United States
Environmental Protection
Agency
Office of Policy, Planning,
and Evaluation
Washington, DC 20460
EPA 230/02-89-042
February 1989
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Methods for Evaluating the
Attainment of Cleanup Standards
Volume 1: Soils and Solid Media
rt S Environmental
" , , 5. Library
'
Statistical Policy Branch (PM-223)
Office of Policy, Planning, and Evaluation
U. S. Environmental Protection Agency
401 M Street, S.W.
Washington, DC 20460
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DISCLAIMER
This report was prepared under contract to an agency of the United States Government.
Neither the United States Government nor any of its employees, contractors,
subcontractors, or their employees makes any warranty, expressed or implied, or assumes
any legal liability or responsibility for any third parry's use or the results of such use of any
information, apparatus, product, model, formula, or process disclosed in this report, or
represents that its use by such third parry would not infringe on privately owned rights.
Publication of the data in this document does not signify that the contents necessarily reflect
the joint or separate views and policies of each co-sponsoring agency. Mention of trade
names or commercial products does not constitute endorsement or recommendation for use.
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TABLE OF CONTENTS
Page
Authors and Contributors xv
Executive Summary xvi
1. INTRODUCTION 1-1
1.1 General Scope and Features of the Guidance Document 1-1
1.1.1 Purpose : 1-1
1.1.2 Intended Audience and Use 1-3
1.1.3 Bibliography, Glossary, Boxes, Worksheets, Examples,
and References to "Consult a Statistician" 1-6
1.2 A Categorization Scheme for Cleanup Standards 1-6
1.2.1 Technology-Based Standards 1-7
1.2.2 Background-Based Standards 1-7
1.2.3 Risk-Based Standards 1-8
1.3 Use of this Guidance in Superfund Program Activities 1-8
1.3.1 Emergency/Removal Action 1-8
1.3.2 Remedial Response Activities 1-9
1.3.3 Superfund Enforcement 1-9
1.4 Treatability Studies and Soils Treatment Technologies 1-9
1.4.1 Laboratory/Bench-Scale Treatability Studies 1-10
1.4.2 Field/Pilot-Scale Treatability Studies 1-10
1.4.3 Soils Treatment by Chemical Modification 1-11
1.4.4 Soils Treatment by In Situ Removal of Contaminants 1-11
1.4.5 Soils Treatment by Incineration 1-13
1.4.6 Soils Removal 1-13
1.4.7 Soils Capping 1-14
1.5 Summary: 1-14
2. INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS 2-1
2.1 Hypothesis Formulation and Uncertainty 2-2
2.2 Power Curves as a Method of Expressing Uncertainty and
Developing Sample Size Requirements 2-6
2.3 Attainment or Compliance Criteria 2-8
2.3.1 Mean 2-9
2.3.2 Proportions or Percentiles 2-9
2.4 Components of a Risk-Based Standard 2-11
2.5 Missing or Unusable Data, Detection Limits, Outliers 2-12
2.5.1 Missing or Unusable Data 2-12
2.5.2 Evaluation of Less-Than-Detection-Limit Data 2-15
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2.5.3 Outliers 2-16
2.6 General Assumptions 2-17
2.7 A Note on Statistical Versus Field Sampling Terminology 2-17
2.8 Summary 2-18
3. SPECIFICATION OF ATTAINMENT OBJECTIVES 3-1
3.1 Specification of Sample Areas 3-1
3.2 Specification of Sample Collection and Handling Procedures 3-4
3.3 Specification of the Chemicals to be Tested 3-5
3.4 Specification of the Cleanup Standard 3-5
3.5 Selection of the Statistical Parameter to Compare with the
Cleanup Standard 3-6
3.5.1 Selection Criteria for the Mean, Median, and Upper
Percentile 3-6
3.5.2 Multiple Attainment Criteria 3-9
3.6 Decision Making With Uncertainty: The Chance of Concluding
the Site Is Protective of Public Health and the Environment
When It Is Actually Not Protective 3-10
3.7 Data Quality Objectives 3-11
3.8 Summary 3-12
4. DESIGN OF THE SAMPLING AND ANALYSIS PLAN 4-1
4.1 The Sampling Plan 4-1
4.1.1 Random Versus Systematic Sampling 4-2
4.1.2 Simple Versus Stratified Sampling 4-4
4.1.3 Sequential Sampling 4-6
4.2 The Analysis Plan 4-6
4.3 Summary 4-7
5. HELD SAMPLING PROCEDURES 5-1
5.1 Determining the General Sampling Location 5-1
5.2 Selecting the Sample Coordinates for a Simple Random Sample 5-3
5.3 Selecting the Sample Coordinates for a Systematic Sample 5-5
5.3.1 An Alternative Method for Locating the Random Start
Position for a Systematic Sample 5-10
5.4 Extension to Stratified Sampling 5-13
5.5 Field Procedures for Determining the Exact Sampling Location 5-13
5.6 Subsampling and Sampling Across Depth 5-14
5.6.1 Depth Discrete Sampling 5-15
5.6.2 Compositing Across Depth 5-15
5.6.3 Random Sampli ng Across Depth 5-17
IV
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5.7 Quality Assurance/Quality Control (QA/QC) in Handling the
Sample During and After Collection 5-18
5.8 Summary 5-18
6. DETERMINING WHETHER THE MEAN CONCENTRATION OF
THE SITE IS STATISTICALLY LESS THAN A CLEANUP
STANDARD 6-1
6.1 Notation Used in This Chapter 6-1
6.2 Calculating the Mean, Variance, and Standard Deviation 6-2
6.3 Methods for Random Samples 6-4
6.3.1 Estimating the Variability of the Chemical Concentration
Measurements 6-4
6.3.1.1 Use of Data from a Prior Study to Estimate a 6-5
6.3.1.2 Obtain Data to Estimate a After a Remedial Action
Pilot 6-5
6.3.1.3 An Alternative Approximation for & 6-6
6.3.2 Formulae for Determining Sample Size 6-7
6.3.3 Calculating the Mean, Standard Deviation, and Confidence
Intervals 6-10
6.3.4 Inference: Deciding Whether the Site Meets Cleanup
Standards 6-11
6.4 Methods for Stratified Random Samples 6-12
6.4.1 Sample Size Determination 6-13
6.4.2 Calculation of the Mean and Confidence Intervals 6-15
6.4.3 Inference: Deciding Whether the Site Meets Cleanup
Standards 6-18
6.5 Methods for Systematic Samples 6-20
6.5.1 Estimating Sample Size 6-20
6.5.2 Concerns Associated with Estimating the Mean, Estimating
the Variance, and Making Inference from a Systematic
Sample 6-21
6.5.2.1 Treating a Systematic Sample as a Random Sample 6-22
6.5.2.2 Treating the Systematic Sample as a Stratified
Sample 6-22
6.5.2.3 Linearization and Estimates from Differences
Between Adjacent Observations of a Systematic
Sample 6-25
6.6 Using Composite Samples When Testing the Mean 6-26
6.7 Summary 6-27
7. DETERMINING WHETHER A PROPORTION OR PERCENTILE OF
THE SITE IS LESS THAN A CLEANUP STANDARD 7-1
7.1 Notation Used in This Chapter 7-2
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7.2 Steps to Correct for Laboratory Error 7-3
7.3 Methods for Simple Random Samples 7-5
7.3.1 Sample Size Determination 7-5
7.3.2 Understanding Sample Size Requirements 7-6
7.3.3 Estimating the Proportion Contaminated and the
Associated Standard Error 7-7
7.3.4 Inference: Deciding Whether a Specified Proportion of the
Site is Less than a Cleanup Standard Using a Large
Sample Normal Approximation 7-8
7.3.5 Deciding Whether a Specified Proportion of the Site is
Less than the Cleanup Standard Using an Exact Test 7-9
7.4 A Simple Exceedance Rule Me'thod for Determining Whether a
Site Attains the Cleanup Standard 7-11
7.5 Methods for Stratified Samples 7-12
7.5.1 Sample Size Determination 7-13
7.5.2 Calculation of Basic Statistics 7-16
7.5.3 Inference: Deciding Whether the Site Meets Cleanup
Standards 7-19
7.6 Testing Percentiles from a Normal or Lognormal Population
Using Tolerance Intervals 7-20
7.6.1 Sample Size Determination 7-21
7.6.2 Testing the Assumption of Normality 7-23
7.6.3 Inference: Deciding Whether the Site Meets Cleanup
Standards Using Tolerance Limits 7-24
7.7 Summary 7-26
8. TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING 8-1
8.1 Notation Used in This Chapter 8-2
8.2 Description of the Sequential Procedure 8-3
8.3 Sampling Considerations in Sequential Testing 8-4
8.4 Computational Aspects of Sequential Testing 8-5
8.5 Inference: Deciding Whether the Site Meets Cleanup Standards 8-7
8.6 Grouping Samples in Sequential Analysis 8-8
8.7 Summary 8-10
9. SEARCHING FOR HOT SPOTS 9-1
9.1 Selected Literature that Describes Methods for Locating Hot
Spots 9-1
9.2 Sampling and Analysis Required to Search for Hot Spots 9-1
9.2.1 Basic Concepts 9-1
9.2.2 Choice of a Sampling Plan 9-4
9.2.3 Analysis Plan; 9-8
9.3 Summary 9-8
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10. THE USE OF GEOSTATISTICAL TECHNIQUES FOR EVALUATING
THE ATTAINMENT OF CLEANUP STANDARDS 10-1
10.1 Background 10-2
10.1.1 What Is Geostatistics and How Does It Operate? 10-2
10.1.2 Introductory Geostatistical References 10-4
10.2 Soils Remediation Technology and the Use of Geostatistical
Methods 10-4
10.2.1 Removal 10-5
10.2.2 Treatment Involving Homogenization 10-6
10.2.3 Flushing 10-7
10.3 Geostatistical Methods that Are Most Useful for Verifying the
Completion of Cleanup 10-8
10.4 Implementation of Geostatistical Methods 10-9
10.5 Summary 10-12
BIBLIOGRAPHY BIB-1
APPENDIX A STATISTICAL TABLES A-l
APPENDIXB EXAMPLE WORKSHEETS B-l
APPENDIX C BLANK WORKSHEETS C-l
APPENDIX D GLOSSARY D-l
INDEX IND
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LIST OF FIGURES
Page
Figure 1.1 Steps in Evaluating Whether a Site Has Attained the Cleanup
Standard 1-2
Figure 2.1 A Statistical Perspective of the Sequence of Ground Water
Monitoring Requirements Under RCRA 2-4
Figure 2.2 Hypothetical Power Curve 2-6
Figure 2.3 Hypothetical Power Curve Showing False Positive and
False Negative Rates 2-7
Figure 2.4 Measures of Location: Mean, Median, 25th Percentile, 75th
Percentile, and 95th Percentile for Three Distributions 2-10
Figure 2.5 Components of a Risk-Based Standard 2-13
Figure 3.1 Steps in Defining the Attainment Objectives 3-2
Figure 3.2 Geographic Areas and Subareas Within the Site 3-4
Figure 4.1 Dlustration of Random, Systematic, and Stratified Sampling 4-3
Figure 5.1 Map of a Sample Area with a Coordinate System 5-2
Figure 5.2 Map of a Sample Area Showing Random Sampling
Locations 5-6
Figure 5.3 Examples of a Square and a Triangular Grid for Systematic
Sampling 5-6
Figure 5.4 Locating a Square Grid Systematic Sample 5-8
Figure 5.5 Map of a Sample Site Showing Systematic Sampling
Locations 5-10
Figure 5.6 Method for Positioning Systematic Sample Locations in the
Field 5-12
Figure 5.7 An Example Illustration of How to Choose an Exact Field
Sampling Location from an Approximate Location 5-14
Figure 5.8 Subsampling and Sampling Across Depth 5-16
Figure 6.1 An Example of How to Group Sample Points from a
Systematic Sample so that the Variance and Mean Can Be
Calculated Using the Methodology for a Stratified Sample 6-24
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Figure 6.2 Example of a Serpentine Pattern 6-25
Figure 8.1 Graphic Example of Sequential Testing 8-4
Figure 9.1 A Square Grid of Systematically Located Grid Points with
Circular and Elliptical Hot Spots Superimposed 9-4
Figure 9.2 Grid Spacing and Ellipse Shape Definitions for the Hot Spot
Search Table in Appendix A 9-5
Figure 10.1 An Example of an Empirical Variogram and a Spherical
Variogram Model -. 10-3
Figure 10.2 Contour Map of the Probability in Percent of Finding the
Value of 1,000 ppm or a Larger Value 10-11
Figure 10.3 Contour Map of the Probability in Percent of a False Positive
in the Remedial Action Areas and the 1,000 Contour Line 10-11
Figure 10.4 Contour Map of the Probability in Percent of a False
Negative in the Remedial Action Areas and the 1,000 ppm
Contour Line 10-11
Figure A.I Power Curves fora= 1% A-12
Figure A.2 Power Curves for a = 5% A-13
Figure A.3 Power Curves for a = 10% A-14
Figure A.4 Power Curves for a = 25% A-15
Figure B.I Example Worksheets: Parameters to Test in Each Sample
Area and Map of the Site B-2
Figure B.2 Example Worksheets: Sequence in Which the Worksheets
are Completed B-3
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TABLE OF CONTENTS
LIST OF TABLES
Page
Table 1.1 EPA guidance documents that present methodologies for
collecting and evaluating soils data 1-5
Table 2.1 A diagrammatic explanation of false positive and false
negative conclusions 2-5
Table 3.1 Points to consider when trying to choose among the mean,
high percentile, or median 3-7
Table 3.2 Recommended parameters to test when comparing the
cleanup standard to the average concentration of a chemical
with chronic effects 3-9
Table 4.1 Where sample designs and analysis methods for soil
sampling are discussed in this document 4-7
Table 7.1 Selected information from Tables A.7-A.9 that can be used
to determine the sample sizes required for zero or few
exceedance rules associated with various levels of statistical
performance and degrees of cleanup 7-14
Table 9.1 Selected references regarding the methodologies for
identifying hot spots at waste sites 9-2
Table 9.2 Factors controlling the design of a hot spot search sampling
plan 9-6
Table 10.1 Selected introductory and advanced references that introduce
and discuss geostatistical concepts 10-5
Table 10.2 Introductory references for indicator, probability, and
nonparametric global estimation kriging 10-10
Table 10.3 Selected geostatistical software 10-13
Table A.I Table of t for selected alpha and degrees of freedom A-l
Table A.2 Table of z for selected alpha or beta A-2
Table A.3 Table of k for selected alpha, PQ, and sample size where
alpha = 0.10 (i.e., 10%) A-3
Table A.4 Table of k for selected alpha, PO, and sample size where
alpha = 0.05 (i.e., 5%) A-4
Table A.5 Table of k for selected alpha, PQ, and sample size where
alpha = 0.01 (i.e., 1%) A-5
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Table A.6 Sample sizes required for detecting a scaled difference tau of
the mean from the cleanup standard for selected values of
alpha and beta A-6
Table A.7 Sample size required for test for proportions with a = .01
and (3 = .20, for selected values of PO and P! A-7
Table A.8 Sample size required for test for proportions with a = .05
and (3 = .20, for selected values of P0 and P! A-8
Table A.9 Sample size required for test for proportions with a = .10
and P = .20, for selected values of PQ and P! A-9
Table A. 10 Tables for determining critical values for the exact binomial
test, with a = 0.01, 0.05, and 0.10 A-10
Table A. 11 The false positive rates associated with hot spot searches as a
function of grid spacing and hot spot shape A-ll
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TABLE OF CONTENTS
LIST OF BOXES
Page
Box 2.1 Estimating the Final Sample Size Required 2-15
Box 5.1 Steps for Generating Random Coordinates that Define
Sampling Locations 5-4
Box 5.2 An Example of Generating Random Sampling Locations 5-5
Box 5.3 Calculating Spacing Between Adjacent Sampling Locations 5-7
Box 5.4 Locating Systematic Coordinates 5-9
Box 5.5 Alternative Method for Locating the Random Stan Position
for a Systematic Sample 5-11
Box 6.1 Calculating Sample Mean, Variance, Standard Deviation,
and Coefficient of Variation 6-3
Box 6.2 An Alternate Approximation for & 6-6
Box 6.3 Formulae for Calculating the Sample Size Needed to
Estimate the Mean 6-7
Box 6.4 Example of Sample Size Calculations 6-8
Box 6.5 Example: Determining Sample Size for Testing the Mean
Using the Power Curves 6-9
Box 6.6 Determining the Approximate Power Curve for a Specified
Sample Size 6-10
Box 6.7 Computing the Upper One-sided Confidence Limit 6-11
Box 6.8 An Example Evaluation of Cleanup Standard Attainment 6-12
Box 6.9 Calculating the Proportion of the Volume of Soil 6-13
Box 6.10 Calculating Desired Sample Size for Each Stratum of a
Stratified Random Sample 6-14
Box 6.11 An Example Sample Size Determination for a Stratified
Sample 6-15
Box 6.12 Formula for the Mean Concentration from a Statified Sample....6-16
Box 6.13 Formula for the Standard Error from a Stratified Sample 6-17
Box 6.14 Formula for Degrees of Freedom from a Stratified Sample 6-17
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Box 6.15 Formula for the Upper One-sided Confidence Interval from a
Stratified Sample 6-18
Box 6.16 An Example Illustrating the Determination of Whether the
Mean from a Stratified Sample Attains a Cleanup Standard 6-19
Box 6.17 Estimating the Mean, the Standard Error of the Mean, and
Degrees of Freedom When a Systematic Sample Is Treated
as a Stratified Sample 6-24
Box 6.18 Formula for Upper One-sided Confidence Interval for the
True Mean Contamination, When a Systematic Sample Is
Treated as a Stratified Sample 6-24
Box 6.19 Computational Formula for Estimating the Standard Error
and Degrees of Freedom from Samples Analyzed in a
Serpentine Pattern 6-26
Box 7.1 Illustration of Multiple Measurement Procedure for Reducing
Laboratory Error 7-4
Box 7.2 Computing the Sample Size When Testing a Proportion or
Percentile 7-6
Box 7.3 Example of How to Determine Sample Sizes When
Evaluating Cleanup Standards Relative to a Proportion 7-7
Box 7.4 Calculating the Proportion Contaminated and the Standard
Error of the Proportion 7-8
Box 7.5 Calculation of the Upper Confidence Limit on a Proportion
Using a Large Sample Normal Approximation 7-9
Box 7.6 An Example of Inference Based on the Exact Test 7-11
Box 7.7 Computing the Sample Size for Stratum h 7-16
Box 7.8 Sample Size Calculations for Stratified Sampling 7-17
Box 7.9 Calculating an Overall Proportion of Exceedances and the
Standard Error of the Proportion from a Stratified Sample 7-18
Box 7.10 Calculating the Upper Limit of the One-sided Confidence
Interval on an Estimate of the Proportion 7-19
Box 7.11 Inference for Proportions Using Stratified Sampling 7-20
Box 7.12 Calculating the Sample Size Requirements for Tolerance
Intervals 7-22
Box 7.13 Calculating Sample Size for Tolerance Intervals—Two
Examples 7-23
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Box 7.14 Calculating the Upper Tolerance Limit 7-24
Box 7.15 Tolerance Intervals: Testing for the 95th Percentile with
Lognormal Data 7-25
Box 8.1 Defining Acceptance and Rejection Criteria for the Sequential
Test of Proportions 8-6
Box 8.2 Deciding When the Site Attains the Cleanup Standard 8-7
Box 8.3 An Example of Sequential Testing 8-9
Box 8.4 Example of Sequential Test'Using Grouped Samples 8-10
Box 9.1 Approximating the Sample Size When Area and Grid
Interval Are Known 9-7
Box 10.1 Steps for Obtaining Geostatistical Software from EMSL-LV ..10-12
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AUTHORS AND CONTRIBUTORS
This manual represents the combined efforts of several organizations and
many individuals. The names of the primary contributors, along with the role of each
organization, are summarized below.
Dynamac Corporation, 11140 Rockville Pike, Rockville, MD 20852 (subcontractor to
Westat) — sampling, treatment, chemical analysis of samples. Key Dynamac Corporation
staff included:
David Lipsky Wayne Tusa
Richard Dorrler
EPA, OPPE, Statistical Policy Branch — project management, technical input, peer
review. Key EPA staff included:
Barnes Johnson
SRA Technologies, Inc., 4700 King Street, Suite 300, Alexandria, VA 22302, EPA
Contract No. 68-01-7379, Task 11 -- editorial and graphics support, final report
preparation. Key SRA staff included:
Marcia Gardner Jocelyn Smith
Sylvia Burns Lori Hidinger
Westat, Inc., 1650 Research Boulevard, Rockville, MD 20850, Contract No. 68-01-
7359, Task 5 - research, statistical procedures, draft report. Key Westat staff included:
John Rogers Jill Braden
Paul Flyer Ed Bryant
AdamChu
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EXECUTIVE SUMMARY
This document provides regional project managers, onsite coordinators, and
their contractors with sampling and analysis methods for evaluating whether a soils
remediation effort has been successful. The verification of cleanup by evaluating a site
relative to a cleanup standard or applicable and relevant or appropriate requirement (ARAR)
is discussed in section 121 of the Superfund Amendments and Reauthorization Act
(SARA). In section 121 of SARA the "attainment" of cleanup standards and ARARs is
mentioned repeatedly. This manual, the first in a series, provides a technical interpretation
of what sampling and data analysis methods are acceptable for verifying "attainment" of a
cleanup standard in soils and solid media.
Statistical methods are emphasized because there is a practical need to make
decisions regarding whether a site has met a cleanup standard in spite of uncertainty. The
uncertainty arises because Superfund managers are faced with being able to sample and
analyze only a small portion of the soil at the site yet having to make a decision regarding
the entire site. Statistical methods are designed to permit this extrapolation from the results
of a few samples to a statement regarding the entire site.
The methods in this document approach cleanup standards as having three
components that influence the overall stringency of the standard. The first component is
the magnitude, level, or concentration that is deemed protective of public health and the
environment. The second component of the standard is the sampling that is done to
evaluate whether a site is above or below the standard. The final component is how the
resulting data are compared with the standard to decide whether the remedial action was
successful. All three of these components are important. Failure to address any of the
three components can result in far less cleanup than desired. Managers must look beyond
the cleanup level and explore the sampling and analysis that will allow evaluation of the site
relative to the cleanup level.
For example, suppose that a cleanup level is chosen and that only a few
samples are acquired. When the results are available, it is found that the mean of those
samples is just below the cleanup level, and therefore, the site is judged as having been
successfully remediated. Under this scenario, there may be a large chance that the average
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EXECUTIVE SUMMARY
of the entire site, as opposed to the samples, is well above the cleanup level. Uncertainty
was not considered, and therefore, there is a large chance that the wrong decision was
made and the site-wide average is not below the cleanup level.
These concepts and solutions to the potential pitfalls are presented in a
sequence that begins with an introduction to the statistical reasoning required to implement
these methods. Then the planning activities are described; these require input from both
nonstatisticians and statisticians. The statistical aspects of field sampling are presented.
Finally, a series of methodological chapters are presented which consider the cleanup
standard as: (1) an average condition; (2) a value to be rarely exceeded; (3) being defined
by small discrete hot spots of contamination that should be found if present; or (4) broad
areas that should be defined and characterized. A more detailed discussion of the
document follows.
Chapter One introduces the need for the guidance and its application with
risk-based standards, under various soils treatment alternatives, and in various parts of the
Superfund program. Standards development and usage depends on certain factors, and the
three categories of standards used by EPA are discussed: technology-based, background-
based, and risk-based standards.
The statistical methods described in this manual are useful in various phases
of treatment, testing, piloting, and full-scale implementation of various treatment
technologies. In addition, the methods in this manual apply in various programmatic
circumstances including both Superfund and Enforcement lead sites and removal actions.
Chapter Two addresses statistical concepts as they relate to the evaluation of
cleanup attainment Discussions of the form of the null and alternative hypothesis, types of
errors, statistical power curves, and special data like less-than-detection-limit values and
outliers are presented.
A site manager inevitably confronts the possibility of error in evaluating the
attainment of the cleanup standard: is the site really contaminated because a few samples
are above the standard? Conversely, is the site really "clean" because the sampling shows
the majority of the samples to be within the cleanup standard? The statistical methods
demonstrated in the guidance document allow decision making under uncertainty and valid
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EXECUTIVE SUMMARY
extrapolation of information that can be defended and used with confidence to determine
whether the site meets the cleanup standard.
The procedures in this guidance document favor protection of the
environment and human health. If uncertainty is large or the sampling inadequate, these
methods conclude that the sample area does not attain the cleanup standard. Therefore, the
null hypothesis, in statistical terminology, is that the site does not attain the cleanup
standard until sufficient data are acquired to prove otherwise.
Chapter Three discusses the steps in specifying attainment objectives.
Definition of the attainment objectives is the first task in the evaluation of whether a site has
attained a cleanup standard. Attainment objectives are not specified by statisticians, but
must be provided by risk assessors, engineers, and soil scientists. Specifying attainment
objectives includes specifying the chemicals of concern and the cleanup levels, as well as
choosing the area to be remediated.
Chapter Four presents approaches to the design of remedial verification
sampling and analysis plans. The specification of this plan requires consideration of how
the environment and human health are to be protected and how the sampling and analysis
are to achieve adequate precision at a reasonable cost.
Sampling designs considered in this guidance document are random
sampling, stratified sampling, systematic sampling, and sequential sampling. Differences
in these approaches, including advantages and disadvantages, are both discussed and
graphically displayed. With any plan, the methods of analysis must be consistent with the
sample design.
A primary objective of the analysis plan involves making a decision
regarding how to treat the applicable cleanup standard. For example, is the cleanup
standard a value that should rarely be exceeded (1 or 5 percent of the time) at a remediated
site? Or, alternatively, should there be high confidence that the mean of the site is below
the cleanup standard? Should there be no hot spots with concentrations in excess of the
cleanup standard? Or should the analysis plan employ a combination of these criteria.
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EXECUTIVE SUMMARY
Chapter Five discusses the statistical aspects of field sampling procedures.
The procedures used to establish random and systematic sample locations are discussed. In
addition to selecting sampling locations, the advantages and disadvantages of methods for
subsampling across depth are discussed and illustrated. Three approaches presented are
depth discrete sampling, compositing across depth, and random sampling across depth.
Chapter Six describes procedures for determining whether there is
confidence, based on the results of a set of samples, that the mean concentration of the
contaminant in a sample area is less than the cleanup standard. Basic formulas are given
and used in examples to illustrate the procedures. The primary point is that to ensure with
confidence that the site mean is below the cleanup standard, the sample mean must be well
below the cleanup standard by a distance determined by a confidence limit.
The following topics—determination of sample size; calculation of the mean,
standard deviation, and confidence interval; and deciding if the sample area attains the
cleanup standard-are discussed for these three sampling plans:
• Simple random sampling;
• Stratified random sampling; and
• Systematic sampling.
Chapter Seven presents several approaches that allow evaluation of whether
a specified proportion or percentage of soil at a hazardous waste site is below the cleanup
standard. The methods described apply if there is interest in verifying that only a small
proportion or percentage of the soil at the site exceeds the cleanup standard.
One way to implement these methods is to use simple exceedance rules. A
sample size and number of exceedances are specified that coincide with an acceptable level
of certainty and level of cleanup. If the prespecified number of samples is obtained and the
number of exceedances is less than or equal to the allowed number of exceedances the site
is judged clean. If there are more exceedances than allowed then cleanup cannot be
verified. The more exceedances allowed, the more soil samples that need to be collected to
maintain the statistical performance of the method.
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EXECUTIVE SUMMARY
Chapter Eight deals with sequential sampling as a method for testing
percentiles. Unlike the fixed-sample-size methods discussed in the two previous chapters,
with sequential sampling, a statistical test is performed after each sample or small batch of
samples is collected and analyzed. The test then makes one of three decisions: the site has
attained the cleanup standard, the site has not attained the cleanup standard, or select
another sample. Sequential sample findings can respond quickly to very clean or very
contaminated sites and therefore offer cost savings. Although these procedures yield a
lower sample size on the average than that for fixed-sample-size procedures, in order to be
practical, they require "rapid turn-around" laboratory methods.
Chapter Nine illustrates the design of sampling plans to search for hot
spots. The conclusions that can be drawn regarding the presence or absence of hot spots
are discussed. Hot spots are generally defined as relatively small, localized, elliptical areas
with contaminant concentrations in excess of the cleanup standard. Tables are provided to
help determine grid spacing and detect hot spots of various sizes with different
probabilities.
Chapter Ten discusses the use of geostatistical methods, which provide a
method for mapping spatial data that enables both interpolation between existing data points
and a method for estimating the precision of the interpolation. Geostatistical applications
are described as a two-step process. First, the spatial relationship is modeled as a
variogram and then the variogram is used by a kriging algorithm to estimate concentrations
at points that were not sampled. Indicator and probability kriging are most useful for
remedial verification purposes.
Geostatistical methods have many applications in soil remediation
technology, especially when the extent of contamination needs to be characterized. This
chapter includes guidance to help decide whether geostatistical data analysis and evaluation
methods are appropriate for use with soils remediation activities that involve removal,
homogenization, and flushing.
Before being applied the kriging techniques will require further study on the
part of the user. Reference documents are listed. Because kriging cannot be conveniently
or practically implemented without a computer and the appropriate software, a first-level
familiarity with the methodology along with use of a software package is desirable to
xx
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EXECUTIVE SUMMARY
explore example applications and data sets. EPA has developed the first version of a
geostatistical software package which can be obtained by following instructions at the end
of Chapter 10.
xxi
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1. INTRODUCTION
Congress revised the Superfund legislation in the Superfund Amendments
and Reauthorization Act of 1986 (SARA). Among other provisions of SARA, section 121,
Cleanup Standards, discusses criteria for selecting applicable and relevant or appropriate
requirements (ARARs) and includes specific language that requires EPA cleanups to attain
ARARs.
Neither SARA nor EPA regulations or guidances specify how to determine
attainment or verify that the cleanup standards have been met. This document offers
procedures that can be used to determine whether, after a remediation action, a site has
attained an appropriate cleanup standard.
1.1 General Scope and Features of the Guidance Document
1.1.1 Purpose
This document describes methods for testing whether soil chemical
concentrations at a site are statistically below a cleanup standard or ARAR. If it can be
reasonably concluded that the remaining soil or treated soil at a site has concentrations that
are statistically less than relevant cleanup standards then the site can be judged protective of
human health and the environment. Figure 1.1 shows the steps involved in this evaluation
which requires specification of attainment objectives, sampling protocols, and analysis
methods.
For example, consider the situation where several samples were taken. The
results indicate that one or two of the samples exceed the standard: How should this
information be used to decide whether the standard has been attained? Some possible
considerations include: the mean of those samples could be compared with the standard;
the magnitude of the two sample values that are larger than the standard might be useful in
making a decision; or the area where the two large sample values were obtained might
provide some insight. The following factors are important in reaching the decision as to
whether a cleanup standard has been attained:
1-1
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CHAPTER 1: INTRODUCTION
Figure 1.1 Steps in Evaluating Whether a Site Has Attained the Cleanup Standard
Clean Up All or Part of
the Sample Area
•
'
Reassess Cleanup
Technology
'
'
Identify Areas of High
Contamination
Chapters 6, 7, 8, 9 & 10
4
k
^
C Start J
4
Define Attainment
Objectives
Chapter 3
4
Specify Sample Design
and Analysis Plan
Determine Sample Size
Chapters 4, 6, 7, 8 & 9
i
Collect Data
Chapter 5
4
Determine If the
Sample Area Attains the
Cleanup Standard
Chapters 6, 7, 8, 9 & 10
/ Is the Ny
/ Cleanup \.
X Standard /
No \ Attained? /
Tves
\~ End J
1-2
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CHAPTER 1: INTRODUCTION
• The spatial extent of the sampling and the size of the sample area;
• The number of samples taken;
• The strategy of taking samples; and
• The way the data are analyzed.
Simply to require that a Superfund site be cleaned until the soil
concentration of a chemical is below 50 mg/kg is incomplete. Statements suggesting that
the site will be remediated until the soil concentration of a chemical is 50 mg/kg reveal little
in terms of the environmental results anticipated, the future exposure expected, the resultant
risk to the local population, or the likelihood that substantial contamination will remain after
a decision is made that the site has been fully remediated. A specific sampling and data
analysis protocol must accompany the risk-based standard for the standard to be
meaningful in terms of benefit or actual risk.
This document does not attempt to suggest which standards apply or when
they apply (i.e., the "How clean is clean?" issue). Other Superfund guidance documents
(e.g., USEPA,1986c and USEPA, 1986d) perform that function.
1.1.2 Intended Audience and Use
Management/supervisory personnel will find the executive summary and
introductory chapters useful. However, this manual is intended primarily for Agency
personnel (primarily onsite coordinators and regional project managers), responsible
parties, and their contractors who are involved with monitoring the progress of soils
remediation at Superfund sites. Although selected introductory statistical concepts are
reviewed, the document is directed toward readers that have had prior training or
experience applying quantitative methods.
This document discusses data analysis and statistical methods for evaluating
the effectiveness of Superfund remedial actions. However, there are many other technical
aspects to this problem. Input from soil scientists, engineers, geologists, hydrologists,
geochemists, and analytical chemists is essential. There must be dialogue among this
group, including the statistician, so that each member understands and considers the point
1-3
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CHAPTER 1: INTRODUCTION
of view of the others. It is only through collaboration that an effective evaluation scheme
can be developed to measure the effectiveness of a remedial action.
This document does not intend to address the issues that the other members
of the team specialize in such as:
• Soil sample acquisition protocols;
• Areas of the vadose zone of concern under different situations;
• The influence of soil chemistry;
• Waste types based on industrial processes;
• Leaching procedures that approximate the expected weathering
processes and risk assessment assumptions;
• Chemical analysis methods useful given particular soils matrices; or
• Approaches to soils remediation.
Table 1.1 lists other relevant EPA guidance documents on sampling and
evaluating soils and solid media that apply to both the statistical and other technical
components of a sampling and analysis program.
The selection of statistical methods for use in assessing the attainment of
cleanup standards depends on the characteristics of the data. In soils, concentrations of
contaminants change relatively slowly, with little variation from season to season. In
ground water, the number of measurements available for spatial characterization is limited
and seasonal patterns may exist in the data. As a result of these differences, separate
procedures are recommended for the differing problems associated with soils and solid
media, and ground water, surface water, and air. These media will be addressed in
separate volumes.
1-4
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CHAPTER 1: INTRODUCTION
Table 1.1
Tide
EPA guidance documents that present methodologies for collecting and
evaluating soils data
Sponsoring
Office
Date
ID
Number
Preparation of Soil
Sampling Protocol:
Techniques and
Strategies
Soil Sampling Quality
Assurance User's Guide
Verification of PCB
Spill Cleanup by
Sampling and Analysis
Guidance Document for
Cleanup of Surface
Impoundment Sites
Test Methods for
Evaluating Solid
Waste
Draft Surface Impoundment
Clean Closure Guidance
Manual
Data Quality Objectives
for Remedial Response
Activities: Development
Process
Data Quality Objectives
for Remedial Response
Activities: Example
Scenario RI/FS Activities
at a Site with Contaminated
Soils and Ground Water
EMSL-LV
ORD
EMSL-LV
ORD
OTS
OPTS
OERR
OSWER
OSW
OSWER
OSW
OSWER
OERR
OSWER
August
1983
May
1984
August
1985
June
1986
November
1987
March
1987
March
1987
EPA 600/
4-83-020
EPA 600/
4-84-043
EPA 5607
5-85-026
OSWER
DIRECTIVE
9380.0-6
SW-846
OSWER
DIRECTIVE
9476.0-8.C
EPA 5407
G-87/003
OERR
OSWER
March
1987
EPA 5407
G-87/004
It must be emphasized that this document is intended to provide flexible
guidance and general dkection. This manual is not a regulation and should not be imposed
as a regulation. Finally, this document should not be used as a "cookbook" or a
replacement for engineering judgment.
1-5
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CHAPTER 1: INTRODUCTION
1.1.3 Bibliography, Glossary, Boxes, Worksheets, Examples, and
References to "Consult a Statistician"
The document includes a bibliography which provides a point of departure
for the user interested in further reading. There are references to primary textbooks,
pertinent journal articles, and related guidances.
The glossary is included to provide short, practical definitions of
terminology used in the manual. The glossary does not use theoretical explanations or
formulae and should not be considered a replacement for more complete discussions in the
text or alternative sources of information.
Boxes are used throughout the document to separate and highlight
calculation methods and example applications of the methods. A listing of all boxes and
their page numbers is provided on pages xii - xiv.
A series of worksheets is included to help organize calculations. Reference
to the pertinent sections of the document appears at the top of each worksheet.
Example data and calculations are presented in the boxes and worksheets.
The data and sites are hypothetical, but elements of the examples correspond closely to
actual sites.
Finally, the document often directs the reader to "consult a statistician"
when more difficult and complicated situations are encountered. A directory of Agency
statisticians is available from the Statistical Policy Branch (PM-223) at EPA Headquarters.
1.2 A Categorization Scheme for Cleanup Standards
Superfund remediations require standards for assessing the success and
completion of the cleanup. The criteria for choosing the type of standard and setting the
magnitude of the standard come from different sources, depending on many factors
including the nature of the contamination, negotiations with potentially responsible parties,
and public comment on alternatives identified by EPA.
1-6
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CHAPTER 1: INTRODUCTION
Many programs throughout EPA use numerical standards variously
described as ARARs, concentration limits, limitations, regulatory thresholds, action levels,
and criteria. These standards are often expressed as concentration measures of chemicals
or chemical indicators. Standards development and usage depends on the media to which
the standard applies, the data used to develop the standard, and the manner of evaluating
compliance with the standard. The following discussion categorizes the standards used by
EPA and compares the features of each category.
1.2.1 Technology-Based Standards
Technology-based standards are developed for the purpose of defining the
effectiveness of pollution abatement technology from an engineering perspective. For
example, waste water treatment plants operating under the National Pollution Discharge
Elimination System (NPDES) must be designed and operated under a numerically
prescribed level of technological performance depending on the particular industrial
category. Technology-based standards such as the NPDES standards are developed and
applied using statistical methods that consider variability in the operation of the treatment
system. The likelihood of exceeding the standards is rare if the technology is installed and
operated properly. Often Superfund sites require the installation of waste water treatment
systems and compliance with NPDES standards.
