.71 (
).81 0.80 0.79 (
).88 0.86 0.85 (
).92 0.91 0.90 (
).96 0.94 0.93 (
.02 1.00 0.99 (
.06 1.05 1.03
.10 1.08 1.07
.12 1.10 1.09
.15 1.13 1.12
.19 1.17 1.16
.22 1.20 1.19
.25 1.23 1.21
.27 1.25 1.23
.29 1.27 1.25
).41 0.40 (
).54 0.54 (
).62 0.61 (
).67 0.66 (
).71 0.70 (
).78 0.78 (
).85 0.84 (
).89 0.89 (
).92 0.92 (
).98 0.98 (
.02 1.02
.06 1.05
.08 1.07
.11 1.10
.15 1.14
.18 1.17
.20 1.19
.22 1.21
.24 1.23
).40 0.40 (
).54 0.53 (
).61 0.60 (
).66 0.65 (
).70 0.69 (
).77 0.77 (
).84 0.83 (
).88 0.87 (
).91 0.90 (
).97 0.96 (
.01 1.00
.04 1.03
.07 1.06
.10 1.09
.14 1.12
.16 1.15
.19 1.17
.21 1.19
.22 1.21
).39
).53
).60
).65
).69
).76
).82
).87
).90
).96
.00
.03
.05
.08
.12
.14
.17
.19
.20
     Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2 (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.30 0.98 0.85 (
.59 1.19 1.04 (
.76 1.31 1.14
.88 1.40 1.21
.97 1.46 1.27
2.15 1.60 1.38
2.31 1.71 1.48
2.42 1.79 1.55
2.50 1.85 1.60
2.65 1.95 1.69
2.75 2.02 1.75
2.83 2.08 1.80
2.90 2.13 1.84
2.97 2.18 1.88
3.07 2.25 1.94
3.14 2.30 1.99
3.20 2.35 2.03
3.25 2.38 2.06
3.30 2.41 2.08
).78 0.73 0.68 (
).95 0.89 0.83 (
.04 0.98 0.91 (
.11 1.05 0.97 (
.16 1.09 1.01 (
.27 1.19 1.10
.35 1.28 1.18
.42 1.33 1.23
.46 1.38 1.27
.54 1.45 1.34
.60 1.51 1.39
.65 1.55 1.43
.68 1.58 1.46
.72 1.62 1.50
.78 1.67 1.55
.82 1.71 1.58
.85 1.74 1.61
.88 1.77 1.64
.91 1.79 1.66
).64 (
).79 (
).87 (
).92 (
).96 (
.05
.12
.17
.21
.28
.33
.36
.39
.43
.47
.51
.53
.56
.58
).62 0.60 (
).75 0.73 (
).83 0.81 (
).88 0.86 (
).92 0.90 (
.01 0.98 (
.08 1.05
.12 1.09
.16 1.13
.23 1.19
.27 1.24
.31 1.27
.33 1.30
.37 1.33
.41 1.37
.44 1.40
.47 1.43
.49 1.45
.51 1.47
).59 (
).72 (
).79 (
).84 (
).88 (
).96 (
.03
.07
.11
.17
.21
.24
.27
.30
.34
.37
.40
.42
.44
).58 0.57 (
).71 0.70 (
).78 0.77 (
).83 0.82 (
).87 0.86 (
).94 0.93 (
.01 1.00 (
.05 1.04
.09 1.07
.15 1.13
.19 1.17
.22 1.21
.25 1.23
.28 1.26
.32 1.30
.35 1.33
.37 1.36
.39 1.38
.41 1.39
).56 0.55 0.55 (
).69 0.68 0.67 (
).76 0.75 0.74 (
).81 0.80 0.79 (
).85 0.83 0.83 (
).92 0.91 0.90 (
).99 0.97 0.96 (
.03 1.01 1.00 (
.06 1.05 1.03
.12 1.10 1.09
.16 1.14 1.13
.19 1.17 1.16
.22 1.20 1.18
.25 1.23 1.21
.29 1.27 1.25
.32 1.30 1.28
.34 1.32 1.30
.36 1.34 1.32
.38 1.36 1.34
).54 0.54 (
).67 0.66 (
).74 0.73 (
).78 0.78 (
).82 0.81 (
).89 0.89 (
).95 0.95 (
).99 0.99 (
.02 1.02
.08 1.07
.12 1.11
.15 1.14
.17 1.17
.20 1.19
.24 1.23
.27 1.26
.29 1.28
.31 1.30
.33 1.32
).54 0.53 (
).66 0.65 (
).73 0.72 (
).77 0.77 (
).81 0.80 (
).88 0.87 (
).94 0.93 (
).98 0.97 (
.01 1.00
.07 1.06
.11 1.09
.14 1.12
.16 1.15
.19 1.17
.22 1.21
.25 1.24
.27 1.26
.29 1.28
.31 1.29
).53
).65
).72
).76
).80
).87
).92
).96
.00
.05
.09
.12
.14
.17
.20
.23
.25
.27
.28
                                                    D-92
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
      Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2 (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.20 0.87 0.74 (
1.56 1.11 0.95 (
1.78 1.26 1.07 (
1.93 1.36 1.15
2.05 1.44 1.21
2.31 1.60 1.35
2.53 1.74 1.46
2.69 1.84 1.54
2.81 1.91 1.60
3.02 2.05 1.71
3.17 2.14 1.78
3.29 2.21 1.84
3.38 2.27 1.89
3.49 2.34 1.95
3.63 2.43 2.02
3.74 2.50 2.07
3.83 2.55 2.12
3.90 2.60 2.16
3.97 2.64 2.19
).67 0.63 0.58 (
).86 0.80 0.74 (
).97 0.90 0.83 (
.04 0.97 0.89 (
.10 1.02 0.94 (
.21 1.13 1.04 (
.31 1.22 1.12
.38 1.29 1.18
.44 1.34 1.22
.53 1.43 1.30
.60 1.49 1.35
.65 1.53 1.40
.69 1.57 1.43
.74 1.62 1.47
.81 1.68 1.53
.86 1.72 1.57
.90 1.76 1.60
.93 1.79 1.63
.96 1.82 1.65
).54 (
).70 (
).78 (
).84 (
).89 (
).98 (
.06
.11
.15
.23
.28
.32
.35
.39
.44
.47
.51
.53
.55
).52 0.50 (
).67 0.65 (
).75 0.73 (
).81 0.78 (
).85 0.82 (
).94 0.91 (
.01 0.98 (
.06 1.03
.10 1.07
.17 1.13
.22 1.18
.26 1.21
.29 1.24
.32 1.28
.37 1.32
.40 1.36
.43 1.38
.46 1.41
.48 1.43
).49 (
).63 (
).71 (
).76 (
).80 (
).89 (
).96 (
.00 (
.04
.11
.15
.19
.21
.25
.29
.32
.35
.37
.39
).48 0.48 (
).62 0.61 (
).70 0.69 (
).75 0.74 (
).79 0.78 (
).87 0.86 (
).94 0.93 (
).99 0.97 (
.02 1.01
.09 1.07
.13 1.11
.16 1.15
.19 1.17
.22 1.21
.27 1.25
.30 1.28
.32 1.31
.35 1.33
.37 1.34
).47 0.46 0.46 (
).61 0.60 0.59 (
).68 0.67 0.66 (
).73 0.72 0.71 (
).77 0.76 0.75 (
).85 0.84 0.83 (
).92 0.90 0.89 (
).96 0.95 0.94 (
.00 0.98 0.97 (
.06 1.04 1.03
.10 1.08 1.07
.13 1.12 1.10
.16 1.14 1.13
.19 1.17 1.16
.23 1.21 1.20
.26 1.24 1.23
.29 1.27 1.25
.31 1.29 1.27
.33 1.30 1.29
).45 0.45 (
).59 0.58 (
).66 0.65 (
).71 0.70 (
).74 0.74 (
).82 0.82 (
).88 0.88 (
).93 0.92 (
).96 0.95 (
.02 1.01
.06 1.05
.09 1.08
.12 1.11
.15 1.14
.18 1.17
.21 1.20
.24 1.23
.26 1.25
.27 1.26
).45 0.44 (
).58 0.57 (
).65 0.64 (
).70 0.69 (
).74 0.73 (
).81 0.80 (
).87 0.86 (
).92 0.91 (
).95 0.94 (
.01 1.00 (
.05 1.03
.08 1.06
.10 1.09
.13 1.12
.17 1.15
.20 1.18
.22 1.20
.24 1.22
.25 1.24
).44
).57
).64
).69
).72
).80
).86
).90
).93
).99
.03
.06
.08
.11
.15
.17
.20
.21
.23
   Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2 (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.56 1.11 0.95 (
1.93 1.36 1.15
2.15 1.50 1.26
2.31 1.60 1.35
2.43 1.68 1.41
2.69 1.84 1.54
2.91 1.97 1.65
3.06 2.07 1.72
3.17 2.14 1.78
3.38 2.27 1.89
3.52 2.36 1.96
3.63 2.43 2.02
3.72 2.49 2.06
3.83 2.55 2.12
3.97 '"2. '64'. 2.19
4.07 -2,71 2.24 ;
4.16 2,76 2.29 ;
4.23 2.80 2.32 ;
4.29 2.84 2.35 ;
).86 0.80 0.74 (
.04 0.97 0.89 (
.14 1.07 0.98 (
.21 1.13 1.04 (
.27 1.18 1.08
.38 1.29 1.18
.48 1.38 1.26
.55 1.44 1.31
.60 1.49 1.35
.69 1.57 1.43
.76 1.63 1.48
.81 1.68 1.53
.85 1.72 1.56
.90 1.76 1.60
.96 1.82 1.65
>.01 1.86 1.69
>.04 1.90 1.72
>.08 1.93 1.75
MO 1.95 1.77
).70 (
).84 (
).92 (
).98 (
.02 (
.11
.19
.24
.28
.35
.40
.44
.47
.51
.55
.59
.62
.64
.67
).67 0.65 (
).81 0.78 (
).88 0.86 (
).94 0.91 (
).98 0.95 (
.06 1.03
.13 1.10
.18 1.14
.22 1.18
.29 1.24
.33 1.29
.37 1.32
.40 1.35
.43 1.38
.48 1.43
.51 1.46
.54 1.49
.56 1.51
.58 1.53
).63 (
).76 (
).84 (
).89 (
).93 (
.00 (
.07
.12
.15
.21
.26
.29
.32
.35
.39
.42
.45
.47
.49
).62 0.61 (
).75 0.74 (
).82 0.81 (
).87 0.86 (
).91 0.90 (
).99 0.97 (
.05 1.04
.10 1.08
.13 1.11
.19 1.17
.23 1.22
.27 1.25
.29 1.27
.32 1.31
.37 1.34
.40 1.38
.42 1.40
.44 1.42
.46 1.44
).61 0.60 0.59 (
).73 0.72 0.71 (
).80 0.79 0.78 (
).85 0.84 0.83 (
).89 0.87 0.86 (
).96 0.95 0.94 (
.03 1.01 1.00 (
.07 1.05 1.04
.10 1.08 1.07
.16 1.14 1.13
.20 1.18 1.17
.23 1.21 1.20
.26 1.24 1.22
.29 1.27 1.25
.33 1.30 1.29
.36 1.33 1.31
.38 1.36 1.34
.40 1.38 1.36
.42 1.39 1.37
).59 0.58 (
).71 0.70 (
).77 0.77 (
).82 0.82 (
).86 0.85 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.12 1.11
.15 1.15
.18 1.17
.21 1.20
.24 1.23
.27 1.26
.30 1.29
.32 1.31
.34 1.33
.36 1.35
).58 0.57 (
).70 0.69 (
).77 0.76 (
).81 0.80 (
).85 0.84 (
).92 0.91 (
).97 0.96 (
.01 1.00
.05 1.03
.10 1.09
.14 1.13
.17 1.15
.19 1.18
.22 1.20
.25 1.24
.28 1.27
.30 1.29
.32 1.31
.34 1.32
).57
).69
).75
).80
).83
).90
).96
.00
.03
.08
.12
.15
.17
.20
.23
.26
.28
.30
.31
                                                    D-93
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                      Unified Guidance
     Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2 (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.93 1.36 1.15
2.31 1.60 1.35
2.53 1.74 1.46
2.69 1.84 1.54
2.81 1.91 1.60
3.06 2.07 1.72
3.27 2.20 1.83
3.41 2.29 1.91
3.52 2.36 1.96
3.72 2.49 2.06
3.86 2.57 2.14
3.97 2.64 2.19
4.05 2.69 2.23 ;
416 2.76 2.29 ;
4.29 2.84 2.35 ;
4.39 '"z.'gj" 2.4i ;
4.47 2.96 2.45 ;
4.54 3.00 2.48 ;
4.60 3.04 2.51 ;
.04 0.97 0.89 (
.21 1.13 1.04 (
.31 1.22 1.12
.38 1.29 1.18
.44 1.34 1.22
.55 1.44 1.31
.64 1.53 1.39
.71 1.59 1.44
.76 1.63 1.48
.85 1.72 1.56
.91 1.77 1.61
.96 1.82 1.65
>.00 1.85 1.68
>.04 1.90 1.72
MO 1.95 1.77
>.15 1.99 1.81
>.19 2.03 1.84
1.22 2.06 1.86
1.24 2.08 1.89
).84 (
).98 (
.06
.11
.15
.24
.31
.36
.40
.47
.52
.55
.58
.62
.67
.70
.73
.75
.77
).81 0.78 (
).94 0.91 (
.01 0.98 (
.06 1.03
.10 1.07
.18 1.14
.25 1.21
.30 1.25
.33 1.29
.40 1.35
.44 1.39
.48 1.43
.51 1.45
.54 1.49
.58 1.53
.62 1.56
.64 1.59
.67 1.61
.69 1.63
).76 (
).89 (
).96 (
.00 (
.04
.12
.18
.22
.26
.32
.36
.39
.42
.45
.49
.52
.55
.57
.58
).75 0.74 (
).87 0.86 (
).94 0.93 (
).99 0.97 (
.02 1.01
.10 1.08
.16 1.14
.20 1.18
.23 1.22
.29 1.27
.33 1.31
.37 1.34
.39 1.37
.42 1.40
.46 1.44
.49 1.47
.52 1.49
.54 1.51
.55 1.53
).73 0.72 0.71 (
).85 0.84 0.83 (
).92 0.90 0.89 (
).96 0.95 0.94 (
.00 0.98 0.97 (
.07 1.05 1.04
.13 1.11 1.10
.17 1.15 1.14
.20 1.18 1.17
.26 1.24 1.22
.30 1.27 1.26
.33 1.30 1.29
.35 1.33 1.31
.38 1.36 1.34
.42 1.39 1.37
.45 1.42 1.40
.47 1.44 1.42
.49 1.46 1.44
.51 1.48 1.46
).71 0.70 (
).82 0.82 (
).88 0.88 (
).93 0.92 (
).96 0.95 (
.03 1.02
.08 1.08
.12 1.12
.15 1.15
.21 1.20
.24 1.23
.27 1.26
.30 1.29
.32 1.31
.36 1.35
.39 1.37
.41 1.40
.43 1.41
.44 1.43
).70 0.69 (
).81 0.80 (
).87 0.86 (
).92 0.91 (
).95 0.94 (
.01 1.00
.07 1.06
.11 1.10
.14 1.13
.19 1.18
.23 1.21
.25 1.24
.28 1.26
.30 1.29
.34 1.32
.37 1.35
.39 1.37
.41 1.39
.42 1.40
).69
).80
).86
).90
).93
.00
.05
.09
.12
.17
.20
.23
.25
.28
.31
.34
.36
.38
.39
      Table 19-7. K-Multipliers for 1-of-3 Interwell Prediction Limits on Means of Order 2 (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.64 1.14 0.96 (
2.07 1.40 1.17
2.34 1.56 1.29
2.53 1.67 1.38
2.68 1.76 1.45
3.00 1.94 1.59
3.27 2.09 1.71
3.47 2.20 1.80
3.62 2.29 1.86
3.88 2.44 1.98
4.07 "£55" 2.06
4.21 2.63 2.13
4.33 2.70 2.18
4.47 2.78 2.24
4.65 2.88 2.32 ;
4.78 2.96 2.38 ;
4.90 3.02 2.43 :
4.99 3.07 2,47 '<
5.07 3.12 2,31 ;
).87 0.81 0.74 (
.05 0.98 0.90 (
.16 1.08 0.98 (
.24 1.15 1.04 (
.29 1.20 1.09
.42 1.31 1.19
.52 1.40 1.27
.59 1.47 1.33
.65 1.52 1.37
.75 1.61 1.45
.82 1.68 1.51
.88 1.73 1.55
.92 1.77 1.59
.97 1.81 1.63
>.04 1.88 1.68
MO 1.92 1.73
>.14 1.96 1.76
>.17 1.99 1.79
1.20 2.02 1.81
).70 (
).85 (
).93 (
).99 (
.03 (
.12
.20
.25
.29
.36
.42
.46
.49
.53
.58
.61
.65
.67
.69
).67 0.65 (
).81 0.78 (
).89 0.86 (
).94 0.91 (
).98 0.95 (
.07 1.03
.14 1.10
.19 1.15
.23 1.19
.30 1.25
.34 1.30
.38 1.33
.41 1.36
.45 1.39
.49 1.44
.53 1.47
.56 1.50
.58 1.52
.60 1.54
).63 (
).77 (
).84 (
).89 (
).93 (
.01 (
.07
.12
.16
.22
.26
.30
.32
.36
.40
.43
.46
.48
.50
).62 0.62 (
).75 0.74 (
).83 0.81 (
).87 0.86 (
).91 0.90 (
).99 0.98 (
.05 1.04
.10 1.08
.13 1.12
.20 1.18
.24 1.22
.27 1.25
.30 1.28
.33 1.31
.37 1.35
.40 1.38
.43 1.41
.45 1.43
.47 1.44
).61 0.60 0.59 (
).74 0.72 0.71 (
).81 0.79 0.78 (
).85 0.84 0.83 (
).89 0.88 0.86 (
).97 0.95 0.94 (
.03 1.01 1.00 (
.07 1.05 1.04
.10 1.09 1.07
.16 1.14 1.13
.21 1.18 1.17
.24 1.21 1.20
.26 1.24 1.22
.29 1.27 1.25
.33 1.31 1.29
.36 1.34 1.32
.39 1.36 1.34
.41 1.38 1.36
.43 1.40 1.38
).59 0.58 (
).71 0.70 (
).78 0.77 (
).82 0.82 (
).86 0.85 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.12 1.11
.16 1.15
.19 1.18
.21 1.20
.24 1.23
.28 1.26
.30 1.29
.33 1.32
.35 1.33
.36 1.35
).58 0.57 (
).70 0.69 (
).77 0.76 (
).81 0.80 (
).85 0.84 (
).92 0.91 (
).98 0.96 (
.02 1.00
.05 1.04
.10 1.09
.14 1.13
.17 1.16
.19 1.18
.22 1.21
.26 1.24
.28 1.27
.31 1.29
.32 1.31
.34 1.32
).57
).69
).75
).80
).83
).90
).96
.00
.03
.08
.12
.15
.17
.20
.23
.26
.28
.30
.31
                                                    D-94
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
   Table 19-7. K-Multipliers for 1-of-3 Interwell Prediction Limits on Means of Order 2 (10 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.07 1.40 1.17
2.53 1.67 1.38
2.80 1.83 1.50
3.00 1.94 1.59
3.15 2.02 1.66
3.47 2.20 1.80
3.74 2.36 1.91
3.93 2.46 2.00
4.07 2.55 2.06
4.33 2.7O 2.18
4.51 2.80 2.26
4.65 2.88 2.32 ;
4.76 2.94 2.37 ;
4.90 3.02 2.43 ;
5.07 3.12 2.51 ;
5.20 3.19 2.57 ;
5.30 3.26 2,61 '<
5.39 3.31 2.65 ;
5.47 3.35 2.69 '4
.05 0.98 0.90 (
.24 1.15 1.04 (
.34 1.24 1.13
.42 1.31 1.19
.47 1.36 1.23
.59 1.47 1.33
.70 1.56 1.41
.77 1.63 1.47
.82 1.68 1.51
.92 1.77 1.59
.99 1.83 1.64
>.04 1.88 1.68
>.09 1.92 1.72
>.14 1.96 1.76
1.20 2.02 1.81
1.26 2.07 1.85
>.30 2.11 1.89
1.33 2.14 1.91
>.36 2.16 1.94
).85 (
).99 (
.06
.12
.16
.25
.32
.38
.42
.49
.54
.58
.61
.65
.69
.73
.76
.79
.81
).81 0.78 (
).94 0.91 (
.01 0.98 (
.07 1.03
.11 1.07
.19 1.15
.26 1.21
.31 1.26
.34 1.30
.41 1.36
.46 1.40
.49 1.44
.52 1.47
.56 1.50
.60 1.54
.64 1.58
.66 1.60
.69 1.62
.71 1.64
).77 (
).89 (
).96 (
.01 (
.04
.12
.18
.23
.26
.32
.37
.40
.43
.46
.50
.53
.56
.58
.60
).75 0.74 (
).87 0.86 (
).94 0.93 (
).99 0.98 (
.03 1.01
.10 1.08
.16 1.15
.21 1.19
.24 1.22
.30 1.28
.34 1.32
.37 1.35
.40 1.38
.43 1.41
.47 1.44
.50 1.47
.52 1.50
.55 1.52
.56 1.54
).74 0.72 0.71 (
).85 0.84 0.83 (
).92 0.90 0.89 (
).97 0.95 0.94 (
.00 0.98 0.97 (
.07 1.05 1.04
.13 1.11 1.10
.17 1.15 1.14
.21 1.18 1.17
.26 1.24 1.22
.30 1.28 1.26
.33 1.31 1.29
.36 1.33 1.31
.39 1.36 1.34
.43 1.40 1.38
.45 1.43 1.40
.48 1.45 1.43
.50 1.47 1.45
.52 1.48 1.46
).71 0.70 (
).82 0.82 (
).88 0.88 (
).93 0.92 (
).96 0.95 (
.03 1.02
.09 1.08
.13 1.12
.16 1.15
.21 1.20
.25 1.24
.28 1.26
.30 1.29
.33 1.32
.36 1.35
.39 1.38
.41 1.40
.43 1.42
.45 1.43
).70 0.69 (
).81 0.80 (
).87 0.86 (
).92 0.91 (
).95 0.94 (
.02 1.00
.07 1.06
.11 1.10
.14 1.13
.19 1.18
.23 1.21
.26 1.24
.28 1.26
.31 1.29
.34 1.32
.37 1.35
.39 1.37
.41 1.39
.42 1.40
).69
).80
).86
).90
).93
.00
.05
.09
.12
.17
.20
.23
.25
.28
.31
.34
.36
.38
.39
    Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2 (10 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.53 1.67 1.38
3.00 1.94 1.59
3.27 2.09 1.71
3.47 2.20 1.80
3.62 2.29 1.86
3.93 2.46 2.00
4.19 2.61 2.11
437 2.72 2.20
451 2.80 2.26
4.76 2.94 2.37 ;
4.93 3.04 2.45 ;
5.07 3.12 2.51 ;
5.17 3.18 2.55 ;
5.30 3.26 2.61 ;
5.47 3.35 2.69 ;
5.60 3.42 2.74 ;
5.70 3.48 2.79 ;
5.78 3.53 2.83 ;
5.86 3.57 2,86 \
.24 1.15 1.04 (
.42 1.31 1.19
.52 1.40 1.27
.59 1.47 1.33
.65 1.52 1.37
.77 1.63 1.47
.87 1.72 1.54
.94 1.78 1.60
.99 1.83 1.64
>.09 1.92 1.72
>.15 1.98 1.77
1.20 2.02 1.81
1.25 2.06 1.85
>.30 2.11 1.89
>.36 2.16 1.94
>.41 2.21 1.98
>.45 2.25 2.01
>.48 2.28 2.04
>.51 2.30 2.06
).99 (
.12
.20
.25
.29
.38
.45
.50
.54
.61
.66
.69
.72
.76
.81
.84
.87
.90
.92
).94 0.91 (
.07 1.03
.14 1.10
.19 1.15
.23 1.19
.31 1.26
.37 1.32
.42 1.37
.46 1.40
.52 1.47
.57 1.51
.60 1.54
.63 1.57
.66 1.60
.71 1.64
.74 1.68
.77 1.70
.79 1.72
.81 1.74
).89 (
.01 (
.07
.12
.16
.23
.29
.33
.37
.43
.47
.50
.53
.56
.60
.63
.65
.67
.69
).87 0.86 (
).99 0.98 (
.05 1.04
.10 1.08
.13 1.12
.21 1.19
.27 1.25
.31 1.29
.34 1.32
.40 1.38
.44 1.41
.47 1.44
.49 1.47
.52 1.50
.56 1.54
.59 1.57
.62 1.59
.64 1.61
.66 1.63
).85 0.84 0.83 (
).97 0.95 0.94 (
.03 1.01 1.00 (
.07 1.05 1.04
.10 1.09 1.07
.17 1.15 1.14
.23 1.21 1.19
.27 1.25 1.23
.30 1.28 1.26
.36 1.33 1.31
.40 1.37 1.35
.43 1.40 1.38
.45 1.42 1.40
.48 1.45 1.43
.52 1.48 1.46
.54 1.51 1.49
.57 1.53 1.51
.59 1.55 1.53
.60 1.57 1.55
).82 0.82 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.13 1.12
.18 1.17
.22 1.21
.25 1.24
.30 1.29
.33 1.32
.36 1.35
.38 1.37
.41 1.40
.45 1.43
.47 1.46
.49 1.48
.51 1.50
.53 1.51
).81 0.80 (
).92 0.91 (
).98 0.96 (
.02 1.00
.05 1.04
.11 1.10
.16 1.15
.20 1.19
.23 1.21
.28 1.26
.31 1.30
.34 1.32
.36 1.35
.39 1.37
.42 1.40
.45 1.43
.47 1.45
.49 1.47
.50 1.48
).80
).90
).96
.00
.03
.09
.14
.18
.20
.25
.29
.31
.33
.36
.39
.42
.44
.45
.47
                                                    D-95
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
     Table 19-7. K-Multipliers for 1-of-3 Interwell Prediction Limits on Means of Order 2 (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.16 1.43 1.19
2.69 1.72 1.40
3.02 1.89 1.53
3.26 2.01 1.62
3.45 2.11 1.70
3.84 2.31 1.85
4.19 2,49 1.98
4.43 2,61 2.07
4.62 2.71 2.14
4.95 2.88 2.27
5.19 3.00 2.36 ;
5.37 3.10 2,43 ;
5.51 3.17 2.48 \
5.69 3.26 2,55 :
5.91 3.38 2,64 '*
6.08 3.47 2.71 ;
6.22 3.54 2.76 ;
6.34 3.60 2.81
6.44 3.66 2.85
.06 0.99 0.90 (
.25 1.16 1.05 (
.36 1.25 1.14
.44 1.32 1.20
.50 1.38 1.24
.62 1.49 1.34
.73 1.59 1.42
.81 1.65 1.48
.87 1.71 1.53
.97 1.80 1.61
>.05 1.87 1.66
Ml 1.92 1.71
>.16 1.96 1.74
1.21 2.01 1.79
1.29 2.08 1.84
>.34 2.13 1.89
2.39 2.17 1.92
2.43. 2.20 1.95
2:46 2.23 1.98
).85 (
).99 (
.07
.12
.17
.26
.33
.38
.43
.50
.55
.59
.62
.66
.71
.75
.78
.81
.83
).81 0.78 (
).94 0.91 (
.02 0.98 (
.07 1.03
.11 1.07
.19 1.15
.26 1.22
.31 1.26
.35 1.30
.42 1.36
.47 1.41
.50 1.44
.53 1.47
.57 1.51
.61 1.55
.65 1.58
.68 1.61
.70 1.63
.72 1.65
).77 (
).89 (
).96 (
.01 (
.05
.12
.19
.23
.27
.33
.37
.40
.43
.46
.51
.54
.56
.59
.61
).75 0.74 (
).88 0.86 (
).94 0.93 (
).99 0.98 (
.03 1.01
.10 1.09
.16 1.15
.21 1.19
.24 1.22
.30 1.28
.34 1.32
.38 1.35
.40 1.38
.43 1.41
.47 1.45
.50 1.48
.53 1.50
.55 1.52
.57 1.54
).74 0.72 0.71 (
).85 0.84 0.83 (
).92 0.90 0.89 (
).97 0.95 0.94 (
.00 0.98 0.97 (
.07 1.05 1.04
.13 1.11 1.10
.17 1.15 1.14
.21 1.18 1.17
.26 1.24 1.22
.30 1.28 1.26
.34 1.31 1.29
.36 1.33 1.31
.39 1.36 1.34
.43 1.40 1.38
.46 1.43 1.41
.48 1.45 1.43
.50 1.47 1.45
.52 1.49 1.46
).71 0.70 (
).82 0.82 (
).89 0.88 (
).93 0.92 (
).96 0.95 (
.03 1.02
.09 1.08
.13 1.12
.16 1.15
.21 1.20
.25 1.24
.28 1.27
.30 1.29
.33 1.32
.36 1.35
.39 1.38
.41 1.40
.43 1.42
.45 1.43
).70 0.69 (
).81 0.80 (
).87 0.86 (
).92 0.91 (
).95 0.94 (
.02 1.00
.07 1.06
.11 1.10
.14 1.13
.19 1.18
.23 1.21
.26 1.24
.28 1.26
.31 1.29
.34 1.32
.37 1.35
.39 1.37
.41 1.39
.42 1.41
).69
).80
).86
).90
).93
.00
.05
.09
.12
.17
.20
.23
.25
.28
.31
.34
.36
.38
.39
   Table 19-7. K-Multipliers for 1-of-3 Interwell Prediction Limits on Means of Order 2  (20 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.69 1.72 1.40
3.26 2.01 1.62
3.60 2.19 1.75
3.84 2.31 1.85
4.03 2.41 1.92
443 2.61 2.07
4.77 2.79 2.20
5.00 2.91 2.29
5.19 3.00 2.36 ;
5.51 3.17 2.48 ;
5.74 3.29 2.57 ;
5.91 3.38 2,64 ;
6.05 3.45 2.69 ;
6.22 3.54 2,70 :
6.44 3.66 2.85 ;
6.61 3.74 2.91 ;
6.74 3.81 2.96 ;
6.85 3.87 3.01 ;
6.95 3.92 3.04 ",
.25 1.16 1.05 (
.44 1.32 1.20
.55 1.42 1.28
.62 1.49 1.34
.68 1.54 1.39
.81 1.65 1.48
.92 1.75 1.56
.99 1.82 1.62
>.05 1.87 1.66
>.16 1.96 1.74
1.23 2.03 1.80
1.29 2.08 1.84
1.33 2.12 1.88
>.39 2.17 1.92
>.46 2.23 1.98
>.51 2.28 2.02
>.56 2.32 2.05
>.60 2.35 2.08
2.63' 2.38 2.10
).99 (
.12
.20
.26
.30
.38
.46
.51
.55
.62
.67
.71
.74
.78
.83
.87
.90
.93
.95
).94 0.91 (
.07 1.03
.14 1.10
.19 1.15
.23 1.19
.31 1.26
.38 1.33
.43 1.38
.47 1.41
.53 1.47
.58 1.52
.61 1.55
.64 1.58
.68 1.61
.72 1.65
.76 1.69
.79 1.71
.81 1.74
.83 1.76
).89 (
.01 (
.08
.12
.16
.23
.29
.34
.37
.43
.47
.51
.53
.56
.61
.64
.66
.68
.70
).88 0.86 (
).99 0.98 (
.06 1.04
.10 1.09
.14 1.12
.21 1.19
.27 1.25
.31 1.29
.34 1.32
.40 1.38
.44 1.42
.47 1.45
.50 1.47
.53 1.50
.57 1.54
.60 1.57
.62 1.59
.64 1.62
.66 1.63
).85 0.84 0.83 (
).97 0.95 0.94 (
.03 1.01 1.00 (
.07 1.05 1.04
.11 1.09 1.07
.17 1.15 1.14
.23 1.21 1.19
.27 1.25 1.23
.30 1.28 1.26
.36 1.33 1.31
.40 1.37 1.35
.43 1.40 1.38
.45 1.42 1.40
.48 1.45 1.43
.52 1.49 1.46
.55 1.51 1.49
.57 1.54 1.51
.59 1.56 1.53
.61 1.57 1.55
).82 0.82 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.13 1.12
.18 1.17
.22 1.21
.25 1.24
.30 1.29
.34 1.32
.36 1.35
.39 1.37
.41 1.40
.45 1.43
.47 1.46
.50 1.48
.51 1.50
.53 1.51
).81 0.80 (
).92 0.91 (
).98 0.96 (
.02 1.00
.05 1.04
.11 1.10
.16 1.15
.20 1.19
.23 1.21
.28 1.26
.31 1.30
.34 1.32
.36 1.35
.39 1.37
.42 1.41
.45 1.43
.47 1.45
.49 1.47
.50 1.48
).80
).90
).96
.00
.03
.09
.14
.18
.20
.25
.29
.31
.33
.36
.39
.42
.44
.45
.47
                                                    D-96
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                      Unified Guidance
    Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2 (20 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.26 2.01 1.62
3.84 2.31 1.85
4.19 2.49 1.98
443 2.61 2.07
4.62 2.71 2.14
5.00 ,2,97 2.29
5.33 3.08 2.42 ;
5.56 3.20 2.50 ;
5.74 3.29 2.57 ;
6.05 3.45 2.69 ;
6.27 3.57 2.78 ;
6.44 3.66 2,85 \
6.57 3.73 2.9O '<
6.74 3.81 2,96 *
6.95 3.92 3.04 ;
7.11 4.01 3.11 ;
7.24 4.07 3.16 ;
7.34 4.13 3.20 ;
7.44 4.18 3.24 ;
.44 1.32 1.20
.62 1.49 1.34
.73 1.59 1.42
.81 1.65 1.48
.87 1.71 1.53
.99 1.82 1.62
MO 1.91 1.70
>.17 1.98 1.76
1.23 2.03 1.80
1.33 2.12 1.88
>.40 2.18 1.93
>.46 2.23 1.98
>.50 2.27 2.01
>.56 2.32 2.05
>.63 2.38 2.10
>.68 2.43 2.15
2.72 2.47 2.18 ;
1.76 2.50 2.21 ;
1.79 2.53 2.23 ;
.12
.26
.33
.38
.43
.51
.58
.63
.67
.74
.79
.83
.86
.90
.95
.99
1.02
>.04
>.06
.07 1.03
.19 1.15
.26 1.22
.31 1.26
.35 1.30
.43 1.38
.50 1.44
.54 1.48
.58 1.52
.64 1.58
.69 1.62
.72 1.65
.75 1.68
.79 1.71
.83 1.76
.87 1.79
.89 1.81
.92 1.84
.94 1.85
.01 (
.12
.19
.23
.27
.34
.40
.44
.47
.53
.57
.61
.63
.66
.70
.73
.76
.78
.80
).99 0.98 (
.10 1.09
.16 1.15
.21 1.19
.24 1.22
.31 1.29
.37 1.35
.41 1.39
.44 1.42
.50 1.47
.54 1.51
.57 1.54
.59 1.57
.62 1.59
.66 1.63
.69 1.66
.72 1.68
.74 1.70
.75 1.72
).97 0.95 0.94 (
.07 1.05 1.04
.13 1.11 1.10
.17 1.15 1.14
.21 1.18 1.17
.27 1.25 1.23
.33 1.30 1.28
.37 1.34 1.32
.40 1.37 1.35
.45 1.42 1.40
.49 1.46 1.44
.52 1.49 1.46
.54 1.51 1.49
.57 1.54 1.51
.61 1.57 1.55
.64 1.60 1.57
.66 1.62 1.60
.68 1.64 1.61
.70 1.66 1.63
).93 0.92 (
.03 1.02
.09 1.08
.13 1.12
.16 1.15
.22 1.21
.27 1.26
.31 1.30
.34 1.32
.39 1.37
.42 1.41
.45 1.43
.47 1.46
.50 1.48
.53 1.51
.55 1.54
.58 1.56
.59 1.58
.61 1.59
).92 0.91 (
.02 1.00
.07 1.06
.11 1.10
.14 1.13
.20 1.19
.25 1.24
.29 1.27
.31 1.30
.36 1.35
.40 1.38
.42 1.41
.44 1.43
.47 1.45
.50 1.48
.53 1.51
.55 1.53
.57 1.54
.58 1.56
).90
.00
.05
.09
.12
.18
.23
.26
.29
.33
.37
.39
.41
.44
.47
.49
.51
.53
.54
      Table 19-7. K-Multipliers for 1-of-3 Interwell Prediction Limits on Means of Order 2 (40 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.81 1.74 1.42
3.46 2.06 1.65
3.87 2.25 1.78
4.17 2.39 1.88
4.40 2, SO 1.96
4.90 2.73 2.12
5.33 2.93 2.26
5.63 3.07 2.36 ;
5.86 3.18 2,44 :
6.29 3.38 2.58 '4
6.58 3.51 2.68 ;
6.81 3.62 2.75 ;
6.99 3.71 2.81 ;
7.21 3.81 2.89
7.49 3.95 2.98
7.71 4.05 3.06
7.89 4.13 3.12
8.03 4.20 3.17
8.16 4.26 3.21
.26 1.16 1.05 (
.45 1.33 1.20
.56 1.43 1.28
.64 1.50 1.35
.70 1.56 1.39
.84 1.67 1.49
.95 1.77 1.57
>.03 1.84 1.63
>.09 1.90 1.68
1.21 2.00 1.76
1.29 2.07 1.82
>.35 2.12 1.86
>.40 2.16 1.90
2.46 2.22 1.95
2.54 2.29 2.00
2,60 2.34 2.05
2,65 2.38 2.08
2.69- 2.42 2.11
2.73 -2.45 2.14
).99 (
.13
.20
.26
.30
.39
.47
.52
.56
.63
.69
.73
.76
.80
.85
.89
.92
.95
.97
).94 0.91 (
.07 1.04
.14 1.10
.19 1.15
.23 1.19
.31 1.27
.38 1.33
.43 1.38
.47 1.41
.54 1.48
.59 1.52
.62 1.56
.65 1.58
.69 1.62
.73 1.66
.77 1.69
.80 1.72
.82 1.74
.84 1.76
).89 (
.01 (
.08
.12
.16
.23
.30
.34
.37
.43
.48
.51
.54
.57
.61
.64
.67
.69
.71
).88 0.86 (
).99 0.98 (
.06 1.04
.10 1.09
.14 1.12
.21 1.19
.27 1.25
.31 1.29
.35 1.32
.40 1.38
.44 1.42
.48 1.45
.50 1.48
.53 1.51
.57 1.54
.60 1.57
.63 1.60
.65 1.62
.67 1.64
).85 0.84 0.83 (
).97 0.95 0.94 (
.03 1.01 1.00 (
.07 1.05 1.04
.11 1.09 1.07
.18 1.15 1.14
.23 1.21 1.19
.27 1.25 1.23
.31 1.28 1.26
.36 1.33 1.31
.40 1.37 1.35
.43 1.40 1.38
.45 1.42 1.40
.48 1.45 1.43
.52 1.49 1.47
.55 1.52 1.49
.57 1.54 1.51
.59 1.56 1.53
.61 1.57 1.55
).82 0.82 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.13 1.12
.18 1.17
.22 1.21
.25 1.24
.30 1.29
.34 1.32
.36 1.35
.39 1.37
.41 1.40
.45 1.43
.47 1.46
.50 1.48
.51 1.50
.53 1.52
).81 0.80 (
).92 0.91 (
).98 0.97 (
.02 1.01
.05 1.04
.11 1.10
.16 1.15
.20 1.19
.23 1.21
.28 1.26
.32 1.30
.34 1.33
.36 1.35
.39 1.37
.42 1.41
.45 1.43
.47 1.45
.49 1.47
.50 1.48
).80
).90
).96
.00
.03
.09
.14
.18
.20
.25
.29
.31
.34
.36
.39
.42
.44
.45
.47
                                                    D-97
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
   Table 19-7. K-Multipliers for 1-of-3 Interwell Prediction Limits on Means of Order 2 (40 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.46 2.06 1.65 .45 1.33 1.20
4.17 2.39 1.88 .64 1.50 1.35
459 2.59 2.02 .76 1.60 1.43
4.90 2,73 2.12 .84 1.67 1.49
5.13 2.84 2.20 .90 1.73 1.54
5.63 3.07 2.36 2.03 1.84 1.63
6.06 3.27 2.50 2.15 1.94 1.72
6.35 3.41 2.60 2.23 2.01 1.77
6.58 3.51 2,68 2.29 2.07 1.82
6.99 3.71 2.81 2.40 2.16 1.90
7.28 3.84 2.91 2.48 2.23 1.96
7.49 3.95 2.98 2.54 2.29 2.00
7.67 4.03 3.04 2.59 2.33 2.04
7.89 4.13 3.12 2,65 2.38 2.08
8.16 4.26 3.21 2,73 2.45 2.14
8.37 4.36 3.28 2,79 2.50 2.18 ;
8.53 4.44 3.34 2.83 2.54 2.22 ;
8.68 4.51 3.39 2,87 2.58 2.25 ;
8.80 4.57 3.43 2.91 2,61 2.27 ;
.13
.26
.34
.39
.43
.52
.59
.65
.69
.76
.81
.85
.88
.92
.97
>.01
>.04
>.07
>.09
.07 1.04
.19 1.15
.27 1.22
.31 1.27
.35 1.30
.43 1.38
.50 1.44
.55 1.49
.59 1.52
.65 1.58
.70 1.63
.73 1.66
.76 1.69
.80 1.72
.84 1.76
.88 1.80
.91 1.82
.93 1.85
.95 1.87
.01 (
.12
.19
.23
.27
.34
.40
.44
.48
.54
.58
.61
.64
.67
.71
.74
.76
.79
.80
).99 0.98 (
.10 1.09
.17 1.15
.21 1.19
.24 1.22
.31 1.29
.37 1.35
.41 1.39
.44 1.42
.50 1.48
.54 1.51
.57 1.54
.60 1.57
.63 1.60
.67 1.64
.70 1.66
.72 1.69
.74 1.71
.76 1.73
).97 0.95 0.94 (
.07 1.05 1.04
.13 1.11 1.10
.18 1.15 1.14
.21 1.18 1.17
.27 1.25 1.23
.33 1.30 1.29
.37 1.34 1.32
.40 1.37 1.35
.45 1.42 1.40
.49 1.46 1.44
.52 1.49 1.47
.55 1.51 1.49
.57 1.54 1.51
.61 1.57 1.55
.64 1.60 1.58
.66 1.62 1.60
.68 1.64 1.61
.70 1.66 1.63
).93 0.92 (
.03 1.02
.09 1.08
.13 1.12
.16 1.15
.22 1.21
.27 1.26
.31 1.30
.34 1.32
.39 1.37
.42 1.41
.45 1.43
.47 1.46
.50 1.48
.53 1.52
.56 1.54
.58 1.56
.59 1.58
.61 1.59
).92 0.91 (
.02 1.01
.07 1.06
.11 1.10
.14 1.13
.20 1.19
.25 1.24
.29 1.27
.32 1.30
.36 1.35
.40 1.38
.42 1.41
.45 1.43
.47 1.45
.50 1.48
.53 1.51
.55 1.53
.57 1.54
.58 1.56
).90
.00
.05
.09
.12
.18
.23
.26
.29
.34
.37
.39
.41
.44
.47
.49
.51
.53
.54
    Table 19-7. K-Multipliers  for 1-of-3 Interwell Prediction Limits on Means of Order 2 (40 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4.17 2.39 1.88 1.64 1.50 1.35
4.90 2.73 2.12 1.84 1.67 1.49
5.33 2,a? 2.26 1.95 1.77 1.57
5.63 3.07 2.36 2.03 1.84 1.63
5.86 3.18 2.44 2.09 1.90 1.68
6.35 3.41 2.60 2.23 2.01 1.77
6.77 3.60 2.74 2.34 2.11 1.86
7.06 3.74 2,83 2.42 2.18 1.91
7.28 3.84 2,9?; 2.48 2.23 1.96
7.67 4.03 3.04 2.59 2.33 2.04
7.95 4.16 3.14 2.67 2.40 2.10
8.16 4.26 3.21 2.73 2.45 2.14
8.33 4.34 3.27 2.77 2.49 2.18 ;
8.53 4.44 3.34 2,83. 2.54 2.22 ;
8.80 4.57 3.43 2,97 2.61 2.27 ;
9.00 4.67 3.50 2,97 2.66 2.32 ;
9.16 4.74 3.56 '3,0-1 2.70 2.35 ;
9.29 4.81 3.60 3,05 2.73 2.38 ;
9.41 4.86 3.64 . 3,08 2.76 2.40 ;
.26
.39
.47
.52
.56
.65
.72
.77
.81
.88
.93
.97
>.oo
>.04
>.09
>.13
>.16 ;
>.19 ;
>.2i ;
.19 1.15
.31 1.27
.38 1.33
.43 1.38
.47 1.41
.55 1.49
.61 1.55
.66 1.59
.70 1.63
.76 1.69
.81 1.73
.84 1.76
.87 1.79
.91 1.82
.95 1.87
.99 1.90
1.02 1.92
>.04 1.95
>.06 1.97
.12
.23
.30
.34
.37
.44
.50
.55
.58
.64
.68
.71
.73
.76
.80
.84
.86
.88
.90
.10 1.09
.21 1.19
.27 1.25
.31 1.29
.35 1.32
.41 1.39
.47 1.44
.51 1.48
.54 1.51
.60 1.57
.64 1.61
.67 1.64
.69 1.66
.72 1.69
.76 1.73
.79 1.75
.81 1.78
.83 1.80
.85 1.81
.07 1.05 1.04
.18 1.15 1.14
.23 1.21 1.19
.27 1.25 1.23
.31 1.28 1.26
.37 1.34 1.32
.42 1.40 1.37
.46 1.43 1.41
.49 1.46 1.44
.55 1.51 1.49
.58 1.55 1.52
.61 1.57 1.55
.63 1.60 1.57
.66 1.62 1.60
.70 1.66 1.63
.73 1.69 1.66
.75 1.71 1.68
.77 1.73 1.69
.78 1.74 1.71
.03 1.02
.13 1.12
.18 1.17
.22 1.21
.25 1.24
.31 1.30
.36 1.35
.39 1.38
.42 1.41
.47 1.46
.50 1.49
.53 1.52
.55 1.54
.58 1.56
.61 1.59
.64 1.62
.66 1.64
.67 1.66
.69 1.67
.02 1.01
.11 1.10
.16 1.15
.20 1.19
.23 1.21
.29 1.27
.34 1.32
.37 1.35
.40 1.38
.45 1.43
.48 1.46
.50 1.48
.52 1.50
.55 1.53
.58 1.56
.61 1.58
.63 1.60
.64 1.62
.66 1.63
.00
.09
.14
.18
.20
.26
.31
.34
.37
.41
.45
.47
.49
.51
.54
.57
.59
.60
.62
                                                    D-98
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
      Table 19-8. K-Multipliers  for 1-of-1  Interwell Prediction Limits on Means of Order 3 (1 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.25 1.04 0.96 (
1.65 1.35 1.24
1.87 1.53 1.39
2.03 1.65 1.50
2.16 1.74 1.58
2.41 1.93 1.75
2.62 2.09 1.89
2.76 2.20 1.98
2.87 2.28 2.05
3.06 2.43 2.18 ;
3.19 2.53 2.27 ;
3.29 2.61 2.34 ;
3.38 2.67 2.39 ;
3.47 2.74 2.46 ;
3.59 2.83 2.54 ;
3.69 2.90 2.60 ;
3.76 2.96 2.65 ;
3.82 3.01 2.69 ;
3.88 3.05 2.73 ;
).91 0.88 0.84 (
.17 1.13 1.08
.32 1.27 1.21
.42 1.36 1.30
.49 1.43 1.37
.65 1.58 1.50
.77 1.70 1.62
.86 1.79 1.70
.93 1.85 1.76
>.05 1.96 1.86
M3 2.04 1.93
M9 2.10 1.99
1.24 2.15 2.03
>.30 2.21 2.09 ;
2.38 2.28 2.15 ;
2.44 2.33 2.21 ;
2.48 2.38 2.25 ;
1.52 2.41 2.28 ;
2.56 2.45 2.31 ;
).82 (
.05
.18
.26
.33
.46
.57
.64
.70
.80
.87
.92
.96
1.02
1.08 ;
2.13 ;
2.17 ;
1.20 ;
2.23 ;
).81 0.79 (
.03 1.01
.15 1.13
.23 1.22
.30 1.28
.42 1.40
.53 1.50
.60 1.57
.65 1.63
.75 1.72
.82 1.78
.87 1.83
.91 1.87
.96 1.92
1.02 1.98
1.07 2.03 ;
Ml 2.07 ;
2.14 2.10 ;
2.17 2.12 ;
).79 (
.00
.12
.20
.26
.38
.48
.55
.60
.70
.76
.81
.85
.89
.95
2.00
2.04 ;
2.07 ;
2.09 ;
).78 0.78 (
.00 0.99 (
.11 1.11
.19 1.18
.25 1.24
.37 1.36
.47 1.46
.54 1.53
.59 1.58
.68 1.67
.74 1.73
.79 1.77
.83 1.81
.87 1.86
.93 1.92
.98 1.96
2.01 1.99
2.04 2.02 ;
2.07 2.05 ;
).77 0.77 0.76 (
).99 0.98 0.97 (
.10 1.09 1.09
.18 1.17 1.16
.24 1.23 1.22
.35 1.34 1.33
.45 1.44 1.43
.52 1.50 1.49
.57 1.55 1.54
.65 1.64 1.63
.72 1.70 1.69
.76 1.74 1.73
.80 1.78 1.77
.84 1.83 1.81
.90 1.88 1.87
.94 1.92 1.91
.98 1.96 1.94
2.01 1.99 1.97
2.03 2.01 1.99
).76 0.76 (
).97 0.97 (
.08 1.08
.16 1.15
.21 1.21
.33 1.32
.42 1.42
.48 1.48
.53 1.53
.62 1.61
.68 1.67
.72 1.71
.76 1.75
.80 1.79
.86 1.85
.90 1.89
.93 1.92
.96 1.95
.98 1.97
).76 0.75 (
).96 0.96 (
.07 1.07
.15 1.14
.21 1.20
.32 1.31
.41 1.40
.47 1.47
.52 1.51
.61 1.60
.66 1.65
.71 1.70
.74 1.73
.79 1.77
.84 1.83
.88 1.87
.91 1.90
.94 1.93
.96 1.95
).75
).96
.07
.14
.20
.31
.40
.46
.51
.59
.65
.69
.72
.77
.82
.86
.89
.92
.94
   Table 19-8. K-Multipliers  for 1-of-1 Interwell Prediction Limits on Means of Order 3 (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.65 1.35 1.24
2.03 1.65 1.50
2.25 1.82 1.65
2.41 1.93 1.75
2.52 2.02 1.82
2.76 2.20 1.98
2.96 2.35 2.11
3.09 2.45 2.20 ;
3.19 2.53 2.27 ;
3.38 2.67 2.39 ;
3.50 2.76 2.48 ;
3.59 2.83 2.54 ;
3.67 2.89 2.59 ;
3.76 2.96 2.65 ;
3.88 3.05 2.73 ;
3.96 3.12 2.79 ;
4.03 3.17 2.84 ;
4.09 3.22 2.88 ;
4.14 3.25 2.91 ;
.17 1.13 1.08
.42 1.36 1.30
.55 1.49 1.42
.65 1.58 1.50
.72 1.65 1.57
.86 1.79 1.70
.98 1.90 1.80
1.07 1.98 1.88
2.13 2.04 1.93
2.24 2.15 2.03
1.32 2.22 2.10 ;
2.38 2.28 2.15 ;
2.43 2.32 2.20 ;
2.48 2.38 2.25 ;
2.56 2.45 2.31 ;
2.61 2.50 2.36 ;
2.66 2.54 2.40 ;
2.69 2.58 2.43 ;
1.73 2.61 2.46 ;
.05
.26
.38
.46
.52
.64
.74
.82
.87
.96
2.03
2.08 ;
2.12 ;
2.17 ;
>.23 :
i.28 :
i.32 :
1.35 :
1.37 ;
.03 1.01
.23 1.22
.35 1.32
.42 1.40
.48 1.46
.60 1.57
.70 1.67
.77 1.73
.82 1.78
.91 1.87
.97 1.94
1.02 1.98
2.06 2.02
2.11 2.07 ;
1.17 2.12 ;
2.21 2.17 ;
1.25 2.20 :
1.28 2.23 :
1.31 2.26 ;
.00
.20
.31
.38
.44
.55
.65
.71
.76
.85
.91
.95
.99
2.04 ;
2.09 ;
2.13 ;
2.17 ;
1.20 :
1.22 ;
.00 0.99 (
.19 1.18
.30 1.29
.37 1.36
.43 1.42
.54 1.53
.63 1.62
.69 1.68
.74 1.73
.83 1.81
.89 1.87
.93 1.92
.97 1.95
2.01 1.99
1.07 2.05 :
l.ll 2.09 ;
2.14 2.12 ;
2.17 2.15 ;
1.20 2.18 ;
).99 0.98 0.97 (
.18 1.17 1.16
.28 1.27 1.26
.35 1.34 1.33
.41 1.39 1.39
.52 1.50 1.49
.61 1.59 1.58
.67 1.65 1.64
.72 1.70 1.69
.80 1.78 1.77
.86 1.84 1.82
.90 1.88 1.87
.94 1.92 1.90
.98 1.96 1.94
2.03 2.01 1.99
2.08 2.05 2.03 ;
Ml 2.08 2.07 ;
M4 2.11 2.09 ;
M6 2.14 2.12 ;
).97 0.97 (
.16 1.15
.26 1.25
.33 1.32
.38 1.37
.48 1.48
.57 1.57
.63 1.62
.68 1.67
.76 1.75
.81 1.80
.86 1.85
.89 1.88
.93 1.92
.98 1.97
1.02 2.01 ;
2.05 2.04 ;
2.08 2.07 ;
MO 2.09 ;
).96 0.96 (
.15 1.14
.25 1.24
.32 1.31
.37 1.36
.47 1.47
.56 1.55
.62 1.61
.66 1.65
.74 1.73
.80 1.79
.84 1.83
.87 1.86
.91 1.90
.96 1.95
2.00 1.99
2.03 2.02 ;
2.06 2.05 ;
2.08 2.07 ;
).96
.14
.24
.31
.36
.46
.54
.60
.65
.72
.78
.82
.85
.89
.94
.98
2.01
2.03
2.06
                                                    D-99
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
     Table 19-8.  K-Multipliers for 1-of-1  Interwell Prediction Limits on Means of Order 3 (1 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
2.03 1.65 1.50 .42 1.36 1.30
2.41 1.93 1.75 .65 1.58 1.50
2.62 2.09 1.89 .77 1.70 1.62
2.76 2.20 1.98 .86 1.79 1.70
2.87 2.28 2.05 .93 1.85 1.76
3.09 2.45 2.20 2.07 1.98 1.88
3.28 2.59 2.33 2.18 2.09 1.98
3.40 2.69 2.41 2.26 2.16 2.05
350 2.76 2.48 2.32 2.22 2.10 ;
3.67 2.89 2.59 2.43 2.32 2.20 ;
379 2.98 2.67 2.50 2.39 2.26 ;
3.88 3.05 2.73 2.56 2.45 2.31 ;
3.95 3.10 2.78 2.60 2.49 2.35 ;
4.03 3.17 2.84 2.66 2.54 2.40 ;
414 3.25 2.91 2.73 2.61 2.46 ;
423 3.32 2.97 2.78 2.66 2.51 ;
429 3.37 3.01 2.82 2.70 2.55 ;
4.35 3.41 3.05 2.86 2.73 2.58 ;
4.40 3.45 3.08 2.89 2.76 2.60 ;
20
.26
.46
.57
.64
.70
.82
.91
.98
2.03
1.12 :
1.18 ;
2.23 ;
2.27 ;
>.32 :
1.37 ;
2.42 ;
2.46 ;
1.49 :
LSI ;
25 30
.23 1.22
.42 1.40
.53 1.50
.60 1.57
.65 1.63
.77 1.73
.86 1.82
.92 1.89
.97 1.94
2.06 2.02
1.12 2.08 ;
2.17 2.12 ;
1.20 2.16 ;
1.25 2.20 ;
2.31 2.26 ;
2.35 2.30 ;
2.38 2.33 ;
2.41 2.36 ;
2.44 2.39 ;
35
.20
.38
.48
.55
.60
.71
.80
.86
.91
.99
2.05 ;
2.09 ;
2.13 ;
2.17 ;
1.22 ;
2.27 ;
2.30 ;
2.33 ;
2.35 ;
40 45
.19 1.18
.37 1.36
.47 1.46
.54 1.53
.59 1.58
.69 1.68
.78 1.77
.84 1.83
.89 1.87
.97 1.95
2.02 2.01
2.07 2.05 ;
2.10 2.08 ;
2.14 2.12 ;
2.20 2.18 ;
2.24 2.22 ;
2.27 2.25 ;
2.30 2.28 ;
2.32 2.30 ;
50 60 70
.18 1.17 1.16
.35 1.34 1.33
.45 1.44 1.43
.52 1.50 1.49
.57 1.55 1.54
.67 1.65 1.64
.75 1.74 1.72
.81 1.79 1.78
.86 1.84 1.82
.94 1.92 1.90
.99 1.97 1.95
2.03 2.01 1.99
2.07 2.04 2.03 ;
2.11 2.08 2.07 ;
2.16 2.14 2.12 ;
2.20 2.17 2.16 ;
2.23 2.21 2.19 ;
2.26 2.23 2.21 ;
2.28 2.25 2.24 ;
80 90
.16 1.15
.33 1.32
.42 1.42
.48 1.48
.53 1.53
.63 1.62
.71 1.71
.77 1.76
.81 1.80
.89 1.88
.94 1.93
.98 1.97
2.01 2.00 ;
2.05 2.04 ;
2.10 2.09 ;
2.14 2.13 ;
2.17 2.16 ;
2.20 2.19 ;
2.22 2.21 ;
100 125
.15 1.14
.32 1.31
.41 1.40
.47 1.47
.52 1.51
.62 1.61
.70 1.69
.76 1.74
.80 1.79
.87 1.86
.92 1.91
.96 1.95
2.00 1.98
2.03 2.02 ;
2.08 2.07 ;
2.12 2.10 ;
2.15 2.13 ;
2.18 2.16 ;
2.20 2.18 ;
150
.14
.31
.40
.46
.51
.60
.68
.74
.78
.85
.90
.94
.97
2.01
2.06
2.09
2.12
2.15
2.17
      Table 19-8. K-Multipliers  for 1-of-1  Interwell Prediction Limits on Means of Order 3 (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.78 1.41 1.27
2.24 1.74 1.55
2.52 1.93 1.71
2.71 2.06 1.82
2.86 2.16 1.91
3.17 2.38 2.09
3.43 2.55 2.24 ;
3.60 2.68 2.34 ;
374 2.77 2.42 ;
3.98 2.94 2.56 ;
4.15 3.05 2.65 ;
4.27 3.14 2.73 ;
4.37 3.21 2.79 ;
4.49 3.29 2.86 ;
4.65 3.40 2.95 ;
4.76 3.48 3.02 ;
4.86 3.55 3.O7 '*
4.94 3.60 3,12 \
5.00 3.65 3,76 ;
.20 1.15 1.09
.45 1.39 1.32
.60 1.53 1.44
.70 1.62 1.53
.78 1.69 1.60
.94 1.84 1.73
2.07 1.97 1.84
2.16 2.05 1.92
2.23 2.12 1.98
2.36 2.24 2.09 ;
2.45 2.32 2.16 ;
2.51 2.38 2.22 ;
2.57 2.43 2.27 ;
2.63 2.49 2.32 ;
2.71 2.57 2.39 ;
2.78 2.63 2.45 ;
2.83 2.67 2.49 ;
2.87 2.71 2.52 ;
2.91 2.75 2.56 ;
.06
.28
.40
.48
.54
.67
.77
.85
.91
2.01
2.08 ;
2.13 ;
2.17 ;
2.22 ;
2.29 ;
2.34 ;
2.38 ;
2.42 ;
2.44 ;
.04 1.02
.24 1.22
.36 1.33
.44 1.41
.50 1.47
.62 1.59
.72 1.68
.79 1.75
.84 1.80
.94 1.90
2.01 1.96
2.06 2.01
2.10 2.05 ;
2.15 2.10 ;
2.21 2.16 ;
2.26 2.20 ;
2.30 2.24 ;
2.33 2.27 ;
2.36 2.30 ;
.01
.21
.32
.39
.45
.56
.66
.73
.78
.87
.93
.98
2.01
2.06 ;
2.12 ;
2.16 ;
2.20 ;
2.23 ;
2.26 ;
.00 0.99 (
.20 1.19
.30 1.29
.38 1.37
.43 1.42
.55 1.53
.64 1.63
.71 1.69
.76 1.74
.84 1.83
.90 1.89
.95 1.93
.99 1.97
2.03 2.01 ;
2.09 2.07 ;
2.13 2.11 ;
2.17 2.15 ;
2.20 2.18 ;
2.23 2.20 ;
).99 0.98 0.98 (
.18 1.17 1.16
.29 1.28 1.27
.36 1.35 1.34
.41 1.40 1.39
.52 1.51 1.50
.61 1.60 1.59
.68 1.66 1.65
.73 1.71 1.69
.81 1.79 1.77
.87 1.85 1.83
.92 1.89 1.87
.95 1.93 1.91
2.00 1.97 1.95
2.05 2.02 2.01
2.09 2.07 2.05 ;
2.13 2.10 2.08 ;
2.16 2.13 2.11 ;
2.18 2.15 2.13 ;
).97 0.97 (
.16 1.15
.26 1.26
.33 1.33
.38 1.38
.49 1.48
.58 1.57
.64 1.63
.68 1.67
.76 1.75
.82 1.81
.86 1.85
.90 1.89
.94 1.93
.99 1.98
2.03 2.02 ;
2.06 2.05 ;
2.09 2.08 ;
2.11 2.10 ;
).97 0.96 (
.15 1.15
.25 1.25
.32 1.31
.37 1.36
.48 1.47
.56 1.55
.62 1.61
.67 1.66
.75 1.74
.80 1.79
.84 1.83
.88 1.86
.92 1.90
.97 1.96
2.01 1.99
2.04 2.03 ;
2.07 2.05 ;
2.09 2.07 ;
).96
.14
.24
.31
.36
.46
.55
.60
.65
.73
.78
.82
.85
.89
.94
.98
2.01
2.04
2.06
                                                    D-100
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means
Unified Guidance
   Table 19-8. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 3 (2 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
2.24 1.74 1.55 1.45 1.39 1.32
2.71 2.06 1.82 1.70 1.62 1.53
2.98 2.25 1.98 1.84 1.75 1.65
3.17 2.38 2.09 1.94 1.84 1.73
3.31 2.47 2.17 2.01 1.91 1.79
3.60 2.68 2.34 2.16 2.05 1.92
3.85 2.84 2.48 2.29 2.17 2.03
4.02 2.96 2.58 2.38 2.25 2.11 ;
415 3.05 2.65 2.45 2.32 2.16 ;
437 3.21 2.79 2.57 2.43 2.27 ;
4.53 3.32 2.88 2.65 2.51 2.34 ;
4.65 3.40 2.95 2.71 2.57 2.39 ;
4.74 3.47 3.01 2.77 2.62 2.44 ;
4.86 3.55 3.07 2.83 2.67 2.49 ;
5.00 3.65 3.16 2.91 2.75 2.56 ;
5.11 3.73 3.23 2.97 2.80 2.61 ;
5.20 3.79 3,28 3.01 2.85 2.65 ;
5.28 3.84 3,32 3.05 2.88 2.68 ;
5.34 3.89 3.36 3.09 2.92 2.71 ;
20
.28
.48
.59
.67
.73
.85
.95
1.02
1.08 :
1.17 :
1.24 ;
1.29 :
1.33 :
i.38 :
1.44 ;
2.49 ;
2.53 ;
2.56 ;
2.59 ;
25 30
.24 1.22
.44 1.41
.54 1.51
.62 1.59
.67 1.64
.79 1.75
.89 1.85
.96 1.91
2.01 1.96
2.10 2.05 ;
2.16 2.11 ;
1.21 2.16 ;
1.25 2.19 ;
2.30 2.24 ;
2.36 2.30 ;
2.40 2.34 ;
2.44 2.38 ;
2.47 2.41 ;
2.50 2.43 ;
35
.21
.39
.49
.56
.62
.73
.82
.88
.93
2.01
2.07 ;
2.12 ;
2.16 ;
1.20 :
i.26 ;
2.30 ;
2.33 ;
2.36 ;
2.39 ;
40 45
.20 1.19
.38 1.37
.48 1.47
.55 1.53
.60 1.59
.71 1.69
.80 1.78
.86 1.84
.90 1.89
.99 1.97
2.05 2.02 ;
2.09 2.07 ;
2.13 2.10 ;
2.17 2.15 ;
2.23 2.20 ;
2.27 2.24 ;
2.30 2.28 ;
2.33 2.30 ;
2.35 2.33 ;
50 60 70
.18 1.17 1.16
.36 1.35 1.34
.46 1.44 1.43
.52 1.51 1.50
.57 1.56 1.55
.68 1.66 1.65
.76 1.74 1.73
.82 1.80 1.79
.87 1.85 1.83
.95 1.93 1.91
2.01 1.98 1.96
2.05 2.02 2.01
2.09 2.06 2.04 ;
2.13 2.10 2.08 ;
2.18 2.15 2.13 ;
2.22 2.19 2.17 ;
2.25 2.22 2.20 ;
2.28 2.25 2.23 ;
2.31 2.27 2.25 ;
80 90
.16 1.15
.33 1.33
.42 1.42
.49 1.48
.54 1.53
.64 1.63
.72 1.71
.78 1.77
.82 1.81
.90 1.89
.95 1.94
.99 1.98
2.02 2.01 ;
2.06 2.05 ;
2.11 2.10 ;
2.15 2.14 ;
2.18 2.17 ;
2.21 2.20 ;
2.23 2.22 ;
100 125
.15 1.15
.32 1.31
.41 1.41
.48 1.47
.53 1.52
.62 1.61
.70 1.69
.76 1.75
.80 1.79
.88 1.86
.93 1.92
.97 1.96
2.00 1.99
2.04 2.03 ;
2.09 2.07 ;
2.13 2.11 ;
2.16 2.14 ;
2.19 2.17 ;
2.21 2.19 ;
150
.14
.31
.40
.46
.51
.60
.68
.74
.78
.85
.91
.94
.98
2.01
2.06
2.10
2.13
2.15
2.17
     Table 19-8. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 3 (2 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20 25 30
2.71 2.06 1.82 1.70 1.62 1.53 .48 .44 1.41
3.17 2.38 2.09 1.94 1.84 1.73 .67 .62 1.59
3.43 2.55 2.24 2.07 1.97 1.84 .77 .72 1.68
3.60 2.68 2.34 2.16 2.05 1.92 .85 .79 1.75
374 2.77 2.42 2.23 2.12 1.98 .91 .84 1.80
402 2.96 2.58 2.38 2.25 2.11 2.02 .96 1.91
425 3.12 2.71 2.50 2.37 2.21 2.12 2.05 2.00
441 3.23 2.81 2.59 2.45 2.28 2.19 2.11 2.06 ;
4.53 3.32 2.88 2.65 2.51 2.34 2.24 2.16 2.11 ;
474 3 47 3.01 2.77 2.62 2.44 2.33 2.25 2.19 ;
4.89 3.57 3.09 2.84 2.69 2.50 2.40 2.31 2.25 ;
5.00 3.65 3.16 2.91 2.75 2.56 2.44 2.36 2.30 ;
5.09 3.71 3.22 2.95 2.79 2.60 2.48 2.39 2.33 ;
5.20 3.79 3.28 3.01 2.85 2.65 2.53 2.44 2.38 ;
5.34 3.89 3.36 3.09 2.92 2.71 2.59 2.50 2.43 ;
5.44 3.96 3.43 3.15 2.97 2.76 2.64 2.54 2.48 ;
5.53 4.02 3,48 3.19 3.01 2.80 2.68 2.58 2.51 ;
5.60 4.07 3,52 3.23 3.05 2.83 2.71 2.61 2.54 ;
5.66 4.11 3.56 3.26 3.08 2.86 2.74 2.63 2.57 ;
35
.39
.56
.66
.73
.78
.88
.97
2.03 ;
2.07 ;
2.16 ;
2.21 ;
2.26 ;
2.29 ;
2.33 ;
2.39 ;
2.43 ;
2.46 ;
2.49 ;
2.52 ;
40 45
.38 1.37
.55 1.53
.64 1.63
.71 1.69
.76 1.74
.86 1.84
.94 1.92
2.00 1.98
2.05 2.02 ;
2.13 2.10 ;
2.18 2.16 ;
2.23 2.20 ;
2.26 2.23 ;
2.30 2.28 ;
2.35 2.33 ;
2.40 2.37 ;
2.43 2.40 ;
2.46 2.43 ;
2.48 2.45 ;
50 60 70
.36 1.35 1.34
.52 1.51 1.50
.61 1.60 1.59
.68 1.66 1.65
.73 1.71 1.69
.82 1.80 1.79
.91 1.88 1.87
.96 1.94 1.92
2.01 1.98 1.96
2.09 2.06 2.04 ;
2.14 2.11 2.09 ;
2.18 2.15 2.13 ;
2.21 2.18 2.16 ;
2.25 2.22 2.20 ;
2.31 2.27 2.25 ;
2.35 2.31 2.29 ;
2.38 2.34 2.32 ;
2.40 2.37 2.34 ;
2.43 2.39 2.37 ;
80 90
.33 1.33
.49 1.48
.58 1.57
.64 1.63
.68 1.67
.78 1.77
.85 1.84
.91 1.90
.95 1.94
2.02 2.01 ;
2.07 2.06 ;
2.11 2.10 ;
2.15 2.13 ;
2.18 2.17 ;
2.23 2.22 ;
2.27 2.26 ;
2.30 2.29 ;
2.32 2.31 ;
2.35 2.33 ;
100 125
.32 1.31
.48 1.47
.56 1.55
.62 1.61
.67 1.66
.76 1.75
.84 1.82
.89 1.88
.93 1.92
2.00 1.99
2.05 2.04 ;
2.09 2.07 ;
2.12 2.10 ;
2.16 2.14 ;
2.21 2.19 ;
2.24 2.22 ;
2.27 2.25 ;
2.30 2.28 ;
2.32 2.30 ;
150
.31
.46
.55
.60
.65
.74
.81
.87
.91
.98
2.02
2.06
2.09
2.13
2.17
2.21
2.24
2.26
2.28
                                                   D-101
                                                                                                  March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means
Unified Guidance
      Table 19-8. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 3 (5 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
2.62 1.92 1.68 1.56 1.49 1.40 .35
3.22 2.29 1.98 1.82 1.73 1.62 .55
358 2.51 2.15 1.97 1.86 1.74 .67
383 2.66 2.27 2.08 1.96 1.82 .75
403 2.78 2.36 2.16 2.03 1.89 .81
444 3.03 2.56 2.32 2.18 2.02 .93
4.79 3.23 2.72 2.47 2.31 2.14 2.04
5.03 3.38 2.84 2.57 2.40 2.22 2.11 ;
5.21 3.49 2.93 2.64 2.47 2.28 2.17 ;
5.53 3.69 3,08 2.78 2.60 2.39 2.27 ;
5.76 3.83 3.19 2.88 2.69 2.46 2.34 ;
5.93 3.93 3.28 2.95 2.75 2.52 2.39 ;
6.07 4.02 3.35 3.O1 2.81 2.57 2.44 ;
6.23 4.12 3.43 3, OS 2.87 2.63 2.49 ;
6.44 4.25 3.53 3,17 2.95 2.70 2.56 ;
6.60 4.35 3.61 3.24 3.O2 2.76 2.61 ;
6.72 4.42 3.67 3.30 3,07 2.80 2.65 ;
6.83 4.49 3.73 3.34 '3,11 2.84 2.69 ;
6.92 4.55 3.77 3.38 3,15 2.87 2.72 ;
25 30
.31 1.29
.51 1.48
.61 1.58
.69 1.65
.75 1.71
.86 1.82
.96 1.91
2.03 1.98
2.08 2.03
2.18 2.12 ;
1.24 2.18 ;
2.30 2.23 ;
2.34 2.27 ;
2.39 2.32 ;
2.45 2.38 ;
2.50 2.42 ;
2.54 2.46 ;
2.57 2.49 ;
2.60 2.52 ;
35
.27
.45
.56
.63
.68
.79
.88
.94
.99
2.08 ;
2.14 ;
2.19 ;
1.22 ;
2.27 ;
2.33 ;
2.37 ;
2.4i ;
2.44 ;
2.46 ;
40 45
.26 1.25
.44 1.43
.54 1.52
.61 1.59
.66 1.64
.76 1.75
.85 1.83
.92 1.89
.96 1.94
2.05 2.02 ;
2.11 2.08 ;
2.15 2.13 ;
2.19 2.16 ;
2.23 2.20 ;
2.29 2.26 ;
2.33 2.30 ;
2.37 2.34 ;
2.40 2.37 ;
2.42 2.39 ;
50 60 70
.24 1.23 1.22
.42 1.40 1.39
.51 1.50 1.49
.58 1.56 1.55
.63 1.61 1.60
.73 1.71 1.70
.82 1.80 1.78
.88 1.85 1.84
.92 1.90 1.88
2.01 1.98 1.96
2.06 2.03 2.01 ;
2.11 2.07 2.05 ;
2.14 2.11 2.09 ;
2.18 2.15 2.13 ;
2.24 2.20 2.18 ;
2.28 2.24 2.22 ;
2.31 2.28 2.25 ;
2.34 2.30 2.28 ;
2.36 2.33 2.30 ;
80 90
.22 1.21
.38 1.38
.48 1.47
.54 1.53
.59 1.58
.69 1.68
.77 1.76
.82 1.81
.87 1.86
.94 1.93
2.00 1.98
2.04 2.02 ;
2.07 2.06 ;
2.11 2.10 ;
2.16 2.15 ;
2.20 2.18 ;
2.23 2.21 ;
2.26 2.24 ;
2.28 2.26 ;
100 125
.21 1.20
.37 1.37
.46 1.46
.53 1.52
.57 1.56
.67 1.66
.75 1.74
.81 1.79
.85 1.83
.92 1.91
.98 1.96
2.01 2.00
2.05 2.03 ;
2.09 2.07 ;
2.13 2.11 ;
2.17 2.15 ;
2.20 2.18 ;
2.23 2.21 ;
2.25 2.23 ;
150
.20
.36
.45
.51
.56
.65
.73
.78
.82
.90
.95
.99
2.02
2.05
2.10
2.14
2.17
2.19
2.21
   Table 19-8. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 3 (5 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.22
3.83
4.19
4.44
4.63
5.03
5.36
5.59
5.76
6.07
6.28
6.44
6.57
6.72
6.92
7.07
7.19
7.29
7.34
6 8 10 12 16 20 25
2.29 1.98 1.82 1.73 1.62 .55 .51
2.66 2.27 2.08 1.96 1.82 .75 .69
2.88 2.44 2.22 2.09 1.94 .86 .79
3.03 2.56 2.32 2.18 2.02 .93 .86
3.14 2.65 2.40 2.26 2.09 .99 .92
3.38 2.84 2.57 2.40 2.22 2.11 2.03
358 3.00 2.71 2.53 2.33 2.21 2.13
3 72 3.11 2.80 2.62 2.41 2.29 2.19
3.83 3.19 2.88 2.69 2.46 2.34 2.24
4.02 3.35 3.01 2.81 2.57 2.44 2.34
4.15 3.45 3.10 2.89 2.65 2.51 2.40
4.25 3.53 3.17 2.95 2.70 2.56 2.45
4.33 3.59 3,23, 3.01 2.75 2.60 2.49
4.42 3.67 3.30 3.07 2.80 2.65 2.54
4.55 3.77 3,38 3.15 2.87 2.72 2.60
4.64 3.85 3.45 3.21 2.93 2.77 2.65
4.72 3.91 3.50 3.26 2.97 2.81 2.68
4.78 3.96 3.55 3.30 3.01 2.84 2.72
4.83 4.00 3.59 3,34 3.04 2.87 2.74
30
1.48
1.65
1.75
1.82
1.87
1.98
2.07
2.13
2.18
2.27
2.33
2.38
2.42
2.46
2.52
2.56
2.60
2.63
2.66
35
.45
.63
.72
.79
.84
.94
2.03
2.09
2.14
2.22
2.28
2.33
2.36
2.41
2.46
2.51
2.54
2.57
2.60
40
.44
.61
.70
.76
.81
.92
2.00
2.06
2.11
2.19
2.25
2.29
2.32
2.37
2.42
2.46
2.50
2.53
2.55
45
1.43
1.59
1.68
1.75
1.79
1.89
1.98
2.04 ;
2.08 ;
2.16 ;
2.22 ;
2.26 ;
2.29 ;
2.34 ;
2.39 ;
2.43 ;
2.46 ;
2.49 ;
2.52 ;
50
.42
.58
.67
.73
.78
.88
.96
2.02
2.06
2.14
2.20
2.24
2.27
2.31
2.36
2.40
2.44
2.46
2.49
60
1.40
1.56
1.65
1.71
1.76
1.85
1.93
1.99
2.03
2.11
2.16
2.20
2.24
2.28
2.33
2.36
2.40
2.42
2.45
70 80
1.39 .38
1.55 .54
1.64 .63
1.70 .69
1.74 .73
1.84 .82
1.92 .90
1.97 .96
2.01 2.00
2.09 2.07
2.14 2.12
2.18 2.16
2.21 2.19
2.25 2.23
2.30 2.28
2.33 2.32
2.37 2.35
2.39 2.37
2.42 2.39
90 100
1.38 .37
1.53 .53
1.62 .61
1.68 .67
1.72 .72
1.81 .81
1.89 .88
1.94 .93
1.98 .98
2.06 2.05
2.11 2.10
2.15 2.13
2.17 2.17
2.21 2.20
2.26 2.25
2.30 2.29
2.33 2.32
2.35 2.34
2.38 2.36
125 150
1.37 .36
1.52 .51
1.60 .59
1.66 .65
1.70 .69
1.79 .78
1.87 .86
1.92 .91
1.96 .95
2.03 2.02
2.08 2.06
2.11 2.10
2.14 2.13
2.18 2.17
2.23 2.21
2.26 2.25
2.29 2.28
2.32 2.30
2.34 2.32
                                                   D-102
                                                                                                 March 2009

-------
Appendix D.  Chapter 19 Interwell K-Tables for Means
Unified Guidance
     Table 19-8. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 3 (5 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.83
4.44
4.79
5.03
5.21
5.59
5.90
6.11
6.28
6.57
6.77
6.92
7.04
7.19
7.34
7.52
7.64
7.73
7.82
6 8 10 12 16 20 25 30 35 40
2.66 2.27 2.08 1.96 1.82 1.75 1.69 1.65 .63 .61
3.03 2.56 2.32 2.18 2.02 1.93 1.86 1.82 .79 .76
3.23 2.72 2.47 2.31 2.14 2.04 1.96 1.91 .88 .85
3.38 2.84 2.57 2.40 2.22 2.11 2.03 1.98 .94 .92
3.49 2.93 2.64 2.47 2.28 2.17 2.08 2.03 .99 .96
3.72 3.11 2.80 2.62 2.41 2.29 2.19 2.13 2.09 2.06
391 3.26 2.94 2.74 2.51 2.38 2.29 2.22 2.18 2.14
405 3.37 3.03 2.83 2.59 2.45 2.35 2.28 2.24 2.20
415 3.45 3.10 2.89 2.65 2.51 2.40 2.33 2.28 2.25
4.33 3.59 3.23 3.01 2.75 2.60 2.49 2.42 2.36 2.32
445 370 3.32 3.09 2.82 2.67 2.55 2.47 2.42 2.38
455 377 3.38 3.15 2.87 2.72 2.60 2.52 2.46 2.42
4.62 3.83 3.44 3.20 2.92 2.76 2.64 2.56 2.50 2.46
4.72 3.91 3,50 3.26 2.97 2.81 2.68 2.60 2.54 2.50
4.83 4.00 3,59 3.34 3.04 2.87 2.74 2.66 2.60 2.55
4.92 4.08 3.65 3.39 3.09 2.92 2.79 2.70 2.64 2.59
5.00 4.13 3.70 3.44 3.14 2.96 2.83 2.74 2.67 2.62
5.06 4.18 3.75 3.48 3.17 2.99 2.86 2.77 2.70 2.65
5.11 4.23 3.78 3,52 3.20 3.02 2.88 2.79 2.73 2.68
45
1.59
1.75
1.83
1.89
1.94
2.04
2.12
2.17
2.22
2.29
2.35
2.39
2.42
2.46
2.52
2.56
2.59
2.61
2.64
50
.58
.73
.82
.88
.92
2.02
2.10
2.15
2.20
2.27
2.32
2.36
2.40
2.44
2.49
2.53
2.56
2.58
2.61
60
1.56
1.71
1.80
1.85
1.90
1.99
2.07
2.12
2.16
2.24
2.29
2.33
2.36
2.40
2.45
2.48
2.51
2.54
2.56
70
1.55
1.70
1.78
1.84
1.88
1.97
2.05
2.10
2.14
2.21
2.26
2.30
2.33
2.37
2.42
2.45
2.48
2.51
2.53
80
.54
.69
.77
.82
.87
.96
2.03
2.08
2.12
2.19
2.24
2.28
2.31
2.35
2.39
2.43
2.46
2.48
2.50
90
1.53
1.68
1.76
1.81
1.86
1.94
2.02
2.07
2.11
2.17
2.23
2.26
2.29
2.33
2.38
2.41
2.44
2.46
2.49
100
.53
.67
.75
.81
.85
.93
2.01
2.06
2.10
2.17
2.21
2.25
2.28
2.32
2.36
2.40
2.43
2.45
2.47
125
1.52
1.66
1.74
1.79
1.83
1.92
1.99
2.04 ;
2.08 ;
2.14 ;
2.19 ;
2.23 ;
2.26 ;
2.29 ;
2.34 ;
2.37 ;
2.40 ;
2.42 ;
2.44 ;
150
.51
.65
.73
.78
.82
.91
.98
>.03
>.06
>.13
>.18
>.21
2.24
2.28
1.32
1.35
1.38
2.40
2.42
     Table 19-8. K-Multipliers  for 1-of-1  Interwell Prediction Limits on Means of Order 3 (10 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.41
4.15
4.60
4.92
5.16
5.68
6.11
6.41
6.64
7.05
7.33
7.55
7.72
7.93
8.19
8.39
8.55
8.68
8.80
6
2.35
2.76
3.00
3.17
3.30
3.58
3.82
3.99
4.11
4.34
4.50
4.62
4.71
4.83
4.98
5.09
5.18
5.26
5.32
8
2.01
2.32
2.50
2.63
2.73
2.94
3,12
3.24
3.34
3.51
3.63
3.73
3.80
3.89
4.00
4.09
4.16
4.22
4.27
10 12 16 20 25 30 35
1.84 1.74 1.62 1.56 .51 1.48 .46
2.10 1.98 1.83 1.75 .69 1.65 .63
2.26 2.12 1.95 1.86 .80 1.75 .72
2.37 2.21 2.04 1.94 .87 1.82 .79
2.45 2.29 2.10 2.00 .93 1.88 .84
2.63 2.45 2.24 2.13 2.04 1.99 .95
2.78 2.58 2.36 2.23 2.14 2.08 2.04
2.89 2.68 2.44 2.31 2.21 2.14 2.10
2.97 2.75 2.50 2.36 2.26 2.19 2.15
3.77 2.88 2.61 2.47 2.35 2.28 2.23
3.22 2.97 2.69 2.54 2.42 2.34 2.29
3.30 3.O4. 2.75 2.59 2.47 2.39 2.34
3.36 3,10 2.80 2.63 2.50 2.43 2.38
3.44 3.17 2.86 2.69 2.56 2.48 2.42
3.53 3.25 2.94 2.76 2.63 2.54 2.48
3.61 3.32 2.99 2.81 2.67 2.58 2.52
3.67 3.38 3,04 2.86 2.71 2.62 2.56
3.72 3.42 3.O8 2.89 2.75 2.65 2.59
3.76 3.46 3.12 2.92 2.78 2.68 2.61
40
.44
.61
.70
.77
.82
.92
2.01
2.07
2.11
2.20
2.25
2.30
2.33
2.38
2.43
2.48
2.51
2.54
2.57
45
1.43
1.59
1.69
1.75
1.80
1.90
1.98
2.04 ;
2.09 ;
2.17 ;
2.22 ;
2.27 ;
2.30 ;
2.34 ;
2.40 ;
2.44 ;
2.47 ;
2.50 ;
2.53 ;
50
.42
.58
.67
.73
.78
.88
.96
1.02
2.07
2.15
1.20
1.24
1.28
1.32
1.37
2.41
2.45
2.47
2.50
60
1.40
1.56
1.65
1.71
1.76
1.86
1.94
1.99
2.04
2.11
2.17
2.21
2.24
2.28
2.33
2.37
2.40
2.43
2.45
70 80
1.39 .39
1.55 .54
1.64 .63
1.70 .69
1.74 .73
1.84 .83
1.92 .90
1.97 .96
2.01 2.00
2.09 2.07
2.14 2.12
2.18 2.16
2.21 2.19
2.25 2.23
2.30 2.28
2.34 2.32
2.37 2.35
2.40 2.38
2.42 2.40
90 100
1.38 .37
1.53 .53
1.62 .61
1.68 .67
1.72 .72
1.82 .81
1.89 .88
1.95 .94
1.99 .98
2.06 2.05
2.11 2.10
2.15 2.14
2.18 2.17
2.22 2.20
2.27 2.25
2.30 2.29
2.33 2.32
2.36 2.34
2.38 2.36
125 150
1.37 .36
1.52 .51
1.60 .59
1.66 .65
1.70 .69
1.79 .78
1.87 .86
1.92 .91
1.96 .95
2.03 2.02
2.08 2.07
2.12 2.10
2.15 2.13
2.18 2.17
2.23 2.21
2.27 2.25
2.29 2.28
2.32 2.30
2.34 2.26
                                                   D-103
                                                                                                  March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-8. K- Multipliers for 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.15
4.92
5.36
5.68
5.92
6.41
6.83
7.11
7.33
7.72
7.99
8.19
8.35
8.55
8.80
8.99
9.14
9.27
9.38
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.92
5.68
6.11
6.41
6.64
7.11
7.51
7.78
7.99
8.35
8.60
8.80
8.95
9.14
9.38
9.56
9.70
9.82
9.93
6
2.76
3.17
3.41
3.58
3.71
3.99
4.22
4.38
4.50
4.71
4.87
4.98
5.07
5.18
5.32
5.43
5.52
5.60
5.66
9-8.
6
3.17
3.58
3.82
3.99
4.11
4.38
4.60
4.75
4.87
5.07
5.22
5.32
5.42
5.52
5.66
5.76
5.85
5.91
5.97
8
2.32
2.63
2.81
2.94
3.04
3,24
3.42
3.54
3.63
3.80
3.92
4.00
4.08
4.16
4.27
4.36
4.42
4.48
4.53
10
2.10
2.37
2.52
2.63
2.71
2.89
3.03
3.14
"3,22"
3, 36
3.46
3.53
3.59
3.67
3.76
3.83
3.89
3.94
3.98
K-Multipliers
8
2.63
2.94
3.12
3.24
3.34
3,54
3.71
3.83
3.92
4.08
4.19
4.27
4.34
4.42
4.53
4.61
4.68
4.73
4.77
10
2.37
2.63
2.78
2.89
2.97
3.14
3.28
3.38
3.46
3,59
3.69
3.76
3.82
3.89
3.98
4.05
4.11
4.16
4.20
12
1.98
2.21
2.35
2.45
2.52
2.68
2.81
2.90
2.97
3.10
3.19
'3.25 '
3,31
3,38
3.46
3.53
3.58
3.62
3.66
for
12
2.21
2.45
2.58
2.68
2.75
2.90
3.03
3.12
3.19
3.31
3.39
3.46
3,51
3,58
3.66
3.72
3.77
3.82
3.85
-of-1
16
1.83
2.04
2.16
2.24
2.30
2.44
2.55
2.63
2.69
2.80
2.88
2.94
2.98
3.04
3.12
3.17
3,22
3,26
3,29
Interwell Prediction Limits on Means of Order 3 (10
20 25 30 35 40 45 50 60 70 80
1.75 1.69 1.65 1.63 .61 1.59 .58 1.56 1.55 .54
1.94 1.87 1.82 1.79 .77 1.75 .73 1.71 1.70 .69
2.05 1.97 1.92 1.88 .86 1.84 .82 1.80 1.78 .77
2.13 2.04 1.99 1.95 .92 1.90 .88 1.86 1.84 .83
2.18 2.09 2.04 2.00 .97 1.95 .93 1.90 1.88 .87
2.31 2.21 2.14 2.10 2.07 2.04 2.02 1.99 1.97 .96
2.41 2.30 2.23 2.19 2.15 2.12 2.10 2.07 2.05 2.03
2.48 2.37 2.30 2.25 2.21 2.18 2.16 2.12 2.10 2.08
2.54 2.42 2.34 2.29 2.25 2.22 2.20 2.17 2.14 2.12
2.63 2.50 2.43 2.38 2.33 2.30 2.28 2.24 2.21 2.19
2.71 2.58 2.49 2.43 2.39 2.36 2.33 2.29 2.26 2.24
2.76 2.63 2.54 2.48 2.43 2.40 2.37 2.33 2.30 2.28
2.80 2.67 2.58 2.51 2.47 2.43 2.41 2.36 2.33 2.31
2.86 2.71 2.62 2.56 2.51 2.47 2.45 2.40 2.37 2.35
2.92 2.78 2.68 2.61 2.57 2.53 2.50 2.45 2.42 2.40
2.98 2.82 2.73 2.66 2.61 2.57 2.54 2.49 2.46 2.43
3.02 2.86 2.76 2.69 2.64 2.60 2.57 2.52 2.49 2.46
3.05 2.90 2.79 2.72 2.67 2.63 2.60 2.55 2.51 2.49
3.08 2.92 2.82 2.75 2.69 2.65 2.62 2.57 2.54 2.51
COC
90
1.53
1.68
1.76
1.82
1.86
1.95
2.02
2.07
2.11
2.18
2.23
2.27
2.30
2.33
2.38
2.42
2.44
2.47
2.49
, Semi-Annual)
100
.53
.67
.75
.81
.85
.94
2.01
2.06
2.10
2.17
2.22
2.25
2.28
2.32
2.36
2.40
2.43
2.45
2.47
125
1.52
1.66
1.74
1.79
1.83
1.92
1.99
2.04
2.08
2.15
2.19
2.23
2.26
2.29
2.34
2.37
2.40
2.42
2.45
150
.51
.65
.73
.78
.82
.91
.98
2.03
2.07
2.13
2.18
2.21
2.24
2.28
2.26
2.36
2.38
2.41
2.43
1-of-1 Interwell Prediction Limits on Means of Order 3 (10 COC, Quarterly)
16
2.04
2.24
2.36
2.44
2.50
2.63
2.74
2.82
2.88
2.98
3.06
3.12
3.16
3.22
3.29
3.34
3.39
3.43
3.46
20 25 30 35 40 45 50 60 70 80
1.94 1.87 1.82 1.79 1.77 1.75 1.73 1.71 1.70 1.69
2.13 2.04 1.99 1.95 1.92 1.90 1.88 1.86 1.84 1.83
2.23 2.14 2.08 2.04 2.01 1.98 1.96 1.94 1.92 1.90
2.31 2.21 2.14 2.10 2.07 2.04 2.02 1.99 1.97 1.96
2.36 2.26 2.19 2.15 2.11 2.09 2.07 2.04 2.01 2.00
2.48 2.37 2.30 2.25 2.21 2.18 2.16 2.12 2.10 2.08
2.58 2.46 2.38 2.33 2.29 2.26 2.24 2.20 2.17 2.16
2.65 2.53 2.44 2.39 2.35 2.31 2.29 2.25 2.23 2.21
2.71 2.58 2.49 2.43 2.39 2.36 2.33 2.29 2.26 2.24
2.80 2.67 2.58 2.51 2.47 2.43 2.41 2.36 2.33 2.31
2.87 2.73 2.64 2.57 2.52 2.49 2.46 2.41 2.38 2.36
2.92 2.78 2.68 2.61 2.57 2.53 2.50 2.45 2.42 2.40
2.97 2.82 2.72 2.65 2.60 2.56 2.53 2.48 2.45 2.43
3.02 2.86 2.76 2.69 2.64 2.60 2.57 2.52 2.49 2.46
3.08 2.92 2.82 2.75 2.69 2.65 2.62 2.57 2.54 2.51
3.13 2.97 2.86 2.79 2.73 2.69 2.66 2.61 2.57 2.54
3.17 3.01 2.90 2.82 2.77 2.72 2.69 2.64 2.60 2.57
3.21 3.04 2.93 2.85 2.80 2.75 2.72 2.66 2.63 2.60
3.24 3.07 2.96 2.88 2.82 2.77 2.74 2.68 2.65 2.62
90
1.68
1.82
1.89
1.95
1.99
2.07
2.14
2.19
2.23
2.30
2.34
2.38
2.41
2.44
2.49
2.52
2.55
2.58
2.60
100
.67
.81
.88
.94
.98
2.06
2.13
2.18
2.22
2.28
2.33
2.36
2.39
2.43
2.47
2.51
2.54
2.56
2.58
125
1.66
1.79
1.87
1.92
1.96
2.04
2.11
2.16
2.19
2.26
2.30
2.34
2.37
2.40
2.45
2.48
2.51
2.53
2.55
150
.65
.78
.86
.91
.95
2.03
2.10
2.14
2.18
2.24
2.29
2.26
2.35
2.38
2.43
2.46
2.49
2.51
2.53

                                                     D-104
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-8.
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.38
5.31
5.86
6.26
6.57
7.22
7.76
8.14
8.43
8.94
9.30
9.57
9.79
10.05
10.38
10.63
10.83
11.00
11.15
Table 19-8
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.31
6.26
6.82
7.22
7.52
8.14
8.66
9.02
9.30
9.79
10.12
10.38
10.58
10.83
11.15
11.38
11.58
11.74
11.88
6
2.82
3.27
3.54
3.74
3.89
4.21
4.48
4.67
4.81
5.07
5.25
5.39
5.50
5.64
5.81
5.94
6.04
6.13
6.20
K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 3 (20 COC, Annual)
8
2.34
2.67
2.87
3.01
3,12
3.35
3.54
3.68
3.78
3.97
4.11
4.21
4.29
4.39
4.52
4.61
4.69
4.76
4.81
10
2.12
2.39
2.55
2.67
2.76
2.95
3,11
3.22
3.31
3.46
3.57
3.66
3.72
3.81
3.91
3.99
4.06
4.11
4.16
. K- Multipliers
6
3.27
3.74
4.01
4.21
4.36
4.67
4.93
5.11
5.25
5.50
5.68
5.81
5.91
6.04
6.20
6.33
6.43
6.52
6.59
8
2.67
3.01
3.21
3.35
3.45
3.68
3.87
4.00
4.11
4.29
4.42
4.52
4.59
4.69
4.81
4.90
4.98
5.04
5.10
10
2.39
2.67
2.83
2.95
3.03
3,22
3,38
3.49
3.57
3.72
3.83
3.91
3.98
4.06
4.16
4.24
4.30
4.35
4.40
12
1.99
2.23
2.37
2.47
2.55
2.71
2.85
2.95
3,03
3,17
3.26
3.34
3.40
3.47
3.56
3.63
3.69
3.74
3.78
for 1
12
2.23
2.47
2.61
2.71
2.79
2.95
3.09
3.19
3,26
3.4O
3.49
3.56
3.62
3.69
3.78
3.85
3.91
3.95
3.99
16
1.84
2.05
2.17
2.25
2.32
2.46
2.57
2.66
2.72
2.84
2.92
2, '98
3.O3
3.O9
3.1 7
3,23
3.28
3.32
3.36
-of-1
16
2.05
2.25
2.37
2.46
2.52
2.66
2.77
2.86
2.92
3.03
3.11
3.17
3,22
3,28
3,36
3,42
3,47
3,51,
3.54
20
1.76
1.95
2.06
2.13
2.19
2.32
2.42
2.50
2.56
2.66
2.73
2.79
2.83
2.89
2.96
3,01
3.O6
3.1O
3,13
25
1.70
1.87
1.97
2.05
2.10
2.21
2.31
2.38
2.43
2.53
2.59
2.64
2.68
2.73
2.80
2.85
2.89
2.92
2.95
Interwell
20
1.95
2.13
2.24
2.32
2.38
2.50
2.60
2.68
2.73
2.83
2.90
2.96
3.00
3.06
3.13
3.18
3,22
3.26
3,29
25
1.87
2.05
2.14
2.21
2.27
2.38
2.47
2.54
2.59
2.68
2.75
2.80
2.84
2.89
2.95
3.00
3.04
3.07
3.10
30
1.66
1.83
1.92
1.99
2.04
2.15
2.24
2.30
2.35
2.44
2.50
2.55
2.59
2.64
2.70
2.74
2.78
2.81
2.84
35
1.63
1.79
1.89
1.95
2.00
2.10
2.19
2.25
2.30
2.38
2.44
2.49
2.52
2.57
2.63
2.67
2.71
2.74
2.76
Prediction
30
1.83
1.99
2.08
2.15
2.20
2.30
2.39
2.45
2.50
2.59
2.65
2.70
2.73
2.78
2.84
2.89
2.92
2.95
2.98
35
1.79
1.95
2.04
2.10
2.15
2.25
2.34
2.40
2.44
2.52
2.58
2.63
2.66
2.71
2.76
2.81
2.84
2.87
2.90
40
.61
.77
.86
.92
.97
2.07
2.15
2.21
2.26
2.34
2.40
2.44
2.48
2.52
2.57
2.62
2.65
2.68
2.70
Limits
40
1.77
1.92
2.01
2.07
2.12
2.21
2.29
2.35
2.40
2.48
2.53
2.57
2.61
2.65
2.70
2.75
2.78
2.81
2.83
45 50 60 70 80 90 100 125
1.59 1.58 .56 1.55 .54 1.53 .53 .52
1.75 1.74 .71 1.70 .69 1.68 .67 .66
1.84 1.82 .80 1.78 .77 1.76 .75 .74
1.90 1.88 .86 1.84 .83 1.82 .81 .79
1.95 1.93 .90 1.88 .87 1.86 .85 .83
2.04 2.02 .99 1.97 .96 1.95 .94 .92
2.13 2.10 2.07 2.05 2.03 2.02 2.01 .99
2.18 2.16 2.13 2.10 2.08 2.07 2.06 2.04
2.23 2.20 2.17 2.14 2.12 2.11 2.10 2.08
2.31 2.28 2.24 2.22 2.20 2.18 2.17 2.15
2.36 2.33 2.29 2.27 2.25 2.23 2.22 2.17
2.40 2.38 2.33 2.31 2.28 2.27 2.25 2.23
2.44 2.41 2.37 2.34 2.31 2.30 2.28 2.26
2.48 2.45 2.41 2.37 2.35 2.33 2.32 2.29
2.53 2.50 2.46 2.42 2.40 2.38 2.37 2.34
2.57 2.54 2.49 2.46 2.44 2.42 2.40 2.37
2.61 2.58 2.53 2.49 2.47 2.45 2.43 2.40
2.64 2.60 2.55 2.52 2.49 2.47 2.45 2.43
2.66 2.63 2.57 2.54 2.51 2.49 2.48 2.45
150
1.51
1.65
1.73
1.78
1.82
1.91
1.98
2.03
2.07
2.13
2.18
2.21
2.24
2.26
2.32
2.36
2.38
2.41
2.43
on Means of Order 3 (20 COC, Semi-Annual)
45 50 60 70 80 90 100 125
1.75 1.74 1.71 1.70 1.69 1.68 .67 .66
1.90 1.88 1.86 1.84 1.83 1.82 .81 .79
1.98 1.97 1.94 1.92 1.90 1.89 .88 .87
2.04 2.02 1.99 1.97 1.96 1.95 .94 .92
2.09 2.07 2.04 2.02 2.00 1.99 .98 .96
2.18 2.16 2.13 2.10 2.08 2.07 2.06 2.04
2.26 2.24 2.20 2.18 2.16 2.14 2.13 2.11
2.32 2.29 2.25 2.23 2.21 2.19 2.18 2.16
2.36 2.33 2.29 2.27 2.25 2.23 2.22 2.17
2.44 2.41 2.37 2.34 2.31 2.30 2.28 2.26
2.49 2.46 2.42 2.39 2.36 2.34 2.33 2.31
2.53 2.50 2.46 2.42 2.40 2.38 2.37 2.34
2.57 2.54 2.49 2.45 2.43 2.41 2.40 2.37
2.61 2.58 2.53 2.49 2.47 2.45 2.43 2.40
2.66 2.63 2.57 2.54 2.51 2.49 2.48 2.45
2.70 2.67 2.61 2.58 2.55 2.53 2.51 2.48
2.73 2.70 2.64 2.60 2.58 2.56 2.54 2.51
2.76 2.72 2.67 2.63 2.60 2.58 2.56 2.53
2.78 2.75 2.69 2.65 2.62 2.60 2.58 2.55
150
1.65
1.78
1.86
1.91
1.95
2.03
2.10
2.14
2.18
2.24
2.29
2.32
2.35
2.38
2.43
2.46
2.49
2.51
2.53

                                                     D-105
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-8. K-Multipliers for 1-of-1 Interwell Prediction Limits on Means of Order 3 (20 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
6.26
7.22
7.76
8.14
8.43
9.02
9.52
9.86
10.12
10.58
10.90
11.15
11.34
11.58
11.88
12.10
12.29
12.44
12.57
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.59
6.75
7.45
7.95
8.33
9.15
9.83
10.31
10.67
11.32
11.77
12.10
12.38
12.71
13.12
13.44
13.70
13.91
14.09
6
3.74
4.21
4.48
4.67
4.81
5.11
5.37
5.54
5.68
5.91
6.08
6.20
6.31
6.43
6.59
6.71
6.80
6.89
6.95
9-8.
6
3.33
3.85
4.16
4.38
4.55
4.91
5.22
5.44
5.60
5.90
6.11
6.27
6.40
6.55
6.75
6.90
7.02
7.12
7.21
8
3.01
3.35
3,54
3.68
3.78
4.00
4.19
4.32
4.42
4.59
4.72
4.81
4.89
4.98
5.10
5.19
5.26
5.32
5.37
10
2.67
2.95
3.11
3.22
3.31
3.49
3.64
3.75
3.83
3.98
4.08
4.16
4.22
4.30
4.40
4.48
4.54
4.59
4.63
12
2.47
2.71
2.85
2.95
3.03
3.19
3.32
3.42
"'3.49'
3,62
3.71
3.78
3.84
3.91
3.99
4.06
4.12
4.16
4.20
K-Multipliers for
8
2.70
3, OS
3.27
3.42
3.54
3.78
4.00
4.15
4.26
4.47
4.62
4.73
4.82
4.93
5.07
5.18
5.26
5.31
5.40
10
2.41
2.69
2.86
2.98
3.O8
3.28
3.45
3.57
3.66
3.83
3.95
4.04
4.11
4.20
4.31
4.40
4.47
4.53
4.58
12
2.24
2.49
2.63
2.74
2.82
2.99
3.13
3,24'
3.32
3.46
3.56
3.64
3.71
3.78
3.88
3.96
4.02
4.07
4.12
16
2.25
2.46
2.57
2.66
2.72
2.86
2.97
3.05
3.11
3.22
3.30
3.36
3.41
3,47
3.54
3.6O
3,64
3,68
3,69
20
2.13
2.32
2.42
2.50
2.56
2.68
2.78
2.85
2.90
3.00
3.07
3.13
3.17
3.22
3.29
3.34
3.39
3.42
3.45
25
2.05
2.21
2.31
2.38
2.43
2.54
2.63
2.70
2.75
2.84
2.90
2.95
2.99
3.04
3.10
3.15
3.19
3.22
3.25
30
1.99
2.15
2.24
2.30
2.35
2.45
2.54
2.60
2.65
2.73
2.79
2.84
2.88
2.92
2.98
3.02
3.06
3.09
3.12
35 40 45 50 60 70 80 90 100 125
1.95 1.92 1.90 1.88 1.86 1.84 1.83 1.82 1.81 1.79
2.10 2.07 2.04 2.02 1.99 1.97 1.96 1.95 1.94 1.92
2.19 2.15 2.13 2.10 2.07 2.05 2.03 2.02 2.01 1.99
2.25 2.21 2.18 2.16 2.13 2.10 2.08 2.07 2.06 2.04
2.30 2.26 2.23 2.20 2.17 2.14 2.12 2.11 2.10 2.08
2.40 2.35 2.32 2.29 2.25 2.23 2.21 2.19 2.18 2.16
2.48 2.43 2.40 2.37 2.31 2.30 2.28 2.26 2.25 2.22
2.54 2.49 2.45 2.42 2.38 2.35 2.33 2.31 2.29 2.27
2.58 2.53 2.49 2.46 2.42 2.39 2.36 2.34 2.33 2.31
2.66 2.61 2.57 2.54 2.49 2.45 2.43 2.41 2.40 2.37
2.72 2.66 2.62 2.59 2.54 2.50 2.48 2.46 2.44 2.41
2.76 2.70 2.66 2.63 2.57 2.54 2.51 2.49 2.48 2.45
2.80 2.74 2.69 2.66 2.61 2.57 2.54 2.52 2.50 2.47
2.84 2.78 2.73 2.70 2.64 2.60 2.58 2.56 2.54 2.51
2.90 2.83 2.78 2.75 2.69 2.65 2.62 2.60 2.58 2.55
2.94 2.87 2.82 2.79 2.73 2.69 2.66 2.63 2.62 2.58
2.97 2.91 2.86 2.82 2.76 2.72 2.69 2.66 2.64 2.61
3.00 2.93 2.88 2.84 2.78 2.74 2.71 2.68 2.67 2.63
3.03 2.96 2.91 2.87 2.80 2.76 2.73 2.70 2.69 2.65
150
1.78
1.91
1.98
2.03
2.07
2.14
2.21
2.25
2.29
2.35
2.39
2.43
2.45
2.49
2.53
2.56
2.59
2.61
2.63
1-of-1 Interwell Prediction Limits on Means of Order 3 (40 COC, Annual)
16
2.05
2.26
2.38
2.47
2.53
2.67
2.79
2.88
2.95
3,06
3,15
3, 2O
3.27-
3.33
3.41
3.47
3.53
3.57
3.61
20
1.95
2.14
2.25
2.32
2.38
2.51
2.61
2.69
2.75
2.85
2.93
'2. '98"
3,03
3.O9
3.16
3,22
3.26
3,30
3.33
25
1.88
2.05
2.15
2.22
2.27
2.38
2.48
2.55
2.60
2.70
2.76
2.81
2.85
2.91
2,97
3,02
3.O6
3.1O
3,13
30
1.83
1.99
2.09
2.15
2.20
2.31
2.40
2.46
2.51
2.60
2.66
2.71
2.74
2.79
2.85
2.90
2.94
2,97
3.OO
35 40 45 50 60 70 80 90 100 125
1.79 1.77 1.75 1.74 1.71 1.70 1.69 1.68 .67 .66
1.95 1.92 1.90 1.88 1.86 1.84 1.83 1.82 .81 .79
2.04 2.01 1.99 1.97 1.94 1.92 1.90 1.89 .88 .87
2.10 2.07 2.05 2.02 2.00 1.97 1.96 1.95 .94 .92
2.15 2.12 2.09 2.07 2.04 2.02 2.00 1.99 .98 .96
2.25 2.21 2.19 2.16 2.13 2.10 2.09 2.07 2.06 2.04
2.34 2.30 2.27 2.24 2.20 2.18 2.16 2.14 2.13 2.11
2.40 2.35 2.32 2.29 2.26 2.23 2.21 2.19 2.18 2.16
2.45 2.40 2.36 2.34 2.30 2.27 2.25 2.23 2.22 2.19
2.53 2.48 2.44 2.41 2.37 2.34 2.32 2.30 2.28 2.26
2.59 2.54 2.50 2.46 2.42 2.39 2.33 2.35 2.33 2.31
2.63 2.58 2.54 2.51 2.46 2.42 2.40 2.38 2.37 2.34
2.67 2.61 2.57 2.54 2.49 2.46 2.43 2.41 2.40 2.37
2.71 2.66 2.61 2.58 2.53 2.49 2.47 2.44 2.43 2.40
2.77 2.71 2.67 2.63 2.58 2.54 2.51 2.49 2.48 2.45
2.81 2.75 2.71 2.67 2.62 2.58 2.55 2.53 2.51 2.48
2.85 2.79 2.74 2.70 2.65 2.61 2.58 2.56 2.52 2.51
2.88 2.82 2.77 2.73 2.67 2.63 2.60 2.58 2.56 2.53
2.91 2.84 2.79 2.75 2.69 2.65 2.62 2.60 2.58 2.54
150
1.65
1.78
1.86
1.91
1.95
2.03
2.10
2.14
2.18
2.20
2.29
2.32
2.34
2.38
2.43
2.46
2.49
2.51
2.53

                                                     D-106
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
Unified Guidance
Table 19-8. K- Multipliers for 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
6.75
7.95
8.65
9.15
9.52
10.31
10.96
11.42
11.77
12.38
12.80
13.12
13.38
13.70
14.09
14.39
14.64
14.84
15.01
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
7.95
9.15
9.83
10.31
10.67
11.42
12.04
12.48
12.80
13.38
13.79
14.09
14.34
14.64
15.01
15.30
15.48
15.73
15.89
6
3.85
4.38
4.69
4.91
5.08
5.44
5.74
5.95
6.11
6.40
6.60
6.75
6.87
7.02
7.21
7.35
7.47
7.56
7.65
9-8.
6
4.38
4.91
5.22
5.44
5.60
5.95
6.24
6.44
6.60
6.87
7.06
7.21
7.32
7.47
7.65
7.78
7.90
7.99
8.07
8
3.05
3.42
3.63
3.78
3.90
4.15
4.36
4.51
4.62
4.82
4.96
5.07
5.16
5.26
5.40
5.50
5.58
5.65
5.71
10
2.69
2.98
3.15
3,28
3,37
3.57
3.74
3.85
3.95
4.11
4.22
4.31
4.38
4.47
4.58
4.66
4.73
4.79
4.84
K-Multipliers
8
3.42
3.78
4.00
4.15
4.26
4.51
4.69
4.85
4.96
5.16
5.29
5.40
5.48
5.58
5.71
5.81
5.89
5.96
6.02
10
2.98
3.28
3.45
3.57
3.66
3.85
4.02
4.14
4.22
4.38
4.50
4.58
4.65
4.73
4.84
4.92
4.99
5.04
5.09
12
2.49
2.74
2.88
2.99
3.07
3.24
3.38
3.49
3.56
3.71
3.81
3.88
3.94
4.02
4.12
4.19
4.25
4.30
4.34
for
12
2.74
2.99
3.13
3.24
3.32
3.49
3,63
3.73
3.81
3.94
4.04
4.12
4.18
4.25
4.34
4.41
4.47
4.52
4.56
-of-1 Interwell Prediction Limits on Means of Order 3
16 20 25 30 35 40 45 50 60 70
2.26 2.14 2.05 1.99 1.95 1.92 1.90 1.88 1.86 1.84
2.47 2.32 2.22 2.15 2.10 2.07 2.05 2.02 2.00 1.97
2.59 2.43 2.32 2.24 2.19 2.16 2.13 2.11 2.07 2.05
2.67 2.51 2.38 2.31 2.25 2.21 2.19 2.16 2.13 2.10
2.74 2.57 2.44 2.36 2.30 2.26 2.23 2.20 2.17 2.14
2.88 2.69 2.55 2.46 2.40 2.35 2.32 2.29 2.26 2.23
3.00 2.79 2.64 2.55 2.48 2.44 2.40 2.37 2.33 2.30
3.08 2.87 2.71 2.61 2.54 2.49 2.45 2.42 2.38 2.35
3.15 2.93 2.76 2.66 2.59 2.54 2.50 2.46 2.42 2.39
3.27 3.03 2.85 2.74 2.67 2.61 2.57 2.54 2.49 2.46
3.35 3.10 2.92 2.81 2.73 2.67 2.62 2.59 2.54 2.50
3.41 3.16 2.97 2.85 2.77 2.71 2.67 2.63 2.58 2.54
3.46 3.2O 3.01 2.89 2.81 2.75 2.70 2.66 2.61 2.57
3.53 3.26 3.06 2.94 2.85 2.79 2.74 2.70 2.65 2.61
3.61 3,33 3.13 3.00 2.91 2.84 2.79 2.75 2.69 2.65
3.67 3.39 3.18 3.04 2.95 2.88 2.83 2.79 2.73 2.69
3.72 -3.43 3.22 3.08 2.99 2.92 2.86 2.82 2.76 2.72
3.76 3,47 3.25 3.11 3.01 2.94 2.89 2.85 2.77 2.74
3.80 -3,50 3:28 3.14 3.04 2.97 2.91 2.87 2.81 2.76
1-of-1 Interwell Prediction Limits on Means of Order
16 20 25 30 35 40 45 50 60 70
2.47 2.32 2.22 2.15 2.10 2.07 2.05 2.02 2.00 1.97
2.67 2.51 2.38 2.31 2.25 2.21 2.19 2.16 2.13 2.10
2.79 2.61 2.48 2.40 2.34 2.30 2.27 2.24 2.20 2.18
2.88 2.69 2.55 2.46 2.40 2.35 2.32 2.29 2.26 2.23
2.95 2.75 2.60 2.51 2.45 2.40 2.36 2.34 2.30 2.27
3.08 2.87 2.71 2.61 2.54 2.49 2.45 2.42 2.38 2.35
3.20 2.97 2.80 2.69 2.62 2.57 2.53 2.50 2.45 2.42
3.28 3.05 2.87 2.76 2.68 2.63 2.58 2.55 2.50 2.47
3.35 3.10 2.92 2.81 2.73 2.67 2.62 2.59 2.54 2.50
3,48 3.20 3.01 2.89 2.81 2.75 2.70 2.66 2.61 2.57
3.54 3.28 3.08 2.95 2.86 2.80 2.75 2.71 2.66 2.62
3.61 3.33 3.13 3.00 2.91 2.84 2.79 2.75 2.69 2.65
3.66 3.38 3.17 3.03 2.94 2.87 2.82 2.78 2.72 2.68
,3,72 3.43 3.22 3.08 2.99 2.92 2.86 2.82 2.76 2.72
3.80 3.50 3.28 3.14 3.04 2.97 2.91 2.87 2.81 2.76
3.86 3.55 3.33 3.18 3.08 3.01 2.95 2.91 2.84 2.80
3.90 .3,60 3.37 3.22 3.12 3.04 2.99 2.94 2.87 2.83
3.95 3.63 3.40 3.25 3.15 3.07 3.01 2.97 2.90 2.85
3.98 3.67 3.43 3.28 3.17 3.09 3.04 2.99 2.92 2.87
(40
80
1.83
1.96
2.03
2.09
2.13
2.21
2.28
2.33
2.33
2.43
2.48
2.51
2.54
2.58
2.62
2.66
2.69
2.71
2.73
COC,
90
1.82
1.95
2.02
2.07
2.11
2.19
2.26
2.31
2.35
2.41
2.46
2.49
2.52
2.56
2.60
2.64
2.66
2.69
2.71
Semi-Annual)
100
1.81
1.94
2.01
2.06
2.10
2.18
2.25
2.30
2.33
2.40
2.44
2.48
2.50
2.52
2.58
2.62
2.64
2.67
2.69
125
1.79
1.92
1.99
2.04
2.08
2.16
2.22
2.26
2.31
2.37
2.41
2.45
2.47
2.51
2.54
2.58
2.57
2.63
2.65
150
1.78
1.91
1.98
2.03
2.07
2.14
2.21
2.25
2.29
2.34
2.39
2.43
2.45
2.46
2.47
2.56
2.59
2.61
2.60
3 (40 COC, Quarterly)
80
1.96
2.09
2.16
2.21
2.25
2.33
2.39
2.44
2.48
2.54
2.58
2.62
2.65
2.69
2.73
2.77
2.79
2.82
2.84
90
1.95
2.07
2.14
2.19
2.23
2.31
2.38
2.42
2.46
2.52
2.57
2.60
2.63
2.66
2.71
2.74
2.77
2.79
2.81
100
1.94
2.06
2.13
2.18
2.22
2.30
2.36
2.41
2.44
2.50
2.55
2.58
2.61
2.64
2.69
2.72
2.75
2.77
2.79
125
1.92
2.04
2.11
2.16
2.19
2.26
2.33
2.38
2.41
2.47
2.52
2.54
2.58
2.61
2.65
2.68
2.71
2.73
2.75
150
1.91
2.03
2.10
2.14
2.18
2.20
2.32
2.32
2.39
2.46
2.50
2.53
2.56
2.59
2.63
2.66
2.68
2.71
2.73

                                                     D-107
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
      Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (1 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
(
(

















).67 (
).95 (
.11 (
.22 (
.30 ]
.47 ]
.62
.72
.79
.92
1.02
1.09
>.14
>.21
1.29
1.36
>.41
>.45
>.49
).53 0.46 (
).75 0.67 (
).88 0.79 (
).97 0.86 (
L.04 0.92 (
L.17 1.05 (
.28 1.15
.36 1.21
.42 1.27
.52 1.36
.59 1.42
.65 1.47
.69 1.51
.74 1.56
.81 1.62
.86 1.66
.90 1.70
.93 1.73
.96 1.75
).42 0.40 0.37 (
).62 0.59 0.55 (
).73 0.69 0.65 (
).80 0.77 0.72 (
).86 0.82 0.77 (
).97 0.93 0.87 (
.07 1.02 0.95 (
.13 1.08 1.01 (
.18 1.13 1.06
.27 1.21 1.13
.33 1.26 1.19
.37 1.31 1.23
.41 1.34 1.26
.45 1.38 1.30
.51 1.44 1.35
.55 1.48 1.39
.58 1.51 1.42
.61 1.53 1.44
.63 1.56 1.46
).35 (
).53 (
).62 (
).69 (
).74 (
).84 (
).92 (
).97 (
.02 (
.09
.14
.18
.21
.25
.30
.33
.36
.38
.41
).34 0.33 (
).51 0.50 (
).60 0.59 (
).67 0.65 (
).71 0.70 (
).81 0.79 (
).89 0.87 (
).94 0.92 (
).98 0.96 (
.05 1.03
.10 1.08
.14 1.11
.17 1.14
.21 1.18
.25 1.22
.29 1.26
.32 1.29
.34 1.31
.36 1.33
).32 (
).49 (
).58 (
).64 (
).68 (
).78 (
).85 (
).90 (
).94 (
.01
.06
.09
.12
.16
.20
.24
.26
.29
.30
).31 0.31 (
).48 0.48 (
).57 0.56 (
).63 0.62 (
).68 0.67 (
).77 0.76 (
).84 0.83 (
).89 0.88 (
).93 0.92 (
.00 0.99 (
.05 1.03
.08 1.07
.11 1.10
.14 1.13
.19 1.17
.22 1.21
.25 1.23
.27 1.25
.29 1.27
).31 0.30 0.30 (
).47 0.47 0.46 (
).56 0.55 0.55 (
).62 0.61 0.61 (
).66 0.65 0.65 (
).75 0.74 0.74 (
).83 0.82 0.81 (
).88 0.86 0.86 (
).91 0.90 0.89 (
).98 0.97 0.96 (
.03 1.01 1.00
.06 1.05 1.04
.09 1.07 1.06
.12 1.11 1.10
.16 1.15 1.14
.20 1.18 1.17
.22 1.21 1.19
.24 1.23 1.21
.26 1.24 1.23
).29 0.29 (
).46 0.45 (
).54 0.54 (
).60 0.60 (
).64 0.64 (
).73 0.73 (
).80 0.80 (
).85 0.85 (
).89 0.88 (
).95 0.95 (
.00 0.99 (
.03 1.02
.06 1.05
.09 1.08
.13 1.12
.16 1.15
.18 1.18
.20 1.20
.22 1.22
).29 0.29 (
).45 0.45 (
).54 0.53 (
).60 0.59 (
).64 0.63 (
).72 0.72 (
).79 0.79 (
).84 0.84 (
).88 0.87 (
).94 0.93 (
).98 0.98 (
.02 1.01
.04 1.04
.08 1.07
.12 1.11
.15 1.14
.17 1.16
.19 1.18
.21 1.20
).29
).45
).53
).59
).63
).71
).78
).83
).87
).93
).97
.00
.03
.06
.10
.13
.15
.17
.19
   Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
(


















