ProUCL Version 3.0
User Guide
April 2004
-------
£ EPA/600/R04/079
* VM/^ % April 2004
ProUCL Version 3.0
User Guide
by
Anita Singh
Lockheed Martin Environmental Services
1050 E. Flamingo Road, Suite E120, Las Vegas, NV89119
Ashok K. Singh
Department of Mathematical Sciences
University of Nevada, Las Vegas, NV 891 54
Robert W. Maichle
Lockheed Martin Environmental Services
1050 E. Flamingo Road, Suite E120, Las Vegas, NV89119
-------
Table of Contents
Authors i
Table of Contents ii
Disclaimer vii
Executive Summary viii
Introduction ix
Installation Instructions 1
Minimum Hardware Requirements 1
A. ProUCL Menu Structure 2
1. File 3
2. View 4
3. Help 5
B. ProUCL Components 6
1. File 7
a. Input File Format 9
b. Result of Opening an Input Data File 10
2. Edit 11
3. View 12
4. Options 13
a. The Data Location Screen 14
5. Summary Statistics 15
a. Summary Statistics 16
b. Results Obtained Using the Summary Statistics Option 17
c. Printing Summary Statistics 17
6. Histogram 18
a. Histogram Screen 19
b. Results of Histogram Option 20
7. Goodness-of-Fit Tests 21
a. Goodness-of-Fit Tests Screen 23
b. Result of Selecting Perform Normality Test Option 24
c. Resulting Normal Q-Q Plot Display to Perform Normality Test 25
d. Result of Selecting Perform Lognormality Test Option 26
e. Resulting Lognormal Q-Q Plot Display to Perform Lognormality Test . . 27
f Result of Selecting Perform Gamma Test Option 28
g. Resulting Gamma Q-Q Plot Display to Perform Gamma Test 29
8. UCLs 30
a. UCLs Computation Screen 32
b. Results After Clicking on Compute UCLs Drop-Down Menu Item 33
c. Display After Selecting the Normal UCLs Option 34
d. Display After Selecting the Gamma UCLs Option 35
-------
e. Display After Selecting the Lognormal UCLs Option 36
f. Display After Selecting the Non-parametric UCLs Option 37
g. Display After Selecting the All UCLs Option 38
h. Result After Clicking on Fixed Excel Format Drop-Down Menu Item 39
i. Result After Clicking the Fixed Excel Format Compute UCLs Button 40
9. Window 43
10. Help 44
Run Time Notes 45
Rules to Remember When Editing or Creating a New Data File 46
C. Recommendation to Compute a 95% UCL of the Population Mean (The Exposure Point
Concentration (EPC) Term) 47
D. Recommendations to Compute a 95% UCL of the Population Mean, //13 Using
Symmetric and Positively Skewed Data Sets 48
1. Normally or Approximately Normally Distributed Data Sets 48
2. Gamma Distributed Skewed Data Sets 49
Table 1 - Summary Table for the Computation of a 95% UCL
of the Unknown Mean, //b of a Gamma Distribution 50
3. Lognormally Distributed Skewed Data sets 51
Table 2 - Summary Table for the Computation of a 95% UCL
of the Unknown Mean, //13 of a Lognormal Population 52
4. Data Sets Without a Discernable Skewed Distribution - Non-parametric Skewed
Data Sets 53
Table 3 - Summary Table for the Computation of a 95% UCL of the
Unknown
Mean, //13 of a Skewed Non-parametric Distribution with all Positive
Values,
Where a is the Sd of Log-transformed Data 54
E. Should the Maximum Observed Concentration be Used as an Estimate of the
EPC Term? 55
F. Left-Censored Data Sets With Non-detects 57
Glossary 58
References 59
in
-------
Appendix A
TECHNICAL BACKGROUND - METHODS FOR COMPUTING THE EPC TERM
((1-a) 100%UCL) AS INCORPORATED IN ProUCL VERSION 3.0 SOFTWARE
1. Introduction A-l
1.1 Non-detects and Missing Data A-5
2. Procedures to Test for Data Distribution A-6
2.1 Test Normality and Lognormality of a Data Set A-7
2.1.1 Normal Quantile-Quantile (Q-Q) Plot A-7
2.1.2 Shapiro-Wilk W Test A-8
2.1.3 Lilliefors Test A-8
2.2 Gamma Distribution A-9
2.2.1 Quantile- Quantile (Q-Q) Plot for a Gamma Distribution A-10
2.2.2 Empirical Distribution Function (EDF) Based Goodness-of-Fit Tests . . A-l 1
3. Estimation of Parameters of the Three Distributions Included in ProUCL A-13
3.1 Normal Distribution A-14
3.2 Lognormal Distribution A-14
3.2.1 MLEs of the Parameters of a Lognormal Distribution A-15
3.2.2 Relationship Between Skewness and Standard Deviation, a A-15
3.2.3 MLEs of the Quantiles of a Lognormal Distribution A-16
3.2.4 MVUEs of Parameters of a Lognormal Distribution A-17
3.3 Estimation of the Parameters of a Gamma Distribution A-18
4. Methods for Computing a UCL of the Unknown Population Mean A-22
4.1 (1-a) 100% UCL of the Mean Based Upon Student's-t Statistic A-24
4.2 Computation of UCL of the Mean of a Gamma, G(k,0) Distribution A-25
4.3 (1-a) 100% UCL of the Mean Based Upon H-Statistic (H-UCL) A-27
4.4 (1-a) 100% UCL of the Mean Based Upon Modified-t Statistic for
Asymmetrical Populations A-28
4.5 (1-a) 100% UCL of the Mean Based Upon the Central Limit Theorem A-29
4.6 (1-a) 100% UCL of the Mean Based Upon the Adjusted Central Limit
Theorem (Adjusted -CLT) A-30
4.7 (1-a) 100% UCL of the Mean Based Upon the Chebyshev Theorem
(Using the Sample Mean and Sample Sd) A-31
4.8 (1-a) 100% UCL of the Mean of a Lognormal Population Based Upon the
Chebyshev Theorem (Using the MVUE of the Mean and its Standard Error) . . A-3 3
4.9 (1-a) 100% UCL of the Mean Using the Jackknife and Bootstrap Methods . . A-35
4.9.1 (1-a) 100% UCL of the Mean Based Upon the Jackknife Method A-36
4.9.2 (1-a) 100% UCL of the Mean Based Upon the Standard Bootstrap
Method A-37
4.9.3 (1-a) 100% UCL of the Mean Based Upon the Simple Percentile
Bootstrap Method A-39
4.9.4 (1-a) 100% UCL of the Mean Based Upon the Bias-Corrected
Accelerated (BCA) Percentile Bootstrap Method A-40
IV
-------
4.9.5 (1-a) 100% UCL of the Mean Based Upon the Bootstrap-t Method A-41
4.9.6 (1-a) 100% UCL of the Mean Based Upon Hall's Bootstrap Method . . . A-43
5. Recommendations and Summary A-45
5.1 Recommendations to Compute a 95% UCL of the Unknown Population
Mean, ^ Using Symmetric and Positively Skewed Data Sets A-46
5.1.1 Normally or Approximately Normally Distributed Data Sets A-46
5.1.2 Gamma Distributed Skewed Data Sets A-47
5.1.3 Lognormally Distributed Skewed Data Sets A-50
5.1.4 Data Sets Without a Discernable Skewed Distribution -
Non-parametric Skewed Data Sets A-55
5.2 Summary of the Procedure to Compute a 95% UCL of the Population Mean . . A-57
5.3 Should the Maximum Observed Concentration be Used as an Estimate of the
EPC Term? A-60
References A-63
Appendix B
CRITICAL VALUES OF ANDERSON-DARLING TEST STATISTIC
AND KOLMOGOROV-SMIRNOV TEST STATISTIC FOR GAMMA DISTRIBUTION
WITH UNKNOWN PARAMETERS
Critical Values for Anderson Darling Test - Significance Level of 0.20 B-l
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.20 B-2
Critical Values for Anderson Darling Test - Significance Level of 0.15 B-3
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.15 B-4
Critical Values for Anderson Darling Test - Significance Level of 0.10 B-5
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.10 B-6
Critical Values for Anderson Darling Test - Significance Level of 0.05 B-7
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.05 B-8
Critical Values for Anderson Darling Test - Significance Level of 0.025 B-9
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.025 B-10
Critical Values for Anderson Darling Test - Significance Level of 0.01 B-l 1
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.01 B-12
Appendix C
GRAPHS SHOWING COVERAGE COMPARISONS FOR THE VARIOUS METHODS
FOR NORMAL, GAMMA, AND LOGNORMAL DISTRIBUTIONS
Figure 1 - Coverage Probabilities by 95% UCL of the Mean of N O=50, o=20) C-l
Figure 2 - Coverage Probabilities by 95% UCLs of the Mean of G(k=0.05,0=50) C-l
Figure 3 - Coverage Probabilities by 95% UCLs of the Mean of G(k=0.10,0=50) C-2
Figure 4 - Coverage Probabilities by 95% UCLs of the Mean of G(k=0.15,0=50) C-2
Figure 5 - Coverage Probabilities by 95% UCLs of the Mean of G(k=0.20,0=50) C-3
-------
Figure 6 - Coverage Probabilities by 95% UCLs of the Mean of G(k=0.50,0=50) C-3
Figure 7 - Coverage Probabilities by 95% UCLs of the Mean of G(k=l .00,0=50) C-4
Figure 8 - Coverage Probabilities by 95% UCLs of the Mean of G(k=2.00,0=50) C-4
Figure 9 - Coverage Probabilities by 95% UCLs of the Mean of G(k=5.00,0=50) C-5
Figure 10 - Coverage Probabilities by 95% UCL of the Mean of LN (// =5, o=0.5) .... C-5
Figure 11 - Coverage Probabilities by 95% UCL of the Mean of LN (// =5, o=l .0) .... C-6
Figure 12 - Coverage Probabilities by 95% UCL of the Mean of LN (// =5, o=1.5) .... C-6
Figure 13 - Coverage Probabilities by 95% UCL of the Mean of LN (// =5, o=2.0) C-7
Figure 14 - Coverage Probabilities by 95% UCL of the Mean of LN (// =5, o=2.5) C-7
Figure 15 - Coverage Probabilities by 95% UCL of the Mean of LN (// =5, o=3.0) C-8
VI
-------
Disclaimer
The United States Environmental Protection Agency (EPA) through its Office of Research and
Development funded and managed the research described here. It has been peer reviewed by the
EPA and approved for publication. Mention of trade names or commercial products does not
constitute endorsement or recommendation by the EPA for use.
ProUCL software was developed by Lockheed Martin under a contract with the EPA and is
made available through the EPA Technical Support Center in Las Vegas, Nevada.
Use of any portion of ProUCL that does not comply with the ProUCL User Guide is not
recommended.
ProUCL contains embedded licensed software. Any modification of the ProUCL source code
may violate the embedded licensed software agreements and is expressly forbidden.
ProUCL software provided by the EPA was scanned with McAfee VirusScan v4.5.1 SP1 and is
certified free of viruses.
With respect to ProUCL distributed software and documentation, neither the EPA nor any of
their employees, assumes any legal liability or responsibility for the accuracy, completeness, or
usefulness of any information, apparatus, product, or process disclosed. Furthermore, software
and documentation are supplied "as-is" without guarantee or warranty, expressed or implied,
including without limitation, any warranty of merchantability or fitness for a specific purpose.
Vll
-------
Executive Summary
Exposure assessment and cleanup decisions in support of U.S. Environmental Protection Agency
(EPA) projects are often made based upon the mean concentrations of the contaminants of
potential concern. A 95% upper confidence limit (UCL) of the unknown population arithmetic
mean (AM), //b is often used to:
Estimate the exposure point concentration (EPC) term,
Determine the attainment of cleanup standards,
Estimate background level mean contaminant concentrations, or
Compare the soil concentrations with site specific soil screening levels.
It is important to compute a reliable, conservative, and stable 95% UCL of the population mean
using the available data. The 95% UCL should approximately provide the 95% coverage for the
unknown population mean, ^ .
The EPA has issued guidance for calculating the UCL of the unknown population mean for
hazardous waste sites, and ProUCL software has been developed to compute an appropriate 95%
UCL of the unknown population mean. All UCL computation methods contained in the EPA
guidance documents are available in ProUCL, Version 3.0. Additionally, ProUCL, Version 3.0
can also compute a 95% UCL of the mean based upon the gamma distribution, which is better
suited to model positively skewed environmental data sets. ProUCL tests for normality,
lognormality, and a gamma distribution of the data set, and computes a conservative and stable
95% UCL of the unknown population mean, ^ . It should be emphasized that the computation
of an appropriate 95% UCL is based upon the assumption that the data set under study consists
of observations only from a single population.
Several parametric and distribution-free non-parametric methods are included in ProUCL. The
UCL computation methods in ProUCL cover a wide range of skewed data distributions arising
from the various environmental applications. For lognormally distributed data sets, the use of
Land's H-statistic many times yields unrealistically large and impractical UCL values. This
occurrence is prevalent when the sample size is small and standard deviation of the log-
transformed data is large. Gamma distribution has been incorporated in ProUCL to model these
types of positively skewed data sets. Singh, Singh, and laci (2002b) observed that a UCL of the
mean based upon a gamma distribution results in reliable and stable values of practical merit. It
is always desirable to test if an environmental data set follows a gamma distribution. For data
sets (of all sizes) which follow a gamma distribution, the EPC term should be computed using an
adjusted gamma UCL (when 0.1 < k < 0.5) of the mean or an approximate gamma UCL (when k
> 0.5) of the mean. These UCLs approximately provide the specified 95% coverage to the
population mean, ^ of a gamma distribution. For values of k < 0.1, a 95% UCL may be obtained
using the bootstrap-t method or Hall's bootstrap method when the sample size is small (n < 15),
and for larger samples, a UCL of the mean should be computed using the adjusted or
approximate gamma UCL.
Vlll
-------
Introduction
The computation of a (1-a) 100% upper confidence limit (UCL) of the population mean depends
upon the data distribution. Typically, environmental data are positively skewed, and a default
lognormal distribution (EPA, 1992) is often used to model such data distributions. The H-
statistic based Land's (Land 1971, 1975) H- UCL of the mean is used in these applications.
Hardin and Gilbert (1993), Singh, Singh, and Engelhardt (1997,1999), Schultz and Griffin, 1999,
Singh et al. (2002a), and Singh, Singh, and laci (2002b) pointed out several problems associated
with the use of the lognormal distribution and the H-UCL of the population AM. In practice, for
lognormal data sets with high standard deviation (sd), a, of the natural log-transformed data
(e.g., o exceeding 2.0), the H-UCL can become unacceptably large, exceeding the 95% and 99%
data quantiles, and even the maximum observed concentration, by orders of magnitude (Singh,
Singh, and Engelhardt, 1997). This is especially true for skewed data sets of smaller sizes (e.g.,
n<50).
The H-UCL is also very sensitive to a few low or high values. For example, the addition of a
sample with below detection limit measurement can cause the H-UCL to increase by a large
amount (Singh, Singh, and laci, 2002b). Realizing that use of the H-statistic can result in
unreasonably large UCL, it is recommended (EPA, 1992) to use the maximum observed value as
an estimate of the UCL (EPC term) in cases where the H-UCL exceeds the maximum observed
value. Recently, Singh, Singh and laci (2002b), and Singh and Singh (2003) studied the
computation of the UCLs based upon a gamma distribution and several non-parametric bootstrap
methods. Those methods have also been incorporated in ProUCL Version 3.0. ProUCL
Version 3.0 contains fifteen UCL computation methods; five are parametric and ten are non-
parametric. The non-parametric methods do not depend upon any of the data distributions.
Both lognormal and gamma distributions can be used to model positively skewed data sets. It
should be noted that it is difficult to distinguish between a lognormal and a gamma distribution,
especially when the sample size is small (e.g., n < 50). Singh, Singh, and laci (2002b) observed
that the UCL based upon a gamma distribution results in reliable and stable values of practical
merit. It is therefore always desirable to test if an environmental data set follows a gamma
distribution. For data sets (of all sizes) which follow a gamma distribution, the EPC term should
be computed using an adjusted gamma UCL (when 0.1 < k < 0.5) of the mean or an approximate
gamma UCL (when k > 0.5) of the mean as these UCLs approximately provide the specified
95% coverage to the population mean, /^ = k6 of a gamma distribution. For values of k < 0.1, a
95% UCL may be obtained using bootstrap-t method or Hall's bootstrap method when the
sample size is small (n < 15), and for larger samples a UCL of the mean should be computed
using the adjusted or approximate gamma UCL. For this application, k is the shape parameter of
a gamma distribution. It should be noted that both bootstrap-t and Hall's bootstrap methods
sometimes result in erratic, inflated, and unstable UCL values especially in the presence of
outliers. Therefore, these two methods should be used with caution. The user should examine
the various UCL results and determine if the UCLs based upon the bootstrap-t and Hall's
IX
-------
bootstrap methods represent reasonable and reliable UCL values of practical merit. If the results
based upon these two methods are much higher than the rest of methods (except for the UCLs
based upon lognormal distribution), then this could be an indication of erratic UCL values. In
case these two bootstrap methods yield erratic and inflated UCLs, the UCL of the mean should
be computed using the adjusted or the approximate gamma UCL computation method for highly
skewed gamma distributed data sets of small sizes.
ProUCL tests for normality, lognormality, and gamma distribution of a data set, and computes a
conservative and stable 95% UCL of the population mean, ^ . It should be emphasized that
throughout this User Guide, and in the ProUCL software, it is assumed that one is dealing with a
single population. If multiple populations (e.g., background and site data mixed together) are
present, it is recommended to separate them out (e.g., using other statistical population
partitioning techniques), and respective appropriate 95% UCLs should be computed for each of
the identified populations. Also, outliers if any should be identified and thoroughly investigated.
Outliers when present distort all statistics (mean, UCLs etc.) of interest. Decisions about their
exclusion (or inclusion) in the data set used to compute the EPC term should be made by all
parties involved (e.g., EPA, local agencies, potentially responsible party etc.). The critical
values of Anderson-Darling test statistic and Kolmogorov-Smirnov test statistic to test for
gamma distribution were generated using Monte Carlo simulation experiments. These critical
values are tabulated in Appendix B for various values of the level of significance. Singh, Singh,
and Engelhardt (1997,1999), Singh, Singh, and laci (2002b), and Singh and Singh (2003) studied
several parametric and non-parametric UCL computation methods which have been included in
ProUCL. Most of the mathematical algorithms and formulas used in the development of
ProUCL to compute the various statistics are summarized in Appendix A. For details, the user is
referred to Singh, Singh, and laci (2002b), and Singh and Singh (2003). ProUCL computes the
various summary statistics for raw, as well as log-transformed data. ProUCL defines log-
transform (log) as the natural logarithm (In) to the base e. ProUCL also computes the maximum
likelihood estimates (MLEs) and the minimum variance unbiased estimates (MVUEs) of various
unknown population parameters of normal, lognormal, and gamma distributions. This, of
course, depends upon the underlying data distribution. Based upon the data distribution,
ProUCL computes the (1-a) 100% UCLs of the unknown population mean, ^ using five
parametric and ten non-parametric methods.
The five parametric UCL computation methods include:
1. Student' s-t UCL,
2. approximate gamma UCL using chi-square approximation,
3. adjusted gamma UCL (adjusted for level significance),
4. Land' sH-UCL, and
5. Chebyshev inequality based UCL (using MVUEs of parameters of a lognormal distribution).
-------
The ten non-parametric methods included in ProUCL are:
1. the central limit theorem (CLT) based UCL,
2. modified-t statistic (adjusted for skewness) bases UCL,
3. adjusted-CLJ (adjusted for skewness) based UCL,
4. Chebyshev inequality based UCL (using sample mean and sample standard deviation),
5. Jackknife method based UCL,
6. UCL based upon standard bootstrap,
7. UCL based upon percentile bootstrap,
8. UCL based upon bias - corrected accelerated (BCA) bootstrap,
9. UCL based upon bootstrap-t, and
10. UCL based upon Hall's bootstrap.
An extensive comparison of these methods has been performed by Singh and Singh (2003) using
Monte Carlo simulation experiments. It is well known that the Jackknife method (with sample
mean as an estimator) and Student's-t method yield identical UCL values. However, a typical
user may be unaware of this fact. It has been suggested that a 95% UCL based upon the
Jackknife method may provide adequate coverage to the population mean of skewed
distributions, which of course is not true (just like a UCL based upon the Student's-t statistic).
For the benefit of all ProUCL users, it was decided to retain the Jackknife UCL computation
method in ProUCL.
The standard bootstrap and the percentile bootstrap UCL computation methods do not perform
well (do not provide adequate coverage to population mean) for skewed data sets. For skewed
distributions, the bootstrap-t and Hall's bootstrap (meant to adjust for skewness) methods do
perform better (in terms of coverage for the population mean) than the various other bootstrap
methods. However, it has been noted (e.g., see Singh, Singh, and laci (2002b), Singh and Singh
(2003)) that these two bootstrap methods sometimes yield erratic and inflated UCL values
(orders of magnitude higher than the other UCLs). This is especially true when outliers may be
present in a data set. Therefore, whenever applicable (e.g., based upon the findings of Singh and
Singh (2003)), ProUCL provides a caution statement regarding the use of these two bootstrap
methods. ProUCL software provides warning messages whenever the use of these methods is
recommended. However, for the sake of completeness, all of the parametric as well as non-
parametric methods have been included in ProUCL.
The use of some other methods (e.g., bias-corrected accelerated bootstrap method) that were not
included in ProUCL Version 2.1 was suggested by some practitioners due to opinions that these
omitted methods may perform better than the various other methods already incorporated in
ProUCL. In order to satisfy all users, ProUCL Version 3.0 has several additional UCL
computation methods which were not included in ProUCL, Version 2.1.
XI
-------
This User Guide contains software installation instructions and brief descriptions for each
window in the ProUCL software menu. A short glossary of terms used in this document and in
the ProUCL program is also provided.
Three appendices listed as follows provide additional information and details of the various
methods and references used in the development of ProUCL Version 3.0.
D Appendix A is a discussion of the methods incorporated into ProUCL for calculating the
exposure point concentration term using the various methods and distributions. Appendix A
represents a stand-alone technical writeup of the various methods incorporated in ProUCL
and is provided for review by statistically advanced users. There is duplication between
some of the information provided in the main body of the User Guide and Appendix A. This
duplication is intentional since Appendix A is designed to be a stand-alone technical
discussion of the methods incorporated into ProUCL.
D Appendix B contains the tables of the critical values of the Anderson-Darling Test statistic
and Kolmogorov-Smirnov Test statistic for gamma distribution for various levels of
significance.
D Appendix C has the graphs from Singh and Singh (2003) showing coverage comparisons
(achieved confidence coefficient) for the various UCL computation methods for normal,
gamma, and lognormal distributions as incorporated in ProUCL software package.
Should the Maximum Observed Concentration be Used as an Estimate of the EPC Term?
Singh and Singh (2003) also included the Max Test (using the maximum observed value as an
estimate of the EPC term) in their simulation studies. In the past (e.g., EPA 1992 RAGS
Document), the use of the maximum observed value has been recommended as a default value to
estimate the EPC term when a 95% UCL (e.g., the H-UCL) exceeded the maximum value.
However, (e.g., EPA 1992), only two 95% UCL computation methods, namely: the Student's-1
UCL and Land's H-UCL were used to estimate the EPC term. Today, ProUCL, Version 3.0 can
compute a 95% UCL of the mean using several methods based upon normal, gamma, lognormal,
and non-parametric distributions. Thus, ProUCL, Version 3.0 has about fifteen 95% UCL
computation methods, at least one of which (depending upon skewness and data distribution) can
be used to compute an appropriate estimate of the EPC term. Furthermore, since the EPC term
represents the average exposure contracted by an individual over an exposure area (EA) during a
long period of time, therefore, the EPC term should be estimated by using an average value (such
as an appropriate 95% UCL of the mean) and not by the maximum observed concentration. With
the availability of the UCL computation methods, the developers of ProUCL Version 3.0 do not
consider it necessary to use the maximum observed value as an estimate of the EPC term. Singh
and Singh (2003) also noted that for skewed data sets of small sizes (e.g., n < 10 - 20), the Max
Test does not provide the specified 95% coverage to the population mean, and for larger data
sets, it overestimates the EPC term. This can also viewed in the graphs presented in Appendix
Xll
-------
C. Also, for the skewed distributions (gamma, lognormal) considered, the maximum value is not
a sufficient statistic for the unknown population mean. The use of the maximum value as an
estimate of the EPC term ignores most (except for the maximum value) of the information
contained in a data set. It is, therefore not desirable to use the maximum observed value as
estimate of the EPC term representing average exposure to an individual over an EA.
It should also be noted that for highly skewed data sets, the sample mean may exceed the upper
90%, 95 %, etc. percentiles, and consequently, a 95% UCL of the mean can exceed the
maximum observed value of a data set. This is especially true when one is dealing with highly
skewed lognormally distributed data sets of small sizes. For such highly skewed data sets which
can not be modeled by a gamma distribution, a 95% UCL of the mean should be computed using
an appropriate non-parametric method. These observations are summarized in Tables 1-3 of this
User Guide.