1.2.2 Background-Based Standards
Background-based standards are developed using site-specific background
data. An example is the background ground water concentration standards that hazardous
waste land disposal facilities use under Resource Conservation and Recovery Act (RCRA)
permits. The background data are used to establish a standard for the facility, which
accounts for the presence of any existing contamination hydraulically upgradient of the
facility. Background standards are applied on a site-specific basis, but because they are
developed using statistical methodologies, the standards can be associated with a known
false positive and false negative rate.
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CHAPTER 1: INTRODUCTION
1.2.3 Risk-Based Standards
A third class of standards, risk-based standards are developed using risk
assessment methodologies. Chemical-specific ARARs adopted from other programs often
include at least a generalized component of risk. However, risk standards may be specific
to a site, developed using a local endangerment evaluation.
Risk-based standards are expressed as a concentration value. However,
cleanup standards based on risk as applied in the Superfund program are not associated
with a standard method of interpretation when applied in the field. Therefore, risk-based
standards, when applied in the field, do not consider false positive and false negative
errors. Although statistical methods are used to develop elements of risk-based standards,
the estimated uncertainties are not carried through the analysis or used to qualify the
standards for use in a field sampling program. Even though risk standards are not
accompanied by measures of uncertainty, field data, collected for the purpose of
representing the entire site and validating cleanup, will be uncertain. This document allows
decision making regarding site cleanup by providing methods that statistically compare risk
standards with field data in a scientifically defensible manner that allows for uncertainty.
1.3 Use of this Guidance in Superfund Program Activities
Standards that apply to Superfund activities normally fall into the third
category of risk-based standards. There are many Superfund activities where risk-based
standards might apply. The following discussion provides suggestions for using the
methods described in this document in the implementation and evaluation of Superfund
activities.
1.3.1 Emergency/Removal Action
Similar to the guidance regarding sampling strategies associated with PCB
spills (USEPA, 1985), cleanup activities associated with the methods in this document will
be useful for circumstances that are encountered during emergency cleanups and removals.
In many cases, because of the time, safety, and exposure constraints associated with
emergency activity, initial cleanup will focus on areas visually or otherwise known to be
1-8
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CHAPTER 1: INTRODUCTION
contaminated. The methods described in this document will, however, be most useful in
verifying the initial cleanup of obvious contamination.
1.3.2 Remedial Response Activities
The objective of remediation is to ensure that release of and exposure to
contaminants is curtailed. Remedial efforts are normally long-term and require diverse,
innovative technology. As discussed in section 1.4, soil or solid media remediation can be
addressed using a variety of technologies. Numerical standards are used to define the
degree of curtailment. The methods described in this document can help to evaluate the
utility of the remediation technology in treating contaminants with respect to a particular
numerical standard.
1.3.3 Superfund Enforcement
The methods described in this document will also be useful for providing
more technically exacting negotiations, consent decree stipulations, and responsible party
oversight. Questions such as "How much is enough?" and "When can I stop cleaning?"
are constantly emerging at the enforcement negotiation table. More specific questions such
as "How much should you sample?", "What sampling pattern or method of sampling
design should be applied?" and "How can I minimize the chance of saying the site is still
dirty when it is basically clean?" are addressed here, as well as the ultimate question: "How
do I know when the standard has been attained at the entire site, knowing that the decision
is based on a body of data that is incomplete and uncertain?"
1.4 Treatability Studies and Soils Treatment Technologies
In addition to discussing the methods described in this document and their
relationship to aspects of the Agency's Superfund program, it is also important to discuss
how the methods will function when applied in treatability studies and under various soils
treatment technologies.
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CHAPTER 1: INTRODUCTION
1.4.1 Laboratory/Bench-Scale Treatabiiity Studies
Feasibility studies often include small bench-scale laboratory evaluations of
how various treatment agents and concentrations of agents will perform. Suppose that the
contaminant and soil characteristics at the site indicate that two fixation media offer a
promising remediation approach. A treatability study examining several concentrations of
the two media is proposed.
Under this scenario, the methods described in this manual could be applied
to the sampling program used to obtain soils material for the treatability study. Treatability
studies require "worst case" material—that is, soils with the highest concentrations or with
the most tightly bound contaminants. Therefore, "worst case" sample areas within the site
must be delineated, using data from prior remedial investigations. Once the "worst case"
sample area is defined, the soils can be sampled as described in this manual, the treatability
study executed, and the resulting data analyzed using the methods described in this
document to examine whether the method has sufficiently treated the soil to allow
attainment of the relevant cleanup standard.
1.4.2 Field/Pilot-Scale Treatability Studies
Once the feasibility study establishes an effective approach to treatment, it
may be implemented as an onsite pilot using the chemical/physical/biological remedy with
construction-scale onsite machinery. The approach favored in the bench-scale laboratory
experiment may be chosen if the cost is reasonable. Machinery such as cement mixers, soil
washers, soil mixing augers, incinerators, vacuum extraction manifold networks, or
infiltration or injection systems are used in a pretest fashion. With an associated
monitoring program, the methods in this guidance can be applied to determine whether the
method will attain the desired level of cleanup.
The primary difference between the laboratory testing results and those
obtained from field-scale pilot application is that far greater variability will be encountered
in the onsite pilot. Unless the treatment method is exceptionally effective relative to risk-
based standards in the laboratory, the variability encountered in the field may obscure the
treatment's effectiveness. This document guides the user to methods that will help in such
1-10
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CHAPTER 1: INTRODUCTION
a situation. In addition, if a reasonable sampling program is conducted at the pilot-scale,
these data can be used to estimate sample sizes for the sampling program associated with
the full-scale implementation of the technology.
1.4.3 Soils Treatment by Chemical Modification
Soils are often treated by chemical fixation or stabilization. This technology
uses a cement or grout-like material mixed with the contaminated soil or sediment. Once
the mixture reacts, it solidifies, and contaminants are retained in the matrix and resist
leaching. When this technology is used, the methods in this manual can be applied,
keeping in mind, however, the following caveats.
Once the material has solidified onsite, it cannot be sampled easily. The
ability to stabilize the site may be compromised if cores were obtained throughout the area.
In addition, the resulting monolith may be capped, which would restrict access to the
solidified matrix. Because it cannot be sampled after fixation, monitoring plans should be
developed before the mixing occurs. The sampling could occur by taking samples at
randomly located positions across the site and then pouring cylindrical casts of the material
immediately after it is mixed prior to setup. Enough casts must be obtained for the initial
evaluations of the site and for monitoring the aging process of the stabilized material.
During analysis, concentrations are measured in leachate obtained from an accepted
extraction procedure. Evaluation of the leachate from the casts allows determination of
whether the lithified material attains or continues to attain the relevant cleanup standard.
1.4.4 Soils Treatment by In Situ Removal of Contaminants
Several soil treatment technologies, including vacuum extraction, soils
leaching, and bioremediation, remove the contaminants without massive soil movement.
The efficacy of these systems can be evaluated using the methods in this document, with
the exceptions noted below.
Vacuum extraction is used to remove volatile compounds. Ambient air is
drawn down through the soil into a well network and then into an adjustable manifold
1-11
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CHAPTER 1: INTRODUCTION
system attached to a vacuum pump. Air is then sent through carbon columns to remove the
volatile compounds.
Soils leaching technologies are generally designed to extract contaminants
that are water soluble. Soils leaching also relies on a network of wells attached to a
manifold system. The system includes infiltration areas where aqueous solutions are
allowed to recharge into the soils system. A pumping system is attached to the manifold
and the water, after migration from the infiltration area to each well, is extracted and sent to
a waste water treatment system.
Soils bioremediation can be used to degrade contaminants. Microorganisms
use the contaminant as an energy source. One or more injection wells introduce water
possibly enriched with oxygen, nutrients, microorganisms, or other essential growth-
promoting materials. The injection wells are installed on one side of the contaminated area
and monitoring wells are installed in various patterns throughout and possibly beyond the
area of contamination. Again, a manifold system might be used for injection or sampling,
and extraction wells may be used to direct or improve water movement.
With these technologies, something other than direct soils sampling may be
used to evaluate effectiveness of the remediation, for example mass balance differencing.
In this case, the methods herein may not always apply. However, monitoring of the soil
relative to a risk-based cleanup standard is the most direct and protective measure of any
soils cleanup technology.
Another concern is that when these systems are in place, the above-ground
or slightly buried piping network will restrict the access of soils sampling equipment. For
example, vehicle-deployed augers may not be able to reach certain areas. Engineering
specifications should call for easy disassembly whenever possible. In cases where this is
not possible, the guidance can still be applied after exclusion of certain soil areas because of
inaccessibility.
A third consideration is that, after implementing the soil remediation
technology, the soils concentration profile may begin to take on a regular spatial pattern.
This occurs because removal wells are often arranged in a grid pattern and each well has a
zone of influence where the concentrations have been reduced substantially. The result is a
1-12
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CHAPTER 1: INTRODUCTION
series of areas with high and low concentrations across the site. As discussed in the
sampling chapter, under these circumstances systematic sampling should not be used
because all or many of the samples may be located in areas with high or low
concentrations. Random sampling is recommended to avoid this problem.
A final concern is that the soils system must be at steady state during the
sampling program. This requires shutting down the extraction process and allowing the
system to return to its original balance. This may take some time depending on
characteristics of the system. In some cases when progress is being measured over time,
methods pertaining to ground water in Volume 2 of this series might be more appropriate.
1.4.5 Soils Treatment by Incineration
Soils incineration involves the burning of soils in a furnace at high
temperatures to degrade the contaminants into a nontoxic form. The product of the
incineration is an ash. If questions arise as to whether the ash material contains chemicals
in excess of applicable standards, then this manual might be useful. Sampling will have to
be designed based on site-specific circumstances. If the treatment is highly effective and
uniform, only a few samples may be necessary to verify effectiveness. However, if the
standard is quite low and the measurement technology is variable at low concentrations,
more samples may be required.
1.4.6 Soils Removal
In the soils removal approach to site cleanup, soils are permanently or
temporarily removed from the site. Sampling must be done to verify that enough soil has
been removed, and to ensure that clean soil is not needlessly removed. Under the
circumstances associated with soils removal, there is no homogenization of the soil through
a fixation process or artificial regularity to the soil profile caused by local extraction. In this
case, geostatistical applications (Chapter 10) are useful for characterizing the contaminant
profile. A new concentration profile can be estimated with each succeeding layer that is
removed. In addition, geostatistical applications can help to identify hot spots that should
be removed and sampling and analysis to detect hot spots might be useful (Chapter 9).
1-13
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CHAPTER 1: INTRODUCTION
Finally, the simpler, more conventional evaluation methods that comprise the bulk of this
manual can also be used. Exner et al. (1985) describe an application of these evaluation
methods to a soils removal scenario at a Superfund site with dioxin contamination.
1.4.7 Soils Capping
A final category of soils remediation is to cap a site with impermeable layers
of clay and synthetic membranes. This prevents surface water from recharging to the
ground water through contaminated soils. Often caps are added as an additional measure in
conjunction with other approaches. The methods in this document can be used to
determine whether caps have met an engineering specification. For example, if the cap is
intended to be constructed with no more than a 10~~7 cms/sec permeability, samples might
be obtained to document that permeability has been attained. Sampling may be difficult
because it might disturb the integrity of the cap; however, it is possible that a pilot-scale
procedure could be implemented to verify attainment of the standard.
1.5 Summary
This document deals with statistical methodology and procedures for use in
assessing whether, after remediation, the treated soil or remaining soil attain the cleanup
standards that are protective of public health and the environment as required by section
121 of SARA.
Use of the document is intended primarily for Agency personnel,
responsible parties, and contractors who are involved with monitoring the progress of soils
and remediation at Superfund sites. Although selected introductory statistical concepts are
reviewed, the document is directed toward users having prior training or experience in
applying quantitative methods.
Important factors in determining whether a cleanup standard has been
attained are:
• The spatial extent of the sampling and the size of the sample area;
• The number of samples taken;
1-14
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CHAPTER 1: INTRODUCTION
• The strategy of taking samples; and
• The way the data are analyzed.
The three types of EPA cleanup standards are technology-based standards,
background-based standards, and risk-based standards. Superfund activities usually
employ risk-based standards. By providing methods that statistically compare risk
standards with field data in a scientifically defensible manner that allows for uncertainty,
this document allows decision making regarding site cleanup. The statistical methods can
be applied to the implementation and evaluation of:
• Emergency/removal action,
• Remedial response activities, and
• Superfund enforcement.
Also discussed are the functions of the statistical methods described in the document in the
context of a variety of treatability studies and soils treatment technologies.
1-15
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-------
2. INTRODUCTION TO STATISTICAL CONCEPTS
AND DECISIONS
When it comes to verifying cleanup, suppose that no exceedances of the
cleanup standard are to be allowed. In that case, one of the most frequently asked
questions regarding the use of statistical techniques in the evaluation of cleanup standards
is:
Why should I use statistical methods and complicate the
remedial verification process?
Allowing no exceedances of a standard is a perfectly acceptable decision
rule to use. In fact, that simple rule is a statistical procedure because errors are possible.
However, there is a chance that no exceedances will be discovered, yet a substantial portion
of the site is above the cleanup standard. This is clearly not a desirable environmental
result. With small sample sizes the chance of missing contamination is greater than with
larger sample sizes. This is intuitive; the more you search for contamination and do not
find it, the more confident you become in your conclusion that the site is clean.
Alternatively, consider the situation where a reasonable number of samples
is taken and (me sample exceeds the cleanup standard. In this case, you would conclude
that the site continues to be dirty under the no exceedance rule. However, the problem is
that this conclusion may be in error. Either laboratory error occurred or some rare and
insignificant parcel of contamination could have been discovered. Revisiting the remedial
method after many years or dollars of implementation is not reasonable because of the
possibility that an error was made. As sample sizes are increased, the chances of finding
one of the few obscure samples above the cleanup standard increases. How can you
balance the two sets of possibilities: the chance that the site is contaminated even when the
sampling shows attainment of the cleanup standard, and the chance of contamination when
the majority of samples taken show the site to be clean?
The answer is to evaluate the potential magnitude of these two errors and
balance them using the statistical strategies described in this manual. Statistical methods
perform a powerful and useful function--they allow extrapolation from a set of samples to
the entire site in a scientifically valid fashion.
2-1
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
Consider the following circumstance. The surface layer of soil from the
bottom of a 4-hectare lagoon at a Superfund site will be sampled using cores with a 4-cm
area. Given the size of the core and lagoon there will be approximately 10 million sample
locations; however, concentration measurements will only be made on 100 of the 10
million. Statistical sampling and analysis methods provide an approach for choosing which
100 of the 10 million locations to sample so that valid results can be presented and
statements can be made regarding the characteristics of the 10 million potential samples or
the entire site.
Clearly, because of the extrapolation exercise, the statements or inferences
regarding the 10 million sample locations have uncertainty. Statistical methods enable
estimation of the uncertainty. Without the statistical methods, uncertainty still exists; but
the uncertainty cannot be estimated validly.
This chapter will elaborate on statistical concepts and their specific
application to the evaluation of cleanup standards. Statistical concepts such as the form of
the null and alternative hypothesis, types of errors, statistical power, and handling peculiar
data structures like less-than-detection-limit values and outliers are discussed to promote
understanding. However, it is not necessary to read this chapter to apply the methods in
this manual.
2.1 Hypothesis Formulation and Uncertainty
With any statistical procedure, conclusions will vary depending on which
soil sample locations are selected. Therefore, based on the data collected, the investigator
may conclude that:
• The site attains the cleanup standard;
The site does not attain the cleanup standard; or
• More information is required to make a decision with a specified
level of confidence.
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
When the results of the investigation are uncertain, the procedures in this
guidance document favor protection of the environment and human health and conclude that
the sample area does not attain the cleanup standard. In the statistical terminology applied
in this document, the null hypothesis is that the site does not attain the cleanup standard.
The null hypothesis is assumed to be true unless substantial evidence shows that it is false.
Let <)> represent the true (but unknown) value of a particular soil property, such as the
mean concentration of a specified chemical over the entire site. The null hypothesis is:
HQ: $ > Cleanup Standard (CONTAMINATED or DIRTY),
and the alternative hypothesis is:
HI: < Cleanup Standard (CLEAN).
This document describes how to gather and analyze data that will provide evidence
necessary to contradict the null hypothesis and demonstrate that the site indeed attains the
cleanup standard. Figure 2.1 shows how the null and alternative hypothesis change as
contamination is detected and subsequently corrected. This illustration specifically pertains
to ground water evaluations for land disposal facilities operating under the Resource
Conservation and Recovery Act (RCRA), but the concept is similar for the soils
contamination situation. Initially, the the null hypothesis is that there is no contamination
(A-C). Once a statistical demonstration can be made that the downgradient concentrations
are first above background-level concentrations (B) and also above a relevant action limit or
other standard (D), then the null hypothesis is that the site is contaminated. Most
Superfund sites that require cleanup are in the situation described by D-E. The site must, at
that point, be remediated (E,F) and proven to be clean (G) before the null hypothesis as
described above can be rejected and the site declared clean.
If the null and alternative hypothesis described above were reversed, then a
situation similar to C would designate a satisfactory cleanup. As can be seen by comparing
C with G, the improper specification of the null and alternative hypothesis during a
corrective action can result in very different levels of cleanup.
2-3
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
Figure 2.1 A Statistical Perspective of the Sequence of Ground Water Monitoring
Requirements Under RCRA
HI
Upgradient Downgradient
(background)
DETECTION
MONITORING
NO RELEASE
TRIGGER
COMPLIANCE
MONITORING
COMPLIANCE
MONITORING
TRIGGER
CORRECTIVE
ACTION
CORRECTIVE
ACTION
BEGINS
CORRECTIVE
ACTION
CONTINUES
RETURN TO
COMPLIANCE
MONITORING
NULL ;
HYPOTHESIS
CLEAN
ALTERNATIVE
HYPOTHESIS
CONTAMINATED
NULL
HYPOTHESIS
CONTAMINATED
ALTERNATIVE
HYPOTHESIS
CLEAN
CONCENTRATION
(Notice that until contamination above a risk standard is documented (D) the null hypothesis is
that the facility is clean. Once the facility has been proven to be in exceedance of a health criteria
then the null hypothesis is that the facility is contaminated until proven otherwise (G).)
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
When specifying simplified Superfund site cleanup objectives in consent
decrees, records of decision, or work plans, it is extremely important to say that the site
shall be cleaned up until the sampling program indicates with reasonable confidence that the
concentrations of the contaminants at the entire site are statisticallv less than the cleanup
standard. This prescription will result in the site being designated clean only after a
situation similar to G is observed. However, attainment is often wrongly described by
saying that concentrations at the site shall not exceed the cleanup standard. This second
prescription can result in a situation similar to C being designated as clean.
As discussed in the introduction to this chapter, variation in sampling and
lab analysis introduces uncertainty into the decision concerning the attainment of a cleanup
standard. As a result of the uncertainty and the null/alternative hypothesis arrangement
discussed above, the site can be determined clean when, in fact, it is not, resulting in a
false positive decision (or Type I error). The converse of a false positive decision is a
false negative decision (or Type II error), the mistake of saying the site needs additional
cleanup when, in fact, it meets the standard. The Greek letter alpha (a) is used to
represent the probability of a false positive decision and beta ((3) is used to represent the
probability of false negative decision. The definitions above are summarized in Table 2.1.
Table 2.1 A diagrammatic explanation of false positive and false negative conclusions
Decisi6n based on
the sample data is:
Clean
Dirty
The true condition is:
Clean
Dirty
Correct
Power (1 - (5)
False negative
(Probability is (3
False positive
(Probability is a)
Correct
Certainty (1 - a)
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
It can be seen that if both a and P can be reduced, the percent of time that
the correct decision will be made will be increased Unfortunately, simultaneous reduction
usually can be achieved only by increasing sample size, which may be expensive.
2.2
Power Curves as a Method of Expressing Uncertainty and
Developing Sample Size Requirements
The probability of declaring the sample area clean will depend on the
population mean concentration. If the true population mean is above the cleanup standard
the sample area will rarely be declared clean (this will only happen if the mean of the
particular set of samples is by chance well below the cleanup standard). If the population
mean is much smaller than the cleanup standard, the sample area will almost always be
judged clean. This relationship can be demonstrated by Figure 2.2. The figure illustrates
a power curve that shows the probability of deciding that the site attains the cleanup
standard on the vertical axis and the true, but always unknown, population mean
concentration on the horizontal axis.
Figure 2.2 Hypothetical Power Curve
Probability
of Deciding
the Site
Attains the
Cleanup
Standard
Cleanup
standard
0.4 0.6 0.8
Population Mean Concentration
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
If the population mean concentration in the sample area is equal to or just
above the cleanup standard (i.e., does not attain the cleanup standard), there is still a small
5-percent probability of declaring the sample area clean; this is the false positive rate
denoted by a.
If the population mean is equal to 0.6 ppm (i.e., attains the cleanup standard
of 1.0 ppm), the probability of declaring the sample area clean is 80 percent. Conversely
the probability of declaring the site dirty, given that it is actually clean, is 20 percent. This
is the false negative rate for a population mean of 0.6 ppm. Note that the probability of
declaring the site clean changes depending on the population mean. These false positive
and false negative rates are shown in Figure 2.3.
Figure 2.3 Hypothetical Power Curve Showing False Positive and False Negative
Rates
1
Probability
of Deciding
the Site
Attains the
Cleanup
Standard
g I False negative rate for
J a mean of .6 ppm = 20%
0.8
0.7 4-
0.6
0.5
0.4
0.3
0.2
0.1
0
Power at M. i=80%
False positive rate
at the cleanup
standard = 5%
=.6
Cleanup
standard
0.2 0.4 0.6 0.8 1
Population Mean Concentration, ppm
1.2
1.4
The following items specify the shape and location of the power curve:
• The population coefficient of variation;
• The method of sample selection (the sampling plan);
• The statistical test to be used;
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
• The false positive rate; and
• The sample size.
In summary, there are two important uses of power curves. The first is to
further facilitate understanding of the concept that, although the site may actually be clean, a
set of samples from the site can be obtained that suggest the site is dirty. The cleaner the
site, the less chance of this happening. Conversely, a site may be dirty, but the particular
set of samples suggest the site is clean. Again the dirtier the site, the less chance of this
occurring. The chances of these errors are controlled by the position and shape of the
power curve relative to the cleanup standard. Figures A.I - A.4 illustrate several families
of power curves. The ideal shape of a power curve is a step function that has a 1.0
probability of declaring the site clean whenever the true concentration is less than the
cleanup standard and a zero probability of declaring the site clean when the concentration is
greater than the cleanup standard.
The second use of a power curve is to help decide on an appropriate sample
size for a sampling program. The lower the variability and the more samples taken, the
closer the power curve will come to approaching the ideal step function described above.
In addition, the trade-off between the false positive and negative rate influences the position
of the power curve. Use the power curves in Appendix A to assist with the sample size
determination process in one of two ways:
• Select the power curve desired for the statistical test and determine
from this the sample size that is required; or
• Select the sample size to be collected and determine what the
. resulting power curve will be for the statistical procedure.
Chapters 6, 7, and 8 provide specific methods for making sample size
determinations.
2.3 Attainment or Compliance Criteria
The characteristic of the chemical concentrations to be compared to the
cleanup standard must be specified in order to define a statistical test to determine whether a
sample area attains the cleanup standard. Such characteristics might be the mean
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
concentration, the median, or the 95th percentile of the concentrations. In other words, it
must be decided whether the cleanup standard is intended to be applied as a mean value
such that the mean of the site must be below the cleanup standard or whether the cleanup
standard is a high percentile value that must rarely be exceeded at only 5 or 10 percent of
the site. Figure 2.4 illustrates these characteristics on three distributions. Section 3.5
offers a more detailed discussion of these parameters.
2.3.1 Mean
The location or general magnitude of a set of data is often characterized by
the mean of the distribution. The mean of the concentration distribution is the value that
corresponds to the "center" of the distribution in the sense of the "center of gravity." In
determining the mean from a highly skewed lognormal distribution, small amounts of soil
with concentrations far above the mean are balanced by large amounts of soil with
concentrations close to, but below, the mean.
Whether the mean is a useful summary of the distribution depends on the
characteristics of the sample area and the objectives of the cleanup. In a sample area with
uniform contamination and very little spread or range in the concentration measurements,
the mean will work well. If the spread in the data is large relative to the mean, the average
conditions will not adequately reflect the most heavily contaminated parts of the population.
If interest is in the average exposure or the chronic risk, the mean may be an appropriate
parameter.
When using the mean, consideration should be given to the number of
measurements that are likely to be recorded as below the detection limit. With many
observations below the detection limit, the simple estimate of the population mean cannot
be calculated (see the discussion in section 2.5.2).
2.3.2 Proportions or Percentiles
High percentiles or proportions pertain to the tail of a distribution and
control against having large concentration values. The 50th percentile, the median, is often
a useful alternative to the mean.
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
Figure 2.4 Measures of Location: Mean, Median, 25th Percentile, 75th Percentile, and
95th Percentile for Three Distributions
-1
0
Hypothetical Distribution
2345
Concentration ppm
12345
Concentration ppm
Lognormal Distribution
12345
Concentration ppm
Legend:
Measures of Location:
25th Percentile
Median (50th Percentile)
Mean
75th Percentile
95th Percentile
Measure of Spread:
± 1 Standard Deviation
Around the Mean
« H
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
Methods are available for proportions that are unaffected by concentrations
below the detection limit, as long as the detection limit is below the cleanup standard. The
likelihood of having many data values below the detection limit makes the proportion of
soil units above the cleanup standard an appealing parameter to use in assessing attainment.
If the cleanup standard is only slightly above the detection limit, then it will always be
possible to calculate the proportion of soil samples above the cleanup standard.
Knowing the maximum concentration of the hazardous contaminant at a
waste site would be helpful in making decisions. Unfortunately, in realistic situations the
maximum cannot be determined from a sample of data. A test of proportions, using an
upper percentile of the concentration distribution, can serve as a reasonable approximation
of the maximum value.
2.4 Components of a Risk-Based Standard
Chapter 1 introduced the concept of a risk-based standard and its application
to Superfund activities. Here we will describe how statistical sampling and analysis
methods can be used to adjust the stringency of a risk-based standard.
A hypothetical example of a risk-based standard is as follows: a soil
concentration of arsenic greater than 20 ug/kg at a specific smelter subjects workers to a 1
in a million chance of oral cancer during a lifetime. It is commonly thought that the only
way to change the stringency of the 20 ug/kg standard is to change the magnitude of the
number, 20. In other words, a less stringent standard is obtained by changing the risk-
based standard to 25 ug/kg with an associated increase in the probability of acquiring oral
cancer. This is true, but there are other ways to influence the stringency of the standard.
There are three components of a risk-based standard that can be used to
adjust its stringency. Bisgaard and Hunter (1986) provide discussion of these components
and their application. The three components are:
1) The magnitude of the Concentration Threshold Level (Cs):
2) The method for obtaining data or the Sampling Plan: and
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
3) The evaluation scheme, Decision Rule, that will be used to compare
the data with the threshold level.
Figure 2.5 illustrates the relationship among these components. The choice of a numerical
level is one element of a risk standard. Other questions must also be answered
regardingsampling: How many samples? In what area will the samples be obtained? In
what pattern will the samples be chosen? In addition, after the data are obtained a decision
framework must be developed to analyze the data. Will no more than one exceedance in 10
samples be permitted or will no more than 10 exceedances in 100 be allowed? That is,
what level of confidence is required to conclude that the site is clean? Answers to these
questions influence the spread of the distribution in Figure 2.1 in D, E, F, and G and,
therefore, the steepness of the curve used for the Decision Rule in Figure 2.5, which is a
power curve similar to Figure 2.2.
The following scenario describes the impact that the sampling plan and
decision rule can have on the actual degree of cleanup. A stringent chemical concentration
level is imposed as a requirement at a site (component 1). In contrast, five samples will be
obtained after remediation to verify attainment of the standard (component 2), and 80
percent confidence that the new site mean is less than the standard will be required
(component 3). The health effect results obtained by imposing a stringent numerical level
standard are weakened because the area has not been thoroughly sampled and the
associated level of confidence in the conclusions is relatively low. In this case, a poor
sampling plan and low required level of confidence have influenced the actual degree of
cleanup in spite of the stringency of the numerical standard.
2.5 Missing or Unusable Data, Detection Limits, Outliers
2.5.1 Missing or Unusable Data
In any sampling program, physical samples will be obtained in the field and
then, some time during processing, a problem develops and a reliable measurement is not
available. Samples can be lost, be labeled incorrectly, exceed holding times, be transcribed
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
Figure 2.5 Components of a Risk-Based Standard
maximum
acceptable
risk
estimate of
true process *2. Sampling Plan
decision
(of whether the
site has attained
cleanup standard)
probability of saying
the site is "dirty" when
rt is really "clean"
probability of saying
the site is "dean" when
it is really "dirty"
1. Concentration
Threshold Level
Decision Rule
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
incorrectly, or not satisfy quality control specifications. Clearly, missing data are not
available and cannot be used in data analysis. Data that do not satisfy the most rigorous
quality control specifications may or may not be usable; however, this depends on the
requirements as specified in the Quality Assurance Project Plan.
One of the primary problems with missing data is the possibility that bias is
imposed on statistical estimates. For example, if the presence of high concentrations of a
specific contaminant causes laboratory-interferences that prevent samples with the
contaminant from satisfying quality control specifications, then the data set will not
adequately reflect the presence of the contaminant Careful attention should be paid to the
pattern of missing data to determine if the missing samples have a similar attribute such as
location, time, or chain of custody. If so, then they may all have a special concentration
profile, and their absence may be affecting or biasing the result summary.
However, the main question is how can planning help to prevent the
problem of an excessive number of missing values. One method can be used to help plan
for missing values. The method can be used if the approximate proportion of missing
values can be anticipated, based on prior experience with or a professional judgment of a
sampling team, laboratory, and data analyst. The number of samples needed to conduct a
particular statistical evaluation is inflated by the expected rate of missing values. More
sample results than needed will not be a problem because precision will increase; on the
other hand, too few sample results will be a problem, and may result in more treatment
being required.
The equation for the simplest situation requires prior estimation of the
sample size for the statistical procedures (nd). This is discussed above and throughout the
document. Also, the rate at which missing or unusable values occur must be determined
(R). The final sample size required (nf) is then estimated using the simple equation in Box
2.1.
Throughout this guidance document, when sample size formulae, tables,
and graphs are used, the resulting sample sizes (nd and nnd) required for a statistical
analysis having a specified precision can be increased using these equations in anticipation
of missing data.
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
Box 2.1
Estimating the Final Sample Size Required
nf=nd/(l-R)
A similar equation is used for each of the h strata in a stratified
sampling plan:
nhf=nhd/d-Rh)
2.5.2 Evaluation of Less-Than-Detection-Limit Data
The science and terminology associated with less-than-detection-limit
chemistry are unstandardized. There are a variety of opinions, methods, and approaches
for reporting chemicals present at low concentration. The problem can be segmented.
First, there is the problem of how a chemist determines the detection limit value and
EXACTLY what it means when values are reported above and below a detection limit.
This question is not the subject of this document, but it is important. There is substantial
literature on this subject and Bishop (1985) and Clayton £t al. (1986) offer useful insight
and access to other references.
The second problem is: How should less-than-detection-limit values be
evaluated along with Other values larger than the detection limit when both are present in a
data set? This subject also is supported by a considerable amount of literature. Examples
include Gilbert and Kinnison (1981); Gilliom and Helsel (1986); Helsel and Gilliom
(1986); and Gleit (1985). This aspect of the detection limit problem is discussed briefly in
the following paragraph.
Fortunately, because of the null and alternative hypothesis arrangement,
having concentrations less then a detection limit is no problem when a proportion is being
tested, provided the detection limit is less than Cs. When the proportion or percentile is
being tested, the important attribute of each data value is whether it is larger or smaller than
the Cs, rather than the magnitude of the value. In fact, a site can be evaluated easily relative
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
to a high percentile in spite of a data set that includes many values less than the detection
limit, which is expected when a cleanup technology has uniformly reduced most
concentration measurements to less than the detection limit
When the mean is being used as the basis of comparison with a cleanup
standard, the magnitude of each value is important. When values are reported as being
less than a detection limit, it is generally recommended that they be included in the analysis
as values at the detection limit. This method accommodates detection limits that vary across
samples, and the method is simple to use. In addition, this approach, although statistically
biased, errs in favor of health and environmental protection because of the construction of
the null and alternative hypothesis described earlier. In some cases a less-than-detection-
limit value may be quite large relative to other measured values in a data set. In this case it
may be best to delete such a value. Other methods are available for statistically addressing
less-than-detection-limit values as described above, but they may not be as conservative
with respect to environmental protection.
2.5.3 Outliers
Measurements that are extremely large or small relative to the rest of the data
gathered and that are suspected of misrepresenting the true concentration at the sample
location are often called "outliers." If a particular observation is suspected to be in error,
the error should be identified and corrected, and the corrected value used in the analysis. If
no such verification is possible, a statistician should be consulted to provide modifications
to the statistical analysis that account for the suspected "outliers." Methods to detect and
accommodate outliers are described in Barnett and Lewis (1978) and Grubbs (1969).
The handling of outliers is a controversial topic. This document
recommends that all data not known to be in error should be considered valid
because:
• The expected distribution of concentration values may be skewed
(i.e., nonsymmetric) so that large concentrations, which look like
"outliers" to some analysts, may be legitimate;
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
The procedures recommended in this document are less sensitive to
extremely low concentrations than to extremely high concentrations;
and
• High concentrations are of particular concern for their potential
health and environmental impact.
2.6 General Assumptions
The statistical procedures recommended in this guidance document must be
applicable to many different field situations; therefore, the procedures that have been
chosen are generally based on few assumptions. Situations in which other statistical
procedures might be used to provide more accurate or more cost-effective results will be
noted with references.
This document assumes that (1) the sources of contamination and
contaminating chemicals are known, (2) the sources of contamination have been removed,
or there is no reason to believe that the concentrations of contaminant in the soil will
increase after treatment, and (3) chemical concentrations do not exhibit short-term
variability over the sampling period. The methods presented can be used if sources of
contamination exist or concentrations are expected to increase. However, sampling may
have to be repeated and the results carefully interpreted and presented to reflect the
possibility of additional contamination.
When statistical tests are repeated to evaluate several chemicals, such as
testing that concentration levels for two chemicals both attain the cleanup standard, it is
assumed that the sample area will be declared to attain the cleanup standard only if all
statistical tests used are consistent with this conclusion. For other procedures that might be
used to combine the results of individual tests, it would be advantageous to consult a
statistician.
2.7 A Note on Statistical Versus Field Sampling Terminology
The term sample is used in two different ways. One refers to a physical
soil sample collected for laboratory analysis, and the other refers to a collection of data
called a statistical sample. To avoid confusion, definitions of several terms follow.
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
Physical sample or soil sample: A portion of material (such as a soil
core, scoop, etc.) gathered at the waste site on which laboratory measurements are to be
made. This may also be called a soil unit.
Statistical sample: A statistical sample consists of the collection of
multiple physical samples obtained for assessing attainment of the cleanup standard. The
units included in a statistical sample are selected by probabilistic means.
Sample: The word "sample" in this manual will generally have the
meaning of "statistical sample."
Sample size: The number of soil units being measured or the size of the
statistical sample. Thus, a sample of size 10 consists of the measurements taken on 10 soil
units.
Size of the physical sample: This term refers to the volume or weight
of a soil unit or the quantity of soil in a single physical sample.
The following terms refer to the manner in which the statistical sample of
physical samples is collected: random sample, systematic sample, stratified
sample, judgment sample. These sample designs are discussed in Chapter 4.
2.8 Summary
Errors are possible in evaluating whether or not a site attains the cleanup
standard. For example, consider the errors associated with an extreme decision rule where
no exceedances of a standard are allowed. The site may be dirty even when substantial
sampling shows no exceedances; however, one sample may exceed the cleanup standard
and the site is judged dirty even when the site is acceptably clean.
Statistical methods provide approaches for balancing these two decision
errors and allow extrapolation in a scientifically valid fashion. This chapter reviews the
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CHAPTER 2: INTRODUCTION TO STATISTICAL CONCEPTS AND DECISIONS
statistical concepts that are assumed and used as part of the procedures described in this
guidance document. These include:
A false positive decision-that the site is thought to be clean when it
is not;
• A false negative decision—that the site is thought to be contaminated
when it is not;
• The factors that specify the shape and location of the power curve
relative to the cleanup standard and to sample size determination;
• The mean--the value that corresponds to the "center" of the
concentration distribution;
• Proportions or percentiles—a value that can be used effectively,
based on the distribution of contaminant concentration, to
approximate the maximum concentration of the hazardous
contaminant.
The components of a risk-based standard and how these components relate
to one another are reviewed and graphically illustrated. Methods to help plan for missing
or unusable data, less-than-detection-limit data, and outliers are discussed, followed by the
general assumptions associated with the statistical procedures explained in this document.
These assumptions are that:
• All of the sources of contamination and contaminating chemicals are
known;
• These sources have been removed, so that the contamination will not
increase after treatment; and
• Chemical concentrations do not exhibit short-term variability over
the sampling period.
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3. SPECIFICATION OF ATTAINMENT OBJECTIVES
The specification of attainment objectives must be completed by personnel
familiar with:
• The engineering aspects of the remediation;
• The nature and extent of contamination present;
• Health and environmental risks of the chemicals involved; and
• The costs of sampling, analysis, and cleanup.
Attainment objectives are the procedures and criteria that must be defined to
guide waste site managers and personnel in the process of sampling and data analysis to
achieve a predetermined cleanup standard. Meeting these objectives and criteria enable the
waste site to be judged sufficiently remediated.
As indicated in Figure 1.1, defining attainment objectives is the first task in
the evaluation of whether a site has attained a cleanup standard. Figure 3.1 divides the box
devoted to the establishment and definition of cleanup objectives into its components.
3.1 Specification of Sample Areas
Three terms describing areas within the waste site are:
• Sample area;
• Strata; and
• Sample location.
These terms are used in establishing the attainment objectives and the
sampling and analysis plans. Sample area specification is discussed below and methods
for defining strata and sample locations are discussed in Chapter 5.
The waste site should be divided into sample areas. Each sample area will
be evaluated separately for attainment of a cleanup standard and will require a separate
statistical sample.
3-1
-------
CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
Figure 3.1 Steps in Defining the Attainment Objectives
r Start J
4
Define the sample areas.
(section 3.1)
1
Specify the sample handling
and collection procedures.
(section 3.2)
1
Specify the chemicals to be tested.
(section 3.3)
*
Establish the cleanup standard.
(section 3.4)
1
Specify the parameter to be compared
to the cleanup standard.