).95 (
.22 (
.37 ]
.47 ]
.55 ]
.72 ]
.85
.94
1.02
>.14
1.23
1.29
2.35
>.41
1.49
1.55 :
1.60 :
1.64 :
1.68 ;
).75 0.67 (
).97 0.86 (
L.09 0.97 (
L.17 1.05 (
L.23 1.10
L.36 1.21
.47 1.31
.54 1.37
.59 1.42
.69 1.51
.76 1.57
.81 1.62
.85 1.65
.90 1.70
.96 1.75
>.01 1.79
>.05 1.83
>.08 1.86
Ml 1.88
).62 0.59 0.55 (
).80 0.77 0.72 (
).91 0.86 0.81 (
).97 0.93 0.87 (
.03 0.98 0.92 (
.13 1.08 1.01 (
.22 1.16 1.09
.28 1.22 1.15
.33 1.26 1.19
.41 1.34 1.26
.46 1.40 1.31
.51 1.44 1.35
.54 1.47 1.38
.58 1.51 1.42
.63 1.56 1.46
.67 1.60 1.50
.71 1.63 1.53
.73 1.65 1.55
.75 1.67 1.57
).53 (
).69 (
).78 (
).84 (
).88 (
).97 (
.05
.10
.14
.21
.26
.30
.33
.36
.41
.44
.47
.49
.51
).51 0.50 (
).67 0.65 (
).75 0.73 (
).81 0.79 (
).85 0.83 (
).94 0.92 (
.01 0.99 (
.06 1.04
.10 1.08
.17 1.14
.22 1.19
.25 1.22
.28 1.25
.32 1.29
.36 1.33
.39 1.36
.42 1.39
.44 1.41
.46 1.43
).49 (
).64 (
).72 (
).78 (
).82 (
).90 (
).97 (
.02
.06
.12
.17
.20
.23
.26
.30
.34
.36
.38
.40
).48 0.48 (
).63 0.62 (
).71 0.70 (
).77 0.76 (
).81 0.80 (
).89 0.88 (
).96 0.95 (
.01 1.00 (
.05 1.03
.11 1.10
.15 1.14
.19 1.17
.21 1.20
.25 1.23
.29 1.27
.32 1.30
.34 1.33
.36 1.35
.38 1.37
).47 0.47 0.46 (
).62 0.61 0.61 (
).70 0.69 0.68 (
).75 0.74 0.74 (
).79 0.78 0.78 (
).88 0.86 0.86 (
).94 0.93 0.92 (
).99 0.98 0.97 (
.03 1.01 1.00
.09 1.07 1.06
.13 1.12 1.10
.16 1.15 1.14
.19 1.17 1.16
.22 1.21 1.19
.26 1.24 1.23
.29 1.27 1.26
.32 1.30 1.29
.34 1.32 1.31
.36 1.34 1.32
).46 0.45 (
).60 0.60 (
).68 0.67 (
).73 0.73 (
).77 0.77 (
).85 0.85 (
).92 0.91 (
).96 0.96 (
.00 0.99 (
.06 1.05
.10 1.09
.13 1.12
.15 1.15
.18 1.18
.22 1.22
.25 1.24
.28 1.27
.30 1.29
.31 1.30
).45 0.45 (
).60 0.59 (
).67 0.67 (
).72 0.72 (
).76 0.76 (
).84 0.84 (
).91 0.90 (
).95 0.94 (
).98 0.98 (
.04 1.04
.08 1.08
.12 1.11
.14 1.13
.17 1.16
.21 1.20
.24 1.23
.26 1.25
.28 1.27
.30 1.29
).45
).59
).66
).71
).75
).83
).89
).94
).97
.03
.07
.10
.12
.15
.19
.22
.24
.26
.28
                                                    D-108
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                      Unified Guidance
     Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.22 (
.47 ]
.62 ]
.72 ]
.79 ]
.94 ]
2.07
2.16
2.23
2.35
2.43
2.49
2.54 ;
2.60 ;
2.68 ;
2.74 ;
2.79 ;
2.82 ;
2.86 ;
).97 0.86 (
L.17 1.05 (
L.28 1.15
L.36 1.21
L.42 1.27
L.54 1.37
.64 1.46
.71 1.52
.76 1.57
.85 1.65
.91 1.71
.96 1.75
>.00 1.79
>.05 1.83
Ml 1.88
>.15 1.92
>.19 1.95
1.22 1.98
1.25 2.00
).80 0.77 0.72 (
).97 0.93 0.87 (
.07 1.02 0.95 (
.13 1.08 1.01 (
.18 1.13 1.06
.28 1.22 1.15
.36 1.30 1.22
.42 1.35 1.27
.46 1.40 1.31
.54 1.47 1.38
.59 1.52 1.43
.63 1.56 1.46
.67 1.59 1.49
.71 1.63 1.53
.75 1.67 1.57
.79 1.71 1.61
.82 1.74 1.63
.85 1.76 1.66
.87 1.78 1.68
).69 (
).84 (
).92 (
).97 (
.02 (
.10
.17
.22
.26
.33
.37
.41
.43
.47
.51
.54
.57
.59
.61
).67 0.65 (
).81 0.79 (
).89 0.87 (
).94 0.92 (
).98 0.96 (
.06 1.04
.13 1.11
.18 1.15
.22 1.19
.28 1.25
.33 1.29
.36 1.33
.39 1.35
.42 1.39
.46 1.43
.49 1.46
.52 1.48
.54 1.50
.56 1.52
).64 (
).78 (
).85 (
).90 (
).94 (
.02
.09
.13
.17
.23
.27
.30
.33
.36
.40
.43
.46
.48
.49
).63 0.62 (
).77 0.76 (
).84 0.83 (
).89 0.88 (
).93 0.92 (
.01 1.00 (
.07 1.06
.12 1.11
.15 1.14
.21 1.20
.26 1.24
.29 1.27
.31 1.30
.34 1.33
.38 1.37
.41 1.40
.44 1.42
.46 1.44
.47 1.46
).62 0.61 0.61 (
).75 0.74 0.74 (
).83 0.82 0.81 (
).88 0.86 0.86 (
).91 0.90 0.89 (
).99 0.98 0.97 (
.05 1.04 1.03
.10 1.08 1.07
.13 1.12 1.10
.19 1.17 1.16
.23 1.21 1.20
.26 1.24 1.23
.29 1.27 1.26
.32 1.30 1.29
.36 1.34 1.32
.38 1.37 1.35
.41 1.39 1.37
.43 1.41 1.39
.44 1.42 1.41
).60 0.60 (
).73 0.73 (
).80 0.80 (
).85 0.85 (
).89 0.88 (
).96 0.96 (
.02 1.02
.06 1.06
.10 1.09
.15 1.15
.19 1.19
.22 1.22
.25 1.24
.28 1.27
.31 1.30
.34 1.33
.36 1.35
.38 1.37
.40 1.39
).60 0.59 (
).72 0.72 (
).79 0.79 (
).84 0.84 (
).88 0.87 (
).95 0.94 (
.01 1.00
.05 1.04
.08 1.08
.14 1.13
.18 1.17
.21 1.20
.23 1.22
.26 1.25
.30 1.29
.33 1.31
.35 1.33
.37 1.35
.38 1.37
).59
).71
).78
).83
).87
).94
.00
.04
.07
.12
.16
.19
.21
.24
.28
.30
.33
.34
.36
      Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
.04 0.80 0.70 (
.36 1.04 0.91 (
.55 1.17 1.02 (
.68 1.27 1.11
.78 1.34 1.17
.99 1.49 1.29
2.17 .61 1.40
2.29 .70 1.47
2.38 .76 1.53
2.54 .88 1.63
2.66 .96 1.70
2.74 2.02 1.75
281 2.07 1.79
2.90 2.13 1.84
3.00 2.21 1.91
3.08 2.26 1.96
3.15 2.31 2.00
3.20 2.35 2.03
3.25 2.38 2.06
).64 0.61 0.56 (
).84 0.79 0.73 (
).94 0.89 0.83 (
.02 0.96 0.89 (
.07 1.01 0.94 (
.19 1.12 1.04 (
.29 1.21 1.12
.35 1.28 1.18
.40 1.32 1.23
.49 1.41 1.30
.56 1.47 1.36
.60 1.51 1.40
.64 1.55 1.43
.69 1.59 1.47
.75 1.65 1.52
.79 1.69 1.56
.83 1.72 1.59
.86 1.75 1.62
.88 1.77 1.64
).54 (
).70 (
).79 (
).85 (
).90 (
).99 (
.07
.13
.17
.24
.29
.33
.36
.40
.45
.49
.52
.54
.56
).52 0.50 (
).67 0.66 (
).76 0.74 (
).82 0.80 (
).87 0.84 (
).96 0.93 (
.03 1.00 (
.08 1.06
.12 1.09
.19 1.16
.24 1.21
.28 1.25
.31 1.27
.35 1.31
.39 1.35
.43 1.39
.46 1.41
.48 1.44
.50 1.46
).49 (
).64 (
).73 (
).78 (
).83 (
).91 (
).99 (
.03
.07
.14
.18
.22
.25
.28
.33
.36
.39
.41
.43
).49 0.48 (
).64 0.63 (
).72 0.71 (
).77 0.76 (
).82 0.81 (
).90 0.89 (
).97 0.96 (
.02 1.01
.06 1.04
.12 1.11
.17 1.15
.20 1.19
.23 1.21
.26 1.25
.30 1.29
.34 1.32
.36 1.35
.38 1.37
.40 1.39
).47 0.47 0.46 (
).62 0.61 0.61 (
).70 0.69 0.69 (
).76 0.75 0.74 (
).80 0.79 0.78 (
).88 0.87 0.86 (
).95 0.94 0.93 (
.00 0.98 0.97 (
.03 1.02 1.01
.10 1.08 1.07
.14 1.12 1.11
.17 1.16 1.14
.20 1.18 1.17
.23 1.21 1.20
.27 1.25 1.24
.31 1.29 1.27
.33 1.31 1.29
.35 1.33 1.32
.37 1.35 1.33
).46 0.46 (
).60 0.60 (
).68 0.68 (
).73 0.73 (
).77 0.77 (
).85 0.85 (
).92 0.91 (
).97 0.96 (
.00 0.99 (
.06 1.05
.10 1.09
.13 1.13
.16 1.15
.19 1.18
.23 1.22
.26 1.25
.28 1.27
.30 1.29
.32 1.31
).45 0.45 (
).60 0.59 (
).67 0.67 (
).73 0.72 (
).77 0.76 (
).84 0.84 (
).91 0.90 (
).95 0.95 (
).99 0.98 (
.05 1.04
.09 1.08
.12 1.11
.15 1.13
.18 1.16
.21 1.20
.24 1.23
.27 1.25
.29 1.27
.30 1.29
).45
).59
).66
).72
).75
).83
).90
).94
).97
.03
.07
.10
.13
.16
.19
.22
.25
.26
.28
                                                    D-109
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
   Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (2 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.36 1.04 0.91 (
1.68 1.27 1.11
1.86 1.40 1.22
1.99 1.49 1.29
2.09 1.56 1.35
2.29 1.70 1.47
2.45 1.82 1.57
2.57 1.90 1.64
2.66 1.96 1.70
2.81 2.07 1.79
2.92 2.15 1.86
3.00 2.21 1.91
3.07 2.25 1.95
3.15 2.31 2.00
3.25 2.38 2.06
3.33 2.44 2.10
3.39 2.48 2.14
344 2.52 2.17
3.49 2.55 2.20 ;
).84 0.79 0.73 (
.02 0.96 0.89 (
.12 1.06 0.98 (
.19 1.12 1.04 (
.24 1.17 1.09
.35 1.28 1.18
.44 1.36 1.26
.51 1.42 1.32
.56 1.47 1.36
.64 1.55 1.43
.70 1.60 1.48
.75 1.65 1.52
.78 1.68 1.55
.83 1.72 1.59
.88 1.77 1.64
.93 1.81 1.68
.96 1.85 1.71
.99 1.87 1.73
>.01 1.90 1.75
).70 (
).85 (
).94 (
).99 (
.04
.13
.20
.25
.29
.36
.41
.45
.48
.52
.56
.60
.63
.65
.67
).67 0.66 (
).82 0.80 (
).90 0.88 (
).96 0.93 (
.00 0.97 (
.08 1.06
.16 1.12
.21 1.17
.24 1.21
.31 1.27
.36 1.32
.39 1.35
.42 1.38
.46 1.41
.50 1.46
.53 1.49
.56 1.52
.58 1.54
.60 1.56
).64 (
).78 (
).86 (
).91 (
).95 (
.03
.10
.15
.18
.25
.29
.33
.35
.39
.43
.46
.48
.51
.52
).64 0.63 (
).77 0.76 (
).85 0.84 (
).90 0.89 (
).94 0.93 (
.02 1.01
.09 1.07
.13 1.12
.17 1.15
.23 1.21
.27 1.26
.30 1.29
.33 1.31
.36 1.35
.40 1.39
.43 1.42
.46 1.44
.48 1.46
.50 1.48
).62 0.61 0.61 (
).76 0.75 0.74 (
).83 0.82 0.81 (
).88 0.87 0.86 (
).92 0.91 0.90 (
.00 0.98 0.97 (
.06 1.05 1.04
.11 1.09 1.08
.14 1.12 1.11
.20 1.18 1.17
.24 1.22 1.21
.27 1.25 1.24
.30 1.28 1.27
.33 1.31 1.29
.37 1.35 1.33
.40 1.38 1.36
.43 1.40 1.39
.45 1.42 1.40
.46 1.44 1.42
).60 0.60 (
).73 0.73 (
).81 0.80 (
).85 0.85 (
).89 0.89 (
).97 0.96 (
.03 1.02
.07 1.06
.10 1.09
.16 1.15
.20 1.19
.23 1.22
.25 1.25
.28 1.27
.32 1.31
.35 1.34
.37 1.36
.39 1.38
.41 1.40
).60 0.59 (
).73 0.72 (
).80 0.79 (
).84 0.84 (
).88 0.87 (
).95 0.95 (
.02 1.01
.06 1.05
.09 1.08
.15 1.13
.18 1.17
.21 1.20
.24 1.23
.27 1.25
.30 1.29
.33 1.32
.35 1.34
.37 1.36
.39 1.37
).59
).72
).78
).83
).87
).94
.00
.04
.07
.13
.16
.19
.22
.25
.28
.31
.33
.35
.36
     Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.68 1.27 1.11
1.99 1.49 1.29
2.17 1.61 1.40
2.29 1.70 1.47
2.38 1.76 1.53
2.57 1.90 1.64
2.73 2.01 1.74
2.84 2.09 1.81
2.92 2.15 1.86
3.07 2.25 1.95
3.17 2.33 2.01
3.25 2.38 2.06
3.31 2.43 2.10
3.39 2.48 2.14
3.49 2.55 2.20 ;
3.56 2.60 2.24 ;
3.62 2.64 2.28 ;
3.67 2.68 2.31 ;
3.71 2.71 2.34 ;
.02 0.96 0.89 (
.19 1.12 1.04 (
.29 1.21 1.12
.35 1.28 1.18
.40 1.32 1.23
.51 1.42 1.32
.60 1.50 1.39
.66 1.56 1.44
.70 1.60 1.48
.78 1.68 1.55
.84 1.73 1.60
.88 1.77 1.64
.92 1.81 1.67
.96 1.85 1.71
>.01 1.90 1.75
>.06 1.94 1.79
>.09 1.97 1.82
1.12 1.99 1.84
>.14 2.01 1.86
).85 (
).99 (
.07
.13
.17
.25
.33
.38
.41
.48
.53
.56
.59
.63
.67
.70
.73
.75
.77
).82 0.80 (
).96 0.93 (
.03 1.00 (
.08 1.06
.12 1.09
.21 1.17
.27 1.24
.32 1.28
.36 1.32
.42 1.38
.47 1.42
.50 1.46
.53 1.48
.56 1.52
.60 1.56
.63 1.59
.66 1.61
.68 1.63
.70 1.65
).78 (
).91 (
).99 (
.03
.07
.15
.21
.26
.29
.35
.39
.43
.45
.48
.52
.55
.58
.60
.62
).77 0.76 (
).90 0.89 (
).97 0.96 (
.02 1.01
.06 1.04
.13 1.12
.19 1.18
.24 1.22
.27 1.26
.33 1.31
.37 1.35
.40 1.39
.43 1.41
.46 1.44
.50 1.48
.53 1.51
.55 1.53
.57 1.55
.59 1.57
).76 0.75 0.74 (
).88 0.87 0.86 (
).95 0.94 0.93 (
.00 0.98 0.97 (
.03 1.02 1.01
.11 1.09 1.08
.17 1.15 1.14
.21 1.19 1.18
.24 1.22 1.21
.30 1.28 1.27
.34 1.32 1.30
.37 1.35 1.33
.40 1.37 1.36
.43 1.40 1.39
.46 1.44 1.42
.49 1.47 1.45
.52 1.49 1.47
.53 1.51 1.49
.55 1.53 1.51
).73 0.73 (
).85 0.85 (
).92 0.91 (
).97 0.96 (
.00 0.99 (
.07 1.06
.13 1.12
.17 1.16
.20 1.19
.25 1.25
.29 1.28
.32 1.31
.34 1.33
.37 1.36
.41 1.40
.44 1.42
.46 1.45
.48 1.47
.49 1.48
).73 0.72 (
).84 0.84 (
).91 0.90 (
).95 0.95 (
).99 0.98 (
.06 1.05
.11 1.10
.15 1.14
.18 1.17
.24 1.23
.28 1.26
.30 1.29
.33 1.31
.35 1.34
.39 1.37
.42 1.40
.44 1.42
.46 1.44
.47 1.46
).72
).83
).90
).94
).97
.04
.10
.14
.16
.22
.25
.28
.30
.33
.36
.39
.41
.43
.44
                                                    D-110
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
      Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.62 1.17 1.01 (
2.02 1.43 1.22
2.26 1.58 1.34
2.43 1.69 1.43
2.57 1.77 1.50
2.84 1.94 1.63
3.07 2.09 1.75
3.24 2.19 1.83
3.36 2.27 1.89
3.58 2.40 2.00
3.73 2.50 2.08
3.85 2.57 2.14
3.94 2.63 2.19
4.06 2.70 2.24 ;
4.20 2,79 2.32 ;
4.31 2,86 2.37 ;
4.40 2.91 2.42 ;
4.47 2.96 2.45 ;
4.53 3.00 2.49 ;
).92 0.86 0.80 (
.11 1.04 0.96 (
.22 1.14 1.05
.29 1.21 1.11
.35 1.26 1.16
.47 1.37 1.26
.57 1.47 1.34
.65 1.53 1.40
.70 1.58 1.44
.80 1.67 1.52
.86 1.73 1.58
.92 1.78 1.62
.96 1.82 1.65
>.01 1.86 1.70
>.07 1.92 1.75
>.12 1.97 1.79
>.16 2.01 1.82
>.19 2.04 1.85
1.22 2.06 1.87
).76 (
).91 (
.00 (
.05
.10
.19
.27
.32
.36
.44
.49
.53
.56
.60
.65
.68
.72
.74
.76
).73 0.71 (
).87 0.85 (
).95 0.93 (
.01 0.98 (
.05 1.02
.14 1.11
.21 1.18
.26 1.22
.30 1.26
.37 1.33
.42 1.37
.46 1.41
.49 1.44
.52 1.47
.57 1.52
.60 1.55
.63 1.58
.66 1.60
.68 1.62
).70 (
).83 (
).91 (
).96 (
.00 (
.08
.15
.20
.23
.30
.34
.38
.40
.44
.48
.51
.54
.56
.58
).68 0.68 (
).82 0.81 (
).89 0.88 (
).95 0.93 (
).98 0.97 (
.06 1.05
.13 1.12
.18 1.16
.21 1.19
.27 1.26
.32 1.30
.35 1.33
.38 1.36
.41 1.39
.45 1.43
.48 1.46
.51 1.49
.53 1.51
.55 1.52
).67 0.66 0.65 (
).80 0.79 0.78 (
).87 0.86 0.85 (
).92 0.91 0.90 (
).96 0.95 0.94 (
.04 1.02 1.01
.10 1.09 1.07
.15 1.13 1.12
.18 1.16 1.15
.24 1.22 1.21
.28 1.26 1.25
.32 1.29 1.28
.34 1.32 1.30
.37 1.35 1.33
.41 1.39 1.37
.44 1.42 1.40
.47 1.44 1.42
.49 1.46 1.44
.51 1.48 1.46
).65 0.64 (
).78 0.77 (
).85 0.84 (
).89 0.89 (
).93 0.92 (
.00 1.00 (
.06 1.06
.11 1.10
.14 1.13
.19 1.19
.23 1.22
.26 1.25
.29 1.28
.32 1.31
.36 1.34
.38 1.37
.41 1.40
.43 1.41
.44 1.43
).64 0.64 (
).77 0.76 (
).84 0.83 (
).88 0.87 (
).92 0.91 (
).99 0.98 (
.05 1.04
.09 1.08
.12 1.11
.18 1.17
.22 1.20
.25 1.23
.27 1.26
.30 1.29
.34 1.32
.36 1.35
.39 1.37
.41 1.39
.42 1.40
).63
).76
).82
).87
).90
).97
.03
.07
.10
.16
.20
.22
.25
.28
.31
.34
.36
.38
.39
   Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.02 1.43 1.22
2.43 1.69 1.43
2.67 1.84 1.55
2.84 1.94 1.63
2.97 2.02 1.70
3.24 2.19 1.83
3.46 2.33 1.94
3.61 2.43 2.02
3.73 2.50 2.08
3.94 2.63 2.19
4.09 2.72 2.26 ;
4.20 2.79 2.32 ;
4.29 2.85 2.36 ;
4.40 2,91 2.42 ;
4.53 3.00 2.49 ;
4.64 3.07 2.54 ;
4.72 3.12 2.58 ;
4.79 3.16 2.62 ;
4.85 3.20 2.65 ;
.11 1.04 0.96 (
.29 1.21 1.11
.40 1.31 1.20
.47 1.37 1.26
.53 1.42 1.30
.65 1.53 1.40
.74 1.62 1.48
.81 1.68 1.53
.86 1.73 1.58
.96 1.82 1.65
1.02 1.88 1.71
>.07 1.92 1.75
Ml 1.96 1.78
>.16 2.01 1.82
1.22 2.06 1.87
1.27 2.11 1.91
>.31 2.14 1.94
2.34 2.17 1.97
2.37 2.19 1.99
).91 (
.05
.13
.19
.23
.32
.40
.45
.49
.56
.61
.65
.68
.72
.76
.80
.83
.85
.87
).87 0.85 (
.01 0.98 (
.09 1.05
.14 1.11
.18 1.14
.26 1.22
.33 1.29
.38 1.34
.42 1.37
.49 1.44
.53 1.48
.57 1.52
.60 1.54
.63 1.58
.68 1.62
.71 1.65
.74 1.68
.76 1.70
.78 1.72
).83 (
).96 (
.03
.08
.12
.20
.26
.31
.34
.40
.45
.48
.51
.54
.58
.61
.64
.66
.68
).82 0.81 (
).95 0.93 (
.02 1.00 (
.06 1.05
.10 1.09
.18 1.16
.24 1.22
.28 1.27
.32 1.30
.38 1.36
.42 1.40
.45 1.43
.48 1.46
.51 1.49
.55 1.52
.58 1.55
.60 1.58
.62 1.60
.64 1.62
).80 0.79 0.78 (
).92 0.91 0.90 (
).99 0.98 0.97 (
.04 1.02 1.01
.07 1.06 1.05
.15 1.13 1.12
.21 1.19 1.17
.25 1.23 1.21
.28 1.26 1.25
.34 1.32 1.30
.38 1.36 1.34
.41 1.39 1.37
.44 1.41 1.39
.47 1.44 1.42
.51 1.48 1.46
.54 1.51 1.49
.56 1.53 1.51
.58 1.55 1.53
.60 1.57 1.54
).78 0.77 (
).89 0.89 (
).96 0.95 (
.00 1.00 (
.04 1.03
.11 1.10
.16 1.16
.20 1.19
.23 1.22
.29 1.28
.33 1.32
.36 1.34
.38 1.37
.41 1.40
.44 1.43
.47 1.46
.49 1.48
.51 1.50
.53 1.51
).77 0.76 (
).88 0.87 (
).95 0.94 (
).99 0.98 (
.02 1.01
.09 1.08
.15 1.14
.19 1.18
.22 1.20
.27 1.26
.31 1.29
.34 1.32
.36 1.34
.39 1.37
.42 1.40
.45 1.43
.47 1.45
.49 1.47
.50 1.49
).76
).87
).93
).97
.01
.07
.13
.17
.20
.25
.28
.31
.33
.36
.39
.42
.44
.46
.47
                                                    D-111
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                      Unified Guidance
     Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.43 1.69 1.43
2.84 1.94 1.63
3.07 2.09 1.75
3.24 2.19 1.83
3.36 2.27 1.89
3.61 2.43 2.02
3.83 2.56 2.13
3.98 2.65 2.20
4.09 2.72 2.26 ;
4.29 2.85 2.36 ;
4.43 2.93 2.43 ;
4.53 3.00 2.49 ;
4.62 3.05 2.53 ;
4.72 3.12 2.58 ;
4.85 3.20 2.65 ;
4.95 3.26 2.70 ;
5.03 3.31 2.74 ;
5.10 3.36 2.78 ;
5.16 3.39 2.81 ;
.29 1.21 1.11
.47 1.37 1.26
.57 1.47 1.34
.65 1.53 1.40
.70 1.58 1.44
.81 1.68 1.53
.91 1.77 1.61
.97 1.83 1.67
1.02 1.88 1.71
Ml 1.96 1.78
M8 2.02 1.83
1.22 2.06 1.87
1.26 2.10 1.90
>.31 2.14 1.94
2.37 2.19 1.99
>.41 2.24 2.03
2.45 2.27 2.06
2.48 2.30 2.08
2.51 2.32 2.11
.05
.19
.27
.32
.36
.45
.52
.57
.61
.68
.73
.76
.79
.83
.87
.91
.94
.96
.98
.01 0.98 (
.14 1.11
.21 1.18
.26 1.22
.30 1.26
.38 1.34
.45 1.40
.50 1.45
.53 1.48
.60 1.54
.64 1.59
.68 1.62
.70 1.65
.74 1.68
.78 1.72
.81 1.75
.84 1.78
.86 1.80
.88 1.82
).96 (
.08
.15
.20
.23
.31
.37
.41
.45
.51
.55
.58
.60
.64
.68
.71
.73
.75
.77
).95 0.93 (
.06 1.05
.13 1.12
.18 1.16
.21 1.19
.28 1.27
.34 1.33
.39 1.37
.42 1.40
.48 1.46
.52 1.49
.55 1.52
.57 1.55
.60 1.58
.64 1.62
.67 1.65
.70 1.67
.72 1.69
.73 1.71
).92 0.91 0.90 (
.04 1.02 1.01
.10 1.09 1.07
.15 1.13 1.12
.18 1.16 1.15
.25 1.23 1.21
.31 1.29 1.27
.35 1.33 1.31
.38 1.36 1.34
.44 1.41 1.39
.48 1.45 1.43
.51 1.48 1.46
.53 1.50 1.48
.56 1.53 1.51
.60 1.57 1.54
.62 1.59 1.57
.65 1.62 1.59
.67 1.63 1.61
.68 1.65 1.63
).89 0.89 (
.00 1.00 (
.06 1.06
.11 1.10
.14 1.13
.20 1.19
.26 1.25
.30 1.29
.33 1.32
.38 1.37
.41 1.40
.44 1.43
.47 1.45
.49 1.48
.53 1.51
.55 1.54
.58 1.56
.59 1.58
.61 1.59
).88 0.87 (
).99 0.98 (
.05 1.04
.09 1.08
.12 1.11
.19 1.18
.24 1.23
.28 1.27
.31 1.29
.36 1.34
.39 1.38
.42 1.40
.44 1.43
.47 1.45
.50 1.49
.53 1.51
.55 1.53
.57 1.55
.58 1.56
).87
).97
.03
.07
.10
.17
.22
.26
.28
.33
.37
.39
.41
.44
.47
.50
.52
.53
.55
      Table 19-9. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 3 (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.15 1.48 1.24
2.64 1.76 1.46
2.94 1.93 1.59
3.15 2.04 1.68
3.32 2.14 1.75
3.66 2.33 1.90
3.95 2.49 2.03
4.15 2.61 2.12
4.31 2.69 2.18
4.58 2,85 2.30 ;
4.77 2.96 2.39 ;
4.92 3.04 2.45 ;
5.04 3.11 2.50 ;
5.18 3.19 2.57 ;
5.36 3.29 2.65 ;
5.50 3.37 2,71 *
5.61 3.43 2.76 '4
5.70 3.49 2.80 \
5.78 3.53 2,83 \
.12 1.05 0.96 (
.31 1.22 1.12
.43 1.32 1.21
.50 1.40 1.27
.56 1.45 1.32
.69 1.56 1.42
.80 1.66 1.50
.87 1.73 1.56
.93 1.78 1.60
2.03 1.87 1.68
Ml 1.94 1.74
M6 1.99 1.78
1.21 2.03 1.82
1.26 2.08 1.86
2.33 2.14 1.92
2.38 2.18 1.96
1.42 2.22 1.99
2.46 2.26 2.02
2.49 2.28 2.05
).91 (
.06
.14
.20
.24
.33
.41
.46
.51
.58
.63
.67
.70
.74
.79
.83
.86
.89
.91
).88 0.85 (
.01 0.98 (
.09 1.06
.14 1.11
.19 1.15
.27 1.23
.34 1.30
.39 1.35
.43 1.38
.50 1.45
.55 1.49
.58 1.53
.61 1.56
.65 1.59
.70 1.63
.73 1.67
.76 1.69
.78 1.72
.81 1.74
).83 (
).96 (
.03
.08
.12
.20
.27
.31
.35
.41
.45
.49
.51
.55
.59
.62
.65
.67
.69
).82 0.81 (
).95 0.94 (
.02 1.00 (
.07 1.05
.10 1.09
.18 1.16
.24 1.23
.29 1.27
.32 1.30
.38 1.36
.43 1.40
.46 1.44
.48 1.46
.52 1.49
.56 1.53
.59 1.56
.61 1.59
.64 1.61
.65 1.63
).80 0.79 0.78 (
).93 0.91 0.90 (
).99 0.98 0.97 (
.04 1.02 1.01
.08 1.06 1.05
.15 1.13 1.12
.21 1.19 1.18
.25 1.23 1.22
.29 1.26 1.25
.35 1.32 1.30
.39 1.36 1.34
.42 1.39 1.37
.44 1.41 1.40
.47 1.44 1.42
.51 1.48 1.46
.54 1.51 1.49
.57 1.53 1.51
.59 1.55 1.53
.60 1.57 1.55
).78 0.77 (
).89 0.89 (
).96 0.95 (
.00 1.00 (
.04 1.03
.11 1.10
.16 1.16
.21 1.20
.24 1.23
.29 1.28
.33 1.32
.36 1.35
.38 1.37
.41 1.40
.45 1.43
.47 1.46
.50 1.48
.51 1.50
.53 1.52
).77 0.76 (
).88 0.87 (
).95 0.94 (
).99 0.98 (
.02 1.01
.09 1.08
.15 1.14
.19 1.18
.22 1.21
.27 1.26
.31 1.29
.34 1.32
.36 1.34
.39 1.37
.42 1.41
.45 1.43
.47 1.45
.49 1.47
.51 1.49
).76
).87
).93
).97
.01
.07
.13
.17
.20
.25
.28
.31
.33
.36
.39
.42
.44
.46
.47
                                                    D-112
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
   Table 19-9. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 3 (10 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.64 1.76 1.46
3.15 2.04 1.68
3.45 2.21 1.81
3.66 2.33 1.90
3.82 2.42 1.97
4.15 2.61 2.12
443 2.76 2.24
4.63 2.87 2.32 ;
4.77 ^,96 2.39 ;
5.04 3.11 2.50 ;
5.22 3.21 2.58 ;
5.36 3.29 2.65 ;
5.47 3.36 2.70 ;
5.61 3.43 2.76 ;
5.78 3.53 2.83 ;
5.91 3.61 2,89 ;
6.02 3.67 2.94 \
6.10 3.72 2,98 ;
6.18 3.76 3,01 ;
.31 1.22 1.12
.50 1.40 1.27
.61 1.49 1.36
.69 1.56 1.42
.75 1.62 1.46
.87 1.73 1.56
.98 1.82 1.64
2.05 1.89 1.70
Ml 1.94 1.74
1.21 2.03 1.82
2.28 2.09 1.87
2.33 2.14 1.92
2.37 2.18 1.95
1.42 2.22 1.99
1.49 2.28 2.05
2.54 2.33 2.09
2.58 2.37 2.12
2.62 2.40 2.15 ;
2.65 2.42 2.17 ;
.06
.20
.28
.33
.38
.46
.54
.59
.63
.70
.75
.79
.82
.86
.91
.95
.98
2.00
2.02
.01 0.98 (
.14 1.11
.22 1.18
.27 1.23
.31 1.27
.39 1.35
.46 1.41
.51 1.46
.55 1.49
.61 1.56
.66 1.60
.70 1.63
.73 1.66
.76 1.69
.81 1.74
.84 1.77
.87 1.80
.89 1.82
.91 1.84
).96 (
.08
.15
.20
.24
.31
.38
.42
.45
.51
.56
.59
.62
.65
.69
.72
.75
.77
.79
).95 0.94 (
.07 1.05
.13 1.12
.18 1.16
.21 1.20
.29 1.27
.35 1.33
.39 1.37
.43 1.40
.48 1.46
.53 1.50
.56 1.53
.58 1.56
.61 1.59
.65 1.63
.68 1.66
.71 1.68
.73 1.70
.75 1.72
).93 0.91 0.90 (
.04 1.02 1.01
.11 1.09 1.07
.15 1.13 1.12
.18 1.16 1.15
.25 1.23 1.22
.31 1.29 1.27
.35 1.33 1.31
.39 1.36 1.34
.44 1.41 1.40
.48 1.45 1.43
.51 1.48 1.46
.54 1.51 1.48
.57 1.53 1.51
.60 1.57 1.55
.63 1.60 1.57
.66 1.62 1.60
.68 1.64 1.62
.69 1.66 1.63
).89 0.89 (
.00 1.00 (
.06 1.06
.11 1.10
.14 1.13
.21 1.20
.26 1.25
.30 1.29
.33 1.32
.38 1.37
.42 1.41
.45 1.43
.47 1.46
.50 1.48
.53 1.52
.56 1.54
.58 1.56
.60 1.58
.61 1.60
).88 0.87 (
).99 0.98 (
.05 1.04
.09 1.08
.12 1.11
.19 1.18
.24 1.23
.28 1.27
.31 1.29
.36 1.34
.40 1.38
.42 1.41
.45 1.43
.47 1.45
.51 1.49
.53 1.51
.55 1.53
.57 1.55
.59 1.56
).87
).97
.03
.07
.10
.17
.22
.26
.28
.33
.37
.39
.42
.44
.47
.50
.52
.54
.55
    Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (10 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.15 2.04 1.68 .50 1.40 1.27
3.66 2.33 1.90 .69 1.56 1.42
3.95 2.49 2.03 .80 1.66 1.50
4.15 2.61 2.12 .87 1.73 1.56
431 2.69 2.18 .93 1.78 1.60
463 2.87 2.32 2.05 1.89 1.70
4.89 3.03 2.44 2.15 1.98 1.78
5.08 3.13 2.52 2.22 2.04 1.83
5.22 3.21 2.58 2.28 2.09 1.87
5.47 3.36 2.70 2.37 2.18 1.95
5.65 3.46 2.77 2.44 2.24 2.00
5.78 3.53 2.83 2.49 2.28 2.05
5.89 3.60 2.88 2.53 2.32 2.08
6.02 3.67 2.94 2.58 2.37 2.12
6.18 3.76 3.01 2.65 2.42 2.17 ;
6.31 3.84 3.O7 2.69 2.47 2.21 ;
6.41 3.90 3, 12 2.73 2.51 2.24 ;
6.49 3.94 3, 16 2.77 2.54 2.27 ;
6.56 3.99 3,19 2.80 2.56 2.29 ;
.20
.33
.41
.46
.51
.59
.66
.72
.75
.82
.87
.91
.94
.98
2.02
2.06
2.09
2.11 ;
2.13 ;
.14 1.11
.27 1.23
.34 1.30
.39 1.35
.43 1.38
.51 1.46
.58 1.52
.62 1.57
.66 1.60
.73 1.66
.77 1.70
.81 1.74
.83 1.76
.87 1.80
.91 1.84
.95 1.87
.97 1.90
2.00 1.92
2.01 1.94
.08
.20
.27
.31
.35
.42
.48
.52
.56
.62
.66
.69
.71
.75
.79
.82
.84
.86
.88
.07 1.05
.18 1.16
.24 1.23
.29 1.27
.32 1.30
.39 1.37
.45 1.43
.49 1.47
.53 1.50
.58 1.56
.62 1.60
.65 1.63
.68 1.65
.71 1.68
.75 1.72
.78 1.75
.80 1.77
.82 1.79
.84 1.81
.04 1.02 1.01
.15 1.13 1.12
.21 1.19 1.18
.25 1.23 1.22
.29 1.26 1.25
.35 1.33 1.31
.41 1.38 1.37
.45 1.42 1.40
.48 1.45 1.43
.54 1.51 1.48
.57 1.54 1.52
.60 1.57 1.55
.63 1.59 1.57
.66 1.62 1.60
.69 1.66 1.63
.72 1.68 1.66
.74 1.71 1.68
.76 1.72 1.70
.78 1.74 1.71
.00 1.00 (
.11 1.10
.16 1.16
.21 1.20
.24 1.23
.30 1.29
.35 1.34
.39 1.38
.42 1.41
.47 1.46
.50 1.49
.53 1.52
.55 1.54
.58 1.56
.61 1.60
.64 1.62
.66 1.64
.68 1.66
.69 1.68
).99 0.98 (
.09 1.08
.15 1.14
.19 1.18
.22 1.21
.28 1.27
.33 1.32
.37 1.35
.40 1.38
.45 1.43
.48 1.46
.51 1.49
.53 1.51
.55 1.53
.59 1.56
.61 1.59
.63 1.61
.65 1.63
.66 1.64
).97
.07
.13
.17
.20
.26
.31
.34
.37
.42
.45
.47
.49
.52
.55
.57
.59
.61
.63
                                                    D-113
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
     Table 19-9. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 3 (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.80 1.80 1.48
3.41 2.12 1.72
3.78 2.30 1.85
4.04 2.44 1.95
4.24 2.54 2.03
4.67 "'2.76' 2.19
5.04 2.94 2.32 ;
5.29 3.07 2.42 ;
5.48 3.17 2.49 ;
5.83 3.35 2.62 ;
6.07 3.47 2,72, '*
6.25 3.57 2.79 \
6.40 3.65 2:84 '<
6.58 3.74 2.9^ ;
6.81 3.86 3.00 ;
6.98 3.95 3.07 ;
7.12 4.02 3.12 ".
7.24 4.08 3.17 ,
7.34 4.13 3.21
.33 1.23 1.12
.52 1.41 1.28
.64 1.51 1.36
.72 1.58 1.43
.78 1.64 1.47
.91 1.75 1.57
>.03 1.85 1.66
Ml 1.92 1.72
>.17 1.98 1.76
>.28 2.08 1.85
>.36 2.14 1.90
>.41 2.20 1.95
>.46 2.24 1.99
1.52 2.29 2.03
>.59 2.36 2.09
2.65 2.41 2.13
?,7O 2.45 2.16 ;
?:74 2.48 2.19 ;
2.77 2.51 2.22 ;
.06
.20
.28
.34
.38
.47
.55
.60
.65
.72
.77
.81
.85
.89
.94
.98
>.01
>.03
>.06
.02 0.99 (
.15 1.11
.22 1.18
.27 1.23
.31 1.27
.40 1.35
.47 1.42
.52 1.46
.56 1.50
.62 1.56
.67 1.61
.71 1.64
.74 1.67
.78 1.71
.82 1.75
.86 1.78
.89 1.81
.91 1.83
.93 1.85
).96 (
.09
.16
.20
.24
.32
.38
.42
.46
.52
.56
.60
.62
.66
.70
.73
.76
.78
.80
).95 0.94 (
.07 1.05
.13 1.12
.18 1.16
.22 1.20
.29 1.27
.35 1.33
.40 1.37
.43 1.41
.49 1.46
.53 1.50
.56 1.54
.59 1.56
.62 1.59
.66 1.63
.69 1.66
.72 1.69
.74 1.71
.75 1.72
).93 0.91 0.90 (
.04 1.02 1.01
.11 1.09 1.07
.15 1.13 1.12
.19 1.17 1.15
.26 1.23 1.22
.32 1.29 1.27
.36 1.33 1.31
.39 1.36 1.34
.45 1.42 1.40
.48 1.46 1.43
.52 1.48 1.46
.54 1.51 1.49
.57 1.54 1.51
.61 1.57 1.55
.64 1.60 1.58
.66 1.62 1.60
.68 1.64 1.62
.70 1.66 1.63
).89 0.89 (
.00 1.00 (
.07 1.06
.11 1.10
.14 1.13
.21 1.20
.26 1.25
.30 1.29
.33 1.32
.38 1.37
.42 1.41
.45 1.43
.47 1.46
.50 1.48
.53 1.52
.56 1.54
.58 1.57
.60 1.58
.61 1.60
).88 0.88 (
).99 0.98 (
.05 1.04
.09 1.08
.12 1.11
.19 1.18
.24 1.23
.28 1.27
.31 1.29
.36 1.34
.40 1.38
.42 1.41
.45 1.43
.47 1.45
.51 1.49
.53 1.51
.55 1.53
.57 1.55
.59 1.57
).87
).97
.03
.07
.10
.17
.22
.26
.28
.33
.37
.39
.42
.44
.47
.50
.52
.54
.55
   Table 19-9. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 3 (20 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
341 2.12 1.72 .52 1.41 1.28
4.04 2.44 1.95 .72 1.58 1.43
441 2.62 2.09 .83 1.68 1.51
4.67 2.76 2.19 .91 1.75 1.57
487 2.86 2.26 .98 1.81 1.62
5.29 3.07 2.42 2.11 1.92 1.72
5.64 3.25 2.55 2.22 2.02 1.80
5.89 3.38 2.64 2.30 2.09 1.86
6.07 3.47 2.72 2.36 2.14 1.90
6.40 3.65 2.84 2.46 2.24 1.99
6.63 3.77 2.93 2.54 2.30 2.04
6.81 3.86 3..QO 2.59 2.36 2.09
6.95 3.93 3.06 2.64 2.40 2.12
7.12 4.02 3.12 2.70 2.45 2.16 ;
7.34 4.13 3.21 2.77 2.51 2.22 ;
7.50 4.22 3.27 2.82 2.56 2.26 ;
7.64 4.29 3.33 2.87 2.60 2.29 ;
7.75 4.35 3.37 2.90 2.63 2.32 ;
7.84 4.40 3.41 2.94 2.66 2.35 ;
.20
.34
.42
.47
.52
.60
.68
.73
.77
.85
.90
.94
.97
>.01
>.06
>.09
>.13 ;
>.15 ;
>.i7 ;
.15 1.11
.27 1.23
.35 1.30
.40 1.35
.44 1.39
.52 1.46
.59 1.53
.64 1.57
.67 1.61
.74 1.67
.79 1.72
.82 1.75
.85 1.78
.89 1.81
.93 1.85
.97 1.89
>.00 1.91
1.02 1.93
>.04 1.95
.09
.20
.27
.32
.35
.42
.49
.53
.56
.62
.67
.70
.72
.76
.80
.83
.85
.87
.89
.07 1.05
.18 1.16
.25 1.23
.29 1.27
.32 1.30
.40 1.37
.46 1.43
.50 1.47
.53 1.50
.59 1.56
.63 1.60
.66 1.63
.69 1.66
.72 1.69
.75 1.72
.79 1.75
.81 1.78
.83 1.80
.85 1.81
.04 1.02 1.01
.15 1.13 1.12
.21 1.19 1.18
.26 1.23 1.22
.29 1.27 1.25
.36 1.33 1.31
.41 1.39 1.37
.45 1.43 1.41
.48 1.46 1.43
.54 1.51 1.49
.58 1.55 1.52
.61 1.57 1.55
.63 1.60 1.57
.66 1.62 1.60
.70 1.66 1.63
.73 1.69 1.66
.75 1.71 1.68
.77 1.73 1.70
.79 1.75 1.72
.00 1.00 (
.11 1.10
.17 1.16
.21 1.20
.24 1.23
.30 1.29
.35 1.34
.39 1.38
.42 1.41
.47 1.46
.50 1.49
.53 1.52
.55 1.54
.58 1.57
.61 1.60
.64 1.62
.66 1.65
.68 1.66
.69 1.68
).99 0.98 (
.09 1.08
.15 1.14
.19 1.18
.22 1.21
.28 1.27
.33 1.32
.37 1.35
.40 1.38
.45 1.43
.48 1.46
.51 1.49
.53 1.51
.55 1.53
.59 1.57
.61 1.59
.63 1.61
.65 1.63
.67 1.64
).97
.07
.13
.17
.20
.26
.31
.34
.37
.42
.45
.47
.49
.52
.55
.58
.60
.61
.63
                                                    D-114
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                      Unified Guidance
    Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (20 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4.04 2.44 1.95 1.72 1.58 1.43
4.67 2.76 2.19 1.91 1.75 1.57
5.04 2.94 2.32 2.03 1.85 1.66
529 3.07 2.42 2.11 1.92 1.72
5.48 3.17 2.49 2.17 1.98 1.76
5.89 3.38 2.64 2.30 2.09 1.86
6.22 3.55 2.77 2.40 2.19 1.94
6.46 3.67 2.86 2.48 2.25 2.00
6.63 3.77 2.93 2.54 2.30 2.04
6.95 3.93 3.O6 2.64 2.40 2.12
7.17 4.05 3.14 2.71 2.46 2.18 ;
7.34 4.13 3,21 2.77 2.51 2.22 ;
7.47 4.20 3.26 2.81 2.55 2.25 ;
7.64 4.29 3.33 2.87 2.60 2.29 ;
7.84 4.40 3.41 2.94 2.66 2.35 ;
8.00 4.48 3.47 2.99 2.71 2.39 ;
8.13 4.55 3.52 3.03 2.75 2.42 ;
8.23 4.61 3.56 3.O7 2.78 2.45 ;
8.33 4.65 3.60 3.1O 2.81 2.47 ;
.34
.47
.55
.60
.65
.73
.81
.86
.90
.97
1.02
>.06
>.09
>.13 ;
>.17 ;
>.2i ;
>.24 ;
2.27 ;
>..29 ;
.27 1.23
.40 1.35
.47 1.42
.52 1.46
.56 1.50
.64 1.57
.70 1.64
.75 1.68
.79 1.72
.85 1.78
.90 1.82
.93 1.85
.96 1.88
>.00 1.91
>.04 1.95
>.07 1.99
>.10 2.01
>.13 2.03
>.15 2.05
.20
.32
.38
.42
.46
.53
.59
.63
.67
.72
.76
.80
.82
.85
.89
.92
.95
.97
.99
.18 1.16
.29 1.27
.35 1.33
.40 1.37
.43 1.41
.50 1.47
.56 1.53
.60 1.57
.63 1.60
.69 1.66
.72 1.69
.75 1.72
.78 1.75
.81 1.78
.85 1.81
.88 1.84
.90 1.87
.92 1.88
.94 1.90
.15 1.13 1.12
.26 1.23 1.22
.32 1.29 1.27
.36 1.33 1.31
.39 1.36 1.34
.45 1.43 1.41
.51 1.48 1.46
.55 1.52 1.49
.58 1.55 1.52
.63 1.60 1.57
.67 1.63 1.61
.70 1.66 1.63
.72 1.68 1.66
.75 1.71 1.68
.79 1.75 1.72
.81 1.77 1.74
.84 1.79 1.76
.86 1.81 1.78
.87 1.83 1.80
.11 1.10
.21 1.20
.26 1.25
.30 1.29
.33 1.32
.39 1.38
.44 1.43
.48 1.46
.50 1.49
.55 1.54
.59 1.57
.61 1.60
.64 1.62
.66 1.65
.69 1.68
.72 1.70
.74 1.72
.76 1.74
.77 1.76
.09 1.08
.19 1.18
.24 1.23
.28 1.27
.31 1.29
.37 1.35
.42 1.40
.45 1.44
.48 1.46
.53 1.51
.56 1.54
.59 1.57
.61 1.59
.63 1.61
.67 1.64
.69 1.67
.71 1.68
.73 1.70
.74 1.72
.07
.17
.22
.26
.28
.34
.39
.42
.45
.49
.53
.55
.57
.60
.63
.65
.67
.68
.70
      Table 19-9. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 3 (40 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.60 2.16 1.74 1.53 1.42 1.28
4.35 2.51 1.98 1.74 1.59 1.43
4.81 2.72 2.13 1.86 1.70 1.52
5.14 2.87 2.24 1.94 1.77 1.58
5.40 2.99 2.32 2.01 1.83 1.63
5.94 3.24 2.49 2.15 1.95 1.73
6.39 3.45 2.64 2.27 2.05 1.82
6.71 3.60 2,74 2.35 2.13 1.88
6.96 3.71 2,82 2.42 2.18 1.92
7.39 3.91 2.97 2.53 2.29 2.01
7.70 4.06 3.07 2.62 2.36 2.07
7.93 4.16 3.15 2,68 2.41 2.12
8.11 4.25 3.21 2,73 2.46 2.15
8.34 4.36 3.29 2,79 2.51 2.20 ;
8.62 4.49 3.38 2.87 2.58 2.26 ;
8.84 4.60 3.46 2,93 2.63 2.30 ;
9.01 4.68 3.52 2.98 2,68 2.34 ;
9.16 4.75 3.57 3.03 2.71 2.37 ;
9.29 4.81 3.61 3.06 2,75 2.39 ;
.21
.34
.42
.48
.52
.61
.69
.74
.78
.86
.91
.95
.99
>.03
>.08
1.12
1.15 ;
>.18 ;
1.20 ;
.15 1.11
.28 1.23
.35 1.30
.40 1.35
.44 1.39
.52 1.47
.59 1.53
.64 1.58
.68 1.61
.75 1.68
.80 1.72
.83 1.76
.86 1.78
.90 1.82
.95 1.86
.98 1.90
>.01 1.92
>.04 1.95
>.06 1.97
.09
.20
.27
.32
.35
.43
.49
.53
.57
.63
.67
.70
.73
.76
.80
.83
.86
.88
.90
.07 1.05
.18 1.17
.25 1.23
.29 1.27
.33 1.31
.40 1.38
.46 1.43
.50 1.48
.53 1.51
.59 1.56
.63 1.60
.66 1.63
.69 1.66
.72 1.69
.76 1.73
.79 1.76
.82 1.78
.84 1.80
.86 1.82
.04 1.03 1.01
.15 1.13 1.12
.21 1.19 1.18
.26 1.23 1.22
.29 1.27 1.25
.36 1.33 1.31
.42 1.39 1.37
.46 1.43 1.41
.49 1.46 1.43
.54 1.51 1.49
.58 1.55 1.52
.61 1.58 1.55
.63 1.60 1.57
.66 1.63 1.60
.70 1.66 1.64
.73 1.69 1.66
.75 1.71 1.68
.77 1.73 1.70
.79 1.75 1.72
.00 1.00 (
.11 1.10
.17 1.16
.21 1.20
.24 1.23
.30 1.29
.35 1.34
.39 1.38
.42 1.41
.47 1.46
.51 1.49
.53 1.52
.55 1.54
.58 1.57
.62 1.60
.64 1.63
.66 1.65
.68 1.66
.70 1.68
).99 0.98 (
.09 1.08
.15 1.14
.19 1.18
.22 1.21
.28 1.27
.33 1.32
.37 1.35
.40 1.38
.45 1.43
.48 1.46
.51 1.49
.53 1.51
.55 1.53
.59 1.57
.61 1.59
.63 1.61
.65 1.63
.67 1.64
).98
.07
.13
.17
.20
.26
.31
.34
.37
.42
.45
.47
.50
.52
.55
.58
.60
.61
.62
                                                    D-115
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Interwell K-Tables for Means
                                                                     Unified Guidance
   Table 19-9. K-Multipliers for 1-of-2 Interwell Prediction Limits on Means of Order 3 (40 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4.35 2.51 1.98 1.74 1.59 1.43
5.14 2.87 2.24 1.94 1.77 1.58
5.61 3.09 2.39 2.06 1.88 1.67
5.94 3.24 2.49 2.15 1.95 1.73
6.19 3.35 2.57 2.21 2.01 1.78
6.71 3.60 2.74 2.35 2.13 1.88
7.15 3.80 2.89 2.47 2.23 1.96
7.46 3.95 2.99 2.55 2.30 2.02
7.70 4.06 3.07 2.62 2.36 2.07
8.11 4.25 3.21 2.73 2.46 2.15
8.40 4.39 3.31 2.81 2.53 2.21 ;
8.62 4.49 3.38 '"2.87" 2.58 2.26 ;
8.80 4.58 3.44 2,92 2.62 2.29 ;
9.01 4.68 3.52 2,98 2.68 2.34 ;
9.29 4.81 3.61 . 3.OB ' 2.75 2.39 ;
9.50 4.91 3.68 3.12 2.80 2.44 ;
9.67 4.99 3.74 3.17 2.84 2.47 ;
9.81 5.06 3.79 3.21 2,87 2.50 ;
9.93 5.12 3.83 3.24 2.90 2.53 ;
.34
.48
.56
.61
.65
.74
.82
.87
.91
.99
2.04
2.08
Ml
2.15 ;
1.20 ;
2.24 ;
2.27 ;
2.30 ;
>..32 ;
.28 1.23
.40 1.35
.47 1.42
.52 1.47
.56 1.50
.64 1.58
.71 1.64
.76 1.69
.80 1.72
.86 1.78
.91 1.83
.95 1.86
.98 1.89
2.01 1.92
2.06 1.97
2.09 2.00
1.12 2.03
2.15 2.05
2.17 2.07 ;
.20
.32
.38
.43
.46
.53
.59
.64
.67
.73
.77
.80
.83
.86
.90
.93
.96
.98
2.00
.18 1.17
.29 1.27
.35 1.33
.40 1.38
.43 1.41
.50 1.48
.56 1.53
.60 1.57
.63 1.60
.69 1.66
.73 1.70
.76 1.73
.78 1.75
.82 1.78
.86 1.82
.89 1.85
.91 1.87
.93 1.89
.95 1.91
.15 1.13 1.12
.26 1.23 1.22
.32 1.29 1.27
.36 1.33 1.31
.39 1.36 1.34
.46 1.43 1.41
.51 1.48 1.46
.55 1.52 1.49
.58 1.55 1.52
.63 1.60 1.57
.67 1.63 1.61
.70 1.66 1.64
.72 1.69 1.66
.75 1.71 1.68
.79 1.75 1.72
.82 1.77 1.74
.84 1.80 1.77
.86 1.82 1.78
.88 1.83 1.80
.11 1.10
.21 1.20
.26 1.25
.30 1.29
.33 1.32
.39 1.38
.44 1.43
.48 1.46
.51 1.49
.55 1.54
.59 1.57
.62 1.60
.64 1.62
.66 1.65
.70 1.68
.72 1.70
.74 1.72
.76 1.74
.77 1.76
.09 1.08
.19 1.18
.24 1.23
.28 1.27
.31 1.29
.37 1.35
.42 1.40
.45 1.44
.48 1.46
.53 1.51
.56 1.54
.59 1.57
.61 1.59
.63 1.61
.67 1.64
.69 1.67
.71 1.69
.73 1.70
.74 1.72
.07
.17
.22
.26
.28
.34
.39
.42
.45
.50
.53
.55
.57
.60
.62
.65
.67
.69
.70
    Table 19-9. K-Multipliers  for 1-of-2 Interwell Prediction Limits on Means of Order 3 (40 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
514 2.87 2.24 1.94 1.77 1.58
5.94 3.24 2.49 2.15 1.95 1.73
6.39 3.45 2.64 2.27 2.05 1.82
6.71 3.60 2.74 2.35 2.13 1.88
6.96 3.71 2.82 2.42 2.18 1.92
7.46 3.95 2.99 2.55 2.30 2.02
7.88 4.14 3.13 2.67 2.40 2.11
8.18 4.28 3.23 2.75 2.47 2.17 ;
8.40 4.39 3.31 2.81 2.53 2.21 ;
8.80 4.58 3.44 2.92 2.62 2.29 ;
9.08 4.71 3.54 3.00 2.69 2.35 ;
9.29 4.81 3.61 3.Q6 2.75 2.39 ;
9.46 4.89 3.67 3,71 2.79 2.43 ;
9.67 4.99 3.74 3,\7' 2.84 2.47 ;
9.93 5.12 3.83 3.24 2.90 2.53 ;
10.12 5.21 3.90 3.30 2.95 2.57 ;
10.29 5.29 3.96 3.35 3.00 2.61 ;
10.42 5.36 4.00 3.39 3.03 2.63 ;
10.54 5.41 4.04 3.42 3.O6 2.66 ;
.48 1.40 1.35
.61 1.52 1.47
.69 1.59 1.53
.74 1.64 1.58
.78 1.68 1.61
.87 1.76 1.69
.95 1.83 1.75
2.00 1.87 1.79
2.04 1.91 1.83
Ml 1.98 1.89
M6 2.02 1.93
1.20 2.06 1.97
1.23 2.09 1.99
1.27 2.12 2.03
2.32 2.17 2.07 ;
2.36 2.20 2.10 ;
2.39 2.23 2.13 ;
2.42 2.25 2.15 ;
2.44 2.27 2.17 ;
.32 ]
.43 ]
.49 ]
.53 ]
.57 ]
.64 ]
.70
.74
.77
.83
.87
.90
.93
.96
2.00
2.03
2.05 ;
2.07 ;
2.09 :
L.29 1.27
L.40 1.38
L.46 1.43
L.50 1.48
L.53 1.51
L.60 1.57
.66 1.63
.70 1.67
.73 1.70
.78 1.75
.82 1.79
.86 1.82
.88 1.84
.91 1.87
.95 1.91
.98 1.94
2.00 1.96
2.02 1.98
2.04 2.00
.26 1.23
.36 1.33
.42 1.39
.46 1.43
.49 1.46
.55 1.52
.60 1.57
.64 1.61
.67 1.63
.72 1.69
.76 1.72
.79 1.75
.81 1.77
.84 1.80
.88 1.83
.90 1.86
.93 1.88
.95 1.90
.96 1.91
.22 1.21 1.20
.31 1.30 1.29
.37 1.35 1.34
.41 1.39 1.38
.43 1.42 1.41
.49 1.48 1.46
.55 1.53 1.51
.58 1.56 1.55
.61 1.59 1.57
.66 1.64 1.62
.69 1.67 1.65
.72 1.70 1.68
.74 1.72 1.70
.77 1.74 1.72
.80 1.77 1.76
.82 1.80 1.78
.85 1.82 1.80
.86 1.84 1.82
.88 1.85 1.83
.19 1.18
.28 1.27
.33 1.32
.37 1.35
.40 1.38
.45 1.44
.50 1.48
.54 1.52
.56 1.54
.61 1.59
.64 1.62
.67 1.64
.69 1.66
.71 1.69
.74 1.72
.77 1.74
.79 1.76
.80 1.77
.82 1.79
.17
.26
.31
.34
.37
.42
.47
.50
.53
.57
.60
.62
.65
.67
.70
.72
.74
.76
.77
                                                    D-116
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations	Unified Guidance
                                        D  STATISTICAL TABLES

D.3TABLES  FROM CHAPTER  19: INTRAWELL  PREDICTION LIMITS FOR  FUTURE OBSERVATIONS

    TABLE 19-10 K-Multipliers for 1 -of-2 Intrawell Prediction Limits on Observations	D-118
    TABLE 19-11 /c-Multipliers for 1 -of-3 Intrawell Prediction Limits on Observations	D-127
    TABLE 19-12 /c-Multipliers for 1 -of-4 Intrawell Prediction Limits on Observations	D-136
    TABLE19-13 /c-Multipliers for Mod. Cal. Intrawell Prediction Limits on Observations	D-145
                                                      D-117                                             March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
Table 19-10. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Observations (1 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.78 0.67 0.61 (
1.27 .05 0.97 (
1.59 .28 1.17
1.82 .45 1.31
2.02 .58 1.41
2.47 .86 1.65
2.90 2.11 1.85
3.24 2.30 1.99
3.52 2.45 2.11
4.09 2.73 2.32 ;
4.54 2.94 2,47 *
4.91 3.11 2.59 ;
5.24 3.26 2.70 t
5.67 3.44 2.82
6.26 3.68 2.99 ;
6.76 3.88 3.12 ;
7.20 4.05 3.23 ;
7.59 4.19 3.33 ;
7.95 4.32 3.41 :
).58 (
).92 (
.10
.23
.33
.54
.72
.84
.94
2.12 ;
2.25 ;
2.35 ;
2,43 ;
?:s4 :
1.61 \
1.18:
2.81
2.94
3.01
).57 0.54 (
).89 0.85 (
.06 1.02 (
.19 1.13
.28 1.22
.47 1.40
.64 1.55
.75 1.65
.84 1.73
2.01 1.88
1.12 1.98
1.21 2.06
1.29 2.12 ;
2.38 2.20 ;
2,49" 2.30 ;
2,59 2.37 ;
2.66 2,44 ;
2.12 2,49 :
2.18 2,54 '„
).53 (
).83 (
).99 (
.10
.19
.36
.50
.60
.67
.80
.90
.97
2.03
>.io ;
2.19 ;
1.26 :
1.32 ;
2.37 ;
?".'4'l" '*
).52 0.51 0.51 (
).81 0.80 0.79 (
).97 0.96 0.95 (
.08 1.06 1.05
.16 1.14 1.13
.32 1.30 1.29
.46 1.43 1.42
.55 1.53 1.51
.62 1.59 1.57
.75 1.72 1.69
.84 1.80 1.78
.91 1.87 1.84
.96 1.92 1.89
2.03 1.98 1.95
2.11 2.07 2.03 ;
2.18 2.13 2.09 ;
1.23 2.18 2.14 ;
2.28 2.22 2.18 ;
1.32 2.26 2.22 ;
).50 0.50 (
).79 0.78 (
).94 0.94 (
.04 1.04
.12 1.11
.28 1.27
.40 1.39
.49 1.48
.56 1.55
.68 1.66
.76 1.74
.82 1.80
.87 1.85
.93 1.91
2.01 1.99
1.07 2.05 ;
Ml 2.09 ;
M5 2.13 ;
M9 2.16 ;
).50 0.50 0.49 (
).78 0.78 0.77 (
).93 0.93 0.92 (
.03 1.03 1.02
.11 1.10 1.09
.26 1.25 1.24
.39 1.38 1.37
.47 1.46 1.45
.54 1.52 1.51
.65 1.64 1.63
.73 1.71 1.70
.79 1.77 1.76
.84 1.82 1.81
.90 1.88 1.86
.97 1.95 1.93
2.03 2.01 1.99
1.07 2.05 2.03 ;
Ml 2.09 2.07 ;
M5 2.12 2.10 ;
).49 0.49 (
).77 0.77 (
).92 0.91 (
.02 1.01
.09 1.09
.24 1.24
.36 1.36
.44 1.44
.51 1.50
.62 1.61
.69 1.69
.75 1.74
.80 1.79
.85 1.84
.92 1.91
.98 1.97
1.02 2.01 ;
2.06 2.05 ;
2.09 2.08 ;
).49 0.49 (
).77 0.76 (
).91 0.91 (
.01 1.01
.08 1.08
.23 1.23
.35 1.35
.43 1.43
.50 1.49
.61 1.60
.68 1.67
.74 1.73
.78 1.77
.84 1.83
.91 1.89
.96 1.95
2.00 1.99
2.04 2.02 ;
2.07 2.05 ;
).49
).76
).91
.00
.08
.22
.34
.42
.48
.59
.66
.72
.76
.82
.89
.94
.98
2.01
2.04
     Table 19-10.  K-Multipliers for 1-of-2 Intrawell Prediction Limits on Observations (1 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4 6 8 10 12 16
1.21 .03 0.95 0.90 0.88 0.84 (
1.76 .42 1.29 .22 .18 1.13
2.11 .66 1.49 .40 .35 1.28
2.39 .83 1.63 .53 .47 1.39
2.62 .97 1.74 .63 .56 1.48
3.15 2.27 1.98 .83 .75 1.65
3.67 2.55 2.19 2.01 .91 1.79
4.08 2.75 2.34 2.14 2.03 1.90
4.42 2.91 2.46 2.24 2.12 1.98
5.11 3.23 2.68 2.43 2.28 2.12 ;
5.65 3.46 2.85 2.56 2.40 2.22 ;
6.11 3.65 2.98 2.67 2.49 2.30 ;
6.50 3.81 3.08 2.75 2.56 2.36 ;
7.02 4.02 3.22 '"2.86" 2.66 2.44 ;
7.75 4.29 3.40 3.00 . 2,78 2.54 ;
8.36 4.51 3.54 3.11 2,87 2.62 ;
8.91 4.70 3.66 3.20 2,95 2.68 ;
9.38 4.86 3.76 3.28 3.02 2.73 ;
9.81 5.01 3.85 3.35 3.07 2, 78 '*

20
).82 (
.10
.25
.35
.43
.59
.73
.82
.90
2.03
M2 ;
M9 ;
1.25 :
1.32 :
2.4i ;
2.48 ;
2.54 ;
2.59 ;
2.63 ;

25 30 35
).81 0.80 0.79 (
.08 1.06 1.05
.22 1.20 1.19
.32 1.30 1.29
.40 1.37 1.36
.55 1.52 1.50
.68 1.65 1.63
.77 1.74 1.71
.84 1.80 1.78
.96 1.92 1.89
2.05 2.00 1.97
Ml 2.06 2.03 ;
M7 2.12 2.08 ;
1.23 2.18 2.14 ;
1.32 2.26 2.22 ;
2.38 2.32 2.27 ;
2.43 2.37 2.32 ;
2.48 2.41 2.36 ;
2.51 2.44 2.40 ;

40 45
).79 0.78 (
.04 1.04
.18 1.17
.27 1.27
.35 1.34
.49 1.48
.61 1.60
.69 1.68
.76 1.74
.87 1.85
.95 1.93
2.01 1.99
2.05 2.03 ;
2.11 2.09 ;
2.19 2.16 ;
2.24 2.22 ;
2.29 2.26 ;
2.33 2.30 ;
2.36 2.33 ;

50 60 70
).78 0.77 0.77 (
.03 1.02 1.02
.17 1.16 1.15
.26 1.25 1.24
.33 1.32 1.31
.47 1.46 1.45
.59 1.57 1.56
.67 1.65 1.64
.73 1.71 1.70
.84 1.82 1.81
.91 1.89 1.88
.97 1.95 1.93
2.02 2.00 1.98
2.07 2.05 2.03 ;
2.15 2.12 2.10 ;
2.20 2.17 2.15 ;
2.24 2.22 2.20 ;
2.28 2.25 2.23 ;
2.31 2.28 2.26 ;

80 90
).77 0.77 (
.02 1.01
.15 1.14
.24 1.23
.31 1.30
.44 1.44
.56 1.55
.63 1.63
.69 1.69
.80 1.79
.87 1.86
.92 1.91
.97 1.96
2.02 2.01 ;
2.09 2.08 ;
2.14 2.13 ;
2.18 2.17 ;
2.22 2.20 ;
2.24 2.23 ;

100 125
).76 0.76 (
.01 1.01
.14 1.14
.23 1.23
.30 1.29
.43 1.43
.55 1.54
.62 1.61
.68 1.67
.78 1.77
.85 1.84
.91 1.89
.95 1.94
2.00 1.99
2.07 2.05 ;
2.12 2.10 ;
2.16 2.14 ;
2.19 2.18 ;
2.22 2.21 ;

150
).76
.00
.13
.22
.29
.42
.53
.61
.66
.76
.83
.89
.93
.98
2.04
2.09
2.13
2.17
2.19

                                                    D-118
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
      Table 19-10. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Observations (1 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
1.65 1.37 1.26 1.20 1.16 1.12
2.27 1.78 1.61 1.51 1.45 1.39
2.68 2.04 1.81 1.69 1.62 1.54
3.00 2.22 1.96 1.82 1.74 1.64
327 2.37 2.07 1.92 1.83 1.72
390 2.70 2.32 2.13 2.02 1.89
4.52 3.00 2.54 2.31 2.18 2.03
5.01 3.23 2.70 2.45 2.30 2.14 ;
5.42 3.41 2.83 2.55 2.39 2.21 ;
6.24 3.76 3.06 2.74 2.56 2.36 ;
6.89 4.02 3.24 2.88 2.68 2.46 ;
7.45 4.23 3.38 2.99 2.77 2.53 ;
7.92 4.41 3.49 3.08 2.85 2.60 ;
8.55 4.64 3.64 3.19 2.94 2.68 ;
9.43 4.95 3.83 3.34 3.07 2.78 ;
10.17 5.20 3.98 3.46 3,17 2.86 ;
10.82 5.41 4.11 3.55 3,g5 '• 2.92 ;
11.40 5.59 4.22 3.64 3,32 2.98 ;
11.92 5.76 4.32 3.71 3.37 3.02 ;
20
.09
.35
.49
.59
.66
.82
.95
2.05
2.12 ;
1.25 :
1.34 ;
2.4i ;
2.47 ;
2.54 ;
2.63 ;
2.70 ;
2.75 ;
2.80 ;
2.84 ;
25
.07
.32
.45
.55
.62
.77
.89
.98
2.05 ;
2.17 ;
>..25 ;
2.32 ;
2.37 ;
2.43 ;
2.5i ;
2.58 ;
2.63 ;
2.67 ;
2.7i ;
30
.06
.30
.43
.52
.59
.73
.85
.94
2.00
2.11 ;
2.19 ;
2.26 ;
2.3i ;
2.37 ;
2.44 ;
2.50 ;
2.55 ;
2.59 ;
2.63 ;
35
.05
.28
.41
.50
.57
.71
.83
.91
.97
2.08 ;
2.16 ;
2.22 ;
2.26 ;
2.32 ;
2.40 ;
2.45 ;
2.50 ;
2.54 ;
2.57 ;
40
.04
.27
.40
.49
.56
.69
.81
.89
.95
2.05 ;
2.13 ;
2.19 ;
2.23 ;
2.29 ;
2.36 ;
2.42 ;
2.46 ;
2.50 ;
2.53 ;
45
.03
.26
.39
.48
.54
.68
.79
.87
.93
2.03 ;
2.11 ;
2.16 ;
2.21 ;
2.26 ;
2.33 ;
2.39 ;
2.43 ;
2.47 ;
2.50 ;
50 60 70 80 90 100
.03 1.02 1.02 1.01 1.01 1.01
.26 1.25 1.24 1.24 1.23 1.23
.38 1.37 1.37 1.36 1.36 1.35
.47 1.46 1.45 1.44 1.44 1.43
.54 1.52 1.51 1.51 1.50 1.50
.67 1.65 1.64 1.63 1.63 1.62
.78 1.76 1.75 1.74 1.73 1.73
.86 1.84 1.82 1.81 1.80 1.80
.91 1.89 1.88 1.87 1.86 1.85
2.02 2.00 1.98 1.97 1.96 1.95
2.09 2.07 2.05 2.04 2.03 2.02 ;
2.15 2.12 2.10 2.09 2.08 2.07 ;
2.19 2.16 2.14 2.13 2.12 2.11 ;
2.24 2.22 2.20 2.18 2.17 2.16 ;
2.31 2.28 2.26 2.24 2.23 2.22 ;
2.37 2.33 2.31 2.29 2.28 2.27 ;
2.41 2.37 2.35 2.33 2.32 2.31 ;
2.44 2.41 2.39 2.37 2.35 2.34 ;
2.48 2.44 2.42 2.40 2.38 2.37 ;
125
.00
.23
.34
.43
.49
.61
.72
.79
.84
.94
2.00
2.05 ;
2.09 ;
2.14 ;
2.21 ;
2.25 ;
2.29 ;
2.32 ;
2.35 ;
150
.00
.22
.34
.42
.48
.61
.71
.78
.83
.93
.99
2.04
2.08
2.13
2.19
2.24
2.28
2.31
2.34
       Table 19-10. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Observations (2 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
1.27 1.05 0.97 0.92 0.89 0.85 (
1.82 1.45 1.31 1.23 1.19 1.13
2.19 1.68 1.50 1.41 1.35 1.29
2.47 1.86 1.65 1.54 1.47 1.40
2.70 1.99 1.76 1.64 1.56 1.48
3.24 2.30 1.99 1.84 1.75 1.65
3.77 2.57 2.20 2.02 1.92 1.80
4.19 2.78 2.35 2.15 2.03 1.90
4.54 2.94 2,47 2.25 2.12 1.98
5.24 3.26 2.70 2,43 2.29 2.12 ;
5.80 3.49 2.86 2,57 2.40 2.22 ;
6.26 3.68 2.99 2.67 '2.49' 2.30 ;
6.67 3.84 3.10 2.76 2,57 2.36 ;
7.20 4.05 3.23 2.87 2.66 2,44 '*
7.95 4.32 3.41 3.01 2.78 2,54 .
8.57 4.55 3.55 3.12 2.88 2,62 ,
9.12 4.74 3.67 3.21 2.95 2.68
9.61 4.90 3.77 3.29 3.02 2.73
10.06 5.04 3.86 3.35 3.08 2.78 ,
20
).83 (
.10
.25
.36
.43
.60
.73
.83
.90
2.03
2.12 ;
2.19 ;
2.25 ;
2.32 ;
2,47 ;
2.48 :
2,54 .
2.59 <
2.63 ..
25
).81 (
.08
.22
.32
.40
.55
.68
.77
.84
.96
2.05 ;
2.11 ;
2.17 ;
2.23 ;
2.32 ;
2.38 ;
2,43 :
2.48 .
2,57
30
).80 (
.06
.20
.30
.38
.53
.65
.74
.80
.92
2.00
2.07 ;
2.12 ;
2.18 ;
2.26 ;
2.32 ;
2.37 ;
2,47 :
2,44 ',
35
).79 (
.05
.19
.29
.36
.51
.63
.71
.78
.89
.97
2.03 ;
2.08 ;
2.14 ;
2.22 ;
2.27 ;
2.32 ;
2.36 ;
2,40 ;
40
).79 (
.04
.18
.28
.35
.49
.61
.69
.76
.87
.95
2.01
2.05 ;
2.11 ;
2.19 ;
2.24 ;
2.29 ;
2.33 ;
2.36 ;
45
).78 (
.04
.17
.27
.34
.48
.60
.68
.74
.85
.93
.99
2.03 ;
2.09 ;
2.16 ;
2.22 ;
2.26 ;
2.30 ;
2.33 ;
50 60 70 80 90 100
).78 0.78 0.77 0.77 0.77 0.77 (
.03 1.03 1.02 1.02 1.01 1.01
.17 1.16 1.15 1.15 1.15 1.14
.26 1.25 1.24 1.24 1.24 1.23
.33 1.32 1.31 1.31 1.30 1.30
.47 1.46 1.45 1.44 1.44 1.43
.59 1.57 1.56 1.56 1.55 1.55
.67 1.65 1.64 1.63 1.63 1.62
.73 1.71 1.70 1.69 1.69 1.68
.84 1.82 1.81 1.80 1.79 1.78
.91 1.89 1.88 1.87 1.86 1.85
.97 1.95 1.93 1.92 1.91 1.91
2.02 2.00 1.98 1.97 1.96 1.95
2.08 2.05 2.03 2.02 2.01 2.00
2.15 2.12 2.10 2.09 2.08 2.07 ;
2.20 2.17 2.15 2.14 2.13 2.12 ;
2.24 2.22 2.20 2.18 2.17 2.16 ;
2.28 2.25 2.23 2.22 2.20 2.19 ;
2.31 2.28 2.26 2.24 2.23 2.22 ;
125
).76 (
.01
.14
.23
.29
.43
.54
.61
.67
.77
.84
.89
.93
.99
2.05 ;
2.10 ;
2.14 ;
2.18 ;
2.21 ;
150
).76
.00
.13
.22
.29
.42
.53
.61
.66
.76
.83
.89
.92
.98
2.04
2.09
2.13
2.17
2.19
                                                   D-119
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
     Table 19-10. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Observations (2 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
1.76 1.42 1.29 1.22 1.18 1.13
2.39 1.83 1.63 1.53 1.47 1.39
2.82 2.08 1.83 1.71 1.63 1.54
315 2.27 1.98 1.83 1.75 1.65
3.43 2.42 2.09 1.93 1.84 1.73
4.08 2.75 2.34 2.14 2.03 1.90
4.72 3.05 2.56 2.33 2.19 2.04
5.22 3.28 2.72 2.46 2.31 2.14 ;
5.65 3.46 2,85 2.56 2.40 2.22 ;
6.50 3.81 3.08 2.75 2.57 2.36 ;
7.18 4.08 3.26 2,89 2.68 2.46 ;
7.75 4.29 3.40 3.00 '2,78" 2.54 ;
8.25 4.47 3.51 3.09 2,86 2.60 ;
8.91 4.70 3.66 3.20 2,95 2.68 ;
9.81 5.01 3.85 3.35 3.07 ,2,78 :
10.59 5.26 4.00 3.47 3.17 2,86 \
11.25 5.47 4.13 3.56 3.25 2.92 '<
11.86 5.66 4.24 3.65 3.32 2,98 <
12.40 5.82 4.34 3.72 3.38 3.03
20
.10
.35
.49
.59
.67
.82
.96
2.05
2.12 ;
1.25 :
1.34 ;
2.4i ;
2.47 ;
2.54 ;
2.63 ;
2.70 ;
2.75 ;
Z8O :
2.84 ;
25
.08
.32
.46
.55
.62
.77
.89
.98
2.05 ;
2.17 ;
>..25 ;
2.32 ;
2.37 ;
2.43 ;
2.5i ;
2.58 ;
2.63 ;
2.67 ;
2.7i ;
30
.06
.30
.43
.52
.59
.74
.86
.94
2.00
2.12 ;
2.20 ;
2.26 ;
2.31 ;
2.37 ;
2.44 ;
2.50 ;
2.55 ;
2.59 ;
2.63 ;
35
.05
.29
.42
.50
.57
.71
.83
.91
.97
2.08 ;
2.16 ;
2.22 ;
2.26 ;
2.32 ;
2.40 ;
2.45 ;
2.50 ;
2.54 ;
2.57 ;
40
.04
.28
.40
.49
.56
.69
.81
.89
.95
2.05 ;
2.13 ;
2.19 ;
2.23 ;
2.29 ;
2.36 ;
2.42 ;
2.46 ;
2.50 ;
2.53 ;
45
.04
.27
.39
.48
.55
.68
.79
.87
.93
2.03 ;
2.11 ;
2.16 ;
2.21 ;
2.26 ;
2.33 ;
2.39 ;
2.43 ;
2.47 ;
2.50 ;
50 60 70 80 90 100
.03 1.02 1.02 1.02 1.01 1.01
.26 1.25 1.24 1.24 1.23 1.23
.39 1.37 1.37 1.36 1.36 1.35
.47 1.46 1.45 1.44 1.44 1.43
.54 1.52 1.51 1.51 1.50 1.50
.67 1.65 1.64 1.63 1.63 1.62
.78 1.76 1.75 1.74 1.73 1.73
.86 1.84 1.82 1.81 1.81 1.80
.91 1.89 1.88 1.87 1.86 1.85
2.02 2.00 1.98 1.97 1.96 1.95
2.09 2.07 2.05 2.04 2.03 2.02 ;
2.15 2.12 2.10 2.09 2.08 2.07 ;
2.19 2.16 2.14 2.13 2.12 2.11 ;
2.24 2.22 2.20 2.18 2.17 2.16 ;
2.31 2.28 2.26 2.24 2.23 2.22 ;
2.37 2.33 2.31 2.29 2.28 2.27 ;
2.41 2.38 2.35 2.33 2.32 2.31 ;
2.45 2.41 2.39 2.37 2.35 2.34 ;
2.48 2.44 2.41 2.40 2.38 2.37 ;
125
.01
.23
.35
.43
.49
.61
.72
.79
.84
.93
2.00
2.05 ;
2.09 ;
2.14 ;
2.21 ;
2.25 ;
2.29 ;
2.32 ;
2.35 ;
150
.00
.22
.34
.42
.48
.61
.71
.78
.83
.92
.99
2.04
2.08
2.13
2.19
2.24
2.28
2.31
2.34
      Table 19-10. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Observations (2 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.27
3.00
3.51
3.90
4.23
5.01
5.78
6.38
6.89
7.92
8.74
9.43
10.03
10.82
11.92
12.85
13.67
14.39
15.04
6 8 10 12 16 20 25 30 35
1.78 1.61 1.51 1.45 1.39 .35 .32 .30 .28
2.22 1.96 1.82 1.74 1.64 .59 .55 .52 .50
2.50 2.17 2.00 1.90 1.79 .73 .68 .65 .63
2.70 2.32 2.13 2.02 1.89 .82 .77 .73 .71
2.87 2.44 2.23 2.11 1.97 .89 .84 .80 .77
3.23 2.70 2.45 2.30 2.14 2.05 .98 .94 .91
357 2.93 2.64 2.47 2.28 2.18 2.10 2.05 2.02
382 3.10 2.77 2.59 2.38 2.27 2.19 2.13 2.10 ;
4.02 3.24 2.88 2.68 2.46 2.34 2.25 2.19 2.16 ;
441 349 3.08 2.85 2.60 2.47 2.37 2.31 2.26 ;
4.71 3.68 '"3. '23" 2.97 2.70 2.56 2.45 2.38 2.34 ;
4.95 3.83 3.34 3.07 2.78 2.63 2.51 2.44 2.40 ;
5.15 3.95 3.44 3.75 2.84 2.68 2.57 2.49 2.44 ;
5.41 4.11 3.55 3.25 2.92 2.75 2.63 2.55 2.50 ;
5.76 4.32 3.71 3.38 3.02 2.84 2.71 2.63 2.57 ;
6.04 4.48 3.83 3.48 3.10 2.91 2.77 2.69 2.63 ;
6.28 4.62 3.93 3.56 3.-17 2.97 2.82 2.73 2.67 ;
6.49 4.74 4.02 3.63 3.23. 3.02 2.87 2.77 2.71 ;
6.68 4.85 4.10 3.69 . 3,27 3.06 2.91 2.81 2.74 ;
40
.27
.49
.61
.69
.76
.89
.99
2.07 ;
2.13 ;
2.23 ;
2.30 ;
2.36 ;
2.4i ;
2.46 ;
2.53 ;
2.58 ;
2.63 ;
2.67 ;
2.70 ;
45
.26
.48
.60
.68
.74
.87
.98
2.05 ;
2.11 ;
2.21 ;
2.28 ;
2.33 ;
2.38 ;
2.43 ;
2.50 ;
2.55 ;
2.60 ;
2.63 ;
2.66 ;
50
.26
.47
.59
.67
.73
.86
.96
2.03
2.09
2.19
2.26
2.31
2.36
2.41
2.48
2.53
2.57
2.60
2.63
60
1.25
1.46
1.57
1.65
1.71
1.84
1.94
2.01
2.07
2.16
2.23
2.28
2.32
2.37
2.44
2.49
2.53
2.56
2.59
70
1.24
1.45
1.56
1.64
1.70
1.82
1.92
1.99
2.05
2.14
2.21
2.26
2.30
2.35
2.41
2.46
2.50
2.54
2.56
80
1.24
1.44
1.56
1.63
1.69
1.81
1.91
1.98
2.04
2.13
2.19
2.24
2.29
2.33
2.40
2.44
2.48
2.51
2.54
90
1.23
1.44
1.55
1.63
1.69
1.80
1.90
1.97
2.03
2.12
2.18
2.23
2.27
2.32
2.38
2.43
2.47
2.50
2.53
100
1.23
1.43
1.55
1.62
1.68
1.80
1.90
1.97
2.02 ;
2.11 ;
2.17 ;
2.22 ;
2.26 ;
2.3i ;
2.37 ;
2.42 ;
2.45 ;
2.49 ;
2.5i ;
125 150
.23 .22
.43 .42
.54 .53
.61 .61
.67 .66
.79 .78
.89 .88
.95 .94
2.00 .99
2.09 2.08
2.16 2.15
2.21 2.19
2.24 2.23
2.29 2.28
2.35 2.34
2.40 2.38
2.43 2.42
2.46 2.45
2.49 2.48
                                                   D-120
                                                                                                  March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
       Table 19-10. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Observations (5 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
2.02 1.58 1.41 1.33 1.28 1.22 .19
2.70 1.99 1.76 1.64 1.56 1.48 .43
316 2.25 1.96 1.81 1.73 1.63 .57
3.52 2.45 2.11 1.94 1.84 1.73 .67
3.83 2.60 2.22 2.04 1.93 1.81 .74
4.54 2.94 2,47 2.25 2.12 1.98 .90
5.24 3.26 2.69 2,43 2.29 2.12 2.03
5.80 3.49 2.86 2,57 2.40 2.22 2.12 ;
6.26 3.68 2.99 2.67 2,49 2.30 2.19 ;
7.20 4.05 3.23 2.87 2.66 2.44 . 2.32 ;
7.95 4.32 3.41 3.01 2.78 2,54 2,41 <
8.57 4.55 3.55 3.12 2.87 2,62 .2,48 \
9.12 4.74 3.67 3.21 2.95 2.68 .2,5.4. ,
9.84 4.97 3.82 3.32 3.05 2.76 2,6.1 .
10.84 5.30 4.02 3.47 3.17 2.86 2.70
11.67 5.57 4.17 3.59 3.27 2.94 2.77
12.45 5.79 4.30 3.69 3.36 3.00 2.83
13.09 5.98 4.42 3.77 3.42 3.06 2.87
13.67 6.15 4.52 3.85 3.49 3.11 2.91
25
.16
.40
.53
.62
.69
.84
.96
2.05 ;
>.n ;
1.23 :
1.32 ;
2.38 ;
2.43 :
2,50 ,
2,58 . ,
2.64 .
2.69 ,
2.73
2.11
30
.14
.38
.50
.59
.66
.80
.92
2.00
2.07 ;
2.18 ;
>..26 ;
2.32 ;
2.37 ;
2,43 ;
2, SO ,
2,56 .
2.61
2,65 ..
2.69
35
.13
.36
.49
.57
.64
.78
.89
.97
2.03 ;
2.14 ;
2.22 ;
2.28 ;
2.32 ;
2.38 ;
2.4S .
2,51
2.56 .
2,59 .
2,63 .
40
.12
.35
.47
.56
.62
.76
.87
.95
2.01
2.11 ;
2.19 ;
2.24 ;
2.29 ;
2.35 ;
2.42: .
2,47 .
2,51 .
2,55 .
2,58 . ,
45
.11
.34
.46
.55
.61
.74
.85
.93
.99
2.09 ;
2.16 ;
2.22 ;
2.26 ;
2.32 ;
2,39 ;
2.44 ',
2:48 .
2,52 .
2,55 ' .
50 60 70 80 90 100
.11 1.10 1.09 1.09 1.09 1.08
.33 1.32 1.31 1.31 1.30 1.30
.45 1.44 1.43 1.43 1.42 1.42
.54 1.52 1.51 1.51 1.50 1.50
.60 1.59 1.58 1.57 1.56 1.56
.73 1.71 1.70 1.69 1.69 1.68
.84 1.82 1.81 1.80 1.79 1.78
.91 1.89 1.88 1.87 1.86 1.85
.97 1.95 1.93 1.92 1.91 1.91
2.08 2.05 2.03 2.02 2.01 2.00
2.15 2.12 2.10 2.09 2.08 2.07 ;
2.20 2.17 2.15 2.14 2.13 2.12 ;
2.24 2.22 2.20 2.18 2.17 2.16 ;
2.30 2.27 2.25 2.23 2.22 2.21 ;
2.37 2.33 2.31 2.29 2.28 2.27 ;
2.42 2,38 2.36 2.34 2.33 2.32 ;
2.46 2,43 2.4O 2,38 2.37 2.36 ;
2.50- 2.46. 2,44 2,42 2.40 . 2.39. ;
2,53 2,49 .2146 2,44 2,43 .2,42 t
125
.08
.29
.41
.49
.55
.67
.77
.84
.89
.99
2.05 ;
2.10 ;
2.14 ;
2.19 ;
2.25 ;
2.30 ;
2.34 ;
2.37 ;
2,40 .
150
.08
.29
.40
.48
.54
.66
.76
.83
.89
.98
2.04
2.09
2.13
2.18
2.24
2.29
2.33
2.36
2,38
     Table 19-10. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Observations (5 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.62
3.43
3.98
4.42
4.79
5.65
6.50
7.18
7.75
8.90
9.81
10.58
11.25
12.13
13.38
14.40
15.33
16.11
16.89
6
1.97
2.42
2.70
2.91
3.08
3.46
3.81
4.08
4.29
4.70
5.01
5.26
5.47
5.74
6.12
6.41
6.67
6.88
7.08
8
1.74
2.09
2.31
2.46
2.58
2.85
3.08
3.26
3.40
3.66
3.85
4.00
4.13
4.29
4.50
4.68
4.82
4.94
5.05
10
1.63
1.93
2.11
2.24
2.35
2.56
2.75
.2.89
3.00
3.20
3.35
3.47
3.56
3.68
3.84
3.97
4.06
4.16
4.24
12
1.56
1.84
2.00
2.12
2.21
2.40
2.57
2.68
2,78
2,95
3.07
3.17
3.25
3.35
3.48
3.58
3.67
3.74
3.80
16 20 25 30 35 40 45 50 60
1.48 .43 .40 .37 .36 .35 .34 .33 1.32
1.73 .67 .62 .59 .57 .56 .55 .54 1.52
1.87 .80 .75 .72 .69 .68 .66 .65 1.64
1.98 .90 .84 .80 .78 .76 .74 .73 1.71
2.05 .97 .91 .87 .84 .82 .80 .79 1.77
2.22 2.12 2.05 2.00 .97 .95 .93 .91 1.89
2.36 2.25 2.17 2.12 2.08 2.05 2.03 2.02 2.00
2.46 2.34 2.25 2.20 2.16 2.13 2.11 2.09 2.07
2.54 2.41 2.32 2.26 2.22 2.19 2.16 2.15 2.12
2.68 2.54 2.43 2.37 2.32 2.29 2.26 2.24 2.22
'2.78' ' 2.63 2.51 2.44 2.40 2.36 2.33 2.31 2.28
2.86 2.70 2.58 2.50 2.45 2.42 2.39 2.37 2.33
2:'92' 2.75 2.63 2.55 2.50 2.46 2.43 2.41 2.38
3,00 2.82 2.69 2.61 2.56 2.51 2.48 2.46 2.43
3.11 2-91 2,77" 2.69 2.63 2.58 2.55 2.53 2.49
3.19 2,98 2,84 2.75 2.68 2.64 2.60 2.58 2.54
3.25 . 3.O4 2,89 2,79 2.73 2.68 2.65 2.62 2.58
3.31 3.09 2,93 2,83 2,76 2.72 2.68 2.66 2.61
3.36 3.13 2,97 2,87 2.8O '•'2.75" 2.71 2.69 2.64
70
1.31
1.51
1.63
1.70
1.76
1.88
1.98
2.05
2.10
2.20
2.26
2.31
2.35
2.40
2.46
2.51
2.55
2.58
2.61
80
1.31
1.51
1.62
1.69
1.75
1.87
1.97
2.04
2.09
2.18
2.25
2.29
2.33
2.38
2.44
2.49
2.53
2.56
2.59
90
1.30
1.50
1.61
1.69
1.74
1.86
1.96
2.03
2.08
2.17
2.23
2.28
2.32
2.37
2.43
2.47
2.51
2.55
2.56
100
1.30
1.50
1.61
1.68
1.74
1.85
1.95
2.02 ;
2.07 ;
2.16 ;
2.22 ;
2.27 ;
2.3i ;
2.36 ;
2.42 ;
2.46 ;
2.50 ;
2.53 ;
2.54 ;
125 150
.29 .29
.49 .48
.60 .59
.67 .66
.73 .72
.84 .83
.93 .93
2.00 .99
2.05 2.04
2.14 2.13
2.21 2.19
2.25 2.24
2.29 2.28
2.34 2.33
2.40 2.38
2.44 2.43
2.48 2.46
2.50 2.48
2.50 2.48
                                                   D-121
                                                                                                  March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
Table 19-10. K-Multipliers for 1-of-2 ntrawell Prediction Limits on Observations (5 COC,
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6
3.27 2.37
4.23 2.87
4.89 3.18
5.42 3.41
5.86 3.60
6.89 4.02
7.93 4.41
8.74 4.71
9.43 4.95
10.82 5.41
11.93 5.76
12.85 6.04
13.67 6.28
14.75 6.59
16.21 6.99
17.48 7.34
18.55 7.62
19.53 7.86
20.51 8.11
Table 19-1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6
2.70 1.99
3.52 2.45
4.09 2.73
4.54 2.94
4.91 3.11
5.79 3.49
6.67 3.84
7.36 4.11
7.95 4.32
9.13 4.73
10.05 5.05
10.84 5.30
11.54 5.51
12.42 5.79
13.71 6.15
14.77 6.45
15.70 6.71
16.52 6.91
17.34 7.12
8 10
2.07 1.92
2.44 2.23
2.66 2.42
2.83 2.55
2.96 2.66
3.24 2.88
3.49 3.08
3.68 3,23
3.83 3.34
4.11 3.55
4.32 3.71
4.48 3.83
4.62 3.93
4.80 4.06
5.03 4.22
5.21 4.36
5.37 4.47
5.51 4.57
5.62 4.65
12 16
1.83 1.72
2.11 1.97
2.27 2.11
2.39 2.21
2.48 2.29
2.68 2.46
2.85 2.60
2.97 2.70
3.07 2.78
3,25 2.92
3.38 3.02
3.48 3.10
3.56 "3.'i7"
3.66 3:25
3.80 3,36
3.91 3.44
3.99 3.50
4.06 3.56
4.14 3.61
0. K-Multipliers for 1
8 10
1.76 1.64
2.11 1.94
2.32 2.12
2,47 2.25
2.59 2.35
2.86 2,57
3.10 2.76
3.27 2.90
3.41 3.01
3.67 3.21
3.86 3.35
4.01 3.47
4.15 3.57
4.31 3.68
4.51 3.85
4.69 3.97
4.83 4.07
4.95 4.16
5.07 4.25
12 16
1.56 1.48
1.84 1.73
2.01 1.88
2.12 1.98
2.21 2.06
2.40 2.22
2,57, 2.36
2.69 2.46
2.78 2,54
2.95 2.68
3.08 2.78
3.18 2.86
3.26 2.93
3.35 3.00
3.49 3.11
3.59 3.19
3.67 3.25
3.74 3.31
3.81 3.35
20 25
1.66 1.62
1.89 1.84
2.03 1.96
2.12 2.05
2.19 2.11
2.34 2.25
2.47 2.37
2.56 2.45
2.63 2.51
2.75 2.63
2.84 2.71
2.91 2.77
2.97 2.82
3.04 2.89
3.13 2.97
3.20 3.03
3,26 3.08
3.31 3,12
3,35 3,18
30 35
1.59 1.57
1.80 1.77
1.92 1.89
2.00 1.97
2.06 2.03
2.19 2.16
2.31 2.26
2.38 2.34
2.44 2.40
2.55 2.50
2.63 2.57
2.69 2.63
2.73 2.67
2.79 2.73
2.87 2.80
2.92 2.85
2.97 2.90
3.02 2.94
3.05 2.97
40 45
1.56 1.54
1.76 1.74
1.87 1.85
1.95 1.93
2.01 1.99
2.13 2.11
2.23 2.21
2.30 2.28
2.36 2.33
2.46 2.43
2.53 2.50
2.58 2.55
2.63 2.59
2.68 2.65
2.75 2.71
2.80 2.76
2.84 2.81
2.88 2.84
2.92 2.87
50 60
1.54 1.52
1.73 1.71
1.84 1.82
1.91 1.89
1.97 1.95
2.09 2.07
2.19 2.16
2.26 2.23
2.31 2.28
2.41 2.38
2.48 2.44
2.53 2.49
2.57 2.53
2.62 2.58
2.68 2.64
2.73 2.69
2.77 2.73
2.81 2.76
2.84 2.79
70
1.51
1.70
1.81
1.88
1.93
2.05
2.14
2.21
2.26
2.35
2.41
2.46
2.50
2.55
2.61
2.65
2.69
2.73
2.76
80 90
1.51 1.50
1.69 1.69
1.80 1.79
1.87 1.86
1.92 1.91
2.04 2.03
2.13 2.12
2.19 2.18
2.24 2.23
2.33 2.32
2.40 2.38
2.44 2.43
2.48 2.47
2.53 2.51
2.59 2.57
2.62 2.62
2.67 2.66
2.70 2.69
2.73 2.71
Quarterly)
100 125
1.50 .49
1.68 .67
1.78 .77
1.85 .84
1.91 .89
2.02 2.00
2.11 2.09
2.17 2.16
2.22 2.21
2.31 2.29
2.37 2.35
2.42 2.40
2.46 2.43
2.50 2.48
2.55 2.51
2.60 2.58
2.64 2.62
2.67 2.64
2.69 2.67
150
1.48
1.66
1.76
1.83
1.89
1.99
2.08
2.15
2.19
2.28
2.34
2.38
2.42
2.46
2.48
2.56
2.60
2.63
2.65
-of-2 ntrawell Prediction Limits on Observations (10 COC, Annual)
20 25
.43 .40
.67 .62
.80 .75
.90 .84
.97 .91
2.12 2.05
2.25 2.17
2.34 2.25
2,41. 2.32
2,54 2.43
2,63 2.51
2.70 2,58
2.75 2,63
2.82 2.69
2.92 2.77
2.98 2.83
3.04 2.89
3.09 2.93
3.13 2.97
30 35
.38 .36
.59 .57
.72 .69
.80 .78
.87 .84
2.00 .97
2.12 2.08
2.20 2.16
2.26 2.22
2.37 2.32
2,44 2,40
2,50 2.45
2.55. 2. SO
2,61 2,56
2.69 2.63
2.75 2.68
2.79 2.73
2.83 2.77
2.87 2.80
40 45
.35 .34
.56 .55
.68 .66
.76 .74
.82 .80
.95 .93
2.05 2.03
2.13 2.11
2.19 2.16
2.29 2.26
2.36 2.33
'2,42 ' ' ' "2.39 '
2,46 2.43
2,52 2,48
' 2,59 ; 2.55-
,2.64 2,60
2.68 2.64
2.72 2.68
2.75 2.71
50 60
.33 1.32
.54 1.52
.65 1.64
.73 1.71
.79 1.77
.92 1.89
2.02 2.00
2.09 2.07
2.15 2.12
2.24 2.22
2.31 2.28
2.37 2.33
2.41 2.37
2,46 2,42
2,53, 2.49
2,58 ,2.54
2.62 2.53.
2.65 2.61
2.68 2.64
70
1.31
1.51
1.63
1.70
1.76
1.88
1.98
2.05
2.10
2.20
2.26
2.31
2.35
2,40
2.46
2,51
2,55
2,59
2,61
80 90
1.31 1.30
1.51 1.50
1.62 1.61
1.69 1.69
1.75 1.74
1.87 1.86
1.97 1.96
2.04 2.03
2.09 2.08
2.18 2.17
2.24 2.23
2.29 2.28
2.33 2.32
2,38. 2.37
2,44 "2.43
,2,49 2,48
2,53, 2,51
2,56 2,54
2.59 2,57
100 125
1.30 .29
1.50 .49
1.61 .60
1.68 .67
1.74 .73
1.85 .84
1.95 .93
2.02 2.00
2.07 2.05
2.16 2.14
2.22 2.21
2.27 2.25
2.31 2.29
2.36 2.34
" '2.4'2" " '2.40"
2,46 2,44
2.50 2.48
2,52 2,51
2,54 2.53
150
.29
.48
.59
.66
.72
.83
.92
.99
2.04
2.13
2.19
2.24
2.28
2.33
'"2.38"
2,43
2,46
2.49
2.52

                                                     D-122
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
Table 19-10. K- Multipliers for 1-of-2 Intrawell Prediction Limits on Observations (10 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.43
4.42
5.11
5.65
6.11
7.18
8.25
9.10
9.81
11.25
12.39
13.36
14.24
15.35
16.88
18.16
19.34
20.39
21.33
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.23
5.42
6.24
6.89
7.45
8.74
10.03
11.06
11.92
13.67
15.06
16.23
17.23
18.57
20.51
22.03
23.44
24.61
25.78
6
2.42
2.91
3.23
3.46
3.65
4.08
4.47
4.77
5.01
5.47
5.82
6.11
6.36
6.67
7.08
7.41
7.71
7.97
8.20
9-10
6
2.87
3.41
3.76
4.02
4.23
4.71
5.15
5.49
5.76
6.28
6.68
7.00
7.27
7.62
8.09
8.47
8.79
9.08
9.32
8
2.09
2.46
2.68
2.85
2.98
3.26
3.51
3.70
3.85
4.13
4.34
4.50
4.64
4.82
5.05
5.24
5.39
5.54
5.65
10
1.93
2.24
2.43
2.56
2.67
"'2.89'
3.09
3.24
3.35
3.56
3.72
3.84
3.94
4.06
4.23
4.37
4.48
4.57
4.66
12
1.84
2.12
2.28
2.40
2.49
2.68
2,86
2.98
3.07
3.25
3.38
3.48
3.56
3.67
3.80
3.91
4.00
4.07
4.13
K-Multipliers
8
2.44
2.83
3.06
3.24
3.38
3.68
3.95
4.16
4.32
4.62
4.85
5.03
5.18
5.37
5.62
5.83
5.99
6.15
6.27
10
2.23
2.55
2.74
2.88
2.99
3,23
3.44
3.59
3.71
3.93
4.10
4.23
4.34
4.47
4.64
4.79
4.91
5.01
5.10
12
2.11
2.39
2.56
2.68
2.77
2.97
3, IS
3.28
3.37
3.56
3.69
3.80
3.89
3.99
4.13
4.25
4.34
4.41
4.48
16
1.73
1.98
2.12
2.22
2.30
2.46
2.60
2.70
2,78
2,92
3.03
3.11
3.17
3.25
3.36
3.44
3.51
3.57
3.62
for 1
16
1.97
2.21
2.36
2.46
2.53
2.70
2.84
2.95
3.02
3,77
3.27
3:35
3.42
3.50
3.61
3.70
3.76
3.82
3.88
20
1.67
1.90
2.03
2.12
2.19
2.34
2.47
2.56
2.63
2, 75
2,84
2,91
2.97
3.O4
3.13
3.20
3.26
3.31
3.35
-of -2
20
1.89
2.12
2.25
2.34
2.41
2.56
2.68
2.77
2.84
2.97
3.06
3.13
3.19
3.26
3,35
3,42
3.48
3.53
3.57
25
1.62
1.84
1.96
2.05
2.11
2.25
2.37
2.45
2.51
2.63
2.71
2.77
2,82
2,89
„ 2,97
3.O3
3.08
3.13
3.16
30
1.59
1.80
1.92
2.00
2.06
2.20
2.31
2.38
2.44
2.55
2.63
2.69
2.74
2,79
2,87
2.93-
2.97
. 3.O2
3.O5
35 40 45 50 60 70 80 90 100 125
1.57 1.56 .55 .54 1.52 1.51 1.51 1.50 1.50 .49
1.78 1.76 .74 .73 1.71 1.70 1.69 1.69 1.68 .67
1.89 1.87 .85 .84 1.82 1.81 1.80 1.79 1.78 .77
1.97 1.95 .93 .91 1.89 1.88 1.87 1.86 1.85 .84
2.03 2.01 .99 .97 1.95 1.93 1.92 1.91 1.91 .89
2.16 2.13 2.11 2.09 2.07 2.05 2.04 2.03 2.02 2.00
2.26 2.23 2.21 2.19 2.16 2.14 2.13 2.12 2.11 2.09
2.34 2.31 2.28 2.26 2.23 2.21 2.19 2.18 2.17 2.16
2.40 2.36 2.33 2.31 2.28 2.26 2.24 2.23 2.22 2.21
2.50 2.46 2.43 2.41 2.37 2.35 2.33 2.32 2.31 2.29
2.57 2.53 2.50 2.48 2.44 2.42 2.40 2.38 2.37 2.35
2.63 2.59 2.55 2.53 2.49 2.46 2.44 2.43 2.42 2.40
2.67 2.63 2.59 2.57 2.53 2.50 2.48 2.47 2.46 2.43
2.73 2.68 2.65 2.62 2.58 2.55 2.53 2.51 2.50 2.48
2.80 2.75 2.71 2.68 2.64 2.61 2.59 2.56 2.54 2.50
'2,85 2,80 2.70" 2.73 2.68 2.65 2.62 2.60 2.60 2.58
2.90 2,85 2.81 2,78 2.72 2.69 2.67 2.66 2.64 2.61
2.94 2.89 2,84 2,81 2,76 2.72 2.70 2.68 2.67 2.64
2.9.7 2:92 2,87 2,84 2,79 2,75 2.73 2.71 2.70 2.67
150
.48
.66
.76
.83
.89
.99
2.08
2.15
2.19
2.28
2.34
2.38
2.42
2.46
2.52
2.56
2.60
2.63
2.65
Intrawell Prediction Limits on Observations (10 COC, Quarterly)
25
1.84
2.05
2.17
2.25
2.32
2.45
2.57
2.65
2.71
2.82
2.90
2.97
3.02
3.08
3. 16
3,23
3.28
3,33
3,36
30
1.80
2.00
2.11
2.19
2.26
2.38
2.49
2.57
2.63
2.73
2.81
2.87
2.92
2.97
3.05
3.11
3.16.
3.19
.3.23
35 40 45 50 60 70 80 90 100 125
1.77 1.76 1.74 1.73 1.71 1.70 1.69 1.69 1.68 1.67
1.97 1.95 1.93 1.91 1.89 1.88 1.87 1.86 1.85 1.84
2.08 2.05 2.03 2.02 2.00 1.98 1.97 1.96 1.95 1.94
2.16 2.13 2.11 2.09 2.07 2.05 2.04 2.03 2.02 2.00
2.22 2.19 2.16 2.15 2.12 2.10 2.09 2.08 2.07 2.05
2.34 2.31 2.28 2.26 2.23 2.21 2.19 2.18 2.17 2.16
2.44 2.41 2.38 2.36 2.32 2.30 2.29 2.27 2.26 2.24
2.52 2.48 2.45 2.42 2.39 2.37 2.35 2.33 2.32 2.30
2.57 2.53 2.50 2.48 2.44 2.42 2.40 2.38 2.37 2.35
2.67 2.63 2.59 2.57 2.53 2.50 2.48 2.47 2.46 2.43
2.74 2.70 2.66 2.63 2.59 2.56 2.54 2.53 2.51 2.49
2.80 2.75 2.71 2.68 2.64 2.61 2.59 2.57 2.55 2.51
2.84 2.79 2.75 2.72 2.68 2.64 2.61 2.61 2.57 2.57
2.90 2.85 2.81 2.77 2.72 2.70 2.67 2.65 2.64 2.61
2.97 2.92 2.87 2.84 2.79 2.75 2.73 2.71 2.70 2.67
3.03 2.97 2.93 2.89 2.83 2.80 2.78 2.75 2.74 2.71
3.07 3.01 2.96 2.93 2.87 2.84 2.81 2.79 2.78 2.75
3.11 3.05 3.00 2.96 2.91 2.87 2.84 2.82 2.81 2.77
'3.14' 3.08 3.02 2.99 2.93 2.89 2.87 2.84 2.82 2.78
150
1.66
1.83
1.93
1.99
2.04
2.15
2.23
2.29
2.34
2.42
2.48
2.48
2.55
2.60
2.65
2.69
2.72
2.75
2.75

                                                     D-123
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
Table
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.52
4.54
5.24
5.79
6.26
7.36
8.45
9.32
10.05
11.53
12.71
13.69
14.57
15.70
17.27
18.59
19.77
20.86
21.80
19-10. K-Multipliers for 1-of-2 Intrawell Prediction
6 8 10 12 16 20 25 30 35 40
2.45 2.11 1.94 1.84 1.73 1.67 1.62 1.59 1.57 1.56
2.94 2,47 2.25 2.12 1.98 1.90 1.84 1.80 1.78 1.76
3.26 2.70 2,43 2.28 2.12 2.03 1.96 1.92 1.89 1.87
3.49 2.86 2,57 2.40 2.22 2.12 2.05 2.00 1.97 1.95
3.68 2.99 2.67 2:49". 2.30 2.19 2.11 2.07 2.03 2.01
4.11 3.27 2.90 2.69 '2.46" 2.34 2.25 2.20 2.16 2.13
4.50 3.52 3.10 2.86 .2,80 ,2,47 2.37 2.31 2.26 2.23
4.80 3.71 3.24 2.98 2.70 2,56 2,45 2.38 2.34 2.31
5.04 3.86 3.36 3.08 2.78 2.63 2.51 2.44 2.4O 2.36
5.51 4.14 3.57 3.25 2.92 2.75 2.63 ,2,55 2,,5O ,2,46
5.86 4.35 3.72 3.38 3.03 2.84 2.71 2,63 2,57 . 2,5,3
6.15 4.52 3.85 3.48 3.11 2.91 2.77 2.69 2.63 2,58
6.39 4.66 3.95 3.57 3.17 2.97 2.82 2.73 2.67 2.63
6.70 4.83 4.07 3.67 3.25 3.04 2.89 2.79 2.73 2.68
7.12 5.06 4.24 3.80 3.36 3.13 2.97 2.87 2.80 2.75
7.46 5.24 4.37 3.91 3.44 3.20 3.03 2.93 2.86 2.80
7.75 5.40 4.48 4.00 3.51 3.26 3.08 2.97 2.90 2.85
8.01 5.54 4.58 4.07 3.56 3.31 3.13 3.02 2.94 2.88
8.24 5.65 4.66 4.14 3.61 3.35 3.16 3.05 2.97 2.92
Limits on Observations (20 COC, Annual)
45
.55
.74
.85
.93
.99
2.11
2.21
2.28
2.33
2,43
2.5O
2,;55,
2,60
••2.65 •
2.71
2.76
2.81
2.84
2.87
50
.54
.73
.84
.92
.97
2.09
2.19
2.26
2.31
2,4?
2.48
2,53
2,57
2,52
.2,68
2.73
2.77
2.81
2.84
60
1.52
1.71
1.82
1.89
1.95
2.07
2.16
2.23
2.28
2.38
,2,44
2.49
•'2,53
.2,58
2,64
2.69
2.72
2.76
2.79
70
1.51
1.70
1.81
1.88
1.93
2.05
2.14
2.21
2.26
2.35
2,42
2,46
,2,50
2,55
2.6O
'2.65
2.69
2.72
2.76
Table 19-10. K- Multipliers for 1-of-2 Intrawell Prediction Limits on Observations
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.42
5.65
6.50
7.18
7.75
9.10
10.44
11.50
12.40
14.22
15.66
16.88
17.93
19.34
21.25
22.97
24.38
25.62
26.88
6 8 10 12 16 20 25 30 35 40
291 2.46 2.24 2.12 1.97 1.90 1.84 1.80 1.78 1.76
3.46 2,35 2.56 2.40 2.22 2.12 2.05 2.00 1.97 1.95
381 308 2.75 2.57 2.36 2.25 2.17 2.12 2.08 2.05
4.08 3.26 2,89 2.68 2.46 2.34 2.25 2.20 2.16 2.13
4.29 3.40 3.00 2,78 2.54 2.41 2.32 2.26 2.22 2.19
4.77 3.70 3.24 2.98 2.70 2.56 2.45 2.38 2.34 2.31
5.21 3.97 3.44 3.15 2,84 2.68 2.57 2.49 2.44 2.41
5.55 4.18 3.60 3.28 2,95 2,77 2.65 2.57 2.52 2.48
5.83 4.34 3.72 3.38 3.03 2,84 2.71 2.63 2.57 2.53
6.35 4.64 3.94 3.56 3.17 2.97, 2,82 2.73 2.67 2.63
6.75 4.87 4.11 3.70 3.28 3.06 2.9T 2,81 ". 2.74 2.70
7.08 5.05 4.23 3.80 3.36 3.13 2,97 2,87 2,80 £75
7.35 5.20 4.34 3.89 3.43 3.19 3.O2 2,92 2,84 2,79
7.71 5.39 4.48 4.00 3.51 3.26 3.08 2,97 2,90 2,85
8.18 5.64 4.65 4.14 3.61 3.35 3.16 3,05 2;97 2,92
8.55 5.85 4.79 4.25 3.70 3.42 3.23 3.11 3,03 , 2,97
8.91 6.02 4.91 4.34 3.77 3.48 3.28 3.15 3,07 3:01
9.18 6.17 5.01 4.42 3.83 3.53 3.33 3.20 3.11 3, OS
9.45 6.29 5.10 4.49 3.88 3.57 3.36 3.23 3.14 3.08
45
1.74
1.93
2.03
2.11
2.16
2.28
2.38
2.45
2.50
2.60
2.66
2.71
2,75
"2.81
2,87,
,2,92
2,97 ,
3,O? ,
3.O4
50
1.73
1.91
2.02
2.09
2.15
2.26
2.36
2.42
2.48
2.57
2.63
2.68
2.72
2.77
2,84
2,89
2,93
2,96
2,99
60
1.71
1.89
2.00
2.07
2.12
2.23
2.32
2.39
2.44
2.53
2.59
2.64
2.68
2.72
2.79
2,83
2,87
,2,91
2,93
70
1.70
1.88
1.98
2.05
2.10
2.21
2.30
2.37
2.42
2.50
2.56
2.61
2.64
2.69
'•2,' 75"
2, SO
2,84
2.87
2,89
80
1.51
1.69
1.80
1.87
1.92
2.04
2.13
2.20
2.25
2.33
2.40
2,44
2,48
2,53
2,58
2.62
2,66
2.70
2.73
(20
80
1.69
1.87
1.97
2.04
2.09
2.20
2.29
2.35
2.40
2.48
2.54
2.59
2.61
2.67
2.73
2,77
2.8f
2,34
2,86
90
1.50
1.69
1.79
1.86
1.91
2.03
2.12
2.18
2.23
2.32
2,38
2,43
2,47
2,57
2.55
2.6O
2.64
2.68
2.71
COC,
90
1.69
1.86
1.96
2.03
2.08
2.18
2.27
2.33
2.38
2.47
2.53
2.56
2.59
2.65
2.71
,2,75
2.79
••2.81
2,84
100
1.50
1.68
1.78
1.85
1.91
2.02
2.11
2.17
2.22
2.31
2.37
2,42
2.45
2,50
2,54,
2,58
2,64
2,67
2.7O .
125
.49
.67
.77
.84
.89
2.00
2.09
2.16
2.21
2.29
2.35
2.4O
2.43
2.43
2.5O
2.58
2,67
2,64
2,67
150
.48
.66
.76
.83
.89
.99
2.08
2.15
2.19
2.28
2.34
2,38
2,42
2,46,
2,48
• 2,56'
2.60 .
2.63
2,65 .•
Semi-Annual)
100
1.68
1.85
1.95
2.02
2.07
2.17
2.26
2.32
2.37
2.46
2.51
2.54
2.57
2.64
2.70
2.74
2.77
2.80
2,83
125
1.67
1.84
1.93
2.00
2.05
2.16
2.24
2.30
2.35
2.43
2.49
2.50
2.54
2.61
2.67
2.71
2.74
2,77"
2.80 .
150
1.66
1.83
1.92
1.99
2.04
2.15
2.23
2.29
2.34
2.42
2.48
2.48
2.55
2.60
2.65
2.69
2.73
2.76
'2.78"

                                                     D-124
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.42
6.89
7.92
8.74
9.43
11.06
12.68
13.96
15.06
17.25
18.98
20.47
21.76
23.44
25.78
27.81
29.53
31.09
32.50
Table
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.54
5.79
6.67
7.36
7.95
9.32
10.70
11.79
12.71
14.56
16.04
17.29
18.37
19.80
21.80
23.47
24.96
26.28
27.42
9-10. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Observations (20
6 8 10 12 16 20 25 30 35 40
341 2.83 2.55 2.39 2.21 2.12 2.05 2.00 1.97 1.95
4.02 3.24 2.88 2.68 2.46 2.34 2.25 2.19 2.16 2.13
441 349 3.08 2.85 2.60 2.47 2.37 2.31 2.26 2.23
4.71 3.68 3,23 2.97 2.70 2.56 2.45 2.38 2.34 2.30
4.95 3.83 3.34 3.07 2.78 2.63 2.51 2.44 2.40 2.36
5.49 4.16 3.59 .'3.28". 2.94 2.77 2.65 2.57 2.52 2.48
5.99 4.45 3.81 3.46 3.09 2.90 2.76 2.68 2.62 2.57
6.37 4.67 3.97 3.59 3,19 2.99 2.84 2.75 2.69 2.64
6.68 4.85 4.10 3.69 3.-27. 3.06 2.90 2.81 2.74 2.70
7.27 5.18 4.33 3.89 3.42 3,W 3.02 2.91 2.84 2.79
7.72 5.42 4.51 4.02 3.53 3,28 3.10 2.99 2.91 2.86
8.10 5.62 4.64 4.13 3.61 3.3S 3,16 3.05 2.97 2.92
8.40 5.79 4.76 4.22 3.68 3,41 3,22 3.10 3.02 2.96
8.81 6.00 4.90 4.34 3.76 3.48 3.28 3,16 3.07 3.01
9.34 6.27 5.09 4.48 3.88 3.57 . 3,36 3,23 3,14 3.08
9.77 6.48 5.24 4.60 3.96 3.65 3,42 3.29 3.2O 3.12
10.16 6.68 5.37 4.70 4.03 3.71 3,48 3,34 3.24 .3.17
10.47 6.84 5.47 4.79 4.09 3.75 3.52 3.38 3,28 3.2O
10.78 6.97 5.57 4.85 4.15 3.80 3.55 3,41 . 3.32 3,25
19-10. K-Multipliers for 1-of-2 Intrawell Prediction
6 8 10 12 16 20 25 30 35 40
2.94 2.47 2.25 2.12 1.98 1.90 1.84 1.80 1.78 1.76
3.49 2.86 2.57 2.40 2.22 2.12 2.05 2.00 1.97 1.95
3.84 3.10 2.76 2.57 2.36 2.25 2.17 2.12 2.08 2.05
4.11 3.27 2.90 2.69 2.46 2.34 2.25 2.20 2.16 2.13
4.32 3.41 3.01 2.78 2,54 .2.41- 2.32 2.26 2.22 2.19
4.80 3.71 3.24 2.98 2.70 2,56 2,45 2.38 2.34 2.31
5.25 3.99 3.45 3.16 2.85 2.68 2.57 2,49 2.44 :2,4l"
5.59 4.19 3.60 3.28 2.95 2.77 2,65 2,57 .2.52.. 2.48
5.86 4.35 3.72 3.38 3.03 2.84 2.71 2.63. 2.57 : 2.53
6.39 4.66 3.95 3.57 3.17 2.97 2.82 2.73 2.67 2,63.
6.80 4.88 4.11 3.70 3.28 3.06 2.91 2.81 2.74 2.70
7.12 5.06 4.24 3.81 3.36 3.13 2.97 2.87 2.80 2.75
7.40 5.21 4.35 3.89 3.42 3.19 3.02 2.92 2.85 2.79
7.76 5.41 4.48 4.00 3.51 3.26 3.08 2.97 2.90 2.85
8.23 5.66 4.66 4.14 3.61 3.35 3.16 3.05 2.97 2.92
8.61 5.86 4.80 4.25 3.70 3.42 3.23 3.11 3.03 2.97
8.94 6.03 4.92 4.35 3.77 3.48 3.28 3.16 3.07 3.01
9.23 6.17 5.02 4.42 3.83 3.53 3.32 3.20 3.11 3.05
9.49 6.31 5.11 4.49 3.88 3.58 3.36 3.23 3.14 3.08
45
1.93
2.11
2.21
2.28
2.33
2.45
2.54
2.61
2.66
2.75
2.82
2.87
2.92
2.96
3.02
3.08
3.12
3, IS
3.18
Limits
45
1.74
1.93
2.03
2.11
2.16
2.28
2.38
"2'.45:""
2.50
2,59
2,66
2.71
2.75
2.81
2.87
2.93
2.97
3.00
3.04
50
1.91
2.09
2.19
2.26
2.31
2.42
2.52
2.58
2.63
2.72
2.79
2.84
2.88
2.92
2.99
3.04
3.08
3.11
3.14
on
50
1.73
1.92
2.02
2.09
2.15
2.26
2.36
"2. ',42
2,48
2,57
2,63
2.68
2.72
2.77
2.84
2.89
2.93
2.97
3.00
60
1.89
2.07
2.16
2.23
2.28
2.39
2.48
2.54
2.59
2.68
2.74
2.79
2.83
2.87
2.93
2.98
3.02
3.05
3.08
70
1.88
2.05
2.14
2.21
2.26
2.37
2.45
2.52
2.56
2.64
2.71
2.76
2.79
2.84
2.90
2.94
2.97
3.01
3.04
80
1.87
2.04
2.13
2.19
2.24
2.35
2.44
2.50
2.54
2.61
2.68
2.73
2.77
2.81
2.87
2.91
2.94
2.98
3.00
COC
90
1.86
2.03
2.12
2.18
2.23
2.33
2.42
2.48
2.53
2.59
2.67
2.71
2.75
2.79
2.84
2.88
2.92
2.95
2.98
, Quarterly)
100
1.85
2.02
2.11
2.17
2.22
2.32
2.41
2.47
2.51
2.57
2.65
2.70
2.73
2.77
2.82
2.86
2.90
2.93
2.96
125
1.84
2.00
2.09
2.16
2.21
2.30
2.39
2.45
2.49
2.57
2.63
2.67
2.70
2.75
2.78
2.82
2.87
2.90
2.92
150
1.83
1.99
2.08
2.15
2.19
2.29
2.37
2.43
2.48
2.55
2.61
2.65
2.69
2.73
2.76
2.80
2.85
2.88
2.90
Observations (40 COC, Annual)
60
1.71
1.89
2.00
2.07
2.12
2.23
2.32
""'2.39'
. 2,44
2,53
2.59
2,64
', 2,68
2.73
2.79
2.84
2.88
2.92
2.95
70
1.70
1.88
1.98
2.05
2.10
2.21
2.30
2.37
2,41 .
2.5O
2.56
2.6O
2,64,
2; 69 .
2.76
2.81
2.85
2.89
2.92
80
1.69
1.87
1.97
2.04
2.09
2.20
2.29
2.35
2,40
2.48
2.54
2.58
2.61
2.66
2.73
2.77
2.80
2.84
2.86
90
1.69
1.86
1.96
2.03
2.08
2.18
2.27
2.33
2.38
.2.47
. 2,53.
2.56
2.59.
2.64
2.71
2.76
2.79
2.82
2.84
100
1.68
1.85
1.95
2.02
2.07
2.17
2.26
2.32
2.37
2.46
2,51
2,54
2:57
2.64
2:70
2.74
2.77
2.80
2.82
125
1.67
1.84
1.93
2.00
2.05
2.16
2.24
2.31
2.35
2:43
2.49
2. SO
2.54
2,61
2.67
2.71
2.75
2.77
2.79
150
1.66
1.83
1.92
1.99
2.04
2.15
2.23
2.29
2.34
2.42
2.46
2,48
2,56
2,60
2,65
2,69
2.72
2.76
2.78

                                                     D-125
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
Table 19-10. K- Multipliers for 1-of-2 Intrawell Prediction Limits on Observations (40 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.65
7.18
8.25
9.10
9.81
11.51
13.19
14.53
15.66
17.94
19.75
21.27
22.63
24.39
26.81
28.92
30.76
32.34
33.75
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
6.89
8.74
10.03
11.06
11.92
13.97
16.00
17.62
19.00
21.75
23.95
25.80
27.42
29.53
32.52
35.07
37.27
39.20
40.96
6
3.46
4.08
4.47
4.77
5.01
5.55
6.06
6.44
6.75
7.35
7.81
8.18
8.49
8.90
9.44
9.89
10.26
10.59
10.88
9-10.
6
4.02
4.71
5.15
5.49
5.76
6.37
6.94
7.37
7.72
8.40
8.92
9.34
9.70
10.15
10.77
11.27
11.69
12.08
12.39
8
2,85
3.26
3.51
3.70
3.85
4.18
4.47
4.69
4.87
5.20
5.45
5.64
5.81
6.02
6.30
6.51
6.70
6.87
7.01
10
2.56
2,89
3.09
3.24
3.35
3.60
3.82
3.98
4.10
4.34
4.52
4.65
4.77
4.91
5.10
5.25
5.37
5.48
5.58
12 16 20 25 30 35 40 45 50 60
2.40 2.22 2.12 2.05 2.00 1.97 1.95 1.93 1.91 1.89
2.68 2.46 2.34 2.25 2.20 2.16 2.13 2.11 2.09 2.07
2,86 2.60 2.47 2.37 2.31 2.26 2.23 2.21 2.19 2.16
298 2.70 2.56 2.45 2.38 2.34 2.31 2.28 2.26 2.23
3.07 2,78 2.63 2.51 2.44 2.40 2.36 2.33 2.31 2.28
3.28 2,95 '2,77.' 2.65 2.57 2.52 2.48 2.45 2.42 2.39
3.46 3.09 '.2,90 2,76 2.68 2.62 2.57 2.54 2.52 2.48
3.59 3.19 2,99 2,84. 2,75 2.69 2.64 2.61 2.58 2.54
3.70 3.28 3.06 2.91 2.81 2.74 2.70 2.66 2.63 2.59
3.89 3.42 3.19 :3.02 2.91 2.84. 2.79 2.75. 2.72 2.68
4.03 3.53 3.28 3.10 2.99 2.92. 2,86 2,82 2.79 2.74
4.14 3.61 3.35 3.16 3.O5, . 2,97 2.92 2,87 '2.84 2.79
4.23 3.68 3.41 3.22 3, 7O 3.O2 2,96 2,92 2,88 2,83
4.34 3.77 3.48 3.28 3.16 .3,07 3,07 2,97 2,93 2,87
4.49 3.88 3.57 3.36 3.23 3.14 3.O8 3,04 2.99 2,93
4.60 3.96 3.65 3.42 3.29 3.20 3.14 .3,09 : 3..O4 2.98
4.70 4.03 3.71 3.48 3.34 3.24 3.18 3.73 3, 7O 3,02
4.78 4.09 3.76 3.52 3.38 3.28 3.22 3.17 3,73, 3.O5
4.86 4.15 3.80 3.56 3.41 3.32 3.25 3.20 3.16 3,00
70
1.88
2.05
2.14
2.21
2.26
2.37
2.45
2.52
2.56
2.64
2.70
-2,' 75"
'2.79
2,84
2,89
2,94
2,97
3.O7
3.O3
K-Multipliers for 1-of-2 Intrawell Prediction Limits on Observations
8
3.24
3.68
3.95
4.16
4.32
4.67
5.00
5.23
5.42
5.79
6.05
6.27
6.45
6.67
6.98
7.22
7.43
7.60
7.76
10
2.88
3,23
3.44
3.59
3.71
3.97
4.20
4.37
4.51
4.76
4.94
5.09
5.21
5.37
5.56
5.72
5.86
5.98
6.08
12 16 20 25 30 35 40 45 50 60
2.68 2.46 2.34 2.25 2.19 2.16 2.13 2.11 2.09 2.07
2.97 2.70 2.56 2.45 2.38 2.34 2.30 2.28 2.26 2.23
3,75 2.84 2.68 2.57 2.49 2.44 2.41 2.38 2.36 2.32
328 2.94 2.77 2.65 2.57 2.52 2.48 2.45 2.42 2.39
338 3.02 2.84 2.71 2.63 2.57 2.53 2.50 2.48 2.44
3.59 3,79 2.99 2.84 2.75 2.69 2.64 2.61 2.58 2.54
3.78 3,34 3,72 2.96 2.86 2.79 2.74 2.70 2.67 2.63
3.92 3.45 3,27 3.04 2.93 2.86 2.81 2.77 2.74 2.69
4.02 3.53 3,28 3.10 2.99 2.91 2.86 2.82 2.79 2.74
4.22 3.68 3:47 3,22 3.10 3.02 2.96 2.92 2.88 2.83
4.37 3.79 3.50 3,30 3,77 3.09 3.03 2.98 2.94 2.89
4.49 3.88 3.57 3,36. 3,23 3.74 3.08 3.02 2.99 2.93
4.58 3.95 3.63 3,47 3,28 3,79 3,72 3.07 3.03 2.97
4.70 4.03 3.71 3,43 3,34 3,25 3,77 3,72 3.08 3.02
4.85 4.14 3.80 3.56 3,47 3,32 3,25 3.78 3,74 3.08
4.98 4.24 3.87 3.62 3,47 3.37 3.3O 3,23 3,79 3,73
5.08 4.31 3.93 3.67 3.52 3,47 3,34 ,3,27 3,23 3,76
5.16 4.37 3.98 3.72 3.56 ,3,45 3,38 3.32 3,26 3,20
5.24 4.42 4.03 3.76 3.59 3.48 3,47 3,35 3.37 3,22
70
2.05
2.21
2.30
2.37
2.41
2.52
2.60
2.66
2.71
2.79
2.85
2.89
2.93
2.98
3.03
3.08
3,72
3.15
3,78
80
1.87
2.04
2.13
2.20
2.25
2.35
2.44
2.50
2.54
2.61
2.68
2.73
'2.W
2,87
2,86
2,97
2,94
2,97
3.0O
(40
80
2.04
2.19
2.29
2.35
2.40
2.50
2.58
2.63
2.68
2.77
2.82
2.87
2.90
2.94
3.00
3.05
3.09
3,72
3,75 ,
90
1.86
2.03
2.12
2.18
2.23
2.33
2.42
2.48
2.53
2.59
2.67
2.71
2.75
2,79
2,84
2,88
2,92
2,95
2,98
COC,
90
2.03
2.18
2.27
2.33
2.38
2.48
2.56
2.62
2.67
2.75
2.80
2.84
2.87
2.92
2.98
3.02
3.05
3.08
3.11
100
1.85
2.02
2.11
2.17
2.22
2.32
2.41
2.47
2.51
2.60
2.65
2.70
2.73
2,77
2,82
2,86
2.90
2,93
2,96
125
1.84
2.00
2.09
2.16
2.21
2.30
2.39
2.45
2.49
2.57
2.63
2.67
2.70
2.75
2,78
2,82
2,86
2,89 ,
2,92
150
1.83
1.99
2.08
2.15
2.19
2.29
2.37
2.43
2.48
2.55
2.61
2.65
2.69
2.73
2,75
2.8O
2,83
2,87
2.90
Quarterly)
100
2.02
2.17
2.26
2.32
2.37
2.47
2.55
2.61
2.65
2.73
2.79
2.82
2.85
2.90
2.96
3.00
3.03
3.06
3.09
125
2.00
2.16
2.24
2.30
2.35
2.45
2.51
2.58
2.63
2.70
2.76
2.78
2.82
2.87
2.93
2.96
2.99
3.03
3.05
150
1.99
2.15
2.23
2.29
2.34
2.43
2.51
2.57
2.61
2.69
2.74
2.76
2.79
2.85
2.90
2.94
2.96
3.00
3.03