Alternatively, for such highly skewed data sets, other measures of central tendency such as the
median (or some higher order quantile such as 70% etc.) and its upper confidence limit may be
considered. The EPA and all other interested agencies and parties need to come to an agreement
on the use of median and its UCL to estimate the EPC term. However, the use of the sample
median and/or its UCL as estimates of the EPC term needs further research and investigation.
It is recommended that the maximum observed value NOT be used as an estimate of the
EPC term. For the sake of interested users, the ProUCL displays a warning message when the
recommended 95% UCL (e.g., Hall's bootstrap UCL etc.) of the mean exceeds the observed
maximum concentration. For such cases (when a 95% UCL does exceed the maximum observed
value), if applicable, an alternative 95% UCL computation method is recommended by ProUCL.
Handling of Non-Detects
ProUCL does not handle left-censored data sets with non-detects, which are inevitable in many
environmental applications. All parametric as well as non-parametric recommendations (as
summarized in Tables 1-3) to compute the mean, standard deviation, 95% UCLs and all other
statistics computed by ProUCL are based upon full data sets without censoring. It should be
noted that for mild to moderate number of non-detects (e.g., < 15%), one may use the commonly
used !/2 detection limit (!/2 DL) proxy method to compute the various statistics. However, the
proxy methods should be used cautiously, especially when one is dealing with lognormally
distributed data sets. For lognormally distributed data sets of small sizes, even a single value
small (e.g., obtained after replacing the non-detects by 1A DL) or large (e.g., an outlier) can have
a drastic influence (can yield an unrealistically large 95% UCL) on the value of the associated
Land's 95% UCL. The issue of estimating the mean, standard deviation, and an appropriate 95%
UCL of the mean based upon left-censored data sets with varying degrees of censoring (e.g.,
15% - 50%, 50% - 75%, greater than 75% etc.) is currently under investigation.
Xlll
-------
Installation Instructions
DCaution: If you have previous versions of the ProUCL which were installed, you should
remove or rename the directory in which that version is currently located.
DDownload the file SETUP.EXE from the EPA website and save to a temporary location.
DRun the SETUP.EXE program. This will create a ProUCL directory and two folders; USER
GUIDE and the DATA (sample data).
DTo run the program, use Windows Explorer to locate the ProUCL application file and double
click on it, or use the RUN command from the start menu to locate and run ProUCL.exe.
D To uninstall the program, use Windows Explorer to locate and delete the ProUCL folder.
Minimum Hardware Requirements
Dlntel Pentium 200MHz
Dl2 MB of hard drive space
D48 MB of memory (RAM)
DCD-ROM drive
DWindows 98 or newer. ProUCL was thoroughly tested on NT-4, Windows 2000, and
Windows XP operating systems. Limited testing has been conducted on Windows ME.
-------
A. ProUCL Menu Structure
ProUCL contains a pull-down menu structure, similar to a typical Windows program.
The screen below appears when the program is executed.
& ProUCL Version 3.0
File View Help
n c?
f
For Help, press Fl
The following menu options appear on the screen
1. File
2. View
3. Help
The options available with these menu items are described on the following pages.
-------
1. File
Click on the File menu item to reveal these drop-down menu options.
- n x
' File View Help
New
Ctrl+N
Ctrl+O
Working directory
Print Setup.,,
1 H:\cdelvl
2 H:\log3
3H:\)og35
4 H:\Jog5
5 H:\test2
6 H:\track
The following File drop-down menu options are available:
DNew option: creates new spreadsheet.
DOpen option: browses the disk for a file. The browse program will start in the working
directory if a directory has been set.
DWorking directory option: select and set a working directory.
Note: A file from the directory must be selected before setting the directory. All subsequent
files are read from and saved in the chosen working directory.
DPrint Setup option: sets printer options. For example, one can choose the landscape format.
DClick on a previously used file to re-open that file.
DExit opti on: exits ProUCL.
-------
2. View
Click on the View menu item to reveal these drop-down options.
File View Help
D * Toolbar
~~ ^ Status Bar
The following View drop-down menu options are available:
Dloolbar: the Toolbar is that row of symbols immediately below the menu items. Clicking on
this option toggles the display. This is useful if the user wants to view more data on the
screen.
DStatus Bar: the Status Bar is the wide bar at the bottom of the screen which displays helpful
information. Clicking on this option toggles the display. This is useful if the user wants to
view more data on the screen.
-------
3. Help
Click on the Help menu item to reveal these drop-down options.
/f ProUCL Version 3.0
File View Help
D c£
f
For Help, press Fl
The following Help drop-down menu options are available:
DHelp Topics: help topics have not been developed for Version 3.0.
DAbout ProUCL: displays the software version number.
-------
B. ProUCL Components
The following menu structure of ProUCL appears after opening or creating a data file.
/? ProUCL Version 3.0
^! File Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help
B
H
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
For Help, press Fl
The following menu items are available.
1. File
2. Edit
3. View
4. Options
5. Summary Statistics
6. Histogram
7. Goodness-of-Fit Tests
8. UCLs
9. Window
10. Help
The options available with these menu items are described on the following pages.
6
-------
1. File
Click on the File menu item to reveal these drop-down options.
rJV ProUCL
File Edit
New
Open...
Close
Save As...
View
Options Summary Statistics
Ctrl+N
Ctrl+O
Histogram Goodness-of-Flt Tests UCLs
-inlxl
Window Help
Working directory
Print... Ctrl+P
Print Preview
Print Setup,.,
1 H:\cdelvl
2H:\)og3
3 H:\Jog35
4 H:\Jog5
5 H:\test2
6 H:\track
Exit
H
B
ife . ,'
A '-
C i D
-ID
E
yde 4, 4 '-DDT Dieldrin Heptachlor Endrin aldehyde
.0018
101
101
85
85
.0019
0.0018 0.0018 0
0.00185 0.00185
0.018 0.018 0
0.0019 0.029 0
0.002
10215
.001 0.0019
1.05 2.05
.105 0.0019
.001 0.00195
0.2 0.2 0.001 0.0019
0.002 0.002 0
10425
n
1
II
111
J!
' > 1 S
III
II II II
II 1 1
ii 5 '.^
I j |
][|
1 i 1
1 1 I
Jit?
42 29 0
ill i rn n n in
in i i n Nil
' 1 1 Ii !M i !i i ! III I
1 1 111 ^ i I 11 II H 111
hi! 11 t it It HISM,
.001 0.24
.125 0.00195
ii 1 i in i in i 'ii'
I 1 1 III III I I III I
ii n i i 11 1 ( us ii i i i ii! i
11 11 iU II 1 [ 111 ill 1 ^ 1 1 i
1 It 'J I til I! U 1
JJKJ
F
Die!'
*
;
,
0 I
O.C I
o {
'
M
1 11
1 II
1 II i
;i lei [
! it
i
i
i
1
j
1
[
|
I
|
i
t
L
The following File drop-down menu options are available:
DNew option: opens a blank spreadsheet screen.
DOpen option: browses the disk and selects a file which is then opened in spreadsheet format.
The browse program will start in the working directory if a working directory has been set.
Recognized input format options:
Excel *.xls
Text *.txt (tab delimited)
Lotus *.wk?
Lotus *.123
Default *.* will be read in Excel format.
>DClose option: closes the active window.
-------
DSave As option: allows the user to save the active window. This option follows the
Windows standard and saves the active window to a file in Excel 95 (or higher) format. All
modified/edited data files, and output screens generated by the software, can be saved in
Excel 95 (or higher) format.
DWorking directory option: selects and sets a working directory for all I/O operations. All
subsequent files are read from and saved in the working directory. You must select a file
before you set the working directory.
DPrint option: sends the active window to the printer.
DPrint Preview option: displays a preview of the output on the screen.
DPrint Setup option: follows Windows standard. The user can choose the landscape format
under this option.
DPreviously opened files: click on a previously used file to re-open that file.
DExit opti on: exits ProUCL.
NOTE: All subsequent screens and examples in this User Guide use the spreadsheets given by
track.xls and Cdelvl.xls to illustrate the various goodness-of-fit test procedures and the UCL
computation methods as incorporated in the software ProUCL, Version 3.0.
-------
la. Input File Format
DData in each column must end with a non-zero value. The last non-zero entry in each
column is considered as the end of that column's data. If your data column ends with a zero
value, that last zero value will be ignored. This may require you to move observations
around if your column ends with zero values.
DThe program can read tab delimited Text (ASCII), Excel, and Lotus files.
DColumns in a Text (ASCII) file should be separated by one tab. Spaces between columns are
not allowed in this format.
DAll input data files should have column labels in the first row and numerical data without text
(e.g., non-numeric characters and blank values) for those variables in the remaining rows.
DThe data file can have multiple variables (columns) with unequal number of observations.
DNon-numeric text may only appear in the header row (first row) of each column. All other
non-numeric data (blank, other characters, and strings) appearing elsewhere in the data file
are treated as zero entries. The user should make sure that his data set does not contain such
non-numeric values.
DA large value, such as 1E31 (IxlO31), can be used for missing (alpha numeric text or blank
values) data. All entries with this value are ignored from the computations.
DNote that all other zero data (in the beginning or middle of a data column) are treated as valid
zero values.
DProUCL does not handle the left-censored data sets with non-detects which are inevitable in
environmental applications. All parametric as well as non-parametric recommendations
made by ProUCL are based upon full data sets without censoring. The issue of estimating
the mean, standard deviation, and a 95% UCL of the mean based upon left-censored data sets
with varying degrees of censoring is currently under investigation. For mild to moderate
number of non-detects (e.g., < 15%), one may use the commonly used !/2 detection limit
(DL) proxy method. However, the proxy methods should be used cautiously, especially
when one is dealing with lognormally distributed data sets. For lognormally distributed data
sets of small sizes, a single value, whether small (e.g., obtained after replacing the non-detect
by l/2 DL) or large (e.g., an outlier), can have a drastic influence (can yield an unrealistically
large 95% UCL) on the value of the associated Land's 95% UCL.
-------
Ib. Result of Opening an Input Data File
Dlhe data screen follows the standard Windows design. It can be resized, or portions of data
can be viewed using scroll bars.
DNote that scroll bars appear when the window is activated and the title bar is highlighted.
File Edit View Options Summary Statistics Histogram Goodness-or-Fit Tests UCLs Window Help
A
14000
14900
14100
9510
9110
13900
21300
9110
E
45100
37600
40450
26500
38600
42700
41000
26700
For Help, press Fl
10
-------
2. Edit
Click on the Edit menu item to reveal the following drop-down options.
A- ProUCL
File
1 D
Edit View Options
Erase Ctrl+E
Copy Ctrl+C
Paste Ctrl+V
Summary Statistics
f M?
Histogram Goodness-of-Fit Tests UCLs
_ ID |
Window Help
1
2
3
4
5
6
7
8
9
10
11
12
Endrin aldehyde
n x,
a
CD E
Dieldrin Heptachlor Endrin aldehyde
>[\,Dataf
0.0018
0.00185
0.00185
0.0019
0.002
0.00215
0.00425
0.018
0.0215
0.08
0.13
0.0018
0.00185
0.018
0.0019
0.2
0.002
42
0.00185
0.0305
0.00215
0.7155
0.0018
0.00185
0.018
0.029
0.2
0.002
29
0.00185
0.37
0.00215
180
0.001
1.05
0.105
0.001
0.001
0.001
0.125
0.001
0.00105
0.00105
0.00095
0.0019
2.05
0.0019
0.00195
0.0019
0.24
0.00195
0.00205
0.0021
0.00185
0.0195
The following Edit drop-down menu options are available:
DErase option: used to remove the highlighted portion of the data. Note that the erased data is
not written to any buffer and cannot be recovered. Therefore, when data is erased, it is gone.
DCopy option: similar to a standard Windows Edit option, such as in Excel. It performs
typical edit functions of identifying highlighted data (similar to a buffer).
DPaste option: similar to a standard Windows Edit option, such as in Excel. It performs
typical edit functions of pasting data identified (highlighted) to the current spreadsheet cell.
DThere is no Cut option available in ProUCL because there is no actual buffer available in the
commercial software(s) used in the development of ProUCL software.
11
-------
3. View
Click on the View menu item to reveal these drop-down options.
/f. ProUCL Version 3.0
D"~M
&
IS
iOfSBi
^ Toolbar
Status Bar
Summary Statistics Histogram
m t
*?
Goodness-of-Fit
^H
Tests UCLs
Window Help
^**S:S^^
P C:\ProUCL\Data\track.xls
eti
Iff
i
PB
m
tm
111
i
fK|
3
4
5
6
7
8
9
10
A
14000
14900
14100
9510
9110
13900
21300
9110
MJatsu/
B
7
5.1
6.15
5.3
4.2
6.9
7
4.4
C D
32
22.7
24.55
17
24.8
17.4
28.2
21
19.5
17.6
20.6
17.3
14.7
21.2
14
10.7
E
45100
37600
40450
26500
38600
42700
41000
26700
F
574
368
671
1120
759
727
409
434
LJ
0.17
0.488
0.4
0.5
0.34
1.1
0.45
I.
li|
(ii
iff!)
iHi
His
Jj£sf»K<»K
jfesSSS!
IsllBBBBBBBiSB
*n »
slsftfill
The following View drop-down menu options are available:
Dloolbar: the Toolbar is that row of symbols immediately below the menu items. Clicking on
this option toggles the display. This is useful if the user wants to view more data on the
screen.
DStatus Bar: the Status Bar is the wide bar at the bottom of the screen which displays helpful
information. Clicking on this option toggles the display. This is useful if the user wants to
view more data on the screen.
12
-------
4. Options
Click on the Options menu item to reveal these drop-down options.
Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help
Set Data
-------
4a. The Data Location Screen
The following Data location screen appears when Set Data option is executed.
Data Location!
Top row
Bottom row
Please specify the location of data
153
Leftmost column
Rightmost column
OK
Cancel
Dlt is recommended to use the default settings for the data screen. This means that all of
the data will be processed.
DCaution: Highlighting a portion of the spreadsheet before invoking the Set Data option may
sometimes cause unpredicted results.
DCaution: Blank cells in the top data row may confuse the automatic sizing algorithm. The
user can avoid this problem by re-setting the Rightmost column value using this option.
DThe first row in the spreadsheet contains the alphanumeric text (column headings), not data.
DThe default Top row of data is row 2. This value can be changed to process a subset of the
data in the spreadsheet.
DThe default Bottom row is the last row in the spreadsheet which contains nonzero data. This
value can be changed to process a subset of the data in the spreadsheet.
DThe selected data must correspond to the same columns as the text in the first row. The
Leftmost column value (column number) cannot be changed by the user.
DThe Rightmost column number can be changed by the user. Note that you must have a
column of data for the selected Rightmost column.
14
-------
5. Summary Statistics
Dlhis option computes general summary statistics for all variables in the data file.
Dlwo Choices are available:
Raw data (the default option)
Log-transformed data (Natural logarithm)
Din ProUCL, Log-transformation means natural logarithm (In).
Dwhen computing summary statistics for raw data, a message will be displayed for each
variable that contains non-numeric values.
Dlhe Summary Statistics option computes log-transformed data only if all of the data values
for the selected variable are positive real numbers. A message will be displayed if non-
numeric characters, zero, or negative values are found in the column corresponding to the
selected variable.
15
-------
5a. Summary Statistics Menu
Click on the Summary Statistics menu item to reveal the following drop-down option.
ProUCL Version 3.0
File Edit View Options ^^^^^fttSjaj^J Histogram Goodness-of-Fit Tests UCLs Window Help
D G? : £ % @ Compute 7
A B C DE F G H
1 Al As Cr Co Fe Mn Se SI Z|
2 12600 6.8 22.4 18.1 39600 501 0.315 0.055
3 14000 7 32 19.5 45100 574 1 0.115
Summary
When the user clicks on the
Compute option button,
the window on the right appears.
-Data
( Raw data
f Log-transformed data
Compute | Cancel
DSelect your data choice, and click on the Compute button to continue or on the Cancel button
to cancel the summary operations.
Dlhe results screen follows the standard Windows design. It can be edited, widened, printed,
resized, or scrolled.
Dlhe resulting Summary Statistics screen can be saved as an Excel file. Right double click on
the screen for additional save options.
16
-------
5b. Results Obtained Using the Summary Statistics Option
f£-_ ProUCL Version 3.0 - [Summary Statistics (Raw Data)]
E^ File
Edit View
D G£ X P
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
zor Help,
A
From File
Options Summary Statistics
* G & t *?
B C
C:\ProUCL\Data\track. xls
Histogram Goodness-of-Fit Tests
D
Variable name NumObs Minimum
Al
As
Cr
Co
Fe
Mn
Se
SI
In
Summary
press Fl
22
22
22
22
22
22
22
19
22
StatistjcsforRawDitay
2520
2.7
12.2
9.4
1400
0.115
0.12
0.05
35.6
E
Maximum
21300
42.8
111
65.6
65300
2400
187
69.5
120
F
Mean
11755.455
9.0181818
32.227273
21 .952273
34947.727
823.89159
14.3815
6.0651579
56.347727
UCLs Window Help
G
Median
12500
6.075
24.675
18.275
37400
699
0.43
0.12
54.65
H
3d
3959.426
8.9541896
24.05552
14.678754
14006.057
508.55278
44.110575
17.421608
19.903652
1
CV
0.3368161
0.9929041
0.7464336
0.6686667
0.4007716
0.6172569
3.0671748
2.872408
0.353229
J
Skewness
-0.209682
3.1282709
2.3223005
1.9385213
-0.094731
1 .4005582
3.4909232
3.2642255
1 .8624475
_ 3 X
K
Variance
15677055
80.177511
578.66803
215.46583
2E-rf]08
258625.93
1945.7428
303.51243
396.15535
"
On the results screen, the following summary statistics are displayed for each variable in the data
file:
S NumObs = Number of Observations
-------
6. Histogram
Dlhis option produces a histogram for the selected variable in the data file.
DFor data sets with more than one variable, the user should select a variable first. The
histogram is computed and displayed for each selected variable, one variable at a time.
- By default, the program selects the first variable.
The user specifies if the data should be transformed.
- The default choice is to display the histogram for raw data.
DTwo Choices are available:
- Raw data (the default option)
- Log-transformed data (natural logarithm, In)
DThe user can select the number of bins for the histogram.
- The default number of bins is 15.
DNote that in order to display and capture the best histogram window, the user may want to
maximize the window before printing.
18
-------
6a. Histogram Screen
DClick on the Histogram menu item and then click on the Draw Histogram option.
& PrpUCL.Version 3,0
File Edit View Options Summary Statistics
0 c£ & Hi © Dlhe window on the right
will appear.
C Raw data
( Log-transformed data
slumber of bins: 15
Display
FnHrir, slrlohwHo ^^^^H
AA'-DDJ
Dieldrin
Heptachlor
Endrin aldehyde
Dieldrin
4,4'-DDE
Aroclor-1 2^8
Aroclor-1 242
Cancel
DSelect Raw data or Log-transformed data.
DYou can change the number of bins to be used in the histogram.
DSelect a variable and then hit the display key to view the histogram for the selected variable.
19
-------
6b. Results of Histogram Option
ft- ProUCL Version 3.0
File Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help _ 5 X
D G? f *?
Histogram of Al
For Help, press Fl
Dlhe Histogram window shown above has been resized for display and reflects the use of
default values displayed in Section 6a (Histogram Screen).
DYou may close the window by using normal windows operations or click on the Close
window button at the bottom left corner of the screen.
Dlhe histogram can be printed or copied by clicking on the right button on the mouse.
DCaution: A right click of the mouse will have options other than print and save. These
options may function but are NOT recommended due to the program disruption that may
occur. Use these other options only at your own risk!
20
-------
7. Goodness-of-Fit Tests
DSeveral goodness-of-fit tests are available in ProUCL which are described in Appendix A.
Dlhroughout this User Guide, and in ProUCL, it is assumed that the user is dealing with a
single population. If multiple populations are present, it is recommended to separate them
out (using other statistical techniques). Appropriate tests and statistics (e.g., Goodness-of-fit
tests, 95% UCLs) should be computed separately for each of the identified populations.
Also, outliers if any should be identified and thoroughly investigated. The presence of
outliers distort all statistics including the UCLs. Decisions about their inclusion (or
exclusion) from the data set to be used to compute the UCLs should be made by all parties
involved.
DFor data sets with more than one variable, the user should select a variable first. The data
distribution is tested using an appropriate goodness-of-fit test and the associated results
are displayed for the selected variable, one variable at a time.
4 By default, the program selects the first variable.
Dlhis option tests for normal, gamma, or lognormal distribution of the selected variable.
DThe user specifies the distribution (normal, gamma, or lognormal) to be tested.
DThe user specifies the level of significance. Three choices are available for the level of
significance: 0.01, 0.05, or 0.1.
4 The default choice for level of significance is 0.05.
DProUCL displays a Quantile-Quantile (Q-Q) plot for the selected variable (or the log-
transformed variable). A Q-Q plot can be generated for each of the three distributions.
DThe linear pattern displayed by the Q-Q plot suggests approximate goodness-of-fit for the
selected distribution.
DThe program computes the intercept, slope, and the correlation coefficient for the linear
pattern displayed by the Q-Q plot. A high value of the correlation coefficient (e.g., > 0.95) is
an indication of approximate goodness-of-fit for that distribution. Note that these statistics
are displayed on the Q-Q plot.
DOn this graph, observations that are well separated from the bulk (central part) of the data
typically are potential outliers needing further investigation.
21
-------
DSignificant and obvious jumps in a Q-Q plot (for any distribution) are indication of the
presence of more than one population which should be partitioned out before estimating an
EPC Term. It is strongly recommended that both graphical and formal goodness-of fit tests
should be used on the same data set to determine the distribution of the data set under study.
Din addition to the graphical normal and lognormal Q-Q plot, two more powerful methods are
also available to test the normality or lognormality of the data set:
4 Lilliefors Test: a test typically used for samples of larger size (> 50). When the
sample size is greater than 50, the program defaults to the Lilliefors test. However,
note that the Lilliefors test is available for samples of all sizes. There is no applicable
upper limit for sample size for the Lilliefors test.
4 Shapiro and Wilk W-Test: a test used for samples of smaller size (< 50). W-Testis
available only for samples of size 50 or less.
4 It should be noted that sometimes, these two tests may lead to different conclusions.
Therefore, the user should exercise caution interpreting the results.
Din addition to the graphical gamma Q-Q plot, two more powerful Empirical Distribution
Function (EDF) procedures are also available to test the gamma distribution of the data set.
These are the Anderson-Darling Test and the Kolmogorov-Smirnov Test.
4 It should be noted that these two tests may also lead to different conclusions.
Therefore, the user should exercise caution interpreting the results.
4 These two tests may be used for samples of size in the range 4-2500. Also, for these
two tests, the value of k (k hat) should lie in the interval [0.01,100.0]. Consult
Appendix A for detailed description of k. Extrapolation beyond these sample sizes
and values of k is not recommended.
DProUCL computes the relevant test statistic and the associated critical value, and prints them
on the associated Q-Q plot. On this Q-Q plot, the program informs the user if the data are
gamma, normally, or lognormally distributed. It highly recommended not to skip the use of
graphical Q-Q plot to determine the data distribution as a Q-Q plot also provides the useful
information about the presence of multiple populations and/or outliers.
The Q-Q plot can be printed or copied by clicking on the right button on the mouse.
DNote: In order to capture the entire graph window, the user should maximize the window
before printing.
22
-------
7a. Goodness-of-Fit Tests Screen
DClick on the Goodness-of-Fit Tests menu item and a drop-down menu list will appear as
shown in the screen below:
& ProUCL Version 3.0 - [C:\ProUCL\Data\track.xls]
D
1
2
3
.......
5
6
7
8
9
10
11
a; & qgi i
A
Al As
12600
14000
14900
14100
9510
9110
13900
21300
9110
14600
a ^ f HI?
B
Cr
6.8
7
5.1
6.15
5.3
4.2
6.9
7
4.4
5.2
C
Co
22.4
32
22.7
24.55
17
24.8
17.4
28.2
21
13.1
D
Fe
18.1
19.5
17.6
20.6
17.3
14.7
21.2
14
10.7
10.4
Perform
Perform
Perform
^Hifili
3 X
Normality Test
Lognormality '
Gamma Test
min oe
39800
45100
37600
40450
26500
38600
42700
41000
26700
31300
501
574
368
671
1120
759
727
409
434
586
rest
JSI
0.315
1
0.17
0.488
0.4
0.5
0.34
1.1
0.45
0.8
H
Zn
0.055
0.115
0.055
0.123
0.05
0.12
1
0.125
0.06
0.11
1
46.3
45.4
61.2
48.3
37.5
36.5
68.7
55
42.6
54.3
>DTo test your variable for normality, click on Perform Normality Test from the drop-down
menu list.
>DTo test your variable for lognormality, click on Perform Lognormality Test from the drop-
down menu list.
>DTo test your variable for gamma distribution, click on Perform Gamma Test from the drop-
down menu list.