(section 3.5)
1
Specify the probability of mistakenly
declaring the sample area clean.
(section 3.6)
4
Review all elements of the
attainment objectives.
/ Are
>^ change
NV ream
Yes
No
(Specify sampling j
I and analysis plan. I
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
Consider the following example, which emphasizes the importance of the
sample area definition. A site consists of an open field with little contamination and a
waste pile covering one-quarter of the site. If sampling and data analysis were executed
without respect to the waste pile, it might be maintained that the mean concentration of the
site was statistically lower than the standard. The site wide mean was "excessively" low
because the waste pile data were "diluted" by many open field measurements. The solution
is to define the waste pile as one sample area and the open field as another. Attainment
decisions will be made independently for each area.
Because of the potential for this problem, it is important to ensure that
sample areas are clearly defined during the design phase. Parties must agree that if the
sample area is judged clean, no more cleanup is required in any part of the sample area.
There are several considerations associated with the definition of sample areas.
1) It is generally useful to define multiple sample areas within a waste
site. These areas should be defined so that they are as homogeneous
as possible with respect to prior waste management activities. For
example, if a PCB transformer disposal area and a lead battery
recycling area are located on the same site, they should not be
included in the same sample area.
2) It may also be useful to define sample areas by batches of material
that will receive a treatment action, for example, dump truck loads
(see Exner et al., 1985) or the minimum sized areas that can be
stabilized or capped.
3) A site may be comprised of areas that require different sampling or
treatment technologies. For example, disturbed versus natural soils,
wetlands versus firm terrain, or sandy versus clay soils may suggest
establishment of different sample areas.
4) Finally, while more (smaller) sample areas provide more flexible
response to changing conditions, sampling costs will increase with
the number of sample areas.
5) Sample area definitions also require that the depth or depth intervals
of interest be specified. This is discussed in greater detail in section
5.6.
Figure 3.2 shows how different geographic sample areas relate to one
another.
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
Figure 3.2 Geographic Areas and Subareas Within the Site
Site boundary
Map of the waste site
Waste site with two sample areas, SA1 and SA2. Separate attainment
decisions are made for each sample area. Sample area SA1 is divided into
two strata, ST1 and ST2. (See Chapter 4 for more on stratified sampling
[multiple strata per sample area].) Stratum ST1 has randomly selected soil
sample locations indicated by "*".
3.2
Specification of Sample Collection and Handling Procedures
Deciding whether a sample area attains the cleanup standard requires that
measurements be made on a statistical sample of soil units, and that these measurements be
compared to the cleanup standard. An important task for any decision procedure is to
define carefully what is being measured; questions that must be answered include:
• What is meant by a soil unit or soil sample?
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
• How is the soil sample collected and what equipment and
procedures are used?
• How is the soil sample handled between collection and
measurement?
• How are the laboratory measurements to be made and what accuracy
is to be achieved?
The above questions are not addressed in this document. Consult the
guidances listed in Table 1.1 for more information.
3.3 Specification of the Chemicals to be Tested
For each sample area, the chemicals to be tested in each soil unit should be
listed. When multiple chemicals are tested, this document assumes that all chemicals must
attain the cleanup standard for the sample area to be declared clean.
3.4 Specification of the Cleanup Standard
Concentration measurements for each physical sample will be compared to
the appropriate, relevant, or applicable cleanup standard chosen for each chemical to be
tested. Cleanup standards are determined by EPA during the site-specific endangerment
assessments. The cleanup standard for each chemical of concern must be stated at the
outset of the remedial verification investigation. Final selection of the cleanup standard
depends on many factors as discussed in USEPA (1986c). Selection of the cleanup
standard depends on the following factors:
• The availability and value of other appropriate criteria;
• Factors related to toxicology and exposure, for example, the effect
of multiple contaminants, potential use of the waste site and
pathways of exposure, population sensitivities to the chemical;
• Factors related to uncertainty, for example, the effectiveness of
treatment alternatives, reliability of exposure data, and the reliability
of institutional controls; and
• Factors related to technical limitations, for example, laboratory
detection limits, background contamination levels, and technical
limitations to restoration.
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
Throughout this document, the cleanup standard will be denoted by Cs.
3.5 Selection of the Statistical Parameter to Compare with the
Cleanup Standard
3.5.1 Selection Criteria for the Mean, Median, and Upper Percentile
Criteria for selecting the parameter to use in the statistical assessment
decision are:
• The criteria used to develop the risk-based standards, if known;
• The lexicological effect of the contaminant being measured (e.g.,
carcinogenic, systemic toxicant, developmental toxicant).
• The relative sample sizes required or the relative ease of calculation;
• The likelihood of concentration measurements below the cleanup
standard; and
• The relative spread of the data.
Table 3.1 presents these criteria and when they support or contradict the use
of the mean, upper percentile, and median. The median may offer a reasonable
compromise because the median is the 50th percentile and a measure of central tendency.
Table 3.2 illustrates the broad potential utility of the median.
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
Table 3.1 Points to consider when trying to choose among the mean, high percentile,
or median
Parameter
Points to Consider
Mean
1) Easy to calculate and estimate a confidence interval.
2) Requires fewer samples than other parameters to achieve similar
confidence.
3) Useful when the cleanup standard has been based on
consideration of carcinogenic or chronic health effects or long-term
average exposure.
4) Useful when the soil is uniform with little spread in the sample
data.
5) Not as useful when contamination exists in small areas within a
larger area that is being sampled because the mean can be "diluted"
or reduced by the inclusion of clean areas in the sample area.
6) Not very representative of highly variable soils because the most
heavily contaminated areas are not characterized by a mean.
7) Not useful when there are a large proportion of less-than-
detection-limit values.
Upper
Proportion/
Percentile
1) Can be expressed in terms that have more meaning than tests of
the mean. Volumes or areas can be expressed relative to the total
volume or area of concern, and this can be a proportion of importance.
For example, if no more than 10,000 m^ in a total volume of
1,000,000 m3 can exceed a cleanup standard, then this becomes a
test to verify with reasonable confidence that no less than 99 percent
of the site is below the cleanup standard.
2) Will provide the best control of extreme values when data are
highly variable.
3) Some methods are unaffected by less-than-detection-limit values,
as long as the detection limit is less than the cleanup standard.
4) If the health effects of the contaminant are acute or worst-case
effects, extreme concentrations are of concern and are best evaluated
by ensuring that a large proportion of the site is below a cleanup
standard.
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
Table 3.1 Points to consider when trying to choose among the mean, high percentile,
or median (continued)
Parameter
Points to Consider
Upper
Proportion/
Percentile
(continued)
5) Similar to the mean, if contamination exists within a small area,
but if the sampling program is conducted to include a much larger
surrounding area with little contamination, the proportion will be
affected or "diluted."
6) The proportion of the site that must be below the cleanup standard
must be chosen.
7) When statistical methods are used that require few assumptions,
a larger sample size will be required than for tests based on the
mean.
Median
1) Has benefits over the mean because it is not as heavily
influenced by outliers and highly variable data, and can be used with
a large number of less-than-detection-limit values.
2) Has many of the positive features of the mean, in particular its
usefulness for evaluating cleanup standards based on carcinogenic
or chronic health effects and long-term average exposure.
3) Has positive features of the proportion, including its reliance on
fewer assumptions.
4) Retains some negative features of the mean in that the median
will not control extreme values.
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
Table 3.2 Recommended parameters to test when comparing the cleanup standard to
the average concentration of a chemical with chronic effects
Data Variability
Large Coefficient
of Variation
(Perhaps cv > .5)
Small Coefficient
of Variation
(Perhaps cv < .5)
Proportion of the data with concentrations
below the detection limit:
Low High
(Perhaps < 50%) (Perhaps > 50%)
Mean
(or Median)
Mean
(or Median)
Upper Percentile
Median
3.5.2
Multiple Attainment Criteria
This guidance document addresses testing for a single parameter--the mean
or a specified percentile of the distribution—that is below the cleanup standard. However,
in some situations two or more parameters can be chosen. The sample area would be
declared clean if all parameters were significantly less than the cleanup standard. For
example, there may be interest in providing protection against excessive extreme and
average concentrations. Therefore, the mean and an upper percentile can be tested using
the rule that the sample area attains the cleanup standard if both parameters are below the
cleanup standard. When testing both parameters, the number of samples collected will be
either the number required for the test of the mean or the number required for the test of the
percentile (whichever number is larger).
Other more complicated criteria may be used to assess the attainment of the
cleanup criteria. Multiple criteria are established in the following examples. In each case it
is desirable that:
• Most of the soil has concentrations below the cleanup standard and
that the concentrations above the cleanup standard are not too large.
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
This may be accomplished by testing whether the 75th percentile is
below the cleanup standard and whether the mean of those
concentrations above the cleanup standard is less than twice the
cleanup standard. This combination of tests can be performed with
minor modifications to the methods presented in this document.
The mean concentration be less than the cleanup standard and that
the standard deviation of the data be small, thus limiting the number
of extreme concentrations. This may be accomplished by testing if
the mean is below the cleanup standard and the coefficient of
variation is below some low level (.5 for example). This document
does not address testing the standard deviation, variance, or
coefficient of variation against a cleanup standard.
The mean concentration be less than the cleanup standard and that
the remaining contamination be uniformly distributed across the
sample area relative to the overall spread of the data. Testing these
criteria may be accomplished by testing for a mean below the
cleanup standard and variability between strata means that is not
large compared to the variability within strata (analysis of variance).
The mean concentration be less than the cleanup standard and that no
area of contaminated soil (assumed to be circular) be larger than a
specified size. Testing these criteria involves testing for hot spots,
which are discussed in Chapter 9 and more extensively in Gilbert
(1987).
3.6 Decision Making With Uncertainty: The Chance of Concluding
the Site Is Protective of Public Health and the Environment
When It Is Actually Not Protective
As discussed in Chapter 2, the validity of the decision that a site meets the
cleanup standard depends on how well the samples of soil represent the site, how
accurately the soil samples are analyzed, and other factors, all of which are subject to
variation. Different sampling patterns will yield different results and repeated
measurements on individual soil samples will yield different concentrations. This variation
introduces uncertainty into the decision concerning the attainment of a cleanup standard.
As a result of this uncertainty, one may decide that the site is clean when it
is not. In the context of this document, this mistaken conclusion can be referred to as a
false positive finding (the chance or probability of a false positive is indicated by the
Greek letter alpha, a). There are two important points surrounding false positives:
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
• First, from an environmental and health protection perspective, it is
imperative to reduce the chance of a false positive. In direct terms a
false positive is the chance of deciding a Superfund site is clean
when it still poses a health or environmental threat. Of course, a
low false positive rate does not come without a cost. The additional
cost required to lower the false positive rate comes from additional
samples and more accurate sampling and analysis methods.
• Second, the definition of a false positive in this document is exactly
opposite the more familiar definition of a false positive under
RCRA detection and compliance monitoring. This is because the
null and alternative hypotheses are reversed, once a site has been
verified to have contamination. Under the RCRA detection
monitoring situation, EPA was concerned about a high false
negative rate; here EPA is concerned about a high false positive rate.
In order to design a statistical test for deciding whether the sample area
attains the cleanup standard, those individuals specifying the sampling and analysis
objectives should select and specify the false positive rate for testing the site. While
different false positive rates can be used for each chemical, it is recommended that all
chemicals in the sample area use the same rates. This rate is the maximum probability that
the sample area will be declared clean by mistake when it is actually dirty. For a further
discussion of false positive rates, see Sokal and Rohlf (1981).
3.7 Data Quality Objectives
The Quality Assurance Management staff within EPA has developed
requirements and procedures for establishing Data Quality Objectives (DQOs) when
environmental data are collected to support regulatory and programmatic decisions. The
DQOs are a clear set of statements addressing the following issues (see USEPA, 1987a and
USEPA, 1987b).
• The decision to be made;
The reasons environmental data are needed and how they will be
used;
• Time and resource constraints on data collection;
• Detailed description of the data to be collected;
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
Specifications regarding the domain of the decision;
The consequences of an incorrect decision attributable to inadequate
environmental data;
The calculations, statistical or otherwise, that will be performed on
the data in order to arrive at the result, including the statistic that will
be used to summarize the data and the "action level" (cleanup
standard) to which the summary statistic will be compared; and
The level of uncertainty that the decision maker is willing to accept
in the results derived from the environmental data.
The specification of attainment objectives that have been discussed in this
chapter and the sampling and analysis plan discussed in the next chapter are an important
part of the Data Quality Objectives process. Completion of the DQO process will provide
the required information for the specification of attainment objectives.
3.8 Summary
The following steps must be taken to evaluate whether a site has attained the
cleanup standard:
• Define the attainment objectives;
• Specify sample design and analysis plan, and determine sample size;
• Collect the data; and
• Determine if the sample area attains the cleanup standard.
This chapter discusses attainment objective specifications. Attainment
objectives are specified by RPMs, RPs, and their contractors. They are not statistically
based decisions.
• Define the sample area. The waste site should be divided into
sample areas. Each sample area will be evaluated separately for
attainment of a cleanup standard and will require a separate statistical
sample. It is important to ensure that sample areas are clearly
defined during the design phase.
• Specify the sample handling and collection procedures.
An important task for any decision procedure is to define carefully
what is being measured.
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CHAPTER 3: SPECIFICATION OF ATTAINMENT OBJECTIVES
Specify the chemicals to be tested. Chemicals to be tested in
each soil unit should be listed.
Establish the cleanup standard. Cleanup standards are
determined by EPA using site-specific risk assessments or ARARs.
The cleanup standard for each chemical of concern must be stated at
the outset of the remedial verification investigation.
Specify the parameter to be compared to the cleanup
standard. In other words: "Does the cleanup standard represent an
average condition (mean) or a level to be rarely exceeded (high
percentile)? Criteria for selecting the parameter to use in the
statistical assessment decision are:
The criteria used to develop the risk-based standards, if
known;
Whether the contaminant being measured has an acute or
long-term chronic effect;
The relative sample sizes required or the relative ease of
calculation;
The likelihood of concentration measurements below the
detection limit; and
The relative spread of the data.
Specify the probability of mistakenly declaring the
sample area clean. Select and specify the false positive rate for
testing the site. It is recommended that all chemicals in the sample
area use the same rates. This rate is the maximum probability that
the sample area will be declared clean by mistake when it is actually
dirty.
Review all elements of the attainment objectives.
3-13
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4. DESIGN OF THE SAMPLING AND ANALYSIS
PLAN
Once the attainment objectives are specified by program and subject matter
personnel, statisticians can be useful for designing important components of sampling and
analysis plans.
The methods of analysis must be consistent with the sample design and the
attainment objectives. For example, data that are collected using stratified sampling cannot
be analyzed using the equations for simple random sampling. The sample design and
analysis plan must coincide. If there appears to be any reason to use different sample
designs or analysis plans than those discussed in this manual, or if there is any reason to
change either the sample design or the analysis plan after field data collection has started, it
is recommended that a statistician be consulted.
This chapter presents some approaches to the design of a sampling and
analysis plan and presents the strengths and weaknesses of various designs.
4.1 The Sampling Plan
The following sections provide background discussion guiding the choice
of sampling plan for each sampling area. Chapter 5 discusses the details of how to
implement a sampling plan. For more details, see Kish (1965), Cochran (1977), Hansen si
al (1953), or the EPA guidances in Table 1.1.
The sample designs considered in this document are:
• Simple random sampling called random sampling in this
document;
• Stratified random sampling called stratified sampling in this
document;
• Simple systematic sampling called systematic sampling in this
document; and
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CHAPTER 4: DESIGN OF THE SAMPLING AND ANALYSIS PLAN
• Sequential random sampling called sequential sampling in this
document.
Randomization is necessary to make probability or confidence statements
about the results of the sampling. Both random and random start systematic sample
locations have random components. In contrast, sample selection using the judgment of
the sampler has no randomization. Results from such samples cannot be generalized to the
whole sample area and no probability statements can be made when judgment sampling is
used. Judgment sampling may be justified, for example, during the preliminary
assessment and site investigation stages if the sampler has substantial knowledge of the
sources and history of contamination. However, judgment samples should not be used to
determine whether the cleanup standard has been attained.
Combinations of the designs referred to above can also be used. For
example, systematic sampling could be used with stratified sampling. In the situation
where cleanup has occurred, if the concentrations across the site are relatively low and
uniform and the site is accessible, the sample designs considered in this document should
be adequate. If other more complicated sample designs are necessary, it is recommended
that a statistician be consulted on the best design, and on the appropriate analysis method
for that design. Figure 4.1 illustrates a random, systematic, and stratified sample.
4.1.1 Random Versus Systematic Sampling
Random selection of sample points requires that each sample point be
selected independent of the location of all other sample points. Figure 4.1 shows a random
sample. Note that under random sampling no pattern is expected in the distribution of the
points. However, it is possible (purely by chance) that all of the sample points will be
clustered in, say, one or two quadrants of the site. This possibility is extremely small for
larger sample sizes.
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CHAPTER 4: DESIGN OF THE SAMPLING AND ANALYSIS PLAN
Figure 4.1 Illustration of Random, Systematic, and Stratified Sampling (axes are
distance in meters)
Random Sampling
0 25 50 75 100 125 150 175
0
o-
o-
Systematic Sampling
25 50 75 100 125 150 175
Stratified Random Sampling
0 25 50 75 100 125 150 175
Legend:
•Sample Area Boundary
"Strata Boundary
Randomly Selected Sample Location
Sample Location Determined
Systematically
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CHAPTER 4: DESIGN OF THE SAMPLING AND ANALYSIS PLAN
An alternative to random sampling is systematic sampling, which distributes
the sample more uniformly over the site. Because the sample points follow a simple pattern
and are separated by a fixed distance, locating the sample points in the field may be easier
using a systematic sample than using a random sample. In many circumstances, estimates
from systematic sampling may be preferred. More discussion of systematic versus random
sampling can be found in Finney (1948), Legg, §1 §L (1985), Cochran (1977), Osborne
(1942), Palley and Horwitz (1961), Peshkova (1970), and Wolter (1984).
4.1.2 Simple Versus Stratified Sampling
The precision of statistical estimates may be improved by dividing a sample
area into more homogeneous strata. In this way, the variability due to soil, location,
characteristics of the terrain, etc. can be controlled, thereby improving the precision of
contamination level estimates. Homogeneous areas from which separate samples are
drawn are referred to as "strata," and the combined sample from all areas is referred to as a
"stratified sample."
Like systematic sampling, stratification provides another way of minimizing
the possibility that important areas of the site will not be represented in the sample. Note in
Figure 4.1 that the two strata represent subareas for which representation in the sample
will be guaranteed under a stratified sampling design.
The main advantage of stratification is that it can result in a more efficient
allocation of resources than would be possible with a simple random sample. For example,
suppose that, based on physical features, the site can be divided into a hilly and a flat area,
and that the hilly area comprises about 75 percent of the total area and is more expensive to
sample than the flat area. If there is no reason to analyze the two subareas separately, we
might consider selecting a simple random sample of soil units across the entire site.
However, with a simple random sample, about 75 percent of the sample would be in the
hilly, and therefore more expensive, areas of the site. With stratified sampling, the sample
can be allocated disproportionately to the two subareas, i.e., sample fewer units from hilly
areas and more from flat areas. In this way, the resulting cost savings (over a simple
random sample) can be used to increase the total sample size and, hence, the precision of
estimates from the sample.
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CHAPTER 4: DESIGN OF THE SAMPLING AND ANALYSIS PLAN
The above illustration is highly simplified. In addition to differential
stratum costs, factors such as the relative sizes of the strata and the variability of the
contaminant under study in the different strata will affect the optimum allocation. The
illustration does, however, point out that stratification can be used to design a more
efficient sample, and is more than simply a device to ensure that particular subareas of the
site are represented in the sample. A formal discussion of stratified sampling, and the cost
and variance considerations used to determine an optimum allocation, is beyond the scope
of this manual. However, sections 5.4 and 6.4 offer a discussion of the basic principles
used to guide the design of a stratified sample.
Although stratified sampling is more difficult to implement in the field and
slightly more difficult to analyze, stratified sampling will provide benefits if differences in
mean concentrations or sampling costs across the sample area exist and can be reasonably
identified using available data. It is important to define strata so that the physical samples
within a stratum are more similar to each other than to samples from different strata.
Factors that can be used to define strata are:
• Sampling depth (see section 5.6 for details);
• Concentration level;
• Physiography/topography;
• The presence of other contaminants that affect the analytical
techniques required at the lab;
• The history and sources of contamination over the site;
• Previous cleanup attempts; or
• Weathering and run-off processes.
There are two fundamental and important points to remember when defining
areas that will become different strata:
• The strata must not overlap-no area within one strata can be within
another strata; and
• The sum of the sizes of the strata must equal the area of the sample
area.
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CHAPTER 4: DESIGN OF THE SAMPLING AND ANALYSIS PLAN
In other words, the strata must collectively account for the entire sample
area of interest—no more, no less.
4.1.3 Sequential Sampling
For most statistical methods, the analysis is performed after the entire
sample has been collected and the laboratory results are complete. In sequential random
sampling, the samples are analyzed as they are collected. A statistical analysis of the data,
after each sample is collected and analyzed, is used to determine if another sample is to be
collected or if the sampling program ten >inates with a decision that the site is clean or dirty.
(Sequential sampling is the subject of Chapter 8.)
4.2 The Analysis Plan
Similar to sampling plan designs, planning an approach to analysis and the
actual analysis begin before the first sample is collected. The first task of the analysis plan
is to determine how the cleanup standard should function. In other words, what is the
cleanup standard: a value that should be rarely exceeded; an average value; or a level that
defines the presence of a hot spot? This must be decided because it determines whai
analysis method will be used to determine attainment.
Second, the analysis plan must be developed in conjunction with the
sampling plan discussed earlier in this chapter. For example, plans to conduct stratified
sampling cannot be analyzed using the equations for random sampling.
Third, the first actual step required in the analysis plan should be a
determination of the appropriate sample size. This requires calculations and evaluation
before the data are collected. Often the number of samples is determined by economics and
budget rather than an evaluation of the required accuracy. Nevertheless, it is important to
evaluate the accuracy associated with a prespecified number of samples.
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CHAPTER 4: DESIGN OF THE SAMPLING AND ANALYSIS PLAN
Fourth, the analysis plan will describe the evaluation of the resulting data.
Chapters 6 through 10 offer various analytical approaches, depending on attainment
objectives and the sampling program. Table 4.1 presents where in this document various
combinations of analysis and sampling plans are discussed.
Table 4.1 Where sample designs and analysis methods for soil sampling are discussed
in this document
Type of
Evaluation
Test of the Mean
Test of
Percentiles
Hot Spot
Evaluation
Geostatistics
Analysis
Method
Test for means
Nonparametric
Tolerance Intervals
Sequential Sampling
Indicator Kriging
Chapter Location
Sample Design
Random
6.3.3
7.3.3
7.3.6
Stratified
6.4.2
7.5.2
Systematic
6.5.2
7.6
9.2.1
10.3
Sequential
8.2
4.3
Summary
Design of the sampling and analysis plan requires specification of attainment
objectives by program and subject matter personnel. The sampling and analysis objectives
can be refined with the assistance of statistical expertise. The sample design and analysis
plans go together, therefore, the following methods of analysis must be consistent with the
sample design:
• Random sampling;
• Stratified sampling;
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CHAPTER 4: DESIGN OF THE SAMPLING AND ANALYSIS PLAN
• Systematic sampling; and
• Sequential sampling.
Random selection of sample points requires that each sample point be selected independent
of the location of all other sample points. An alternative to random sampling is systematic
sampling, which distributes the sample more uniformly over the site. Systematic sampling
is preferred in hot spot searches and in geostatistical studies.
Like systematic sampling, stratified sampling minimizes the possibility that
important areas of the site will not be represented by dividing a sample area into
homogeneous subareas. The main advantage of stratification is that it can result in a more
efficient allocation of resources than would be possible with a random sample.
Sequential sampling (Chapter 8) requires that the samples be analyzed as
they are collected.
Decisions required to plan an approach to analysis are:
• Determine the analysis method that is most useful;
• Develop the plan in conjunction with the sampling plan;
• Determine the appropriate sample size; and
• Describe how the resulting data will be evaluated.
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5. FIELD SAMPLING PROCEDURES
The procedures discussed in this chapter ensure that:
• The method of establishing soil sample locations in the field is
consistent with the planned sample design;
• Each sample location is selected in a nonjudgmental and unbiased
way; and
• Complete documentation of all sampling steps is maintained.
The procedures discussed in this chapter assume that the sampling plan has
been selected; the boundaries of the waste site, the sample areas, and any strata have been
defined; a detailed map of the waste site is available; and the sample size is known. Sample
size determination is discussed in Chapters 6, 7, 8, and 9. Also, if sequential sampling or
hot spot searches are planned, the reader should refer to Chapters 8 and 9, respectively, for
additional guidance on field sampling.
5.1 Determining the General Sampling Location
Locating the soil samples is accomplished using a detailed map of the waste
site with a coordinate system to identify sampling locations. Recording and automation of
station-specific data should retain coordinate information, especially if geostatistical
manipulations are performed (see Chapter 10) or a geographic information system will be
used.
Soil sample locations will be identified by X and Y coordinates within the
grid system. It is not necessary to draw a grid for the entire waste site; it is only necessary
to identify the actual coordinates selected. Figure 5.1 is an example of a map with a
coordinate system. In this example, the origin of the coordinate system is at the lower
lefthand corner of the map; however, this may not be true for coordinate systems based on
measurements from a reference point on the ground, i.e., a benchmark or a standard
coordinate system such as latitude and longitude.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Figure 5.1 Map of a Sample Area with a Coordinate System
100
75
50
Boundary of
the sample area
0
25 50
75
100 125 150 175
X Coordinate
The boundaries of the sample areas (areas within the site for which separate
attainment decisions are to be made) and strata within the sample areas (if stratified
sampling is required) should be shown on the map. The map should also include other
important features that will be useful in identifying sample locations in the field.
Accurate location of sampling points can be expensive and time consuming.
Therefore, a method is suggested which uses the coordinate system to identify the general
area within which the soil sample is to be collected, followed by a second stage of
sampling, described in section 5.5, to identify the sample point accurately.
The X and Y coordinates of each sample location must be specified. This
distance between coordinates on each axis represents a reasonable accuracy for measuring
distance in the field, and is represented by M. If distances can be measured easily to within
2 m, but not to within 1 meter, the coordinates should be provided to the nearest 2 m (M =
2 m). The sampling coordinates can be identified with greater accuracy when the distances
to be measured between reference points are short, the measuring equipment is accurate or
easy to use, or there are few obstructions to line-of-sight measuring such as hills, trees, or
bushy vegetation. For example, the location within a small lagoon, say, 30 by 30 m, can
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CHAPTER 5: FIELD SAMPLING PROCEDURES
be established to within 5 cm. On the other hand, in a 10 hectare field it may only be
reasonable to identify a location to within 10 m.
5.2 Selecting the Sample Coordinates for a Simple Random Sample
A random sample of soil units within the sample area or stratum will be
selected by generating a series of random (X,Y) coordinates, finding the. location in the
field associated with these (X,Y) coordinates, and following the field procedures described
in section 5.5 for collecting soil samples. If the waste site contains multiple sample areas
and/or strata, the same procedure described above is used to generate random pairs of
coordinates with the appropriate range until the specified sample size for the particular
portion of the site has been met. In other words, a separate simple random sample of
locations should be drawn for each sample area or stratum. To simplify the discussion, the
procedures below discuss selection of a random sample in a sample area.
The number of soil samples to be collected must be specified for each
sample area. In what follows, the term nf will be used to denote the number of samples to
be collected in the sample area.
To generate the nf random coordinates (Xj,Yj), i = 1 to nf, for the sample
area, determine the range of X and Y coordinates that will completely cover the sample
area. These coordinate ranges will define a rectangle that circumscribes the sample area.
Let the coordinate ranges be Xmin to Xmax and Ymin to Ymax. Thus, the point (Xmin,
Ymin) represents the lower lefthand corner of the rectangle, and (Xmax, Ymax) represents
the upper righthand corner of the rectangle. The nf sample coordinates (Xj.Yj) can be
generated using a random number generator and the steps described in Box 5.1. Box 5.2
gives an example of generating random sample locations.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Box 5.1
Steps for Generating Random Coordinates That
Define Sampling Locations
1) Generate a set of coordinates (X,Y) using the following
equations:
X = Xmin + (Xmax - Xmin)*RND (5.1)
Y = Ymin + (Ymax - Ymin)*RND (5.2)
RND is the next unused random number between 0 and 1 in a
sequence of random numbers. Random numbers can be
obtained from calculators, computer software, or tables of
random numbers.
2) If (X,Y) is outside the sample area, return to step 1 to generate
another random coordinate; otherwise go to step 3.
3) Define (Xj,Yj) using the following steps:
Round X to the nearest unit that can be located easily in the field
(see section 5.1); set this equal to Xj
Round Y to the nearest unit that can be located easily in the field
(see section 5.1); set this equal to Yj.
4) Continue to generate the next random coordinate, (Xj+i,Yj+i).
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Box 5.2
An Example of Generating Random Sampling Locations
To illustrate the selection of simple random sample of locations,
assume that seven soil units will be selected from the site in Figure 5.2. Pairs of
random numbers (one X coordinate and Y coordinate for each pair) identify each
sample point. X will be measured on the map's coordinate system in the
horizontal direction and Y in the vertical direction. It is assumed for this example
that selected coordinates can be identified to the nearest meter. The first number of
pair, Xj[, must be between 0 and 190 (i.e., Xmjn = 0 and Xmax = 190) and the
second, Yj, between 0 and 100 (Ymjn = 0 and Ymax = 100) for this example. If
the X and Y coordinates for any pair identify a location outside the area of interest,
they are ignored and the process is continued until the sample size nf has been
achieved.
XYpair
1
2
3
4
5
6
7
8
9
Random
X coordinate
67
97
190
17
94
123
25
35
152
Random
Y coordinate
80
4
88 (outside of sample
15 (outside of sample
76
49
52
39
14
area)
area)
It took nine attempts to secure seven coordinates that fall within the
sample area. The randomly selected coordinates for pairs 3 and 4 fall outside the
waste site and are to be discarded. The remaining seven locations are randomly
distributed throughout the site.
These locations can now be plotted on the map, as shown in Figure 5.2.
5.3
Selecting the Sample Coordinates for a Systematic Sample
A square grid and a triangular grid are two common patterns used in
systematic or grid sampling. These patterns are shown in Figure 5.3. Note that the rows
of points in the triangular grid are closer (.866L) than the distance between points in a row
(L) and that the points in every other row are offset by half a grid width.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Figure 5.2 Map of a Sample Area Showing Random Sampling Locations
Locations of the random samples are indicated by a •. The numbers
reference the XY pairs in Box 5.2.
100
0 25 50 75 100 125 150 175
0
X Coordinate
Figure 5.3 Examples of a Square and a Triangular Grid for Systematic Sampling
Square Grid Triangular Grid
f 1
1 J
• '
» 4
k . A 4h
' T
L
, . .1
* .*. .* *
* /\ /\ * * T
\ .- \ -866L
• ¥ • • -L
t -r t
h-L— 4
The size of the sample area must be determined in order to calculate the
distance, L, between the sampling locations in the systematic grid. The area can be
measured on a map using a planimeter. The units of the area measurement (such as square
feet, hectares, square meters) should be recorded.
Denote the surface area of the sample area by A. Use the equations in Box
5.3 to calculate the spacing between adjacent sampling locations.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Box 5.3
Calculating Spacing Between Adjacent Sampling Locations
for the Square Grid in Figure 5.3
(5.3)
for the Triangular Grid in Figure 5.3
(5.4)
The distance between adjacent points, L, should be rounded to the nearest unit that can be
easily measured in the field.
After computing L, the actual location of one point in the grid should be
chosen by a random procedure. First, select a random coordinate (X,Y) following the
procedure in Box 5.1. Using this location as one intersection of two gridlines, construct
gridlines running parallel to the coordinate axes and separated by a distance L. The
sampling locations are the points at the intersections of the gridlines that are within the
sample area boundaries. Figure 5.4 illustrates this procedure. Using this procedure, the
grid will always be oriented parallel to the coordinate axes. The grid intersections that lie
outside the sample area are ignored. There will be some variation in sample size,
depending on the location of the initial randomly drawn point. However, the relative
variation in number of sample points becomes small as the number of desired sample points
increases. For unusually shaped sample areas (or strata), the number of sample points can
vary considerably from the desired number.
The coordinates for the sample points will be all coordinates (Xj.Yj) such
that
• (X,,Yj) is inside the sample area or stratum;
• Xj = X + j*L, for some positive or negative integer j, and;
• Yj = Y + k*L, for some positive or negative integer k.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Figure 5.4 Locating a Square Grid Systematic Sample
(1) S elect initial random point.
100
(2) Construct coordinate axis going
through initial point.
100
25 50 75 100 125 ISO 175
X
(3) Construct lines parallel to
vertical axis, separated by
a distance of L.
(4) Construct lines parallel to
horizontal axis, separated by
a distance of L.
25 50 75 100 125 150 175
25 SO 75 100 125 150 175
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Box 5.4 and Figure 5.5 give an example of locating systematic coordinates
and the resultant sampling locations plotted on a map of the site.
Box 5.4
Locating Systematic Coordinates
Using the map in Figure 5.1 and a planimeter, the area of the sample
area is determined to be 14,025 sq. m. If the sample size is 12, the spacing
between adjacent points is:
[JT 1
= A/ — = M
14025 ,,, ...
= 34 m, rounded to the nearest meter
Using the procedure in Box 5.1, a random coordinate (X,Y) = (11,60)
is generated. Starting from this point, the following sampling points can be
calculated:
(79,94) (113,94) (147,94)
(11,60) (45,60) (79,60) (113,60) (147,60) (181,60)
(45,26) (79,26) (113,26) (147,26) (181,26)
These points are shown in Figure 5.5. The intended sample size was
12; however, because of the random selection process and the irregularity of the
sample area boundary, there are 14 sample points within the sample area. A
sample will be collected at all 14 locations.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Figure 5.5 Map of a Sample Site Showing Systematic Sampling Locations
0
0
I
25 50 75 100 125 150 175
X Coordinate
Initial Randomly Selected Sample Point
5.3.1 An Alternative Method for Locating the Random Start Position
for a Systematic Sample
An alternative method may be used to locate the random start position for a
systematic triangular grid sample (J. Barich, Pers. Com., 1988). This approach, as
detailed in Box 5.5, determines a random start location by choosing a random angle A and
a random distance Y from point X. This approach is useful under circumstances where a
transit and stadia rod are available for turning angles, measuring distances, and establishing
transects. This method is essentially equivalent to the method described above.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Box 5.5
Alternative Method for Locating the Random
Start Position for a Systematic Sample
Figure 5.6 and the following steps explain how to implement the
sequence.
1) Establish the main transect with endpoints X and X' using any
convenient reference line (e.g., established boundary). Notice that the transect X-
X' must be longer than the line indicated in Figure 5.6 in order to site all of the
transects that intersect the sample area.
2) Randomly choose a point Y between X and X'.
3) Randomly choose an angle A between 0° and 90°.
4) Locate transect with endpoints Y and Y', A degrees from transect X
and X1. If this transect intersects the boundary of the sample area, mark the
transect.
5) Locate another transect beginning at point Y and 90° +A (i.e.,
perpendicular) from that transect that intersects the boundary of the sample area;
then mark the transect Y-Y'. If this transect intersects the boundary of the sample
area then mark the transect.
6) Move away from point Y on transect X-X' a distance D, where
D=L/sin(A). L is the desired interval between sampling points along the grid
pattern.
7) At the point D units away from Y, establish two more transects:
one A degrees from transect X-X' and parallel to transect Y-Y', and the other
90°+A degrees from X-X' also beginning at the point D units from point Y.
8) Continue to move intervals of distance D along the transect X-X'
until two transects intersect within the boundary of the sample area. Establish the
first sample location at that point. Then measure along that transect from the first
sampling location a distance of L and establish more transects and grid points
using the approach described in the previous method for systematic samples.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Figure 5.6 Method for Positioning Systematic Sample Locations in the Field
Sample Locations
Sample Area
Boundary
A
A A A
Y+D Y+2D
A
Xf
Where D = L/sin A
Y is chosen randomly
A is chosen randomly
L is determined from sample size calculations
D is a physical sampling location
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CHAPTER 5: FIELD SAMPLING PROCEDURES
5.4 Extension to Stratified Sampling
The extension of these procedures to stratified sampling is straightforward.
Each stratum is sampled separately using the methods discussed above. Different random
sequences (or random numbers for locating the grids) should be used in each stratum
within the sample area. The sampling approach chosen for one stratum does not have to be
used in another stratum. For example, if a sample area is made up of a small waste pile and
a large 200-acre hillside, then it would be possible to use systematic sampling for the
hillside and random sampling for the waste pile.
5.5 Field Procedures for Determining the Exact Sampling Location
The grid points specified for the coordinate system or other reference points
(e.g., trees, boulders, or other landmarks) provide the starting point for locating the sample
points in the field. The location of a sample point in the field will be approximate because
the sampling coordinates were rounded to distances that are easy to measure, the
measurement has some inaccuracies, and there is judgment on the part of the field staff in
locating the sample point.
A procedure to locate the exact sample collection point is recommended to
avoid subjective factors that may affect the results. Without this precaution, subtle factors
such as the difficulty in collecting a sample, the presence of vegetation, or the color of the
soil may affect where the sample is taken, and thus bias the results.
To locate the exact sample collection point in the field, use one of the
following procedures (or a similar procedure) to move from the location identified when
measuring from the reference points to the final sample collection point. In the methods
below, M is the accuracy to which distances can be easily measured in the field.
• Choose a random compass direction (0 to 360 degrees or N, NE, E,
SE, etc.) and a random distance (from zero to M meters) to go to the
sample location (as illustrated in Figure 5.7).
• Choose a random distance (from -M meters to M meters) to go in the
X direction and a random distance (from -M meters to M meters) to
go in the Y direction, based on the coordinate system.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Figure 5.7 An Example Illustration of How to Choose an Exact Field Sampling
Location from an Approximate Location
Approximate Location
Exact
Sampling
Location
Randomly chosen
angle between
0°and360°
Randomly chosen, accurately
measured distance from the
approximate sampling
location
For either of these procedures, the random numbers can be generated in the
field using a hand-held calculator or by generating the random numbers prior to sampling.
The sample should be collected as close to this exact sampling location as possible.