                                                     D-126
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
       Table 19-11. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Observations (1 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
468
0.33 0.25 0.21 (
0.71 0.57 0.50 (
0.95 0.75 0.67 (
1.13 0.88 0.78 (
1.27 0.98 0.87 (
1.60 1.20 1.05 (
1.91 1.39 1.21
2.15 1.54 1.33
2.35 1.65 1.42
2.75 1.86 1.58
3.06 2.02 1.70
3.32 2.15 1.80
3.55 2,26 1.87
3.84 2.39 1.97
4.26 2.57 2.10
4.60 2.72 2.2Q
4.90 2.84 2.28 '*
5.17 2.95 2.35 ;
5.42 3.04 2.42 ;
10 12 16
).18 0.17 0.15 (
).46 0.44 0.41 (
).62 0.59 0.55 (
).73 0.69 0.65 (
).81 0.77 0.72 (
).98 0.93 0.87 (
.12 1.06 1.00 (
.22 1.16 1.08
.30 1.23 1.15
.44 1.36 1.27
.55 1.45 1.35
.63 1.53 1.41
.69 1.59 1.47
.77 1.66 1.53
.88 1.75 1.61
.96 1.82 1.67
>.03 1.88 1.72
>.09 1.93 1.77
>.14 1.98 1.80
20
).14 (
).39 (
).53 (
).63 (
).70 (
).84 (
).96 (
.04
.10
.21
.29
.35
.40
.46
.53
.59
.64
.68
.71
25 30
).13 0.12 (
).38 0.37 (
).52 0.51 (
).61 0.60 (
).68 0.66 (
).82 0.80 (
).93 0.91 (
.01 0.99 (
.07 1.05
.17 1.15
.25 1.22
.30 1.27
.35 1.32
.40 1.37
.48 1.44
.53 1.49
.57 1.53
.61 1.57
.64 1.60
35
).12 (
).37 (
).50 (
).59 (
).65 (
).79 (
).90 (
).97 (
.03
.13
.20
.25
.30
.35
.41
.46
.50
.54
.57
40 45
).ll 0.11 (
).36 0.36 (
).49 0.49 (
).58 0.58 (
).65 0.64 (
).78 0.77 (
).89 0.88 (
).96 0.95 (
.02 1.01
.12 1.11
.19 1.17
.24 1.23
.28 1.27
.33 1.32
.39 1.38
.44 1.43
.48 1.47
.52 1.50
.54 1.53
50 60 70
).ll 0.11 0.10 (
).35 0.35 0.35 (
).48 0.48 0.47 (
).57 0.57 0.56 (
).64 0.63 0.62 (
).77 0.76 0.75 (
).87 0.86 0.86 (
).95 0.94 0.93 (
.00 0.99 0.98 (
.10 1.09 1.08
.17 1.15 1.14
.22 1.20 1.19
.26 1.24 1.23
.31 1.29 1.28
.37 1.35 1.34
.42 1.40 1.39
.45 1.44 1.42
.49 1.47 1.45
.51 1.49 1.48
80 90
).10 0.10 (
).34 0.34 (
).47 0.47 (
).56 0.56 (
).62 0.62 (
).75 0.75 (
).85 0.85 (
).92 0.92 (
).98 0.97 (
.07 1.07
.14 1.13
.19 1.18
.22 1.22
.27 1.27
.33 1.32
.38 1.37
.41 1.41
.44 1.44
.47 1.46
100 125
).10 0.10 (
).34 0.34 (
).47 0.46 (
).55 0.55 (
).62 0.61 (
).74 0.74 (
).85 0.84 (
).92 0.91 (
).97 0.96 (
.06 1.06
.13 1.12
.17 1.17
.21 1.21
.26 1.25
.32 1.31
.36 1.35
.40 1.39
.43 1.42
.46 1.44
150
).10
).34
).46
).55
).61
).74
).84
).91
).96
.05
.11
.16
.20
.25
.30
.35
.38
.41
.44
     Table 19-11.  K-Multipliers for 1-of-3 Intrawell Prediction Limits on Observations (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1 0.67 0.54 0.49 (
2 1.08 0.86 0.77 (
3 1.35 1.05 0.93 (
4 1.55 1.18 1.04 (
5 1.71 1.29 1.13
8 2.09 1.52 1.32
12 2.46 1.73 1.48
16 2.75 1.88 1.60
20 2.99 2.00 1.69
30 3.47 2.24 1.87
40 3.84 2.41 1.99
50 4.16 2,55 2.09
60 4.44 2.67 2.17
75 4.80 2.82 2.28 ;
100 5.30 3.02 2.41 ;
125 5.73 3.19 2.5?. :
150 6.10 3.32 2.61 '<
175 6.43 3.44 2.69 ;
200 6.73 3.55 2.75 ;
).45 0.43 0.41 (
).72 0.69 0.65 (
).87 0.83 0.78 (
).97 0.92 0.87 (
.05 1.00 0.94 (
.22 1.15 1.08
.36 1.29 1.20
.46 1.38 1.28
.54 1.45 1.35
.69 1.58 1.46
.79 1.68 1.55
.87 1.75 1.61
.94 1.81 1.66
>.03 1.88 1.72
>.13 1.98 1.80
2.22 2.05 1.87
2.29 2.11 1.92
2.35 2.16 1.96
>.40 2.21 2.00
).39 (
).62 (
).75 (
).84 (
).90 (
.04
.15
.23
.29
.40
.47
.53
.58
.64
.71
.77
.81
.85
.89
).38 0.37 (
).61 0.59 (
).73 0.71 (
).81 0.80 (
).88 0.86 (
.01 0.99 (
.12 1.09
.19 1.16
.25 1.22
.35 1.32
.42 1.39
.47 1.44
.52 1.48
.57 1.53
.64 1.60
.69 1.65
.74 1.69
.77 1.72
.80 1.75
).36 (
).59 (
).70 (
).79 (
).85 (
).97 (
.07
.15
.20
.30
.36
.41
.45
.50
.57
.61
.65
.69
.72
).36 0.36 (
).58 0.57 (
).70 0.69 (
).78 0.77 (
).84 0.83 (
).96 0.95 (
.06 1.05
.13 1.12
.19 1.17
.28 1.27
.34 1.33
.39 1.38
.43 1.42
.48 1.47
.54 1.53
.59 1.57
.63 1.61
.66 1.64
.69 1.67
).35 0.35 0.35 (
).57 0.56 0.56 (
).69 0.68 0.68 (
).77 0.76 0.75 (
).83 0.82 0.81 (
).95 0.94 0.93 (
.05 1.03 1.03
.11 1.10 1.09
.17 1.15 1.14
.26 1.24 1.23
.32 1.30 1.29
.37 1.35 1.34
.41 1.39 1.38
.45 1.44 1.42
.51 1.49 1.48
.56 1.54 1.52
.60 1.57 1.56
.63 1.60 1.59
.65 1.63 1.61
).34 0.34 (
).56 0.55 (
).67 0.67 (
).75 0.75 (
).81 0.80 (
).92 0.92 (
.02 1.02
.09 1.08
.14 1.13
.22 1.22
.28 1.28
.33 1.32
.37 1.36
.41 1.41
.47 1.46
.51 1.50
.55 1.54
.58 1.57
.60 1.59
).34 0.34 (
).55 0.55 (
).67 0.66 (
).74 0.74 (
).80 0.80 (
).92 0.91 (
.01 1.01
.08 1.07
.13 1.12
.21 1.21
.27 1.26
.32 1.31
.36 1.35
.40 1.39
.46 1.44
.50 1.49
.53 1.52
.56 1.55
.59 1.57
).34
).55
).66
).74
).79
).91
.00
.07
.11
.20
.26
.30
.34
.38
.44
.48
.51
.54
.56
                                                    D-127
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
      Table 19-11. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Observations (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.01 0.82 0.75 (
1.47 1.15 1.02 (
1.77 1.34 1.19
2.00 1.49 1.30
2.19 1.60 1.39
2.64 1.85 1.59
3.07 2.08 1.76
3.42 2.24 1.88
3.70 2.38 1.98
4.28 2.64 2.16
4.74 2,83. 2.29 ;
5.12 2.99 2.40 ;
5.45 3.12 2.49 ;
5.89 3.29 2.60 ;
6.50 3.51 2.74 ;
7.02 3.70 2,86 ;
7.46 3.85 2.95 ;
7.87 3.98 3.04 ;
8.23 4.10 3.11 ;
).70 0.67 0.64 (
).96 0.91 0.86 (
.10 1.05 0.99 (
.21 1.15 1.08
.29 1.22 1.14
.45 1.37 1.28
.60 1.50 1.40
.70 1.60 1.48
.78 1.67 1.54
.93 1.80 1.66
>.04 1.90 1.74
>.13 1.97 1.80
1.20 2.03 1.85
>.28 2.11 1.92
>.40 2.21 2.00
>.49 2.28 2.06
2.56 2.34 2.11
>.63 2.40 2.15 ;
>.68 2.44 2.19 ;
).62 (
).83 (
).95 (
.04
.10
.23
.34
.41
.47
.58
.65
.71
.76
.81
.89
.94
.99
>.03
>.06
).60 0.59 (
).81 0.79 (
).93 0.91 (
.00 0.98 (
.06 1.04
.19 1.16
.29 1.26
.36 1.33
.42 1.39
.52 1.48
.59 1.55
.64 1.60
.68 1.64
.74 1.69
.80 1.75
.85 1.80
.90 1.84
.93 1.87
.96 1.90
).58 (
).78 (
).89 (
).97 (
.03
.14
.24
.31
.36
.45
.52
.57
.61
.65
.72
.76
.80
.83
.86
).58 0.57 (
).78 0.77 (
).89 0.88 (
).96 0.95 (
.02 1.01
.13 1.12
.23 1.22
.29 1.28
.34 1.33
.43 1.42
.50 1.48
.54 1.53
.58 1.56
.63 1.61
.69 1.67
.73 1.71
.77 1.75
.80 1.78
.83 1.81
).57 0.56 0.56 (
).76 0.76 0.75 (
).87 0.86 0.86 (
).95 0.94 0.93 (
.00 0.99 0.98 (
.11 1.10 1.09
.21 1.19 1.18
.27 1.26 1.25
.32 1.30 1.29
.41 1.39 1.38
.47 1.45 1.44
.51 1.49 1.48
.55 1.53 1.52
.60 1.57 1.56
.65 1.63 1.61
.70 1.67 1.66
.73 1.71 1.69
.76 1.74 1.72
.79 1.76 1.74
).56 0.55 (
).75 0.74 (
).85 0.85 (
).92 0.92 (
).98 0.97 (
.09 1.08
.18 1.17
.24 1.23
.28 1.28
.37 1.36
.43 1.42
.47 1.46
.51 1.50
.55 1.54
.60 1.59
.64 1.63
.68 1.67
.71 1.69
.73 1.72
).55 0.55 (
).74 0.74 (
).85 0.84 (
).92 0.91 (
).97 0.96 (
.08 1.07
.17 1.16
.23 1.22
.27 1.26
.36 1.35
.41 1.40
.46 1.44
.49 1.48
.53 1.52
.59 1.57
.63 1.61
.66 1.64
.69 1.67
.71 1.69
).55
).73
).84
).91
).96
.07
.15
.21
.26
.34
.39
.44
.47
.51
.56
.60
.64
.66
.69
        Table 19-11. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Observations (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.71 0.57 0.50 (
1.13 0.88 0.78 (
1.39 1.07 0.94 (
1.60 1.20 1.05 (
1.76 1.31 1.14
2.15 1.54 1.33
2.52 1.74 1.49
2.81 1.90 1.61
3.06 2.02 1.70
3.55 2,26 1.87
3.93 2.43 2.00
4.26 2.57 2.10
4.54 2.69 2, 18
4.90 2.84 2,28 '*
5.42 3.04 2.42 ;
5.85 3.20 2.53
6.23 3.34 2.61
6.56 3.46 2.69
6.87 3.57 2.76
).46 0.44 0.41 (
).73 0.69 0.65 (
).87 0.83 0.78 (
).98 0.93 0.87 (
.06 1.00 0.94 (
.22 1.16 1.08
.37 1.29 1.20
.47 1.38 1.29
.55 1.45 1.35
.69 1.59 1.47
.80 1.68 1.55
.88 1.75 1.61
.95 1.81 1.66
>.03 1.88 1.72
>.14 1.98 1.80
2,22 2.05 1.87
2,29 2.11 1.92
?,35 2.17 1.96
>.41 &21" 2.00
).39 (
).63 (
).75 (
).84 (
).91 (
.04
.15
.23
.29
.40
.47
.53
.58
.64
.71
.77
.81
.85
.89
).38 0.37 (
).61 0.60 (
).73 0.72 (
).82 0.80 (
).88 0.86 (
.01 0.99 (
.12 1.09
.19 1.16
.25 1.22
.35 1.32
.42 1.39
.48 1.44
.52 1.48
.57 1.53
.64 1.60
.69 1.65
.74 1.69
.77 1.72
.80 1.75
).37 (
).59 (
).71 (
).79 (
).85 (
).97 (
.08
.15
.20
.30
.36
.41
.45
.50
.57
.61
.65
.69
.72
).36 0.36 (
).58 0.58 (
).70 0.69 (
).78 0.77 (
).84 0.83 (
).96 0.95 (
.06 1.05
.13 1.12
.19 1.17
.28 1.27
.34 1.33
.39 1.38
.43 1.42
.48 1.47
.54 1.53
.59 1.57
.63 1.61
.66 1.64
.69 1.67
).35 0.35 0.35 (
).57 0.57 0.56 (
).69 0.68 0.68 (
).77 0.76 0.75 (
).83 0.82 0.81 (
).95 0.94 0.93 (
.05 1.03 1.03
.11 1.10 1.09
.17 1.15 1.14
.26 1.24 1.23
.32 1.30 1.29
.37 1.35 1.34
.41 1.39 1.38
.45 1.44 1.42
.51 1.49 1.48
.56 1.54 1.52
.60 1.57 1.56
.63 1.60 1.59
.65 1.63 1.61
).34 0.34 (
).56 0.56 (
).67 0.67 (
).75 0.75 (
).81 0.80 (
).92 0.92 (
.02 1.02
.09 1.08
.14 1.13
.22 1.22
.28 1.28
.33 1.32
.37 1.36
.41 1.41
.47 1.46
.51 1.50
.55 1.54
.58 1.57
.60 1.59
).34 0.34 (
).55 0.55 (
).67 0.66 (
).74 0.74 (
).80 0.80 (
).92 0.91 (
.01 1.01
.08 1.07
.13 1.12
.21 1.21
.27 1.26
.32 1.31
.36 1.35
.40 1.39
.46 1.44
.50 1.49
.53 1.52
.56 1.55
.59 1.57
).34
).55
).66
).74
).79
).91
.00
.07
.11
.20
.26
.30
.34
.38
.44
.48
.51
.54
.56
                                                    D-128
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
     Table 19-11. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Observations (2 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.08 0.86 0.77 (
1.55 1.18 1.04 (
1.85 1.38 1.20
2.09 1.52 1.32
2.29 1.63 1.41
2.75 1.88 1.60
3.19 2.11 1.77
3.55 2.28 1.89
3.84 2.41 1.99
4.44 2.67 2.17
4.91 2.87 2.31 ;
5.30 3.02 2.41 ;
5.65 3.15 ^,50 ;
6.10 3.32 2,61 :
6.73 3.55 2.75 ;
7.27 3.73 2.87
7.73 3.89 2.97
8.14 4.02 3.05
8.53 4.15 3.12
).72 0.69 0.65 (
).97 0.92 0.87 (
.11 1.06 0.99 (
.22 1.15 1.08
.30 1.23 1.15
.46 1.38 1.28
.61 1.51 1.40
.71 1.60 1.48
.79 1.68 1.55
.94 1.81 1.66
>.05 1.90 1.74
>.13 1.98 1.80
1.20 2.04 1.85
1.29 2.11 1.92
>.40 2.21 2.00
2.49 2.29 2.06
?,57 2.35 2.11
2,63 2.40 2.16 ;
2,69 2.45 2.19 ;
).62 (
).84 (
).96 (
.04
.10
.23
.34
.42
.47
.58
.65
.71
.76
.81
.89
.94
.99
>.03
>.06
).61 0.59 (
).81 0.80 (
).93 0.91 (
.01 0.99 (
.07 1.04
.19 1.16
.29 1.26
.37 1.33
.42 1.39
.52 1.48
.59 1.55
.64 1.60
.68 1.64
.74 1.69
.80 1.75
.85 1.80
.90 1.84
.93 1.87
.96 1.90
).59 (
).79 (
).90 (
).97 (
.03
.15
.24
.31
.36
.45
.52
.57
.61
.65
.72
.76
.80
.83
.86
).58 0.57 (
).78 0.77 (
).89 0.88 (
).96 0.95 (
.02 1.01
.13 1.12
.23 1.22
.29 1.28
.34 1.33
.43 1.42
.50 1.48
.54 1.53
.58 1.57
.63 1.61
.69 1.67
.73 1.71
.77 1.75
.80 1.78
.83 1.81
).57 0.56 0.56 (
).77 0.76 0.75 (
).87 0.86 0.86 (
).95 0.94 0.93 (
.00 0.99 0.98 (
.11 1.10 1.09
.21 1.19 1.18
.27 1.26 1.25
.32 1.30 1.29
.41 1.39 1.38
.47 1.45 1.44
.51 1.49 1.48
.55 1.53 1.52
.60 1.57 1.56
.65 1.63 1.61
.70 1.67 1.66
.73 1.71 1.69
.76 1.74 1.72
.79 1.76 1.74
).56 0.55 (
).75 0.75 (
).85 0.85 (
).92 0.92 (
).98 0.97 (
.09 1.08
.18 1.17
.24 1.23
.28 1.28
.37 1.36
.43 1.42
.47 1.46
.51 1.50
.55 1.54
.60 1.59
.64 1.63
.68 1.67
.70 1.69
.73 1.72
).55 0.55 (
).74 0.74 (
).85 0.84 (
).92 0.91 (
).97 0.96 (
.08 1.07
.17 1.16
.23 1.22
.27 1.26
.36 1.35
.41 1.40
.46 1.44
.49 1.48
.53 1.52
.59 1.57
.63 1.61
.66 1.64
.69 1.67
.71 1.69
).55
).74
).84
).91
).96
.07
.15
.21
.26
.34
.39
.44
.47
.51
.56
.60
.64
.66
.68
      Table 19-11. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Observations (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.47 .15 1.02 (
2.00 .49 1.30
2.36 .70 1.47
2.64 .85 1.59
2.87 .97 1.68
3.42 2.24 1.88
3.95 2.49 2.06
4.38 2.68 2.19
4.74 2,83 2.29 ;
5.45 3.12 2.49 ;
6.02 3.34 2.63 ;
6.50 3.51 2.74 ;
6.92 3.66 2,84 ;
7.46 3.85 2.95 2
8.23 4.10 3.11 ;
8.88 4.31 3.23
9.43 4.48 3.34
9.95 4.64 3.43
10.40 4.78 3.51
).96 (
.21
.35
.45
.53
.70
.85
.96
>.04
1.20 ;
>.31 ;
>.40 ;
2.47 ;
>.56 ;
>.68 ;
2,78 ;
2,85 ;
2.92 ;
2.98 ;
).91 0.86 0.83 (
.15 1.08 1.04
.28 1.20 1.15
.37 1.28 1.23
.45 1.34 1.29
.60 1.48 1.41
.73 1.59 1.52
.82 1.68 1.59
.90 1.74 1.65
>.03 1.85 1.76
>.13 1.93 1.83
1.21 2.00 1.89
1.27 2.05 1.93
>.34 2.11 1.99
>.44 2.19 2.06
1.52 2.26 2.12 ;
>.59 2.31 2.16 ;
>.64 2.35 2.20 ;
>.69 2.39 2.23 ;
).81 0.79 0.78 (
.00 0.98 0.97 (
.11 1.09 1.07
.19 1.16 1.14
.25 1.22 1.20
.36 1.33 1.31
.46 1.43 1.40
.53 1.49 1.47
.59 1.55 1.52
.68 1.64 1.61
.75 1.70 1.67
.80 1.75 1.72
.85 1.79 1.75
.90 1.84 1.80
.96 1.90 1.86
>.01 1.95 1.91
>.06 1.99 1.94
>.09 2.02 1.98
1.12 2.05 2.00
).78 0.77 0.76 (
).96 0.95 0.95 (
.06 1.05 1.04
.13 1.12 1.11
.18 1.17 1.16
.29 1.28 1.27
.38 1.37 1.36
.45 1.43 1.42
.50 1.48 1.47
.58 1.56 1.55
.64 1.62 1.61
.69 1.67 1.65
.73 1.71 1.69
.77 1.75 1.73
.83 1.81 1.79
.87 1.85 1.83
.91 1.89 1.87
.94 1.92 1.90
.97 1.94 1.92
).76 0.75 (
).94 0.93 (
.03 1.03
.10 1.09
.15 1.14
.26 1.25
.34 1.33
.40 1.39
.45 1.44
.53 1.52
.59 1.57
.63 1.61
.67 1.65
.71 1.69
.76 1.74
.80 1.78
.84 1.82
.86 1.84
.89 1.87
).75 0.74 (
).92 0.92 (
.02 1.02
.09 1.08
.14 1.13
.24 1.23
.32 1.32
.38 1.37
.43 1.42
.51 1.50
.56 1.55
.60 1.59
.64 1.63
.68 1.67
.73 1.72
.77 1.76
.80 1.79
.83 1.82
.85 1.84
).74 (
).92 (
.01
.08
.13
.23
.31
.37
.41
.49
.54
.59
.62
.66
.71
.75
.78
.81
.83
).74 0.73
).91 0.91
.01 1.00
.07 1.07
.12 1.11
.22 1.21
.30 1.29
.36 1.35
.40 1.39
.48 1.47
.53 1.52
.57 1.56
.60 1.60
.64 1.64
.69 1.69
.73 1.72
.76 1.75
.79 1.78
.81 1.80
                                                    D-129
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
        Table 19-11. K-Multipliers for 1-of-3  Intrawell Prediction Limits on Observations (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.27 0.98 0.87 (
1.76 1.31 1.14
2.09 1.50 1.30
2.35 1.65 1.42
2.56 1.77 1.51
3.06 2.02 1.70
3.55 2,28 1.87
3.93 2.43 2.00
4.26 2.57 2.10
4.90 2.84 2,28 :
5.42 3.04 2.42 ;
5.85 3.20 2.53
6.23 3.34 2.61
6.72 3.52 2.73
7.41 3.76 2.88
8.00 3.95 3.00
8.50 4.11 3.09
8.96 4.25 3.18
9.38 4.38 3.25
).81 0.77 0.72 (
.06 1.00 0.94 (
.20 1.14 1.06
.30 1.23 1.15
.38 1.30 1.21
.55 1.45 1.35
.69 1.59 1.47
.80 1.68 1.55
.88 1.75 1.61
>.03 1.88 1.72
M4 1.98 1.80
2.22' 2.05 1.87
2,29 2.11 1.92
2.38 2,19 1.98
2.50 2.29 2.06
2.59 2.36 2.12 ;
2.66 2.43 2,18 \
1.12 2.48 2,22' :
2.18 2.53 2.28' '4
).70 (
).91 (
.02 (
.10
.16
.29
.40
.47
.53
.64
.71
.77
.81
.87
.94
>.oo
>.04
>.08
M2 ;
).68 0.66 (
).88 0.86 (
).99 0.97 (
.07 1.05
.13 1.10
.25 1.22
.35 1.32
.42 1.39
.48 1.44
.57 1.53
.64 1.60
.69 1.65
.74 1.69
.79 1.74
.85 1.80
.91 1.85
.95 1.89
.98 1.92
>.01 1.95
).65 (
).85 (
).96 (
.03
.09
.20
.30
.36
.41
.50
.57
.61
.65
.70
.76
.81
.85
.88
.91
).65 0.64 (
).84 0.83 (
).95 0.94 (
.02 1.01
.07 1.06
.19 1.17
.28 1.27
.34 1.33
.39 1.38
.48 1.47
.54 1.53
.59 1.57
.63 1.61
.68 1.66
.74 1.71
.78 1.76
.82 1.79
.85 1.82
.88 1.85
).64 0.63 0.62 (
).83 0.82 0.81 (
).93 0.92 0.91 (
.00 0.99 0.98 (
.06 1.04 1.04
.17 1.15 1.14
.26 1.24 1.23
.32 1.30 1.29
.37 1.35 1.34
.45 1.44 1.42
.51 1.49 1.48
.56 1.54 1.52
.60 1.57 1.56
.64 1.62 1.60
.70 1.67 1.66
.74 1.72 1.70
.78 1.75 1.73
.81 1.78 1.76
.83 1.80 1.78
).62 0.62 (
).81 0.80 (
).91 0.90 (
).98 0.97 (
.03 1.03
.14 1.13
.22 1.22
.28 1.28
.33 1.32
.41 1.41
.47 1.46
.51 1.50
.55 1.54
.59 1.58
.64 1.63
.68 1.67
.72 1.71
.74 1.74
.77 1.76
).62 0.61 (
).80 0.80 (
).90 0.90 (
).97 0.96 (
.02 1.01
.13 1.12
.21 1.21
.27 1.26
.32 1.31
.40 1.39
.46 1.44
.50 1.49
.53 1.52
.57 1.56
.63 1.61
.67 1.65
.70 1.68
.72 1.71
.75 1.73
).61
).79
).89
).96
.01
.11
.20
.26
.30
.38
.44
.48
.51
.55
.60
.64
.67
.70
.72
     Table 19-11. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Observations (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.71 1.29 1.13
2.29 1.63 1.41
2.68 1.85 1.57
2.99 2.00 1.69
3.24 2.13 1.79
3.84 2.41 1.99
4.44 2.67 2.17
4.91 2.87 2.31 ;
5.30 3.02 2.41 ;
6.10 3.32 2,61 ;
6.73 3.55 2.75 2
7.27 3.73 2.87
7.73 3.89 2.97
8.33 4.09 3.09
9.20 4.35 3.25 ;
9.90 4.57 3.38 ;
10.55 4.75 3.49 ;
11.07 4.91 3.57 :
11.60 5.05 3.65 :
.05
.30
.44
.54
.62
.79
.94
>.05
>.13
1.29 ;
>.40 ;
?,49 :
2,57 :
2,68 :
2.78
2.88
2.96 •<
3.02
3.08
.00 0.94 0.90 (
.23 1.15 1.10
.36 1.27 1.21
.45 1.35 1.29
.52 1.41 1.35
.68 1.55 1.47
.81 1.66 1.58
.90 1.74 1.65
.98 1.80 1.71
Ml 1.92 1.81
>.21 2.00 1.89
1.29 2.06 1.94
>.35 2.11 1.99
>.42 2.18 2.05
2,53" 2.26 2.12 ;
2,6.1 2.32 2.17 ;
2,67 2.37 2.22 ;
2.72 2.42 2.26 ;
2.78 2,45 2.29 ;
).88 0.86 0.85 (
.07 1.04 1.03
.17 1.15 1.13
.25 1.22 1.20
.30 1.27 1.25
.42 1.39 1.36
.52 1.48 1.45
.59 1.55 1.52
.64 1.60 1.57
.74 1.69 1.65
.80 1.75 1.72
.85 1.80 1.76
.90 1.84 1.80
.95 1.89 1.85
>.01 1.95 1.91
1.07 2.00 1.95
Ml 2.04 1.99
M4 2.07 2.02
M7 2.10 2.05 ;
).84 0.83 0.83 (
.02 1.01 1.00 (
.12 1.11 1.10
.19 1.17 1.17
.24 1.23 1.22
.34 1.33 1.32
.43 1.42 1.41
.50 1.48 1.47
.54 1.53 1.51
.63 1.61 1.60
.69 1.67 1.65
.73 1.71 1.70
.77 1.75 1.73
.82 1.79 1.78
.88 1.85 1.83
.92 1.89 1.87
.96 1.93 1.91
.98 1.96 1.94
>.01 1.98 1.96
).82 0.81 (
).99 0.98 (
.09 1.08
.15 1.14
.20 1.19
.30 1.29
.39 1.38
.45 1.44
.49 1.48
.57 1.56
.63 1.61
.67 1.66
.71 1.69
.75 1.73
.80 1.78
.84 1.82
.88 1.86
.90 1.88
.93 1.91
).81 0.80 (
).98 0.97 (
.07 1.07
.14 1.13
.18 1.18
.28 1.28
.37 1.36
.43 1.42
.47 1.46
.55 1.54
.60 1.59
.64 1.63
.68 1.67
.72 1.71
.77 1.76
.81 1.80
.84 1.83
.87 1.85
.89 1.88
).80 (
).97 (
.06
.13
.17
.27
.36
.41
.46
.53
.59
.63
.66
.70
.75
.79
.82
.85
.87
).80 0.79
).96 0.96
.06 1.05
.12 1.11
.17 1.16
.26 1.26
.35 1.34
.40 1.39
.44 1.44
.52 1.51
.57 1.56
.61 1.60
.64 1.64
.68 1.67
.73 1.72
.77 1.76
.80 1.79
.83 1.82
.85 1.84
                                                    D-130
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
      Table 19-11. K-Multipliers for 1-of-3  Intrawell Prediction Limits on Observations (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.19 1.60 1.39 .29
2.87 1.97 1.68 .53
3.34 2.21 1.85 .68
3.70 2.38 1.98 .78
4.01 2.52 2.08 .87
4.74 '"2.83" 2.29 2.04
5.45 3.12 2.49 2.20 ;
6.02 3.34 2.63 2.31 ;
6.50 3.51 2.74 2.40 ;
7.46 3.85 2.95 2.56 ;
8.23 4.10 3.11 2.68 ;
8.88 4.31 3.23 2, 78 \
9.43 4.48 3.34 2.85 '<
10.18 4.71 3.47 2,95. :
11.22 5.01 3.64 3.08
12.07 5.25 3.78 3.18
12.83 5.46 3.90 3.26
13.54 5.64 4.00 3.33
14.18 5.80 4.09 3.40
.22 1.14 1.10
.45 1.34 1.29
.58 1.46 1.40
.67 1.54 1.47
.74 1.61 1.53
.90 1.74 1.65
>.03 1.85 1.76
M3 1.93 1.83
>.21 2.00 1.89
1.34 2.11 1.99
>.44 2.19 2.06
1.52 2.26 2.12 ;
>.59 2.31 2.16 ;
1.67 2.37 2.22 ;
2,77 2.45 2.29 ;
2,88 2.52 2.35 ;
2.92 2.57 2.40 ;
2,98 2.62 2.43 ;
3.03 2.66 2.47 ;
.06 1.04 1.03
.25 1.22 1.20
.35 1.32 1.29
.42 1.39 1.36
.47 1.44 1.41
.59 1.55 1.52
.68 1.64 1.61
.75 1.70 1.67
.80 1.75 1.72
.90 1.84 1.80
.96 1.90 1.86
>.01 1.95 1.91
>.06 1.99 1.94
Ml 2.04 1.99
M7 2.10 2.05 ;
1.22 2.15 2.09 ;
1.26 2.18 2.13 ;
>.30 2.22 2.16 ;
>.33 2.24 2.19 ;
.02 1.01 1.00 (
.18 1.17 1.16
.28 1.27 1.26
.34 1.33 1.32
.39 1.38 1.37
.50 1.48 1.47
.58 1.56 1.55
.64 1.62 1.61
.69 1.67 1.65
.77 1.75 1.73
.83 1.81 1.79
.87 1.85 1.83
.91 1.89 1.87
.95 1.93 1.91
>.01 1.98 1.96
>.05 2.03 2.00
>.09 2.06 2.04 ;
1.12 2.09 2.06 ;
M5 2.11 2.09 ;
).99 0.98 (
.15 1.14
.24 1.23
.30 1.29
.35 1.34
.45 1.44
.53 1.52
.59 1.57
.63 1.61
.71 1.69
.76 1.74
.80 1.78
.84 1.82
.88 1.86
.93 1.91
.97 1.95
>.00 1.98
>.03 2.00
>.05 2.03 ;
).98 0.97 (
.14 1.13
.22 1.22
.28 1.28
.33 1.32
.43 1.42
.51 1.50
.56 1.55
.60 1.59
.68 1.67
.73 1.72
.77 1.76
.80 1.79
.84 1.83
.89 1.88
.93 1.92
.96 1.95
.98 1.97
>.01 1.99
).97 (
.13
.21
.27
.32
.41
.49
.54
.59
.66
.71
.75
.78
.82
.87
.90
.94
.96
.98
).96 0.96
.12 1.11
.20 1.20
.26 1.26
.31 1.30
.40 1.39
.48 1.47
.53 1.52
.57 1.56
.64 1.64
.69 1.69
.73 1.72
.76 1.75
.80 1.79
.85 1.84
.89 1.87
.92 1.90
.94 1.93
.96 1.95
       Table 19-11. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Observations (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.76 1.31 1.14
2.35 1.65 1.42
2.75 1.86 1.58
3.06 2.02 1.70
3.32 2.15 1.80
3.93 2.43 2.00
4.53 2.69 2. 18
5.02 2.88 2.31 \
5.42 3.04 2.42 2
6.23 3.34 2.62
6.87 3.57 2.76 ;
7.42 3.75 2.87 ;
7.89 3.91 2.97 ;
8.50 4.11 3.09 ;
9.38 4.37 3.25 ;
10.11 4.59 3.38 ;
10.74 4.79 3.49 ;
11.33 4.93 3.59 :
11.82 5.08 3.66 :
.06
.30
.44
.55
.63
.80
.95
>.05
M4
?,29 ;
2.41
2.50 •„
1.51 .
2.66
1.18
1.8:8:
2.95
3.03
3.09
.00 0.94 0.91 (
.23 1.15 1.10
.36 1.27 1.21
.45 1.35 1.29
.53 1.41 1.35
.68 1.55 1.47
.81 1.66 1.58
.90 1.74 1.65
.98 1.80 1.71
Ml 1.92 1.81
2.21 2.00 1.89
2,29 2.06 1.94
2:35 2.11 1.99
2.43 2, 18 2.04
2.53 2.26 2.12 ;
2.61 2.32 2,17 :
2.67 .2.37 2,22 ;
2.73 2.42 2,26 .
2.78 2.45 2,29 ,
).88 0.86 0.85 (
.07 1.05 1.03
.17 1.15 1.13
.25 1.22 1.20
.30 1.27 1.25
.42 1.39 1.36
.52 1.48 1.45
.59 1.55 1.52
.64 1.60 1.57
.74 1.69 1.65
.80 1.75 1.72
.86 1.80 1.76
.90 1.84 1.80
.95 1.89 1.85
>.01 1.95 1.91
1.07 2.00 1.95
Ml 2.04 1.99
2.14 2.07 2.02
2.17 2.10 2.04 ;
).84 0.83 0.83 (
.02 1.01 1.00 (
.12 1.11 1.10
.19 1.17 1.17
.24 1.23 1.22
.34 1.33 1.32
.43 1.42 1.41
.50 1.48 1.47
.54 1.53 1.51
.63 1.61 1.60
.69 1.67 1.65
.73 1.71 1.70
.77 1.75 1.73
.82 1.79 1.78
.87 1.85 1.83
.92 1.89 1.87
.95 1.93 1.91
.98 1.96 1.93
>.01 1.98 1.96
).82 0.81 (
).99 0.98 (
.09 1.08
.15 1.14
.20 1.19
.30 1.29
.39 1.38
.45 1.44
.49 1.48
.57 1.56
.63 1.61
.67 1.66
.71 1.69
.75 1.73
.80 1.78
.84 1.82
.88 1.86
.90 1.88
.93 1.90
).81 0.80 (
).98 0.97 (
.07 1.07
.14 1.13
.19 1.18
.29 1.28
.37 1.36
.43 1.42
.47 1.46
.55 1.54
.60 1.59
.64 1.63
.68 1.67
.72 1.71
.77 1.76
.81 1.80
.84 1.83
.87 1.86
.89 1.88
).80 (
).97 (
.06
.13
.17
.27
.36
.41
.46
.53
.59
.63
.66
.70
.75
.79
.82
.84
.87
).80 0.79
).96 0.96
.06 1.05
.12 1.11
.17 1.16
.26 1.26
.35 1.34
.40 1.39
.44 1.44
.52 1.51
.57 1.56
.61 1.60
.64 1.64
.68 1.67
.73 1.72
.77 1.76
.80 1.79
.83 1.82
.85 1.84
                                                    D-131
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
    Table 19-11. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Observations (10 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.29 1.63 1.41 .30
2.99 2.00 1.69 .54
3.47 2.24 1.87 .69
3.84 2.41 1.99 .79
4.16 ^,55 2.09 .87
4.91 2.87 2.31 2.05
5.65 3.16 Z.5O 2.20 ;
6.24 3.37 2,64 2.32 ;
6.73 3.55 2.75 2.40 ;
7.73 3.89 2.97 2,57 :
8.52 4.14 3.12 2,69 '4
9.18 4.35 3.25 2.78
9.77 4.53 3.35 2.86 .,
10.55 4.75 3.48 2.95
11.62 5.05 3.66 3.08
12.50 5.30 3.80 3.19
13.28 5.52 3.91 3.27
13.96 5.69 4.00 3.34
14.65 5.86 4.10 3.41
.23 1.15 1.10
.45 1.35 1.29
.58 1.46 1.40
.68 1.55 1.47
.75 1.61 1.53
.90 1.74 1.65
>.04 1.85 1.76
>.13 1.94 1.83
>.21 2.00 1.89
>.35 2.11 1.99
>.45 2.19 2.06
2,53 2.26 2.12 ;
2,59 2.31 2.16 ;
2,67 2.37 2.22 ;
2.18 2,46 2.29 ;
2.86 2,52 2.35 ;
>.93 2,58 2.39 ;
2.98 2,62 2.44 ;
3.04 2,66 2,47 :
.07 1.04 1.03
.25 1.22 1.20
.35 1.32 1.30
.42 1.39 1.36
.47 1.44 1.41
.59 1.55 1.52
.68 1.64 1.61
.75 1.70 1.67
.80 1.75 1.72
.90 1.84 1.80
.96 1.90 1.86
>.01 1.95 1.91
>.06 1.99 1.94
Ml 2.04 1.99
>.17 2.10 2.05 ;
1.22 2.15 2.09 ;
1.26 2.19 2.13 ;
>.30 2.22 2.16 ;
>.33 2.25 2.19 ;
.02 1.01 1.00 (
.19 1.17 1.17
.28 1.27 1.26
.34 1.33 1.32
.39 1.38 1.37
.50 1.48 1.47
.58 1.57 1.55
.64 1.62 1.61
.69 1.67 1.65
.77 1.75 1.73
.83 1.81 1.79
.87 1.85 1.83
.91 1.89 1.87
.95 1.93 1.91
>.01 1.98 1.96
>.05 2.03 2.00
>.09 2.06 2.04 ;
1.12 2.09 2.06 ;
>.14 2.11 2.09 ;
).99 0.98 (
.15 1.14
.24 1.23
.30 1.29
.35 1.34
.45 1.44
.53 1.52
.59 1.57
.63 1.61
.71 1.69
.76 1.74
.80 1.78
.84 1.82
.88 1.86
.93 1.91
.97 1.95
>.00 1.98
>.03 2.00
>.05 2.03 ;
).98 0.97 (
.14 1.13
.22 1.22
.28 1.28
.33 1.32
.43 1.42
.51 1.50
.56 1.55
.60 1.59
.68 1.67
.73 1.72
.77 1.76
.80 1.79
.84 1.83
.89 1.88
.93 1.92
.96 1.95
.98 1.97
>.01 2.00
).97 (
.13
.21
.27
.32
.41
.49
.54
.59
.66
.71
.75
.78
.82
.87
.90
.93
.96
.98
).96 0.96
.12 1.11
.21 1.20
.26 1.26
.31 1.30
.40 1.39
.48 1.47
.53 1.52
.57 1.56
.64 1.63
.70 1.68
.73 1.72
.76 1.75
.80 1.79
.85 1.84
.89 1.87
.92 1.90
.94 1.93
.96 1.95
      Table 19-11. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Observations (10 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.87 1.97 1.68 1.53 .45 1.34 1.29
3.70 2.38 1.98 1.78 .67 1.54 1.47
4.28 2.64 2.16 1.93 .80 1.66 1.58
4.74 "Zi33': 2.29 2.04 .90 1.74 1.65
5.12 2.99 2.40 2.13 .97 1.80 1.71
6.02 3.34 2.63 2.31 2.13 1.93 1.83
6.92 3.66 2,84 2.47 2.27 2.05 1.93
7.63 3.90 2.99 2.59 2.37 2.13 2.00
8.23 4.10 3.11 2.68 2.44 2.19 2.06
9.44 4.49 3.34 2,85 2.59 2.31 2.16 ;
10.40 4.77 3.51 2.98 2.69 2.39 2.23 ;
11.21 5.00 3.64 3.08 2,77 2.46 2.29 ;
11.91 5.21 3.75 3.16 2.84 2.51 2.34 ;
12.84 5.46 3.89 3.26 2.92 2.57 2.39 ;
14.16 5.80 4.09 3.40 3,03 2.66 2.47 ;
15.23 6.08 4.24 3.50 3.12 2.72 2.52 ;
16.21 6.32 4.37 3.59 3.19 2.78 2.57 ;
17.09 6.52 4.47 3.67 3.25 2.82 2.61 ;
17.77 6.69 4.57 3.74 3.31 -2.86 2.64 ;
.25 1.22 1.20
.42 1.39 1.36
.52 1.48 1.45
.59 1.55 1.52
.64 1.60 1.57
.75 1.70 1.67
.85 1.79 1.75
.91 1.85 1.81
.96 1.90 1.86
>.06 1.99 1.94
1.12 2.05 2.00
>.17 2.10 2.05 ;
1.21 2.14 2.08 ;
1.26 2.18 2.13 ;
2.33 2.24 2.19 ;
>.38 2.29 2.23 ;
>.42 2.33 2.26 ;
>.45 2.36 2.29 ;
>.48 2.39 2.33 ;
.18 1.17 1.16
.34 1.33 1.32
.43 1.42 1.41
.50 1.48 1.47
.54 1.53 1.51
.64 1.62 1.61
.73 1.71 1.69
.78 1.76 1.75
.83 1.81 1.79
.91 1.89 1.87
.97 1.94 1.92
>.01 1.98 1.96
>.05 2.02 1.99
>.09 2.06 2.04 ;
>.15 2.11 2.09 ;
>.19 2.15 2.13 ;
1.22 2.19 2.16 ;
1.25 2.22 2.19 ;
>.28 2.24 2.21 ;
.15 1.14
.30 1.29
.39 1.38
.45 1.44
.49 1.48
.59 1.57
.67 1.65
.72 1.70
.76 1.74
.84 1.82
.89 1.87
.93 1.91
.96 1.94
>.00 1.98
>.05 2.03 ;
>.09 2.06 ;
1.12 2.09 ;
>.15 2.12 ;
>.17 2.14 ;
.14 1.13
.28 1.28
.37 1.36
.43 1.42
.47 1.46
.56 1.55
.64 1.63
.69 1.68
.73 1.72
.80 1.79
.85 1.84
.89 1.88
.92 1.91
.96 1.95
>.01 1.99
>.04 2.03 ;
1.07 2.06 ;
>.10 2.08 ;
>.12 2.11 ;
.13
.27
.36
.41
.46
.54
.62
.67
.71
.78
.83
.87
.90
.93
.98
1.02 :
1.05 ;
>.07 ;
>.09 ;
.12 1.11
.26 1.26
.35 1.34
.40 1.39
.44 1.44
.53 1.52
.61 1.60
.66 1.65
.69 1.68
.76 1.75
.81 1.80
.85 1.84
.88 1.87
.92 1.90
.96 1.95
>.00 1.98
1.02 2.01
>.05 2.03
>.07 2.05
                                                    D-132
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
       Table 19-11. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Observations (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.35 1.65 1.42 .30
3.06 2.02 1.70 .55
3.55 2,26 1.87 .69
3.93 2.43 2.00 .80
4.25 2.57 2.10 .88
5.02 2.88 2,31 2.05
5.77 3.17 2.51 2,21 :
6.37 3.39 2.65 2,32 .
6.87 3.57 2.76 2.41
7.89 3.91 2.97 2.57
8.69 4.17 3.13 2.69
9.38 4.38 3.25 2.78
9.97 4.55 3.36 2.86
10.74 4.78 3.49 2.96
11.84 5.08 3.66 3.09
12.74 5.32 3.80 3.19
13.57 5.54 3.92 3.27
14.26 5.71 4.02 3.34
14.94 5.88 4.11 3.41
.23 1.15 1.10
.45 1.35 1.29
.59 1.47 1.40
.68 1.55 1.47
.75 1.61 1.53
.90 1.74 1.65
>.04 1.86 1.76
2.14 1.94 1.83
2,27 2.00 1.89
2,35 2.11 1.99
2.45 2.19 2.06
2.53 2.26 2.12 ;
2.59 2.31 Z'.IS' :
2.67 2,37 '2.22 '*
2.18 2.46 2.29 .
2.86 2.52 '2:35 .
2.93 2.58 2.39 ,
2.98 2.62 2.43 ,
3.04 2.66 2.47
.07 1.05 1.03
.25 1.22 1.20
.35 1.32 1.30
.42 1.39 1.36
.48 1.44 1.41
.59 1.55 1.52
.68 1.64 1.61
.75 1.70 1.67
.80 1.75 1.72
.90 1.84 1.80
.96 1.90 1.86
>.01 1.95 1.91
>.06 1.99 1.94
Ml 2.04 1.99
2,77 2.10 2.05 ;
2,22 2.15 2.09 ;
2,26. 2,13 2.13 :
2:30 2.22' 2,16 .
3.33 2.24 2.19 .
.02 1.01 1.00 (
.19 1.17 1.17
.28 1.27 1.26
.34 1.33 1.32
.39 1.38 1.37
.50 1.48 1.47
.58 1.57 1.55
.64 1.62 1.61
.69 1.67 1.65
.77 1.75 1.73
.83 1.81 1.79
.87 1.85 1.83
.91 1.89 1.87
.95 1.93 1.91
>.01 1.98 1.96
>.05 2.02 2.00
>.09 2.06 2.04 ;
2,72 2.09 2.06 ;
2,75 2,77, 2.09 ;
).99 0.98 (
.15 1.14
.24 1.23
.30 1.29
.35 1.34
.45 1.44
.53 1.52
.59 1.57
.63 1.61
.71 1.69
.76 1.74
.80 1.78
.84 1.82
.88 1.86
.93 1.91
.97 1.95
>.00 1.98
>.03 2.00
>.05 2.03 ;
).98 0.97 (
.14 1.13
.22 1.22
.29 1.28
.33 1.32
.43 1.42
.51 1.50
.56 1.55
.60 1.59
.68 1.67
.73 1.72
.77 1.76
.80 1.79
.84 1.83
.89 1.88
.93 1.91
.96 1.95
.98 1.97 ]
>.01 1.99 1
).97 (
.13
.21
.27
.32
.41
.49
.54
.59
.66
.71
.75
.78
.82
.87
.90
.93
L.96
L.98
).96 0.96
.12 1.11
.21 1.20
.26 1.26
.31 1.30
.40 1.39
.48 1.47
.53 1.52
.57 1.56
.64 1.64
.69 1.68
.73 1.72
.76 1.75
.80 1.79
.85 1.84
.89 1.87
.92 1.90
.94 1.93
.96 1.95
    Table 19-11. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Observations (20 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.99 2.00 1.69 1.54 .45 1.35 1.29
3.84 2.41 1.99 1.79 .68 1.55 1.47
4.44 2.67 2.17 1.94 .81 1.66 1.58
4.91 2.87 2.31 2.05 .90 1.74 1.65
5.30 3.02 2.41 2.13 .98 1.80 1.71
6.24 3.37 2,64 2.32 2.13 1.94 1.83
7.16 3.70 2.85 2,48 2.27 2.05 1.93
7.90 3.94 3.00 2.59 2.37 2.13 2.00
8.52 4.14 3.12 2,69 2,45 2.19 2.06
9.77 4.53 3.35 2.86 2,59 2.31 2.16 ;
10.77 4.82 3.52 2.98 2.69 2.39 2.24 ;
11.60 5.05 3.66 3.08 2.78 2.46 2.29 ;
12.33 5.26 3.77 3.16 2.84 2,57 2.34 ;
13.28 5.51 3.91 3.27 2.93 2,57 2.39 ;
14.65 5.85 4.10 3.40 3.04 2.66 2,47 :
15.77 6.14 4.25 3.51 3.12 2,73 2,52 :
16.80 6.37 4.38 3.60 3.19 2.78 2,57 :
17.68 6.58 4.49 3.67 3.25 2.83 2,67 ,
18.46 6.76 4.58 3.74 3.31 2.86 2,64 .
.25 1.22 1.20
.42 1.39 1.36
.52 1.48 1.45
.59 1.55 1.52
.64 1.60 1.57
.75 1.70 1.67
.85 1.79 1.75
.91 1.85 1.81
.96 1.90 1.86
>.06 1.99 1.94
>.12 2.05 2.00
>.17 2.10 2.05 ;
>.21 2.14 2.08 ;
1.26 2.18 2.13 ;
2.33 2.24 2.19 ;
>.38 2.29 2.23 ;
>.42 2.33 2.27 ;
3.46 2.36 2.30 ;
2,48 2.39 2.32 ;
.19 1.17 1.17
.34 1.33 1.32
.43 1.42 1.41
.50 1.48 1.47
.54 1.53 1.51
.64 1.62 1.61
.73 1.71 1.69
.79 1.76 1.75
.83 1.81 1.79
.91 1.89 1.87
.97 1.94 1.92
>.01 1.98 1.96
>.05 2.02 2.00
>.09 2.06 2.04 ;
>.14 2.11 2.09 ;
>.19 2.15 2.13 ;
>.22 2.19 2.16 ;
1.25 2.22 2.19 ;
>.28 2.24 2.21 ;
.15 1.14
.30 1.29
.39 1.38
.45 1.44
.49 1.48
.59 1.57
.67 1.65
.72 1.70
.76 1.74
.84 1.82
.89 1.87
.93 1.91
.96 1.94
>.00 1.98
>.05 2.03 ;
>.09 2.06 ;
>.12 2.09 ;
>.15 2.12 ;
>.17 2.14 ;
.14 1.13
.28 1.28
.37 1.36
.43 1.42
.47 1.46
.56 1.55
.64 1.63
.69 1.68
.73 1.72
.80 1.79
.85 1.84
.89 1.88
.92 1.91
.96 1.95
>.01 1.99
>.04 2.03 ;
1.07 2.06 ;
>.10 2.08 ;
>.12 2.10 ;
.13
.27
.36
.41
.46
.54
.62
.67
.71
.78
.83
.87
.90
.93
.98
1.02
1.05 ;
>.07 ;
>.09 ;
.12 1.11
.26 1.26
.35 1.34
.40 1.39
.44 1.44
.53 1.52
.61 1.60
.66 1.65
.69 1.68
.76 1.75
.81 1.80
.85 1.84
.88 1.87
.92 1.90
.96 1.95
.99 1.98
>.03 2.01
>.05 2.03
>.07 2.05
                                                    D-133
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                         Unified Guidance
Table 19-11. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Observations (20 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
3.70 2.38 1.98 1.78 1.67 1.54 1.47
4.74 "Zi33': 2.29 2.04 1.90 1.74 1.65
5.45 3.12 2.49 2.20 2.03 1.85 1.76
6.02 3.34 2.63 2.31 2.13 1.93 1.83
6.50 3.51 2.74 2.40 2.21 2.00 1.89
7.63 3.91 2.99 2.59 2.37 2.13 2.00
25 30 35
.42 1.39 1.36
.59 1.55 1.52
.68 1.64 1.61
.75 1.70 1.67
.80 1.75 1.72
.91 1.85 1.81
8.76 4.27 3.21 2.76 2.51 2.24 2.11 2.00 1.94 1.90
9.65 4.55 3.38 2,88 2.61 2.33 2.18 2.07 2.00 1.96
10.40 4.77 3.51 2.98 2.69 2.39 2.23 2.12 2.05 2.00
40 45 50 60 70 80 90 100 125 150
.34 1.33 1.32
.50 1.48 1.47
.58 1.56 1.55
.64 1.62 1.61
.69 1.67 1.65
.78 1.76 1.75
.87 1.84 1.82
.92 1.90 1.88
.97 1.94 1.92
11.93 5.21 3.76 3.16 2.84 2.51 2.34 2.21 2.14 2.08 2.05 2.02 1.99
.30 1.29
.45 1.44
.53 1.52
.59 1.57
.63 1.61
.72 1.70
.80 1.78
.85 1.83
.89 1.87
.96 1.94
13.13 5.54 3.94 3.29 2.95 2.59 2.41 2.28 2.20 2.14 2.10 2.07 2.05 2.01 1.99
.28 1.28
.43 1.42
.50 1.50
.56 1.55
.60 1.59
.69 1.68
.76 1.75
.81 1.80
.85 1.84
.92 1.91
.97 1.96
14.16 5.80 4.09 3.40 3.03 2.66 2.47 2.33 2.24 2.19 2.14 2.11 2.09 2.05 2.03 2.01 1.99
.27
.41
.49
.54
.59
.67
.74
.79
.83
.90
.95
.98
15.04 6.03 4.21 3.49 3.10 2.71 2.51 2.37 2.28 2.22 2.18 2.15 2.12 2.08 2.06 2.04 2.02 2.01
.26 1.26
.40 1.39
.48 1.47
.53 1.52
.57 1.56
.66 1.65
.73 1.72
.78 1.76
.81 1.80
.88 1.87
.93 1.91
.96 1.95
.99 1.98
16.21 6.32 4.36 3.59 3.19 2.78 2.57 2.42 2.33 2.27 2.22 2.19 2.16 2.12 2.09 2.07 2.06 2.04 2.02 2.00
17.87 6.70 4.57 3.74 3.30 2,8.6. 2.64 2.49 2.39 2.32 2.28 2.24 2.21 2.17 2.14 2.12 2.10 2.09 2.06 2.04
19.24 7.03 4.74 3.85 3.39 2.93 2.70 2.54 2.44 2.37 2.32 2.28 2.25 2.21 2.18 2.16 2.14 2.13 2.10 2.09
20.41 7.30 4.87 3.94 3.47 2.98 2.75 2.58 2.47 2.40 2.35 2.31 2.28 2.24 2.21 2.19 2.17 2.15 2.13 2.12
21.48 7.53 4.99 4.03 3.53 3.Q3: 2.79 2.61 2.51 2.43 2.38 2.34 2.31 2.27 2.24 2.21 2.19 2.18 2.16 2.14
22.46 7.74 5.09 4.10 3.58 3.O8 2.82 2.64 2.53 2.46 2.40 2.37 2.33 2.29 2.26 2.23 2.22 2.20 2.18 2.16
Table 19-11. K-Multipliers for 1-of-3
ntrawell Prediction Limits on Observations (40 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                         45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.06 2.02 1.70 1.55 .45 1.35 1.29
3.93 2.43 2.00 1.80 .68 1.55 1.47
4.54 2.69 2.18, 1.95 .81 1.66 1.58
5.02 2.88 2.31 2.05 .90 1.74 1.65
5.42 3.04 2.42 2.14 .98 1.80 1.71
6.37 3.39 2.65 2.32 2.14 1.94 1.83
7.31 3.72 2.86 2.48 ' 2,-27 2.05 1.93
8.06 3.97 3.01 2.60 2'.37 2.13 2.01
8.69 4.17 3.13 2.69 2.45 2. 19 2.06
9.97 4.55 3.36 2.86 2.59 2.31 2.16 '*
10.98 4.84 3.53 2.99 2.70 2,39 2.24 ,
11.84 5.08 3.66 3.09 2.78 2.46 2.29 ,
12.58 5.28 3.77 3.17 2.84 2.51 2.34 :
13.55 5.54 3.92 3.27 2.93 2.57 2.39 ,
14.92 5.88 4.11 3.41 3.04 2.66 2.47
16.09 6.17 4.26 3.52 3.12 2.72 2.52
17.11 6.41 4.38 3.60 3.19 2.78 2.57
17.97 6.60 4.49 3.67 3.25 2.82 2.61
18.75 6.80 4.59 3.75 3.30 2.86 2.65
.25 1.22 1.20
.42 1.39 1.36
.52 1.48 1.45
.59 1.55 1.52
.64 1.60 1.57
.75 1.70 1.67
.85 1.79 1.75
.91 1.85 1.81
.96 1.90 1.86
>.06 1.99 1.94
2, 12 2.05 2.00
2,17 2.10 2.05 ;
2.21 2.14 2.08 ;
2,26 2.18, 2,13 ;
2,33' 2.24 2.19 ,
2.38.: 2,29 2.23 \ ,
2,42 2.33 2.27 , .
2.46 2.36 . 2.29 ,
2.49 2.39 2.32 .
.19 1.17 1.17
.34 1.33 1.32
.43 1.42 1.41
.50 1.48 1.47
.54 1.53 1.51
.64 1.62 1.61
.73 1.71 1.69
.79 1.76 1.75
.83 1.81 1.79
.91 1.89 1.87
.97 1.94 1.92
>.01 1.98 1.96
>.05 2.02 1.99
>.09 2.06 2.04 ;
2.14 2.11 2.09 ;
2.19 2.1S 2.13 :
2.22 2,19 2.16 .
2.25 . 2:22 2.19 t
2.28 2.24 2,21 ,
.15 1.14
.30 1.29
.39 1.38
.45 1.44
.49 1.48
.59 1.57
.67 1.65
.72 1.70
.76 1.74
.84 1.82
.89 1.87
.93 1.91
.96 1.94
>.00 1.98
>.05 2.03 ;
>.09 2.06 ;
2,12 2.09 ;
2,15 '2,'i'2"' :
2,17 2.14 ,
.14 1.13
.29 1.28
.37 1.36
.43 1.42
.47 1.46
.56 1.55
.64 1.63
.69 1.68
.73 1.72
.80 1.79
.85 1.84
.89 1.88
.92 1.91
.96 1.95
>.01 1.99
>.04 2.03 ;
1.07 2.06 ;
MO 2.08 ;
2.12 2.10 ;
.13
.27
.36
.41
.46
.54
.62
.67
.71
.78
.83
.87
.90
.93
.98
1.02
1.05 ;
>.07 ;
>.09 ;
.12 1.11
.26 1.26
.35 1.34
.40 1.39
.44 1.44
.53 1.52
.61 1.60
.66 1.65
.69 1.68
.76 1.75
.81 1.80
.85 1.84
.88 1.87
.92 1.90
.96 1.94
.99 1.98
>.03 2.01
>.05 2.04
>.09 2.06
                                                      D-134
                                                                                                        March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
    Table 19-11. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Observations (40 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20 25 30 35
3.84 2.41 1.99 1.79 1.68 1.55 1.47 .42 1.39 1.36
4.91 2.87 2.31 2.05 1.90 1.74 1.65 .59 1.55 1.52
5.65 3.16 2.50 2.20 2.04 1.85 1.76 .68 1.64 1.61
6.24 3.37 2,64 2.32 2.13 1.94 1.83 .75 1.70 1.67
6.73 3.55 2.75 2.40 2.21 2.00 1.89 .80 1.75 1.72
7.90 3.94 3.00 '2-59 2.37 2.13 2.00 .91 1.85 1.81
9.06 4.31 3.22 2.76 2, SI 2.25 2.11 2.01 1.94 1.90
9.98 4.59 3.39 2.89 2,£1 2.33 2.18 2.07 2.00 1.96
10.76 4.82 3.52 2.98 2.69 2.39 2.24 2.12 2.05 2.00
12.33 5.25 3.77 3.17 2.84 2,51 2.34 2.21 2.14 2.08 ;
13.59 5.59 3.95 3.30 2.95 2,59 2.41 2.28 2.20 2.14 ;
14.65 5.85 4.10 3.40 3.03 2,66 2,47 2.33 2.24 2.19 ;
15.55 6.08 4.22 3.49 3.11 2,71 2,51 2.37 2.28 2.22 ;
16.80 6.37 4.38 3.60 3.19 2.78 2,57 2.42 2.33 2.27 ;
18.44 6.76 4.58 3.74 3.31 2.87 2.64 2,49 2.39 2.32 ;
19.84 7.09 4.75 3.86 3.40 2.93 .2, 70 2,53 2.44 2.37 ;
21.09 7.34 4.88 3.95 3.47 2.99 2.74 2,58 2,47 2.40 ;
22.19 7.58 5.00 4.03 3.54 3.04 2,79 •. 2,61 ' • 2.5O. 2.43 ;
23.28 7.81 5.12 4.10 3.59 3.08 2.82 2.64 2.53 2.46 ;
40 45 50
.34 1.33 1.32
.50 1.48 1.47
.58 1.57 1.55
.64 1.62 1.61
.69 1.67 1.65
.79 1.76 1.75
.87 1.84 1.82
.92 1.90 1.88
.97 1.94 1.92
>.05 2.02 1.99
MO 2.07 2.05 ;
>.14 2.11 2.09 ;
>.18 2.15 2.12 ;
1.22 2.19 2.16 ;
>.28 2.24 2.21 ;
1.32 2.28 2.25 ;
2.35 2.31 2.29 ;
>.38 2.34 2.31 ;
>.41 2.36 2.33 ;
60 70
.30 1.29
.45 1.44
.53 1.52
.59 1.57
.63 1.61
.72 1.70
.80 1.78
.85 1.83
.89 1.87
.96 1.94
>.01 1.99
>.05 2.03 ;
>.08 2.06 ;
1.12 2.09 ;
>.17 2.14 ;
1.21 2.18 ;
>.24 2.21 ;
1.27 2.24 ;
1.29 2.26 ;
80 90
.28 1.28
.43 1.42
.51 1.50
.56 1.55
.60 1.59
.69 1.68
.76 1.75
.81 1.80
.85 1.84
.92 1.91
.97 1.96
>.01 1.99
>.04 2.02 ;
1.07 2.06 ;
1.12 2.10 ;
>.16 2.14 ;
>.19 2.17 ;
1.21 2.19 ;
1.24 2.22 ;
100
.27
.41
.49
.54
.59
.67
.74
.79
.83
.90
.95
.98
>.01
>.04 ;
>.09 ;
>.i2 ;
>.ie ;
>.is ;
1.20 ;
125 150
.26 1.26
.40 1.39
.48 1.47
.53 1.52
.57 1.56
.66 1.65
.73 1.72
.78 1.76
.81 1.80
.88 1.87
.93 1.91
.96 1.95
.99 1.97
1.02 2.00
>.06 2.04
>.10 2.08
>.13 2.12
>.16 2.14
>.18 2.16
      Table 19-11. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Observations (40 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.74
6.02
6.92
7.63
8.23
9.65
11.06
12.19
13.13
15.04
16.56
17.85
18.98
20.47
22.50
24.22
25.78
27.19
28.44
6
2. 83
3.34
3.66
3.91
4.10
4.55
4.96
5.28
5.53
6.03
6.40
6.71
6.96
7.29
7.73
8.11
8.40
8.67
8.91
8
2.29
2.63
'2,84' '
2.99
3.11
3.38
3.62
3.80
3.94
4.21
4.41
4.57
4.70
4.87
5.10
5.27
5.43
5.55
5.66
10
2.04
2.31
2.47
2.59
2.68
2.88
3.06
3.19
3.29
3.48
3.63
3.74
3.83
3.95
4.10
4.22
4.32
4.40
4.47
12
1.90
2.13
2.27
2.37
2.44
2.61
2.76
. 2,86
2.95
3.10
3.22
3.30
3.38
3.47
3.58
3.68
3.76
3.83
3.89
16
1.74
1.93
2.05
2.13
2.19
2.33
2.44
2.53
2.59
2.71
2.80
2.86
2.92
2,99
3,08
3.14
3.20
3.25
3.29
20
1.65
1.83
1.93
2.00
2.06
2.18
2.28
2.35
2.41
2.51
2.59
2.64
2.69
2.75
2,82
2,88
2,92
2.97
3.OO
25
.59
.75
.85
.91
.96
2.07
2.16
2.23
2.28
2.37
2.44
2.49
2.53
2.58
2.64
2.69
2.73
2.77 •
2.8O
30
1.55
1.70
1.79
1.85
1.90
2.00
2.09
2.15
2.20
2.28
2.34
2.39
2.43
2.47
2.53
2.58
2.62
2.65
2.68
35 40 45 50 60 70 80 90 100 125
1.52 .50 1.48 1.47 .45 1.44 .43 1.42 .41 .40
1.67 .64 1.62 1.61 .59 1.57 .56 1.55 .54 .53
1.75 .73 1.71 1.69 .67 1.65 .64 1.63 .62 .61
1.81 .78 1.76 1.75 .72 1.70 .69 1.68 .67 .66
1.86 .83 1.81 1.79 .76 1.74 .73 1.72 .71 .69
1.96 .92 1.90 1.88 .85 1.83 .81 1.80 .79 .78
2.04 2.00 1.98 1.95 .92 1.90 .88 1.87 .86 .84
2.10 2.06 2.03 2.01 .97 1.95 .93 1.92 .91 .89
2.14 2.10 2.07 2.05 2.01 1.99 .97 1.96 .95 .93
2.22 2.18 2.15 2.12 2.08 2.06 2.04 2.02 2.01 .99
2.28 2.23 2.20 2.17 2.13 2.10 2.08 2.07 2.05 2.03
2.32 2.28 2.24 2.21 2.17 2.14 2.12 2.10 2.09 2.06
2.36 2.31 2.27 2.24 2.20 2.17 2.15 2.13 2.12 2.10
2.40 2.35 2.31 2.28 2.24 2.21 2.19 2.17 2.16 2.13
2.46 2.40 2.37 2.33 2.29 2.26 2.23 2.22 2.20 2.18
2.50 2.45 2.41 2.37 2.33 2.29 2.27 2.25 2.24 2.21
2.54 2.48 2.44 2.40 2.36 2.32 2.30 2.28 2.26 2.24
2.57 2.51 2.47 2.43 2.38 2.35 2.32 2.31 2.29 2.26
2.59 2.53 2.49 2.46 2.41 2.37 2.35 2.33 2.31 2.28
150
1.39
1.52
1.60
1.65
1.68
1.76
1.83
1.88
1.91
1.98
2.01
2.04
2.08
2.12
2.16
2.19
2.22
2.24
2.26
                                                   D-135
                                                                                                  March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
       Table 19-12. K-Multipliers  for 1-of-4 Intrawell Prediction Limits on Observations (1 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6
0.06 -0.01
0.39 0.28
0.59 0.44
0.74 0.55
0.85 0.63
1.11 0.82
1.36 0.98
1.54 .10
1.70 .19
2.01 .37
2.25 .49
2.45 .60
2.62 .68
2.85 .79
3.16 .94
3.42 2, OS
3.65 2,15
3.85 2.24
4.04 2.31
8 10 12
-0.04 -0.07 -0.08
0.22 0.19 0.17
0.37 0.33 0.30
0.47 0.42 0.39
0.54 0.49 0.46
0.70 0.64 0.60
0.84 0.76 0.72
0.94 0.85 0.80
1.01 0.92 0.86
1.15
1.25
1.33
1.39
1.47
1.58
1.66
1.73
1.79
1.84
.04 0.98
.13 .06
.20 .12
.25 .17
.32 .23
.41 .31
.48 .37
.54 .42
.58 .46
.63 .50
16
20
-0.10 -0.11
0.14 0.12
0.27 0.25
0.36 0.33
0.42 0.40
0.55 0.52
0.66 0.63
0.74 0.70
0.80 0.76
0.90 0.86
0.97 0.92
1.03 0.98
1.07
1.13
1.20
1.25
1.29
1.33
1.36
.02
.07
.13
.18
.22
.26
.29
25 30 35 40 45 50 60 70
-0.12 -0.13 -0.13 -0.13 -0.14 -0.14 -0.14 -0.14
0.11 0.10 0.10 0.09 0.09 0.09 0.08 0.08
0.23 0.22 0.22 0.21 0.21 0.20 0.20 0.20
0.32 0.31 0.30 0.29 0.29 0.28 0.28 0.27
0.38 0.37 0.36 0.35 0.35 0.34 0.34 0.33
0.50 0.49 0.48 0.47 0.46 0.46 0.45 0.45
0.60 0.59 0.58 0.57 0.56 0.56 0.55 0.54
0.67 0.66 0.64 0.63 0.63 0.62 0.61 0.61
0.73 0.71 0.69 0.68 0.68 0.67 0.66 0.65
0.82 0.80 0.78 0.77 0.76 0.76 0.75 0.74
0.89 0.86 0.85 0.83 0.82 0.82 0.81 0.80
0.94 0.91 0.89 0.88 0.87 0.86 0.85 0.84
0.98 0.95 0.93 0.92 0.91 0.90 0.89 0.88
.02 1.00 0.98 0.96 0.95 0.94 0.93 0.92
.09 1.06 1.03 .02 1.01 .00 0.98 0.97
.13 1.10 1.08 .06 1.05 .04 1.02 1.01
.17 1.14 1.11 .10 1.08 .07 1.06 1.05
.20 1.17 1.14 .13 1.11 .10 1.09 1.07
.23 1.20 1.17 .15 1.14 .13 1.11 1.10
80
-0.15
0.08
0.19
0.27
0.33
0.44
0.54
0.60
0.65
0.73
0.79
0.84
0.87
0.91
0.97
1.01
1.04
1.07
1.09
90
-0.15
0.07
0.19
0.27
0.33
0.44
0.53
0.60
0.65
0.73
0.79
0.83
0.87
0.91
0.96
1.00
1.03
1.06
1.08
100
-0.15
0.07
0.19
0.27
0.32
0.44
0.53
0.59
0.64
0.73
0.78
0.83
0.86
0.90
0.96
1.00
1.03
1.05
1.08
125
-0.15
0.07
0.19
0.26
0.32
0.43
0.53
0.59
0.64
0.72
0.78
0.82
0.85
0.90
0.95
0.99
1.02
1.04
1.07
150
-0.15
0.07
0.18
0.26
0.32
0.43
0.52
0.59
0.63
0.72
0.77
0.81
0.85
0.89
0.94
0.98
1.01
1.04
1.06
     Table 19-12. K-Multipliers for 1-of-4 Intrawell Prediction Limits on Observations (1 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
468
0.36 0.25 0.21 (
0.70 0.53 0.46 (
0.91 0.69 0.60 (
1.07 0.80 0.69 (
1.21 0.89 0.77 (
1.50 1.08 0.93 (
1.79 .26 1.07 (
2.01 .38 1.17
2.20 .48 1.24
2.56 .67 1.39
2.85 .81 1.49
3.10 .92 1.57
3.31 2.02 1.64
3.58 2.14 1.73
3.96 2.3O 1.84
4.28 2,43 1.93
4.57 2.54 2.00
4.82 2.63 2.06
5.04 2.72 2.12
10 12 16
).18 0.16 0.13 (
).41 0.39 0.35 (
).54 0.51 0.47 (
).63 0.60 0.55 (
).70 0.66 0.61 (
).85 0.80 0.74 (
).97 0.91 0.84 (
.06 0.99 0.91 (
.13 1.05 0.97 (
.25 1.17 1.07
.34 1.25 1.14
.41 1.31 1.19
.46 1.36 1.24
.53 1.42 1.29
.62 1.50 1.36
.70 1.56 1.42
.75 1.61 1.46
.80 1.66 1.50
.85 1.70 1.53
20
).12 (
).33 (
).44 (
).52 (
).58 (
).70 (
).80 (
).87 (
).92 (
.02 (
.08
.13
.17
.22
.29
.34
.38
.41
.44
25 30
).ll 0.10 (
).31 0.30 (
).43 0.41 (
).50 0.49 (
).56 0.54 (
).67 0.66 (
).77 0.75 (
).84 0.81 (
).89 0.86 (
).98 0.95 (
.04 1.01 (
.09 1.06
.12 1.09
.17 1.14
.23 1.19
.28 1.24
.31 1.27
.35 1.30
.37 1.33
35
).09 (
).30 (
).40 (
).48 (
).53 (
).64 (
).73 (
).80 (
).85 (
).93 (
).99 (
.03
.07
.11
.17
.21
.25
.28
.30
40 45
).09 0.09 (
).29 0.29 (
).40 0.39 (
).47 0.46 (
).52 0.52 (
).63 0.63 (
).72 0.72 (
).79 0.78 (
).83 0.82 (
).92 0.91 (
).97 0.96 (
.02 1.01
.05 1.04
.10 1.08
.15 1.14
.19 1.18
.23 1.21
.25 1.24
.28 1.26
50 60 70
).08 0.08 0.08 (
).28 0.28 0.27 (
).39 0.38 0.38 (
).46 0.45 0.45 (
).51 0.50 0.50 (
).62 0.61 0.61 (
).71 0.70 0.69 (
).77 0.76 0.75 (
).82 0.81 0.80 (
).90 0.89 0.88 (
).95 0.94 0.93 (
.00 0.98 0.97 (
.03 1.02 1.01
.07 1.06 1.05
.13 1.11 1.10
.17 1.15 1.14
.20 1.18 1.17
.23 1.21 1.19
.25 1.23 1.22
80 90 100 125 150
).08 0.07 0.07 0.07 0.07
).27 0.27 0.27 0.26 0.26
).37 0.37 0.37 0.36 0.36
).44 0.44 0.44 0.43 0.43
).50 0.49 0.49 0.49 0.48
).60 0.60 0.59 0.59 0.59
).69 0.68 0.68 0.67 0.67
).75 0.74 0.74 0.73 0.73
).79 0.79 0.78 0.78 0.77
).87 0.86 0.86 0.85 0.85
).92 0.92 0.91 0.91 0.90
).97 0.96 0.96 0.95 0.94
.00 0.99 0.99 0.98 0.97
.04 1.03 .03 1.02 .01
.09 1.08 .08 1.07 .06
.13 1.12 .11 1.10 .10
.16 1.15 .15 1.14 .13
.18 1.18 .17 1.16 .15
.21 1.20 .19 1.18 .17
                                                   D-136
                                                                                                  March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
      Table 19-12. K-Multipliers for 1-of-4 Intrawell Prediction Limits on Observations (1  COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
0.64
.01
.25
.44
.59
.93
2.27
2.53
2.75
3.20
3.54
3.83
4.09
4.42
4.89
5.27
5.62
5.92
6.19
Table
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
0.39
0.74
0.95
1.11
1.24
1.54
1.83
2, OB
2.25
2.62
2.91
3.16
3.37
3.65
4.04
4.37
4.65
4.91
5.14
6
0.50
0.78
0.94
1.06
1.15
1.36
1.54
1.68
1.79
2.00
2.15
2.27
2.38
2.51
2.69
2.84
2.96
3.06
3.16
19-1
6
0.28
0.55
0.70
0.82
0.91
1.10
.27
.39
.49
.68
.82
.94
2.Q3
2.15
2.31
2.44
2.55
2.65
2.73
8
0.44
0.68
0.82
0.92
0.99
1.16
1.30
1.40
1.48
1.63
1.74
1.83
1.90
1.99
2.11
2.20
2.28
2.35
2.41
10
0.40
0.62
0.75
0.84
0.91
.05
.17
.26
.33
.46
.55
.62
.68
.75
.84
.92
.98
2.03
2.08
12
0.37
0.59
0.71
0.79
0.85
0.99
1.10
1.18
1.24
1.35
1.43
1.50
1.55
1.61
1.69
1.76
1.81
1.86
1.89
2. K-Multipliers
8
0.22
0.47
0.61
0.70
0.78
0.94
1.07
1.17
1.25
1.39
1.50
1.58
1.65
1.73
1.84
1.93
2.O1
2.O7
2,12
10
0.19
0.42
0.55
0.64
0.71
0.85
0.97
.06
.13
.25
.34
.41
.47
.54
.63
.70
.76
.81
.85
12
0.17
0.39
0.51
0.60
0.67
0.80
0.91
0.99
1.06
1.17
1.25
1.31
1.36
1.42
1.50
1.56
1.62
1.66
1.70
16
0.34
0.54
0.65
0.73
0.79
0.91
1.01
1.08
1.14
1.24
1.31
1.36
1.40
1.46
1.53
1.58
1.63
1.66
1.70
for
16
0.14
0.36
0.47
0.55
0.61
0.74
0.84
0.92
0.97
1.07
1.14
1.20
1.24
1.29
1.36
1.42
1.46
1.50
1.53
20
0.32
0.52
0.62
0.70
0.75
0.87
0.96
.03
.08
.17
.24
.29
.33
.38
.44
.49
.53
.56
.59
1-of-4
20
0.12
0.33
0.45
0.52
0.58
0.70
0.80
0.87
0.92
.02
.08
.13
.17
.22
.29
.34
.38
.41
.44
25 30 35 40 45 50 60 70 80
0.31 0.30 0.29 0.29 0.28 0.28 0.27 0.27 0.27
0.50 0.48 0.47 0.47 0.46 0.46 0.45 0.45 0.44
0.60 0.58 0.57 0.57 0.56 0.55 0.55 0.54 0.54
0.67 0.65 0.64 0.63 0.63 0.62 0.61 0.61 0.60
0.72 0.71 0.69 0.68 0.68 0.67 0.66 0.65 0.65
0.83 0.81 0.80 0.79 0.78 0.77 0.76 0.75 0.75
0.93 0.90 0.88 0.87 0.86 0.85 0.84 0.83 0.83
0.99 0.96 0.94 0.93 0.92 0.91 0.90 0.89 0.88
.04 1.01 0.99 0.97 0.96 0.95 0.94 0.93 0.92
.12 1.09 .07 .05 1.04 .03 1.02 1.01 .00
.18 1.15 .13 .11 1.10 .08 1.07 1.06 .05
.23 1.19 .17 .15 1.14 .13 1.11 1.10 .09
.27 1.23 .20 .19 1.17 .16 1.14 1.13 .12
.31 1.27 .25 .23 1.21 .20 1.18 1.17 .16
.37 1.33 .30 .28 1.26 .25 1.23 1.22 .21
.42 1.37 .34 .32 1.30 .29 1.27 1.25 .24
.45 1.41 .38 .35 1.33 .32 1.30 1.28 .27
.48 1.44 .40 .38 1.36 .35 1.32 1.31 .30
.51 1.46 .43 .40 1.38 .37 1.35 1.33 .32
Intrawell Prediction Limits on Observations (2
25 30 35 40 45 50 60 70 80
0.11 0.10 0.10 0.09 0.09 0.09 0.08 0.08 0.08
0.32 0.31 0.30 0.29 0.29 0.28 0.28 0.27 0.27
0.43 0.41 0.41 0.40 0.39 0.39 0.38 0.38 0.37
0.50 0.49 0.48 0.47 0.46 0.46 0.45 0.45 0.44
0.56 0.54 0.53 0.52 0.52 0.51 0.51 0.50 0.50
0.67 0.66 0.64 0.63 0.63 0.62 0.61 0.61 0.60
0.77 0.75 0.74 0.72 0.72 0.71 0.70 0.69 0.69
0.84 0.81 0.80 0.79 0.78 0.77 0.76 0.75 0.75
0.89 0.86 0.85 0.83 0.82 0.82 0.81 0.80 0.79
0.98 0.95 0.93 0.92 0.91 0.90 0.89 0.88 0.87
.04 1.01 0.99 0.97 0.96 0.95 0.94 0.93 0.92
.09 1.06 .03 .02 1.01 .00 0.98 0.97 0.97
.12 1.09 .07 .05 1.04 .03 1.02 1.01 .00
.17 1.14 .11 .10 1.08 .07 1.06 1.05 .04
.23 1.19 .17 .15 1.14 .13 1.11 1.10 .09
.28 1.24 .21 .19 1.18 .17 1.15 1.14 .13
.31 1.27 .25 .23 1.21 .20 1.18 1.17 .16
.35 1.30 .28 .25 1.24 .23 1.21 1.19 .18
.37 1.33 .30 .28 1.26 .25 1.23 1.22 .21
90
0.27
0.44
0.53
0.60
0.64
0.74
0.82
0.88
0.92
0.99
1.04
1.08
1.11
1.15
1.20
1.24
1.26
1.29
1.31
COC
90
0.07
0.27
0.37
0.44
0.49
0.60
0.68
0.74
0.79
0.87
0.92
0.96
0.99
1.03
1.08
1.12
1.15
1.18
1.20
100
0.26
0.44
0.53
0.59
0.64
0.74
0.82
0.87
0.91
0.99
.04
.08
.11
.15
.19
.23
.26
.28
.30
125
0.26
0.43
0.53
0.59
0.64
0.73
0.81
0.87
0.91
0.98
1.03
1.07
1.10
1.14
1.18
1.22
1.25
1.27
1.29
150
0.26
0.43
0.52
0.59
0.63
0.73
0.81
0.86
0.90
0.97
.02
.06
.09
.13
.17
.21
.24
.26
.28
, Annual)
100
0.07
0.27
0.37
0.44
0.49
0.59
0.68
0.74
0.78
0.86
0.91
0.96
0.99
.03
.08
.11
.15
.17
.19
125
0.07
0.26
0.37
0.43
0.49
0.59
0.67
0.73
0.78
0.85
0.91
0.95
0.98
1.02
1.07
1.11
1.14
1.16
1.18
150
0.07
0.26
0.36
0.43
0.48
0.59
0.67
0.73
0.77
0.85
0.90
0.94
0.97
.01
.06
.10
.13
.15
.17
                                                    D-137
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
     Table 19-12.  K-Multipliers for 1-of-4  Intrawell Prediction Limits on Observations (2 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
468
0.70 0.53 0.46 (
1.07 0.80 0.69 (
1.32 0.97 0.83 (
1.50 1.08 0.93 (
1.66 1.18 1.01 (
2.01 1.38 1.17
2.36 1.57 1.31
2.63 1.70 1.41
2.85 1.81 1.49
3.31 2.02 1.64
3.66 2.17 1.75
3.96 "'2.3O' 1.84
4.22 £.41 1.91
4.56 2.54 2.00
5.04 2.72 2.12
5.45 2.86 2.21
5.79 2.99 2,29
6.11 3.09 2,.36 :
6.39 3.19 2.42 '<
10 12 16
).41 0.39 0.35 (
).63 0.60 0.55 (
).76 0.71 0.66 (
).85 0.80 0.74 (
).92 0.86 0.79 (
.06 0.99 0.91 (
.18 1.10 1.01 (
.27 1.18 1.09
.34 1.25 1.14
.46 1.36 1.24
.55 1.44 1.31
.62 1.50 1.36
.68 1.55 1.41
.75 1.61 1.46
.85 1.70 1.53
.92 1.76 1.58
.98 1.81 1.63
>.04 1.86 1.66
>.08 1.90 1.70
20
).33 (
).52 (
).63 (
).70 (
).76 (
).87 (
).96 (
.03 (
.08
.17
.24
.29
.33
.38
.44
.49
.53
.56
.59
25 30
).31 0.30 (
).50 0.49 (
).60 0.59 (
).67 0.66 (
).73 0.71 (
).84 0.81 (
).93 0.90 (
).99 0.96 (
.04 1.01 (
.12 1.09
.18 1.15
.23 1.19
.27 1.23
.31 1.27
.37 1.33
.42 1.37
.45 1.41
.48 1.44
.51 1.46
35
).30 (
).48 (
).58 (
).64 (
).69 (
).80 (
).88 (
).94 (
).99 (
.07
.13
.17
.20
.25
.30
.34
.38
.40
.43
40 45
).29 0.29 (
).47 0.46 (
).57 0.56 (
).63 0.63 (
).68 0.68 (
).79 0.78 (
).87 0.86 (
).93 0.92 (
).97 0.96 (
.05 1.04
.11 1.10
.15 1.14
.19 1.17
.23 1.21
.28 1.26
.32 1.30
.35 1.33
.38 1.36
.40 1.38
50 60 70
).28 0.28 0.27 (
).46 0.45 0.45 (
).55 0.55 0.54 (
).62 0.61 0.61 (
).67 0.66 0.65 (
).77 0.76 0.75 (
).85 0.84 0.83 (
).91 0.90 0.89 (
).95 0.94 0.93 (
.03 1.02 1.01
.09 1.07 1.06
.13 1.11 1.10
.16 1.14 1.13
.20 1.18 1.17
.25 1.23 1.22
.29 1.27 1.25
.32 1.30 1.28
.35 1.32 1.31
.37 1.35 1.33
80 90
).27 0.27 (
).44 0.44 (
).54 0.53 (
).60 0.60 (
).65 0.65 (
).75 0.74 (
).83 0.82 (
).88 0.88 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.12 1.11
.16 1.15
.21 1.20
.24 1.24
.27 1.26
.30 1.29
.32 1.31
100 125
).27 0.26 (
).44 0.43 (
).53 0.53 (
).59 0.59 (
).64 0.64 (
).74 0.73 (
).82 0.81 (
).87 0.87 (
).91 0.91 (
).99 0.98 (
.04 1.03
.08 1.07
.11 1.10
.15 1.14
.19 1.18
.23 1.22
.26 1.25
.28 1.27
.30 1.29
150
).26
).43
).52
).59
).63
).73
).81
).86
).90
).97
.02
.06
.09
.13
.17
.21
.24
.26
.28
      Table 19-12. K-Multipliers for 1-of-4 Intrawell Prediction Limits on Observations (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.01 0.78 0.68 (
1.44 1.06 0.92 (
1.72 1.23 1.06 (
1.93 1.36 1.16
2.11 1.46 1.23
2.53 1.68 1.40
2.95 1.88 1.55
3.27 2.03 1.66
3.54 2.15 1.74
4.09 2.38 1.90
4.52 2.55 2.02
4.89 2.69 2.11
5.20 2.81 2.19
5.62 2.96 2.28
6.20 3.16 2.41 ;
6.69 3.32 2.51 ;
7.11 3.46 2.59 ;
7.49 3.58 2,67 ;
7.84 3.69 2.73 \
).62 0.59 0.54 (
).84 0.79 0.73 (
).96 0.91 0.84 (
.05 0.99 0.91 (
.12 1.05 0.97 (
.26 1.18 1.08
.39 1.29 1.18
.48 1.37 1.25
.55 1.43 1.31
.68 1.55 1.40
.77 1.63 1.47
.84 1.69 1.53
.90 1.75 1.57
.98 1.81 1.63
>.08 1.89 1.69
>.16 1.96 1.75
1.22 2.01 1.79
>.28 2.06 1.83
1.32 2.10 1.86
).52 (
).70 (
).80 (
).87 (
).92 (
.03 (
.12
.19
.24
.33
.39
.44
.48
.53
.59
.64
.68
.71
.74
).50 0.48 (
).67 0.65 (
).77 0.75 (
).83 0.81 (
).88 0.86 (
).99 0.96 (
.08 1.05
.14 1.11
.18 1.15
.27 1.23
.33 1.29
.37 1.33
.41 1.36
.45 1.41
.51 1.46
.56 1.50
.59 1.54
.62 1.57
.65 1.59
).47 (
).64 (
).73 (
).80 (
).84 (
).94 (
.03
.08
.13
.20
.26
.30
.33
.38
.43
.47
.50
.53
.55
).47 0.46 (
).63 0.63 (
).72 0.72 (
).79 0.78 (
).83 0.82 (
).93 0.92 (
.01 1.00 (
.07 1.05
.11 1.10
.19 1.17
.24 1.22
.28 1.26
.31 1.29
.35 1.33
.40 1.38
.44 1.42
.47 1.45
.50 1.48
.52 1.50
).46 0.45 0.45 (
).62 0.61 0.61 (
).71 0.70 0.69 (
).77 0.76 0.75 (
).82 0.80 0.80 (
).91 0.90 0.89 (
).99 0.98 0.97 (
.04 1.03 1.02
.08 1.07 1.06
.16 1.14 1.13
.21 1.19 1.18
.25 1.23 1.22
.28 1.26 1.25
.32 1.30 1.28
.37 1.35 1.33
.41 1.38 1.37
.44 1.41 1.40
.46 1.44 1.42
.48 1.46 1.44
).44 0.44 (
).60 0.60 (
).69 0.68 (
).75 0.74 (
).79 0.79 (
).88 0.88 (
).96 0.95 (
.01 1.00
.05 1.04
.12 1.11
.17 1.16
.21 1.20
.24 1.23
.27 1.26
.32 1.31
.35 1.35
.38 1.37
.41 1.40
.43 1.42
).44 0.43 (
).59 0.59 (
).68 0.67 (
).74 0.73 (
).78 0.78 (
).87 0.87 (
).95 0.94 (
.00 0.99 (
.04 1.03
.11 1.10
.16 1.15
.19 1.18
.22 1.21
.26 1.25
.30 1.29
.34 1.33
.37 1.35
.39 1.38
.41 1.40
).43
).59
).67
).73
).77
).86
).93
).98
.02
.09
.14
.17
.20
.24
.28
.32
.34
.37
.39
                                                    D-138
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
       Table 19-12. K-Multipliers for 1-of-4 Intrawell Prediction Limits on Observations (5 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
468
0.85 0.63 0.54 (
1.24 0.91 0.78 (
1.50 1.07 0.91 (
1.70 1.19 1.01 (
1.86 1.29 1.09 (
2.25 1.49 1.25
2.62 1.68 1.39
2.91 1.82 1.50
3.16 1.94 1.58
3.65 2,15 1.73
4.04 2.31 1.84
4.37 2.44 1.93
4.65 2.55 2.O1
5.02 2.69 2. TO
5.55 2.88 2.22
5.99 3.03 2.31
6.37 3.16 2.40 ,
6.71 3.27 2.46
7.03 3.37 2.52
10 12 16
).49 0.46 0.42 (
).71 0.67 0.61 (
).83 0.78 0.72 (
).92 0.86 0.80 (
).99 0.92 0.85 (
.13 1.06 0.97 (
.25 1.17 1.07
.34 1.25 1.14
.41 1.31 1.20
.54 1.42 1.29
.63 1.50 1.36
.70 1.56 1.42
.76 1.62 1.46
.83 1.68 1.51
.92 1.76 1.58
Z.QO 1.83 1.64
2,06 1.88 1.68
2.12 1.92 1.72
2, 16 1,96 1.75
20
).40 (
).58 (
).69 (
).76 (
).81 (
).92 (
.02 (
.08
.13
.22
.29
.34
.38
.43
.49
.54
.58
.61
.64
25 30
).38 0.37 (
).56 0.54 (
).66 0.64 (
).73 0.71 (
).78 0.76 (
).89 0.86 (
).98 0.95 (
.04 1.01 (
.09 1.06
.17 1.14
.23 1.19
.28 1.24
.31 1.27
.36 1.32
.42 1.37
.46 1.42
.50 1.45
.53 1.48
.56 1.50
35
).36 (
).53 (
).63 (
).69 (
).74 (
).85 (
).93 (
).99 (
.03
.11
.17
.21
.25
.29
.34
.38
.42
.44
.47
40 45
).35 0.35 (
).52 0.52 (
).62 0.61 (
).68 0.68 (
).73 0.73 (
).83 0.82 (
).92 0.91 (
).97 0.96 (
.02 1.01
.10 1.08
.15 1.14
.19 1.18
.23 1.21
.27 1.25
.32 1.30
.36 1.34
.39 1.37
.42 1.40
.44 1.42
50 60 70
).34 0.34 0.33 (
).51 0.51 0.50 (
).61 0.60 0.59 (
).67 0.66 0.65 (
).72 0.71 0.70 (
).82 0.81 0.80 (
).90 0.89 0.88 (
).95 0.94 0.93 (
.00 0.98 0.97 (
.07 1.06 1.05
.13 1.11 1.10
.17 1.15 1.14
.20 1.18 1.17
.24 1.22 1.21
.29 1.27 1.25
.33 1.31 1.29
.36 1.34 1.32
.38 1.36 1.35
.41 1.38 1.37
80 90
).33 0.33 (
).50 0.49 (
).59 0.58 (
).65 0.65 (
).70 0.69 (
).79 0.79 (
).87 0.87 (
).92 0.92 (
).97 0.96 (
.04 1.03
.09 1.08
.13 1.12
.16 1.15
.20 1.19
.24 1.24
.28 1.27
.31 1.30
.33 1.32
.35 1.35
100 125
).32 0.32 (
).49 0.49 (
).58 0.58 (
).64 0.64 (
).69 0.68 (
).78 0.78 (
).86 0.85 (
).91 0.91 (
).96 0.95 (
.03 1.02
.08 1.07
.11 1.11
.15 1.14
.18 1.17
.23 1.22
.26 1.25
.29 1.28
.32 1.30
.34 1.33
150
).32
).48
).57
).63
).68
).77
).85
).90
).94
.01
.06
.10
.13
.16
.21
.25
.27
.30
.32
     Table 19-12.  K-Multipliers for 1-of-4 Intrawell Prediction Limits on Observations (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.21 0.89 0.77 (
1.66 1.18 1.01 (
1.96 1.35 1.14
2.20 1.48 1.24
2.39 1.58 1.32
2.85 1.81 1.49
3.31 2.02 1.64
3.66 2.17 1.75
3.96 2,3O 1.84
4.56 2.54 2.00
5.04 2.72 2.12
5.45 2.86 2.21
5.79 2.99 2,29
6.25 3.14 2,39 '*
6.90 3.35 2.52 ;
7.44 3.52 2.62 ;
7.91 3.67 2.71
8.33 3.79 2.78
8.73 3.90 2.85 ,
).70 0.66 0.61 (
).92 0.86 0.79 (
.04 0.97 0.90 (
.13 1.05 0.97 (
.19 1.11 1.02 (
.34 1.25 1.14
.46 1.36 1.24
.55 1.44 1.31
.62 1.50 1.36
.75 1.61 1.46
.85 1.70 1.53
.92 1.76 1.58
.98 1.81 1.63
>.06 1.88 1.68
>.16 1.96 1.75
1.24 2.03 1.80
2,30 2.08 1.85
2.36 2.13 1.89
2.41 2.17 1.92
).58 (
).76 (
).85 (
).92 (
).97 (
.08
.17
.24
.29
.38
.44
.49
.53
.58
.64
.69
.72
.76
.79
).56 0.54 (
).73 0.71 (
).82 0.80 (
).89 0.86 (
).94 0.91 (
.04 1.01 (
.12 1.09
.18 1.15
.23 1.19
.31 1.27
.37 1.33
.42 1.37
.45 1.41
.50 1.45
.56 1.50
.60 1.55
.64 1.58
.67 1.61
.69 1.63
).53 (
).69 (
).78 (
).85 (
).89 (
).99 (
.07
.13
.17
.25
.30
.34
.38
.42
.47
.51
.54
.57
.59
).52 0.52 (
).68 0.68 (
).77 0.76 (
).83 0.82 (
).88 0.87 (
).97 0.96 (
.05 1.04
.11 1.10
.15 1.14
.23 1.21
.28 1.26
.32 1.30
.35 1.33
.39 1.37
.44 1.42
.48 1.46
.51 1.49
.54 1.52
.56 1.54
).51 0.50 0.50 (
).67 0.66 0.65 (
).76 0.75 0.74 (
).82 0.81 0.80 (
).86 0.85 0.84 (
).95 0.94 0.93 (
.03 1.02 1.01
.09 1.07 1.06
.13 1.11 1.10
.20 1.18 1.17
.25 1.23 1.22
.29 1.27 1.25
.32 1.30 1.28
.36 1.34 1.32
.41 1.38 1.37
.44 1.42 1.40
.47 1.45 1.43
.50 1.47 1.45
.52 1.49 1.48
).50 0.49 (
).65 0.65 (
).73 0.73 (
).79 0.79 (
).84 0.83 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.16 1.15
.21 1.20
.24 1.24
.27 1.26
.31 1.30
.35 1.35
.39 1.38
.42 1.41
.44 1.43
.46 1.45
).49 0.49 (
).64 0.64 (
).73 0.72 (
).78 0.78 (
).83 0.82 (
).91 0.91 (
).99 0.98 (
.04 1.03
.08 1.07
.15 1.14
.19 1.18
.23 1.22
.26 1.25
.29 1.28
.34 1.32
.37 1.36
.40 1.39
.42 1.41
.44 1.43
).48
).63
).72
).77
).81
).90
).97
.02
.06
.13
.17
.21
.24
.27
.32
.35
.38
.40
.42
                                                    D-139
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
      Table 19-12. K-Multipliers for 1-of-4  Intrawell Prediction Limits on Observations (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.59 .15 0.99 (
2.11 .46 1.23
2.47 .65 1.38
2.75 .79 1.48
2.99 .90 1.56
3.54 2.15 1.74
4.09 2.38 1.90
4.52 2.55 2.02
4.89 2.69 2.11
5.62 2.96 2.28
6.20 3.16 2.41 ;
6.69 3.32 2.51 ;
7.11 3.46 2,59 ;
7.68 3.64 2,70 ;
8.45 3.87 2.84 ;
9.11 4.06 2.95 ;
9.70 4.23 3.05
10.20 4.37 3.13
10.66 4.50 3.20
).91 (
.12
.24
.33
.40
.55
.68
.77
.84
.98
>.08
>.16
1.22 ;
>.30 ;
>.41 ;
>.49 ;
2,56 :
7.61 ;
2,67 :
).85 0.79 0.75 (
.05 0.97 0.92 (
.16 1.07 1.01 (
.24 1.14 1.08
.30 1.19 1.13
.43 1.31 1.24
.55 1.40 1.33
.63 1.47 1.39
.69 1.53 1.44
.81 1.63 1.53
.89 1.69 1.59
.96 1.75 1.64
>.01 1.79 1.68
>.08 1.85 1.72
>.17 1.92 1.79
2.24 1.97 1.83
1.29 2.02 1.88
1.34 2.05 1.91
>.38 2.09 1.94
).72 0.71 0.69 (
).88 0.86 0.84 (
).97 0.95 0.93 (
.04 1.01 0.99 (
.08 1.05 1.03
.18 1.15 1.13
.27 1.23 1.20
.33 1.29 1.26
.37 1.33 1.30
.45 1.41 1.38
.51 1.46 1.43
.56 1.50 1.47
.59 1.54 1.50
.64 1.58 1.54
.69 1.63 1.59
.73 1.67 1.63
.77 1.71 1.66
.80 1.73 1.69
.83 1.76 1.71
).68 0.68 0.67 (
).83 0.82 0.82 (
).92 0.91 0.90 (
).97 0.96 0.95 (
.02 1.01 1.00 (
.11 1.10 1.08
.19 1.17 1.16
.24 1.22 1.21
.28 1.26 1.25
.35 1.33 1.32
.40 1.38 1.37
.44 1.42 1.41
.47 1.45 1.44
.51 1.49 1.47
.56 1.54 1.52
.60 1.57 1.56
.63 1.60 1.59
.66 1.63 1.61
.68 1.65 1.63
).66 0.65 (
).80 0.80 (
).89 0.88 (
).94 0.93 (
).98 0.97 (
.07 1.06
.14 1.13
.19 1.18
.23 1.22
.30 1.28
.35 1.33
.38 1.37
.41 1.40
.45 1.43
.49 1.48
.53 1.51
.56 1.54
.58 1.56
.60 1.58
).65 0.64 (
).79 0.79 (
).87 0.86 (
).92 0.92 (
).97 0.96 (
.05 1.04
.12 1.11
.17 1.16
.21 1.20
.27 1.26
.32 1.31
.35 1.35
.38 1.37
.42 1.41
.46 1.45
.50 1.48
.52 1.51
.55 1.53
.57 1.55
).64 (
).78 (
).86 (
).91 (
).96 (
.04
.11
.16
.19
.26
.30
.34
.37
.40
.44
.48
.50
.53
.54
).64 0.63
).78 0.77
).85 0.85
).91 0.90
).95 0.94
.03 1.02
.10 1.09
.15 1.14
.18 1.17
.25 1.24
.29 1.28
.33 1.32
.35 1.34
.39 1.38
.43 1.42
.46 1.45
.49 1.48
.51 1.50
.53 1.52
       Table 19-12. K-Multipliers  for 1-of-4 Intrawell Prediction Limits on Observations (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1 1.24 0.91 0.78 (
2 1.70 1.19 1.01 (
3 2.01 1.37 1.15
4 2.25 1.49 1.25
5 2.45 1.60 1.33
8 2.91 1.82 1.50
12 3.37 2.O3 1.65
16 3.74 2.19 1.76
20 4.04 2.31 1.84
30 4.65 2.55 2.OO
40 5.14 2.73 2.12
50 5.55 2.88 2.22
60 5.90 3.00 2.30
75 6.37 3.16 2.39
100 7.03 3.37 2.52
125 7.59 3.54 2.63
150 8.06 3.68 2.71
175 8.50 3.81 2.79
200 8.88 3.93 2.86
).71 0.67 0.61 (
).92 0.86 0.80 (
.04 0.98 0.90 (
.13 1.06 0.97 (
.20 1.12 1.03 (
.34 1.25 1.14
.47 1.36 1.24
.56 1.44 1.31
.63 1.50 1.36
.76 1.62 1.46
.85 1.70 1.53
.92 1.76 1.58
1.99 1.81 1.63
2,06 1.88 1.68
2.10 1,38 1.75
>.24 2,03 1.80
>.31 2.08 1.85
2.36 2,13 1.89
>.41 . 2.17 1.92
).58 (
).76 (
).86 (
).92 (
).98 (
.08
.17
.24
.29
.38
.44
.49
.53
.58
.64
.69
.72
.76
.79
).56 0.54 (
).73 0.71 (
).82 0.80 (
).89 0.86 (
).94 0.91 (
.04 1.01 (
.12 1.09
.18 1.15
.23 1.19
.31 1.27
.37 1.33
.42 1.37
.45 1.41
.50 1.45
.56 1.50
.60 1.55
.64 1.58
.67 1.61
.69 1.63
).53 (
).69 (
).78 (
).85 (
).89 (
).99 (
.07
.13
.17
.25
.30
.34
.38
.42
.47
.51
.54
.57
.59
).52 0.52 (
).68 0.68 (
).77 0.76 (
).83 0.82 (
).88 0.87 (
).97 0.96 (
.05 1.04
.11 1.10
.15 1.14
.23 1.21
.28 1.26
.32 1.30
.35 1.33
.39 1.37
.44 1.42
.48 1.46
.51 1.49
.54 1.52
.56 1.54
).51 0.51 0.50 (
).67 0.66 0.65 (
).76 0.75 0.74 (
).82 0.81 0.80 (
).86 0.85 0.84 (
).95 0.94 0.93 (
.03 1.02 1.01
.09 1.07 1.06
.13 1.11 1.10
.20 1.18 1.17
.25 1.23 1.22
.29 1.27 1.25
.32 1.30 1.28
.36 1.34 1.32
.41 1.38 1.37
.44 1.42 1.40
.47 1.45 1.43
.50 1.47 1.45
.52 1.49 1.48
).50 0.49 (
).65 0.65 (
).73 0.73 (
).79 0.79 (
).84 0.83 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.16 1.15
.21 1.20
.24 1.24
.27 1.26
.31 1.30
.35 1.35
.39 1.38
.42 1.41
.44 1.43
.46 1.45
).49 0.49 (
).64 0.64 (
).73 0.72 (
).78 0.78 (
).83 0.82 (
).91 0.91 (
).99 0.98 (
.04 1.03
.08 1.07
.15 1.14
.19 1.18
.23 1.22
.26 1.25
.29 1.28
.34 1.32
.37 1.36
.40 1.38
.42 1.41
.44 1.43
).48
).63
).72
).77
).81
).90
).97
.02
.06
.13
.17
.21
.24
.27
.32
.35
.38
.40
.42
                                                    D-140
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                        Unified Guidance
    Table 19-12. K-Multipliers for 1-of-4 Intrawell Prediction Limits on Observations (10 COC, Semi-Annual)
w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.66 1.18
2.20 1.48
2.56 1.67
2.85 1.81
3.10 1.92
3.66 2.17
4.22 2.41 ]
4.67 2.58 ;
5.04 2.72 ;
5.79 2.99
6.39 3.19 ,
6.90 3.35 ;
7.34 3.49 ;
7.91 3.67 ;
8.72 3.90 ;
9.40 4.10 ;
9.99 4.26 :
10.55 4.41 :
11.02 4.54 :
.01 (
.24
.39
.49
.57
.75
L.91
>.03
>.12
2,29
2.42 ;
1.52 ;
i.Qo ;
>.71
2.85
2.96 \
3.05
3.13
3.21
).92 0.86 (
.13 1.05 (
.25 1.17
.34 1.25
.41 1.31
.55 1.44
.68 1.55
.78 1.63
.85 1.70
.98 1.81
>.08 1.90
>.16 1.96
1.22 2.02
2.3O 2.08
2,4.1 2.17
2.49 2.24
2.56 2.3O :
2.62 2.34 ;
2.67 2.39 ;
).79 (
).97 (
.07
.14
.19
.31
.41
.48
.53
.63
.70
.75
.79
.85
.92
.97
1.02
>.05
>.09
).76 0.73 0.71 (
).92 0.89 0.86 (
.02 0.98 0.95 (
.08 1.04 1.01 (
.13 1.09 1.06
.24 1.18 1.15
.33 1.27 1.23
.39 1.33 1.29
.44 1.37 1.33
.53 1.45 1.41
.59 1.51 1.46
.64 1.56 1.50
.68 1.59 1.54
.72 1.64 1.58
.79 1.69 1.63
.83 1.74 1.67
.88 1.77 1.71
.91 1.80 1.73
.94 1.83 1.76
).69 0.68 0.68 (
).85 0.83 0.82 (
).93 0.92 0.91 (
).99 0.97 0.96 (
.03 1.02 1.01
.13 1.11 1.10
.20 1.19 1.17
.26 1.24 1.22
.30 1.28 1.26
.38 1.35 1.33
.43 1.40 1.38
.47 1.44 1.42
.50 1.47 1.45
.54 1.51 1.49
.59 1.56 1.54
.63 1.60 1.57
.66 1.63 1.60
.69 1.66 1.63
.71 1.68 1.65
).67 0.66 (
).82 0.81 (
).90 0.89 (
).95 0.94 (
.00 0.98 (
.09 1.07
.16 1.14
.21 1.19
.25 1.23
.32 1.30
.37 1.35
.41 1.38
.44 1.41
.47 1.45
.52 1.49
.56 1.53
.59 1.56
.61 1.58
.63 1.60
).65 0.65 0.65 (
).80 0.79 0.79 (
).88 0.87 0.86 (
).93 0.92 0.92 (
).97 0.97 0.96 (
.06 1.05 1.04
.13 1.12 1.11
.18 1.17 1.16
.22 1.21 1.20
.28 1.27 1.26
.33 1.32 1.31
.37 1.35 1.35
.40 1.38 1.37
.43 1.42 1.41
.48 1.46 1.45
.51 1.50 1.48
.54 1.52 1.51
.56 1.55 1.53
.58 1.57 1.55
).64 0.64 (
).78 0.78 (
).86 0.85 (
).91 0.91 (
).96 0.95 (
.04 1.03
.11 1.10
.16 1.15
.19 1.18
.26 1.25
.30 1.29
.34 1.32
.37 1.35
.40 1.39
.44 1.43
.48 1.46
.50 1.49
.53 1.51
.55 1.53
).63
).77
).85
).90
).94
.02
.09
.14
.17
.24
.28
.32
.34
.38
.42
.45
.48
.50
.52
      Table 19-12. K-Multipliers  for 1-of-4 Intrawell Prediction Limits on Observations (10 COC, Quarterly)
w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.11 1.46 .23
2.75 1.79 .48
3.20 2.00 .63
3.54 2.15 .74
3.83 2.27 .83
4.52 2.55 2.02
5.20 2.81 2.19
5.74 3.00 2.31 ;
6.20 3.16 2.41 ;
7.11 3.46 2.59 '4
7.84 3.69 2.73 \
8.45 3.87 2.84 ;
8.99 4.03 2.93 ;
9.70 4.23 3.05
10.66 4.50 3.20 ,
11.51 4.72 3.32
12.25 4.91 3.42
12.89 5.07 3.51
13.48 5.21 3.59
.12 1.05 (
.33 1.24
.46 1.35
.55 1.43
.62 1.50
.77 1.63
.90 1.75
>.00 1.83
>.08 1.89
1.22 2.01
1.32 2.10
>.41 2.17
1.47 2.22
2.56 2.29 ;
2.67 2.38 ;
2.75 ' 2.45 ;
2.83 2.51 :
2.89 2. SB ;
2.94 2.61 :
).97 (
.14
.24
.31
.36
.47
.57
.64
.69
.79
.86
.92
.96
1.02
>.09
>.14
>.19 ;
2.23 ;
1.26 :
).92 0.88 0.86 (
.08 1.04 1.01 (
.17 1.12 1.09
.24 1.18 1.15
.29 1.23 1.19
.39 1.33 1.29
.48 1.41 1.36
.54 1.47 1.42
.59 1.51 1.46
.68 1.59 1.54
.74 1.65 1.59
.79 1.69 1.63
.83 1.73 1.66
.87 1.77 1.71
.94 1.83 1.76
.98 1.87 1.80
>.03 1.90 1.83
>.06 1.94 1.86
>.09 1.96 1.88
).84 0.83 0.82 (
).99 0.97 0.96 (
.07 1.05 1.04
.13 1.11 1.10
.17 1.15 1.14
.26 1.24 1.22
.33 1.31 1.29
.39 1.36 1.34
.43 1.40 1.38
.50 1.47 1.45
.55 1.52 1.50
.59 1.56 1.54
.62 1.59 1.57
.66 1.63 1.60
.71 1.68 1.65
.75 1.71 1.69
.78 1.74 1.72
.81 1.77 1.74
.83 1.79 1.76
).82 0.80 (
).95 0.94 (
.03 1.02
.08 1.07
.13 1.11
.21 1.19
.28 1.26
.33 1.31
.37 1.35
.44 1.41
.48 1.46
.52 1.49
.55 1.52
.59 1.56
.63 1.60
.67 1.64
.70 1.66
.72 1.69
.74 1.71
).80 0.79 0.79 (
).93 0.92 0.92 (
.01 1.00 0.99 (
.06 1.05 1.04
.10 1.09 1.08
.18 1.17 1.16
.25 1.24 1.23
.29 1.28 1.28
.33 1.32 1.31
.40 1.38 1.37
.44 1.43 1.42
.48 1.46 1.45
.50 1.49 1.48
.54 1.52 1.51
.58 1.57 1.55
.61 1.60 1.59
.64 1.62 1.61
.66 1.65 1.64
.68 1.67 1.65
).78 0.78 (
).91 0.91 (
).99 0.98 (
.04 1.03
.08 1.07
.16 1.15
.22 1.21
.27 1.26
.30 1.29
.37 1.35
.41 1.40
.44 1.43
.47 1.45
.50 1.49
.54 1.53
.58 1.56
.60 1.58
.62 1.61
.64 1.62
).77
).90
).97
.02
.06
.14
.20
.25
.28
.34
.39
.42
.44
.48
.52
.55
.57
.59
.61
                                                    D-141
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
       Table 19-12. K-Multipliers  for 1-of-4 Intrawell Prediction Limits on Observations (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.70 .19 1.01 (
2.25 .49 1.25
2.62 .68 1.39
2.91 .82 1.50
3.16 .94 1.58
3.74 2.19 1.76
4.31 2.42 1.91
4.76 2.59 2.O3
5.14 2.73 2.12
5.90 3.00 2.30
6.51 3.20 2.42
7.02 3.37 2.52
7.47 3.51 2.61 ;
8.06 3.68 2.71 ;
8.88 3.93 2.85 ;
9.58 4.12 2.97 ;
10.20 4.28 3.06 ;
10.72 4.42 3.14 ;
11.19 4.56 3.21 ;
).92 (
.13
.25
.34
.41
.56
.68
.78
.85
T.99
2.O8
2.16.
2.23
>.31
>.41
2.49
2.56
2.62
1.61
).86 0.80 0.76 (
.06 0.97 0.92 (
.17 1.07 1.02 (
.25 1.14 1.08
.31 1.20 1.13
.44 1.31 1.24
.55 1.41 1.33
.63 1.48 1.39
.70 1.53 1.44
.81 1.63 1.53
.90 1.70 1.59
7,96 1.75 1.64
2,O2 1.79 1.68
?.08' 1.85 1.72
2. 17 1.92 1.79
2.24 1,97 . 1.83
2.30 2.02 1.88
2.34 2.O6 1.91
2.39 2.09 1.94
).73 0.71 0.69 (
).89 0.86 0.85 (
).98 0.95 0.93 (
.04 1.01 0.99 (
.09 1.06 1.03
.18 1.15 1.13
.27 1.23 1.20
.33 1.29 1.26
.37 1.33 1.30
.45 1.41 1.38
.51 1.46 1.43
.56 1.50 1.47
.59 1.54 1.50
.64 1.58 1.54
.69 1.63 1.59
.74 1.67 1.63
.77 1.71 1.66
.80 1.73 1.69
.83 1.76 1.71
).68 0.68 0.67 (
).83 0.82 0.82 (
).92 0.91 0.90 (
).97 0.96 0.95 (
.02 1.01 1.00 (
.11 1.10 1.09
.19 1.17 1.16
.24 1.22 1.21
.28 1.26 1.25
.35 1.33 1.32
.40 1.38 1.37
.44 1.42 1.41
.47 1.45 1.44
.51 1.49 1.47
.56 1.54 1.52
.60 1.57 1.56
.63 1.60 1.59
.66 1.63 1.61
.68 1.65 1.63
).66 0.65 (
).81 0.80 (
).89 0.88 (
).94 0.93 (
).98 0.97 (
.07 1.06
.14 1.13
.19 1.18
.23 1.22
.30 1.28
.35 1.33
.38 1.37
.41 1.40
.45 1.43
.49 1.48
.53 1.51
.56 1.54
.58 1.56
.60 1.58
).65 0.65 (
).79 0.79 (
).87 0.87 (
).92 0.92 (
).97 0.96 (
.05 1.04
.12 1.11
.17 1.16
.21 1.20
.27 1.26
.32 1.31
.35 1.35
.38 1.37
.42 1.41
.46 1.45
.50 1.48
.52 1.51
.55 1.53
.57 1.55
).64 (
).78 (
).86 (
).91 (
).96 (
.04
.11
.16
.19
.26
.30
.34
.37
.40
.44
.48
.50
.53
.55
).64 0.63
).78 0.77
).85 0.85
).91 0.90
).95 0.94
.03 1.02
.10 1.09
.15 1.14
.18 1.17
.25 1.24
.29 1.28
.32 1.32
.35 1.34
.39 1.37
.43 1.42
.46 1.45
.49 1.48
.51 1.50
.53 1.52
    Table 19-12. K-Multipliers  for 1-of-4 Intrawell Prediction Limits on Observations (20 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.20 1.48 1.24
2.85 1.81 1.49
3.31 2.02 1.64
3.66 2.17 1.75
3.96 2.3Q 1.84
4.67 2.58 2.03
5.37 2.84 2.19
5.93 3.03 2.32 \
6.39 3.19 2.42 :
7.34 3.49 2.60 2
8.09 3.72 2.74 '„
8.72 3.91 2.85 ,
9.27 4.06 2.94
9.99 4.26 3.05 ;
11.02 4.53 3.21 ;
11.87 4.75 3.33 ;
12.60 4.94 3.43 ;
13.30 5.10 3.52 ;
13.89 5.24 3.60 ;
.13
.34
.46
.55
.62
.78
.91
>.01
>.08
>.22 ;
2,33 ' ;
2.41 ;
?,4B :
2.56 .,
2.67
2.76
2.83
2.89 ,
2.95
.05 0.97 0.92 (
.25 1.14 1.08
.36 1.24 1.17
.44 1.31 1.24
.50 1.36 1.29
.63 1.48 1.39
.75 1.57 1.48
.83 1.64 1.54
.90 1.70 1.59
1.02 1.79 1.68
MO 1.86 1.74
>.17 1.92 1.79
1.23 1.96 1.83
2.29 2.02 1.88
2,39 2.09 1.94
2,46 2.14 1.98
2.52 2.19 2.03
2.56 2.23 2.06
2.61 2.26 2.09
).89 0.86 0.85 (
.04 1.01 0.99 (
.12 1.09 1.07
.18 1.15 1.13
.23 1.19 1.17
.33 1.29 1.26
.41 1.36 1.33
.47 1.42 1.39
.51 1.46 1.43
.59 1.54 1.50
.65 1.59 1.55
.69 1.63 1.59
.73 1.67 1.62
.77 1.71 1.66
.83 1.76 1.71
.87 1.80 1.75
.90 1.83 1.78
.94 1.86 1.81
.96 1.88 1.83
).83 0.82 0.82 (
).97 0.96 0.95 (
.05 1.04 1.03
.11 1.10 1.09
.15 1.14 1.13
.24 1.22 1.21
.31 1.29 1.28
.36 1.34 1.33
.40 1.38 1.37
.47 1.45 1.44
.52 1.50 1.48
.56 1.54 1.52
.59 1.57 1.55
.63 1.60 1.59
.68 1.65 1.63
.71 1.69 1.67
.74 1.72 1.70
.77 1.74 1.72
.79 1.76 1.74
).81 0.80 (
).94 0.93 (
.02 1.01
.07 1.06
.11 1.10
.19 1.18
.26 1.25
.31 1.29
.35 1.33
.41 1.40
.46 1.44
.49 1.48
.52 1.50
.56 1.54
.60 1.58
.64 1.61
.66 1.64
.69 1.66
.71 1.68
).79 0.79 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.17 1.16
.24 1.23
.28 1.28
.32 1.31
.38 1.37
.43 1.42
.46 1.45
.49 1.48
.52 1.51
.57 1.55
.60 1.59
.62 1.61
.65 1.63
.67 1.65
).78 (
).91 (
).99 (
.04
.08
.16
.22
.27
.30
.37
.41
.44
.47
.50
.54
.58
.60
.62
.64
).78 0.77
).91 0.90
).98 0.97
.03 1.02
.07 1.06
.15 1.14
.21 1.20
.26 1.25
.29 1.28
.35 1.34
.40 1.39
.43 1.42
.45 1.44
.49 1.48
.53 1.52
.56 1.55
.58 1.57
.61 1.59
.62 1.61
                                                    D-142
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
      Table 19-12. K-Multipliers  for 1-of-4 Intrawell Prediction Limits on Observations (20 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.75 1.79 1.48 .33
3.54 2.15 1.74 .55
4.09 2.38 1.90 .68
452 2.55 2.02 .77
4.88 2.69 2.11 .84
5.74 3.00 2.31 2.00
6.59 3.29 2.49 2.14
7.27 3.51 2,62 2.24 ;
7.84 3.69 2.73 2.32 ;
8.99 4.03 2.93 2.47 ;
9.90 4.29 3.08 2,88 '*
10.68 4.50 3.20 2.67, \
11.35 4.68 3.30 2.74 \
12.23 4.90 3.42 2.83 ;
13.48 5.21 3.59 2.94 ,
14.53 5.46 3.72 3.04
15.41 5.67 3.83 3.11
16.23 5.86 3.93 3.18 • '„
16.99 6.02 4.01 3.24 ,
.24 1.14 1.08
.43 1.31 1.24
.55 1.40 1.33
.63 1.47 1.39
.69 1.53 1.44
.83 1.64 1.54
.95 1.74 1.63
>.03 1.81 1.69
MO 1.86 1.74
1.22 1.96 1.83
>.31 2.03 1.89
>.38 2.09 1.94
>.44 2.13 1.98
>.51 2.19 2.02
2,61 2.26 2.09
2.68 2.32 2.14 ;
2.74 2.37 2.18 ;
2.79 2.41 2.21 ;
2.84 2.44 2.24 ;
.04 1.01 0.99 (
.18 1.15 1.13
.27 1.23 1.20
.33 1.29 1.26
.37 1.33 1.30
.47 1.42 1.39
.55 1.50 1.46
.60 1.55 1.51
.65 1.59 1.55
.73 1.66 1.62
.78 1.72 1.67
.83 1.76 1.71
.86 1.79 1.74
.91 1.83 1.78
.96 1.88 1.83
>.00 1.92 1.87
>.04 1.96 1.90
1.07 1.98 1.92
>.09 2.01 1.95
).97 0.96 0.95 (
.11 1.10 1.08
.19 1.17 1.16
.24 1.22 1.21
.28 1.26 1.25
.36 1.34 1.33
.43 1.41 1.40
.48 1.46 1.45
.52 1.50 1.48
.59 1.57 1.55
.64 1.62 1.60
.68 1.65 1.63
.71 1.68 1.66
.74 1.72 1.70
.79 1.76 1.74
.83 1.80 1.77
.86 1.83 1.80
.88 1.85 1.83
.90 1.87 1.85
).94 0.93 (
.07 1.06
.14 1.13
.19 1.18
.23 1.22
.31 1.29
.38 1.36
.42 1.41
.46 1.44
.52 1.50
.57 1.55
.60 1.58
.63 1.61
.66 1.64
.71 1.68
.74 1.72
.77 1.74
.79 1.76
.81 1.78
).92 0.92 (
.05 1.04
.12 1.11
.17 1.16
.21 1.20
.28 1.28
.35 1.34
.39 1.38
.43 1.42
.49 1.48
.53 1.52
.57 1.55
.59 1.58
.63 1.61
.67 1.65
.70 1.68
.72 1.71
.74 1.73
.76 1.75
).91 (
.04
.11
.16
.19
.27
.33
.38
.41
.47
.51
.54
.57
.60
.64
.67
.70
.72
.74
).91 0.90
.03 1.02
.10 1.09
.15 1.14
.18 1.17
.26 1.25
.32 1.31
.36 1.35
.40 1.39
.45 1.44
.50 1.49
.53 1.52
.55 1.54
.58 1.57
.62 1.61
.65 1.64
.68 1.67
.70 1.69
.72 1.70
       Table 19-12. K-Multipliers  for 1-of-4 Intrawell Prediction Limits on Observations (40 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.25 1.49 1.25
2.91 1.82 1.50
3.37 "'2.03' 1.65
3.74 2.19 1.76
4.04 2.31 1.84
4.76 2.59 2.O3.
5.47 2.85 2, 2O
6.04 3.05 2.32
6.51 3.20 2.42
7.47 3.51 2.61 ;
8.23 3.74 2.74 ;
8.88 3.93 2.85 ;
9.43 4.08 2.94 ;
10.20 4.28 3.06 ;
11.19 4.56 3.21 ;
12.07 4.78 3.33 ;
12.89 4.95 3.44 ;
13.48 5.13 3.52 ;
14.18 5.27 3.60 ;
.13
.34
.47
.56
.63
.78
.91
2,01
2.08
2.23
2.33
>.41
2.48
2.56
1.61
2.16
2.83
2.90
2.94
.06 0.97 0.92 (
.25 1.14 1.08
.36 1.24 1.17
.44 1.31 1.24
.50 1.36 1.29
.63 1.48 1.39
.75 1.57 1.48
.83 1.64 1.54
.90 1.70 1.59
2.02 1.79 1.68
2.10 1.86 1.74
2.17. 1.92 1.79
2.23 1.96 1.83
2.30 2. 02 1.88
2.39 .2.O9 1.94
2.46 2.15 1.98
2.52 2,19 . 2.O3
2.56 2.23 2.06
2.61 2.26'.- 2.O9
).89 0.86 0.85 (
.04 1.01 0.99 (
.12 1.09 1.07
.18 1.15 1.13
.23 1.19 1.17
.33 1.29 1.26
.41 1.37 1.33
.47 1.42 1.39
.51 1.46 1.43
.59 1.54 1.50
.65 1.59 1.55
.69 1.63 1.59
.73 1.66 1.62
.77 1.71 1.66
.83 1.76 1.71
.87 1.80 1.75
.90 1.83 1.78
.93 1.86 1.81
1,96 1.88 1.83
).83 0.82 0.82 (
).97 0.96 0.95 (
.05 1.04 1.03
.11 1.10 1.09
.15 1.14 1.13
.24 1.22 1.21
.31 1.29 1.28
.36 1.34 1.33
.40 1.38 1.37
.47 1.45 1.44
.52 1.50 1.48
.56 1.54 1.52
.59 1.57 1.55
.63 1.60 1.59
.68 1.65 1.63
.71 1.69 1.67
.74 1.72 1.70
.77 1.74 1.72
.79 1.77 1.74
).81 0.80 (
).94 0.93 (
.02 1.01
.07 1.06
.11 1.10
.19 1.18
.26 1.25
.31 1.29
.35 1.33
.41 1.40
.46 1.44
.49 1.48
.52 1.50
.56 1.54
.60 1.58
.64 1.61
.66 1.64
.69 1.66
.71 1.68
).79 0.79 (
).92 0.92 (
.00 0.99 (
.05 1.04
.09 1.08
.17 1.16
.24 1.23
.28 1.28
.32 1.31
.38 1.37
.43 1.42
.46 1.45
.49 1.48
.52 1.51
.57 1.55
.60 1.59
.63 1.61
.65 1.63
.67 1.65
).78 (
).91 (
).99 (
.04
.08
.16
.22
.27
.30
.37
.41
.44
.47
.50
.55
.58
.60
.62
.64
).78 0.77
).91 0.90
).98 0.97
.03 1.02
.07 1.06
.15 1.14
.21 1.20
.26 1.25
.29 1.28
.35 1.34
.39 1.38
.43 1.42
.45 1.44
.49 1.48
.53 1.52
.56 1.55
.59 1.57
.60 1.59
.62 1.61
                                                    D-143
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
    Table 19-12. K-Multipliers for 1-of-4 Intrawell Prediction Limits on Observations (40 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.85 1.81 1.49 .34
3.66 2.17 1.75 .55
4.22 2.41 1.91 .68
4.67 2.58 2.03 .78
5.04 2.72 2.12 .85
5.93 3.03 2,32 2.01
6.80 3.32 2.50 2.15
7.50 3.54 2.63 2.25 ;
8.09 3.72 2.74 2,33 \
9.27 4.06 2.94 2,48 :
10.21 4.32 3.09 2.59
11.02 4.53 3.21 2.67 ,
11.72 4.72 3.31 2.74 ' ,
12.60 4.94 3.43 2.83
13.89 5.24 3.60 2.95
15.00 5.51 3.73 3.04
15.94 5.71 3.84 3.12
16.76 5.89 3.94 3.18
17.58 6.06 4.01 3.24
.25 1.14 1.08
.44 1.31 1.24
.55 1.41 1.33
.63 1.48 1.39
.70 1.53 1.44
.83 1.64 1.54
.95 1.74 1.63
>.04 1.81 1.69
MO 1.86 1.74
1.23 1.96 1.83
2.31 2.03 1.89
2,39 2.09 1.94
2,44 2.14 1.98
2.52 2.19 2.03
2.61 2,26 2.09
2.68 2.32 2.14 ;
1.15 .2.37 2.18 ;
2.80 '2.40 2.21 ;
2.84 2,44 2.24 ;
.04 1.01 0.99 (
.18 1.15 1.13
.27 1.23 1.20
.33 1.29 1.26
.37 1.33 1.30
.47 1.42 1.39
.55 1.50 1.46
.60 1.55 1.51
.65 1.59 1.55
.73 1.67 1.62
.78 1.72 1.67
.83 1.76 1.71
.86 1.79 1.74
.90 1.83 1.78
.96 1.88 1.83
>.00 1.92 1.87
>.04 1.96 1.90
1.07 1.98 1.93
>.09 2.01 1.95
).97 0.96 0.95 (
.11 1.10 1.09
.19 1.17 1.16
.24 1.22 1.21
.28 1.26 1.25
.36 1.34 1.33
.43 1.41 1.40
.48 1.46 1.45
.52 1.50 1.48
.59 1.57 1.55
.64 1.61 1.60
.68 1.65 1.63
.71 1.68 1.66
.74 1.72 1.70
.79 1.76 1.74
.83 1.80 1.77
.86 1.83 1.80
.88 1.85 1.83
.90 1.87 1.85
).94 0.93 (
.07 1.06
.14 1.13
.19 1.18
.23 1.22
.31 1.29
.38 1.36
.42 1.41
.46 1.44
.52 1.50
.57 1.55
.60 1.58
.63 1.61
.66 1.64
.71 1.68
.74 1.72
.77 1.74
.79 1.77
.81 1.78
).92 0.92 (
.05 1.04
.12 1.11
.17 1.16
.21 1.20
.28 1.28
.35 1.34
.39 1.38
.43 1.42
.49 1.48
.53 1.52
.57 1.55
.59 1.58
.62 1.61
.67 1.65
.70 1.68
.72 1.71
.75 1.73
.77 1.75
).91 (
.04
.11
.16
.19
.27
.33
.38
.41
.47
.51
.54
.57
.60
.64
.67
.70
.72
.74
).91 0.90
.03 1.02
.10 1.09
.15 1.14
.18 1.17
.26 1.25
.32 1.31
.36 1.35
.40 1.39
.45 1.44
.50 1.49
.53 1.52
.55 1.54
.58 1.57
.62 1.61
.65 1.64
.68 1.67
.70 1.69
.72 1.70
      Table 19-12. K-Multipliers  for 1-of-4 Intrawell Prediction Limits on Observations (40 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.54 2.15 1.74 1.55 .43 1.31 1.24
452 2.55 2.02 1.77 .63 1.47 1.39
5.20 2.81 2.19 1.90 .75 1.57 1.48
5.74 3.00 2.31 2.00 .83 1.64 1.54
6.20 3.16 2.41 2.08 .89 1.69 1.59
7.27 3.51 2.62 2.24 2.03 1.81 1.69
8.34 3.84 2.82 2.39 2.16 1.91 1.78
9.18 4.09 2.96 2.50 2.24 1.98 1.84
9.90 4.29 3.08 2,58 2.31 2.03 1.89
11.35 4.68 3.30 2,74 2.44 2.13 1.98
12.51 4.97 3.46 2.85 2.53 2.21 2.04
13.48 5.21 3.59 2.94 2.6T 2.26 2.09
14.33 5.41 3.69 3.02 2,87 2.31 2.13 ;
15.41 5.67 3.83 3.11 . 2, 74 2.37 2.18 ;
16.99 6.02 4.01 3.24 2,84 2.44 2.24 ;
18.28 6.30 4.15 3.34 2.92 2.50 2.29 ;
19.45 6.56 4.28 3.42 2.98 2,55 2.33 ;
20.51 6.77 4.38 3.49 3.03 2,59 2.36 ;
21.33 6.94 4.47 3.54 3.08 2.62 2.39 ;
.18 1.15 1.13
.33 1.29 1.26
.41 1.36 1.33
.47 1.42 1.39
.51 1.46 1.43
.60 1.55 1.51
.68 1.62 1.58
.74 1.68 1.63
.78 1.72 1.67
.86 1.79 1.74
.92 1.84 1.79
.96 1.88 1.83
>.00 1.92 1.86
>.04 1.96 1.90
>.09 2.01 1.95
>.14 2.05 1.98
>.18 2.08 2.01
1.20 2.11 2.04
2.23 2.13 2.07 ;
.11 1.10 1.08
.24 1.22 1.21
.31 1.29 1.28
.36 1.34 1.33
.40 1.38 1.37
.48 1.46 1.45
.55 1.53 1.51
.60 1.58 1.56
.64 1.62 1.60
.71 1.68 1.66
.76 1.73 1.71
.79 1.76 1.74
.82 1.79 1.77
.86 1.83 1.80
.90 1.87 1.85
.94 1.91 1.88
.97 1.93 1.91
.99 1.96 1.93
>.01 1.98 1.95
.07 1.06
.19 1.18
.26 1.25
.31 1.29
.35 1.33
.42 1.41
.49 1.47
.53 1.51
.57 1.55
.63 1.61
.67 1.65
.71 1.68
.73 1.71
.77 1.74
.81 1.78
.84 1.81
.87 1.84
.89 1.86
.91 1.88
.05 1.04
.17 1.16
.24 1.23
.28 1.28
.32 1.31
.39 1.38
.46 1.45
.50 1.49
.53 1.52
.59 1.58
.63 1.62
.67 1.65
.69 1.68
.72 1.71
.76 1.75
.79 1.78
.82 1.81
.84 1.83
.86 1.85
.04
.16
.22
.27
.30
.38
.44
.48
.51
.57
.61
.64
.67
.70
.74
.77
.79
.81
.83
.03 1.02
.15 1.14
.21 1.20
.26 1.25
.29 1.28
.36 1.35
.42 1.41
.46 1.45
.50 1.49
.55 1.54
.59 1.58
.62 1.61
.65 1.63
.68 1.66
.72 1.70
.74 1.73
.77 1.76
.79 1.78
.81 1.79
                                                    D-144
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
    Table 19-13. K-Multipliers  for Modified Calif.  Intrawell Prediction Limits on Observations (1 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.71 0.59 0.53 (
1.14 0.92 0.83 (
1.42 1.11 0.99 (
1.63 1.25 1.11
1.80 1.36 1.20
2.19 1.60 1.40
2.58 1.82 1.57
2.88 1.98 1.69
3.13 2.11 1.78
3.63 2.35 1.96
4.02 2.53 2.09
4.36 2.68 2.20
4.65 2.80 2.28 ;
5.02 2.96 2.39 ;
5.55 3.17 2.53 ;
5.99 3.34 2,64 ;
6.38 3.48 2.73 ;
6.72 3.60 2.81 ;
7.04 3.72 2.88 ;
).50 0.48 0.45 (
).78 0.74 0.71 (
).93 0.89 0.84 (
.04 0.99 0.94 (
.12 1.07 1.01 (
.29 1.23 1.15
.44 1.36 1.27
.54 1.46 1.36
.63 1.53 1.42
.78 1.67 1.54
.88 1.76 1.63
.97 1.84 1.69
>.04 1.90 1.74
>.12 1.97 1.81
>.24 2.07 1.89
1.32 2.15 1.95
>.40 2.21 2.00
>.46 2.26 2.05
>.51 2.31 2.09
).44 (
).68 (
).81 (
).90 (
).97 (
.11
.22
.30
.37
.47
.55
.61
.66
.72
.79
.85
.89
.93
.97
).43 0.42 (
).66 0.65 (
).79 0.78 (
).88 0.86 (
).94 0.93 (
.08 1.05
.19 1.16
.26 1.23
.32 1.29
.42 1.39
.49 1.46
.55 1.51
.59 1.55
.65 1.60
.72 1.67
.77 1.72
.81 1.76
.85 1.80
.88 1.83
).41 (
).64 (
).77 (
).85 (
).91 (
.04
.14
.22
.27
.37
.43
.48
.53
.58
.64
.69
.73
.76
.79
).41 0.41 (
).64 0.63 (
).76 0.75 (
).84 0.84 (
).90 0.90 (
.03 1.02
.13 1.12
.20 1.19
.26 1.24
.35 1.34
.41 1.40
.46 1.45
.50 1.49
.55 1.54
.62 1.60
.66 1.64
.70 1.68
.73 1.71
.76 1.74
).40 0.40 0.40 (
).63 0.62 0.62 (
).75 0.74 0.74 (
).83 0.82 0.82 (
).89 0.88 0.88 (
.01 1.00 1.00 (
.11 1.10 1.09
.18 1.17 1.16
.23 1.22 1.21
.33 1.31 1.30
.39 1.37 1.36
.44 1.42 1.41
.48 1.46 1.45
.52 1.50 1.49
.58 1.56 1.55
.63 1.61 1.59
.67 1.64 1.63
.70 1.67 1.66
.72 1.70 1.68
).40 0.39 (
).62 0.61 (
).73 0.73 (
).81 0.81 (
).87 0.87 (
).99 0.99 (
.09 1.08
.15 1.15
.20 1.20
.29 1.29
.35 1.35
.40 1.39
.44 1.43
.48 1.47
.54 1.53
.58 1.57
.61 1.61
.64 1.63
.67 1.66
).39 0.39 (
).61 0.61 (
).73 0.73 (
).81 0.80 (
).87 0.86 (
).98 0.98 (
.08 1.07
.14 1.14
.19 1.19
.28 1.27
.34 1.33
.39 1.38
.42 1.41
.47 1.46
.52 1.51
.56 1.55
.60 1.58
.63 1.61
.65 1.64
).39
).61
).72
).80
).86
).97
.07
.13
.18
.27
.32
.37
.40
.45
.50
.54
.58
.60
.63
 Table 19-13. K-Multipliers  for Modified Calif.  Intrawell Prediction Limits on Observations (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.07 0.88 0.81 (
1.55 1.22 1.09
1.86 1.42 1.26
2.10 1.57 1.38
2.30 1.69 1.47
2.77 1.95 1.67
3.23 2.18 1.85
3.59 2.36 1.98
3.89 2.50 2.08
4.49 2.77 2.27 ;
4.97 2.97 2.40 ;
5.37 3.13 2.51 ;
5.72 3.27 2.60 ;
6.17 3.44 2.72 ;
6.81 3.68 2,87 '*
7.35 3.87 2.99 ;
7.82 4.03 3.09 ;
8.25 4.17 3.17 ;
8.63 4.29 3.25 ;
).76 0.73 0.70 (
.02 0.98 0.93 (
.18 1.12 1.06
.28 1.22 1.15
.36 1.29 1.21
.54 1.45 1.36
.69 1.59 1.47
.79 1.68 1.56
.88 1.76 1.62
>.03 1.89 1.74
>.14 1.99 1.82
1.23 2.07 1.89
>.30 2.13 1.94
>.39 2.20 2.00
>.51 2.30 2.08
>.60 2.38 2.15 ;
>.68 2.45 2.20 ;
1.74 2.50 2.24 ;
>.80 2.55 2.28 ;
).68 (
).90 (
.02 (
.11
.17
.30
.41
.49
.55
.66
.73
.79
.84
.89
.97
1.02
1.07
>.n ;
>.14 ;
).66 0.65 (
).88 0.86 (
).99 0.97 (
.07 1.05
.13 1.11
.26 1.23
.36 1.33
.44 1.40
.49 1.46
.59 1.55
.66 1.62
.72 1.67
.76 1.71
.81 1.76
.88 1.83
.93 1.87
.97 1.91
>.01 1.95
>.04 1.98
).64 (
).85 (
).96 (
.04
.10
.21
.31
.38
.43
.53
.59
.64
.68
.73
.79
.83
.87
.91
.93
).64 0.63 (
).84 0.83 (
).95 0.94 (
.03 1.02
.08 1.08
.20 1.19
.30 1.29
.36 1.35
.41 1.40
.50 1.49
.57 1.55
.61 1.60
.65 1.63
.70 1.68
.76 1.74
.81 1.78
.84 1.82
.87 1.85
.90 1.88
).63 0.62 0.62 (
).83 0.82 0.82 (
).94 0.93 0.92 (
.01 1.00 1.00 (
.07 1.06 1.05
.18 1.17 1.16
.28 1.26 1.25
.34 1.32 1.31
.39 1.37 1.36
.48 1.46 1.45
.54 1.52 1.50
.58 1.56 1.55
.62 1.60 1.58
.67 1.64 1.63
.72 1.70 1.68
.77 1.74 1.72
.80 1.78 1.76
.83 1.80 1.78
.86 1.83 1.81
).62 0.61 (
).81 0.81 (
).92 0.91 (
).99 0.99 (
.04 1.04
.15 1.15
.24 1.24
.31 1.30
.35 1.35
.44 1.43
.49 1.48
.54 1.53
.57 1.56
.61 1.61
.67 1.66
.71 1.70
.74 1.73
.77 1.76
.79 1.78
).61 0.61 (
).81 0.80 (
).91 0.91 (
).98 0.98 (
.04 1.03
.14 1.14
.23 1.22
.29 1.29
.34 1.33
.42 1.41
.48 1.47
.52 1.51
.56 1.54
.60 1.58
.65 1.64
.69 1.68
.72 1.71
.75 1.74
.77 1.76
).61
).80
).90
).97
.03
.13
.22
.28
.32
.40
.46
.50
.54
.58
.63
.67
.70
.72
.75
                                                    D-145
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
   Table 19-13. K-Multipliers  for Modified Calif.  Intrawell Prediction Limits on Observations (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.43 1.17 1.06
1.97 1.52 1.35
2.33 1.74 1.52
2.61 1.90 1.65
2.84 2.02 1.74
3.39 2.31 1.95
3.93 2.56 2.14
435 2.76 2.27 ;
4.71 2.91 2.38 ;
5.42 3.21 2.58 ;
5.99 3.43 2.73 ;
6.47 3.61 2.84 ;
6.89 3.77 2.94 ;
7.43 3.96 3.06 ;
8.20 4.22 3.23 ;
8.84 4.43 3.35 ;
9.40 4.62 3.46 ;
9.90 4.77 3.55 C
10.36 4.91 3.64
.00 (
.26
.41
.52
.60
.78
.93
>.04
>.13
1.29 :
>.40 ;
>.so ;
2.57 ;
>.66 ;
2.79 ;
>.88 ;
2.96 ;
5.03 ;
3..Q9 ;
).96 0.92 0.89 (
.20 1.14 1.10
.34 1.26 1.22
.44 1.35 1.30
.52 1.41 1.36
.67 1.55 1.49
.81 1.67 1.59
.91 1.75 1.67
.98 1.82 1.73
1.12 1.93 1.83
1.22 2.02 1.91
2.30 2.08 1.96
2.36 2.13 2.01
2.44 2.20 2.07
2.54 2.28 2.14 ;
1.62 2.35 2.20 ;
2.69 2.40 2.25 ;
1.75 2.44 2.29 ;
2.80 2.48 2.32 ;
).87 0.85 0.84 (
.07 1.05 1.04
.18 1.16 1.14
.26 1.23 1.21
.32 1.29 1.27
.44 1.40 1.38
.54 1.50 1.47
.61 1.57 1.54
.66 1.62 1.59
.76 1.71 1.68
.83 1.77 1.74
.88 1.82 1.79
.92 1.86 1.83
.97 1.91 1.87
2.04 1.98 1.93
2.09 2.02 1.98
2.13 2.06 2.02
1.17 2.10 2.05 ;
1.20 2.13 2.07 ;
).84 0.83 0.83 (
.02 1.02 1.01
.13 1.12 1.11
.20 1.19 1.18
.25 1.24 1.23
.36 1.35 1.34
.45 1.44 1.43
.52 1.50 1.49
.57 1.55 1.54
.65 1.63 1.62
.71 1.69 1.68
.76 1.74 1.72
.80 1.78 1.76
.84 1.82 1.80
.90 1.88 1.86
.95 1.92 1.90
.98 1.95 1.93
2.01 1.98 1.96
2.04 2.01 1.99
).82 0.81 (
.00 0.99 (
.10 1.09
.17 1.16
.22 1.21
.32 1.31
.41 1.40
.47 1.46
.52 1.50
.60 1.58
.65 1.64
.70 1.68
.73 1.72
.77 1.76
.83 1.81
.87 1.85
.90 1.88
.93 1.91
.96 1.93
).81 0.81 (
).99 0.99 (
.09 1.08
.15 1.15
.20 1.20
.31 1.30
.39 1.38
.45 1.44
.49 1.48
.57 1.56
.63 1.62
.67 1.66
.70 1.69
.74 1.73
.79 1.78
.83 1.82
.87 1.85
.89 1.88
.92 1.90
).81 (
).98 (
.08
.14
.19
.29
.38
.43
.48
.56
.61
.65
.68
.72
.77
.81
.84
.87
.89
).80 0.80
).98 0.97
.07 1.07
.14 1.13
.19 1.18
.28 1.28
.37 1.36
.42 1.42
.47 1.46
.54 1.54
.60 1.59
.64 1.63
.67 1.66
.71 1.70
.76 1.75
.80 1.79
.83 1.82
.85 1.84
.88 1.86
    Table 19-13. K-Multipliers  for Modified Calif. Intrawell Prediction Limits on Observations (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.14 0.92 0.83 (
1.63 1.25 1.11
1.95 1.46 1.28
2.19 1.60 1.40
2.40 1.72 1.49
2.88 1.98 1.69
3.35 2.22 1.86
3.72 2.39 1.99
4.02 2.53 2.09
4.65 2.80 2.28 ;
5.14 3.00 2.42 ;
5.55 3.17 2.53 ;
5.91 3.31 2,62 :
6.38 3.48 2.73 ;
7.04 3.72 2.88 ;
7.60 3.91 3.00 <
8.09 4.07 3.10 ,
8.51 4.21 3.19
8.91 4.34 3.26
).78 0.74 0.71 (
.04 0.99 0.94 (
.19 1.13 1.06
.29 1.23 1.15
.37 1.30 1.22
.54 1.46 1.36
.69 1.59 1.48
.80 1.69 1.56
.88 1.76 1.63
2.04 1.90 1.74
2.15 1.99 1.82
2.24 2.07 1.89
2.31 2.13 1.94
2.40 2.21 2.00
2.51 2.31 2.09
2,61 2.39 2.15 ;
2,68 2.45 2.20 ;
2.75 2.50 2.25 ;
>.81 2,55" 2.29 ;
).68 (
).90 (
.03
.11
.17
.30
.41
.49
.55
.66
.73
.79
.84
.89
.97
2.03
2.07
Ml ;
2.14 ;
).66 0.65 (
).88 0.86 (
.00 0.98 (
.08 1.05
.14 1.11
.26 1.23
.37 1.34
.44 1.41
.49 1.46
.59 1.55
.66 1.62
.72 1.67
.76 1.71
.81 1.76
.88 1.83
.93 1.88
.98 1.92
2.01 1.95
2.04 1.98
).64 (
).85 (
).96 (
.04
.10
.22
.31
.38
.43
.53
.59
.64
.68
.73
.79
.83
.87
.91
.93
).64 0.63 (
).84 0.84 (
).95 0.95 (
.03 1.02
.09 1.08
.20 1.19
.30 1.29
.36 1.35
.41 1.40
.50 1.49
.57 1.55
.62 1.60
.65 1.64
.70 1.68
.76 1.74
.81 1.78
.84 1.82
.87 1.85
.90 1.88
).63 0.62 0.62 (
).83 0.82 0.82 (
).94 0.93 0.92 (
.01 1.00 1.00 (
.07 1.06 1.05
.18 1.17 1.16
.28 1.26 1.25
.34 1.32 1.31
.39 1.37 1.36
.48 1.46 1.45
.54 1.52 1.50
.58 1.56 1.55
.62 1.60 1.58
.67 1.64 1.63
.72 1.70 1.68
.77 1.74 1.72
.80 1.78 1.76
.83 1.80 1.79
.86 1.83 1.81
).62 0.61 (
).81 0.81 (
).92 0.92 (
).99 0.99 (
.04 1.04
.15 1.15
.24 1.24
.31 1.30
.35 1.35
.44 1.43
.49 1.48
.54 1.53
.57 1.56
.61 1.61
.67 1.66
.71 1.70
.74 1.73
.77 1.76
.79 1.78
).61 0.61 (
).81 0.80 (
).91 0.91 (
).98 0.98 (
.04 1.03
.14 1.14
.23 1.23
.29 1.29
.34 1.33
.42 1.41
.48 1.47
.52 1.51
.56 1.54
.60 1.58
.65 1.64
.69 1.68
.72 1.71
.75 1.74
.77 1.76
).61
).80
).90
).97
.03
.13
.22
.28
.32
.40
.46
.50
.54
.58
.63
.67
.70
.72
.75
                                                    D-146
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                        Unified Guidance
 Table 19-13. K-Multipliers  for Modified Calif. Intrawell Prediction Limits on Observations (2 COC, Semi-Annual)
w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.55 1.22
2.10 1.57
2.48 1.79
2.77 1.95
3.01 2.08
3.59 2.36
4.15 2.62 ;
4.59 2.81 ;
4.97 2.97 ;
5.72 3.27 ;
6.31 3.50 ;
6.81 3.68
7.25 3.83
7.82 4.03 :
8.63 4.29 :
9.30 4.51 :
9.89 4.69 :
10.43 4.85 :
10.90 5.00 :
.09
.38
.55
.67
.77
.98
2.16
2.30 ;
2.40 ;
2.60 ;
2.75 ;
?,8? :
2:9? :
3.09 ;
3.25 ;
3.38 ",
3.49 ,
3.58
3.66
.02 0.98 (
.28 1.22
.43 1.36
.54 1.45
.62 1.53
.79 1.68
.95 1.82
2.06 1.91
2.14 1.99
2.30 2.13
2.42 2.23 ;
2.51 2.30 ;
2.58 2.37 ;
2.68 2.45 ;
2.80 2.55 ;
2.89' 2.63 ;
2,98 2.70 ;
3.04 2.75 :
3.11 2.80 :
).93 (
.15
.27
.36
.42
.56
.68
.76
.82
.94
1.02
2.08
2.14 ;
1.20 :
1.28 ;
2.35 ;
2.40 ;
2.45 ;
2.48 ;
).90 0.88 0.86 (
.11 1.07 1.05
.22 1.18 1.16
.30 1.26 1.23
.36 1.32 1.29
.49 1.44 1.40
.60 1.54 1.50
.67 1.61 1.57
.73 1.66 1.62
.84 1.76 1.71
.91 1.83 1.78
.97 1.88 1.83
2.01 1.92 1.87
2.07 1.97 1.91
2.14 2.04 1.98
2.20 2.09 2.03
2.25 2.14 2.06 ;
2.29 2.17 2.10 ;
2.32 2.20 2.13 ;
).85 0.84 0.83 (
.04 1.03 1.02
.14 1.13 1.12
.21 1.20 1.19
.27 1.25 1.24
.38 1.36 1.35
.47 1.46 1.44
.54 1.52 1.50
.59 1.57 1.55
.68 1.65 1.63
.74 1.71 1.69
.79 1.76 1.74
.83 1.80 1.78
.87 1.84 1.82
.93 1.90 1.88
.98 1.95 1.92
2.02 1.98 1.95
2.05 2.01 1.98
2.07 2.04 2.01
).83 0.82 (
.01 1.00
.11 1.10
.18 1.17
.23 1.22
.34 1.32
.43 1.41
.49 1.47
.54 1.52
.62 1.60
.68 1.65
.72 1.70
.76 1.73
.80 1.78
.86 1.83
.90 1.87
.93 1.90
.96 1.93
.99 1.96
).82 0.81 0.81 (
.00 0.99 0.99 (
.09 1.09 1.08
.16 1.15 1.15
.21 1.20 1.20
.31 1.31 1.30
.40 1.39 1.38
.46 1.45 1.44
.50 1.49 1.48
.58 1.57 1.56
.64 1.63 1.62
.68 1.67 1.66
.72 1.70 1.69
.76 1.74 1.73
.81 1.79 1.78
.85 1.83 1.82
.88 1.87 1.85
.91 1.89 1.88
.93 1.92 1.90
).81 0.80 (
).98 0.98 (
.08 1.07
.14 1.14
.19 1.19
.29 1.29
.38 1.37
.43 1.42
.48 1.47
.56 1.54
.61 1.60
.65 1.64
.68 1.67
.72 1.71
.77 1.76
.81 1.80
.84 1.83
.87 1.85
.89 1.88
).80
).97
.07
.13
.18
.28
.36
.42
.46
.54
.59
.63
.66
.70
.75
.79
.82
.84
.86
   Table 19-13. K-Multipliers  for Modified Calif. Intrawell Prediction Limits on Observations (2 COC, Quarterly)
w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.97 1.52 1.35 .26 1.20
2.61 1.90 1.65 .52 1.44
3.04 2.13 1.82 .67 1.58
3.39 2.31 1.95 .78 1.67
3.68 2.45 2.05 .86 1.75
435 2.76 2.27 2.04 1.91
5.02 3.04 2.47 2.20 2.04
5.55 3.26 2.61 2.31 2.14
5.99 3.43 2.73 2.40 2.22 ;
6.89 3.77 2.94 2.57 2.36 ;
7.60 4.02 3. 10 2.69 2.46 ;
8.20 4.22 3.23 2.79 2.54 ;
8.72 4.40 3.33 2.87 2.61 ;
9.40 4.62 3.46 2.96 2.69 ;
10.36 4.91 3.64 3.O9 2.80 ;
11.18 5.16 3.78 3,20 2.88 ;
11.88 5.36 3.89 3,28 2.95 ;
12.51 5.54 4.00 3.35 3.01 2
13.10 5.70 4.08 3.42 3.06 2
.14
.35
.47
.55
.62
.75
.87
.95
2.02
2.13 ;
2.22 ;
2.28 ;
2.33 ;
2.40 ;
2.48 ;
2.55 ;
2.60 ;
2.65 ;
2.69 ;
.10 1.07 1.05
.30 1.26 1.23
.41 1.36 1.33
.49 1.44 1.40
.55 1.49 1.46
.67 1.61 1.57
.78 1.70 1.66
.85 1.77 1.72
.91 1.83 1.77
2.01 1.92 1.86
2.09 1.99 1.93
2.14 2.04 1.98
2.19 2.08 2.02
2.25 2.13 2.06 ;
2.32 2.20 2.13 ;
2.38 2.25 2.17 ;
2.42 2.29 2.21 ;
2.46 2.33 2.24 ;
2.50 2.36 2.27 ;
.04 1.02 1.02
.21 1.20 1.19
.31 1.30 1.28
.38 1.36 1.35
.43 1.41 1.40
.54 1.52 1.50
.63 1.61 1.59
.69 1.67 1.65
.74 1.71 1.69
.83 1.80 1.78
.89 1.86 1.83
.93 1.90 1.88
.97 1.94 1.91
2.02 1.98 1.95
2.07 2.04 2.01
2.12 2.08 2.05 ;
2.16 2.12 2.09 ;
2.19 2.15 2.11 ;
2.21 2.17 2.14 ;
.01 1.00 (
.18 1.17
.27 1.26
.34 1.32
.39 1.37
.49 1.47
.57 1.55
.63 1.61
.68 1.65
.76 1.73
.81 1.79
.86 1.83
.89 1.86
.93 1.90
.99 1.96
2.03 1.99
2.06 2.03 ;
2.09 2.05 ;
2.11 2.08 ;
).99 0.99 0.99 (
.16 1.15 1.15
.25 1.24 1.24
.31 1.31 1.30
.36 1.35 1.35
.46 1.45 1.44
.54 1.53 1.52
.60 1.58 1.58
.64 1.63 1.62
.72 1.70 1.69
.77 1.75 1.74
.81 1.79 1.78
.84 1.83 1.82
.88 1.87 1.85
.93 1.92 1.90
.97 1.95 1.94
2.00 1.98 1.97
2.03 2.01 2.00
2.05 2.03 2.02 ;
).98 0.98 (
.14 1.14
.23 1.22
.29 1.28
.34 1.33
.43 1.42
.51 1.50
.57 1.56
.61 1.60
.68 1.67
.73 1.72
.77 1.76
.81 1.79
.84 1.83
.89 1.88
.93 1.91
.96 1.94
.99 1.97
2.01 1.99
).97
.13
.22
.28
.32
.42
.49
.55
.59
.66
.71
.75
.78
.82
.86
.90
.93
.95
.97
                                                    D-147
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
    Table 19-13. K-Multipliers  for Modified Calif. Intrawell Prediction Limits on Observations (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.80 1.36 1.20
2.40 1.72 1.49
2.81 1.94 1.66
3.13 2.11 1.78
3.40 2.24 1.88
4.02 2.53 2.09
4.65 2.80 2.28 ;
5.14 3.00 2.42 ;
5.55 3.17 2.53 ;
6.38 3.48 2.73 2
7.04 3.72 2.88 2
7.60 3.91 3.00 ,
8.09 4.07 3.10 ,
8.71 4.28 3.23 ;
9.61 4.55 3.39 ;
10.35 4.78 3.53 :
11.02 4.96 3.63 :
11.60 5.14 3.73 :
12.11 5.27 3.82 :
.12
.37
.52
.63
.71
.88
>.04
>.15
2.24 ;
>.40 ;
>.51 ;
?,eo :
2.68 :
1.18:
2.90 ,
3.00 .
3.08
3.15
3.21
.07 1.01 0.97 (
.30 1.22 1.17
.44 1.34 1.29
.53 1.42 1.37
.61 1.49 1.43
.76 1.63 1.55
.90 1.74 1.66
.99 1.82 1.73
>.07 1.89 1.79
1.21 2.00 1.89
>.31 2.09 1.97
2.39 2.15 2.03
2.45 2.20 2.07
?.53 2.27 2.13 ;
3,83 2.35 2.20 ;
2.71 2.41 2.26 ;
2.78 2.47 2.30 ;
2.84 2.51 2.34 ;
2.89 2.SS 2.38 ;
).94 0.93 0.91 (
.14 1.11 1.10
.25 1.22 1.20
.32 1.29 1.27
.38 1.35 1.32
.49 1.46 1.43
.59 1.55 1.53
.66 1.62 1.59
.72 1.67 1.64
.81 1.76 1.73
.88 1.83 1.79
.93 1.88 1.83
.98 1.91 1.87
2.03 1.96 1.92
2.09 2.03 1.98
2.14 2.07 2.02
2.19 2.11 2.06 ;
1.22 2.14 2.09 ;
1.25 2.17 2.12 ;
).90 0.90 0.89 (
.09 1.08 1.07
.19 1.18 1.17
.26 1.24 1.23
.31 1.30 1.29
.41 1.40 1.39
.50 1.49 1.48
.57 1.55 1.54
.62 1.60 1.58
.70 1.68 1.67
.76 1.74 1.72
.81 1.78 1.77
.84 1.82 1.80
.89 1.86 1.84
.95 1.92 1.90
.99 1.96 1.94
2.03 2.00 1.98
2.06 2.03 2.00
2.08 2.05 2.03
).88 0.88 (
.06 1.05
.15 1.15
.22 1.21
.27 1.26
.37 1.36
.46 1.45
.52 1.50
.56 1.55
.64 1.63
.70 1.68
.74 1.72
.77 1.76
.82 1.80
.87 1.85
.91 1.89
.94 1.92
.97 1.95
.99 1.97
).87 0.87 (
.04 1.04
.14 1.13
.20 1.20
.25 1.25
.35 1.35
.44 1.43
.49 1.48
.54 1.53
.61 1.61
.67 1.66
.71 1.70
.74 1.73
.78 1.77
.83 1.82
.87 1.86
.90 1.89
.93 1.92
.95 1.94
).87 (
.04
.13
.19
.24
.34
.42
.48
.52
.60
.65
.69
.72
.76
.81
.85
.88
.91
.93
).86 0.86
.03 1.03
.12 1.12
.19 1.18
.23 1.23
.33 1.32
.41 1.40
.47 1.46
.51 1.50
.59 1.58
.64 1.63
.68 1.67
.71 1.70
.75 1.74
.80 1.78
.83 1.82
.87 1.85
.89 1.88
.91 1.90
 Table 19-13. K-Multipliers  for Modified Calif.  Intrawell Prediction Limits on Observations (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.30 1.69 1.47 .36
301 2.08 1.77 .62
3.50 2.32 1.95 .77
3.89 2.50 2.08 .88
4.21 2.65 2.18 .96
4.97 2.97 2.40 2.14
5.72 3.27 2.60 2.30 ;
6.31 3.49 2.75 2.42 ;
6.81 3.68 2.87 2.51 ;
7.82 4.03 3.09 2.68 ;
8.62 4.29 3.25 2.80 ;
9.30 4.51 3.38 2.9O \
9.88 4.69 3.49 2.98 :
10.66 4.92 3.62 3.08 ;
11.76 5.23 3.80 3.21
12.66 5.49 3.95 3.31
13.44 5.70 4.06 3.40
14.14 5.90 4.17 3.47
14.84 6.05 4.26 3.54
.29 1.21 1.17
.53 1.42 1.36
.66 1.54 1.47
.76 1.62 1.55
.83 1.69 1.61
.99 1.82 1.73
2.13 1.94 1.84
1.23 2.02 1.91
2.30 2.08 1.97
2.45 2.20 2.07
2.55 2.28 2.14 ;
2.63 2.35 2.20 ;
1.70 2.40 2.25 ;
2.78 2.47 2.30 ;
?,89 2.55 2.38 ;
2.97 2.62 2.43 ;
3,04 2.67 2.48 ;
3.10 2.71 2.52 ;
3.15 2.75 2.55 ;
.13 1.11 1.10
.32 1.29 1.27
.42 1.39 1.37
.49 1.46 1.43
.55 1.51 1.48
.66 1.62 1.59
.76 1.71 1.68
.83 1.78 1.74
.88 1.83 1.79
.97 1.91 1.87
2.04 1.98 1.93
2.09 2.03 1.98
2.14 2.06 2.02
2.19 2.11 2.06 ;
2.25 2.17 2.12 ;
2.30 2.22 2.17 ;
2.34 2.26 2.20 ;
2.38 2.29 2.23 ;
2.41 2.32 2.26 ;
.08 1.08 1.07
.25 1.24 1.23
.35 1.34 1.33
.41 1.40 1.39
.46 1.45 1.44
.57 1.55 1.54
.65 1.63 1.62
.71 1.69 1.68
.76 1.74 1.72
.84 1.82 1.80
.90 1.88 1.86
.95 1.92 1.90
.98 1.95 1.93
2.03 2.00 1.98
2.08 2.05 2.03
1.12 2.09 2.07 ;
2.16 2.13 2.10 ;
2.19 2.16 2.13 ;
1.21 2.18 2.16 ;
.06 1.05
.22 1.21
.31 1.30
.37 1.36
.42 1.41
.52 1.50
.60 1.58
.65 1.64
.70 1.68
.77 1.76
.83 1.81
.87 1.85
.90 1.88
.94 1.92
.99 1.97
2.03 2.01
1.07 2.04 ;
2.09 2.07 ;
1.12 2.09 ;
.04 1.04
.20 1.20
.29 1.29
.35 1.35
.40 1.39
.49 1.48
.57 1.56
.63 1.62
.67 1.66
.74 1.73
.79 1.78
.83 1.82
.87 1.85
.90 1.89
.95 1.94
.99 1.98
1.02 2.01 ;
2.05 2.03 ;
1.07 2.06 ;
.04
.19
.28
.34
.39
.48
.56
.61
.65
.72
.77
.81
.84
.88
.93
.97
2.00
1.02 ;
2.04 ;
.03 1.03
.19 1.18
.27 1.27
.33 1.32
.38 1.37
.47 1.46
.54 1.54
.60 1.59
.64 1.63
.71 1.70
.76 1.75
.80 1.79
.83 1.82
.86 1.85
.91 1.90
.95 1.93
.98 1.96
2.00 1.99
1.02 2.01
                                                    D-148
                                                                                                    March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                      Unified Guidance
   Table 19-13. K-Multipliers  for Modified Calif. Intrawell Prediction Limits on Observations (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.84 2.02 1.74 1.60 1.52 1.41 1.36
3.68 2.45 2.05 1.86 1.75 1.62 1.55
425 2.71 2.24 2.02 1.88 1.74 1.65
471 2.91 2.38 2.13 1.98 1.82 1.73
509 3.07 2.49 2.22 2.06 1.88 1.79
5.99 3.43 2.73 2.40 2.22 2.02 1.91
6.89 3.77 2.94 2.57 2.36 2.13 2.01
7.60 4.02 3, W 2.69 2.46 2.22 2.09
8.20 4.22 3.23 2.79 2.54 2.28 2.14 ;
9.40 4.62 3.46 2.97 2.69 2.40 2.25 ;
10.36 4.91 3.64 3,09 2.80 2.48 2.32 ;
11.17 5.16 3.78 3, 2O 2.88 2.55 2.38 ;
11.88 5.36 3.89 3.28 2.95 2.60 2.42 ;
12.81 5.62 4.04 3.39 3.03 2.67 2.48 ;
14.10 5.98 4.23 3.53 3,15, 2.75 2.55 ;
15.20 6.26 4.39 3.63 3,23 2.82 2.61 ;
16.17 6.50 4.52 3.73 3,31 •. 2.88 2.66 ;
17.03 6.72 4.63 3.81 3.37 2.92 2.70 ;
17.81 6.91 4.74 3.88 3.43 2.96 2.73 ;
.32 1.29 1.27
.49 1.46 1.43
.59 1.55 1.52
.66 1.62 1.59
.71 1.67 1.64
.83 1.77 1.74
.92 1.86 1.83
.99 1.93 1.89
2.04 1.98 1.93
2.13 2.06 2.02
1.20 2.13 2.07 ;
2.25 2.17 2.12 ;
1.29 2.21 2.16 ;
1.34 2.26 2.20 ;
2.41 2.32 2.26 ;
2.46 2.37 2.30 ;
2.50 2.40 2.34 ;
2.54 2.44 2.37 ;
2.57 2.47 2.40 ;
.25 1.24 1.23
.41 1.40 1.39
.50 1.49 1.48
.57 1.55 1.54
.61 1.60 1.58
.71 1.69 1.68
.80 1.78 1.76
.86 1.83 1.81
.90 1.88 1.86
.98 1.95 1.93
2.04 2.01 1.99
2.08 2.05 2.03 ;
2.12 2.09 2.06 ;
2.16 2.13 2.10 ;
2.21 2.18 2.16 ;
2.26 2.22 2.19 ;
2.29 2.26 2.23 ;
2.32 2.29 2.26 ;
2.35 2.31 2.28 ;
.22 1.21
.37 1.36
.46 1.44
.52 1.50
.56 1.55
.65 1.64
.73 1.71
.79 1.77
.83 1.81
.90 1.88
.96 1.93
2.00 1.97
2.03 2.00
2.07 2.04 ;
2.12 2.09 ;
2.16 2.13 ;
2.19 2.16 ;
2.21 2.18 ;
2.24 2.21 ;
.20 1.20
.35 1.35
.44 1.43
.49 1.48
.54 1.53
.63 1.62
.70 1.69
.75 1.74
.79 1.78
.87 1.85
.92 1.90
.95 1.94
.98 1.97
2.02 2.01 ;
2.07 2.06 ;
2.11 2.09 ;
2.14 2.12 ;
2.16 2.15 ;
2.18 2.17 ;
.19
.34
.42
.48
.52
.61
.68
.73
.77
.84
.89
.93
.96
2.00
2.04 ;
2.08 ;
2.11 ;
2.13 ;
2.15 ;
.19 1.18
.33 1.32
.41 1.40
.47 1.46
.51 1.50
.60 1.59
.67 1.66
.72 1.71
.76 1.75
.83 1.82
.88 1.86
.91 1.90
.94 1.93
.98 1.96
2.02 2.00
2.05 2.04
2.08 2.07
2.11 2.09
2.13 2.11
   Table 19-13. K-Multipliers for Modified Calif. Intrawell Prediction Limits on Observations (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                        45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.40 1.72 1.49 .37
3.13 2.11 1.78 .63
3.63 2.35 1.96 .78
4.02 2.53 2.09 .88
4.36 2.68 2.20 .97
5.14 3.00 2.42 2.15
5.91 3.31 2,62 2.31 ;
6.52 3.53 2.76 2.42 ;
7.04 3.72 2.88 2.51 ;
8.08 4.07 3.10 2,68 '*
8.91 4.33 3.26 2.80
9.62 4.55 3.39 2.90 ,
10.21 4.74 3.50 2.98
11.04 4.97 3.64 3.08
12.11 5.27 3.82 3.21
13.09 5.54 3.96 3.32
13.87 5.76 4.08 3.41
14.65 5.96 4.17 3.48
15.23 6.10 4.27 3.54
.30 1.22 1.17
.53 1.42 1.37
.67 1.54 1.47
.76 1.63 1.55
.84 1.69 1.61
.99 1.82 1.73
2.13 1.94 1.84
2.23 2.02 1.91
2.31 2.09 1.97
2.45 2.20 2.07
2.55 2.29 2.15 ;
2.63 2.35 2.20 ;
2,70 2.40 2.25 ;
2.18 2.47 2.30 ;
>.89 2.55 2.38 ;
>.97 2,62 2.44 ;
3.04 2.67 2.48 ;
3.10 2,72 2,52' ;
3.15 2,76 2,55 ;
.14 1.11 1.10
.32 1.29 1.27
.42 1.39 1.37
.49 1.46 1.43
.55 1.51 1.48
.66 1.62 1.59
.76 1.71 1.68
.83 1.78 1.74
.88 1.83 1.79
.97 1.91 1.87
2.04 1.98 1.93
2.09 2.02 1.98
2.13 2.06 2.02
2.19 2.11 2.06 ;
2.25 2.17 2.12 ;
2.30 2.22 2.17 ;
2.34 2.26 2.20 ;
2.38 2.29 2.23 ;
2.41 2.32 2.26 ;
.09 1.08 1.07
.26 1.24 1.23
.35 1.34 1.33
.41 1.40 1.39
.46 1.45 1.44
.57 1.55 1.54
.65 1.63 1.62
.71 1.69 1.68
.76 1.74 1.72
.84 1.82 1.80
.90 1.88 1.86
.95 1.92 1.90
.98 1.95 1.93
2.03 2.00 1.97
2.08 2.05 2.03 ;
2.12 2.09 2.07 ;
2.16 2.13 2.11 ;
2.19 2.15 2.13 ;
2.22 2.18 2.15 ;
.06 1.05
.22 1.21
.31 1.30
.37 1.36
.42 1.41
.52 1.50
.60 1.58
.65 1.64
.70 1.68
.78 1.76
.83 1.81
.87 1.85
.90 1.88
.94 1.92
2.00 1.97
2.04 2.01
2.07 2.04 ;
2.09 2.07 ;
2.12 2.09 ;
.04 1.04
.20 1.20
.29 1.29
.35 1.35
.40 1.39
.49 1.49
.57 1.56
.63 1.62
.67 1.66
.74 1.73
.79 1.78
.83 1.82
.87 1.85
.90 1.89
.95 1.94
.99 1.98
2.02 2.01 ;
2.05 2.03 ;
2.07 2.05 ;
.04
.19
.28
.34
.39
.48
.56
.61
.65
.72
.77
.81
.84
.88
.93
.97
2.00
2.02 ;
2.04 ;
.03 1.03
.19 1.18
.27 1.27
.33 1.32
.38 1.37
.47 1.46
.54 1.54
.60 1.59
.64 1.63
.71 1.70
.76 1.75
.80 1.79
.83 1.82
.86 1.85
.91 1.90
.95 1.93
.98 1.96
2.00 1.99
2.02 2.01
                                                    D-149
                                                                                                     March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
Table 19-13. K-Multipliers for Modified Calif. Intrawell Prediction Limits on Observations (10 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.01 2.08 1.77 1.62 1.53 1.42 1.36
3.89 2.50 2.08 1.88 1.76 1.62 1.55
449 2.77 2.27 2.03 1.89 1.74 1.66
4.97 2.97 2.40 2.14 1.99 1.82 1.73
5.37 3.13 2.51 2.23 2.07 1.89 1.79
6.31 3.50 2.75 2.42 2.23 2.02 1.91
7.25 3.83 2.97 2.58 2.37 2.14 2.01
8.00 4.09 3.12 2.70 2.47 2.22 2.09
8.63 4.29 3.25 2.80 2.55 2.28 2.14 ;
9.89 4.69 3.49 2,98 2.70 2.40 2.25 ;
10.89 4.99 3.66 3.10 2.8O 2.48 2.32 ;
11.77 5.24 3.80 3.21 2,89 2.55 2.38 ;
12.50 5.44 3.92 3.29 2,95 2.60 2.42 ;
13.48 5.71 4.06 3.40 3.O4 2.67 2.48 ;
14.84 6.05 4.26 3.54 3.15 2.76 2.55 ;
16.02 6.35 4.42 3.65 3.24 2.82 2.61 ;
16.99 6.59 4.54 3.74 3.31 2.88 2.66 ;
17.97 6.84 4.66 3.81 3.37 2.93 2.70 ;
18.75 7.03 4.76 3.88 3.43 2.97 2.73 ;
.32 1.29 1.27
.49 1.46 1.43
.59 1.55 1.53
.66 1.62 1.59
.72 1.67 1.64
.83 1.78 1.74
.92 1.87 1.83
.99 1.93 1.89
2.04 1.98 1.93
2.13 2.06 2.02
1.20 2.13 2.08 ;
2.25 2.17 2.12 ;
1.29 2.21 2.16 ;
1.34 2.26 2.20 ;
2.41 2.32 2.26 ;
2.46 2.37 2.30 ;
2.50 2.40 2.34 ;
2.54 2.44 2.37 ;
2.56 2.47 2.39 ;
.25 1.24 1.23
.41 1.40 1.39
.50 1.49 1.48
.57 1.55 1.54
.61 1.60 1.58
.71 1.69 1.68
.80 1.78 1.76
.86 1.83 1.81
.90 1.88 1.86
.98 1.95 1.93
2.04 2.01 1.99
2.08 2.05 2.03 ;
2.12 2.09 2.06 ;
2.16 2.13 2.10 ;
2.22 2.18 2.15 ;
2.26 2.22 2.20 ;
2.29 2.26 2.23 ;
2.32 2.28 2.26 ;
2.34 2.31 2.28 ;
.22 1.21
.37 1.36
.46 1.45
.52 1.50
.56 1.55
.65 1.64
.73 1.72
.79 1.77
.83 1.81
.90 1.88
.96 1.93
2.00 1.97
2.03 2.00
2.07 2.04 ;
2.12 2.09 ;
2.15 2.13 ;
2.19 2.16 ;
2.22 2.19 ;
2.23 2.20 ;
.20 1.20
.35 1.35
.44 1.43
.49 1.48
.54 1.53
.63 1.62
.70 1.69
.75 1.74
.79 1.78
.87 1.85
.92 1.90
.95 1.94
.99 1.97
2.02 2.01 ;
2.07 2.06 ;
2.11 2.09 ;
2.14 2.12 ;
2.16 2.15 ;
2.19 2.17 ;
.19
.34
.42
.48
.52
.61
.68
.73
.77
.84
.89
.93
.96
2.00
2.04 ;
2.08 ;
2.11 ;
2.13 ;
2.15 ;
.19 1.18
.33 1.32
.41 1.40
.47 1.46
.51 1.50
.60 1.59
.67 1.66
.72 1.71
.76 1.75
.83 1.82
.88 1.86
.91 1.90
.94 1.93
.98 1.96
2.02 2.00
2.05 2.04
2.09 2.07
2.11 2.09
2.13 2.11
  Table 19-13. K-Multipliers  for Modified Calif. Intrawell Prediction Limits on Observations (10 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
3.68
4.71
5.42
5.99
6.47
7.60
8.72
9.61
10.36
11.89
13.09
14.11
14.99
16.16
17.77
19.14
20.31
21.48
22.46