23
-------
7b. Result of Selecting Perform Normality Test Option
The following window will appear:
Normality Test
Select Variables
As
Cr
Co
Fe
Mn
Se
SI
Zn
Lilliefors Test
Level of Significance
r 0.01
ff 0.05
r 0.10
Shapiro - Wilk Test
Cancel
DSelect a variable.
DSelect a Level of Significance.
DClick on either Lilliefors Test or Shapiro-Wilk Test.
24
-------
7c. Resulting Q-Q Plot Display to Perform Normality Test
ff. ProUCL Version 3.0
File Edit View Options Summary Statistics Histogram Goodness-oF-Fit Tests UCLs Window Help - 3 X
D L* [(: . f *?
2 0 -
S 15-
£ 1 n -
£
O 05
£
O
J -1 5 .
-2 0 -
Normal Q-Q Plot for Al
^^~^
*
^^<~"
*
>^-^~*~
t *
^**~~' '
_^>->
t~~*~~*~~~*
^*~^~^
*
*
^^~*^'
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Theoretical Quantiles (Standard Normal)
N=22, Mean = 11755. 4545, Sd = 3959. 4260
Slope = 0.9977, Intercept = 0.0000, Correlation, R = 0.96359690
Liliiefors Statistic = 0.1 68, Critical Value(O.OS) = 0.1 89, Data are Normal
' .' >> .. ,t., >
:or Help, press Fl
Dlhe Q-Q plot window shown above has been resized for display.
Dlwo different Q-Q plot windows are produced for each Normality test request. The first
graph plots the raw data along the vertical axis, and the second plot (as shown above) uses
the standardized data along the vertical axis. These two Q-Q plots convey the same
information about the data distribution and potential outliers, and therefore they also look
very similar, but they do represent two separate (not duplicate) plots. It is the user's
preference to pick one of these two Q-Q plots to assess approximate normality of the data set
under study.
DRight click on a graph to print or save that graph.
DCaution: A right click of the mouse will have options other than print and save. These
options may function but are NOT recommended due to the program disruption that may
occur. Use these other options only at your own risk!
25
-------
7d. Result of Selecting Perform Lognormality Test Option
The following window will appear:
Lognormality Test
Al
As
Cr
Fe
Mn
Se
SI
Zn
Select Variables
LillieforsTest
Level of Significance
r 0.01
& 0.05
r 0.10
Shapiro - Wilks Test
Cancel
DSelect a variable.
DSelect a Level of Significance.
DClick on either Lilliefors Test or Shapiro-Wilk Test.
26
-------
7e. Resulting Lognormal Q-Q Plot Display to Perform Lognormality Test
/f ProUCL Version 3.0
_J File Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help _ n1 X
D c£ . r- . t *?
Lognormal Q-Q Plot for Co
£ 15-
j2
O
T3 05"
a>
T3
ID
4s -0 5 -
1
= -10-
Vf
2 0 -
*
^*^
^-^^
^**^
^>*^
_^*r<*^'
^ r"^
+
*
^^~^
*
^-~~~
2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Theoretical Quantiles (Standard Normal)
N = 22, Mean = 2.9320, Sd = 0.5393
Slope - 0.9891 , Intercept = 0.0000, Correlation, R - 0.95530053
Lilliefors Statistic = 0.1 61 , Critical Value(0.05) = 0.1 89, Data are Lognormal
1 'I; ' .i.l ,,
:or Help, press Fl
Dlhe Q-Q plot window shown above has been resized for display.
Dlwo different Q-Q plot windows are produced for each Lognormality test request. The first
plot uses the log-transformed data along the vertical axis, and the second plot (shown above)
uses the standardized data. As mentioned before, these two plots provide the same
information about the data distribution and potential outliers, but they do represent two
separate (not duplicate) plots. The user can pick any of these two Q-Q plots to assess
approximate lognormality of the data set under study.
DRight click on a graph to print or save that graph.
DCaution: As before, a right click of the mouse will have options other than print and save.
These options may function but are NOT recommended due to the program disruption that
may occur. Use these other options only at your own risk!
27
-------
7f. Result of Selecting Perform Gamma Test Option
The following window will appear:
Gemma Test
Select Variables
Al
As
Cr
Co
Fe
Mn
Se
SI
Anderson - Darling
Level of Significance
r 0.01
fS" 0.05
C 0.10
Kolmogorov - Smirnov
Cancel
DSelect a variable.
DSelect a Level of Significance.
DClick on either the Anderson - Darling Test or Kolmogorov - Smirnov Test.
28
-------
7g. Resulting Gamma Q-Q Plot Display to Perform Gamma Test
ff, ProllCL Version 3.0 ]
_ File Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help _ 3 X 1
D a: . f tff 1
110 -
100 -
1
o
T3
£ fin -
i bu
"E
O 5fj
30 -
20 -
Gamma Q-Q PlotfotZn
-*
--
J^**^*^
**~**~^r
^^^^
^^^^
*
^^
"
*
^^
^^
20 30 40 50 GO 70 80 90 100
Theoretical Puantiles of Gamma Distribution
N - 22, Mean - 56.348, k hat - 1 0.4668
Slope = 1.033, Intercept = -1.759, Correlation, R =0.942
A-D Test Statistic= 0.632, Critical Value(0.05) = 0.743, Data are Gamma Distributed
''-..: > ' i ' ! ! ' . . 5 ' -'
:or Helpj press Fl
Dlhe Q-Q plot window shown above has been resized for display.
DOnly one Q-Q plot window is produced for each Gamma test request: the display using the
original raw data (as shown above).
DRight click on the graph to print or save the graph.
DCaution: A right click of the mouse will have options other than print and save. These
options may function but are NOT recommended due to the program disruption that may
occur. Use these other options only at your own risk!
29
-------
8. UCLs
Dlhis option computes the UCLs for the selected variable.
Dlhe program can compute UCLs using all available methods. For details regarding the
various distributions and methods, refer to Appendix A.
Dlhe user specifies the confidence level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
Dlhe program computes several non-parametric UCLs using the Central Limit Theorem,
Chebyshev inequality, Jackknife, and the various Bootstrap methods.
DFor the bootstrap method, the user can specify the number of bootstrap runs. The default
choice for the number of bootstrap runs is 2000.
DThe user is responsible for selecting an appropriate choice for the data distribution: normal,
gamma, lognormal, or non-parametric. The user determines the data distribution using the
Goodness-of-Fit Test option prior to using the UCLs option. The UCLs option will also
inform the user if the data are normal, gamma, lognormal, or non-parametric. The program
computes relevant statistics depending on the user selection.
DFor data sets which are not normal, one should try the gamma UCLs next. The program will
offer you advice if you chose the wrong UCLs option.
DFor data sets which are neither normal nor gamma, you should try the lognormal UCLs next.
The program will offer you advice if you chose the wrong UCLs option.
DData sets that are not normal, gamma, or lognormal are classified as non-parametric data sets.
The user should use non-parametric UCLs option for such data sets. The program will offer
you advice if you chose the wrong UCLs option.
DFor lognormal data sets, ProUCL can compute only a 90% or a 95% Land's statistic based H-
UCL of the mean. For all other methods, ProUCL can compute a UCL for any confidence
coefficient in the interval [0.5,1.0), 0.5 inclusive.
Dlf you have selected a proper distribution, ProUCL will provide a recommended UCL
computation method for the 0.95 confidence coefficient. Even though ProUCL can compute
UCLs for confidence coefficients in the interval [0.5, 1.0), recommendations are provided
only for 95% UCL computation methods as the EPC term is estimated by a 95% UCL of the
mean.
30
-------
DProUCL can compute the H-UCL for sample sizes up to 1000 using the critical values as
given by Land (1975).
DFor lognormal data sets, ProUCL also computes the Maximum Likelihood Estimates (MLEs)
of the population percentiles, and the minimum variance unbiased estimates (MVUEs) of the
population mean, median, standard deviation, and the standard error (SE) of the mean. Note
that for lognormally distributed background data sets, these MLEs of the population
percentiles (e.g., 95% percentile) can be used as estimates of the background level threshold
values.
Dlhe detailed theory and formulas used to compute these gamma and lognormal statistics are
given by Land (1971, 1975), Gilbert (1987), Singh, Singh, and Engelhardt (1997, 1999),
Singh et al. (2002a), Singh et al. (2002b), and Singh and Singh (2003).
DFormulas, methods, and cited references used in the development of ProUCL are summarized
in Appendix A.
31
-------
8a. UCLs Computation Screen
Click on the UCLs menu item and the drop down menu shown below will appear.
F^ProUCL Version 3.0 - [C:V>rollCL\Data\track.xls]
D a;
1 Al
2
3
4
5
6
7
8
9
10
11
12
13
14
15
(I/*
K Da
& ^ 1
A
As
12600
14000
14900
14100
9510
9110
13900
21300
9110
14600
5270
14900
14600
10400
A ._k|-^(_||_|
t§7
s m
B
Cr
6.8
7
5.1
6.15
5.3
4.2
6.9
7
4.4
5.2
26.2
2.7
7.1
5.15
r~ -7
f Iff
C
Co
22.4
32
22.7
24.55
17
24.8
17.4
28.2
21
13.1
85.8
18.6
46.2
16.25
-H-- --i
Goodness-of-Fit Tests Window Help
3 X
mmmmm
Compute UCLs
D
Fe
18.1
19.5
17.6
20.6
17.3
14.7
21.2
14
10.7
10.4
24.5
9.6
24.6
18.45
-ij-
E
Mn
39800
45100
37600
40450
26500
38600
42700
41000
26700
31300
13600
31500
46200
29100
J-l-TJ-lf-i(^i
F
Se
501
574
368
671
1120
759
727
409
434
586
1060
950
1280
527.5
4 A 4 (-f
Fixed Exce
o
SI
0.315
1
0.17
0.488
0.4
0.5
0.34
1.1
0.45
0.8
100
0.265
0.12
0.41
(-1 --i--i
Format
ii
Zn
0.055
0.115
0.055
0.123
0.05
0.12
1
0.125
0.06
0.11
35.7
0.12
0.12
0.125
~T J~
1
46.3
45.4
61.2
48.3
37.5
36.5
68.7
55
42.6
54.3
95.3
53.7
68.1
38.45
j-(-i J-
"
Dlhe Compute UCLs option is intended for general use. It displays results in a format
suitable for review by all users. The output results can be printed or saved for subsequent
use. Saved results can be imported into other documents and reports.
D The Fixed Excel Format option produces a results screen that can be exported to another
program written for production purposes. Therefore, UCL results are stored in specific cells
and no attempt has been made to accommodate human review. These fixed format results
are not formatted to be printed.
32
-------
8b. Results After Clicking on Compute UCLs Drop-Down Menu Item
Upper Confidence Limits
Select Variables
As
Cr
Co
Fe
Mn
Se
SI
Zn
Select UCL Type
f~ Normal
f~ Gamma
(" Lognormal
r~ Non-Parametric
ff All
Confidence Coefficent [0.5,1.0) Number of Bootstrap Runs
0.95
2000
Compute UCLs
Cancel
DNote that the UCLs are computed for one variable at a time. The user selects a variable from
the variable list.
Dlhe user may change the Confidence Coefficient (default is 0.95). The range allowed is
between 0.5 and 1.0, 0.5 inclusive.
Dlhe user may adjust the number of bootstrap runs (default is 2,000).
Dlhe user selects one of the options: Normal, Gamma, Lognormal, Non-parametric, or All
option. The All option is the default choice. The All option automatically determines the
data distribution without checking for outliers and/or the presence of multiple populations.. It
is highly recommended to verify the data distribution (for outliers and multiple populations)
using an appropriate Q-Qplot under the Goodness-of-Fit Tests option.
DThe All option computes and displays the UCLs using all parametric and non-parametric
methods available in ProUCL. Finally, the user clicks on the Compute UCLs button.
33
-------
8c. Display After Selecting the Normal UCLs Option
e3» ProUCL Version 3.0 - [Normal UCL Statistics for A I]
y File Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help
_ 3 X
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
r>
A B C D
Data File C:\ProUCL\Data\track xls
Number of Valid Samples 22
Number of Distinct Samples 18
Minimum 2520
Maximum 21300
Mean 11755.455
Median 12500
Standard Deviation 3959.426
Variance 15677055
Coefficient of Variation 0.3368161
Skewness -0.209682
Shapiro-Wilk Test Statisitic 0.9437594
Shapiro-Wilk 5% Critical Value 0.911
Data are normal at 5% significance level
95% UCL (Assuming Normal Distribution)
Student's-t 13208.024
Data are normal (0.05)
Recommended UCLto use:
Use Student's-t UCL
[\ Normal Statistics /
ij press Fl
F G
Variable: Al
H
>D This data does not follow the normal distribution for the selected variable.
>D The program notes that the data follow an approximate gamma distribution and suggests in
blue that the user should try Gamma UCLs.
>D This output spreadsheet is easily saved by using the Save As option under the File menu.
>D Double right click on the UCL output spreadsheet to view a screen with more options to
save, print, or write this output sheet to a file.
34
-------
8d. Display After Selecting the Gamma UCLs Option
l^ProJ
1^! File
D G
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
i\
JCLVei'sioiri 3,0 -pammaMCL Statistics for Zn]
Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs
A B C D E F G
Data File C:\ProUCL\Data\track.xls Variable: Zn
Number of Valid Samples 22
Number of Distinct Samples 22
Minimum 35.6
Maximum 120
Mean 56.347727
Standard Deviation 19.903652
Variance 396.15535
k hat 10.466773
k star (bias corrected) 9.0697887
Theta hat 5.3834862
Theta star 6.2126835
nu hat 460.53801
nu star 399.0707
Approx.Chi Square Value (.05) 353.75646
Adjusted Level of Significance 0.0386
Adjusted Chi Square Value 350.57725
A-D Test Statistic 0.6315704
A-D 5% Critical Value 0.7434474
K-S Test Statistic 0.1313278
K-S 5% Critical Value 0.1852904
Data follow gamma distribution
at 5% signifcance level
95% UCL (Adjusted for Skewness)
Adjusted-CLT UCL 65.128042
Modified-t UCL 63.930481
95% Non-parametric UCL
Bootstrap-t UCL 66.748464
Hall's Bootstrap UCL 98.979436
95% Gamnna UCLs (Assuming Gamma Distribution)
Approximate Gamma UCL 63.565559
Adjusted Gamma UCL 64.142003
Data follow gamma distribution (O.05)
Recommended UCL to use:
Use Approximate Gamma UCL
G a mjQTa^tatjstjcs/
^^^^i^Jtal
Window Help _ 51 xl
H 1
||
DSave this output spreadsheet by using the Save As option under the File menu.
DDouble right click on the UCL output spreadsheet to view a screen with more options to
save, print, or write this output sheet to a file.
-------
8e. Display After Selecting the Lognormal UCLs Option
^JPingiy£L/!fei^
|^J File Edit View Options Summary Statistics Histogram Goodness-oF-Fit Tests UCLs Window Help a1 X
A B C D
1 ;Data File C:\ProUCL\Data\1rack.xls
2
3 Number of Valid Samples
4 Number of Distinct Samples
5 Minimum of log data
6 Maximum of log data
7 Mean of log data
8 Standard Deviation of log data
9 , Variance of log data
10
11 Shapiro-Wilk Test Statisitic
12 Shapiro-Wilk 5% Critical Value
13 Data are lognormal at 5% significance level
14
E F G H 1
Variable: Cr
22
22
2.501436
4.7095302
3.2958229
0.5602537
0.3138842
0.9149804
0.911
15 95% UCL (Assuming Normal Distribution)
16 Student's-t 41.052366
17
18 Estimates Assuming Lognormal Distribution
19 MLE Mean 31.587611
20 MLE Standard Deviation
21 ;MLE Coefficient ofVariation
22 MLE Skewness
23 MLE Median
24 MLE 80% Quantile
25 MLE 90% Quantile
26 MLE 95% Quantile
27 MLE 99% Quantile
28
29 MVU Estimate of Median
30 MVU Estimate of Mean
31 MVU Estimate of 3d
32 MVU Estimate of SE of Mean
33
34 95% Non-parametric UCL
35 Adjusted-CLT UCL (Adjusted for Skewness)
36 Modified-t UCL (Adjusted for Skewness)
37 Hall's Bootstrap UCL
38 95% Chebyshev (Mean, Sd) UCL
39 97.5% Chebyshev (Mean, Sd) UCL
40 99% Chebyshev (Mean, Sd) UCL
41
19.18102
0.6072324
2.0456027
26.999623
43.346989
55.464815
67.859553
99.382188
26.807641
31 .332966
18.480569
3.9208996
43.376415
41 .47558
80.951244
54.582557
64.255707
83.256736
42 UCLs (Assuming Lognormal Distribution)
43 95% H-UCL 40.527674
44 95% Chebyshev (MVUE) UCL
45 97.5% Chebyshev (MVUE) UCL
46 99% Chebyshev (MVUE) UCL
47
48 Data are lognormal (0.05)
49
50 Recommended UCLto use:
51 Use H-UCL
ri
» |\ Loqnormal Statistics /
48.423771
55.818976
70.345424
II
>DUse the Print or Save As option under File menu or double right click on the UCL output
spreadsheet to view a screen with more options to save, print, or write this output sheet to a
file.
36
-------
8f. Display After Selecting the Non-Parametric UCLs Option
"wf""^^
L^.ProUCL Version 3.0 - [No n- parametric UCL Statistics for SI]
f5! File
D Li
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
i\
For Help,
Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help _ S1 X
A B C D
Data File C:\ProUCL\Data\track.xls
Number of Valid Samples
Number of Unique Samples
Minimum
Maximum
Mean
Median
Standard Deviation
Variance
Coefficient of Variation
Skewness
Mean of log data
Standard Deviation of log data
95% UCL (Adjusted for Skewness)
Adjusted-CLT UCL
Modified-t UCL
95% Non-parametric UCL
CLT UCL
Jackknife UCL
Standard Bootstrap UCL
Bootstrap-t UCL
Hall's Bootstrap UCL
Percentile Bootstrap UCL
BCA Bootstrap UCL
95% Chebyshev (Mean, Sd) UCL
97.5% Chebyshev (Mean, 3d) UCL
99% Chebyshev (Mean, Sd) UCL
Data are Non-parametric (0.05)
Recommended UCL to use:
Use 99% Chebyshev (Mean, Sd)
Non-parametric Statistics /
press Fl
E F G H 1
Variable: SI
19
13
0.05
69.5
6.0651579
0.12
17.421608
303.51243
2.872408
3.2642255
-1 .322622
2.1718122
15.837417
13.49469
12.639294
12.995847
12.472003
63.261944
74.990748
13.367789
18.762789
23.486766
31.02511
45.832726
UCL
II
Dlhe program notes that the data follow an approximate gamma distribution, and suggests in
blue that the user should try Gamma UCLs.
DSave this output spreadsheet by using the Save As option under the File menu.
DDouble right click on the UCL output spreadsheet to view a screen with more options to
save, print, or write this output sheet to a file.
37
-------
8g. Display After Selecting the All UCLs Option
3 File Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help _ 31 X
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
A
Data File
BCD
C:\ProUCL\Data\track.xls
E F G
Variable: Zn
H
Raw Statistics
Number of Valid Samples 22
Number of Unique Samples 22
Minimum 35.6
Maximum 120
Mean 56.347727
Median 54.65
Standard Deviation 19.903652
Variance 396.15535
Coefficient of Variation 0.353229
Skewness 1.8624475
Gamrna Statistics
k hat 10.466773
k star (bias corrected) 9.0697887
Theta hat 5.3834862
.Theta star 6.2126835
nu hat 460.53801
nu star 399.0707
Approx.Chi Square Value (.05) 353.75646
Adjusted Level of Significance 0.0386
Adjusted Chi Square Value 350.57725
Log-transformed Statistics
Minimum of log data 3.5723456
Maximum of log data 4.7874917
Mean of log data 3.9830117
Standard Deviation of log data 0.30625
Variance of log data 0.0937891
RECOMMENDATION
Data follow gamma distribution (0.05)
Use Approximate Gamrna UCL
' JV_GeneraJ_Statistics_/~
Normal Distribution Test
Shapiro-Wilk Test Statisitic 0.8179533
Shapiro-Wilk 5% Critical Value 0.911
Data not normal at 5% significance level
95% UCL (Assuming Normal Distribution)
Student's-t UCL 63.649652
Gamma Distribution Test
A-D Test Statistic 0.6315704
A-D 5% Critical Value 0.7434474
K-S Test Statistic 0.1313278
K-S 5% Critical Value 0.1852904
Data follow gamma distribution
at 5% significance level
95% UCLs (Assuming Gamma Distribution)
Approximate Gamma UCL 63.565559
Adjusted Gamma UCL 64.142003
Lognormal Distribution Test
Shapiro-Wilk Test Statisitic 0.9260604
Shapiro-Wilk 5% Critical Value 0.911
Data are lognormal at 5% significance level
95% UCLs (Assuming Lognormal Distribution)
95% H-UCL 63.587309
95% Chebyshev (MVUE) UCL 72.348122
97.5% Chebyshev (MVUE) UCL 79.36529
99% Chebyshev (MVUE) UCL 93.149158
95% Non-pararnetric UCLs
CLTUCL 63.327619
Adj-CLT UCL (Adjusted for skewness) 65.128042
Mod-t UCL (Adjusted for skewness) 63.930481
Jackknife UCL 63.649652
Standard Bootstrap UCL 62.968288
Bootstrap-t UCL 67.192818
Hall's Bootstrap UCL 78.089743
Percentile Bootstrap UCL 63.509091
BCA Bootstrap UCL 67.022727
95% Chebyshev (Mean, Sd) UCL 74.844596
97.5% Chebyshev (Mean, Sd) UCL 82.848206
99% Chebyshev (Mean, Sd) UCL 98.569749
>D For explanations of the methods and statistics used, refer to Appendix A.
>D Use the Print or Save As option under File menu or double right click on the UCL output
spreadsheet to view a screen with more options to save, print, or write this output to a file.
38
-------
8h. Result After Clicking on Fixed Excel Format Drop-Down Menu Item
Fixed
Select Variable
Endrin aldehvde
4.4'-DDT
Dieldrin
Heptachlor
Endrin aldehyde
Dieldrin
4,4'-DDE
Aroclor-1 248
Aroclor-1 242
Number of Bootstrap Runs
2000
Compute UCLs
Cancel
DNote that the UCLs are computed for one variable at a time. The user selects a variable from
the variable list.
DFor this Fixed Format option, the 0.95 Confidence Coefficient is used in all UCL
computations.
Dlhe user may adjust the number of bootstrap runs (default is 2,000).
DClick on the Compute UCLs button to display the results.
Dlhis option will display all statistics computed by ProUCL for each of the three parametric
distributions and also for all non-parametric methods including the five bootstrap methods.
39
-------
8i. Results After Clicking the Fixed Excel Format Compute UCLs Button
File Edit View Options Summary Statistics Histogram Goodness-of-FitTests UCLs Window Help
IDC* £ na> e s | ? *?
A B
1 Data File
2 iVariable:
3 Raw Statistics
4 Number of Observations
5 Number of Missing Data
C D | E F I G [ H [ I
"D:\ProUCL\DATA\CDELV1.XLS
Endrin aldehyde
17
0
6 INumber of Valid Samples 17
7 Number of Unique Samples 16
8 jMinimum
9 jMaximum
10 [Mean
11 Standard Deviation
12 Variance
13 iCoefficient of Variation
14 Skewness
15 jToo Few Observations?
16 Normal Statistics
17 jLilliefors Test Statisitic
0.0018
120
7.7820765
28.960933
838.73566
3.7214917
4.1026919
NO
N/R Shapiro Wilk method yields a more
18 Lilliefors 5% Critical Value N/R Shapiro Wilk method yields a more
19 IShapiro-Wilk Test Statisitic 0.2945067
20 iShapiro-Wilk 5% Critical Value 0.892
21 J5% Normality Test Result NOT NORMAL Data not normal at 5% significance
22 J95% Student's-t UCL
23 Gamma Statistics
20.045264
DNote that the output is not sized to fit a printed page.
Dlhis option can be omitted by all users who are not planning to import the ProUCL
calculation results into some other software to automate the calculations of exposure point
concentration terms. That is, all users who are not planning to use ProUCL as a production
tool to produce UCLs for several variables and data files may skip the use of this option.
DOn Fixed Format output spreadsheet, each row contains a single item description or
calculated statistic.
Dlhree primary columns contain information:
4 Column A is a description of the various results and statistics.
4 Column E contains all appropriate calculated results.
4 Column G contains additional descriptive information as needed.
4 Note that information from the primary columns (e.g., A, E, and G) may overflow
into the columns to the right.
40
-------
>DFor column E:
4 N/A means that the calculation for the associated statistic is not available.
4 N/R means that the calculations for the associated statistic may not be reliable.
4 Row 15 displays YES if there are too few observations to calculate appropriate UCL
statistics and displays NO if enough observations are available to compute all
relevant statistics and UCLs.
4 Row 35 displays AD GAMMA (if data are gamma distributed using A-D test) or
NOT AD GAMMA (if data are not gamma distributed using A-D test) using the
Anderson-Darling Gamma Test for 0.05 level of significance.