5.6
Subsampling and Sampling Across Depth
Methods for deciding how and where to subsample a soil core are important
to understand and include in a sampling plan. These methods should be executed
consistently throughout the site. The field methods that are used will depend on many
things including the soil sampling device, the quantity of material needed for analysis, the
contaminants that are present, and the consistency of the solid or soils media that is being
sampled. The details of how these considerations influence field procedures are not the
subject of this discussion, but they are important and related to the discussion. More detail
can be obtained in the Soil Sampling Quality Assurance User's Guide (USEPA, 1984).
This discussion describes methods for soil acquisition across depth once an
exact auguring or coring position has been determined and describes how these approaches
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CHAPTER 5: FIELD SAMPLING PROCEDURES
influence the interpretation of sampling results. There are several approaches that might be
considered each with advantages and disadvantages; these are outlined in Figure 5.8.
5.6.1 Depth Discrete Sampling
The first approach is to decide before sampling on an exact position or
positions across depth that will be retained for analysis. For example, it may be decided
that throughout the site a split spoon will be driven so that the soil within the following
intervals is retained and sent to the laboratory for separate analysis: at elevations 1.5 m to
1.4 m, -0.5m to -0.6 m, and -4.5 m to -4.6 m (relative to a geodetic or site standard
elevation). The size of the interval would depend on the volume required by the
laboratory. In this example, all the soils material within each interval is extracted and
analyzed. Advantages of this approach are that each depth can be considered a different
sample area and conclusions regarding the attainment of cleanup standards can be made
independently for each soil horizon. This is also a preferred method when the presence of
volatiles in the soils media prevents the application of compositing methods.
5.6.2 Compositing Across Depth
Other approaches to sample acquisition within a core are based on
compositing methods. Compositing methods are generally to be approached with caution
unless the statistical parameter of interest is the mean concentration. If the mean is the
statistic of interest, then the variance of the mean contributed by differences in location
across the site from composited samples will be lower than the same variance associated
with the mean from noncomposited samples. However, compositing will restrict the
evaluation of the proportion of soil above an established cleanup standard because of the
physical averaging that occurs in the compositing process. Clearly compositing is not
recommended if the compositing process will influence the mass of material in the sample
as in the case of volatile organics within a soils matrix. Numerous authors have
contributed to the understanding of the effects of compositing (Duncan, 1962; Elder ejaL,
1980; Rohde, 1976; Schaeffer and Janardan, 1978; and Schaeffer & flL, 1980), and these
references or a statistician should be consulted if complicated compositing strategies are
planned.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Figure 5.8 Subsampling and Sampling Across Depth
Subsample
at the same
depth(s) in
each soil
sample core
Random
subsamples
from each
core are
mixed
Entire core
is mixed
A single
randomly
selected
location is
sampled in
each core
Soil Sample
Core
Relative Possible
Depth in Subsampling
Laboratory Sample
Samples Results
Possible
Subsampling
Xdi
Xd2
Xd3
Xi
Xi
Mixed
Xi
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CHAPTER 5: FIELD SAMPLING PROCEDURES
Under one compositing method, segments of the soil core are retained from
randomly or systematically identified locations. Then only the sampled portions are
homogenized and then subsampled. Another approach calls for retaining the entire core
and homogenizing all of the material and then subsampling. The latter approach is
preferred from a statistical point of view because the subsampling variance will be lower.
However, the second method may present difficulties if the soil samples are obtained to
considerable depth or by split spoon. In these situations, it is clearly not reasonable or cost
effective to acquire a core from the entire soil profile. On the other hand, if a hand-held
core or continuous coring device such as a vibra-corer is being used, then homogenization
of the entire core may be possible. In general, large amounts of material, material that is
difficult to manipulate because of its physical properties, material containing analytes that
will volatilize, or hazardous soil make thorough mixing more difficult, which may
eventually defeat the positive features associated with homogenization of the entire core.
5.6.3 Random Sampling Across Depth
A final approach involves randomly sampling a single location within each
core. At first, this approach appears to have many difficulties, but if the interest is in
verifying that the proportion of soil above a cleanup standard is low, this approach will
work quite well.
Suppose that an in situ soils stabilization method was used to treat all of the
overburden soils within a former lagoon. The treatment was previously found to yield
effective and homogeneous results over depth and space. It would clearly not be
appropriate to sample at a single depth of, say, 3m. Since depth homogeneity is expected,
it may also not be necessary to evaluate several specific depths by sampling 1-m, 3-m, 7-
m, and 15-m horizons in each boring. Finally and most importantly, it would not be
recommended to perform compositing because the statistical parameter of interest is the
proportion of soil at the site above the cleanup standard and not the mean concentration.
In this situation it may be useful to pick a random depth at each location. In
this way, many depths will be represented across the lagoon. Also, cost may be reduced
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CHAPTER 5: FIELD SAMPLING PROCEDURES
because at many locations the auger will not have to drill to bedrock because the sample
will be obtained from a random location that, in some samples, will be near the surface.
5.7 Quality Assurance/Quality Control (QA/QC) in Handling the
Sample During and After Collection
Data resulting from a sampling program can only be evaluated and
interpreted with confidence when adequate quality assurance methods and procedures have
been incorporated into the design. An adequate quality assurance program requires
awareness of the sources of error associated with each step of the sampling effort.
A full discussion of this topic is beyond the scope of the document;
however, the implementation of a QA program is important. For additional details, see Soil
Sampling Quality Assurance User's Guide (USEPA, 1984), Brown and Black (1983), and
Garner (1985).
5.8 Summary
Locating soil samples is accompli shed using a detailed map of the waste site
with a coordinate system to identify sampling locations. The boundaries of the sample
areas (areas within the site for which separate cleanup verification decisions are to be made)
and strata within the sample areas should be shown on the map. It is not necessary to draw
a grid for the entire waste site, only to identify the actual coordinates selected.
A random sample of soil units within the sample area or stratum will be
selected by generating a series of random (X,Y) coordinates and identifying the location
associated with these coordinates.
When selecting the sample coordinates for a systematic sample, two
common patterns of systematic or grid samples are a square grid and a triangular grid.
Various methods can be used to select a systematic sample; however, the most important
point is that one of the systematic sample locations must be identified randomly.
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CHAPTER 5: FIELD SAMPLING PROCEDURES
A separate random or systematic sample is selected for each sample area. In
addition, the extension of these procedures to stratified sampling is straightforward. Each
stratum is sampled separately. The sampling approach chosen for one stratum, or sample
area does not have to be used in another stratum.
Once a horizontal position is chosen, the method of acquiring samples
across depth must be decided. Methods for subsampling and sampling across depth should
be executed consistently throughout the site. The methods discussed are:
• Depth discrete sampling;
• Compositing across depth; and
• Random sampling across depth.
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6. DETERMINING WHETHER THE MEAN
CONCENTRATION OF THE SITE IS LESS THAN A
CLEANUP STANDARD
This chapter describes statistical procedures for determining whether the
mean concentration in the sample area attains the cleanup standard. Testing whether the
mean attains the cleanup standard is appropriate if the mean (or average) concentration is of
particular interest and if the higher concentrations found in limited areas are not of concern.
If the median concentration or the more extreme concentrations (e.g., the concentration for
which 95 percent of the site is lower and 5 percent of the site is higher) are of interest, then
see Chapter 7 for appropriate statistical techniques.
The statistical procedures given in this chapter for deciding if the mean
concentration attains the cleanup standard are called "paramedic" procedures. They usually
require certain assumptions about the underlying distribution of the data. Fortunately, the
procedures perform well even when these assumptions are not strictly true, and thus they
are applicable in many different field conditions (see Conover, 1980).
The following topics-determination of sample size; calculation of the mean,
standard deviation, and confidence interval; and deciding if the sample area attains the
cleanup standard-are discussed for each of the following sample plans in the sections
indicated:
• Simple random sampling (section 6.3);
• Stratified random sampling (section 6.4); and
• Systematic sampling (section 6.5).
6.1 Notation Used in This Chapter
The following notation is used throughout this chapter:
Cs The cleanup standard relevant to the sample area and the contaminant
being tested.
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CHAPTER 6: DETERMINING WHETHER THE MEAN CONCENTRATION OF THE
SITE IS LESS THAN A CLEANUP STANDARD
}i The "true" but unknown mean contaminant concentrations across the
sample area, the population mean.
HO The null hypothesis, which is assumed to be true in the absence of
significant contradictory data. When testing the mean, the null
hypothesis is that the sample area does not attain the cleanup
standard: HQ: |i ^ Cs.
a The desired false positive rate for the statistical test. The false
positive rate for the statistical procedure is the probability that the
sample area will be declared to be clean when it is actually dirty.
HI The alternative hypothesis, which is declared to be true only if the
null hypothesis is shown to be false based on significant
contradictory data. When testing the mean, the alternative hypothesis
is that the sample area attains the cleanup standard: HI: |i< Cs.
[LI The value of |i under the alternative hypothesis for which a specified
false negative rate is to be controlled (\i\ < (i).
|3 The false negative rate for the statistical procedure is the probability
that the sample area will be declared to be dirty when it is actually
clean and the true mean is m. The desired sample size n^ is selected
so that the statistical procedure has a false negative rate of (i at \i\.
n^ The desired sample size for the statistical calculations.
n The final sample size, i.e., the number of data values available for
statistical analysis including the concentrations that are below the
detection level.
xj The contaminant concentration measured for soil sample i,
i = 1 to n. For measurements reported as below detection, x, =
the detection limit. See section 2.5.2 for more details.
6.2 Calculating the Mean, Variance, and Standard Deviation
For many purposes in this chapter it is necessary to calculate the mean,
variance, or standard deviation for a sample of data. The basic formulas are provided in
Box 6.1 for use in later sections.
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CHAPTER 6: DETERMINING WHETHER THE MEAN CONCENTRATION OF THE
SITE IS LESS THAN A CLEANUP STANDARD
Box 6.1
Calculating Sample Mean, Variance, Standard Deviation,
and Coefficient of Variation
If the data are a random sample of n observations (i.e., the sample size
is n), designate the data as xj, X2...,xj,... to xn. The sample mean (or average),
indicated by x, is calculated as:
x==£- (6.1)
The formula for the sample variance, s2, is:
n-1
The formula for the standard deviation is:
n_
s=
n-1
The formula for the coefficient of variation is:
cv=§ (6.4)
x
The standard deviation provides a measure of the variability of the sample
data. In particular, it is used to obtain estimates of standard errors and confidence limits.
Degrees of freedom, denoted by df, provide a measure of how much
information the variance or standard deviation is based on. The variance and the standard
deviation calculated above for simple random samples have "n-1 degrees of freedom." The
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CHAPTER 6: DETERMINING WHETHER THE MEAN CONCENTRATION OF THE
SITE IS LESS THAN A CLEANUP STANDARD
degrees of freedom are used in calculating confidence intervals and performing hypothesis
tests to determine whether the sample area has attained the cleanup standard.
6.3 Methods for Random Samples
Methods in this section are applicable when the criterion for deciding
whether the site attains the cleanup standard is based on the mean concentration and the
samples are collected using simple random sampling. The steps involved in the data
collection and analysis are:
• Determine the required sample size (section 6.3.2);
• Identify the locations within the site from which the soil samples are to be
collected and collect the physical samples for analysis (Chapter 5);
• Perform appropriate statistical analysis using the procedures described in
section 6.3.3 and on the basis of the decision rule given in section 6.3.4,
decide whether the site requires additional cleanup.
6.3.1 Estimating the Variability of the Chemical Concentration
Measurements
Before sample collection, determine the number of samples needed to
achieve the desired confidence in the findings. The number of soil samples depends on the
anticipated variability of the soil measurements. Therefore, an estimate of the standard
deviation of the underlying contamination levels must be obtained. The true value of the
standard deviation is denoted by the Greek letter sigma, o. Estimation of o is discussed
in the next section.
To estimate the required sample size, some information about the standard
deviation, a (or equivalently the variance a2), is needed. Unfortunately, the standard
deviation is usually unknown, and steps must be taken to estimate this quantity for the
purpose of determining sample size. The symbol "A" is used to denote that 6 is an estimate
of a. In practice, & is either obtained from prior data or by conducting a small preliminary
investigation such as a pilot-scale treatability study. Cochran (1977) discusses aspects of
determining a preliminary value for &.
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CHAPTER 6: DETERMINING WHETHER THE MEAN CONCENTRATION OF THE
SITE IS LESS THAN A CLEANUP STANDARD
6.3.1.1 Use of Data from a Prior Study to Estimate a
If there are data on contamination levels for the site under investigation from
a previously selected sample of soil units or a treatability study, this information can be
used to obtain 6\ Note that the characteristics of physical samples used in the previous
study should be roughly the same as those planned for the present evaluation. For best
results, the sample from the prior study should be a simple random sample. If not, the
sample should at least be "representative" in the sense that the measurements are
distributed evenly across the cleanup area. In particular, measurements that tend to be
located within a specific subarea would generally be inappropriate for estimating the
variability across the entire area.
To obtain & from the existing sample, calculate the variance of the chemical
observations. It is best to have at least 20 observations for the variance calculations. The
sample standard deviation, s, can be calculated using equation (6.3) in Box 6.1. Use the
calculated value of s for 6\
6.3.1.2 Obtain Data to Estimate a After a Remedial Action Pilot
This approach will be best implemented as part of a pilot scale treatability
study.
1) Using the sampling procedures described in Chapter 5, select a preliminary
(simple random) sample of nj = 20 soil units. Determine the concentrations
for these 20 units.
2) From this preliminary sample, compute the standard deviation, s, of the
contaminant levels. Using s for &, determine the required sample size, n,
using equation (6.6) in Box 6.3.
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CHAPTER 6: DETERMINING WHETHER THE MEAN CONCENTRATION OF THE
SITE IS LESS THAN A CLEANUP STANDARD
3) If the sample size determined is less than or equal to 20, proceed with the
statistical analysis as outlined in sections 6.3.2, 6.3.3, and 6.3.4, using the
preliminary sample as the complete sample. Otherwise, select enough
additional soil samples so that the preliminary sample plus the additional
samples add up to the required sample size. In this case, the results for the
initial sample and the supplement should be combined for the statistical
analysis.
6.3.1.3 An Alternative Approximation for &
If there are no existing data to estimate o, and a preliminary study is not
feasible, a crude approximation for & can be obtained. The approximation is based on
speculations and judgments concerning the range within which the soil measurements are
likely to fall. The approximation is based on virtually no data, so the sample sizes
computed from these approximations may not satisfy the specified level of precision.
Consequently, it should only be used if no other alternative is available.
The approximation described in Box 6.2 uses the range of possible soil
measurements (i.e., the largest possible value minus the smallest value). The range
provides a measure of the variability of the data. Moreover, if the frequency distribution
of the soil measurements of interest is approximately bell-shaped, then over 99 percent of
the measurements can be expected to lie within three standard deviations of the mean.
Box 6.2
An Alternative Approximation for &
An estimate of CT is given by:
& = RANGE/6 (6.5)
Where RANGE = the expected spread between the smallest and largest values.
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CHAPTER 6: DETERMINING WHETHER THE MEAN CONCENTRATION OF THE
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6.3.2 Formulae for Determining Sample Size
The equations for determining sample size require the specification of
equations 6.6 and 6.7, given in Box 6.3 and the following quantities: cleanup standard
(Cs), the mean concentration where the site should be declared clean with a high probability
([LI), the false positive rate (a), the false negative rate (P), and the standard deviation
Box 6.3
Formulae for Calculating the Sample Size
Needed to Estimate the Mean
(6.6)
where zj.p and zi_a are the critical values for the normal distribution with
probabilities of 1- a and 1 - 13 (Table A.2).
The sample size may also be written in the following equivalent form:
- -a L
nd = P 2 « where T = - -. (6.7)
The term i (Greek letter tau) expresses the difference in units of
standard deviation. For convenience, the values of n as computed from this
formula are given in Table A.6 for selected values of a, 3, and ?.
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CHAPTER 6: DETERMINING WHETHER THE MEAN CONCENTRATION OF THE
SITE IS LESS THAN A CLEANUP STANDARD
Box 6.4 gives an example of calculating sample size.
Box 6.4
Example of Sample Size Calculations
Suppose it is desirable to verify cleanup when the mean concentration
is .1 ppm below the cleanup standard of .5 ppm (Cs = .5, fi^ = .3) with a power
of .80 (i.e., |3 = .20). Also suppose o = .43, a = .05, and 99 percent of the
sample points will result in analyzable samples, then
•=.465
a .43
From Table A.6 with P = .20, a = .05, and T = .465, the desired
sample size is between 25 and 30. Using linear interpolation gives a sample size
of about 29. From Table A.2,
Zj_a= 1.645, Z!_P = 0.842.
Using formula 6.7,
(zl-B + zl-a)2 (.842 +
and
28.6
Rounding up, the sample size is 29.
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CHAPTER 6: DETERMINING WHETHER THE MEAN CONCENTRATION OF THE
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Box 6.5 gives an example of determining sample size for testing the mean
using power curves.
Box 6.5
Example: Determining Sample Size for Testing the Mean
Using the Power Curves
At a former wood processing plant it is desirable to determine if the
average concentrations of PAH compounds in the surface soil are below 50 ppm
(the cleanup standard Cs). The project managers have decided that the dangers
from long-term exposure can be reasonably controlled if the mean concentration in
the sample area is less than the cleanup standard. The false positive rate for the test
is to be at most 5 percent (i.e., a = .05). The coefficient of variation of the data is
thought to be about 1.2. After reviewing the power curves in Figure A.2 and the
approximate sample sizes for random sampling, the managers decide:
1) While it would be desirable to have a test with power curves similar
to curves E and F, the samples sizes of more than 100 will cost too much.
2) Power curves A, B, and C have unacceptably low power when the
mean concentration is roughly 75 percent of the cleanup standard (i.e., 37 ppm),
the expected mean based on a few preliminary samples.
3) Thus the test should have power similar to that in curve D.
Based on the specifications above and the table at bottom of the Figure
A.2, the information needed to calculate the sample size is:
a = .05;
P = .20; and
14 = Cs * .69 = 34.5 ppm.
These values can be used to determine the sample size using the
equations described earlier.
If the sample size has been specified in advance, perhaps based on cost
considerations, Figures A.I through A.4 can be used to determine the approximate shape
of the power curve for the associated test. See Box 6.6 for an example.
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SITE IS LESS THAN A CLEANUP STANDARD
Box 6.6
Determining the Approximate Power Curve
for a Specified Sample Size
Suppose that after review of the budget and analytical costs, the
managers had decided that 40 samples would be collected. What is the
approximate shape of the power curve for the associated test assuming a = .05,
P = .20, and a systematic sample is used?
Based on previous samples the managers believe that the coefficient of
variation of the concentration measurement will be around 1.1. Assuming that a
systematic sample will behave statistically like a random sample (a reasonable
assumption of a site which has been cleaned up) and looking at the bottom of
Figure A.2 at the sample sized for testing the mean:
1) If the cv were 1.0, the power curve for a sample size of 40 would be
between curves C (sample size = 34) and curve D (sample size = 65), and closer to
curve C.
2) If the cv were 1.5, the power curve for sample size of 40 would be
between curves A (sample size = 25) and curve B (sample size = 43), and closer to
curve B.
3) Since the cv is about 1.2, the power curve for the test will be
between curves B and C.
6.3.3 Calculating the Mean, Standard Deviation, and Confidence
Intervals
This section describes the computational procedures used to calculate the
mean concentration and related quantities necessary to evaluate attainment of the cleanup
standard based on a random sample. For concentrations below the detection limit, as
discussed in section 2.5.2, substitute the detection limit in the calculations below.
The mean of the sampling data is an estimate of the mean contamination of
the entire sample area, but does not convey information regarding the reliability of the
estimate. Through the use of a "confidence interval," it is possible to provide a range of
values within which the true mean is located.
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The formula for an upper one-sided lOO(l-oc) percent confidence
limit around the population mean is presented in Box 6.7. The one-sided confidence
interval should be used to test whether the site has attained the cleanup standard. The
corresponding decision rule is given in section 6.3.4.
Box 6.7
Computing the Upper One-sided Confidence Limit
HUa = x + t^tf— (6.8)
where x is the computed mean level of contamination, and s is the corresponding
standard deviation. The appropriate value of tj a ft can be obtained from Table
A.I.
6.3.4 Inference: Deciding Whether the Site Meets Cleanup Standards
To determine whether the site meets a specified cleanup standard, use the
upper one-sided confidence limit \i\ja, defined above in equation (6.8). Use the following
rule to decide whether or not the site attains the cleanup standard:
If |J.ua < Cs, conclude that the area is clean (i.e., }i < Cs).
If Hua ^ Cs, conclude that the area is dirty (i.e., jj. > Cs).
Box 6.8 presents an example of an evaluation of cleanup standard
attainment.
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Box 6.8
An Example Evaluation of Cleanup Standard Attainment
From Box 6.4 the required sample size is 29. Assume for this example
that all 29 field samples were collected and analyzed. Six values were below the
detection level; these values were included in the analysis at the detection limit.
Based on these data, the mean is .29 with a standard deviation of .41. (Note that
41
this gives a coefficient of variation of-W = 1.48).
The upper one-sided 99 percent confidence interval goes to
= * + 'l-a.df = -29 + 2.467 = .478 ppm
Since 0.478 < 0.5, there is a 99 percent confidence that the mean
concentration of the sample area attains the cleanup standard of 0.5 ppm.
6.4 Methods for Stratified Random Samples
The following sections discuss methods of obtaining an overall estimate of
the mean contamination from a stratified sample. The steps in data collection and analysis
are:
• Determine the required sample sizes for each stratum (Chapter 6.4.1);
• Within each stratum, identify the sampling locations (Chapter 5). Collect
the physical samples and send the soil samples to the laboratory for
analysis;
• Perform statistical analysis using the procedures described in section 6.4.2,
and, on the basis of the decision rule given in section 6.4.3, decide whether
the site attains the cleanup standard.
The calculations for stratified samples require knowledge of the proportion
of the surface area or volume of soil represented in each stratum. The proportion of the
volume of soil can be calculated using the formula in Box 6.9.
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Box 6.9
Calculating the Proportion of the Volume of Soil
Suppose there are L strata designated by h = 1, 2, 3, ..., L. Compute
the volume of soil in stratum h as:
Vj, = Surface area of stratum h * Depth of sampling in stratum h
Then the proportion of the volume in stratum h is:
Wh=-- (6.9)
h=l
6.4.1 Sample Size Determination
The determination of the appropriate sample size for each stratum is
complicated. There are methods (Cochran, 1977 or Kish, 1965) for determining the
"optimum" allocation, but these require considerable advance knowledge about the relative
costs and variability within each strata. Consequently, general guidelines, rather than rigid
rules, are given in this guidance document to assist in planning the sample sizes for a
stratified sample. These guidelines are expected to cover most situations likely to occur in
the field. For more complex situations, consult Cochran (1977) and a statistician.
The formulas for sample size use the following notation, where h indicates
the stratum number:
n^ The desired sample size for the statistical calculations in stratum h.
nj, The final sample size in stratum h, the number of data values available for
statistical analysis including the concentrations that are below the detection
level.
Wh Proportion of the volume or area of soil in the sample area that is in stratum
h.
&, The estimated standard deviation of measurements from stratum h. See
h
section 6.3.1 on estimating & within a strata or sample area. If only an
overall estimate, ft, is available, use this for all strata.
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Ch The relative cost of collecting, processing, and analyzing a soil sample in
strata h. If all strata are assumed to have the same relative cost for an
additional sample, it will be easiest to use C^ = 1 for all strata.
L The number of strata.
xhj The reported concentration of the chemical for the ith sample unit in
stratum h.
h The stratum number.
After strata are defined, it is necessary to decide how many soil units should
be collected in each stratum. The recommendations below are based on the following
factors:
• The physical size of the stratum;
• The cost of sampling and processing a soil unit selected from the
stratum; and
• The underlying variability of the chemical concentration of the soil
units in the stratum.
The "optimum" sample allocation will produce the most accurate estimate of
the overall mean across strata for a fixed total cost. In Boxes 6.10 and 6.11, n^ will
denote the desired sample size to be selected from stratum h. Thus, for a total of L strata,
the overall desired sample size is n^ = ni
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CHAPTER 6: DETERMINING WHETHER THE MEAN CONCENTRATION OF THE
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Box 6.11
An Example Sample Size Determination for a Stratified Sample
A site consists of two strata (L = 2). Stratum 1 includes loose sand
soil, while stratum 2 consists of dense hard clay soil mixed with large rocks, but
the same kind of contamination is present in both strata. Three meters of soil have
been excavated from both areas. Stratum 1 comprises 10 percent of the sample
area (W^ = .10, W2 = .90). The sample and analysis costs are considerably
different in the two strata. The cost of sampling and analysis in stratum 2 is
estimated to be 10 times that in stratum 1 (C\ = 1, €2 = 10) because of the cost
associated with extracting a soil core. The estimated standard deviation of the
measurements, based on previous sampling, is b\ = 25 in stratum 1, while in
stratum 2, &2 = 13.1. Using a cleanup standard of 40, a = .01, \LI = 35 and (3 =
.20, the sample size in each strata can be calculated as follows:
2.326 + .842J2 =
40- 35 J
= (.10 * 25 * VI) + (.90 * 13.1 * VTO") = 39.78
Using equation (6.10),
and
10 * 25
nld = 39.78 *.401) " ^ =399
VI
90 * 1^1
n2d = 39.78 * .401) '*U -J3'1 = 59.5
V10
Rounding up, and assuming that all samples will be collected and
analyzed, the final sample sizes are nlf = 40 and n2f = 60.
When multiple statistical tests are used, or multiple chemicals tested, use the
field sample size in a stratum that is the largest field sample size for any statistical test or
chemical. Although this procedure for multiple tests will always provide an adequate
sample size, it may not be the most cost efficient
6.4.2 Calculation of the Mean and Confidence Intervals
If the number of values below the detection limit is moderate, procedures
and formulae presented in this chapter and in the following boxes based on the sample
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average are applicable. Section 2.5.2 discusses the adjustment for values below the
detection limit. If the proportion of values in the data set that have values below the
detection limit in any stratum is large, the procedures in Chapter 7 for testing proportions
may be preferred.
Box 6.12
Formula for the Mean Concentration from a Stratified Sample
The overall mean concentration, xst, should be computed as:
"h
(6.11)
or
Xst =
(6.12)
The equations in Box 6.13 give the formula for the standard error of xst.
The standard error is required for establishing confidence limits around the actual
population mean and deciding if the site attains the cleanup standard.
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Box 6.13
Formula for the Standard Error from a Stratified Sample
The standard error of xst, denoted by s^ is calculated as follows:
ST = <6'13'
where
nh _ 2
s2 = i?i(Xhi'Xh) (6.14)
Sh V1
and
,
x, = 1=1
h """ (6.15)
The approximate degrees of freedom for the standard error can be
calculated using the formula in Box 6.14. The degrees of freedom should be rounded to
the closest integer.
Box 6.14
Formula for Degrees of Freedom from a Stratified Sample
(616)
h=l
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The mean, standard error, and degrees of freedom are used to estimate an
upper one-sided confidence interval with a confidence of 1-a (see Box 6.15).
Box 6.15
Formula for the Upper One-sided Confidence Interval
from a Stratified Sample
Compute the upper one-sided confidence limit as:
(6.17)
where xst is the mean level of contamination from Box 6.12, and s* is the
SI
corresponding standard error from Box 6.13. The appropriate value of ti_a df can
be obtained from Table A.I.
The value M-ua^8 a 100(1 -a) percent confidence interval for the
population mean.
6.4.3 Inference: Deciding Whether the Site Meets Cleanup Standards
The test statistic to be used for testing the hypothesis that the site meets
specified cleanup standards is the upper one-sided confidence M-uct' defined above in
equation (6.17). Use the following rules to decide whether or not the site attains the
cleanup standard. An example illustrating the procedure is in Box 6.16.
If (lua < Cs, conclude that the area is clean (i.e., p. < Cs).
If H-ua - Cs, conclude that the area is dirty (i.e., [0. > Cs).
If the upper one-sided confidence interval of the sample is below the Cs,
then there is 1-a certainty that the mean of the sample area is below the Cs and the site
attains the Cs.
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Box 6.16
An Example Illustrating the Determination of Whether
the Mean from a Stratified Sample Attains a Cleanup Standard
Following with the example in Box 6.11, the sample area has two
strata. Stratum 1 comprises only 10 percent of the total site in terms of surface
area. The sample consists of 40 units from stratum 1, and 60 units from stratum
2. After the soil units were analyzed in the laboratory, it was learned that two of
the units in stratum 1 were below the detection limit. Hence, the chemical
concentration for these two cases was set to the minimum detectable level.
1) Calculate stratum means: Suppose that for the 40 data values from
stratum 1 the average concentration of the chemical under study was computed to
be KI = 23 ppm. Similarly, for the 60 data values from stratum 2, suppose that the
average concentration was determined to be x2 = 35 ppm.
2) Calculate stratum variances: Using equation (6.2) the stratum
standard deviations are: sj = 18.2 and s2 = 20.5. Note that the 38 observations
in stratum 1 that were above the detection limit, plus the two observations that
were set to the minimum detectable level, were used in the calculation of Sj.
3) Calculate overall mean: Since 10 percent of the site is contained in
stratum 1, we have Wj = .10, and W2 = .90. Thus, from equation (6.11), the
overall mean for the entire site is:
xst = W! Xl + W2x2 = (.1)(23) + (.9X35) = 33.8 ppm.
4) Calculate standard error: The standard error of the estimate
computed from the equation (6.13) is:
I^+
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6.5 Methods for Systematic Samples
If systematic sampling is chosen, some changes in statistical methodology
may be considered and are discussed in this section. One concern is that systematic
sampling should be avoided when the pattern of contamination is likely to have a cyclical or
periodic pattern across the sample area. Such a situation might occur if waste was placed in
trenches, if contamination blew into windows, or if a remediation technology is used such
as vacuum extraction, which creates a regular pattern caused by well induced zones of
influence. In such a case, a systematic sampling pattern may capture only high (or low)
values of the contaminant and therefore yield biased results. It is presumed that the
likelihood of this pattern will be known in advance, and be used to create strata and the
need to sample randomly.
6.5.1 Estimating Sample Size
Systematic sampling can result in an increase in the precision of the
statistical estimates and a corresponding decrease in the required sample size (Cochran,
1977). Unfortunately, the possible advantages of systematic sampling are difficult to
predict before the sample is collected. The sampling precision of an estimated mean from a
systematic sample depends on the pattern of contamination at the site and how the
systematic sample is constructed. However, the standard error of a mean based on a
systematic sample will usually be comparable to or less than the standard error of a mean
based on a random sample of the same size. Therefore, using the sample size formulas for
a random sample when the sample was collected systematically usually will be as or more
protective of human health and the environment.
Use the procedures in section 6.3.2 to determine the sample size required
for a systematic sample. If other procedures for calculating sample size for a systematic
sample are considered, a statistician should be consulted.
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6.5.2 Concerns Associated with Estimating the Mean, Estimating the
Variance, and Making Inference from a Systematic Sample
When a systematic sample is obtained, apply the same methods used for a
random sample. As with a simple random sample, the simple average of the sample points
is an unbiased estimate of the population average. Note, however, that the number of
sample points in a systematic sample of an irregularly shaped area may vary from the
targeted sample size. A smaller sample size will produce estimates that have less precision
than larger samples, but will not introduce bias. The loss in precision tends to be negligible
except for small sample sizes.
In general, an unbiased estimate of the standard error of a mean based on a
systematic sample is not available. In the special case where contamination is distributed
randomly over the sample area, unbiased estimates of the standard error can be constructed.
This situation may be approximately the case after a careful cleanup has been done where
the cleanup has effectively removed the contaminated soil from all of the high
contamination areas or the soil is being mixed, fixed, or incinerated.
Several methods are commonly used to estimate the standard error of a
mean from a systematic sample (Koop, 1976; Wolter, 1984; Tornqvist, 1963; Yates,
1981). These methods treat the systematic observations as:
• A random sample;
• A stratified sample; and
• A serpentine pattern of observations that employs a special variance
calculation procedure.
It is suggested that the serpentine pattern be used with overlapping pairs of
points as the principal method of estimating the standard errors in a systematic design.
However, if the boundaries of the sample area are so irregular as to make this approach
difficult, the stratification approach is recommended. The random sample estimate should
seldom be used. These approaches are discussed below.
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6.5.2.1 Treating a Systematic Sample as a Random Sample
The simplest method of estimating the standard error for a systematic
sample is to use the variance formulas in Box 6.1 for a simple random sample (section
6.3). This method is appropriate if the contamination is distributed randomly across the
sample area. If there are gradients of contamination, or if there are substantial contiguous
areas that have higher (or lower) than average contamination, this method can be biased
(Osborne, 1942). In this case, the actual standard error of the mean will, on average, be
smaller than that computed from the simple random sample formulas. Thus, the sample
estimate will appear to be less precise than it really is and there will be a tendency to take
more observations than are necessary or to do more cleanup work than is necessary.
6.5.2.2 Treating the Systematic Sample as a Stratified Sample
An estimate of the standard error that is less subject to bias than the random
sample estimate can be obtained by aggregating adjacent points in the systematic design into
groups, and treating these groups as though they were strata (Yates, 1981) as depicted in
Figure 6.1. It should be noted that this grouping can be done whether or not the sample
area was previously stratified. If stratification was used, grouping for purposes of
estimating standard errors would be done within strata (see Box 6.17).
A commonly used group size consists of four observations. The groups
need not be the same size, but efficiency is gained if they are nearly the same size and if
they are small. Points in a group should be adjacent and the groups must cover the sample
area comprehensively. One must not form the groups on the basis of the observed data--
this would add bias. Instead, they should be formed strictly on the basis of geographic
adjacency and boundary restrictions without regard to their observed values. If the sample
locations form a square grid, the recommended grouping will be four adjacent sample
locations forming a square. (At the edge of the strata or sample area, the clusters of four
points might not form squares due to irregular boundaries.)
Although the average contamination measure is computed in the usual
manner as the sum of all observations divided by the number of them, the average may be
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considered as a weighted sum of the group means, where the weights are the number of
observations in the group.
Figure 6.1 An Example of How to Group Sample Points from a Systematic Sample so
that the Variance and Mean Can Be Calculated Using the Methodology for a
Stratified Sample
The tests described in section 6.3 for simple random samples can be adapted
for systematic samples by simply replacing the quantities s/Vn in equation (6.8) with the
expression for the standard error given in equation (6.19). Box 6.18 gives a formula for
an upper one-sided confidence interval for the true mean contamination when a systematic
sample is treated as a stratified sample.
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Box 6.17
Estimating the Mean, the Standard Error of the Mean,
and Degrees of Freedom When a Systematic Sample
Is Treated as a Stratified Sample
L
- 1
(6-18)
h=l
where h denotes the h* group, L is the total number of groups, nh is the number
of observations in group h, xh is the mean of the observation in group h, and n is
the total number of observations in the sample.
The estimated standard error of the mean, s^, can then be computed as:
(6.19)
where sj,2 is the variance of the observations in group h as computed from the
equations in section 6.2. The degrees of freedom are computed as: df = n-L.
Box 6.18
Formula for Upper One-sided Confidence Interval for the True Mean
Contamination When a Systematic Sample Is Treated as a Stratified Sample
For example, the upper one-sided confidence interval for the true mean
contamination is:
= * +tl-a,dfSx (6.20)
In this case, the sample area would be declared to be clean if
less than the cleanup standard; otherwise the sample area would be declared to be
dirty.
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6.5.2.3 Linearization and Estimates from Differences Between
Adjacent Observations of a Systematic Sample
Another commonly used method is to linearize the systematic pattern by
forming a serpentine association between each observation and the one preceding it in a
serpentine pattern. Consider the example pattern in Figure 6.2. The numbers represent the
sample points and their location in the sample area.
The numbers string the pattern into a linear sequence. The difference
between the observations of an adjacent pair contain a systematic component that represents
the "true" difference between them plus a random component. The systematic component
represents bias but, since the two members of the pair are adjacent geographically, one
would expect the systematic component of the difference to be small compared to, for
example, comparing point 1 with point 19.
Figure 6.2 Example of a Serpentine Pattern
Numbers indicate the sequence (i) required for the calculations in Box 6.19.
To estimate the standard error from a serpentine pattern makes use of
overlapping pairs. That is, point 1 is compared with point 2, 2 with 3, 3 with 4, and so
on. The method gives a somewhat more precise estimate of the standard error. The
method is shown in Box 6.19.
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Box 6.19
Computational Formula for Estimating the Standard Error and
Degrees of Freedom from Samples Analyzed in a Serpentine Pattern
= A/
\
(l/2n) I(xi-xi_i)2/(n-l) (6.21)
i=2
df = 2n/3
The associated number of degrees of freedom in Box 6.19 is given
approximately by DuMouchel el aL (1973).
It should be noted that the serpentine pattern can be constructed by moving
from top to bottom, from right to left, or diagonally within the systematic pattern. The
pattern should be planned prior to sampling. If it is suspected that there will be a gradient
in the data, say from top to bottom, then the serpentine pattern should be formed so that it
follows the contours of the gradient to the extent that it is feasible to do so.
6.6 Using Composite Samples When Testing the Mean
"Compositing" refers to the process of physically combining and mixing
several individual soil samples to form a single "composite" sample (see Rohde, 1976 and
1979; Duncan, 1962; Elder el al, 1980; Gilbert, 1987, Gilbert el al, 1989, and section
5.6.2 of this document). A primary advantage of compositing is that it reduces the number
of lab analyses that must be performed.
Composite samples can be created using the following procedure:
• Collect the samples using a random or systematic sample design, collecting
n soil samples from the field;
• Physically mix randomly selected groups of ten samples to create n/10 = m
samples, which are sent to the lab for analysis; and
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• Perform the statistical analysis on the m lab results to determine if the mean
attains the cleanup standard.
In the procedure above, each soil sample sent to the lab was composited
from 10 original samples; the compositing factor was 10. The compositing was
accomplished by mixing the first 10 randomly selected samples, the second 10 randomly
selected samples, etc. to get the final m samples to send to the lab. To specify how
compositing is done, both the method of selecting the samples that get mixed together and
the compositing factor must be specified. In addition, compositing requires that each
original sample is the same or known physical size in terms of volume or weight and that
the samples are very well mixed. These criteria may be difficult to achieve. This possible
advantage will be reduced if the mixing is not complete or uses soil samples of different
physical sizes. Nevertheless, as mentioned above, the number of lab analyses that must be
performed may be greatly reduced.