6
2.45
2.91
3.21
3.43
3.61
4.02
4.40
4.68
4.91
5.36
5.70
5.97
6.20
6.49
6.91
7.23
7.52
7.76
7.96

8 10 12 16 20 25 30
2.05 1.86 1.75 1.62 1.55 .49 1.46
2.38 2.13 1.98 1.82 1.73 .66 1.62
2.58 2.29 2.12 1.93 1.83 .76 1.71
2.73 2.40 2.22 2.02 1.91 .83 1.77
2.84 2.50 2.30 2.08 1.96 .88 1.82
3.10 2.69 2.46 2.22 2.09 .99 1.93
333 2.87 2.61 2.33 2.19 2.08 2.02
350 2.99 2.71 2.42 2.26 2.15 2.08
3.64 3.09 2.80 2.48 2.32 2.20 2.13
3.89 3.28 2.95 2.60 2.42 2.29 2.21
4.08 3.42 3.06 2.69 2.50 2.36 2.27
4.24 3.52 . "3.15' 2.75 2.55 2.41 2.32
4.36 3.61 3.22 2.81 2.60 2.45 2.36
4.52 3.73 3.3.1 2.87 2.66 2.50 2.40
4.74 3.88 3.42 2.97 2.73 2.57 2.47
4.91 3.99 3.52 3.03 2.79 2.62 2.51
5.05 4.09 3.59 3.O9 2.84 2.66 2.55
5.18 4.17 3.66 3.-14 2.88 2.69 2.58
5.27 4.25 3.71 3,77 2.92 2.72 2.61

35
1.43
1.59
1.68
1.74
1.79
1.89
1.97
2.03
2.07 ;
2.16 ;
2.21 ;
2.26 ;
2.29 ;
2.34 ;
2.40 ;
2.44 ;
2.48 ;
2.50 ;
2.53 ;

40
.41
.57
.65
.71
.76
.86
.94
.99
2.04
2.12
2.17
2.21
2.25
2.29
2.35
2.39
2.42
2.45
2.48

45
1.40
1.55
1.63
1.69
1.74
1.83
1.91
1.97
2.01
2.09
2.14
2.18
2.22
2.26
2.31
2.35
2.38
2.41
2.44

50 60
1.39 .37
1.54 .52
1.62 .60
1.68 .65
1.72 .70
1.81 .79
1.89 .86
1.95 .91
1.99 .96
2.06 2.03
2.11 2.08
2.15 2.12
2.19 2.15
2.23 2.19
2.28 2.24
2.32 2.27
2.35 2.30
2.38 2.33
2.40 2.35

70 80
1.36 .35
1.50 .49
1.58 .57
1.64 .63
1.68 .67
1.77 .75
1.84 .83
1.89 .88
1.93 .92
2.00 .99
2.05 2.03
2.09 2.07
2.12 2.10
2.16 2.14
2.21 2.19
2.24 2.22
2.27 2.25
2.29 2.27
2.32 2.29

90
1.35
1.48
1.56
1.62
1.66
1.74
1.82
1.87
1.90
1.97
2.02 ;
2.06 ;
2.08 ;
2.12 ;
2.17 ;
2.20 ;
2.23 ;
2.26 ;
2.28 ;

100 125
.34 .33
.48 .47
.56 .54
.61 .60
.65 .64
.73 .72
.81 .79
.86 .84
.89 .87
.96 .94
2.01 .99
2.04 2.02
2.07 2.04
2.11 2.09
2.15 2.13
2.19 2.16
2.22 2.19
2.24 2.22
2.26 2.23

150
1.32
1.46
1.54
1.59
1.63
1.71
1.78
1.83
1.86
1.93
1.97
2.00
2.04
2.07
2.11
2.15
2.17
2.20
2.22

                                                    D-150
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
                                                                     Unified Guidance
   Table 19-13. K-Multipliers  for Modified Calif. Intrawell Prediction Limits on Observations (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.13 2.11 1.78 1.63 1.53 1.42 1.37
4.02 2.53 2.09 1.88 1.76 1.63 1.55
4.65 2.80 2.28 2.04 1.90 1.74 1.66
5.14 3.00 2.42 2.15 1.99 1.82 1.73
5.55 3.17 2.53 2.24 2.07 1.89 1.79
6.52 3.53 2.76 2.42 2.23 2.02 1.91
7.49 3.87 2.98 2,59 2.37 2.14 2.01
8.26 4.13 3.14 2. 71 2.47 2.22 2.09
8.91 4.34 3.26 2.81 '2,55" 2.29 2.14 ;
10.22 4.74 3.50 2.98 2,70 2.40 2.25 ;
11.25 5.04 3.68 3.11 2.81 2.49 2.32 ;
12.13 5.29 3.82 3.21 2.89 "2,'55" 2.38 ;
12.89 5.49 3.93 3.30 2.96 2.6O 2.42 ;
13.89 5.76 4.08 3.41 3.04 2,67 2.48 ;
15.35 6.12 4.28 3.54 3.16 .. 2,76 . 2,56 ;
16.52 6.42 4.42 3.65 3.24 2.83 ,2-67 ;
17.58 6.65 4.57 3.75 3.31 2.88 2.66 '.',
18.52 6.86 4.69 3.82 3.38 2.93 2,70 ,
19.22 7.09 4.78 3.90 3.43 2.97 2,73 ,
.32 1.29 1.27
.49 1.46 1.43
.59 1.55 1.53
.66 1.62 1.59
.72 1.67 1.64
.83 1.78 1.74
.92 1.87 1.83
.99 1.93 1.89
>.04 1.98 1.93
M4 2.06 2.02
>.20 2.13 2.07 ;
2.25 2.17 2.12 ;
1.29 2.21 2.16 ;
1.34 2.26 2.20 ;
>.41 2.32 2.26 ;
2.46 2.37 2.30 ;
2.50 •", 2.40 2.34 ;
2,53 2.44 2.37 ;
2,56 2.47 2.40 ;
.26 1.24 1.23
.41 1.40 1.39
.50 1.49 1.48
.57 1.55 1.54
.62 1.60 1.58
.71 1.69 1.68
.80 1.78 1.76
.86 1.83 1.81
.90 1.88 1.86
.98 1.95 1.93
>.04 2.01 1.99
>.08 2.05 2.03 ;
1.12 2.09 2.06 ;
M6 2.13 2.10 ;
1.22 2.18 2.16 ;
1.26 2.22 2.20 ;
1.29 2.26 2.23 ;
1.32 2.29 2.26 ;
1.34 2.31 2.28 ;
.22 1.21
.37 1.36
.46 1.45
.52 1.50
.56 1.55
.66 1.64
.73 1.72
.79 1.77
.83 1.81
.90 1.88
.96 1.93
>.00 1.97
>.03 2.00
1.07 2.04 ;
1.12 2.09 ;
M5 2.13 ;
M9 2.16 ;
1.21 2.18 ;
1.23 2.20 ;
.20 1.20
.35 1.35
.44 1.43
.49 1.49
.54 1.53
.63 1.62
.70 1.69
.75 1.74
.79 1.78
.87 1.85
.92 1.90
.95 1.94
.98 1.97
1.02 2.01 :
1.07 2.05 ;
Ml 2.09 ;
M4 2.12 ;
M6 2.15 ;
M8 2.17 ;
.19
.34
.42
.48
.52
.61
.68
.73
.77
.84
.89
.93
.96
>.oo
>.04 ;
>.os ;
>.n ;
M3 ;
MS ;
.19 1.18
.33 1.32
.41 1.40
.47 1.46
.51 1.50
.60 1.59
.67 1.66
.72 1.71
.76 1.75
.83 1.82
.88 1.86
.91 1.90
.94 1.93
.98 1.96
1.02 2.00
1.05 2.03
>.08 2.07
Ml 2.09
M3 2.12
Table 19-13. K-Multipliers for Modified Calif.  Intrawell Prediction Limits on Observations (20 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
3.89
4.97
5.72
6.31
6.81
8.00
9.18
10.11
10.90
12.48
13.77
14.82
15.76
16.99
18.75
20.16
21.33
22.50
23.44

6
2.50
2.97
3.27
3.50
3.68
4.09
4.47
4.76
4.99
5.44
5.79
6.06
6.30
6.59
7.00
7.32
7.62
7.85
8.09

8 10 12 16 20 25 30
2.08 1.88 1.76 1.62 1.55 .49 1.46
2.40 2.14 1.99 1.82 1.73 .66 1.62
2.60 2.30 2.13 1.94 1.84 .76 1.71
2.75 2.42 2.23 2.02 1.91 .83 1.78
2,87 2.51 2.30 2.08 1.97 .88 1.83
312 2.70 2.47 2.22 2.09 .99 1.93
3.35 2.88 2.61 2.34 2.19 2.08 2.02
3.52 3.00 2.72 2.42 2.26 2.15 2.08
3.66 3.10 2.8O 2.48 2.32 2.20 2.13
3.92 3.29 2.96 2.60 2.42 2.29 2.21
4.11 3.43 3.07 2.69 2.50 2.36 2.27
4.26 3.54 3.15 2.76 2.55 2.41 2.32
4.39 3.63 3.22 2,81 2.60 2.45 2.36
4.55 3.74 3.31 2.88 2.66 2.50 2.41
4.76 3.88 3.43 2,97 . 2.73 2.57 2.46
4.94 4.00 3.52 3,03 2,79 2.62 2.51
5.07 4.10 3.60 .3.O9 2.84 2.66 2.55
5.20 4.19 3.66 3.13 2,88 2.70 2.58
5.30 4.25 3.72 3.18 2,92 2.72 2.61

35
1.43
1.59
1.68
1.74
1.79
1.89
1.97
2.03
2.07 ;
2.16 ;
2.21 ;
2.26 ;
2.30 ;
2.34 ;
2.40 ;
2.44 ;
2.48 ;
2.50 ;
2.53 ;

40
.41
.57
.65
.71
.76
.86
.94
.99
>.04
M2
M7
1.22
1.25
1.29
1.35
1.39
1.42
1.45
1.48

45
1.40
1.55
1.63
1.69
1.74
1.83
1.91
1.97
2.01
2.09
2.14
2.18
2.22
2.26
2.31
2.35
2.38
2.41
2.43

50 60
1.39 .37
1.54 .52
1.62 .60
1.68 .65
1.72 .70
1.81 .79
1.89 .86
1.95 .92
1.99 .96
2.06 2.03
2.11 2.08
2.16 2.12
2.19 2.15
2.23 2.19
2.28 2.24
2.32 2.27
2.35 2.31
2.38 2.33
2.40 2.35

70 80
1.36 .35
1.50 .49
1.58 .57
1.64 .63
1.68 .67
1.77 .75
1.84 .83
1.89 .88
1.93 .92
2.00 .98
2.05 2.03
2.09 2.07
2.12 2.10
2.16 2.14
2.20 2.18
2.24 2.22
2.27 2.25
2.30 2.27
2.32 2.29

90
1.35
1.48
1.56
1.62
1.66
1.74
1.82
1.86
1.90
1.97
2.02 ;
2.06 ;
2.08 ;
2.12 ;
2.17 ;
2.20 ;
2.23 ;
2.26 ;
2.28 ;

100 125
.34 .33
.48 .47
.56 .54
.61 .60
.65 .64
.73 .72
.81 .79
.86 .84
.89 .88
.96 .94
>.01 .99
>.04 2.02
1.07 2.04
Ml 2.09
M5 2.13
M9 2.16
1.22 2.19
1.24 2.22
1.26 2.23

150
1.32
1.46
1.54
1.59
1.63
1.71
1.78
1.83
1.86
1.93
1.97
2.00
2.04
2.07
2.11
2.15
2.18
2.20
2.22

                                                    D-151
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
  Table 19-13. K-Multipliers  for Modified Calif. Intrawell Prediction Limits on Observations (20 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.71
5.99
6.89
7.60
8.20
9.61
11.02
12.14
13.08
15.00
16.52
17.81
18.93
20.39
22.50
24.14
25.78
27.19
28.12
6
2.91
3.43
3.77
4.02
4.22
4.68
5.11
5.43
5.70
6.20
6.59
6.90
7.18
7.50
7.97
8.35
8.67
8.96
9.20
8 10 12 16 20 25 30 35 40
2.38 2.13 1.98 1.82 1.73 1.66 1.62 1.59 .57
2.73 2.40 2.22 2.02 1.91 1.83 1.77 1.74 .71
2.94 2.57 2.36 2.13 2.01 1.92 1.86 1.83 .80
3.70 2.69 2.46 2.22 2.09 1.99 1.93 1.89 .86
3.23 2.79 2.54 2.28 2.14 2.04 1.98 1.93 .90
350 2.99 2.71 2.42 2.26 2.15 2.08 2.03 .99
3.75 3.18 2.86 2.54 2.37 2.24 2.16 2.11 2.07
3.93 3.31 2.97 2.62 2.44 2.31 2.23 2.17 2.13
408 3.42 3.06 2.69 2.50 2.36 2.27 2.21 2.17
4.36 3.61 3,22 2.81 2.60 2.45 2.36 2.30 2.25
4.57 3.76 3.33 2.90 2.68 2.52 2.42 2.35 2.30
4.73 3.87 3.42 2.96 2.73 2.57 2.46 2.40 2.35
4.87 3.97 3.50 3.02 2.78 2.61 2.50 2.43 2.38
5.05 4.09 3.59 3.O9 2.84 2.66 2.55 2.48 2.42
5.27 4.25 3.71 3.18 2.92 2.72 2.61 2.53 2.48
5.46 4.37 3.81 3,25 2.97 2.78 2.66 2.57 2.52
5.62 4.48 3.90 3,31 3.02 2.82 2.70 2.61 2.55
5.77 4.57 3.96 3,35 3.06 2.85 2.72 2.64 2.58
5.86 4.64 4.01 3,40 3.10 2.89 2.75 2.67 2.60
45
1.55
1.69
1.78
1.83
1.88
1.97
2.04
2.10
2.14
2.21
2.27
2.31
2.34
2.38
2.44
2.48
2.50
2.53
2.56
50
1.54
1.68
1.76
1.81
1.86
1.95
2.02
2.07 ;
2.11 ;
2.19 ;
2.24 ;
2.28 ;
2.3i ;
2.35 ;
2.40 ;
2.44 ;
2.47 ;
2.50 ;
2.52 ;
60
.52
.65
.73
.79
.83
.91
.99
>.04
>.08
>.15
1.20
1.24
1.27
1.30
1.35
1.39
1.42
1.45
2.47
70
1.50
1.64
1.71
1.77
1.81
1.89
1.96
2.01 ;
2.05 ;
2.12 ;
2.17 ;
2.21 ;
2.24 ;
2.27 ;
2.32 ;
2.35 ;
2.38 ;
2.4i ;
2.43 ;
80
.49
.63
.70
.75
.79
.88
.95
>.oo
>.03
>.10
>.15
>.18
>.21
1.25
1.29
1.33
1.36
1.38
1.40
90
1.48
1.62
1.69
1.74
1.78
1.86
1.93
1.98
2.02 ;
2.08 ;
2.13 ;
2.17 ;
2.20 ;
2.23 ;
2.27 ;
2.3i ;
2.34 ;
2.36 ;
2.38 ;
100 125
.48 .47
.61 .60
.68 .67
.73 .72
.77 .76
.86 .84
.92 .90
.97 .95
>.01 .99
1.07 2.04
1.12 2.10
>.15 2.13
>.18 2.16
1.22 2.19
1.26 2.23
1.30 2.27
1.32 2.29
1.34 2.32
1.37 2.34
150
1.46
1.59
1.66
1.71
1.75
1.83
1.89
1.94
1.97
2.04
2.08
2.11
2.14
2.18
2.22
2.25
2.27
2.30
2.32
   Table 19-13. K-Multipliers  for Modified Calif. Intrawell Prediction Limits on Observations (40 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.02
5.14
5.91
6.52
7.04
8.26
9.48
10.44
11.25
12.90
14.20
15.31
16.27
17.54
19.30
20.78
22.11
23.28
24.38
6
2.53
3.00
3.31
3.53
3.72
4.13
4.51
4.80
5.04
5.49
5.84
6.12
6.36
6.66
7.07
7.40
7.70
7.93
8.16
8
2.09
2.42
2.82
2.76
2.88
3.14
3.37
3.54
3.68
3.93
4.12
4.27
4.40
4.57
4.78
4.95
5.09
5.21
5.32
10
1.88
2.15
2.31
2.42
2.51
2,71
2.88
3.01
3.11
3.30
3.43
3.54
3.63
3.74
3.89
4.01
4.11
4.19
4.26
12
1.76
1.99
2.13
2.23
2.31
2.47
2.62
2.72
2.81
2.96
3.07
3.15
3.23
3.32
3.43
3.53
3.60
3.66
3.72
16
1.63
1.82
1.94
2.02
2.09
2.22
2.34
2.42
2.49
2,60
2.69
. 2.78
2.81
2.88
2.97
3.03
3.09
3.14
3.18
20
1.55
1.73
1.84
1.91
1.97
2.09
2.19
2.26
2.32
2.42
2.50
'2,55"
2.60
2,66'-
2,73
2,79
2.84
2.88
2.92
25
.49
.66
.76
.83
.88
.99
2.08
2.15
2.20
2.29
2.36
2.41
2.45
2. SO
2:57
,2,62
2,66
2.70
2,72:-
30
1.46
1.62
1.71
1.78
1.83
1.93
2.02
2.08
2.13
2.21
2.27
2.32
2.36
2.40
2.46
2,5-1
-2,55
2.58
2,61
35 40 45 50 60 70 80 90 100 125
1.43 .41 1.40 1.39 .37 1.36 .35 1.35 .34 .33
1.59 .57 1.55 1.54 .52 1.50 .49 1.49 .48 .47
1.68 .65 1.64 1.62 .60 1.58 .57 1.56 .56 .54
1.74 .71 1.69 1.68 .66 1.64 .63 1.62 .61 .60
1.79 .76 1.74 1.72 .70 1.68 .67 1.66 .65 .64
1.89 .86 1.83 1.81 .79 1.77 .75 1.74 .74 .72
1.97 .94 1.91 1.89 .86 1.84 .83 1.82 .81 .79
2.03 .99 1.97 1.95 .92 1.89 .88 1.86 .86 .84
2.07 2.04 2.01 1.99 .96 1.93 .92 1.90 .89 .87
2.16 2.12 2.09 2.06 2.03 2.00 .99 1.97 .96 .94
2.21 2.17 2.14 2.11 2.08 2.05 2.03 2.02 2.01 .99
2.26 2.21 2.18 2.16 2.12 2.09 2.07 2.05 2.04 2.02
2.29 2.25 2.21 2.19 2.15 2.12 2.10 2.08 2.07 2.04
2.34 2.29 2.26 2.23 2.19 2.16 2.14 2.12 2.11 2.08
2.40 2.35 2.31 2.28 2.24 2.21 2.18 2.17 2.15 2.13
2.44 2.39 2.35 2.32 2.27 2.24 2.22 2.20 2.19 2.16
2.48 2.42 2.38 2.35 2.30 2.27 2.25 2.23 2.22 2.19
2,50 2.45 2.41 2.38 2.33 2.30 2.28 2.26 2.25 2.21
2,53 2.48 2.43 2.40 2.35 2.32 2.30 2.28 2.27 2.23
150
1.32
1.46
1.54
1.59
1.63
1.71
1.78
1.83
1.86
1.93
1.97
2.00
2.02
2.07
2.11
2.15
2.18
2.20
2.22
                                                   D-152
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations
Unified Guidance
Table 19-13. K- Multipliers for Modified Calif. Intrawell Prediction Limits on
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6
4.97 2.97
6.31 3.49
7.25 3.83
8.00 4.09
8.63 4.29
10.11 4.76
11.59 5.19
12.76 5.52
13.76 5.79
15.76 6.30
17.34 6.69
18.71 7.01
19.88 7.28
21.41 7.62
23.59 8.09
25.47 8.48
27.03 8.79
28.44 9.06
29.69 9.30
8 10
2.40 2.14
2.75 2.42
2.97 2.58
3.12 2.70
3.25 2.80
3.53 3.OO
3.78 3.19
3.96 3.32
4.11 3.43
4.39 3.63
4.59 3.77
4.76 3.89
4.90 3.98
5.08 4.10
5.31 4.26
5.49 4.38
5.64 4.49
5.78 4.57
5.90 4.65
Table 19-13. K-Multipliers
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6
5.99 3.43
7.60 4.02
8.72 4.40
9.61 4.68
10.36 4.91
12.14 5.43
13.91 5.92
15.32 6.29
16.50 6.59
18.91 7.17
20.82 7.61
22.42 7.97
23.83 8.28
25.70 8.67
28.28 9.18
30.47 9.61
32.34 10.00
34.06 10.31
35.62 10.59
8 10
2.73 2.40
3.W 2.69
3.33 2.87
3.50 2.99
3.64 3.O9
3.93 3.31
4.21 3.51
4.41 3.65
4.57 3.76
4.87 3.97
5.10 4.12
5.28 4.25
5.43 4.35
5.62 4.47
5.88 4.64
6.07 4.78
6.25 4.88
6.41 4.98
6.52 5.06
12 16
1.99 1.82
2.23 2.02
2.37 2.14
2.47 2.22
2.55 2.28
2.72 2.42
2,87 2.54
2,98 2.62
3.07 2.69
3.22 2.81
3.34 2.9O
3.43 2,37
3.51 3,02
3.60 3.O9
3.72 3.18
3.82 3.25
3.90 3.31
3.96 3.36
4.02 3.40
20 25
1.73 1.66
1.91 1.83
2.01 1.92
2.09 1.99
2.14 2.04
2.26 2.15
2.37 2.24
2.44 2.31
2.50 2.36
2.60 2.45
2.68 2.52
2.73 2.57
2.78 2.61
2,84 2.66
2.92 2.72
.- 2,97 2.77
3,02 2,82
3.O6 2.85
3. 1O 2,88
30 35
1.62 1.59
1.78 1.74
1.87 1.83
1.93 1.89
1.98 1.93
2.08 2.03
2.16 2.11
2.23 2.17
2.27 2.21
2.36 2.29
2.42 2.35
2.46 2.40
2.50 2.43
2.55 2.48
2.61 2.53
2.66 2.57
2.70 2.61
2.72 2.64
2.75 2.67
40
.57
.71
.80
.86
.90
.99
2.07
2.13
2.17
2.25
2.30
2.35
2.38
2.42
2.48
2.52
2.55
2.58
2.60
45 50
1.55 1.54
1.69 1.68
1.78 1.76
1.83 1.81
1.88 1.86
1.97 1.95
2.04 2.02
2.10 2.07
2.14 2.11
2.21 2.19
2.27 2.24
2.31 2.28
2.34 2.31
2.38 2.35
2.43 2.40
2.47 2.44
2.50 2.47
2.53 2.50
2.56 2.52
Observations (40 COC, Semi-Annual)
60 70
.52 1.50
.65 1.64
.73 1.72
.79 1.77
.83 1.81
.91 1.89
.99 1.96
2.04 2.01
2.08 2.05
2.15 2.12
2.20 2.17
2.24 2.21
2.27 2.24
2.30 2.27
2.35 2.32
2.39 2.36
2.42 2.39
2.45 2.41
2.47 2.43
80 90
.49 1.48
.63 1.62
.70 1.69
.75 1.74
.79 1.78
.88 1.86
.95 1.93
2.00 1.98
2.03 2.02
2.10 2.08
2.15 2.13
2.18 2.17
2.21 2.20
2.25 2.23
2.29 2.28
2.33 2.31
2.36 2.34
2.39 2.36
2.41 2.39
for Modified Calif. Intrawell Prediction Limits on Observations (40
12 16
2.22 2.02
2.46 2.22
2.61 2.33
2.71 2.42
2.80 2.48
2.97 2.62
3'. 13' 2.74
3,24 2.83
3,33 2.90
3.50 3.02
3.62 3.11
3.72 3.-18'
3.79 3.24.
3.89 3,31
4.02 3,40
4.12 3.47
4.20 3.54
4.28 3.58
4.34 3.62
20 25
1.91 1.83
2.09 1.99
2.19 2.08
2.26 2.15
2.32 2.20
2.44 2.31
2.54 2.40
2.62 2.47
2.67 2.52
2.78 2.61
2.85 2.67
2.91 2.72
2.96 2.77
3.02 2.82
3, 70 2.88
3/15 2.93
3.2Q 2.97
3,25 3.01
3.28 3.04
30 35
1.77 1.74
1.93 1.89
2.02 1.97
2.08 2.03
2.13 2.07
2.23 2.17
2.31 2.25
2.37 2.31
2.42 2.35
2.50 2.43
2.56 2.49
2.61 2.53
2.65 2.57
2.69 2.61
2.75 2.67
2.80 2.71
2.84 2.74
2.87 2.77
2.90 2.80
40
1.71
1.86
1.94
1.99
2.04
2.13
2.21
2.26
2.30
2.38
2.43
2.48
2.51
2.55
2.60
2.64
2.68
2.71
2.73
45 50
1.69 1.68
1.83 1.81
1.91 1.89
1.97 1.95
2.01 1.99
2.10 2.07
2.17 2.15
2.23 2.20
2.27 2.24
2.34 2.31
2.39 2.36
2.43 2.40
2.47 2.43
2.51 2.47
2.56 2.52
2.60 2.56
2.63 2.59
2.66 2.62
2.68 2.64
60 70
.65 1.64
.79 1.77
.86 1.84
.91 1.89
.96 1.93
2.04 2.01
2.11 2.08
2.16 2.13
2.20 2.17
2.27 2.24
2.32 2.28
2.35 2.32
2.38 2.35
2.42 2.38
2.47 2.43
2.50 2.47
2.53 2.50
2.56 2.52
2.58 2.54
80 90
.63 .62
.75 .74
.83 .82
.88 .86
.92 .90
2.00 .98
2.06 2.05
2.11 2.09
2.15 2.13
2.21 2.20
2.26 2.24
2.29 2.28
2.32 2.30
2.36 2.34
2.40 2.38
2.44 2.41
2.47 2.44
2.49 2.47
2.51 2.49
100 125
.48 .47
.61 .60
.68 .67
.73 .72
.77 .76
.86 .84
.92 .90
.97 .95
2.01 .99
2.07 2.04
2.12 2.10
2.15 2.13
2.18 2.16
2.22 2.19
2.26 2.23
2.29 2.27
2.32 2.29
2.34 2.32
2.36 2.34
150
1.46
1.59
1.66
1.71
1.75
1.83
1.89
1.94
1.97
2.02
2.08
2.11
2.14
2.18
2.22
2.25
2.28
2.30
2.32
COC, Quarterly)
100 125
1.61 .60
1.73 .72
1.81 .79
1.86 .84
1.89 .87
1.97 .95
2.04 2.02
2.08 2.06
2.12 2.10
2.18 2.16
2.23 2.20
2.26 2.23
2.29 2.26
2.32 2.29
2.36 2.33
2.40 2.37
2.42 2.39
2.45 2.42
2.47 2.44
150
1.59
1.71
1.78
1.83
1.86
1.94
2.00
2.05
2.08
2.14
2.18
2.22
2.24
2.27
2.31
2.34
2.37
2.40
2.41

                                                     D-153
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Observations                                         Unified Guidance
                                    This page intentionally left blank
                                                     D-154                                            March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means	Unified Guidance





                                          D  STATISTICAL TABLES








D.3 TABLES FROM CHAPTER 19:  INTRAWELL PREDICTION LIMITS  FOR FUTURE  MEANS








    TABLE 19-14 /c-Multipliers for 1 -of-1 Intrawell Prediction Limits on Mean Order 2	D-156



    TABLE 19-15 /c-Multipliers for 1 -of-2 Intrawell Prediction Limits on Mean Order 2	 D-165



    TABLE 19-16 K-Multipliers for 1 -of-3 Intrawell Prediction Limits on Mean Order 2	D-174



    TABLE 19-17 /c-Multipliers for 1-of-1 Intrawell Prediction Limits on Mean Order 3	 D-183



    TABLE 19-18 /c-Multipliers for 1 -of-2 Intrawell Prediction Limits on Mean Order 3	 D-192
                                                        D-1 55                                               March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-14. K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 2 (1 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
1.42 1.20 1.12 1.07 .04 .01 0.98 (
2.01 1.63 1.48 1.41 .36 .30 .27
2.41 1.88 1.70 1.60 .54 .47 .43
2.71 2.07 1.85 1.73 .66 .58 .54
2.97 2.22 1.97 1.84 .76 .67 .62
355 2.55 2.22 2.06 .96 .85 .79
413 2.85 2.45 2.25 2.13 2.00 .93
4.59 3.07 2.61 2.39 2.26 2.11 2.03
4.97 3.25 2.74 2.49 2.35 2.19 2.10 ;
5.74 3.59 2.98 2.69 2.52 2.34 2.24 ;
6.34 3.85 3.15 2.83 2.65 2.45 2.34 ;
6.85 4.06 3.29 2.94 2.75 2.53 2.41 ;
7.30 4.23 3.41 3.O4 2.83 2.59 2.47 ;
7.88 4.46 3.56 3,15 2.92 2.68 2.54 ;
8.69 4.76 3.75 3.30 3, OS 2.78 2.64 ;
9.38 5.00 3.90 3.42 '3,15 2.86 2.71 ;
9.98 5.21 4.03 3.52 3.23 2.93 2.77 ;
10.51 5.39 4.14 3.60 3.30 "'2.99' 2.82 ;
11.00 5.54 4.24 3.67 3.36 • 3;Q4 2.86 ;
25
).97 (
.25
.40
.50
.58
.74
.88
.97
2.04 ;
2.17 ;
1.26 :
1.32 :
i.38 ;
2.45 ;
2.53 ;
2.60 ;
2.65 ;
2.70 ;
2.74 ;
30
).96 (
.23
.38
.48
.56
.71
.84
.93
2.00
2.12 ;
1.20 ;
2.27 ;
2.32 ;
2.38 ;
2.47 ;
2.53 ;
2.58 ;
2.62 ;
2.66 ;
35
).95 (
.22
.37
.46
.54
.69
.82
.90
.97
2.09 ;
2.17 ;
2.23 ;
2.28 ;
2.34 ;
2.42 ;
2.48 ;
2.53 ;
2.57 ;
2.60 ;
40
).94 (
.21
.35
.45
.53
.67
.80
.88
.95
2.06 ;
2.14 ;
2.20 ;
2.25 ;
2.3i ;
2.39 ;
2.44 ;
2.49 ;
2.53 ;
2.56 ;
45
).94 (
.20
.35
.44
.52
.66
.78
.87
.93
2.04 ;
2.12 ;
2.18 ;
2.23 ;
2.29 ;
2.36 ;
2.42 ;
2.46 ;
2.50 ;
2.53 ;
50 60 70 80 90 100
).94 0.93 0.93 0.93 0.92 0.92 (
.20 1.19 1.19 1.18 1.18 1.18
.34 1.33 1.32 1.32 1.32 1.31
.44 1.43 1.42 1.41 1.41 1.40
.51 1.50 1.49 1.48 1.48 1.47
.65 1.64 1.63 1.62 1.62 1.61
.77 1.76 1.75 1.74 1.73 1.73
.86 1.84 1.83 1.82 1.81 1.80
.92 1.90 1.89 1.88 1.87 1.86
2.03 2.01 1.99 1.98 1.97 1.97
2.11 2.08 2.07 2.05 2.05 2.04 ;
2.16 2.14 2.12 2.11 2.10 2.09 ;
2.21 2.19 2.17 2.15 2.14 2.14 ;
2.27 2.24 2.22 2.21 2.20 2.19 ;
2.34 2.31 2.29 2.28 2.26 2.26 ;
2.40 2.37 2.34 2.33 2.32 2.31 ;
2.44 2.41 2.39 2.37 2.36 2.35 ;
2.48 2.45 2.42 2.41 2.39 2.38 ;
2.51 2.48 2.45 2.44 2.42 2.41 ;
125
).92 (
.17
.31
.40
.47
.60
.72
.79
.85
.95
2.02 ;
2.08 ;
2.12 ;
2.17 ;
2.24 ;
2.29 ;
2.33 ;
2.36 ;
2.39 ;
150
).92
.17
.30
.39
.46
.60
.71
.79
.84
.95
2.02
2.07
2.11
2.16
2.23
2.28
2.32
2.35
2.38
   Table 19-14. K-Multipliers  for 1-of-1  Intrawell Prediction Limits on Means of Order 2 (1  COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
1.89
2.57
3.03
3.39
3.68
4.38
5.07
5.61
6.07
6.99
7.72
8.33
8.87
9.57
10.55
11.38
12.10
12.74
13.33

6
1.58
2.02
2.29
2.50
2.66
3.02
3.35
3.59
3.79
4.17
4.46
4.69
4.89
5.14
5.47
5.75
5.98
6.18
6.36

8 10 12 16 20 25 30 35 40 45 50 60 70 80 90 100 125 150
1.45 1.38 1.34 .29 .26 .24 .23 .22 .21 .20 .20 1.19 1.18 1.18 1.18 1.18 .17 .17
1.82 1.71 1.65 .57 .53 .50 .48 .46 .45 .44 .43 1.42 1.42 1.41 1.41 1.40 .40 .39
2.04 1.91 1.83 .73 .68 .64 .61 .60 .58 .57 .56 1.55 1.54 1.54 1.53 1.53 .52 .52
2.19 2.04 1.95 .84 .78 .74 .71 .69 .67 .66 .65 1.64 1.63 1.62 1.62 1.61 .60 .60
2.32 2.15 2.04 .93 .86 .81 .78 .76 .74 .73 .72 1.70 1.69 1.69 1.68 1.67 .67 .66
2.58 2.37 2.25 2.10 2.03 .97 .93 .90 .88 .87 .86 1.84 1.83 1.82 1.81 1.80 .79 .79
2.82 2.57 2.42 2.25 2.16 2.09 2.05 2.02 2.00 .98 .97 1.95 1.93 1.92 1.92 1.91 .90 .89
2.99 2.71 2.54 2.36 2.26 2.18 2.14 2.10 2.08 2.06 2.05 2.02 2.01 2.00 1.99 1.98 .97 .96
3.13 2.82 2.64 2.44 2.33 2.25 2.20 2.17 2.14 2.12 2.11 2.08 2.07 2.05 2.05 2.04 2.02 2.02
339 3.02 2.82 2.59 2.47 2.38 2.32 2.28 2.25 2.23 2.21 2.19 2.17 2.15 2.14 2.14 2.12 2.11
357 3.17 2.94 2.70 2.56 2.46 2.40 2.36 2.33 2.30 2.28 2.26 2.24 2.22 2.21 2.20 2.19 2.18
3.72 3,29 3.04 2.78 2.64 2.53 2.46 2.42 2.39 2.36 2.34 2.31 2.29 2.28 2.26 2.26 2.24 2.23
385 3,39 3.13 2.84 2.70 2.59 2.52 2.47 2.43 2.41 2.39 2.36 2.33 2.32 2.31 2.30 2.28 2.27
401 351 3.23 2.93 2.77 2.65 2.58 2.53 2.49 2.46 2.44 2.41 2.39 2.37 2.36 2.35 2.33 2.32
4.21 3.66 3,36 3.03 2.86 2.74 2.66 2.60 2.56 2.53 2.51 2.48 2.45 2.44 2.42 2.41 2.39 2.38
4.38 3.79 3.46 3.12 2.93 2.80 2.72 2.66 2.62 2.59 2.57 2.53 2.50 2.48 2.47 2.46 2.44 2.43
452 389 355 3.18 2.99 2.85 2.77 2.71 2.67 2.63 2.61 2.57 2.54 2.53 2.51 2.50 2.48 2.46
4.64 3.98 3.62 3.24 3.04 2.90 2.81 2.75 2.70 2.67 2.64 2.61 2.58 2.56 2.54 2.53 2.51 2.50
4.74 4.06 3.68 3,29 3.09 2.94 2.85 2.78 2.74 2.70 2.68 2.64 2.61 2.59 2.57 2.56 2.54 2.52

                                                   D-156
                                                                                                 March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
Unified Guidance
    Table 19-14. K-Multipliers  for 1-of-1  Intrawell Prediction Limits on Means of Order 2 (1 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.37
3.14
3.67
4.09
4.43
5.25
6.06
6.70
7.24
8.32
9.18
9.90
10.54
11.36
12.52
13.50
14.35
15.11
15.80
6
1.94
2.41
2.71
2.93
3.10
3.50
3.86
4.13
4.35
4.78
5.10
5.35
5.57
5.85
6.23
6.53
6.79
7.02
7.22
8
1.77
2.15
2.37
2.54
2.67
2.95
3.20
3.39
3.53
3.81
4.01
4.17
4.30
4.47
4.70
4.88
5.03
5.16
5.27
10
1.68
2.01
2.21
2.34
2.45
2.68
2.89
3.03
3.15
3.36
3.52
3,64
3.74
3.87
4.04
4.17
4.28
4.37
4.45
12
1.62
1.93
2.10
2.23
2.32
2.53
2.70
2.83
2.93
3.11
3.24
3.35
3.43
3.54
3,67
3.78
3.87
3.94
4.01
16
1.56
1.83
1.98
2.09
2.18
2.35
2.50
2.61
2.69
2.84
2.94
3.03
3.09
3.18
3.29
3.37
3.44
3.50
3.55
20
1.52
1.77
1.92
2.02
2.09
2.25
2.39
2.48
2.56
2.69
2.79
2.86
2.92
2.99
3.08
3.16
3.22
3.27
3.31
25
1.49
1.73
1.87
1.96
2.03
2.18
2.31
2.39
2.46
2.58
2.67
2.73
2.79
2.85
2.94
3.00
3.06
3.10
3.14
30
.47
.70
.83
.92
.99
2.13
2.25
2.34
2.40
2.51
2.59
2.66
2.71
2.77
2.85
2.91
2.96
3.00
3.03
35
.45
.68
.81
.90
.96
2.10
2.22
2.30
2.36
2.47
2.54
2.60
2.65
2.71
2.78
2.84
2.89
2.93
2.96
40
.44
.67
.79
.88
.94
2.08
2.19
2.27
2.33
2.43
2.51
2.56
2.61
2.67
2.74
2.79
2.84
2.87
2.91
45
.44
.66
.78
.86
.93
2.06
2.17
2.24
2.30
2.41
2.48
2.53
2.58
2.63
2.70
2.76
2.80
2.84
2.87
50
.43
.65
.77
.85
.92
2.05
2.15
2.23
2.28
2.39
2.46
2.51
2.55
2.61
2.68
2.73
2.77
2.81
2.84
60
1.42
1.64
1.75
1.84
1.90
2.02
2.13
2.20
2.26
2.36
2.42
2.48
2.52
2.57
2.64
2.69
2.73
2.76
2.79
70
1.41
1.63
1.74
1.82
1.88
2.01
2.11
2.18
2.24
2.33
2.40
2.45
2.49
2.54
2.61
2.66
2.70
2.73
2.76
80
1.41
1.62
1.74
1.82
1.88
2.00
2.10
2.17
2.22
2.32
2.38
2.43
2.48
2.53
2.59
2.64
2.67
2.71
2.74
90
1.40
1.61
1.73
1.81
1.87
1.99
2.09
2.16
2.21
2.31
2.37
2.42
2.46
2.51
2.57
2.62
2.66
2.69
2.72
100
1.40
1.61
1.72
1.80
1.86
1.98
2.08
2.15
2.20
2.30
2.36
2.41
2.45
2.50
2.56
2.61
2.65
2.68
2.70
125
.40
.60
.72
.79
.85
.97
2.07
2.14
2.19
2.28
2.34
2.39
2.43
2.48
2.54
2.58
2.62
2.65
2.68
150
.39
.60
.71
.79
.84
.96
2.06
2.13
2.18
2.27
2.33
2.38
2.42
2.46
2.52
2.57
2.60
2.63
2.66
     Table 19-14. K-Multipliers  for 1-of-1  Intrawell Prediction Limits on Means of Order 2 (2 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
2.01
2.71
3.18
3.55
3.86
4.59
5.30
5.87
6.34
7.30
8.05
8.69
9.25
9.98
11.00
11.86
12.61
13.28
13.89

6
1.63
2.07
2.35
2.55
2.71
3.07
3.40
3.65
3.85
4.23
4.52
4.76
4.95
5.21
5.54
5.82
6.05
6.26
6.44

8
1.48
1.85
2.06
2.22
2.34
2.61
2.84
3.O2
3.15
3.41
3.60
3.75
3.87
4.03
4.24
4.40
4.54
4.66
4.77

10
1.41
1.73
1.92
2.06
2.16
2.39
2.58
2.72
2.83
. 3.O4
3.18
3.30
3.40
3.52
3.67
3.80
3.90
3.99
4.07

12
1.36
1.66
1.84
1.96
2.06
2.26
2.43
2.55
2.65
2.83
2.95
3. OS
3,13.
3.23
3.36
3.47
3.55
3.63
3.69

16
.30
.58
.74
.85
.93
2.11
2.26
2.36
2.45
2.59
2.70
2.78
2.85
2.93
3.04
3,12
3,19
3.24
3.29

20
.27
.54
.69
.79
.87
2.03
2.17
2.26
2.34
2.47
2.56
2.64
2.70
2.77
2.86
2.94
2,99
3,04
3,09

25 30 35 40 45 50 60 70 80 90
.25 .23 .22 .21 .20 .20 1.19 1.19 1.18 1.18
.50 .48 .46 .45 .44 .44 1.43 1.42 1.41 1.41
.64 .62 .60 .58 .57 .57 1.55 1.54 1.54 1.53
.74 .71 .69 .67 .66 .65 1.64 1.63 1.62 1.62
.82 .78 .76 .74 .73 .72 1.71 1.69 1.69 1.68
.97 .93 .90 .88 .87 .86 1.84 1.83 1.82 1.81
2.10 2.05 2.02 2.00 .98 .97 1.95 1.93 1.92 1.92
2.19 2.14 2.11 2.08 2.06 2.05 2.03 2.01 2.00 1.99
2.26 2.20 2.17 2.14 2.12 2.11 2.08 2.07 2.05 2.05
2.38 2.32 2.28 2.25 2.23 2.21 2.19 2.17 2.15 2.14
2.47 2.40 2.36 2.33 2.30 2.29 2.26 2.24 2.22 2.21
2.53 2.47 2.42 2.39 2.36 2.34 2.31 2.29 2.28 2.26
2.59 2.52 2.47 2.43 2.41 2.39 2.36 2.33 2.32 2.31
2.65 2.58 2.53 2.49 2.46 2.44 2.41 2.39 2.37 2.36
2.74 2.66 2.60 2.56 2.53 2.51 2.48 2.45 2.44 2.42
2.80 2.72 2.66 2.62 2.59 2.57 2.53 2.50 2.48 2.47
2.86 2.77 2.71 2.67 2.63 2.61 2.57 2.55 2.53 2.51
2.90 2.81 2.75 2.70 2.67 2.65 2.61 2.58 2.56 2.54
2.94 2.85 2.78 2.74 2.70 2.68 2.64 2.61 2.59 2.57

100
1.18
1.40
1.53
1.61
1.68
1.80
1.91
1.98
2.04 ;
2.14 ;
2.20 ;
2.26 ;
2.30 ;
2.35 ;
2.4i ;
2.46 ;
2.50 ;
2.53 ;
2.56 ;

125
.17
.40
.52
.60
.67
.79
.90
.97
2.02 ;
2.12 ;
2.19 ;
2.24 ;
2.28 ;
1.33 ;
2.39 ;
2.44 ;
2.48 ;
2.5i ;
2.54 ;

150
.17
.39
.52
.60
.66
.79
.89
.96
1.02
2.11
2.18
2.23
2.27
1.32
2.38
2.43
2.46
2.50
2.51

                                                   D-157
                                                                                                  March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-14. K-Multipliers
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.57
3.39
3.94
4.38
4.75
5.61
6.47
7.15
7.72
8.87
9.78
10.55
11.22
12.10
13.33
14.36
15.25
16.05
16.80
6
2.02
2.50
2.80
3.02
3.20
3.59
3.96
4.24
4.46
4.89
5.21
5.47
5.70
5.98
6.36
6.67
6.93
7.16
7.37
Table 19-14.
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.14
4.09
4.74
5.25
5.68
6.70
7.71
8.51
9.18
10.54
11.61
12.52
13.32
14.35
15.80
17.02
18.10
19.02
19.90
6
2.41
2.93
3.25
3.50
3.69
4.13
4.54
4.85
5.10
5.57
5.93
6.23
6.48
6.79
7.22
7.57
7.87
8.12
8.35
8
1.82
2.19
2.42
2.58
2.71
2.99
3.24
3.43
3.57
3.85
4.05
4.21
4.35
4.52
4.74
4.92
5.07
5.20
5.31
10
1.71
2.04
2.23
2.37
2.48
2.71
2.91
3.06
3.17
3,39
3.54
3.66
3.76
3.89
4.06
4.19
4.30
4.39
4.47
for 1
12
1.65
1.95
2.12
2.25
2.34
2.54
2.72
2.85
2.94
3.13
3.26
3.36"
3,44
3.55
3.68
3.79
3.88
3.96
4.02
K-Multipliers for
8
2.15
2.54
2.78
2.95
3.09
3.39
3.65
3.85
4.01
4.30
4.52
4.70
4.84
5.03
5.27
5.46
5.63
5.77
5.89
10
2.01
2.34
2.54
2.68
2.80
3.03
3.25
3.40
3.52
3.74
3.91
4.04
4.14
4.28
4.45
4.59
4.71
4.81
4.89
12
1.93
2.23
2.40
2.53
2.62
2.83
3.01
3.14
3.24
3.43
3.57
"'3.67'
3.76
3.87
4.01
4.12
4.22
4.29
4.36
-of-1 Intrawell Prediction Limits on Means of Order 2 (2 COC
16
1.57
1.84
2.00
2.10
2.19
2.36
2.51
2.61
2.70
2.84
2.95
3.03
3.10
3.18
3.29
3,37
3,44
3, SO
3.55
20 25 30 35 40 45 50 60 70 80
1.53 1.50 1.48 .46 .45 .44 .43 1.42 1.42 1.41
1.78 1.74 1.71 .69 .67 .66 .65 1.64 1.63 1.62
1.93 1.87 1.84 .81 .80 .78 .77 1.76 1.75 1.74
2.03 1.97 1.93 .90 .88 .87 .86 1.84 1.83 1.82
2.10 2.04 2.00 .97 .95 .93 .92 1.90 1.89 1.88
2.26 2.18 2.14 2.10 2.08 2.06 2.05 2.02 2.01 2.00
2.39 2.31 2.26 2.22 2.19 2.17 2.15 2.13 2.11 2.10
2.49 2.40 2.34 2.30 2.27 2.25 2.23 2.20 2.18 2.17
2.56 2.46 2.40 2.36 2.33 2.30 2.28 2.26 2.24 2.22
2.70 2.59 2.52 2.47 2.43 2.41 2.39 2.36 2.33 2.32
2.79 2.67 2.60 2.54 2.51 2.48 2.46 2.42 2.40 2.38
2.86 2.74 2.66 2.60 2.56 2.53 2.51 2.48 2.45 2.44
2.92 2.79 2.71 2.65 2.61 2.58 2.56 2.52 2.49 2.48
2.99 2.85 2.77 2.71 2.67 2.63 2.61 2.57 2.54 2.53
3.09 2.94 2.85 2.78 2.74 2.70 2.68 2.64 2.61 2.59
3.16 3.00 2.91 2.84 2.79 2.76 2.73 2.69 2.66 2.64
3.22 3.06 2.96 2.89 2.84 2.80 2.77 2.73 2.70 2.68
3.27 3.10 3.00 2.93 2.87 2.84 2.81 2.76 2.73 2.71
3,31 3.14 3.03 2.96 2.91 2.87 2.84 2.79 2.76 2.74
1-of-1 Intrawell Prediction Limits on Means of Order 2
16
1.83
2.09
2.24
2.35
2.43
2.61
2.76
2.86
2.94
3.09
3.20
3.29
3.35
3.44
3.55
3,83,
3,70
3.76
3,82
20 25 30 35 40 45 50 60 70 80
1.77 1.73 1.70 1.68 1.67 1.66 1.65 1.64 1.63 1.62
2.02 1.96 1.92 1.90 1.88 1.86 1.85 1.84 1.82 1.82
2.16 2.09 2.05 2.02 2.00 1.98 1.97 1.95 1.93 1.92
2.25 2.18 2.13 2.10 2.08 2.06 2.05 2.02 2.01 2.00
2.33 2.25 2.20 2.17 2.14 2.12 2.10 2.08 2.07 2.05
2.48 2.39 2.34 2.30 2.27 2.24 2.23 2.20 2.18 2.17
2.62 2.52 2.45 2.41 2.37 2.35 2.33 2.30 2.28 2.27
2.71 2.60 2.53 2.48 2.45 2.42 2.40 2.37 2.35 2.33
2.79 2.67 2.59 2.54 2.51 2.48 2.46 2.42 2.40 2.38
2.92 2.79 2.71 2.65 2.61 2.58 2.55 2.52 2.49 2.48
3.01 2.87 2.79 2.73 2.68 2.65 2.62 2.59 2.56 2.54
3.08 2.94 2.85 2.78 2.74 2.70 2.68 2.64 2.61 2.59
3.14 2.99 2.90 2.83 2.78 2.75 2.72 2.68 2.65 2.63
3.22 3.06 2.96 2.89 2.84 2.80 2.77 2.73 2.70 2.67
3.31 3.14 3.03 2.96 2.91 2.87 2.84 2.79 2.76 2.74
3.38 3.20 3.09 3.02 2.96 2.92 2.89 2.84 2.81 2.78
3.44 3.26 3.14 3.06 3.01 2.96 2.93 2.88 2.84 2.82
3.49 3.30 3.18 3.10 3.04 3.00 2.96 2.91 2.88 2.85
3.54 3.34 3.22 3.13 3.07 3.03 2.99 2.94 2.91 2.88
90
1.41
1.62
1.73
1.81
1.87
1.99
2.09
2.16
2.21
2.31
2.37
2.42
2.46
2.51
2.57
2.62
2.66
2.69
2.72
Semi-Annual)
100
1.40
1.61
1.73
1.80
1.86
1.98
2.08
2.15
2.20
2.30
2.36
2.41
2.45
2.50
2.56
2.61
2.65
2.68
2.70
125
.40
.60
.72
.79
.85
.97
2.07
2.14
2.19
2.28
2.34
2.39
2.43
2.48
2.54
2.58
2.62
2.65
2.68
150
.39
.60
.71
.79
.84
.96
2.06
2.13
2.18
2.27
2.33
2.38
2.42
2.46
2.52
2.57
2.60
2.63
2.66
(2 COC, Quarterly)
90
1.61
1.81
1.92
1.99
2.04
2.16
2.25
2.32
2.37
2.46
2.52
2.57
2.61
2.66
2.72
2.76
2.80
2.83
2.85
100
1.61
1.80
1.91
1.98
2.04
2.15
2.25
2.31
2.36
2.45
2.51
2.56
2.60
2.65
2.70
2.75
2.78
2.82
2.84
125
1.60
1.79
1.90
1.97
2.02
2.14
2.23
2.29
2.34
2.43
2.49
2.54
2.57
2.62
2.68
2.72
2.76
2.79
2.81
150
1.60
1.79
1.89
1.96
2.02
2.13
2.22
2.28
2.33
2.42
2.48
2.52
2.56
2.60
2.66
2.70
2.74
2.77
2.79

                                                     D-158
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-14
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
2.97
3.86
4.48
4.97
5.38
6.34
7.30
8.05
8.69
9.98
10.99
11.86
12.61
13.59
14.97
16.14
17.14
18.05
18.87
Table 19-1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.68
4.75
5.49
6.07
6.56
7.72
8.87
9.78
10.55
12.10
13.33
14.36
15.26
16.44
18.11
19.51
20.74
21.80
22.85
6
2.22
2.71
3.02
3.25
3.44
3.85
4.23
4.52
4.76
5.21
5.54
5.82
6.05
6.35
6.75
7.08
7.35
7.59
7.81
. K-Multipliers for1-of-1 I
8
1.97
2.34
2.57
2.74
2.87
3.15
3.41
3.60
3.75
4.03
4.24
4.40
4.54
4.71
4.94
5.13
5.28
5.41
5.53
10
1.84
2.16
2.36
2.49
2.60
2.83
3.O4
3.18
3.30
3.52
3.67
3.80
3.90
4.03
4.20
4.33
4.44
4.53
4.61
4. K-Multipliers
6
2.66
3.20
3.54
3.79
4.00
4.46
4.89
5.21
5.47
5.98
6.36
6.67
6.93
7.27
7.72
8.09
8.41
8.67
8.91
8
2.32
2.71
2.95
3.13
3.27
3.57
3.85
4.05
4.21
4.52
4.74
4.92
5.07
5.26
5.51
5.71
5.88
6.03
6.15
10
2.15
2.48
2.68
2.82
2.93
3.17
3,39
3.54
3.66
3.89
4.06
4.19
4.30
4.43
4.61
4.75
4.87
4.97
5.06
12
1.76
2.06
2.23
2.35
2.45
2.65
2.83
2.95
3, OS
3.23
3.36
3.47
3.55
3.66
3.80
3.90
4.00
4.06
4.13
for
12
2.04
2.34
2.52
2.64
2.74
2.94
3.13
3.26
3.36
3.55
3.68
3.79
3.88
3.99
4.13
4.24
4.34
4.42
4.49
16
1.67
1.93
2.09
2.19
2.27
2.45
2.59
2.70
2.78
2.93
3,O 4
3.12
3.19
3.27
3.38
3.46
3.53
3.59
3.64
20
1.62
1.87
2.01
2.10
2.18
2.34
2.47
2.56
2.64
2.77
2.86
2.94
2.99
. 3.07
3,16
3.23
3.29
3.34
3.39
ntrawell Prediction Limits on Means of Order 2 (5 COC, Annual)
25
1.58
1.82
1.95
2.04
2.11
2.26
2.38
2.47
2.53
2.65
2.74
2.80
2.86
2.92
3.0O
3.O7
3,12
3,16
3.20
1-of-1 Intrawell
16
1.93
2.19
2.34
2.44
2.52
2.70
2.84
2.95
3.03
3.18
3.29
3.37
3,44
3.53
3.64
3.72
3.79
3.85
3.91
20
1.86
2.10
2.24
2.33
2.41
2.56
2.70
2.79
2.86
2.99
3.09
3.16
3.22
3.29
3.39
3,46
3.52
3.57
3.61
25
1.81
2.04
2.16
2.25
2.32
2.46
2.59
2.67
2.74
2.85
2.94
3.00
3.06
3.12
3.20
3,27
3,32
3,36
3:40
30 35 40 45 50 60 70 80 90
1.56 1.54 1.53 .52 .51 1.50 1.49 1.48 1.48
1.78 1.76 1.74 .73 .72 1.71 1.69 1.69 1.68
1.91 1.88 1.86 .85 .84 1.82 1.81 1.80 1.79
2.00 1.97 1.95 .93 .92 1.90 1.89 1.88 1.87
2.07 2.03 2.01 .99 .98 1.96 1.95 1.94 1.93
2.20 2.17 2.14 2.12 2.11 2.08 2.07 2.05 2.05
2.32 2.28 2.25 2.23 2.21 2.19 2.17 2.15 2.14
2.40 2.36 2.33 2.30 2.29 2.26 2.24 2.22 2.21
2.47 2.42 2.39 2.36 2.34 2.31 2.29 2.28 2.26
2.58 2.53 2.49 2.46 2.44 2.41 2.39 2.37 2.36
2.66 2.60 2.56 2.54 2.51 2.48 2.45 2.44 2.42
2.72 2.66 2.62 2.59 2.57 2.53 2.50 2.48 2.47
2.77 2.71 2.67 2.63 2.61 2.57 2.55 2.53 2.51
2.83 2.77 2.72 2.69 2.66 2.62 2.59 2.57 2.56
2.91 2.84 2.79 2.76 2.73 2.69 2.66 2.64 2.62
2,97 2.90 2.85 2.81 2.78 2.74 2.71 2.68 2.67
3,01 '2. as" 2.89 2.85 2.82 2.78 2.75 2.72 2.70
3, OS 2,98 2.92 2.89 2.86 2.81 2.78 2.75 2.74
3.09 3,02 2.96 2.92 2.89 2.84 2.81 2.78 2.76
Prediction Limits on Means of Order 2 (5 COC,
30 35 40 45 50 60 70 80 90
1.78 1.76 1.74 1.73 1.72 1.70 1.69 1.69 1.68
2.00 1.97 1.95 1.93 1.92 1.90 1.89 1.88 1.87
2.12 2.09 2.06 2.04 2.03 2.01 1.99 1.98 1.97
2.20 2.17 2.14 2.12 2.11 2.08 2.07 2.05 2.05
2.27 2.23 2.20 2.18 2.16 2.14 2.12 2.11 2.10
2.40 2.36 2.33 2.30 2.28 2.26 2.24 2.22 2.21
2.52 2.47 2.43 2.41 2.39 2.36 2.33 2.32 2.31
2.60 2.54 2.51 2.48 2.46 2.42 2.40 2.38 2.37
2.66 2.60 2.56 2.53 2.51 2.48 2.45 2.44 2.42
2.77 2.71 2.67 2.63 2.61 2.57 2.54 2.53 2.51
2.85 2.78 2.74 2.70 2.68 2.64 2.61 2.59 2.57
2.91 2.84 2.79 2.76 2.73 2.69 2.66 2.64 2.62
2.96 2.89 2.84 2.80 2.77 2.73 2.70 2.68 2.66
3.02 2.94 2.89 2.85 2.82 2.78 2.75 2.72 2.70
3.09 3.02 2.96 2.92 2.89 2.84 2.81 2.78 2.76
3.15 3.07 3.01 2.97 2.94 2.89 2.85 2.83 2.81
3.20 3.12 3.06 3.01 2.98 2.93 2.89 2.87 2.85
3.24 3.15 3.09 3.05 3.01 2.96 2.92 2.90 2.88
3.'28" 3.19 3.13 3.08 3.04 2.99 2.95 2.92 2.90
100
1.47
1.68
1.79
1.86
1.92
2.04
2.14
2.20
2.26
2.35
2.41
2.46
2.50
2.55
2.60
2.65
2.69
2.72
2.75
125
.47
.67
.78
.85
.91
2.02
2.12
2.19
2.24
2.33
2.39
2.44
2.48
2.52
2.58
2.62
2.66
2.70
2.72
150
.46
.66
.77
.84
.90
2.02
2.11
2.18
2.23
2.32
2.38
2.43
2.46
2.51
2.57
2.61
2.65
2.68
2.70
Semi-Annual)
100
1.67
1.86
1.97
2.04
2.09
2.20
2.30
2.36
2.41
2.50
2.56
2.61
2.65
2.68
2.75
2.79
2.83
2.86
2.89
125
1.67
1.85
1.95
2.02
2.08
2.19
2.28
2.34
2.39
2.48
2.54
2.58
2.62
2.66
2.72
2.77
2.80
2.83
2.86
150
1.66
1.84
1.95
2.02
2.07
2.18
2.27
2.33
2.38
2.46
2.52
2.57
2.60
2.65
2.70
2.75
2.78
2.81
2.84

                                                     D-159
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-14.
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.43
5.68
6.55
7.24
7.81
9.18
10.54
11.61
12.52
14.35
15.81
17.02
18.09
19.48
21.45
23.09
24.55
25.84
27.07
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.86
4.97
5.74
6.34
6.85
8.05
9.25
10.20
11.00
12.61
13.89
14.96
15.90
17.15
18.91
20.31
21.64
22.81
23.83
6
3.10
3.69
4.07
4.35
4.58
5.10
5.57
5.93
6.23
6.79
7.22
7.57
7.87
8.24
8.75
9.15
9.49
9.79
10.03
9-14.
6
2.71
3.25
3.59
3.85
4.06
4.52
4.95
5.28
5.54
6.05
6.44
6.75
7.01
7.35
7.82
8.19
8.51
8.78
9.03
K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 2 (5 COC, Quarterly)
8 10 12 16 20 25 30 35 40 45
2.67 2.45 2.32 2.18 2.09 2.03 1.99 1.96 1.94 1.93
3.09 2.80 2.62 2.43 2.33 2.25 2.20 2.17 2.14 2.12
3.34 3.00 2.80 2.58 2.46 2.37 2.32 2.28 2.25 2.23
3.53 3.15 2.93 2.69 2.56 2.46 2.40 2.36 2.33 2.30
368 3.27 3.03 2.77 2.63 2.53 2.46 2.42 2.39 2.36
401 3.52 3.24 2.94 2.79 2.67 2.59 2.54 2.51 2.48
430 374 3.43 3.09 2.92 2.79 2.71 2.65 2.61 2.58
452 391 3.57 3.20 3.01 2.87 2.79 2.73 2.68 2.65
4.70 4.04 3.67 3.29 3.08 2.94 2.85 2.78 2.74 2.70
5.03 4.28 3.87 3.44 3.22 3.06 2.96 2.89 2.84 2.80
5.27 4.45 4.01 3.55 3.31 3.14 3.03 2.96 2.91 2.87
5.46 4.59 4.12 "'3,63' 3.38 3.20 3.09 3.02 2.96 2.92
5.63 4.71 4.22 3.70 3.44 3.26 3.14 3.06 3.01 2.96
5.83 4.85 4.33 3. 79 3.52 3.32 3.20 3.12 3.06 3.01
6.10 5.04 4.48 3.90 3.61 3.40 3.28 3.19 3.13 3.08
6.32 5.19 4.60 3.99 3.68 3.47 3.34 3.25 3.18 3.13
6.50 5.32 4.69 4.06 3.74 3.52 3.39 3.29 3.22 3.18
6.67 5.43 4.78 4.12 3.80 3.57 3.43 3.33 3.26 3.21
6.80 5.52 4.85 4.18 3.84 3.61 3.46 3.36 3.29 3.24
K-Multipliers for 1-of-1 Intrawell Prediction Limits
8 10 12 16 20 25 30 35 40 45
2.34 2.16 2.06 1.93 1.87 1.82 1.78 1.76 1.74 1.73
2.74 2.49 2.35 2.19 2.10 2.04 2.00 1.97 1.95 1.93
2.98 2.69 2.52 2.34 2.24 2.17 2.12 2.09 2.06 2.04
315 2.83 2.65 2.45 2.34 2.26 2.20 2.17 2.14 2.12
329 2.94 2.75 2.53 2.41 2.32 2.27 2.23 2.20 2.18
3.60 3.18 2.95 2.70 2.56 2.47 2.40 2.36 2.33 2.30
3.87 3.40 '"3'.'i'3" 2.85 2.70 2.59 2.52 2.47 2.43 2.41
4.08 3.55 3.26 2.95 2.79 2.67 2.60 2.55 2.51 2.48
4.24 3.67 3.36 3. 04 2.86 2.74 2.66 2.60 2.56 2.53
4.54 3.90 3.55 3.19 2,99 2.86 2.77 2.71 2.67 2.63
4.77 4.07 3.69 3.29 3.09 2.94 2.85 2.78 2.74 2.70
4.94 4.20 3.80 3.38 .'3.16 3.OO 2.91 2.84 2.79 2.76
5.09 4.31 3.88 3.44 3.22 3.06 2.96 2.89 2.84 2.80
5.28 4.44 3.99 3.53 3.29 3.12 3.O2 2.95 2.89 2.85
5.53 4.61 4.13 3.64 3.39 3.2O 3.O9 3.02 :2.96:' 2.92
5.74 4.76 4.25 3.73 3.46 3.27 3,75, 3.07 3.02^2.97
5.91 4.88 4.35 3.80 3.52 3.32 3.20 3.12 3.06 3,O1
6.05 4.98 4.42 3.86 3.57 3.37 . -3.24, 3.16 3.10 3.Q5
6.18 5.07 4.49 3.91 3.61 3.41 3.28 3. 19 3.13 3.O8
50
1.92
2.10
2.21
2.28
2.34
2.46
2.55
2.62
2.68
2.77
2.84
2.89
2.93
2.98
3.04
3.09
3.13
3.17
3.20
60
1.90
2.08
2.19
2.26
2.31
2.42
2.52
2.59
2.64
2.73
2.79
2.84
2.88
2.93
2.99
3.04
3.07
3.11
3.13
on Means
50
1.72
1.92
2.03
2.11
2.16
2.29
2.39
2.46
2.51
2.61
2.68
2.73
2.77
2.82
2.89
2.94
2,98
3.01
.3.04
60
1.71
1.90
2.01
2.08
2.14
2.26
2.36
2.42
2.48
2.57
2.64
2.69
2.73
2.78
2.84
2.89
2.93
2.96
2.99
70
1.88
2.07
2.17
2.24
2.29
2.40
2.49
2.56
2.61
2.70
2.76
2.81
2.84
2.89
2.95
3.00
3.03
3.07
3.09
80
1.88
2.05
2.15
2.22
2.28
2.38
2.48
2.54
2.59
2.67
2.74
2.78
2.82
2.87
2.93
2.97
3.00
3.04
3.06
of Order 2
70
1.69
1.89
1.99
2.07
2.12
2.24
2.33
2.40
2.45
2.55
2.61
2.66
2.69
2.75
2.81
2.85
2.89
2.91
2,95
80
1.69
1.88
1.98
2.05
2.11
2.22
2.32
2.38
2.44
2.53
2.59
2.64
2.67
2.72
2.78
2.83
2.87
2.90
2.92
90
1.87
2.04
2.14
2.21
2.26
2.37
2.46
2.52
2.57
2.66
2.72
2.76
2.80
2.85
2.90
2.95
2.98
3.01
3.04
100
1.86
2.04
2.14
2.20
2.25
2.36
2.45
2.51
2.56
2.65
2.70
2.75
2.78
2.83
2.89
2.93
2.96
2.99
3.02
(10 COC,
90
1.68
1.87
1.97
2.05
2.10
2.21
2.31
2.37
2.42
2.51
2.57
2.62
2.66
2.70
2.76
2.81
2.85
2.88
2.90
100
1.68
1.86
1.97
2.04
2.09
2.20
2.30
2.36
2.41
2.50
2.56
2.60
2.64
2.69
2.75
2.79
2.83
2.86
2.89
125
1.85
2.02
2.12
2.19
2.24
2.34
2.43
2.49
2.54
2.62
2.68
2.72
2.76
2.80
2.86
2.90
2.93
2.96
2.99
150
1.84
2.02
2.11
2.18
2.23
2.33
2.42
2.48
2.52
2.60
2.66
2.70
2.74
2.78
2.84
2.88
2.91
2.94
2.96
Annual)
125
1.67
1.85
1.95
2.02
2.08
2.19
2.28
2.34
2.39
2.48
2.54
2.58
2.62
2.66
2.72
2.77
2.80
2.83
2.86
150
1.66
1.84
1.95
2.02
2.07
2.18
2.27
2.33
2.38
2.46
2.52
2.57
2.60
2.65
2.70
2.75
2.78
2.81
2.84

                                                     D-160
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.75
6.07
6.99
7.72
8.33
9.78
11.22
12.36
13.33
15.25
16.80
18.09
19.22
20.70
22.81
24.61
26.09
27.50
28.75
4. K- Multipliers for 1
6
3.20
3.79
4.17
4.46
4.69
5.21
5.70
6.06
6.36
6.93
7.37
7.72
8.03
8.40
8.91
9.30
9.65
9.96
10.23
8
2.71
3.13
3.39
3.57
3.72
4.05
4.35
4.57
4.74
5.07
5.31
5.51
5.67
5.88
6.15
6.38
6.56
6.72
6.86
10
2.48
2.82
3.02
3.17
3,29
3.54
3.76
3.93
4.06
4.30
4.47
4.61
4.73
4.87
5.06
5.21
5.34
5.45
5.55
Table 19-14. K-Multipliers
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.68
7.24
8.32
9.18
9.90
11.61
13.32
14.67
15.80
18.10
19.90
21.45
22.81
24.53
27.03
29.14
30.94
32.66
34.06
6
3.69
4.35
4.78
5.10
5.35
5.93
6.48
6.89
7.22
7.87
8.35
8.75
9.08
9.49
10.04
10.51
10.90
11.25
11.56
8
3.09
3.53
3.81
4.01
4.17
4.52
4.84
5.08
5.27
5.63
5.89
6.10
6.28
6.50
6.81
7.03
7.21
7.36
7.46
10
2.80
3.15
3.36
3.52
3,84
3.91
4.14
4.32
4.45
4.71
4.89
5.04
5.16
5.32
5.52
5.68
5.81
5.93
6.04
12
2.34
2.64
2.82
2.94
3.04
3.26
3.44
3.58
3.68
3.88
4.02
4.13
4.22
4.34
4.49
4.61
4.71
4.79
4.86
for
12
2.62
2.93
3.11
3.24
3.35
3.57
3.76
3.90
4.01
4.22
4.36
4.48
4.58
4.70
4.85
4.98
5.08
5.17
5.24
-of-1
16
2.19
2.44
2.59
2.70
2.78
2.95
3.10
3.21
3 ,,29
3.44
3.55
3.64
3.71
3.79
3.91
3.99
4.07
4.13
4.18
Intrawell Prediction Limits on Means of Order 2 (10 COC, Semi-Annual)
20
2.10
2.33
2.47
2.56
2.64
2.79
2.92
3.01
3.09
3.22
3.31
3.39
3.44
3,52
3.61
3.69
3.75
3.80
3.84
25
2.04
2.25
2.38
2.46
2.53
2.67
2.79
2.87
2.94
3.06
3.14
3.20
'3. 26"
••3*32
3.40
3.47
3,52
3,57
3.61
1-of-1 Intrawell
16
2.43
2.69
2.84
2.94
3.03
3.20
3.35
3.46
3.55
3.7O
3.82
3.90
3.97
4.06
4.18
4.27
4.35
4.40
4.46
20
2.33
2.56
2.69
2.79
2.86
3.01
3.14
3.24
3.31
3.44
3.54
3.61
3,67
3.75
3.84
3.92
3.97
4.03
4.07
25
2.25
2.46
2.58
2.67
2.73
2.87
2.99
3.07
3.14
3.26
3.34
3.40
3.46
3.52
'3.81 '
. 3.67
3.73
3,77
3,81
30
2.00
2.20
2.32
2.40
2.46
2.60
2.71
2.79
2.85
2.96
3.03
3.09
3.14
3.20
3.28
3.34
3.39
3.43
3.46
35 40 45 50 60 70 80 90 100 125
1.97 1.95 1.93 1.92 1.90 1.89 1.88 1.87 1.86 1.85
2.17 2.14 2.12 2.11 2.08 2.07 2.05 2.05 2.04 2.02
2.28 2.25 2.23 2.21 2.19 2.17 2.15 2.14 2.14 2.12
2.36 2.33 2.30 2.28 2.26 2.24 2.22 2.21 2.20 2.19
2.42 2.39 2.36 2.34 2.31 2.29 2.28 2.26 2.26 2.24
2.54 2.51 2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34
2.65 2.61 2.58 2.56 2.52 2.49 2.48 2.46 2.45 2.43
2.73 2.68 2.65 2.62 2.59 2.56 2.54 2.52 2.51 2.49
2.78 2.74 2.70 2.68 2.64 2.61 2.59 2.57 2.56 2.54
2.89 2.84 2.80 2.77 2.73 2.70 2.68 2.66 2.65 2.62
2.96 2.91 2.87 2.84 2.79 2.76 2.74 2.72 2.70 2.68
3.02 2.96 2.92 2.89 2.84 2.81 2.78 2.76 2.75 2.72
3.06 3.00 2.96 2.93 2.88 2.85 2.82 2.80 2.78 2.76
3.12 3.06 3.01 2.98 2.93 2.89 2.87 2.85 2.83 2.80
3.19 3.12 3.08 3.04 2.99 2.95 2.92 2.90 2.89 2.86
3.25 3.17 3.13 3.09 3.04 3.00 2.97 2.95 2.93 2.90
3.30 3.22 3.17 3.13 3.07 3.03 3.00 2.98 2.96 2.93
3.33 3.26 3.21 3.17 3.11 3.07 3.04 3.01 2.99 2.96
3.35 3.29 3.24 3.20 3.13 3.09 3.06 3.04 3.02 2.99
150
1.84
2.02
2.11
2.18
2.23
2.33
2.42
2.48
2.52
2.60
2.66
2.70
2.74
2.78
2.84
2.88
2.91
2.94
2.96
Prediction Limits on Means of Order 2 (10 COC, Quarterly)
30
2.20
2.40
2.51
2.59
2.66
2.79
2.90
2.97
3.03
3.14
3.22
3.28
3.33
3.39
3.46
3.52
3.57
3.6O
3.64
35 40 45 50 60 70 80 90 100 125
2.17 2.14 2.12 2.10 2.08 2.07 2.05 2.04 2.04 2.02
2.36 2.33 2.30 2.28 2.26 2.24 2.22 2.21 2.20 2.19
2.47 2.43 2.41 2.39 2.36 2.33 2.32 2.31 2.30 2.28
2.54 2.51 2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34
2.60 2.56 2.53 2.51 2.48 2.45 2.43 2.42 2.41 2.39
2.73 2.68 2.65 2.62 2.59 2.56 2.54 2.52 2.51 2.49
2.83 2.78 2.75 2.72 2.68 2.65 2.63 2.61 2.60 2.57
2.90 2.85 2.81 2.79 2.74 2.71 2.69 2.67 2.66 2.63
2.96 2.91 2.87 2.84 2.79 2.76 2.74 2.72 2.70 2.68
3.06 3.01 2.96 2.93 2.88 2.84 2.82 2.80 2.78 2.76
3.13 3.07 3.03 2.99 2.94 2.91 2.88 2.86 2.84 2.81
3.19 3.13 3.08 3.04 2.99 2.95 2.92 2.90 2.89 2.86
3.23 3.17 3.12 3.08 3.03 2.99 2.96 2.94 2.92 2.89
3.29 3.22 3.17 3.13 3.07 3.03 3.00 2.98 2.96 2.93
3.36 3.29 3.24 3.20 3.13 3.09 3.06 3.04 3.02 2.99
3.42 3.34 3.29 3.24 3.18 3.14 3.11 3.08 3.06 3.03
3.46 3.38 3.33 3.28 3.22 3.17 3.14 3.12 3.10 3.06
3.50 3.42 3.36 3.32 3.25 3.20 3.17 3.14 3.12 3.09
3.53 3.45 3.39 3.35 3.28 3.23 3.20 3.17 3.15 3.11
150
2.02
2.18
2.27
2.33
2.38
2.48
2.56
2.62
2.66
2.74
2.79
2.84
2.87
2.91
2.96
3.00
3.04
3.06
3.09

                                                     D-161
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-14.
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.97
6.34
7.30
8.05
8.69
10.20
11.69
12.89
13.89
15.92
17.53
18.90
20.12
21.68
23.83
25.59
27.34
28.71
30.08
Table 19-1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
6.07
7.72
8.87
9.78
10.55
12.37
14.17
15.61
16.82
19.29
21.24
22.85
24.32
26.17
28.91
31.05
33.01
34.77
36.33
6
3.25
3.85
4.23
4.52
4.76
5.28
5.77
6.14
6.44
7.02
7.46
7.81
8.12
8.50
9.03
9.47
9.81
10.16
10.45
K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 2 (20 COC, Annual)
8
2.74
3.15
3.41
3.60
3.75
4.08
4.37
4.59
4.77
5.10
5.34
5.54
5.70
5.91
6.18
6.40
6.59
6.74
6.88
10
2.49
2.83
3.O4
3.18
3.30
3.55
3.78
3.94
4.07
4.31
4.48
4.62
4.74
4.88
5.08
5.22
5.35
5.46
5.54
4. K- Multipliers
6
3.79
4.46
4.89
5.21
5.47
6.06
6.61
7.03
7.37
8.02
8.52
8.92
9.27
9.72
10.30
10.79
11.18
11.52
11.87
8
3.13
3.57
3.85
4.05
4.21
4.57
4.89
5.12
5.32
5.68
5.94
6.15
6.34
6.56
6.86
7.10
7.30
7.47
7.62
10
2.82
3.17
3,39
3.54
3.66
3.93
4.16
4.34
4.47
4.73
4.91
5.06
5.19
5.33
5.54
5.70
5.83
5.96
6.05
12
2.35
2.65
2.82
2.95
3, OS
3.26
3.45
3.58
3.69
3.88
4.03
4.14
4.23
4.35
4.49
4.61
4.71
4.80
4.86
for 1
12
2.64
2.94
3.13
3.26
3,36
3.58
3.77
3.91
4.02
4.22
4.37
4.49
4.58
4.71
4.86
4.98
5.09
5.18
5.25
16
2.19
2.45
2.59
2.70
2.78
2.95
3.1O
3,21
3.29
3.45
3.56
3.64
3.71
3.80
3.91
4.00
4.06
4.13
4.19
-of-1
16
2.44
2.70
2.84
2.95
3.03
3.21
3.36
3.47
3.55
3.71
3.82
3.91
3.98
4.06
4.18
4.27
4.35
4.41
4.47
20
2.10
2.34
2.47
2.56
2.64
2.79
2.92
3.02
3.O9
3.22
3.31
3.39
3.45
3.52
3.61
3.69
3.75
3.80
3.85
25
2.04
2.26
2.38
2.47
2.53
2.67
2.79
2.87
2.94
3.06
3,14
3. 2O
3.26
3.32
3.41
3.47
3.52
3.56
3.60
Intrawell
20
2.33
2.56
2.70
2.79
2.86
3.01
3.15
3.24
3.31
3,45.
3,54
3.61
3.67
3.75
3.85
3.92
3.98
4.03
4.08
25
2.25
2.46
2.59
2.67
2.74
2.87
2.99
3.08
3.14
.3.26
3.34
3,41
3.46
3:52
3.61
3.67
3.72
3.77
3.81
30
2.00
2.20
2.32
2.40
2.47
2.60
2.71
2.79
2.85
2,96
3.03
3.O9
3, 14
3.20
3.23.
3.33
3.38
3.42
3.45
35
1.97
2.17
2.28
2.36
2.42
2.55
2.65
2.73
2.78
2.89
2,96
3. 02
3.06
3.12
3:19
3,25
3.29
3.33
3.36
Prediction
30
2.20
2.40
2.52
2.60
2.66
2.79
2.90
2.97
3.03
3.14
3.22
'3. 28"
3.32
3.38
3.46
3.52
3.56
3,61
3.64
35
2.17
2.36
2.47
2.54
2.60
2.73
2.83
2.90
2.96
3.06
3.13
3.19
3.23
3.29
3.36
3.42
3.46
3.50
'3.53
40
1.95
2.14
2.25
2.33
2.39
2.51
2.61
2.68
2.74
2.84
2.91
2.96
3.Q1 .
3.06
3.12
3.18
3.22
3.26
3.29.
Limits
40
2.14
2.33
2.43
2.51
2.56
2.68
2.78
2.85
2.91
3.01
3.07
3.13
3.17
3.22
3.29
3.34 .,
3.39
3.42
3.45
45 50 60 70 80 90 100 125
1.93 1.92 1.90 1.89 1.88 1.87 1.86 1.85
2.12 2.11 2.08 2.07 2.05 2.05 2.04 2.02
2.23 2.21 2.19 2.17 2.15 2.14 2.14 2.12
2.30 2.29 2.26 2.24 2.22 2.21 2.20 2.19
2.36 2.34 2.31 2.29 2.28 2.26 2.26 2.24
2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34
2.58 2.56 2.52 2.49 2.48 2.46 2.45 2.43
2.65 2.62 2.59 2.56 2.54 2.52 2.51 2.49
2.70 2.68 2.64 2.61 2.59 2.57 2.56 2.54
2.80 2.77 2.73 2.70 2.67 2.66 2.64 2.62
2.87 2.84 2.79 2.76 2.74 2.72 2.70 2.68
2.92 2.89 2.84 2.81 2.78 2.76 2.75 2.72
2,90 2.93 2.88 2.84 2.82 2.80 2.78 2.76
3,02 2.98 2.93 2.89 2.87 2.84 2.83 2.80
3,08 3.04 2.99 2.95 2.92 2.90 2.89 2.86
3.'13 3.09 3.04 3.00 2.97 2.95 2.93 2.90
3.17 3.13 3.O8 3.03. '3.OO 2.98 2.97 2.93
3,21 3.17 .3.11 3.06 3.O3 3.02 2.99. 2.96
3.23 3,20 3,14 3.O9 3,06 . 3.04 3.02 2.99
150
1.84
2.02
2.11
2.18
2.23
2.33
2.42
2.48
2.52
2.60
2.66
2.70
2.74
2.78
2.84
2.88
2.91
2;94
2.96
on Means of Order 2 (20 COC, Semi-Annual)
45 50 60 70 80 90 100 125
2.12 2.11 2.08 2.07 2.05 2.05 2.04 2.02
2.30 2.29 2.26 2.24 2.22 2.21 2.20 2.19
2.41 2.39 2.36 2.33 2.32 2.31 2.30 2.28
2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34
2.53 2.51 2.48 2.45 2.44 2.42 2.41 2.39
2.65 2.62 2.59 2.56 2.54 2.52 2.51 2.49
2.75 2.72 2.68 2.65 2.63 2.61 2.60 2.57
2.82 2.79 2.74 2.71 2.69 2.67 2.66 2.63
2.87 2.84 2.79 2.76 2.74 2.72 2.70 2.68
2.96 2.93 2.88 2.85 2.82 2.80 2.78 2.76
3.03 2.99 2.94 2.91 2.88 2.86 2.84 2.81
3.08 3.04 2.99 2.95 2.92 2.90 2.89 2.86
3.12 3.08 3.03 2.99 2.96 2.94 2.92 2.89
3.17 3.13 3.07 3.03 3.01 2.98 2.96 2.93
3.23 3.20 3.13 3.09 3.06 3.04 3.01 2.99
3.29 3.24 3.18 3.14 3.11 3.08 3.06 3.03
3.33 3.28 3.22 3.17 3.14 3.12 3.10 3.06
3.36 3.31 3.25 3.20 3.17 3.14 3.12 3.09
3.39 3.34 3.28 3.23 3.20 3.17 3.15 3.11
150
2.02
2.18
2.27
2.33
2.38
2.48
2.56
2.62
2.66
2.74
2.78
2.84
2.87
2.91
2.96
3.00
3.04
3.06
3.09