4 Similarly, Row 38 displays KS GAMMA or NOT KS GAMMA using the
Kolmogorov-Smirnov Gamma Test for 0.05 level of significance.
4 As mentioned before, it should be noted that these two goodness-of-fit tests may lead
to different conclusion (as is the case with other goodness-of-fit tests) about the data
distribution. In that case, ProUCL leads to the conclusion that the data follow an
approximate gamma distribution.
4 Row 39 displays NOT GAMMA, APPROX GAMMA, or GAMMA depending on the
results of the two Gamma goodness-of-fit tests.
4 Row 52 displays LOGNORMAL or NOT LOGNORMAL depending on the result of
the appropriate lognormality test for 0.05 level of significance.
4 Row 86 displays YES if user inspection is recommended and displays NO if no
potential problems requiring manual inspection needed with the selected variable.
4 Row 87 displays NORMAL, GAMMA, LOGNORMAL, or NON-PARAMETRIC as
the distribution used in determining 95% UCL computation recommendations.
4 Row 88 displays a recommended UCL value to use as an estimate of the EPC term.
4 Row 89 displays a second recommended UCL (e.g., use of either Hall's bootstrap or
bootstrap-t method may be recommended on the same data set). These cells will be
blank if only one UCL is recommended for the selected variable.
4 Row 90 displays a third recommended UCL. These cells will be blank if only one or
two UCLs are recommended for the selected variable.
4 Row 91 displays YES if the recommended 95% UCL exceeds the maximum value in
the data set.
4 Row 92 displays PLEASE CHECK if the recommended bootstrap UCLs are subject
to erratic or inflated values due to possible presence of outliers. Otherwise, row 92
displays NONE.
4 Row 93 displays IN CASE if the recommended bootstrap UCL has an inflated value
due to the presence of outliers. Otherwise, row 93 displays NONE.
>DFor column G:
4 Row 88 displays the name of the recommended 95% UCL.
4 Row 89 displays the name of the second recommended 95% UCL. These cells will
be blank if only one UCL is recommended for the selected variable.
41
-------
Row 90 displays the name of the third recommended 95% UCL. These cells will be
left blank if only one UCL is recommended for the selected variable.
Row 93 displays the name of the alternative UCL to utilize if the recommended
bootstrap (e.g., bootstrap-t or Hall's bootstrap) 95% UCL has an inflated value due to
presence of potential outliers.
42
-------
9. Window
Click on the Window menu to reveal these drop-down options.
D
,
File Edit View Options Summary Statistics Histogram Goodness-of-RtTests UCLs Window Help
I G New Window
Cascade
Tile
Arrange Icons
A I _ B | _^C
12 Skewness
13 Mean of log data
14 Standard Deviation of log data
15
16 95% UCL (Adjusted for Skewness)
17 Adjusted-CLT UCL
18 Modified-tUCL
19
20 95% Non-parametric UCL
21 CLT UCL
E
TT026919'
-2.867611
3.4991462
26.803772
21.210144
19.335624
n]_x]
]
L
1 H:\cdelvl.XLS
2 Normal UCL Statistics for Endrin aldehyde
3 Gamma UCL Statistics for Endrin aldehyde
4 Lognormal UCL Statistics for Endrin aldehyde
5 Non-parametric UCL Statistics for Endrin aldehyde
The following Window drop-down menu options are available:
DNew Window option: opens a blank spreadsheet window.
DCascade option: arranges windows in a cascade format. This is similar to a typical Windows
program option.
DTile option: resizes each window and then displays all open windows. This is similar to a
typical Windows program option.
DArrange Icons: similar to a typical Windows program option.
DThe drop-down options include a list of all open windows with a check mark in front of the
active window. Click on any of the windows listed to make that window active.
43
-------
10. Help
Click on the Help menu item to reveal these drop-down options.
File Edit View Options Summary Statistics Histogram Goodness-of-FitTests UCLs Window | Help .-Jg,l..xJ
Help Topics
About ProUCL...
12
13
14
15
16
17
18
19
20
21
A I B [ C C
Skewness
Mean of log data
Standard Deviation of log data
95% UCL (Adjusted for Skewness)
Adjusted-CLT UCL
Modified-t UCL
95% Non-parametric UCL
CLT UCL
22 Jackknife UCL
23 Standard Bootstrap U(
24 Bootstrap-t UCL
25 [Hall's Bootstrap UCL
26 Percentile Bootstrap L
pT r>,^ A n~~4-~4-.-~~ I l^--l
* I > |\ Non-parametric Statist!
:L
CL
cs /
) E I F G | H
471026919
-2.867611
3.4991462
26.803772
21.210144
19.335624
20.045264
18.987354
207,52895
186.10293
21.769762
'"S-l A ^i A C;r~i A
The following Help drop-down menu options are available:
DHelp Topics option: ProUCL version 3.0 does not have an online help program.
DAbout ProUCL: displays the software version number.
44
-------
Run Time Notes
DCell size can be changed. The user can change the size of a cell by moving the mouse to the
top row (the gray shaded row with a letter), then moving the mouse to the right side until the
cursor changes to an arrow symbol (^X depress the left mouse button.
Dlhis can be used to reveal additional precision or hidden text.
/f ProUCL Version
.jDJxJ
15 File Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help _ |g| x|
A B | C | D
1 'Data File HAcdelvlTXLS
2 ;
3 Number of Valid Samples 17
4 Number of Unique Samples 16
5 Minimum 0.0018
6 .Maximum 120
7 'Mean 7782076471
8 'Standard Deviation 28.96093326
9 Variance 838.7356553
10 ~'khat 0.155424908
11 k star (bias corrected) 0.1672126693
12 Theta hat 50.06968684
13 Theta star 46.53999306
14 nu hat 5.284446872
15 nu star 5.685230757
16 .Approx.Chi Square Value (.05) 1.480711891
17 Adjusted Level of Significance 0.03461
18 Adjusted Chi Square Value 1.269069171
4 ^ (\ G^miD§^lMsticsV
E I F G [ H
Variable: Endrin aldehyde
li
45
-------
Rules to Remember When Editing or Creating a New Data File
S^^^^Sl
ti.ji 1 i i
Dl File
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
< .>J\
_- lull x j
Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help _ \&\ x)
A [ B | C j D E F
test
1.3
2.9
4.1
0
2
5
3.2
0.5
1.9
4.3
SheetL/
G I H I I
1 i.
||
> Text may appear in the first row only. This row has column headers (variable names) for
your data.
>D All alphanumeric text (including blanks, strings) appearing elsewhere (other than first row)
will be treated as zero data.
>D Missing data (alphanumeric text, blanks) can be set to a large value such as IxlO31. All
entries with this value will be ignored from the computations.
>D The last data entry for each column must be non-zero. The program determines the number
of observations by working backwards up the data until a non-zero value is encountered.
Data in each column must end with a non-zero entry as shown above otherwise that zero
value will be ignored. All intermediate zero entries are treated as valid data.
>D It is recommended to use the default settings of the Data location screen when working with
your data sets.
46
-------
C. Recommendations to Compute a 95% UCL of the Population Mean (The
Exposure Point Concentration Term)
This section describes the recommendations on the computation of a 95% UCL of the unknown
population arithmetic mean, //b of a contaminant data distribution. These recommendations are
based upon the findings of Singh, Singh, and Engelhardt (1997, 1999); Singh et al. ( 2002a);
Singh, Singh, and laci (2002b); and Singh and Singh (2003). These recommendations are
applicable to full data sets without censoring and non-detect observations.
Recommendations have been summarized for:
1) normally distributed data sets,
2) gamma distributed data sets,
3) lognormally distributed data sets, and
4) data sets which are non-parametric and do not follow any of the above mentioned three
distributions included in ProUCL.
A detailed description of the recommendations can be found in Section 5 of Appendix A. Also,
a list of all cited references is given in Appendix A.
For skewed parametric as well as non-parametric data sets, there is no simple solution to
compute a 95% UCL of the population mean, //;. Contrary to the general conjecture, Singh et al.
(2002a), Singh, Singh, and laci (2002b), and Singh and Singh (2003) noted that the UCLs based
upon the skewness adjusted methods, such as the Johnson's modified-t and Chen's adjusted-CLJ
do not provide the specified coverage (e.g., 95 %) to the population mean even for mildly to
moderately skewed (e.g.,
-------
D. Recommendations to Compute a 95% UCL of the Population Mean, //7
Using Symmetric and Positively Skewed Data Sets
Graphs from Singh and Singh (2003) showing coverage comparisons (e.g., attainment of the
specified confidence coefficient) for normal, gamma, and lognormal distributions for the various
methods considered are given in Appendix C. The user may want to consult those graphs for a
better understanding of the recommendations summarized in this section.
1. Normally or Approximately Normally Distributed Data Sets
For normally distributed data sets, a UCL based upon the Student's-t statistic as given by
equation (32) of Appendix A provides the optimal UCL of the population mean. Therefore,
for normally distributed data sets, one should always use a 95% UCL based upon the
Student's-t statistic.
The 95% UCL of the mean given by equation (32) based upon Student's-t statistic may also
be used when the sd, sy of the log-transformed data is less than 0.5, or when the data set
approximately follows a normal distribution. A data set is approximately normal when the
normal Q-Q plot displays a linear pattern (without outliers and significant jumps) and the
resulting correlation coefficient is quite high (e.g., 0.95 or higher).
Student's-t UCL may also be used when the data set is symmetric (but possibly not normally
yv
distributed). A measure of symmetry (or skewness) is &3, which is given by equation (43) of
Appendix A. As a rule of thumb, a value of k3 close to zero (e.g., \k3\< 0.2 - 0.3) suggests
approximate symmetry. The approximate symmetry of a data distribution can also be judged
by evaluating the histogram of the data set.
48
-------
2. Gamma Distributed Skewed Data Sets
In practice, many skewed data sets can be modeled both by a lognormal distribution and a
gamma distribution, especially when the sample size is smaller than 100. Land's H-statistic
based, 95% H-UCL of the mean based upon a lognormal model often results in an unjustifiably
large and impractical 95% UCL value. In such cases, a gamma model, G (k,0), may be used to
compute a reliable 95% UCL of the unknown population mean, //;.
Many skewed data sets follow a lognormal as well as a gamma distribution. It should be
noted that the population means based upon the two models can differ significantly. The
lognormal model, based upon a highly skewed (e.g., a > 2.5) data set, will have an
unjustifiably large and impractical population mean, //13 and its associated UCL. The gamma
distribution is better suited to model positively skewed environmental data sets.
One should always first check if a given skewed data set follows a gamma distribution. If a
data set does follow a gamma distribution or an approximate gamma distribution, one should
compute a 95% UCL based upon a gamma distribution. Use of highly skewed (e.g., a > 2.5-
3.0) lognormal distributions should be avoided. For such highly skewed lognormally
distributed data sets that can not be modeled by a gamma or an approximate gamma
distribution, non-parametric UCL computation methods based upon the Chebyshev
inequality may be used. ProUCL prints out at least one recommended UCL associated with
each data set.
The five bootstrap methods do not perform better than the two gamma UCL computation
methods. It is noted that the performances (in terms of coverage probabilities) of bootstrap-t
and Hall's bootstrap methods are very similar. Out of the five bootstrap methods, bootstrap-t
and Hall's bootstrap methods perform the best (with coverage probabilities for the population
mean closer to the nominal level of 0.95). This is especially true when skewness is quite
high (e.g., k < 0.1) and sample size is small (e.g., n < 10-15). This is illustrated in the
graphs given in Appendix C. As mentioned before, whenever the use of Hall's UCL or
bootstrap-t UCL is recommended, an informative warning message about their use is also
printed.
Also, contrary to the conjecture, the bootstrap BCA method does not perform better than the
Hall's method or the bootstrap-t method. The coverage for the population mean, ^ provided
by the BCA method is much lower than the specified 95% coverage. This is especially true
when the skewness is high (e.g., k < 1) and sample size is small (Singh and Singh (2003)).
From the results presented in Singh, Singh, and laci (2002b) and in Singh and Singh (2003),
it is concluded that for data sets which follow a gamma distribution, a 95% UCL of the mean
should be computed using the adjusted gamma UCL when the shape parameter, k, is:
49
-------
0.1 < k < 0.5, and for values ofk > 0.5, a 95% UCL can be computed using an approximate
gamma UCL of the mean, ^.
For highly skewed gamma distributed data sets with k < 0.1, bootstrap-t UCL or Hall's
bootstrap (Singh and Singh (2003)) may be used when the sample size is small (e.g., n < 15)
and adjusted gamma UCL should be used when sample size starts approaching and exceeding
15. The small sample size requirement increases as skewness increases (that is as k
decreases, n is required to increase).
It should be pointed out that the bootstrap-t and Hall's bootstrap methods should be used
with caution as some times these methods yield erratic, unreasonably inflated, and unstable
UCL values, especially in the presence of outliers. In case Hall's bootstrap and bootstrap-t
methods yield inflated and erratic UCL results, the 95% UCL of the mean should be
computed based upon adjusted gamma UCL.
These recommendations for the use of gamma distribution are summarized in Table 1.
Table 1
Summary Table for the Computation of a 95% UCL
of the Unknown Mean, jul of a Gamma Distribution
k
k>0.5
0.1 < k < 0.5
; 15
Recommendation
Approximate Gamma 95%UCL
Adjusted Gamma 95% UCL
95% UCL Based Upon Bootstrap-t or Hall's
Bootstrap Method *
Adjusted Gamma 95% UCL if available,
otherwise use Approximate Gamma 95% UCL
* If bootstrap-t or Hall's bootstrap methods yield erratic, inflated, and unstable UCL values
(which often happens when outliers are present), the UCL of the mean should be computed using
adjusted gamma UCL.
50
-------
3. Lognormally Distributed Skewed Data Sets
For lognormally distributed data sets, LN(/u, a2), the H-statistic based UCL provides the
specified 0.95 coverage for the population mean for all values of a. However, the H-statistic
often results in unjustifiably large UCL values which do not occur in practice. This is especially
true when skewness is high (e.g., a > 2.0). The use of a lognormal model unjustifiably
accommodates large and impractical values of the mean concentration and its UCLs. The
problem associated with the use of a lognormal distribution is that the population mean, ^ of a
lognormal model becomes impractically large for larger values of a, which in turn results in
inflated H-UCL of the population mean, //;. Since the population mean of a lognormal model
becomes too large, none of the other methods except for the inflated H-UCL provides the
specified 95% coverage for that inflated population mean, //;. This is especially true when the
sample size is small and skewness is high. For extremely skewed data sets (with a > 2.5-3.0) of
sizes (e.g., < 70-100), the use of a lognormal distribution based H-UCL should be avoided (e.g.,
see Singh et al. (2002a), Singh and Singh (2003)). Therefore, alternative UCL computation
methods such as the use of a gamma distribution, or the use of a UCL based upon non-parametric
bootstrap methods or Chebyshev inequality based methods, are desirable. All skewed data sets
should first be tested for a gamma distribution. For lognormally distributed data sets (that can
not be modeled by a gamma distribution), the method as summarized in Table 2 on the following
page, may be used to compute a 95% UCL of the mean. The details can be found in Appendix
A.
ProUCL can compute an H-UCL for samples of sizes up to 1000. For highly skewed
lognormally distributed data sets of smaller sizes, some alternative methods to compute a 95%
UCL of the population mean, //13 are summarized in Table 2. Since skewness (as defined in
Section 3.2.2, Appendix A) is a function of a (or
-------
Table 2
Summary Table for the Computation of a 95% UCL
of the Unknown Mean, //1 of a Lognormal Population
a
a <0.5
0.5 < 6< 1.0
1.0 < a < 1.5
1.5 < 6<2.0
2.0 < 6<2.5
2.5 < 6<3.0
3.0 <6< 3.5
6>3.5
Sample Size, n
For all n
For all n
n<25
n> 25
n<20
20 < n < 50
n> 50
n<20
20 < n<50
50 < n < 70
n> 70
n<30
30 < n<70
70 < n<100
n > 100
n< 15
15 < n<50
50 < n<100
100 < n< 150
n > 150
For all n
Recommendation
Student' s-t, modified-t, or H-UCL
H-UCL
95% Chebyshev (MVUE) UCL
H-UCL
99% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
99% Chebyshev (MVUE) UCL
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Larger of (99% Chebyshev (MVUE) UCL,
99% Chebyshev(Mean, Sd))
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Hall's bootstrap method *
Larger of (99% Chebyshev (MVUE) UCL,
99% Chebyshev(Mean, Sd))
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Use non-parametric methods *
* If Hall's bootstrap method yields an erratic unrealistically large UCL value, then the UCL of
the mean may be computed based upon the Chebyshev inequality.
52
-------
4. Data Sets Without a Discernable Skewed Distribution - Non-parametric Skewed Data
Sets
The use of gamma and lognormal distributions as discussed here will cover a wide range of
skewed data distributions. For skewed data sets which are neither gamma nor lognormal, one
can use a non-parametric Chebyshev UCL or Hall's bootstrap UCL (for small data sets) of the
mean to estimate the EPC term.
For skewed non-parametric data sets with negative and zero values, use a 95%
Chebyshev (Mean, Sd) UCL of the mean, fa to estimate the EPC term.
For all other non-parametric data sets with only positive values, the following method
may be used to estimate the EPC term:
For mildly skewed data sets with a < 0.5, one can use the Student's-t statistic or
modified-t statistic to compute a 95% UCL of the mean, fa.
For non-parametric moderately skewed data sets (e.g., a or its estimate, a in the interval
(0.5, 1]), one may use a 95% Chebyshev (Mean, Sd) UCL of the population mean, fa.
For non-parametric moderately to highly skewed data sets (e.g., 6 in the interval (1.0,
2.0]), one may use a 99% Chebyshev (Mean, Sd) UCL or 97.5% Chebyshev (Mean, Sd)
UCL of the population mean, fa, to obtain an estimate of the EPC term.
For highly skewed to extremely highly skewed data sets with 6 in the interval (2.0, 3.0],
one may use Hall's UCL or 99% Chebyshev (Mean, Sd) UCL to compute the EPC term.
Extremely skewed non-parametric data sets with a exceeding 3.0 are badly behaved and
UCLs based upon such data sets often provide poor coverage to the population mean.
For such highly skewed data distributions, none of the methods considered provide the
specified 95% coverage for the population mean, fa. The coverages provided by the
various methods decrease as a increases. For such highly skewed data sets of sizes (e.g.,
< 30), a 95% UCL can be computed based upon Hall's bootstrap method or bootstrap-t
method. Hall's bootstrap method provides the highest coverage (but less than 0.95) when
the sample size is small. It is noted that the coverage for the population mean provided
by Hall's method (and bootstrap-t method) does not increase much as the sample size, n
increases. However, as the sample size increases, coverage provided by 99% Chebyshev
(Mean, Sd) UCL method increases. Therefore, for larger samples, a UCL should be
computed based upon 99% Chebyshev (Mean, Sd) method. This large sample size
requirement increases as
-------
Note: As mentioned before, the Hall's bootstrap method (and also bootstrap-t method)
sometimes yields erratic and unstable UCL values, especially when the outliers are present. If
Hall's bootstrap UCL represents an erratic and unstable value, a UCL of the population mean
may be computed using the 99% Chebyshev (Mean, Sd) method.
Table 3
Summary Table for the Computation of a 95% UCL of the Unknown Mean,
//! of a Skewed Non-parametric Distribution with all Positive Values,
Where a is the Sd of Log-transformed Data
yv
3.5
Sample Size, n
For all n
For all n
n<50
n > 50
n<10
n > 10
n<30
n > 30
n< 100
n> 100
Recommendation
95% UCL based upon Student' s-t statistic or
Modified-t statistic
95% Chebyshev (Mean, Sd) UCL
99% Chebyshev (Mean, Sd) UCL
97.5% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL*
99% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL *
99% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL *
99% Chebyshev (Mean, Sd) UCL
* If the Hall's bootstrap method yields an erratic and unstable UCL value (e.g., this tends to
happen when outliers are present), the EPC term may be computed using the 99% Chebyshev
(Mean, Sd) UCL.
54
-------
E. Should the Maximum Observed Concentration be Used as an Estimate of
the EPC Term?
Singh and Singh (2003) also included the Max Test (using the maximum observed value as an
estimate of the EPC term) in their simulation study. Previous (e.g., EPA 1992 RAGS Document)
use of the maximum observed value has been recommended as a default value to estimate the
EPC term when a 95% UCL (e.g., the H-UCL) exceeded the maximum value. Only two 95%
UCL computation methods, namely: the Student's-1 UCL and Land's H-UCL were used
previously to estimate the EPC term (e.g., EPA 1992). ProUCL can compute a 95% UCL of
mean using several methods based upon normal, Gamma, lognormal, and non-parametric
distributions. Thus, ProUCL has about fifteen (15) 95% UCL computation methods, at least one
of which (depending upon skewness and data distribution) can be used to compute an
appropriate estimate of the EPC term. Furthermore, since the EPC term represents the average
exposure contracted by an individual over an exposure area (EA) during a long period of time;
therefore, the EPC term should be estimated by using an average value (such as an appropriate
95% UCL of the mean) and not by the maximum observed concentration. With the availability
of so many UCL computation methods, the developers of ProUCL, Version 3.0 do not feel any
need to use the maximum observed value as an estimate of the EPC term. Singh and Singh
(2003) also noted that for skewed data sets of small sizes (e.g., <10-20), the Max Test does not
provide the specified 95% coverage to the population mean, and for larger data sets, it
overestimates the EPC term which may require unnecessary further remediation. This can also
be viewed in the graphs presented in Appendix C. Also, for the distributions considered, the
maximum value is not a sufficient statistic for the unknown population mean. The use of the
maximum value as an estimate of the EPC term ignores most (except for the maximum value) of
the information contained in a data set. It is, therefore not desirable to use the maximum
observed value as an estimate of the EPC term representing average exposure by an individual
over an EA It is recommended that the maximum observed value NOT be used as an
estimate of the EPC term. However, for the sake of interested users, ProUCL displays a
warning message when the recommended 95% UCL (e.g., Hall's bootstrap UCL etc.) of the
mean exceeds the observed maximum concentration. For such cases (when a 95% UCL does
exceed the maximum observed value), if applicable, an alternative UCL computation method is
recommended by ProUCL.
It should also be noted that for highly skewed data sets, the sample mean indeed can even exceed
the upper 90%, 95 % etc. percentiles, and consequently, a 95% UCL of mean can exceed the
maximum observed value of a data set. This is especially true when one is dealing with
lognormally distributed data sets of small sizes. For such highly skewed data sets which can not
be modeled by a gamma distribution, a 95% UCL of the mean should be computed using an
appropriate non-parametric method. These recommendations are summarized in Tables 1
through 3 of this User Guide.
55
-------
Alternatively, for such highly skewed data sets, other measures of central tendency such as the
median (or some higher order quantile such as 70% etc.) and its upper confidence limit may be
considered. The EPA, all other interested agencies and parties need to come to an agreement on
the use of median and its UCL to estimate the EPC term. However, the use of the sample
median and/or its UCL as estimates of the EPC term needs further research and investigation.
56
-------
F. Left-Censored Data Sets with Non-detects
ProUCL does not handle the left-censored data sets with non-detects, which are inevitable in
many environmental studies. All parametric as well as non-parametric recommendations to
compute the mean, standard deviation, and a 95% UCL of the mean made by ProUCL software
are based upon full data sets without censoring. For mild to moderate number of non-detects
(e.g., < 15%), one may compute these statistics based upon the commonly used rule of thumb of
using !/2 detection limit (DL) proxy method. However, the proxy methods should be used
cautiously, especially when one is dealing with lognormally distributed data sets. For
lognormally distributed data sets of small sizes, even a single value - small (e.g., obtained after
replacing the non-detect by 1A DL) or large (e.g., an outlier) can have a drastic influence (can
yield an unrealistically large 95% UCL) on the value of the associated Land's 95% UCL. The
issue of estimating the mean, standard deviation, and a 95% UCL of the mean based upon left-
censored data sets of varying degrees (e.g., <15%, 15%-50%, 50%-75%, or greater than 75%
etc.) of censoring is currently under investigation.
57
-------
Glossary
This glossary defines selected words in this User Guide to describe impractically large UCL
values of the unknown population mean, //,. In practice, the UCLs based upon Land's H-statistic
(H-UCL), and some bootstrap methods such as the bootstrap-t and Hall's bootstrap methods
(especially when outliers are present) can become impractically large. The UCLs based upon
these methods often become larger than the UCLs based upon all other methods by several
orders of magnitude. Such large UCL values are not achievable as they do not occur in practice.
Words like unstable and unrealistic have been used to describe such impractically large UCL
values.
UCL: Upper Confidence Limit of the unknown population mean.
Coverage = Coverage Probability: The coverage probability (e.g., = 0.95) of a UCL of the
population mean represents the confidence coefficient associated with the UCL.
Optimum: An interval is optimum if it possesses optimal properties as defined in the statistical
literature. This may mean that it is the shortest interval providing the specified coverage (e.g.,
0.95) to the population mean. For example, for normally distributed data sets, the UCL of the
population mean based upon Student's t distribution is optimum.