Other considerations are the decisions related to how best to composite the
original samples, the number of soil samples to collect, and the number of soil samples to
send to the lab. If the laboratory error is large, compositing may provide little benefit. The
specification of which samples to combine will be affected by the sample design and the
variability across the sample area, among other things. For some types of soil or chemicals
being tested, mixing will affect the laboratory analysis. For example, mixing samples with
volatile organics may release contaminants.
Compositing can be a useful technique if the mean is to be tested, but must
always be considered and implemented with caution. Compositing should never be used if
percentiles or proportions are used as the attainment criteria. Other methods of compositing
are discussed by Gilbert (1987). If compositing is considered, consultation with a
statistician is recommended.
6.7 Summary
The methods in this chapter apply when the cleanup standard is intended to
control the average conditions at the site, not simply the average of the sample. The mean
estimated from a sample must be sufficiently below the cleanup standard to ensure with
confidence that the entire site is below the cleanup standard.
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Basic formulas are provided to calculate the mean, variance, or standard
deviation for a sample of data. The standard deviation provides a measure of the variability
of the sample data and is used to obtain estimates of standard errors and confidence limits.
These statistics help determine how far the sample mean must be below the cleanup
standard to ensure with reasonable confidence that the site mean is below the cleanup
standard.
For a random sampling, the number of soil samples required depends on the
anticipated variability of the soil measurements. To estimate the required sample size, some
information about the standard deviation, a, or the variance CT^, is needed. Steps to
estimate a are discussed. Equations for determining sample size require the following
quantities: cleanup standard (Cs), the mean concentration where the site should be declared
clean with a high probability (m), the false positive rate (a), the false negative rate (P), and
the standard deviation (a). The mean of the sampling data is an estimate of the mean
contamination of the entire sample area. The use of an upper "confidence interval"
provides an upper bound on the true sample area mean. When a one-sided 100 (1-ot)
percent upper confidence limit of the mean is less than the Cs, the site is judged clean.
Estimating the mean contamination from a stratified sample requires
considerable advance knowledge about the relative costs and variability within each strata.
Guidelines and formulae are given to assist in planning the sample sizes for a stratified
sample and how many soil units should be collected in each stratum. They are also given
for establishing the standard error, the approximate degrees of freedom for the standard
error, and the upper one-sided confidence interval. If the upper one-sided confidence
interval on the sample mean is below the cleanup standard (Cs), cleanup is verified.
If systematic sampling is used, special methods are required; these
procedures are discussed and illustrated. To estimate the standard error for a systematic
sample, formulae used for a simple random sample and a stratified sample may be applied,
as can be the method of linearizing the systematic pattern into a serpentine pattern. Two
estimates of the standard error are common when the points (sampling locations) have been
linearized; these are discussed.
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Compositing samples—the act of physically combining and mixing several
individual soil samples to form a single composite sample—is discussed. Its primary
advantage is that it reduces the number of lab analyses that must be performed.
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7. DETERMINING WHETHER A PROPORTION OR
PERCENTILE OF THE SITE IS LESS THAN A
CLEANUP STANDARD
This chapter describes statistical procedures for determining with confidence
whether a specified proportion of the soil is less than a cleanup standard. The extreme
concentrations at a hazardous waste site are often of primary concern. In this case, an
appropriate statistical test can be based on either a high percentile of the distribution of
chemical measurements over the area, or on a large proportion of the area that has
concentrations less than the cleanup standard. For example, the methods in this chapter
apply if there is interest in verifying that a large percentage (e.g., 90,95, or 99 percent) of
the soil at the site has concentrations below the cleanup standard.
Throughout Chapter 7 the statistical evaluations are designed to detect when
a large proportion of the site is less than a cleanup standard. However, there is another
equivalent way of stating this objective: these evaluations are designed to ensure that no
more than a small proportion of the site is above the cleanup standard. The numerical
methods in this chapter are designed and presented in the context of the second approach.
Therefore, we will be testing to verify that only a small proportion or percentage of the site
(e.g., 10, 5, or 1 percent) exceeds the cleanup standard.
Two approaches to testing percentiles and proportions are discussed in this
chapter
• Exact and large sample nonparametric tests for proportions based on
the binomial distribution; and
• A parametric test for percentiles based on tolerance intervals, which
assumes the data have a normal distribution.
In the nonparametric approach, each soil sample measurement is designated
as either equal to or above the cleanup standard, Cs and coded as "1," or below Cs and
coded as "0." The analysis is based on the resulting data set of O's and 1's. The
proportion of the soil (or equivalently, the percentage of the area under investigation) at or
above the cleanup standard can be estimated from the coded data. If the proportion of 1's
is high, the site will be declared contaminated. On the other hand, if the proportion of O's
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is sufficiently large, the area is considered to have attained an acceptable level of cleanup.
A test based on proportions works with any concentration distribution and requires only
that the cleanup standard be greater than the analytical method detection limit. However,
this method has limitations because it does not consider how far above or below the Cs the
data value is, only if it is above or below.
The second approach for testing percentiles of the concentration
distribution, which does not require coding the data as above, is based on estimating a
confidence interval for a percentile of the normal distribution. These intervals are called
tolerance intervals (Guttman, 1970). The assumption that the data have a normal
distribution (or that a suitable transformation of the data is approximately normal) is critical
to this test. In addition, this method may be biased if more than 10 percent of the
observations are below the detection limit.
The following sampling and analysis plans are discussed in the sections
indicated:
• Simple random sampling for proportions (section 7.3);
• Stratified random sampling for proportions (section 7.5); and
• Simple random sampling for testing percentiles of a normal
distribution (section 7.6).
7.1 Notation Used in This Chapter
The following notation is used throughout this chapter:
Cs The cleanup standard relevant to the sample area and the contaminant
being tested (see section 3.4 for more details).
P The "true" but unknown proportion of the sample area with
contaminant concentrations greater than the cleanup standard.
PQ The criterion for defining whether the sample area is clean or dirty.
According to the attainment objectives, the sample area attains the
cleanup standard if the proportion of the sample area with
contaminant concentrations greater than the cleanup standard is less
than PQ, i.e., the sample area is clean if P < PQ.
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HO The null hypothesis, which is assumed to be true in the absence of
significant contradictory data. When testing proportions, the null
hypothesis is that the sample area does not attain the cleanup
standard; HQ: P > PQ.
a The desired false positive rate for the statistical test to be used. The
false positive rate for the statistical procedure is the probability that
the sample area will be declared to be clean when it is actually dirty.
HI The alternative hypothesis, which is declared to be true only if the
null hypothesis is shown to be false based on significant
contradictory data. When testing proportions, the alternative
hypothesis is that the sample area attains the cleanup standard; HI: P
Cs),
thenyj= 1.
7.2 Steps to Correct for Laboratory Error
All of the procedures for estimating proportions and percentiles assume that
the chemical concentrations can be measured with little or no error. If there is substantial
variability in the measurement process, the corresponding estimates of proportions may be
biased (Mee el aj.., 1986 and Schwartz, 1985). If an upper percentile (greater then the
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median) is being tested, the bias may be conservative. In other words, the sample area may
be cleaner than the statistical test would indicate. This bias will be more important in some
situations than in others:
• The measurement error causes no problems if the median (50th
percentile) is tested;
• The measurement error is likely to be the greatest problem when the
percentile to be tested is between the 75th and 99th; and
• The measurement error is likely to be the greatest problem if the true
proportion of contaminated soil samples is close to the proportion
being tested, i.e., the sample area just attains the cleanup standard.
There are three possible ways to reduce the bias:
• Use a more precise analytical method that has a smaller measurement
error,
• Perform multiple laboratory measurements on each soil sample and
use the average or median measurement in the statistical analyses
(see the example in Box 7.1); or
• Perform more cleanup of the sample area than is required to attain
the cleanup standard.
Box 7.1
Illustration of Multiple Measurement Procedure for
Reducing Laboratory Error
Soil
unit
1
2
3
Measurement (ppb)
1
<50
75
<50
2
95
105
<50
3
101
102
55
Sample
median
(ppb)
95
102
<50
Coded
result
0
1
0
Detection limit - 50ppb
Cs - 100 ppb
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7.3 Methods for Simple Random Samples
This section describes statistical analysis procedures that apply when the
criterion for deciding whether the site attains the cleanup standard is based on the
proportion of contaminated soil units and when the soil samples are selected by simple
random sampling. The basic steps involved in the data collection and analysis are:
• Determine the required sample size (section 7.3.1);
• Identify the locations within the site from which the soil units are to be
collected, collect the physical samples, and send sampled material to
laboratory for analysis (Chapter 5);
• Perform appropriate statistical analysis using the procedures described in
sections 7.3.3 and 7.3.4 and, on the basis of the statistical analysis, decide
whether the site requires additional cleanup.
Although the use of random samples is recommended, random sampling
may not be practicable. An alternative is to select a systematic or grid sample using the
procedures described in Chapter 5. Systematic samples may be easier to collect and will
provide valid estimates of proportions, but may produce a poor estimate of sampling error.
7.3.1 Sample Size Determination
The sample sizes as computed in Box 7.2 are summarized in Tables A.7
through A.9 for selected values of P0 and PT and for the following values of a and 0:
a =0.01, 0.05, and 0.10, and (3 = 0.20. In most cases, Tables A.7 - A.9 will be
adequate for practical application. However, for values not in the tables, use equation
(7.1) below. Notice that the cleanup standard is not required in order to determine the
sample size.
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Box 7.2
Computing the Sample Size When Testing a Proportion or Percentile
Given the quantities, PQ, PI, a, and (3, the sample size can be
computed from the following formula:
^--^-o
nd=t — E - p . p - ) (7.1)
ro ri
where zj.p and zj. aare the critical values for the normal distribution with
probabilities of 1-oc and l-(3 (Table A.2).
7.3.2 Understanding Sample Size Requirements
To illustrate the use of the sample size tables, consider the following
scenario (also see Box 7.3). A sample area will be considered clean if less than 20 percent
of the area has concentrations of mercury greater than 1,5 ppm. That is, PQ = .20 in this
example. The null hypothesis is HQ:? > .20 and specifies that if 20 percent or more of the
sample area has concentrations exceeding 1.5 ppm, the area is still considered dirty and
requires further remedial action.
Further suppose that the site manager wants no more than a 5 percent
chance of declaring the sample area to be clean when it is dirty (i.e., a = .05). Moreover,
the site manager wants to be 80 percent certain that if only 10 percent of the area has
concentrations exceeding 1.5 ppm the site will be found clean. That is, for PI = .10, he
wants the false negative rate to be moderately low, say 20 percent (i.e., (3 = .20). From
Table A.8 (corresponding to values of a = .05 and P = .20), the required sample size for
PQ = .20 and PI = .10 is rid = 83.
It is evident from Tables A.7 - A.9 that as the value of PI approaches PQ,
the required sample sizes become larger. For example, if the manager in the above example
wanted the false negative rate to be 20 percent for PI = .15 (instead of PI = .10), the
required sample size would be 368. Such a large sample size may be impractical for many
waste site investigations. If the cleanup technology is designed to achieve levels that are
only slightly less (Pi) than the cleanup objective (Po), then many samples will be required
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to verify attainment of PQ. If the cleanup is highly effective and PI is well below PQ, then
few samples will be required to verify cleanup.
Box 7.3
Example of How to Determine Sample Sizes
When Evaluating Cleanup Standards Relative to a Proportion
Soil has been removed from a lagoon bottom that previously contained
corrosive waste. The exposed soil will be sampled to determine whether more
excavation is required. Wanting to minimize the possibility of future health
effects, the site will be judged in attainment of the cleanup standard if there is 90
percent confidence (a = . 10) that less than 10 percent (P0 = . 10) of the topsoil has
concentrations exceeding the cleanup standard. The expected proportion of
contaminated soil is low, less than 5 percent The manager wants to be 80 percent
confident (P = .20) that the sample area will be declared clean if the proportion of
contaminated soil is less than 2 percent (Pi = 2 percent).
From Table A.9, for P0 = 0.10 and PI = 0.02, the required sample
size is n = 39.
Using formula (7.1), from Table A.2, zi_a = 1.282 and zt.p = .842
and:
fzi
=l
nd — - p . p
ro ri
r.842V.02(.98) + 1 282V.10(.9Q) 12
~l fm7V5 i
.10 - .02
= 39.4
7.3.3 Estimating the Proportion Contaminated and the Associated
Standard Error
This section describes the computational procedures to be used to calculate
the proportion contaminated (see Box 7.4) and related quantities necessary to evaluate
attainment of the cleanup standard.
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Box 7.4
Calculating the Proportion Contaminated
and the Standard Error of the Proportion
Set yj = 1 if the concentration in sample i is greater than the cleanup
standard and yj = 0 otherwise. If n = the total number of samples available for
statistical analysis, the proportion of samples, p, above the cleanup standard can be
calculated using the following equations:
r = yi (7.2)
i=l
where yi = 1 if x{ > Cs or yi = 0 if xj < Cs
P = £ (7.3)
The standard error, Sp, of the proportion p is
Sp = V n n v' • (7.4)
These results are used to estimate upper one-sided confidence intervals,
which allow determination of whether the site has attained the prescribed cleanup standard.
If the sample size is sufficiently large, an approximate confidence interval
may be constructed using the normal approximation (see Box 7.5, section 7.3.4). If the
sample size is small, an "exact" procedure should be used to calculate the confidence
interval (see Box 7.6, section 7.3.5).
7.3.4 Inference: Deciding Whether a Specified Proportion of the
Site is Less than a Cleanup Standard Using a Large Sample
Normal Approximation
When np > 10 and n(l-p) > 10, the large sample normal approximation can
be used for evaluating the statistical significance of the number of sample values equal to or
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above Cs. This condition will generally only be met for tests of percentiles between 10 and
90. If the condition is not met, the exact test should be used.
Box 7.5
Calculation of the Upper Confidence Limit on a Proportion
Using a Large Sample Normal Approximation
Compute the following:
PU = P + zl-a sp (7-5)
If PU < PQ, conclude that the area has attained the cleanup standard.
> PQ, conclude that the area has not attained the cleanup standard.
7.3.5 Deciding Whether a Specified Proportion of the Site is Less
Than the Cleanup Standard Using an Exact Test
If the normal approximation is not appropriate, the "exact" procedure
described below should be used to test whether the proportion meets the cleanup standard.
However, if the sample size is too small, it may not be possible to construct a useful
decision rule with the stated false positive rate. These instances are indicated in the tables
(Tables A.7 - A.9) used to perform the tests.
Use the following to perform the exact test:
• Given n, a, and P0, determine the "critical value" of the test, ra;n,
by referring to Table A. 10. To use this table, a must be .01, .05 or
.10, respectively. To determine the critical value, select the column
for PQ specified in the attainment objectives. Reading down the
column find the first number greater than the sample size n. Move
up one row and read ra:n, the critical value, in the leftmost column.
If the number in the first row of the selected column is greater than
the sample size, there are not enough data to perform the given test.
If the bottom number in the selected column is less than the sample
size, use the normal approximation above.
• From the sample, determine the number, r, of soil units that have
chemical concentrations exceeding Cs. Compare r with ra;n.
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• If r < ra;n, conclude that the area has attained the cleanup standard.
• If r > ra;n, conclude that the area has not attained the cleanup
standard.
For values of n, a, and P0 that are not given in the tables, the critical value
for the "exact" test may be determined directly using the algorithm below or using an
equivalent procedure from Brownlee (1965, p. 148-150) based on the F distribution.
Step 1 Compute f(0) = (1 - P0)n.
Step la If f(0) > a, then set ra;n = 0 and stop. Note that if f(0) >
a, a test with the specified false positive rate is not possible;
the actual false positive rate would be f(0).
Step Ib If f(0) < a, go to Step 2.
Step 2 Compute
f(l) = n (r-V-) f(0) (7.6)
1 " ro
where f(0) is computed in Step 1.
Step 3 Next, compare f(0) + f(l) with a. If f(0) + f(l) > a, set ra;n = 0, and
stop. If f(0) + f(l) < a, define a "temporary" variable, y, and set y = 1.
Go to Step 4.
Step 4 For the given value stored in the temporary variable, y, compute f(y) using
the recursion formula below:
f(y) = ) (Tr) f(y-D- (7-7)
Step 5 Compare f(0) + f(l) + ... + f(y) with a. If f(0) + f(l) + ... + f(r) > a, set
ra;n = y and stop. If f(0) + f(l) + ... + f(r) < a, increment the temporary
variable by 1, i.e., set y = y+1, and go to Step 3. Repeat Steps 4 and 5
until the process stops and ra;n has been determined.
Box 7.6 gives an example of an inference based on the "exact" test
described above.
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Box 7.6
An Example of Inference Based on the Exact Test
Assume that only 9 samples collected from 203 sample locations have
concentrations greater than the cleanup standard, i.e., r = 9, and remember that n
= 191, a = .05, and P0 = .05.
Using Table A. 10 read down the column headed by PQ = ,05 and find
the first number greater than the sample size, in this case 208 in row 6. Go up one
row and read ra:n from the lefthand column. The value in the left column and fifth
row is 4 = ra:n.
Since r > 4, the sample area does not attain the cleanup standard.
7.4 A Simple Exceedance Rule Method for Determining Whether a
Site Attains the Cleanup Standard
One of the most straightforward applications of the methods in this chapter
involves the design of zero or few exceedance rules. To apply this method, simply require
that a number of samples be acquired and that zero or a small number of the concentration
measurements be allowed to exceed the cleanup standard. This kind of rule is easy to
implement and evaluate once the data are collected; it only requires specification of the
sample size and number of exceedances as indicated in Table 7.1.
In addition, these rules also have statistical properties. For example, the
more samples collected, the more likely that one sample will exceed a cleanup standard.
That is, it is more likely to measure a rare high value with a larger sample. In addition, the
larger the proportion of the site that must have concentrations below the cleanup standard,
the more soil samples that will be required to document this with certainty. Finally,
because of the chance of outliers, it may be that the rule that allows one or more
exceedances would be preferred in order to still have the site judged in attainment of the
cleanup standard. If more exceedances are allowed, more soil samples are required to
maintain the same statistical performance and proportion of the site that is clean.
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Table 7.1 illustrates these tendencies and offers selected sample sizes and
exceedance rules as a function of statistical performance criteria. For example, if there is
interest in: verifying that 99 percent of the site is below a cleanup standard; keeping the
chance of saying the site is clean when it is dirty at 1 percent; and allowing no exceedances
of the cleanup standard, then 459 soil samples would be required. If 459 samples were
obtained and none of them exceeded the cleanup standard, there is 99 percent confidence
that 99 percent of the site is less than the cleanup standard. If three exceedances were
allowed and the same statistical performance criteria were required then 1001 soil samples
would be required and 998 of the measurements would have to be less than the cleanup
standard.
On the other hand, if the statistical performance criteria are relaxed, sample
size requirements decrease. For example, if there is interest in allowing no exceedances
and a false positive rate of 90 percent that 90 percent of the site is less than the cleanup
standard, then 22 samples would have to be obtained and all results would have to be less
than the cleanup standard. If three exceedances were permitted and the same statistical
criteria were applied, then 65 samples would be required and 62 of the measurements
would have to be less than the cleanup standard.
7.5 Methods for Stratified Samples
In some circumstances it may be useful to establish a stratified sampling
regime as discussed in Chapter 5. If the waste can be divided into homogeneous subareas,
the precision of an estimated proportion can often be improved through the use of a
stratified sample. These homogeneous areas from which separate samples are drawn are
referred to as "strata," and the combined sample from all areas is referred to as a "stratified
sample."
The statistical procedures discussed here apply when the criterion for
deciding whether the site attains the cleanup standard is based on the proportion of
contaminated soil units. The basic steps involved in the data collection and analysis are:
• Determine the required sample sizes for each stratum using the equation in
section 7.5.1;
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• Within each stratum, identify the locations within the site from which the
soil units are to be selected, collect the physical samples, and send sampled
material to laboratory for analysis (Chapter 5); and
• Perform appropriate statistical analysis using the procedures that follow
(sections 7.5.2 and 7.5.3) and, on the basis of the statistical results, decide
whether the site has attained the cleanup standard.
The tests described in this section assume that soil samples within each
stratum are collected randomly. Although the use of simple random samples is
recommended, simple random sampling may not always be practicable. An alternative
method would be to select a systematic (grid) sample; however, this type of sampling
should be approached with caution as described in section 7.3 and Chapter 6.
7.5.1 Sample Size Determination
Determination of the appropriate sample size is complicated in stratified
sampling because there are many ways the sample can be allocated to strata. For example,
if 100 soil units will be sampled, a decision must be made on whether to allocate the sample
equally among strata, in proportion to the relative size of each strata, or according to some
rules. There are methods for determining the "optimum" allocation; however, these require
considerable advance knowledge about the underlying variability of each strata.
Consequently, the equations below are general guidelines to assist in planning the sample
sizes for a stratified sample. These guidelines will cover many field situations. For more
complex situations, a text such as Cochran (1977) should be consulted.
The formulas for sample size use the following notation, where h indicates
the stratum number:
h As a subscript, indicates a value for a stratum within the sample area
rather than for the entire sample area.
njxj The desired sample size for the statistical calculations.
nh The final sample size, the number of data values available for
statistical analysis including the concentrations that are below the
detection level.
Wh Proportion of the volume of soil in the sample area which is in
stratum h
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Cost of collecting, processing, and analyzing one additional soil
sample, on a relative scale.
The number of strata.
The scored concentration data, where yhi = 1 if the measured
concentration is greater than the cleanup standard and 0 otherwise.
Table 7.1 Selected information from Tables A.7 - A.9 that can be used to determine
the sample sizes required for zero or few exceedance rules associated with
various levels of statistical performance and degrees of cleanup
Chance of Saying the
Site is Clean When
It is Dirty
(Certainty)
Proportion of
the Site That
Is Clean
Sample Size Requirements
Under Various Numbers of
Allowed Exceedances of the
Cleanup Standard
False Positive Rate, Alpha
(1 - Alpha)
1-PO
Number of Allowed Exceedances
0135
.01
(.99)
.99
.95
.90
459 662 1001 1307
90 130 198 259
44 64 97 127
.05
(.95)
.99
.95
.90
299 473 773 1049
59 93 153 208
29 46 76 103
.10
(.90)
.99
.95
.90
230 388 667 926
45 77 132 184
22 38 65 91
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Once the sample area has been divided into strata, it is necessary to decide
how many soil units should be collected in each stratum. The equations below will provide
an "optimal" sample size for each stratum provided that the following information is
available:
• The physical size of the stratum;
• The cost of sampling and processing a soil unit selected from the
stratum;
• The underlying proportion of the soil units in the stratum that are
contaminated, i.e., have chemical concentrations exceeding the
specified cutoff, Cs; and
• The overall desired accuracy of the test.
An optimum sample allocation to each stratum will produce the most
accurate measure of the proportion of soil contaminated across strata in the entire sample
area for a fixed total cost. In what follows, n^ will denote the corresponding sample size to
be selected from stratum h. Thus, the total sample size n, is calculated as follows: n =
n1+n2+...+nL.
Although the sample size equations assume that the quantities C^ and P^ are
known, reasonable assumptions can be used, following the rules below (see Box 7.7):
• If the relative sampling costs, Cn, are not known or all strata are assumed to
have the same cost for an additional sample, set Cn =1 for all strata;
• If data are not available to provide an estimate of P^ in some strata, set PI, =
P0 for those strata.
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The expected number of contaminated samples in stratum h is Ph* n^. It is
recommended that the expected number of contaminated samples in each stratum be at least
5 for calculation of reliable confidence intervals. Occasionally this may require increasing
the sample size in one or more strata.
Box 7.7
Computing the Sample Size for Stratum h
Given Ch, PH. and WH, the sample size for stratum h should be
computed as:
W A/~Tr *•! l'a ^-f * ——
hVSj \P0-P1 j
7.5.2 Calculation of Basic Statistics
This section describes the computational procedures to be used to calculate
the quantities necessary to evaluate attainment of the cleanup standard on the basis of the
overall proportion of contaminated samples. Box 7.8 gives sample size calculations for
stratified sampling.
Use the formula below in Box 7.9 for calculating an overall proportion of
exceedance from a stratified sample. Note that the overall sample proportion, denoted by
pst, is simply a weighted average of the individual stratum means.
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Box 7.8
Sample Size Calculations for Stratified Sampling
At a site with heavy metal contamination, the sample area has been
divided into two strata, one consisting of high elevation areas, another of low
elevation areas which received most of the historical runoff. The strata are the
same volume (W^ = .5, W2 = .5) The expected proportion of contaminated soil is
5 percent on the higher ground and 10 percent in the lower area (Pi = .05, P2 =
.10). Due to difficult access and low trafficability in the lower area, the cost of
sampling is twice what it is on the high ground (Cj = 1, €2 = 2). EPA has
decided that less than 10 percent of the soil can have concentrations over the
cleanup standard (with a confidence of 90 percent, a = .10). The site manager
must be able to conclude that the site is clean with a confidence of 80 percent (P =
.20) at an overall contamination proportion of 4 percent.
To determine the sample size, the site manager first determines:
zla = 1.282, ZI_P = .842
from Appendix A. Then, following equation (7.8):
-a+zl-Bl2 _ J1.282+.84212 _
-P ~l -10 - .04 J -
1
= {(.5 * VT) + (.5 * VI)} = 1.207
J
f W. ]
nhd = Phd - Ph) * 1-207 * 1,253 * -
and
nld = .05(1 - .05) * 1.207 * 1,253 * -^ = 35.9
VI
n2d = .10(1 - .10) * 1.207 * 1,253 * ~ = 48.1
V2
Rounding up, the samples sizes of the strata are:
= 36, and n2f = 48
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Box 7.9
Calculating an Overall Proportion of Exceedances and
the Standard Error of the Proportion From a Stratified Sample
Ph = the sample proportion of units in stratum h that have chemical
concentrations exceeding Cs.
The estimated overall proportion of soil units that have chemical
concentrations exceeding Cs is given by the formula below:
IWhPh (7.10)
h=l
Use equation (7.11) to estimate the standard error of p ,. The
SI
standard error is required for constructing an approximate decision rule and also
for establishing confidence limits around the actual population proportion.
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In Box 7.10, use the equation (7.12) to compute the upper limit of the
one-sided confidence interval.
Box 7.10
Calculating the Upper Limit of the One-sided
Confidence Interval on an Estimate of the Proportion
PUa = Pst + zl-a Spst (7-12)
where pst is the computed overall proportion of contaminated units, and Sp is the
*st
corresponding standard error. The value of z\.a can be obtained from Table A.2.
The value pUa designates an upper 100(1-a) percent one-sided confidence limit for the
population proportion.
7.5.3 Inference: Deciding Whether the Site Meets Cleanup Standards
The upper one-sided confidence limit, P\ja, is used for testing the
hypothesis that 1 - PQ of the site attains the specified cleanup standard. Use the following
rules to decide whether or not the site attains the cleanup standard:
If pua < PO> conclude that the site meets the cleanup standard.
If puct ^ PO, conclude that the site does not meet the cleanup standard.
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See Box 7.11 for an example of an inference for proportions using stratified
sampling.
Box 7.11
Inference for Proportions Using Stratified Sampling
Following the example in Box 7.8, all 434 samples from stratum 2
were collected; however, of the 324 samples in stratum 1, four were lost due to a
lab error, leaving 320 samples for the analysis. The proportion of samples
collected in each strata that had concentrations greater than or equal to the cleanup
standard are: .0531 in strata 1 (the higher ground) and .0922 in strata 2.
Using equation (7.10)
L
Pst = XWh Ph = -5 * -0531 + -5 * .0922 = .0727
h=l
Using equation (7.11)
=A/LWh2
X h=l
Ph(l - Ph)
_ / .25 * .0531(1 - .0531) . .25 * .0922(1 - .0922) ___.
~ \ 320 + 454-.0094
Using equation (7.12)
PUa = Pst + zl-a Sp = -O727 + 1-282 * .0094 = .0848
Since .0848 is less than P0 (.10), based on the proportion of
contaminated samples, the sample area attains the cleanup standard.
7.6 Testing Percentiles from a Normal or Lognormal Population
Using Tolerance Intervals
Tolerance intervals assume that the distribution of concentration
measurements follows a normal distribution. Tolerance interval techniques are sensitive to
the assumption that the data follow a normal distribution. This procedure is not robust to
departures from the normality assumption.
7-20
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CHAPTER 7: DETERMINING WHETHER A PROPORTION OR PERCENTILE OF
THE SITE IS LESS THAN A CLEANUP STANDARD
If it is suspected that the data do not approximately follow a normal
distribution, then either:
• Do not use the tolerance interval procedure and instead use
the nonparametric procedures described in section 7.4; or
• Transform the data so that the transformed data more nearly
approximate a normal distribution.
An approach that may be used to evaluate the assumption that the data follows a normal
distribution is discussed in section 7.6.2. If the data are not normal and a transformation is
being used then the transformation should be applied in the following manner. First,
transform both the data and the cleanup standard. Then calculate the upper confidence limit
on the percentile estimate of the transformed data. Compare the transformed upper limit
with the transformed cleanup standard. Do not reverse; transform the upper confidence
limit on the percentile for comparison with the untransformed cleanup standard. If
stratified random sampling is used then consult Mee (1989).
7.6.1 Sample Size Determination
To determine the required sample size, the following terms need to be
defined, PQ, Pr a, p. Once these terms have been established, the following are obtained
from Table A.2 and the equation in Box 7.12 is used to estimate the sample sizes:
zl p the upper p-percentage point of a z distribution;
zi-a *e UPP61" tt-percentage point of a z distribution;
z1 p the upper P0-percentage point of a z distribution; and
z j p the upper Pj -percentage point of a z distribution.
7-21
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CHAPTER 7: DETERMINING WHETHER A PROPORTION OR PERCENTTLE OF
THE SITE IS LESS THAN A CLEANUP STANDARD
Box 7.12
Calculating the Sample Size Requirements for Tolerance Intervals
(Guttman, 1970)
1-P0 zl-p
(7,3,
This sample size equation (7.13) requires smaller sample sizes than the
corresponding formula in section 7.3.1. This happens because the tolerance intervals gain
efficiency over the other methods in this chapter from the assumption that the data follow a
normal distribution.
If the normal distribution is not followed, even after transformation, the
procedure in this section is inappropriate. However, distributional form will not be
evaluated until after the sample is collected and the data analyzed. At this point it may be
decided to use the nonparametric procedures presented earlier in this chapter, but the
sample size may not be sufficient to ensure the desired false negative rate and, therefore,
may not be as sensitive as required.
Two example sample size calculations for tolerance intervals are shown in
Box 7.13. The reduction in the required sample size between the nonparametric test and
the tolerance interval test can be compared. The comparable sample sizes for the
nonparametric test are 1990 samples for example #1 and 315 samples for example #2. In
both examples, the tolerance interval method requires fewer samples, provided that it can
be reasonably concluded that the data follow a normal distribution.
7-22
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CHAPTER 7: DETERMINING WHETHER A PROPORTION OR PERCENTILE OF
THE SITE IS LESS THAN A CLEANUP STANDARD
7.6.2 Testing the Assumption of Normality
The statistical tests used for evaluating whether or not the data follow a
specified distribution are called goodness-of-fit tests. There are many different tests and
references demonstrating the evaluation of normality (e.g., Conover, 1980; D'Agostino,
1970; Filliben, 1975; Mage, 1982; and Shapiro and Wilk, 1965). If a choice is available,
the Shapiro-Wilk or the Kolmogorov-Smirnov test with Lilliefors critical values are
suggested. For easy application, Geary's test described by D'Agostino (1970) can be
used.
Box 7.13
Calculating Sample Size for Tolerance Intervals—Two Examples
Following are two examples of the computation required to calculate
the sample size when testing percentiles using confidence intervals.
Example #1 PQ=.010 Zj p = z99Q= 2.326
Pj=.005 Zj p = z955 = 2.576
a=.05 z = z= 1.645
= z 800 = °-842
f.842 + 1.64512 f7 48712
nd=|2.326-2.576J "^S = 98'96
Example #2 PQ=. 10 zl p =2^=1 .282
p
P^.05 zlpi = z95= 1.645
a=.05 zla =z95= 1.645
P=.05 z =z = 1.645
1.645 + 1.64515
1.282- 1.645 J
3.29
= 82.14 = 82
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CHAPTER 7: DETERMINING WHETHER A PROPORTION OR PERCENTILE OF
THE SITE IS LESS THAN A CLEANUP STANDARD
7.6.3 Inference: Deciding Whether the Site Meets Cleanup Standards
Using Tolerance Limits
The test of significance will be performed by estimating the upper
confidence interval on the point below which at least (1-P0)*100 percent of the data falls:
the [(l-P0)*100]th percentile. For example, the concentration measurement associated with
PO = .05 is the value below which at least 95 percent of the data falls. The concentration
measurement associated with PQ = .05 will be calculated from the sample mean and
standard deviation, x and s, as well as the constant k. The constant, k, necessary for
finding the upper tolerance limit, Tu is found using values of a, PQ, n, and T in Table A.3.
For values of k not shown in Table A.3, see Guttman (1970). With these three quantities
an estimated upper tolerance limit will be calculated for the desired percentile using the
equation in Box 7.14.
Box 7.14
Calculating the Upper Tolerance Limit
Tu = x + ks. (7.14)
If Tu is greater than the cleanup standard, then it is concluded that the
site fails to meet the cleanup standard.
Box 7.15 presents data and calculations that illustrate use of tolerance
intervals to test for percentiles with lognormal data.
7-24
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CHAPTER 7: DETERMINING WHETHER A PROPORTION OR PERCENTILE OF
THE SITE IS LESS THAN A CLEANUP STANDARD
Box 7.15
Tolerance Intervals: Testing for the 95th Percentile with Lognormal Data
The following data were collected to determine if the 95th percentile of
the concentrations was below the cleanup standard of 100 ppm (with a false
positive rate of 1 percent). The data is assumed to follow a lognormal distribution,
therefore logarithm of the data (the transformed data) are analyzed. In the
following, x refers to the original data and y refers to the transformed data.
Because the log of the data is used, the upper confidence interval on the 95th
percentile of the data must be compared to the log of the cleanup standard
(ln(100)=4.605). Twenty samples were obtained.
X
34
79
38
62
6
14
20
31
42
36
57
24
57
188
26
45
46
83
25
33
Total
ln(x)=y
3.526
4.369
3.638
4.127
1.792
2.639
2.996
3.434
3.738
3.584
4.043
3.178
4.043
5.236
3.258
3.807
3.829
4.419
3.219
3.497
72.372
V2
12.433
19.088
13.235
17.032
3.211
6.964
8.976
11.792
13.973
12.845
16.346
10.100
16.346
27.416
10.615
14.493
14.661
19.528
10.362
12.229
271.645
(6.3):
Using the logarithms as the data to analyze, the sample mean is:
_ 72.372 _,1Q
y =^-=3.619
The standard deviation, s, can be calculated using equations (6.2) and
2_ 271.645-20(3.619) .„
s — , a = .jil
s = V3TT = 0.715
For a sample size of 20, a = .01 and P0 = 5 percent, k = 2.808 (from
Table A.5). Finally, Tu can be calculated using equation (7.14):
TU = 3.619 + 2.808(.715) = 5.627
Since 5.627 is greater than 4.605 (the cleanup standard in logged
units), the sample area does not attain the cleanup standard.
7-25
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CHAPTER 7: DETERMINING WHETHER A PROPORTION OR PERCENTILE OF
THE SITE IS LESS THAN A CLEANUP STANDARD
7.7 Summary
These methods can apply to the 50th percentile or median as an alternative to
the mean or to a high percentile such as the 90th, 95th, or 99th. High percentile criteria
apply when the clean up standard is viewed as a value that should be rarely exceeded at the
site. Similar to testing the mean, the proportion of soil samples above the cleanup standard
must be sufficiently low to ensure with confidence that the proportion of soil at the site
meets the established percentile.
Two approaches to testing whether proportions or percentiles of the soil at a
site are less than the cleanup standard are discussed:
• Exact and large sample nonparametric tests for proportions based on
the binomial distribution; and
• A parametric test for percentiles based on tolerance intervals, which
assumes the data have a normal distribution.
The first approach, or test, works with any concentration distribution and
requires only that the cleanup standard(s) be greater than the analytical method detection
limit. For testing proportions, simple random and stratified random sampling are
discussed.
All of the procedures discussed assume that the chemical concentrations can
be measured with little or no error; variability in measurement may bias the corresponding
estimates of proportions. Ways to reduce the potential bias are discussed.
For simple random samples, the basic steps involved and that are discussed
include the following:
• Determine the required sample size;
• Identify locations within the site from which soil units are to be
collected, collect the samples, and send them to the laboratory; and
• Perform the statistical procedures described in this chapter, and then
decide whether the site needs additional cleanup.
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CHAPTER 7: DETERMINING WHETHER A PROPORTION OR PERCENTILE OF
THE SITE IS LESS THAN A CLEANUP STANDARD
The implementation of simple exceedance rules in the statistical plan design
requires that a certain number of samples be acquired and that none or a few of the
concentration measurements be allowed to exceed the cleanup standard. The more
exceedances allowed, the more soil samples that need to be collected to maintain the
statistical performance and proportion of the site that is clean. Sample sizes and exceedance
rules as a function of statistical performance criteria are presented in the chapter.
If stratified sampling is chosen, the basic steps involved include the
following:
• Determine the required sample sizes for each stratum;
• Within each stratum, identify the locations within the site from
which the soil units are to be collected, collect the samples, and send
them to the laboratory; and
Perform the statistical procedures described in this chapter, and then
decide whether the site needs additional cleanup.
The use of tolerance intervals, which is discussed next in this chapter,
assumes that the distribution of concentration measurements follows a normal distribution.
Techniques for using tolerance intervals, including the transformation of lognormal data to
a normal distribution, are included with two examples of sample size calculation and other
relevant equations.
7-27
-------
-------
8. TESTING PERCENTILES AND PROPORTIONS
USING SEQUENTIAL SAMPLING
This chapter discusses sequential sampling as a method for testing
percentiles. With sequential sampling, a statistical test is performed after each sample or
small batch of samples is collected and analyzed. The statistical test determines whether an
additional sample should be collected or whether the sample area is judged to have or have
not attained the cleanup standard.
Chapters 6 and 7 dealt with statistical tests that are based on samples of a
predetermined size. Fixed sample size methods will sometimes require that an
unnecessarily large sample size be used in order to meet the stated precision requirements.
This can be avoided by using a sequential procedure. Sequential procedures terminate
when enough evidence is obtained to either accept or reject the null hypothesis, and thus,
sequential tests can respond quickly to very clean or very contaminated sites. Sequential
procedures will also yield a lower sample size on the average than the fixed sample size
procedure even when the true level of P is not greatly different from PQ.