                                                     D-162
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-14. K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 2 (20 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
7.24
9.18
0.54
11.61
12.52
14.67
16.81
18.52
19.95
22.85
25.20
27.15
28.81
31.05
34.18
36.72
39.06
41.41
42.97
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
6.34
8.05
9.25
10.20
11.00
12.89
14.77
16.27
17.53
20.08
22.12
23.82
25.31
27.30
30.06
32.34
34.45
36.21
37.85
6
4.35
5.10
5.57
5.93
6.23
6.89
7.50
7.97
8.35
9.09
9.64
10.11
10.47
10.99
11.62
12.21
12.65
13.09
13.38
9-14.
6
3.85
4.52
4.95
5.28
5.55
6.14
6.70
7.12
7.46
8.12
8.62
9.03
9.38
9.81
10.42
10.90
11.31
11.69
12.01
8
3.53
4.01
4.30
4.52
4.70
5.08
5.43
5.69
5.89
6.28
6.57
6.80
6.99
7.25
7.57
7.84
8.06
8.25
8.40
10
3.15
3.52
3.74
3.91
4.04
4.32
4.57
4.75
4.89
5.16
5.36
5.52
5.65
5.81
6.03
6.20
6.35
6.47
6.59
12 16 20 25 30 35 40 45 50 60
2.93 2.69 2.56 2.46 2.40 2.36 2.33 2.30 2.28 2.26
3.24 2.94 2.79 2.67 2.59 2.54 2.51 2.48 2.46 2.42
3.43 3.09 2.92 2.79 2.71 2.65 2.61 2.58 2.55 2.52
3.57 3.20 3.01 2.87 2.79 2.73 2.68 2.65 2.62 2.59
3.67 3.29 3.08 2.94 2.85 2.78 2.74 2.70 2.68 2.64
390 3.46 3.24 3.07 2.97 2.90 2.85 2.81 2.79 2.74
4.10 3,62 3.37 3.19 3.08 3.01 2.95 2.91 2.88 2.83
4.25 3.-J3' 3.47 3.27 3.16 3.08 3.02 2.98 2.94 2.89
4.36 3,82 3.54 3.34 3.22 3.13 3.07 3.03 2.99 2.94
4.57 3.98 3,67 3.46 3.32 3.23 3.17 3.12 3.08 3.03
4.73 4.09 3,77 3.54 3.40 3.31 3.24 3.19 3.15 3.09
4.85 4.18 .3,84 3.6O 3.46 3.36 3.29 3.24 3.20 3.13
4.96 4.25 3.91 3.66 3.51 3.41 3.33 3.28 3.23 3.17
5.08 4.35 3.98 3,72 3.56 3.46 3.38 3.33 3.28 3.22
5.25 4.46 4.08 3.81 3,64 3.53 3.45 3.39 3.34 3.28
5.37 4.55 4.15 3,87 3.7Q • • 3.58 , 3.50 3.44 3.39 3.32
5.48 4.63 4.21 3,92 :3.7S 3,63 3.55 3.48 3.43 3.36
5.57 4.70 4.27 3.97 3,78 3,67 3,58 3.52 3.47 3.39
5.66 4.76 4.32 4.00 3,82 . 3.7O 3,61 3.55 3.49 3.42
K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means
8
3.15
3.60
3.87
4.08
4.24
4.59
4.91
5.15
5.34
5.70
5.97
6.18
6.36
6.58
6.88
7.13
7.32
7.50
7.66
10
2.83
3.18
3.40
3.55
3.67
3.94
4.17
4.35
4.48
4.74
4.92
5.07
5.20
5.35
5.55
5.71
5.84
5.96
6.06
12 16 20 25 30 35 40 45 50 60
2.65 2.45 2.34 2.26 2.20 2.17 2.14 2.12 2.11 2.08
2.95 2.70 2.56 2.47 2.40 2.36 2.33 2.30 2.29 2.26
3.13. 2.85 2.70 2.59 2.52 2.47 2.43 2.41 2.39 2.36
326 2.95 2.79 2.67 2.60 2.55 2.51 2.48 2.46 2.42
3.36 3,04 2.86 2.74 2.66 2.60 2.56 2.54 2.51 2.48
3.58 '3.21 3.02 2.87 2.79 2.73 2.68 2.65 2.62 2.59
3.78 3.36 3,15 '"2.99'. 2.90 2.83 2.78 2.75 2.72 2.68
3.92 3.47 3,24 3,08 '2,97. 2.90 2.85 2.82 2.79 2.74
4.03 3.55 3.31 3, 14 3.O3 . 2,96 2.91 2.87 2.84 2.79
4.23 3.71 3.45 . 3,26 3.14 3.Q6 3,01 2,96 2.93 2.88
4.38 3.82 3.54 3.34 3.22 3.13 3.07 3,03 2,99 """2.94
4.50 3.91 3.61 3.40 3.28 3.19 3.13 3.08 3.O4 2.99
4.59 3.98 3.67 3.46 3.33 . 3.23 3,17 3.12 3, OS 3,O3
4.71 4.07 3.75 3.52 3.38 3.29 3,22 3,17 .3,13 3.O7
4.87 4.18 3.85 3.61 3.46 3.36 3,29 3.24 3.20 3,13
4.99 4.27 3.92 3.67 3.52 3.42 3.34 3,29 3:24 3:18
5.10 4.35 3.98 3.72 3.57 3.46 3.38 3.33 3,28 3.22
5.19 4.41 4.04 3.77 3.61 3.50 3.42 3.36 3,32 3. 25
5.26 4.47 4.08 3.81 3.64 3.53 3.45 3.39 3.35 3,28
70
2.24
2.40
2.49
2.56
2.61
2.71
2.80
2.86
2.91
2.99
3.05
3.09
3.13
3.17
3.23
3.27
3.31
3.34
3.37
80
2.22
2.38
2.48
2.54
2.59
2.69
2.77
2.83
2.88
2.96
3.02
3.06
3.10
3.14
3.20
3.24
3.27
3.30
3.33
of Order 2
70
2.07
2.24
2.33
2.40
2.45
2.56
2.65
2.71
2.76
2.85
2.90
'2.95"
2,39
3.O3
.3,09
3,14
3,17
'• 3,2.0
3.23
80
2.05
2.22
2.32
2.38
2.44
2.54
2.63
2.69
2.74
2.82
2.88
2.92
.2,96
3.OO
3,06.
3,11
3.14
3,17
,3;2Q
90
2.21
2.37
2.46
2.52
2.57
2.67
2.75
2.81
2.86
2.94
3.00
3.04
3.07
3.11
3.17
3.21
3.25
3.27
3.30
100
2.20
2.36
2.45
2.51
2.56
2.66
2.74
2.80
2.84
2.92
2.98
3.02
3.05
3.10
3.15
3.19
3.22
3.25
3.28
(40 COC,
90
2.05
2.21
2.31
2.37
2.42
2.52
2.61
2.67
2.72
2.80
2.86
2.90
2,94
2,98
3,04
3,08
3,12
3.15
3.17
100
2.04
2.20
2.30
2.36
2.41
2.51
2.60
2.66
2.70
2.79
2.84
2.89
2.92
2.96
3.O2
3.0(3
3.1Q
3,12
3,15
125
2.19
2.34
2.43
2.49
2.54
2.63
2.71
2.77
2.81
2.89
2.94
2.99
3.02
3.06
3.11
3.15
3.19
3.21
3.24
150
2.18
2.33
2.42
2.48
2.52
2.62
2.70
2.75
2.79
2.87
2.92
2.96
3.00
3.04
3.09
3.13
3.16
3.19
3.21
Annual)
125
2.02
2.19
2.28
2.34
2.39
2.49
2.57
2.63
2.68
2.76
2.81
2.86
2.89
2.93
2.99
3,03
3,06
3.O9
3,11
150
2.02
2.18
2.27
2.33
2.38
2.48
2.56
2.62
2.66
2.74
2.79
2.84
2.87
2.91
"2. '98"
3,'oa
3,O 4
3.Q6
3,09

                                                     D-163
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
7.72
9.78
11.22
12.36
13.33
15.61
17.89
19.70
21.23
24.32
26.78
28.83
30.64
33.05
36.33
39.14
41.60
43.83
45.70
4. K- Multipliers for 1
6
4.46
5.21
5.70
6.06
6.36
7.03
7.66
8.13
8.52
9.27
9.83
10.30
10.69
11.19
11.87
12.42
12.89
13.30
13.65
8
3.57
4.05
4.35
4.57
4.74
5.13
5.47
5.73
5.94
6.33
6.62
6.86
7.05
7.30
7.62
7.90
8.12
8.31
8.47
10
3.17
3.54
3.76
3.93
4.06
4.34
4.59
4.77
4.91
5.18
5.38
5.54
5.67
5.83
6.05
6.23
6.36
6.49
6.60
Table 19-14. K-Multipliers
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
9.18
11.61
13.32
14.67
15.81
18.52
21.21
23.36
25.17
28.81
31.73
34.16
36.33
39.14
43.12
46.41
49.34
52.03
54.38
6
5.10
5.93
6.48
6.89
7.22
7.97
8.68
9.21
9.65
10.48
11.13
11.65
12.08
12.64
13.42
14.03
14.56
15.03
15.44
8
4.01
4.52
4.84
5.08
5.27
5.69
6.07
6.35
6.57
7.00
7.32
7.57
7.79
8.05
8.41
8.70
8.94
9.14
9.33
10
3.52
3.91
4.14
4.32
4.45
4.75
5.01
5.21
5.36
5.65
5.86
6.03
6.17
6.35
6.58
6.76
6.91
7.05
7.16
12
2.94
3.26
3,44
3.58
3.68
3.91
4.11
4.26
4.37
4.59
4.74
4.86
4.96
5.09
5.26
5.38
5.49
5.59
5.67
for
12
3.24
3.57
3.76
3.90
4.01
4.25
4.46
4.61
4.73
4.95
5.12
5.24
5.35
5.48
5.66
5.79
5.91
6.01
6.09
-of-1
16
2.70
2.95
3.10
3.21
3,29
3,47
3.62
3.73
3.82
3.98
4.09
4.18
4.26
4.35
4.46
4.56
4.64
4.70
4.76
Intrawell Prediction Limits on Means of Order 2 (40 COC, Semi-Annual)
20
2.56
2.79
2.92
3.01
3.09
3.24
3,37
3,47
3,54
3.67
3.77
3.85
3.91
3.98
4.08
4.15
4.22
4.27
4.31
25
2.46
2.67
2.79
2.87
2.94
3.08
3.19
3, 28 .
3.34
3.46 .
3,54
3.61
3.66
3.72
3.81
3.87
3.93
3.97
4.01
1-of-1 Intrawell
16
2.94
3.20
3.35
3.46
3.55
3,73
3.89
4.00
4.09
4.25
4.37
4.46
4.54
4.63
4.75
4.85
4.93
5.00
5.05
20
2.79
3.01
3.14
3.24
3.31
3.47
3,60
3,69
3,77
3.90
4.00
4.08
4.14
4.22
4.31
4.39
4.46
4.51
4.56
25
2.67
2.87
2.99
3.07
3.14
3.27
3.39
3.48
3.54
3,66
3.74
3.81
3,86
3,92
4.01
4.07
4.13
4.17
4.21
30
2.40
2.60
2.71
2.79
2.85
2.97
3.08
3.16
3.22
3,32
3,40
3,46
3,51
3,57
3.64
3.70
3.75
3.79
3.82
35
2.36
2.54
2.65
2.73
2.78
2.90
3.01
3.08
3.13
3.23
'3,31
3.36
3.41
3,46
3,53
3,59,
3,63
3.67
3.70
Prediction
30
2.59
2.79
2.89
2.97
3.03
3.16
3.26
3.34
3.40
3.51
3,58
3,64
3,69
3.75
3,82
3,88
3,93
3.97
4.01
35
2.54
2.73
2.83
2.90
2.96
3.08
3.18
3.25
3.31
3.41
3.48
3.53
3.58
3,63
3.7O
3,75
3,80
3,83
3,87
40 45 50 60 70 80 90 100 125 150
2.33 2.30 2.29 2.26 2.24 2.22 2.21 2.20 2.19 2.18
2.51 2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34 2.33
2.61 2.58 2.56 2.52 2.49 2.48 2.46 2.45 2.43 2.42
2.68 2.65 2.62 2.59 2.56 2.54 2.52 2.51 2.49 2.48
2.74 2.70 2.68 2.64 2.61 2.59 2.57 2.56 2.54 2.52
2.85 2.82 2.79 2.74 2.71 2.69 2.67 2.66 2.63 2.62
2.95 2.91 2.88 2.83 2.80 2.77 2.75 2.74 2.71 2.70
3.02 2.98 2.94 2.89 2.86 2.83 2.81 2.80 2.77 2.75
3.07 3.03 2.99 2.94 2.90 2.88 2.86 2.84 2.81 2.78
3.17 3.12 3.08 3.03 2.99 2.96 2.94 2.92 2.88 2.87
3.24 3.19 3.15 3.09 3.05 3.02 2.99 2.98 2.94 2.92
3,29 3.24 3.20 3.13 3.09 3.06 3.04 3.01 2.99 2.96
3,33 3,28 3.23 3.17 3.13 3.10 3.07 3.05 3.02 3.00
3,38 3,33 3,28 3.22 3.17 3.14 3.12 3.10 3.06 3.04
3,45 3.39 3.35 3.28 3.23 3.20 3.17 3.15 3.11 3.09
3. SO 3.44 ,3.39 3.32 3.27 3.24 3.21 3.19 3.15 3.13
3,54 3.48, 3.43 , 3,36 3,,31 '3.2?'' 3.25 3.22 3.19 3.16
3,58 3,52 3,46 3,39 3.34 , 3.30 3.27 3.25 3.21 3.19
3,61 3.54 3.49 3.42 3.37 '3,33' ,3.30 3,28 3.24 3.21
Limits on Means of Order 2 (40 COC, Quarterly)
40 45 50 60 70 80 90 100 125 150
2.51 2.48 2.46 2.42 2.40 2.38 2.37 2.36 2.34 2.33
2.68 2.65 2.62 2.59 2.56 2.54 2.52 2.51 2.49 2.48
2.78 2.75 2.72 2.68 2.65 2.63 2.61 2.60 2.57 2.56
2.85 2.81 2.79 2.74 2.71 2.69 2.67 2.66 2.63 2.62
2.91 2.87 2.84 2.79 2.76 2.74 2.72 2.70 2.68 2.66
3.02 2.98 2.94 2.89 2.86 2.83 2.81 2.80 2.77 2.75
3.12 3.07 3.03 2.98 2.94 2.92 2.89 2.88 2.85 2.83
3.19 3.14 3.10 3.04 3.00 2.97 2.95 2.93 2.90 2.88
3.24 3.19 3.15 3.09 3.05 3.02 2.99 2.98 2.94 2.92
3.33 3.28 3.24 3.17 3.13 3.10 3.07 3.04 3.02 3.00
3.40 3.34 3.30 3.23 3.19 3.15 3.12 3.11 3.07 3.05
3.45 3.39 3.35 3.28 3.23 3.20 3.17 3.15 3.11 3.09
3.49 3.43 3.38 3.32 3.27 3.23 3.20 3.18 3.15 3.12
3.54 3.48 3.43 3.36 3.31 3.27 3.25 3.22 3.19 3.16
3,61 3.54 3.49 3.42 3.37 3.33 3.30 3.28 3.24 3.21
3,66 3,59 3.54 3.46 3.41 3.37 3.34 3.32 3.28 3.25
3.70 ,3,63 3.58 3.50 3.44 3.40 3.37 3.35 3.31 3.28
3.74 3.67 3.61 3.53 3.47 3.43 3.40 3.38 3.33 3.30
3.77 3.70 3.64 3.56 3.50 3.46 3.43 3.40 3.36 3.33

                                                     D-164
                                                                                                      March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
     Table 19-15. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (1 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.69 0.56 0.50 (
1.10 0.87 0.77 (
1.36 1.05 0.93 (
1.56 1.18 1.04 (
1.72 1.28 1.12
2.10 1.51 1.30
2.46 1.71 1.46
2.75 1.86 1.57
2.99 1.98 1.66
3.46 2.21 1.82
3.84 2.38 1.94
4.16 2.51 2.04
4.43 2.63 2.12
4.79 2.77 2.22
5.29 2.97 2.35 ;
5.71 3.13 2.45 ;
6.08 3.26 2.54 ;
6.42 3.38 2.61 ;
6.71 3.48 2,88 :
).47 (
).72 (
).86 (
).96 (
.04 (
.19
.33
.43
.50
.64
.74
.82
.88
.96
>.06
>.14
>.21 ;
2.27 ;
2.32 ;
).45 0.42 (
).69 0.65 (
).82 0.77 (
).91 0.86 (
).98 0.92 (
.13 1.05
.25 1.16
.34 1.24
.41 1.30
.53 1.41
.62 1.49
.69 1.54
.74 1.59
.81 1.65
.90 1.72
.97 1.78
>.03 1.83
>.08 1.87
>.12 1.90
).40 (
).62 (
).74 (
).82 (
).88 (
.01 (
.11
.19
.24
.34
.41
.46
.51
.56
.63
.68
.72
.76
.79
).39 0.38 (
).60 0.59 (
).72 0.70 (
).80 0.78 (
).86 0.84 (
).98 0.95 (
.08 1.05
.14 1.12
.20 1.17
.29 1.26
.36 1.32
.40 1.37
.45 1.41
.49 1.45 ]
.56 1.51 ]
.60 1.56 ]
.64 1.59 ]
.68 1.62 ]
.70 1.65 ]
).37 (
).58 (
).69 (
).77 (
).83 (
).94 (
.03
.10
.15
.23
.29
.34
.38
L.42
L.48
L.52
L.56
L.59
L.61
).37 0.37 (
).58 0.57 (
).69 0.68 (
).76 0.75 (
).82 0.81 (
).93 0.92 (
.02 1.01
.08 1.07
.13 1.12
.22 1.20
.28 1.26
.32 1.31
.36 1.34
.40 1.38
.46 1.44
.50 1.48
.53 1.51
.56 1.54
.59 1.57
).36 (
).57 (
).68 (
).75 (
).80 (
).91 (
.00 (
.06
.11
.19
.25
.29
.33
.37
.42
.47
.50
.53
.55
).36 0.36 (
).56 0.56 (
).67 0.66 (
).74 0.73 (
).79 0.79 (
).90 0.89 (
).99 0.98 (
.05 1.04
.10 1.09
.18 1.17
.23 1.22
.28 1.26
.31 1.30
.35 1.34
.40 1.39
.44 1.43
.48 1.46
.50 1.49
.53 1.51
).35 0.35 (
).55 0.55 (
).66 0.66 (
).73 0.73 (
).78 0.78 (
).89 0.88 (
).98 0.97 (
.04 1.03
.08 1.07
.16 1.15
.21 1.21
.25 1.25
.29 1.28
.33 1.32
.38 1.37
.42 1.41
.45 1.44
.47 1.46
.50 1.49
).35 0.35 0.35
).55 0.55 0.54
).65 0.65 0.65
).72 0.72 0.72
).78 0.77 0.77
).88 0.88 0.87
).97 0.96 0.96
.03 1.02 1.01
.07 1.06 1.06
.15 1.14 1.13
.20 1.19 1.19
.24 1.23 1.23
.27 1.26 1.26
.31 1.30 1.30
.36 1.35 1.34
.40 1.39 1.38
.43 1.42 1.41
.46 1.44 1.43
.48 1.46 1.46
   Table 19-15.  K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.02 0.83 0.75 (
1.47 1.14 1.02 (
1.77 1.33 1.17
2.00 1.47 1.28
2.19 1.58 1.37
2.63 1.82 1.55
3.06 2.04 1.72
3.40 2.21 1.84
3.68 2.34 1.93
4.26 2.59 2.10
471 2.78 2.23
5.09 2,93 2.33 ;
5.42 3.06 2.42 ;
5.85 3.22 2.52 ;
6.46 3.44 2.66 ;
6.97 3.62 2.77 ;
7.41 3.77 2,86 ;
7.81 3.90 2,94 :
8.17 4.02 3.01 ;
).70 (
).95 (
.09
.18
.26
.42
.56
.65
.73
.87
.98
>.06
>.12
1.20 :
1.31 :
i.40 :
1.47 :
1.53 :
i.58 ;
).67 0.64 (
).90 0.85 (
.03 0.97 (
.12 1.05
.19 1.11
.33 1.24
.46 1.35
.55 1.42
.61 1.48
.74 1.59
.83 1.66
.90 1.72
.95 1.77
1.02 1.83
1.12 1.90
>.19 1.96
1.25 2.01
1.30 2.05
1.34 2.08
).62 (
).82 (
).93 (
.01 (
.06
.18
.29
.36
.41
.51
.57
.63
.67
.72
.79
.84
.88
.92
.95
).60 0.59 (
).79 0.78 (
).90 0.88 (
).97 0.95 (
.03 1.01 (
.14 1.12
.24 1.21
.30 1.27
.35 1.32
.44 1.40
.51 1.46
.56 1.51 ]
.59 1.55 ]
.64 1.59 ]
.70 1.65 ]
.75 1.69 ]
.79 1.73 ]
.82 1.76 1
.85 1.79 ]
).58 (
).77 (
).87 (
).94 (
).99 (
.10
.19
.25
.29
.38
.43
L.48
L.51
L.56
L.61
L.66
L.69
L.72
L.74
).57 0.57 (
).76 0.75 (
).86 0.85 (
).93 0.92 (
).98 0.97 (
.08 1.07
.17 1.16
.23 1.22
.28 1.26
.36 1.34
.41 1.40
.46 1.44
.49 1.47
.53 1.51
.59 1.57
.63 1.61
.66 1.64
.69 1.67
.71 1.69
).56 (
).75 (
).84 (
).91 (
).96 (
.06
.15
.21
.25
.33
.38
.42
.46
.50
.55
.59
.62
.65
.67
).56 0.55 (
).74 0.73 (
).84 0.83 (
).90 0.89 (
).95 0.94 (
.05 1.04
.13 1.12
.19 1.18
.23 1.22
.31 1.30
.36 1.35
.40 1.39
.44 1.42
.48 1.46
.53 1.51
.56 1.55
.59 1.58
.62 1.60
.64 1.62
).55 0.55 (
).73 0.73 (
).82 0.82 (
).89 0.88 (
).94 0.93 (
.04 1.03
.12 1.11
.17 1.17
.21 1.21
.29 1.28
.34 1.33
.38 1.37
.41 1.40
.45 1.44
.50 1.49
.53 1.52
.56 1.55
.59 1.58
.61 1.60
).55 0.54 0.54
).72 0.72 0.72
).82 0.81 0.81
).88 0.87 0.87
).93 0.92 0.92
.03 1.02 1.01
.11 1.10 1.09
.16 1.15 1.14
.20 1.19 1.19
.27 1.26 1.26
.32 1.31 1.31
.36 1.35 1.34
.39 1.38 1.37
.43 1.42 1.41
.48 1.46 1.46
.51 1.50 1.49
.54 1.53 1.52
.57 1.55 1.54
.59 1.57 1.56
                                                   D-165
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-15. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.34 1.09 0.98 (
1.85 1.42 1.25
2.19 1.62 1.41
2.45 1.77 1.52
2.67 1.89 1.61
3.19 2.15 1.81
3.69 2.39 1.98
4.09 2.57 2.11
4.43 2.71 2.20
5.10 2.99 2.39 ;
5.64 3.20 2.53 ;
6.09 3.37 2.63 ;
6.48 3.51 2.72 ;
7.00 3.69 2.84 ;
7.71 3.94 2.99 ;
8.32 4.14 3,11 ;
8.85 4.31 3.21 '<
9.33 4.45 3.29 ;
9.76 4.58 3.37 ;
).92 (
.16
.30
.40
.48
.64
.78
.88
.96
Ml
1.22 ;
2.30 ;
2.37 ;
2.45 ;
2.57 ;
2.66 ;
2.73 ;
2.80 ;
2.85 ;
).88 0.84 (
.10 1.04
.23 1.15
.32 1.23
.39 1.29
.54 1.42
.66 1.52
.75 1.60
.82 1.66
.95 1.77
2.04 1.84
2.11 1.90
2.17 1.95
2.24 2.01
2.33 2.08
2.41 2.14 ;
2.47 2.19 ;
2.52 2.23 ;
2.57 2.26 ;
).81 (
.00 (
.11
.18
.24
.35
.45
.52
.57
.67
.73
.79
.83
.88
.95
2.00
2.04
2.08
2.11
).79 0.77 (
).97 0.95 (
.07 1.05
.14 1.11
.19 1.16
.30 1.27
.39 1.36
.46 1.42
.51 1.46
.59 1.55
.65 1.60 ]
.70 1.65 ]
.74 1.69 ]
.79 1.73 ]
.85 1.79 ]
.89 1.83 1
.93 1.86 1
.96 1.89 ]
.99 1.92 ]
).76 (
).93 (
.03
.09
.14
.25
.33
.39
.43
.51
L.57
L.61
L.65
L.69
L.74
L.78
L.82
L.85
L.87
).75 0.75 (
).92 0.92 (
.02 1.01
.08 1.07
.13 1.12
.23 1.22
.31 1.30
.37 1.35
.41 1.40
.49 1.47
.54 1.52
.59 1.57
.62 1.60
.66 1.64
.71 1.69
.75 1.73
.78 1.76
.81 1.79
.84 1.81
).74 (
).91 (
.00 (
.06
.11
.21
.28
.34
.38
.46
.51
.55
.58
.62
.67
.71
.74
.76
.79
).74 0.73 (
).90 0.89 (
).99 0.98 (
.05 1.04
.10 1.09
.19 1.18
.27 1.26
.32 1.31
.36 1.35
.44 1.42
.49 1.47
.52 1.51
.56 1.54
.59 1.58
.64 1.62
.68 1.66
.71 1.69
.73 1.71
.75 1.73
).73 0.72 (
).89 0.88 (
).97 0.97 (
.03 1.03
.08 1.07
.17 1.16
.25 1.24
.30 1.29
.34 1.33
.41 1.40
.46 1.45
.50 1.49
.53 1.52
.56 1.55
.61 1.60
.64 1.63
.67 1.66
.70 1.68
.72 1.70
).72 0.72 0.71
).88 0.87 0.87
).97 0.96 0.96
.03 1.02 1.01
.07 1.06 1.06
.16 1.15 1.14
.23 1.22 1.22
.29 1.28 1.27
.32 1.31 1.31
.39 1.38 1.37
.44 1.43 1.42
.48 1.46 1.46
.51 1.49 1.48
.54 1.53 1.52
.59 1.57 1.56
.62 1.61 1.60
.65 1.63 1.62
.67 1.66 1.65
.69 1.68 1.67
      Table 19-15. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.10 0.87 0.77 (
1.56 1.18 1.04 (
1.86 1.37 1.19
2.10 1.51 1.30
2.29 1.62 1.39
2.75 1.86 1.57
3.19 2.08 1.73
3.54 2.24 1.85
3.84 2.38 1.94
4.43 2.63 2.12
4.90 2.82 2.25
5.29 2.97 2.35 ;
5.64 3.10 2.43 ;
6.09 3.26 2.54 ;
6.71 3.48 2,68 :
7.24 3.66 2,79 :
7.71 3.82 2.88 ;
8.12 3.95 2.96 ;
8.50 4.06 3.03 ;
).72 (
).96 (
.10
.19
.27
.43
.56
.66
.74
.88
.98
2.06
2.13
2.21 ;
2.32 ;
2.40 ;
2.48 ;
2.53 ;
2.59 ;
).69 0.65 (
).91 0.86 (
.04 0.97 (
.13 1.05
.20 1.11
.34 1.24
.46 1.35
.55 1.43
.62 1.49
.74 1.59
.83 1.67
.90 1.72
.96 1.77
2.03 1.83
2.12 1.90
2.19 1.96
2.25 2.01
2.30 2.05
2.34 2.09
).62 (
).82 (
).93 (
.01 (
.07
.19
.29
.36
.41
.51
.58
.63
.67
.72
.79
.84
.88
.92
.95
).60 0.59 0.58 (
).80 0.78 0.77 (
).90 0.88 0.87 (
).98 0.95 0.94 (
.03 1.01 0.99 (
.14 1.12 .10
.24 1.21 .19
.30 1.27 .25
.36 1.32 .29
.45 1.41 .38
.51 1.46 1.44
.56 1.51 1.48
.60 1.55 1.51
.64 1.59 1.56
.70 1.65 1.61
.75 1.69 1.66
.79 1.73 1.69
.82 1.76 1.72
.85 1.79 1.74
).58 0.57 (
).76 0.75 (
).86 0.85 (
).93 0.92 (
).98 0.97 (
.08 1.07
.17 1.16
.23 1.22
.28 1.26
.36 1.34
.41 1.40
.46 1.44
.49 1.47
.53 1.51
.59 1.57
.63 1.61
.66 1.64
.69 1.67
.71 1.69
).57 (
).75 (
).85 (
).91 (
).96 (
.06
.15
.21
.25
.33
.38
.42
.46
.50
.55
.59
.62
.65
.67
).56 0.56 (
).74 0.73 (
).84 0.83 (
).90 0.89 (
).95 0.94 (
.05 1.04
.13 1.12
.19 1.18
.23 1.22
.31 1.30
.36 1.35
.40 1.39
.44 1.42
.48 1.46
.53 1.51
.56 1.55
.59 1.58
.62 1.60
.64 1.62
).55 0.55 (
).73 0.73 (
).82 0.82 (
).89 0.88 (
).94 0.93 (
.04 1.03
.12 1.11
.17 1.17
.21 1.21
.29 1.28
.34 1.33
.38 1.37
.41 1.40
.45 1.44
.50 1.49
.53 1.52
.56 1.55
.59 1.58
.61 1.60
).55 0.55 0.54
).72 0.72 0.72
).82 0.81 0.81
).88 0.88 0.87
).93 0.92 0.92
.03 1.02 1.01
.11 1.10 1.09
.16 1.15 1.15
.20 1.19 1.19
.27 1.26 1.26
.32 1.31 1.31
.36 1.35 1.34
.39 1.38 1.37
.43 1.42 1.41
.48 .46 1.46
.51 .50 1.49
.54 .53 1.52
.57 .55 1.54
.59 .57 1.56
                                                    D-166
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
   Table 19-15.  K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (2 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.47 1.14 1.02 0.95 (
2.00 1.47 1.28 1.18
2.35 1.68 1.44 1.32
2.63 1.82 1.55 1.42
2.86 1.94 1.64 1.49
3.40 2.21 1.84 1.65
3.93 2.45 2.01 1.79
4.35 2.63 2.13 1.90
471 2.78 2.23 1.98
5.42 3.06 2.42 2.12
5.98 3.27 2.55 2.23 ;
6.46 3.44 2.66 2.31 ;
6.87 3.59 2.75 2.38 ;
7.41 3.77 2.86 2.47 ;
8.17 4.02 3.01 2.58 ;
8.82 4.22 3.13 2.67 ;
9.38 4.39 3.23 2.74 ;
9.87 4.53 3.32 2.81 ;
10.34 4.67 3.39 2.88 '4
).90 (
.12
.25
.33
.40
.55
.67
.76
.83
.95
2.04
2.12
2.17
1.25 :
1.34 ;
2.42 ;
2.48 ;
2.53 ;
2.57 ;
).85 (
.05
.16
.24
.30
.42
.53
.61
.66
.77
.84
.90
.95
2.01
2.08
2.14 ;
2.19 ;
1.23 :
1.27 ;
).82 (
.01 (
.11
.18
.24
.36
.45
.52
.57
.67
.74
.79
.83
.88
.95
2.00
2.04
2.08
Ml
).79 (
).97 (
.07
.14
.20
.30
.39
.46
.51
.59
.66
.70
.74
.79
.85
.90
.93
.96
.99
).78 (
).95 (
.05
.12
.17
.27
.36
.42
.46
.55
.61
.65
.69
.73
.79
.83
.86
.90
.92
).77 (
).94 (
.03
.10
.15
.25
.33
.39
.43
.51
.57
.61
.65
.69
.74
.79
.82
.85
.87
).76 (
).93 (
.02
.08
.13
.23
.31
.37
.41
.49
.54
.59
.62
.66
.71
.75
.78
.81
.83
).75 (
).92 (
.01
.07
.12
.22
.30
.35
.40
.47
.52
.57
.60
.64
.69
.73
.76
.79
.81
).75 0.74 0.73 0.73 0.73 0.72 (
).91 0.90 0.89 0.89 0.88 0.88 (
.00 0.99 0.98 0.98 0.97 0.97 (
.06 1.05 1.04 1.04 1.03 1.03
.11 1.10 1.09 1.08 1.07 1.07
.21 1.19 1.18 1.17 1.17 1.16
.29 1.27 1.26 1.25 1.24 1.23
.34 1.32 1.31 1.30 1.29 1.29
.38 1.36 1.35 1.34 1.33 1.32
.46 1.44 1.42 1.41 1.40 1.39
.51 1.49 1.47 1.46 1.45 1.44
.55 1.53 1.51 1.50 1.49 1.48
.58 1.56 1.54 1.53 1.52 1.51
.62 1.59 1.58 1.56 1.55 1.54
.67 1.64 1.62 1.61 1.60 1.59
.71 1.68 1.66 1.64 1.63 1.62
.74 1.71 1.69 1.67 1.66 1.65
.76 1.73 1.71 1.70 1.68 1.67
.79 1.76 1.73 1.72 1.70 1.69
).72 (
).87 (
).96 (
.02
.06
.15
.22
.28
.31
.38
.43
.46
.49
.53
.57
.61
.63
.66
.68
).72
).87
).96
.01
.06
.14
.22
.27
.31
.37
.42
.46
.48
.52
.56
.60
.62
.65
.66
    Table 19-15. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.85 1.42 1.25 1.16
2.45 1.77 1.52 1.40
2.86 1.99 1.69 1.54
3.19 2.15 1.81 1.64
3.46 2.28 1.90 1.72
4.09 2.57 2.11 1.88
4.72 2.84 2.29 2.03
5.22 3.04 2.42 2.13
5.64 3.20 2.53 2.22 ;
6.48 3.51 2.72 2.37 ;
7.15 3.75 2.87 2.48 ;
7.72 3.94 2.99 2.57 ;
8.21 4.10 3.O8 2.64 ;
8.85 4.31 3,21 2.73 ;
9.76 4.58 3.37 2.85 ;
10.52 4.81 3.50 2.94 2
11.19 5.00 3.61 3.02 2
11.78 5.16 3.70 3.O9 \
12.30 5.32 3.78 3.T5 \
.10
.32
.45
.54
.60
.75
.88
.97
2.04
2.17
1.26 :
1.33 :
1.39 :
2.47 ;
2.57 ;
2.64 ;
2.7i ;
2.76 ;
2.8i ;
.04
.23
.34
.42
.48
.60
.71
.78
.84
.95
2.02
2.08
2.13
2.19 ;
2.27 ;
2.32 ;
2.37 ;
2.42 ;
2.45 ;
.00 (
.18
.28
.35
.41
.52
.61
.68
.73
.83
.90
.95
.99
2.04
2.11
2.16 ;
2.20 ;
2.24 ;
2.27 ;
).97 (
.14
.23
.30
.35
.46
.54
.61
.65
.74
.80
.85
.89
.93
.99
2.04
2.08 ;
2.11 ;
2.14 ;
).95 (
.11
.20
.27
.32
.42
.50
.56
.60
.69
.74
.79
.82
.86
.92
.96
2.00
2.03
2.05 ;
).93 (
.09
.18
.25
.29
.39
.47
.53
.57
.65
.70
.74
.78
.82
.87
.91
.94
.97
2.00
).92 (
.08
.17
.23
.27
.37
.45
.50
.54
.62
.67
.71
.74
.78
.84
.87
.91
.93
.96
).92 (
.07
.16
.22
.26
.35
.43
.48
.52
.60
.65
.69
.72
.76
.81
.85
.88
.90
.92
).91 0.90 0.89 0.89 0.88 0.88 (
.06 1.05 1.04 1.03 1.03 1.02
.15 1.13 1.12 1.12 1.11 1.10
.21 1.19 1.18 1.17 1.16 1.16
.25 1.23 1.22 1.21 1.21 1.20
.34 1.32 1.31 1.30 1.29 1.29
.42 1.40 1.38 1.37 1.36 1.36
.47 1.45 1.43 1.42 1.41 1.40
.51 1.49 1.47 1.46 1.45 1.44
.58 1.56 1.54 1.53 1.52 1.51
.63 1.60 1.59 1.57 1.56 1.55
.67 1.64 1.62 1.61 1.60 1.59
.70 1.67 1.65 1.64 1.63 1.62
.74 1.71 1.69 1.67 1.66 1.65
.79 1.76 1.73 1.72 1.70 1.69
.82 1.79 1.77 1.75 1.74 1.73
.85 1.82 1.80 1.78 1.77 1.75
.88 1.84 1.82 1.80 1.79 1.78
.90 1.86 1.84 1.82 1.81 1.80
).87 (
.02
.10
.15
.19
.28
.34
.39
.43
.49
.54
.57
.60
.63
.68
.71
.74
.76
.78
).87
.01
.09
.14
.19
.27
.34
.38
.42
.48
.53
.56
.59
.62
.66
.70
.72
.74
.76
                                                    D-167
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
     Table 19-15. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.72 1.28 1.12 1.04 (
2.29 1.62 1.39 1.27
2.68 1.82 1.55 1.41
2.99 1.98 1.66 1.50
3.24 2.10 1.75 1.58
3.84 2.38 1.94 1.74
4.43 2.63 2.12 1.88
4.90 2.82 2.25 1.98
5.29 2.97 2.35 2.06
6.08 3.26 2.54 2.21 ;
6.71 3.48 2,68 2.32 ;
7.25 3.66 2,79 2.40 ;
7.71 3.81 2.88 2.48 ;
8.30 4.00 3.00 2.56 ;
9.16 4.27 3.15 2,68 '*
9.88 4.47 3.27 2,77 \
10.51 4.66 3.37 2,84 \
11.05 4.80 3.47 2.91
11.56 4.96 3.54 2.96
).98 (
.20
.32
.41
.48
.62
.74
.83
.90
2.03
1.12
2.19
1.25 :
1.32 :
2.42 ;
2.49 ;
2.55 ;
?,8i :
2,65 ;
).92 (
.11
.23
.30
.36
.49
.59
.67
.72
.83
.90
.96
2.01
2.07
2.14 ;
1.20 ;
1.25 :
1.29 :
1.33 ;
).88 (
.07
.17
.24
.30
.41
.51
.58
.63
.72
.79
.84
.88
.93
2.00
2.05
2.09
2.13 ;
2.16 ;
).86 (
.03
.13
.20
.25
.36
.45
.51
.56
.64
.70
.75
.79
.84
.90
.94
.98
2.01
2.04
).84 (
.01 (
.10
.17
.22
.32
.41
.46
.51
.59
.65
.69
.73
.77
.83
.87
.91
.94
.96
).83 (
).99 (
.08
.15
.20
.29
.38
.43
.48
.56
.61
.66
.69
.73
.78
.83
.86
.89
.91
).82 (
).98 (
.07
.13
.18
.28
.36
.41
.46
.53
.59
.63
.66
.70
.75
.79
.82
.85
.88
).81 (
).97 (
.06
.12
.17
.26
.34
.40
.44
.51
.57
.61
.64
.68
.73
.77
.80
.82
.85
).80 0.79 0.79 0.78 0.78 0.78 (
).96 0.95 0.94 0.94 0.93 0.93 (
.05 1.04 1.03 1.02 1.02 1.01
.11 1.10 1.09 1.08 1.07 1.07
.16 1.14 1.13 1.12 1.12 1.11
.25 1.23 1.22 1.21 1.21 1.20
.33 1.31 1.30 1.29 1.28 1.27
.38 1.36 1.35 1.34 1.33 1.33
.42 1.40 1.39 1.38 1.37 1.36
.50 1.48 1.46 1.45 1.44 1.43
.55 1.53 1.51 1.50 1.49 1.48
.59 1.56 1.55 1.53 1.52 1.51
.62 1.59 1.58 1.56 1.55 1.54
.66 1.63 1.61 1.60 1.59 1.58
.71 1.68 1.66 1.64 1.63 1.62
.75 1.72 1.69 1.68 1.67 1.66
.77 1.75 1.72 1.71 1.69 1.68
.80 1.77 1.75 1.73 1.72 1.71
.82 1.79 1.77 1.75 1.74 1.73
).77 (
).92 (
.01
.06
.10
.19
.26
.31
.35
.42
.46
.50
.53
.56
.61
.64
.67
.69
.71
).77
).92
.00
.06
.10
.19
.26
.31
.34
.41
.46
.49
.52
.55
.60
.63
.66
.68
.70
   Table 19-15.  K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.19 1.58 1.37 1.26
2.86 1.94 1.64 1.49
3.32 2.17 1.81 1.63
3.68 2.34 1.93 1.73
3.99 2.48 2.02 1.81
471 2.78 2.23 1.98
5.42 3.06 2.42 2.12
5.98 3.27 2.55 2.23 ;
6.46 3.44 2.66 2.31 ;
7.41 3.77 2,86 2.47 ;
8.17 4.02 3.01 2.58 ;
8.82 4.22 3.13 2.67 ;
9.38 4.39 3.23 2.74 ;
10.12 4.60 3.36 2,84 '*
11.13 4.89 3.53 2,96 \
11.99 5.14 3.66 3.OS ;
12.73 5.33 3.77 3.13 ;
13.44 5.51 3.87 3.20 ,
14.06 5.66 3.96 3.26 „
.19
.40
.53
.61
.68
.83
.95
2.04
1.12
1.25 ;
2.34 ;
2.41 ;
2.48 ;
2.55 ;
2.65 ;
2.73 ;
2.79 ;
2,85 :
2,89 :
.11
.30
.41
.48
.54
.66
.77
.84
.90
2.01
2.08
2.14 ;
2.19 ;
2.25 ;
2.33 ;
2.39 ;
2.44 ;
2.48 ;
2.5i ;
.06
.24
.34
.41
.46
.57
.67
.74
.79
.88
.95
2.00
2.04
2.09
2.16 ;
2.21 ;
2.25 ;
2.29 ;
2.32 ;
.03
.20
.29
.35
.40
.51
.59
.66
.70
.79
.85
.90
.93
.98
2.04
2.08 ;
2.12 ;
2.15 ;
2.18 ;
.01 (
.17
.26
.32
.37
.46
.55
.61
.65
.73
.79
.83
.87
.91
.96
2.01
2.04
2.07 ;
2.09 ;
).99 (
.15
.23
.29
.34
.43
.51
.57
.61
.69
.74
.78
.82
.86
.91
.95
.98
2.01
2.04
).98 (
.13
.22
.28
.32
.41
.49
.54
.59
.66
.71
.75
.78
.82
.88
.91
.94
.97
.99
).97 (
.12
.20
.26
.31
.40
.47
.52
.57
.64
.69
.73
.76
.80
.85
.88
.91
.94
.96
).96 0.95 0.94 0.94 0.93 0.93 (
.11 1.10 1.09 1.08 1.07 1.07
.19 1.18 1.17 1.16 1.15 1.15
.25 1.23 1.22 1.21 1.21 1.20
.29 1.28 1.26 1.25 1.25 1.24
.38 1.36 1.35 1.34 1.33 1.32
.46 1.44 1.42 1.41 1.40 1.39
.51 1.49 1.47 1.46 1.45 1.44
.55 1.53 1.51 1.50 1.49 1.48
.62 1.59 1.58 1.56 1.55 1.54
.67 1.64 1.62 1.61 1.60 1.59
.71 1.68 1.66 1.64 1.63 1.62
.74 1.71 1.69 1.67 1.66 1.65
.78 1.74 1.72 1.71 1.69 1.68
.82 1.79 1.77 1.75 1.74 1.73
.86 1.83 1.80 1.78 1.77 1.76
.89 1.86 1.83 1.81 1.80 1.79
.91 1.88 1.85 1.84 1.82 1.81
.94 1.90 1.88 1.86 1.84 1.83
).92 (
.06
.14
.19
.23
.31
.38
.43
.46
.53
.57
.61
.63
.67
.71
.74
.77
.79
.80
).92
.06
.13
.19
.23
.31
.37
.42
.46
.52
.56
.60
.62
.66
.70
.73
.75
.77
.79
                                                   D-168
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-15. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.67 1.89 1.61 1.48 .39
3.46 2.28 1.90 1.72 .60
4.00 2.53 2.08 1.86 .73
443 2.71 2.20 1.96 .82
479 2.87 2.31 2.04 .89
5.64 3.20 2.53 2.22 2.04
6.48 3.51 2.72 2.37 2.17
7.15 3.75 2.87 2.48 2.26 ;
7.71 3.94 2.99 2.57 2.33 ;
8.85 4.31 3,21 2.73 2.47 ;
9.76 4.58 3.37 2.85 2.57 ;
10.52 4.81 3.50 2.94 2.64 ;
11.19 5.00 3.61 3.02 2.71 ;
12.05 5.24 3.74 3. 12 2.79 ;
13.28 5.57 3.92 3,25 2.89 ;
14.30 5.84 4.07 3.35 2.97 ;
15.23 6.07 4.19 3.43 3.04 ;
16.02 6.27 4.30 3.51 3.-O9 '*
16.72 6.45 4.38 3.57 3,74 ;
.29
.48
.58
.66
.72
.84
.95
2.02
2.08
2.19 ;
2.26 ;
2.32 ;
2.37 ;
1.43 :
LSI ;
2.57 ;
2.62 ;
2.67 ;
2.7i ;
.24
.41
.50
.57
.62
.73
.83
.90
.95
2.04
2.11
2.16 ;
2.20 ;
2.25 ;
2.32 ;
2.37 ;
2.42 ;
2.45 ;
2.48 ;
.19
.35
.44
.51
.55
.65
.74
.80
.85
.93
.99
2.04
2.08 ;
2.12 ;
2.18 ;
2.23 ;
2.26 ;
2.30 ;
2.32 ;
.16
.32
.40
.46
.51
.60
.69
.74
.79
.86
.92
.96
2.00
2.04
2.09 ;
2.14 ;
2.17 ;
2.20 ;
2.23 ;
.14
.29
.38
.43
.48
.57
.65
.70
.74
.82
.87
.91
.94
.98
2.04
2.08 ;
2.11 ;
2.14 ;
2.16 ;
.13
.27
.36
.41
.46
.54
.62
.67
.71
.78
.84
.87
.91
.94
.99
2.03 ;
2.06 ;
2.09 ;
2.11 ;
.12
.26
.34
.40
.44
.52
.60
.65
.69
.76
.81
.85
.88
.91
.96
2.00
2.03 ;
2.06 ;
2.08 ;
.11 1.10 1.09 1.08 1.07 1.07
.25 1.23 1.22 1.21 1.21 1.20
.33 1.31 1.30 1.29 1.28 1.27
.38 1.36 1.35 1.34 1.33 1.32
.42 1.40 1.39 1.38 1.37 1.36
.51 1.49 1.47 1.46 1.45 1.44
.58 1.56 1.54 1.53 1.52 1.51
.63 1.60 1.59 1.57 1.56 1.55
.67 1.64 1.62 1.61 1.60 1.59
.74 1.71 1.69 1.67 1.66 1.65
.79 1.75 1.73 1.72 1.70 1.69
.82 1.79 1.77 1.75 1.74 1.73
.85 1.82 1.80 1.78 1.77 1.75
.89 1.85 1.83 1.81 1.80 1.79
.94 1.90 1.87 1.85 1.84 1.83
.97 1.93 1.91 1.88 1.87 1.85
2.00 1.96 1.93 1.91 1.90 1.89
2.03 1.99 1.96 1.94 1.92 1.91
2.05 2.01 1.98 1.96 1.94 1.93
.06
.19
.26
.31
.35
.43
.49
.54
.57
.63
.68
.71
.74
.77
.80
.84
.86
.88
.90
.06
.19
.26
.31
.34
.42
.48
.53
.56
.62
.67
.70
.72
.75
.78
.82
.85
.87
.89
     Table 19-15. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.29 1.62 1.39 1.27
2.99 1.98 1.66 1.50
3.46 2.21 1.82 1.64
3.84 2.38 1.94 1.74
4.15 2.51 2.04 1.82
4.90 2.82 2.25 1.98
5.63 3.10 2.43 2.13
6.22 3.31 2.57 2.24 ;
6.71 3.48 2,68 2.32 ;
7.70 3.81 2.88 2.47 ;
8.50 4.06 3.03 2.59 ;
9.16 4.27 3.15 2,68 \
9.74 4.44 3.25 2,75 :
10.50 4.66 3.38 2,84 '*
11.57 4.96 3.54 2.97
12.50 5.20 3.67 3.06
13.28 5.40 3.78 3.14 ,
13.96 5.57 3.88 3.21
14.65 5.71 3.98 3.27 ;
.20
.41
.53
.62
.69
.83
.96
2.05
2.12
2.25 ;
2.34 ;
2.42 ;
2.48 ;
2.55 ;
y.'ee'' ;
2,73 ;
z.ao ;
2,84 ;
?.89 ;
.11
.30
.41
.49
.54
.67
.77
.85
.90
2.01
2.09
2.15 ;
2.19 ;
2.25 ;
2.33 ;
2.39 ;
2.44 ;
2.48 ;
2.5i ;
.07
.24
.34
.41
.46
.58
.67
.74
.79
.88
.95
2.00
2.04
2.09
2.16 ;
2.21 ;
2.26 ;
2.29 ;
2.32 ;
.03
.20
.29
.36
.40
.51
.60
.66
.70
.79
.85
.90
.93
.98
2.04
2.09 ;
2.12 ;
2.15 ;
2.19 ;
.01 (
.17
.26
.32
.37
.46
.55
.61
.65
.73
.79
.83
.86
.91
.96
2.01
2.04
2.07 ;
2.10 ;
).99 (
.15
.23
.29
.34
.44
.51
.57
.61
.69
.74
.79
.82
.86
.91
.95
.98
2.01
2.04 ;
).98 (
.13
.22
.28
.32
.41
.49
.54
.59
.66
.71
.75
.79
.82
.87
.91
.94
.97
2.00
).97 (
.12
.20
.26
.31
.40
.47
.53
.57
.64
.69
.73
.76
.80
.85
.88
.91
.94
.96
).96 0.95 0.94 0.94 0.93 0.93 (
.11 1.10 1.09 1.08 1.07 1.07
.19 1.18 1.17 1.16 1.15 1.15
.25 1.23 1.22 1.21 1.21 1.20
.29 1.28 1.26 1.25 1.25 1.24
.38 1.36 1.35 1.34 1.33 1.33
.46 1.44 1.42 1.41 1.40 1.39
.51 1.49 1.47 1.46 1.45 1.44
.55 1.53 1.51 1.50 1.49 1.48
.62 1.59 1.58 1.56 1.55 1.54
.67 1.64 1.62 1.61 1.60 1.59
.71 1.68 1.66 1.64 1.63 1.62
.74 1.71 1.69 1.67 1.66 1.65
.78 1.75 1.72 1.71 1.69 1.68
.82 1.79 1.77 1.75 1.74 1.73
.86 1.82 1.80 1.79 1.77 1.76
.89 1.86 1.83 1.81 1.80 1.79
.92 1.88 1.86 1.84 1.82 1.81
.93 1.90 1.87 1.85 1.84 1.83
).92 (
.06
.14
.19
.23
.31
.38
.43
.46
.53
.57
.61
.63
.67
.71
.74
.77
.79
.81
).92
.06
.13
.19
.23
.31
.37
.42
.46
.52
.56
.59
.62
.66
.70
.73
.75
.77
.79
                                                    D-169
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
  Table 19-15. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (10 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.86 1.94 1.64 1.49 .40
3.68 2.34 1.93 1.73 .61
426 2.59 2.10 1.87 .74
471 2.78 2.23 1.98 .83
5.09 2,93 2.33 2.06 .90
5.98 3.27 2.55 2.23 2.05
6.87 3.59 2.75 2.38 2.17
7.58 3.82 2.9O 2.49 2.27 ;
8.18 4.02 3.01 2.58 2.34 ;
9.38 4.39 3.23 2.74 2.47 ;
10.33 4.67 3.40 2.88 2.57 ;
11.13 4.90 3.53 2.-98 2.65 ;
11.84 5.09 3.63 3.O3 2.71 ;
12.74 5.33 3.77 3.13 2.79 ;
14.06 5.66 3.96 3.26 2,89 '*
15.14 5.96 4.10 3.36 2.97 '<
16.11 6.18 4.22 3.44 3,04 '<
16.99 6.40 4.32 3.52 3.1O :
17.77 6.54 4.42 3.59 3.15 ;
.30
.48
.59
.66
.72
.84
.95
1.02
2.08
2.19 ;
2.27 ;
2.33 ;
2.38 ;
2.44 ;
2.5i ;
2.58 ;
i.62 :
i.67 :
i.7i ;
.24
.41
.51
.57
.63
.74
.83
.90
.95
2.04
Ml
2.16 ;
1.20 ;
>.26 :
1.32 ;
2.37 ;
2.42 ;
2.45 ;
2.48 ;
.20
.35
.44
.51
.56
.66
.74
.80
.85
.93
.99
2.04
2.08 ;
2.12 ;
2.18 ;
2.23 ;
2.26 ;
2.29 ;
2.33 ;
.17
.32
.40
.46
.51
.61
.69
.74
.79
.87
.92
.96
2.00
2.04
2.09 ;
2.14 ;
2.17 ;
2.20 ;
2.23 ;
.15
.29
.38
.43
.48
.57
.65
.70
.74
.82
.87
.91
.94
.98
2.04
2.08 ;
2.11 ;
2.14 ;
2.16 ;
.13
.28
.36
.41
.46
.54
.62
.67
.71
.78
.84
.87
.91
.94
.99
2.03 ;
2.06 ;
2.09 ;
2.11 ;
.12
.26
.34
.40
.44
.52
.60
.65
.69
.76
.81
.85
.88
.91
.96
2.00
2.03 ;
2.06 ;
2.08 ;
.11 1.10 1.09 1.08 1.07 1.07
.25 1.23 1.22 1.21 1.21 1.20
.33 1.31 1.30 1.29 1.28 1.27
.38 1.36 1.35 1.34 1.33 1.32
.42 1.40 1.39 1.38 1.37 1.36
.51 1.49 1.47 1.46 1.45 1.44
.58 1.56 1.54 1.53 1.52 1.51
.63 1.60 1.59 1.57 1.56 1.55
.67 1.64 1.62 1.61 1.60 1.59
.74 1.71 1.69 1.67 1.66 1.65
.79 1.75 1.73 1.72 1.70 1.69
.82 1.79 1.77 1.75 1.74 1.73
.85 1.82 1.80 1.78 1.77 1.75
.89 1.86 1.83 1.81 1.80 1.79
.94 1.90 1.87 1.86 1.84 1.83
.97 1.93 1.90 1.88 1.87 1.85
2.00 1.96 1.93 1.91 1.89 1.89
2.03 1.99 1.96 1.93 1.92 1.91
2.05 2.01 1.98 1.95 1.94 1.93
.06
.19
.26
.31
.35
.43
.49
.54
.57
.63
.68
.71
.73
.77
.80
.84
.86
.89
.90
.06
.19
.26
.31
.34
.42
.48
.53
.56
.62
.66
.70
.72
.75
.77
.82
.85
.87
.89
    Table 19-15.  K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (10 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1 3.46 2.28 1.90 1.72 1.60 .48
2 443 2.71 2.20 1.96 1.82 .66
3 511 2.99 2.39 2.11 1.95 .77
4 5.64 3.20 2.53 2.22 2.04 .84
5 6.09 3.37 2.63 2.30 2.11 .90
8 7 15 375 2.87 2.48 2.26 2.02
12 8.21 4.10 3.O8 2.64 2.39 2.13
16 9.05 4.37 3.24 2.76 2.49 2.20 ;
20 9.75 4.58 3.37 2.85 2.57 2.26 ;
30 11.18 5.00 3.61 3.02 2.71 2.37 ;
40 12.30 5.32 3.78 3,15 2.81 2.45 ;
50 13.28 5.57 3.92 3.25'. 2.89 2.51 ;
60 14.11 5.79 4.04 3.33 2.95 2.56 ;
75 15.23 6.07 4.19 3.44 3.03 2.62 ;
100 16.80 6.45 4.38 3.57 3,74 2.70 ;
125 18.07 6.74 4.54 3.67 3,22 2.76 :
150 19.14 7.01 4.68 3.77 3.3O 2.82 ;
175 20.12 7.23 4.79 3.85 3.36 2.86 ;
200 21.09 7.42 4.88 3.91 3.41 2.91 ;
.41
.57
.67
.73
.79
.90
.99
2.06
2.11
2.20 ;
2.27 ;
2.32 ;
2.36 ;
2.42 ;
2.48 ;
2.53 ;
2.58 ;
2.6i ;
2.65 ;
.35
.51
.59
.65
.70
.80
.89
.95
.99
2.08 ;
2.13 ;
2.18 ;
2.22 ;
2.26 ;
2.33 ;
2.37 ;
2.4i ;
2.44 ;
2.47 ;
.32
.46
.55
.60
.65
.74
.82
.88
.92
2.00
2.05 ;
2.10 ;
2.13 ;
2.17 ;
2.23 ;
2.27 ;
2.30 ;
2.33 ;
2.36 ;
.29
.43
.51
.57
.61
.70
.78
.83
.87
.94
2.00
2.04
2.07 ;
2.11 ;
2.16 ;
2.20 ;
2.23 ;
2.26 ;
2.28 ;
.27
.41
.49
.54
.59
.67
.74
.80
.83
.91
.96
.99
2.02
2.06 ;
2.11 ;
2.15 ;
2.19 ;
2.21 ;
2.23 ;
.26
.40
.47
.52
.56
.65
.72
.77
.81
.88
.92
.96
.99
2.03 ;
2.08 ;
2.11 ;
2.15 ;
2.17 ;
2.19 ;
.25 1.23 1.22 1.21 1.21 1.20
.38 1.36 1.35 1.34 1.33 1.32
.46 1.44 1.42 1.41 1.40 1.39
.51 1.49 1.47 1.46 1.45 1.44
.55 1.52 1.51 1.50 1.49 1.48
.63 1.60 1.59 1.57 1.56 1.55
.70 1.67 1.65 1.64 1.63 1.62
.75 1.72 1.70 1.68 1.67 1.66
.79 1.75 1.73 1.72 1.70 1.69
.85 1.82 1.80 1.78 1.76 1.75
.90 1.86 1.84 1.82 1.81 1.80
.94 1.90 1.87 1.85 1.84 1.83
.97 1.93 1.90 1.88 1.86 1.85
2.00 1.96 1.93 1.91 1.90 1.89
2.05 2.01 1.98 1.96 1.94 1.93
2.09 2.04 2.01 1.99 1.97 1.96
2.12 2.07 2.04 2.01 2.00 1.98
2.14 2.09 2.06 2.03 2.02 2.00
2.15 2.11 2.08 2.05 2.04 2.02
.19
.31
.38
.43
.46
.54
.60
.64
.68
.74
.78
.80
.83
.86
.90
.93
.96
.98
.99
.19
.31
.37
.42
.46
.53
.59
.63
.66
.72
.76
.78
.82
.85
.89
.92
.94
.96
.98
                                                   D-170
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
     Table 19-15. K-Multipliers for 1-of-2  Intrawell Prediction Limits on Means of Order 2 (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.99 1.98 1.66 1.50 .41
3.84 2.38 1.94 1.74 .62
4.43 2.63 2.12 1.88 .74
4.90 2.82 2.25 1.98 .83
5.29 2.97 2.35 2.06 .90
6.22 3.31 2.57 2.24 2.05
7.14 3.63 2,77 2.39 2.18
7.88 3.87 2.91 2.50 2.27 ;
8.49 4.06 3.03 2.59 2.34 ;
9.74 4.44 3.25 2,75 2.48 ;
10.73 4.72 3.41 2.87 2.58 ;
11.55 4.96 3.54 2.97 2,55 ;
12.30 5.14 3.65 3.04 .' 2.72 '<
13.26 5.40 3.79 3.14 2,79 '*
14.63 5.74 3.96 3.26 2.90 ;
15.72 6.02 4.12 3.37 2.97 ;
16.68 6.22 4.24 3.45 3.04 "\
17.64 6.43 4.34 3.52 3.10 ,
18.32 6.63 4.44 3.59 3.14
.30
.49
.59
.67
.72
.85
.95
2.03
2.08
2.19 ;
2.27 ;
2.33 ;
2.38 ;
2.44 ;
>..52 ;
2.58 ;
'i'.eg" ;
2.67 :
2,71 ;
.24
.41
.51
.58
.63
.74
.83
.90
.95
2.04
Ml
2.16 ;
1.20 ;
1.26 :
1.32 ;
2.38 ;
2.42 ;
2.45 ;
1.49 ;
.20
.36
.45
.51
.56
.66
.74
.80
.85
.93
.99
2.04
2.08 ;
1.12 :
i.is ;
2.23 ;
2.26 ;
2.30 ;
2.32 ;
.17
.32
.41
.46
.51
.61
.69
.74
.79
.86
.92
.96
2.00
2.04
2.10 ;
2.14 ;
2.17 ;
2.20 ;
2.23 ;
.15
.29
.38
.44
.48
.57
.65
.70
.74
.82
.87
.91
.94
.98
2.04 ;
2.08 ;
2.11 ;
2.14 ;
2.16 ;
.13
.28
.36
.41
.46
.54
.62
.67
.71
.78
.84
.87
.91
.94
2.00
2.03 ;
2.06 ;
2.09 ;
2.11 ;
.12
.26
.34
.40
.44
.53
.60
.65
.69
.76
.81
.85
.88
.91
.96
2.00
2.03 ;
2.06 ;
2.08 ;
.11 1.10 1.09 1.08 1.07 1.07
.25 1.23 1.22 1.21 1.21 1.20
.33 1.31 1.30 1.29 1.28 1.27
.38 1.36 1.35 1.34 1.33 1.32
.42 1.40 1.39 1.38 1.37 1.36
.51 1.49 1.47 1.46 1.45 1.44
.58 1.56 1.54 1.53 1.52 1.51
.63 1.60 1.59 1.57 1.56 1.55
.67 1.64 1.62 1.61 1.60 1.59
.74 1.71 1.69 1.67 1.66 1.65
.79 1.75 1.73 1.72 1.70 1.69
.82 1.79 1.77 1.75 1.74 1.73
.85 1.82 1.80 1.78 1.76 1.75
.89 1.85 1.83 1.81 1.80 1.79
.94 1.90 1.87 1.85 1.83 1.82
.97 1.93 1.91 1.88 1.87 1.85
2.00 1.96 1.93 1.91 1.89 1.88
2.03 1.99 1.96 1.94 1.92 1.91
2.05 2.01 1.98 1.96 1.94 1.93
.06
.19
.26
.31
.35
.43
.49
.54
.57
.63
.68
.71
.73
.77
.79
.82
.86
.88
.91
.06
.19
.26
.31
.34
.42
.48
.53
.56
.62
.67
.70
.72
.75
.77
.80
.85
.87
.89
  Table 19-15. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (20 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.68 2.34 1.93 1.73 1.61 .48
471 2.78 2.23 1.98 1.83 .66
5.42 3.06 2.42 2.12 1.95 .77
5.98 3.27 2.55 2.23 2.04 .84
6.46 3.44 2.66 2.31 2.12 .90
7.58 3.82 2.9O 2.49 2.27 2.03
8.70 4.18 3.11 2.65 2.40 2.13
9.59 4.45 3.27 2.77 2.50 2.21 ;
10.33 4.67 3.40 2.86 2.57 2.27 ;
11.84 5.09 3.64 3.Q4 2.71 2.38 ;
13.06 5.41 3.81 3.16 2.81 2.45 ;
14.05 5.67 3.95 3.26 2.89 2.51 ;
14.94 5.90 4.07 3.34 2,96 2.56 ;
16.13 6.17 4.22 3.44 3.O4 2.63 ;
17.77 6.56 4.41 3.58 3.14 2.70 ;
19.14 6.87 4.58 3.69 3.23 2.77 ;
20.23 7.11 4.72 3.78 3.30 2.82 '<
21.33 7.38 4.82 3.86 3.37 2,86 ;
22.42 7.59 4.92 3.93 3.42 2.91 \
.41
.57
.67
.74
.79
.90
.99
2.06
2.11
2.20 ;
2.27 ;
2.32 ;
2.36 ;
2.42 ;
2.48 ;
2.54 ;
2.58 ;
2.6i ;
2.65 ;
.35
.51
.59
.66
.70
.80
.89
.95
.99
2.08 ;
2.14 ;
2.18 ;
2.22 ;
2.26 ;
2.32 ;
2.37 ;
2.4i ;
2.44 ;
2.47 ;
.32
.46
.55
.61
.65
.74
.82
.88
.92
2.00
2.05 ;
2.10 ;
2.13 ;
2.17 ;
2.23 ;
2.27 ;
2.3i ;
2.33 ;
2.36 ;
.29
.43
.51
.57
.61
.70
.78
.83
.87
.95
2.00
2.04
2.07 ;
2.11 ;
2.16 ;
2.20 ;
2.23 ;
2.26 ;
2.29 ;
.28
.41
.49
.54
.59
.67
.75
.80
.84
.91
.96
.99
2.03
2.06 ;
2.11 ;
2.15 ;
2.18 ;
2.21 ;
2.23 ;
.26
.40
.47
.52
.57
.65
.72
.77
.81
.88
.92
.96
.99
2.03 ;
2.08 ;
2.11 ;
2.14 ;
2.17 ;
2.20 ;
.25 1.23 1.22 1.21 1.21 1.20
.38 1.36 1.35 1.34 1.33 1.32
.46 1.44 1.42 1.41 1.40 1.39
.51 1.49 1.47 1.46 1.45 1.44
.55 1.53 1.51 1.50 1.49 1.48
.63 1.60 1.59 1.57 1.56 1.55
.70 1.67 1.65 1.64 1.63 1.62
.75 1.72 1.70 1.68 1.67 1.66
.79 1.75 1.73 1.72 1.70 1.69
.85 1.82 1.80 1.78 1.77 1.75
.90 1.86 1.84 1.82 1.81 1.80
.94 1.90 1.87 1.85 1.84 1.83
.97 1.93 1.90 1.88 1.86 1.85
2.00 1.96 1.93 1.91 1.89 1.89
2.05 2.01 1.98 1.95 1.94 1.93
2.08 2.04 2.01 1.99 1.97 1.96
2.11 2.07 2.03 2.01 2.00 1.98
2.14 2.09 2.06 2.03 2.01 2.00
2.16 2.11 2.08 2.05 2.03 2.02
.19
.31
.38
.43
.46
.54
.60
.64
.68
.74
.78
.80
.82
.86
.90
.93
.96
.98
.99
.19
.31
.37
.42
.46
.53
.59
.63
.67
.72
.76
.77
.82
.85
.89
.92
.94
.96
.98
                                                   D-171
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-15. K-Multipliers for 1-of-2  Intrawell Prediction Limits on Means of Order 2 (20 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4.43 2.71 2.20 1.96 1.82 1.66 .57
5.64 3.20 2.53 2.22 2.04 1.84 .73
6.48 3.51 2.72 2.37 2.17 1.95 .83
7.15 375 2.87 2.48 2.26 2.02 .90
7.72 3.94 2.99 2.57 2.33 2.08 .95
9.05 4.37 3.24 2.76 2.49 2.21 2.06
10.37 4.77 3.47 2.93 2.63 2.31 2.15 ;
11.43 5.07 3.64 3.05 2.73 2.39 2.22 ;
12.32 5.31 3.78 3,15 2.81 2.45 2.27 ;
14.12 5.79 4.04 3.33 2.95 2.56 2.36 ;
15.55 6.15 4.23 3.46 3.06 2.64 2.43 ;
16.75 6.44 4.38 3.57 3,74 2.70 2.48 ;
17.77 6.68 4.51 3.66 3.21- 2.76 2.53 ;
19.14 7.01 4.67 3.77 3<3O 2.82 2.58 ;
21.05 7.43 4.89 3.91 3.41 2.90 2.65 ;
22.70 7.79 5.06 4.03 3.49 2.97 2.70 ;
24.20 8.07 5.21 4.12 3.57 3.02 2.74 ;
25.43 8.34 5.33 4.20 3.63 3.06 2.79 ;
26.52 8.54 5.43 4.27 3.69 3, 1O 2.81 ;
.51
.65
.74
.80
.85
.95
2.03
2.09 ;
2.14 ;
1.22 :
i.28 ;
2.32 ;
2.36 ;
2.4i ;
2.47 ;
2.5i ;
2.55 ;
2.58 ;
2.6i ;
.46
.60
.69
.74
.79
.88
.96
2.01
2.05 ;
2.13 ;
2.18 ;
2.23 ;
2.26 ;
2.30 ;
2.36 ;
2.40 ;
2.44 ;
2.47 ;
2.49 ;
.43
.57
.65
.70
.74
.83
.90
.96
2.00
2.07 ;
2.12 ;
2.16 ;
2.19 ;
2.23 ;
2.29 ;
2.32 ;
2.36 ;
2.38 ;
2.4i ;
.41
.54
.62
.67
.71
.80
.87
.92
.96
2.03
2.07 ;
2.11 ;
2.14 ;
2.18 ;
2.23 ;
2.27 ;
2.30 ;
2.33 ;
2.35 ;
.40
.52
.60
.65
.69
.77
.84
.89
.92
.99
2.04 ;
2.08 ;
2.11 ;
2.14 ;
2.19 ;
2.23 ;
2.26 ;
2.29 ;
2.3i ;
.38 1.36 1.35 1.34 1.33 1.32
.51 1.49 1.47 1.46 1.45 1.44
.58 1.56 1.54 1.53 1.52 1.51
.63 1.60 1.59 1.57 1.56 1.55
.67 1.64 1.62 1.61 1.60 1.59
.75 1.72 1.70 1.68 1.67 1.66
.82 1.78 1.76 1.74 1.73 1.72
.86 1.83 1.81 1.79 1.77 1.76
.90 1.86 1.84 1.82 1.81 1.80
.97 1.93 1.90 1.88 1.86 1.85
2.01 1.97 1.94 1.92 1.91 1.89
2.05 2.01 1.98 1.95 1.94 1.93
2.08 2.03 2.00 1.98 1.96 1.95
2.11 2.07 2.04 2.01 2.00 1.98
2.16 2.11 2.08 2.05 2.04 2.02
2.19 2.14 2.11 2.08 2.06 2.05 ;
2.22 2.17 2.14 2.11 2.09 2.07 ;
2.24 2.19 2.16 2.13 2.11 2.10 ;
2.26 2.21 2.18 2.15 2.13 2.11 ;
.31
.43
.49
.54
.57
.64
.70
.74
.78
.83
.87
.90
.93
.96
.99
2.02 ;
2.05 ;
2.07 ;
2.08 ;
.31
.42
.48
.53
.56
.63
.69
.73
.76
.82
.86
.89
.91
.94
.98
2.00
2.03
2.05
2.07
     Table 19-15. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (40 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.84 2.38 1.94 1.74 1.62 .49
4.90 2.82 2.25 1.98 1.83 .67
5.63 3.10 2.43 2.13 1.96 .77
6.22 3.31 2.57 2.24 2.05 .85
6.71 3.48 2,68 2.32 2.12 .90
7.88 3.87 2.91 2.50 2.27 2.03
9.03 4.23 3.13 "2,66" 2.40 2.13
9.96 4.50 3.29 2, 78 2.50 2.21 ;
10.74 4.72 3.41 2.87 2.58 2.27 ;
12.30 5.15 3.65 3.04 .2.71 2.38 ;
13.52 5.47 3.83 3.16 2,82 2.46 ;
14.61 5.74 3.96 3.27 2.90 2.51 ;
15.47 5.96 4.08 3.35 2.96 2.56 ;
16.72 6.25 4.24 3.46 3.05 2,63 '*
18.44 6.64 4.43 3.59 3.14 2,71 :
19.69 6.95 4.59 3.69 3.24 2,77 ;
21.25 7.19 4.73 3.79 3.30 2,81 .
22.19 7.42 4.84 3.87 3.36 2,87 ,
23.12 7.66 4.92 3.95 3.42 -2.91 ,
.41
.58
.67
.74
.79
.90
.99
2.06
2.11
2.20 ;
2.27 ;
2.32 ;
2.36 ;
2.42 ;
2.48 ;
2.54 ;
2.58 ;
2,62 :
2,65 :
.36
.51
.60
.66
.70
.80
.89
.95
.99
2.08 ;
2.14 ;
2.18 ;
2.22 ;
2.27 ;
2.32 ;
2.37 ;
2.40 ;
2.44 ;
2.46 ;
.32
.46
.55
.61
.65
.74
.82
.88
.92
2.00
2.05 ;
2.09 ;
2.13 ;
2.17 ;
2.23 ;
2.27 ;
2.30 ;
2.32 ;
2.35 ;
.29
.43
.51
.57
.61
.70
.78
.83
.87
.95
2.00
2.04
2.07 ;
2.11 ;
2.16 ;
2.20 ;
2.23 ;
2.26 ;
2.29 ;
.28
.41
.49
.54
.59
.67
.74
.80
.84
.91
.96
.99
2.03
2.06 ;
2.11 ;
2.15 ;
2.18 ;
2.21 ;
2.23 ;
.26
.40
.47
.53
.57
.65
.72
.77
.81
.88
.92
.96
.99
2.03 ;
2.08 ;
2.11 ;
2.14 ;
2.17 ;
2.19 ;
.25 1.23 1.22 1.21 1.21 1.20
.38 1.36 1.35 1.34 1.33 1.33
.46 1.44 1.42 1.41 1.40 1.39
.51 1.49 1.47 1.46 1.45 1.44
.55 1.53 1.51 1.50 1.49 1.48
.63 1.61 1.59 1.57 1.56 1.55
.70 1.67 1.65 1.64 1.63 1.62
.75 1.72 1.70 1.68 1.67 1.66
.79 1.76 1.73 1.72 1.70 1.69
.85 1.82 1.80 1.78 1.77 1.75
.90 1.87 1.84 1.82 1.81 1.80
.94 1.90 1.88 1.86 1.84 1.83
.97 1.93 1.90 1.88 1.87 1.86
2.00 1.96 1.93 1.91 1.90 1.88
2.05 2.01 1.98 1.95 1.94 1.92
2.08 2.04 2.01 1.99 1.97 1.96
2.11 2.07 2.04 2.01 1.99 1.98
2.14 2.09 2.06 2.03 2.02 2.00
2.16 2.11 2.08 2.05 2.03 2.02
.19
.31
.38
.43
.46
.54
.60
.64
.68
.74
.78
.81
.83
.87
.90
.93
.96
.98
.99
.19
.31
.37
.42
.46
.53
.59
.63
.67
.72
.76
.79
.82
.85
.89
.92
.94
.96
.98
                                                    D-172
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                    Unified Guidance
  Table 19-15. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (40 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
471 2.78 2.23 1.98 1.83 1.66 .57
5.98 3.27 2.55 2.23 2.04 1.84 .74
6.87 3.59 2.75 2.38 2.17 1.95 .83
7.58 3.82 2.9O 2.49 2.27 2.03 .90
8.17 4.02 3.01 2.58 2.34 2.08 .95
9.58 4.45 3.27 2.77 2.50 2.21 2.06
10.98 4.85 3.50 2.94 2.63 2.32 2.15 ;
12.11 5.16 3.67 3.06 2.73 2.39 2.22 ;
13.05 5.41 3.81 3.16 2.81 2.45 2.27 ;
14.92 5.90 4.07 3.34 2,96 2.56 2.36 ;
16.48 6.25 4.26 3.48 3.07 2.64 2.43 ;
17.73 6.56 4.41 3.58 3.14 2.71 2.48 ;
18.91 6.80 4.55 3.67 3.21 2.75 2.52 ;
20.31 7.11 4.71 3.78 3.30 2.82 2.58 ;
22.34 7.58 4.92 3.93 3.42 2.9O 2.65 ;
24.06 7.89 5.08 4.04 3.50 -2.97 2.70 ;
25.62 8.20 5.23 4.14 3.57 3.O2 2.74 ;
26.88 8.44 5.39 4.22 3.63 3.O7 2.77 ;
28.12 8.75 5.47 4.30 3.69 3.11 2.81 ;
.51
.66
.74
.80
.85
.95
2.03
2.09 ;
2.14 ;
1.22 :
i.28 ;
2.32 ;
2.36 ;
2.4i ;
2.47 ;
2.5i ;
2.55 ;
2.58 ;
2.6i ;
.46
.61
.69
.74
.79
.88
.96
2.01
2.05 ;
2.13 ;
2.18 ;
2.23 ;
2.26 ;
2.30 ;
2.36 ;
2.39 ;
2.43 ;
2.46 ;
2.48 ;
.43
.57
.65
.70
.74
.83
.90
.96
2.00
2.07 ;
2.12 ;
2.16 ;
2.19 ;
2.23 ;
2.28 ;
2.32 ;
2.35 ;
2.38 ;
2.40 ;
.41
.54
.62
.67
.71
.80
.87
.92
.96
2.02
2.07 ;
2.11 ;
2.14 ;
2.18 ;
2.23 ;
2.27 ;
2.29 ;
2.32 ;
2.34 ;
.40
.52
.60
.65
.69
.77
.84
.89
.93
.99
2.04 ;
2.08 ;
2.10 ;
2.14 ;
2.19 ;
2.22 ;
2.25 ;
2.28 ;
2.29 ;
.38 1.36 1.35 1.34 1.33 1.32
.51 1.49 1.47 1.46 1.45 1.44
.58 1.56 1.54 1.53 1.52 1.51
.63 1.60 1.59 1.57 1.56 1.55
.67 1.64 1.62 1.61 1.60 1.59
.75 1.72 1.70 1.68 1.67 1.66
.82 1.78 1.76 1.74 1.73 1.72
.86 1.83 1.81 1.79 1.77 1.76
.90 1.87 1.84 1.82 1.81 1.80
.97 1.93 1.90 1.88 1.87 1.85
2.01 1.97 1.94 1.92 1.91 1.89
2.05 2.00 1.98 1.96 1.94 1.93
2.08 2.03 2.00 1.98 1.97 1.95
2.11 2.07 2.04 2.01 2.00 1.98
2.15 2.11 2.08 2.05 2.04 2.02 ;
2.19 2.14 2.11 2.08 2.07 2.05 ;
2.22 2.17 2.14 2.11 2.09 2.08 ;
2.25 2.19 2.16 2.13 2.11 2.10 ;
2.27 2.21 2.18 2.15 2.13 2.11 ;
.31
.43
.49
.54
.57
.64
.70
.74
.78
.83
.87
.90
.93
.96
2.00
2.03 ;
2.05 ;
2.07 ;
2.09 ;
.31
.42
.48
.53
.56
.63
.69
.73
.76
.82
.86
.89
.91
.94
.98
2.01
2.03
2.05
2.07
    Table 19-15. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 2 (40 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
5.64
7.15
8.21
9.05
9.76
11.43
13.11
14.41
15.55
17.81
19.61
21.09
22.50
24.22
26.56
28.75
30.62
31.88
33.75

6
3.20
3.75
4.10
4.37
4.58
5.07
5.52
5.87
6.15
6.69
7.11
7.42
7.73
8.09
8.59
8.98
9.30
9.61
9.84

8 10 12 16 20 25 30 35 40
2.53 2.22 2.04 1.84 1.73 .65 .60 .57 .54
2.87 2.48 2.26 2.02 1.90 .80 .74 .70 .67
3,08 2.64 2.39 2.13 1.99 .89 .82 .78 .74
324 2.76 2.49 2.21 2.06 .95 .88 .83 .80
337 2.85 2.57 2.26 2.11 .99 .92 .87 .84
365 3.05 2.73 2.39 2.22 2.09 2.01 .96 .92
3.90 3,23 2.87 2.50 2.31 2.17 2.09 2.03 .99
4.08 3.36 2.98 2.58 2.38 2.23 2.14 2.08 2.04
4.23 3.46 3.06 2.64 2.43 2.28 2.18 2.12 2.07
4.51 3.66 3.21 2.75 2.52 2.36 2.26 2.19 2.14
4.73 3.80 3,32 2.84 2.59 2.42 2.31 2.24 2.19
4.88 3.92 3.41 2.90 2.65 2.47 2.36 2.28 2.23
5.03 4.00 3.48 2.95 2.69 2.50 2.39 2.31 2.26
5.21 4.12 3.56 3.02 2.74 2.55 2.43 2.35 2.29
5.43 4.28 3.69 3,W 2.81 2.61 2.49 2.40 2.34
5.62 4.39 3.77 -3,16 2.86 2.66 2.53 2.44 2.38
5.78 4.49 3.87 3.22 2.91 2.70 2.56 2.47 2.41
5.94 4.59 3.93 3.26 2.95 2.72 2.59 2.50 2.43
6.02 4.69 3.98 3,30 2.99 2.75 2.62 2.52 2.46

45
.52
.65
.72
.77
.81
.89
.95
2.00
2.04 ;
2.10 ;
2.15 ;
2.19 ;
2.22 ;
2.25 ;
2.29 ;
2.33 ;
2.36 ;
2.38 ;
2.40 ;

50
.51
.63
.70
.75
.79
.86
.93
.98
2.01
2.08
2.12
2.16
2.18
2.22
2.26
2.29
2.32
2.34
2.36

60
1.49
1.60
1.67
1.72
1.76
1.83
1.89
1.94
1.97
2.03
2.08
2.11
2.14
2.17
2.21
2.25
2.27
2.29
2.31

70
1.47
1.59
1.65
1.70
1.73
1.81
1.87
1.91
1.94
2.00
2.05
2.08
2.10
2.13
2.18
2.21
2.23
2.26
2.28

80
1.46
1.57
1.64
1.68
1.72
1.79
1.85
1.89
1.92
1.98
2.02
2.05
2.08
2.11
2.15
2.18
2.21
2.23
2.25

90
1.45
1.56
1.63
1.67
1.70
1.77
1.83
1.88
1.91
1.96
2.00
2.04
2.06
2.09
2.13
2.16
2.18
2.21
2.22

100
1.44
1.55
1.62
1.66
1.69
1.76
1.82
1.86
1.89
1.95
1.99
2.02
2.05 ;
2.08 ;
2.11 ;
2.14 ;
2.17 ;
2.19 ;
2.21 ;

125 150
.43 .42
.54 .53
.60 .59
.64 .63
.68 .67
.74 .73
.80 .79
.84 .83
.87 .86
.93 .91
.97 .95
.99 .98
2.02 2.00
2.05 2.03
2.08 2.07
2.11 2.09
2.14 2.12
2.16 2.14
2.18 2.15

                                                   D-173
                                                                                                  March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
Unified Guidance
     Table 19-16. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (1 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6
0.37 0.27
0.71 0.53
0.92 0.68
.07 0.79
.20 0.87
.49 .05
.77 .22
.98 .34
2.17 .43
2.52 .61
2.81 .74
3.04 .85
3.25 .94
3.52 2.06
3.89 2.21
4.21 2.33
4.49 2.43
4.73 "'2.52'
4.96 2.61
8
0.22 (
0.45 (
0.58 (
0.67 (
0.74 (
0.89 (
1.02 (
1.11
1.19
1.32
1.42
1.49
1.56
1.63
1.74
1.82
1.89
1.94
2.00
10 12
).19 0.17
).41 0.38
).53 0.49
).61 0.57
).67 0.63
).81 0.75
).92 0.86
.00 0.93
.06 0.99
.18 .09
.26 .16
.32 .22
.37 .26
.44 .32
.52 .39
.58 .45
.64 .49
.68 .53
.72 .57
16 20
0.14 0.13 (
0.34 0.32 (
0.45 0.42 (
0.52 0.49 (
0.58 0.54 (
0.69 0.65 (
0.78 0.74 (
0.85 0.80 (
0.90 0.85 (
0.99 0.93 (
1.05 0.99 (
1.10 .04 (
1.14 .07
1.19 .12
1.25 .17
1.30 .21
1.34 .25
1.37 .28
1.40 .30
25
).ll
).30
).40
).47
).52
).62
).71
).77
).81
).89
).94
).99
.02
.06
.11
.15
.18
.21
.24
30 35 40 45 50 60 70
0.11 0.10 0.10 0.09 0.09 0.08 0.08
0.29 0.28 0.28 0.27 0.27 0.26 0.26
0.39 0.38 0.37 0.37 0.36 0.35 0.35
0.45 0.44 0.44 0.43 0.42 0.42 0.41
0.50 0.49 0.48 0.48 0.47 0.46 0.46
0.60 0.59 0.58 0.57 0.57 0.56 0.55
0.69 0.67 0.66 0.65 0.64 0.63 0.62
0.74 0.73 0.71 0.70 0.70 0.68 0.68
0.78 0.77 0.75 0.74 0.74 0.72 0.72
0.86 0.84 0.83 0.81 0.81 0.79 0.78
0.91 0.89 0.88 0.86 0.85 0.84 0.83
0.95 0.93 0.91 0.90 0.89 0.88 0.87
0.99 0.96 0.94 0.93 0.92 0.91 0.89
1.02 1.00 0.98 0.97 0.96 0.94 0.93
1.07 1.05 .03 1.01 .00 0.98 0.97
1.11 1.08 .06 1.05 .04 1.02 1.00
1.14 1.11 .09 1.08 .06 1.04 1.03
1.17 1.14 .12 1.10 .09 1.07 1.05
1.19 1.16 .14 1.12 .11 1.09 1.07
80
0.08
0.25
0.35
0.41
0.45
0.54
0.62
0.67
0.71
0.78
0.82
0.86
0.89
0.92
0.96
0.99
1.02
1.04
1.06
90
0.08
0.25
0.34
0.40
0.45
0.54
0.61
0.67
0.70
0.77
0.82
0.85
0.88
0.91
0.96
0.99
1.01
1.04
1.05
100
0.08
0.25
0.34
0.40
0.45
0.54
0.61
0.66
0.70
0.77
0.81
0.85
0.87
0.91
0.95
0.98
1.01
1.03
1.05
125
0.07
0.25
0.34
0.40
0.44
0.53
0.61
0.65
0.69
0.76
0.80
0.84
0.86
0.90
0.94
0.97
1.00
1.02
1.04
150
0.07
0.24
0.33
0.39
0.44
0.53
0.60
0.65
0.69
0.75
0.80
0.83
0.86
0.89
0.93
0.96
0.99
1.01
1.03
   Table 19-16. K-Multipliers  for 1-of-3  Intrawell Prediction Limits on Means of Order 2 (1 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

468
0.64 0.50 0.43 (
.00 0.76 0.66 (
.23 0.91 0.79 (
.41 1.03 0.88 (
.56 1.11 0.95 (
.89 1.31 1.10 (
2.22 1.48 1.23
2.48 1.61 1.33
2.69 1.72 1.40
3.12 1.91 1.54
3.46 2.06 1.65
3.75 2.18 1.73
3.99 2.28 1.79
4.32 2.40 1.88
477 2.57 1.99
5.16 "£77" 2.08
5.47 2.82 2.15
5.78 2.93 2.21
6.05 3.02 2.27

10 12 16
).39 0.36 0.33 (
).60 0.56 0.51 (
).72 0.67 0.62 (
).80 0.75 0.68 (
).86 0.80 0.74 (
).99 0.92 0.85 (
.11 1.03 0.94 (
.19 1.10 1.00 (
.25 1.16 1.05 (
.36 1.26 1.14
.45 1.33 1.20
.51 1.39 1.25
.57 1.43 1.29
.63 1.49 1.33
.72 1.56 1.40
.79 1.62 1.44
.84 1.67 1.48
.89 1.71 1.52
.93 1.75 1.54

20
).31 (
).49 (
).58 (
).65 (
).70 (
).80 (
).88 (
).94 (
).99 (
.07
.13
.17
.21
.25
.30
.35
.38
.41
.44

25 30
).30 0.29 (
).47 0.45 (
).56 0.54 (
).62 0.60 (
).67 0.65 (
).76 0.74 (
).84 0.82 (
).90 0.87 (
).94 0.91 (
.02 0.99 (
.07 1.04
.11 1.07
.14 1.10
.18 1.14
.24 1.19
.27 1.23
.31 1.26
.33 1.28
.36 1.30

35
).28 (
).44 (
).53 (
).59 (
).63 (
).72 (
).80 (
).85 (
).89 (
).96 (
.01 (
.05
.08
.11
.16
.20
.23
.25
.27

40 45
).27 0.27 (
).43 0.43 (
).52 0.51 (
).58 0.57 (
).62 0.61 (
).71 0.70 (
).79 0.78 (
).84 0.83 (
).88 0.86 (
).94 0.93 (
).99 0.98 (
.03 1.01
.06 1.04
.09 1.08
.14 1.12
.17 1.15
.20 1.18
.22 1.20
.25 1.22

50 60 70 80 90 100 125 150
).26 0.26 0.25 0.25 0.25 0.25 0.24 0.24
).42 0.41 0.41 0.41 0.40 0.40 0.40 0.39
).51 0.50 0.49 0.49 0.48 0.48 0.48 0.47
).56 0.56 0.55 0.54 0.54 0.54 0.53 0.53
).61 0.60 0.59 0.59 0.58 0.58 0.57 0.57
).70 0.68 0.68 0.67 0.67 0.66 0.65 0.65
).77 0.75 0.75 0.74 0.73 0.73 0.72 0.72
).82 0.80 0.79 0.79 0.78 0.78 0.77 0.76
).85 0.84 0.83 0.82 0.82 0.81 0.80 0.80
).92 0.90 0.89 0.89 0.88 0.87 0.86 0.86
).97 0.95 0.94 0.93 0.92 0.92 0.91 0.90
.00 0.98 0.97 0.96 0.96 0.95 0.94 0.93
.03 1.01 1.00 0.99 0.98 0.98 0.96 0.96
.06 1.04 1.03 .02 1.01 .01 1.00 0.99
.11 1.09 1.07 .06 1.05 .05 1.04 .03
.14 1.12 1.10 .09 1.08 .08 1.07 .06
.17 1.15 1.13 .12 1.11 .10 1.09 .08
.19 1.17 1.15 .14 1.13 .12 1.11 .10
.21 1.19 1.17 .16 1.15 .14 1.13 .12