Stable UCL: The UCL of a population mean is a stable UCL if it represents a number of
practical merit, which also has some physical meaning. That is, a stable UCL represents a
realistic number (e.g., contaminant concentration) that can occur in practice. Also, a stable UCL
provides the specified (at least approximately, as much as possible, as close as possible to the
specified value) coverage (e.g., -0.95) to the population mean.
Reliable UCL: This is similar to a stable UCL.
Unstable UCL = Unreliable UCL = Unrealistic UCL: The UCL of a population mean is
unstable, unrealistic, or unreliable if it is orders of magnitude higher than the various other UCLs
of population mean. It represents an impractically large value that cannot be achieved in
practice. For example, the use of Land's H statistic often results in impractically large inflated
UCL value. Some other UCLs such as the bootstrap-t UCL and Hall's UCL, can be inflated by
outliers resulting in an impractically large and unstable value. All such impractically large UCL
values are called unstable, unrealistic, unreliable, or inflated UCLs in this User Guide.
58
-------
References
EPA (1992), "Supplemental Guidance to RAGS: Calculating the Concentration Term,"
Publication EPA 9285.7-081, May 1992.
Gilbert, R.O. (1987), Statistical Methods for Environmental Pollution Monitoring, New York:
Van Nostrand Reinhold.
Hardin, J.W., and Gilbert, R.O. (1993), "Comparing Statistical Tests for Detecting Soil
Contamination Greater Than Background," Pacific Northwest Laboratory, Battelle, Technical
Report #DE 94-005498.
Land, C. E. (1971), "Confidence Intervals for Linear Functions of the Normal Mean and
Variance," Annals of Mathematical Statistics, 42, 1187-1205.
Land, C. E. (1975), "Tables of Confidence Limits for Linear Functions of the Normal Mean and
Variance," in Selected Tables in Mathematical Statistics, Vol. Ill, American Mathematical
Society, Providence, R.I., 385-419.
Schulz, T. W., and Griffin, S. (1999), Estimating Risk Assessment Exposure Point
Concentrations when Data are Not Normal or Lognormal. Risk Analysis, Vol. 19, No. 4, 1999.
Scout: A Data Analysis Program, Technology Support Project. EPA, NERL -LV, Las Vegas,
NV 89193-3478.
Singh, A. K., Singh, Anita, and Engelhardt, M., "The Lognormal Distribution in Environmental
Applications," EPA/600/R-97/006, December 1997.
Singh, A. K., Singh, Anita, and Engelhardt, M., "Some Practical Aspects of Sample Size and
Power Computations for Estimating the Mean of Positively Skewed Distributions in
Environmental Applications," EPA/600/S-99/006, November 1999.
Singh, A,. Singh, A.K., Engelhardt, M., and Nocerino, J.M. (2002a), " On the Computation of
the Upper Confidence Limit of the Mean of Contaminant Data Distributions." Under EPA
Review.
Singh, A., Singh, A. K., and laci, R. J. (2002b). " Estimation of the Exposure Point
Concentration Term Using a Gamma Distribution." EPA/600/R-02/084.
59
-------
Singh, A. and Singh, A.K. (2003). Estimation of the Exposure Point Concentration Term (95%
UCL) using Bias-Corrected Accelerated (BCA) Bootstrap Method and Several Other Methods
for Normal, Lognormal, and Gamma Distributions. Draft EPA Internal Report.
60
-------
APPENDIX A
TECHNICAL BACKGROUND
METHODS FOR COMPUTING
THE EPC TERM ((1-a) 100%UCL)
AS INCORPORATED IN
ProUCL VERSION 3.0 SOFTWARE
-------
METHODS FOR COMPUTING THE EPC TERM ((1-cc) 100%UCL)
AS INCORPORATED IN ProlJCL VERSION 3.0 SOFTWARE
1. Introduction
Exposure assessment and cleanup decisions in support of U.S. EPA projects are often made
based upon the mean concentrations of the contaminants of potential concern. A 95% upper
confidence limit (UCL) of the unknown population arithmetic mean (AM), fa, is often used to:
estimate the exposure point concentration (EPC) term (EPA, 1992, EPA, 2002), determine the
attainment of cleanup standards (EPA, 1989 and EPA, 1991), estimate background level
contaminant concentrations, or compare the soil concentrations with site specific soil screening
levels (EPA, 1996). It is, therefore, important to compute a reliable, conservative, and stable
95% UCL of the population mean using the available data. The 95% UCL should
approximately provide the 95% coverage for the unknown population mean, fa. EPA (2002) has
developed a guidance document for calculating upper confidence limits for hazardous waste
sites. All of the UCL computation methods as described in the EPA (2002) guidance document
are available in ProUCL, Version 3.0. Additionally, ProUCL, Version 3.0 can also compute a
95% UCL of the mean based upon the gamma distribution which is better suited to model
positively skewed environmental data sets.
Computation of a (1-a) 100% UCL of the population mean depends upon the data
distribution. Typically, environmental data are positively skewed, and a default lognormal
distribution (EPA, 1992) is often used to model such data distributions. The H-statistic based
Land's (Land 1971, 1975) H-UCL of the mean is used in these applications. Hardin and Gilbert
(1993), Singh, Singh, and Engelhardt (1997,1999), Schultz and Griffin, 1999, Singh et al.
(2002a), and Singh, Singh, and laci (2002b) pointed out several problems associated with the
A-l
-------
use of the lognormal distribution and the H-UCL of the population AM. In practice, for
lognormal data sets with high standard deviation (Sd), a of the natural log-transformed data
(e.g., a exceeding 2.0), the H-UCL can become unacceptably large, exceeding the 95% and
99% data quantiles, and even the maximum observed concentration, by orders of magnitude
(Singh, Singh, and Engelhardt, 1997). This is especially true for skewed data sets of sizes
smaller than n < 50 - 70.
The H-UCL is also very sensitive to a few low or high values. For example, the addition of a
sample with below detection limit measurement can cause the H-UCL to increase by a large
amount (Singh, Singh, and laci, (2002b)). Realizing that the use of H-statistic can result in
unreasonably large UCL, it has been recommended (EPA, 1992) to use the maximum observed
value as an estimate of the UCL (EPC term) in cases where the H-UCL exceeds the maximum
observed value. Recently, Singh, Singh and laci (2002b), and Singh and Singh (2003) studied
the computation of the UCLs based upon a gamma distribution and several non-parametric
bootstrap methods. Those methods have also been incorporated in ProUCL, Version 3.0. There
are fifteen UCL computation methods available in ProUCL; five are parametric and ten are non-
parametric. The non-parametric methods do not depend upon any of the data distributions.
Graphs from Singh and Singh (2003) showing coverage comparisons for normal, gamma, and
lognormal distributions for the various methods are given in Appendix C.
Both lognormal and gamma distributions can be used to model positively skewed data sets.
It should be noted that it is hard to distinguish between a lognormal and a gamma distribution,
especially when the sample size is small such as n < 50 - 70. In practice many skewed data sets
follow a lognormal as well as a gamma distribution. Singh, Singh, and laci (2002b) observed
that the UCL based upon a gamma distribution results in reliable and stable values of practical
merit. It is therefore, always desirable to test if an environmental data set follows a gamma
distribution. For data sets (of all sizes) which follow a gamma distribution, EPC should be
A-2
-------
computed using an adjusted gamma UCL (when 0.1 < k < 0.5) of the mean or an approximate
gamma UCL (when k > 0.5) of the mean as these UCLs approximately provide the specified
95% coverage to the population mean,//j = kO of a gamma distribution. For values of k< 0.1,
a 95% UCL may be obtained using bootstrap-t method or Hall's bootstrap method when the
sample size, n is less than 15, and for larger samples, a UCL of the mean should be computed
using the adjusted or approximate gamma UCL. Here, k is the shape parameter of a gamma
distribution as described in Section 2.2. It should be pointed out that both bootstrap-t and Hall's
bootstrap methods sometimes result in erratic, inflated, and unstable UCL values especially in
the presence of outliers. Therefore, these two methods should be used with caution. The user
should examine the various UCL results and determine if the UCLs based upon the bootstrap-t
and Hall's bootstrap methods represent reasonable and reliable UCL values of practical merit. If
the results based upon these two methods are much higher than the rest of methods (except for
the UCLs based upon lognormal distribution), then this could be an indication of erratic UCL
values. ProUCL prints out a warning message whenever the use of these two bootstrap methods
is recommended. In case these two bootstrap methods yield erratic and inflated UCLs, the UCL
of the mean should be computed using the adjusted or the approximate gamma UCL computation
method.
ProUCL has been developed to test for normality, lognormality, and a gamma distribution of
a data set, and to compute a conservative and stable 95% UCL of the population mean, //;. The
critical values of Anderson-Darling test statistic and Kolmogorov-Smirnov test statistic to test
for gamma distribution were generated using Monte Carlo simulation experiments. These
critical values are tabulated in Appendix B for various levels of significance. Singh, Singh, and
Engelhardt (1997,1999), Singh, Singh, and laci (2002b), and Singh and Singh (2003) studied
several parametric and non-parametric UCL computation methods which have been included in
ProUCL. Most of the mathematical algorithms and formulae used in ProUCL to compute the
various statistics are summarized in this Appendix A. For details, the user is referred to Singh
gh,
A-3
-------
Singh, and laci (2002b), and Singh and Singh (2003). Some graphs from Singh and Singh
(2003) showing coverage comparisons for normal, gamma, and lognormal distributions for the
various methods are given in Appendix C. ProUCL computes the various summary statistics for
raw, as well as log-transformed data. In this User Guide and in ProUCL, log-transform (log)
stands for the natural logarithm (In) to the base e. ProUCL also computes the maximum
likelihood estimates (MLEs) and the minimum variance unbiased estimates (MVUEs) of various
unknown population parameters of normal, lognormal, and gamma distributions. This, of
course, depends upon the underlying data distribution. Based upon the data distribution,
ProUCL computes the (1-a) 100% UCLs of the unknown population mean, ^ using five (5)
parametric and ten (10) non-parametric methods.
The five parametric UCL computation methods include:
1) Student's-tf/CZ,
2) approximate gamma UCL,
3) adjusted gamma UCL,
4) Land' sH-UCL, and
5) Chebyshev inequality based UCL (using MVUE of parameters of a lognormal distribution).
The ten non-parametric methods included in ProUCL are:
1) the central limit theorem (CLT) based UCL,
2) modified-t statistic (adjusted for skewness),
3) adjusted-CLT (adjusted for skewness),
4) Chebyshev inequality based UCL (using sample mean and sample standard deviation),
5) Jackknife UCL,
6) standard bootstrap,
7) percentile bootstrap,
8) bias - corrected accelerated (BCA) bootstrap,
A-4
-------
9) bootstrap-t, and
10) Hall's bootstrap.
An extensive comparison of these methods have been performed by Singh and Singh (2003)
using Monte Carlo simulation experiments. It is well known that the Jackknife method (with
sample mean as an estimator) and Student's-t method yield identical UCL values. It is also well
known that the standard bootstrap method and the percentile bootstrap method do not perform
well (do not provide adequate coverage) for skewed data sets. However, for the sake of
completeness all of the parametric as well as non-parametric methods have been included in
ProUCL. Also, it has been noted that the omission of a method (e.g., bias-corrected accelerated
bootstrap method) triggers the curiosity of some of the users as they start thinking that the
omitted method may perform better than the various other methods already incorporated in
ProUCL. In order to satisfy all users, ProUCL Version 3.0 has additional UCL computation
methods which were not included in ProUCL Version 2.1.
1.1 Non-detects and Missing Data
ProUCL does not handle non-detects. All parametric as well as non-parametric
recommendations to compute the mean, standard deviation, and a 95% UCL of the mean made
by ProUCL software are based upon full data sets without censoring. The program can be
modified to incorporate methods which can be used to compute appropriate estimates of the
population mean and standard deviation, and a UCL of the mean for left-censored data sets with
non-detects. For now, for data sets with mild to moderate number of non-detects (e.g., < 15%),
one may replace non-detects by half of the detection limit (as often done in practice) and use
ProUCL on the resulting data set to compute an appropriate 95% UCL of the mean, //;. However,
the proxy methods such as replacing non-detects by !/2 of the detection limit (DL) should be used
cautiously, especially when one is dealing with lognormally distributed data sets. For
A-5
-------
lognormally distributed data sets of small sizes, even a single value small (e.g., obtained after
replacing the non-detect by 1A DL) or large (e.g., an outlier) can have a drastic influence (can
yield an unrealistically large 95% UCL) on the value of the associated Land's 95% UCL. The
issue of estimating the mean, standard deviation, and a 95% UCL of the mean based upon left-
censored data sets of varying degrees of censoring (e.g., < 15%, 15% - 50%, 50% - 75%, and
greater than 75%) is currently under investigation.
However, it should be noted that ProUCL can handle missing data. Missing data value can be
entered as a very large value in scientific notation, such as 1.0 E 31. All entries with this value
will be treated as missing data.
2. Procedures to Test for Data Distribution
Let xl3 x2, ..., xn be a random sample (e.g., representing lead concentrations) from the
underlying population (e.g, remediated part of a site) with unknown mean, //b and variance, a,2.
Let n and a represent the population mean and the population standard deviation (Sd) of the log-
transformed (natural log to the base e) data. Let y and sy (= o) be the sample mean and sample
Sd, respectively, of the log-transformed data,_y, = log (x,); / = 1, 2, ..., n. Specifically, let
- 1 "
y = - E yf > (!)
7
Similarly, let x and sx be the sample mean and Sd of the raw data, x, , x2 , .. , xn, obtained by
replacing y by x in equations (1) and (2), respectively. In this User Guide, irrespective of the
underlying distribution, fa, and o, 2 represent the mean and variance of the random variable X
A-6
-------
(in original units), whereas ^ and o2 represent the mean and variance of its logarithm, given by
Y = loge(X) = natural logarithm.
Three data distributions have been considered. These include the normal and lognormal
distributions, and the gamma distribution. Shapiro - Wilk (n < 50) and Lilliefors (n > 50) test
statistics are used to test for normality or lognormality of a data set. The empirical distribution
function (EDF) based methods: the Kolmogorov-Smirnov (K-S) test and the Anderson-Darling
(A-D) test are used to test for a gamma distribution. Extensive critical values for these two test
statistics have been obtained via Monte Carlo simulation experiments. For interested users,
these critical values are given in Appendix B for various levels of significance. In addition to
these formal tests, the informal histogram and quantile-quantile (Q-Q) plot are also available to
test data distributions. A brief description of these tests follows.
2.1 Test Normality and Lognormality of a Data Set
ProUCL tests the normality or lognormality of the data set using the three different
methods described below. The program tests normality or lognormality at three different levels
of significance, namely, 0.01, 0.05, and 0.1. The details of these methods can be found in the
cited references.
2.1.1 Normal Quantile-Quantile (Q-Q) Plot
This is a simple informal graphical method to test for an approximate normality or
lognormality of a data distribution (Hoaglin, Mosteller, and Tukey (1983), Singh (1993)). A
linear pattern displayed by the bulk of the data suggests approximate normality or lognormality
(performed on log-transformed data) of the data distribution. For example, a high value (e.g.,
0.95 or greater) of the correlation coefficient of the linear pattern may suggest approximate
A-7
-------
normality (or lognormality) of the data set under study. However, it should be noted that on this
graphical display, observations well separated (sticking out) from the linear pattern displayed by
the bulk data represent the outlying observations. Also, apparent jumps and breaks in the Q-Q
plot suggest the presence of multiple populations. The correlation coefficient of such a Q-Q plot
can still be high, which does not necessarily imply that the data follow a normal (or lognormal)
distribution. Therefore, the informal graphical Q-Q plot test should always be accompanied by
other more powerful tests, such as the Shapiro-Wilk test or the Lilliefors test. The goodness-of-
fit test of a data set should be judged based upon the formal more powerful tests. The normal Q-
Q plot may be used as an aid to identify outliers and/or to identify multiple populations.
ProUCL performs the graphical Q-Q plot test on raw data as well as on standardized data. All
relevant statistics such as the correlation coefficient are also displayed on the Q-Q plot.
2.1.2 Shapiro-Wilk W Test
This is a powerful test and is often used to test the normality or lognormality of the data set
under study (Gilbert, 1987). ProUCL performs this test for samples of size 50 or smaller. Based
upon the selected level of significance and the computed test statistic, ProUCL also informs the
user if the data are normally (or lognormally) distributed. This information should be used to
obtain an appropriate UCL of the mean. The program prints the relevant statistics on the Q-Q
plot of the data (or the standardized data). For convenience, the normality, lognormality, or
gamma distribution test results at 0.05 level of significance are also displayed on the UCL Excel-
type output summary sheets.
2.1.3 Lilliefors Test
This test is useful for data sets of larger size (Dudewicz and Misra, 1988). ProUCL performs
this test for samples of sizes up to 1000. Based upon the selected level of significance and the
-------
computed test statistic, ProUCL informs the user if the data are normally (or lognormally)
distributed. The user should use this information to obtain an appropriate UCL of the mean.
The program prints the relevant statistics on the Q-Q plot of data (or standardized data). For
convenience, the normality, lognormality, or gamma distribution test results at 0.05 level of
significance are also displayed on the UCL output summary sheets. It should be pointed out
that sometimes, in practice, these two goodness-of-fit tests can lead to different conclusions.
2.2 Gamma Distribution
Singh, Singh, and laci (2002b) studied gamma distribution to model positively skewed
environmental data sets and to compute a UCL of the mean based upon a gamma distribution.
They studied several UCL computation methods using Monte Carlo simulation experiments. A
continuous random variable, X (e.g., concentration of a contaminant), is said to follow a gamma
distribution, G (k,0) with parameters k > 0 (shape parameter) and 0 > 0 (scale parameter), if its
probability density function is given by the following equation:
,,
6ri{k)
and zero otherwise. The parameter k is the shape parameter, and 6 is the scale parameter. Many
positively skewed data sets follow a lognormal as well as a gamma distribution. Gamma
distribution can be used to model positively skewed environmental data sets. It is observed that
the use of a gamma distribution results in reliable and stable 95% UCL values. It is therefore,
desirable to test if an environmental data set follows a gamma distribution. If a skewed data set
does follow a gamma model, then a 95% UCL of the population mean should be computed using
a gamma distribution. For details of the two gamma goodness-of-fit tests, maximum likelihood
estimation of gamma parameters, and the computation of a 95% UCL of the mean based upon a
gamma distribution, refer to D'Agostino and Stephens (1986), and Singh, Singh, and laci
A-9
-------
(2002b). These methods are briefly described as follows.
For data sets which follow a gamma distribution, the adjusted 95% UCL of the mean based
upon a gamma distribution is optimal and approximately provides the specified 95% coverage to
population mean, /^ = k6 (Singh, Singh, and laci (2002b)). Moreover, this adjusted gamma
UCL yields reasonable numbers of practical merit. The two test statistics used for testing for a
gamma distribution are based upon the empirical distribution function (EDF). The two EDF
tests included in ProUCL are the Kolmogorov-Smirnov (K-S) test and Anderson - Darling (A-D)
test which are described in D'Agostino and Stephens (1986) and Stephens (1970). The
graphical Q-Q plot for gamma distribution has also been included in ProUCL. The critical
values for the two EDF tests are not easily available, especially when the shape parameter, k is
small (k < 1). Therefore, the associated critical values have been obtained via extensive Monte
Carlo simulation experiments. These critical values for the two test statistics are given in
Appendix B. The 1%, 5%, and 10% critical values of these two test statistics have been
incorporated in ProUCL, Version 3.0. A brief description of the three goodness-of-fit tests for
gamma distribution is given as follows. It should be noted that the goodness-of-fit tests for
gamma distribution depend upon the MLEs of gamma parameters, k and Q which should be
computed first before performing the goodness-of-fit tests.
2.2.1 Quantile - Quantile (Q-Q) Plot for a Gamma Distribution
Let Xj, x2, ..., xn be a random sample from the gamma distribution, G(k,0). Let
JC(1) < JC(2) <....< JC(M) represent the ordered sample. Letk and 9 represent the maximum
likelihood estimates (MLEs) of k and 6, respectively. For details of the computation of MLEs of
k and 6, refer to Singh, Singh, and laci (2002b). Estimation of gamma parameters is also briefly
described later in this User Guide. The Q-Q plot for gamma distribution is obtained by plotting
the scatter plot of pairs (x0i, jc(;) );/':= 1,2,..., n. The quantiles, X0j are given by the
A-10
-------
equation x0i = z0i& / 2;i:= 1,2,..., n , where the quantiles z0i (already ordered) are obtained
by using the inverse chi-square distribution and are given as follows.
(4)
In (4), X f represents a chi-square random variable with 2k degrees of freedom (d.f.). The
2, K
program, PPCHI2 (Algorithm AS91) as given in Best and Roberts (1975), Applied Statistics
(1975, Vol. 24, No. 3) has been used to compute the inverse chi-square percentage points, z0/. as
given by the above equation given by (4). This is an informal graphical test to test for a gamma
distribution. This informal test should always be accompanied by the formal Anderson-Darling
test or Kolmogorov- Smirnov test. A linear pattern displayed by the scatter plot of bulk of the
data may suggest approximate gamma distribution. For example, a high value (e.g., 0.95 or
greater) of the correlation coefficient of the linear pattern may suggest approximate gamma
distribution of the data set under study. However, on this Q-Q plot points well separated from
the bulk of data may represent outliers. Also, apparent breaks and jumps in the gamma Q-Q plot
suggest the presence of multiple populations. The correlation coefficient of such a Q-Q plot can
still be high which does not necessarily imply that the data follow a gamma distribution.
Therefore, the graphical Q-Q plot test should always be accompanied by the other more
powerful formal EDF tests, such as the Anderson-Darling test or the Kolmogorov-Smirnov test.
The final conclusion about the data distribution should be based upon the formal goodness-of-fit
tests. The Q-Q plot may be used to identify outliers and/or presence of multiple populations. All
relevant statistics including theMLE of k are also displayed on the gamma Q-Q plot.
2.2.2 Empirical Distribution Function (EDF) Based Goodness-of -Fit Tests
Next, the two formal EDF test statistics used to test for a gamma distribution are described
briefly. Let F(x) be the cumulative distribution function (CDF) of the gamma random variable
A-ll
-------
X. Let Z=F(X), then Z represents a uniform U(0,l) random variable. For each xt , compute z.
using the incomplete gamma function given by the equation z. = F(x.);/:= 1,2, ...,«. The
algorithm as given in Numerical Recipes book (Press et al., 1990) has been used to compute the
incomplete gamma function. Arrange the resulting, zi in ascending order as
z(1) < z(2) <...< z(M) . Let z = 2_t zt / n be the mean of the z. ;/:= 1,2,..., « . Compute the
following two test statistics.
D+ = max. (1 / n - z( .} } and D~ = max. (z( .} - (i - \) / n} (5)
The Kolmogorov - Smirnov test statistic is given by D = max(Z)+ ,D~ ) .
Anderson Darling test statistic is given by the following equation.
A2 = -n- (1/«)X {(2/- l)[logzl + log(l- zwf !_,)]} (6)
The critical values for these two statistics D and A2 are not readily available. For the Anderson-
Darling test, only asymptotic critical values are available in the statistical literature (D' Agostino
and Stephens (1986)). Some raw critical values for K-S test are given in Schneider (1978), and
Schneider and Clickner (1976). For these two tests, ExpertFit (2001) software and Law and
Kelton (2000) use generic critical values for all completely specified distributions as given in
D' Agostino and Stephens (1986). It is observed that the conclusions derived using these generic
critical values for completely specified distributions and the simulated critical values for gamma
distribution with unknown parameters can be different. Therefore, to test for a gamma
distribution, it is preferred and advised to use the critical values of these test statistics
specifically obtained for gamma distributions with unknown parameters.
A-12
-------
In practice, the distributions are not completely specified and exact critical values for these
two test statistics are needed. It should be noted that the distributions of the K-S test statistic, D
and A-D test statistic, A2 do not depend upon the scale parameter, 9, therefore, the scale
parameter, 9 has been set equal to 1 in all of the simulation experiments. The critical values for
these two statistics have been obtained via extensive Monte Carlo simulation experiments for
several small and large values of the shape parameter, k and with 0=1. These critical are
included in Appendix B. In order to generate the critical values, random samples from gamma
distributions were generated using the algorithm as given in Whittaker (1974). It is observed
that the critical values thus obtained are in close agreement with all available published critical
values. The generated critical values for the two test statistics have been incorporated in
ProUCL for three levels of significance, 0.1, 0.05, and 0.01. For each of the two tests, if the test
statistic exceeds the corresponding critical value, then the hypothesis that the data follow a
gamma distribution is rejected. ProUCL computes these test statistics and prints them on the
gamma Q-Q plot and also on the UCL summary output sheets generated by ProUCL. The
estimation of the parameters of the three distributions as incorporated in ProUCL is discussed
next. It should be pointed out that sometimes, in practice, these two goodness-of-fit tests can
lead to different conclusions.