Decisions based on sequential sampling methods will be particularly useful
in conjunction with the "rapid turnaround" analytical methodologies that are being used
more often at Superfund sites. Devices that measure volatile soil gases, H-NU's, ion
specific probes, or onsite scanning laboratories can be used much more rapidly and
extensively than conventional intensive laboratory extraction, identification, and
quantification methods. Without rapid turnaround and the potential for additional sampling
within a day or two, sequential methods are not useful because of the cost to remobilize a
sampling team and the time required for laboratory processing. Nevertheless, "rapid
turnaround" technology is typically less accurate than conventional methods and therefore,
despite the larger sample sizes that are possible, should be applied in an orderly and
thoughtful manner.
References on sequential analysis include: Armitage (1947), Wetherill
(1975), Siegmund (1985), Sirjaev (1973), and Wald (1973).
8-1
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
8.1 Notation Used in This Chapter
The following notation is used throughout this chapter:
Cs The cleanup standard relevant to the sample area and the contaminant
being tested (see section 3.4 for more details).
P The "true" but unknown proportion of the sample area with
contaminant concentrations greater then the cleanup standard.
PQ The criterion for defining whether the sample area is clean or dirty.
According to the attainment objectives, the sample area attains the
cleanup standard if the proportion of the sample area with
contaminant concentrations greater than the cleanup standard is less
than PQ, i.e., the sample area is clean if P < PQ.
HO The null hypothesis, which is assumed to be true in the absence of
significant contradictory data. When testing proportions, the null
hypothesis is that the sample area does not attain the cleanup
standard; HQ: P > PQ-
a The desired false positive rate for the statistical test to be used. The
false positive rate for the statistical procedure is the probability that
the sample area will be declared to be clean when it is actually dirty.
HI The alternative hypothesis, which is declared to be true only if the
null hypothesis is shown to be false based on significant
contradictory data. When testing proportions, the alternative
hypothesis is that the sample area attains the cleanup standard; HI: P
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
8.2 Description of the Sequential Procedure
In the sequential testing procedures developed by Wald (1973), sampling is
performed by analyzing one soil unit at a time until enough data have been collected to
either reject the null hypothesis in favor of the alternative hypothesis, or accept the null
hypothesis.1 The expected sample size, using this sequential procedure, will be
approximately 30 to 60 percent lower than the corresponding fixed sample size test with the
same a, (3, PQ, and PI. The sequential procedure will be especially helpful in situations
where contamination at the site is very high or very low. In these situations the sequential
procedure will quickly accumulate enough evidence to conclude that the site either fails to
meet or meets the cleanup standard. However, it must be emphasized that the actual sample
size of the sequential procedure for a given site could be larger than for the fixed sample
size methods (see section 8.3).
Wald's sequential procedure consists of forming an acceptance and rejection
region for the cumulative number of contaminated soil units relative to the total number of
soil units evaluated. Figure 8.1 shows graphically how the procedure operates. The
horizontal axis, denoted by n, represents the number of soil units evaluated. The vertical
axis represents the cumulative number of contaminated soil units after n soil unit
evaluations. The two lines in the graph establish the boundaries of the acceptance and
rejection regions for the test. The intersection of these lines, CA and CB, with the vertical
axis and their slope, are important parameters of this sequential procedure.
The sampled soil units are evaluated one at a time, and after each evaluation,
the cumulative number or sum of contaminated units (i.e., soil units with concentrations
exceeding the cleanup standard, Cs) is determined. If the cumulative sum crosses the
topmost line into the acceptance region, the hypothesis of contamination is accepted. If the
cumulative sum stays low and enters the rejection region below the second (lowermost)
line, it is concluded that the site is not contaminated (i.e., the null hypothesis of
contamination is rejected). Otherwise, the process continues; that is, another soil unit is
evaluated, and the new cumulative sum is compared with the boundary values to determine
whether to accept or reject the null hypothesis or to continue evaluating soil units. In
lpThe procedure in Wald's book is for a test of PI > PQ. In the present situation this has been reversed. To
adapt the sequential procedure to this situation, the roles of a and P were reversed. The corresponding
acceptance and rejection regions of the graphs were also reversed.
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
Figure 8.1 the process terminates after 22 soil units have been evaluated, at which time the
null hypothesis that the area is contaminated is accepted.
Note that several soil samples can be collected and analyzed at the beginning
of the sequential process, since some minimum number of results must be available before
a decision can be reached
Figure 8.1 Graphic Example of Sequential Testing
Cumulative
Sum of
Concentrations
Exceeding the
Cleanup
Standard
9 •
7 • site is dirty - accept
5
3
1
-ID
-XX XX
xxxx
XX
x continue sampling
x xxx xx
xx
site is clean - reject
'B
n
Number of Soil Units Evaluated
8.3
Sampling Considerations in Sequential Testing
It may be impractical to randomly collect a soil unit, chemically analyze the
soil unit, and then decide whether or not to acquire the next unit. Instead, multiple soil
units can be selected initially using the simple random sampling procedures described in
section 5.2. The sampled soil units can then be chemically analyzed and each result
evaluated individually in random order, until the sequential procedure terminates. It may
also be possible, provided that the holding times or other analytical criteria are not violated,
to chemically analyze samples one at a time.
In situations where contaminant concentrations at the site are marginally
different from the cleanup standard, the sequential procedure can be expected to require
more samples until the sample size approaches the sample size required for the fixed sample
procedure. However, this is only an expectation, so in some situations where the actual
8-4
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
contamination is close to the cleanup standard, the sequential procedure can require a
substantially larger sample than the fixed sample procedure. In this situation, a cutoff rule
is suggested. If the sequential procedure requires a sample size twice the sample size
requked for the fixed procedure, then the sequential sampling should be stopped and a
decision made on the data collected up to that point. Procedures for accommodating this
situation are discussed in Box 8.2.
Also, as with all of the procedures in this manual, the site is assumed to be
at steady state during sampling. During the sequence of sampling the soil concentrations
should not be changing. Sequential sampling and analysis does not imply that changes
over time are being evaluated or that the progress of cleanup is being monitored.
Sequential sampling is performed during steady state conditions, only to reduce the sample
size required for a decision.
8.4 Computational Aspects of Sequential Testing
As was the case for the fixed sample tests described in earlier chapters, the
following quantities must be defined to implement the sequential testing procedure: Cs,
PO, PI, a, and p. Box 8.1 describes the method for establishing the acceptance and
rejection boundaries described in Figure 8.1.
Denote the Qth percentile of chemical concentrations by XQ. To test
whether XQ > Cs or greater (i.e., the site fails to meet the cleanup standard) against the
hypothesis that XQ < Cs (the site meets cleanup standards), set PQ = 1 - Q, and set the
maximum allowable error rate for falsely rejecting that the true percentile is Cs (i.e., false
positive rate) to a. If the Qth percentile is really less than Cs (indicating that fewer than PQ
of the area is contaminated), specify the minimum value of this percentile, PI < PO, that
should be detected with at least a probability of 1 - (3.
To test whether the Qth percentile is equal to Cs, the sequential procedure is
formatted by calculating the sequential procedure acceptance and rejection criteria as
described in Box 8.1. Then follow the steps in Box 8.2 to decide whether the site attains
the cleanup standard.
8-5
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
Box 8.1
Defining Acceptance and Rejection Criteria for the Sequential
Tests of Proportions
Let ln(x) denote the natural logarithm of x. Given a, (3, P0, and P1?
compute:
1) B = ln(— ) and A = ln(-^);
2) *i = (\~_ p") and R2=pY5
3) Use these values computed in (1) and (2) to determine the slope
of the two lines defining the rejection and acceptance regions,
and the points at which the two lines cross the vertical axis,
4) CA = -4- and CB -- !U.
5) Finally, compute the desired sample size for the corresponding
fixed sample size procedure,
f l-u
= 1 c p . p J
nd~l— p . p
rO rl
8-6
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
8.5 Inference: Deciding Whether the Site Meets Cleanup Standards
Box 8.2
Deciding When the Site Attains the Cleanup Standard
1) Calculate the sequential procedure acceptance and rejection
criteria described in Box 8.1.
2) After each evaluation calculate the cumulative number of soil
units that exceed the cleanup standard, Cs:
(8.2)
where yj = 1 if the i-th sample was above the cleanup
standard, and yi = 0 otherwise; and where n is the number of
soil units evaluated up to this point. Compare the current
value of k against the current critical value to decide whether to
accept or reject the null hypothesis or to continue sampling.
3) Starting with n = 1, if k > nM + CA, then stop evaluating
samples and accept HO: P > PQ. Conclude that the site is dirty
and requires additional cleanup.
4) If k < nM + CB, then stop evaluating samples and reject HO in
favor of P < PI. Conclude that the site is clean.
5) If neither of the two conditions above is met, continue
sampling and evaluation.
6) If the number of soil units that has currently been evaluated
exceeds 2.0*nf, stop the sampling and:
accept HQ: P > PO if k > nM + °A2+CB Or
accept HI: P < PI if k < nM + CA^" °B.
Rule 6 provides an approximate test and will have only a small effect on the actual levels of
a and p (see Wald, 1973).
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
If the conclusion in step 3 is reached, this means that the cumulative sum
has exceeded the line with intercept CA in Figure 8.1 and the site is judged contaminated.
However, if the conclusion in step 4 is made, then the cumulative sum has fallen below the
line with intercept CB in Figure 8.1 and the site is found to be clean. Notice that the
intercept values depend on the error rates (a and (3), the proportion that is being tested
(PO), and the proportion where the false negative error rate is estimated (Pi). The slope of
these lines is determined strictly by PO and PI.
Box 8.3 presents an example application of sequential testing.
8.6 Grouping Samples in Sequential Analysis
Under the random sampling approach discussed in section 7.3, a large
number of soil units are selected from the site at one time, and the laboratory analysis is
conducted on each unit, one at a time. In many situations it will be more efficient for the
laboratory to analyze the soil units in small batches or groups rather than one at a time. The
sequential procedure can be modified easily to account for this type of laboratory analysis.
The quantities Cs, PO, PI, ot, and |3 are defined in exactly the same way as
for stratified sampling. Similarly, the stopping rules are also defined in exactly the same
way. The only modification to the previously discussed procedures is in the calculation of
k. Previously, after each soil unit was analyzed, k was calculated as the cumulative
number of soil units that exceeded the cleanup standard, Cs. To modify k to take into
account the grouped nature of the data, k should be computed as the cumulative number of
soil units that exceed Cs after each batch has been analyzed. This minor modification is
illustrated in Box 8.4 for groups of five, using the example of Box 8.3. In the example,
after 4 groups of 5 or a total of 20 soil units, k = 4 > nM + CA = 3.0324, so sampling is
terminated and the site is considered contaminated.
8-8
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
Box 8.3
An Example of Sequential Testing
Assume that for the chemical under investigation, the following values
have been specified in the objectives worksheet: a = .05, P = .10, PO = .05, and
P! = .02. In this case, the quantities necessary to construct the acceptance and
rejection regions are:
B = ln(-) = -2.8904, and A = InpJ) = 2.2513;
Rl=||=.9694, and R2=:§=2.5; M = = .0328;
-2 8904 2 2513
- and CA - = 2'3764-
Below is a sequence of outcomes that might be observed for a
particular chemical. Note that the values of the boundary limits use the values of
M, CA, and CB, computed above. In the table, k = the cumulative number of soil
units that are found to have excessive levels of the contaminant. The process
terminates after the 18th soil unit has been analyzed. Prior to the 18th observation,
the value of k falls between the computed values of nM+CA and nM+CB.
However, with the 18th soil unit, k = 3 > nM+CA = 2.9668, and hence the null
hypothesis is accepted, i.e., the site fails to meet the cleanup standard.
Soil Sample
unit outcome k nM+CA nM+CB Decision
1 0 0 2.4092 -3.0182 continue
20 0 2.4420 -2.9854 continue
30 0 2.4748 -2.9526 continue
40 0 2.5076 -2.9198 continue
50 0 2.5404 -2.8870 continue
60 0 2.5732 -2.8542 continue
70 0 2.6060 -2.8214 continue
80 0 2.6388 -2.7886 continue
90 0 2.6716 -2.7558 continue
10 0 0 2.7044 -2.7230 continue
11 1 1 2.7372 -2.6902 continue
12 0 1 2.7700 -2.6574 continue
13 0 1 2.8028 -2.6246 continue
14 0 1 2.8356 -2.5918 continue
15 0 1 2.8684 -2.5590 continue
16 1 2 2.9012 -2.5262 continue
17 0 2 2.9340 -2.4934 continue
18 1 3 2.9668 -2.4606 accept
8-9
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
Box 8.4
Example of Sequential Test Using Grouped Samples
Example
Soil
unit
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
using the data
Group
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
of Box 8.3
Sample
outcome
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
1
0
1
after grouping soil units
k nM + CA
2.4092
2.4420
2.4748
2.5076
0 2.5404
2.5732
2.6060
2.6388
2.6716
0 2.7044
2.7372
2.7700
2.8028
2.8356
1 2.8684
2.9012
2.9340
2.9668
2.9996
4 3.0324
into groups
nM + Ce
-3.0182
-2.9854
-2.9526
-2.9198
-2.8870
-2.8542
-2.8214
-2.7886
-2.7558
-2.7230
-2.6902
-2.6574
-2.6246
-2.5918
-2.5590
-2.5262
-2.4934
-2.4606
-2.4278
-2.3950
of 5.
Decision
continue
continue
continue
accept
8.7
Summary
Sequential sampling means that a statistical test is performed after each
sample or small batch of samples is collected and analyzed. Sequential testing does not
imply that a time dynamic phenomenon is being monitored. Volume 2, which discusses
ground water, considers sampling and analysis over time. Sequential sampling is
performed during steady state conditions and is used only to reduce the sample size
required for a decision.
Sequential sampling procedures terminate when enough evidence is
obtained to either accept or reject the null hypothesis. Thus, sequential tests can respond
8-10
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CHAPTER 8: TESTING PERCENTILES AND PROPORTIONS USING SEQUENTIAL
SAMPLING
quickly to very clean or very contaminated sites and in these cases require far less sampling
than the conventional methods discussed in Chapter 7. In situations where contaminant
concentrations at the site are only marginally different from the cleanup standard, the
sequential procedure can be expected to require more samples until the sample size
approaches the sample size required for the fixed sample procedure.
The procedure and some computational aspects of sequential testing are
discussed. Sequential sampling and testing are treated separately from the discussions of
other similar evaluation methods because of the distinct differences in sampling approach.
However, the chapter makes comparisons with Chapter 7 procedures for sample size
determination.
8-11
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9. SEARCHING FOR HOT SPOTS
As suggested by Barth el M- (1989), it may be desirable to verify cleanups
by documenting that no hot spots could be identified provided that a sampling plan was
used that had an acceptably large probability of finding hot spots. This chapter discusses
how to conduct a valid sampling program to search for hot spots and the conclusions that
can be drawn regarding the presence or absence of hot spots. In general, the methods in
this chapter are presented so they are easy to understand and apply.
This chapter first describes the literature that discusses methods for locating
hot spots. This will provide the interested reader with an avenue into discussions regarding
specific applications and details. A simple approach, useful under two different sampling
designs, is summarized. This enables application of selected basic methods without having
to obtain and study the literature.
9.1 Selected Literature that Describes Methods for Locating Hot
Spots
Table 9.1 lists several references regarding hot spots and their identification.
Gilbert (1987) offers a general overview of the hot spot searching technique, including
example applications of the simplest methods as well as more advanced application.
Zirschky and Gilbert (1984) offer applications of these methods at hazardous waste sites.
9.2 Sampling and Analysis Required to Search for Hot Spots
9.2.1 Basic Concepts
The term hot spot is used frequently in discussions regarding the sampling
of hazardous waste sites, yet there is no universal definition of what constitutes a hot spot.
The methods in this chapter model hot spots as localized elliptical areas with concentrations
in excess of the cleanup standard. Hot spots are generally small relative to the area being
sampled. The hot spot must either be considered a volume defined by the projection of the
surface area through the soil zone that will be sampled or a discrete horizon within the soil
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CHAPTER 9: SEARCHING FOR HOT SPOTS
zone that will be sampled. When a sampl aken and the concentration of a chemical
exceeds the cleanup standard for that chemk .t is concluded that the sampling position in
the field was located within a hot spot.
Table 9.1 Selected references regarding the methodologies for identifying hot spots at
waste sites
Gilbert, R.O. (1982) Some Statistical Aspects of Finding Hot
Spots and Buried Radioactivity
Gilbert, R.O. (1987) Statistical Methods for Environmental
Pollution Monitoring
Parkhurst, D.F. (1984) Optimal Sampling Geometry for Hazardous
Waste Sites
Singer, D.A. (1972) Elipgrid: A Fortran IV Program for
Calculating the Probability of Success in
Locating Elliptical Targets with Square,
Rectangular, and Hexagonal Grids
Singer, D.A. (1975) Relative Efficiencies of Square and
Triangular Grids in the Search for Elliptically
Shaped Resource Targets
USEPA (1985) Verification of PCB Spill Cleanup by
Sampling and Analysis
Zirschky, J. and Detecting Hot Spots at Hazardous Waste
Gilbert, R.O. (1984) Sites
Hot spot location techniques involve systematic sampling from a grid of
sampling points arranged in a particular pattern. If a systematic sample is taken and none
of the samples yield concentrations in excess of the cleanup standard, then no hot spots
were found and the site is judged clean. However, what does this mean in terms of the
chances of contaminant residuals remaining at the site? Since all of the soil could not be
sampled, hot spots could still be present. An important question is: What level of certainty
is there that no hot spots exist at the site? The answer to this question requires that several
other questions be answered. For example:
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CHAPTER 9: SEARCHING FOR HOT SPOTS
• What shape hot spot is of concern: circular, fat-elliptical, skinny-
elliptical?
• What is the length of the longest axis of the hot spot: 1 cm, 10m,
or 100 m?
• What sampling pattern was used: square, triangular?
• What was the distance between sampling points in the grid: 0.1 m,
1 m, 100 m?
If these questions are answered; a sampling plan implemented; and no hot
spots are found, it is possible to conclude with an associated level of confidence that no hot
spots of a certain size are present. In general, there is a smaller chance of detecting hot
spots and less confidence in conclusions when:
• Hot spot sizes of interest become smaller,
• Hot spots are likely to be narrow;
• A square rather than a triangular grid is used; and
• The spacing between grid points is increased.
Figure 9.1 illustrates a sampling grid with hot spots of various sizes and shapes. Hot spots
B and D were "hit" with sampling points and hot spots A anc C were missed.
If one of the samples results in concentrations in excess of the applicable
cleanup standard, a hot spot has been identified. The conclusion is that the site is not clean.
The normal, reasonable action will be to continue remediation in the areas identified as hot
spots. However, once these locations are remediated, another systematic sample, over the
entire site, with a new random start must be taken in order to conclude with confidence that
no hot spots of a specified size and shape are present at the site. Because of this
requirement it may be advisable, after identifying the presence of a single hot spot, to
continue less formal searching followed by treatment throughout the entire sample area.
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CHAPTER 9: SEARCHING FOR HOT SPOTS
9.2.2
Choice of a Sampling Plan
The sampling plan requires no calculations. Instead all the information is
obtained from tables. Figure 9.2 describes the grid spacing definition for two grid
configurations and how to calculate the parameter for defining the ellipse shape.
The sampling plan for hot spot detection can be approached in three ways.
The three factors listed in Table 9.2 control the performance of a hot spot detection
sampling episode. Two of these factors are chosen and fixed. The third factor is
determined by the choice of the first two factors. Table A. 11 includes information that
allows choice of two factors while providing the resulting third parameter.
Figure 9.1 A Square Grid of Systematically Located Grid Points with Circular and
Elliptical Hot Spots Superimposed
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CHAPTER 9: SEARCHING FOR HOT SPOTS
Figure 9.2 Grid Spacing and Ellipse Shape Definitions for the Hot Spot Search Table
in Appendix A (Table A. 11)
Square
Trianglar
G
• Sample point location
G Grid spacing
Ellipse Shape
Long axis
Short axis
Length of the long axis = L
Length of the short axis = S
S/L = Ellipse Shape (ES)
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CHAPTER 9: SEARCHING FOR HOT SPOTS
Table 9.2 Factors controlling the design of a hot spot search sampling plan
GRID PATTERN
Spacing between sample points.
Geometry of the sample point locations.
HOT SPOT SHAPE
The length of the long axis of the hot spot.
FALSE POSITIVE RATE
An acceptable false positive probability; concluding that no hot spots are present
when there is at least one present.
Three examples are offered that describe the approaches to sample plan
design. First, suppose that the size of the hot spot is known or assumed. The shape and
size of the hot spots that are being searched for are elliptical with a long axis of L = 5 m and
a short axis of S = 2 m. Therefore, the ellipse shape, ES = S/L = 2/5 = 0.4. In addition,
the sampling team decided that they could accept no more than a 10 percent chance of
missing a hot spot if a hot spot was present the false positive rate. A triangular grid pattern
was chosen because the probability of detection was better with an elliptical shaped hot spot
and the sampling team had experience laying out a triangular coordinate system. The
triangular grid pattern in Table A.I 1 is entered for a value of ES = 0.4 across the top and a
false positive rate of a = .10 or less within the table. This corresponds to an L/G value of
0.9, since L = 5, and 0.9 = 5/G, G = 5.55, or a grid spacing in a triangular pattern of 5.6
m. The density of the grid spacing must be evaluated with respect to the size of the sample
area.
Once the grid spacing density has been determined it is important to
estimate for the sample area how many samples would be required given sampling intervals
of 5.6 m on a triangular grid as specified in Figure 9.2. The following method in Box 9.1
can be used to approximate the sample size necessary when area and grid interval are
known.
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CHAPTER 9: SEARCHING FOR HOT SPOTS
Box 9.1
Approximating the Sample Size
When Area and Grid Interval Are Known
n = A/G2
Where: n = total number of samples required
A = size of the area to be sampled (in the same units of
measures as G)
G = grid spacing as defined in Figure 9.2
For example, suppose that a lagoon will be sampled that is 45 m by 73 m.
This is a 3285-m2 lagoon. The number of samples required is:
3285 m2 / (5.6 m)2 = 104
On the other hand, a lagoon that is 17 m by 20 m or 340-m2 would require
the following number of samples:
340m2/(5.6m)2 =11
If the size of the area is relatively small, then the level of confidence
described above may be affordable and acceptable. However, if the area is large and the
number of samples required excessive, alternatives are available.
For example, a second approach can be considered that limits the samples
from the 3285-m2 lagoon. Suppose that no more than 40 samples are available because of
cost, time, or logistics. The minimum grid spacing is estimated to be:
3285m2/G2<40
G>9.1m
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CHAPTER 9: SEARCHING FOR HOT SPOTS
The question becomes: what probability statement can be made with a 9.1 m
grid spacing searching for the same size hot spot. Review of Table A. 11 indicates that if
L/G = 5/9.1 = .55, and ES = S/L = 2/5 = 0.4 then .33 < a < .63. Reference to Gilbert
(1987) indicates that a -.55. This means first that the cost has been reduced by taking 64
fewer samples from the 3285-sq. m lagoon. This was accomplished by increasing the grid
spacing from 5.6 m to 9.1 m. However, the sampling cost reduction increases the chance
of missing contamination. Specifically, the chance of missing a hot spot and concluding
that the site is clean when a hot spot with an ES of 0.4 and a long axis of 5 m is really
present increases from 10 percent to 55 percent when the sample size is reduced from 104
to 40. If this chance is unacceptably high, there is a third approach.
The third approach involves fixing the false positive rate, fixing the sample
size or grid spacing, and searching for hot spots that are larger or have a different shape.
Suppose it could be safely assumed that the hot spot of concern was not as elliptically
shaped or as skinny as the ellipse with an ES = 0.4. Instead, the ES = L = 4/5 = 0.8. The
long axis remained at 5 m, but the short axis doubled from 2 m to 4 m. For the grid
spacing of G = 5.6 m, the L/G = 5/5.6 = 0.9. From Table A.ll it is clear that the false
positive rate is low, a = .01. A willingness to search for a larger sized or fatter shaped hot
spot improves the performance of the hot spot search technique from a 10 percent false
positive rate to a less than 1 percent false positive rate with no increase in sample intensity
above 104 samples.
9.2.3 Analysis Plan
The analysis is straightforward. Establish a grid of sampling points as
described in Chapter 5 with density and pattern determined using the methods in section
9.2.2 and Figure 9.2. If one of the chemical measurement results exceeds the cleanup
standard then conclude that a hot spot has been found and the completion of remediation
can not be verified. If none of the samples exceeds the cleanup standard, assume that the
site is clean and conclude with the level of confidence associated with the sampling plan
that it is unlikely a hot spot exists at the site.
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9.3 Summary
Hot spots are generally defined as relatively small, localized, elliptical areas
with contaminant concentrations in excess of the cleanup standard. Samples that are taken
and found to exceed the cleanup standard are defined as being located within a hot spot.
Locating hot spots involves systematic sampling from a grid of sampling
points arranged in a specific pattern. Several questions must be answered to conclude with
a level of confidence that no hot spots of a certain size are present:
• What size hot spot is of concern?
• What sampling pattern was used?
• What was the distance between sampling points in the grid?
The sampling plan for hot spot detection is guided by the dimensions and
shape of the grid pattern, the hot spot shape of interest, and the false positive rate. The
information needed is contained in Table A.ll. Three illustrative examples present
sampling plans for these cases:
• The size of the hot spot and false positive rate are known or
assumed, and the grid spacing/sample size is determined;
• Sample size/grid spacing and ellipse shape are fixed, and the false
positive rate is determined;
• The false positive rate and sample size or grid spacing are fixed, and
hot spot size is determined.
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10. THE USE OF GEOSTATISTICAL TECHNIQUES
FOR EVALUATING THE ATTAINMENT OF CLEANUP
STANDARDS
The science of gee-statistics involves the analysis of spatially correlated data.
There are several features of geostatistics that are important to any potential user.
• Geostatistical methods provide a powerful and attractive method for
mapping spatial data. Geostatistical methods provide for
interpolation between existing data points that have been collected in
a spatial array and a method for estimating the precision of the
interpolation.
• Geostatistical methods are complicated mathematically, and the
procedures required to contour an area cannot be practically
implemented by hand and calculator.
• New users of geostatistics will need to devote time to understanding
the basic approach, concepts and the unique vocabulary associated
with geostatistical methods.
• To help explore applications, PC-based geostatistical computer
software is now readily available to the EPA community (USEPA,
1988). However, some preliminary study should be completed, and
then the software can be used as an educational and exploratory tool
to better understand how geostatistical methods perform.
This chapter:
• Explains fundamental concepts regarding geostatistical methods;
• Offers a point of departure into the literature that will provide more
details;
• Discusses which cleanup scenarios can benefit the most from a
geostatistical evaluation;
• Describes which geostatistical methods are most appropriate for
evaluating the completion of cleanup; and
• Lists software available for implementing geostatistical methods.
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10.1 Background
10.1.1 What Is Geostatistics and How Does It Operate?
Many view the science of geostatistics in a broad context as the use of
statistical methods applied to the geographic and geological sciences. Others refer to
geostatistics as a science that strictly applies to the family of methods that enable the
analysis, evaluation, or characterization of spatially correlated data. Regardless, kriging
and variogram modeling are primary tools of geostatistical analysis.
In simple terms, a geostatistical analysis can be viewed as a two-step
process. First, a model is developed that predicts the spatial relationship between a location
where a concentration will be estimated and the existing data obtained from sample points
which are various distances away from the location. Existing data points nearer to the
location will tend to be closely related and have a large influence on the estimate, and points
far away will tend to be less related and, therefore, impose less influence. This relationship
function, which describes how influential nearby existing data will be, is modeled and
called a variogram or semi-variogram.
Figure 10.1 illustrates the general form of a standard or typical variogram
model. The X or horizontal axis measures the distance between sample points. The
vertical or Y axis measures the degree of relationship between points. When there is little
distance between points it is expected that there will be little variability between points. As
the distance between points increases, the difference or variability between points
increases. The form of this relationship depends on what the variogram modeler knows
about characteristics of the site and the data, and what assumptions are reasonable to make
regarding spatial relationships at the site.
The second step of the geostatistical analysis is kriging. This involves
estimating chemical concentrations for each point or block in the area of concern. For each
point to be estimated, the surrounding points provide a weighted contribution to the
estimate. The weightings are determined by using the variogram model, the location of the
point that is being estimated, and the proximity of other nearby data values, enabling
chemical concentration estimation for locations within the sample area that were not
sampled and therein lies the true value of a geostatistical analysis. In addition to estimates
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THE ATTAINMENT OF CLEANUP STANDARDS
of the concentration, kriging allows estimation of the precision associated with the estimate.
If the surrounding data are highly variable, or if the closest data points are relatively far
away, the precision may be low.
Figure 10.1 An Example of an Empirical Variogram and a Spherical Variogram Model
2.5
2 -
Square of the
Difference 1.5 -
Between Points
at Distance h
Apart 1
0.5 -
• •
0 10 20 30 40 50 60
Distance h
70 80 90 100
Kriging provides concentration and associated precision estimates across the
site at all possible points or blocks within the site. The concentration and precision
estimates can then be graphically contoured across the site. Maps, plotting concentration
isopleths, are the final product. In addition, a precision map that provides isopleths of the
kriging variance or some function of the kriging variance is generated. These sorts of maps
are illustrated in Flatman and Yfantis (1984) and USEPA (1987b).
As a slightly more technical conclusion to this section, consider the
following discussion of kriging and variogram modeling. Kriging is an interpolation
method based on a weighted moving average where the weights are assigned to samples in
a way that minimizes the variance associated with interpolated estimates. The estimation
variance is computed as a function of the spatial relationship model known as the
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THE ATTAINMENT OF CLEANUP STANDARDS
variogram, the location of the sampling points relative to each other, and to the location
being estimated (USEPA, 1988).
10.1.2 Introductory Geostatistical References
The discussion in section 10.1.1 is intended to provide a simple notion of
how kriging operates. The next level of understanding requires that the reader consult
specialized literature and a practicing geostatistician. Several general discussions of
geostatistics are available and are listed in Table 10.1. In addition to the references in Table
10.1, there is a wide range of refereed journal literature supporting the theory and
application of geostatistics. Finally, the EPA's Environmental Monitoring Systems
Laboratory in Las Vegas, Nevada (EMSL-LV), includes a group of researchers specializing
in the application of geostatistical methods to environmental monitoring problems. The
group is responsible for the development of the GEOEAS software referenced in Tables
10.1, 10.3, and Box 10.1. In addition, the EMSL-LV has produced refereed literature and
funded university researchers. The researchers operating under cooperative agreement with
the EMSL-LV have produced a series of reports that also provide insights regarding
application of geostatistical methods to environmental problems.
10.2 Soils Remediation Technology and the Use of Geostatistical
Methods
As recognized in Chapter 1, there are a variety of soils remediation
methods. Geostatistical methods have many applications, and are especially useful during
remedial investigations where a primary objective is to characterize the extent of
contamination. Geostatistical techniques, particularly specialized kriging techniques
referenced in section 10.3, will also be useful for evaluating certain soils remediation
efforts.
This section provides guidance that will help in deciding whether
geostatistical methods are most appropriate for use under different types of soils
remediation methods. The reader should note that in cases where geostatistical approaches
are not necessarily called for if they are used then the geostatistical approaches will give the
same result as the classical approaches used throughout the document. The choice of
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CHAPTER 10: THE USE OF GEOSTATISTICAL TECHNIQUES FOR EVALUATING
THE ATTAINMENT OF CLEANUP STANDARDS
whether to use geostatistical data analysis and evaluation methods depends on the physical
arrangement of the cleanup system, its mode of operation, and the effect that the
remediation technology will have on the soils environment
Table 10.1 Selected introductory and advanced references that introduce and discuss
geostatistical concepts
INTRODUCTORY
Clark, I.
(1979)
Davis, J.C.
(1986)
USEPA
(1987a)
USEPA
(1987b)
USEPA
(1988)
ADVANCED
Journel, A.G. and
Huijbregts, C.H.
(1978)
David, M.
(1984)
Verly, G.
(1984)
Practical Geostatistics
Statistical and Data Analysis in Geology
Data Quality Objectives for Remedial
Response Activities: Development Process
Data Quality Objectives for Remedial
Response Activities: Example
Scenario RI/FS Activities at a Site with
Contaminated Soils and Ground Water
GEOEAS (Geostatistical Environmental
Assessment Software) User's Guide
Mining Geostatistics
Geostatistical Ore Reserve Estimation
Geostatistics for Natural Resources
Characterization
10.2.1
Removal
Soils remediation may involve either permanent or temporary removal of
soils. Soils may be permanently transported away from the site. However, soils may be
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THE ATTAINMENT OF CLEANUP STANDARDS
temporarily removed to undergo treatment and then returned. In these situations,
geostatistical methods may be useful for efficiently directing the removal.
For example, although a single three-dimensional geostatistical study or a
series of two-dimensional geostatistical studies at various depth horizons would have been
preferred during the site characterization phase, this may not have been done. Therefore,
during removal, as the surface material is skimmed off and new layers are exposed, the
areas of greatest concentration may change. This changing condition with depth could be
characterized via a geostatistical study. However, there are practical requirements in this
situation. In order to be most successful and efficient onsite rapid chemical analysis and
geostatistical data analysis must take place.
A geostatistical analysis will permit the estimation of concentrations between
the sampled points and allow prediction of which areas should and should not be removed.
As horizons are reached that are below the cleanup standard, they can be avoided. The
sampling program and data analysis have the ability to operate in a useful and constructive
way that will help direct the cleanup effort and minimize costs. Indicator and probability
kriging, discussed below, are ideal candidates for evaluating areas that are above and below
cleanup standards.
10.2.2 Treatment Involving Homogenization
Many soils remediation technologies homogenize the soils media. This
occurs during soils fixation or chemical modification when soil mixers are used to blend
materials with the soil media. Sampling this type of process could occur at a discharge
point of the mixing apparatus. In this instance, samples may be taken, placed in canisters,
and allowed to solidify or undergo the chemical reaction. After an established period of
time, the media in the canisters can be extracted and the leachate concentrations tested
relative to the appropriate cleanup standard. Samples may also be acquired onsite after the
mixing equipment such as banks of steam injection augers, has passed over each location
that has been pre-selected for sampling to test attainment of the cleanup standard.
Regardless of how the sampling is conducted, from a statistical perspective,
there are several anticipated results. First, there should be reduction in the variability of
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THE ATTAINMENT OF CLEANUP STANDARDS
chemical contaminants across the site. One way of viewing the effect of treatment is that it
has reduced the magnitude of the large values in the distribution of values at the site. This
can be thought of as "bringing in" the upper tail of the distribution such that the distribution
becomes less lognormal-like and more bell-shaped or normal-like. In practical terms, this
is the same as reducing the variance. In short, the site should be more homogenous, and
there should be a more random behavior of contaminants across the site. Finally, the
degree of spatial relationship will be reduced because of the homogeneity. That is, a point
1m away from a point of concern should be just as similar as a point 50m away.
Because of these anticipated results, geostatistical applications are less
useful when remediation results in a homogenization. First, it is likely that the spatial
correlation has been grossly disturbed by the treatment process. Also, sampling may occur
at a discharge point or in association with the operation of a mixing device, rather than in a
spatial framework. If the treatment technology is operating as anticipated, the effectiveness
will be high; the extractable concentrations will be low relative to the cleanup standard and
will have a small variance. Under this scenario, a sampling and analysis program as
discussed in Chapters 4-9 can be implemented with a minimum of samples to verify the
effectiveness of treatment rather than require an elaborate geostatistical study.
10.2.3 Flushing
There is a family of soils remediation techniques that can be thought of as
flushing methods. They rely on surface manifolds attached to extraction wells on one end
and to suction pumps on the other end. These systems can be designed to remove
infiltrated water, artificial liquids, or air. In either case, the liquid or air is the media used
to transport the contaminants. The liquid can flush out soluble contaminants, and the air
can flush out volatile contaminants. Often extraction systems have to contend with both air
and liquid.
A system of extraction wells, screened at appropriate depths, are installed
across the contaminated area. Each of the wells is linked by a manifold or piping system,
which is connected to a pump system that provides the vacuum for withdrawal. The
dynamics of removal differ depending on many factors including the makeup of the soils
media, the degree of infiltration, the surrounding ground water system, the type of
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contaminants, and the media that is being extracted. Regardless of these factors, there is a
tendency with these systems to create zones of influence around each well. Depending on
how long the system has operated and many other factors, the zone of influence will have
much higher or lower concentrations than the surrounding area. The site will then tend to
have a series of zones of influence across the site. Some of the zones will overlap; others
will be irregular in shape because of irregularities in the soils media or the turning on and
off of banks of wells in the system.
Geostatistical methods are generally not practical for characterizing sites that
have been remediated using flushing technologies because of the highly complex structure
associated with the many overlapping zones of influence around each of the extraction
wells that are distributed across the site. Although it may be technically possible to
geostatistically model this structure, many samples would be required to provide sufficient
resolution of the many complex gradients across the site.
However, it may be that by the time verification sampling is conducted the
zones of influence are not likely to be apparent and the site is anticipated to be uniformly
below the relevant cleanup standard. If extraction has been completed to this point and
there is interest in characterizing the concentration profile across the site, a geostatistical
study may be warranted. However, the main objective at this stage will normally not be to
characterize the extent of the remaining contaminants that have concentrations below the
cleanup standard, but instead to simply document that the site has met its cleanup
objectives.
10.3 Geostatistical Methods that Are Most Useful for Verifying the
Completion of Cleanup
As previously described, there are many methods of variogram modeling
and many approaches to kriging. Each technique requires different assumptions or has
advantages in a particular application. The traditional forms of kriging allow estimates of
central tendency and variance throughout an area. These forms, which include simple,
ordinary, and universal kriging, require different assumptions regarding the model used to
make the kriging estimates. These types of kriging methods can be used to describe the
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extent of contamination remaining and the precision associated with the concentration
estimates. In this way, the traditional forms of kriging are useful for cleanup verification.
In addition to the more common methods of kriging described above, there
are several forms of nonparametric kriging, such as indicator and probability kriging, that
have been developed relatively recently and are directly useful for evaluating attainment of
cleanup standards. These types of kriging are the best forms of kriging for demonstrating
that a particular area is less than a cleanup standard, and unlike the conventional forms of
kriging, these forms are distribution-free.
