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
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Unified Guidance
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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
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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
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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.
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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
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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.
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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.
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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
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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.
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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.
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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.
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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
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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.
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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.
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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].
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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
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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
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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-
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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
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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).
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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.
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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.
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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
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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.
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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
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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).
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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.
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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
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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).
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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
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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
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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
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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
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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.
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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
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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
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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).
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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
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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
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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.
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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?
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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?
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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?
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*»* 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?
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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
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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
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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
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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).
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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
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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
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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.
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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
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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.
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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
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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
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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
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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.
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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.
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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
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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
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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).
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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.
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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.
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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.
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»«» 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.
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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.
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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
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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.
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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,
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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.
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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).
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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.
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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
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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
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Chapter 6. Detection Monitoring Design
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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.
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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.
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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
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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
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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).
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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
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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
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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.
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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.
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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.
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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
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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)
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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.
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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.
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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
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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.
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Chapter 6. Detection Monitoring Design
Unified Guidance
Figure 6-4. Landfill Site Configuration
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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
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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
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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.
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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
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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:
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»«» 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).
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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.
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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
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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.
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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,
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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
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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.
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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.
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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
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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.
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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
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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.
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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.
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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.
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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. ^
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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
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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.
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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:®
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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'. &
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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-
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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
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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
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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
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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.
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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.
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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.
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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.
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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
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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.
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Chapter 7. Compliance Monitoring Strategies
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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.
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Chapter 7. Compliance Monitoring Strategies
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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
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Chapter 7. Compliance Monitoring Strategies
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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.
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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
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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.
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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.
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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.
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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).
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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
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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.
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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
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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-
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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:
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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.
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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
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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
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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.
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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
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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
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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
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March 2009
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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
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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
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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.
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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
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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.
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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.
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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.
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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
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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.
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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.
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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
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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
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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
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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.
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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.
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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.
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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.
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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.
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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
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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).
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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).
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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.
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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
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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.
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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.
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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.
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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
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'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
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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
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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.
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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
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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
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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:
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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. -^
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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.
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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.
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^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>
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Chapter 9. Exploratory Tools
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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
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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
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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
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Chapter 9. Exploratory Tools
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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
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Chapter 9. Exploratory Tools
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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
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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.
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Chapter 9. Exploratory Tools
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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
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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 !
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Chapter 9. Exploratory Tools
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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
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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
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Chapter 9. Exploratory Tools
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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.
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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.
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Chapter 9. Exploratory Tools
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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
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March 2009
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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).
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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
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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.
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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.
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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
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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:
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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.
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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.
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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
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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
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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.
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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. -^
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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]
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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.
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Chapter 10. Fitting Distributions
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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.
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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]
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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.
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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
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March 2009
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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].
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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
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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
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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
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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.
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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. -^
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March 2009
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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.
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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.
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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. -^
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Chapter 11. Testing Equality of Variance
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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:
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March 2009
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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.
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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.
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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]:
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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
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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.
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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
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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.
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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.
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Chapter 12. Identifying Outliers
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^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
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Chapter 12. Identifying Outliers
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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
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Chapter 12. Identifying Outliers
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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
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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):
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Chapter 12. Identifying Outliers
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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
#
#
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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):
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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
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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
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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:
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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
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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)
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March 2009
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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
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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
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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
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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.
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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
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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
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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)
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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.
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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Chapter 14. Temporal Variability
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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.
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Chapter 14. Temporal Variability
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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
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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
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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
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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
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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)
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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
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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 .
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Chapter 14. Temporal Variability
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^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
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Chapter 14. Temporal Variability
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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.
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March 2009
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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.
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Chapter 14. Temporal Variability
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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March 2009
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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.
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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.
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March 2009
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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.
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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
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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.
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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
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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.
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Chapter 15. Managing Non-Detect Data
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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
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Chapter 15. Managing Non-Detect Data
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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
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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.
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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
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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.
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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:
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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
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Chapter 15. Managing Non-Detect Data
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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)
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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). ^
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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
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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
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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.
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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
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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.
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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
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PART III. DETECTION MONITORING TESTS Unified Guidance
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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
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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
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March 2009
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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
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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.
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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.
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Chapter 16. Two-Sample Tests
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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.
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Chapter 16. Two-Sample Tests
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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:
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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
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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
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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 (
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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 (
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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.
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Chapter 16. Two-Sample Tests
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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
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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
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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
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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
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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:
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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.
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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
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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.
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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:
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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-
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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. -^
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Chapter 16. Two-Sample Tests Unified Guidance
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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
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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
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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
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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:
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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
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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.
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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 '
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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. -^
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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.
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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-
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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]:
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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:
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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.
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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. -^
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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?
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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
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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.
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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.
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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:
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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.,
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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.
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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:
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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
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Figure 17-7. Time Series Plot of Sodium Concentrations (ppm)
CL
90
91
Year (2-Digit Format)
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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
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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
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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
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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
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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.
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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.
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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 (
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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.
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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.
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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. -^
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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.
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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.
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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.
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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]:
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[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
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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
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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
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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.
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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
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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
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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.
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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
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March 2009
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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
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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
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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.
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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.
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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.
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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.
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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
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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).
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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
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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.
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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
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March 2009
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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.)
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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Chapter 19. Prediction Limits with Retesting Unified Guidance
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19-36 March 2009
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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
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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
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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
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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
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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
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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
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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.
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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).
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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
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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.
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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:
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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.
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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
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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
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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
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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
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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.
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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:
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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.
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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:
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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.
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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.
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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!=\.
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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
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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:
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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.
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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)
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March 2009
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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:
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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.
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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
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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.
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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
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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
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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
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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.
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Chapter 21. Confidence Intervals
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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)
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Chapter 21. Confidence Intervals
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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
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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.
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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
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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
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Chapter 21. Confidence Intervals
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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
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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.
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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
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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.
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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.
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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
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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 = *
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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APPENDICES Unified Guidance
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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
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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
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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
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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
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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
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Appendix A—References, Glossary & Index Unified Guidance
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A^4
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Appendix A—References, Glossary & Index Unified Guidance
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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,
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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)
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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
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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.
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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.
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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.
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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
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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
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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
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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.
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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
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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
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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
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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.
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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
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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
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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
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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.
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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
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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
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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
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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.
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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.
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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.
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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
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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
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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.
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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
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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.
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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)
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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)
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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
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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
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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
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Appendix D. Chapter 21 Tables Unified Guidance
This page intentionally left blank
D-244 March 2009
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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
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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
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Appendix D. Chapter 21 Tables Unified Guidance
This page intentionally left blank
D-254 March 2009
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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
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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
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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
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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
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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
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Appendix D. Chapter 22 Tables Unified Guidance
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D-268
March 2009
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