3. Estimation of Parameters of the Three Distributions Included in ProUCL
Through out this User Guide, fa and Oj2 are the mean and variance of the random variable X,
and // and a2 are the mean and variance of the random variable Y = log(X). Also, a represents
the standard deviation of the log-transformed data. It should be noted that for both lognormal
and gamma distributions, the associated random variable can take only positive values. This is
typical of environmental data sets to consist of only positive values.
A-13
-------
3.1 Normal Distribution
Let X be a continuous random variable (e.g., concentration of COPC), which follows a
normal distribution, N( //b Oj2) with mean, //b and variance, a2. The probability density function
of a normal distribution is given by the following equation:
(x- ^)2 /2c712);-Go < x < GO (7)
For normally distributed data sets, it is well known (Hogg and Craig, 1978) that the minimum
variance unbiased estimates (MVUEs) of mean, //13 and variance, of are respectively given by
the sample mean, x and sample variance, sx . It is also well known that for normally distributed
data sets, a UCL of the unknown mean, ^ based upon Student' s-t distribution is optimal. It is
observed via Monte Carlo simulation experiments (Singh and Singh (2003) Draft EPA Report)
that for normally distributed data sets, the modified-t UCL and UCL based upon bootstrap-t
method also provide the exact 95% coverage to the population mean. For normally distributed
data sets, the UCLs based upon these three methods are very similar.
3.2 Lognormal Distribution
If 7= log(JQ is normally distributed with the mean // and variance cr2, X is said to be
lognormally distributed with parameters // and o2 and is denoted by LN(//, a2) . It should be
noted that // and o2 are not the mean and variance of the lognormal random variable, X, but they
are the mean and variance of the log-transformed random variable 7, whereas //;, and ^ 2
represent the mean and variance of X. Some parameters of interest of a two-parameter
lognormal distribution, LN(//, <72), are given as follows:
Mean = ^ = exp(/u + 0.5cr2) (8)
A-14
-------
Median = M = exp(/u) (9)
Variance = a\ = exp(2//+ <72)(exp(o2)- 1) (10)
Coefficient of Variation = CV = o/^ = \/(exp(a2) - 1) (11)
Skewness = (CV)3 + 3(CF) (12)
3.2.1 MLEs of the Parameters of a Lognormal Distribution
For lognormal distributions, note that y and sy (= 6 ) are the maximum likelihood
estimators (MLEs) of// and o, respectively. TheMLE of any function of the parameters // and
<72 is obtained by simply substituting these MLEs in place of the parameters (Hogg and Craig,
1978). Therefore, replacing // and a by their MLEs in equations (8) through (12) will result in
the MLEs (but biased) of the respective parameters of the lognormal distribution. The program
ProUCL computes all of these MLEs for lognormally distributed data sets. These MLEs are also
printed on the Excel-type output spread sheets generated by ProUCL.
3.2.2 Relationship Between Skewness and Standard Deviation, a
Note that for a lognormal distribution, the CV (given by equation (11) above) and the
skewness (given by equation (12)) depend only on o. Therefore, in this User Guide and also in
ProUCL, the standard deviation, o (Sd of log-transformed variable, Y), or its MLE, sy(=d) has
been used as a measure of skewness of lognormal and also of other skewed data sets with
positive values. The larger is the Sd, the larger are the CFand the skewness. For example, for a
lognormal distribution: with o = 0.5, the skewness = 1.75; with o =1.0, the skewness = 6.185;
with a =1.5, the skewness = 33.468; and with a = 2.0, the skewness = 414.36. Thus, the
skewness of a lognormal distribution becomes unreasonably large as <7 starts approaching and
A-15
-------
exceeding 2.0. Note that for gamma distribution, skewness is a function of the gamma
parameter, k. As k decreases, skewness increases.
It is observed (Singh, Singh, Engelhardt (1997), and Singh et al. (2002a)) that for smaller
sample sizes (such as smaller than 50), and for values of a approaching 2.0 (and skewness
approaching 414), the use of the H-statistic based UCL results in impractical and unacceptably
large values. For simplicity, the various levels of skewness of a positive data set as used in
ProUCL and in this User Guide are summarized as follows:
Skewness as a Function of a (or its MLE, s = 6), Sd of log(X)
Standard Deviation
o <0.5
0.5 3.0
Skewness
Symmetric to mild skewness
Mild Skewness to Moderate Skewness
Moderate Skewness to High Skewness
High skewness
Extremely high skewness
Provides poor coverage
These values of a (or its estimate, Sd of log-transformed data) are used to define skewness levels
of lognormal and skewed non-parametric data distributions as used in Tables A2 and A3.
3.2.3 MLEs of the Quantiles of a Lognormal Distribution
For highly skewed (e.g., a exceeding 1.5), lognormally distributed populations, the
population mean, //l3often exceeds the higher quantiles (e.g., 80%, 90%, 95%) of the
distribution. Therefore, the computation of these quantiles is also of interest. This is especially
true when one may want to use the MLEs of the higher order quantiles (e.g., 95%, 97.5% etc.) as
A-16
-------
an estimate of the EPC term. The formulae to compute these quantiles are briefly described
here.
The pth quantile (or WO pth percentile), xp, of the distribution of a random variable, X, is
defined by the probability statement, P(X < xp) =p. lfzp is the pth quantile of the standard
normal random variable, Z, with P(Z < zp) =p, then the/rth quantile of a lognormal distribution
is given by xp = exp(// + zpd). Thus the MLE of the pth quantile is given by
X = exp(// + zn<7) (13)
rj 1 \t fj / \ /
p
For example, on the average, 95% of the observations from a lognormal LN(//, cr2) distribution
would lie below exp(// + 1.65 a). The 0.5th quantile of the standard normal distribution is z05 =
0, and the 0.5th quantile (or median) of a lognormal distribution is M = exp(//), which is
obviously smaller than the mean, //b as given by equation (8). Also note that the mean, //b is
greater than xp if and only ifo> 2zp. For example, when p = 0.80, zp = 0.845, ^ exceeds x 080,
the 80th percentile if and only if a > 1.69, and, similarly, the mean, //13 will exceed the 95th
percentile if and only if a > 3.29. ProUCL computes the MLEs of the 50% (median), 90%,
95%, and 99% percentiles of lognormally distributed data sets. For lognormally distributed
background data sets, a 95% or 99% percentile may be used as an estimate of the background
threshold value, that is background level contaminant concentration.
3.2.4 MVUEs of Parameters of a Lognormal Distribution
Even though the sample AM, x , is an unbiased estimator of the population AM, //;, it does
not have the minimum variance (MV). The MV unbiased estimates (MVUEs) of ^ and ot of a
lognormal distribution are given as follows:
A-17
-------
= exp(jT)g>;/2), (14)
2s2)-gn((n-2)S2J(n-m (15)
where the series expansion of the function gn( ju ) is given in Bradu and Mundlak (1970), and
Aitchison and Brown (1976). Tabulations of this function are also provided by Gilbert (1987).
Bradu and Mundlak (1970) give \heMVUE of the variance of the estimate fa,
- gn((n-2)s2yl(n- 1))] (16)
The square root of the variance given by equation (16) is called the standard error (SE) of the
estimate, /ul3 given by equation (14). Similarly, &MVUE of the median of a lognormal
distribution is given by
M= expCy^C- *,/(2(»-l))). (17)
For lognormally distributed data set, ProUCL also computes these MVUEs given by equations
(14) through (17).
3.3 Estimation of the Parameters of a Gamma Distribution
Next, we consider the estimation of parameters of a gamma distribution. Since the
estimation of gamma parameters is typically not included in standard statistical text books, this
has been described in some detail in this User Guide. The population mean and variance of a
gamma distribution, G(k,0), are functions of both parameters, k and 6. In order to estimate the
mean, one has to obtain estimates ofk and 6. The computation of the maximum likelihood
estimate (MLE) of & is quite complex and requires the computation of Digamma and Trigamma
A-18
-------
functions. Several authors (Choi and Wette, 1969, Bowman and Shenton, 1988, Johnson, Kotz,
and Balakrishnan, 1994) have studied the estimation of shape and scale parameters of a gamma
distribution. The maximum likelihood estimation method to estimate shape and scale parameters
of a gamma distribution is described below.
Let xl^c2,...^xn be a random sample (e.g., representing contaminant concentrations) of size n
from a gamma distribution, G(k,0), with unknown shape and scale parameters k and 6,
respectively. The log likelihood function (obtained using equation (3)) is given as follows:
logL(*l5*25... ,*;£,#) = - «Hog(0) - H logr (£)+(£ -1)2 log *,.--2>,. (18)
V
To find the MLEs of & and 6, we differentiate the log likelihood function as given in (18) with
respect to k and 0, and set the derivatives to zero. This results in the following two equations:
log(0)+ 7-= -logO,) ,and (19)
kO = -2*; = * (2°)
n
yv.
Solving equation (20) for 9 and substituting the result in equation (19), we get the following
equation:
(2D
There does not exist a closed form solution of equation (21). This equation needs to be solved
numerically for k , which requires the use of Digamma and Trigamma functions. This is quite
easy to do using a personal computer. An estimate ofk can be computed iteratively by using the
Newton-Raphson (Faires and Burden, 1993) method leading to the following iterative equation:
A-19
-------
log(£, , ) - Y (k, , ) - A/
, ,- av *-y ^^ - (22)
^v
The iterative process stops when k starts to converge. In practice, convergence is typically
achieved in fewer than 10 iterations. In equation (22)
M = log(J) - -
and
where ^F (£) is the Digamma function, and ^F ' (£) is the Trigamma function. In order to obtain
the MLEs of & and 6, one needs to compute the Digamma and Trigamma functions. Good
approximate values for these two functions (Choi and Wette, 1969) can be obtained using the
following approximations. For k > 8, these functions are approximated by
(23)
and
- (1/5- l/(7k2))/ k2]/(3k)}/(2k)}/ k (24)
For k < 8, one can use the following recurrence relation to compute these functions:
k,
and ₯'(&)= ^"(k+V+l/k2 (26)
A-20
-------
In ProUCL, equations (23) - (26) have been used to estimate k. The iterative process requires an
initial estimate of k. A good starting value for k in this iterative process is given
by k0 = 1 / (2 M) . Thorn (1968) suggested the following approximation as an estimate ofk:
l+Jl+-Af\ (27)
Bowman and Shenton (1988) suggested using k as given by (27) to be a starting value of k for
an iterative procedure, calculating kt at the Ith iteration from the following formula:
(28)
Both equations (22) and (28) have been used to compute the MLE ofk. It is observed that the
estimate, k based upon Newton-Raphson method as given by equation (22) is in close
agreement with that obtained using equation (28) with Thorn's approximation as an initial
estimate. Choi and Wette (1969) further concluded that the MLE ofk, k , is biased high. A
bias-corrected (Johnson, Kotz, and Balakrishnan, 1994) estimate of & is given by:
k* = (n-3)k/n+2/(3n) (29)
^.
In (29), k is the MLE ofk obtained using either (22) or (28). Substitution of equation (29) in
equation (20) yields an estimate of the scale parameter, 0 given as follows:
0* = x I k* (30)
ProUCL computes simple MLE ofk and 6, and also bias- corrected estimates ofk and 6. The
bias-corrected estimate ofk as given by (29) has been used in the computation of the UCLs (as
given by equations (34) and (35)) of the mean of a gamma distribution.
A-21
-------
4. Methods for Computing a UCL of the Unknown Population Mean
ProUCL computes a (1-u) 100 % UCL of the population mean, ^ using the following five
parametric and ten non-parametric methods. Five of the ten non-parametric methods are based
upon the bootstrap method. Modified-t and adjusted central limit theorem adjust for skewness
for skewed data sets. However, it is noted that (Singh, Singh, and laci (2002b) and Singh and
Singh (2003)) this adjustment is not adequate enough for moderately skewed to highly skewed
data sets. Some graphs from Singh and Singh (2003) showing coverage comparisons for normal,
gamma, and lognormal distributions for the various methods are given in Appendix C. The
methods as included in ProUCL are listed as follows.
Parametric Methods
1. Student's-t statistic - assumes normality or approximate normality
2. Approximate Gamma UCL - assumes gamma distribution of the data set
3. Adjusted Gamma UCL - assumes gamma distribution of the data set
4. Land's H-Statistic - assumes lognormality
5. Chebyshev Theorem using theAfVUE of the parameters of a lognormal distribution
(denoted by Chebyshev (MVUE)) - assumes lognormality
Non-parametric Methods
1. Modified-1 statistic - modified for skewed distributions
2. Central Limit Theorem (CLT) - to be used for large samples
3. Adjusted Central Limit Theorem (Adjusted-CZT) - adjusted for skewed distributions and
to be used for large samples
A-22
-------
4. Chebyshev Theorem using the sample arithmetic mean and Sd (denoted by Chebyshev
(Mean, Sd))
5. Jackknife method - yields the same result as Student's-t statistic for the UCL of the
population mean
6. Standard bootstrap
7. Percentile bootstrap
8. Bias-corrected accelerated (BCA) bootstrap
9. Bootstrap-t
10. Hall's bootstrap
Even though it is well known that some of the non-parametric methods (e.g., CLT method,
UCL based upon Jackknife method (same as Student's-t UCL), standard bootstrap and percentile
bootstrap methods) do not perform well to provide the adequate coverage to the population mean
of skewed distributions, these methods have been included in ProUCL to satisfy the curiosity of
all users.
ProUCL can compute a (1-a) 100 % UCL (except for the H-UCL and adjusted gamma UCL)
of the mean for any confidence coefficient (1-a.) value lying in the interval [0.5, 1.0). For the
computation of the H-UCL, only two confidence levels, namely, 0.90 and 0.95 are supported by
ProUCL. For adjusted gamma UCL, three confidence levels namely, 0.90, 0.95, and 0.99 are
supported by ProUCL. An approximate gamma UCL can be computed for any level of
significance in the interval [0.5,1). Based upon the sample size, n, skewness, and the data
distribution, the program also makes recommendations on how to obtain an appropriate 95%
UCL of the unknown population mean, ^ . These recommendations are summarized in the
Recommendations and Summary Section 5 of this appendix. The various algorithms and
methods used to compute a (1-a) 100% UCL of the mean as incorporated in ProUCL are
described in section 4.1.
A-23
-------
4.1 (1-a) 100% UCL of the Mean Based Upon Student's-t Statistic
The widely used well-known Student's-t statistic is given by,
A 1*1
t = - £1, (31)
.? iJvt
where x and sx are, respectively, the sample mean and sample standard deviation obtained
using the raw data. If the data are a random sample from a normal population with mean, //13 and
standard deviation, o^ then the distribution of this statistic is the familiar Student's-t distribution
with (n-\) degrees of freedom (df). Let ta ^^ be the upper ath quantile of the Student's-t
distribution with (n- 1) df.
A (1-a) 100 % UCL of the population mean, //l3 is given by,
UCL = x + tn_lSjJn. (32)
For a normally (when the skewness is about ~0) distributed population, equation (32) provides
the best (optimal) way of computing a UCL of the mean. Equation (32) may also be used to
compute a UCL of the mean based upon very mildly skewed (e.g., |skewness|<0.5) data sets,
where skewness is given by equation (43). It should be pointed out that even for mildly to
moderately skewed data sets (e.g., when o, Sd of log-transformed data starts approaching and
exceeding 0.5), the UCL given by (32) may not provide the desired coverage (e.g., =0.95) to the
population mean. This is especially true when the sample size is smaller than 20-25 (Singh et al.
(2002a), and Singh and Singh (2003)). The situation gets worse (coverage much smaller than
0.95) for higher values of the Sd, a, or itsMLE, sy.
A-24
-------
4.2 Computation of UCL of the Mean of a Gamma, G(k,0) Distribution
In statistical literature, even though methods exist to compute a UCL of the mean of a gamma
distribution (Grice and Bain, 1980, Wong, 1993), those methods have not become popular due
to their computational complexity. Those approximate and adjusted methods depend upon the
Chi-square distribution and an estimate of the shape parameter, k. As seen above, computation
of an MLE of k is quite involved, and this works as a deterrent to the use of a gamma
distribution-based UCL of the mean. However, the computation of a gamma UCL currently
should not be a problem due to easy availability of personal computers.
Given a random sample, xl,x2,...,xno£ size n from a gamma, G(&,0) distribution, it can be
shown that 2nX I 9 follows a Chi-square distribution, x^nk > witn ^nk degrees of freedom (df).
When the shape parameter, k, is known, a uniformly most powerful test of size a of the null
hypothesis, £!<,://! >CS, against the alternative hypothesis, H^ ^ < Cs, is to reject H0 if
X I Cs < ^2nk (a ) / 2/2& . The corresponding (l-oc)100% uniformly most accurate UCL for
the mean, //b is then given by the probability statement.
22nk (a)>Jul)=\-a (33)
where xl (a ) denotes the a cumulative percentage point of the Chi-square distribution (e.g.,
a is the area in the left tail). That is, if Y follows xl > then P(Y < xl («))=«. In practice,
k is not known and needs to be estimated from data. A reasonable method is to replace k by its
bias -corrected estimate, k" , as given by equation (29). This results in the following approximate
(1-a) 100% UCL of the mean, /*, .
^* 9
Approximate - UCL = 2nk x I % r (a ) (34)
"Ink
A-25
-------
It should be pointed out that the UCL given by equation (34) is an approximate UCL and
there is no guarantee that the confidence level of (1-oc) will be achieved by this UCL. However,
it does provide a way of computing a UCL of the mean of a gamma distribution. Simulation
studies conducted in Singh, Singh, and laci (2002b) and in Singh and Singh (2003) suggest that
an approximate gamma UCL thus obtained provides the specified coverage (95%) as the shape
parameter, k approaches 0.5. Thus when k > 0.5, one can always use the approximate UCL
given by (34). This approximation is good even for smaller (e.g., n = 5) sample sizes as shown
in Singh, Singh, and laci (2002b), and in Singh and Singh (2003).
Grice and Bain (1980) computed an adjusted probability level, p (adjusted level of
significance), which can be used in (34) to achieve the specified confidence level of (1-oc). For
a = 0.05 (confidence coefficient of 0.95), a = 0.1, and a = 0.01, these probability levels are given
below in Table 1 for some values of the sample size n. One can use interpolation to obtain an
adjusted p for values of n not covered in the table. The adjusted (1-a) 100% UCL of the
gamma mean, ^ = k0 is given by the following equation.
Adjusted - UCL = 2nk*x I x\^ (A (35)
where p is given in Table 1 for a = 0.05, 0.1, and 0.01. Note that as the sample size, n, becomes
large, the adjusted probability level, p, approaches the specified level of significance, a. Except
for the computation of theMLE of k, equations (34) and (35) provide simple Chi-square-
distribution-based UCLs of the mean of a gamma distribution. It should also be noted that the
UCLs as given by (34) and (35) only depend upon the estimate of the shape parameter, k, and are
independent of the scale parameter, 0, and its ML estimate. Consequently, as expected, it is
observed that coverage probabilities for the mean associated with these UCLs do not depend
upon the values of the scale parameter, 0. It should also be noted that gamma UCLs do not
depend upon the standard deviation of data which gets distorted by the presence of outliers.
A-26
-------
Thus, outliers will have reduced influence on the computation of the gamma distribution based
UCLs of the mean, fa.
Table 1. Adjusted Level of Significance, p
n
5
10
20
40
a = 0.05
probability level, p
0.0086
0.0267
0.0380
0.0440
0.0500
a = 0.1
probability level, p
0.0432
0.0724
0.0866
0.0934
0.1000
a = 0.01
probability level, p
0.0000
0.0015
0.0046
0.0070
0.0100
4.3 (1-oc) 100% UCL of the Mean Based Upon H-Statistic (H-UCL)
The one-sided (1 -a) 100% UCL for the mean, fa, of a lognormal distribution as derived by
Land (1971, 1975) is given as follows:
UCL =
exp(>;
+ 0.5s ,
(36)
Tables of H-statistic critical values can be found in Land (1975) and also in Gilbert (1987).
Theoretically, when the population is lognormal, Land (1971) showed that the UCL given by
equation (36) possesses optimal properties and is the uniformly most accurate unbiased
confidence limit. However, it is noticed that in practice, the H-statistic based results can be
quite disappointing and misleading especially when the data set consists of outliers, or is a
mixture from two or more distributions (Singh, Singh, and Engelhardt, 1997, 1999), Singh,
Singh, and laci (2002b)). Even a minor increase in the Sd, s drastically inflates the MVUE of
A-27
-------
//; and the associated H-UCL. The presence of low as well as high data values increases the Sd,
Sy, which in turn inflates the H-UCL. Furthermore, it is observed (Singh, Singh, Engelhardt, and
Nocerino (2002a)) that for samples of sizes smaller than 15-25, and for values of a approaching
1.0 and higher (for moderately skewed to highly skewed data sets), the use of H-statistic based
UCL results in impractical and unacceptably large UCL values.
In practice many data sets follow a lognormal as well as gamma model. However, the
population mean based upon a lognormal model can be significantly greater (often unrealistically
large) than the population mean based upon a gamma model. In order to provide the specified
95% coverage for an inflated mean based upon a lognormal model, the resulting UCL based
upon H-statistic also yield impractical UCL values. Use of a gamma model results in practical
estimates (e.g., UCL) of the population mean. Therefore, for positively skewed data sets, it is
recommended to test for a gamma model first. If data follow a gamma distribution, then the
UCL of the mean should be computed using a gamma distribution. The gamma distribution is
better suited to model positively skewed environmental data sets.
4.4 (1-a) 100% UCL of the Mean Based Upon Modified-t Statistic for Asymmetrical
Populations
Chen (1995), Johnson (1978), Kleijnen, Kloppenburg, and Meeuwsen (1986), and Sutton
(1993) suggested the use of the modified-t statistic for testing the mean of a positively skewed
distribution (including the lognormal distribution). The (1 -a) 100 % UCL of the mean thus
obtained is given by
UCL= x +|I3/(65» + f^.^/V (37)
where p,3 , an unbiased moment estimate (Kleijnen, Kloppenburg, and Meeuwsen, 1986) of the
A-28
-------
third central moment, is given as follows,
Pa = «E (*;.-F)3/[(«-i)(«-2)]. (38)
I =1
It should be pointed out that this modification for a skewed distribution does not perform well
even for mildly to moderately skewed data sets (e.g., when o starts approaching and exceeding
0.75). Specifically, it is observed that the UCL given by equation (37) may not provide the
desired coverage of the population mean, //13 when o starts approaching and exceeding 0.75
(Singh, Singh, and laci (2002b)). This is especially true when the sample size is smaller than
20-25. This small sample size requirement increases as (/increases. For example, when o starts
approaching and exceeding 1.5, the UCL given by equation (37) does not provide the specified
coverage (e.g., 95%), even for samples as large as 100. Since this method does not require any
distributional assumptions, it is a non-parametric method.
4.5 (1-a) 100% UCL of the Mean Based Upon the Central Limit Theorem
The Central Limit Theorem (CLT) states that the asymptotic distribution, as n approaches
infinity, of the sample mean, xn is normally distributed with mean, //b and variance, o^ln .
More precisely, the sequence of random variables given by
^ (39)
has a standard normal limiting distribution. In practice, for large sample sizes, n, the sample
mean, jc, has an approximate normal distribution irrespective of the underlying distribution
function. Since the CLT method requires no distributional assumptions, this is a non-parametric
method.
A-29
-------
As noted by Hogg and Craig (1978), if ol is replaced by the sample standard deviation, sx, the
normal approximation for large n is still valid. This leads to the following approximate large
sample non-parametric (1-u) 100% UCL of the mean,
UCL = x + zjjjn. (40)
An often cited rule of thumb for a sample size associated with the CZrmethod is n > 30.
However, this may not be adequate enough if the population is skewed, specifically when, a (Sd
of log-transformed variable) starts exceeding 0.5 (Singh, Singh, laci 2002b). In practice for
skewed data sets, even a sample as large as 100 is not large enough to provide adequate coverage
to the mean of skewed populations (even for mildly skewed populations). A refinement of the
CLTapproach, which makes an adjustment for skewness as discussed by Chen (1995), is given
as follows.
4.6 (1-a) 100% UCL of the Mean Based Upon the Adjusted Central Limit Theorem
(Adjusted -CLT)
The "adjusted-CLT" UCL is obtained if the standard normal quantile, za in the upper limit of
equation (40) is replaced by (Chen, 1995)
* .
6\/«
Thus, the adjusted (1 -a) 100 % UCL for the mean, //13 is given by
UCL = x+ [za + (l + 2za)/(6)K/. (42)
f\
Here k3 , the coefficient of skewness (raw data) is given by
A-30
-------
>v 3
Skewness (raw data) &3 = $3/sx (43)
where p,3, an unbiased estimate of the third moment, is given by equation (38). This is another
large sample approximation for the UCL of the mean of skewed distributions. This is a non-
parametric method as it does not depend upon any of the distributional assumptions.
As with the modified-t UCL, it is observed that this adjusted-CLJ UCL does not provide
adequate coverage to the population mean when the population is skewed, specifically when o
starts approaching and exceeding 0.75 (Singh, Singh, and laci (2002b), Singh and Singh (2003)).