Indicator kriging operates basically by kriging data that have been
transformed into zeros and ones. For each measurement, the value is transformed to a zero
if the measurement was less than or equal to the cleanup standard, and transformed to a one
if the measurement was greater than the cleanup standard. The data set of zeros and ones is
then used to produce kriging estimates of the probability of exceeding the cleanup standard
across the site. It then becomes possible to produce a map that contours the probabilities of
having concentrations in excess of the cleanup standard. Extensions of indicator kriging to
probability kriging allow the development of false positive and false negative error maps.
That is, probability kriging can be used to estimate where there is a chance that an area that
appears to be clean is actually dirty and where there is a chance that areas that might be
indicated dirty are actually clean. Figures 10.2, 10.3, and 10.4 were adapted from the
probability kriging study of a lead smelter (Flatman el M-, 1985).
Although these forms of kriging are directly applicable to the cleanup
verification problem, they are relatively new methods. Nonparametric and Baysian kriging
are currently an active area of research. Understanding and application of these kriging
methods will require a substantial investment of time and study. Table 10.2 offers some
initial references.
10.4 Implementation of Geostatistical Methods
As mentioned in the introduction to this chapter, kriging cannot be
conveniently or practically implemented without a computer and the appropriate software.
Even with the appropriate software, it will take an interested individual a considerable
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investment of time to understand the jargon and mathematics associated with geostatistical
methods.
Table 10.2 Introductory references for indicator, probability, and nonparametric global
estimation kriging
Buxton, B.E. Geostatistical Construction of Confidence
(1985) Intervals for Global Reserve Estimation
Isaaks, E.H. Risk Qualified Mappings for Hazardous
(1984) Waste Sites: A Case Study in Distribution
Free Geostatistics
Journel, A.G. Nonparametric Estimation of Spatial
(1983) Distributions
Sullivan, J. Conditional Recovery Estimation Through
(1984) Probability Kriging
In many cases, it is best to recognize the power and utility of a geostatistical
study and acquire, or at least have available, the expertise of a geostatistician. An
alternative is to obtain a first-level familiarity with the methodology and then use a
softwarepackage along with example data sets to explore the practical dynamics and effects
of different modeling decisions.
The EMSL-LV has recently produced the first version of a geostatistical
software package that provides a convenient environment for exploring the application of
geostatistical methods to hazardous waste site sampling problems (USEPA, 1988). The
software operates on a PC and is provided in an executable form. It is entirely in the public
domain and can be obtained using the information in Box 10.1.
The software does not support indicator and probability kriging at this point;
however, as the software undergoes development, it is anticipated that these will be added.
There are other geostatistical software packages available in the public
domain that can be purchased. Table 10.3 lists some examples and sources of software.
10-10
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CHAPTER 10: THE USE OF GEOSTATISTICAL TECHNIQUES FOR EVALUATING
THE ATTAINMENT OF CLEANUP STANDARDS
Figure 10.2 Contour Map of the Probability in
Percent of Finding the Value of 1,000
ppm or a Larger Value
10000
7500
sooo
2500
Figure 10.4 Contour Map of the Probability in
Percent of a False Negative in the
Remedial Action Areas and the 1,000
ppm Contour Line
2500
5000
7500
10000
Figure 10.3 Contour Map of the Probability in
Percent of a False Positive in the
Remedial Action Areas and the 1,000
ppm Contour Line
10OOO
750O
5000
2500
N
I
10000
7500
5000
2500
l
2500
5000
7500 1000O
2500 5000 7500 10000
10-11
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CHAPTER 10: THE USE OF GEOSTATISTICAL TECHNIQUES FOR EVALUATING
THE ATTAINMENT OF CLEANUP STANDARDS
Box 10.1
Steps for Obtaining Geostatistical Software from EMSL-LV
The software:
• Operates on a PC;
• Is provided in an executable form;
• Is entirely in the public domain; and
• Can be obtained by writing to:
Evan Englund (GEO-EAS)
USEPA, EMSL-LV, BAD
P.O. Box 93478
Las Vegas, NV 89193-3478
PLEASE, YOU MUST DO THE FOLLOWING TO OBTAIN
THE SOFTWARE!:
1) PRE-FORMAT ALL DISKETTES.
2) SEND ENOUGH DISKETTES FOR 3 MEGABYTES
OF STORAGE AS FOLLOWS:
TYPE NUMBER
5 1/4" 1.2MB 3
51/4" 360KB 9
3 1/2" 1.44MB 3
3 1/2" 722KB 6
10.5 Summary
Geostatistical methods provide a method for mapping spatial data that
enables both interpolation between existing data points and a method for estimating the
precision of the interpolation.
Geostatistical applications normally involve a two-step process. First, a
spatial correlation model is developed that predicts how much spatial relationship exists
among sample points various distances apart
10-12
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CHAPTER 10: THE USE OF GEOSTATISTICAL TECHNIQUES FOR EVALUATING
THE ATTAINMENT OF CLEANUP STANDARDS
Table 10.3 Selected geostatistical software
Program
Geo-EAS SYSTEM
(Geostatistical Environ-
mental Assessment
Software)
USGS Statpac Programs
TOXIPAC
GEOBASE and GEORES
Source
See Box 10.1
COGS (Computer Oriented Geological Society)
P.O. Box 1317
Denver, Colorado 80201-1317
Geostat Systems International, Inc.
P.O.Box 1193
Golden, CO 80402
GEOMATH
4860 Ward Road
Wheat Ridge, CO 80033
The second step, kriging, involves estimating chemical concentrations for
locations within the sample area that were not sampled. For each point to be estimated, the
surrounding points provide a weighted contribution to the estimate based on the variogram
model, the location of the point being estimated, and the proximity of other nearby data
values. Kriging also allows estimation of the precision associated with the estimated
chemical concentrations. Maps that plot concentration isopleths are the final product of the
geostatistical analysis.
Geostatistical methods have many applications in soil remediation
technology, especially when the extent of contamination needs to be characterized. This
chapter includes guidance to help decide whether geostatistical data analysis and evaluation
methods are appropriate for use with three types of soils remediation activities: removal,
treatment involving renamed homogenization, and flushing.
Of the many methods of variogram modeling and many approaches to
kriging, each requires different assumptions or has advantages in certain applications. The
10-13
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CHAPTER 10: THE USE OF GEOSTATISTICAL TECHNIQUES FOR EVALUATING
THE ATTAINMENT OF CLEANUP STANDARDS
traditional forms of kriging, including simple, ordinary, and universal, are primarily useful
for characterization but may also be used for cleanup verification. Nonparametric,
indicator, and probability kriging are the best forms for demonstrating probabilistically that
an area is less than a cleanup standard and, unlike the traditional forms, are distribution-
free.
Geostatistical techniques referred to in the chapter will need in-depth study
by the intended user before being applied. References are provided to help familiarize the
interested reader. Because kriging cannot be conveniently or practically implemented
without a computer and the appropriate software, a first-level familiarity with the
methodology along with use of a software package is a practical way of exploring example
applications and data sets. EPA has developed the first version of a geostatistical software
for the novice, available by following instructions at the end of this chapter.
10-14
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Remedial Response Activities: Development Process, Washington D.C.,
March 1987 (EPA 540/G-87/003).
U.S. Environmental Protection Agency. 1987b. Data Quality Objectives For
Remedial Response Activities: Example Senarip RI/FS Activities at a Site with
Contaminated Soils and Ground Water, Washington D.C., March 1987 (EPA
540/G-87/004).
U.S. Environmental Protection Agency. 1987c. EPA Journal, The New
Superfund: Protecting People and the Environment, Washington D.C., Vol.
13, No. 1.
U.S. Environmental Protection Agency. 1987d. Surface Impoundment Clean
Closure Guidance Manual [draft], Washington D.C., March 1987, (OSWER
Directive 9476.0-8.C).
U.S. Environmental Protection Agency. 1987e. Test Methods for Evaluating Solid
Waste, Washington D.C., November 1987 (SW-846).
U.S. Environmental Protection Agency. 1988. GEOEAS (Geostatistical
Environmental Assessment Software User's Guide, EMSL-Las Vegas, NV,
(EPA 600/4-88/033).
Verhagen, E. J. H. 1988. "Review of Thermal and Extraction Soil Treatment Plants in
the Netherlands," National Institute of Public Health and Environmental
Protection (RTVM) Laboratory for Waste and Emission Research (LAE), In K.
Wolf, W. J. vanden Brink, F. J. Colon (eds.), Contaminated Soil '88, 797-
808, Kluwer Academic Publishers.
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Geostatistics for Natural Resources Characterization, Parts 1 and 2, D. Reidel
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BIB-10
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BIBLIOGRAPHY
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BIB-11
-------
-------
APPENDIX A: STATISTICAL TABLES
Table A. 1 Table of t for selected alpha and degrees of freedom
Use alpha to determine which column to use based on the desired parameter, ti-
-------
APPENDIX A: STATISTICAL TABLES
Table A.2 Table of z for selected alpha or beta
Use alpha or beta to determine which row to read. Obtain the z value from the zi-
-------
APPENDIX A: STATISTICAL TABLES
Table A.3 Table of k for selected alpha, PQ, and sample size where alpha = 0.10
(i.e., 10%)
Use alpha to determine which table to read. The k for use in a tolerance interval test is at
the intersection of the column with the specified PO and the row with the sample size, n.
n
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
50
70
100
200
500
infinity
0.25
5.842
2.603
1.972
1.698
1.540
1.435
1.360
1.302
1.257
1.219
1.188
1.162
1.139
1.119
1.101
1.085
1.071
1.058
1.046
1.035
1.025
1.016
1.007
1.000
0.992
0.985
0.979
0.973
0.967
0.942
0.923
0.894
0.857
0.825
0.779
0.740
0.674
PO
0.1
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
1.624
1.598
1.559
1.511
1.470
1.411
1.362
1.282
0.05
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
2.041
2.010
1.965
1.909
1.861
1.793
1.736
1.645
0.010
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
2.833
2.793
2.735
2.662
2.601
2.514
2.442
2.326
A-3
-------
APPENDIX A: STATISTICAL TABLES
Table A.4 Table of k for selected alpha, PQ, and sample size where alpha = 0.05
(i.e., 5%)
n
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
50
70
100
200
500
infinity
0.25
11.763
3.806
2.618
2.150
1.895
1.732
1.618
1.532
1.465
1.411
1.366
1.328
1.296
1.268
1.243
1.220
1.201
1.183
1.166
1.152
1.138
1.125
1.114
1.103
1.093
1.083
1.075
1.066
1.058
1.025
0.999
0.960
0.911
0.870
0.809
0.758
0.674
PO
0.1
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
1.732
1.697
1.646
1.581
1.527
1.450
1.385
1.282
0.05
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
2.167
2.125
2.065
1.990
1.927
1.837
1.763
1.645
0010
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
2.995
2.941
2.862
2.765
2.684
2.570
2.475
2.326
A-4
-------
APPENDIX A: STATISTICAL TABLES
Table A.5 Table of k for selected alpha, PQ, and sample size where alpha = 0.01 (i.e.,
1%)
n PO
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
50
70
100
200
500
infinity
0.25
58.939
8.728
4.715
3.454
2.848
2.491
2.253
2.083
1.954
1.853
1.771
1.703
1.645
1.595
1.552
1.514
1.481
1.450
1.423
1.399
1.376
1.355
1.336
1.319
1.303
1.287
1.273
1.260
1.247
1.195
1.154
1.094
1.020
0.957
0.868
0.794
0.674
0.1
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
2.105
2.085
2.065
2.047
2.030
1.957
1.902
1.821
1.722
1.639
1.524
1.430
1.282
0.05
131.426
17.370
9.083
6.578
5.406
4.728
4.258
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
2.269
2.153
2.056
1.923
1.814
1.645
0.010
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
3.334
3.249
3.125
2.974
2.850
2.679
2.540
2.326
A-5
-------
APPENDIX A: STATISTICAL TABLES
Table A.6 Sample sizes required for detecting a scaled difference tau of the mean from
the cleanup standard for selected values of alpha and beta*
1
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
P = 0.20
a
0.10
1,798
449
200
112
72
50
37
28
22
18
15
12
11
9
8
7
6
6
5
4
0.05
2,470
618
274
154
99
69
50
39
30
25
20
17
15
13
11
10
9
8
7
6
0.01
4,020
1,005
447
251
161
112
82
63
50
40
33
28
24
21
18
16
14
12
11
10
3 = 0.10
a
0.10
2,621
655
291
164
105
73
53
41
32
26
22
18
16
13
12
10
9
8
7
7
0.05
3,422
856
380
214
137
95
70
53
42
34
28
24
20
17
15
13
12
11
9
9
0.01
5,213
1,303
579
326
209
145
106
81
64
52
43
36
31
27
23
20
18
16
14
13
*See section 6.1 and Box 6.3 for definitions of alpha (a), beta ((3), and tau (T).
A-6
-------
APPENDIX A: STATISTICAL TABLES
Table A.7 Sample size required for test for proportions with a = .01 and (3 = .20, for
selected values of PO and PI
PO
0.005
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
0.002
4,519
1,131
407
241
169
129
103
86
73
64
56
0.005
3,383
659
333
217
158
124
101
85
73
64
Value of P under the alternative hypothesis, PI
0.010 0.020 0.030 0.040 0.050 0.060
1,676
577
323
218
162
127
104
88
75
2,649
823
434
281
202
156
125
104
3,593
1,058
538
340
240
182
144
4,515
1,287
639
396
276
207
5,416
1,509
737
451
311
6,295
1,726
833
504
0.070 0.080
7,155
1,938 7,994
925 2,145
0.090
8,813
Value of P under the alternative hypothesis, PI
PO
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.010
218
75
43
28
21
16
12
10
8
6
0.020
434
104
53
34
24
18
14
11
9
7
0.050
311
103
55
35
25
19
14
11
9
0.100
469
140
71
43
30
22
16
13
0.150
606
171
83
50
33
24
17
0.200
723
197
93
54
36
25
0.250
819
217
101
58
37
0.300
894
233
106
60
0.350 0.400
950
243 986
109 248
0.450
1,001
A-7
-------
APPENDIX A: STATISTICAL TABLES
Table A.8 Sample size required for test for proportions with a = .05 and P = .20, for
selected values of PO and PI
Value of P under the alternative hypothesis, PI
PO
0.005
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
0.002
2,623
633
222
129
90
68
55
45
38
33
29
0.005
1,990
373
185
119
86
67
54
45
39
34
0.010
986
332
183
122
90
70
57
48
41
0.020
1,588
485
252
162
116
88
71
58
0.030
2,171
630
317
198
139
105
83
0.040
2,741
772
380
234
162
120
0.050
3,297
910
441
268
183
0.060
3,840
1,044
500
301
0.070 0.080
4,371
1,175 4,889
557 1,303
0.090
5,394
Value of P under the alternative hypothesis, PI
PO
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.010
122
41
23
15
11
8
7
5
4
3
0.020
252
58
29
19
13
10
7
6
5
4
0.050
183
59
31
20
14
10
8
6
5
0.100
282
83
41
25
17
12
9
7
0.150
368
103
49
29
20
14
10
0.200
440
119
56
33
21
15
0.250
500
132
61
35
23
0.300
548
142
64
37
0.350
583
149
67
0.400 0.450
606
153 616
A-8
-------
APPENDIX A: STATISTICAL TABLES
Table A.9 Sample size required for test for proportions with a = .10 and (3 = .20, for
selected values of PQ and PI
Value of P under the alternative hypothesis, PI
PO
0.005
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
0.002
1,822
426
145
84
58
44
35
29
24
21
19
0.005
1,398
254
124
79
57
44
35
29
25
22
0.010
693
229
125
82
60
47
38
32
27
0.020
1,133
341
175
111
79
60
48
39
0.030
1,559
447
223
138
97
72
57
0.040
1,975
551
269
164
113
84
0.050
2,381
652
314
189
129
0.060
2,778
750
357
214
0.070
3,166
846
399
0.080 0.090
3,544
940 3,913
Value of P under the alternative hypothesis, PI
PO
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.010
82
27
15
10
7
5
4
3
3
2
0.020
175
39
20
12
9
6
5
4
3
2
0.050
129
41
22
14
10
7
5
4
3
0.100
202
59
29
18
12
9
6
5
0.150
265
73
35
21
14
10
7
0.200
318
85
40
23
15
11
0.250
363
95
44
25
16
0.300
398
103
47
26
0.350
424
108
48
0.400 0.450
441
111 449
A-9
-------
APPENDIX A: STATISTICAL TABLES
Table A. 10 Tables for determining critical values for the exact binomial test, with a =
0.01, 0.05, and 0.10
To determine the critical value, select the column for PQ specified in the attainment
objectives, reading down the column finding the first number greater than the sample size,
n, move up one row, read ra:n. the critical value, in the leftmost column.
Alpha =.01
PQ, Proportion of contaminated soil units under
rft;n
0
1
2
3
4
5
6
7
8
9
10
0,01
459
662
838
1001
1157
1307
1453
1596
1736
1874
2010
0,02
228
330
418
499
577
652
725
796
866
935
1003
0.03
152
219
277
332
383
433
482
529
576
622
667
0.04
113
164
207
248
287
324
360
396
431
465
499
0.05
90
130
165
198
229
259
288
316
344
371
398
0.06 0.07
75 64
108 92
137 117
164 140
190 162
215 184
239 204
263 225
286 244
309 264
331 283
Alpha = .05
PQ, Proportion of contaminated soil units under
r«:n
0
1
2
3
4
5
6
7
8
9
10
0.01
299
473
628
773
913
1049
1182
1312
1441
1568
1693
0,02
149
236
313
386
456
523
590
655
719
782
845
0.03
99
157
208
257
303
348
392
436
478
521
562
0.04
74
117
156
192
227
261
294
326
358
390
421
0.05
59
93
124
153
181
208
234
260
286
311
336
0.06 0.07
49 42
78 66
103 88
127 109
150 129
173 148
195 167
217 185
238 203
259 221
280 239
Alpha =.10
the null hypothesis
0.08
56
81
102
122
142
160
178
196
213
230
247
the null
0.08
36
58
77
95
112
129
146
162
178
193
209
PQ, Proportion of contaminated soil units under the null
ra;n
0
1
2
3
4
5
6
7
8
9
10
0,01
230
388
531
667
798
926
1051
1175
1297
1418
1538
0,02
114
194
265
333
398
462
525
587
648
708
768
0.03
76
129
176
221
265
308
349
390
431
471
511
0.04
57
%
132
166
198
230
262
292
323
353
383
0.05
45
77
105
132
158
184
209
234
258
282
306
0.06 0.07
38 32
64 55
88 75
110 94
132 113
153 131
174 149
194 166
215 184
235 201
255 218
0.08
28
48
65
82
98
114
130
145
160
175
190
0.09
49
71
91
109
126
142
158
174
189
204
219
0.10
44
64
81
97
113
127
142
156
170
183
197
0.11
40
58
74
88
102
116
129
141
154
166
178
0,12
37
53
67
81
93
106
118
129
141
152
163
hypothesis
0.09
32
51
68
84
100
115
129
143
158
172
185
0.10
29
46
61
76
89
103
116
129
142
154
167
0.11
26
42
56
69
81
93
105
117
128
140
151
0.12
24
38
51
63
74
85
96
107
117
128
138
hypothesis
0.09
25
42
58
73
87
101
115
129
142
156
169
0.10
22
38
52
65
78
91
104
116
128
140
152
0.11
20
34
47
59
71
83
94
105
116
127
138
0.12
19
31
43
54
65
76
86
96
106
116
126
A-10
-------
APPENDIX A: STATISTICAL TABLES
Table A. 11 The false positive rates associated with hot spot searches as a function of
grid spacing and hot spot shape
False Positive Rates
ES
Triangular Grid Pattern
Square Grid Pattern
L/G
0.1
0.3
0.5
0.7
0.9
1.0
0.1
0.3
0.5
0.7
0.9
1.0
1.0
.95
.66
.08
.00
.00
.00
.97
.72
.21
.00
.00
.00
.80
.96
.74
.27
.00
.00
.00
.97
.77
.38
.02
.00
.00
.60
.97
.80
.44
.08
.00
.00
.98
.80
.54
.16
.00
.00
.40
.98
.86
.63
.33
.10
.04
.98
.88
.69
.42
.17
.08
.20
.98
.93
.82
.65
.47
.37
.98
.94
.85
.70
.53
.44
.10
.99
.96
.91
.83
.72
.66
.99
.97
.92
.85
.76
.70
Source: These tables were extracted from the graphs in Gilbert (1987).
A-ll
-------
APPENDIX A: STATISTICAL TABLES
Figure A. 1 Power Curves for a = 1 %
0.9-.
0.8-.
Probability 0.7 •.
of Deciding
the Site
Attains the 0-5 • •
Cleanup
Standard
Cs or Po
A
B
r*
•— F
True parameter as a fraction of Cs or
Parameters for the
Power Curves
a =
P =
Hl =
Pl-
A J
.01
.20
.19*Cs
.19*P0
B
.01
.20
.36*Cs
.36*P0
Powei
C
.01
.20
.53*Cs
.53*P0
r Curve:
D
.01
.20
.65*Cs
.65*P0
E
.01
.20
.75*Cs
.75*P0
F
.01
.20
.81*Cs
.81*P0
Approximate sample sizes for simple random sampling for testing the parameters indicated
Power Curve:
Parameters being tested | A j B | C I D I E I F
Mean
with cv(data) = .5
with cv(data) = 1
withcv(data) = 1.5
Proportions
P0 = 10%
Non-parametric test
Tolerance Intervals
P0 = 20%
Non-parametric test
Tolerance Intervals
4
16
35
101
16
46
12
7
25
56
179
38
81
26
12
46
103
356
89
161
60
21
82
185
670
184
301
122
41
161
362
1353
399
607
261
70
279
626
2384
728
1066
473
Note: a = saying the site is clean when dirty, (3 = saying the site is dirty when clean, 1-(J = saying
the site is clean when clean.
A-12
-------
APPENDIX A: STATISTICAL TABLES
Figure A.2 Power Curves for a = 5%
1 -
0.9 ..
0.8 ..
Probability 0.7 • .
of Deciding 0.6 ..
V Cs or Po
the Site
Attains the
Cleanup
Standard
0.5 ..
0.4 ..
0.3 ••
0.2 -.
0.1 ••
0
A
B
«-« F
Parameters for the
Power Curves
True parameter as a fraction of Cs or PQ.
Power Curve:
I A | B | C | D
a =
P =
Hi —
P ^
.05
.20
.25*Cs
.25*P0
.05
.20
.43*Cs
.43*P0
.05
.20
.57*Cs
.57*P0
.05
.20
.69*Cs
.69*P0
.05
.20
.77*Cs
.77*P0
.05
.20
.84*Cs
.84*P0
Approximate sample sizes for simple random sampling for testing the parameters indicated
Power Curve:
Parameters being tested I A | B I C I D I E | F
Mean
with cv(data) = .5
with cv(data) = 1
withcv(data) =1.5
Proportions
P0 = 10%
Non-parametric test
Tolerance Intervals
P0 = 20%
Non-parametric test
Tolerance Intervals
4
11
25
70
14
32
10
5
20
43
136
33
62
23
9
34
76
257
69
116
47
17
65
145
520
151
234
100
30
117
264
975
296
438
193
61
242
544
2065
649
925
420
Note: a = saying the site is clean when dirty, (i = saying the site is dirty when clean, l-(3 = saying the site
is clean when clean.
A-13
-------
APPENDIX A: STATISTICAL TABLES
Figure A.3 Power Curves for a = 10%
Probability
of Deciding
the Site
Attains the
Cleanup
Standard
Parameters for the
Power Curves
True parameter as a fraction of Cs or PQ.
Power Curve:
I A | B I C | D
A
B
'-* F
a =
P =
Hi =
PI =
.10
.20
.30*Cs
.30*P0
.10
.20
.46*Cs
.46*P0
.10
.20
.60*Cs
.60*P0
.10
.20
.71*Cs
.71*P0
.10
.20
.79*Cs
.79*P0
.10
.20
.85*Cs
.85*P0
Approximate sample sizes for simple random sampling for testing the parameters indicated
Power Curve:
Parameters being tested I A I B I C I D I E I F
Mean
with cv(data) = .5
with cv(data) = 1
withcv(data) =1.5
Proportions
P0 = 10%
Non-parametric test
Tolerance Intervals
P0 = 20%
Non-parametric test
Tolerance Intervals
3
10
21
57
13
26
9
4
16
35
108
28
50
19
8
29
64
214
60
97
40
14
54
121
430
129
194
85
26
103
231
849
264
382
172
51
201
452
1706
544
764
351
Note: a = saying the site is clean when dirty, P = saying the site is dirty when clean, 1-(J = saying the site
is clean when clean.
A-14
-------
APPENDIX A: STATISTICAL TABLES
Figure A.4 Power Curves for a = 25%
CsorPo
Probability
of Deciding
the Site
Attains the
Cleanup
Standard
— A
— B
Q
~- F
True parameter as a fraction of Cs or
Parameters for the
Power Curves
a =
P =
M| =
P —
A
.25
.20
.19*Cs
.19*P0
B
.25
.20
.40*Cs
.40*P0
Powe
C
.25
.20
.54*Cs
.54*P0
r Curve:
D
.25
.20
.76*Cs
.76*P0
E |
.25
.20
.83*Cs
.83*P0
F
.25
.20
.87*Cs
.87*P0
Approximate sample sizes for simple random sampling for testing the parameters indicated
Power Curve:
Parameters being tested I A I B I C I D I E I F
Mean
with cv(data) = .5
with cv(data) = 1
with cv(data) =1.5
Proportions
P0 = 10%
Non-parametric test
Tolerance Intervals
P0 = 20%
Non-parametric test
Tolerance Intervals
2
7
15
38
11
18
8
3
11
25
73
22
34
15
5
20
45
147
46
67
30
10
40
90
315
100
142
66
20
80
179
654
212
294
138
34
136
306
1143
375
513
242
Note: a = saying the site is clean when dirty, p = saying the site is dirty when clean, 1-|3 = saying the site
is clean when clean.
A-15
-------
-------
APPENDIX B: EXAMPLE WORKSHEETS
The worksheets in this appendix have been completed to serve as an example in
understanding the forms and making the necessary calculations.
The numbers and situations represented on the worksheets are hypothetical.
The example situation consists of a waste site that is divided into two sample areas. The
first uses random sampling to test the mean and proportion of contaminated soil for two
chemicals. The second uses stratified sampling to test the mean and proportion of
contaminated soil for one chemical. In this example, the different chemicals, labeled only
Chemical #1, #2, and #3, are tested in the different sample areas; in most applications, the
same chemicals will be tested in all or most of the sample areas. Two statistical parameters
are tested for two chemicals to show how to complete the worksheet under a variety of
conditions.
The following figures show the 1) the parameters being tested, 2) a hypothetical
map of the site, and 3) the sequence in which the worksheets are completed. The
worksheets for sample area #2 follow those for sample area #1 in this appendix.
In actual use, these worksheets would be accompanied by additional
documentation such as maps, background material, justification of different choices, field
notes, and copies of the results as reported by the laboratory.
B-l
-------
APPENDIX B: EXAMPLE WORKSHEETS
Figure B. 1 Example Worksheets: Parameters to Test in Each Sample Area and Map of the Site
Waste Site
Old XYZ Disposal Site
a
a.
a
Sample area #1
Field used for storing batteries
Random sampling
Sample area #2
Old Lagoon Area
Stratified sampling
e
I
Mean concentration
of Chemical #1
Proportion of soil
contaminated with
Chemical #1
Mean concentration
of Chemical #3
a>
6
Proportion of soil
contaminated with
Chemical #2
Proportion of soil
contaminated with
Chemical #3
Waste Site
Sample area#l
Field used for
storing batteries
Stratum #1
Center of Lagoon
^ Stratum #2 Edge of Lagoon
^ .......
B-2
-------
APPENDIX B: EXAMPLE WORKSHEETS
Figure B.2 Example Worksheets: Sequence in Which the Worksheets Are Completed
Worksheet 1
Define sample areas
Worksheet 2
Sample area #1
Random Sample
Attainment Objectives
I
Worksheet 3
Sample Design
and Analysis Plan
Worksheet 4
Sample Size for
testing the mean
Worksheet 6
Chemical #1
Data Sheet and
Calculations
Worksheet 7
Analysis and
Inference
Worksheet 5
Sample Size for
testing proportion
Worksheet 6
Chemical #2
Data Sheet and
Calculations
Worksheet 7
Analysis and
Inference
Worksheet 2
Sample area #2
Stratified Sample
Attainment Objectives
I
Worksheet 3
Sample Design
and Analysis Plan
Worksheet 8
Define Strata
Worksheet 9
Chemical #3
Sample Sizes
for the mean
Worksheet 10
Chemical #3
Sample Sizes
for proportions
Worksheet 11
Field Sample Sizes
Worksheet 12
Stratum #1
Chemical #3
Data Sheet and
Calculations
Worksheet 12
Stratum #2
Chemical #3
Data Sheet and
Calculations
Worksheet 13
Chemical #3
Analysis and
Inference for
the mean
Worksheet 14
Chemical #3
Analysis and
Inference for
proportions
B-3
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 1 Sample Areas
See Section 3.1 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE: Former XYZ Disposal Site
Sample
Area
Number Describe the sample areas and the reasons for treating each area separately.
g
1
Field used for storing batteries
Old Lagoon Area
Use the Sample Area Number (g) to refer on other sheets to the sample areas described above.
Attach a map showing the sample areas within the waste site.
Date Completed: EXAMPLE Completed by EXAMPLE
Use additional sheets if necessary. Page of.
Continue to WORKSHEET 2
B-4
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 2 Attainment Objectives
See Section 3.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 1. Field used for storing batteries
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Sample Collection Procedures to be used (attach separate sheet if necessary):
For example: .5 liter scoop of soil from the top 5 cm of soil, etc.
Probability of mistakenly declaring the site clean = a =
.05
Chemical
to be tested
Number
j
Chemical
Name
Cleanup
Standard
(with units)
Cs
Parameter to test:
Mean
Yes/No
Proportion
PO
1
2
Chemical #1
Chemical #2
20
2
Yes
No
25%
50%
Secondary Objectives/ Other purposes for which the data is to be collected:
Use the Chemical Number (j) to refer on other sheets to the chemical described above.
Attach documentation describing the lab analysis procedure for each chemical.
Date Completed: EXAMPLE Completed by EXAMPLE
Use additional sheets if necessary. Page of
Continue to WORKSHEET 3
B-5
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 3 Sampling Design and Analysis Plan
See Chapter 4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 1. Field used for storing batteries
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Sample Design:
Simple Random Sample
Systematic Random Sample
Stratified Sample
Chemical Comments on the Prob of Type n error Alternate Parameter value
to be tested Sample Design and Chance of concluding the for the specified P
Number [2] Analysis Plan site is dirty when it is clean Mean Proportion
j p m P!
1
2
.20
.20
15
5%
20%
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by
EXAMPLE
Page
of
Continue to WORKSHEET 4 for random or systematic sampling and WORKSHEET 8 for stratified sampling.
B-6
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 4 Sample Size for Testing the Mean Using Simple Random
Sampling
See Section 6.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
If the mean concentration is not to be tested for this chemical, continue to WORKSHEET 5
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [ 1 ]
SAMPLE AREA: 1. Field used for storing batteries
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
From z -Table, Appendix A
Probability of mistakenly declaring the site clean [2] = a
zl-a =
Chemical
Number [3]
[2]
From
z table
Appendix A
Zl.p
Calculate:
[2]
Cs
[3]
A=
Cs-m
nj=.
1
.20
.842
20
15
49
4.042
Column Maximum, Max n; =
12.12
12.12
Fraction of samples expected to be analyzable = R =
Max n[
T, '' = B =
R
B rounded up = Sample Size for Testing Means = nf =
Date Completed: EXAMPLE
Use additional sheets if necessary.
Continue to WORKSHEET 5
Completed by
EXAMPLE
Page
of
B-7
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 5 Sample Size for Testing Proportions Using Simple Random
Sampling
See Section 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
If the mean concentration is not to be tested for this chemical, continue to WORKSHEET 6
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 1. Field used for storing batteries
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
From z -Table, Appendix A
Probability of mistakenly declaring the site clean [2] = a
Calculate:
.05
Chemical
Number [3]
[2]
j P
From
z table
[2]
PO
[3]
Pi
I
2
.20
.20
.842
.842
.25
.50
.05
.20
.712
.823
.184
.337
Column Maximum, Max HJ =
20.06
14.93
20.06
Fraction of samples expected to be collectible = B =
Max nj _ p _
B ~~
C rounded up = Sample Size for Testing Proportions =
Date Completed: EXAMPLE
Use additional sheets if necessary.
Continue to WORKSHEET 6
Completed by
EXAMPLE
Page
of
B-8
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 6 Data Calculations for a Simple Random Sample, by Chemical
See Section 7.3 or 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 1. Field used for storing batteries
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
1. Chemical #1
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Maximum Sample Size from Worksheets 4 and 5 = Sample Size =
Cleanup standard[2] = Cs
Method Detection Limit: =
Concentration used when it is reported as less than the method detection limit =
22
20
Was the Reported Is \[ Greater
Sample Concentration Concentration than Cs?
Sample Sample Collectible? If Corrected for l = Yes
Number ID 0 = No Collectible Detection Limit 0 = No
i l = Yes x; y
(Xi)2
1
2
3
4
5
6
7
8
9
10
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
Total from previous page
Column Totals:
1
1
1
1
1
1
1
0
1
1
14.7
17.7
22.8
2.9
35.5
28.6
4.9
#N/A
5.2
17.2
14.7
17.7
22.8
4
35.5
28.6
4.9
0
5.2
17.2
A 9
B 150.6
0
0
1
0
1
1
0
0
0
0
C 3
216.09
313.29
519.84
16
1260.25
817.96
24.01
0
27.04
295.84
D 3490.32
A = n
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by
C = r
EXAMPLE
Page
of
Complete WORKSHEET 6 for other chemicals or continue to WORKSHEET 7
B-9
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 6 Data Calculations for a Simple Random Sample, by Chemical
See Section 6.3 or 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [ 1 ]
1. Field used for storing batteries
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
1. Chemical #1
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Maximum Sample Size from Worksheets 4 and 5 = Sample Size =
Cleanup standard[2] = Cs
Method Detection Limit: =
Concentration used when it is reported as less than the detection limit =
22
20
Sample
Number
Was the Reported
Sample Concentration
Sample Collectible? If
ID 0 = No Collectible
l = Yes
Is Xj Greater
Concentration than Cs?
Corrected for 1 = Yes
Detection Limit 0 = No
1
11
12
13
14
15
16
17
18
19
20
2243
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
Total from previous page
Column Totals:
1
1
1
1
1
1
1
1
1
1
1
14.7
10.9
7.7
12.4
15.2
14.9
10.2
17.4
11.6
12.4
19.1
14.7
10.9
7.7
12.4
15.2
14.9
10.2
17.4
11.6
12.4
19.1
0
0
0
0
0
0
0
0
0
0
0
9
A 19
150.6
B 282.4
3
216.09
118.81
59.29
153.76
231.04
222.01
104.04
302.76
134.56
153.76
364.81
3490.3
C 3
D 5335.2
A = n
B =
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by
C = r
EXAMPLE
Page
of
Complete WORKSHEET 6 for other chemicals or continue to WORKSHEET 7
B-10
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 6 Data Calculations for a Simple Random Sample, by Chemical
See Section 6.3 or 7.3 in "Statistical Methods for Evaluating the Attainment of Superfund Cleanup Standards",
Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 1. Field used for storing batteries
CHEMICAL:
NUMBER(J) AND DESCRIPTION [2]
1. Chemical #1
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Maximum Sample Size from Worksheets 4 and 5 = Sample Size =
Cleanup standard [2] = Cs
Method Detection Limit: =
Concentration used when it is reported as less than the detection limit =
Was the Reported
Sample Concentration
Is Xj Greater
Concentration than Cs?
22
20
Sample Sample Collectible? If Corrected for l = Yes
Number ID 0 = No Collectible Detection Limit 0 = No
i l = Yes Xj yj (xO2
21
22
2263
2264
Total from previous page
Column Totals:
1
1
8.9
16.5
8.9
16.5
19
282.4
A 21
B 307.8
0
0
3
79.21
272.25
5335.2
C 3
D 5686.6
A = n
C = r
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by
EXAMPLE
Page
of
Complete WORKSHEET 6 for other chemicals or continue to WORKSHEET 7
B-ll
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 7 Inference for Simple Random Samples by Chemical
Sec Section 6.3 or 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 1. Field used for storing batteries
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
1. Chemical #1
.05
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Testing the Mean [2] a =
[2]
Number of Collectible Samples [6] = n =
Total of the concentration measurements [6] = V xj = B =
Total for x^ [6] =
. B _
Mean concentration = — = x =
Standard Deviation of the Data = 'V ——j— = s =
Degrees of Freedom for s = n - 1 = df =
Standard Error for the Mean concentration = -7=- =
Vn
_ §
Upper One Sided Confidence Interval = x + ti_a>(jf -r=-
If \i\j(,< Cs then circle Clean, otherwise circle Dirty:
__ Based on the mean concentration, the sample area is:
=
=
=
=
-
1
1 «
DC""
20
21
307.8
5687
14.66
7.67
20
1.73
1.67
17.54
Clean Dirty
Testing Percentiles [2] P0 =
[4 or 5] zi.a =
Number of Samples with Concentration Greater than Cs [6] = r =
Proportion of Contaminated Samples = — = p =
.25
Standard Error for the Proportion = "V n = sp =
Test Statistic = p + zj.a 'V ^ „ =
If UL < PQ then circle Clean, otherwise circle Dirty:
Based on the proportion of contaminated samples, the sample area is:
1.645
3
.143
.0764
.298
Clean Dirty
Date Completed: EXAMPLE
Complete WORKSHEET 7 for other chemicals
Completed by
EXAMPLE
Page_
of
B-12
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 6 Data Calculations for a Simple Random Sample, by Chemical
See Section 6.3 or 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [ 1 ]
SAMPLE AREA: 1. Field used for storing batteries
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
2. Chemical #2
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Maximum Sample Size from Worksheets 4 and 5 = Sample Size =
Cleanup standard [2] = Cs
Method Detection Limit: =
Concentration used when it is reported as less than the detection limit =
22
20
1.2
1.2
Sample
Number
Was the Reported
Sample Concentration
Sample Collectible? If
ID 0 = No Collectible
l = Yes
Is Xj Greater
Concentration than Cs?