                                                  D-174
                                                                                                 March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-16. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.90 0.71 0.62 (
1.30 0.98 0.85 (
1.56 1.14 0.98 (
1.76 1.26 1.08 (
1.93 1.36 1.15
2.32 1.57 1.31
2.71 1.76 1.44
3.01 1.90 1.54
3.26 2.01 1.62
3.76 2.23 1.77
4.17 2.39 1.88
4.50 2.52 1.97
4.79 2.63 2.04
518 2.77 2.13
5.72 2.96 2.25
6.17 3.12 2.34
6.56 3.24 2.42 ;
6.91 3.36 2.49 ;
7.23 3.46 2.54 ;
).57 0.54 0.50 (
).78 0.73 0.68 (
).89 0.84 0.77 (
).98 0.91 0.84 (
.04 0.97 0.89 (
.17 1.09 0.99 (
.29 1.19 1.08
.37 1.27 1.15
.44 1.32 1.20
.56 1.43 1.28
.64 1.50 1.35
.71 1.56 1.39
.76 1.61 1.43
.83 1.67 1.48
.92 1.74 1.54
.99 1.80 1.59
>.06 1.85 1.63
MO 1.89 1.66
>.15 1.93 1.69
).48 (
).64 (
).73 (
).79 (
).84 (
).94 (
.02 (
.08
.12
.20
.26
.30
.34
.38
.44
.48
.51
.54
.57
).46 0.45 (
).62 0.60 (
).70 0.68 (
).76 0.74 (
).81 0.78 (
).90 0.87 (
).97 0.94 (
.03 1.00 (
.07 1.03
.14 1.10
.19 1.15
.23 1.19
.27 1.22
.30 1.26
.35 1.30
.39 1.34
.43 1.37
.45 1.39
.47 1.42
).44 (
).59 (
).67 (
).72 (
).76 (
).85 (
).92 (
).97 (
.01 (
.08
.12
.16
.19
.22
.27
.30
.33
.35
.38
).43 0.42 (
).58 0.57 (
).66 0.65 (
).71 0.70 (
).75 0.74 (
).84 0.82 (
).91 0.89 (
).95 0.94 (
).99 0.98 (
.06 1.04
.10 1.09
.14 1.12
.17 1.15
.20 1.18
.24 1.22
.28 1.26
.30 1.28
.33 1.31
.35 1.32
).42 0.41 0.41 (
).56 0.55 0.55 (
).64 0.63 0.62 (
).69 0.68 0.67 (
).73 0.72 0.71 (
).82 0.80 0.79 (
).88 0.87 0.86 (
).93 0.91 0.90 (
).97 0.95 0.94 (
.03 1.01 1.00 (
.07 1.05 1.04
.11 1.09 1.07
.13 1.11 1.10
.17 1.15 1.13
.21 1.19 1.17
.24 1.22 1.20
.27 1.24 1.22
.29 1.26 1.25
.31 1.28 1.26
).40 0.40 (
).54 0.54 (
).62 0.61 (
).67 0.66 (
).71 0.70 (
).79 0.78 (
).85 0.84 (
).90 0.89 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.09 1.08
.12 1.11
.16 1.15
.19 1.18
.21 1.20
.23 1.22
.25 1.24
).40 0.39 (
).54 0.53 (
).61 0.60 (
).66 0.65 (
).70 0.69 (
).78 0.77 (
).84 0.83 (
).88 0.87 (
).92 0.91 (
).98 0.96 (
.02 1.01
.05 1.04
.07 1.06
.10 1.09
.14 1.13
.17 1.16
.19 1.18
.21 1.20
.23 1.22
).39
).53
).60
).65
).69
).76
).83
).87
).90
).96
.00
.03
.05
.08
.12
.15
.17
.19
.21
      Table 19-16. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.71 0.53 0.45 (
.07 0.79 0.67 (
.31 0.94 0.80 (
.49 1.05 0.89 (
.64 1.14 0.96 (
.98 1.34 1.11
2.32 1.51 1.25
2.59 1.64 1.34
2.81 1.74 1.42
3.25 1.94 1.56
3.60 2.09 1.66
3.89 2.21 1.74
4.15 2.30 1.80
4.49 2.43 1.89
4.96 '"2.61" 2.00
5.35 2.74 2.09
5.70 2.85 2.16
6.02 2.97 2.23
6.25 3.05 2.28
).41 0.38 0.34 (
).61 0.57 0.52 (
).73 0.68 0.62 (
).81 0.75 0.69 (
).87 0.81 0.74 (
.00 0.93 0.85 (
.11 1.03 0.94 (
.19 1.10 1.00 (
.26 1.16 1.05 (
.37 1.26 1.14
.45 1.34 1.20
.52 1.39 1.25
.57 1.44 1.29
.64 1.49 1.34
.72 1.57 1.40
.79 1.63 1.45
.85 1.67 1.48
.89 1.72 1.52
.93 1.75 1.54
).32 (
).49 (
).59 (
).65 (
).70 (
).80 (
).89 (
).95 (
).99 (
.07
.13
.17
.21
.25
.30
.35
.38
.41
.44
).30 0.29 (
).47 0.45 (
).56 0.54 (
).62 0.60 (
).67 0.65 (
).77 0.74 (
).85 0.82 (
).90 0.87 (
).94 0.91 (
.02 0.99 (
.07 1.04
.11 1.07
.15 1.11
.18 1.14
.24 1.19
.27 1.23
.31 1.26
.33 1.28
.36 1.30
).28 (
).44 (
).53 (
).59 (
).63 (
).73 (
).80 (
).85 (
).89 (
).96 (
.01 (
.05
.08
.11
.16
.20
.23
.25
.27
).28 0.27 (
).44 0.43 (
).52 0.51 (
).58 0.57 (
).62 0.62 (
).71 0.70 (
).79 0.78 (
).84 0.83 (
).88 0.86 (
).94 0.93 (
).99 0.98 (
.03 1.01
.06 1.04
.09 1.08
.14 1.12
.17 1.15
.20 1.18
.23 1.21
.25 1.23
).27 0.26 0.26 (
).42 0.42 0.41 (
).51 0.50 0.49 (
).57 0.56 0.55 (
).61 0.60 0.59 (
).70 0.68 0.68 (
).77 0.76 0.75 (
).82 0.80 0.79 (
).85 0.84 0.83 (
).92 0.91 0.89 (
).97 0.95 0.94 (
.00 0.98 0.97 (
.03 1.01 1.00 (
.06 1.04 1.03
.11 1.09 1.07
.14 1.12 1.10
.17 1.15 1.13
.19 1.17 1.15
.21 1.19 1.17
).25 0.25 (
).41 0.40 (
).49 0.49 (
).54 0.54 (
).59 0.58 (
).67 0.67 (
).74 0.73 (
).79 0.78 (
).82 0.82 (
).89 0.88 (
).93 0.92 (
).96 0.96 (
).99 0.98 (
.02 1.01
.06 1.05
.09 1.08
.12 1.11
.14 1.13
.16 1.15
).25 0.25 0.24
).40 0.40 0.39
).48 0.48 0.47
).54 0.53 0.53
).58 0.57 0.57
).66 0.65 0.65
).73 0.72 0.72
).78 0.77 0.76
).81 0.80 0.80
).87 0.86 0.86
).92 0.91 0.90
).95 0.94 0.93
).98 0.97 0.96
.01 1.00 0.99
.05 1.04 .03
.08 1.06 .06
.10 1.09 .08
.12 1.11 .10
.14 1.13 .12
                                                    D-175
                                                                                                    March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
   Table 19-16.  K-Multipliers for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (2 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.00 0.76 0.66 (
1.41 1.03 0.88 (
1.68 1.19 1.01 (
1.89 1.31 1.10 (
2.07 1.40 1.17
2.48 1.61 1.33
2.88 1.80 1.47
3.20 1.94 1.57
3.46 2.06 1.65
3.99 2.28 1.79
4.41 2.44 1.90
4.77 2.57 1.99
5.08 "'2.69' 2.06
5.47 2.82 2.15
6.05 3.02 2.27
6.52 3.16 2.36 ;
6.95 3.30 2.44 ;
7.34 3.42 2.50 ;
7.66 3.52 2.56 ;
).60 0.56 0.51 (
).80 0.75 0.68 (
).91 0.85 0.78 (
).99 0.92 0.85 (
.05 0.98 0.90 (
.19 1.10 1.00 (
.30 1.20 1.09
.38 1.27 1.15
.45 1.33 1.20
.57 1.43 1.29
.65 1.51 1.35
.72 1.56 1.40
.77 1.61 1.44
.84 1.67 1.48
.93 1.75 1.54
>.00 1.81 1.59
>.06 1.86 1.63
Ml 1.89 1.67
>.16 1.93 1.69
).49 (
).65 (
).74 (
).80 (
).85 (
).94 (
.03 (
.08
.13
.21
.26
.30
.34
.38
.44
.48
.51
.54
.57
).47 0.45 (
).62 0.60 (
).71 0.68 (
).76 0.74 (
).81 0.78 (
).90 0.87 (
).98 0.95 (
.03 1.00 (
.07 1.04
.14 1.10
.20 1.15
.24 1.19
.27 1.22
.31 1.26
.36 1.30
.40 1.34
.43 1.37
.45 1.40
.47 1.42
).44 (
).59 (
).67 (
).72 (
).77 (
).85 (
).92 (
).97 (
.01 (
.08
.12
.16
.19
.23
.27
.30
.33
.36
.38
).43 0.43 (
).58 0.57 (
).66 0.65 (
).71 0.70 (
).75 0.74 (
).84 0.83 (
).91 0.89 (
).96 0.94 (
).99 0.98 (
.06 1.04
.10 1.09
.14 1.12
.17 1.15
.20 1.18
.25 1.22
.28 1.26
.30 1.28
.33 1.30
.35 1.32
).42 0.41 0.41 (
).56 0.56 0.55 (
).64 0.63 0.62 (
).70 0.68 0.68 (
).74 0.72 0.71 (
).82 0.80 0.79 (
).88 0.87 0.86 (
).93 0.91 0.90 (
).97 0.95 0.94 (
.03 1.01 1.00 (
.07 1.05 1.04
.11 1.09 1.07
.13 1.11 1.10
.17 1.15 1.13
.21 1.19 1.17
.24 1.22 1.20
.26 1.24 1.23
.29 1.26 1.25
.31 1.28 1.26
).41 0.40 (
).54 0.54 (
).62 0.61 (
).67 0.67 (
).71 0.70 (
).79 0.78 (
).85 0.84 (
).90 0.89 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.09 1.08
.12 1.11
.16 1.15
.19 1.18
.21 1.20
.23 1.22
.25 1.24
).40 0.40 (
).54 0.53 (
).61 0.60 (
).66 0.65 (
).70 0.69 (
).78 0.77 (
).84 0.83 (
).88 0.87 (
).92 0.91 (
).98 0.96 (
.02 1.01
.05 1.04
.07 1.06
.10 1.09
.14 1.13
.17 1.16
.19 1.18
.21 1.20
.23 1.22
).39
).53
).60
).65
).69
).76
).83
).87
).90
).96
.00
.03
.05
.08
.12
.15
.17
.19
.21
    Table 19-16. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.30 0.98 0.85 (
1.76 1.26 1.08 (
2.08 1.44 1.21
2.32 1.57 1.31
2.53 1.67 1.38
3.01 1.90 1.54
3.48 2.11 1.69
3.85 2.26 1.80
4.17 2.39 1.88
4.79 2.63 2.04
5.29 2.81 2.15
5.72 2.96 2.25
6.09 3.09 2.32
6.56 3.24 2.42 ;
7.23 3.46 2.54 ;
7.81 3.63 2.65 ;
8.28 3.77 2.73 ;
8.75 3.91 2.80 ;
9.14 4.02 2,87 :
).78 0.73 0.68 (
).98 0.91 0.84 (
.09 1.02 0.93 (
.17 1.09 0.99 (
.24 1.15 1.04 (
.37 1.27 1.15
.49 1.37 1.24
.57 1.44 1.30
.64 1.50 1.35
.76 1.61 1.43
.85 1.68 1.49
.92 1.74 1.54
.98 1.79 1.58
>.06 1.85 1.63
>.15 1.93 1.69
2.23 1.99 1.74
1.29 2.04 1.78
>.34 2.08 1.82
>.38 2.12 1.85
).64 (
).79 (
).88 (
).94 (
).99 (
.08
.16
.22
.26
.34
.39
.44
.47
.51
.57
.61
.65
.67
.70
).62 0.60 (
).76 0.74 (
).84 0.82 (
).90 0.87 (
).94 0.91 (
.03 1.00 (
.10 1.07
.15 1.12
.19 1.15
.27 1.22
.32 1.27
.35 1.30
.39 1.33
.43 1.37
.47 1.42
.51 1.45
.54 1.48
.57 1.50
.59 1.52
).59 (
).72 (
).80 (
).85 (
).89 (
).97 (
.04
.09
.12
.19
.23
.27
.30
.33
.38
.41
.44
.46
.48
).58 0.57 (
).71 0.70 (
).78 0.77 (
).84 0.82 (
).87 0.86 (
).95 0.94 (
.02 1.01 (
.07 1.05
.10 1.09
.17 1.15
.21 1.19
.24 1.22
.27 1.25
.30 1.28
.35 1.32
.38 1.35
.41 1.38
.43 1.40
.45 1.42
).56 0.55 0.55 (
).69 0.68 0.67 (
).77 0.75 0.74 (
).82 0.80 0.79 (
).85 0.84 0.83 (
).93 0.91 0.90 (
).99 0.98 0.97 (
.04 1.02 1.01
.07 1.05 1.04
.13 1.11 1.10
.18 1.15 1.14
.21 1.19 1.17
.23 1.21 1.19
.27 1.24 1.22
.31 1.28 1.26
.34 1.31 1.29
.36 1.33 1.31
.38 1.35 1.33
.40 1.37 1.35
).54 0.54 (
).67 0.66 (
).74 0.73 (
).79 0.78 (
).82 0.82 (
).90 0.89 (
).96 0.95 (
.00 0.99 (
.03 1.02
.09 1.08
.13 1.12
.16 1.15
.18 1.17
.21 1.20
.25 1.24
.28 1.27
.30 1.29
.32 1.31
.34 1.32
).54 0.53 (
).66 0.65 (
).73 0.72 (
).78 0.77 (
).81 0.80 (
).88 0.87 (
).94 0.93 (
).98 0.97 (
.02 1.01
.07 1.06
.11 1.10
.14 1.13
.16 1.15
.19 1.18
.23 1.22
.26 1.24
.28 1.26
.30 1.28
.31 1.30
).53
).65
).72
).76
).80
).87
).93
).97
.00
.05
.09
.12
.14
.17
.21
.23
.25
.27
.29
                                                    D-176
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
     Table 19-16. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.20 0.87 0.74 (
1.64 1.14 0.96 (
1.93 1.31 1.09 (
2.16 1.43 1.19
2.36 1.53 1.26
2.81 1.74 1.42
3.25 1.94 1.56
3.60 2.09 1.66
3.89 2.21 1.74
4.48 2.43 1.89
4.95 2.6O 2.00
5.34 2.74 2.08
5.68 2.86 2.16
6.13 3.01 2.25
6.76 3.21 2.37 ;
7.30 3.37 2,47 '<
7.77 3.51 2.SS '<
8.16 3.63 2,62 :
8.55 3.73 2.68 ;
).67 0.63 0.58 (
).87 0.81 0.74 (
).98 0.91 0.83 (
.06 0.99 0.90 (
.12 1.04 0.95 (
.26 1.16 1.05 (
.37 1.26 1.14
.45 1.34 1.20
.52 1.39 1.25
.64 1.49 1.34
.72 1.57 1.40
.79 1.63 1.45
.85 1.67 1.48
.92 1.73 1.53
2.01 1.81 1.59
2.08 1.87 1.64
2.14 1.92 1.68
2.19 1.96 1.71
1.24 2.00 1.74
).54 (
).70 (
).79 (
).85 (
).90 (
).99 (
.07
.13
.17
.25
.30
.35
.38
.42
.48
.52
.56
.59
.61
).52 0.50 (
).67 0.65 (
).75 0.73 (
).81 0.78 (
).85 0.83 (
).94 0.91 (
.02 0.99 (
.07 1.04
.11 1.07
.18 1.14
.24 1.19
.27 1.23
.31 1.26
.35 1.30
.39 1.34
.43 1.38
.46 1.41
.49 1.43
.51 1.45
).49 (
).63 (
).71 (
).77 (
).81 (
).89 (
).96 (
.01 (
.05
.11
.16
.20
.22
.26
.30
.34
.37
.39
.41
).48 0.48 (
).62 0.62 (
).70 0.69 (
).75 0.74 (
).79 0.78 (
).88 0.86 (
).94 0.93 (
).99 0.98 (
.03 1.01
.09 1.08
.14 1.12
.17 1.15
.20 1.18
.23 1.21
.28 1.26
.31 1.29
.34 1.32
.36 1.34
.38 1.35
).47 0.46 0.46 (
).61 0.60 0.59 (
).68 0.67 0.66 (
).74 0.72 0.71 (
).77 0.76 0.75 (
).85 0.84 0.83 (
).92 0.91 0.89 (
).97 0.95 0.94 (
.00 0.98 0.97 (
.06 1.04 1.03
.11 1.09 1.07
.14 1.12 1.10
.17 1.15 1.13
.20 1.18 1.16
.24 1.22 1.20
.27 1.25 1.23
.30 1.27 1.25
.32 1.29 1.27
.34 1.31 1.29
).45 0.45 (
).59 0.58 (
).66 0.65 (
).71 0.70 (
).75 0.74 (
).82 0.82 (
).89 0.88 (
).93 0.92 (
).96 0.96 (
.02 1.01
.06 1.05
.09 1.08
.12 1.11
.15 1.14
.19 1.18
.22 1.21
.24 1.23
.26 1.25
.28 1.27
).45 0.44 (
).58 0.57 (
).65 0.64 (
).70 0.69 (
).74 0.73 (
).81 0.80 (
).87 0.86 (
).92 0.91 (
).95 0.94 (
.01 1.00 (
.05 1.04
.08 1.07
.10 1.09
.13 1.12
.17 1.16
.20 1.18
.22 1.21
.24 1.23
.26 1.24
).44
).57
).64
).69
).72
).80
).86
).90
).93
).99
.03
.06
.08
.11
.15
.17
.20
.22
.23
   Table 19-16. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (5 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.56 .11 0.95 (
2.07 .40 1.17
2.42 .58 1.31
2.69 .72 1.40
2.92 .82 1.48
3.46 2.06 1.65
3.99 2.28 1.79
4.41 2.44 1.90
477 2.57 1.99
5.48 2.83 2.15
6.04 3.02 2.27
6.52 3.17 2.36 ;
6.93 3.30 2.44 ;
7.48 3.47 2.54 ;
8.24 3.69 2,67 '<
8.91 3.88 2,77 '*
9.45 4.03 2.86 ;
9.96 4.17 2.93 ;
10.39 4.30 3.00 ;
).86 (
.05 (
.17
.25
.31
.45
.57
.65
.72
.84
.93
2.00
2.06
2.14
2.23
2.31 ;
2.37 ;
2.43 ;
2.47 ;
).80 0.74 0.70 (
).98 0.90 0.85 (
.08 0.99 0.93 (
.16 1.05 0.99 (
.21 1.10 1.03 (
.33 1.20 1.13
.43 1.29 1.21
.51 1.35 1.26
.57 1.40 1.30
.67 1.48 1.38
.75 1.54 1.44
.81 1.59 1.48
.86 1.63 1.51
.92 1.68 1.56
.99 1.74 1.61
2.06 1.79 1.65
2.11 1.83 1.69
2.15 1.87 1.72
2.19 1.89 1.74
).67 0.65 0.63 (
).81 0.78 0.77 (
).89 0.86 0.84 (
).94 0.91 0.89 (
).98 0.95 0.93 (
.07 1.04 1.01 (
.14 1.11 1.08
.20 1.15 1.12
.24 1.19 1.16
.31 1.26 1.22
.36 1.30 1.27
.39 1.34 1.30
.43 1.37 1.33
.46 1.41 1.37
.51 1.45 1.41
.55 1.49 1.44
.58 1.52 1.47
.61 1.54 1.49
.63 1.56 1.51
).62 0.61 0.61 (
).75 0.74 0.74 (
).83 0.81 0.81 (
).88 0.86 0.85 (
).91 0.90 0.89 (
).99 0.98 0.97 (
.06 1.04 1.03
.10 1.09 1.07
.14 1.12 1.11
.20 1.18 1.17
.24 1.22 1.21
.28 1.26 1.24
.30 1.28 1.27
.34 1.31 1.30
.38 1.36 1.34
.41 1.39 1.37
.44 1.41 1.39
.46 1.43 1.41
.48 1.45 1.43
).60 0.59 (
).72 0.71 (
).79 0.78 (
).84 0.83 (
).88 0.87 (
).95 0.94 (
.01 1.00 (
.05 1.04
.09 1.07
.15 1.13
.19 1.17
.22 1.20
.24 1.22
.27 1.25
.31 1.29
.34 1.32
.36 1.34
.38 1.36
.40 1.38
).59 0.58 (
).71 0.70 (
).78 0.77 (
).82 0.82 (
).86 0.85 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.12 1.11
.16 1.15
.19 1.18
.21 1.20
.24 1.23
.28 1.27
.31 1.29
.33 1.32
.35 1.34
.36 1.35
).58 (
).70 (
).77 (
).81 (
).85 (
).92 (
).98 (
.02
.05
.10
.14
.17
.19
.22
.26
.29
.31
.33
.34
).57 0.57
).69 0.69
).76 0.75
).80 0.80
).84 0.83
).91 0.90
).97 0.96
.01 1.00
.04 1.03
.09 1.08
.13 1.12
.16 1.15
.18 1.17
.21 1.20
.24 1.23
.27 1.26
.29 1.28
.31 1.30
.33 1.31
                                                   D-177
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-16. K-Multipliers for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.93 1.36 1.15
2.53 1.67 1.38
2.94 1.86 1.52
3.26 2.01 1.62
3.53 2.13 1.71
4.17 2.39 1.88
4.79 2.63 2.04
5.30 2.81 2.15
5.72 2.96 2.24
6.56 3.24 2.42 ;
7.24 3.45 2.54 ;
7.80 3.63 2.65 ;
8.30 3.77 2.73 ;
8.95 3.96 2,84 ;
9.84 4.22 2,98 '*
10.62 4.42 3.09 ;
11.29 4.59 3.18 ;
11.88 4.75 3.27 ;
12.42 4.88 3.34 ;
.04 (
.24
.35
.44
.50
.64
.76
.85
.92
2.05
2.15
1.22
1.29 ;
2.36 ;
2.46 ;
2.54 ;
2.6i ;
2.67 ;
2.7i ;
).97 0.89 0.84 (
.15 1.04 0.99 (
.25 1.13 1.07
.32 1.20 1.12
.38 1.24 1.17
.50 1.35 1.26
.61 1.43 1.34
.68 1.49 1.39
.74 1.54 1.44
.85 1.63 1.51
.93 1.69 1.57
.99 1.74 1.61
2.04 1.78 1.64
2.10 1.83 1.69
2.19 1.89 1.74
2.25 1.94 1.78
2.30 1.98 1.82
2.35 2.02 1.85
2.39 2.05 1.87
).81 0.78 0.76 (
).94 0.91 0.89 (
.02 0.98 0.96 (
.07 1.03 1.01 (
.11 1.07 1.05
.19 1.15 1.12
.27 1.22 1.19
.32 1.27 1.23
.36 1.30 1.27
.43 1.37 1.33
.47 1.42 1.38
.51 1.45 1.41
.54 1.48 1.44
.58 1.52 1.47
.63 1.56 1.51
.67 1.60 1.55
.70 1.62 1.57
.72 1.65 1.60
.75 1.67 1.62
).75 0.74 0.73 (
).87 0.86 0.85 (
).94 0.93 0.92 (
).99 0.98 0.97 (
.03 1.01 1.00 (
.10 1.09 1.07
.17 1.15 1.13
.21 1.19 1.18
.24 1.22 1.21
.30 1.28 1.27
.35 1.32 1.31
.38 1.36 1.34
.41 1.38 1.36
.44 1.41 1.39
.48 1.45 1.43
.51 1.48 1.46
.54 1.51 1.49
.56 1.53 1.51
.58 1.55 1.52
).72 0.71 (
).84 0.83 (
).90 0.89 (
).95 0.94 (
).98 0.97 (
.05 1.04
.11 1.10
.15 1.14
.19 1.17
.24 1.22
.28 1.26
.31 1.29
.33 1.31
.36 1.34
.40 1.38
.43 1.41
.45 1.43
.47 1.45
.49 1.47
).71 0.70 (
).82 0.82 (
).89 0.88 (
).93 0.92 (
).96 0.96 (
.03 1.02
.09 1.08
.13 1.12
.16 1.15
.21 1.20
.25 1.24
.28 1.27
.30 1.29
.33 1.32
.36 1.35
.39 1.38
.41 1.40
.43 1.42
.45 1.44
).70 (
).81 (
).87 (
).92 (
).95 (
.02
.07
.11
.14
.19
.23
.26
.28
.31
.34
.37
.39
.41
.43
).69 0.69
).80 0.80
).86 0.86
).91 0.90
).94 0.93
.01 1.00
.06 1.05
.10 1.09
.13 1.12
.18 1.17
.21 1.21
.24 1.23
.26 1.25
.29 1.28
.33 1.31
.35 1.34
.37 1.36
.39 1.38
.41 1.39
     Table 19-16. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.64 .14 0.96 (
2.16 .43 1.19
2.52 .61 1.32
2.81 .74 1.42
3.04 .85 1.49
3.60 2.09 1.66
4.15 2.31 1.80
4.58 ^,47 1.91
4.95 2.6O 2.00
5.68 2.86 2.16
6.27 3.05 2.28
6.76 3.21 2.37 ;
7.19 3.34 2,45 ;
7.77 3.51 2,55 '*
8.55 3.73 2.68 ;
9.22 3.93 2.78 ;
9.84 4.08 2.87 ;
10.31 4.22 2.95
10.78 4.34 3.01
).87 (
.06 (
.18
.26
.32
.45
.57
.66
.72
.85
.94
2.01
2.07
2.14
2.24 ;
2.31 ;
2.37 ;
?,43 :
2,48 :
).81 0.74 0.70 (
).99 0.90 0.85 (
.09 0.99 0.93 (
.16 1.05 0.99 (
.22 1.10 1.04 (
.34 1.20 1.13
.44 1.29 1.21
.51 1.35 1.26
.57 1.40 1.30
.67 1.48 1.38
.75 1.55 1.44
.81 1.59 1.48
.86 1.63 1.51
.92 1.68 1.56
2.00 1.74 1.61
2.06 1.79 1.66
2.11 1.83 1.69
2.16 1.87 1.72
2.19 1.89 1.74
).67 0.65 0.63 (
).81 0.78 0.77 (
).89 0.86 0.84 (
).94 0.91 0.89 (
).99 0.95 0.93 (
.07 1.04 1.01 (
.15 1.11 1.08
.20 1.15 1.12
.24 1.19 1.16
.31 1.26 1.22
.36 1.30 1.27
.39 1.34 1.30
.43 1.37 1.33
.46 1.41 1.37
.51 1.45 1.41
.55 1.49 1.44
.58 1.52 1.47
.61 1.54 1.49
.63 1.56 1.51
).62 0.62 0.61 (
).75 0.74 0.74 (
).83 0.82 0.81 (
).88 0.86 0.85 (
).91 0.90 0.89 (
).99 0.98 0.97 (
.06 1.04 1.03
.10 1.09 1.07
.14 1.12 1.11
.20 1.18 1.17
.24 1.22 1.21
.28 1.26 1.24
.30 1.28 1.27
.34 1.32 1.30
.38 1.35 1.34
.41 1.39 1.37
.44 1.41 1.39
.46 1.44 1.42
.48 1.46 1.43
).60 0.59 (
).72 0.71 (
).79 0.78 (
).84 0.83 (
).88 0.87 (
).95 0.94 (
.01 1.00 (
.05 1.04
.09 1.07
.15 1.13
.19 1.17
.22 1.20
.24 1.22
.27 1.25
.31 1.29
.34 1.32
.36 1.34
.38 1.36
.40 1.38
).59 0.58 (
).71 0.70 (
).78 0.77 (
).82 0.82 (
).86 0.85 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.12 1.11
.16 1.15
.19 1.18
.21 1.20
.24 1.23
.28 1.27
.31 1.29
.33 1.32
.35 1.34
.36 1.35
).58 (
).70 (
).77 (
).81 (
).85 (
).92 (
).98 (
.02
.05
.10
.14
.17
.19
.22
.26
.28
.31
.33
.34
).57 0.57
).69 0.69
).76 0.75
).80 0.80
).84 0.83
).91 0.90
).97 0.96
.01 1.00
.04 1.03
.09 1.08
.13 1.12
.16 1.15
.18 1.17
.21 1.20
.24 1.23
.27 1.26
.29 1.28
.31 1.30
.33 1.31
                                                    D-178
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
  Table 19-16. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (10 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.07 1.40
2.69 1.72
3.12 1.91
3.46 2.06
3.75 2.18
4.41 2.44
5.08 2,68 ;
5.60 2.87 ;
6.04 3.02 ;
6.93 3.30 ;
7.66 3.52 ;
8.24 3.69 ",
8.77 3.85 ,
9.45 4.03
10.39 4.30
11.25 4.49
11.88 4.67
12.50 4.84
13.12 4.96
.17 1.05 0.98 0.90 0.85 0.81 (
.40 1.25 1.16 1.05 0.99 0.94 (
.54 1.36 1.26 1.14 1.07 1.02 (
.65 1.45 1.33 1.20 1.13 1.07
.73 1.51 1.39 1.25 1.17 1.11
.90 1.65 1.51 1.35 1.26 1.20
2.06 1.77 1.61 1.43 1.34 1.27
2.17 1.86 1.69 1.50 1.39 1.32
2.27 1.93 1.75 1.54 1.44 1.36
2.44 2.06 1.86 1.63 1.51 1.43
2.56 2.16 1.93 1.69 1.57 1.47
2,87' 2.23 1.99 1.74 1.61 1.51
2,75 2.29 2.05 1.78 1.65 1.54
>.86 2.37 2.11 1.83 1.69 1.58
3.00 2.47 2.19 1.89 1.74 1.63
3.12 2.55 2.26 1.94 1.78 1.67
3.20 2.62 2.30 1.98 1.82 1.70
3.28 2,68 2.35 2.02 1.85 1.72
3.36 2.73 2.39 2.05 1.88 1.75
).78 (
).91 (
).98 (
.04
.07
.15
.22
.27
.30
.37
.42
.45
.48
.52
.56
.60
.63
.65
.67
).77 (
).89 (
).96 (
.01 (
.05
.12
.19
.23
.27
.33
.38
.41
.44
.47
.51
.55
.57
.60
.62
).75 (
).88 (
).94 (
).99 (
.03
.10
.17
.21
.24
.30
.35
.38
.41
.44
.48
.51
.54
.56
.58
).74 (
).86 (
).93 (
).98 (
.01
.09
.15
.19
.22
.28
.32
.36
.38
.41
.45
.48
.51
.53
.55
).74 0.72 0.71 0.71 0.70 (
).85 0.84 0.83 0.82 0.82 (
).92 0.90 0.89 0.89 0.88 (
).97 0.95 0.94 0.93 0.92 (
.00 0.98 0.97 0.96 0.96 (
.07 1.05 1.04 1.03 1.02
.13 1.11 1.10 1.09 1.08
.18 1.15 1.14 1.13 1.12
.21 1.19 1.17 1.16 1.15
.27 1.24 1.22 1.21 1.20
.31 1.28 1.26 1.25 1.24
.34 1.31 1.29 1.28 1.27
.36 1.33 1.31 1.30 1.29
.39 1.36 1.34 1.33 1.32
.43 1.40 1.38 1.36 1.35
.46 1.43 1.41 1.39 1.38
.48 1.46 1.43 1.41 1.40
.51 1.47 1.45 1.43 1.42
.52 1.49 1.46 1.45 1.44
).70 (
).81 (
).87 (
).92 (
).95 (
.02
.07
.11
.14
.19
.23
.26
.28
.31
.34
.37
.39
.41
.43
).69 (
).80 (
).86 (
).91 (
).94 (
.01
.06
.10
.13
.18
.21
.24
.26
.29
.33
.35
.37
.39
.41
).69
).80
).86
).90
).93
.00
.05
.09
.12
.17
.20
.23
.25
.28
.31
.34
.36
.38
.39
    Table 19-16.  K-Multipliers for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (10 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100  125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.53 1.67 1.38
3.26 2.01 1.62
3.77 2.23 1.77
4.17 2.39 1.88
4.50 2.52 1.97
5.30 2.81 2.15
6.08 3.08 2.32
6.71 3.29 2.45 ;
7.24 3.45 2.54 ;
8.30 3.77 2.73 ;
9.14 4.02 2,87 ;
9.84 4.22 2.98 \
10.47 4.38 3.07 2
11.29 4.59 3.18 2
12.42 4.88 3.34 ;
13.44 5.12 3.46 ;
14.22 5.31 3.56
15.00 5.49 3.65
15.62 5.62 3.73
.24
.44
.56
.64
.71
.85
.98
2.08
2.15
1.29 ;
2.39 ;
2.46 ;
2.53 ;
2.6i ;
2.7i ;
2.80 ;
?,87 ;
2.93 :
2,99 ;
.15 1.04 0.99 (
.32 1.20 1.12
.43 1.28 1.20
.50 1.35 1.26
.56 1.39 1.30
.68 1.49 1.39
.79 1.58 1.47
.87 1.64 1.52
.93 1.69 1.57
2.04 1.78 1.64
1.12 1.84 1.70
2.19 1.89 1.74
2.24 1.93 1.77
2.30 1.98 1.82
2.39 2.05 1.87
2.46 2.10 1.91
2.51 2.14 1.95
2.56 2.18 1.98
2.60 2.21 2.01
).94 0.91 0.89 (
.07 1.03 1.01 (
.14 1.10 1.08
.19 1.15 1.12
.23 1.19 1.16
.32 1.27 1.23
.39 1.33 1.30
.44 1.38 1.34
.47 1.42 1.38
.54 1.48 1.44
.59 1.53 1.48
.63 1.56 1.51
.66 1.59 1.54
.70 1.62 1.57
.75 1.67 1.62
.78 1.70 1.65
.82 1.73 1.67
.84 1.75 1.69
.87 1.78 1.71
).87 0.86 0.85 (
).99 0.98 0.97 (
.06 1.04 1.03
.10 1.09 1.07
.14 1.12 1.11
.21 1.19 1.18
.27 1.25 1.23
.31 1.29 1.28
.35 1.32 1.31
.41 1.38 1.36
.45 1.42 1.40
.48 1.45 1.43
.50 1.48 1.46
.54 1.51 1.49
.58 1.55 1.52
.61 1.58 1.55
.63 1.60 1.58
.66 1.62 1.60
.67 1.64 1.62
).84 0.83 (
).95 0.94 (
.01 1.00 (
.05 1.04
.09 1.07
.15 1.14
.21 1.19
.25 1.23
.28 1.26
.33 1.31
.37 1.35
.40 1.38
.42 1.40
.45 1.43
.49 1.47
.52 1.49
.54 1.52
.56 1.53
.58 1.55
).82 0.82 (
).93 0.92 (
).99 0.98 (
.03 1.02
.06 1.05
.13 1.12
.18 1.17
.22 1.21
.25 1.24
.30 1.29
.34 1.32
.36 1.35
.39 1.37
.41 1.40
.45 1.44
.47 1.46
.50 1.48
.51 1.50
.53 1.51
).81 (
).92 (
).98 (
.02
.05
.11
.16
.20
.23
.28
.32
.34
.36
.39
.43
.45
.47
.49
.50
).80 0.80
).91 0.90
).97 0.96
.01 1.00
.04 1.03
.10 1.09
.15 1.14
.19 1.18
.21 1.21
.26 1.25
.30 1.29
.33 1.31
.35 1.34
.37 1.36
.41 1.39
.43 1.42
.45 1.44
.46 1.45
.48 1.47
                                                   D-179
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
     Table 19-16. K-Multipliers for 1-of-3  Intrawell Prediction Limits on Means of Order 2 (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100  125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.16 1.43 1.19
2.81 1.74 1.42
3.25 1.94 1.56
3.60 2.09 1.66
3.89 2.21 1.74
4.58 '"2.47" 1.91
5.27 2.72 2.07
5.82 2.90 2.19
6.27 3.05 2.28
7.20 3.34 2.45 :
7.93 3.56 2,58 '*
8.54 3.74 2.68 2
9.08 3.88 2.76 2
9.81 4.08 2.87 2
10.79 4.33 3.02
11.62 4.54 3.12
12.40 4.71 3.22
13.09 4.88 3.30
13.67 5.03 3.37 ;
.06 (
.26
.37
.45
.52
.66
.78
.87
.94
>.07
>.16
>.24 ;
>.30 ;
>.37 ;
?,48 ;
2,56 :
2,62 :
2,69 ;
1.13 ;
).99 0.90 0.85 (
.16 1.05 0.99 (
.26 1.14 1.07
.34 1.20 1.13
.39 1.25 1.17
.51 1.35 1.26
.62 1.44 1.34
.69 1.50 1.39
.75 1.55 1.44
.86 1.63 1.51
.94 1.70 1.57
>.00 1.74 1.61
>.05 1.78 1.64
Ml 1.83 1.69
>.19 1.90 1.74
2.26 1.95 1.79
>.31 1.98 1.82
>.36 2.02 1.85
>.39 2.05 1.87
).81 0.78 0.77 (
).94 0.91 0.89 (
.02 0.99 0.96 (
.07 1.04 1.01 (
.11 1.07 1.05
.20 1.15 1.12
.27 1.22 1.19
.32 1.27 1.24
.36 1.31 1.27
.43 1.37 1.33
.48 1.42 1.38
.51 1.45 1.41
.54 1.48 1.44
.58 1.52 1.47
.63 1.56 1.51
.67 1.60 1.55
.70 1.62 1.57
.73 1.65 1.60
.75 1.67 1.62
).75 0.74 0.74 (
).88 0.86 0.85 (
).94 0.93 0.92 (
).99 0.98 0.97 (
.03 1.01 1.00 (
.10 1.09 1.07
.17 1.15 1.13
.21 1.19 1.18
.24 1.22 1.21
.30 1.28 1.27
.35 1.32 1.31
.38 1.36 1.34
.41 1.38 1.36
.44 1.41 1.39
.48 1.45 1.43
.51 1.48 1.46
.54 1.51 1.49
.56 1.53 1.51
.57 1.55 1.53
).72 0.71 (
).84 0.83 (
).91 0.89 (
).95 0.94 (
).98 0.97 (
.05 1.04
.11 1.10
.15 1.14
.19 1.17
.24 1.22
.28 1.26
.31 1.29
.34 1.32
.36 1.34
.40 1.38
.43 1.41
.45 1.43
.47 1.45
.49 1.46
).71 0.70 (
).82 0.82 (
).89 0.88 (
).93 0.92 (
).96 0.96 (
.03 1.02
.09 1.08
.13 1.12
.16 1.15
.21 1.20
.25 1.24
.28 1.27
.30 1.29
.33 1.32
.36 1.35
.39 1.38
.41 1.40
.43 1.42
.45 1.43
).70 (
).81 (
).87 (
).92 (
).95 (
.02
.07
.11
.14
.19
.23
.26
.28
.31
.34
.37
.39
.41
.42
).69 0.69
).80 0.80
).86 0.86
).91 0.90
).94 0.93
.01 1.00
.06 1.05
.10 1.09
.13 1.12
.18 1.17
.21 1.21
.24 1.23
.26 1.25
.29 1.28
.32 1.32
.35 1.34
.37 1.36
.39 1.38
.41 1.39
  Table 19-16. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (20 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.69 1.72 1.40 1.25 1.16 1.05 0.99 0.94 (
3.46 2.06 1.65 1.45 1.33 1.20 1.13 1.07
3.99 2.28 1.79 1.57 1.43 1.29 1.21 1.14
4.41 2.44 1.90 1.65 1.51 1.35 1.26 1.20
477 2.57 1.99 1.72 1.57 1.40 1.30 1.24
5.60 2.87 2.17 1.86 1.69 1.50 1.39 1.32
6.43 3.14 2.34 1.99 1.80 1.58 1.47 1.39
7.09 3.35 2.47 2.08 1.87 1.65 1.53 1.44
7.65 3.52 2.57 2.16 1.93 1.69 1.57 1.47
8.76 3.85 2,75 2.29 2.05 1.78 1.64 1.54
9.67 4.09 2.89 2.39 2.13 1.84 1.70 1.59
10.40 4.28 3.00 2.47 2.19 1.90 1.74 1.63
11.08 4.46 3.09 2.54 2.24 1.93 1.78 1.66
11.91 4.68 3.21 2,62 2.31 1.98 1.82 1.70
13.13 4.96 3.36 2,73 2.39 2.05 1.87 1.75
14.16 5.20 3.49 2,81 2.46 2.10 1.92 1.79
15.04 5.40 3.59 2,88 2.51 2.14 1.95 1.82
15.82 5.57 3.69 2.94 2.56 2.17 1.98 1.84
16.60 5.74 3.76 3.00 2,60 2.21 2.01 1.86
).91 (
.04
.11
.15
.19
.27
.33
.38
.42
.48
.53
.56
.59
.62
.67
.70
.73
.75
.78
).89 (
.01 (
.08
.12
.16
.23
.30
.34
.38
.44
.48
.51
.54
.57
.61
.65
.67
.70
.72
).88 (
).99 (
.06
.10
.14
.21
.27
.31
.35
.41
.45
.48
.50
.54
.58
.61
.63
.65
.67
).86 (
).98 (
.04
.09
.12
.19
.25
.29
.32
.38
.42
.45
.48
.51
.55
.58
.60
.62
.64
).85 0.84 0.83 0.82 0.82 (
).97 0.95 0.94 0.93 0.92 (
.03 1.01 1.00 0.99 0.98 (
.07 1.05 1.04 1.03 1.02
.11 1.09 1.07 1.06 1.05
.18 1.15 1.14 1.13 1.12
.23 1.21 1.19 1.18 1.17
.28 1.25 1.23 1.22 1.21
.31 1.28 1.26 1.25 1.24
.36 1.33 1.32 1.30 1.29
.40 1.37 1.35 1.34 1.32
.43 1.40 1.38 1.36 1.35
.46 1.43 1.40 1.39 1.37
.49 1.45 1.43 1.41 1.40
.52 1.49 1.47 1.45 1.43
.55 1.52 1.49 1.47 1.46
.58 1.54 1.52 1.50 1.48
.60 1.56 1.54 1.51 1.50
.61 1.57 1.55 1.53 1.51
).81 (
).92 (
).98 (
.02
.05
.11
.16
.20
.23
.28
.32
.34
.36
.39
.43
.45
.47
.49
.50
).80 (
).91 (
).97 (
.01
.04
.10
.15
.19
.21
.26
.30
.33
.35
.37
.41
.43
.45
.47
.48
).80
).90
).96
.00
.03
.09
.14
.18
.21
.25
.29
.31
.34
.36
.39
.42
.44
.45
.47
                                                   D-180
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-16. K-Multipliers for 1-of-3  Intrawell Prediction Limits on Means of Order 2 (20 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100  125  150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.26 2.01 1.62 1.44
4.17 2.39 1.88 1.64
4.80 2.63 2.04 1.76
5.29 2.81 2.15 1.85
5.72 2.96 2.24 1.92
6.71 3.29 2.45 2.08
7.70 3.60 2.63 2.21
8.48 3.83 2.76 2.31 ;
9.14 4.02 2,87 2.38 ;
10.47 4.38 3.07 2.53 ;
11.55 4.66 3.22 2.63 ;
12.45 4.88 3.34 2.72 ;
13.23 5.07 3.44 2.79 ;
14.26 5.31 3.56 2,87 :
15.72 5.64 3.74 2. -99 '4
16.89 5.91 3.87 3.O8 \
17.97 6.15 3.98 3.15 ;
18.95 6.35 4.08 3.22 ;
19.73 6.49 4.15 3.27 ;
.32 1.20 1.12
.50 1.35 1.26
.61 1.43 1.34
.68 1.49 1.39
.74 1.54 1.44
.87 1.64 1.52
.98 1.73 1.60
>.06 1.79 1.66
>.12 1.84 1.70
1.24 1.93 1.78
1.32 2.00 1.83
2.39 2.05 1.87
>.44 2.09 1.91
>.51 2.14 1.95
>.60 2.20 2.01
1.67 2.26 2.05
1.73 2.30 2.08
>.78 2.33 2.11
>.82 2.37 2.14
.07 1.03 1.01 (
.19 1.15 1.12
.27 1.22 1.19
.32 1.27 1.23
.36 1.30 1.27
.44 1.38 1.34
.51 1.45 1.40
.55 1.49 1.45
.59 1.53 1.48
.66 1.59 1.54
.71 1.63 1.58
.75 1.67 1.61
.78 1.70 1.64
.82 1.73 1.67
.86 1.78 1.72
.90 1.81 1.75
.93 1.84 1.77
.96 1.86 1.79
.98 1.88 1.81
).99 0.98 0.97 (
.10 1.09 1.07
.17 1.15 1.13
.21 1.19 1.18
.24 1.22 1.21
.31 1.29 1.28
.37 1.35 1.33
.41 1.39 1.37
.45 1.42 1.40
.50 1.48 1.46
.54 1.52 1.49
.58 1.55 1.52
.60 1.57 1.55
.63 1.60 1.58
.67 1.64 1.61
.70 1.67 1.64
.73 1.69 1.67
.75 1.72 1.68
.77 1.73 1.70
).95 0.94 (
.05 1.04
.11 1.10
.15 1.14
.19 1.17
.25 1.23
.30 1.29
.34 1.32
.37 1.35
.43 1.40
.46 1.44
.49 1.47
.51 1.49
.54 1.52
.58 1.55
.61 1.58
.63 1.60
.64 1.62
.66 1.63
).93 0.92 (
.03 1.02
.09 1.08
.13 1.12
.16 1.15
.22 1.21
.27 1.26
.31 1.30
.34 1.32
.39 1.37
.42 1.41
.45 1.44
.47 1.46
.50 1.48
.53 1.52
.56 1.54
.58 1.56
.60 1.58
.61 1.60
).92 (
.02
.07
.11
.14
.20
.25
.29
.32
.36
.40
.43
.45
.47
.50
.53
.55
.57
.58
).91 0.90
.01 1.00
.06 1.05
.10 1.09
.13 1.12
.19 1.18
.24 1.23
.27 1.26
.30 1.29
.35 1.34
.38 1.37
.41 1.39
.43 1.41
.45 1.43
.48 1.46
.51 1.49
.53 1.51
.54 1.53
.56 1.54
     Table 19-16. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (40 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.81 1.74 1.42 1.26 1.16 1.05 0.99 0.94 (
3.60 2.09 1.66 1.45 1.34 1.20 1.13 1.07
4.15 2.31 1.80 1.57 1.44 1.29 1.21 1.15
4.58 '"2.47" 1.91 1.66 1.51 1.35 1.26 1.20
4.95 2,60 2.00 1.72 1.57 1.40 1.31 1.24
5.82 2.90 2.18 1.87 1.69 1.50 1.39 1.32
6.68 3.18 2.35 2.00 1.80 1.59 1.47 1.39
7.36 3.39 ""2.48" 2.09 1.88 1.65 1.53 1.44
7.93 3.56 2,58. 2.16 1.94 1.70 1.57 1.48
9.10 3.89 2.76 2.30 2.05 1.78 1.65 1.54
10.02 4.13 2.90 2.40 2.13 1.85 1.70 1.59
10.81 4.34 3.01 2.48 2.19 1.90 1.74 1.63
11.48 4.50 3.11 2,54 2.24 1.94 1.78 1.66
12.36 4.72 3.22 2,63 2.31 1.98 1.82 1.70
13.59 5.01 3.38 2.73 2.4O 2.05 1.88 1.75
14.65 5.24 3.50 2.82 2.46 2.10 1.92 1.78
15.59 5.45 3.60 2.89 2,52 2.14 1.95 1.82
16.41 5.62 3.69 2.94 2,56 2.18 1.98 1.84
17.11 5.80 3.76 3.00 2.61 2.20 2.01 1.86
).91 (
.04
.11
.15
.19
.27
.33
.38
.42
.48
.53
.56
.59
.62
.67
.70
.73
.75
.78
).89 (
.01 (
.08
.12
.16
.24
.30
.34
.38
.44
.48
.51
.54
.57
.61
.65
.67
.70
.71
).88 (
).99 (
.06
.10
.14
.21
.27
.31
.35
.41
.45
.48
.51
.54
.58
.61
.63
.66
.67
).86 (
).98 (
.04
.09
.12
.19
.25
.29
.32
.38
.42
.45
.48
.51
.55
.58
.60
.62
.64
).85 0.84 0.83 0.82 0.82 (
).97 0.95 0.94 0.93 0.92 (
.03 1.01 1.00 0.99 0.98 (
.07 1.05 1.04 1.03 1.02
.11 1.09 1.07 1.06 1.05
.18 1.15 1.14 1.13 1.12
.23 1.21 1.19 1.18 1.17
.28 1.25 1.23 1.22 1.21
.31 1.28 1.26 1.25 1.24
.36 1.33 1.32 1.30 1.29
.40 1.37 1.35 1.34 1.32
.43 1.40 1.38 1.36 1.35
.46 1.42 1.40 1.39 1.37
.49 1.45 1.43 1.41 1.40
.52 1.49 1.47 1.45 1.44
.55 1.52 1.49 1.48 1.46
.58 1.54 1.52 1.50 1.48
.60 1.56 1.53 1.52 1.50
.61 1.57 1.55 1.53 1.52
).81 (
).92 (
).98 (
.02
.05
.11
.16
.20
.23
.28
.32
.34
.36
.39
.42
.45
.47
.49
.51
).80 (
).91 (
).97 (
.01
.04
.10
.15
.19
.21
.26
.30
.33
.35
.37
.40
.43
.45
.47
.48
).80
).90
).96
.00
.03
.09
.14
.18
.21
.25
.29
.31
.33
.36
.39
.42
.44
.45
.47
                                                    D-181
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
  Table 19-16. K-Multipliers  for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (40 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.46 2.06 1.65 1.45 1.33 1.20 1.13 1.07
4.41 2.44 1.90 1.65 1.51 1.35 1.26 1.20
5.08 2,68 2.06 1.77 1.61 1.44 1.34 1.27
5.60 2.87 2.17 1.86 1.69 1.50 1.39 1.32
6.05 3.02 2.27 1.93 1.75 1.54 1.44 1.36
7.09 3.35 2.47 2.08 1.87 1.65 1.53 1.44
8.13 3.66 2,65 2.22 1.98 1.73 1.60 1.51
8.96 3.90 2,78 2.32 2.06 1.80 1.66 1.56
9.66 4.09 2.89 2.39 2.13 1.85 1.70 1.59
11.07 4.46 3.09 2.54 2.24 1.93 1.78 1.66
12.19 4.74 3.24 2,64 2.33 2.00 1.83 1.71
13.12 4.97 3.36 2,73 . 2.39 2.05 1.87 1.75
13.95 5.16 3.46 2.8O 2.45 2.09 1.91 1.78
15.06 5.41 3.59 2,88 2.52 2.14 1.95 1.81
16.58 5.74 3.76 3.00 2,80 2.21 2.01 1.86
17.81 6.01 3.90 3.09 2,67 2.26 2.05 1.90
18.98 6.24 4.00 3.16 2,73 2.30 2.08 1.93
19.92 6.45 4.10 3.22 2.78 2.34 2.12 1.96
20.86 6.62 4.19 3.28 2,83 2.37 2.14 1.98
.04
.15
.22
.27
.30
.38
.45
.49
.53
.59
.63
.67
.70
.73
.78
.81
.84
.86
.88
.01 (
.12
.19
.23
.27
.34
.40
.45
.48
.54
.58
.61
.64
.67
.72
.75
.77
.79
.82
).99 (
.10
.17
.21
.24
.31
.37
.41
.45
.50
.55
.58
.60
.63
.67
.70
.73
.75
.77
).98 (
.09
.15
.19
.22
.29
.35
.39
.42
.48
.52
.55
.57
.60
.64
.67
.69
.71
.73
).97 0.95 0.94 0.93 0.92 (
.07 1.05 1.04 1.03 1.02
.13 1.11 1.10 1.09 1.08
.18 1.15 1.14 1.13 1.12
.21 1.19 1.17 1.16 1.15
.28 1.25 1.23 1.22 1.21
.33 1.30 1.29 1.27 1.26
.37 1.34 1.32 1.31 1.30
.40 1.37 1.35 1.34 1.32
.46 1.43 1.40 1.39 1.37
.49 1.46 1.44 1.42 1.41
.52 1.49 1.47 1.45 1.44
.55 1.51 1.49 1.47 1.46
.58 1.54 1.52 1.50 1.48
.61 1.58 1.55 1.53 1.52
.64 1.60 1.58 1.56 1.54
.67 1.63 1.60 1.58 1.56
.68 1.64 1.62 1.60 1.58
.70 1.66 1.63 1.61 1.60
).92 (
.02
.07
.11
.14
.20
.25
.29
.32
.36
.40
.42
.44
.47
.50
.53
.55
.57
.58
).91 (
.01
.06
.10
.13
.19
.24
.27
.30
.35
.38
.41
.42
.45
.48
.51
.53
.55
.56
).90
.00
.05
.09
.12
.18
.23
.26
.29
.34
.37
.39
.41
.43
.47
.49
.51
.53
.55
    Table 19-16.  K-Multipliers for 1-of-3 Intrawell Prediction Limits on Means of Order 2 (40 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100  125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
417 2.39 1.88 1.64 1.50 1.35 1.26
5.29 2.81 2.15 1.85 1.68 1.49 1.39
6.08 3.08 2.32 1.98 1.79 1.58 1.47
6.71 3.29 2.45 2.08 1.87 1.64 1.52
7.23 3.46 2.54 2.15 1.93 1.69 1.57
8.48 3.83 2.76 2.31 2.06 1.79 1.66
9.72 4.18 2,96 2.45 2.17 1.88 1.73
10.71 4.44 3.10 2.55 2.26 1.95 1.79
11.54 4.66 3.22 2.63 2.32 2.00 1.83
13.21 5.07 3.44 2.79 2.44 2.09 1.91
14.56 5.38 3.60 2.9O 2.53 2.15 1.96
15.70 5.64 3.73 2.99. 2.60 2.20 2.01
16.67 5.86 3.84 3.O6 2.66 2.25 2.04
17.99 6.14 3.98 3.15 2.73 2.30 2.08
19.80 6.50 4.16 3.27 2.82 2.37 2.14
21.33 6.83 4.31 3.37 2,89 2.42 2.18 ;
22.62 7.09 4.42 3.45 2.96 2.46 2.22 ;
23.91 7.29 4.54 3.52 3,07 2.50 2.25 ;
24.84 7.50 4.63 3.57 3.05 2.53 2.28 ;
.19 1.15 1.12
.32 1.27 1.23
.39 1.33 1.30
.44 1.38 1.34
.47 1.42 1.38
.55 1.49 1.45
.62 1.55 1.51
.67 1.60 1.55
.71 1.63 1.58
.78 1.70 1.64
.83 1.74 1.68
.86 1.78 1.72
.89 1.80 1.74
.93 1.84 1.77
.98 1.88 1.81
1.02 1.92 1.85
>.05 1.94 1.87
>.07 1.97 1.89
>.09 1.99 1.91
.10 1.09 1.07
.21 1.19 1.18
.27 1.25 1.23
.31 1.29 1.28
.35 1.32 1.31
.41 1.39 1.37
.47 1.45 1.43
.51 1.49 1.46
.54 1.52 1.49
.60 1.57 1.55
.64 1.61 1.58
.67 1.64 1.61
.70 1.66 1.64
.73 1.69 1.67
.77 1.73 1.70
.80 1.76 1.73
.82 1.78 1.75
.84 1.80 1.77
.86 1.82 1.79
.05 1.04
.15 1.14
.21 1.19
.25 1.23
.28 1.26
.34 1.32
.40 1.37
.43 1.41
.46 1.44
.51 1.49
.55 1.52
.58 1.55
.60 1.57
.63 1.60
.66 1.63
.69 1.66
.71 1.68
.73 1.70
.74 1.71
.03 1.02
.13 1.12
.18 1.17
.22 1.21
.25 1.24
.31 1.30
.36 1.35
.39 1.38
.42 1.41
.47 1.46
.50 1.49
.53 1.52
.55 1.54
.58 1.56
.61 1.59
.64 1.62
.66 1.64
.68 1.66
.69 1.67
.02
.11
.16
.20
.23
.29
.34
.37
.40
.45
.48
.50
.52
.55
.58
.61
.63
.64
.66
.01 1.00
.10 1.09
.15 1.14
.19 1.18
.21 1.21
.27 1.26
.32 1.31
.35 1.34
.38 1.37
.43 1.41
.46 1.44
.48 1.47
.50 1.49
.53 1.51
.56 1.54
.58 1.57
.60 1.59
.62 1.60
.64 1.62
                                                    D-182
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
     Table 19-17. K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 3 (1 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.25 1.04 0.96 (
1.78 1.41 1.27
2.12 1.63 1.45
2.39 1.79 1.58
2.62 1.92 1.68
3.14 2.21 1.90
3.64 2.47 2.09
4.05 2.66 2.23 ;
4.38 2.82 2.34 ;
5.06 3.11 2.55 ;
5.59 3.33 2.70 ;
6.04 3.51 2.82 ;
6.43 3.67 2.92 ;
6.95 3.86 3.O4 :
7.66 4.12 3.21 ;
8.27 4.33 3.34 ;
8.80 4.51 3.45 ;
9.27 4.66 3.54
9.70 4.80 3.63
).91 (
.20
.36
.47
.56
.75
.91
2.03
1.12
1.29 :
2.41 ;
2.50 ;
2.58 ;
2.68 ;
2.8i ;
2.9i ;
2.99 ;
3,oe" :
3.12 ' :
).88 0.84 (
.15 1.09
.30 1.23
.41 1.33
.49 1.40
.66 1.55
.80 1.68
.91 1.77
.99 1.84
2.13 1.96
2.24 2.05
1.32 2.12 ;
2.39 2.18 ;
2.47 2.24 ;
2.58 2.33 ;
2.66 2.40 ;
1.73 2.46 ;
2.79 2.50 ;
2.84 2.55 ;
).82 (
.06
.19
.28
.35
.49
.61
.69
.76
.87
.95
2.01
2.06
2.12 ;
1.20 :
1.26 :
i.3i ;
2.35 ;
2.39 ;
).81 0.79 (
.04 1.02
.16 1.14
.25 1.23
.31 1.29
.45 1.42
.56 1.53
.64 1.60
.70 1.66
.80 1.76 ]
.88 1.83 ]
.93 1.88 ]
.98 1.92 ]
2.03 1.98 1
Ml 2.04 I
2.16 2.10 I
1.21 2.14 I
2.24 2.17 ;
2.28 2.20 ;
).79 (
.01
.13
.21
.27
.40
.50
.57
.63
L.73
L.79
L.85
L.89
L.94
2.00
2.05 ;
2.09 ;
2.13 ;
2.15 ;
).78 0.78 (
.00 0.99 (
.12 1.11
.20 1.19
.26 1.25
.38 1.37
.49 1.47
.56 1.54
.61 1.59
.70 1.69
.77 1.75
.82 1.80
.86 1.84
.91 1.89
.97 1.95
1.02 2.00
1.06 2.03 ;
2.09 2.06 I
1.12 2.09 I
).77 (
).99 (
.11
.18
.24
.36
.46
.53
.58
.67
.74
.78
.82
.87
.93
.98
2.01
2.04 ;
2.07 ;
).77 0.76 (
).98 0.98 (
.10 1.09
.17 1.17
.23 1.22
.35 1.34
.45 1.44
.51 1.50
.56 1.55
.65 1.64
.71 1.70
.76 1.75
.80 1.78
.85 1.83
.90 1.88
.95 1.93
.98 1.96
2.01 1.99
2.04 2.02 ;
).76 0.76 (
).97 0.97 (
.08 1.08
.16 1.16
.22 1.21
.33 1.33
.43 1.42
.49 1.49
.54 1.53
.63 1.62
.69 1.68
.73 1.72
.77 1.76
.81 1.80
.87 1.86
.91 1.90
.95 1.94
.98 1.96
2.00 1.99
).76 0.75 0.75
).97 0.96 0.96
.08 1.07 1.07
.15 1.15 1.14
.21 1.20 1.20
.32 1.31 1.31
.42 1.41 1.40
.48 1.47 1.46
.53 1.52 1.51
.61 1.60 1.59
.67 1.66 1.65
.72 1.70 1.69
.75 1.74 1.73
.80 1.78 1.77
.85 1.84 1.82
.89 1.88 1.87
.93 1.91 1.90
.95 1.94 1.93
.98 1.96 1.95
   Table 19-17.  K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 3 (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.65 1.35 1.24 1.17
2.24 1.74 1.55 1.45
2.64 1.98 1.74 1.62
2.96 2.15 1.87 1.73
3.22 2.29 1.98 1.82
3.83 2.60 2.21 2.01
4.43 2.89 2.41 2.18 ;
4.91 3.10 2.56 2.30 ;
5.31 3.27 2.67 2.39 ;
6.11 3.60 2.89 2.57 ;
6.75 3.85 3.05 2.69 ;
7.28 4.05 3.18 2.79 ;
7.75 4.22 3,29 2.87 ;
8.36 4.43 3.42 2.98 ;
9.22 4.72 3.60 3.11 ;
9.95 4.96 3.74 3,22. \
10.58 5.16 3.86 3. 3D C
11.14 5.33 3.96 . 3,38 , C
11.65 5.49 4.05 3.44 C
.13
.39
.54
.64
.73
.90
2.04
2.15
2.23 ;
2.38 ;
2.49 ;
2.57 ;
2.64 ;
2.73 ;
2.84 ;
2.92 ;
5.00 ;
5.06 ;
5.11 ;
.08
.32
.45
.54
.62
.76
.89
.98
2.05
2.17 ;
2.26 ;
2.33 ;
2.39 ;
2.45 ;
2.54 ;
2.6i ;
2.67 ;
2.72 ;
2.76 ;
.05
.28
.40
.49
.55
.69
.81
.89
.95
2.06
2.14 ;
2.20 ;
2.25 ;
2.31 ;
2.39 ;
2.45 ;
2.50 ;
2.54 ;
2.58 ;
.03
.24
.36
.44
.51
.63
.74
.82
.87
.98
2.05
2.10 ;
2.15 ;
2.20 ;
2.27 ;
2.33 ;
2.37 ;
2.4i ;
2.44 ;
.01
.22
.34
.42
.48
.60
.70
.77
.83
.92
.99
2.04 ;
2.09 ;
2.14 ;
2.20 ;
2.25 ;
2.30 ;
2.33 ;
2.36 ;
.00
.21
.32
.40
.45
.57
.67
.74
.79
.89
.95
2.00
2.04 ;
2.09 ;
2.15 ;
2.20 ;
2.24 ;
2.27 ;
2.30 ;
.00 (
.20
.31
.38
.44
.55
.65
.72
.77
.86
.92
.97
2.01
2.06 ;
2.12 ;
2.17 ;
2.20 ;
2.23 ;
2.26 ;
).99 (
.19
.30
.37
.43
.54
.64
.70
.75
.84
.90
.95
.99
2.03 ;
2.09 ;
2.14 ;
2.17 ;
2.20 ;
2.23 ;
).99 0.98 0.97 0.97 0.97 0.96 (
.18 1.17 1.16 1.16 1.15 1.15
.29 1.28 1.27 1.26 1.26 1.25
.36 1.35 1.34 1.33 1.33 1.32
.42 1.40 1.39 1.38 1.38 1.37
.53 1.51 1.50 1.49 1.48 1.48
.62 1.60 1.59 1.58 1.57 1.57
.69 1.67 1.65 1.64 1.63 1.63
.74 1.71 1.70 1.69 1.68 1.67
.82 1.80 1.78 1.77 1.76 1.75
.88 1.86 1.84 1.83 1.82 1.81
.93 1.90 1.88 1.87 1.86 1.85
.97 1.94 1.92 1.90 1.89 1.88
2.01 1.98 1.96 1.95 1.94 1.93
2.07 2.04 2.02 2.00 1.99 1.98
2.11 2.08 2.06 2.04 2.03 2.02 ;
2.15 2.12 2.09 2.07 2.06 2.05 ;
2.18 2.14 2.12 2.10 2.09 2.08 ;
2.21 2.17 2.14 2.13 2.11 2.10 ;
).96 (
.15
.25
.31
.37
.47
.56
.61
.66
.74
.79
.84
.87
.91
.96
2.00
2.03 ;
2.06 ;
2.08 ;
).96
.14
.24
.31
.36
.46
.55
.61
.65
.73
.78
.82
.86
.90
.95
.99
2.02
2.04
2.07
                                                   D-183
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-17. K-Multipliers  for 1-of-1 Intrawell Prediction Limits on Means of Order 3 (1 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16
2.03 1.65 1.50 1.42 .36 .30
2.71 2.06 1.82 1.70 .62 .53
3.17 2.32 2.02 1.86 .77 .66
3.53 2.51 2.16 1.98 .88 .75
383 2.66 2.27 2.08 .96 .82
454 3.00 2.51 2.27 2.13 .97
5.24 3.31 2.73 2.45 2.28 2.09
5.80 3.55 2.88 2.57 2.39 2.18 ;
6.26 3.74 3.01 2.67 2.47 2.25 ;
7.20 4.10 3.24 2.85 2.63 2.38 ;
7.95 4.38 3.42 2.98 2.74 2.47 ;
8.58 4.60 3.56 3.09 2.82 2.54 ;
9.12 4.79 3.67 3.17 2.90 2.59 ;
9.84 5.03 3.81 3.28 2.98 2.66 ;
10.84 5.35 4.01 3.42 3.10 2.75 ;
11.69 5.62 4.16 3,53 3.19 2.83 ;
12.43 5.84 4.29 3.63 3.27 2.88 ;
13.09 6.03 4.40 3.71 3.33 2.93 ;
13.68 6.21 4.50 3.78 3.39 2.97 ;
20
.26
.48
.60
.68
.75
.88
.99
2.07
2.13 ;
2.25 ;
1.32 ;
2.39 ;
2.43 ;
2.50 ;
2.57 ;
2.63 ;
2.68 ;
2.73 ;
2.76 ;
25
.23
.44
.55
.63
.69
.81
.92
.99
2.05
2.15 ;
2.22 ;
2.27 ;
2.32 ;
2.37 ;
2.44 ;
2.50 ;
2.54 ;
2.58 ;
2.6i ;
30
.22
.41
.52
.59
.65
.77
.87
.94
.99
2.08 ;
2.15 ;
2.20 ;
2.24 ;
2.29 ;
2.36 ;
2.4i ;
2.45 ;
2.48 ;
2.5i ;
35
.20
.39
.50
.57
.62
.74
.83
.90
.95
2.04 ;
2.10 ;
2.15 ;
2.19 ;
2.24 ;
2.30 ;
2.35 ;
2.39 ;
2.42 ;
2.45 ;
40
.19
.38
.48
.55
.61
.72
.81
.87
.92
2.01
2.07 ;
2.12 ;
2.16 ;
2.20 ;
2.26 ;
2.3i ;
2.34 ;
2.37 ;
2.40 ;
45
.18
.37
.47
.54
.59
.70
.79
.85
.90
.99
2.05 ;
2.09 ;
2.13 ;
2.17 ;
2.23 ;
2.27 ;
2.3i ;
2.34 ;
2.37 ;
50 60 70 80 90 100
.18 1.17 1.16 1.16 1.15 1.15
.36 1.35 1.34 1.33 1.33 1.32
.46 1.44 1.43 1.43 1.42 1.41
.53 1.51 1.50 1.49 1.48 1.48
.58 1.56 1.55 1.54 1.53 1.53
.69 1.67 1.65 1.64 1.63 1.63
.77 1.75 1.74 1.72 1.72 1.71
.84 1.81 1.79 1.78 1.77 1.76
.88 1.86 1.84 1.83 1.82 1.81
.97 1.94 1.92 1.90 1.89 1.88
2.02 2.00 1.97 1.96 1.95 1.94
2.07 2.04 2.02 2.00 1.99 1.98
2.11 2.07 2.05 2.03 2.02 2.01
2.15 2.12 2.09 2.07 2.06 2.05 ;
2.21 2.17 2.14 2.13 2.11 2.10 ;
2.25 2.21 2.18 2.17 2.15 2.14 ;
2.28 2.24 2.22 2.20 2.18 2.17 ;
2.31 2.27 2.24 2.22 2.21 2.20 ;
2.34 2.30 2.27 2.25 2.23 2.22 ;
125
.14
.31
.41
.47
.52
.61
.70
.75
.79
.87
.92
.96
.99
2.03 ;
2.08 ;
2.12 ;
2.15 ;
2.17 ;
2.20 ;
150
.14
.31
.40
.46
.51
.61
.69
.74
.78
.86
.91
.95
.98
2.02
2.07
2.10
2.13
2.16
2.18
     Table 19-17. K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 3 (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.78 1.41 1.27 1.20
2.39 1.79 1.58 1.47
2.81 2.03 1.77 1.63
3.14 2.21 1.90 1.75
3.41 2.35 2.01 1.84
4.05 2.66 2.23 2.03
4.68 2.95 2.44 2.19 ;
5.17 3.16 2.58 2.31 ;
5.59 3.33 2.70 2.41 ;
6.43 3.67 2.92 2.58 ;
7.10 3.92 3.O8 2.71 ;
7.66 4.12 3.21 2.81 ;
8.15 4.29 3.32 2.89 ;
8.80 4.51 3.45 2.99 ;
9.70 4.80 3.63 3,12 '<
10.46 5.04 3.77 3.23 ;
11.12 5.24 3.89 3.32
11.71 5.42 3.99 3.39
12.26 5.58 4.08 3.46
.15
.41
.55
.66
.74
.91
2.05
2.16
2.24 ;
2.39 ;
2.49 ;
2.58 ;
2.65 ;
2.73 ;
2.84 ;
2.93 ;
?,oo ;
3,06 :
3,12 :
.09
.33
.46
.55
.62
.77
.89
.98
2.05
2.18 ;
2.26 ;
2.33 ;
2.39 ;
2.46 ;
2.55 ;
2.62 ;
2.67 ;
2.72 ;
2.76 ;
.06
.28
.41
.49
.56
.69
.81
.89
.95
2.06
2.14 ;
2.20 ;
2.25 ;
2.31 ;
2.39 ;
2.45 ;
2.50 ;
2.54 ;
2.58 ;
.04
.25
.37
.45
.51
.64
.74
.82
.88
.98
2.05
2.11 ;
2.15 ;
2.21 ;
2.28 ;
2.33 ;
2.37 ;
2.4i ;
2.44 ;
.02
.23
.34
.42
.48
.60
.70
.77
.83
.92
.99
2.04 ;
2.09 ;
2.14 ;
2.20 ;
2.25 ;
2.30 ;
2.33 ;
2.36 ;
.01
.21
.32
.40
.46
.57
.67
.74
.79
.89
.95
2.00
2.04 ;
2.09 ;
2.15 ;
2.20 ;
2.24 ;
2.28 ;
2.30 ;
.00 (
.20
.31
.38
.44
.56
.65
.72
.77
.86
.92
.97
2.01
2.06 ;
2.12 ;
2.17 ;
2.20 ;
2.23 ;
2.26 ;
).99 (
.19
.30
.37
.43
.54
.64
.70
.75
.84
.90
.95
.99
2.03 ;
2.09 ;
2.14 ;
2.17 ;
2.20 ;
2.23 ;
).99 0.98 0.98 0.97 0.97 0.97 (
.18 1.17 1.17 1.16 1.16 1.15
.29 1.28 1.27 1.26 1.26 1.25
.36 1.35 1.34 1.33 1.33 1.32
.42 1.40 1.39 1.39 1.38 1.37
.53 1.51 1.50 1.49 1.49 1.48
.62 1.60 1.59 1.58 1.57 1.57
.69 1.67 1.65 1.64 1.63 1.63
.74 1.71 1.70 1.69 1.68 1.67
.82 1.80 1.78 1.77 1.76 1.75
.88 1.86 1.84 1.83 1.82 1.81
.93 1.90 1.88 1.87 1.86 1.85
.97 1.94 1.92 1.90 1.89 1.88
2.01 1.98 1.96 1.95 1.94 1.93
2.07 2.04 2.02 2.00 1.99 1.98
2.11 2.08 2.06 2.04 2.03 2.02 ;
2.15 2.12 2.09 2.07 2.06 2.05 ;
2.18 2.15 2.12 2.10 2.09 2.08 ;
2.21 2.17 2.14 2.13 2.11 2.10 ;
).96 (
.15
.25
.31
.37
.47
.56
.62
.66
.74
.79
.84
.87
.91
.96
2.00
2.03 ;
2.06 ;
2.08 ;
).96
.14
.24
.31
.36
.46
.55
.61
.65
.73
.78
.82
.86
.90
.95
.99
2.02
2.04
2.07
                                                   D-184
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
Unified Guidance
   Table 19-17. K-Multipliers  for 1-of-1  Intrawell Prediction Limits on Means of Order 3 (2 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
2.24 1.74 1.55 1.45 .39 .32 .28
2.96 2.15 1.87 1.73 .64 .54 .49
3.44 2.41 2.07 1.89 .79 .67 .61
383 2.60 2.21 2.01 .90 .76 .69
415 2.76 2.32 2.10 .98 .83 .75
491 3.10 2.56 2.30 2.15 .98 .89
565 342 2.77 2.47 2.30 2.10 2.00
6.25 3.65 2.93 2.59 2.40 2.19 2.08
6.75 3.85 3.05 2.69 2.49 2.26 2.14 ;
7.75 4.22 3,29 2.87 2.64 2.39 2.25 ;
8.55 4.50 3.46 3.01 2.75 2.47 2.33 ;
9.22 4.72 3.60 3.11 2.84 2.54 2.39 ;
9.81 4.92 3.72 3.20 2.91 2.60 2.44 ;
10.58 5.16 3.86 3,30 3.00 2.67 2.50 ;
11.65 5.49 4.05 3.44 3.11 2.76 2.58 ;
12.55 5.76 4.21 3.56 3.20 2.83 2.64 ;
13.33 5.98 4.34 3.65 "'3.28' 2.89 2.69 ;
14.04 6.18 4.45 3.73 3,34 2.94 2.73 ;
14.69 6.36 4.54 3.80 3, .40 2.98 2.76 ;
25
.24
.44
.56
.63
.69
.82
.92
.99
2.05
2.15 ;
1.22 ;
2.27 ;
2.32 ;
2.37 ;
2.44 ;
2.50 ;
2.54 ;
2.58 ;
2.6i ;
30
.22
.42
.52
.60
.65
.77
.87
.94
.99
2.09 ;
2.15 ;
2.20 ;
2.24 ;
2.30 ;
2.36 ;
2.4i ;
2.45 ;
2.48 ;
2.5i ;
35
.21
.40
.50
.57
.63
.74
.84
.90
.95
2.04 ;
2.11 ;
2.15 ;
2.19 ;
2.24 ;
2.30 ;
2.35 ;
2.39 ;
2.42 ;
2.45 ;
40
.20
.38
.48
.55
.61
.72
.81
.87
.92
2.01
2.07 ;
2.12 ;
2.16 ;
2.20 ;
2.26 ;
2.3i ;
2.34 ;
2.38 ;
2.40 ;
45
.19
.37
.47
.54
.59
.70
.79
.85
.90
.99
2.05 ;
2.09 ;
2.13 ;
2.17 ;
2.23 ;
2.27 ;
2.3i ;
2.34 ;
2.37 ;
50 60 70 80 90 100
.18 1.17 1.16 1.16 1.15 1.15
.36 1.35 1.34 1.33 1.33 1.32
.46 1.45 1.43 1.43 1.42 1.42
.53 1.51 1.50 1.49 1.48 1.48
.58 1.56 1.55 1.54 1.53 1.53
.69 1.67 1.65 1.64 1.63 1.63
.78 1.75 1.74 1.72 1.72 1.71
.84 1.81 1.80 1.78 1.77 1.76
.88 1.86 1.84 1.83 1.82 1.81
.97 1.94 1.92 1.90 1.89 1.88
2.03 2.00 1.97 1.96 1.95 1.94
2.07 2.04 2.02 2.00 1.99 1.98
2.11 2.07 2.05 2.03 2.02 2.01
2.15 2.12 2.09 2.07 2.06 2.05 ;
2.21 2.17 2.14 2.13 2.11 2.10 ;
2.25 2.21 2.19 2.17 2.15 2.14 ;
2.28 2.24 2.22 2.20 2.18 2.17 ;
2.31 2.27 2.24 2.22 2.21 2.20 ;
2.34 2.30 2.27 2.25 2.23 2.22 ;
125
.15
.31
.41
.47
.52
.61
.70
.75
.79
.87
.92
.96
.99
2.03 ;
2.08 ;
2.12 ;
2.15 ;
2.17 ;
2.20 ;
150
.14
.31
.40
.46
.51
.61
.69
.74
.78
.86
.91
.95
.98
2.02
2.07
2.10
2.13
2.16
2.18
    Table 19-17. K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 3 (2 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
2.71
3.53
4.10
4.54
4.92
5.80
6.67
7.36
7.95
9.12
10.06
10.84
11.53
12.43
13.68
14.75
15.66
16.49
17.24

6 8 10 12 16 20 25 30 35
2.06 1.82 1.70 1.62 1.53 .48 .44 .41 .39
2.51 2.16 1.98 1.88 1.75 .68 .63 .59 .57
2.79 2.36 2.15 2.02 1.88 .80 .74 .70 .67
3.00 2.51 2.27 2.13 1.97 .88 .81 .77 .74
3.17 2.63 2.37 2.21 2.04 .94 .87 .82 .79
355 2.88 2.57 2.39 2.18 2.07 .99 .94 .90
3.90 3.11 2.75 2.54 2.31 2.18 2.09 2.03 .99
416 3.28 2.88 2.65 2.40 2.26 2.16 2.10 2.06
438 3.42 2.98 2.74 2.47 2.32 2.22 2.15 2.10
4.79 3.67 3.17 2.90 2.59 2.43 2.32 2.24 2.19
5.10 3.86 3.31 3.01 2.68 2.51 2.39 2.31 2.26
5.35 4.01 3.42 3.10 2.75 2.57 2.44 2.36 2.30
5.57 4.13 3,5? 3.17 2.81 2.62 2.49 2.40 2.34
5.84 4.29 3.63 3.27 2.88 2.68 2.54 2.45 2.39
6.21 4.50 3.78 3.39 2.97 2.76 2.61 2.51 2.45
6.51 4.66 3.89 3,48 3.05 2.82 2.66 2.56 2.50
6.76 4.80 3.99 3,58 3.11 2.87 2.71 2.60 2.53
6.98 4.92 4.08 .'3,62. 3.16 2.92 2.74 2.64 2.57
7.18 5.03 4.15 3.68 3.20 2.95 2.77 2.67 2.59

40
.38
.55
.65
.72
.77
.87
.96
2.02
2.07
2.16
2.22
2.26
2.30
2.34
2.40
2.45
2.48
2.51
2.54

45
.37
.54
.63
.70
.75
.85
.94
2.00
2.05 ;
2.13 ;
2.19 ;
2.23 ;
2.27 ;
2.3i ;
2.37 ;
2.4i ;
2.44 ;
2.47 ;
2.50 ;

50
.36
.53
.62
.69
.73
.84
.92
.98
2.03
2.11
2.16
2.21
2.24
2.28
2.34
2.38
2.41
2.44
2.47

60
1.35
1.51
1.60
1.67
1.71
1.81
1.89
1.95
2.00
2.07
2.13
2.17
2.20
2.24
2.30
2.34
2.37
2.40
2.42

70
1.34
1.50
1.59
1.65
1.70
1.79
1.88
1.93
1.97
2.05
2.10
2.14
2.18
2.22
2.27
2.31
2.34
2.37
2.39

80
1.33
1.49
1.58
1.64
1.69
1.78
1.86
1.92
1.96
2.03
2.09
2.13
2.16
2.20
2.25
2.29
2.32
2.34
2.36

90
1.33
1.48
1.57
1.63
1.68
1.77
1.85
1.91
1.95
2.02
2.07
2.11
2.14
2.18
2.23
2.27
2.30
2.32
2.35

100
1.32
1.48
1.57
1.63
1.67
1.76
1.84
1.90
1.94
2.01
2.06 ;
2.10 ;
2.13 ;
2.17 ;
2.22 ;
2.25 ;
2.29 ;
2.3i ;
2.33 ;

125 150
.31 .31
.47 .46
.56 .55
.61 .61
.66 .65
.75 .74
.83 .82
.88 .87
.92 .91
.99 .98
2.04 2.03
2.08 2.07
2.11 2.10
2.15 2.13
2.20 2.18
2.23 2.22
2.26 2.24
2.28 2.27
2.31 2.29

                                                   D-185
                                                                                                 March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
Unified Guidance
     Table 19-17. K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 3 (5 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4 6 8 10 12 16 20
2.62 1.92 1.68 1.56 1.49 .40 .35
3.41 2.35 2.01 1.84 1.74 .62 .56
395 2.62 2.20 2.00 1.88 .75 .68
4.38 2.82 2.34 2.12 1.99 .84 .76
474 2.98 2.46 2.21 2.07 .91 .82
5.59 3.33 2.70 2.41 2.24 2.05 .95
6.43 3.67 2.92 2.58 2.39 2.18 2.06
7.10 3.92 3,08 2.71 2.49 2.26 2.14 ;
7.66 4.12 3.21 2.81 2.58 2.33 2.20 ;
8.80 4.51 3.45 2.99 2.73 2.46 2.31 ;
9.70 4.80 3.63 3,12 2.84 2.55 2.39 ;
10.46 5.04 3.77 3.23 2.93 2.62 2.45 ;
11.12 5.24 3.89 3.32 ."3,00. 2.67 2.50 ;
11.99 5.50 4.04 3.43 3,09 2.74 2.56 ;
13.21 5.85 4.23 3.57 3,21 2.83 2.64 ;
14.23 6.13 4.39 3.68 3.30 2.90 2.70 ;
15.14 6.36 4.52 3.77 3.37 2.96 2.75 ;
15.92 6.57 4.63 3.85 3.44 3.O1 2.79 ;
16.65 6.75 4.73 3.92 3.49 3,05 2.83 ;
25
.31
.51
.62
.70
.75
.88
.98
2.05
>.n ;
1.21 :
1.28 ;
2.33 ;
2.37 ;
2.43 ;
2.50 ;
2.55 ;
2.60 ;
2.63 ;
2.66 ;
30
.29
.48
.58
.66
.71
.83
.92
.99
2.04 ;
2.14 ;
2.20 ;
2.25 ;
2.30 ;
2.35 ;
2.4i ;
2.46 ;
2.50 ;
2.53 ;
2.56 ;
35
.27
.46
.56
.63
.68
.79
.89
.95
2.00
2.09 ;
2.15 ;
2.20 ;
2.24 ;
2.29 ;
2.35 ;
2.40 ;
2.44 ;
2.47 ;
2.49 ;
40
.26
.44
.54
.61
.66
.77
.86
.92
.97
2.06 ;
2.12 ;
2.17 ;
2.20 ;
2.25 ;
2.3i ;
2.35 ;
2.39 ;
2.42 ;
2.45 ;
45
.25
.43
.53
.59
.65
.75
.84
.90
.95
2.03 ;
2.09 ;
2.14 ;
2.17 ;
2.22 ;
2.27 ;
2.32 ;
2.35 ;
2.38 ;
2.4i ;
50 60 70 80 90 100
.24 1.23 1.22 1.22 1.21 1.21
.42 1.40 1.39 1.39 1.38 1.37
.51 1.50 1.49 1.48 1.47 1.47
.58 1.56 1.55 1.54 1.53 1.53
.63 1.61 1.60 1.59 1.58 1.58
.74 1.71 1.70 1.69 1.68 1.67
.82 1.80 1.78 1.77 1.76 1.75
.88 1.86 1.84 1.83 1.82 1.81
.93 1.90 1.88 1.87 1.86 1.85
2.01 1.98 1.96 1.95 1.94 1.93
2.07 2.04 2.02 2.00 1.99 1.98
2.11 2.08 2.06 2.04 2.03 2.02 ;
2.15 2.12 2.09 2.07 2.06 2.05 ;
2.19 2.16 2.13 2.11 2.10 2.09 ;
2.25 2.21 2.19 2.17 2.15 2.14 ;
2.29 2.25 2.24 2.19 2.19 2.18 ;
2.33 2.29 2.26 2.24 2.22 2.21 ;
2.36 2.31 2.28 2.26 2.25 2.23 ;
2.38 2.34 2.31 2.29 2.27 2.25 ;
125
.20
.37
.46
.52
.56
.66
.74
.79
.84
.91
.96
2.00
2.03 ;
2.07 ;
2.12 ;
2.15 ;
2.18 ;
2.21 ;
2.23 ;
150
.20
.36
.45
.51
.56
.65
.73
.78
.82
.90
.95
.99
2.02
2.06
2.10
2.14
2.17
2.19
2.22
   Table 19-17. K-Multipliers  for 1-of-1  Intrawell Prediction Limits on Means of Order 3 (5 COC, Semi-Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
3.22
4.15
4.80
5.31
5.74
6.75
7.75
8.55
9.22
10.58
11.65
12.55
13.33
14.38
15.82
17.04
18.12
19.09
19.97

6
2.29
2.76
3.05
3.27
3.45
3.85
4.22
4.50
4.72
5.16
5.49
5.76
5.98
6.27
6.67
6.98
7.25
7.47
7.67

8
1.98
2.32
2.52
2.67
2.79
3.05
3,29
3.46
3.60
3.86
4.05
4.21
4.34
4.50
4.71
4.88
5.03
5.15
5.26

10
1.82
2.10
2.27
2.39
2.49
2.69
2.87
3.01
3.11
3.3O
3.44
3.56
3.65
3.76
3.92
4.04
4.14
4.22
4.30

12
1.73
1.98
2.12
2.23
2.31
2.49
2.64
2.75
2.84
3.00
3.11
3.20
3,28
3,37
3.49
3.59
3.67
3.74
3.79

16
1.62
1.83
1.96
2.05
2.12
2.26
2.39
2.47
2.54
2.67
2.76
2.83
2.89
2.96
3.05
3.12
3.18
'"3,23"
3,28

20
1.55
1.75
1.87
1.95
2.01
2.14
2.25
2.33
2.39
2.50
2.58
2.64
2.69
2.75
2.83
2.89
2.94
2.98
3.02

25 30 35 40 45 50 60 70 80 90 100 125 150
.51 .48 .45 .44 .43 .42 1.40 1.39 1.38 1.38 1.37 .37 .36
.69 .65 .63 .61 .59 .58 1.56 1.55 1.54 1.53 1.53 .52 .51
.80 .76 .73 .70 .69 .67 1.65 1.64 1.63 1.62 1.61 .60 .59
.87 .83 .79 .77 .75 .74 1.71 1.70 1.69 1.68 1.67 .66 .65
.93 .88 .84 .82 .80 .78 1.76 1.75 1.73 1.72 1.72 .70 .69
2.05 .99 .95 .92 .90 .88 1.86 1.84 1.83 1.82 1.81 .79 .78
2.15 2.09 2.04 2.01 .99 .97 1.94 1.92 1.90 1.89 1.88 .87 .86
2.22 2.15 2.11 2.07 2.05 2.03 2.00 1.97 1.96 1.95 1.94 .92 .91
2.27 2.20 2.15 2.12 2.09 2.07 2.04 2.02 2.00 1.99 1.98 .96 .95
2.37 2.30 2.24 2.20 2.17 2.15 2.12 2.09 2.07 2.06 2.05 2.03 2.02
2.44 2.36 2.30 2.26 2.23 2.21 2.17 2.14 2.13 2.11 2.10 2.08 2.07
2.50 2.41 2.35 2.31 2.27 2.25 2.21 2.19 2.17 2.15 2.14 2.12 2.10
2.54 2.45 2.39 2.34 2.31 2.28 2.24 2.22 2.20 2.18 2.17 2.15 2.13
2.59 2.50 2.44 2.39 2.35 2.33 2.29 2.26 2.24 2.22 2.21 2.18 2.17
2.66 2.56 2.50 2.45 2.41 2.38 2.34 2.31 2.29 2.27 2.25 2.23 2.22
2.72 2.61 2.54 2.49 2.45 2.42 2.38 2.35 2.31 2.31 2.29 2.27 2.25
2.76 2.65 2.58 2.53 2.49 2.46 2.41 2.37 2.35 2.34 2.32 2.30 2.28
2.80 2.69 2.61 2.56 2.51 2.48 2.44 2.40 2.38 2.36 2.35 2.32 2.30
2.83 2.72 2.64 2.59 2.54 2.51 2.46 2.43 2.40 2.38 2.37 2.34 2.32

                                                  D-186
                                                                                                 March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
Unified Guidance
    Table 19-17. K-Multipliers  for 1-of-1 Intrawell Prediction Limits on Means of Order 3 (5 COC, Quarterly)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
3.83
4.92
5.67
6.26
6.76
7.95
9.12
10.06
10.84
12.43
13.68
14.75
15.66
16.87
18.58
20.02
21.29
22.41
23.44
6
2.66
3.17
3.49
3.74
3.94
4.38
4.79
5.10
5.35
5.84
6.21
6.51
6.76
7.08
7.52
7.86
8.15
8.40
8.64
8 10 12 16 20 25 30 35 40 45
2.27 2.08 1.96 1.82 1.75 1.69 1.65 1.62 .61 .59
2.63 2.37 2.21 2.04 1.94 1.87 1.82 1.79 .77 .75
2.85 2.54 2.36 2.16 2.06 1.97 1.92 1.89 .86 .84
3.01 2.67 2.47 2.25 2.13 2.05 1.99 1.95 .92 .90
3.14 2.77 2.56 2.32 2.20 2.10 2.04 2.00 .97 .95
3.42 2.98 2.74 2.47 2.32 2.22 2.15 2.10 2.07 2.05
367 3.17 2.90 2.59 2.43 2.32 2.24 2.19 2.16 2.13
386 3.31 3.01 2.68 2.51 2.39 2.31 2.26 2.22 2.19
401 3.42 3.10 2.75 2.57 2.44 2.36 2.30 2.26 2.23
4.29 3.63 3.27 2.88 2.68 2.54 2.45 2.39 2.34 2.31
450 378 3.39 2.97 2.76 2.61 2.51 2.45 2.40 2.37
4.66 3.89 '"3.48" 3.05 2.82 2.66 2.56 2.50 2.45 2.41
4.80 3.99 3.-S6 3.11 2.87 2.71 2.60 2.53 2.48 2.44
4.98 4.11 3,65 3.18 2.94 2.76 2.65 2.58 2.53 2.49
5.21 4.28 3.78 3.27 3.01 2.83 2.72 2.64 2.58 2.54
5.40 4.40 3.88 3.35 3.08 2.88 2.77 2.69 2.63 2.58
5.55 4.51 3.96 3.41 3.13 2.93 2.81 2.73 2.66 2.62
5.69 4.60 4.03 3.46 3.17 2.96 2.84 2.75 2.69 2.64
5.80 4.68 4.10 3.5O 3.21 2.99 2.87 2.78 2.72 2.67
50
.58
.73
.82
.88
.93
2.03
2.11
2.16
2.21
2.28
2.34
2.38
2.41
2.45
2.51
2.55
2.58
2.60
2.63
60
1.56
1.71
1.80
1.86
1.90
2.00
2.07
2.13
2.17
2.24
2.30
2.34
2.37
2.41
2.46
2.49
2.53
2.56
2.58
70
1.55
1.70
1.78
1.84
1.88
1.97
2.05
2.10
2.14
2.22
2.27
2.31
2.34
2.38
2.43
2.46
2.49
2.52
2.54
80
1.54
1.69
1.77
1.83
1.87
1.96
2.03
2.09
2.13
2.20
2.25
2.29
2.32
2.35
2.40
2.44
2.47
2.49
2.52
90
1.53
1.68
1.76
1.82
1.86
1.95
2.02
2.07
2.11
2.18
2.23
2.27
2.30
2.34
2.38
2.42
2.45
2.47
2.49
100
1.53
1.67
1.75
1.81
1.85
1.94
2.01
2.06 ;
2.10 ;
2.17 ;
2.22 ;
2.25 ;
2.29 ;
2.32 ;
2.37 ;
2.40 ;
2.43 ;
2.46 ;
2.48 ;
125
.52
.66
.74
.79
.83
.92
.99
>.04 ;
>.os ;
>.is ;
>..20 ;
2.23 ;
i.26 :
i.30 :
1.34 ;
>.38 ;
>.40 ;
1.43 :
1.45 ;
150
.51
.65
.73
.78
.82
.91
.98
>.03
>.07
>.13
>.18
1.22
1.24
1.28
1.32
1.36
1.39
1.41
1.43
     Table 19-17. K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 3 (10 COC, Annual)
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200

4
3.41
4.38
5.06
5.59
6.04
7.10
8.16
8.99
9.70
11.12
12.25
13.20
14.03
15.12
16.66
17.96
19.09
20.10
21.01

6
2.35
2.82
3.11
3.33
3.51
3.92
4.29
4.57
4.80
5.24
5.57
5.84
6.07
6.36
6.77
7.09
7.37
7.61
7.82

8 10 12 16 20 25 30 35 40 45 50 60 70 80 90 100 125 150
2.01 1.84 1.74 1.62 1.56 .51 .48 .46 .44 .43 .42 1.40 1.39 1.39 1.38 1.37 .37 .36
2.34 2.12 1.99 1.84 1.76 .70 .66 .63 .61 .59 .58 1.56 1.55 1.54 1.53 1.53 .52 .51
2.55 2.29 2.13 1.96 1.87 .80 .76 .73 .70 .69 .67 1.65 1.64 1.63 1.62 1.61 .60 .59
2.70 2.41 2.24 2.05 1.95 .88 .83 .79 .77 .75 .74 1.71 1.70 1.69 1.68 1.67 .66 .65
2.82 2.50 2.32 2.12 2.01 .93 .88 .85 .82 .80 .78 1.76 1.75 1.73 1.72 1.72 .70 .69
3,08 2.71 2.49 2.26 2.14 2.05 .99 .95 .92 .90 .88 1.86 1.84 1.83 1.82 1.81 .79 .78
332 2.89 2.65 2.39 2.25 2.15 2.09 2.04 2.01 .99 .97 1.94 1.92 1.90 1.89 1.88 .87 .86
349 3.O2 2.76 2.48 2.33 2.22 2.15 2.11 2.07 2.05 2.03 2.00 1.97 1.96 1.95 1.94 .92 .91
363 3.12 2.84 2.55 2.39 2.28 2.20 2.15 2.12 2.09 2.07 2.04 2.02 2.00 1.99 1.98 .96 .95
3.89 3.32 3.00 2.67 2.50 2.37 2.30 2.24 2.20 2.17 2.15 2.12 2.09 2.07 2.06 2.05 2.03 2.02
408 346 3.12 2.76 2.58 2.44 2.36 2.30 2.26 2.23 2.21 2.17 2.15 2.13 2.11 2.10 2.08 2.07
4.23 3.57 3,21 2.83 2.64 2.50 2.41 2.35 2.31 2.27 2.25 2.21 2.19 2.17 2.15 2.14 2.12 2.10
4.36 3.66 3.28 2.89 2.69 2.54 2.45 2.39 2.34 2.31 2.28 2.24 2.22 2.19 2.18 2.17 2.15 2.13
453 378 337 2.96 2.75 2.59 2.50 2.44 2.39 2.35 2.33 2.29 2.26 2.24 2.22 2.21 2.18 2.17
4.74 3.93 3.50 "'s'.OS' 2.83 2.66 2.56 2.49 2.45 2.41 2.38 2.34 2.31 2.29 2.27 2.25 2.23 2.22
4.91 4.05 3.59 3.13 2.89 2.70 2.61 2.54 2.49 2.45 2.42 2.38 2.35 2.32 2.31 2.29 2.27 2.25
5.06 4.15 3.67 3. 18 2.94 2.76 2.65 2.58 2.53 2.49 2.46 2.41 2.38 2.35 2.34 2.32 2.30 2.28
5.18 4.24 3.74 3.24 2,98 2.80 2.69 2.61 2.56 2.52 2.48 2.44 2.40 2.38 2.36 2.35 2.32 2.30
5.29 4.31 3.80 3,28 3.O2 2.83 2.72 2.64 2.58 2.54 2.51 2.46 2.43 2.40 2.38 2.37 2.34 2.32

                                                   D-187
                                                                                                  March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.15
5.31
6.11
6.75
7.28
8.55
9.81
10.81
11.65
13.33
14.68
15.82
16.82
18.13
19.98
21.50
22.85
24.02
25.20
7. K- Multipliers
6
2.76
3.27
3.60
3.85
4.05
4.50
4.92
5.23
5.49
5.98
6.36
6.67
6.92
7.25
7.66
8.03
8.33
8.61
8.85
Table 19-17.
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.92
6.26
7.20
7.95
8.57
10.06
11.53
12.70
13.68
15.66
17.24
18.57
19.75
21.27
23.44
25.25
26.84
28.24
29.53
6
3.17
3.74
4.10
4.38
4.60
5.10
5.57
5.92
6.21
6.76
7.18
7.51
7.80
8.16
8.64
9.04
9.38
9.70
9.96
8
2.32
2.67
2.89
3.05
3.18
3.46
3.72
3.90
4.05
4.34
4.54
4.71
4.85
5.03
5.27
5.45
5.61
5.74
5.80
10
2.10
2.39
2.57
2.69
2.79
3.01
3.20
3,33
3.44
3.65
3.80
3.91
4.01
4.14
4.30
4.42
4.53
4.63
4.70
for 1
12
1.98
2.23
2.38
2.49
2.57
2.75
2.91
3.02
3.11
3,28
3,4Q
3.49
3.57
3.67
3.79
3.89
3.98
4.04
4.11
K-Multipliers for
8
2.63
3.01
3.24
3.42
3.56
3.86
4.13
4.33
4.50
4.80
5.03
5.21
5.36
5.55
5.80
5.99
6.14
6.27
6.39
10
2.37
2.67
2.85
2.98
3.09
3.31
3,51
3.66
3.78
3.99
4.15
4.28
4.38
4.51
4.68
4.82
4.94
5.03
5.11
12
2.21
2.47
2.63
2.74
2.82
3.01
3.17
3.29
3.39
3,56
3.68
3.78
3.86
3.97
4.10
4.20
4.29
4.37
4.43
-of-1
16
1.83
2.05
2.17
2.26
2.33
2.47
2.60
2.69
2.76
2.89
2.98
3.05
3.11
3.18
3,28
3,35
3,41..
3,46
3,51
Intrawell Prediction Limits on Means of Order 3 (10 COC, Semi-Annual)
20
1.75
1.95
2.06
2.14
2.20
2.33
2.44
2.52
2.58
2.69
2.76
2.83
2.88
2.94
3.02
3.08
3.13
3.17
3,21
25 30 35 40 45 50 60 70 80 90
1.69 1.65 1.63 .61 .59 .58 1.56 1.55 1.54 1.53
1.87 1.83 1.79 .77 .75 .74 1.71 1.70 1.69 1.68
1.98 1.92 1.89 .86 .84 .82 1.80 1.78 1.77 1.76
2.05 1.99 1.95 .92 .90 .88 1.86 1.84 1.83 1.82
2.10 2.04 2.00 .97 .95 .93 1.90 1.88 1.87 1.86
2.22 2.15 2.11 2.07 2.05 2.03 2.00 1.97 1.96 1.95
2.32 2.24 2.19 2.16 2.13 2.11 2.07 2.05 2.03 2.02
2.39 2.31 2.26 2.22 2.19 2.16 2.13 2.10 2.09 2.07
2.44 2.36 2.30 2.26 2.23 2.21 2.17 2.14 2.13 2.11
2.54 2.45 2.39 2.34 2.31 2.28 2.24 2.22 2.20 2.18
2.61 2.51 2.45 2.40 2.37 2.34 2.30 2.27 2.25 2.23
2.66 2.56 2.50 2.45 2.41 2.38 2.34 2.31 2.29 2.27
2.71 2.60 2.53 2.48 2.44 2.41 2.37 2.34 2.32 2.30
2.76 2.65 2.58 2.53 2.49 2.45 2.41 2.38 2.35 2.34
2.83 2.72 2.64 2.59 2.54 2.51 2.46 2.43 2.40 2.38
2.88 2.77 2.69 2.63 2.58 2.55 2.50 2.46 2.44 2.42
2.93 2.81 2.72 2.66 2.62 2.58 2.53 2.49 2.47 2.45
2.96 2.84 2.76 2.69 2.65 2.61 2.56 2.52 2.49 2.47
3.00 2.87 2.79 2.72 2.67 2.63 2.58 2.54 2.52 2.49
100
1.53
1.67
1.75
1.81
1.85
1.94
2.01
2.06
2.10
2.17
2.22
2.25
2.29
2.32
2.37
2.40
2.43
2.46
2.48
125
.52
.66
.74
.79
.83
.92
.99
2.04
2.08
2.15
2.20
2.23
2.26
2.30
2.34
2.38
2.40
2.43
2.45
150
.51
.65
.73
.78
.82
.91
.98
2.03
2.07
2.13
2.18
2.22
2.24
2.28
2.32
2.36
2.39
2.41
2.43
1-of-1 Intrawell Prediction Limits on Means of Order 3 (10 COC, Quarterly)
16
2.04
2.25
2.38
2.47
2.54
2.68
2.81
2.90
2.98
3.11
3.20
3.27
3.33
3.41
3, 5 O
3,58
3,64
3,70
3.74
20
1.94
2.13
2.25
2.32
2.39
2.51
2.62
2.70
2.76
2.87
2.95
3.01
3.07
3.13
3.21
3.27
3.32
3.36
3.40
25 30 35 40 45 50 60 70 80 90
1.87 1.82 1.79 1.77 1.75 1.73 1.71 1.70 1.69 1.68
2.05 1.99 1.95 1.92 1.90 1.88 1.86 1.84 1.83 1.82
2.15 2.08 2.04 2.01 1.99 1.97 1.94 1.92 1.90 1.89
2.22 2.15 2.10 2.07 2.05 2.03 2.00 1.97 1.96 1.95
2.27 2.20 2.15 2.12 2.09 2.07 2.04 2.02 2.00 1.99
2.39 2.31 2.26 2.22 2.19 2.16 2.13 2.10 2.09 2.07
2.49 2.40 2.34 2.30 2.27 2.24 2.20 2.18 2.16 2.14
2.56 2.46 2.40 2.36 2.32 2.30 2.26 2.23 2.21 2.19
2.61 2.51 2.45 2.40 2.37 2.34 2.30 2.27 2.25 2.23
2.71 2.60 2.53 2.48 2.44 2.41 2.37 2.34 2.32 2.30
2.77 2.67 2.59 2.54 2.50 2.47 2.42 2.39 2.36 2.35
2.83 2.72 2.64 2.58 2.54 2.51 2.46 2.43 2.40 2.38
2.87 2.76 2.68 2.62 2.57 2.54 2.49 2.46 2.43 2.41
2.92 2.81 2.72 2.66 2.62 2.58 2.53 2.49 2.47 2.45
2.99 2.87 2.78 2.72 2.67 2.63 2.58 2.54 2.52 2.49
3.05 2.92 2.83 2.76 2.71 2.67 2.62 2.58 2.55 2.53
3.11 2.96 2.86 2.80 2.75 2.71 2.65 2.61 2.58 2.56
3.14 3.00 2.89 2.83 2.77 2.73 2.67 2.63 2.60 2.58
3.18 3.02 2.92 2.85 2.80 2.76 2.70 2.66 2.63 2.60
100
1.67
1.81
1.88
1.94
1.98
2.06
2.13
2.18
2.22
2.29
2.33
2.37
2.40
2.43
2.48
2.51
2.54
2.56
2.58
125
.66
.79
.87
.92
.96
2.04
2.11
2.16
2.20
2.26
2.31
2.34
2.37
2.40
2.45
2.48
2.51
2.53
2.55
150
.65
.78
.86
.91
.95
2.03
2.10
2.14
2.18
2.24
2.29
2.32
2.35
2.39
2.43
2.46
2.49
2.51
2.53

                                                     D-188
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-17.
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
4.38
5.59
6.44
7.10
7.67
8.99
10.32
11.37
12.25
14.04
15.47
16.66
17.71
19.07
21.01
22.68
24.08
25.31
26.37
Table 19-1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.31
6.75
7.75
8.55
9.22
10.81
12.39
13.66
14.72
16.85
18.54
20.00
21.27
22.85
25.22
27.07
28.83
30.23
31.64
6
2.82
3.33
3.67
3.92
4.12
4.57
5.00
5.32
5.58
6.08
6.46
6.77
7.03
7.36
7.82
8.17
8.53
8.79
9.05
K-Multipliers for 1-of-1 Intrawell Prediction Limits
8 10 12 16 20 25 30 35 40
2.34 2.12 1.99 1.84 1.76 1.70 1.66 1.63 .61
2.70 2.41 2.24 2.05 1.95 1.88 1.83 1.79 .77
2.92 2.58 2.39 2.18 2.06 1.98 1.92 1.89 .86
3,08 2.71 2.49 2.26 2.14 2.05 1.99 1.95 .92
321 2.81 2.58 2.33 2.20 2.11 2.04 2.00 .97
3.49 3.O2 2.76 2.48 2.33 2.22 2.15 2.11 2.07
3.74 3.21' 2.92 2.60 2.44 2.32 2.25 2.19 2.16
3.93 3.35 3.03 2.69 2.52 2.39 2.31 2.26 2.22
4.08 3.46 3.12 2.76 2.58 2.44 2.36 2.30 2.26
4.36 3.66 3.28 2.89 2.69 2.54 2.45 2.39 2.34
4.58 3.81 3.40 2.98 2.77 2.61 2.51 2.45 2.40
4.74 3.93 3.50 3.05 2.83 2.66 2.57 2.50 2.45
4.88 4.03 3.58 3.11 2.88 2.71 2.60 2.54 2.48
5.05 4.15 3.67 3: 19 2.94 2.76 2.65 2.58 2.53
5.30 4.31 3.80 3.28 3.02 2.83 2.72 2.64 2.58
5.48 4.44 3.90 3.35 3.08 2.88 2.77 2.69 2.63
5.65 4.55 3.98 3.42 3,13 2.93 2.81 2.72 2.66
5.78 4.64 4.05 3.46 3. 18 2.97 2.84 2.75 2.69
5.89 4.72 4.11 3.52 3.21 3.0O 2.87 2.78 2.72
45
.59
.75
.84
.90
.95
2.05
2.13
2.19
2.23
2.31
2.37
2.41
2.44
2.49
2.54
2.58
2.61
2.65
2.67
7. K- Multipliers for 1-of-1 Intrawell Prediction Limits on
6
3.27
3.85
4.22
4.50
4.72
5.23
5.71
6.07
6.36
6.93
7.35
7.70
8.00
8.37
8.88
9.32
9.67
9.98
10.28
8 10 12 16 20 25 30 35 40
2.67 2.39 2.23 2.05 1.95 1.87 1.83 1.79 1.77
3.05 2.69 2.49 2.26 2.14 2.05 1.99 1.95 1.92
'"3.29" 2.87 2.64 2.39 2.25 2.15 2.09 2.04 2.01
346 3.01 2.75 2.47 2.33 2.22 2.15 2.11 2.07
360 3.11 2.84 2.54 2.39 2.27 2.20 2.15 2.12
3.90 3.34 3.02 2.69 2.52 2.39 2.31 2.26 2.22
4.18 3.54 3.19 2.82 2.63 2.49 2.40 2.34 2.30
4.38 3.68 3.3O 2.91 2.70 2.56 2.47 2.40 2.36
4.55 3.80 3.40 2.98 2.77 2.61 2.51 2.45 2.40
4.85 4.02 3.57 3.11 2.88 2.71 2.60 2.53 2.48
5.08 4.17 3.69 3.20 2.96 2.78 2.67 2.59 2.54
5.26 4.30 3.79 3.28 3.02 2.83 2.72 2.64 2.58
5.42 4.41 3.87 3.34 3.07 2.88 2.76 2.68 2.62
5.60 4.54 3.98 3,4? 3.13 2.93 2.80 2.72 2.66
5.87 4.70 4.11 3,57 3.21 3.00 2.87 2.78 2.72
6.06 4.84 4.22 3.58 . 3.27 3.05 2.92 2.83 2.76
6.24 4.95 4.31 3.65 .3,32 3.10 2.96 2.86 2.80
6.39 5.05 4.37 3.70 3,37 3.13 2.99 2.89 2.82
6.53 5.14 4.44 3.75 3,4? 3.16 3.02 2.92 2.85
45
1.75
1.90
1.99
2.05
2.09
2.19
2.27
2.32
2.37
2.44
2.50
2.54
2.58
2.62
2.67
2.71
2.75
2.77
2.80
on Means of Order 3 (20 COC, Annual)
50
1.58
1.74
1.82
1.88
1.93
2.03
2.11
2.16
2.21
2.28
2.34
2.38
2.41
2.46
2.51
2.55
2.58
2.61
2.64
Means
50
1.74
1.88
1.97
2.03
2.07
2.16
2.24
2.30
2.34
2.41
2.47
2.51
2.54
2.58
2.63
2.68
2.71
2.74
2.76
60
1.56
1.71
1.80
1.86
1.90
2.00
2.07
2.13
2.17
2.24
2.30
2.34
2.37
2.41
2.46
2.50
2.53
2.56
2.58
70
1.55
1.70
1.78
1.84
1.88
1.97
2.05
2.10
2.14
2.22
2.27
2.31
2.34
2.38
2.43
2.46
2.49
2.52
2.54
of Order
60
1.71
1.86
1.94
2.00
2.04
2.13
2.20
2.26
2.30
2.37
2.42
2.46
2.49
2.53
2.58
2.62
2.65
2.68
2.70
70
1.70
1.84
1.92
1.97
2.02
2.10
2.18
2.23
2.27
2.34
2.39
2.43
2.46
2.49
2.54
2.58
2.61
2.64
2.66
80
1.54
1.69
1.77
1.83
1.87
1.96
2.03
2.09
2.13
2.20
2.25
2.29
2.32
2.35
2.40
2.44
2.47
2.49
2.52
3 (20
80
1.69
1.83
1.90
1.96
2.00
2.09
2.16
2.21
2.25
2.32
2.36
2.40
2.43
2.47
2.52
2.55
2.58
2.60
2.63
90
1.53
1.68
1.76
1.82
1.86
1.95
2.02
2.07
2.11
2.18
2.23
2.27
2.30
2.33
2.38
2.42
2.45
2.47
2.49
COC
90
1.68
1.82
1.89
1.95
1.99
2.07
2.14
2.19
2.23
2.30
2.35
2.38
2.41
2.45
2.49
2.53
2.56
2.58
2.60
100
1.53
1.67
1.75
1.81
1.85
1.94
2.01
2.06
2.10
2.17
2.22
2.25
2.29
2.32
2.37
2.40
2.43
2.46
2.48
125
.52
.66
.74
.79
.84
.92
.99
2.04
2.08
2.15
2.19
2.23
2.26
2.30
2.34
2.38
2.40
2.43
2.45
150
.51
.65
.73
.78
.82
.91
.98
2.03
2.07
2.13
2.18
2.22
2.24
2.28
2.32
2.36
2.38
2.41
2.43
, Semi-Annual)
100
1.67
1.81
1.88
1.94
1.98
2.06
2.13
2.18
2.22
2.29
2.33
2.37
2.40
2.43
2.48
2.51
2.54
2.56
2.58
125
1.66
1.79
1.87
1.92
1.96
2.04
2.11
2.16
2.20
2.26
2.31
2.34
2.37
2.40
2.45
2.48
2.51
2.53
2.55
150
.65
.78
.86
.91
.95
2.03
2.10
2.14
2.18
2.24
2.29
2.32
2.35
2.39
2.43
2.46
2.49
2.51
2.53

                                                     D-189
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-17. K-Multipliers for
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
6.26
7.95
9.12
10.06
10.85
12.71
14.57
16.04
17.29
19.80
21.80
23.47
24.96
26.89
29.53
31.99
33.75
35.86
37.27
Table 1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
5.59
7.10
8.16
8.99
9.70
11.37
13.03
14.35
15.46
17.71
19.51
21.01
22.32
24.08
26.48
28.54
30.35
31.88
33.40
6
3.74
4.38
4.79
5.10
5.35
5.92
6.45
6.86
7.18
7.81
8.29
8.68
9.01
9.45
10.02
10.46
10.90
11.25
11.51
9-17.
6
3.33
3.92
4.29
4.57
4.80
5.32
5.80
6.16
6.46
7.03
7.46
7.82
8.12
8.50
9.02
9.43
9.80
10.11
10.40
8
3.01
3.42
3.67
3.86
4.01
4.33
4.63
4.85
5.03
5.36
5.61
5.81
5.98
6.19
6.46
6.68
6.88
7.03
7.16
10
2.67
2.98
3.17
3.31
3.42
3.66
3.87
4.03
4.15
4.38
4.55
4.68
4.80
4.93
5.12
5.26
5.38
5.49
5.58
12
2.47
2.74
2.90
3.01
3.10
3.29
3.46
3,59
3.68
3.86
3.99
4.10
4.19
4.29
4.43
4.54
4.64
4.70
4.77
1-of-1 Intrawell Prediction Limits on Means of Order 3 (20 COC, Quarterly)
16
2.25
2.47
2.59
2.68
2.75
2.90
3.03
3.13
3.20
3.33
3.43
3.5O
•3.57.
3,64
3,74
3.82
3.89
3.94
3.99
20
2.13
2.32
2.44
2.51
2.57
2.70
2.81
2.89
2.95
3.07
3.14
3.21
3.26
3.32
3.41
3.47' '
•3.52
3,56
3,60
25 30 35 40 45 50 60 70 80 90
2.05 1.99 1.95 1.92 1.90 1.88 1.86 1.84 1.83 1.82
2.22 2.15 2.10 2.07 2.05 2.03 2.00 1.97 1.96 1.95
2.32 2.24 2.19 2.16 2.13 2.11 2.07 2.05 2.03 2.02
2.39 2.31 2.25 2.22 2.19 2.16 2.13 2.10 2.09 2.07
2.44 2.36 2.30 2.26 2.23 2.21 2.17 2.14 2.13 2.11
2.56 2.46 2.40 2.36 2.32 2.30 2.26 2.23 2.21 2.19
2.65 2.55 2.49 2.44 2.40 2.37 2.33 2.30 2.28 2.26
2.72 2.62 2.55 2.50 2.46 2.43 2.38 2.35 2.33 2.31
2.78 2.67 2.59 2.54 2.50 2.47 2.42 2.39 2.36 2.35
2.87 2.76 2.68 2.62 2.57 2.54 2.49 2.46 2.43 2.41
2.94 2.82 2.74 2.68 2.63 2.59 2.54 2.50 2.48 2.46
3.00 2.87 2.78 2.72 2.67 2.63 2.58 2.54 2.51 2.49
3.04 2.91 2.82 2.75 2.71 2.67 2.61 2.57 2.54 2.52
3.10 2.96 2.86 2.80 2.75 2.71 2.65 2.61 2.58 2.56
3.16 3.02 2.92 2.85 2.80 2.76 2.70 2.66 2.63 2.60
3.22 3.07 2.97 2.89 2.84 2.80 2.74 2.69 2.66 2.64
3.26 3.11 3.00 2.93 2.87 2.83 2.77 2.72 2.69 2.66
3.30 3.14 3.03 2.96 2.90 2.86 2.79 2.75 2.71 2.69
3.33 3.17 3.06 2.98 2.92 2.88 2.81 2.77 2.73 2.71
100
1.81
1.94
2.01
2.06
2.10
2.18
2.25
2.30
2.33
2.40
2.44
2.48
2.50
2.54
2.58
2.62
2.65
2.67
2.69
K-Multipliers for 1-of-1 Intrawell Prediction Limits on Means of Order 3 (40 COC,
8
2.70
3.-O8
3.32
3.49
3.63
3.93
4.21
4.41
4.57
4.88
5.11
5.30
5.45
5.64
5.90
6.11
6.28
6.43
6.56
10
2.41
2.71
2.89
3.Q2"
3.12
3.35
3.55
3.69
3.81
4.03
4.19
4.31
4.42
4.54
4.72
4.86
4.97
5.07
5.16
12
2.24
2.49
2.65
2.76
2.84
3,03,
3,19
3.31
3.40
3.58
3.70
3.80
3.88
3.98
4.11
4.22
4.31
4.38
4.45
16
2.05
2.26
2.39
2.48
2.55
2.69
2.82
2.91
2,98,
3.1 T.
3,21
3,28
3.34
3.41
3.51
3.59
3.65
3.70
3.75
20
1.95
2.14
2.25
2.33
2.39
2.52
2.63
2.71
2.77
2.88
2.96
. 3.Q2
3,07
3,13
3,21
3,27
3.33
3.37
3.41
25 30 35 40 45 50 60 70 80 90
1.88 1.83 1.79 1.77 1.75 1.74 1.71 1.70 1.69 1.68
2.05 1.99 1.95 1.92 1.90 1.88 1.86 1.84 1.83 1.82
2.15 2.09 2.04 2.01 1.99 1.97 1.94 1.92 1.90 1.89
2.22 2.15 2.11 2.07 2.05 2.03 2.00 1.97 1.96 1.95
2.28 2.20 2.15 2.12 2.09 2.07 2.04 2.02 2.00 1.99
2.39 2.31 2.26 2.22 2.19 2.16 2.13 2.10 2.09 2.07
2.49 2.40 2.34 2.30 2.27 2.24 2.20 2.18 2.16 2.14
2.56 2.47 2.40 2.36 2.32 2.30 2.26 2.23 2.21 2.19
2.61 2.51 2.45 2.40 2.37 2.34 2.30 2.27 2.25 2.23
2.71 2.60 2.53 2.48 2.44 2.41 2.37 2.34 2.32 2.30
2.78 2.67 2.59 2.54 2.50 2.47 2.42 2.39 2.36 2.35
2.83 2.72 2.64 2.58 2.54 2.51 2.46 2.43 2.40 2.38
2.87 2.76 2.68 2.62 2.58 2.54 2.49 2.46 2.43 2.41
2.93 2.81 2.72 2.66 2.62 2.58 2.53 2.49 2.47 2.45
3,00 2.87 2.78 2.72 2.67 2.63 2.58 2.54 2.51 2.49
3,05 2.92 2.83 2.76 2.71 2.67 2.62 2.58 2.55 2.53
3,09 2,90 2.86 2.80 2.75 2.71 2.65 2.61 2.58 2.56
3,73 2.99 2.89 2.83 2.77 2.73 2.68 2.63 2.60 2.58
3.16 3,02 2.92 2.85 2.80 2.76 2.70 2.66 2.63 2.60
100
1.67
1.81
1.88
1.94
1.98
2.06
2.13
2.18
2.22
2.28
2.33
2.37
2.40
2.43
2.48
2.51
2.54
2.56
2.58
125
1.79
1.92
1.99
2.04
2.08
2.16
2.22
2.27
2.31
2.37
2.41
2.45
2.48
2.51
2.55
2.58
2.61
2.63
2.65
150
1.78
1.91
1.98
2.03
2.07
2.14
2.21
2.25
2.29
2.35
2.39
2.43
2.46
2.49
2.53
2.56
2.59
2.61
2.63
Annual)
125
1.66
1.79
1.87
1.92
1.96
2.04
2.11
2.16
2.19
2.26
2.31
2.34
2.37
2.40
2.45
2.48
2.51
2.53
2.55
150
.65
.78
.86
.91
.95
2.03
2.10
2.14
2.18
2.24
2.29
2.32
2.35
2.39
2.43
2.46
2.49
2.51
2.53

                                                     D-190
                                                                                                      March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
Unified Guidance
Table 19-1
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
6.75
8.55
9.81
10.81
11.66
13.65
15.64
17.23
18.56
21.25
23.41
25.22
26.81
28.89
31.76
34.22
36.33
38.32
40.08
7. K- Multipliers for 1
6
3.85
4.50
4.92
5.23
5.49
6.07
6.61
7.02
7.35
8.00
8.49
8.89
9.23
9.66
10.24
10.72
11.13
11.48
11.81
8
3.05
3.46
3.72
3.90
4.05
4.38
4.68
4.90
5.08
5.42
5.67
5.86
6.03
6.24
6.53
6.75
6.94
7.10
7.25
10
2.69
3.01
3.20
3,33
3.44
3.68
3.89
4.05
4.17
4.40
4.57
4.71
4.82
4.95
5.14
5.29
5.41
5.51
5.60
Table 19-17. K-Multipliers
w/n
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4
7.95
10.06
11.53
12.71
13.70
16.04
18.37
20.23
21.80
24.96
27.48
29.59
31.46
33.87
37.32
40.20
42.66
45.00
47.11
6
4.38
5.10
5.57
5.92
6.21
6.85
7.46
7.92
8.29
9.02
9.57
10.01
10.39
10.88
11.53
12.07
12.51
12.92
13.27
8
3.42
3.86
4.13
4.33
4.50
4.85
5.18
5.42
5.61
5.97
6.24
6.46
6.64
6.87
7.18
7.43
7.63
7.81
7.97
10
2.98
3.31
3,51
3.66
3.78
4.03
4.25
4.42
4.55
4.79
4.97
5.12
5.24
5.38
5.58
5.73
5.87
5.98
6.08
12
2.49
2.75
2.91
3.02
3.11
3,30
3.47
3.60
3.69
3.87
4.00
4.11
4.19
4.30
4.44
4.55
4.64
4.72
4.79
for
12
2.74
3.01
3.17
3.29
3.39
3,59
3.76
3.89
3.99
4.18
4.32
4.43
4.52
4.63
4.78
4.89
4.99
5.07
5.14
-of-1
16
2.26
2.47
2.60
2.69
2.76
2.91
3.04
3.13
3.20
3,34
3,43
3,51
3.57
3.65
3.75
3.82
3.89
3.94
3.99
Intrawell Prediction Limits on Means of Order 3 (40 COC, Semi-Annual)
20 25 30 35 40 45 50 60 70 80
2.14 2.05 1.99 1.95 1.92 1.90 1.88 1.86 1.84 1.83
2.33 2.22 2.15 2.11 2.07 2.05 2.03 2.00 1.97 1.96
2.44 2.32 2.24 2.19 2.16 2.13 2.11 2.07 2.05 2.03
2.52 2.39 2.31 2.26 2.22 2.19 2.16 2.13 2.10 2.09
2.58 2.44 2.36 2.30 2.26 2.23 2.21 2.17 2.14 2.13
2.70 2.56 2.47 2.40 2.36 2.32 2.30 2.26 2.23 2.21
2.82 2.65 2.56 2.49 2.44 2.40 2.37 2.33 2.30 2.28
2.89 2.72 2.62 2.55 2.50 2.46 2.43 2.38 2.35 2.33
2.96 2.78 2.67 2.59 2.54 2.50 2.47 2.42 2.39 2.36
3.07 2.87 2.76 2.68 2.62 2.58 2.54 2.49 2.46 2.43
3.15 2.94 2.82 2.74 2.68 2.63 2.59 2.54 2.51 2.48
3.21 3.00 2.87 2.78 2.72 2.67 2.63 2.58 2.54 2.51
3,26 3.04 2.91 2.82 2.75 2.70 2.67 2.61 2.57 2.54
3,32 3.10 2.96 2.86 2.80 2.75 2.71 2.65 2.61 2.58
3,40 3.16 3.02 2.92 2.85 2.80 2.76 2.70 2.66 2.63
3.47 '3.22' 3.07 2.97 2.89 2.84 2.80 2.74 2.69 2.66
3,52 3.26 3.11 3.00 2.93 2.87 2.83 2.76 2.72 2.69
3,50 3,30 3.14 3.03 2.96 2.90 2.86 2.79 2.75 2.71
3.60 3,33 3.17 3.06 2.98 2.93 2.88 2.81 2.77 2.73
90
1.82
1.95
2.02
2.07
2.11
2.19
2.26
2.31
2.35
2.41
2.46
2.49
2.52
2.56
2.60
2.64
2.66
2.69
2.71
100
1.81
1.94
2.01
2.06
2.10
2.18
2.25
2.30
2.33
2.40
2.44
2.48
2.51
2.54
2.58
2.62
2.65
2.67
2.69
125
1.79
1.92
1.99
2.04
2.08
2.16
2.22
2.27
2.31
2.37
2.41
2.45
2.48
2.51
2.55
2.58
2.61
2.63
2.65
150
1.78
1.91
1.98
2.03
2.07
2.14
2.21
2.25
2.29
2.35
2.39
2.43
2.46
2.49
2.53
2.56
2.59
2.61
2.63
1-of-1 Intrawell Prediction Limits on Means of Order 3 (40 COC, Quarterly)
16
2.47
2.68
2.81
2.90
2.98
3.13
3.26
3.35
3.43
3,57
3,66
3,74
3.81
3.88
3.99
4.07
4.13
4.19
4.24
20 25 30 35 40 45 50 60 70 80
2.32 2.22 2.15 2.10 2.07 2.05 2.03 2.00 1.97 1.96
2.51 2.39 2.31 2.26 2.22 2.19 2.16 2.13 2.10 2.09
2.62 2.49 2.40 2.34 2.30 2.27 2.24 2.20 2.18 2.16
2.70 2.56 2.46 2.40 2.36 2.32 2.30 2.26 2.23 2.21
2.76 2.61 2.51 2.45 2.40 2.37 2.34 2.30 2.27 2.25
2.89 2.72 2.62 2.55 2.50 2.46 2.43 2.38 2.35 2.33
3.00 2.82 2.71 2.63 2.58 2.53 2.50 2.45 2.42 2.39
3.08 2.89 2.77 2.69 2.63 2.59 2.55 2.50 2.47 2.44
3.15 2.94 2.82 2.74 2.67 2.63 2.59 2.54 2.51 2.48
3.26 3.04 2.91 2.82 2.75 2.70 2.67 2.61 2.57 2.54
3.34 3.11 2.97 2.88 2.81 2.76 2.72 2.66 2.62 2.59
3.40 3.16 3.02 2.92 2.85 2.80 2.76 2.70 2.66 2.63
3.46 3.21 3.06 2.96 2.89 2.83 2.79 2.73 2.69 2.65
3,52 3.26 3.11 3.00 2.93 2.87 2.83 2.77 2.72 2.69
3,60 3.33 3.17 3.06 2.98 2.93 2.88 2.81 2.77 2.73
3,67 3.39 3.22 3.11 3.02 2.96 2.92 2.85 2.80 2.77
3,72 3.43 3.26 3.14 3.06 3.00 2.95 2.88 2.83 2.80
3,76 3,47" 3.29 3.18 3.09 3.02 2.98 2.91 2.86 2.82
3,80 3.50 3.32 3.20 3.11 3.05 3.00 2.93 2.89 2.84
90
1.95
2.07
2.14
2.19
2.23
2.31
2.38
2.42
2.46
2.52
2.57
2.60
2.63
2.66
2.71
2.74
2.77
2.79
2.81
100
1.94
2.06
2.13
2.18
2.22
2.30
2.36
2.41
2.44
2.51
2.55
2.58
2.61
2.65
2.69
2.72
2.75
2.77
2.79
125
1.92
2.04
2.11
2.16
2.19
2.27
2.33
2.38
2.41
2.48
2.52
2.55
2.58
2.61
2.65
2.68
2.71
2.73
2.75
150
1.91
2.03
2.10
2.14
2.18
2.25
2.32
2.36
2.39
2.46
2.50
2.53
2.56
2.59
2.63
2.66
2.68
2.71
2.73

                                                     D-191
                                                                                                      March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                    Unified Guidance
     Table 19-18. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (1 COC, Annual)
w/n 468
1 0.67 0.53 0.46 (
2 .04 0.80 0.70 (
3 .28 0.96 0.84 (
4 .47 1.08 0.93 (
5 .62 1.17 1.01 (
8 .97 1.38 1.17
12 2.31 1.56 1.31
16 2.58 1.69 1.41
20 2.80 1.80 1.48
30 3.24 2.01 1.63
40 3.60 2.16 1.74
50 3.89 2.28 1.82
60 415 2.39 1.89
75 4.48 2.52 1.98
100 4.95 2.70 2.09
125 5.35 2,84 2.19
150 5.69 2.96 2.26
175 6.00 3.07 2.33 ;
200 6.28 3.16 2.39 ;
10
).42 (
).64 (
).77 (
).85 (
).92 (
.06 (
.18
.26
.33
.45
.53
.60
.66
.73
.82
.89
.95
>.oo
>.04
12 16
).40 0.37 (
).61 0.56 (
).72 0.67 (
).80 0.74 (
).86 0.80 (
).99 0.91 (
.10 1.01 (
.17 1.07
.23 1.12
.34 1.22
.42 1.28
.47 1.33
.52 1.37
.58 1.42
.66 1.48
.72 1.53
.77 1.58
.81 1.61
.85 1.64
20 25 30 35
).35 0.34 0.33 0.32 (
).54 0.52 0.50 0.49 (
).64 0.61 0.60 0.59 (
).71 0.68 0.66 0.65 (
).76 0.73 0.71 0.70 (
).87 0.83 0.81 0.79 (
).95 0.91 0.89 0.87 (
.02 0.97 0.94 0.92 (
.06 1.02 0.99 0.96 (
.15 1.09 1.06 .04
.21 1.15 1.11 .09
.25 1.19 1.15 .13
.29 1.23 1.18 .16
.33 1.27 1.22 .19
.39 1.32 1.27 .24
.43 1.36 1.31 1.28
.47 1.39 1.34 1.31
.50 1.42 1.37 1.33
.53 1.44 1.39 1.35
40 45
).31 0.31 (
).49 0.48 (
).58 0.57 (
).64 0.63 (
).68 0.68 (
).78 0.77 (
).86 0.84 (
).91 0.90 (
).95 0.94 (
.02 1.01
.07 1.05
.11 1.09
.14 1.12
.17 1.16
.22 1.20
.25 1.24
.28 1.26
.31 1.29
.33 1.31
50
).31 (
).47 (
).56 (
).62 (
).67 (
).76 (
).84 (
).89 (
).93 (
.00 (
.04
.08
.11
.14
.19
.22
.25
.27
.29
60 70
).30 0.30 (
).47 0.46 (
).56 0.55 (
).62 0.61 (
).66 0.65 (
).75 0.74 (
).82 0.81 (
).87 0.86 (
).91 0.90 (
).98 0.97 (
.03 1.01
.06 1.05
.09 1.08
.12 1.11
.17 1.15
.20 1.18
.23 1.21
.25 1.23
.27 1.25
80 90
).29 0.29 (
).46 0.46 (
).55 0.54 (
).60 0.60 (
).65 0.64 (
).74 0.73 (
).81 0.80 (
).86 0.85 (
).89 0.89 (
).96 0.95 (
.00 1.00 (
.04 1.03
.07 1.06
.10 1.09
.14 1.13
.17 1.16
.20 1.19
.22 1.21
.24 1.23
100 125 150
).29 0.29 0.29
).45 0.45 0.45
).54 0.54 0.53
).60 0.59 0.59
).64 0.64 0.63
).73 0.72 0.72
).80 0.79 0.79
).85 0.84 0.83
).88 0.88 0.87
).95 0.94 0.93
).99 0.98 0.98
.03 1.01 1.01
.05 1.04 1.03
.08 1.07 1.07
.12 1.11 1.11
.16 1.14 1.14
.18 1.17 1.16
.20 1.19 1.18
.22 1.21 1.20
   Table 19-18. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (1 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
0.95 0.75 0.67 (
1.36 1.04 0.91 (
1.64 1.21 1.05 (
1.85 1.33 1.14
2.02 1.43 1.22
2.43 1.65 1.38
2.83 1.85 1.53
3.14 2.00 1.63
3.41 2.12 1.72
3.94 2.34 1.87
4.35 2.51 1.98
470 2.65 2.07
5.01 2.76 2.15
5.41 ,2,97 2.24
5.97 3.11 2.37 ;
6.45 3.27 2.46 ;
6.86 3.40 2.55 ;
7.22 3.52 2.62 ;
7.56 3.63 2.68 ;
).62 (
).84 (
).96 (
.04 (
.11
.25
.37
.46
.52
.65
.74
.81
.87
.94
>.03
Ml
>.17
1.22 :
1.27 ;
).59 0.55 (
).79 0.73 (
).90 0.83 (
).98 0.90 (
.04 0.96 (
.17 1.07
.27 1.16
.35 1.23
.41 1.28
.52 1.37
.59 1.43
.65 1.48
.70 1.52
.76 1.57
.84 1.64
.91 1.69
.96 1.73
>.00 1.76
>.04 1.79
).53 (
).70 (
).80 (
).86 (
).91 (
.01 (
.10
.16
.20
.29
.34
.39
.42
.47
.53
.57
.61
.64
.66
).51 0.50 (
).67 0.66 (
).76 0.74 (
).83 0.80 (
).87 0.85 (
).97 0.94 (
.05 1.02
.11 1.07
.15 1.11
.22 1.18
.28 1.23
.32 1.27
.35 1.30
.39 1.34 ]
.44 1.39 ]
.48 1.43 ]
.52 1.46 ]
.54 1.48 ]
.57 1.51 ]
).49 (
).64 (
).73 (
).79 (
).83 (
).92 (
.00 (
.05
.09
.16
.20
.24
.27
L.31
L.35
L.39
L.42
L.44
L.46
).48 0.48 (
).64 0.63 (
).72 0.71 (
).78 0.77 (
).82 0.81 (
).91 0.90 (
).98 0.97 (
.03 1.02
.07 1.05
.14 1.12
.18 1.17
.22 1.20
.25 1.23
.28 1.26
.33 1.31
.36 1.34
.39 1.37
.41 1.39
.43 1.41
).47 (
).62 (
).70 (
).76 (
).80 (
).89 (
).96 (
.01 (
.04
.11
.15
.19
.21
.25
.29
.32
.35
.37
.39
).47 0.46 (
).61 0.61 (
).69 0.69 (
).75 0.74 (
).79 0.78 (
).87 0.86 (
).94 0.93 (
).99 0.98 (
.02 1.01
.09 1.08
.13 1.12
.17 1.15
.19 1.18
.23 1.21
.27 1.25
.30 1.28
.32 1.31
.34 1.33
.36 1.34
).46 0.45 (
).60 0.60 (
).68 0.68 (
).74 0.73 (
).78 0.77 (
).86 0.85 (
).92 0.92 (
).97 0.96 (
.00 1.00 (
.07 1.06
.11 1.10
.14 1.13
.17 1.16
.20 1.19
.24 1.23
.27 1.26
.29 1.28
.31 1.30
.33 1.32
).45 0.45 0.45
).60 0.59 0.59
).67 0.67 0.66
).73 0.72 0.72
).77 0.76 0.76
).85 0.84 0.83
).91 0.90 0.90
).96 0.95 0.94
).99 0.98 0.97
.05 1.04 1.03
.09 1.08 1.07
.12 1.11 1.10
.15 1.14 1.13
.18 1.17 1.16
.22 1.21 1.20
.25 1.24 1.23
.27 1.26 1.25
.29 1.28 1.27
.31 1.30 1.28
                                                   D-192
                                                                                                  March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-18. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (1 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.22 0.97 0.86 (
1.68 1.27 1.11
1.99 1.45 1.25
2.23 1.58 1.35
2.43 1.69 1.43
2.91 1.93 1.60
3.37 2.14 1.75
3.73 2.31 1.86
4.04 2.44 1.95
4.66 2.69 2.12
514 2.87 2.24
5.55 3.02 2.33 ;
5.91 3.15 2.41 ;
6.38 3.32 2.51 ;
7.04 3.53 2.65 ;
7.59 3.71 2.75 ;
8.07 3.86 2.84 ;
8.51 4.00 2.92 ;
8.90 4.11 2.98 ;
).80 (
.02 (
.14
.23
.29
.44
.56
.65
.72
.85
.94
1.02
>.08
>.15
1.25 ;
2.33 ;
>.40 ;
>.45 ;
>.so ;
).77 0.72 (
).96 0.89 (
.07 0.99 (
.15 1.06
.21 1.11
.34 1.22
.44 1.31
.52 1.38
.58 1.43
.69 1.52
.77 1.58
.83 1.63
.89 1.67
.95 1.72
>.03 1.79
>.09 1.84
>.15 1.88
>.19 1.92
1.23 1.95
).69 (
).85 (
).94 (
.01 (
.05
.15
.24
.30
.34
.42
.48
.52
.56
.60
.66
.70
.74
.77
.80
).67 0.65 (
).82 0.80 (
).91 0.88 (
).96 0.94 (
.01 0.98 (
.10 1.07
.18 1.14
.23 1.19
.27 1.23
.35 1.30
.40 1.35
.44 1.39
.47 1.42 ]
.51 1.46 ]
.56 1.50 ]
.60 1.54 ]
.64 1.57 ]
.66 1.60 ]
.69 1.62 ]
).64 (
).78 (
).86 (
).92 (
).96 (
.05
.12
.17
.20
.27
.32
.35
L.38
L.42
L.46
L.50
L.53
L.55
L.57
).63 0.62 (
).77 0.76 (
).85 0.84 (
).90 0.89 (
).95 0.93 (
.03 1.01
.10 1.08
.14 1.13
.18 1.16
.25 1.23
.29 1.27
.33 1.31
.35 1.33
.39 1.37
.43 1.41
.47 1.44
.49 1.47
.52 1.49
.53 1.51
).62 (
).76 (
).83 (
).89 (
).92 (
.00 (
.07
.12
.15
.21
.26
.29
.32
.35
.39
.42
.45
.47
.49
).61 0.61 (
).75 0.74 (
).82 0.81 (
).87 0.86 (
).91 0.90 (
).99 0.98 (
.05 1.04
.10 1.08
.13 1.12
.19 1.18
.23 1.22
.27 1.25
.29 1.27
.32 1.31
.36 1.34
.39 1.37
.42 1.40
.44 1.42
.46 1.44
).60 0.60 (
).73 0.73 (
).81 0.80 (
).86 0.85 (
).89 0.89 (
).97 0.96 (
.03 1.02
.07 1.07
.11 1.10
.17 1.16
.21 1.20
.24 1.23
.26 1.25
.29 1.28
.33 1.32
.36 1.35
.38 1.37
.40 1.39
.42 1.41
).60 0.59 0.59
).73 0.72 0.72
).80 0.79 0.79
).85 0.84 0.83
).88 0.87 0.87
).96 0.95 0.94
.02 1.01 1.00
.06 1.05 1.04
.09 1.08 1.07
.15 1.14 1.13
.19 1.18 1.17
.22 1.21 1.20
.24 1.23 1.22
.27 1.26 1.25
.31 1.30 1.28
.34 1.32 1.31
.36 1.35 1.33
.38 1.36 1.35
.40 1.38 1.37
      Table 19-18. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (2 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.04 0.80 0.70 (
1.47 1.08 0.93 (
1.75 1.25 1.07 (
1.97 1.38 1.17
2.15 1.48 1.24
2.58 1.69 1.41
2.99 1.89 1.55
3.32 2.04 1.65
3.60 2.16 1.74
4.15 2.39 1.89
4.58 2.56 2.00
4.95 2.70 2.09
5.27 2,81 2.17
5.69 2.96 2.26
6.28 3.16 2.39 ;
6.77 3.33 2.48 ;
7.21 3.46 2.57 ;
7.59 3.58 2.64 ;
7.94 3.69 2.7Q '<
).64 (
).85 (
).97 (
.06 (
.12
.26
.38
.47
.53
.66
.75
.82
.88
.95
>.04
1.12
1.18
1.23 :
1.28 :
).61 0.56 (
).80 0.74 (
).91 0.84 (
).99 0.91 (
.05 0.96 (
.17 1.07
.28 1.17
.36 1.23
.42 1.28
.52 1.37
.60 1.44
.66 1.48
.71 1.53
.77 1.58
.85 1.64
.91 1.69
.96 1.73
>.01 1.77
>.04 1.79
).54 (
).71 (
).80 (
).87 (
).91 (
.02 (
.10
.16
.21
.29
.35
.39
.43
.47
.53
.57
.61
.64
.66
).52 0.50 (
).68 0.66 (
).77 0.75 (
).83 0.81 (
).88 0.85 (
).97 0.94 (
.05 1.02
.11 1.07
.15 1.11
.23 1.18
.28 1.24
.32 1.27
.35 1.30 ]
.39 1.34 ]
.44 1.39 ]
.48 1.43 ]
.52 1.46 ]
.54 1.48 ]
.57 1.51 ]
).49 (
).65 (
).73 (
).79 (
).83 (
).92 (
.00 (
.05
.09
.16
.20
.24
L.27
L.31
L.35
L.39
L.42
L.44
L.46
).49 0.48 (
).64 0.63 (
).72 0.71 (
).78 0.77 (
).82 0.81 (
).91 0.90 (
).98 0.97 (
.03 1.02
.07 1.05
.14 1.12
.18 1.17
.22 1.20
.25 1.23
.28 1.26
.33 1.31
.36 1.34
.39 1.37
.41 1.39
.43 1.41
).47 (
).62 (
).71 (
).76 (
).80 (
).89 (
).96 (
.01 (
.04
.11
.15
.19
.21
.25
.29
.32
.35
.37
.39
).47 0.46 (
).62 0.61 (
).70 0.69 (
).75 0.74 (
).79 0.78 (
).87 0.86 (
).94 0.93 (
).99 0.98 (
.03 1.01
.09 1.08
.13 1.12
.17 1.15
.19 1.18
.23 1.21
.27 1.25
.30 1.28
.32 1.31
.34 1.33
.36 1.34
).46 0.46 (
).60 0.60 (
).68 0.68 (
).74 0.73 (
).78 0.77 (
).86 0.85 (
).92 0.92 (
).97 0.96 (
.00 1.00 (
.07 1.06
.11 1.10
.14 1.13
.17 1.16
.20 1.19
.24 1.23
.27 1.26
.29 1.28
.31 1.30
.33 1.32
).45 0.45 0.45
).60 0.59 0.59
).68 0.67 0.67
).73 0.72 0.72
).77 0.76 0.76
).85 0.84 0.83
).91 0.90 0.90
).96 0.95 0.94
).99 0.98 0.98
.05 1.04 1.03
.09 1.08 1.07
.12 1.11 1.11
.15 1.14 1.13
.18 1.17 1.16
.22 1.21 1.20
.25 1.24 1.23
.27 1.26 1.25
.29 1.28 1.27
.31 1.30 1.28
                                                    D-193
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
   Table 19-18.  K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (2 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.36 1.04 0.91 (
1.85 1.33 1.14
2.18 1.52 1.28
2.43 1.65 1.38
2.64 1.76 1.46
3.14 2.00 1.63
3.64 2.22 1.79
4.03 2.38 1.90
4.35 2.51 1.98
5.01 2.76 2.15
5.53 2,95 2.27
5.97 3.11 2.37 ;
6.36 3.24 2.45 ;
6.86 3.40 2.55 ;
7.56 3.63 2.68 ;
8.15 3.81 2.79 ;
8.67 3.96 2,87 '<
9.14 4.10 2,95 :
9.55 4.22 3.02 '4
).84 (
.04 (
.16
.25
.31
.46
.58
.67
.74
.87
.96
2.03
2.09
2.17
2.27 ;
2.35 ;
2.41 ;
2.47 ;
2.52 ;
).79 0.73 (
).98 0.90 (
.09 1.00 (
.17 1.07
.22 1.12
.35 1.23
.46 1.32
.53 1.38
.59 1.43
.70 1.52
.78 1.59
.84 1.64
.89 1.68
.96 1.73
2.04 1.79
2.10 1.84
2.16 1.88
2.20 1.92
2.24 1.95
).70 (
).86 (
).95 (
.01 (
.06
.16
.24
.30
.34
.42
.48
.53
.56
.61
.66
.71
.74
.77
.80
).67 0.66 0.64 (
).83 0.80 0.79 (
).91 0.89 0.87 (
).97 0.94 0.92 (
.01 0.98 0.96 (
.11 1.07 1.05
.18 1.14 1.12
.24 1.20 1.17
.28 1.23 1.20
.35 1.30 1.27
.40 1.35 1.32
.44 1.39 1.35
.48 1.42 1.38
.52 1.46 1.42
.57 1.51 1.46
.60 1.54 1.50
.64 1.57 1.53
.66 1.60 1.55
.69 1.62 1.57
).64 0.63 (
).78 0.77 (
).85 0.84 (
).91 0.90 (
).95 0.94 (
.03 1.02
.10 1.08
.15 1.13
.18 1.17
.25 1.23
.29 1.27
.33 1.31
.35 1.33
.39 1.37
.43 1.41
.47 1.44
.49 1.47
.52 1.49
.54 1.51
).62 (
).76 (
).84 (
).89 (
).93 (
.01 (
.07
.12
.15
.21
.26
.29
.32
.35
.39
.42
.45
.47
.49
).61 0.61 (
).75 0.74 (
).82 0.81 (
).87 0.86 (
).91 0.90 (
).99 0.98 (
.05 1.04
.10 1.08
.13 1.12
.19 1.18
.23 1.22
.27 1.25
.29 1.27
.32 1.31
.36 1.34
.39 1.37
.42 1.40
.44 1.42
.46 1.44
).60 0.60 (
).74 0.73 (
).81 0.80 (
).86 0.85 (
).89 0.89 (
).97 0.96 (
.03 1.02
.07 1.07
.11 1.10
.17 1.16
.21 1.20
.24 1.23
.26 1.25
.29 1.28
.33 1.32
.36 1.35
.38 1.37
.40 1.39
.42 1.41
).60 0.59 0.59
).73 0.72 0.72
).80 0.79 0.79
).85 0.84 0.83
).88 0.87 0.87
).96 0.95 0.94
.02 1.01 1.00
.06 1.05 1.04
.09 1.08 1.07
.15 1.14 1.13
.19 1.18 1.17
.22 1.21 1.20
.24 1.23 1.22
.27 1.26 1.25
.31 1.30 1.28
.34 1.32 1.31
.36 1.35 1.33
.38 1.36 1.35
.40 1.38 1.37
    Table 19-18. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (2 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.68 1.27 1.11 1.02 (
2.23 1.58 1.35 1.23
2.61 1.78 1.49 1.35
2.91 1.93 1.60 1.44
3.15 2.04 1.68 1.50
3.73 2.31 1.86 1.65
4.31 2.55 2.03 1.78
4.76 2.73 2.14 1.87
514 2.87 2.24 1.94
5.91 3.15 2.41 2.08
6.52 3.36 2.54 2.17
7.04 3.53 2.65 2.25 ;
7.49 3.68 2.73 2.32 ;
8.07 3.86 2.84 2.40 ;
8.90 4.11 2.98 2.50 ;
9.59 4.31 3.1O 2.58 ;
10.20 4.49 3, 19 2.65 ;
10.75 4.64 3.28 2.71 2
11.25 4.77 3.35 2.76 2
).96 (
.15
.26
.34
.40
.52
.63
.71
.77
.89
.97
2.03
2.08
2.15
2.23
2.30 ;
2.35 ;
2.40 ;
2.44 ;
).89 (
.06
.15
.22
.27
.38
.47
.53
.58
.67
.74
.79
.83
.88
.95
2.00
2.04
2.08
2.11
).85 (
.01 (
.09
.15
.20
.30
.38
.43
.48
.56
.62
.66
.70
.74
.80
.84
.88
.91
.93
).82 (
).96 (
.05
.10
.14
.23
.31
.36
.40
.47
.53
.56
.60
.64
.69
.72
.76
.78
.81
).80 (
).94 (
.02 (
.07
.11
.19
.26
.31
.35
.42
.47
.50
.53
.57
.62
.65
.68
.71
.73
).78 (
).92 (
).99 (
.05
.08
.17
.23
.28
.32
.38
.43
.46
.49
.53
.57
.60
.63
.65
.67
).77 (
).90 (
).98 (
.03
.07
.14
.21
.26
.29
.35
.40
.43
.46
.49
.53
.57
.59
.62
.64
).76 (
).89 (
).97 (
.01
.05
.13
.19
.24
.27
.33
.38
.41
.44
.47
.51
.54
.57
.59
.61
).76 0.75 0.74 0.73 0.73 0.73 (
).89 0.87 0.86 0.86 0.85 0.85 (
).96 0.94 0.93 0.92 0.92 0.91 (
.00 0.99 0.98 0.97 0.96 0.96 (
.04 1.02 1.01 1.00 1.00 0.99 (
.12 1.10 1.08 1.07 1.07 1.06
.18 1.16 1.14 1.13 1.13 1.12
.22 1.20 1.19 1.17 1.17 1.16
.26 1.23 1.22 1.21 1.20 1.19
.32 1.29 1.27 1.26 1.25 1.24
.36 1.33 1.31 1.30 1.29 1.28
.39 1.36 1.34 1.33 1.32 1.31
.42 1.39 1.37 1.35 1.34 1.33
.45 1.42 1.40 1.38 1.37 1.36
.49 1.46 1.44 1.42 1.41 1.40
.52 1.49 1.46 1.45 1.44 1.43
.54 1.51 1.49 1.47 1.46 1.45
.56 1.53 1.51 1.49 1.48 1.47
.58 1.55 1.52 1.51 1.49 1.48
).72 (
).84 (
).90 (
).95 (
).98 (
.05
.11
.15
.18
.23
.27
.30
.32
.35
.38
.41
.43
.45
.46
).72
).83
).90
).94
).97
.04
.10
.14
.17
.22
.26
.28
.31
.33
.37
.40
.42
.43
.45
                                                    D-194
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
     Table 19-18. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (5 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
1.62 1.17 1.01 0.92 (
2.15 1.48 1.24 1.12
2.51 1.66 1.38 1.24
2.80 1.80 1.48 1.33
3.04 1.91 1.56 1.39
3.60 2.16 1.74 1.53
415 2.39 1.89 1.66
458 2.56 2.00 1.75
495 2.70 2.09 1.82
5.69 2.96 2.26 1.95
6.28 3.16 2.39 2.04
6.78 3.33 2.48 2.12
7.21 3.46 2.57 2.18
7.77 3.64 2.67 2.26 ;
8.55 3.87 2.81 2.35 ;
9.22 4.06 2,92 2.44 ;
9.84 4.22 3.01 2.50 ;
10.35 4.37 3.09 2.56 2
10.82 4.49 3.15 2.61 2
).86 (
.05 (
.16
.23
.29
.42
.52
.60
.66
.77
.85
.91
.96
>.03
Ml
M7
2.23
>.27
>.31 ;
).80 (
).96 (
.06
.12
.17
.28
.37
.44
.48
.58
.64
.69
.73
.78
.85
.89
.94
.97
>.oo
).76 (
).91 (
.00 (
.06
.11
.21
.29
.35
.39
.47
.53
.57
.61
.65
.71
.75
.79
.82
.84
).73 (
).88 (
).96 (
.02 (
.06
.15
.23
.28
.32
.39
.44
.48
.52
.56
.61
.65
.68
.70
.73
).71 (
).85 (
).93 (
).99 (
.03
.11
.18
.24
.27
.34
.39
.43
.46
.49
.54
.58
.61
.63
.65
).70 (
).83 (
).91 (
).96 (
.00 (
.09
.16
.20
.24
.31
.35
.39
.42
.45
.50
.53
.56
.58
.60
).68 (
).82 (
).90 (
).95 (
).99 (
.07
.14
.18
.22
.28
.33
.36
.39
.42
.47
.50
.53
.55
.57
).68 (
).81 (
).89 (
).94 (
).97 (
.05
.12
.17
.20
.26
.31
.34
.37
.40
.44
.47
.50
.52
.54
).67 0.66 0.65 0.65 0.64 0.64 (
).80 0.79 0.78 0.78 0.77 0.77 (
).88 0.86 0.85 0.85 0.84 0.84 (
).93 0.91 0.90 0.89 0.89 0.88 (
).96 0.95 0.94 0.93 0.92 0.92 (
.04 1.03 1.01 1.00 1.00 0.99 (
.11 1.09 1.08 1.07 1.06 1.05
.15 1.13 1.12 1.11 1.10 1.09
.19 1.17 1.15 1.14 1.13 1.12
.25 1.23 1.21 1.20 1.19 1.18
.29 1.27 1.25 1.24 1.23 1.22
.32 1.30 1.28 1.27 1.26 1.25
.35 1.32 1.31 1.29 1.28 1.27
.38 1.35 1.34 1.32 1.31 1.30
.42 1.39 1.37 1.36 1.35 1.34
.45 1.42 1.40 1.39 1.38 1.37
.48 1.45 1.43 1.41 1.40 1.39
.50 1.47 1.45 1.43 1.42 1.41
.52 1.49 1.46 1.45 1.44 1.43
).64 (
).76 (
).83 (
).88 (
).91 (
).98 (
.04
.08
.11
.17
.21
.24
.26
.29
.32
.35
.37
.39
.41
).63
).76
).82
).87
).90
).98
.03
.07
.11
.16
.20
.23
.25
.28
.31
.34
.36
.38
.40
   Table 19-18.  K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (5 COC,  Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.02 1.43 1.22 1.11
2.64 1.76 1.46 1.31
3.07 1.96 1.61 1.44
3.41 2.12 1.72 1.52
3.69 2.24 1.80 1.59
4.35 2.51 1.98 1.74
5.01 2.76 2.15 1.87
5.53 2,95 2.27 1.96
5.97 3.11 2.37 2.03
6.86 3.40 2.55 2.17
7.56 3.63 2.68 2.27 ;
8.15 3.81 2.79 2.35 ;
8.67 3.96 2,88 2.41 ;
9.34 4.16 2.99 2.49 ;
10.31 4.42 3.13 2.60 2
11.09 4.64 3.25 2.68 2
11.80 4.82 3.35 2.75 2
12.42 4.98 3.44 2.81 2
12.97 5.12 3.52 2,86 \
.04 (
.22
.33
.41
.47
.59
.70
.78
.84
.96
>.04
MO
M6
1.22
>.31 ;
>.37 ;
>.43 ;
>.48 ;
i.52 :
).96 (
.12
.21
.28
.33
.43
.52
.59
.64
.73
.79
.84
.88
.94
>.oo
>.05
>.09
M3
M6
).91 (
.06
.14
.20
.25
.34
.42
.48
.53
.61
.66
.71
.74
.79
.84
.89
.92
.95
.98
).87 (
.01 (
.09
.15
.19
.28
.35
.40
.44
.51
.57
.61
.64
.68
.73
.77
.80
.82
.85
).85 (
).98 (
.06
.11
.15
.23
.30
.35
.39
.46
.51
.54
.57
.61
.65
.69
.72
.74
.76
).83 (
).96 (
.04
.09
.12
.20
.27
.32
.35
.42
.46
.50
.53
.56
.60
.64
.67
.69
.71
).82 (
).95 (
.02
.07
.11
.18
.25
.29
.33
.39
.43
.47
.49
.53
.57
.60
.63
.65
.67
).81 (
).94 (
.01 (
.05
.09
.17
.23
.27
.31
.37
.41
.44
.47
.50
.54
.57
.60
.62
.64
).80 0.79 0.78 0.78 0.77 0.77 (
).93 0.91 0.90 0.89 0.89 0.88 (
).99 0.98 0.97 0.96 0.95 0.95 (
.04 1.02 1.01 1.00 1.00 0.99 (
.08 1.06 1.05 1.04 1.03 1.02
.15 1.13 1.12 1.11 1.10 1.09
.21 1.19 1.18 1.17 1.16 1.15
.26 1.23 1.22 1.21 1.20 1.19
.29 1.27 1.25 1.24 1.23 1.22
.35 1.32 1.31 1.29 1.28 1.27
.39 1.36 1.34 1.33 1.32 1.31
.42 1.39 1.37 1.36 1.35 1.34
.45 1.42 1.40 1.38 1.37 1.36
.48 1.45 1.43 1.41 1.40 1.39
.52 1.49 1.46 1.45 1.44 1.42
.55 1.52 1.49 1.48 1.46 1.45
.57 1.54 1.52 1.50 1.48 1.47
.59 1.56 1.54 1.52 1.50 1.49
.61 1.58 1.55 1.53 1.52 1.51
).76 (
).87 (
).94 (
).98 (
.01
.08
.14
.18
.21
.26
.30
.32
.35
.37
.41
.43
.46
.47
.49
).76
).87
).93
).97
.01
.07
.13
.17
.20
.25
.28
.31
.33
.36
.40
.42
.44
.46
.47
                                                   D-195
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-18. K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (5 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.43 1.69 1.43 1.29
3.15 2.04 1.68 1.50
3.65 2.27 1.84 1.63
4.04 2.44 1.95 1.72
4.37 2.57 2.04 1.79
5.14 2.87 2.24 1.94
5.91 3.15 2.41 2.08
6.52 3.36 2.54 2.17
7.04 3.54 2.65 2.25 ;
8.08 3.86 2.84 2.40 ;
8.90 4.11 2.98 2.50 ;
9.59 4.32 3, W 2.58 ;
10.20 4.49 3.19 2.65 ;
11.00 4.71 3.32 2.74 ;
12.11 5.00 3.48 2.85 '4
13.05 5.24 3.60 2.94 2
13.91 5.45 3.71 3.01 2
14.61 5.62 3.80 3.O8 :
15.31 5.78 3.89 3.13 '4
.21
.40
.50
.58
.64
.77
.89
.97
>.03
M5
1.23
1.30 :
1.35 :
1.42 :
>.si ;
>.58 ;
>.64 ;
>.69 ;
2.73 ;
.11
.27
.36
.43
.48
.58
.67
.74
.79
.88
.95
>.oo
>.04
>.09
M6
>.21 ;
1.26 :
1.29 :
1.32 ;
.05
.20
.28
.34
.39
.48
.56
.62
.66
.74
.80
.84
.88
.92
.98
1.02
>.06
>.09
1.12
.01 (
.14
.22
.27
.32
.40
.47
.53
.56
.64
.69
.72
.76
.80
.85
.88
.92
.94
.97
).98 (
.11
.18
.23
.27
.35
.42
.47
.50
.57
.62
.65
.68
.72
.76
.80
.83
.85
.88
).96 (
.08
.15
.20
.24
.32
.38
.43
.46
.53
.57
.60
.63
.67
.71
.74
.77
.79
.81
).95 (
.07
.13
.18
.22
.29
.35
.40
.43
.49
.53
.57
.59
.63
.67
.70
.72
.75
.77
).93 (
.05
.12
.16
.20
.27
.33
.38
.41
.47
.51
.54
.57
.60
.64
.67
.69
.71
.73
).92 0.91 0.90 0.89 0.89 0.88 (
.04 1.02 1.01 1.00 1.00 0.99 (
.11 1.09 1.07 1.06 1.06 1.05
.15 1.13 1.12 1.11 1.10 1.09
.19 1.17 1.15 1.14 1.13 1.12
.26 1.23 1.22 1.21 1.20 1.19
.32 1.29 1.27 1.26 1.25 1.24
.36 1.33 1.31 1.30 1.29 1.28
.39 1.36 1.34 1.33 1.32 1.31
.45 1.42 1.40 1.38 1.37 1.36
.49 1.46 1.44 1.42 1.41 1.40
.52 1.49 1.46 1.45 1.43 1.43
.54 1.51 1.49 1.47 1.46 1.45
.57 1.54 1.52 1.50 1.48 1.47
.61 1.58 1.55 1.53 1.52 1.51
.64 1.61 1.58 1.56 1.54 1.53
.67 1.63 1.60 1.58 1.57 1.56
.69 1.65 1.62 1.60 1.58 1.57
.71 1.67 1.64 1.62 1.60 1.59
).87 (
).98 (
.04
.08
.11
.18
.23
.27
.30
.35
.38
.41
.43
.46
.49
.51
.53
.55
.57
).87
).97
.03
.07
.10
.17
.22
.26
.28
.33
.37
.40
.42
.44
.47
.50
.52
.54
.55
     Table 19-18. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (10 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.15 1.48 1.24 1.12
2.80 1.80 1.48 1.33
3.24 2.01 1.63 1.45
3.60 2.16 1.74 1.53
3.89 2.28 1.82 1.60
459 2.56 2.00 1.75
5.27 2,81 2.17 1.88
5.82 3.01 2.29 1.97
6.28 3.16 2.39 2.04
7.21 3.46 2.57 2.18
7.95 3.69 2.7O 2.28 ;
8.57 3.87 2.81 2.36 ;
9.11 4.03 2,90 2.42 ;
9.81 4.22 3.01 2.50 ;
10.82 4.49 3.16 2.61 2
11.67 4.71 3.27 2,69 :
12.40 4.90 3.37 2,76 '4
13.09 5.05 3.45 2,82 '4
13.67 5.20 3.53 2.87 '4
.05 (
.23
.34
.42
.47
.60
.71
.79
.85
.96
>.04
Ml
M6
1.23
>.31 ;
>.38 ;
>.44 ;
>.48 ;
i.52 :
).96 (
.12
.22
.28
.33
.44
.53
.59
.64
.73
.79
.84
.89
.94
>.oo
>.05
MO
M3
M6
).91 (
.06
.15
.21
.25
.35
.43
.48
.53
.61
.66
.71
.74
.79
.84
.89
.92
.95
.98
).88 (
.02 (
.09
.15
.19
.28
.35
.40
.44
.52
.57
.61
.64
.68
.73
.77
.80
.82
.85
).85 (
).99 (
.06
.11
.15
.23
.30
.35
.39
.46
.51
.54
.57
.61
.65
.69
.72
.74
.76
).83 (
).96 (
.04
.09
.13
.20
.27
.32
.35
.42
.46
.50
.53
.56
.60
.64
.67
.69
.71
).82 (
).95 (
.02
.07
.11
.18
.25
.29
.33
.39
.43
.47
.49
.53
.57
.60
.63
.65
.67
).81 (
).94 (
.01 (
.05
.09
.17
.23
.27
.31
.37
.41
.44
.47
.50
.54
.57
.60
.62
.64
).80 0.79 0.78 0.78 0.77 0.77 (
).93 0.91 0.90 0.89 0.89 0.88 (
).99 0.98 0.97 0.96 0.95 0.95 (
.04 1.03 1.01 1.00 1.00 0.99 (
.08 1.06 1.05 1.04 1.03 1.03
.15 1.13 1.12 1.11 1.10 1.09
.21 1.19 1.18 1.17 1.16 1.15
.26 1.23 1.22 1.21 1.20 1.19
.29 1.27 1.25 1.24 1.23 1.22
.35 1.32 1.31 1.29 1.28 1.27
.39 1.36 1.34 1.33 1.32 1.31
.42 1.39 1.37 1.36 1.35 1.34
.45 1.42 1.40 1.38 1.37 1.36
.48 1.45 1.43 1.41 1.40 1.39
.52 1.49 1.46 1.45 1.44 1.43
.55 1.52 1.49 1.48 1.46 1.45
.57 1.54 1.52 1.50 1.48 1.47
.59 1.56 1.54 1.52 1.50 1.49
.61 1.58 1.55 1.53 1.52 1.51
).76 (
).87 (
).94 (
).98 (
.01
.08
.14
.18
.21
.26
.30
.32
.35
.37
.41
.43
.46
.47
.49
).76
).87
).93
).98
.01
.07
.13
.17
.20
.25
.28
.31
.33
.36
.40
.42
.44
.46
.47
                                                    D-196
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
  Table 19-18. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (10 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.64 1.76 1.46 1.31
3.41 2.12 1.72 1.52
3.94 2.34 1.87 1.65
4.35 2.51 1.98 1.74
4.71 2.65 2.07 1.81
5.53 2,95 2.27 1.96
6.36 3.24 2.45 2.09
7.01 3.45 2.58 2.19
7.56 3.63 2.68 2.27 ;
8.67 3.96 2.87 2.41 ;
9.55 4.22 3.Q2 2.51 ;
10.30 4.43 3.13 2.60 ;
10.94 4.60 3.23 2.67 ;
11.79 4.82 3.35 2.75 ;
12.99 5.12 3.51 2.86 '*
14.01 5.37 3.64 2,95. '<
14.89 5.58 3.75 3,03 \
15.67 5.76 3.85 3.09 :
16.41 5.91 3.92 3.15 ;
.22
.41
.52
.59
.65
.78
.89
.98
2.04
2.16
2.24
2.31 ;
2.36 ;
1.43 :
1.52 ;
2.59 ;
2.65 ;
2.70 ;
2.74 ;
.12
.28
.37
.43
.48
.59
.68
.74
.79
.88
.95
2.00
2.04
MO
2.16
1.22 ;
1.26 :
1.29 :
1.33 ;
.06
.20
.29
.34
.39
.48
.56
.62
.66
.74
.80
.84
.88
.92
.98
1.02
2.06
2.09
1.12
.01 (
.15
.22
.28
.32
.40
.48
.53
.57
.64
.69
.73
.76
.80
.85
.89
.92
.94
.97
).98 (
.11
.18
.23
.27
.35
.42
.47
.50
.57
.62
.65
.68
.72
.76
.80
.83
.85
.87
).96 (
.09
.16
.20
.24
.32
.38
.43
.46
.53
.57
.60
.63
.66
.71
.74
.77
.79
.81
).95 (
.07
.14
.18
.22
.29
.35
.40
.43
.49
.54
.57
.59
.63
.67
.70
.73
.75
.77
).94 (
.05
.12
.17
.20
.27
.33
.38
.41
.47
.51
.54
.57
.60
.64
.67
.69
.72
.73
).93 0.91 0.90 0.89 0.89 0.88 (
.04 1.02 1.01 1.00 1.00 0.99 (
.11 1.09 1.08 1.07 1.06 1.05
.15 1.13 1.12 1.11 1.10 1.09
.19 1.17 1.15 1.14 1.13 1.12
.26 1.23 1.22 1.21 1.20 1.19
.32 1.29 1.27 1.26 1.25 1.24
.36 1.33 1.31 1.30 1.29 1.28
.39 1.36 1.34 1.33 1.32 1.31
.45 1.42 1.40 1.38 1.37 1.36
.49 1.46 1.44 1.42 1.41 1.40
.52 1.49 1.46 1.45 1.44 1.43
.54 1.51 1.49 1.47 1.46 1.45
.57 1.54 1.52 1.50 1.48 1.47
.61 1.58 1.55 1.53 1.52 1.51
.64 1.61 1.58 1.56 1.54 1.53
.67 1.63 1.60 1.58 1.57 1.55
.69 1.65 1.62 1.60 1.59 1.57
.71 1.67 1.64 1.62 1.60 1.59
).87 (
).98 (
.04
.08
.11
.18
.23
.27
.30
.35
.38
.41
.43
.45
.49
.51
.54
.55
.57
).87
).97
.03
.07
.10
.17
.22
.26
.28
.33
.37
.40
.42
.44
.46
.49
.52
.54
.55
    Table 19-18.  K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (10 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.15 2.04 1.68 1.50
4.04 2.44 1.95 1.72
4.66 2.68 2.12 1.85
5.14 2.87 2.24 1.94
555 3.02 2.33 2.02
6.52 3.36 2.54 2.17
7.49 3.68 2.73 2.32 ;
8.25 3.92 2.87 2.42 ;
8.90 4.11 2.98 2.50 ;
10.21 4.49 3.2O 2.65 ;
11.23 4.77 3.35 2.76 ;
12.11 5.00 3.48 2.85 ;
12.87 5.19 3.58 2.92 ;
13.87 5.44 3.71 3.01 ;
15.28 5.79 3.88 3,13 \
16.46 6.05 4.03 3,22 :
17.48 6.30 4.14 3.30 ;
18.46 6.49 4.25 3.37 ;
19.24 6.67 4.33 3.43 ;
.40
.58
.69
.77
.83
.97
2.08
2.17
1.23
1.35 ;
2.44 ;
2.51 ;
2.57 ;
>..64 ;
2.73 ;
>.8i ;
2.87 ;
2.92 ;
2.97 ;
.27
.43
.52
.58
.63
.74
.83
.90
.95
2.04
2.11
2.16
2.20 ;
2.26 ;
2.32 ;
2.38 ;
2.42 ;
2.46 ;
2.49 ;
.20
.34
.42
.48
.52
.62
.70
.75
.80
.88
.93
.98
2.01
2.06
2.12
2.16 ;
2.20 ;
2.23 ;
2.26 ;
.14
.27
.35
.40
.44
.53
.60
.65
.69
.76
.81
.85
.88
.92
.97
2.01
2.04
2.06
2.09
.11
.23
.30
.35
.39
.47
.53
.58
.62
.68
.73
.76
.79
.83
.88
.91
.94
.96
.98
.08
.20
.27
.32
.35
.43
.49
.54
.57
.63
.67
.71
.73
.77
.81
.85
.87
.90
.92
.07
.18
.25
.29
.33
.40
.46
.50
.53
.59
.63
.67
.69
.72
.77
.80
.82
.85
.86
.05
.16
.23
.27
.31
.38
.44
.48
.51
.57
.61
.64
.66
.69
.73
.76
.79
.81
.83
.04 1.02 1.01 1.00 1.00 0.99 (
.15 1.13 1.12 1.11 1.10 1.09
.21 1.19 1.18 1.17 1.16 1.15
.26 1.23 1.22 1.21 1.20 1.19
.29 1.27 1.25 1.24 1.23 1.22
.36 1.33 1.31 1.30 1.29 1.28
.42 1.39 1.37 1.35 1.34 1.33
.46 1.43 1.41 1.39 1.38 1.37
.49 1.46 1.44 1.42 1.41 1.40
.54 1.51 1.49 1.47 1.46 1.45
.58 1.55 1.52 1.51 1.49 1.48
.61 1.58 1.55 1.53 1.52 1.51
.64 1.60 1.57 1.56 1.54 1.53
.67 1.63 1.60 1.58 1.57 1.56
.71 1.67 1.64 1.62 1.60 1.59
.74 1.70 1.66 1.64 1.63 1.61
.76 1.72 1.69 1.66 1.65 1.63
.78 1.73 1.70 1.68 1.66 1.65
.80 1.75 1.72 1.70 1.68 1.67
).98 (
.08
.14
.18
.21
.27
.32
.35
.38
.43
.46
.49
.50
.53
.57
.59
.61
.63
.64
).97
.07
.13
.17
.20
.26
.31
.34
.37
.42
.45
.47
.50
.52
.55
.58
.60
.61
.63
                                                   D-197
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
     Table 19-18. K-Multipliers for 1-of-2  Intrawell Prediction Limits on Means of Order 3 (20 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
2.80 1.80 1.48 1.33
3.60 2.16 1.74 1.53
415 2.39 1.89 1.66
458 2.56 2.00 1.75
495 2.70 2.09 1.82
5.82 3.01 2.29 1.97
6.68 3.29 2.47 2.10
7.37 3.51 2.60 2.20
7.95 3.69 2,7O 2.28 ;
9.11 4.03 2.89 2.42 ;
10.05 4.28 3.04 2.52 ;
10.81 4.49 3.16 2.61 ;
11.48 4.67 3.25 2.67 ;
12.42 4.89 3.37 2,76 ;
13.65 5.20 3.53 2,87 '*
14.77 5.45 3.66 2.98 \
15.70 5.65 3.78 3.03 ;
16.41 5.86 3.87 3.11
17.11 6.01 3.96 3.15
.23
.42
.52
.60
.66
.79
.90
.98
2.04
2.16
2.24
2.31 ;
2.37 ;
2.44 ;
1.52 ;
2.59 ;
2.65 ;
2,70 :
2,74 ;
.12
.28
.37
.44
.48
.59
.68
.74
.79
.89
.95
2.00
2.05
MO
2.16
1.22 ;
1.26 :
1.29 :
1.33 ;
.06
.21
.29
.35
.39
.48
.56
.62
.66
.74
.80
.84
.88
.92
.98
2.03
2.06
2.09
1.12
.02 (
.15
.23
.28
.32
.40
.48
.53
.57
.64
.69
.73
.76
.80
.85
.89
.92
.94
.97
).99 (
.11
.18
.23
.27
.35
.42
.47
.51
.57
.62
.65
.68
.72
.77
.80
.83
.85
.88
).96 (
.09
.16
.20
.24
.32
.38
.43
.46
.53
.57
.60
.63
.67
.71
.74
.77
.79
.81
).95 (
.07
.14
.18
.22
.29
.36
.40
.43
.49
.54
.57
.59
.63
.67
.70
.72
.75
.77
).94 (
.05
.12
.17
.20
.27
.33
.38
.41
.47
.51
.54
.57
.60
.64
.67
.69
.71
.73
).93 0.91 0.90 0.89 0.89 0.88 (
.04 1.03 1.01 1.00 1.00 0.99 (
.11 1.09 1.08 1.07 1.06 1.05
.15 1.13 1.12 1.11 1.10 1.09
.19 1.17 1.15 1.14 1.13 1.12
.26 1.23 1.22 1.21 1.20 1.19
.32 1.29 1.27 1.26 1.25 1.24
.36 1.33 1.31 1.30 1.29 1.28
.39 1.36 1.34 1.33 1.32 1.31
.45 1.42 1.40 1.38 1.37 1.36
.49 1.46 1.44 1.42 1.41 1.40
.52 1.49 1.46 1.45 1.44 1.42
.54 1.51 1.49 1.47 1.46 1.45
.57 1.54 1.52 1.50 1.48 1.47
.61 1.58 1.55 1.53 1.52 1.51
.64 1.60 1.58 1.56 1.54 1.53
.67 1.63 1.60 1.58 1.57 1.55
.69 1.65 1.62 1.60 1.59 1.57
.71 1.67 1.64 1.62 1.60 1.59
).87 (
).98 (
.04
.08
.11
.18
.23
.27
.30
.35
.38
.41
.43
.46
.49
.51
.53
.55
.57
).87
).98
.03
.07
.11
.17
.22
.26
.28
.33
.37
.40
.42
.44
.47
.50
.52
.54
.55
  Table 19-18. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (20 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3 41 2.12 1.72 1.52
4.35 2.51 1.98 1.74
5.01 2.76 2.15 1.87
5.53 2,95 2.27 1.96
5.97 3.11 2.37 2.03
7.01 3.45 2.58 2.19
8.04 3.78 2.77 2.33 ;
8.86 4.02 2,91 2.43 ;
9.55 4.22 3.O2 2.52 ;
10.96 4.60 3.23 2.67 ;
12.07 4.89 3.39 2.78 ;
13.01 5.12 3.51 2,86 \
13.83 5.32 3.62 2,94 ;
14.88 5.58 3.75 3,02 '4
16.41 5.92 3.93 3.15 ;
17.70 6.21 4.06 3.24 ;
18.75 6.45 4.19 3.33
19.69 6.65 4.28 3.38 ,
20.62 6.86 4.37 3.44
.41
.59
.70
.78
.84
.98
2.09
2.18
1.24
1.36 ;
2.45 ;
1.52 :
1.58 :
i.65 ;
2.74 ;
2.8i ;
2,87 :
2,93 :
2.97 :
.28
.43
.52
.59
.64
.74
.83
.90
.95
2.04
Ml
M6
1.20 :
1.26 ;
2.33 ;
2.38 ;
2.42 ;
2.46 ;
2.49 ;
.20
.34
.42
.48
.53
.62
.70
.75
.80
.88
.94
.98
2.02
2.06
2.12
2.16 ;
2.20 ;
2.23 ;
2.26 ;
.15
.28
.35
.40
.44
.53
.60
.65
.69
.76
.81
.85
.88
.92
.97
2.01
2.04
2.07
2.09
.11
.23
.30
.35
.39
.47
.53
.58
.62
.68
.73
.76
.79
.83
.88
.91
.94
.96
.98
.09
.20
.27
.32
.35
.43
.49
.54
.57
.63
.67
.71
.74
.77
.81
.85
.88
.90
.92
.07
.18
.25
.29
.33
.40
.46
.50
.53
.59
.64
.67
.69
.73
.77
.80
.82
.85
.87
.05
.17
.23
.27
.31
.38
.44
.48
.51
.57
.61
.64
.66
.69
.73
.77
.79
.81
.83
.04 1.02 1.01 1.00 1.00 0.99 (
.15 1.13 1.12 1.11 1.10 1.09
.21 1.19 1.18 1.17 1.16 1.15
.26 1.23 1.22 1.21 1.20 1.19
.29 1.27 1.25 1.24 1.23 1.22
.36 1.33 1.31 1.30 1.29 1.28
.42 1.39 1.37 1.35 1.34 1.33
.46 1.43 1.41 1.39 1.38 1.37
.49 1.46 1.44 1.42 1.41 1.40
.54 1.51 1.49 1.47 1.46 1.45
.58 1.55 1.52 1.51 1.49 1.48
.61 1.58 1.55 1.53 1.52 1.51
.64 1.60 1.57 1.55 1.54 1.53
.67 1.63 1.60 1.58 1.57 1.55
.71 1.67 1.64 1.62 1.60 1.59
.74 1.70 1.67 1.64 1.63 1.61
.76 1.72 1.69 1.66 1.65 1.63
.78 1.74 1.70 1.68 1.66 1.65
.80 1.76 1.72 1.70 1.68 1.67
).98 (
.08
.14
.18
.21
.27
.32
.35
.38
.43
.46
.49
.50
.53
.57
.59
.61
.63
.64
).97
.07
.13
.17
.20
.26
.31
.34
.37
.42
.45
.46
.50
.52
.55
.58
.60
.61
.63
                                                   D-198
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
    Table 19-18. K-Multipliers for 1-of-2  Intrawell Prediction Limits on Means of Order 3 (20 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4.04 2.44 1.95 1.72 1.58
5.14 2.87 2.24 1.94 1.77
5.91 3.15 2.41 2.08 1.89
6.52 3.36 2.54 2.17 1.97
7.04 3.53 2.65 2.25 2.03
8.25 3.92 2.87 2.42 2.17
9.46 4.28 3.O8 2.57 2.29
10.43 4.55 3.23 2.68 2.37 ;
11.24 4.77 3.35 2.76 2.44 ;
12.88 5.20 3.58 2.92 2.57 ;
14.18 5.52 3.75 3.04 2.66 ;
15.29 5.79 3.89 3,13 2.73 ;
16.23 6.01 4.00 3,21 2.79 ;
17.46 6.30 4.15 3.30 2.87 ;
19.22 6.68 4.34 3.43 2.97 ;
20.74 6.97 4.48 3.53 3.04 ;
22.03 7.27 4.61 3.62 3,17 :
23.20 7.50 4.72 3.69 3,16 :
24.38 7.68 4.83 3.75 3,21 '4
.43
.58
.67
.74
.79
.90
.99
2.05
Ml
1.20 :
1.27 ;
2.32 ;
2.37 ;
1.42 :
1.49 ;
2.55 ;
2.59 ;
2.63 ;
2.67 ;
.34
.48
.56
.62
.66
.75
.83
.89
.93
2.01
2.07
2.12
2.15 ;
2.20 ;
2.26 ;
2.30 ;
2.34 ;
2.37 ;
2.40 ;
.27
.40
.47
.53
.56
.65
.72
.77
.81
.88
.93
.97
2.00
2.04
2.09
2.13 ;
2.16 ;
2.19 ;
2.21 ;
.23
.35
.42
.47
.50
.58
.65
.69
.73
.79
.84
.88
.90
.94
.98
2.02
2.05
2.07 ;
2.09 ;
.20
.32
.38
.43
.46
.54
.60
.64
.67
.73
.78
.81
.84
.87
.92
.95
.98
2.00
2.02
.18
.29
.35
.40
.43
.50
.56
.60
.64
.69
.73
.77
.79
.83
.87
.90
.92
.94
.96
.16
.27
.33
.38
.41
.48
.53
.57
.61
.66
.70
.73
.76
.79
.83
.86
.88
.90
.92
.15 1.13 1.12 1.11 1.10 1.09
.26 1.23 1.22 1.21 1.20 1.19
.32 1.29 1.27 1.26 1.25 1.24
.36 1.33 1.31 1.30 1.29 1.28
.39 1.36 1.34 1.33 1.32 1.31
.46 1.43 1.41 1.39 1.38 1.37
.51 1.48 1.46 1.44 1.43 1.42
.55 1.52 1.50 1.48 1.47 1.45
.58 1.55 1.52 1.51 1.49 1.48
.64 1.60 1.57 1.56 1.54 1.53
.68 1.64 1.61 1.59 1.57 1.56
.71 1.67 1.64 1.62 1.60 1.59
.73 1.69 1.66 1.64 1.62 1.61
.76 1.72 1.69 1.66 1.65 1.64
.80 1.75 1.72 1.70 1.68 1.67
.83 1.78 1.75 1.72 1.71 1.69
.85 1.80 1.77 1.74 1.73 1.71
.88 1.82 1.79 1.77 1.74 1.73
.89 1.84 1.80 1.78 1.76 1.74
.08
.18
.23
.27
.30
.35
.40
.44
.46
.50
.54
.57
.59
.61
.64
.67
.68
.70
.72
.07
.17
.22
.26
.28
.34
.39
.42
.45
.50
.53
.55
.57
.60
.63
.65
.67
.68
.70
     Table 19-18. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (40 COC, Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
3.60 2.16 1.74 1.53
459 2.56 2.00 1.75
5.27 2,81 2.17 1.88
5.82 3.01 2.29 1.97
6.28 3.16 2.39 2.04
7.37 3.51 2.60 2.20
8.45 3.84 2,79 2.34 ;
9.31 4.08 2.93* 2.44 ;
10.03 4.28 3.04 2.52 ;
11.50 4.67 3.25 2.67 ;
12.66 4.96 3.41 2,79 '<
13.64 5.20 3.53 2.87 '<
14.49 5.40 3.64 2,94 :
15.65 5.66 3.77 3.03 ;
17.23 6.02 3.95 3.15 '",
18.59 6.29 4.08 3.25
19.69 6.53 4.20 3.33
20.78 6.73 4.31 3.40 ,
21.60 6.94 4.41 3.45 '<
.42
.60
.71
.79
.85
.98
2.10
2.18
2.25
2.37 ;
2.45 ;
2.52 ;
2.58 ;
2.65 ;
'i'.H" :
?,82 :
1,88 ;
2.93 :
?,97 :
.28
.44
.53
.59
.64
.74
.84
.90
.95
2.05
2.11
2.16
2.21 ;
2.26 ;
2.33 ;
2.38 ;
2.43 ;
2.46 ;
2.50 ;
.21
.35
.43
.48
.53
.62
.70
.76
.80
.88
.94
.98
2.02
2.06
2.12
2.16 ;
2.20 ;
2.23 ;
2.26 ;
.15
.28
.35
.40
.44
.53
.60
.65
.69
.76
.81
.85
.88
.92
.97
2.01
2.04
2.06
2.09
.11
.23
.30
.35
.39
.47
.54
.58
.62
.68
.73
.76
.79
.83
.88
.91
.94
.97
.99
.09
.20
.27
.32
.35
.43
.49
.54
.57
.63
.67
.71
.74
.77
.81
.85
.87
.90
.91
.07
.18
.25
.29
.33
.40
.46
.50
.54
.59
.64
.67
.69
.73
.77
.80
.82
.85
.87
.05
.17
.23
.27
.31
.38
.44
.48
.51
.57
.61
.64
.66
.69
.73
.76
.79
.81
.83
.04 1.03 1.01 1.00 1.00 0.99 (
.15 1.13 1.12 1.11 1.10 1.09
.21 1.19 1.18 1.17 1.16 1.15
.26 1.23 1.22 1.21 1.20 1.19
.29 1.27 1.25 1.24 1.23 1.22
.36 1.33 1.31 1.30 1.29 1.28
.42 1.39 1.37 1.35 1.34 1.33
.46 1.43 1.41 1.39 1.38 1.37
.49 1.46 1.44 1.42 1.41 1.40
.54 1.51 1.49 1.47 1.46 1.45
.58 1.55 1.52 1.51 1.49 1.48
.61 1.58 1.55 1.53 1.52 1.50
.64 1.60 1.57 1.55 1.54 1.53
.67 1.63 1.60 1.58 1.57 1.55
.71 1.67 1.64 1.62 1.60 1.59
.74 1.70 1.67 1.65 1.63 1.62
.76 1.72 1.69 1.67 1.66 1.64
.78 1.74 1.71 1.69 1.67 1.66
.80 1.76 1.73 1.71 1.69 1.68
).98 (
.08
.14
.18
.21
.27
.32
.35
.38
.43
.46
.48
.50
.53
.57
.59
.61
.63
.64
).98
.07
.13
.17
.20
.26
.31
.34
.37
.42
.45
.46
.48
.52
.55
.58
.60
.61
.62
                                                    D-199
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Intrawell K-Tables for Means
                                                                     Unified Guidance
  Table 19-18. K-Multipliers  for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (40 COC, Semi-Annual)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
4.35 2.51 1.98 1.74 1.59
5.53 2,95 2.27 1.96 1.78
6.36 3.24 2.45 2.09 1.89
7.01 3.45 2.58 2.19 1.98
7.56 3.63 2.68 2.27 2.04
8.86 4.02 2,91 2.43 2.18
10.16 4.38 3.11 2.58 2.29
11.19 4.66 3.27 2.69 2.38 ;
12.06 4.89 3.39 2.78 2.45 ;
13.81 5.32 3.62 2.94 2.58 ;
15.21 5.65 3.79 3.QS 2.67 ;
16.41 5.92 3.92 3.14 2.74 ;
17.43 6.15 4.04 3.22 2.80 ;
18.80 6.44 4.18 3.32 2,88 :
20.64 6.84 4.38 3.44 2,97 '*
22.29 7.14 4.53 3.55 3,05 \
23.65 7.42 4.66 3.63 3.11 '<
24.88 7.66 4.77 3.71 3.16 ;
25.98 7.86 4.87 3.76 3.21 ;
.43
.59
.68
.74
.79
.90
.99
2.06
Ml
1.20 :
1.27 ;
2.33 ;
2.37 ;
1.42 :
i.so ;
2.55 ;
2.59 ;
2.63 ;
2.67 ;
.34
.48
.56
.62
.66
.75
.83
.89
.94
2.02
2.07
2.12
2.15 ;
2.20 ;
2.26 ;
2.30 ;
2.34 ;
2.37 ;
2.40 ;
.28
.40
.48
.53
.57
.65
.72
.77
.81
.88
.93
.97
2.00
2.04
2.09
2.13 ;
2.16 ;
2.19 ;
2.21 ;
.23
.35
.42
.47
.50
.58
.65
.69
.73
.79
.84
.88
.90
.94
.98
2.02
2.05
2.07 ;
2.09 ;
.20
.32
.38
.43
.46
.54
.60
.64
.67
.74
.78
.81
.84
.87
.92
.95
.98
2.00
2.02
.18
.29
.36
.40
.43
.50
.56
.60
.64
.69
.74
.77
.79
.82
.86
.90
.92
.94
.97
.17
.27
.33
.38
.41
.48
.53
.57
.61
.66
.70
.73
.76
.79
.83
.86
.88
.91
.92
.15 1.13 1.12 1.11 1.10 1.09
.26 1.23 1.22 1.21 1.20 1.19
.32 1.29 1.27 1.26 1.25 1.24
.36 1.33 1.31 1.30 1.29 1.28
.39 1.36 1.34 1.33 1.32 1.31
.46 1.43 1.41 1.39 1.38 1.37
.51 1.48 1.46 1.44 1.43 1.42
.55 1.52 1.50 1.48 1.47 1.46
.58 1.55 1.52 1.51 1.49 1.48
.64 1.60 1.57 1.55 1.54 1.53
.68 1.64 1.61 1.59 1.57 1.56
.71 1.67 1.64 1.62 1.60 1.59
.73 1.69 1.66 1.64 1.62 1.61
.76 1.72 1.69 1.66 1.65 1.63
.80 1.76 1.72 1.70 1.68 1.67
.83 1.79 1.75 1.72 1.70 1.69
.85 1.81 1.78 1.75 1.73 1.71
.87 1.83 1.80 1.76 1.74 1.73
.89 1.85 1.82 1.78 1.76 1.75
.08
.18
.23
.27
.30
.35
.40
.44
.46
.50
.54
.57
.59
.61
.64
.67
.69
.70
.72
.07
.17
.22
.26
.28
.34
.39
.42
.45
.48
.53
.55
.57
.60
.62
.65
.67
.69
.70
    Table 19-18.  K-Multipliers for 1-of-2 Intrawell Prediction Limits on Means of Order 3 (40 COC, Quarterly)
   w/n
10
12
16
20
25
30
35
40
                                       45
50
60
70
80
90
100   125   150
1
2
3
4
5
8
12
16
20
30
40
50
60
75
100
125
150
175
200
514 2.87 2.24 1.94 1.77 .58
6.52 3.36 2.54 2.17 1.97 .74
7.49 3.68 2.73 2.32 2.08 .83
8.25 3.92 2.87 2.42 2.17 .90
8.90 4.11 2.98 2.50 2.23 .95
10.42 4.55 3.23 2.68 2.37 2.06
11.95 4.96 3.45 2.83 2.50 2.15
13.16 5.27 3.62 2.95 2.59 2.22 ;
14.18 5.52 3.75 3.04 2.66 2.27 ;
16.24 6.01 4.00 3,21 2.79 2.37 ;
17.88 6.37 4.18 3.33 2.89 2.44 ;
19.28 6.67 4.33 3.43 2.96 2.49 ;
20.47 6.93 4.46 3.51 3.02 2.54 ;
22.08 7.26 4.61 3.61 3.1O 2.59 ;
24.34 7.69 4.82 3.75 3.2O 2.67 ;
26.11 8.07 4.99 3.86 3,29 2.72 ;
27.89 8.37 5.13 3.95 3.35 2.77 ;
29.26 8.61 5.25 4.02 3.41 2.81 ;
30.62 8.89 5.37 4.09 3.46 2.85 ;
.48
.62
.70
.75
.80
.89
.97
2.03
2.07
2.15 ;
2.21 ;
2.26 ;
2.29 ;
2.34 ;
2.40 ;
2.44 ;
2.48 ;
2.5i ;
2.54 ;
.40
.53
.60
.65
.69
.77
.84
.89
.93
2.00
2.05
2.09
2.12 ;
2.16 ;
2.21 ;
2.25 ;
2.28 ;
2.3i ;
2.33 ;
.35
.47
.53
.58
.62
.69
.76
.80
.84
.90
.95
.98
2.01
2.05
2.09 ;
2.13 ;
2.16 ;
2.18 ;
2.20 ;
.32
.43
.49
.54
.57
.64
.70
.74
.78
.84
.88
.92
.94
.98
2.02
2.05 ;
2.08 ;
2.10 ;
2.12 ;
.29
.40
.46
.50
.53
.60
.66
.70
.73
.79
.83
.87
.89
.92
.96
2.00
2.02
2.04 ;
2.06 ;
.27
.38
.44
.48
.51
.57
.63
.67
.70
.76
.80
.83
.85
.88
.92
.95
.98
2.00
2.01
.26 1.23 1.22 1.21 1.20 1.19
.36 1.33 1.31 1.30 1.29 1.28
.42 1.39 1.37 1.35 1.34 1.33
.46 1.43 1.41 1.39 1.38 1.37
.49 1.46 1.44 1.42 1.41 1.40
.55 1.52 1.50 1.48 1.47 1.45
.61 1.57 1.55 1.53 1.51 1.50
.64 1.61 1.58 1.56 1.55 1.53
.68 1.64 1.61 1.59 1.57 1.56
.73 1.69 1.66 1.64 1.62 1.61
.77 1.72 1.69 1.67 1.66 1.64
.80 1.75 1.72 1.70 1.68 1.67
.82 1.77 1.74 1.72 1.70 1.69
.85 1.80 1.77 1.75 1.73 1.71
.89 1.84 1.80 1.78 1.76 1.75
.92 1.86 1.83 1.81 1.79 1.77
.94 1.89 1.85 1.83 1.81 1.79
.96 1.91 1.87 1.85 1.83 1.81
.98 1.93 1.90 1.86 1.84 1.83
.18
.27
.32
.35
.38
.44
.48
.52
.54
.59
.62
.64
.66
.68
.72
.74
.76
.77
.79
.17
.26
.31
.34
.37
.42
.47
.50
.53
.57
.60
.62
.64
.67
.70
.72
.74
.76
.77
                                                    D-200
                                                                                                   March 2009