This is especially true when the sample size is smaller than 20-25. This small sample size
requirement increases as o increases. For example, when o starts approaching and exceeding
1.5, the UCL given by equation (42) does not provide the specified coverage (e.g., 95%), even
for samples as large as 100. Also, it is noted that the UCL as given by (42) does not provide
adequate coverage to the mean of a gamma distribution, especially when k < 1.0 and sample size
is small. Some graphs from Singh and Singh (2003) showing coverage comparisons for normal,
gamma, and lognormal distributions for the various methods are given in Appendix C.
Thus, the UCLs based upon these skewness adjusted methods, such as the Johnson's
modified-t and Chen's adjusted-CLTdo not provide the specified coverage to the population
mean for mildly to moderately skewed (e.g., o in (0.5, 1.0)) data sets, even for samples as large
as 100 (Singh, Singh, and laci (2002b)). The coverage of the population mean provided by
these UCLs becomes worse (much smaller than the specified coverage) for highly skewed data
sets.
4.7 (1-a) 100% UCL of the Mean Based Upon the Chebyshev Theorem (Using the Sample
Mean and Sample Sd)
A-31
-------
The Chebyshev inequality can be used to obtain a reasonably conservative but stable
estimate of the UCL of the mean, //;. The two-sided Chebyshev theorem (Hogg and Craig, 1978)
states that given a random variable, X, with finite mean and standard deviation, ^ and o b we
have
1- I/A:2. (44)
This result can be applied on the sample mean,;*7 (with mean, ^ and variance, of / « ) to
obtain a conservative UCL for the population mean, //;. For example, if the right side of equation
(44) is equated to 0.95, then k = 4.47, and UCL = x+ 4.47o1/\/« is a conservative 95% upper
confidence limit for the population mean, //;. Of course, this would require the user to know the
value of a j. The obvious modification would be to replace o l with the sample standard
deviation, sx, but since this is estimated from data, the result is no longer guaranteed to be
conservative. In general, the following equation can be used to obtain a (1-u) 100% UCL of the
population mean, //^
UCL = x + -^(\/a)s
A slight refinement of equation (45) is given (suggested by S. Person) as follows,
la)-\)s
ProUCL computes the Chebyshev (1-a) 100% UCL of the population mean using equation
(46). This UCL is denoted by Chebyshev (Mean, Sd) on the output sheets generated by
ProUCL. Since this Chebyshev method requires no distributional assumptions about the data set
under study, this is a non-parametric method. This UCL may be used as an estimate of the
upper confidence limit of the population mean, ^ when data are not normal, lognormal, or
gamma distributed especially when Sd, a (or its estimate, s ) starts approaching and exceeding
A-32
-------
1.5. Recommendations on its use to a compute an estimate of the EPC term are summarized in
Section 5.
4.8 (1-oc) 100% UCL of the Mean of a Lognormal Population Based Upon the Chebyshev
Theorem (Using the MVVE of the Mean and its Standard Error)
ProUCL uses equation (44) on the MVUEs of the lognormal mean and Sd to compute a UCL
(denoted by (l-a)100 % Chebyshev (MVUE) ) of the population mean of a lognormal population.
In general, if fa is an unknown mean, fa is an estimate, and 0(fa) is an estimate of the standard
error of fa, then the following equation,
UCL = fa +((l/a) -If2 a(fa) (47)
will give an approximate (1-a) 100 % UCL for //l3 which should tend to be conservative, but this
is not assured. For example, for a lognormally distributed data set, a 95% (with a =0.05)
Chebyshev (MVUE) UCL of the mean can be obtained using the following equation,
UCL = fa + (4359) a(fa) (48)
where, fa and &(fa) are given by equations (14) and (16), respectively. Thus, for lognormally
distributed data sets, ProUCL also uses equation (48) to compute a (1-a) 100% Chebyshev
(MVUE) UCL of the mean. It should be noted that for lognormally distributed data sets, some
recommendations to compute a 95% UCL of the population mean are summarized in Table A2
of the Recommendations and Summary Section 5.0. It should however be pointed out that
goodness-of-fit test for a gamma distribution should be performed first. If data follow a gamma
distribution (irrespective of the lognormality of the data set), then the UCL of mean, //j should be
computed using a gamma distribution as described in Section 4.2.
A-33
-------
From Monte-Carlo results discussed in Singh, Singh, and laci (2002b) and in Singh and
Singh (2003), it is observed that for highly skewed gamma distributed data sets (with k < 0.5),
the coverage provided by the Chebyshev 95% UCL (given by (46)) is smaller than the specified
coverage of 0.95. This is especially true when the sample size is smaller than 10-20. As
expected, for larger samples sizes, the coverage provided by the 95% Chebyshev UCL is at least
95%. For larger samples, the Chebyshev 95% UCL will result in a higher (but stable) UCL of
the mean of positively skewed gamma distributions.
It is observed (Singh and Singh (2003)) that for moderately skewed to highly skewed
lognormally distributed data sets (e.g., with a exceeding 1), 95% Chebyshev MVUE UCL does
not provide the specified coverage to the population mean. This is true when the sample size is
less than 10-50. Some graphs from Singh and Singh (2003) showing coverage comparisons for
normal, gamma, and lognormal distributions for the various methods are given in Appendix C.
For highly skewed (e.g., a > 2), lognormal data sets of sizes, n less than 50-70, the H-UCL
results in unstable (impractical values which are orders of magnitude higher than other UCLs)
unjustifiably large UCL values (Singh et al., (2002a)). For such highly skewed lognormally
distributed data sets of sizes less than 50 - 70, one may want to use 97.5% or 99% Chebyshev
MVUE UCL of the mean as an estimate of the EPC term (Singh and Singh (2003)). These
recommendations are summarized in Table A2.
It should also be noted that for skewed data sets, the coverage provided by a 95% UCL based
upon Chebyshev inequality is higher than those based upon the percentile bootstrap method or
the BCA bootstrap method. Thus for skewed data sets, the Chebyshev inequality based 95%
UCL of the mean (samples of all sizes from both lognormal and gamma distributions) performs
better than the 95% UCL based upon the BCA bootstrap method. Also, when data are
lognormally distributed, the coverage provided by Chebyshev MVUE UCL (Singh and Singh
(2003)) is better than the one based upon Hall's bootstrap or bootstrap-t method. This is
A-34
-------
especially true when the sample size starts exceeding 10-15. However, for highly skewed data
sets of sizes less than 10-15, it is noted that Hall's bootstrap method provides slightly better
coverage than the Chebyshev MVUE UCL method. Just as for the gamma distribution, it is
observed that for lognormally distributed data sets, the coverage provided by Hall's and
bootstrap-t methods do not increase much with the sample size.
4.9 (1-a) 100% UCL of the Mean Using the Jackknife and Bootstrap Methods
Bootstrap and jackknife methods as discussed by Efiron (1982) are non-parametric statistical
resampling techniques which can be used to reduce the bias of point estimates and construct
approximate confidence intervals for parameters, such as the population mean. These two
methods require no assumptions regarding the statistical distribution (e.g., normal, lognormal, or
gamma) of the underlying population, and can be applied to a variety of situations no matter how
complicated. There exists in the literature of statistics an extensive array of different bootstrap
methods for constructing confidence intervals for the population mean, //j. In the ProUCL,
Version 3.0 software package, five bootstrap methods have been incorporated:
1) the standard bootstrap method,
2) bootstrap-t method (Efron, 1982, Hall, 1988),
3) Hall's bootstrap method (Hall, 1992, Manly, 1997),
4) simple bootstrap percentile method (Manly, 1997), and
5) bias-corrected accelerated (BCA) percentile bootstrap method (Efron and Tibshirani,
1993, Many, 1997).
Let Xj, x2, ... , xn be a random sample of size n from a population with an unknown parameter,
6 (e.g., 0= ft) , and let 6 be an estimate of 6, which is a function of all n observations. For
example, the parameter, 6, could be the population mean, and a reasonable choice for the
A-35
-------
A ^ A
estimate, 0, might be the sample mean, x . Another choice for 0 is the MVUE of the mean of a
lognormal population, especially when dealing with lognormal data sets.
4.9.1 (1-a) 100% UCL of the Mean Based Upon the Jackknife Method
In the jackknife approach, n estimates of 6 are computed by deleting one observation at a
time (Dudewicz and Misra (1988)). Specifically, for each index, /, denote by $(/), the estimate of
6 (computed similarly as 6) when the rth observation is omitted from the original sample of size
w, and let the arithmetic mean of these estimates be given by
*HF> <49)
A quantity known as the rth "pseudo-value" is defined by
J,= n6- (»-l). (50)
The jackknife estimator of 6 is given by the following equation.
J(0) = -v./.= n6- (n- 1)0. (51)
ni=i
If the original estimate $ is biased, then under certain conditions, part of the bias is removed by
the jackknife method, and an estimate of the standard error of the jackknife estimate, J(6\ is
given by
\ / >. L "i
\j n(n- l),-=r
Next, consider the t-type statistic given by
(52)
A-36
-------
t = ^-. (53)
The t-type statistic given by (53) has an approximate Student's-t distribution with n-1 degrees of
freedom, which can be used to derive the following approximate (1 -a)100% UCL for 0,
UCL = J()+ _1. (54)
If the sample size, n, is large, then the upper dh ^-quantile in equation (54) can be replaced with
the corresponding upper ath standard normal quantile, za. Observe, also, that when $ is the
sample mean,jc, then the jackknife estimate is also the sample mean, J(x) = x , and the estimate
of the standard error given by equation (52) simplifies to sjn112, and the UCL in equation (54)
reduces to the familiar t- statistic based UCL given by equation (32). ProUCL uses the jackknife
estimate as the sample mean leading to J(x) = x, which in turn translates equation (54) to the
UCL given by equation (32). This method has been included in ProUCL to satisfy the curiosity
of those users who do not recognize that this jackknife method (with sample mean as the
estimator) yields a UCL of the population mean identical to the UCL based upon the Student's-t
statistic as given by equation (32).
4.9.2 (1-cc) 100% UCL of the Mean Based Upon Standard Bootstrap Method
In bootstrap resampling methods, repeated samples of size n are drawn with replacement
from a given set of observations. The process is repeated a large number of times (e.g., 2000
times), and each time an estimate, $7, of 6 is computed. The estimates thus obtained are used to
compute an estimate of the standard error of 0. A description of the bootstrap method,
illustrated by application to the population mean, yul5 and the sample mean,jc, is given as
follows.
A-37
-------
Step 1. Let (xu, xi2, ..., xir) represent the ith sample of size n with replacement from the
original data set (xlt x2, ..., xj. Then compute the sample mean and denote it by x~{.
Step 2. Perform Step 1 independently TV times (e.g., 1000-2000), each time calculating a new
estimate. Denote those estimates by x^x^....,^. The bootstrap estimate of the
population mean is the arithmetic mean, XB, of the N estimates xt:i= 1,2, ...,7V. The
bootstrap estimate of the standard error of the estimate, x, is given by,
\
(55)
If some parameter, 6 (say, the population median), other than the mean is of concern with an
associated estimate (e.g., the sample median), then the same steps described above could be
applied with the parameter and its estimate used in place of fa and Jc. Specifically, the estimate,
dj, would be computed, instead of xf, for each of the TV bootstrap samples. The general
bootstrap estimate, denoted by 0^, is the arithmetic mean of the TV estimates. The
difference, 0B - 0\ provides an estimate of the bias of the estimate, $, and an estimate of the
standard error of 6 is given by
yid- 0R\. (56)
y- 1 £( ' B> ^ '
The (l-a)100% standard bootstrap UCL for 6is given by
UCL = 0+ zadB. (57)
ProUCL computes the standard bootstrap UCL by using the population AM and sample AM,
respectively given by fa and x . It is observed that the UCL obtained using the standard
bootstrap method is quite similar to the UCL obtained using the Student's-t statistic as given by
A-38
-------
equation (32), and, as such, does not adequately adjust for skewness. For skewed data sets, the
coverage provided by standard bootstrap UCL is much lower than the specified coverage.
Note: For lognormally distributed data sets, one may want to use the jackknife and the standard
bootstrap methods on theMPT/E of the population mean, //;, given by equation (14). However,
the performance of these methods have not been studied. Also, these methods have not been
included in ProUCL.
4.9.3 (1-oc) 100% UCL of the Mean Based Upon Simple Percentile Bootstrap Method
Bootstrap resampling of the original data set is used to generate the bootstrap distribution of
the unknown population mean (Manly, 1997). In this method, xt , the sample mean is computed
from the /th resampling (i=l,2,..., N) of the original data. These xi , i:=l,2,...,N are arranged in
ascending order as J(1) < 3c(2) <....< x(N) . The (l-a)100% UCL of the population mean, ^ is
given by the value, that exceeds the (1-oc) 100% of the generated mean values. The 95% UCL of
the mean is the 95th percentile of the generated means and is given by:
95% Percentile - UCL = 95th%xi;i = 1,2, ...,N (58)
For example, when N=1000, a simple bootstrap 95% percentile- UCL is given by the 950th
ordered mean value given by
Singh and Singh (2003) observed that for skewed data sets, the coverage provided by this
simple percentile bootstrap method is much lower than the coverage provided by the bootstrap-t
and Hall's bootstrap methods. It is observed that for skewed (lognormal and gamma) data sets,
the BCA bootstrap method performs slightly better than the simple percentile method. Some
graphs from Singh and Singh (2003) showing coverage comparisons for normal, gamma, and
A-39
-------
lognormal distributions for the various methods are provided in Appendix C.
4.9.4 (l-a)100% UCL of the Mean Based Upon Bias - Corrected Accelerated (BCA)
Percentile Bootstrap Method
The BCA bootstrap method is also a percentile bootstrap method which adjusts for bias in
the estimate (Efron and Tibshirani, 1993, Manly, 1997). The performance of this method for
skewed distributions (e.g., lognormal and gamma) is not well studied. It was conjectured that
the BCA method would perform better than the various other methods. Singh and Singh (2003)
investigated and compare its performance (in terms of coverage probabilities) with parametric
methods and other bootstrap methods. For skewed data sets, this method does represent a slight
improvement (in terms of coverage probability) over the simple percentile method. However,
this improvement is not adequate enough and yields UCLs with coverage probability much lower
than the specified coverage of 0.95. The BCA upper confidence limit of intended (1-a) 100%
coverage is given by the following equation:
BCA- UCL= x(a*\ (59)
where x 2 is the oc2 100th percentile of the distribution of the xt',i = 1,2,.. .,7V. For example,
C(a2N)
when N=2000, x("2} = (a2N)th ordered statistic of xf;i = 1,2,..., TV given by x. N). Here a2
is given by the following probability statement.
oc2 = 0 (£0 + t_ ~°(~Z+z(i-g))) (60)
Where <&(.) is the standard normal cumulative distribution function and z(1"a) is the 100*(l-a)th
percentile of a standard normal distribution. For example, z (095) = 1.645, and O(1.645) = 0.95.
A-40
-------
Also in equation (60), z0 (bias correction)and a (acceleration factor) are given as follows.
, #(X; < J)
-' '
where O"1 (.) is the inverse function of a standard normal cumulative distribution function, e.g.,
O"1 (0.95)=1.645. a is the acceleration factor and is given by the following equation.
a-
where summation in (62) is being carried from i = 1 to I = n, the sample size, x is the sample
mean based upon all n observations, and x_t is the mean of (n-1) observations without the ith
observation, i = 1,2,. ..,n.
Singh and Singh (2003) observed that for skewed data sets (e.g., gamma and lognormal), the
coverage provided by this BCA percentile method is much lower than the coverage provided by
the bootstrap-t and Hall's bootstrap methods. This is especially true when the sample size is
small. The BCA method does provide an improvement over the simple percentile method and
the standard bootstrap method. However, bootstrap-t and Hall's bootstrap methods perform
better (in terms of coverage probabilities) than the BCA method. For skewed data sets, the BCA
method also performs better than the modified-t UCL. For gamma distributions, the coverage
provided by BCA 95% UCL approaches 0.95 as the sample size increases. For lognormal
distributions, the coverage provided by the BCA 95% UCL is much lower than the specified
coverage of 0.95.
4.9.5 (1-cc) 100% UCL of the Mean Based Upon Bootstrap-t Method
Another variation of the bootstrap method, called the "bootstrap-t" by Efron (1982), is a non-
A-41
-------
parametric method which uses the bootstrap methodology to estimate quantiles of the pivotal
quantity, t statistic, given by equation (31). Rather than using the quantiles of the familiar
Student's-t statistic, Hall (1988) proposed to compute estimates of the quantiles of the statistic
given by equation (31) directly from the data.
Specifically, in Steps 1 and 2 described above in Section 4.9.2, if x is the sample mean
computed from the original data, and xt and sxj are the sample mean and sample standard
deviation computed from the rth resampling of the original data, the TV quantities
ft= (Jn)(xi~ *)/sxi are computed and sorted, yielding ordered quantities, ?(1) < t(2) < < t(N). The
estimate of the lower ath quantile of the pivotal quantity in equation (31) is tajB = t(aN}. For
example, ifW= 1000 bootstrap samples are generated, then the 50th ordered value, t(50), would be
the bootstrap estimate of the lower 0.05th quantile of the pivotal quantity in equation (31). Then
a (1-a) 100% UCL of the population mean based upon the bootstrap-t method is given by
UCL = x - t(aN)sx/fi. (63)
Note the '-' sign in equation (63). ProUCL computes the Bootstrap-t UCL based upon the
quantiles obtained using the sample mean, x. It is observed that the UCL based upon the
bootstrap-t method is more conservative than the other UCLs obtained using the Student's-1,
modified -t, adjusted -CLT, and the standard bootstrap methods. This is specially true for
skewed data sets. This method seems to adjust for skewness to some extent.
It is observed that for skewed data sets (e.g., gamma, lognormal), the 95% UCL based upon
bootstrap-t method performs better than the 95% UCLs based upon the simple percentile and the
BCA percentile methods (Singh and Singh (2003)). For highly skewed (k < 0.1 or a > 2.5-3.0)
data sets of small sizes (e.g., n < 10) the bootstrap-t method performs better than other (adjusted
gamma UCL, or Chebyshev inequality UCL) UCL computation methods. It is noted that for
A-42
-------
gamma distribution, the performances (coverages provided by the respective UCLs) of bootstrap-
t and Hall's bootstrap methods are very similar. It is also noted that for larger samples, these two
methods (bootstrap-t and Hall's bootstrap) approximately provide the specified 95% coverage to
the mean, k0, of the gamma distribution. For gamma distributed data sets, the coverage provided
by a bootstrap-t (and Hall's bootstrap UCL) 95% UCL approaches 95% as sample size increases
for all values of k considered (k = 0.05-5.0) in Singh and Singh (2003). However, it is noted that
the coverage provided by these two bootstrap methods is slightly lower than 0.95 for samples of
smaller sizes.
For lognormally distributed data sets, the coverage provided by bootstrap-t 95% UCL is a
little bit lower than the coverage provided by the 95% UCL based upon Hall's bootstrap method.
However, it should be noted that for lognormally distributed data sets, for samples of all sizes,
the coverage provided by these two methods (bootstrap-t and Hall's bootstrap) is significantly
lower than the specified 0.95 coverage. This is especially true for moderately skewed to highly
skewed (e.g., o>1.0) lognormally distributed data sets. This can be seen from the graphs
presented in Appendix C.
It should be pointed out that the bootstrap-t and Hall's bootstrap methods sometimes result in
unstable, erratic, and unreasonably inflated UCL values especially in the presence of outliers
(Efron and Tibshirani, 1993). Therefore, these two methods should be used with caution. In case
these two methods result in erratic and inflated UCL values, then an appropriate Chebyshev
inequality based UCL may be used to estimate the EPC term for non-parametric skewed data
sets.
4.9.6 (1-oc) 100% UCL of the Mean Based Upon Hall's Bootstrap Method
Hall (1992) proposed a bootstrap method which adjusts for bias as well as skewness. This
A-43
-------
method has been included in UCL guidance document (EPA 2002). For highly skewed data sets
(e.g., LN(5,4)), it performs slightly better (higher coverage) than the bootstrap-t method. In this
yv
method, xt , sx _/; and k3i , the sample mean, sample standard deviation, and sample skewness
are computed from the rth resampling (I = 1, 2,..., N) of the original data. Let x be the sample
^,
mean, sx be the sample standard deviation, and k3 be the sample skewness (as given by
equation (43)) computed from the original data. The quantities W; and Q; given as follows are
computed for each of the N bootstrap samples, where
The quantities Qt (Wt) given above are arranged in ascending order. For a specified (1-oc)
confidence coefficient, compute the (ocN)th ordered value, qa of quantities Qt(Wt) Next,
compute W(qa ) using the inverse function, which is given as follows:
l/3
\ + k,(qa-k,l(6n)} -l/*3. (64)
yv
In equation (64), k3 is computed using equation (43). Finally, the (1-a) 100% UCL of the
population mean based upon Hall's bootstrap method (Manly, 1997) is given as follows:
UCL = x-W(qa}*sx. (65)
For gamma distribution, Singh and Singh (2003) observed that the coverage probabilities
provided by the 95% f/CLs based upon bootstrap-t and Hall's bootstrap methods are in close
agreement. For larger samples these two methods approximately provide the specified 95%
coverage to the population mean, k0 of a gamma distribution. For smaller sample sizes (from
gamma distribution), the coverage provided by these two methods is slightly lower than the
specified level of 0.95. For both lognormal and gamma distributions, these two methods
A-44
-------
(bootstrap-t and Hall's bootstrap) perform better than the other bootstrap methods, namely, the
standard bootstrap method, simple percentile, and bootstrap BCA percentile methods. This can
be seen from graphs presented in Appendix C.
Just like the gamma distribution, for lognormally distributed data sets, it is noted that Hall's
UCL and bootstrap-t UCL provide similar coverages. However, for highly skewed lognormal
data sets, the coverages based upon Hall's method and bootstrap-t method are significantly lower
than the specified 0.95 coverage (Singh and Singh ( 2003)). This is true even in samples of
larger sizes(e.g., n=100). For lognormal data sets, the coverages provided by Hall's bootstrap
and bootstrap-t methods do not increase much with the sample size, n. For highly skewed (e.g.,
a > 2.0) data sets of small sizes (e.g., n < 15), Hall's bootstrap method (and also bootstrap-t
method) performs better than Chebyshev UCL, and for larger samples, Chebyshev UCL performs
better than Hall's bootstrap method. Similar to the bootstrap-t method, it should be noted that
Hall's bootstrap method sometimes results in unstable, inflated, and erratic values especially in
the presence of outliers (Efiron and Tibshirani, 1993). Therefore, these two methods should be
used with caution. If outliers are present in a data set, then a 95% UCL of the mean should be
computed using alternative UCL computation methods.
5. Recommendations and Summary
This section describes the recommendations and summary on the computation of a 95% UCL
of the unknown population arithmetic mean, //b of a contaminant data distribution without
censoring. These recommendations are based upon the findings of Singh, Singh, and
Engelhardt (1997, 1999); Singh et al. ( 2002a); Singh, Singh, and laci (2002b); and Singh and
Singh (2003). Recommendations have been summarized for: 1) normally distributed data sets,
2) gamma distributed data sets, 3) lognormally distributed data sets, and 4) data sets which are
non-parametric and do not follow any of the three distributions included in ProUCL.
A-45
-------
For skewed parametric as well as non-parametric data sets, there is no simple solution to
compute a 95% UCL of the population mean, ^ . Singh et al. (2002a), Singh, Singh, and laci
(2002b), and Singh and Singh (2003) noted that the UCLs based upon the skewness adjusted
methods, such as the Johnson's modified-t and Chen's adjusted-CLrdo not provide the specified
coverage (e.g., 95%) to the population mean even for mildly to moderately skewed (e.g., a in
interval [0.5, 1.0)) data sets for samples of size as large as 100. The coverage of the population
mean by these skewness-adjusted UCLs gets poorer (much smaller than the specified coverage
of 0.95) for highly skewed data sets, where the skewness levels are defined in Section 3.2.2 as a
function of
-------
details refer to Singh and Singh (2003).
For normally distributed data sets, a UCL based upon the Student's-t statistic as given by
equation (32) provides the optimal UCL of the population mean. Therefore, for normally
distributed data sets, one should always use a 95% UCL based upon the Student's-t statistic.
The 95% UCL of the mean given by equation (32) based upon Student's-t statistic may also
be used when the Sd, sy of the log-transformed data is less than 0.5, or when the data set
approximately follows a normal distribution. A data set is approximately normal when the
normal Q-Q plot displays a linear pattern (without outliers and jumps) and the resulting
correlation coefficient is high (e.g., 0.95 or higher).
Student's-t UCL may also be used when the data set is symmetric (but possibly not normally
distributed). A measure of symmetry (or skewness) is k3 which is given by equation (43).
^.
A value of &3 close to zero (e.g., if absolute value of skewness is roughly less than 0.2 or
0.3) suggests approximate symmetry. The approximate symmetry of a data distribution can
also be judged by looking at the histogram of the data set.