Corrected for l = Yes
Detection Limit 0 = No
Xi
(Xi)2
1
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
1
0
1
1
1.2
2.1
0.9
0.1
0.5
0.3
0.3
#N/A
1.9
8.3
1.2
2.1
1.2
1.2
1.2
1.2
1.2
0
1.9
8.3
0
1
0
0
0
0
0
0
0
1
Total from previous page
Column Totals:
B
19.5
C 2
D
A = n
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by
C = r
EXAMPLE
Page
of
Complete WORKSHEET 6 for other chemicals or continue to WORKSHEET 7
B-13
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 6 Data Calculations for a Simple Random Sample, by Chemical
See Section 6.3 or 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [Ij
SAMPLE AREA: 1. Field used for storing batteries
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
2. Chemical #2
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Maximum Sample Size from Worksheets 4 and 5 = Sample Size =
Cleanup standard [2] = Cs
Method Detection Limit: =
Concentration used when it is reported as less than the detection limit =
22
20
1.2
1.2
Sample
Number
Was the Reported
Sample Concentration
Sample Collectible? If
ID 0 = No Collectible
l=Yes
Is X{ Greater
Concentration than Cs?
Corrected for l = Yes
Detection Limit 0 = No
Xi
(Xi)2
11
12
13
14
15
16
17
18
19
20
1
1
1
1
1
1
1
1
1
1
0.5
0.7
2.2
0.7
1.7
2.3
0.3
3.7
0.1
5.6
1.2
1.2
2.2
1.2
1.7
2.3
1.2
3.7
1.2
5.6
0
0
1
0
0
1
0
1
0
1
Total from previous page
Column Totals:
A 19 I
19.5
B 41
2
C 6
D
A = n
B
•I"
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by
C = r
EXAMPLE
Page
of
Complete WORKSHEET 6 for other chemicals or continue to WORKSHEET 7
B-14
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 6 Data Calculations for a Simple Random Sample, by Chemical
See Section 6.3 or 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards." Volume 1 _
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION 11]
SAMPLE AREA: 1. Field used for storing batteries
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
2. Chemical #2
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Maximum Sample Size from Worksheets 4 and 5 = Sample Size =
Cleanup standard[2] = Cs
Method Detection Limit: =
Concentration used when it is reported as less than the detection limit =
Sample
Number
Was the Reported Is xj Greater
Sample Concentration Concentration than Cs?
Sample Collectible? If Corrected for l = Yes
ID 0 = No Collectible Detection Limit 0 = No
1 = Yes x; y.
22
20
21
22
Total from previous page
Column Totals:
1
1
1.3
1.8
1.3
1.8
19
A 21
41
0
0
6
B 44.1
C 6
D
A = n B = Yx: C = r D = Y(xi)2
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by EXAMPLE
Page of.
Complete WORKSHEET 6 for other chemicals or continue to WORKSHEET 7
B-15
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 7 Inference for Simple Random Samples by Chemical
See Section 6.3 or 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [1J
1. Field used for storing batteries
CHEMICAL:
NUMBERQ) AND DESCRIPTION [2]
2. Chemical #2
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Testing the Mean [2]
[2]
Number of Collectible Samples [6] = n =
Total of the concentration measurements [6] =
Total for Xi2 [6] =
B _
Mean concentration = — = x =
Standard Deviation of the Data = *\l "^ = s =
Degrees of Freedom for s = n - 1 = df =
tl-cc.df =
c
Standard Error for the Mean concentration = -:=•=
Vn
n = |iija=
Upper One Sided Confidence Interval = x + tj_a
If M-ua< Cs then circle Clean, otherwise circle Dirty:
Based on the mean concentration, the sample area is:
=
=
=
r=
r =
a=
.05
2
21
44.1
Clean Dirty
Testing Percentiles [2] P0
[4 or 5] Zj.a
Number of Samples with Concentration Greater than Cs [6] = r
Proportion of Contaminated Samples = — = p
Standard Error for the Proportion =
Test Statistic = p +
= U =
.5
1.645
.286
.0986
.465
If UL < PQ then circle Clean, otherwise circle Dirty:
Based on the proportion of contaminated samples, the sample area is:
Clean Dirty
Date Completed: EXAMPLE
Complete WORKSHEET 7 for other chemicals
Completed by
EXAMPLE
Page
of
B-16
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 2 Attainment Objectives
See Section 3.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 1. Old Lagoon Area
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Sample Collection Procedures to be used (attach separate sheet if necessary):
For example: One foot soil core, 2 inches in diameter, etc.
Probability of mistakenly declaring the site clean = a = .05
Chemical
to be tested Chemical
Number Name
j
Cleanup
Standard
(with units)
Cs
Parameter to test:
Mean
Yes/No
Proportion
PO
1
Chemical #3
30
Yes
25%
Secondary Objectives/ Other purposes for which the data is to be collected:
Use the Chemical Number (j) to refer on other sheets to the chemical described above.
Attach documentation describing the lab analysis procedure for each chemical.
Date Completed: EXAMPLE Completed by EXAMPLE
Use additional sheets if necessary. Page of
Continue to WORKSHEET 3
B-17
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 3 Sampling Design and Analysis Plan
See Chapter 4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [ 1 ]
SAMPLE AREA: 1. Old Lagoon Area
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Sample Design: | | simple Random
Systematic Random Sample
Stratified Sample
X
Chemical Comments on the Prob of Type n error Alternate Parameter value
to be tested Sample Design and Chance of concluding the for the specified (3
Number [2] Analysis Plan site is dirty when it is clean Mean Proportion
j P Hi Pi
3
Use non-parametric estimation of
proportions
.20
15
10%
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by
EXAMPLE
Page
of
Continue to WORKSHEET 4 for random or systematic sampling and WORKSHEET 8 for stratified sampling.
B-18
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 8 Definition of Strata Within Sample Area
See Section 4.1 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NTJMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 2. Old Lagoon Area
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Stratum Describe the stratum and the reason
Number for interest in this area
Volume =
Surface Area * Sample depth
vh
Vh
1
2
Center of Lagoon
Edge of Lagoon
Total Volume = Zvn =
141,000 cu. ft.
94,000 cu. ft.
235,000 cu. ft.
.60
.40
Use the Stratum Number (h) to refer on other worksheets to the stratum described above
Attach a map showing the stratum within the sample area.
Date Completed: EXAMPLE Completed by EXAMPLE
Use additional sheets if necessary. Page of
Continue to WORKSHEET 9
B-19
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 9 Desired Sample Sizes for Testing the Mean Using Stratified
Sampling, by Chemical
See Section 6.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1 ]
SAMPLE AREA: 2. Old Lagoon Area
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
Chemical #3
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
From z -Table, Appendix A
Probability of mistakenly declaring the site clean [2] = cc .05
For the Cleanup Standard = Cs = 30
Probability of mistakenly declaring the site dirty [3] = (} = .20
(
Proportion
of Sample
Stratum Area in
Number[8] Stratum[8]
h Wh
1
2
.6
.4
If the true concentration is [3] = |ii = 15
Calculate:
Stratum
Standard
Deviation
35
22
C/-i_i.,_
— coiun
B
o 2
J Cs - |ii lz
1 1 =
Unit
Sample
Cost
1
1
in Sum =
21
8.8
29.8
: A ~ 36.38
zl-a =
2
,,=
Desired final
sample size
nhd =
VcJT
21
8.9
= C/A =
B .819
VciT
17.2
7.21
A =
1.645
.842
Calculation
check
%
25.64
10.74
36.38
Date Completed: EXAMPLE
Use additional sheets if necessary.
Continue to WORKSHEET 10
Completed by
EXAMPLE
Page
of
B-20
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 10 Desired Sample Sizes for Testing a Percentile Using Stratified
Sampling, by Chemical
See Section 7.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 2. Old Lagoon Area
NUMBERS) AND DESCRIPTION [2]
CHEMICAL: 3. Chemical #3
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Probability of mistakenly declaring the site clean [2] = a
Proportion Exceeding Cleanup Standard [2] = PQ =
Probability of mistakenly declaring the site dirty [3] = (3 =
If the true proportion is [2] = Pj =
Calculate:
P P
P0'P1
.05
.25
.2
.1
.00364
From z -Table, Appendix A
zl-a =
Z1-P =
Proportion Proportion Stratum
of Sample of dirty Standard Unit
Stratum Area in Samples Deviation Sample
Number[3] Stratum[3] &K Cost
Desired final
sample size
Calculation
check
Wh
Ph
"h
1
2
.6
.4
.145
.036
.352
.184
1
1
C = Column Sum =
B = C
.211
.074
.285
Divided
by A =
/A =
B 78.3
.211
.074
16.54
6.76
A
Date Completed: EXAMPLE
Use additional sheets if necessary.
Continue to WORKSHEET 11
Completed by
EXAMPLE
Page
of
B-21
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 11 Desired Sample Sizes for All Chemicals and Parameters
See Section 6.4 or 7.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 2. Old Lagoon Area
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Enter the Desired Sample Size (nncj)by Stratum for each combination of chemical
and parameter to be tested, from WORKSHEETS 9 and 10.
Desired Sample Size
Stratum number h
12345
by stratum and:
chemical
Chemical 3
Worksheet 9
Chemical 3
Worksheet 10
Chemical
Worksheet
Chemical
Worksheet
Chemical
Worksheet
Maximum n^
for all Chemicals
and Parameters
nhdmax
Fraction of
Collectible
Field Samples Rf,
A nhdmax
A~ Rh
A Rounded up to the
Next Integer = n^f ;
the field sample size
17.2
16.54
17.2
.95
18.1
19
7.2
5.75
7.2
.95
7.6
8
Date Completed: EXAMPLE
Use additional sheets if necessary.
Continue to WORKSHEET 12
Completed by
EXAMPLE
Page
of
B-22
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 12 Data Calculations, by Stratum and Chemical
See Section 6.4 or 7.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 2. Old Lagoon Area
NUMBER AND DESCRIPTION [3]
STRATUM: 1. Center of Lagoon
NUMBER(j) AND DESCRIPTION [2]
CHEMICAL: 3. Chemical #3
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Samples Size [11] = nnf =
Cleanup standard[2] = Cs =
Method Detection Limit: =
Concentration used when no concentration is reported =
Was the Reported
Sample Concentration
Sample Sample Collectible? If
Number ID 0 = No Collectible
i l = Yes
Is Xj Greater
Concentration than Cs?
Corrected for l = Yes
Detection Limit 0 = No
Yi
19
30
(xi)
1
2
3
4
5
6
7
8
9
10
9301
9302
9303
9304
9305
9306
9307
9308
9309
9310
Total from previous page
Column Totals:
1
1
0
1
1
1
1
1
1
1
17
40
#NA
19
31
0
7
10
15
21
A 9
17
40
0
19
31
5
7
10
15
21
0
1
0
0
1
0
0
0
0
0
B 165
C 2
289
1600
0
361
961
25
49
100
225
441
D 4051
= A
Completed by
Date Completed: EXAMPLE
Use additional sheets if necessary.
Complete WORKSHEET 12 for other chemicals or to WORKSHEET 13
rh=C
EXAMPLE
Page
of
B-23
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 12 Data Calculations, by Stratum and Chemical
See Section 6.4 or 7.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER® AND DESCRIPTION [1]
SAMPLE AREA: 2. Old Lagoon Area
NUMBER AND DESCRIPTION [3]
STRATUM: 1. Center of Lagoon
NUMBER(j) AND DESCRIPTION [21
CHEMICAL: 3. Chemical #3
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Samples Size [11] = nhf =
Cleanup standard[2] = Cs =
Method Detection Limit: =
Concentration used when no concentration is reported =
19
30
Was the Reported Is xj Greater
Sample Concentration Concentration than Cs?
Sample Sample Collectible? If Corrected for l = Yes
Number ID 0 = No Collectible Detection Limit 0 = No
i 1 = Yes x;
(Xi)
11
12
13
14
15
16
17
18
19
9311
9312
9313
9314
9315
9316
9317
9318
9319
1
1
1
1
1
0
1
1
1
18
27
12
28
94
#NA
13
22
23
18
27
12
28
94
0
13
22
23
0
0
0
0
1
0
0
0
0
324
729
144
784
8836
0
169
484
529
Total from previous page
Column Totals:
A 17
165
2
B 402
C 3
405
D 16050
nh = A
rh =
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by
EXAMPLE
Page
of
Complete WORKSHEET 12 for other chemicals or to WORKSHEET 13
B-24
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 12 Data Calculations, by Stratum and Chemical
See Section 6.4 or 7.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume. 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 2. Old Lagoon Area
NUMBER AND DESCRIPTION [3]
STRATUM: 2. Edge of Lagoon
NUMBER(j) AND DESCRIPTION [2]
CHEMICAL: 3. Chemical #3
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Samples Size [11] = %f =
Cleanup standard[2] = Cs =
Method Detection Limit: =
Concentration used when no concentration is reported =
19
30
Was the Reported Is xj Greater
Sample Concentration Concentration than Cs?
Sample Sample Collectible? If Corrected for l = Yes
Number ID 0 = No Collectible Detection Limit 0 = No
i 1 = Yes Xj yj
(Xi)"
1
2
3
4
5
6
7
8
9
9320
9321
9322
9323
9324
9325
9326
9327
9328
Total from previous page
1
1
0
1
1
1
1
1
1
9
16
#NA
0
0
5
8
10
9
9
16
0
5
5
5
8
10
9
1
0
0
0
0
0
0
0
0
0
81
256
0
25
25
25
64
100
81
Column Totals:
8
nn = A
67
|c o |D
657
rh =
Completed by
Date Completed: EXAMPLE
Use additional sheets if necessary.
Complete WORKSHEET 12 for other chemicals or to WORKSHEET 13
EXAMPLE
Page.
of
B-25
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 13 Sample Area Analysis for the Mean Using Stratified
Sampling, by Chemical
See Section 6.4 in " Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1 ]
SAMPLE AREA: 2. Old Lagoon Area
NUMBERS) AND DESCRIPTION [21
CHEMICAL: 3. Chemical #3
Stratum
Number[3] [3]
h Wh
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
[12]
% = ,
[12] [12]
- =B_ , ^ D-xh2A
Whxh F =
(nh-D
1
2
0.6
0.4
17
8
23.65
8.38
Grand Totals:
408.99
13.70
14.19
3.35
G 17.54
8.661
0.274
H 8.935
4.6883
0.0107
I 4.6990
[2] cc =
[2] Cs =
Mean concentration = G = "x =
H2
Degrees of Freedom = -r- Rounded to an integer = df =
tl-a,df =
Standard Error for the Mean concentration =
Upper One Sided Confidence Interval = x +
.05
30
17.54
17
1.74
2.99
22.74
If |iua< Cs men circle Clean, otherwise circle Dirty:
Based on the mean concentration, the sample area is:
Clean Dirty
Date Completed: EXAMPLE
Use additional sheets if necessary.
Continue to WORKSHEET 14
Completed by
EXAMPLE
Page
of
B-26
-------
APPENDIX B: EXAMPLE WORKSHEETS
WORKSHEET 14 Sample Area Analysis for a Percentile Using Stratified
Sampling, by Chemical
See Section 7.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Former XYZ Disposal Site
NUMBER(g) AND DESCRIPTION [1]
SAMPLE AREA: 2. Old Lagoon Area
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Stratum
Number[3] [3]
[9]
[9]
[9]
p=Whph
1
2
Grand 1
0.6
0.4
17
8
0.18
0.00
totals:
0.1453
0
0.1059
0
0.00308
0
G 0.1059
H .00308
[2]
[4 or 5]
Proportion of Contaminated Samples = G = p =
Standard Error for the Proportion =
Test Statistic = p+ sp zi_a = T =
If T < PQ then circle Clean, otherwise circle Dirty:
Based on the mean concentration, the sample area is:
» =
,
.25
1.645
.106
.055
.197
Clean Dirty
Date Completed: EXAMPLE
Use additional sheets if necessary.
Completed by
EXAMPLE
Page
of
B-27
-------
-------
APPENDIX C: BLANK WORKSHEETS
The worksheets in this appendix may be used or modified to document the
decisions, record data, and make calculations to determine if the waste site attains the
cleanup standard. These worksheets are referred to in the document. Appendix B provides
examples of how to fill out the worksheets.
C-l
-------
APPENDIX C: BLANK WORKSHEETS
WORKSHEET 1 Sample Areas
See Section 3.1 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
Sample
Area
Number
g
Describe the sample areas and the reasons for treating each area separately.
Use the Sample Area Number (g) to refer on other sheets to the sample areas described above.
Attach a map showing the sample areas within the waste site.
Date Completed: Completed by
Use additional sheets if necessary. Page of.
Continue to WORKSHEET 2
C-2
-------
APPENDIX C: BLANK WORKSHEETS
WORKSHEET 2 Attainment Objectives
See Section 3.3 in "Methods for Evaluating the Attainment of Cleanup Standards." Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [1J
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Sample Collection Procedures to be used (attach separate sheet if necessary):
Probability of mistakenly declaring the site clean = a =
Chemical Cleanup Parameter to test:
to be tested Chemical Standard
Number Name (with units) Mean Proportion
j Cs Yes/No P0
Secondary Objectives/ Other purposes for which the data is to be collected:
Use the Chemical Number (j) to refer on other sheets to the chemical described above.
Attach documentation describing the lab analysis procedure for each chemical.
Date Completed: Completed by
Use additional sheets if necessary. Page.
of
Continue to WORKSHEET 3
C-3
-------
APPENDIX C: BLANK WORKSHEETS
WORKSHEET 3 Sampling Design and Analysis Plan
See Chapter 4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [1]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Sample Design: | | Simpl{, Random Sample
Systematic Random Sample
Stratified Sample
Chemical Comments on the Prob of Type n error Alternate Parameter value
to be tested Sample Design and Chance of concluding the for the specified (3
Number [2] Analysis Plan site is dirty when it is clean Mean Proportion
j P m PI
Date Completed:
Use additional sheets if necessary.
Completed by
Page
of
Continue to WORKSHEET 4 for random or systematic sampling and WORKSHEET 8 for stratified sampling.
C-4
-------
APPENDIX C: BLANK WORKSHEETS
WORKSHEET 4 Sample Size for Testing the Mean Using Simple Random
Sampling
See Section 6.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
If the mean concentration is not to be tested for this chemical, continue to WORKSHEET 5
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [1]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Probability of mistakenly declaring the site clean [2] = a
[2] [3]
Appendix A
j P z\.n Cs \LI !
From z -Table, Appendix A
] *•«-[
Chemical
Number [3]
[2]
From
z table
Appendix A
A=l
Calculate:
Cs-m
Fraction of samples expected to be analyzable = R =
Max nj _
R =B =
B rounded up = Sample Size for Testing Means = nf =
Date Completed:
Use additional sheets if necessary.
Continue to WORKSHEET 5
Completed by
Column Maximum, Max n: =
Page of.
C-5
-------
APPENDIX C: BLANK WORKSHEETS
WORKSHEET 5 Sample Size for Testing Proportions Using Simple Random
Sampling
See Section 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
If the mean concentration is not to be tested for this chemical, continue to WORKSHEET 6
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [1]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
From z -Table, Appendix A
Probability of mistakenly declaring the site clean [2] = a
Chemical From Calculate:
Number [3] z table [2] [3]
[2]
j P ZI-B PO PI A = ZI-
zi-ct=[
Fraction of samples expected to be collectible = B =
Max nj _ r _
B ~^~
C rounded up to the next integer = Sample Size for Testing Proportions =
Column Maximum, Max n: =
Date Completed:
Use additional sheets if necessary.
Continue to WORKSHEET 6
Completed by
Page of.
C-6
-------
APPENDIX C: BLANK WORKSHEETS
WORKSHEET 6 Data Calculations for a Simple Random Sample, by Chemical
See Section 6.3 or 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards." Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [1J
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Maximum Sample Size from Worksheets 4 and 5 = Sample Size =
Cleanup standard [2] = Cs
Method Detection Limit: =
Concentration used when it is reported as less than the method detection limit=
Was the Reported Is xj Greater
Sample Concentration Concentration thanCs?
Sample Sample Collectible? If Corrected for l = Yes
Number ID 0 = No Collectible Detection Limit 0 = No
i l=Yes Xj yj (xj)2
Total from previous page
Column Totals:
A
B
C
D
A = n
C = r
Date Completed:
Completed by
Use additional sheets if necessary. Page _ of .
Complete WORKSHEET 6 for other chemicals or continue to WORKSHEET 7
C-7
-------
APPENDIX C: BLANK WORKSHEETS
WORKSHEET 7 Inference for Simple Random Samples by Chemical
See Section 6.3 or 7.3 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [1]
CHEMICAL:
NUMBER(j) ANDDESCRIPnON[2]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Testing the Mean [2]
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APPENDIX C: BLANK WORKSHEETS
WORKSHEET 8 Definition of Strata Within Sample Area
See Section 4.1 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [1]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Stratum Describe the stratum and the reason
Number for interest in this area
Volume =
Surface Area * Sample depth
Vh
Vh
Svh
Total Volume =
Use the Stratum Number (h) to refer on other worksheets to the stratum described above
Attach a map showing the stratum within the sample area.
Date Completed: Completed by
Use additional sheets if necessary.
Continue to WORKSHEET 9
Page.
of
C-9
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APPENDIX C: BLANK WORKSHEETS
WORKSHEET 9 Desired Sample Sizes for Testing the Mean Using Stratified
Sampling, by Chemical
See Section 6.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION!!]
CHEMICAL:
NUMBER® AND DESCRIPTION [2]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Probability of mistakenly declaring the site clean [2] = a
For the Cleanup Standard = Cs =
Probability of mistakenly declaring the site dirty [3] = ji =
If the true concentration is [3] = Ui =
Calculate:
Proportion
of Sample Stratum
Stratum Area in Standard
Number[8] Stratum[8] Deviation
h Wh &h
Unit
Sample
Cost
Ch
C = Column Sum =
B = C/A =
Date Completed:
Use additional sheets if necessary.
Continue to WORKSHEET 10
From z -Table^ Appendix A
Desired final
sample size
nhd =
B*Wh*6-h
Completed by.
Calculation
check
A =
Page.
of
C-10
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APPENDIX C: BLANK WORKSHEETS
WORKSHEET 10 Desired Sample Sizes for Testing a Percentile Using Stratified
Sampling, by Chemical
See Section 7.4 in "Methods for Evaluating the Attainment of Cleanup Standards." Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [lj
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Probability of mistakenly declaring the site clean [2] = a
Proportion Exceeding Cleanup Standard [2] = PQ =
Probability of mistakenly declaring the site dirty [3] = [3 =
If the true proportion is [2] = PI =
Calculate:
po-pi
= A =
Proportion Proportion Stratum
of Sample of dirty Standard Unit
Stratum Area in Samples Deviation Sample
Number[3] Stratum[3] &n Cost
Ph
= Column Sum =
B = C/A =
From z -
Appendix A
Desired final
sample size
"hd =
B*Wh*o-h
Calculation
check
A =
Date Completed:
Use additional sheets if necessary.
Continue to WORKSHEET 11
Completed by.
Page
nh
of
C-ll
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APPENDIX C: BLANK WORKSHEETS
WORKSHEET 11 Desired Sample Sizes for AH Chemicals and Parameters
See Section 6.4 or 7.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRlPnON[lJ
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Enter the Desired Sample Size (nj^by Stratum for each combination of chemical
and parameter to be tested, from WORKSHEETS 9 and 10.
Desired Sample Size
Stratum number h
12345
by stratum and:
chemical
Chemical
Worksheet
Chemical
Worksheet
Chemical
Worksheet
Chemical
Worksheet
Chemical
Worksheet
Maximum n^
for all Chemicals
and Parameters
nhdmax
Fraction of
Collectible
Field Samples Rn
A nhdmax
A %
A Rounded up to the
Next Integer = nnf 5
the field sample size
Date Completed:
Use additional sheets if necessary.
Continue to WORKSHEET 12
Completed by.
Page
of
C-12
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APPENDIX C: BLANK WORKSHEETS
WORKSHEET 12 Data Calculations, by Stratum and Chemical
See Section 6.4 or 7.4 in " Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [1 ]
STRATUM:
NUMBER AND DESCRIPTION [3]
CHEMICAL:
NUMBER(j) AND DESCRIPTION [2]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Samples Size [11] = n^f =
Cleanup standard[2] = Cs =
Method Detection Limit: =
Concentration used when no concentration is reported =
Was the Reported Is Xj Greater
Sample Concentration Concentration thanCs?
nh = A
Date Completed:
Completed by
Use additional sheets if necessary.
Complete WORKSHEET 12 for other chemicals or to WORKSHEET 13
Page
Sample Sample Collectible? If Corrected for l = Yes
Number ID 0 = No Collectible Detection Limit 0 = No
i l = Yes Xj yi (xO2
Total from previous page
Column Totals:
A
B
C
D
of
C-13
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APPENDIX C: BLANK WORKSHEETS
WORKSHEET 13 Sample Area Analysis for the Mean Using Stratified
Sampling, by Chemical
See Section 6.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SUE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTIONil J
CHEMICAL:
NUMBERQ AND DESCRIPTION [2]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Stratum
Number[3] [3]
h Wh
[12] [12] [12]
- _B , D-xh2A
Whxh F =
(A-l)
Grand Totals:
H
[2] cc = [
[2] Cs = f
Mean concentration = G = x =
H2
Degrees of Freedom = -r- Rounded to an integer = df =
tl-a,df =
Standard Error for the Mean concentration =
Upper One Sided Confidence Interval = x + s^ ti_a>(jf = |iua =
If |iTja< Cs then circle Clean, otherwise circle Dirty:
Based on the mean concentration, the sample area is:
Date Completed: Completed by
Use additional sheets if necessary.
Continue to WORKSHEET 14
Clean Dirty
Page
of
C-14
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APPENDIX C: BLANK WORKSHEETS
WORKSHEET 14 Sample Area Analysis for a Percentile Using Stratified
Sampling, by Chemical
See Section 7.4 in "Methods for Evaluating the Attainment of Cleanup Standards," Volume 1
SITE:
SAMPLE AREA:
NUMBER(g) AND DESCRIPTION [Ij
CHEMICAL:
NUMBERfj) AND DESCRIPTION [2]
Numbers in square brackets [] refer to the Worksheet from which the information may be obtained.
Stratum
Number[3] [3] [9] [9] [9]
C 2 2 Sh2
nh
Grand 1
"otals:
G
H
[2] P0 =
[4 or 5] Z!.a =
Proportion of Contaminated Samples = G = p =
= sp =
Standard Error for the Proportion =
Test Statistic = p+ sp zi_a = T =
If T < PQ then circle Clean, otherwise circle Dirty:
Based on the mean concentration, the sample area is:
Clean Dirty
Date Completed:
Use additional sheets if necessary.
Completed by
Page of.
C-15
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APPENDIX D: GLOSSARY
Alpha (a) In the context of a statistical test, a is probability of a Type I error.
Alternative Hypothesis See hypothesis.
Analysis Plan The plan specifying how the data are to be analyzed once they are
collected, including what estimates are to be made from the data* how the
estimates are to be calculated, and how the results of the analysis will be
reported.
Attainment The achievement of a prescribed standard/level of concentration.
Attainment Objectives Specifying chemicals to be tested, specifying the cleanup
standard to be attained, specifying the measure or parameter to be compared to
the cleanup standard, and specifying the level of confidence required if the
environment and human health are to be protected
Beta (P) In the context of a statistical test, p is probability of a Type II error.
Binomial Distribution A probability distribution used to describe the number of
occurrences of a specified event in n independent trials. In this manual, the
binomial distribution is used to develop statistical tests concerned with testing
the proportion of soil units in a simple random sample having excessive
concentrations of a contaminant (see Chapter 7). For additional details about
the binomial distribution, consult Conover (1980).
Coefficient of Variation The ratio of the standard deviation to the mean for a set of
data or distribution, abbreviated cv. For data that can only have positive
values, such as concentration measurements, the coefficient of variation
provides a crude measure of skewness.
Confidence Interval A sample-based estimate of a population parameter expressed as a
range or interval of values, rather than as a single value (point estimate).
D-l
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APPENDIX D: GLOSSARY
Confidence Level The degree of confidence associated with an interval estimate. For
example, with a 95% confidence interval, we would be 95% certain that the
interval contains the true value being estimated. The confidence level is equal to
1 minus the Type I error (false positive rate).
Conservative Test A statistical test for which the Type I error rate (false positive rate) is
actually less than that specified for the test. For a conservative test there will be
a greater tendency to accept the null hypothesis when it is not true than for a
non-conservative test.
Distribution The frequencies (either relative or absolute) with which measurements in a
data set fall within specified classes. A graphical display of a distribution is
referred to as a histogram.
Estimate Any numerical quantity computed from a sample of data. For example, a
sample mean is an estimate of the corresponding population mean.
False Positive Rate The probability of mistakenly concluding that the sample area is
clean when it is duty. It is the probability of making a Type I error.
False Negative Rate The probability of mistakenly concluding that the sample area is
dirty when it is clean. It is the probability of making a Type II error.
Geostatistics A methodology for the analysis of spatially correlated data. The
characteristic feature is the use of variograms or related techniques to quantify
and model the spatial correlation structure. Also includes the various techniques
such as kriging, which utilize spatial correlation models.
Histogram A graphical display of a frequency distribution.
Hot Spot Localized elliptical areas with concentrations in excess of the cleanup standard,
either a volume defined by the projection of the surface area through the soil
zone that will be sampled or a discrete horizon within the soil zone that will be
sampled.
D-2
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APPENDIX D: GLOSSARY
Hypothesis An assumption about a property or characteristic of a population under
study. The goal of statistical inference is to decide which of two
complementary hypotheses is likely to be true. In the context of this guidance
document, the null hypothesis is that the sample area is "dirty" and the
alternative hypothesis is that the sample area is "clean."
Inference The process of generalizing (extrapolating) results from a sample to a larger
population.
Judgment sample A sample of data selected according to non-probabilistic methods.
Kriging A weighted-moving-average interpolation method where the set of weights
assigned to samples minimizes the estimation variance, which is computed as a
function of the variogram model and locations of the samples relative to each
other, and to the point or block being estimated. This technique is used to
model the contours of contamination levels at a waste site (see Chapter 10).
Less-Than-Detection-Limit A concentration value that is below the detection limit. It
is generally recommended that these values be included in the analysis as values
at the detection limit
Lognormal Distribution A family of positive-valued, skewed distributions commonly
used in environmental work. See Gilbert (1987, p. 152) for a detailed
discussion of lognormal distributions.
Mean The arithmetic average of a set of data values. Specifically, the mean of a data set,
n
\i, x2,..., Xn, is defined by x = £ xj/n.
i=l
Median The "middle" value of a set of data, after the values have been arranged in
ascending order. If the number of data points is even, the median is defined to
be the average of the two middle values.
Nonparametric Test A test based on relatively few assumptions about the underlying
process generating the data. In particular, few assumptions are made about the
exact form of the underlying probability distribution. As a consequence,
nonparametric tests are valid for a fairly broad class of distributions.
D-3
-------
APPENDIX D: GLOSSARY
Normal Distribution A family of "bell-shaped" distributions described by the mean and
variance, p. and a2. Refer to a statistical text [(e.g., Sokal and Rohlf (1973)]
for a formal definition. The standard normal distribution has u, = 0 and a2 = 1.
Null Hypothesis See hypothesis.
Ordinary Kriging A variety of kriging which assumes that local means are not
necessarily closely related to the population mean, and which therefore uses
only the samples in the local neighborhood for the estimate. Ordinary kriging is
the most commonly used method for environmental situations.
Outlier A measurement that is extremely large or small relative to the rest of the data
gathered and that is suspected of misrepresenting the true concentration at the
sample location.
Parameter A statistical property or characteristic of a population of values. Statistical
quantities such as means, standard deviations, percentiles, etc. are parameters if
they refer to a population of values, rather than to a sample of values.
Parametric Test A test based on relatively strong assumptions about the underlying
process generating the data. For example, most parametric tests assume that the
underlying data are normally distributed. As a consequence, parametric tests
are not valid unless the underlying assumptions are met See robust test.
Percentile The specific value of a distribution that divides the set of measurements in
such a way that P percent of the measurements fall below (or are equal to) this
value, and 1-P percent of the measurements exceed this value. For specificity,
a percentile is described by the value of P (expressed as a percentage). For
example, the 95th percentile (P=0.95) is that value X such that 95 percent of the
data have values less than X, and 5 percent have values exceeding X. By
definition, the median is the 50th percentile.
Physical sample or soil sample A portion of material (such as a soil core, scoop, etc.)
gathered at the waste site on which measurements are to be made. This may
also be called a soil unit. A soil sample may be mixed, subsampled, or
otherwise handled to obtain the sample of soil that is sent for laboratory
analysis.
D-4
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APPENDIX D: GLOSSARY
Point Estimate See estimate.
Population The totality of soil units at a waste site for which inferences regarding
attainment of cleanup standards are to be made.
Power The probability that a statistical test will result in rejecting the null hypothesis
when the null hypothesis is false. Power = 1 - p, where p is the Type II error
rate associated with the test. The term "power function" is more accurate
because it reflects the fact that power is a function of a particular value of the
parameter of interest under the alternative hypothesis.
Precision See standard error.
Proportion The number of soil units in a set of soil units that have a specified
characteristic, divided by the total number of soil units in the set This may also
be expressed as a proportion of area or proportion of volume that has a
specified characteristic.
Random Sample A sample of soil units selected using the simple random sampling
procedures described in Chapter 5.
Range The difference between the maximum and minimum values of measurements in a
data set.
Robust Test A statistical test that is approximately valid under a wide range of
conditions.
Sample Any collection of soil samples taken from a waste site.
Sample Area The specific area within a waste site for which a separate decision on
attainment is to be reached.
Sample Design The procedures used to select the sample of soil units.
Sample Size The number of lab samples (i.e., the size of the statistical sample). Thus, a
sample of size 10 consists of the measurements taken on 10 lab samples.
D-5
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APPENDIX D: GLOSSARY
Sequential Test A statistical test in which the decision to accept or reject the null
hypothesis is made in a sequential fashion. A sequential test for proportions is
described in Chapter 8 of this guidance document.
Semi-variogram Identical to the term "variogram." There is disagreement in the
geostatistical literature as to which term should be used.
Significance Level The probability of a Type I error associated with a statistical test.
In the context of the statistical tests presented in this document, it is the
probability that the sample area is declared to be clean when it is dirty. The
significance level is often denoted by the symbol a (Greek letter alpha).
Size of the physical sample This term refers to the dimensions of a physical sample or
soil unit.
Skewed Distribution Any nonsymmetric distribution.
Soil Sample See physical sample.
Standard Deviation A measure of dispersion of a set of data. Specifically, given a set
of measurements, xl5 x2, ..., x^ the standard deviation is defined to be the
quantity, s = V —j , where x is the sample mean.
Standard Error A measure of the variability (or precision) of a sample estimate.
Standard errors are often used to construct confidence intervals.
Statistical Sample A collection of chemical concentration measurements reported by the
lab for one or more lab samples.
Statistical Test A formal statistical procedure and decision rule for deciding whether a
sample area attains the specified cleanup standard.
Stratified Sample A sample comprised of a number of separate samples from different
strata.
D-6
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APPENDIX D: GLOSSARY
Stratum A subset of a sample area within which a random or systematic sample is
selected. The primary purpose of creating strata for sampling is to improve the
precision of the sample design.
Symmetric Distribution A distribution of measurements for which the two sides of its
overall shape are mirror images of each other about a center line.
Systematic Sample A "grid" sample with a random start position.
Tolerance Interval A confidence interval around a percentile of a distribution of
concentrations.
Type I Error The error made when the sample area is declared to be clean when it is
contaminated. This is also referred to as a false positive.
Type II Error The error made when the sample area is declared to be dirty when it is
clean. This is also referred to as a false negative.
Variance The square of the standard deviation.
Variogram A plot of the variance (one-half the mean squared difference) of paired sample
measurements as a function of the distance (and optionally of the direction)
between samples. Typically, all possible sample pairs are examined, distance
and direction. Variograms provide a means of quantifying the commonly
observed relationship that samples close together will tend to have more similar
values than samples far apart.
Waste Site The entire area being investigated for contamination.
Z Value Percentage point of a standard normal distribution.
D-7
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INDEX OF KEY WORDS
alpha, 2-5
analysis plan, 4-6,9-8
ARAR (applicable and relevant or appropriate
requirements), 1-1,1-7
beta, 2-5
Cs (cleanup standard), 3-5,6-1
compositing methods, 5-15, 6-26
data quality objectives, 3-11
degrees of freedom, 6-3,6-24,6-28
exact test for proportions, 7-9,7-10
false negative, 2-5,2-7
false positive, 2-5, 2-7, 3-11
geostatistical methods, 10-1
grids, 5-5
homogenization, 10-6
hot spots, 9-1
kriging, 10-2
indicator kriging, 10-9, 10-10
probability kriging, 10-9,10-10
mean concentration, 2-8,2-9, 3-3,
3-10, 6-1, 6-16
median, 3-6
null hypothesis, 2-3
outliers, 2-16
parametric procedures, 6-1
proportions or percentiles, 2-9, 2-15,7-1
quality assurance/quality control, 5-18
SARA (Superfund Amendments and
Reauthorization Act of 1986), 1-1
sample
judgment sample, 2-18,4-2
random sample, 2-18,4-2,4-4, 5-3
stratified sample, 2-18,4-4
systematic sample, 2-18,4-2
sample area, 3-1
sample size determination, 2-14,6-7,6-8
for random samples (mean), 6-7
for stratified random samples (mean), 6-13,
for systematic samples (mean), 6-20
for simple random samples (proportions/
percentiles), 7-5
for stratified samples (proportions/
percentiles) 7-13,7-17
for a normal or lognormal population
using tolerance intervals (proportions/
percentiles), 7-21
sampling, 4-1
random sampling, 4-1,4-2,7-5
sequential sampling, 4-2,4-6, 8-1
stratified sampling, 4-1,4-4,6-12, 7-12
systematic sampling, 4-1,4-2, 6-21, 6-22,
6-23
sampling location, 5-1
random samples, 5-3
stratified samples, 5-13
systematic samples, 5-5
sampling plan, 4-1
serpentine pattern, 6-21,6-25,6-26,6-27
simple exceedance rule method, 7-11
soil unit, 2-18
spatial characterization, 1-4
standard deviation, 6-2,6-4
standard error
estimation of, 6-17, 6-18, 6-21, 6-23,
6-25,7-7,7-18
standards, 1-6
background-based, 1-7
cleanup, 1-6, 3-5
risk-based, 1-8,2-11,2-12
technology-based, 1-7
strata, 4-4
subsampling, 5-14
Superfund remediation, 1-6
upper percentile, 3-6
vadosezone, 1-4
IND
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K» .
Cover Photos: McKin Site, Gray, Maine, before and after cleanup
Maine Department of Environmental Protection
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