-------
Appendix D.  Chapter 19 Non-Parametric Prediction Limit Significance Levels	Unified Guidance
                                       D  STATISTICAL TABLES
D.4 TABLES FROM  CHAPTER 19:  NONPARAMETRIC RETESTING PLANS
D.4.1 PLANS ON OBSERVATIONS

           TABLE 19-19 Par-Constituent Significance Levels for Non-Parametric l-of-2 Plan	D-202
           TABLE 19-20 Per-Constituent Significance Levels for Non-Parametric l-of-3 Plan	D-206
           TABLE 19-21 Per-Constituent Significance Levels for Non-Parametric l-of-4 Plan	D-210
           TABLE 19-22 Per-Constituent Significance Levels for Non-Parametric Mod. Cal. Plan	D-214
D.4.2 PLANS ON MEDIANS
           TABLE 19-23 Per-Constituent Significance Levels for Non-Param. 1-of-l Median Plan	D-219
           TABLE 19-24 Per-Constituent Significance Levels for Non-Param. l-of-2 Median Plan	D-223
                                                        D-201
                                                                                                           March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
           Table 19-19. Per-Constituent Significance Levels (a) for Non-Parametric l-of-2 Plan (PL=Xn)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
4
0.6667-1
0.1190
0.1619
0.1980
0.2290
0.3016
0.3386
0.3696
0.4080
0.4576
0.4955
0.5259
0.5509
0.5721
0.5904
0.6063
0.6330
0.6546
0.6726
0.6880
0.7012
0.7231
0.7407
0.7552
0.7674
0.7778
6
0.3571-1
0.6667-1
0.9394-1
0.1183
0.1402
0.1954
0.2255
0.2519
0.2859
0.3320
0.3690
0.3997
0.4257
0.4483
0.4680
0.4856
0.5155
0.5403
0.5613
0.5795
0.5954
0.6221
0.6438
0.6619
0.6773
0.6907
8
0.2222-1
0.4242-1
0.6094-1
0.7802-1
0.9387-1
0.1355
0.1594
0.1808
0.2094
0.2496
0.2830
0.3116
0.3363
0.3582
0.3776
0.3951
0.4256
0.4512
0.4733
0.4927
0.5098
0.5389
0.5629
0.5832
0.6007
0.6160
10
0.1515-
0.2930-
0.4258-
0.5509-
0.6691-
0.9890-1
0.1178
0.1352
0.1589
0.1932
0.2225
0.2482
0.2708
0.2911
0.3095
0.3262
0.3556
0.3809
0.4029
0.4224
0.4399
0.4699
0.4950
0.5165
0.5351
0.5516
12
0.1099-1
0.2143-1
0.3137-1
0.4087-1
0.4995-1
0.7507-1
0.9028-1
0.1045
0.1241
0.1532
0.1787
0.2014
0.2218
0.2403
0.2572
0.2727
0.3005
0.3246
0.3460
0.3651
0.3823
0.4122
0.4376
0.4595
0.4787
0.4958
16 20 25 30 35 40
0.6536-2 0.4329-2 0.2849-2 0.2016-2 0.1502-2 0.1161-2
0.1287-1 0.8564-2 0.5656-2 0.4011-2 0.2991-2 0.2316-2
0.1900-1 0.1271-1 0.8422-2 0.5984-2 0.4468-2 0.3462-2
0.2496-1 0.1677-1 0.1115- 0.7937-2 0.5934-2 0.4602-2
0.3074-1 0.2075-1 0.1384- 0.9870-2 0.7388-2 0.5735-2
0.4717-1 0.3222-1 0.2168- 0.1555- 0.1168- 0.9091-2
0.5744-1 0.3952-1 0.2674- 0.1925- 0.1449- 0.1130-
0.6721-1 0.4656-1 0.3168- 0.2287- 0.1726- 0.1347-
0.8105-1 0.5667-1 0.3885- 0.2819- 0.2134- 0.1669-
0.1022 0.7248-1 0.5025- 0.3673- 0.2794- 0.2194-
0.1214 0.8714-1 0.6102- 0.4491- 0.3433- 0.2704-
0.1390 0.1008 0.7125- 0.5276- 0.4051- 0.3201-
0.1551 0.1136 0.8097- 0.6030- 0.4649- 0.3685-
0.1701 0.1257 0.9024- 0.6757- 0.5230- 0.4157-
0.1840 0.1370 0.9910- 0.7458- 0.5793- 0.4618-
0.1970 0.1478 0.1076 0.8134- 0.6341- 0.5068-
0.2208 0.1677 0.1235 0.9422- 0.7393- 0.5938-
0.2420 0.1858 0.1383 0.1063 0.8391- 0.6770-
0.2611 0.2024 0.1521 0.1177 0.9340- 0.7568-
0.2785 0.2178 0.1650 0.1285 0.1025 0.8334-
0.2944 0.2320 0.1771 0.1387 0.1111 0.9072-
0.3226 0.2576 0.1992 0.1577 0.1274 0.1047
0.3471 0.2802 0.2191 0.1751 0.1424 0.1177
0.3686 0.3004 0.2372 0.1910 0.1564 0.1300
0.3877 0.3185 0.2537 0.2057 0.1694 0.1415
0.4049 0.3351 0.2689 0.2194 0.1817 0.1524
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                      D-202
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
          Table 19-19. Per-Constituent Significance Levels (a) for Non-Parametric l-of-2 Plan (PL=Xn)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.7541-3
0.1505-2
0.2253-2
0.2998-2
0.3739-2
0.5946-2
0.7403-2
0.8848-2
0.1099-1
0.1451-1
0.1797-1
0.2136-1
0.2469-1
0.2796-1
0.3118-1
0.3434-1
0.4051-1
0.4648-1
0.5227-1
0.5789-1
0.6335-1
0.7382-1
0.8376-1
0.9322-1
0.1022
0.1109
60
0.5288-3
0.1056-2
0.1582-2
0.2106-2
0.2628-2
0.4187-2
0.5219-2
0.6244-2
0.7772-2
0.1029-1
0.1277-1
0.1522-1
0.1764-1
0.2002-1
0.2238-1
0.2470-1
0.2927-1
0.3372-1
0.3807-1
0.4232-1
0.4648-1
0.5453-1
0.6225-1
0.6967-1
0.7682-1
0.8371-1
70
0.3912-3
0.7816-3
0.1171-2
0.1560-2
0.1948-2
0.3106-2
0.3874-2
0.4638-2
0.5779-2
0.7665-2
0.9530-2
0.1138-1
0.1320-1
0.1501-1
0.1680-1
0.1858-1
0.2207-1
0.2551-1
0.2887-1
0.3218-1
0.3543-1
0.4176-1
0.4789-1
0.5382-1
0.5957-1
0.6515-1
80
0.3011-3
0.6017-3
0.9018-3
0.1201-2
0.1500-2
0.2395-2
0.2988-2
0.3580-2
0.4463-2
0.5926-2
0.7377-2
0.8816-2
0.1024-1
0.1166-1
0.1307-1
0.1446-1
0.1722-1
0.1993-1
0.2261-1
0.2524-1
0.2784-1
0.3293-1
0.3788-1
0.4270-1
0.4741-1
0.5199-1
90
0.2389-3
0.4775-3
0.7157-3
0.9536-3
0.1191-2
0.1902-2
0.2374-2
0.2845-2
0.3549-2
0.4717-2
0.5876-2
0.7028-2
0.8173-2
0.9310-2
0.1044-1
0.1156-1
0.1379-1
0.1598-1
0.1815-1
0.2030-1
0.2242-1
0.2659-1
0.3066-1
0.3464-1
0.3854-1
0.4236-1
100
0.1941-3
0.3881-3
0.5818-3
0.7752-3
0.9685-3
0.1547-2
0.1932-2
0.2315-2
0.2889-2
0.3842-2
0.4789-2
0.5732-2
0.6669-2
0.7601-2
0.8529-2
0.9451-2
0.1128-1
0.1309-1
0.1489-1
0.1666-1
0.1842-1
0.2189-1
0.2529-1
0.2863-1
0.3191-1
0.3513-1
120
0.1355-3
0.2709-3
0.4061-3
0.5413-3
0.6764-3
0.1081-2
0.1350-2
0.1619-2
0.2021-2
0.2690-2
0.3356-2
0.4019-2
0.4680-2
0.5338-2
0.5994-2
0.6647-2
0.7947-2
0.9237-2
0.1052-1
0.1179-1
0.1305-1
0.1555-1
0.1801-1
0.2044-1
0.2284-1
0.2520-1
140
0.9989-4
0.1997-3
0.2995-3
0.3992-3
0.4989-3
0.7975-3
0.9963-3
0.1195-2
0.1492-2
0.1987-2
0.2480-2
0.2972-2
0.3462-2
0.3951-2
0.4439-2
0.4925-2
0.5893-2
0.6856-2
0.7814-2
0.8766-2
0.9713-2
0.1159-1
0.1345-1
0.1529-1
0.1711-1
0.1892-1
160
0.7668-4
0.1533-3
0.2299-3
0.3065-3
0.3831-3
0.6125-3
0.7653-3
0.9179-3
0.1147-2
0.1527-2
0.1907-2
0.2286-2
0.2664-2
0.3041-2
0.3417-2
0.3793-2
0.4541-2
0.5287-2
0.6029-2
0.6768-2
0.7504-2
0.8966-2
0.1042-1
0.1185-1
0.1328-1
0.1470-1
180
0.6071-4
0.1214-3
0.1821-3
0.2427-3
0.3033-3
0.4851-3
0.6062-3
0.7271-3
0.9084-3
0.1210-2
0.1511-2
0.1812-2
0.2112-2
0.2412-2
0.2711-2
0.3009-2
0.3605-2
0.4199-2
0.4790-2
0.5380-2
0.5967-2
0.7136-2
0.8297-2
0.9451-2
0.1060-1
0.1173-1
200
0.4926-4
0.9850-4
0.1477-3
0.1969-3
0.2462-3
0.3937-3
0.4919-3
0.5902-3
0.7374-3
0.9825-3
0.1227-2
0.1472-2
0.1716-2
0.1959-2
0.2203-2
0.2446-2
0.2931-2
0.3414-2
0.3896-2
0.4377-2
0.4857-2
0.5812-2
0.6762-2
0.7706-2
0.8646-2
0.9580-2
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                      D-203
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
          Table 19-19. Per-Constituent Significance Levels (a) for Non-Parametric l-of-2 Plan (PL=Xn-i)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
4
0.2000
0.3286
0.4190
0.4866
0.5391
0.6450
0.6911
0.7261
0.7653
0.8099
0.8398
0.8613
0.8776
0.8903
0.9006
0.9091
0.9222
0.9320
0.9395
0.9455
0.9505
0.9580
0.9635
0.9677
0.9710
0.9737
6
0.1071
0.1905
0.2576
0.3130
0.3598
0.4651
0.5163
0.5577
0.6071
0.6672
0.7103
0.7429
0.7686
0.7893
0.8064
0.8208
0.8438
0.8613
0.8751
0.8864
0.8957
0.9102
0.9211
0.9296
0.9364
0.9419
8
0.6667-1
0.1232
0.1720
0.2147
0.2524
0.3436
0.3914
0.4317
0.4820
0.5466
0.5955
0.6340
0.6652
0.6910
0.7129
0.7316
0.7621
0.7860
0.8053
0.8212
0.8345
0.8558
0.8720
0.8847
0.8951
0.9037
10
0.4545-1
0.8591-1
0.1223
0.1551
0.1851
0.2612
0.3030
0.3396
0.3867
0.4499
0.4997
0.5402
0.5739
0.6024
0.6270
0.6484
0.6840
0.7125
0.7359
0.7555
0.7722
0.7991
0.8200
0.8367
0.8505
0.8619
12
0.3297-1
0.6319-1
0.9104-1
0.1168
0.1408
0.2038
0.2397
0.2719
0.3144
0.3733
0.4214
0.4616
0.4957
0.5252
0.5510
0.5738
0.6123
0.6438
0.6700
0.6923
0.7115
0.7430
0.7678
0.7879
0.8046
0.8186
16
0.1961-1
0.3818-1
0.5582-1
0.7260-1
0.8860-1
0.1325
0.1588
0.1831
0.2163
0.2648
0.3063
0.3424
0.3743
0.4026
0.4280
0.4509
0.4909
0.5246
0.5535
0.5787
0.6008
0.6380
0.6681
0.6931
0.7143
0.7325
20
0.1299-1
0.2550-1
0.3758-1
0.4924-1
0.6052-1
0.9224-1
0.1118
0.1303
0.1561
0.1950
0.2295
0.2605
0.2884
0.3138
0.3370
0.3584
0.3965
0.4295
0.4584
0.4841
0.5070
0.5464
0.5792
0.6069
0.6308
0.6516
25
0.8547-2
0.1688-
0.2502-
0.3296-
0.4072-
0.6297-
0.7702-
0.9048-
0.1097
0.1394
0.1665
0.1914
0.2145
0.2358
0.2558
0.2744
0.3084
0.3385
0.3656
0.3901
0.4124
0.4515
0.4849
0.5138
0.5391
0.5615
30
0.6048-2
0.1199-
0.1783-
0.2356-
0.2920-
0.4556-
0.5603-
0.6617-
0.8082-
0.1039
0.1253
0.1454
0.1643
0.1821
0.1989
0.2149
0.2443
0.2711
0.2955
0.3179
0.3386
0.3756
0.4079
0.4363
0.4616
0.4844
35
0.4505-;
0.8948-;
0.1333-
0.1766-
0.2193-
0.3442-
0.4248-
0.5036-
0.6182-
0.8006-
0.9732-
0.1137
0.1293
0.1441
0.1582
0.1717
0.1971
0.2205
0.2421
0.2622
0.2810
0.3151
0.3454
0.3725
0.3970
0.4192
40
> 0.3484-2
> 0.6932-2
0.1034-
0.1372-
0.1706-
0.2688-
0.3327-
0.3953-
0.4871-
0.6344-
0.7752-
0.9101-
0.1039
0.1164
0.1283
0.1398
0.1616
0.1820
0.2010
0.2189
0.2358
0.2668
0.2947
0.3201
0.3432
0.3644
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-204
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
         Table 19-19. Per-Constituent Significance Levels (a) for Non-Parametric l-of-2 Plan (PL=Xn-i)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.2262-2
0.4509-2
0.6740-2
0.8956-2
0.1116-1
0.1767-1
0.2194-1
0.2615-1
0.3237-1
0.4247-1
0.5226-1
0.6176-1
0.7098-1
0.7994-1
0.8866-1
0.9714-1
0.1134
0.1289
0.1437
0.1577
0.1712
0.1963
0.2196
0.2410
0.2610
0.2797
60
0.1586-2
0.3165-2
0.4736-2
0.6299-2
0.7854-2
0.1248-1
0.1552-1
0.1853-1
0.2300-1
0.3032-1
0.3746-1
0.4445-1
0.5129-1
0.5799-1
0.6454-1
0.7096-1
0.8342-1
0.9540-1
0.1069
0.1181
0.1288
0.1492
0.1683
0.1863
0.2032
0.2192
70
0.1174-2
0.2343-2
0.3508-2
0.4669-2
0.5825-2
0.9270-2
0.1155-1
0.1380-1
0.1716-1
0.2268-1
0.2811-1
0.3344-1
0.3869-1
0.4384-1
0.4892-1
0.5391-1
0.6367-1
0.7313-1
0.8231-1
0.9123-1
0.9990-1
0.1165
0.1323
0.1473
0.1616
0.1753
80
0.9033-:
0.1804-;
0.2702-;
0.3598-;
0.4491-;
0.7155-;
0.8919-;
0.1067-
0.1328-
0.1759-
0.2184-
0.2603-
0.3017-
0.3425-
0.3828-
0.4226-
0.5006-
0.5768-
0.6512-
0.7238-
0.7948-
0.9320-]
0.1063
0.1189
0.1310
0.1427
90
J 0.7167-3
> 0.1432-2
> 0.2145-2
> 0.2857-2
> 0.3567-2
> 0.5688-2
> 0.7094-2
0.8494-2
0.1058-1
0.1403-1
0.1744-1
0.2082-1
0.2415-1
0.2746-1
0.3073-1
0.3396-1
0.4033-1
0.4658-1
0.5270-1
0.5870-1
0.6459-1
L 0.7604-1
0.8708-1
0.9774-1
0.1080
0.1180
100
0.5824-3
0.1164-2
0.1744-2
0.2323-2
0.2901-2
0.4629-2
0.5776-2
0.6918-2
0.8624-2
0.1145-1
0.1424-1
0.1702-1
0.1976-1
0.2249-1
0.2519-1
0.2786-1
0.3315-1
0.3835-1
0.4346-1
0.4849-1
0.5344-1
0.6310-1
0.7248-1
0.8158-1
0.9041-1
0.9900-1
120
0.4064-3
0.8124-3
0.1218-2
0.1623-2
0.2027-2
0.3237-2
0.4041-2
0.4843-2
0.6041-2
0.8029-2
0.1000-1
0.1197-1
0.1392-1
0.1586-1
0.1778-1
0.1970-1
0.2349-1
0.2724-1
0.3094-1
0.3460-1
0.3821-1
0.4532-1
0.5227-1
0.5907-1
0.6572-1
0.7222-1
140
0.2997-:
0.5990-:
0.8981-:
0.1197-;
0.1495-;
0.2389-;
0.2984-;
0.3577-;
0.4465-;
0.5939-;
0.7406-;
0.8866-;
0.1032-
0.1177-
0.1320-
0.1464-
0.1748-
0.2030-
0.2310-
0.2587-
0.2861-
0.3403-
0.3935-
0.4459-
0.4973-
0.5480-
160
5 0.2300-:
5 0.4599-:
5 0.6896-:
> 0.9191-:
> 0.1149-;
> 0.1836-;
> 0.2293-;
> 0.2749-;
> 0.3433-;
> 0.4568-;
> 0.5700-;
> 0.6827-;
0.7951-;
0.9070-;
0.1019-
0.1130-
0.1351-
0.1570-
0.1788-
0.2004-
0.2219-
0.2645-
0.3064-
0.3478-
0.3886-
0.4289-
180
J 0.1821-3
J 0.3642-3
J 0.5461-3
J 0.7279-3
> 0.9096-3
1 0.1454-2
I 0.1817-2
I 0.2179-2
I 0.2721-2
I 0.3622-2
I 0.4521-2
I 0.5418-2
I 0.6311-2
I 0.7202-2
0.8091-2
0.8977-2
0.1074-1
0.1250-1
0.1424-1
0.1597-1
0.1770-1
0.2112-1
0.2450-1
0.2785-1
0.3116-1
0.3443-1
200
0.1478-3
0.2955-3
0.4431-3
0.5907-3
0.7382-3
0.1180-2
0.1475-2
0.1769-2
0.2209-2
0.2942-2
0.3673-2
0.4403-2
0.5130-2
0.5856-2
0.6580-2
0.7303-2
0.8743-2
0.1018-1
0.1160-1
0.1302-1
0.1444-1
0.1724-1
0.2002-1
0.2278-1
0.2551-1
0.2822-1
Footnote. PL = Prediction Limit; Xn_! = 2nd largest order statistic
                                                      D-205
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
          Table 19-20.  Per-Constituent Significance Levels (a) for Non-Parametric l-of-3 Plan (PL=Xn)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
4
0.2857-1
0.5238-1
0.7283-1
0.9076-1
0.1067
0.1463
0.1678
0.1865
0.2107
0.2437
0.2704
0.2928
0.3121
0.3289
0.3438
0.3573
0.3805
0.4001
0.4171
0.4319
0.4451
0.4677
0.4865
0.5026
0.5166
0.5289
6
0.1190-
0.2273-
0.3267-
0.4187-
0.5045-
0.7320-
0.8643-
0.9846-
0.1147
0.1380
0.1580
0.1754
0.1908
0.2046
0.2172
0.2287
0.2492
0.2669
0.2826
0.2966
0.3093
0.3314
0.3502
0.3666
0.3811
0.3941
8
0.6061-;
0.1179-
0.1722-
0.2240-
0.2735-
0.4101-
0.4929-
0.5704-
0.6780-
0.8386-
0.9810-
0.1109
0.1225
0.1332
0.1431
0.1523
0.1689
0.1837
0.1970
0.2091
0.2202
0.2399
0.2570
0.2722
0.2858
0.2980
10
> 0.3497-2
0.6868-2
0.1013-1
0.1328-1
0.1633-1
0.2499-1
0.3038-1
0.3552-1
0.4279-1
0.5396-1
0.6412-1
0.7348-1
0.8215-1
0.9024-1
0.9783-1
0.1050
0.1181
0.1301
0.1410
0.1510
0.1603
0.1771
0.1920
0.2054
0.2175
0.2286
12
0.2198-;
0.4342-;
0.6435-;
0.8481-;
0.1048-
0.1624-
0.1990-
0.2342-
0.2848-
0.3640-
0.4376-
0.5065-
0.5713-
0.6326-
0.6907-
0.7461-
0.8494-
0.9445-
0.1033
0.1115
0.1192
0.1333
0.1459
0.1574
0.1680
0.1777
16
> 0.1032-;
> 0.2051-;
> 0.3056-;
> 0.4049-;
0.5031-;
0.7906-;
0.9769-;
0.1159-
0.1426-
0.1853-
0.2261-
0.2652-
0.3028-
0.3389-
0.3738-
0.4075-
0.4717-
0.5323-
0.5895-
0.6439-
0.6957-
0.7926-
0.8819-
0.9648-
0.1042
0.1115
20
> 0.5647-:
> 0.1125-;
> 0.1681-;
> 0.2233-;
> 0.2781-;
> 0.4401-;
> 0.5462-;
0.6509-;
0.8054-;
0.1056-
0.1300-
0.1537-
0.1767-
0.1991-
0.2210-
0.2423-
0.2836-
0.3231-
0.3610-
0.3975-
0.4327-
0.4996-
0.5624-
0.6216-
0.6777-
0.7311-
25
5 0.3053-3
> 0.6091-3
> 0.9117-3
> 0.1213-2
> 0.1513-2
> 0.2405-2
> 0.2994-2
> 0.3577-2
) 0.4444-2
0.5866-2
0.7262-2
0.8632-2
0.9978-2
0.1130-1
0.1260-1
0.1388-1
0.1639-1
0.1882-1
0.2118-1
0.2347-1
0.2571-1
0.3003-1
0.3415-1
0.3810-1
0.4189-1
0.4554-1
30
0.1833-3
0.3661-3
0.5483-3
0.7301-3
0.9113-3
0.1452-2
0.1810-2
0.2167-2
0.2697-2
0.3573-2
0.4438-2
0.5292-2
0.6136-2
0.6971-2
0.7796-2
0.8612-2
0.1022-1
0.1179-1
0.1333-1
0.1484-1
0.1632-1
0.1921-1
0.2200-1
0.2470-1
0.2732-1
0.2987-1
35
0.1185-3
0.2369-3
0.3550-3
0.4728-3
0.5905-3
0.9422-3
0.1176-2
0.1408-2
0.1755-2
0.2330-2
0.2900-2
0.3465-2
0.4025-2
0.4581-2
0.5132-2
0.5679-2
0.6760-2
0.7824-2
0.8874-2
0.9908-2
0.1093-1
0.1293-1
0.1488-1
0.1678-1
0.1864-1
0.2045-1
40
0.8103-4
0.1620-3
0.2428-3
0.3235-3
0.4041-3
0.6453-3
0.8056-3
0.9654-3
0.1204-2
0.1601-2
0.1995-2
0.2386-2
0.2776-2
0.3162-2
0.3547-2
0.3929-2
0.4688-2
0.5437-2
0.6179-2
0.6913-2
0.7639-2
0.9069-2
0.1047-1
0.1185-1
0.1320-1
0.1453-1
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                      D-206
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
          Table 19-20.  Per-Constituent Significance Levels (a) for Non-Parametric l-of-3 Plan (PL=Xn)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.4269-4
0.8534-4
0.1280-3
0.1706-3
0.2131-3
0.3406-3
0.4255-3
0.5102-3
0.6371-3
0.8480-3
0.1058-2
0.1268-2
0.1476-2
0.1684-2
0.1892-2
0.2098-2
0.2509-2
0.2918-2
0.3324-2
0.3727-2
0.4128-2
0.4922-2
0.5707-2
0.6482-2
0.7249-2
0.8007-2
60
0.2518-4
0.5035-4
0.7551-4
0.1007-3
0.1258-3
0.2011-3
0.2513-3
0.3015-3
0.3766-3
0.5016-3
0.6263-3
0.7507-3
0.8749-3
0.9989-3
0.1123-2
0.1246-2
0.1492-2
0.1737-2
0.1981-2
0.2224-2
0.2466-2
0.2948-2
0.3425-2
0.3900-2
0.4370-2
0.4837-2
70
0.1608-4
0.3215-4
0.4822-4
0.6429-4
0.8035-4
0.1285-3
0.1606-3
0.1926-3
0.2407-3
0.3207-3
0.4006-3
0.4804-3
0.5600-3
0.6396-3
0.7191-3
0.7984-3
0.9568-3
0.1115-2
0.1272-2
0.1429-2
0.1586-2
0.1898-2
0.2208-2
0.2517-2
0.2825-2
0.3130-2
80
0.1088-4
0.2177-4
0.3264-4
0.4352-4
0.5440-4
0.8701-4
0.1087-3
0.1305-3
0.1630-3
0.2173-3
0.2715-3
0.3256-3
0.3797-3
0.4337-3
0.4877-3
0.5416-3
0.6493-3
0.7568-3
0.8641-3
0.9712-3
0.1078-2
0.1291-2
0.1504-2
0.1715-2
0.1926-2
0.2136-2
90
0.7706-5
0.1541-4
0.2312-4
0.3082-4
0.3852-4
0.6162-4
0.7701-4
0.9240-4
0.1155-3
0.1539-3
0.1923-3
0.2307-3
0.2691-3
0.3074-3
0.3457-3
0.3840-3
0.4605-3
0.5369-3
0.6131-3
0.6893-3
0.7654-3
0.9172-3
0.1069-2
0.1220-2
0.1370-2
0.1520-2
100
0.5654-5
0.1131-4
0.1696-4
0.2261-4
0.2827-4
0.4522-4
0.5652-4
0.6782-4
0.8476-4
0.1130-3
0.1412-3
0.1694-3
0.1976-3
0.2257-3
0.2539-3
0.2820-3
0.3382-3
0.3944-3
0.4505-3
0.5066-3
0.5626-3
0.6744-3
0.7860-3
0.8974-3
0.1009-2
0.1120-2
120
0.3304-5
0.6609-5
0.9913-5
0.1322-4
0.1652-4
0.2643-4
0.3304-4
0.3964-4
0.4955-4
0.6605-4
0.8255-4
0.9905-4
0.1155-3
0.1320-3
0.1485-3
0.1650-3
0.1979-3
0.2308-3
0.2637-3
0.2966-3
0.3294-3
0.3951-3
0.4607-3
0.5262-3
0.5916-3
0.6569-3
140
0.2096-5
0.4191-5
0.6287-5
0.8382-5
0.1048-4
0.1676-4
0.2095-4
0.2514-4
0.3143-4
0.4190-4
0.5237-4
0.6283-4
0.7330-4
0.8376-4
0.9422-4
0.1047-3
0.1256-3
0.1465-3
0.1674-3
0.1883-3
0.2092-3
0.2509-3
0.2926-3
0.3343-3
0.3759-3
0.4175-3
160
0.1411-5
0.2823-5
0.4234-5
0.5645-5
0.7056-5
0.1129-4
0.1411-4
0.1693-4
0.2117-4
0.2822-4
0.3527-4
0.4232-4
0.4937-4
0.5642-4
0.6347-4
0.7052-4
0.8461-4
0.9870-4
0.1128-3
0.1269-3
0.1409-3
0.1691-3
0.1972-3
0.2253-3
0.2534-3
0.2815-3
180
0.9953-6
0.1991-5
0.2986-5
0.3981-5
0.4976-5
0.7962-5
0.9952-5
0.1194-4
0.1493-4
0.1990-4
0.2488-4
0.2985-4
0.3482-4
0.3980-4
0.4477-4
0.4974-4
0.5968-4
0.6962-4
0.7956-4
0.8950-4
0.9944-4
0.1193-3
0.1392-3
0.1590-3
0.1788-3
0.1987-3
200
0.7280-6
0.1456-5
0.2184-5
0.2912-5
0.3640-5
0.5823-5
0.7279-5
0.8735-5
0.1092-4
0.1456-4
0.1820-4
0.2183-4
0.2547-4
0.2911-4
0.3275-4
0.3639-4
0.4366-4
0.5093-4
0.5820-4
0.6548-4
0.7275-4
0.8728-4
0.1018-3
0.1163-3
0.1309-3
0.1454-3
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                      D-207
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
         Table 19-20. Per-Constituent Significance Levels (a) for Non-parametric l-of-3 Plan (PL=Xn-i)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
4
0.1143
0.1952
0.2568
0.3059
0.3464
0.4353
0.4780
0.5124
0.5536
0.6046
0.6419
0.6708
0.6940
0.7132
0.7294
0.7433
0.7661
0.7842
0.7989
0.8112
0.8216
0.8385
0.8518
0.8625
0.8713
0.8788
6
0.4762-1
0.8766-1
0.1221
0.1523
0.1791
0.2448
0.2799
0.3101
0.3485
0.3997
0.4399
0.4728
0.5004
0.5240
0.5445
0.5626
0.5932
0.6182
0.6392
0.6571
0.6727
0.6987
0.7195
0.7367
0.7512
0.7637
8
0.2424-1
0.4615-1
0.6615-1
0.8453-1
0.1015
0.1460
0.1714
0.1942
0.2244
0.2670
0.3023
0.3323
0.3584
0.3814
0.4018
0.4202
0.4521
0.4790
0.5021
0.5223
0.5402
0.5705
0.5955
0.6165
0.6346
0.6504
10
0.1399-1
0.2710-1
0.3944-1
0.5111-1
0.6218-1
0.9230-1
0.1103
0.1269
0.1496
0.1827
0.2114
0.2366
0.2591
0.2793
0.2977
0.3145
0.3443
0.3701
0.3927
0.4128
0.4309
0.4621
0.4885
0.5111
0.5308
0.5482
12
0.8791-2
0.1721-
0.2528-
0.3303-
0.4050-
0.6139-
0.7421-
0.8628-
0.1032
0.1286
0.1512
0.1716
0.1901
0.2072
0.2229
0.2375
0.2638
0.2871
0.3079
0.3266
0.3437
0.3738
0.3996
0.4221
0.4421
0.4599
16
0.4128-;
0.8162-;
0.1211-
0.1597-
0.1975-
0.3064-
0.3757-
0.4424-
0.5384-
0.6883-
0.8274-
0.9573-
0.1079
0.1194
0.1302
0.1405
0.1596
0.1771
0.1931
0.2080
0.2218
0.2468
0.2690
0.2889
0.3069
0.3233
20
> 0.2259-;
> 0.4487-;
0.6686-;
0.8856-;
0.1100-
0.1727-
0.2133-
0.2529-
0.3107-
0.4031-
0.4910-
0.5749-
0.6550-
0.7319-
0.8058-
0.8768-
0.1011
0.1137
0.1255
0.1367
0.1472
0.1667
0.1844
0.2006
0.2156
0.2295
25
> 0.1221-2
> 0.2432-2
> 0.3635-2
> 0.4828-2
0.6012-2
0.9512-2
0.1180-1
0.1406-1
0.1739-1
0.2280-1
0.2804-1
0.3312-1
0.3805-1
0.4285-1
0.4751-1
0.5206-1
0.6082-1
0.6917-1
0.7715-1
0.8480-1
0.9214-1
0.1060
0.1189
0.1310
0.1424
0.1531
30
0.7331-3
0.1463-2
0.2189-2
0.2911-2
0.3630-2
0.5767-2
0.7175-2
0.8570-2
0.1064-1
0.1403-1
0.1735-1
0.2060-1
0.2378-1
0.2690-1
0.2997-1
0.3298-1
0.3884-1
0.4450-1
0.4998-1
0.5529-1
0.6045-1
0.7032-1
0.7968-1
0.8857-1
0.9705-1
0.1052
35
0.4742-3
0.9468-3
0.1418-2
0.1887-2
0.2355-2
0.3750-2
0.4673-2
0.5590-2
0.6955-2
0.9202-2
0.1142-1
0.1360-1
0.1575-1
0.1787-1
0.1997-1
0.2203-1
0.2608-1
0.3004-1
0.3389-1
0.3766-1
0.4135-1
0.4848-1
0.5532-1
0.6190-1
0.6824-1
0.7436-1
40
0.3241-3
0.6475-3
0.9701-3
0.1292-2
0.1613-2
0.2572-2
0.3208-2
0.3841-2
0.4786-2
0.6346-2
0.7889-2
0.9417-2
0.1093-1
0.1243-1
0.1391-1
0.1537-1
0.1827-1
0.2110-1
0.2389-1
0.2663-1
0.2932-1
0.3457-1
0.3966-1
0.4459-1
0.4937-1
0.5403-1
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-208
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
         Table 19-20. Per-Constituent Significance Levels (a) for Non-parametric l-of-3 Plan (PL=Xn-i)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.1708-3
0.3413-3
0.5116-3
0.6817-3
0.8516-3
0.1360-2
0.1698-2
0.2035-2
0.2539-2
0.3375-2
0.4206-2
0.5032-2
0.5853-2
0.6669-2
0.7481-2
0.8288-2
0.9888-2
0.1147-
0.1304-
0.1458-
0.1612-
0.1914-
0.2209-
0.2500-
0.2784-
0.3064-
60
0.1007-3
0.2014-3
0.3020-3
0.4024-3
0.5029-3
0.8037-3
0.1004-2
0.1204-2
0.1503-2
0.2000-2
0.2495-2
0.2989-2
0.3481-2
0.3971-2
0.4459-2
0.4945-2
0.5913-2
0.6874-2
0.7828-2
0.8776-2
0.9717-2
0.1158-
0.1342-
0.1524-
0.1703-
0.1880-
70
0.6431-4
0.1286-3
0.1928-3
0.2571-3
0.3212-3
0.5136-3
0.6417-3
0.7697-3
0.9614-3
0.1280-2
0.1598-2
0.1916-2
0.2232-2
0.2548-2
0.2863-2
0.3177-2
0.3804-2
0.4427-2
0.5048-2
0.5665-2
0.6280-2
0.7501-2
0.8712-2
0.9912-2
0.1110-1
0.1228-1
80
0.4353-4
0.8705-4
0.1306-3
0.1740-3
0.2175-3
0.3479-3
0.4347-3
0.5214-3
0.6515-3
0.8679-3
0.1084-2
0.1300-2
0.1515-2
0.1730-2
0.1945-2
0.2159-2
0.2586-2
0.3012-2
0.3437-2
0.3860-2
0.4282-2
0.5122-2
0.5957-2
0.6786-2
0.7610-2
0.8429-2
90
0.3082-4
0.6164-4
0.9245-4
0.1233-3
0.1540-3
0.2464-3
0.3079-3
0.3694-3
0.4616-3
0.6151-3
0.7684-3
0.9215-3
0.1074-2
0.1227-2
0.1380-2
0.1532-2
0.1836-2
0.2140-2
0.2443-2
0.2745-2
0.3046-2
0.3647-2
0.4244-2
0.4840-2
0.5432-2
0.6022-2
100
0.2262-4
0.4523-4
0.6784-4
0.9045-4
0.1130-3
0.1808-3
0.2260-3
0.2711-3
0.3388-3
0.4516-3
0.5642-3
0.6768-3
0.7892-3
0.9015-3
0.1014-2
0.1126-2
0.1350-2
0.1573-2
0.1797-2
0.2019-2
0.2242-2
0.2685-2
0.3128-2
0.3568-2
0.4007-2
0.4445-2
120
0.1322-4
0.2643-4
0.3965-4
0.5286-4
0.6608-4
0.1057-3
0.1321-3
0.1585-3
0.1981-3
0.2641-3
0.3300-3
0.3959-3
0.4618-3
0.5276-3
0.5934-3
0.6592-3
0.7906-3
0.9218-3
0.1053-2
0.1184-2
0.1315-2
0.1576-2
0.1837-2
0.2097-2
0.2357-2
0.2616-2
140
0.8382-5
0.1676-4
0.2515-4
0.3353-4
0.4191-4
0.6704-4
0.8380-4
0.1006-3
0.1257-3
0.1675-3
0.2094-3
0.2512-3
0.2930-3
0.3348-3
0.3766-3
0.4184-3
0.5019-3
0.5854-3
0.6688-3
0.7521-3
0.8354-3
0.1002-2
0.1168-2
0.1334-2
0.1500-2
0.1665-2
160
0.5645-5
0.1129-4
0.1693-4
0.2258-4
0.2822-4
0.4515-4
0.5644-4
0.6773-4
0.8465-4
0.1129-3
0.1411-3
0.1692-3
0.1974-3
0.2256-3
0.2538-3
0.2819-3
0.3382-3
0.3945-3
0.4508-3
0.5070-3
0.5632-3
0.6756-3
0.7878-3
0.8999-3
0.1012-2
0.1124-2
180
0.3981-5
0.7962-5
0.1194-4
0.1592-4
0.1990-4
0.3185-4
0.3981-4
0.4777-4
0.5970-4
0.7960-4
0.9949-4
0.1194-3
0.1393-3
0.1591-3
0.1790-3
0.1989-3
0.2386-3
0.2784-3
0.3181-3
0.3578-3
0.3975-3
0.4768-3
0.5561-3
0.6353-3
0.7145-3
0.7936-3
200
0.2912-5
0.5824-5
0.8735-5
0.1165-4
0.1456-4
0.2329-4
0.2912-4
0.3494-4
0.4367-4
0.5822-4
0.7277-4
0.8732-4
0.1019-3
0.1164-3
0.1310-3
0.1455-3
0.1746-3
0.2037-3
0.2327-3
0.2618-3
0.2908-3
0.3489-3
0.4070-3
0.4650-3
0.5230-3
0.5810-3
Footnote. PL = Prediction Limit; Xn_! = 2nd largest order statistic
                                                      D-209
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
                                                       Unified Guidance
          Table 19-21. Per-Constituent Significance Levels (a) for Non-Parametric l-of-4 Plan (PL=Xn)
    w\n
10
12
16
20
25
30
35
40
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
0.1429-
0.2655-
0.3735-
0.4701-
0.5578-
0.7815-
0.9067-
0.1018
0.1165
0.1372
0.1545
0.1693
0.1823
0.1939
0.2043
0.2138
0.2306
0.2451
0.2578
0.2692
0.2794
0.2973
0.3125
0.3258
0.3375
0.3480
0.4762-;
0.9191-;
0.1334-
0.1725-
0.2096-
0.3105-
0.3710-
0.4271-
0.5045-
0.6193-
0.7203-
0.8109-
0.8931-
0.9684-
0.1038
0.1103
0.1220
0.1325
0.1419
0.1505
0.1584
0.1725
0.1849
0.1959
0.2058
0.2148
> 0.2020-;
> 0.3963-;
0.5835-;
0.7645-;
0.9397-;
0.1435-
0.1744-
0.2038-
0.2454-
0.3095-
0.3680-
0.4221-
0.4724-
0.5195-
0.5639-
0.6059-
0.6837-
0.7547-
0.8201-
0.8807-
0.9374-
0.1041
0.1133
0.1217
0.1294
0.1365
> 0.9990-3
> 0.1975-2
> 0.2930-2
> 0.3865-2
> 0.4781-2
0.7427-2
0.9115-2
0.1075-1
0.1310-1
0.1682-1
0.2030-1
0.2358-1
0.2670-1
0.2966-1
0.3249-1
0.3521-1
0.4032-1
0.4508-1
0.4954-1
0.5373-1
0.5770-1
0.6506-
0.7177-
0.7796-
0.8370-
0.8907-
0.5495-:
0.1091-;
0.1625-;
0.2152-;
0.2671-;
0.4192-;
0.5176-;
0.6137-;
0.7542-;
0.9790-;
0.1194-
0.1399-
0.1597-
0.1787-
0.1971-
0.2149-
0.2489-
0.2810-
0.3115-
0.3405-
0.3683-
0.4204-
0.4688-
0.5140-
0.5564-
0.5965-
5 0.2064-:
> 0.4114-:
> 0.6151-:
> 0.8176-:
> 0.1019-;
> 0.1615-;
> 0.2006-;
> 0.2394-;
> 0.2966-;
> 0.3901-;
0.4812-;
0.5702-;
0.6571-;
0.7423-;
0.8256-;
0.9074-;
0.1066-
0.1220-
0.1368-
0.1512-
0.1651-
0.1919-
0.2174-
0.2416-
0.2649-
0.2872-
5 0.9411-4
5 0.1879-3
5 0.2814-3
J 0.3745-3
> 0.4674-3
> 0.7441-3
> 0.9271-3
> 0.1109-2
> 0.1380-2
> 0.1826-2
> 0.2265-2
> 0.2698-2
> 0.3126-2
> 0.3548-2
> 0.3965-2
> 0.4376-2
0.5186-2
0.5977-2
0.6751-2
0.7509-2
0.8253-2
0.9700-2
0.1110-1
0.1245-1
0.1376-1
0.1503-1
0.4210-4
0.8413-4
0.1261-3
0.1680-3
0.2098-3
0.3348-3
0.4179-3
0.5006-3
0.6242-3
0.8290-3
0.1032-2
0.1234-2
0.1434-2
0.1633-2
0.1830-2
0.2026-2
0.2414-2
0.2796-2
0.3174-2
0.3548-2
0.3917-2
0.4642-2
0.5352-2
0.6047-2
0.6728-2
0.7396-2
0.2156-4
0.4311-4
0.6463-4
0.8613-4
0.1076-3
0.1719-3
0.2147-3
0.2574-3
0.3213-3
0.4275-3
0.5331-3
0.6383-3
0.7431-3
0.8474-3
0.9512-3
0.1055-2
0.1260-2
0.1464-2
0.1667-2
0.1867-2
0.2067-2
0.2461-2
0.2850-2
0.3234-2
0.3612-2
0.3986-2
0.1216-4
0.2431-4
0.3645-4
0.4859-4
0.6072-4
0.9707-4
0.1213-3
0.1454-3
0.1817-3
0.2419-3
0.3019-3
0.3618-3
0.4215-3
0.4811-3
0.5405-3
0.5998-3
0.7179-3
0.8353-3
0.9522-3
0.1069-2
0.1184-2
0.1414-2
0.1642-2
0.1868-2
0.2092-2
0.2313-2
0.7366-5
0.1473-4
0.2209-4
0.2945-4
0.3681-4
0.5886-4
0.7355-4
0.8822-4
0.1102-3
0.1468-3
0.1834-3
0.2199-3
0.2563-3
0.2926-3
0.3289-3
0.3652-3
0.4375-3
0.5095-3
0.5813-3
0.6529-3
0.7242-3
0.8662-3
0.1007-2
0.1148-2
0.1287-2
0.1426-2
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                      D-210
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
          Table 19-21.  Per-Constituent Significance Levels (a) for Non-Parametric l-of-4 Plan (PL=Xn)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.3162-5
0.6324-5
0.9485-5
0.1265-4
0.1581-4
0.2528-4
0.3160-4
0.3791-4
0.4738-4
0.6314-4
0.7890-4
0.9464-4
0.1104-3
0.1261-3
0.1418-3
0.1575-3
0.1888-3
0.2201-3
0.2514-3
0.2825-3
0.3137-3
0.3758-3
0.4378-3
0.4996-3
0.5612-3
0.6226-3
60
0.1574-5
0.3148-5
0.4721-5
0.6295-5
0.7868-5
0.1259-4
0.1573-4
0.1888-4
0.2359-4
0.3145-4
0.3931-4
0.4716-4
0.5501-4
0.6285-4
0.7069-4
0.7853-4
0.9420-4
0.1098-3
0.1255-3
0.1411-3
0.1567-3
0.1879-3
0.2191-3
0.2501-3
0.2812-3
0.3122-3
70
0.8691-6
0.1738-5
0.2607-5
0.3476-5
0.4345-5
0.6952-5
0.8689-5
0.1043-4
0.1303-4
0.1737-4
0.2171-4
0.2605-4
0.3039-4
0.3473-4
0.3907-4
0.4340-4
0.5207-4
0.6073-4
0.6939-4
0.7805-4
0.8670-4
0.1040-3
0.1213-3
0.1385-3
0.1558-3
0.1730-3
80
0.5183-6
0.1037-5
0.1555-5
0.2073-5
0.2591-5
0.4146-5
0.5182-5
0.6218-5
0.7772-5
0.1036-4
0.1295-4
0.1554-4
0.1813-4
0.2072-4
0.2331-4
0.2589-4
0.3107-4
0.3624-4
0.4141-4
0.4658-4
0.5175-4
0.6208-4
0.7241-4
0.8273-4
0.9304-4
0.1033-3
90
0.3279-6
0.6558-6
0.9837-6
0.1312-5
0.1640-5
0.2623-5
0.3279-5
0.3935-5
0.4918-5
0.6557-5
0.8196-5
0.9835-5
0.1147-4
0.1311-4
0.1475-4
0.1639-4
0.1966-4
0.2294-4
0.2621-4
0.2949-4
0.3276-4
0.3931-4
0.4585-4
0.5239-4
0.5892-4
0.6546-4
100
0.2175-6
0.4350-6
0.6524-6
0.8699-6
0.1087-5
0.1740-5
0.2175-5
0.2610-5
0.3262-5
0.4349-5
0.5436-5
0.6523-5
0.7610-5
0.8697-5
0.9784-5
0.1087-4
0.1304-4
0.1522-4
0.1739-4
0.1956-4
0.2173-4
0.2608-4
0.3042-4
0.3476-4
0.3910-4
0.4344-4
120
0.1066-6
0.2132-6
0.3198-6
0.4264-6
0.5330-6
0.8527-6
0.1066-5
0.1279-5
0.1599-5
0.2132-5
0.2665-5
0.3198-5
0.3730-5
0.4263-5
0.4796-5
0.5329-5
0.6394-5
0.7460-5
0.8525-5
0.9591-5
0.1066-4
0.1279-4
0.1492-4
0.1705-4
0.1918-4
0.2131-4
140
0.5821-7
0.1164-6
0.1746-6
0.2328-6
0.2911-6
0.4657-6
0.5821-6
0.6985-6
0.8731-6
0.1164-5
0.1455-5
0.1746-5
0.2037-5
0.2328-5
0.2619-5
0.2910-5
0.3492-5
0.4074-5
0.4656-5
0.5238-5
0.5820-5
0.6984-5
0.8147-5
0.9311-5
0.1047-4
0.1164-4
160
0.3442-7
0.6884-7
0.1033-6
0.1377-6
0.1721-6
0.2754-6
0.3442-6
0.4131-6
0.5163-6
0.6884-6
0.8605-6
0.1033-5
0.1205-5
0.1377-5
0.1549-5
0.1721-5
0.2065-5
0.2409-5
0.2754-5
0.3098-5
0.3442-5
0.4130-5
0.4818-5
0.5507-5
0.6195-5
0.6883-5
180
0.2164-7
0.4327-7
0.6491-7
0.8655-7
0.1082-6
0.1731-6
0.2164-6
0.2596-6
0.3245-6
0.4327-6
0.5409-6
0.6491-6
0.7573-6
0.8654-6
0.9736-6
0.1082-5
0.1298-5
0.1515-5
0.1731-5
0.1947-5
0.2164-5
0.2596-5
0.3029-5
0.3462-5
0.3894-5
0.4327-5
200
0.1427-7
0.2855-7
0.4282-7
0.5709-7
0.7137-7
0.1142-6
0.1427-6
0.1713-6
0.2141-6
0.2855-6
0.3568-6
0.4282-6
0.4996-6
0.5709-6
0.6423-6
0.7137-6
0.8564-6
0.9991-6
0.1142-5
0.1285-5
0.1427-5
0.1713-5
0.1998-5
0.2284-5
0.2569-5
0.2854-5
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                      D-211
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
         Table 19-21. Per-Constituent Significance Levels (a) for Non-Parametric l-of-4 Plan (PL=Xn-i)
w\n 4
1 0.7143-1
2 0.1247
3 0.1669
4 0.2017
5 0.2312
8 0.2991
10 0.3333
12 0.3619
15 0.3972
20 0.4427
25 0.4776
30 0.5057
35 0.5289
40 0.5487
45 0.5658
50 0.5807
60 0.6060
70 0.6266
80 0.6439
90 0.6587
100 0.6715
120 0.6930
140 0.7103
160 0.7247
180 0.7370
200 0.7476
6
0.2381-
0.4462-
0.6314-
0.7983-
0.9503-
0.1339
0.1557
0.1750
0.2004
0.2357
0.2649
0.2897
0.3112
0.3301
0.3470
0.3623
0.3888
0.4114
0.4309
0.4480
0.4633
0.4895
0.5113
0.5300
0.5462
0.5605
8
0.1010-
0.1950-
0.2831-
0.3660-
0.4444-
0.6571-
0.7837-
0.9006-
0.1061
0.1296
0.1500
0.1681
0.1843
0.1991
0.2125
0.2250
0.2472
0.2667
0.2840
0.2996
0.3137
0.3386
0.3599
0.3785
0.3950
0.4097
10
0.4995-2
0.9784-2
0.1439-1
0.1882-1
0.2310-1
0.3514-1
0.4258-1
0.4962-1
0.5954-1
0.7463-1
0.8825-1
0.1007
0.1121
0.1228
0.1327
0.1420
0.1590
0.1743
0.1882
0.2009
0.2127
0.2338
0.2523
0.2688
0.2837
0.2972
12 16 20 25 30 35 40
0.2747-2 0.1032-2 0.4705-3 0.2105-3 0.1078-3 0.6079-4 0.3683-4
0.5423-2 0.2052-2 0.9382-3 0.4204-3 0.2154-3 0.1215-3 0.7364-4
0.8032-2 0.3060-2 0.1403-2 0.6296-3 0.3229-3 0.1822-3 0.1104-3
0.1058- 0.4056-2 0.1865-2 0.8382-3 0.4302-3 0.2428-3 0.1472-3
0.1307- 0.5042-2 0.2324-2 0.1046-2 0.5372-3 0.3033-3 0.1839-3
0.2021- 0.7935-2 0.3686-2 0.1666-2 0.8574-3 0.4846-3 0.2940-3
0.2474- 0.9815-2 0.4582-2 0.2077-2 0.1070-2 0.6051-3 0.3673-3
0.2909- 0.1166- 0.5467-2 0.2485-2 0.1282-2 0.7254-3 0.4404-3
0.3534- 0.1436- 0.6778-2 0.3093-2 0.1598-2 0.9054-3 0.5500-3
0.4508- 0.1870- 0.8917-2 0.4094-2 0.2123-2 0.1204-2 0.7322-3
0.5411- 0.2286- 0.1100- 0.5083-2 0.2643-2 0.1502-2 0.9138-3
0.6255- 0.2686- 0.1304- 0.6058-2 0.3159-2 0.1798-2 0.1095-2
0.7048- 0.3071- 0.1504- 0.7021-2 0.3671-2 0.2092-2 0.1275-2
0.7795- 0.3443- 0.1699- 0.7973-2 0.4179-2 0.2385-2 0.1455-2
0.8504- 0.3803- 0.1889- 0.8912-2 0.4684-2 0.2677-2 0.1635-2
0.9177- 0.4152- 0.2077- 0.9841-2 0.5185-2 0.2968-2 0.1814-2
0.1043 0.4818- 0.2440- 0.1167-1 0.6177-2 0.3545-2 0.2170-2
0.1158 0.5449- 0.2791- 0.1345-1 0.7155-2 0.4117-2 0.2524-2
0.1265 0.6048- 0.3130- 0.1520-1 0.8120-2 0.4684-2 0.2876-2
0.1364 0.6618- 0.3457- 0.1692-1 0.9073-2 0.5247-2 0.3226-2
0.1457 0.7163- 0.3775- 0.1860-1 0.1001-1 0.5805-2 0.3574-2
0.1626 0.8187- 0.4384- 0.2187-1 0.1186-1 0.6907-2 0.4265-2
0.1778 0.9135- 0.4960- 0.2502-1 0.1367-1 0.7993-2 0.4948-2
0.1916 0.1002 0.5508- 0.2807-1 0.1543-1 0.9062-2 0.5624-2
0.2042 0.1085 0.6031- 0.3103-1 0.1716-1 0.1012-1 0.6293-2
0.2158 0.1163 0.6531- 0.3390-1 0.1886-1 0.1115-1 0.6956-2
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-212
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
         Table 19-21. Per-Constituent Significance Levels (a) for Non-Parametric l-of-4 Plan (PL=Xn-i)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.1581-4
0.3162-4
0.4742-4
0.6321-4
0.7900-4
0.1264-3
0.1579-3
0.1894-3
0.2367-3
0.3153-3
0.3939-3
0.4723-3
0.5506-3
0.6288-3
0.7069-3
0.7849-3
0.9405-3
0.1096-2
0.1250-2
0.1405-2
0.1559-2
0.1865-2
0.2170-2
0.2474-2
0.2775-2
0.3075-2
60
0.7869-5
0.1574-4
0.2360-4
0.3147-4
0.3933-4
0.6292-4
0.7864-4
0.9435-4
0.1179-3
0.1572-3
0.1964-3
0.2356-3
0.2747-3
0.3138-3
0.3529-3
0.3920-3
0.4700-3
0.5480-3
0.6258-3
0.7035-3
0.7810-3
0.9359-3
0.1090-2
0.1244-2
0.1398-2
0.1551-2
70
0.4345-5
0.8691-5
0.1304-4
0.1738-4
0.2172-4
0.3475-4
0.4344-4
0.5212-4
0.6514-4
0.8684-4
0.1085-3
0.1302-3
0.1519-3
0.1735-3
0.1952-3
0.2168-3
0.2601-3
0.3033-3
0.3464-3
0.3896-3
0.4327-3
0.5188-3
0.6047-3
0.6905-3
0.7761-3
0.8616-3
80
0.2591-5
0.5183-5
0.7774-5
0.1036-4
0.1296-4
0.2073-4
0.2591-4
0.3109-4
0.3886-4
0.5180-4
0.6474-4
0.7768-4
0.9061-4
0.1035-3
0.1165-3
0.1294-3
0.1552-3
0.1811-3
0.2069-3
0.2327-3
0.2584-3
0.3100-3
0.3614-3
0.4129-3
0.4642-3
0.5155-3
90
0.1640-5
0.3279-5
0.4919-5
0.6558-5
0.8197-5
0.1312-4
0.1639-4
0.1967-4
0.2459-4
0.3278-4
0.4097-4
0.4916-4
0.5735-4
0.6554-4
0.7373-4
0.8191-4
0.9828-4
0.1146-3
0.1310-3
0.1473-3
0.1637-3
0.1963-3
0.2290-3
0.2616-3
0.2942-3
0.3268-3
100
0.1087-5
0.2175-5
0.3262-5
0.4349-5
0.5437-5
0.8698-5
0.1087-4
0.1305-4
0.1631-4
0.2174-4
0.2718-4
0.3261-4
0.3804-4
0.4348-4
0.4891-4
0.5434-4
0.6520-4
0.7606-4
0.8691-4
0.9776-4
0.1086-3
0.1303-3
0.1520-3
0.1737-3
0.1953-3
0.2170-3
120
0.5330-6
0.1066-5
0.1599-5
0.2132-5
0.2665-5
0.4264-5
0.5329-5
0.6395-5
0.7994-5
0.1066-4
0.1332-4
0.1599-4
0.1865-4
0.2131-4
0.2398-4
0.2664-4
0.3197-4
0.3729-4
0.4262-4
0.4794-4
0.5327-4
0.6391-4
0.7456-4
0.8520-4
0.9583-4
0.1065-3
140
0.2911-6
0.5821-6
0.8732-6
0.1164-5
0.1455-5
0.2328-5
0.2910-5
0.3493-5
0.4366-5
0.5821-5
0.7276-5
0.8731-5
0.1019-4
0.1164-4
0.1310-4
0.1455-4
0.1746-4
0.2037-4
0.2328-4
0.2619-4
0.2910-4
0.3491-4
0.4073-4
0.4654-4
0.5236-4
0.5817-4
160
0.1721-6
0.3442-6
0.5163-6
0.6884-6
0.8605-6
0.1377-5
0.1721-5
0.2065-5
0.2582-5
0.3442-5
0.4303-5
0.5163-5
0.6023-5
0.6884-5
0.7744-5
0.8605-5
0.1033-4
0.1205-4
0.1377-4
0.1549-4
0.1721-4
0.2065-4
0.2409-4
0.2753-4
0.3097-4
0.3441-4
180
0.1082-6
0.2164-6
0.3246-6
0.4327-6
0.5409-6
0.8655-6
0.1082-5
0.1298-5
0.1623-5
0.2164-5
0.2705-5
0.3245-5
0.3786-5
0.4327-5
0.4868-5
0.5409-5
0.6491-5
0.7572-5
0.8654-5
0.9735-5
0.1082-4
0.1298-4
0.1514-4
0.1731-4
0.1947-4
0.2163-4
200
0.7137-7
0.1427-6
0.2141-6
0.2855-6
0.3568-6
0.5709-6
0.7137-6
0.8564-6
0.1071-5
0.1427-5
0.1784-5
0.2141-5
0.2498-5
0.2855-5
0.3211-5
0.3568-5
0.4282-5
0.4996-5
0.5709-5
0.6423-5
0.7136-5
0.8563-5
0.9990-5
0.1142-4
0.1284-4
0.1427-4
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-213
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
                                                       Unified Guidance
    Table 19-22. Per-Constituent Significance Levels (a) for Non-Parametric Modified California Plan (PL=Xn)
    w\n
10
12
16
20
25
30
35
40
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
0.5714-1
0.9971-1
0.1335
0.1614
0.1852
0.2402
0.2682
0.2917
0.3211
0.3593
0.3891
0.4133
0.4336
0.4510
0.4662
0.4796
0.5025
0.5214
0.5374
0.5514
0.5636
0.5842
0.6011
0.6154
0.6277
0.6384
0.2619-
0.4830-
0.6746-
0.8438-
0.9954-
0.1374
0.1580
0.1761
0.1996
0.2319
0.2581
0.2802
0.2992
0.3158
0.3307
0.3440
0.3671
0.3867
0.4036
0.4185
0.4318
0.4545
0.4734
0.4896
0.5038
0.5162
0.1414-
0.2684-
0.3838-
0.4898-
0.5879-
0.8449-
0.9925-
0.1126
0.1304
0.1558
0.1772
0.1958
0.2121
0.2267
0.2399
0.2520
0.2732
0.2916
0.3077
0.3221
0.3350
0.3575
0.3765
0.3931
0.4076
0.4206
0.8492-2
0.1638-1
0.2377-1
0.3072-1
0.3728-1
0.5507-1
0.6565-1
0.7541-1
0.8878-1
0.1084
0.1255
0.1406
0.1541
0.1665
0.1778
0.1882
0.2069
0.2233
0.2379
0.2510
0.2630
0.2841
0.3023
0.3183
0.3324
0.3452
0.5495-;
0.1071-
0.1568-
0.2043-
0.2498-
0.3762-
0.4532-
0.5255-
0.6263-
0.7777-
0.9125-
0.1034
0.1145
0.1247
0.1342
0.1431
0.1591
0.1734
0.1863
0.1981
0.2089
0.2281
0.2449
0.2598
0.2732
0.2853
> 0.2683-;
0.5289-;
0.7824-;
0.1029-
0.1270-
0.1958-
0.2391-
0.2807-
0.3401-
0.4323-
0.5173-
0.5962-
0.6701-
0.7395-
0.8051-
0.8672-
0.9825-
0.1088
0.1185
0.1275
0.1359
0.1512
0.1649
0.1772
0.1885
0.1988
> 0.1506-;
> 0.2985-;
> 0.4438-;
0.5867-;
0.7272-;
0.1136-
0.1398-
0.1653-
0.2023-
0.2610-
0.3163-
0.3687-
0.4186-
0.4662-
0.5118-
0.5555-
0.6380-
0.7148-
0.7868-
0.8545-
0.9185-
0.1037
0.1145
0.1244
0.1335
0.1421
> 0.8315-3
> 0.1654-2
> 0.2468-2
> 0.3274-2
> 0.4071-2
0.6416-2
0.7943-2
0.9442-2
0.1164-1
0.1519-1
0.1859-1
0.2187-1
0.2504-1
0.2811-1
0.3107-1
0.3395-1
0.3947-1
0.4470-1
0.4968-1
0.5442-1
0.5897-1
0.6752-1
0.7545-1
0.8286-1
0.8981-1
0.9637-1
0.5067-3
0.1010-2
0.1510-2
0.2006-2
0.2499-2
0.3959-2
0.4916-2
0.5862-2
0.7260-2
0.9535-2
0.1175-1
0.1390-1
0.1600-1
0.1805-1
0.2006-1
0.2201-1
0.2580-1
0.2944-1
0.3295-1
0.3633-1
0.3959-1
0.4581-1
0.5167-1
0.5721-1
0.6248-1
0.6749-1
0.3313-3
0.6610-3
0.9892-3
0.1316-2
0.1641-2
0.2607-2
0.3244-2
0.3876-2
0.4813-2
0.6349-2
0.7854-2
0.9330-2
0.1078-1
0.1220-1
0.1360-1
0.1497-1
0.1766-1
0.2025-1
0.2278-1
0.2523-1
0.2762-1
0.3221-1
0.3658-1
0.4077-1
0.4478-1
0.4863-1
0.2284-3
0.4559-3
0.6827-3
0.9088-3
0.1134-2
0.1805-2
0.2249-2
0.2691-2
0.3347-2
0.4428-2
0.5492-2
0.6542-2
0.7576-2
0.8596-2
0.9603-2
0.1060-1
0.1255-1
0.1445-1
0.1631-1
0.1812-1
0.1990-1
0.2335-1
0.2667-1
0.2986-1
0.3295-1
0.3594-1
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                      D-214
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
    Table 19-22. Per-Constituent Significance Levels (a) for Non-Parametric Modified California Plan (PL=Xn)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.1217-3
0.2432-3
0.3645-3
0.4856-3
0.6064-3
0.9674-3
0.1207-2
0.1446-2
0.1802-2
0.2392-2
0.2977-2
0.3556-2
0.4131-2
0.4700-2
0.5266-2
0.5826-2
0.6934-2
0.8025-2
0.9100-2
0.1016-
0.1120-
0.1325-
0.1524-
0.1719-
0.1909-
0.2094-
60
0.7240-4
0.1447-3
0.2169-3
0.2891-3
0.3611-3
0.5768-3
0.7202-3
0.8632-3
0.1077-2
0.1432-2
0.1785-2
0.2136-2
0.2485-2
0.2832-2
0.3178-2
0.3521-2
0.4203-2
0.4878-2
0.5546-2
0.6207-2
0.6863-2
0.8155-2
0.9424-2
0.1067-1
0.1190-1
0.1310-1
70
0.4650-4
0.9296-4
0.1394-3
0.1858-3
0.2321-3
0.3710-3
0.4633-3
0.5556-3
0.6937-3
0.9231-3
0.1152-2
0.1379-2
0.1606-2
0.1832-2
0.2058-2
0.2282-2
0.2729-2
0.3172-2
0.3612-2
0.4050-2
0.4484-2
0.5344-2
0.6194-2
0.7032-2
0.7861-2
0.8680-2
80
0.3161-4
0.6321-4
0.9479-4
0.1264-3
0.1579-3
0.2524-3
0.3154-3
0.3782-3
0.4724-3
0.6291-3
0.7853-3
0.9411-3
0.1097-2
0.1252-2
0.1406-2
0.1560-2
0.1868-2
0.2173-2
0.2478-2
0.2780-2
0.3081-2
0.3680-2
0.4272-2
0.4860-2
0.5442-2
0.6018-2
90
0.2246-4
0.4492-4
0.6736-4
0.8980-4
0.1122-3
0.1795-3
0.2242-3
0.2690-3
0.3360-3
0.4476-3
0.5590-3
0.6701-3
0.7810-3
0.8918-3
0.1002-2
0.1113-2
0.1333-2
0.1552-2
0.1770-2
0.1988-2
0.2205-2
0.2636-2
0.3064-2
0.3490-2
0.3912-2
0.4332-2
100
0.1653-4
0.3305-4
0.4957-4
0.6608-4
0.8259-4
0.1321-3
0.1651-3
0.1980-3
0.2474-3
0.3297-3
0.4118-3
0.4938-3
0.5757-3
0.6574-3
0.7391-3
0.8206-3
0.9833-3
0.1146-2
0.1307-2
0.1469-2
0.1630-2
0.1950-2
0.2269-2
0.2586-2
0.2902-2
0.3216-2
120
0.9700-5
0.1940-4
0.2910-4
0.3879-4
0.4848-4
0.7755-4
0.9693-4
0.1163-3
0.1453-3
0.1937-3
0.2420-3
0.2903-3
0.3385-3
0.3867-3
0.4348-3
0.4830-3
0.5790-3
0.6750-3
0.7707-3
0.8664-3
0.9618-3
0.1152-2
0.1342-2
0.1531-2
0.1720-2
0.1908-2
140
0,6170-5
0.1234-4
0.1851-4
0.2468-4
0.3084-4
0.4934-4
0.6167-4
0.7400-4
0.9248-4
0.1233-3
0.1541-3
0.1848-3
0.2156-3
0.2463-3
0.2770-3
0.3077-3
0.3690-3
0.4303-3
0.4915-3
0.5526-3
0.6136-3
0.7356-3
0.8572-3
0.9786-3
0.1100-2
0.1221-2
160
0.4165-5
0.8330-5
0.1249-4
0.1666-4
0.2082-4
0.3331-4
0.4164-4
0.4996-4
0.6244-4
0.8324-4
0.1040-3
0.1248-3
0.1456-3
0.1664-3
0.1871-3
0.2079-3
0.2493-3
0.2908-3
0.3322-3
0.3736-3
0.4149-3
0.4975-3
0.5800-3
0.6624-3
0.7446-3
0.8268-3
180
0.2943-5
0.5885-5
0.8827-5
0.1177-4
0.1471-4
0.2354-4
0.2942-4
0.3530-4
0.4412-4
0.5882-4
0.7352-4
0.8821-4
0.1029-3
0.1176-3
0.1323-3
0.1469-3
0.1763-3
0.2056-3
0.2349-3
0.2642-3
0.2935-3
0.3520-3
0.4104-3
0.4688-3
0.5271-3
0.5854-3
200
0.2155-5
0.4311-5
0.6466-5
0.8621-5
0.1078-4
0.1724-4
0.2155-4
0.2586-4
0.3232-4
0.4309-4
0.5386-4
0.6462-4
0.7538-4
0.8615-4
0.9690-4
0.1077-3
0.1292-3
0.1507-3
0.1722-3
0.1936-3
0.2151-3
0.2580-3
0.3009-3
0.3438-3
0.3866-3
0.4293-3
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                      D-215
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
   Table 19-22. Per-Constituent Significance Levels (a) for Non-Parametric Modified California Plan (PL=Xn-i)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
4
0.2000
0.3182
0.3981
0.4567
0.5019
0.5932
0.6335
0.6647
0.7004
0.7422
0.7715
0.7934
0.8105
0.8244
0.8358
0.8455
0.8611
0.8732
0.8829
0.8909
0.8976
0.9083
0.9165
0.9231
0.9285
0.9330
6
0.9524-1
0.1663
0.2221
0.2677
0.3059
0.3920
0.4343
0.4689
0.5107
0.5632
0.6022
0.6327
0.6574
0.6780
0.6954
0.7104
0.7352
0.7549
0.7710
0.7845
0.7961
0.8149
0.8297
0.8417
0.8516
0.8601
8
0.5253-1
0.9619-1
0.1334
0.1658
0.1943
0.2635
0.3000
0.3313
0.3707
0.4228
0.4635
0.4964
0.5239
0.5474
0.5677
0.5855
0.6155
0.6400
0.6604
0.6778
0.6929
0.7179
0.7379
0.7543
0.7682
0.7801
10
0.3197-1
0.6018-1
0.8541-1
0.1082
0.1290
0.1821
0.2116
0.2377
0.2717
0.3185
0.3566
0.3884
0.4157
0.4393
0.4602
0.4789
0.5108
0.5374
0.5600
0.5796
0.5968
0.6257
0.6492
0.6689
0.6857
0.7002
12
0.2088-
0.3998-
0.5760-
0.7393-
0.8916-
0.1295
0.1528
0.1739
0.2022
0.2424
0.2761
0.3051
0.3304
0.3528
0.3729
0.3910
0.4227
0.4496
0.4728
0.4931
0.5112
0.5421
0.5676
0.5893
0.6079
0.6243
16
0.1032-
0.2014-
0.2950-
0.3846-
0.4705-
0.7085-
0.8534-
0.9889-
0.1177
0.1458
0.1705
0.1926
0.2126
0.2308
0.2475
0.2630
0.2907
0.3150
0.3366
0.3559
0.3735
0.4042
0.4304
0.4531
0.4731
0.4909
20
0.5835-;
0.1149-
0.1698-
0.2232-
0.2751-
0.4230-
0.5157-
0.6042-
0.7300-
0.9234-
0.1100
0.1262
0.1412
0.1552
0.1683
0.1806
0.2032
0.2234
0.2419
0.2587
0.2742
0.3020
0.3262
0.3477
0.3669
0.3843
25
> 0.3242-2
0.6424-2
0.9550-2
0.1262-1
0.1564-1
0.2440-1
0.3001-1
0.3545-1
0.4333-1
0.5576-1
0.6742-1
0.7840-1
0.8880-1
0.9866-1
0.1080
0.1170
0.1338
0.1492
0.1636
0.1770
0.1895
0.2124
0.2329
0.2514
0.2684
0.2839
30
0.1984-2
0.3944-2
0.5881-2
0.7795-2
0.9688-2
0.1524-1
0.1885-1
0.2238-1
0.2754-1
0.3582-1
0.4373-1
0.5131-1
0.5858-1
0.6558-1
0.7232-1
0.7882-1
0.9119-1
0.1028
0.1138
0.1241
0.1339
0.1522
0.1689
0.1843
0.1985
0.2118
35
0.1301-2
0.2591-2
0.3871-2
0.5140-2
0.6400-2
0.1012-1
0.1255-1
0.1495-1
0.1848-1
0.2421-1
0.2974-1
0.3510-1
0.4030-1
0.4535-1
0.5026-1
0.5504-1
0.6422-1
0.7296-1
0.8130-1
0.8928-1
0.9693-1
0.1114
0.1248
0.1373
0.1490
0.1601
40
0.8987-3
0.1792-2
0.2680-2
0.3563-2
0.4441-2
0.7046-2
0.8758-2
0.1045-1
0.1296-1
0.1705-1
0.2104-1
0.2494-1
0.2874-1
0.3246-1
0.3610-1
0.3967-1
0.4658-1
0.5322-1
0.5962-1
0.6580-1
0.7177-1
0.8315-1
0.9386-1
0.1040
0.1136
0.1227
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-216
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
   Table 19-22. Per-Constituent Significance Levels (a) for Non-Parametric Modified California Plan (PL=Xn-i)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.4806-:
0.9597-:
0.1437-;
0.1913-;
0.2387-;
0.3801-;
0.4736-;
0.5665-;
0.7048-;
0.9325-;
0.1157-
0.1378-
0.1596-
0.1810-
0.2022-
0.2231-
0.2642-
0.3041-
0.3432-
0.3813-
0.4185-
0.4906-
0.5598-
0.6262-
0.6902-
0.7520-
60
5 0.2864-:
J 0.5723-:
> 0.8576-:
> 0.1142-;
> 0.1426-;
> 0.2275-;
> 0.2838-;
> 0.3399-;
> 0.4236-;
> 0.5621-;
0.6992-;
0.8351-;
0.9696-;
0.1103-
0.1235-
0.1366-
0.1625-
0.1879-
0.2129-
0.2375-
0.2617-
0.3091-
0.3551-
0.3998-
0.4433-
0.4856-
70
5 0.1842-:
5 0.3682-:
5 0.5520-:
> 0.7355-:
> 0.9187-:
> 0.1467-;
> 0.1831-;
> 0.2195-;
> 0.2738-;
> 0.3639-;
> 0.4533-;
> 0.5423-;
> 0.6306-;
0.7184-;
0.8057-;
0.8924-;
0.1064-
0.1234-
0.1402-
0.1568-
0.1732-
0.2055-
0.2372-
0.2681-
0.2985-
0.3283-
80
5 0.1254-3
J 0.2507-3
J 0.3759-3
5 0.5010-3
5 0.6259-3
> 0.1000-2
> 0.1249-2
> 0.1497-2
> 0.1869-2
> 0.2486-2
> 0.3101-2
> 0.3713-2
> 0.4322-2
> 0.4928-2
> 0.5531-2
> 0.6132-2
0.7327-2
0.8511-2
0.9685-2
0.1085-1
0.1200-1
0.1429-1
0.1653-1
0.1875-1
0.2093-1
0.2308-1
90
0.8919-4
0.1783-3
0.2674-3
0.3564-3
0.4454-3
0.7119-3
0.8892-3
0.1066-2
0.1332-2
0.1773-2
0.2212-2
0.2650-2
0.3086-2
0.3522-2
0.3955-2
0.4388-2
0.5248-2
0.6103-2
0.6953-2
0.7798-2
0.8637-2
0.1030-
0.1195-
0.1357-
0.1518-
0.1677-
100
0.6568-4
0.1313-3
0.1969-3
0.2625-3
0.3281-3
0.5245-3
0.6553-3
0.7860-3
0.9817-3
0.1307-2
0.1632-2
0.1956-2
0.2279-2
0.2602-2
0.2923-2
0.3244-2
0.3883-2
0.4520-2
0.5153-2
0.5783-2
0.6411-2
0.7657-2
0.8891-2
0.1012-1
0.1133-1
0.1253-1
120
0.3859-4
0.7716-4
0.1157-3
0.1543-3
0.1928-3
0.3084-3
0.3853-3
0.4623-3
0.5776-3
0.7695-3
0.9612-3
0.1153-2
0.1344-2
0.1534-2
0.1725-2
0.1915-2
0.2295-2
0.2673-2
0.3051-2
0.3427-2
0.3802-2
0.4550-2
0.5293-2
0.6032-2
0.6766-2
0.7497-2
140
0.2457-4
0.4913-4
0.7368-4
0.9823-4
0.1228-3
0.1964-3
0.2454-3
0.2945-3
0.3680-3
0.4904-3
0.6127-3
0.7349-3
0.8569-3
0.9789-3
0.1101-2
0.1222-2
0.1465-2
0.1708-2
0.1950-2
0.2192-2
0.2433-2
0.2914-2
0.3393-2
0.3871-2
0.4347-2
0.4821-2
160
0.1659-4
0.3318-4
0.4977-4
0.6635-4
0.8294-4
0.1327-3
0.1658-3
0.1990-3
0.2486-3
0.3314-3
0.4141-3
0.4968-3
0.5794-3
0.6619-3
0.7444-3
0.8269-3
0.9916-3
0.1156-2
0.1320-2
0.1484-2
0.1648-2
0.1975-2
0.2302-2
0.2627-2
0.2951-2
0.3275-2
180
0.1173-
0.2345-
0.3518-
0.4690-
0.5862-
0.9379-
0.1172-3
0.1407-3
0.1758-3
0.2343-3
0.2928-3
0.3513-3
0.4098-3
0.4682-3
0.5266-3
0.5850-3
0.7017-3
0.8182-3
0.9347-3
0.1051-2
0.1167-2
0.1399-2
0.1631-2
0.1862-2
0.2093-2
0.2324-2
200
0.8593-5
0.1718-
0.2578-
0.3437-
0.4296-
0.6873-
0.8590-
0.1031-3
0.1288-3
0.1717-3
0.2146-3
0.2575-3
0.3004-3
0.3432-3
0.3861-3
0.4289-3
0.5145-3
0.6000-3
0.6855-3
0.7709-3
0.8563-3
0.1027-2
0.1197-2
0.1367-2
0.1537-2
0.1707-2
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-217
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels	Unified Guidance



D.4.2 PLANS ON MEDIANS OF ORDER 3
                                                   D-218
                                                                                                   March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
     Table 19-23. Per-Constituent Significance Levels (a) for Non-Parametric 1-of-l Plan for Median (PL=Xn)
w/n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
4
0.1429
0.2333
0.2979
0.3473
0.3867
0.4704
0.5094
0.5404
0.5770
0.6218
0.6543
0.6794
0.6996
0.7162
0.7303
0.7424
0.7623
0.7781
0.7911
0.8019
0.8112
0.8264
0.8383
0.8481
0.8562
0.8632
6
0.8333-1
0.1455
0.1945
0.2347
0.2686
0.3458
0.3841
0.4157
0.4544
0.5035
0.5405
0.5698
0.5938
0.6139
0.6312
0.6462
0.6711
0.6912
0.7079
0.7220
0.7342
0.7542
0.7702
0.7833
0.7944
0.8039
8
0.5455-1
0.9890-1
0.1361
0.1681
0.1961
0.2630
0.2980
0.3277
0.3650
0.4139
0.4520
0.4828
0.5084
0.5303
0.5492
0.5658
0.5939
0.6167
0.6359
0.6523
0.6665
0.6902
0.7092
0.7249
0.7382
0.7497
10
0.3846-1
0.7143-1
0.1002
0.1258
0.1487
0.2057
0.2367
0.2636
0.2982
0.3448
0.3821
0.4128
0.4388
0.4613
0.4809
0.4984
0.5280
0.5526
0.5733
0.5912
0.6068
0.6330
0.6543
0.6721
0.6872
0.7003
12
0.2857-1
0.5392-1
0.7669-1
0.9735-1
0.1162
0.1647
0.1918
0.2158
0.2473
0.2907
0.3262
0.3560
0.3816
0.4039
0.4236
0.4413
0.4716
0.4970
0.5187
0.5375
0.5540
0.5820
0.6049
0.6242
0.6407
0.6550
16
0.1754-1
0.3377-1
0.4885-1
0.6295-1
0.7619-1
0.1116
0.1323
0.1512
0.1767
0.2132
0.2441
0.2708
0.2943
0.3153
0.3341
0.3511
0.3810
0.4065
0.4286
0.4481
0.4655
0.4952
0.5200
0.5411
0.5594
0.5754
20
0.1186-1
0.2308-1
0.3372-1
0.4386-1
0.5353-1
0.8012-1
0.9614-1
0.1110
0.1315
0.1618
0.1882
0.2115
0.2324
0.2513
0.2685
0.2844
0.3125
0.3369
0.3585
0.3777
0.3949
0.4249
0.4503
0.4721
0.4912
0.5081
25
0.7937-2
0.1557-
0.2293-
0.3003-
0.3690-
0.5621-
0.6815-
0.7943-
0.9527-
0.1192
0.1407
0.1601
0.1779
0.1942
0.2092
0.2232
0.2486
0.2710
0.2910
0.3091
0.3256
0.3547
0.3797
0.4015
0.4207
0.4380
30
0.5682-2
0.1120-1
0.1658-1
0.2181-1
0.2691-1
0.4147-1
0.5062-1
0.5937-1
0.7184-1
0.9105-1
0.1086
0.1248
0.1397
0.1537
0.1667
0.1790
0.2014
0.2216
0.2398
0.2566
0.2719
0.2993
0.3232
0.3443
0.3632
0.3802
35
0.4267-2
0.8443-2
0.1253-1
0.1653-1
0.2046-1
0.3178-1
0.3898-1
0.4593-1
0.5592-1
0.7154-1
0.8604-1
0.9956-1
0.1122
0.1242
0.1354
0.1461
0.1658
0.1838
0.2003
0.2155
0.2296
0.2551
0.2776
0.2977
0.3158
0.3323
40
0.3322-2
0.6588-2
0.9798-2
0.1296-
0.1606-
0.2509-
0.3089-
0.3652-
0.4467-
0.5756-
0.6965-
0.8106-
0.9185-
0.1021
0.1118
0.1211
0.1385
0.1545
0.1693
0.1831
0.1959
0.2194
0.2404
0.2593
0.2764
0.2922
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                     D-219
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
     Table 19-23. Per-Constituent Significance Levels (a) for Non-Parametric 1-of-l Plan for Median (PL=Xn)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.2177-2
0.4329-2
0.6456-2
0.8560-2
0.1064-1
0.1675-1
0.2071-1
0.2460-1
0.3028-1
0.3939-1
0.4809-1
0.5642-1
0.6441-1
0.7209-1
0.7948-1
0.8661-1
0.1001
0.1128
0.1247
0.1359
0.1466
0.1662
0.1841
0.2006
0.2157
0.2297
60
0.1536-;
0.3059-;
0.4570-;
0.6069-;
0.7555-;
0.1194-
0.1481-
0.1764-
0.2180-
0.2854-
0.3505-
0.4134-
0.4743-
0.5334-
0.5907-
0.6464-
0.7531-
0.8544-
0.9507-
0.1043
0.1130
0.1295
0.1447
0.1588
0.1719
0.1843
70
> 0.1142-2
> 0.2276-2
> 0.3403-2
> 0.4524-2
> 0.5637-2
0.8938-2
0.1111-1
0.1325-1
0.1642-1
0.2158-1
0.2660-1
0.3150-1
0.3627-1
0.4092-1
0.4546-1
0.4990-1
0.5847-1
0.6668-1
0.7456-1
0.8213-1
0.8942-1
0.1032
0.1161
0.1282
0.1397
0.1504
80
0.8816-3
0.1759-2
0.2632-2
0.3501-2
0.4365-2
0.6935-2
0.8628-2
0.1030-1
0.1279-1
0.1686-1
0.2085-1
0.2475-1
0.2857-1
0.3231-1
0.3598-1
0.3958-1
0.4659-1
0.5334-1
0.5986-1
0.6617-1
0.7228-1
0.8395-1
0.9496-1
0.1054
0.1153
0.1247
90
0.7013-:
0.1400-;
0.2096-;
0.2789-;
0.3479-;
0.5535-;
0.6892-;
0.8239-;
0.1024-
0.1353-
0.1676-
0.1993-
0.2306-
0.2612-
0.2914-
0.3211-
0.3792-
0.4355-
0.4901-
0.5433-
0.5949-
0.6943-
0.7888-
0.8789-
0.9651-
0.1048
100
J 0.5711-3
> 0.1140-2
> 0.1708-2
> 0.2273-2
> 0.2837-2
> 0.4518-2
> 0.5630-2
> 0.6735-2
0.8380-2
0.1109-1
0.1376-1
0.1639-1
0.1898-1
0.2153-1
0.2406-1
0.2654-1
0.3142-1
0.3617-1
0.4080-1
0.4532-1
0.4974-1
0.5827-1
0.6644-1
0.7428-1
0.8181-1
0.8905-1
120
0.3998-3
0.7988-3
0.1197-2
0.1594-2
0.1990-2
0.3174-2
0.3958-2
0.4739-2
0.5904-2
0.7829-2
0.9734-2
0.1162-1
0.1348-1
0.1533-1
0.1716-1
0.1897-1
0.2253-1
0.2603-1
0.2946-1
0.3282-1
0.3613-1
0.4258-1
0.4881-1
0.5484-1
0.6068-1
0.6635-1
140
0.2955-:
0.5905-:
0.8849-:
0.1179-;
0.1472-;
0.2350-;
0.2933-;
0.3513-;
0.4381-;
0.5817-;
0.7241-;
0.8655-;
0.1006-
0.1145-
0.1283-
0.1420-
0.1691-
0.1957-
0.2220-
0.2480-
0.2735-
0.3236-
0.3723-
0.4198-
0.4661-
0.5113-
160
J 0.2272-3
5 0.4541-3
5 0.6808-3
> 0.9071-3
> 0.1133-2
> 0.1809-2
> 0.2259-2
> 0.2707-2
> 0.3378-2
> 0.4489-2
> 0.5593-2
> 0.6691-2
0.7782-2
0.8866-2
0.9943-2
0.1101-1
0.1314-1
0.1523-1
0.1731-1
0.1936-1
0.2138-1
0.2537-1
0.2927-1
0.3309-1
0.3683-1
0.4050-1
180
0.1801-3
0.3601-3
0.5399-3
0.7195-3
0.8989-3
0.1436-2
0.1793-2
0.2149-2
0.2683-2
0.3568-2
0.4448-2
0.5325-2
0.6196-2
0.7064-2
0.7927-2
0.8786-2
0.1049-1
0.1218-1
0.1385-1
0.1551-1
0.1715-1
0.2039-1
0.2358-1
0.2670-1
0.2978-1
0.3280-1
200
0.1463-3
0.2925-3
0.4386-3
0.5845-3
0.7304-3
0.1167-2
0.1458-2
0.1748-2
0.2182-2
0.2903-2
0.3621-2
0.4337-2
0.5049-2
0.5758-2
0.6465-2
0.7169-2
0.8567-2
0.9955-2
0.1133-
0.1270-
0.1405-
0.1674-
0.1938-
0.2198-
0.2455-
0.2708-
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                     D-220
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
    Table 19-23. Per-Constituent Significance Levels (a) for Non-Parametric 1-of-l Plan for Median (PL=Xn-i)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
4
0.3714
0.5381
0.6336
0.6957
0.7395
0.8173
0.8474
0.8689
0.8916
0.9158
0.9311
0.9416
0.9493
0.9552
0.9599
0.9637
0.9694
0.9736
0.9767
0.9792
0.9812
0.9843
0.9864
0.9881
0.9894
0.9904
6
0.2262
0.3636
0.4570
0.5251
0.5771
0.6796
0.7232
0.7559
0.7922
0.8329
0.8600
0.8792
0.8937
0.9051
0.9141
0.9216
0.9332
0.9417
0.9482
0.9535
0.9577
0.9642
0.9689
0.9726
0.9754
0.9777
8
0.1515
0.2587
0.3394
0.4026
0.4537
0.5622
0.6120
0.6509
0.6957
0.7483
0.7847
0.8115
0.8321
0.8484
0.8618
0.8729
0.8903
0.9034
0.9137
0.9219
0.9286
0.9391
0.9468
0.9527
0.9575
0.9613
10
0.1084
0.1923
0.2597
0.3152
0.3620
0.4672
0.5183
0.5595
0.6087
0.6685
0.7115
0.7439
0.7694
0.7900
0.8071
0.8214
0.8443
0.8617
0.8755
0.8867
0.8960
0.9105
0.9213
0.9297
0.9365
0.9420
12
0.8132-1
0.1480
0.2041
0.2520
0.2935
0.3913
0.4410
0.4822
0.5327
0.5962
0.6432
0.6796
0.7087
0.7326
0.7527
0.7697
0.7972
0.8185
0.8356
0.8495
0.8612
0.8797
0.8936
0.9045
0.9134
0.9206
16
0.5057-1
0.9501-1
0.1345
0.1699
0.2019
0.2821
0.3257
0.3634
0.4115
0.4754
0.5251
0.5653
0.5984
0.6264
0.6503
0.6711
0.7054
0.7328
0.7551
0.7737
0.7895
0.8149
0.8346
0.8502
0.8630
0.8737
20
0.3444-1
0.6589-1
0.9476-1
0.1214
0.1461
0.2107
0.2474
0.2801
0.3232
0.3827
0.4310
0.4713
0.5054
0.5348
0.5604
0.5831
0.6213
0.6523
0.6782
0.7002
0.7191
0.7500
0.7743
0.7940
0.8103
0.8241
25
0.2320-1
0.4497-1
0.6546-1
0.8481-1
0.1031
0.1527
0.1820
0.2088
0.2450
0.2971
0.3410
0.3788
0.4117
0.4407
0.4665
0.4896
0.5295
0.5628
0.5912
0.6156
0.6370
0.6726
0.7012
0.7248
0.7447
0.7616
30
0.1668-1
0.3259-1
0.4781-1
0.6238-1
0.7634-1
0.1151
0.1385
0.1604
0.1907
0.2353
0.2741
0.3083
0.3387
0.3660
0.3907
0.4131
0.4526
0.4862
0.5153
0.5409
0.5635
0.6018
0.6331
0.6594
0.6818
0.7011
35
0.1257-1
0.2469-1
0.3639-1
0.4771-1
0.5866-1
0.8950-1
0.1086
0.1266
0.1519
0.1899
0.2238
0.2542
0.2817
0.3068
0.3298
0.3509
0.3887
0.4215
0.4503
0.4759
0.4988
0.5382
0.5711
0.5990
0.6230
0.6440
40
0.9805-2
0.1933-
0.2860-
0.3762-
0.4640-
0.7144-
0.8714-
0.1021
0.1234
0.1559
0.1854
0.2122
0.2369
0.2596
0.2807
0.3003
0.3357
0.3670
0.3948
0.4198
0.4425
0.4819
0.5153
0.5440
0.5690
0.5910
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-221
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
    Table 19-23. Per-Constituent Significance Levels (a) for Non-Parametric 1-of-l Plan for Median (PL=Xn-i)
w/n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.6446-2
0.1277-1
0.1897-1
0.2507-1
0.3105-1
0.4837-1
0.5942-1
0.7012-1
0.8553-1
0.1097
0.1321
0.1531
0.1727
0.1911
0.2085
0.2249
0.2552
0.2826
0.3075
0.3304
0.3514
0.3889
0.4214
0.4500
0.4754
0.4981
60
0.4558-;
0.9053-;
0.1349-
0.1786-
0.2218-
0.3480-
0.4295-
0.5090-
0.6247-
0.8088-
0.9829-
0.1148
0.1305
0.1454
0.1596
0.1732
0.1987
0.2222
0.2439
0.2641
0.2830
0.3172
0.3475
0.3747
0.3992
0.4214
70 80 90 100 120 140 160 180 200
> 0.3393-2 0.2623-2 0.2088-2 0.1702-2 0.1193-2 0.8822-3 0.6788-3 0.5385-3 0.4375-3
> 0.6750-2 0.5225-2 0.4163-2 0.3395-2 0.2381-2 0.1762-2 0.1356-2 0.1076-2 0.8744-3
0.1007-1 0.7806-2 0.6225-2 0.5079-2 0.3565-2 0.2639-2 0.2032-2 0.1613-2 0.1311-2
0.1336-1 0.1037-1 0.8273-2 0.6754-2 0.4745-2 0.3514-2 0.2707-2 0.2148-2 0.1746-2
0.1662-1 0.1291-1 0.1031- 0.8420-2 0.5920-2 0.4387-2 0.3380-2 0.2683-2 0.2181-2
0.2620-1 0.2041-1 0.1634- 0.1337-1 0.9419-2 0.6989-2 0.5390-2 0.4282-2 0.3483-2
0.3243-1 0.2531-1 0.2029- 0.1662-1 0.1173-1 0.8713-2 0.6723-2 0.5343-2 0.4347-2
0.3854-1 0.3015-1 0.2420- 0.1985-1 0.1403-1 0.1043- 0.8050-2 0.6401-2 0.5210-2
0.4750-1 0.3726-1 0.2998- 0.2462-1 0.1744-1 0.1298- 0.1003-1 0.7980-2 0.6499-2
0.6189-1 0.4878-1 0.3937- 0.3242-1 0.2304-1 0.1719- 0.1330-1 0.1060-1 0.8635-2
0.7566-1 0.5989-1 0.4850- 0.4003-1 0.2855-1 0.2134- 0.1654-1 0.1319-1 0.1076-
0.8887-1 0.7063-1 0.5738- 0.4747-1 0.3396-1 0.2544- 0.1975-1 0.1576-1 0.1286-
0.1015 0.8102-1 0.6601- 0.5473-1 0.3928-1 0.2949- 0.2292-1 0.1831-1 0.1496-
0.1137 0.9107-1 0.7441- 0.6183-1 0.4451-1 0.3348- 0.2606-1 0.2084-1 0.1704-
0.1254 0.1008 0.8259- 0.6877-1 0.4966-1 0.3743- 0.2917-1 0.2335-1 0.1910-
0.1367 0.1103 0.9056- 0.7556-1 0.5472-1 0.4132- 0.3225-1 0.2584-1 0.2115-
0.1582 0.1283 0.1059 0.8872-1 0.6460-1 0.4897- 0.3832-1 0.3076-1 0.2521-
0.1782 0.1454 0.1206 0.1013 0.7419-1 0.5644- 0.4427-1 0.3560-1 0.2922-
0.1969 0.1616 0.1345 0.1135 0.8348-1 0.6373- 0.5011-1 0.4037-1 0.3318-
0.2146 0.1769 0.1479 0.1252 0.9251-1 0.7085- 0.5585-1 0.4507-1 0.3709-
0.2312 0.1915 0.1607 0.1364 0.1013 0.7782- 0.6148-1 0.4970-1 0.4095-
0.2619 0.2187 0.1848 0.1577 0.1181 0.9129- 0.7244-1 0.5875-1 0.4853-
0.2895 0.2436 0.2070 0.1776 0.1341 0.1042 0.8302-1 0.6754-1 0.5593-
0.3146 0.2665 0.2277 0.1963 0.1492 0.1166 0.9326-1 0.7609-1 0.6315-
0.3375 0.2876 0.2470 0.2138 0.1636 0.1285 0.1032 0.8441-1 0.7021-
0.3586 0.3072 0.2651 0.2304 0.1774 0.1400 0.1128 0.9251-1 0.7711-
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-222
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
                                                       Unified Guidance
     Table 19-24. Per-Constituent Significance Levels (a) for Non-Parametric l-of-2 Plan for Median (PL=Xn)
    w\n
10
12
16
20
25
30
35
40
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
0.5238-1 0.2121-1 0.1019-1 0.5495-;
0.8898-1 0.3853-1 0.1918-1 0.1055-
0.1171 0.5324-1 0.2725-1 0.1526-
0.1400 0.6608-1 0.3461-1 0.1966-
0.1593 0.7747-1 0.4138-1 0.2381-
0.2035 0.1057 0.5901-1 0.3502-
0.2259 0.1210 0.6910-1 0.4167-
0.2448 0.1344 0.7820-1 0.4781-
0.2683 0.1518 0.9038-1 0.5623-
0.2991 0.1758 0.1078 0.6863-
0.3232 0.1954 0.1226 0.7947-
0.3430 0.2120 0.1354 0.8913-
0.3597 0.2264 0.1468 0.9786-
0.3741 0.2391 0.1571 0.1058
0.3867 0.2505 0.1664 0.1132
0.3980 0.2608 0.1749 0.1200
0.4173 0.2787 0.1901 0.1323
0.4334 0.2941 0.2034 0.1432
0.4473 0.3075 0.2151 0.1530
0.4593 0.3194 0.2257 0.1620
0.4700 0.3300 0.2352 0.1702
0.4882 0.3484 0.2520 0.1847
0.5034 0.3640 0.2665 0.1975
0.5163 0.3775 0.2792 0.2087
0.5274 0.3893 0.2904 0.2189
0.5373 0.3999 0.3006 0.2281
> 0.3221-;
0.6265-;
0.9157-;
0.1191-
0.1455-
0.2187-
0.2633-
0.3052-
0.3638-
0.4521-
0.5312-
0.6030-
0.6690-
0.7301-
0.7871-
0.8404-
0.9382-
0.1026
0.1106
0.1180
0.1248
0.1371
0.1480
0.1578
0.1666
0.1748
> 0.1321-;
> 0.2605-;
> 0.3853-;
0.5069-;
0.6255-;
0.9651-;
0.1180-
0.1386-
0.1682-
0.2143-
0.2571-
0.2972-
0.3349-
0.3706-
0.4044-
0.4367-
0.4972-
0.5530-
0.6049-
0.6535-
0.6993-
0.7835-
0.8598-
0.9296-
0.9941-
0.1054
> 0.6385-:
> 0.1266-;
> 0.1884-;
> 0.2492-;
> 0.3092-;
> 0.4839-;
0.5965-;
0.7062-;
0.8661-;
0.1121-
0.1363-
0.1594-
0.1815-
0.2028-
0.2233-
0.2430-
0.2806-
0.3160-
0.3494-
0.3812-
0.4115-
0.4683-
0.5207-
0.5695-
0.6151-
0.6582-
5 0.3002-:
> 0.5976-:
> 0.8923-:
> 0.1185-;
> 0.1474-;
> 0.2329-;
> 0.2888-;
> 0.3438-;
> 0.4248-;
0.5562-;
0.6834-;
0.8067-;
0.9265-;
0.1043-
0.1157-
0.1268-
0.1482-
0.1687-
0.1884-
0.2074-
0.2257-
0.2606-
0.2934-
0.3245-
0.3540-
0.3822-
J 0.1592-:
J 0.3174-:
J 0.4749-:
> 0.6315-:
> 0.7873-:
> 0.1250-;
> 0.1555-;
> 0.1856-;
> 0.2304-;
> 0.3036-;
> 0.3753-;
> 0.4456-;
> 0.5145-;
0.5821-;
0.6486-;
0.7139-;
0.8413-;
0.9649-;
0.1085-
0.1202-
0.1315-
0.1535-
0.1744-
0.1945-
0.2138-
0.2324-
5 0.9207-4 0.5690-4
J 0.1838-3 0.1137-3
J 0.2753-3 0.1703-3
5 0.3664-3 0.2268-3
J 0.4573-3 0.2832-3
> 0.7280-3 0.4517-3
> 0.9071-3 0.5634-3
> 0.1085-2 0.6746-3
> 0.1350-2 0.8406-3
> 0.1786-2 0.1115-2
> 0.2216-2 0.1387-2
> 0.2640-2 0.1656-2
> 0.3059-2 0.1922-2
> 0.3472-2 0.2186-2
> 0.3879-2 0.2448-2
> 0.4282-2 0.2707-2
> 0.5074-2 0.3219-2
> 0.5849-2 0.3723-2
0.6607-2 0.4220-2
0.7349-2 0.4708-2
0.8078-2 0.5190-2
0.9495-2 0.6133-2
0.1086-1 0.7051-2
0.1219-1 0.7947-2
0.1347-1 0.8822-2
0.1472-1 0.9677-2
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                      D-223
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
     Table 19-24. Per-Constituent Significance Levels (a) for Non-Parametric l-of-2 Plan for Median (PL=Xn)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.2513-4
0.5024-4
0.7531-4
0.1004-3
0.1254-3
0.2003-3
0.2501-3
0.2998-3
0.3741-3
0.4975-3
0.6203-3
0.7424-3
0.8640-3
0.9849-3
0.1105-2
0.1225-2
0.1463-2
0.1699-2
0.1933-2
0.2164-2
0.2394-2
0.2848-2
0.3295-2
0.3735-2
0.4169-2
0.4596-2
60
0.1276-
0.2550-
0.3825-
0.5098-
0.6370-
0.1018-3
0.1272-3
0.1526-3
0.1905-3
0.2537-3
0.3167-3
0.3794-3
0.4420-3
0.5045-3
0.5667-3
0.6288-3
0.7525-3
0.8755-3
0.9979-3
0.1120-2
0.1241-2
0.1481-2
0.1720-2
0.1956-2
0.2189-2
0.2421-2
70
0.7145-5
0.1429-
0.2143-
0.2856-
0.3570-
0.5709-
0.7134-
0.8557-
0.1069-3
0.1424-3
0.1779-3
0.2133-3
0.2486-3
0.2839-3
0.3191-3
0.3542-3
0.4244-3
0.4943-3
0.5639-3
0.6334-3
0.7026-3
0.8405-3
0.9775-3
0.1114-2
0.1249-2
0.1384-2
80
0.4307-5
0.8613-5
0.1292-4
0.1722-4
0.2153-4
0.3443-4
0.4303-4
0.5162-4
0.6451-4
0.8596-4
0.1074-3
0.1288-3
0.1502-3
0.1715-3
0.1929-3
0.2142-3
0.2568-3
0.2992-3
0.3416-3
0.3839-3
0.4261-3
0.5103-3
0.5941-3
0.6776-3
0.7608-3
0.8437-3
90
0.2749-5
0.5497-5
0.8244-5
0.1099-4
0.1374-4
0.2198-4
0.2747-4
0.3296-4
0.4119-4
0.5489-4
0.6859-4
0.8228-4
0.9596-4
0.1096-3
0.1233-3
0.1369-3
0.1642-3
0.1914-3
0.2186-3
0.2458-3
0.2729-3
0.3270-3
0.3810-3
0.4348-3
0.4885-3
0.5421-3
100
0.1835-5
0.3671-5
0.5506-5
0.7340-5
0.9175-5
0.1468-4
0.1835-4
0.2201-4
0.2751-4
0.3667-4
0.4583-4
0.5498-4
0.6413-4
0.7327-4
0.8241-4
0.9155-4
0.1098-3
0.1280-3
0.1463-3
0.1645-3
0.1826-3
0.2190-3
0.2552-3
0.2914-3
0.3275-3
0.3635-3
120
0.9090-6
0.1818-5
0.2727-5
0.3636-5
0.4545-5
0.7271-5
0.9088-5
0.1090-4
0.1363-4
0.1817-4
0.2271-4
0.2725-4
0.3179-4
0.3632-4
0.4086-4
0.4539-4
0.5446-4
0.6352-4
0.7257-4
0.8162-4
0.9067-4
0.1087-3
0.1268-3
0.1449-3
0.1629-3
0.1809-3
140
0.5001-6
0.1000-5
0.1500-5
0.2000-5
0.2501-5
0.4001-5
0.5001-5
0.6001-5
0.7501-5
0.1000-4
0.1250-4
0.1500-4
0.1750-4
0.1999-4
0.2249-4
0.2499-4
0.2998-4
0.3497-4
0.3997-4
0.4495-4
0.4994-4
0.5991-4
0.6988-4
0.7984-4
0.8979-4
0.9974-4
160
0,2974-6
0.5949-6
0.8923-6
0.1190-5
0.1487-5
0.2379-5
0.2974-5
0.3569-5
0.4461-5
0.5948-5
0.7434-5
0.8921-5
0.1041-4
0.1189-4
0.1338-4
0.1487-4
0.1784-4
0.2081-4
0.2378-4
0.2675-4
0.2972-4
0.3565-4
0.4159-4
0.4752-4
0.5345-4
0.5938-4
180
0.1878-6
0.3756-6
0.5634-6
0.7511-6
0.9389-6
0.1502-5
0.1878-5
0.2253-5
0.2817-5
0.3755-5
0.4694-5
0.5633-5
0.6571-5
0.7510-5
0.8448-5
0.9387-5
0.1126-4
0.1314-4
0.1502-4
0.1689-4
0.1877-4
0.2252-4
0.2627-4
0.3002-4
0.3377-4
0.3752-4
200
0.1243-6
0.2487-6
0.3730-6
0.4973-6
0.6216-6
0.9946-6
0.1243-5
0.1492-5
0.1865-5
0.2486-5
0.3108-5
0.3729-5
0.4351-5
0.4972-5
0.5594-5
0.6215-5
0.7458-5
0.8701-5
0.9943-5
0.1119-4
0.1243-4
0.1491-4
0.1740-4
0.1988-4
0.2236-4
0.2485-4
Footnote. PL = Prediction Limit; Xn = Maximum order statistic
                                                     D-224
                                                                                                       March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
                                                        Unified Guidance
    Table 19-24. Per-Constituent Significance Levels (a) for Non-Parametric l-of-2 Plan for Median (PL=Xn-i)
    w\n
10
12
16
20
25
30
35
40
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
0.2048 0.8874-1 0.4429-1 0.2448- 0.1460- 0.6125-;
0.3138 0.1515 0.8008-1 0.4576- 0.2786- 0.1195-
0.3849 0.1996 0.1101 0.6462- 0.4003- 0.1751-
0.4363 0.2383 0.1360 0.8158- 0.5129- 0.2283-
0.4758 0.2705 0.1586 0.9698- 0.6178- 0.2794-
0.5558 0.3429 0.2135 0.1363 0.8957- 0.4215-
0.5914 0.3785 0.2425 0.1582 0.1057 0.5085-
0.6192 0.4079 0.2674 0.1776 0.1204 0.5902-
0.6515 0.4437 0.2990 0.2031 0.1401 0.7045-
0.6902 0.4892 0.3413 0.2386 0.1684 0.8764-
0.7178 0.5237 0.3748 0.2678 0.1925 0.1030
0.7389 0.5511 0.4024 0.2925 0.2135 0.1169
0.7557 0.5736 0.4257 0.3140 0.2321 0.1296
0.7694 0.5926 0.4459 0.3329 0.2487 0.1413
0.7810 0.6090 0.4635 0.3498 0.2638 0.1522
0.7910 0.6232 0.4792 0.3650 0.2775 0.1623
0.8072 0.6471 0.5061 0.3915 0.3019 0.1808
0.8201 0.6665 0.5283 0.4139 0.3229 0.1973
0.8306 0.6827 0.5473 0.4334 0.3414 0.2121
0.8394 0.6965 0.5637 0.4504 0.3579 0.2257
0.8469 0.7085 0.5781 0.4656 0.3727 0.2381
0.8592 0.7283 0.6025 0.4917 0.3985 0.2604
0.8688 0.7443 0.6224 0.5135 0.4204 0.2797
0.8767 0.7575 0.6392 0.5320 0.4393 0.2969
0.8832 0.7687 0.6536 0.5481 0.4560 0.3123
0.8888 0.7783 0.6662 0.5623 0.4708 0.3263
> 0.3001-;
0.5917-;
0.8754-;
0.1152-
0.1421-
0.2193-
0.2679-
0.3146-
0.3814-
0.4852-
0.5810-
0.6702-
0.7538-
0.8324-
0.9067-
0.9771-
0.1108
0.1228
0.1339
0.1441
0.1537
0.1712
0.1868
0.2009
0.2138
0.2257
> 0.1427-;
> 0.2832-;
> 0.4216-;
0.5579-;
0.6922-;
0.1084-
0.1337-
0.1583-
0.1942-
0.2514-
0.3057-
0.3574-
0.4068-
0.4542-
0.4997-
0.5436-
0.6268-
0.7047-
0.7781-
0.8475-
0.9134-
0.1036
0.1148
0.1252
0.1349
0.1439
> 0.7629-:
> 0.1519-;
> 0.2267-;
> 0.3009-;
> 0.3745-;
0.5913-;
0.7329-;
0.8722-;
0.1077-
0.1409-
0.1729-
0.2039-
0.2339-
0.2631-
0.2914-
0.3190-
0.3721-
0.4227-
0.4711-
0.5175-
0.5621-
0.6464-
0.7252-
0.7993-
0.8692-
0.9354-
5 0.4439-:
> 0.8852-:
> 0.1324-;
> 0.1760-;
> 0.2194-;
> 0.3480-;
> 0.4326-;
> 0.5163-;
0.6402-;
0.8427-;
0.1040-
0.1233-
0.1422-
0.1607-
0.1789-
0.1967-
0.2312-
0.2646-
0.2969-
0.3281-
0.3584-
0.4166-
0.4717-
0.5242-
0.5743-
0.6223-
J 0.2755-3
5 0.5500-3
> 0.8234-3
> 0.1096-2
> 0.1367-2
> 0.2175-2
> 0.2708-2
> 0.3238-2
> 0.4025-2
> 0.5319-2
0.6592-2
0.7844-2
0.9076-2
0.1029-
0.1149-
0.1266-
0.1497-
0.1722-
0.1942-
0.2156-
0.2365-
0.2770-
0.3158-
0.3532-
0.3892-
0.4240-
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-225
                                                                                                        March 2009

-------
Appendix D. Chapter 19 Non-Parametric Prediction Limit Significance Levels
Unified Guidance
    Table 19-24. Per-Constituent Significance Levels (a) for Non-Parametric l-of-2 Plan for Median (PL=Xn-i)
w\n
1
2
3
4
5
8
10
12
15
20
25
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
50
0.1225-3
0.2447-3
0.3667-3
0.4884-3
0.6099-3
0.9731-3
0.1214-2
0.1454-2
0.1813-2
0.2406-2
0.2994-2
0.3577-2
0.4155-2
0.4728-2
0.5297-2
0.5861-2
0.6976-2
0.8074-2
0.9156-2
0.1022-1
0.1128-
0.1334-
0.1535-
0.1731-
0.1923-
0.2111-
60
0.6242-4
0.1248-3
0.1870-3
0.2493-3
0.3114-3
0.4975-3
0.6212-3
0.7447-3
0.9294-3
0.1236-2
0.1541-2
0.1845-2
0.2147-2
0.2448-2
0.2747-2
0.3045-2
0.3636-2
0.4223-2
0.4804-2
0.5380-2
0.5951-2
0.7080-2
0.8190-2
0.9284-2
0.1036-1
0.1142-1
70
0.3507-4
0.7011-4
0.1051-3
0.1401-3
0.1751-3
0.2799-3
0.3497-3
0.4194-3
0.5237-3
0.6972-3
0.8702-3
0.1043-2
0.1215-2
0.1386-2
0.1557-2
0.1727-2
0.2067-2
0.2404-2
0.2740-2
0.3074-2
0.3406-2
0.4065-2
0.4717-2
0.5362-2
0.6001-2
0.6634-2
80
0.2119-4
0.4236-4
0.6353-4
0.8470-4
0.1058-3
0.1693-3
0.2115-3
0.2537-3
0.3169-3
0.4222-3
0.5272-3
0.6320-3
0.7367-3
0.8411-3
0.9453-3
0.1049-2
0.1257-2
0.1464-2
0.1670-2
0.1875-2
0.2079-2
0.2486-2
0.2891-2
0.3292-2
0.3691-2
0.4087-2
90
0.1354-4
0.2708-4
0.4062-4
0.5416-4
0.6769-4
0.1083-3
0.1353-3
0.1623-3
0.2028-3
0.2702-3
0.3376-3
0.4048-3
0.4720-3
0.5391-3
0.6061-3
0.6730-3
0.8066-3
0.9398-3
0.1073-2
0.1205-2
0.1338-2
0.1601-2
0.1864-2
0.2125-2
0.2385-2
0.2643-2
100
0.9057-5
0.1811-4
0.2717-4
0.3622-4
0.4527-4
0.7241-4
0.9050-4
0.1086-3
0.1357-3
0.1808-3
0.2260-3
0.2710-3
0.3161-3
0.3610-3
0.4060-3
0.4509-3
0.5406-3
0.6302-3
0.7196-3
0.8088-3
0.8980-3
0.1076-2
0.1253-2
0.1429-2
0.1605-2
0.1781-2
120
0.4495-5
0.8991-5
0.1349-4
0.1798-4
0.2247-4
0.3595-4
0.4494-4
0.5392-4
0.6739-4
0.8983-4
0.1123-3
0.1347-3
0.1571-3
0.1795-3
0.2019-3
0.2243-3
0.2690-3
0.3137-3
0.3583-3
0.4030-3
0.4475-3
0.5366-3
0.6254-3
0.7141-3
0.8027-3
0.8911-3
140
0.2477-5
0.4954-5
0.7431-5
0.9908-5
0.1239-4
0.1981-4
0.2477-4
0.2972-4
0.3715-4
0.4952-4
0.6189-4
0.7426-4
0.8663-4
0.9899-4
0.1114-3
0.1237-3
0.1484-3
0.1731-3
0.1978-3
0.2224-3
0.2471-3
0.2964-3
0.3456-3
0.3947-3
0.4439-3
0.4929-3
160
0.1475-5
0.2950-5
0.4425-5
0.5900-5
0.7374-5
0.1180-4
0.1475-4
0.1770-4
0.2212-4
0.2949-4
0.3686-4
0.4423-4
0.5160-4
0.5896-4
0.6633-4
0.7369-4
0.8841-4
0.1031-3
0.1178-3
0.1326-3
0.1473-3
0.1767-3
0.2060-3
0.2354-3
0.2647-3
0.2941-3
180
0.9321-6
0.1864-5
0.2796-5
0.3728-5
0.4660-5
0.7456-5
0.9320-5
0.1118-4
0.1398-4
0.1864-4
0.2330-4
0.2795-4
0.3261-4
0.3727-4
0.4192-4
0.4658-4
0.5589-4
0.6520-4
0.7451-4
0.8381-4
0.9311-4
0.1117-3
0.1303-3
0.1489-3
0.1675-3
0.1860-3
200
0.6175-6
0.1235-5
0.1853-5
0.2470-5
0.3088-5
0.4940-5
0.6175-5
0.7410-5
0.9262-5
0.1235-4
0.1544-4
0.1852-4
0.2161-4
0.2469-4
0.2778-4
0.3087-4
0.3704-4
0.4321-4
0.4938-4
0.5554-4
0.6171-4
0.7404-4
0.8637-4
0.9870-4
0.1110-3
0.1233-3
Footnote. PL = Prediction Limit; Xn_i = 2nd largest order statistic
                                                      D-226
                                                                                                        March 2009

-------
Appendix D.  Chapter 21 Tables	Unified Guidance





                                          D  STATISTICAL TABLES





D.5 TABLES FROM  CHAPTER  21








            TABLE 21-1  Land(H) Factors for 1%LCI_on a Lognormal Arithmetic Mean	D-228



            TABLE 21-2  Land (H) Factors for 2.5% LCL on a Lognormal Arithmetic Mean	D-230



            TABLE 21-3  Land (H) Factors for 5% LCL on a Lognormal Arithmetic Mean	D-232



            TABLE 21-4  Land (H) Factors for 10% LCL on a Lognormal Arithmetic Mean	D-234



            TABLE 21 -5  Land (H) Factors for 90% UCL on a Lognormal Arithmetic Mean	D-236



            TABLE 21 -6  Land (H) Factors for 95% UCL on a Lognormal Arithmetic Mean	D-238



            TABLE 21-7  Land (H) Factors for 97.5%UCL on a Lognormal Arithmetic Mean	D-240



            TABLE 21 -8  Land (H) Factors for 99% UCL on a Lognormal Arithmetic Mean	D-242



            TABLE 21-9  Factors (r) for Parametric Upper Confidence Bounds on Percentiles (P)	D-245



            TABLE 21-10 Factors (r) for Parametric Lower Confidence Bounds on Percentiles (P)	D-247



            TABLE 21 -11  One-sided Non-Parametric Conf. Bnds.on Median, 95th & 99th Percentiles	D-249
                                                             D-227                                                   March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
         Table 21-1. Land's Factors (H.0i) for Confidence Bounds on Lognormal Arithmetic Mean for n
                                            3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
3
-4.435
-3.720
-3.260
-2.943
-2.714
-2.544
-2.415
-2.317
-2.242
-2.185
-2.099
-2.069
-2.075
-2.106
-2.217
-2.371
-2.553
-2.756
-2.973
-3.202
-3.683
-4.185
-4.700
-5.223
-5.753
4
-3.437
-3.089
-2.836
-2.649
-2.508
-2.402
-2.321
-2.260
-2.216
-2.184
-2.147
-2.153
-2.190
-2.247
-2.408
-2.610
-2.839
-3.087
-3.349
-3.622
-4.189
-4.775
-5.374
-5.980
-6.593
5
-3.047
-2.819
-2.646
-2.514
-2.414
-2.338
-2.282
-2.242
-2.214
-2.196
-2.189
-2.220
-2.277
-2.355
-2.552
-2.788
-3.050
-3.331
-3.626
-3.930
-4.559
-5.208
-5.868
-6.536
-7.211
6
-2.849
-2.677
-2.544
-2.442
-2.364
-2.307
-2.266
-2.238
-2.221
-2.214
-2.227
-2.275
-2.348
-2.440
-2.665
-2.927
-3.216
-3.523
-3.842
-4.171
-4.850
-5.548
-6.258
-6.975
-7.698
7
-2.730
-2.590
-2.482
-2.399
-2.337
-2.292
-2.261
-2.241
-2.232
-2.232
-2.260
-2.322
-2.407
-2.511
-2.758
-3.042
-3.352
-3.680
-4.020
-4.370
-5.089
-5.827
-6.577
-7.334
-8.098
8
-2.653
-2.534
-2.441
-2.371
-2.320
-2.283
-2.260
-2.247
-2.244
-2.249
-2.290
-2.362
-2.457
-2.571
-2.836
-3.140
-3.467
-3.812
-4.170
-4.537
-5.291
-6.064
-6.847
-7.639
-8.437
9
-2.598
-2.494
-2.413
-2.353
-2.309
-2.279
-2.262
-2.255
-2.256
-2.265
-2.316
-2.397
-2.501
-2.623
-2.904
-3.223
-3.566
-3.926
-4.299
-4.681
-5.465
-6.267
-7.081
-7.902
-8.730
10
-2.558
-2.465
-2.393
-2.340
-2.302
-2.278
-2.265
-2.262
-2.268
-2.280
-2.339
-2.428
-2.540
-2.668
-2.964
-3.296
-3.652
-4.026
-4.412
-4.808
-5.618
-6.446
-7.286
-8.133
-8.987
11
-2.527
-2.442
-2.378
-2.330
-2.298
-2.278
-2.269
-2.270
-2.279
-2.295
-2.361
-2.456
-2.574
-2.709
-3.017
-3.361
-3.729
-4.115
-4.513
-4.920
-5.754
-6.605
-7.468
-8.339
-9.215
12
-2.503
-2.425
-2.366
-2.324
-2.295
-2.279
-2.274
-2.277
-2.289
-2.308
-2.380
-2.481
-2.605
-2.746
-3.064
-3.419
-3.799
-4.195
-4.603
-5.021
-5.875
-6.748
-7.632
-8.523
-9.420
13
-2.484
-2.411
2.357
-2.319
-2.294
-2.281
-2.278
-2.284
-2.298
-2.320
-2.398
-2.504
-2.633
-2.778
-3.107
-3.472
-3.861
-4.267
-4.685
-5.112
-5.986
-6.877
-7.780
-8.690
-9.607
14
-2.467
-2.400
-2.350
-2.315
-2.293
-2.283
-2.283
-2.291
-2.308
-2.331
-2.414
-2.525
-2.659
-2.809
-3.147
-3.521
-3.918
-4.333
-4.760
-5.195
-6.087
-6.995
-7.916
-8.843
-9.776
Source: Land (1975)
Footnote. Notation n = 3(1)19(2)25(3)31(5)36 is shorthand for n from 3 to 19 by unit steps, from 19 to 25 by 2's, from 25 to 31 by
3's, and from 31 to 36 by 5's
                                                     D-228
       March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
        Table 21-1. Land's Factors (H.0i) for Confidence Bounds on Lognormal Arithmetic Mean for n =
                                         3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
15
-2.454
-2.390
-2.344
-2.312
-2.293
-2.285
-2.287
-2.298
-2.316
-2.341
-2.429
2.545
-2.682
-2.836
-3.183
-3.564
-3.970
-4.393
-4.828
-5.272
-6.179
-7.104
-8.040
-8.983
-9.932
16
-2.442
-2.383
-2.339
-2.310
-2.293
-2.288
-2.292
-2.304
-2.324
-2.351
-2.443
-2.563
-2.704
-2.862
-3.216
-3.605
-4.019
-4.449
-4.891
-5.343
-6.264
-7.204
-8.154
-9.113
-10.080
17
-2.432
-2.376
-2.335
-2.308
-2.294
-2.290
-2.296
-2.310
-2.332
-2.360
-2.456
-2.579
-2.724
-2.886
-3.247
-3.643
-4.063
-4.500
-4.950
-5.408
-6.343
-7.297
-8.261
-9.232
-10.210
18
-2.424
-2.370
-2.332
-2.307
-2.294
-2.292
-2.300
-2.315
-2.339
-2.369
-2.468
-2.595
-2.743
-2.908
-3.275
-3.679
-4.105
-4.549
-5.005
-5.469
-6.418
-7.383
-8.360
-9.344
-10.330
19
-2.416
-2.365
-2.329
-2.306
-2.295
-2.295
-2.304
-2.321
-2.346
-2.377
-2.479
-2.609
-2.760
-2.929
-3.302
-3.711
-4.144
-4.593
-5.055
-5.526
-6.486
-7.465
-8.453
-9.449
-10.450
21
-2.404
-2.357
-2.325
-2.306
-2.298
-2.300
-2.312
-2.331
-2.358
-2.392
-2.500
-2.635
-2.792
-2.966
-3.351
-3.771
-4.215
-4.676
-5.148
-5.630
-6.612
-7.611
-8.621
-9.640
-10.660
23
-2.395
-2.351
-2.322
-2.305
-2.300
-2.305
-2.319
-2.341
-2.370
-2.406
-2.519
-2.659
-2.821
-3.000
-3.394
-3.825
-4.279
-4.749
-5.231
-5.723
-6.724
-7.742
-8.772
-9.809
-10.850
25
-2.386
-2.346
2.320
-2.305
-2.302
-2.309
-2.325
-2.349
-2.380
-2.418
-2.535
-2.680
-2.847
-3.030
-3.434
-3.873
-4.335
-4.814
-5.305
-5.805
-6.824
-7.860
-8.906
-9.961
-11.020
28
-2.377
-2.340
-2.317
-2.306
-2.306
-2.316
-2.334
-2.361
-2.394
-2.434
-2.558
-2.709
-2.881
-3.070
-3.486
-3.936
-4.410
-4.901
-5.404
-5.916
-6.958
-8.017
-9.086
-10.160
-11.250
31
-2.369
-2.336
-2.316
-2.308
-2.310
-2.322
-2.342
-2.373
-2.406
-2.449
-2.578
-2.734
-2.911
-3.105
-3.531
-3.992
-4.476
-4.977
-5.491
-6.012
-7.075
-8.154
-9.244
-10.340
-11.440
36
-2.361
-2.331
-2.315
-2.310
-2.316
-2.330
-2.354
-2.386
-2.425
-2.470
-2.606
-2.769
-2.954
-3.155
-3.569
-4.071
-4.570
-5.086
-5.614
-6.150
-7.241
-8.348
-9.467
-10.590
-11.720
                                                  D-229
      March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
         Table 21-2. Land's Factors (H.025) for Confidence Bounds on Lognormal Arithmetic Mean for n
                                            3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
3
-2.988
-2.639
-2.396
-2.220
-2.090
-1.992
-1.919
-1.864
-1.823
-1.794
-1.759
-1.761
-1.789
-1.834
-1.960
-2.118
-2.299
-2.496
-2.706
-2.925
-3.382
-3.856
-4.341
-4.832
-5.328
4
-2.504
-2.316
-2.176
-2.070
-1.989
-1.929
-1.885
-1.854
-1.833
-1.820
-1.819
-1.849
-1.899
-1.965
-2.132
-2.331
-2.552
-2.789
-3.037
-3.294
-3.826
-4.372
-4.929
-5.492
-6.061
5
-2.314
-2.183
-2.083
-2.007
-1.950
-1.908
-1.879
-1.860
-1.850
-1.848
-1.867
-1.914
-1.981
-2.062
-2.259
-2.487
-2.736
-3.001
-3.276
-3.560
-4.145
-4.744
-5.354
-5.971
-6.592
6
-2.215
-2.113
-2.034
-1.975
-1.932
-1.901
-1.882
-1.871
-1.869
-1.873
-1.907
-1.966
-2.045
-2.138
-2.357
-2.607
-2.879
-3.164
-3.461
-3.766
-4.393
-5.033
-5.685
-6.343
-7.006
7
-2.157
-2.071
-2.006
-1.958
-1.923
-1.900
-1.887
-1.830
-1.885
-1.894
-1.939
-2.009
-2.097
-2.200
-2.438
-2.706
-2.994
-3.298
-3.612
-3.934
-4.594
-5.269
-5.955
-6.646
-7.343
8
-2.117
-2.044
-1.988
-1.948
-1.919
-1.902
-1.894
-1.894
-1.901
-1.913
-1.967
-2.045
-2.141
-2.252
-2.505
-2.788
-3.091
-3.409
-3.738
-4.074
-4.763
-5.467
-6.181
-6.901
-7.626
9
-2.090
-2.025
-1.976
-1.941
-1.918
-1.905
-1.901
-1.904
-1.915
-1.931
-1.992
-2.076
-2.179
-2.296
-2.562
-2.858
-3.174
-3.505
-3.846
-4.194
-4.908
-5.637
-6.375
-7.120
-7.869
10
-2.070
-2.012
-1.968
-1.938
-1.918
-1.908
-1.908
-1.914
-1.927
-1.946
-2.013
-2.104
-2.212
-2.335
-2.612
-2.919
-3.246
-3.588
-3.940
-4.300
-5.035
-5.785
-6.545
-7.311
-8.082
11
-2.055
-2.001
-1.962
-1.935
-1.919
-1.913
-1.914
-1.923
-1.939
-1.959
-2.032
-2.128
-2.242
-2.369
-2.656
-2.973
-3.310
-3.661
-4.023
-4.392
-5.147
-5.916
-6.695
-7.480
-8.270
12
-2.042
-1.994
-1.958
-1.934
-1.920
-1.917
-1.921
-1.932
-1.949
-1.972
-2.049
-2.150
-2.268
-2.400
-2.696
-3.022
-3.367
-3.727
-4.097
-4.475
-5.247
-6.033
-6.829
-7.631
-8.438
13
-2.032
-1.987
-1.954
-1.933
-1.922
-1.920
-1.926
-1.939
-1.958
-1.983
-2.064
-2.169
-2.291
-2.428
-2.731
-3.065
-3.418
-3.786
-4.164
-4.550
-5.337
-6.139
-6.950
-7.768
-8.590
14
-2.025
-1.982
-1.952
-1.933
-1.924
-1.924
-1.932
-1.946
-1.967
-1.993
-2.079
-2.187
-2.313
-2.452
-2.764
-3.105
-3.465
-3.840
-4.226
-4.618
-5.419
-6.235
-7.060
-7.892
-8.728
Source: Land (1975)
Footnote. Notation n = 3(1)19(2)25(3)31(5)36 is shorthand for n from 3 to 19 by unit steps, from 19 to 25 by 2's, from 25 to 31 by
3's, and from 31 to 36 by 5's
                                                     D-230
       March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
        Table 21-2. Land's Factors (H.025) for Confidence Bounds on Lognormal Arithmetic Mean for n =
                                         3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
15
-2.018
-1.978
-1.950
-1.933
-1.926
-1.928
-1.937
-1.953
-1.975
-2.003
-2.091
-2.203
2.332
-2.476
-2.793
-3.141
-3.508
-3.889
-4.281
-4.680
-5.494
-6.324
-7.161
-8.006
-8.855
16
-2.012
-1.974
-1.949
-1.934
-1.928
-1.931
-1.942
-1.959
-1.983
-2.012
-2.104
-2.218
-2.351
-2.496
-2.821
-3.174
-3.547
-3.935
-4.332
-4.738
-5.564
-6.404
-7.254
-8.111
-8.972
17
-2.008
-1.972
-1.947
-1.934
-1.930
-1.934
-1.946
-1.965
-1.990
-2.109
-2.114
-2.232
-2.367
-2.516
-2.845
-3.205
-3.583
-3.976
-4.380
-4.790
-5.628
-6.480
-7.340
-8.208
-9.079
18
-2.003
-1.969
-1.946
-1.935
-1.932
-1.938
-1.951
-1.971
-1.996
-2.027
-2.125
-2.245
-2.383
-2.534
-2.869
-3.233
-3.617
-4.015
-4.424
-4.840
-5.687
-6.549
-7.420
-8.298
-9.179
19
-2.000
-1.967
-1.946
-1.935
-1.933
-1.940
-1.955
-1.976
-2.003
-2.024
-2.134
-2.257
-2.396
-2.551
-2.890
-3.260
-3.649
-4.052
-4.465
-4.886
-5.743
-6.614
-7.495
-8.382
-9.273
21
-1.993
-1.964
-1.945
-1.936
-1.937
-1.946
-1.906
-1.985
-2.014
-2.047
-2.151
-2.278
-2.423
-2.581
-2.930
-3.308
-3.706
-4.118
-4.539
-4.969
-5.844
-6.732
-7.630
-8.535
-9.443
23
-1.989
-1.961
-1.945
-1.938
-1.941
-1.951
-1.969
-1.993
-2.023
-2.059
-2.167
-2.298
-2.446
-2.608
-2.956
-3.351
-3.757
-4.176
-4.606
-5.043
-5.933
-6.837
-7.750
-8.670
-9.594
25
-1.985
-1.959
-1.945
-1.940
-1.944
-1.956
-1.975
-2.001
-2.003
-2.069
-2.181
-2.315
-2.467
-2.633
-2.997
-3.389
-3.802
-4.229
-4.665
-5.110
-6.013
-6.931
-7.858
-8.791
-9.729
28
-1.980
-1.957
-1.945
-1.942
-1.948
-1.962
-1.983
-2.011
-2.044
-2.083
-2.199
-2.338
-2.495
-2.665
-3.038
-3.440
-3.862
-4.298
-4.744
-5.197
-6.119
-7.056
-8.001
-8.952
-9.908
31
-1.977
-1.956
-1.945
-1.944
-1.952
-1.968
-1.991
-2.020
-2.055
-2.095
-2.215
-2.358
-2.518
-2.693
-3.074
-3.484
-3.914
-4.358
-4.812
-5.273
-6.212
-7.164
-8.125
-9.092
-10.060
36
-1.972
-1.954
-1.946
-1.948
-1.958
-1.976
-2.001
-2.032
-2.069
-2.112
-2.237
-2.386
-2.552
-2.733
-3.125
-3.547
-3.988
-4.444
-4.910
-5.382
-6.343
-7.318
-8.301
-9.292
-10.290
                                                  D-231
      March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
         Table 21-3. Land's Factors (H.os) for Confidence Bounds on Lognormal Arithmetic Mean for n
                                            3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
3
-2.130
-1.969
-1.816
-1.717
-1.644
-1.589
-1.549
-1.521
-1.502
-1.490
-1.486
-1.508
-1.547
-1.598
-1.727
-1.880
-2.051
-2.237
-2.434
-2.638
-3.062
-3.499
-3.945
-4.397
-4.852
4
-1.898
-1.791
-1.710
-1.650
-1.605
-1.572
-1.550
-1.537
-1.530
-1.530
-1.549
-1.590
-1.647
-1.714
-1.877
-2.065
-2.272
-2.491
-2.720
-2.957
-3.444
-3.943
-4.451
-4.965
-5.483
5
-1.806
-1.729
-1.669
-1.625
-1.594
-1.573
-1.560
-1.555
-1.556
-1.562
-1.596
-1.650
-1.719
-1.799
-1.986
-2.199
-2.429
-2.672
-2.924
-3.183
-3.715
-4.260
-4.812
-5.371
-5.933
6
-1.759
-1.697
-1.650
-1.615
-1.592
-1.578
-1.572
-1.572
-1.577
-1.588
-1.632
-1.696
-1.774
-1.864
-2.070
-2.301
-2.550
-2.810
-3.080
-3.356
-3.923
-4.502
-5.090
-5.684
-6.280
7
-1.731
-1.678
-1.639
-1.611
-1.594
-1.584
-1.582
-1.586
-1.595
-1.610
-1.662
-1.733
-1.819
-1.917
-2.138
-2.384
-2.647
-2.922
-3.206
-3.497
-4.092
-4.699
-5.315
-5.936
-6.560
8
-1.712
-1.667
-1.633
-1.610
-1.596
-1.591
-1.592
-1.599
-1.611
-1.628
-1.687
-1.764
-1.857
-1.960
-2.193
-2.452
-2.727
-3.015
-3.310
-3.613
-4.231
-4.862
-5.502
-6.146
-6.795
9
-1.699
-1.658
-1.629
-1.610
-1.599
-1.597
-1.600
-1.610
-1.625
-1.644
-1.708
-1.791
-1.889
-1.998
-2.241
-2.510
-2.795
-3.093
-3.399
-3.712
-4.351
-5.002
-5.661
-6.326
-6.994
10
-1.690
-1.653
-1.627
-1.611
-1.603
-1.602
-1.608
-1.620
-1.637
-1.658
-1.727
-1.814
-1.916
-2.029
-2.283
-2.560
-2.855
-3.161
-3.476
-3.798
-4.455
-5.123
-5.800
-6.482
-7.168
11
-1.683
-1.649
-1.626
-1.612
-1.606
-1.608
-1.615
-1.629
-1.647
-1.670
-1.743
-1.834
-1.940
-2.058
-2.319
-2.604
-2.907
-3.221
-3.544
-3.873
-4.546
-5.230
-5.922
-6.620
-7.321
12
-1.677
-1.646
-1.625
-1.613
-1.609
-1.612
-1.622
-1.636
-1.656
-1.681
-1.758
-1.853
-1.962
-2.083
-2.351
-2.644
-2.953
-3.275
-3.605
-3.941
-4.627
-5.325
-6.031
-6.742
-7.458
13
-1.673
-1.644
-1.625
-1.614
-1.612
-1.617
-1.628
-1.644
-1.665
-1.690
-1.770
-1.869
-1.981
-2.106
-2.380
-2.679
-2.995
-3.323
-3.659
-4.001
-4.700
-5.411
-6.129
-6.853
-7.581
14
-1.669
-1.642
-1.624
-1.615
-1.615
-1.621
-1.633
-1.651
-1.673
-1.699
-1.782
-1.883
-1.998
-2.126
-2.406
-2.711
-3.033
-3.366
-3.708
-4.056
-4.766
-5.488
-6.218
-6.954
-7.592
Source: Land (1975)
Footnote. Notation n = 3(1)19(2)25(3)31 (5)36 is shorthand for n from 3 to 19 by unit steps, from 19 to 25 by 2's, from 25 to 31 by
3's, and from 31 to 36 by 5's
                                                     D-232
       March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
        Table 21-3. Land's Factors (H.os) for Confidence Bounds on Lognormal Arithmetic Mean for n =
                                         3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
15
-1.666
-1.640
-1.625
-1.617
-1.618
-1.625
-1.638
-1.656
-1.680
-1.707
-1.793
-1.896
-2.015
-2.144
-2.430
-2.740
-3.067
-3.406
-3.753
-4.107
-4.827
-5.559
-6.300
-7.045
-7.794
16
-1.663
-1.639
-1.625
-1.618
-1.620
-1.629
-1.643
-1.662
-1.686
-1.715
-1.803
-1.909
-2.029
-2.162
-2.452
-2.767
-3.099
-3.443
-3.794
-4.153
-4.882
-5.624
-6.374
-7.129
-7.888
17
-1.661
-1.638
-1.625
-1.620
-1.622
-1.632
-1.647
-1.667
-1.692
-1.722
-1.812
-1.920
-2.043
-2.177
-2.472
2.792
3.128
3.476
3.833
4.195
4.934
5.685
6.443
7.207
7.974
18
-1.659
-1.638
-1.626
-1.622
-1.625
-1.635
-1.651
-1.672
-1.698
-1.728
-1.820
-1.930
-2.055
-2.192
-2.491
-2.815
-3.155
-3.507
-3.868
-4.235
-4.981
-5.741
-6.507
-7.278
-8.054
19
-1.658
-1.637
-1.626
-1.622
-1.627
-1.638
-1.654
-1.677
-1.703
-1.734
-1.828
-1.940
-2.067
-2.205
-2.508
-2.836
-3.180
-3.536
-3.901
-4.272
-5.026
-5.793
-6.566
-7.346
-8.129
21
-1.655
-1.636
-1.627
-1.625
-1.631
-1.643
-1.661
-1.685
-1.713
-1.745
-1.842
-1.958
-2.088
-2.230
-2.540
-2.874
-3.226
-3.589
-3.960
-4.338
-5.106
-5.886
-6.674
-7.468
-8.264
23
-1.653
-1.636
-1.628
-1.627
-1.634
-1.648
-1.667
-1.691
-1.721
-1.755
-1.854
-1.973
-2.107
-2.251
-2.568
-2.908
-3.266
-3.635
-4.013
-4.397
-5.177
-5.970
-6.770
-7.575
-8.385
25
-1.651
-1.635
-1.629
-1.629
-1.638
-1.652
-1.672
-1.698
-1.728
-1.763
-1.866
-1.987
-2.123
-2.271
-2.593
-2.939
-3.302
-3.677
-4.060
-4.449
-5.241
-6.045
-6.855
-7.672
-8.491
28
-1.649
-1.636
-1.630
-1.632
-1.642
-1.658
-1.679
-1.706
-1.738
-1.774
-1.880
-2.005
-2.145
-2.269
-2.625
-2.979
-3.349
-3.731
-4.122
-4.518
-5.325
-6.142
-6.968
-7.798
-8.632
31
-1.648
-1.636
-1.632
-1.635
-1.646
-1.662
-1.686
-1.714
-1.747
-1.784
-1.893
-2.020
-2.164
-2.318
-2.654
-3.014
-3.391
-3.779
-4.176
-4.579
-5.397
-6.227
-7.066
-7.909
-10.060
36
-1.647
-1.636
-1.633
-1.639
-1.651
-1.659
-2.694
-1.724
-1.759
-1.798
-1.911
-2.043
-2.190
-2.349
-2.694
-3.063
-3.448
-3.846
-4.252
-4.664
-5.500
-6.348
-7.204
-8.064
-8.928
                                                  D-233
      March 2009

-------
Appendix D. Chapter 21 Tables
                        Unified Guidance
         Table 21-4. Land's Factors (H.io) for Confidence Bounds on Lognormal Arithmetic Mean for n
                                            3(1)19(2)25(3)31(5)36
 Sy\n
10
         11
12
13
                                   14
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
-1.431
-1.350
-1.289
-1.245
-1.213
-1.190
-1.176
-1.168
-1.165
-1.166
-1.184
-1.217
-1.260
-1.310
-1.426
-1.560
-1.710
-1.871
-2.041
-2.217
-2.581
-2.955
-3.336
-3.721
-4.109
-1.351
-1.299
-1.260
-1.233
-1.214
-1.202
-1.197
-1.197
-1.201
-1.208
-1.240
-1.285
-1.341
-1.403
-1.547
-1.712
-1.889
-2.078
-2.274
-2.475
-2.889
-3.314
-3.744
-4.180
-4.618
-1.320
-1.281
-1.252
-1.233
-1.221
-1.215
-1.215
-1.219
-1.227
-1.239
-1.280
-1.334
-1.398
-1.470
-1.634
-1.817
-2.014
-2.221
-2.435
-2.654
-3.104
-3.564
-4.030
-4.500
-4.973
-1.305
-1.273
-1.251
-1.236
-1.228
-1.226
-1.229
-1.237
-1.248
-1.262
-1.310
-1.371
-1.442
-1.521
-1.700
-1.897
-2.108
-2.329
-2.557
-2.789
-3.267
-3.753
-4.246
-4.742
-5.243
-1.296
-1.268
-1.250
-1.239
-1.234
-1.235
-1.241
-1.251
-1.264
-1.281
-1.334
-1.400
-1.477
-1.562
-1.751
-1.960
-2.183
-2.415
-2.653
-2.897
-3.396
-3.904
-4.418
-4.937
-5.459
-1.291
-1.267
-1.251
-1.243
-1.240
-1.243
-1.251
-1.262
-1.277
-1.296
-1.353
-1.424
-1.505
-1.595
-1.794
-2.013
-2.244
-2.485
-2.733
-2.986
-3.503
-4.029
-4.561
-5.098
-5.638
-1.287
-1.266
-1.253
-1.246
-1.245
-1.250
-1.259
-1.272
-1.289
-1.309
-1.370
-1.444
-1.530
-1.623
-1.830
-2.057
-2.296
-2.545
-2.801
-3.061
-3.593
-4.135
-4.683
-5.234
-5.789
-1.285
-1.266
-1.254
-1.249
-1.250
-1.256
-1.266
-1.280
-1.298
-1.320
-1.384
-1.462
-1.551
-1.647
-1.862
-2.095
-2.341
-2.596
-2.858
-3.126
-3.671
-4.226
-4.787
-5.352
-5.920
-1.283
-1.266
-1.255
-1.252
-1.254
-1.261
-1.273
-1.288
-1.307
-1.329
-1.396
-1.477
-1.569
1.669
-1.889
-2.128
-2.380
-2.641
-2.910
-3.183
-3.740
-4.306
-4.879
-5.455
-6.035
-1.281
-1.266
-1.257
-1.254
-1.257
-1.266
-1.278
-1.294
-1.314
-1.337
-1.407
-1.491
-1.585
-1.688
-1.913
-2.157
-2.415
-2.681
-2.955
-3.233
-3.800
-4.377
-4.960
-5.547
-6.137
-1.281
-1.266
-1.258
-1.257
-1.261
-1.270
-1.283
-1.301
-1.321
-1.345
-1.471
-1.503
-1.599
-1.704
-1.934
-2.183
-2.446
-2.717
-2.995
-3.278
-3.855
-4.441
-5.033
-5.629
-6.228
-1.280
-1.266
-1.259
-1.258
-1.264
-1.274
-1.288
-1.306
-1.327
-1.353
-1.426
-1.514
-1.612
-1.719
-1.953
-2.207
-2.473
-2.749
-3.031
-3.319
-3.904
-4.498
-5.099
-5.703
-6.311
Source: Land (1975)
Footnote. Notation n = 3(1)19(2)25(3)31(5)36 is shorthand for n from 3 to 19 by unit steps, from 19 to 25 by 2's, from 25 to 31 by
3's, and from 31 to 36 by 5's
                                                     D-234
                               March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
        Table 21-4. Land's Factors (H.io) for Confidence Bounds on Lognormal Arithmetic Mean for n =
                                         3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
15
-1.279
-1.266
-1.260
-1.261
-1.266
-1.277
-1.292
-1.311
-1.333
-1.358
-1.434
-1.523
-1.624
-1.733
-1.971
-2.229
-2.499
-2.778
-3.064
-3.356
-3.949
-4.549
-5.159
-5.771
-6.386
16
-1.278
-1.267
-1.261
-1.262
-1.269
-1.280
-1.296
-1.315
-1.338
-1.364
-1.441
-1.533
-1.634
-1.746
-1.987
-2.248
-2.522
-2.805
-3.095
-3.390
-3.989
-4.599
-5.213
-5.833
-6.455
17
-1.278
-1.267
-1.262
-1.264
-1.271
-1.283
-1.299
-1.319
-1.342
-1.369
-1.448
-1.541
-1.645
-1.757
-2.002
2.266
2.544
2.830
3.123
3.421
4.027
4.642
5.264
5.890
6.518
18
-1.278
-1.267
-1.263
-1.266
-1.273
-1.286
-1.302
-1.323
-1.346
-1.374
-1.455
-1.548
-1.654
-1.767
-2.016
-2.283
-2.563
-2.853
-3.149
-3.450
-4.062
-4.683
-5.311
-5.942
-6.578
19
-1.278
-1.268
-1.265
-1.267
-1.275
-1.288
-1.305
-1.326
-1.351
-1.378
-1.460
-1.555
-1.662
-1.777
-2.029
-2.298
-2.581
-2.874
-3.173
-3.477
-4.094
-4.721
-5.354
-5.992
-6.632
21
-1.277
-1.268
-1.266
-1.270
-1.279
-1.292
-1.310
-1.332
-1.358
-1.387
-1.470
-1.568
-1.677
-1.795
-2.051
-2.326
-2.615
-2.913
-3.217
-3.525
-4.153
-4.790
-5.433
-6.080
-6.730
23
-1.277
-1.270
-1.268
-1.272
-1.281
-1.296
-1.315
-1.338
-1.364
-1.393
-1.479
-1.579
-1.690
-1.810
-2.072
-2.351
-2.644
-2.946
-3.255
-3.567
-4.204
-4.850
-5.002
-6.158
-6.817
25
-1.277
-1.270
-1.269
-1.274
-1.284
-1.299
-1.319
-1.342
-1.369
-1.399
-1.487
-1.589
-1.703
-1.825
-2.090
-2.373
-2.670
-2.976
-3.288
-3.605
-4.250
-4.604
-5.564
-6.228
-6.894
28
-1.277
-1.271
-1.271
-1.277
-1.288
-1.304
-1.324
-1.349
-1.377
-1.408
-1.498
-1.602
-1.718
-1.843
-2.113
-2.402
-2.704
-3.015
-3.333
-3.655
-4.311
-4.975
-5.645
-6.319
-6.996
31
-1.277
-1.272
-1.272
-1.279
-1.291
-1.307
-1.329
-1.354
-1.383
-1.414
-1.507
-1.613
-1.732
-1.859
-2.133
-2.427
-2.733
-3.050
-3.372
-3.698
-4.363
-5.037
-5.715
-6.399
-8.755
36
-1.277
-1.272
-1.275
-1.282
-1.295
-1.313
-1.336
-1.361
-1.391
-1.424
-1.519
-1.629
-1.750
-1.881
-2.161
-2.461
-2.775
-3.097
-3.426
-3.759
-4.436
-5.122
-5.815
-6.510
-7.208
                                                  D-235
      March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
         Table 21-5. Land's Factors (H.90) for Confidence Bounds on Lognormal Arithmetic Mean for n
                                            3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
3
1.686
1.885
2.156
2.521
2.990
3.542
4.136
4.742
5.349
5.955
7.466
8.973
10.480
11.980
14.990
18.000
21.000
24.000
27.010
30.010
36.020
42.020
48.030
54.030
60.040
4
1.506
1.620
1.763
1.942
2.160
2.417
2.708
3.023
3.353
3.691
4.558
5.436
6.319
7.206
8.986
10.770
12.560
14.340
16.130
17.920
21.490
25.070
28.650
32.230
35.810
5
1.438
1.522
1.627
1.755
1.907
2.084
2.284
2.503
2.736
2.980
3.617
4.276
4.944
5.619
6.979
8.346
9.717
11.090
12.470
13.840
16.600
19.350
22.110
24.870
27.630
6
1.403
1.472
1.558
1.662
1.785
1.926
2.085
2.260
2.447
2.644
3.167
3.713
4.273
4.842
5.990
7.147
8.312
9.480
10.650
11.820
14.170
16.510
18.860
21.210
23.560
7
1.381
1.442
1.517
1.607
1.712
1.834
1.970
2.119
2.280
2.450
2.904
3.383
3.877
4.380
5.401
6.434
7.473
8.516
9.562
10.610
12.710
14.810
16.910
19.020
21.120
8
1.367
1.422
1.489
1.569
1.664
1.773
1.894
2.027
2.171
2.324
2.732
3.166
3.615
4.075
5.010
5.958
6.913
7.873
8.836
9.800
11.740
13.670
15.610
17.550
19.490
9
1.356
1.407
1.469
1.543
1.630
1.729
1.849
1.962
2.094
2.234
2.610
3.012
3.429
3.857
4.730
5.617
6.511
7.411
8.314
9.219
11.030
12.850
14.670
16.500
18.320
10
1.349
1.396
1.453
1.523
1.604
1.696
1.800
1.914
2.036
2.167
2.518
2.896
3.289
3.693
4.518
5.359
6.208
7.062
7.919
8.779
10.500
12.230
13.960
15.700
17.430
11
1.343
1.387
1.441
1.507
1.583
1.671
1.768
1.876
1.992
2.115
2.448
2.806
3.180
3.564
4.353
5.157
5.970
6.788
7.610
8.434
10.090
11.750
13.410
15.070
16.730
12
1.338
1.380
1.432
1.494
1.567
1.650
1.743
1.845
1.955
2.073
2.391
2.733
3.092
3.461
4.220
4.994
5.778
6.566
7.360
8.155
9.751
11.350
12.960
14.560
16.170
13
1.334
1.374
1.424
1.483
1.553
1.633
1.722
1.820
1.926
2.038
2.344
2.674
3.109
3.376
4.110
4.860
5.619
6.384
7.154
7.924
9.473
11.030
12.580
14.140
15.700
14
1.330
1.369
1.417
1.474
1.542
1.619
1.705
1.799
1.901
2.010
2.305
2.623
2.959
3.305
4.017
4.746
5.486
6.299
6.978
7.729
9.238
10.750
12.270
13.790
15.310
Source: Land (1975)
Footnote. Notation n = 3(1)19(2)25(3)31(5)36 is shorthand for n from 3 to 19 by unit steps, from 19 to 25 by 2's, from 25 to 31 by
3's, and from 31 to 36 by 5's
                                                     D-236
       March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
        Table 21-5. Land's Factors (H.90) for Confidence Bounds on Lognormal Arithmetic Mean for n =
                                         3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
15
1.328
1.365
1.411
1.467
1.532
1.606
1.690
1.781
1.880
1.985
2.271
2.581
2.907
3.244
3.938
4.650
5.370
6.097
6.829
7.563
9.037
10.520
12.000
13.480
14.970
16
1.325
1.361
1.406
1.460
1.524
1.596
1.766
1.765
1.861
1.963
2.242
2.544
2.862
3.191
3.870
4.565
5.271
5.983
6.699
7.418
8.862
10.310
11.770
13.220
14.680
17
1.323
1.358
1.402
1.455
1.516
1.586
1.666
1.752
1.845
1.945
2.217
2.512
2.823
3.145
3.810
4.492
5.184
5.883
6.586
7.292
8.710
10.130
11.560
12.990
14.420
18
1.322
1.355
1.398
1.449
1.509
1.578
1.655
1.739
1.831
1.929
2.195
2.483
2.788
3.104
3.757
4.427
5.107
5.794
6.485
7.179
8.575
9.975
11.380
12.780
14.190
19
1.320
1.353
1.394
1.444
1.503
1.570
1.646
1.728
1.819
1.914
2.174
2.458
2.757
3.069
3.710
4.369
5.039
5.715
6.396
7.080
8.454
9.833
11.220
12.600
13.990
21
1.317
1.348
1.388
1.437
1.494
1.558
1.631
1.710
1.797
1.889
2.141
2.415
2.705
3.005
3.629
4.270
4.921
5.580
6.243
6.909
8.248
9.592
10.940
12.290
13.640
23
1.315
1.345
1.383
1.430
1.485
1.548
1.618
1.695
1.779
1.868
2.113
2.379
2.662
2.954
3.562
4.188
4.825
5.468
6.116
6.767
8.076
9.391
10.710
12.030
13.350
25
1.313
1.342
1.379
1.425
1.478
1.539
1.607
1.682
1.764
1.851
2.089
2.349
2.625
2.911
3.506
4.119
4.743
5.374
6.009
6.648
-7.933
9.222
10.520
11.810
13.110
28
1.310
1.338
1.374
1.417
1.469
1.528
1.594
1.667
1.745
1.830
2.060
2.312
2.579
2.858
3.463
4.033
4.641
5.257
5.876
6.500
7.753
9.013
10.280
11.540
12.810
31
1.308
1.335
1.370
1.412
1.462
1.519
1.583
1.654
1.731
1.812
2.036
2.282
2.543
2.814
3.380
3.964
4.559
5.161
5.769
6.379
7.607
8.842
10.080
11.320
12.560
36
1.306
1.332
1.364
1.404
1.452
1.507
1.568
1.636
1.710
1.789
2.005
2.242
2.494
2.758
3.305
3.872
4.450
5.036
5.626
6.219
7.415
8.616
9.821
11.030
12.240
                                                  D-237
      March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
         Table 21-6. Land's Factors (H.95) for Confidence Bounds on Lognormal Arithmetic Mean for n
                                            3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
3
2.750
3.295
4.109
5.220
6.495
7.807
9.120
10.430
11.740
13.050
16.330
19.600
22.870
26.140
32.690
39.230
45.770
52.310
58.850
65.390
78.470
91.550
104.600
117.700
130.800
4
2.222
2.463
2.777
3.175
3.658
4.209
4.801
5.414
6.038
6.669
8.265
9.874
11.490
13.110
16.350
19.600
22.850
26.110
29.360
32.620
39.130
45.650
52.160
58.680
65.200
5
2.035
2.198
2.402
2.651
2.947
3.287
3.662
4.062
4.478
4.905
6.001
7.120
8.250
9.387
11.670
13.970
16.270
18.580
20.880
23.190
27.810
32.430
37.060
41.680
46.310
6
1.942
2.069
2.226
2.415
2.638
2.892
3.173
3.477
3.796
4.127
4.990
5.880
6.786
7.701
9.546
11.400
13.270
15.140
17.010
18.880
22.630
26.390
30.140
33.900
37.660
7
1.886
1.992
2.125
2.282
2.465
2.673
2.904
3.155
3.420
3.698
4.426
5.184
5.960
6.747
8.339
9.945
11.560
13.180
14.800
16.430
19.680
22.940
26.200
29.460
32.730
8
1.849
1.943
2.058
2.195
2.354
2.534
2.735
2.952
3.184
3.426
4.069
4.741
5.432
6.135
7.563
9.006
10.460
11.920
13.380
14.840
17.780
20.720
23.660
26.600
29.540
9
1.822
1.908
2.011
2.134
2.277
2.439
2.618
2.813
3.021
3.239
3.820
4.433
5.065
5.710
7.021
8.350
9.688
11.030
12.380
13.730
16.440
19.160
21.870
24.590
27.310
10
1.802
1.881
1.977
2.089
2.220
2.368
2.532
2.710
2.902
3.103
3.639
4.207
4.795
5.396
6.621
7.864
9.118
10.380
11.640
12.910
15.450
18.000
20.550
23.100
25.660
11
1.787
1.860
1.949
2.054
2.176
2.314
2.466
2.632
2.810
2.998
3.500
4.033
4.587
5.154
6.312
7.489
8.677
9.872
11.070
12.270
14.690
17.100
19.530
21.950
24.380
12
1.775
1.843
1.927
2.026
2.141
2.271
2.414
2.570
2.738
2.915
3.389
3.896
4.422
4.962
6.067
7.191
8.326
9.469
10.620
11.770
14.080
16.390
18.710
21.030
23.350
13
1.763
1.830
1.909
2.003
2.112
2.235
2.371
2.520
2.679
2.848
3.300
3.784
4.288
4.805
5.866
6.947
8.039
9.140
10.240
11.350
13.580
15.810
18.040
20.280
22.510
14
1.756
1.818
1.894
1.984
2.088
2.206
2.336
2.479
2.631
2.792
3.226
3.691
4.176
4.675
5.698
6.743
7.799
8.864
9.933
11.010
13.160
15.320
17.480
19.650
21.820
Source: Land (1975)
Footnote. Notation n = 3(1)19(2)25(3)31(5)36 is shorthand for n from 3 to 19 by unit steps, from 19 to 25 by 2's, from 25 to 31 by
3's, and from 31 to 36 by 5's
                                                     D-238
       March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
        Table 21-6. Land's Factors (H.95) for Confidence Bounds on Lognormal Arithmetic Mean for n =
                                         3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
15
1.749
1.809
1.882
1.968
2.068
2.181
2.306
2.443
2.589
2.744
3.163
3.612
4.081
4.564
5.557
6.570
7.596
8.630
9.669
10.710
12.810
14.900
17.010
19.110
21.220
16
1.743
1.800
1.871
1.954
2.050
2.160
2.280
2.412
2.554
2.704
3.109
3.544
4.000
4.470
5.435
6.422
7.422
8.429
9.442
10.460
12.500
14.550
16.600
18.650
20.710
17
1.738
1.793
1.861
1.942
2.035
2.141
2.258
2.386
2.523
2.669
3.062
3.485
3.929
4.387
5.328
6.293
7.269
8.254
9.244
10.240
12.230
14.240
16.240
18.250
20.260
18
1.733
1.787
1.853
1.931
2.021
2.124
2.238
2.362
2.496
2.638
3.021
3.434
3.867
4.314
5.236
6.179
7.136
8.100
9.070
10.040
12.000
13.960
15.930
17.900
19.870
19
1.729
1.781
1.845
1.921
2.009
2.110
2.221
2.342
2.472
2.611
2.984
3.388
3.812
4.251
5.153
6.078
7.016
7.963
8.916
9.872
11.790
13.720
15.650
17.590
19.520
21
1.722
1.771
1.833
1.905
1.989
2.085
2.191
2.307
2.432
2.564
2.923
3.311
3.719
4.141
5.013
5.907
6.815
7.731
8.652
9.579
11.440
13.310
15.180
17.050
18.930
23
1.716
1.763
1.822
1.892
1.973
2.065
2.167
2.279
2.399
2.526
2.873
3.248
3.643
4.052
4.898
5.766
6.649
7.540
8.437
9.338
11.150
12.970
14.790
16.620
18.440
25
1.711
1.756
1.813
1.881
1.959
2.048
2.147
2.255
2.371
2.495
2.830
3.195
3.579
3.977
4.802
5.649
6.510
7.380
8.257
9.137
10.910
12.680
14.470
16.250
18.040
28
1.706
1.749
1.802
1.867
1.942
2.027
2.122
2.225
2.337
2.456
2.779
3.130
3.501
3.886
4.683
5.504
6.340
7.184
8.034
8.889
10.610
12.330
14.060
15.800
17.530
31
1.701
1.742
1.793
1.856
1.928
2.010
2.102
2.202
2.310
2.423
2.737
3.077
3.437
3.812
4.588
5.388
6.201
7.024
7.854
8.688
10.360
12.050
13.740
15.430
12.560
36
1.695
1.734
1.783
1.841
1.910
1.988
2.075
2.171
2.273
2.383
2.682
3.008
3.355
3.715
4.463
5.234
6.020
6.816
7.618
8.424
10.050
11.680
13.310
14.950
16.590
                                                  D-239
      March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
         Table 21-7. Land's Factors (H.975) for Confidence Bounds on Lognormal Arithmetic Mean for n
                                            3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
3
4.367
5.849
8.166
10.860
13.590
16.310
19.040
21.760
24.490
27.210
34.020
40.830
47.630
54.440
68.050
81.660
95.270
108.900
122.500
136.100
163.300
190.600
217.800
245.000
272.200
4
3.100
3.571
4.210
5.031
5.989
7.019
8.083
9.164
10.250
11.350
14.110
16.880
19.650
22.430
28.000
33.580
39.160
44.740
50.320
55.900
67.070
78.240
89.410
100.600
111.800
5
2.703
2.987
3.348
3.794
4.322
4.914
5.548
6.208
6.885
7.572
9.320
11.090
12.880
14.670
18.270
21.870
25.490
29.110
32.730
36.350
43.590
50.840
58.100
65.350
72.600
6
2.513
2.723
2.982
3.296
3.664
4.081
4.534
5.014
5.512
6.024
7.339
8.684
10.050
11.420
14.180
16.960
19.740
22.530
25.320
28.120
33.710
39.310
44.910
50.510
56.110
7
2.403
2.573
2.781
3.030
3.319
3.647
4.005
4.389
4.791
5.206
6.285
7.397
8.528
9.671
11.980
14.300
16.640
18.980
21.320
23.670
28.370
33.070
37.770
42.480
47.190
8
2.330
2.476
2.653
2.864
3.107
3.382
3.684
4.009
4.351
4.707
5.636
6.602
7.588
8.588
10.610
12.650
14.710
16.770
18.830
20.890
25.030
29.180
33.330
37.470
41.620
9
2.879
2.409
2.565
2.750
2.963
3.204
3.469
3.754
4.056
4.371
5.199
6.064
6.951
7.853
9.681
11.530
13.390
15.260
17.130
19.000
22.760
26.520
30.280
34.050
37.820
10
2.242
2.359
2.501
2.667
2.859
3.076
3.314
3.572
3.844
4.130
4.884
5.676
6.490
7.320
9.006
10.710
12.430
14.160
15.890
17.630
21.100
24.590
28.080
31.570
35.060
11
2.212
2.321
2.451
2.604
2.780
2.979
3.198
3.434
3.685
3.949
4.647
5.383
6.142
6.916
8.493
10.090
11.700
13.320
14.950
16.580
19.850
23.120
26.390
29.670
32.950
12
2.190
2.291
2.411
2.554
2.718
2.903
3.106
3.327
3.561
3.807
4.461
5.153
5.869
6.599
8.091
9.605
11.130
12.670
14.210
15.750
18.850
21.960
25.070
28.180
31.290
13
2.169
2.265
2.380
2.514
2.668
2.842
3.033
3.240
3.461
3.693
4.312
4.968
5.648
6.344
7.765
9.210
10.670
12.140
13.610
15.090
18.050
21.020
23.990
26.970
29.950
14
2.155
2.245
2.353
2.480
2.626
2.791
2.973
3.169
3.379
3.599
4.189
4.815
5.466
6.133
7.497
8.884
10.290
11.700
13.110
14.540
17.390
20.250
23.110
25.970
28.840
Source: Land (1975)
Footnote. Notation n = 3(1)19(2)25(3)31(5)36 is shorthand for n from 3 to 19 by unit steps, from 19 to 25 by 2's, from 25 to 31 by
3's, and from 31 to 36 by 5's
                                                     D-240
       March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
        Table 21-7. Land's Factors (H.975) for Confidence Bounds on Lognormal Arithmetic Mean for n =
                                         3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
15
2.141
2.227
2.331
2.452
2.592
2.749
2.922
3.109
3.310
3.521
4.086
4.688
5.314
5.956
7.271
8.610
9.964
11.330
12.700
14.070
16.830
19.590
22.360
25.130
27.900
16
2.130
2.212
2.311
2.428
2.562
2.712
2.879
3.059
3.251
3.454
3.998
4.579
5.183
5.804
7.078
8.376
9.689
11.010
12.340
13.670
16.350
19.030
21.720
24.410
27.100
17
2.120
2.199
2.295
2.407
2.536
2.681
2.841
3.015
3.200
3.397
3.922
4.485
5.070
5.674
6.911
8.174
9.451
10.740
12.030
13.330
15.930
18.550
21.170
23.790
26.410
18
2.112
2.188
2.280
2.388
2.513
2.653
2.808
2.976
3.157
3.347
3.856
4.402
4.972
5.559
6.765
7.996
9.242
10.500
11.760
13.030
15.570
18.130
20.680
23.240
25.800
19
2.104
2.178
2.267
2.372
2.493
2.630
2.780
2.943
3.117
3.302
3.798
4.330
4.887
5.461
6.636
7.840
9.058
10.290
11.520
12.760
15.250
17.750
20.250
22.760
25.270
21
2.091
2.161
2.246
2.345
2.460
2.588
2.731
2.886
3.052
3.227
3.700
4.209
4.740
5.289
6.419
7.576
8.748
9.930
11.120
12.310
14.710
17.120
19.530
21.940
24.360
23
2.081
2.147
2.228
2.323
2.432
2.555
2.692
2.840
2.999
3.167
3.621
4.109
4.622
5.151
6.243
7.361
8.495
9.639
10.790
11.950
14.270
16.600
18.940
21.280
23.620
25
2.072
2.135
2.213
2.305
2.409
2.528
2.659
2.802
2.955
3.116
3.555
4.027
4.524
5.037
6.096
7.182
8.284
9.397
10.520
11.640
13.900
16.170
18.450
20.720
23.000
28
2.062
2.121
2.194
2.281
2.381
2.494
2.619
2.755
2.901
3.056
3.474
3.927
4.404
4.897
5.916
6.963
8.027
9.101
10.180
11.270
13.450
15.650
17.840
20.050
22.250
31
2.053
2.110
2.180
2.263
2.359
2.467
2.587
2.717
2.858
3.007
3.410
3.847
4.307
4.784
5.772
6.787
7.820
8.863
9.913
10.970
13.090
15.220
17.360
19.500
17.130
36
2.043
2.096
2.161
2.239
2.329
2.432
2.545
2.668
2.801
2.943
3.327
3.743
4.183
4.639
5.585
6.559
7.551
8.554
9.564
10.580
12.620
14.670
16.730
18.790
20.850
                                                  D-241
      March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
         Table 21-8. Land's Factors (H.99) for Confidence Bounds on Lognormal Arithmetic Mean for n
                                            3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
3
8.328
13.940
20.880
27.850
34.820
41.780
48.750
55.710
62.580
69.650
87.060
104.500
121.900
139.300
174.100
208.900
243.800
278.600
313.400
348.200
417.900
487.500
557.200
626.900
696.500
4
4.665
5.768
7.336
9.244
11.290
13.390
15.520
17.650
19.800
21.950
27.350
32.770
38.190
43.610
54.470
65.340
76.210
87.080
97.960
108.800
130.600
152.300
174.100
195.900
217.600
5
3.760
4.310
5.035
5.934
6.966
8.077
9.231
10.410
11.600
12.810
15.850
18.920
22.010
25.100
31.290
37.500
43.720
49.940
56.160
62.380
74.840
87.290
99.750
112.200
124.700
6
3.360
3.731
4.199
4.771
5.434
6.167
6.947
7.757
8.856
9.430
11.580
13.760
15.950
18.160
22.600
27.050
31.520
35.980
40.450
44.930
53.880
62.840
71.790
80.750
89.720
7
3.137
3.422
3.775
4.199
4.691
5.240
5.831
6.452
7.095
7.753
9.442
11.170
12.920
14.680
18.220
21.790
25.360
28.940
32.530
36.120
43.300
50.490
57.680
64.870
72.070
8
2.994
3.231
3.519
3.862
4.258
4.702
5.183
5.693
6.225
6.772
8.186
9.641
11.120
12.610
15.630
18.660
21.710
24.760
27.820
30.880
37.010
43.140
49.280
55.430
61.570
9
2.897
3.101
3.348
3.640
3.976
4.353
4.764
5.201
5.659
6.133
7.365
8.640
9.940
11.260
13.920
16.600
19.300
22.000
24.710
27.420
32.860
38.300
43.740
49.190
54.640
10
2.825
3.006
3.225
3.482
3.778
4.109
4.471
4.858
5.264
5.686
6.789
7.936
9.109
10.300
12.710
15.140
17.590
20.050
22.510
24.980
29.920
34.870
39.820
44.770
49.730
11
2.770
2.935
3.132
3.364
3.631
3.929
4.255
4.604
4.973
5.357
6.363
7.414
8.492
9.587
11.810
14.060
16.320
18.590
20.870
23.150
27.730
32.310
36.890
41.480
46.070
12
2.727
2.878
3.060
3.273
3.517
3.790
4.089
4.110
4.750
5.103
6.036
7.102
8.016
9.039
11.120
13.220
15.340
17.470
19.600
21.740
26.030
30.330
34.630
38.930
43.240
13
2.691
2.833
3.002
3.200
3.426
3.680
3.958
4.256
4.572
4.903
5.775
6.693
7.638
8.602
10.560
12.540
14.560
16.570
18.590
20.620
24.680
28.750
32.820
36.900
40.980
14
2.663
2.796
2.955
3.140
3.353
3.590
3.851
4.131
4.428
4.740
5.564
6.432
7.330
8.245
10.110
12.010
13.910
15.840
17.760
19.700
23.570
27.450
31.340
35.230
39.130
Source: Land (1975)
Footnote. Notation n = 3(1)19(2)25(3)31(5)36 is shorthand for n from 3 to 19 by unit steps, from 19 to 25 by 2's, from 25 to 31 by
3's, and from 31 to 36 by 5's
                                                     D-242
       March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
        Table 21-8. Land's Factors (H.99) for Confidence Bounds on Lognormal Arithmetic Mean for n =
                                         3(1)19(2)25(3)31(5)36
Sy\ll
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
15
2.638
2.764
2.914
3.090
3.291
3.515
3.762
4.027
4.309
4.605
5.388
6.217
7.074
7.949
9.735
11.550
13.380
15.230
17.070
18.930
22.650
26.380
30.110
33.840
37.580
16
2.618
2.737
2.880
3.047
3.239
3.453
3.687
3.940
4.209
4.491
5.240
6.034
6.857
7.699
9.415
11.170
12.930
14.710
16.490
18.280
21.870
25.460
29.060
32.670
36.280
17
2.600
2.714
2.851
3.011
3.194
3.398
3.623
3.865
4.123
4.394
5.114
5.878
6.671
7.483
9.145
10.840
12.540
14.260
15.990
17.720
21.190
24.680
28.170
31.660
35.150
18
2.584
2.694
2.826
2.979
3.155
3.351
3.567
3.800
4.049
4.309
5.004
5.743
6.510
7.297
8.907
10.550
12.210
13.880
15.550
17.240
20.610
24.000
27.390
30.780
34.180
19
2.571
2.676
2.803
2.951
3.121
3.311
3.519
3.744
3.983
4.235
4.908
5.625
6.369
7.134
8.700
10.300
11.910
13.540
15.170
16.810
20.100
23.400
26.700
30.010
33.320
21
2.548
2.647
2.767
2.904
3.064
3.242
3.438
3.649
3.875
4.112
4.749
5.426
6.134
6.861
8.353
9.875
11.420
12.970
14.350
16.100
19.240
22.390
25.550
28.720
31.880
23
2.529
2.623
2.735
2.867
3.017
3.186
3.372
3.573
3.787
4.013
4.620
5.267
5.944
6.641
8.073
9.536
11.020
12.510
14.010
15.520
18.550
21.580
24.630
27.670
30.720
25
2.514
2.602
2.710
2.836
2.979
3.141
3.318
3.510
3.716
3.931
4.513
5.136
5.788
6.460
7.842
9.256
10.690
12.130
13.590
15.050
17.980
20.920
23.860
26.810
29.770
28
2.495
2.579
2.679
2.798
2.933
3.085
3.253
3.434
3.628
3.833
4.385
4.978
5.599
6.241
7.562
8.916
10.290
11.670
13.070
14.470
17.280
20.100
22.930
25.760
28.600
31
2.480
2.559
2.655
2.767
2.896
3.041
3.200
3.373
3.559
3.755
4.283
4.852
5.449
6.066
7.339
8.645
9.970
11.310
12.660
14.010
16.730
19.450
22.190
24.930
21.640
36
2.462
2.534
2.623
2.729
2.849
2.984
3.134
3.296
3.471
3.655
4.143
4.691
5.256
5.842
7.052
8.269
9.560
10.840
12.120
13.420
16.010
18.620
21.230
23.800
26.470
                                                  D-243
      March 2009

-------
Appendix D. Chapter 21 Tables                                                                   Unified Guidance
                                   This page intentionally left blank
                                                     D-244                                             March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
   Table 21-9. Factors (T) for Parametric Upper Conf. Bounds on Percentiles (P)

n\(l-a)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
55
60
65
70
75
80
85
90
95
100
P = 0.80
0.80
3.417
2.016
1.675
1.514
1.417
1.352
1.304
1.266
1.237
1.212
1.192
1.174
1.159
1.145
1.133
1.123
1.113
1.104
1.096
1.089
1.082
1.076
1.070
1.065
1.060
1.055
1.051
1.047
1.043
1.039
1.035
1.032
1.029
1.026
1.023
1.020
1.017
1.015
1.013
1.010
1.008
1.006
1.004
1.002
1.000
0.998
0.996
0.994
0.993
0.985
0.978
0.972
0.967
0.963
0.959
0.955
0.951
0.948
0.945
0.90
6.987
3.039
2.295
1.976
1.795
1.676
1.590
1.525
1.474
1.433
1.398
1.368
1.343
1.321
1.301
1.284
1.268
1.254
1.241
1.229
1.218
1.208
1.199
1.190
1.182
1.174
1.167
1.160
1.154
1.148
1.143
1.137
1.132
1.127
1.123
1.118
1.114
1.110
1.106
1.103
1.099
1.096
1.092
1.089
1.086
1.083
1.080
1.078
1.075
1.063
1.052
1.043
1.035
1.028
1.022
1.016
1.011
1.006
1.001
0.95
14.051
4.424
3.026
2.483
2.191
2.005
1.875
1.779
1.703
1.643
1.593
1.551
1.514
1.483
1.455
1.431
1.409
1.389
1.371
1.355
1.340
1.326
1.313
1.302
1.291
1.280
1.271
1.262
1.253
1.245
1.237
1.230
1.223
1.217
1.211
1.205
1.199
1.194
1.188
1.183
1.179
1.174
1.170
1.165
1.161
1.157
1.154
1.150
1.146
1.130
1.116
1.104
1.094
1.084
1.076
1.068
1.061
1.055
1.049
0.975
28.140
6.343
3.915
3.058
2.621
2.353
2.170
2.036
1.933
1.851
1.784
1.728
1.681
1.639
1.603
1.572
1.543
1.518
1.495
1.474
1.455
1.437
1.421
1.406
1.392
1.379
1.367
1.355
1.344
1.334
1.325
1.316
1.307
1.299
1.291
1.284
1.277
1.270
1.263
1.257
1.251
1.246
1.240
1.235
1.230
1.225
1.220
1.216
1.211
1.191
1.174
1.159
1.146
1.135
1.124
1.115
1.106
1.098
1.091
0.99
70.376
10.111
5.417
3.958
3.262
2.854
2.584
2.391
2.246
2.131
2.039
1.963
1.898
1.843
1.795
1.753
1.716
1.682
1.652
1.625
1.600
1.577
1.556
1.537
1.519
1.502
1.486
1.472
1.458
1.445
1.433
1.422
1.411
1.400
1.391
1.381
1.372
1.364
1.356
1.348
1.341
1.333
1.327
1.320
1.314
1.308
1.302
1.296
1.291
1.266
1.245
1.226
1.210
1.196
1.183
1.171
1.161
1.151
1.142
P = 0.90
0.80
5.049
2.871
2.372
2.145
2.012
1.923
1.859
1.809
1.770
1.738
1.711
1.689
1.669
1.652
1.637
1.623
1.611
1.600
1.590
1.581
1.572
1.564
1.557
1.550
1.544
1.538
1.533
1.528
1.523
1.518
1.514
1.510
1.506
1.502
1.498
1.495
1.492
1.489
1.486
1.483
1.480
1.477
1.475
1.472
1.470
1.468
1.465
1.463
1.461
1.452
1.444
1.437
1.430
1.425
1.420
1.415
1.411
1.408
1.404
0.90
10.253
4.258
3.188
2.742
2.494
2.333
2.219
2.133
2.066
2.011
1.966
1.928
1.895
1.867
1.842
1.819
1.800
1.782
1.765
1.750
1.737
1.724
1.712
1.702
1.691
1.682
1.673
1.665
1.657
1.650
1.643
1.636
1.630
1.624
1.618
1.613
1.608
1.603
1.598
1.593
1.589
1.585
1.581
1.577
1.573
1.570
1.566
1.563
1.559
1.545
1.532
1.521
1.511
1.503
1.495
1.488
1.481
1.475
1.470
0.95
20.581
6.155
4.162
3.407
3.006
2.755
2.582
2.454
2.355
2.275
2.210
2.155
2.109
2.068
2.033
2.002
1.974
1.949
1.926
1.905
1.886
1.869
1.853
1.838
1.824
1.811
1.799
1.788
1.777
1.767
1.758
1.749
1.740
1.732
1.725
1.717
1.710
1.704
1.697
1.691
1.685
1.680
1.674
1.669
1.664
1.659
1.654
1.650
1.646
1.626
1.609
1.594
1.581
1.570
1.559
1.550
1.542
1.534
1.527
0.975
41.201
8.797
5.354
4.166
3.568
3.206
2.960
2.783
2.647
2.540
2.452
2.379
2.317
2.264
2.218
2.177
2.141
2.108
2.079
2.053
2.028
2.006
1.985
1.966
1.949
1.932
1.917
1.903
1.889
1.877
1.865
1.853
1.843
1.833
1.823
1.814
1.805
1.797
1.789
1.781
1.774
1.767
1.760
1.753
1.747
1.741
1.735
1.730
1.724
1.700
1.679
1.661
1.645
1.630
1.618
1.606
1.596
1.586
1.578
0.99
103.029
13.995
7.380
5.362
4.411
3.859
3.497
3.240
3.048
2.898
2.777
2.677
2.593
2.521
2.459
2.405
2.357
2.314
2.276
2.241
2.209
2.180
2.154
2.129
2.106
2.085
2.065
2.047
2.030
2.014
1.998
1.984
1.970
1.957
1.945
1.934
1.922
1.912
1.902
1.892
1.883
1.874
1.865
1.857
1.849
1.842
1.835
1.828
1.821
1.790
1.764
1.741
1.722
1.704
1.688
1.674
1.661
1.650
1.639
Source: Hahn & Meeker (1991)
                                      D-245
       March 2009

-------
Appendix D.  Chapter 21 Tables
Unified Guidance
   Table 21-9. Factors (T) for Parametric Upper Conf. Bounds on Percentiles (P)

n\(l-a)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
55
60
65
70
75
80
85
90
95
100
P = 0.95
0.80
6.464
3.604
2.968
2.683
2.517
2.407
2.328
2.268
2.220
2.182
2.149
2.122
2.098
2.078
2.059
2.043
2.029
2.016
2.004
1.993
1.983
1.973
1.965
1.957
1.949
1.943
1.936
1.930
1.924
1.919
1.914
1.909
1.904
1.900
1.895
1.891
1.888
1.884
1.880
1.877
1.874
1.871
1.868
1.865
1.862
1.859
1.857
1.854
1.852
1.841
1.832
1.823
1.816
1.810
1.804
1.799
1.794
1.790
1.786
0.90
13.090
5.311
3.957
3.400
3.092
2.894
2.754
2.650
2.568
2.503
2.448
2.402
2.363
2.329
2.299
2.272
2.249
2.227
2.208
2.190
2.174
2.159
2.145
2.132
2.120
2.109
2.099
2.089
2.080
2.071
2.063
2.055
2.048
2.041
2.034
2.028
2.022
2.016
2.010
2.005
2.000
1.995
1.990
1.986
1.981
1.977
1.973
1.969
1.965
1.948
1.933
1.920
1.909
1.899
1.890
1.882
1.874
1.867
1.861
0.95
26.260
7.656
5.144
4.203
3.708
3.399
3.187
3.031
2.911
2.815
2.736
2.671
2.614
2.566
2.524
2.486
2.453
2.423
2.396
2.371
2.349
2.328
2.309
2.292
2.275
2.260
2.246
2.232
2.220
2.208
2.197
2.186
2.176
2.167
2.158
2.149
2.141
2.133
2.125
2.118
2.111
2.105
2.098
2.092
2.086
2.081
2.075
2.070
2.065
2.042
2.022
2.005
1.990
1.976
1.964
1.954
1.944
1.935
1.927
0.975
52.559
10.927
6.602
5.124
4.385
3.940
3.640
3.424
3.259
3.129
3.023
2.936
2.861
2.797
2.742
2.693
2.650
2.611
2.576
2.544
2.515
2.489
2.465
2.442
2.421
2.402
2.384
2.367
2.351
2.336
2.322
2.308
2.296
2.284
2.272
2.262
2.251
2.241
2.232
2.223
2.214
2.206
2.198
2.190
2.183
2.176
2.169
2.163
2.156
2.128
2.103
2.082
2.063
2.047
2.032
2.019
2.006
1.995
1.985
0.99
131.426
17.370
9.083
6.578
5.406
4.728
4.285
3.972
3.738
3.556
3.410
3.290
3.189
3.102
3.028
2.963
2.905
2.854
2.808
2.766
2.729
2.694
2.662
2.633
2.606
2.581
2.558
2.536
2.515
2.496
2.478
2.461
2.445
2.430
2.415
2.402
2.389
2.376
2.364
2.353
2.342
2.331
2.321
2.312
2.303
2.294
2.285
2.277
2.269
2.233
2.202
2.176
2.153
2.132
2.114
2.097
2.082
2.069
2.056
P = 0.99
0.80
9.156
5.010
4.110
3.711
3.482
3.331
3.224
3.142
3.078
3.026
2.982
2.946
2.914
2.887
2.863
2.841
2.822
2.804
2.789
2.774
2.761
2.749
2.738
2.727
2.718
2.708
2.700
2.692
2.684
2.677
2.671
2.664
2.658
2.652
2.647
2.642
2.637
2.632
2.627
2.623
2.619
2.615
2.611
2.607
2.604
2.600
2.597
2.594
2.590
2.576
2.564
2.554
2.544
2.536
2.528
2.522
2.516
2.510
2.505
0.90
18.500
7.340
5.438
4.666
4.243
3.972
3.783
3.641
3.532
3.443
3.371
3.309
3.257
3.212
3.172
3.137
3.105
3.077
3.052
3.028
3.007
2.987
2.969
2.952
2.937
2.922
2.909
2.896
2.884
2.872
2.862
2.852
2.842
2.833
2.824
2.816
2.808
2.800
2.793
2.786
2.780
2.773
2.767
2.761
2.756
2.750
2.745
2.740
2.735
2.713
2.694
2.677
2.662
2.649
2.638
2.627
2.618
2.609
2.601
0.95
37.094
10.553
7.042
5.741
5.062
4.642
4.354
4.143
3.981
3.852
3.747
3.659
3.585
3.520
3.464
3.414
3.370
3.331
3.295
3.263
3.233
3.206
3.181
3.158
3.136
3.116
3.098
3.080
3.064
3.048
3.034
3.020
3.007
2.995
2.983
2.972
2.961
2.951
2.941
2.932
2.923
2.914
2.906
2.898
2.890
2.883
2.876
2.869
2.862
2.833
2.807
2.785
2.765
2.748
2.733
2.719
2.706
2.695
2.684
0.975
74.234
15.043
9.018
6.980
5.967
5.361
4.954
4.662
4.440
4.265
4.124
4.006
3.907
3.822
3.749
3.684
3.627
3.575
3.529
3.487
3.449
3.414
3.382
3.353
3.325
3.300
3.276
3.254
3.233
3.213
3.195
3.178
3.161
3.145
3.131
3.116
3.103
3.090
3.078
3.066
3.055
3.044
3.034
3.024
3.014
3.005
2.996
2.988
2.980
2.943
2.911
2.883
2.859
2.838
2.819
2.802
2.786
2.772
2.759
0.99
185.617
23.896
12.387
8.939
7.335
6.412
5.812
5.389
5.074
4.829
4.633
4.472
4.337
4.222
4.123
4.037
3.960
3.892
3.832
3.777
3.727
3.681
3.640
3.601
3.566
3.533
3.502
3.473
3.447
3.421
3.398
3.375
3.354
3.334
3.315
3.297
3.280
3.264
3.249
3.234
3.220
3.206
3.193
3.180
3.168
3.157
3.146
3.135
3.125
3.078
3.038
3.004
2.974
2.947
2.924
2.902
2.883
2.866
2.850
                                     D-246
       March 2009

-------
Appendix D. Chapter 21 Tables
      Unified Guidance
   Table 21-10. Factors (T) for Parametric Lower Conf. Bounds on Percentiles (P)
                      P = 0.80
n\(l-a)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
55
60
65
70
75
80
85
90
95
100
0.80
0.288
0.377
0.432
0.470
0.499
0.522
0.540
0.556
0.569
0.580
0.591
0.599
0.608
0.615
0.621
0.627
0.633
0.638
0.643
0.647
0.651
0.655
0.659
0.662
0.665
0.669
0.671
0.674
0.677
0.679
0.682
0.684
0.686
0.688
0.690
0.692
0.694
0.696
0.698
0.699
0.701
0.702
0.704
0.705
0.707
0.708
0.709
0.711
0.712
0.718
0.723
0.727
0.731
0.735
0.738
0.741
0.743
0.746
0.748
0.90
-0.084
0.111
0.209
0.272
0.319
0.355
0.384
0.408
0.428
0.446
0.461
0.475
0.487
0.498
0.508
0.518
0.526
0.534
0.541
0.548
0.554
0.560
0.565
0.570
0.575
0.580
0.584
0.588
0.592
0.596
0.600
0.603
0.606
0.610
0.613
0.616
0.618
0.621
0.624
0.626
0.629
0.631
0.633
0.635
0.637
0.640
0.642
0.643
0.645
0.654
0.661
0.668
0.674
0.679
0.684
0.689
0.693
0.697
0.700
0.95
-0.521
-0.127
0.021
0.110
0.173
0.220
0.258
0.290
0.316
0.339
0.359
0.376
0.392
0.406
0.419
0.430
0.441
0.451
0.460
0.468
0.476
0.484
0.491
0.497
0.503
0.509
0.515
0.520
0.525
0.530
0.534
0.539
0.543
0.547
0.551
0.554
0.558
0.561
0.565
0.568
0.571
0.574
0.577
0.579
0.582
0.585
0.587
0.590
0.592
0.603
0.612
0.621
0.628
0.635
0.641
0.647
0.652
0.657
0.661
0.975
-1.229
-0.380
-0.158
-0.038
0.043
0.103
0.150
0.188
0.220
0.247
0.271
0.292
0.310
0.327
0.342
0.356
0.369
0.380
0.391
0.401
0.410
0.419
0.427
0.435
0.442
0.449
0.456
0.462
0.468
0.473
0.479
0.484
0.489
0.494
0.498
0.502
0.507
0.511
0.514
0.518
0.522
0.525
0.529
0.532
0.535
0.538
0.541
0.544
0.547
0.559
0.571
0.581
0.589
0.597
0.605
0.611
0.618
0.623
0.628
0.99
-3.204
-0.792
-0.405
-0.227
-0.117
-0.040
0.020
0.067
0.107
0.140
0.169
0.194
0.216
0.236
0.254
0.271
0.286
0.299
0.312
0.324
0.335
0.345
0.355
0.364
0.373
0.381
0.388
0.396
0.403
0.409
0.416
0.422
0.427
0.433
0.438
0.443
0.448
0.453
0.457
0.462
0.466
0.470
0.474
0.478
0.481
0.485
0.488
0.492
0.495
0.510
0.523
0.535
0.545
0.554
0.563
0.571
0.578
0.584
0.591
0.80
0.737
0.799
0.847
0.883
0.911
0.933
0.952
0.968
0.981
0.993
1.004
1.013
1.022
1.029
1.036
1.043
1.049
1.054
1.059
1.064
1.068
1.073
1.076
1.080
1.084
1.087
1.090
1.093
1.096
1.099
1.101
1.104
1.106
1.108
1.111
1.113
1.115
1.117
1.119
1.120
1.122
1.124
1.126
1.127
1.129
1.130
1.132
1.133
1.134
1.141
1.146
1.151
1.156
1.160
1.163
1.167
1.170
1.172
1.175
0.90
0.403
0.535
0.617
0.675
0.719
0.755
0.783
0.808
0.828
0.847
0.863
0.877
0.890
0.901
0.912
0.921
0.930
0.939
0.946
0.953
0.960
0.966
0.972
0.978
0.983
0.988
0.993
0.997
1.002
1.006
1.010
1.013
1.017
1.020
1.024
1.027
1.030
1.033
1.036
1.038
1.041
1.044
1.046
1.048
1.051
1.053
1.055
1.057
1.059
1.069
1.077
1.085
1.091
1.097
1.103
1.108
1.112
1.116
1.120
0.95
0.138
0.334
0.444
0.519
0.575
0.619
0.655
0.686
0.712
0.734
0.754
0.772
0.788
0.802
0.815
0.827
0.839
0.849
0.858
0.867
0.876
0.884
0.891
0.898
0.904
0.911
0.917
0.922
0.928
0.933
0.938
0.942
0.947
0.951
0.955
0.959
0.963
0.967
0.970
0.974
0.977
0.980
0.983
0.986
0.989
0.992
0.995
0.997
1.000
1.012
1.022
1.032
1.040
1.048
1.054
1.061
1.066
1.072
1.077
0.975
-0.143
0.159
0.298
0.389
0.455
0.507
0.550
0.585
0.615
0.642
0.665
0.685
0.704
0.721
0.736
0.750
0.763
0.775
0.786
0.796
0.806
0.815
0.823
0.831
0.839
0.846
0.853
0.860
0.866
0.872
0.878
0.883
0.888
0.893
0.898
0.903
0.907
0.911
0.916
0.920
0.923
0.927
0.931
0.934
0.938
0.941
0.944
0.947
0.950
0.964
0.976
0.987
0.997
1.006
1.014
1.021
1.028
1.034
1.040
0.99
-0.707
-0.072
0.123
0.238
0.319
0.381
0.431
0.472
0.508
0.538
0.565
0.589
0.610
0.629
0.647
0.663
0.678
0.692
0.705
0.716
0.728
0.738
0.748
0.757
0.766
0.774
0.782
0.790
0.797
0.804
0.810
0.817
0.823
0.828
0.834
0.839
0.844
0.849
0.854
0.859
0.863
0.867
0.872
0.876
0.880
0.883
0.887
0.891
0.894
0.910
0.924
0.937
0.948
0.958
0.968
0.976
0.984
0.991
0.998
P = 0.90
Source: Adapted from Hahn & Meeker (1991)
                                      D-247
             March 2009

-------
Appendix D. Chapter 21 Tables
      Unified Guidance
   Table 21-10. Factors (T) for Parametric Lower Conf. Bounds on Percentiles (P)
                      P = 0.95
n\(l-a)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
55
60
65
70
75
80
85
90
95
100
0.80
1.077
1.126
1.172
1.209
1.238
1.261
1.281
1.298
1.313
1.325
1.337
1.347
1.356
1.364
1.372
1.379
1.385
1.391
1.397
1.402
1.407
1.412
1.416
1.420
1.424
1.427
1.431
1.434
1.437
1.440
1.443
1.446
1.449
1.451
1.453
1.456
1.458
1.460
1.462
1.464
1.466
1.468
1.470
1.472
1.473
1.475
1.477
1.478
1.480
1.487
1.493
1.498
1.503
1.508
1.512
1.515
1.519
1.522
1.525
0.90
0.717
0.840
0.922
0.982
1.028
1.065
1.096
1.122
1.144
1.163
1.180
1.196
1.210
1.222
1.234
1.244
1.254
1.263
1.271
1.279
1.286
1.293
1.300
1.306
1.311
1.317
1.322
1.327
1.332
1.336
1.341
1.345
1.349
1.352
1.356
1.360
1.363
1.366
1.369
1.372
1.375
1.378
1.381
1.383
1.386
1.389
1.391
1.393
1.396
1.406
1.415
1.424
1.431
1.438
1.444
1.449
1.454
1.459
1.463
0.95
0.475
0.639
0.743
0.818
0.875
0.920
0.958
0.990
1.017
1.041
1.062
1.081
1.098
1.114
1.128
1.141
1.153
1.164
1.175
1.184
1.193
1.202
1.210
1.217
1.225
1.231
1.238
1.244
1.250
1.255
1.261
1.266
1.271
1.276
1.280
1.284
1.289
1.293
1.297
1.300
1.304
1.308
1.311
1.314
1.317
1.321
1.324
1.327
1.329
1.343
1.354
1.364
1.374
1.382
1.390
1.397
1.403
1.409
1.414
0.975
0.273
0.478
0.601
0.687
0.752
0.804
0.847
0.884
0.915
0.943
0.967
0.989
1.008
1.026
1.042
1.057
1.071
1.084
1.095
1.107
1.117
1.127
1.136
1.145
1.153
1.161
1.168
1.175
1.182
1.189
1.195
1.201
1.206
1.212
1.217
1.222
1.227
1.232
1.236
1.241
1.245
1.249
1.253
1.257
1.260
1.264
1.267
1.271
1.274
1.289
1.303
1.315
1.326
1.335
1.344
1.352
1.360
1.367
1.373
0.99
0.000
0.295
0.443
0.543
0.618
0.678
0.727
0.768
0.804
0.835
0.862
0.887
0.909
0.929
0.948
0.965
0.980
0.995
1.008
1.021
1.033
1.044
1.054
1.064
1.074
1.083
1.091
1.099
1.107
1.114
1.121
1.128
1.135
1.141
1.147
1.153
1.158
1.164
1.169
1.174
1.179
1.183
1.188
1.192
1.197
1.201
1.205
1.209
1.212
1.230
1.245
1.259
1.272
1.283
1.293
1.302
1.311
1.319
1.326
0.80
1.672
1.710
1.760
1.801
1.834
1.862
1.885
1.904
1.922
1.937
1.950
1.962
1.973
1.983
1.992
2.000
2.008
2.015
2.022
2.028
2.034
2.039
2.045
2.049
2.054
2.058
2.063
2.067
2.070
2.074
2.078
2.081
2.084
2.087
2.090
2.093
2.096
2.098
2.101
2.103
2.106
2.108
2.110
2.112
2.114
2.116
2.118
2.120
2.122
2.131
2.138
2.145
2.151
2.156
2.161
2.166
2.170
2.174
2.177
0.90
1.225
1.361
1.455
1.525
1.578
1.622
1.658
1.688
1.715
1.738
1.758
1.776
1.793
1.808
1.822
1.834
1.846
1.857
1.867
1.876
1.885
1.893
1.901
1.908
1.915
1.922
1.928
1.934
1.940
1.945
1.951
1.956
1.960
1.965
1.970
1.974
1.978
1.982
1.986
1.989
1.993
1.996
2.000
2.003
2.006
2.009
2.012
2.015
2.018
2.031
2.042
2.052
2.061
2.069
2.077
2.083
2.090
2.095
2.101
0.95
0.954
1.130
1.246
1.331
1.396
1.449
1.493
1.530
1.563
1.591
1.616
1.638
1.658
1.677
1.694
1.709
1.724
1.737
1.749
1.761
1.772
1.782
1.791
1.801
1.809
1.817
1.825
1.833
1.840
1.846
1.853
1.859
1.865
1.871
1.876
1.882
1.887
1.892
1.896
1.901
1.905
1.910
1.914
1.918
1.922
1.925
1.929
1.933
1.936
1.952
1.966
1.979
1.990
2.000
2.010
2.018
2.026
2.033
2.040
0.975
0.761
0.958
1.088
1.182
1.256
1.315
1.364
1.406
1.442
1.474
1.502
1.528
1.551
1.572
1.591
1.608
1.625
1.640
1.654
1.667
1.680
1.691
1.702
1.713
1.723
1.732
1.741
1.749
1.757
1.765
1.773
1.780
1.787
1.793
1.799
1.806
1.811
1.817
1.823
1.828
1.833
1.838
1.843
1.847
1.852
1.856
1.860
1.865
1.869
1.887
1.903
1.918
1.931
1.943
1.954
1.964
1.973
1.981
1.989
0.99
0.564
0.782
0.924
1.027
1.108
1.173
1.227
1.273
1.314
1.349
1.381
1.409
1.434
1.458
1.479
1.499
1.517
1.534
1.550
1.565
1.579
1.592
1.605
1.616
1.627
1.638
1.648
1.658
1.667
1.676
1.684
1.692
1.700
1.708
1.715
1.722
1.728
1.735
1.741
1.747
1.753
1.758
1.764
1.769
1.774
1.779
1.784
1.789
1.793
1.815
1.833
1.850
1.865
1.879
1.891
1.903
1.913
1.923
1.932
P = 0.99
                                     D-248
            March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
 Table 21-11. Achievable Conf. Levels for One-Sided Non-Parametric Conf. Bounds
    Around Median, Upper 95th Percentile, and Upper 99th Percentile (n < 20)

n
4
4
4
4
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
10
10
10
10
10
10
10
10
10
10
Footnote. LCL =
Rank of
Bound
4
3
2
1
5
4
3
2
1
6
5
4
3
2
1
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
9
8
7
6
5
4
3
2
1
10
9
8
7
6
5
4
3
2
1
Confidence Level
UCL 50th LCL 50th
0.9375
0.6875
0.3125
0.0625
0.9688
0.8125
0.5000
0.1875
0.0312
0.9844
0.8906
0.6562
0.3438
0.1094
0.0156
0.9922
0.9375
0.7734
0.5000
0.2266
0.0625
0.0078
0.9961
0.9648
0.8555
0.6367
0.3633
0.1445
0.0352
0.0039
0.9980
0.9805
0.9102
0.7461
0.5000
0.2539
0.0898
0.0195
0.0020
0.9990
0.9893
0.9453
0.8281
0.6230
0.3770
0.1719
0.0547
0.0107
0.0010
lower confidence
0.0625
0.3125
0.6875
0.9375
0.0312
0.1875
0.5000
0.8125
0.9688
0.0156
0.1094
0.3438
0.6562
0.8906
0.9844
0.0078
0.0625
0.2266
0.5000
0.7734
0.9375
0.9922
0.0039
0.0352
0.1445
0.3633
0.6367
0.8555
0.9648
0.9961
0.0020
0.0195
0.0898
0.2539
0.5000
0.7461
0.9102
0.9805
0.9980
0.0010
0.0107
0.0547
0.1719
0.3770
0.6230
0.8281
0.9453
0.9893
0.9990
limit; UCL =
UCL 95th
0.1855
0.0140
0.0005
0.0000
0.2262
0.0226
0.0012
0.0000
0.0000
0.2649
0.0328
0.0022
0.0001
0.0000
0.0000
0.3017
0.0444
0.0038
0.0002
0.0000
0.0000
0.0000
0.3366
0.0572
0.0058
0.0004
0.0000
0.0000
0.0000
0.0000
0.3698
0.0712
0.0084
0.0006
0.0000
0.0000
0.0000
0.0000
0.0000
0.4013
0.0861
0.0115
0.0010
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
LCL 95th
0.8145
0.9860
0.9995
1.0000
0.7738
0.9774
0.9988
1.0000
1.0000
0.7351
0.9672
0.9978
0.9999
1.0000
1.0000
0.6983
0.9556
0.9962
0.9998
1.0000
1.0000
1.0000
0.6634
0.9428
0.9942
0.9996
1.0000
1.0000
1.0000
1.0000
0.6302
0.9288
0.9916
0.9994
1.0000
1.0000
1.0000
1.0000
1.0000
0.5987
0.9139
0.9885
0.9990
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
upper confidence limit;
UCL 99th
0.0394
0.0006
0.0000
0.0000
0.0490
0.0010
0.0000
0.0000
0.0000
0.0585
0.0015
0.0000
0.0000
0.0000
0.0000
0.0679
0.0020
0.0000
0.0000
0.0000
0.0000
0.0000
0.0773
0.0027
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0865
0.0034
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0956
0.0043
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
50th =
LCL 99th
0.9606
0.9994
1.0000
1.0000
0.9510
0.9990
1.0000
1.0000
1.0000
0.9415
0.9985
1.0000
1.0000
1.0000
1.0000
0.9321
0.9980
1.0000
1.0000
1.0000
1.0000
1.0000
0.9227
0.9973
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
0.9135
0.9966
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9044
0.9957
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
median
                                    D-249
      March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
 Table 21-11. Achievable Conf. Levels for One-Sided Non-Parametric Conf. Bounds
    Around Median, Upper 95th Percentile, and Upper 99th Percentile (n < 20)
n
11
11
11
11
11
11
11
11
11
11
11
12
12
12
12
12
12
12
12
12
12
12
12
13
13
13
13
13
13
13
13
13
13
13
13
13
14
14
14
14
14
14
14
14
14
14
14
14
14
14
Rank of
Bound
11
10
9
8
7
6
5
4
3
2
1
12
11
10
9
8
7
6
5
4
3
2
1
13
12
11
10
9
8
7
6
5
4
3
2
1
14
13
12
11
10
9
8
7
6
5
4
3
2
1
UCL 50th
0.9995
0.9941
0.9673
0.8867
0.7256
0.5000
0.2744
0.1133
0.0327
0.0059
0.0005
0.9998
0.9968
0.9807
0.9270
0.8062
0.6128
0.3872
0.1938
0.0730
0.0193
0.0032
0.0002
0.9999
0.9983
0.9888
0.9539
0.8666
0.7095
0.5000
0.2905
0.1334
0.0461
0.0112
0.0017
0.0001
0.9999
0.9991
0.9935
0.9713
0.9102
0.7880
0.6047
0.3953
0.2120
0.0898
0.0287
0.0065
0.0009
0.0001
LCL 50th
0.0005
0.0059
0.0327
0.1133
0.2744
0.5000
0.7256
0.8867
0.9673
0.9941
0.9995
0.0002
0.0032
0.0193
0.0730
0.1938
0.3872
0.6128
0.8062
0.9270
0.9807
0.9968
0.9998
0.0001
0.0017
0.0112
0.0461
0.1334
0.2905
0.5000
0.7095
0.8666
0.9539
0.9888
0.9983
0.9999
0.0001
0.0009
0.0065
0.0287
0.0898
0.2120
0.3953
0.6047
0.7880
0.9102
0.9713
0.9935
0.9991
0.9999
Confidence Level
UCL 95th LCL 95th
0.4312
0.1019
0.0152
0.0016
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.4596
0.1184
0.0196
0.0022
0.0002
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.4867
0.1354
0.0245
0.0031
0.0003
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.5123
0.1530
0.0301
0.0042
0.0004
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.5688
0.8981
0.9848
0.9984
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.5404
0.8816
0.9804
0.9978
0.9998
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.5133
0.8646
0.9755
0.9969
0.9997
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.4877
0.8470
0.9699
0.9958
0.9996
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
UCL 99th
0.1047
0.0052
0.0002
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.1136
0.0062
0.0002
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.1225
0.0072
0.0003
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.1313
0.0084
0.0003
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
LCL 99th
0.8953
0.9948
0.9998
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.8864
0.9938
0.9998
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.8775
0.9928
0.9997
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.8687
0.9916
0.9997
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
                                    D-250
      March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
 Table 21-11. Achievable Conf. Levels for One-Sided Non-Parametric Conf. Bounds
    Around Median, Upper 95th Percentile, and Upper 99th Percentile (n < 20)
n
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
17
17
17
17
17
17
17
17
17
17
17
17
17
17
17
17
17
Rank of
Bound
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
UCL 50th
1.0000
0.9995
0.9963
0.9824
0.9408
0.8491
0.6964
0.5000
0.3036
0.1509
0.0592
0.0176
0.0037
0.0005
0.0000
1.0000
0.9997
0.9979
0.9894
0.9616
0.8949
0.7728
0.5982
0.4018
0.2272
0.1051
0.0384
0.0106
0.0021
0.0003
0.0000
1.0000
0.9999
0.9988
0.9936
0.9755
0.9283
0.8338
0.6855
0.5000
0.3145
0.1662
0.0717
0.0245
0.0064
0.0012
0.0001
0.0000
LCL 50th
0.0000
0.0005
0.0037
0.0176
0.0592
0.1509
0.3036
0.5000
0.6964
0.8491
0.9408
0.9824
0.9963
0.9995
1.0000
0.0000
0.0003
0.0021
0.0106
0.0384
0.1051
0.2272
0.4018
0.5982
0.7728
0.8949
0.9616
0.9894
0.9979
0.9997
1.0000
0.0000
0.0001
0.0012
0.0064
0.0245
0.0717
0.1662
0.3145
0.5000
0.6855
0.8338
0.9283
0.9755
0.9936
0.9988
0.9999
1.0000
Confidence Level
UCL 95th LCL 95th
0.5367
0.1710
0.0362
0.0055
0.0006
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.5599
0.1892
0.0429
0.0070
0.0009
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.5819
0.2078
0.0503
0.0088
0.0012
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.4633
0.8290
0.9638
0.9945
0.9994
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.4401
0.8108
0.9571
0.9930
0.9991
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.4181
0.7922
0.9497
0.9912
0.9988
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
UCL 99th
0.1399
0.0096
0.0004
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.1485
0.0109
0.0005
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.1571
0.0123
0.0006
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
LCL 99th
0.8601
0.9904
0.9996
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.8515
0.9891
0.9995
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.8429
0.9877
0.9994
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
                                    D-251
      March 2009

-------
Appendix D. Chapter 21 Tables
Unified Guidance
 Table 21-11. Achievable Conf. Levels for One-Sided Non-Parametric Conf. Bounds
    Around Median, Upper 95th Percentile, and Upper 99th Percentile (n < 20)
n
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
Rank of
Bound
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
UCL 50th
1.0000
0.9999
0.9993
0.9962
0.9846
0.9519
0.8811
0.7597
0.5927
0.4073
0.2403
0.1189
0.0481
0.0154
0.0038
0.0007
0.0001
0.0000
1.0000
1.0000
0.9996
0.9978
0.9904
0.9682
0.9165
0.8204
0.6762
0.5000
0.3238
0.1796
0.0835
0.0318
0.0096
0.0022
0.0004
0.0000
0.0000
LCL 50th
0.0000
0.0001
0.0007
0.0038
0.0154
0.0481
0.1189
0.2403
0.4073
0.5927
0.7597
0.8811
0.9519
0.9846
0.9962
0.9993
0.9999
1.0000
0.0000
0.0000
0.0004
0.0022
0.0096
0.0318
0.0835
0.1796
0.3238
0.5000
0.6762
0.8204
0.9165
0.9682
0.9904
0.9978
0.9996
1.0000
1.0000
Confidence Level
UCL 95th LCL 95th
0.6028
0.2265
0.0581
0.0109
0.0015
0.0002
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.6226
0.2453
0.0665
0.0132
0.0020
0.0002
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.3972
0.7735
0.9419
0.9891
0.9985
0.9998
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3774
0.7547
0.9335
0.9868
0.9980
0.9998
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
UCL 99th
0.1655
0.0138
0.0007
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.1738
0.0153
0.0009
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
LCL 99th
0.8345
0.9862
0.9993
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.8262
0.9847
0.9991
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
                                    D-252
      March 2009

-------
Appendix D. Chapter 21 Tables
                                           Unified Guidance
 Table 21-11.  Achievable Conf. Levels for One-Sided Non-Parametric Conf. Bounds
    Around Median, Upper 95th Percentile, and Upper 99th Percentile (n < 20)
              Rank of
               Bound
                   Confidence Level
UCL 50th   LCL 50th  UCL 95th   LCL 95th  UCL 99th   LCL 99th
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1.0000
1.0000
0.9998
0.9987
0.9941
0.9793
0.9423
0.8684
0.7483
0.5881
0.4119
0.2517
0.1316
0.0577
0.0207
0.0059
0.0013
0.0002
0.0000
0.0000
0.0000
0.0000
0.0002
0.0013
0.0059
0.0207
0.0577
0.1316
0.2517
0.4119
0.5881
0.7483
0.8684
0.9423
0.9793
0.9941
0.9987
0.9998
1.0000
1.0000
0.6415
0.2642
0.0755
0.0159
0.0026
0.0003
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.3585
0.7358
0.9245
0.9841
0.9974
0.9997
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.1821
0.0169
0.0010
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.8179
0.9831
0.9990
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
                                      D-253
                                                  March 2009

-------
Appendix D. Chapter 21 Tables                                     Unified Guidance
                     This page intentionally left blank
                                       D-254                              March 2009

-------
Appendix D. Chapter 22 Tables	Unified Guidance


                                        D  STATISTICAL TABLES


D.5 TABLES  FROM  CHAPTER 22




           TABLE 22-1 Combs, of nand a Achieving Power to Detect Increases of 1 .SxGWPS	D-256

           TABLE 22-2 Combs, of nand a Atiieving Power to Detect Increases of 2.0xGWPS	D-257

           TABLE 22-3 Minimum I ndividual Test a Meeting Power criteria, given n and CV	D-258

           TABLE 22-4 Minimum nto Detect Increases of .75xGWPS, given CV, 1-/3, and a	D-259

           TABLE 22-5 Minimum nto Detect Increases of .SxGWPS, given CK1-/3, and a	D-261

           TABLE 22-6 Minimum nto Detect Increases of .25xGWPS, given CV, 1-/3, and a	D-263

           TABLE22-7 Minimum n to Detect kpo I ncr. over Percentile1-p0, with 1-/3 and a,k> 1	D-265

           TABLE 22-8 Minimum n to Detect kpo I ncr. over Percentile1-p0, with 1-/3 and a,k< 1	D-267
                                                          D-255
                                                                                                                March 2009

-------
Appendix D. Chapter 22 Tables
Unified Guidance
 Table 22-1. Combinations of n (< 40) and a (< .20) Achieving (1-p) Power to Detect Increases of 1.5 x GWPS
l-p = 0.50
n a
4 0.177
5 0.149
6 0.127
7 0.108
8 0.093
9 0.080
10 0.069
11 0.060
12 0.052
13 0.045
14 0.039
15 0.034
16 0.030
17 0.026
18 0.023
19 0.020
20 0.018
21 0.015
22 0.014
23 0.012
24 0.010
25 0.009
26 0.008
27 0.007
28 0.006
29 0.006
30 0.005
31 0.004
32 0.004
33 0.003
34 0.003
35 0.003
36 0.002
37 0.002
38 0.002
39 0.002
40 0.002
1-p = 0.60
n a
6 0.179
7 0.156
8 0.136
9 0.119
10 0.104
11 0.092
12 0.081
13 0.071
14 0.063
15 0.056
16 0.049
17 0.043
18 0.038
19 0.034
20 0.030
21 0.027
22 0.024
23 0.021
24 0.019
25 0.017
26 0.015
27 0.013
28 0.012
29 0.010
30 0.009
31 0.008
32 0.007
33 0.007
34 0.006
35 0.005
36 0.005
37 0.004
38 0.004
39 0.003
40 0.003


1-p = 0.70
n a
8 0.197
9 0.175
10 0.156
11 0.139
12 0.124
13 0.111
14 0.099
15 0.089
16 0.079
17 0.071
18 0.064
19 0.057
20 0.051
21 0.046
22 0.041
23 0.037
24 0.033
25 0.030
26 0.027
27 0.024
28 0.022
29 0.020
30 0.018
31 0.016
32 0.014
33 0.013
34 0.011
35 0.010
36 0.009
37 0.008
38 0.007
39 0.007
40 0.006




1-p = 0.80
n a
12 0.195
13 0.177
14 0.160
15 0.146
16 0.132
17 0.120
18 0.109
19 0.099
20 0.090
21 0.082
22 0.074
23 0.068
24 0.061
25 0.056
26 0.051
27 0.046
28 0.042
29 0.038
30 0.034
31 0.031
32 0.029
33 0.026
34 0.023
35 0.021
36 0.019
37 0.018
38 0.016
39 0.015
40 0.013








1-p = 0.90
n a
19 0.191
20 0.177
21 0.163
22 0.151
23 0.139
24 0.129
25 0.119
26 0.110
27 0.101
28 0.093
29 0.086
30 0.079
31 0.073
32 0.067
33 0.062
34 0.057
35 0.053
36 0.049
37 0.045
38 0.041
39 0.038
40 0.035















1-p = 0.95
n a
26 0.188
27 0.176
28 0.164
29 0.153
30 0.143
31 0.133
32 0.124
33 0.116
34 0.108
35 0.101
36 0.094
37 0.087
38 0.081
39 0.076
40 0.070






















1-p = 0.99
n a





































                                                  D-256
                                                                                                March 2009

-------
Appendix D. Chapter 22 Tables
Unified Guidance
  Table 22-2. Combinations of n (< 40) and a (< .20) Achieving (1-p) Power to Detect Increases of 2 x GWPS
l-p = 0.50
n a
3 0.091
4 0.057
5 0.037
6 0.024
7 0.016
8 0.011
9 0.007
10 0.005
11 0.003
12 0.002
13 0.002
14 0.001
>15 <0.001












1-p = 0.60
n a
3 0.123
4 0.080
5 0.054
6 0.036
7 0.025
8 0.017
9 0.012
10 0.008
11 0.006
12 0.004
13 0.003
14 0.002
15 0.001
>16 <0.001











1-p = 0.70
n a
3 0.164
4 0.113
5 0.079
6 0.055
7 0.039
8 0.027
9 0.019
10 0.014
11 0.010
12 0.007
13 0.005
14 0.004
15 0.003
16 0.002
17 0.001
>18 <0.001









1-p = 0.80
n a
4 0.163
5 0.119
6 0.086
7 0.063
8 0.046
9 0.034
10 0.024
11 0.018
12 0.013
13 0.010
14 0.007
15 0.005
16 0.004
17 0.003
18 0.002
19 0.002
20 0.001
>21 <0.001







1-p = 0.90
n a
5 0.199
6 0.152
7 0.116
8 0.088
9 0.067
10 0.051
11 0.039
12 0.029
13 0.022
14 0.017
15 0.013
16 0.010
17 0.007
18 0.005
19 0.004
20 0.003
21 0.002
22 0.002
23 0.001
>24 <0.001





1-p = 0.95
n a
7 0.183
8 0.144
9 0.113
10 0.089
11 0.069
12 0.054
13 0.042
14 0.033
15 0.025
16 0.020
17 0.015
18 0.012
19 0.009
20 0.007
21 0.005
22 0.004
23 0.003
24 0.002
25 0.002
26 0.002
27 0.001
>28 <0.001



1-p = 0.99
n a
11 0.180
12 0.148
13 0.121
14 0.099
15 0.080
16 0.065
17 0.053
18 0.043
19 0.034
20 0.027
21 0.022
22 0.018
23 0.014
24 0.011
25 0.009
26 0.007
27 0.006
28 0.004
29 0.004
30 0.003
31 0.002
32 0.002
33 0.002
34 0.001
>35 <0.001
                                                  D-257
                                                                                               March 2009

-------
Appendix D.  Chapter 22 Tables
Unified Guidance
                Table 22-3. Minimum Individual Test a Meeting Power Criteria Given n and CV

CV
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0

n=4
0,003
0.022
0.056
0.097
0.137
0.174
0.206
0.233
0.256
0.276
0.309
0.333
0.352
0.368
0.380
0.391
0.400
0.407
0.414
0.419
50%
n=6
O.OOO
0.005
0.021
0.048
0.082
0.116
0.148
0.177
0.203
0.226
0.263
0.293
0.316
0.335
0.350
0.363
0.374
0.383
0.391
0.398
Power at R =
n=8
0,000
0.001
0.008
0.025
0.051
0.080
0.110
0.139
0.165
0.189
0.229
0.261
0.287
0.308
0.326
0.341
0.353
0.364
0.373
0.381
1.5
n = 10
0,000
0.000
0,003
0.014
0.032
0.056
0.083
0.110
0.136
0.160
0.201
0.235
0.263
0.286
0.305
0.322
0.335
0.347
0.358
0.367

n = 12
O.OOO
0.000
0,001
0,007
0.021
0.040
0.064
0.088
0.113
0.136
0.178
0.214
0.243
0.267
0.288
0.305
0.320
0.333
0.344
0.354

n=4
0,007
0.014
0.050
0.113
0.191
0.270
0.342
0.402
0.451
0.492
0.553
0.596
0.626
0.650
0.667
0.682
0.693
0.703
0.711
0.718
80%
n=6
O.OOO
0,002
0.013
0.043
0.093
0.156
0.222
0.284
0.339
0.386
0.462
0.517
0.558
0.590
0.614
0.634
0.650
0.664
0.675
0.685
Power at R
n=8
0,000
0,000
0,00,3
0.017
0.047
0.094
0.149
0.206
0.261
0.310
0.393
0.456
0.505
0.542
0.572
0.596
0.616
0.632
0.646
0.658
= 2
n = 10
O.OOO
0,000
0.001
0.007
0.024
0.057
0.101
0.151
0.203
0.251
0.337
0.406
0.459
0.502
0.536
0.564
0.586
0.605
0.621
0.635

n = 12
O.OOO
0,000
0,000
0,003
0.013
0.035
0.069
0.112
0.158
0.205
0.291
0.362
0.420
0.466
0.504
0.534
0.560
0.581
0.599
0.614
                                                  D-258
                                                                                                 March 2009

-------
Appendix D. Chapter 22 Tables
Unified Guidance
  Table 22-4. Minimum n (> 4) to Detect Decreases of .75 x GWPS for Given CV, Power (1-p), & Error Rate (a)
CV = 0.2
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
6
0.60
4
4
4
4
6
0.70
4
4
4
4
6
0.80
4
4
4
4
7
0.90
4
4
4
5
8
0.95
4
5
5
6
9
CV = 0.6
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
5
7
11
21
0.60
5
7
9
14
25
0.70
7
9
12
17
30
0.80
11
13
16
22
36
0.90
16
19
23
30
45
0.95
22
25
30
37
54
CV = 1.0
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
8
11
17
27
52
0.60
12
16
23
35
63
0.70
18
23
31
44
76
0.80
27
33
42
58
93
0.90
42
50
61
79
120
0.95
58
67
79
100
145
CV = 1.4

-------
Appendix D. Chapter 22 Tables
Unified Guidance
  Table 22-4. Minimum n (> 4) to Detect Decreases of .75 x GWPS for Given CV, Power (1-p), & Error Rate (a)
CV = 1.8
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
22
33
50
81
162
0.60
36
50
70
107
198
0.70
56
72
97
139
240
0.80
84
104
133
182
296
0.90
133
158
193
252
383
0.95
182
212
252
318
463
CV = 2.2
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
32
48
73
120
239
0.60
54
74
104
159
293
0.70
83
108
144
207
357
0.80
125
155
198
271
440
0.90
198
236
288
375
570
0.95
271
315
375
474
690
CV = 2.6
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
44
67
102
167
333
0.60
74
103
145
221
408
0.70
115
150
200
288
498
0.80
174
216
276
378
614
0.90
276
329
402
523
795
0.95
378
439
523
661
963
CV = 2.0
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
27
40
61
100
199
0.60
44
61
87
132
243
0.70
68
89
119
171
296
0.80
103
128
164
225
364
0.90
164
195
238
310
472
0.95
225
261
310
392
571
CV = 2.4
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
38
57
87
143
284
0.60
63
88
124
189
348
0.70
98
128
171
246
425
0.80
148
184
235
323
523
0.90
235
280
342
446
678
0.95
323
375
446
563
821
CV = 2.8
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
51
77
118
193
386
0.60
86
119
168
256
473
0.70
133
173
232
334
577
0.80
201
250
320
438
711
0.90
320
381
465
606
922
0.95
438
509
606
766
1116
                                                  D-260
                                                                                                March 2009

-------
Appendix D. Chapter 22 Tables
Unified Guidance
  Table 22-5. Minimum n (> 4) to Detect Decreases of .5 x GWPS for Given CV, Power (1-p), & Error Rate (a)
CV = 0.2
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
4
0.60
4
4
4
4
4
0.70
4
4
4
4
4
0.80
4
4
4
4
4
0.90
4
4
4
4
4
0.95
4
4
4
4
4
CV = 0.6
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
6
0.60
4
4
4
4
6
0.70
4
4
4
4
6
0.80
4
4
4
4
7
0.90
4
4
4
5
8
0.95
4
5
5
6
9
CV = 1.0
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
5
9
0.60
4
4
4
6
10
0.70
4
4
5
7
11
0.80
4
5
6
8
13
0.90
6
7
9
11
16
0.95
8
9
11
13
19
CV = 1.4
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
5
8
14
0.60
4
5
6
9
17
0.70
5
6
8
11
19
0.80
7
8
11
14
23
0.90
11
12
15
19
29
0.95
14
16
19
24
34
CV = 0.4
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
4
0.60
4
4
4
4
4
0.70
4
4
4
4
5
0.80
4
4
4
4
5
0.90
4
4
4
4
5
0.95
4
4
4
4
6
CV = 0.8
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
7
0.60
4
4
4
5
8
0.70
4
4
4
5
9
0.80
4
4
5
6
10
0.90
5
5
6
8
11
0.95
6
7
8
9
13
CV = 1.2
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
6
11
0.60
4
4
5
8
13
0.70
4
5
6
9
15
0.80
6
7
8
11
18
0.90
8
10
11
15
22
0.95
11
12
15
18
26
CV = 1.6

-------
Appendix D. Chapter 22 Tables
Unified Guidance
  Table 22-5. Minimum n (> 4) to Detect Decreases of .5 x GWPS for Given CV, Power (1-p), & Error Rate (a)
CV = 1.8
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
5
7
11
21
0.60
5
7
9
14
25
0.70
7
9
12
17
30
0.80
11
13
16
22
36
0.90
16
19
23
30
45
0.95
22
25
30
37
54
CV = 2.2
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
5
7
10
16
30
0.60
7
10
13
20
36
0.70
10
13
17
25
43
0.80
15
19
23
32
52
0.90
23
28
34
44
66
0.95
32
37
44
55
79
CV = 2.6
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
6
9
13
21
40
0.60
9
13
18
27
48
0.70
14
18
24
34
58
0.80
21
25
32
44
71
0.90
32
38
46
60
91
0.95
44
51
60
76
110
CV = 2.0
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
6
9
13
25
0.60
6
8
11
17
30
0.70
9
11
15
21
36
0.80
13
16
20
27
43
0.90
20
23
28
36
55
0.95
27
31
36
46
66
CV = 2.4
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
6
8
11
18
35
0.60
8
11
15
23
42
0.70
12
15
20
29
50
0.80
18
22
28
38
61
0.90
28
33
40
51
78
0.95
38
43
51
65
94
CV = 2.8
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
7
10
15
24
46
0.60
11
15
20
30
56
0.70
16
21
27
39
67
0.80
24
29
37
51
82
0.90
37
44
53
69
105
0.95
51
58
69
87
127
                                                  D-262
                                                                                                March 2009

-------
Appendix D. Chapter 22 Tables
Unified Guidance
  Table 22-6. Minimum n (> 4) to Detect Decreases of .25 x GWPS for Given CV, Power (1-p), & Error Rate (a)
CV = 0.2
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
4
0.60
4
4
4
4
4
0.70
4
4
4
4
4
0.80
4
4
4
4
4
0.90
4
4
4
4
4
0.95
4
4
4
4
4
CV = 0.6
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
4
0.60
4
4
4
4
4
0.70
4
4
4
4
4
0.80
4
4
4
4
4
0.90
4
4
4
4
4
0.95
4
4
4
4
4
CV = 1.0
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
4
0.60
4
4
4
4
4
0.70
4
4
4
4
4
0.80
4
4
4
4
4
0.90
4
4
4
4
5
0.95
4
4
4
4
5
CV = 1.4
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
5
0.60
4
4
4
4
5
0.70
4
4
4
4
5
0.80
4
4
4
4
5
0.90
4
4
4
4
6
0.95
4
4
4
5
7
CV = 0.4
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
4
0.60
4
4
4
4
4
0.70
4
4
4
4
4
0.80
4
4
4
4
4
0.90
4
4
4
4
4
0.95
4
4
4
4
4
CV = 0.8
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
4
0.60
4
4
4
4
4
0.70
4
4
4
4
4
0.80
4
4
4
4
4
0.90
4
4
4
4
4
0.95
4
4
4
4
4
CV = 1.2
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
4
0.60
4
4
4
4
4
0.70
4
4
4
4
5
0.80
4
4
4
4
5
0.90
4
4
4
4
5
0.95
4
4
4
4
6
CV = 1.6

-------
Appendix D. Chapter 22 Tables
Unified Guidance
  Table 22-6. Minimum n (> 4) to Detect Decreases of .25 x GWPS for Given CV, Power (1-p), & Error Rate (a)
CV = 1.8
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
6
0.60
4
4
4
4
6
0.70
4
4
4
4
6
0.80
4
4
4
4
7
0.90
4
4
4
5
8
0.95
4
5
5
6
9
CV = 2.2
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
7
0.60
4
4
4
4
7
0.70
4
4
4
5
8
0.80
4
4
4
6
9
0.90
4
5
6
7
10
0.95
6
6
7
8
12
CV = 2.6
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
5
8
0.60
4
4
4
5
8
0.70
4
4
4
6
9
0.80
4
4
5
7
11
0.90
5
6
7
9
13
0.95
7
8
9
11
15
CV = 2.0
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
6
0.60
4
4
4
4
6
0.70
4
4
4
4
7
0.80
4
4
4
5
8
0.90
4
4
5
6
9
0.95
5
5
6
7
10
CV = 2.4
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
4
7
0.60
4
4
4
5
8
0.70
4
4
4
5
9
0.80
4
4
5
6
10
0.90
5
5
6
8
11
0.95
6
7
8
9
13
CV = 2.8
a\l-p
0.20
0.15
0.10
0.05
0.01
0.50
4
4
4
5
8
0.60
4
4
4
6
9
0.70
4
4
5
6
10
0.80
4
5
6
8
12
0.90
6
7
8
10
14
0.95
8
8
10
12
17
                                                  D-264
                                                                                                March 2009

-------
Appendix D. Chapter 22 Tables
Unified Guidance
 Table 22-7. Minimum A? to Detect kp0 Exceedances Over Percentile (1-po) with Power (1-p) and Error Rate (a)
Percentile


-------
Appendix D. Chapter 22 Tables
Unified Guidance
 Table 22-7. Minimum A? to Detect kp0 Exceedences Over Percentile (1-po) with Power (1-p) and Error Rate (a)
Percentile


-------
Appendix D. Chapter 22 Tables
Unified Guidance
 Table 22-8. Minimum A? to Detect kp0 Exceedences Over Percentile (1-po) with Power (1-p) and Error Rate (a)
Percentile

a\l-p
0.20
0.10
0.05
0.02
0.01
= 90th

0.50
26
60
98
152
195


0.60
38
78
121
181
227


0.70
54
100
148
214
264

k = .50
0.80
77
130
184
256
311


0.90
114
177
239
321
382


0.95
150
221
291
380
447


0.99
231
318
401
505
581
k= .25
a\l-p
0.20
0.10
0.05
0.02
0.01
Percentile

a\l-p
0.20
0.10
0.05
0.02
0.01
0.50
12
27
44
68
87
= 98th

0.50
139
322
531
827
1061
0.60
16
32
51
77
97


0.60
205
419
653
979
1232
0.70
20
39
59
87
109


0.70
290
537
798
1154
1428
0.80
27
48
70
100
123

k = .50
0.80
407
693
987
1379
1677
0.90
37
61
86
119
144


0.90
602
943
1281
1723
2054
0.95
47
74
101
136
163


0.95
793
1178
1552
2036
2395
0.99
68
100
131
171
201


0.99
1221
1689
2133
2694
3105
k= .25
a\l-p
0.20
0.10
0.05
0.02
0.01
0.50
62
144
236
368
472
0.60
82
173
274
415
525
0.70
107
209
318
469
585
0.80
140
254
373
535
659
0.90
193
324
458
635
770
0.95
244
388
533
724
868
0.99
354
525
692
907
1067
Percentile

a\l-p
0.20
0.10
0.05
0.02
0.01
= 95th

0.50
54
125
206
321
412


0.60
80
163
254
380
478


0.70
113
209
311
449
555

k = .50
0.80
159
270
384
537
653


0.90
236
368
500
672
800


0.95
311
460
606
794
934


0.99
479
661
834
1052
1212
k= .25
a\l-p
0.20
0.10
0.05
0.02
0.01
Percentile

a\l-p
0.20
0.10
0.05
0.02
0.01
0.50
24
56
92
143
183
_ ggth

0.50
281
651
1072
1671
2144
0.60
32
68
107
161
204


0.60
413
846
1319
1976
2487
0.70
42
82
124
182
228


0.70
584
1083
1611
2330
2883
0.80
55
99
146
209
257

k = .50
0.80
820
1397
1990
2782
3384
0.90
76
127
179
248
300


0.90
1213
1900
2582
3475
4144
0.95
96
152
209
283
339


0.95
1597
2373
3129
4106
4830
0.99
139
206
271
355
417


0.99
2457
3402
4297
5430
6259
k= .25
a\l-p
0.20
0.10
0.05
0.02
0.01
0.50
125
290
477
743
953
0.60
166
350
553
838
1060
0.70
215
420
641
945
1181
0.80
282
512
753
1080
1330
0.90
389
653
922
1281
1553
0.95
490
782
1075
1460
1749
0.99
711
1056
1393
1827
2149
                                                  D-267
                                                                                                 March 2009

-------
Appendix D. Chapter 22 Tables                                                                  Unified Guidance
                                   This page intentionally left blank
                                                     D-268
                                                                                                      March 2009

-------