5.1.2 Gamma Distributed Skewed Data Sets
In practice, many skewed data sets can be modeled both by a lognormal distribution and a
gamma distribution especially when the sample size is smaller than 70-100. As well known, the
95% H-UCL of the mean based upon a lognormal model often results in unjustifiably large and
impractical 95% UCL value. In such cases, a gamma model, G(k,0) may be used to compute a
reliable 95% UCL of the unknown population mean, //;.
Many skewed data sets follow a lognormal as well as a gamma distribution. It should be
A-47
-------
noted that the population means based upon the two models can differ significantly.
Lognormal model based upon a highly skewed (e.g., a > 2.5 )data set will have an
unjustifiably large and impractical population mean, ^ and its associated UCL. The gamma
distribution is better suited to model positively skewed environmental data sets.
One should always first check if a given skewed data set follows a gamma distribution. If a
data set does follow a gamma distribution or an approximate gamma distribution, one should
compute a 95% UCL based upon a gamma distribution. Use of highly skewed (e.g., a > 2.5-
3.0) lognormal distributions should be avoided. For such highly skewed lognormally
distributed data sets which can not be modeled by a gamma or an approximate gamma
distribution, non-parametric UCL computation methods based upon the Chebyshev
inequality may be used.
The five bootstrap methods do not perform better than the two gamma UCL computation
methods. It is noted that the performances (in terms of coverage probabilities) of bootstrap-t
and Hall's bootstrap methods are very similar. Out of the five bootstrap methods, bootstrap-t
and Hall's bootstrap methods perform the best (with coverage probabilities for the population
mean closer to the nominal level of 0. 95). This is especially true when skewness is quite
high (e.g., k < 0.1) and sample size is small (e.g., n < 10-15). This can be seen from graphs
given in Appendix C.
The bootstrap BCA method does not perform better than the Hall's method or the bootstrap-t
method. The coverage for the population mean, ^ provided by the BCA method is much
lower than the specified 95% coverage. This is especially true when the skewness is high
(e.g., k <1) and sample size is small (Singh and Singh (2003)).
From the results presented in Singh, Singh, and laci (2002b) and in Singh and Singh (2003),
A-48
-------
it is concluded that for data sets which follow a gamma distribution, a 95% UCL of the mean
should be computed using the adjusted gamma UCL when the shape parameter, k is: 0.1 < k
< 0.5, and for values of k > 0.5, a 95% UCL can be computed using an approximate gamma
UCL of the mean, //;.
For highly skewed gamma distributed data sets with k < 0.1, bootstrap-t UCL or Hall's
bootstrap (Singh and Singh (2003)) may be used when the sample size is smaller than 15, and
the adjusted gamma UCL should be used when sample size starts approaching and exceeding
15. The small sample size requirement increases as skewness increases (that is as k
decreases, the required sample size, n increases).
The bootstrap-t and Hall's bootstrap methods should be used with caution as some times
these methods yield erratic, unreasonably inflated, and unstable UCL values especially in the
presence of outliers. In case Hall's bootstrap and bootstrap-t methods yield inflated and
erratic UCL results, the 95% UCL of the mean should be computed based upon the adjusted
gamma 95% UCL. ProUCL prints out a warning message associated with the recommended
use of the UCLs based upon the bootstrap-t method or Hall's bootstrap method.
These recommendations for the use of gamma distribution are summarized in Table Al.
Table Al.
Summary Table for the Computation of a 95% UCL of the Unknown Mean, ju^
of a Gamma Distribution
yv
k
k> 0.5
0.1 < k < 0.5
Sample Size, n
For all n
For all n
Recommendation
Approximate Gamma 95%UCL
Adjusted Gamma 95% UCL
A-49
-------
k< 0.1
n<15
95% UCL Based Upon Bootstrap-t
or Hall's Bootstrap Method *
k< 0.1
n> 15
Adjusted Gamma 95% UCL if available,
otherwise use Approximate Gamma 95% UCL
* In case bootstrap-t or Hall's bootstrap methods yield erratic, inflated, and unstable UCL values,
the UCL of the mean should be computed using adjusted gamma UCL.
5.1.3 Lognormally Distributed Skewed Data Sets
For lognormally, LN(//, a2) distributed data sets, the H-statistic based UCL does provide the
specified 0.95 coverage for the population mean for all values of a. However, the H-statistic
often results in unjustifiably large UCL values which do not occur in practice. This is especially
true when skewness is high (e.g., a > 2.0). The use of a lognormal model unjustifiably
accommodates large and impractical values of the mean concentration and its UCLs. The
problem associated with the use of a lognormal distribution is that the population mean, fa, of a
lognormal model becomes impractically large for larger values of a which in turn results in
inflated H-UCL of the population mean, fa. Since the population mean of a lognormal model
becomes too large, none of the other methods except for H-UCL provides the specified 95%
coverage for that inflated population mean, fa. This is especially true when the sample size is
small and skewness is high. For extremely highly skewed data sets (with a > 2.5-3.0) of smaller
sizes (e.g., < 70-100), the use of a lognormal distribution based H-UCL should be avoided (e.g.,
see Singh et al. (2002a), Singh and Singh (2003)). Therefore, alternative UCL computation
methods such as the use of a gamma distribution or use of a UCL based upon non-parametric
bootstrap methods or Chebyshev inequality based methods are desirable.
As expected for skewed (e.g., with a (or a ) > 0.5) lognormally distributed data sets, the
Student's-t UCL, modified-t UCL, adjusted -CLT UCL, standard bootstrap method all fail to
A-50
-------
provide the specified 0.95 coverage for the unknown population mean for samples of all sizes.
Just like the gamma distribution, the performances (in terms of coverage probabilities) of
bootstrap-t and Hall's bootstrap methods are very similar (Singh and Singh (2003)). However, it
is noted that the coverage provided by Hall's bootstrap (and also by bootstrap-t) is much lower
than the specified 95% coverage for the population mean, //;, for samples of all sizes of varying
skewness. Moreover, the coverages provided by Hall's bootstrap or bootstrap-t method do not
increase much with the sample size.
Also the coverage provided by the BCA method is much lower than the coverage provided
by Hall's method or bootstrap-t method. Thus the BCA bootstrap method can not be
recommended to compute a 95% UCL of the mean of a lognormal population. For highly
skewed data sets of small sizes (e.g., < 15) with a exceeding 2.5-3.0, even the Chebyshev
inequality based C/CLs fail to provide the specified 0.95 coverage for the population. However,
as the sample size increases, the coverages provided by the chebyshev inequality based C/CLs
also increase. For such highly skewed data sets ( a > 2.5 ) of sizes less than 10-15, Hall's
bootstrap or bootstrap-t methods provide larger coverage than the coverage provided by the 99%
Chebyshev (MVUE) UCL. Therefore, for highly skewed lognormally distributed data sets of
small sizes, one may use Hall's method to compute an estimate of the EPC term. The small
sample size requirement increases with a. Graphs from Singh and Singh (2003) showing
coverage comparisons for normal, gamma, and lognormal distributions for the various methods
are given in Appendix C.
It should be noted that even a small increase in the Sd, o, increases skewness considerably.
For example, for a lognormal distribution, when a = 2.5, skewness ~ 11825.1; and when o = 3,
skewness ~ 729555. In practice, the occurrence of such highly skewed data sets (e.g., o > 3) is
not very common. Nevertheless, these highly skewed data sets can arise occasionally and,
therefore, require separate attention. Singh et al. (2002a) observed that when the Sd, a, starts
A-51
-------
approaching 2.5 (that is, for lognormal data, when CV > 22.74 and skewness > 11825.1), even
a 99% Chebyshev (MVUE) UCL fails to provide the desired 95% coverage for the population
mean,//;. This is especially true when the sample size, n is smaller than 30. For such
extremely skewed data sets, the larger of the two UCLs: the 99% Chebyshev (MVUE) UCL and
the non-parametric 99% Chebyshev (Mean, Sd) UCL, may be used as an estimate of the EPC.
It is also noted that, as the sample size increases, the H-UCL starts behaving in a stable
manner. Therefore, depending upon the Sd, a (actually its MLE d), for lognormally
distributed data sets, one can use the H-UCL for samples of larger sizes such as greater than
70-100. This large sample size requirement increases as the Sd, 6, increases, as can be seen in
Table A2. ProUCL can compute an H-UCL for samples of sizes up to 1000. For lognormally
distributed data sets of smaller sizes, some alternative methods to compute a 95% UCL of the
population mean, ^ are summarized in Table A2.
Furthermore, it is noted that for larger sample sizes (e.g., n > 150), the H-UCL becomes even
smaller than the Student's-t UCL and various other UCLs. It should be pointed out that the large
sample behavior of H-UCL has not been investigated rigorously. For confirmation purposes
(that is H-UCL does provide the 95% coverage for larger samples also), it is desirable to conduct
such a study for samples of larger sizes.
Since skewness (as defined in Section 3.2.2) is a function of a (or a ), the recommendations
for the computation of the UCL of the population mean are also summarized in Table A2 for
various values of the MLE a of a and the sample size, n. Here a is an MLE of a, and is given
by the Sd of log-transformed data given by equation (2). Note that Table A2 is applicable to the
computation of a 95% UCL of the population mean based upon lognormally distributed data sets
without non-detect observations. A method to compute a 95% UCL of the mean of a lognormal
distribution is summarized as follows:
A-52
-------
Skewed data sets should be first tested for a gamma distribution. For lognormally distributed
data sets (which can not be modeled by a gamma distribution), the method as summarized in
Table A2 may be used to compute a 95% UCL of the mean.
Specifically, for highly skewed (e.g., 1.5 < o < 2.5) data sets of small sizes (e.g., n < 50-70),
the EPC term may be estimated by using a 97.5% or 99%MVUE Chebyshev UCL of the
population mean. For larger samples (e.g., n > 70), H-UCL may be used to estimate the EPC.
For extremely highly skewed (e.g., a > 2.5) lognormally distributed data sets, the population
mean becomes unrealistically large. Therefore, the use of H-UCL should be avoided
especially when the sample size is less thanlOO. For such highly skewed data sets, Hall's
bootstrap UCL may be used when the sample size is less than 10-15 (Singh and Singh
(2003)). The small sample size requirement increases with a . For example, n = 10 is
considered small when a =3.0, and n = 15 is considered small when a =3.5.
Hall's bootstrap methods should be used with caution as some times it yields erratic, inflated,
and unstable UCL values, especially in the presence of outliers. For these highly skewed
data sets of size, n (e.g., less than 10-15), in case Hall's bootstrap method yields an erratic
and inflated UCL value, the 99% Chebyshev MVUE UCL may be used to estimate the EPC
term. ProUCL displays a warning message associated with the recommended use of Hall's
bootstrap method.
A-53
-------
Table A2. Summary Table for the Computation of a
95% UCL of the Unknown Mean, fjj of a Lognormal Population
6
6 <0.5
0.5 < o < 1.0
1.0 < 6 < 1.5
1.5 < 6<2.0
1.5 < 6<2.0
2.5 < 6<3.0
3.0 < a< 3.5
a>3.5
Sample Size, n
For all n
For all n
n<25
n > 25
n<20
20 < n < 50
n > 50
n<20
20 < n < 50
50 < n < 70
n > 70
n<30
30 < n<70
70 < n< 100
n > 100
n< 15
15< n<50
50 < n< 100
100 < n< 150
n> 150
For all n
Recommendation
Student' s-t, modified-t, orH-UCL
H-UCL
95% Chebyshev (MVUE) UCL
H-UCL
99% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
99% Chebyshev (MVUE) UCL
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Larger of (99% Chebyshev (MVUE) UCL or
99% Chebyshev (Mean, Sd))
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Hall's bootstrap method *
Larger of (99% Chebyshev (MVUE) UCL,
99% Chebyshev(Mean, Sd))
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Use non-parametric methods *
* In case Hall's bootstrap method yields an erratic unrealistically large UCL value, then the UCL
of the mean may be computed based upon the Chebyshev inequality.
A-54
-------
5.1.4 Data Sets Without a Discernable Skewed Distribution - Non-parametric Skewed
Data Sets
The use of gamma and lognormal distributions as discussed here will cover a wide range of
skewed data distributions. For skewed data sets which are neither gamma nor lognormal, one can
use a non-parametric Chebyshev UCL or Hall's bootstrap UCL (for small samples) of the mean
to estimate the EPC term.
For skewed non-parametric data sets with negative and zero values, use a 95% Chebyshev
(Mean, Sd) UCL for the population mean, //;.
For all other non-parametric data sets with only positive values, the following method may be
used to estimate the EPC term.
For mildly skewed data sets with a < 0.5, one can use Student's-t statistic or modified-t
statistic to compute a 95% UCL of mean, //;.
For non-parametric moderately skewed data sets (e.g., a or its estimate, a in the interval
(0.5, 1]), one may use a 95% Chebyshev (Mean, Sd) UCL of the population mean, //;.
For non-parametric moderately to highly skewed data sets (e.g., 6 in the interval (1.0,
2.0]), one may use a 99% Chebyshev (Mean, Sd) UCL or 97.5% Chebyshev (Mean, Sd) UCL
of the population mean, //b to obtain an estimate of the EPC term.
For highly skewed to extremely highly skewed data sets with a in the interval (2.0, 3.0], one
may use Hall's UCL or 99% Chebyshev (Mean, Sd) UCL to compute the EPC term.
A-55
-------
Extremely skewed non-parametric data sets with a exceeding 3.0, provide poor coverage.
For such highly skewed data distributions, none of the methods considered provide the
specified 95% coverage for the population mean, //;. The coverages provided by the various
methods decrease as a increases. For such data sets of sizes less than 30, a 95% UCL can be
computed based upon Hall's bootstrap method or bootstrap-t method. Hall's bootstrap
method provides highest coverage (but less than 0.95) when the sample size is small. It is
noted that the coverage for the population mean provided by Hall's method (and bootstrap-t
method) does not increase much as the sample size, n increases. However, as the sample size
increases, coverage provided by 99% Chebyshev (Mean, Sd) UCL method also increases.
Therefore, for larger samples, a UCL should be computed based upon 99% Chebyshev
(Mean, Sd) method. This large sample size requirement increases as a increases. These
recommendations are summarized in Table A3.
A-56
-------
Table A3.
Summary Table for the Computation of a 95% UCL of the Unknown Mean,
Skewed Non-parametric Distribution with all Positive Values,
Where 3.5
Sample Size, n
For all n
For all n
n<50
n > 50
n<10
n > 10
n<30
n > 30
n< 100
n> 100
Recommendation
95% UCL based on Student' s-t or Modified-t statistic
95% Chebyshev (Mean, Sd) UCL
99% Chebyshev (Mean, Sd) UCL
97.5% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL*
99% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL*
99% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL *
99% Chebyshev (Mean, Sd) UCL
* If Hall's bootstrap method yields an erratic and unstable UCL value (e.g., happens when outliers are present), a
UCL of the population mean may be computed based upon the 99% Chebyshev (Mean, Sd) method.
5.2 Summary of the Procedure to Compute a 95% UCL of the Unknown Population
Mean, /jj Based Upon Data Sets Without Non-detect Observations
The first step in computing a 95% UCL of a population arithmetic mean, ^ is to perform
goodness-of-fit tests to test for normality, lognormality, or gamma distribution of the data set
under study. ProUCL has three methods to test for normality or lognormality: the informal
graphical test based upon a Q-Q plot, the Lilliefors test, and the Shapiro-Wilk W test.
ProUCL also has three methods to test for a gamma distribution: the informal graphical Q-Q
plot based upon gamma quantiles, the Kolmogorov-Smirnov (K-S) EDF test, and the
A-57
-------
Anderson-Darling (A-D) EDF test.
ProUCL generates a quantile-quantile (Q-Q) plot to graphically test the normality,
lognormality, or gamma distribution of the data. There is no substitute for graphical displays
of a data set. On this graph, a linear pattern (e.g., with high correlation such as 0.95 or
higher) displayed by bulk of data suggests approximate normality, lognormality, or gamma
distribution. On this graph, points well-separated from the majority of data may be potential
outliers requiring special attention. Also, any visible jumps and breaks of significant
magnitudes on a Q-Q plot suggest that more than one population may be present. In that
case, each of the populations should be considered separately. That is a separate EPC term
should be computed for each of the populations. It is, therefore, recommended to always use
the graphical Q-Q plot as it provides useful information about the presence of multiple
populations (e.g., site and background data mixed together) and/or outliers. Both graphical
Q-Q plot and formal goodness-of-fit tests should be used on the same data set.
A single test statistic such as the Shapiro-Wilk test (or the A-D test etc.) may lead to the
incorrect conclusion that the data are normally (or gamma) distributed even when there are
more than one population present. Only a graphical display such as an appropriate Q-Q can
provide this information. Obviously, when multiple populations are present, those should be
separated out and the EPC terms (the UCLs) should be computed separately for each of those
populations Therefore, it is strongly recommended not to skip the Goodness-of-Fit Tests
Option in ProUCL. Since the computation of an appropriate UCL depends upon data
distribution, it is advisable that the user should take his time (instead of blindly using a
numerical value of a test statistic in an effort to automate the distribution selection process)
to determine the data distribution. Both graphical (e.g., Q-Q plots) and analytical procedures
(Shapiro-Wilk test, K-S test etc.) should be used on the same data set to determine the most
appropriate distribution of the data set under study.
A-58
-------
After performing the Goodness-of-Fit test, ProUCL informs the user about the data
distribution: normal, lognormal, gamma distribution, or non-parametric.
For a normally distributed (or approximately normally distributed) data set, the user is
advised to use Student's-t distribution based UCL of the mean. Student's-t distribution (or
modified-t statistic) may also be used to compute the EPC term when the data set is
^,
symmetric (e.g., k3 is smaller than 0.2-0.3) or mildly skewed, that is when 0.5; use the adjusted gamma UCL for 0.1 < k <
0.5; use bootstrap-t method (or Hall's method) when k < 0.1 and the sample size, n < 15;
yv
and use the adjusted gamma UCL (if available) for k < 0.1 and sample size, n > 15. If the
adjusted gamma UCL is not available then use the approximate gamma UCL as an estimate
of the EPC term. In case bootstrap-t method or Hall's bootstrap method yields an erratic
inflated UCL (e.g., when outliers are present) result, the UCL should be computed using the
adjusted gamma UCL (if available) or the approximate gamma UCL. Some graphs from
Singh and Singh (2003) showing coverage comparisons for normal, gamma, and lognormal
distributions for the various methods considered are given in Appendix C.
For lognormal data sets, ProUCL recommends (as summarized in Table A2, Section 5.1.3) a
method to estimate the EPC term based upon the sample size and standard deviation of the
log-transformed data, a. ProUCL can compute a H-UCL of the mean for samples of size up
to 1000.
Non-parametric UCL computation methods such as the modified-t, CLTmethod, adjusted-
CLT method, bootstrap and jackknife methods are also included in ProUCL. However, it is
A-59
-------
noted that non-parametric UCLs based upon most of these methods do not provide adequate
coverage to the population mean for moderately skewed to highly skewed data sets (e.g.,
see Singh et al. (2002a), and Singh and Singh (2003)).
For data sets which are not normally, lognormally, or gamma distributed, a non-parametric
UCL of the mean based upon the Chebyshev inequality is preferred. The Chebyshev (Mean,
Sd) UCL does not depend upon any distributional assumptions and can be used for
moderately to highly skewed data sets which do not follow any of the three data
distributions incorporated in ProUCL.
It should be noted that for extremely skewed data sets (e.g., with a exceeding 3.0), even a
Chebyshev inequality based 99% UCL of the mean fails to provide the desired coverage
(e.g., 0.95) of the population mean. A method to compute the EPC term for non-parametric
distributions is summarized in Table A3 of Section 5.1.4. It should be pointed out that in
case Hall's bootstrap method appears to yield erratic and inflated results (typically happens
when outliers are present), the 99% Chebyshev UCL may be used as an estimate of the EPC
term.
5.3 Should the Maximum Observed Concentration be Used as an Estimate of the EPC
Term?
Singh and Singh (2003) also included the Max Test (using the maximum observed value as
an estimate of the EPC term) in their simulation study. Previous (e.g., EPA 1992 RAGS
Document) use of the maximum observed value has been recommended as a default value to
estimate the EPC term when a 95% UCL (e.g., the H-UCL) exceeded the maximum value.
However, in past (e.g., EPA 1992), only two 95% UCL computation methods, namely: the
Student's-1 UCL and Land's H-UCL were used to estimate the EPC term. ProUCL, Version
A-60
-------
3.0 can compute a 95% UCL of mean using several methods based upon normal, Gamma,
lognormal, and non-parametric distributions. Thus, ProUCL, Version 3.0 has about fifteen
(15) 95% UCL computation methods, one of which (depending upon skewness and data
distribution) can be used to compute an appropriate estimate of the EPC term. Furthermore,
since the EPC term represents the average exposure contracted by an individual over an
exposure area (EA) during a long period of time, therefore, the EPC term should be estimated
by using an average value (such as an appropriate 95% UCL of the mean) and not by the
maximum observed concentration.
With the availability of so many UCL computation methods (15 of them), the developers of
ProUCL Version 3.0 do not feel any need to use the maximum observed value as an estimate
of the EPC term. Singh and Singh (2003) also noted that for skewed data sets of small sizes
(e.g., < 10-20), the Max Test does not provide the specified 95% coverage to the population
mean, and for larger data sets, it overestimates the EPC term. This can also viewed in the
graphs presented in Appendix C. Also, for the distributions considered, the maximum value
is not a sufficient statistic for the unknown population mean. The use of the maximum value
as an estimate of the EPC term ignores most (except for the maximum value) of the
information contained in a data set. It is not desirable to use the maximum observed value as
estimate of the EPC term representing average exposure over an EA.
It should also be noted that for highly skewed data sets, the sample mean indeed can even
exceed the upper 90%, 95 % etc. percentiles, and consequently, a 95% UCL of mean can
exceed the maximum observed value of a data set. This is especially true when one is dealing
with lognormally distributed data sets of small sizes. As mentioned before, for such highly
skewed data sets which can not be modeled by a gamma distribution, a 95% UCL of the
mean should be computed using an appropriate non-parametric method. These observations
are summarized in Tables A1-A3 of this Appendix A.
A-61
-------
Alternatively, for such highly skewed data sets, other measures of central tendency such as
the median (or some other upper percentile such as 70% percentile) and its upper confidence
limit may be considered. The EPA and all other interested agencies and parties need to come
to an agreement upon the use of the median and its UCL to estimate the EPC term for a
contaminant of concern at a polluted site. It should be mentioned that the use of the sample
median and/or its UCL as estimates of the EPC term needs further research and investigation.
It is recommended that the maximum observed value NOT be used as an estimate of
the EPC term. For the sake of interested users, ProUCL displays a warning message when
the recommended 95% UCL (e.g., Hall's bootstrap UCL etc.) of the mean exceeds the
observed maximum concentration. For such cases (when a 95% UCL does exceed the
maximum observed value), if applicable, an alternative 95% UCL computation method is
recommended by ProUCL.
A-62
-------
References
Aitchison, J., and Brown, J.A.C. (1969), The LognormalDistribution, Cambridge: Cambridge
University Press.
Best, D.J., and Roberts, D.E. (1975). " The Percentage Points of the Chi-square Distribution."
Applied Statistics, 24: 385-388.
Bowman, K. O., and Shenton, L.R. (1988), Properties of Estimators for the Gamma
Distribution, Volume 89. Marcel Dekker, Inc. New York.
Bradu, D., and Mundlak, Y. (1970), "Estimation in Lognormal Linear Models," Journal of the
American Statistical Association, 65, 198-211.
Chen, L. (1995), "Testing the Mean of Skewed Distributions," Journal of the American
Statistical Association., 90, 767-772.
Choi, S. C., and Wette, R. (1969), Maximum Likelihood Estimation of the Parameters of the
Gamma Distribution and Their Bias. Technometrics, Vol. 11, 683-690.
Dudewicz, E.D., and Misra, S.N. (1988), Modern Mathematical Statistics. John Wiley, New
York.
D'Agostino, R.B., and Stephens, M.A. (1986), Goodness-of-Fit-Techniques, Marcel Dekker,
Inc.
A-63
-------
Efron, B. (1982), The Jackknife, the Bootstrap, and Other Resampling Plans, Philadelphia:
SIAM.
Efron, B., and Tibshirani, RJ. (1993), An Introduction to the Bootstrap, Chapman & Hall,
New York.
EPA(1989), "Methods for Evaluating the Attainment of Cleanup Standards, Vol. 1, Soils and
Solid Media," Publication EPA 230/2-89/042.
EPA (1991), "A Guide: Methods for Evaluating the Attainment of Cleanup Standards for
Soils and Solid Media," Publication EPA/540/R95/128.
EPA (1992), "Supplemental Guidance to RAGS: Calculating the Concentration Term,"
Publication EPA 9285.7-081, May 1992.
EPA (1996), "A Guide: Soil Screening Guidance: Technical Background Document," Second
Edition, Publication 9355.4-04FS.
EPA (2002), Calculating Upper Confidence Limits for Exposure Point Concentrations at
Hazardous Waste Sites, OSWER 9285.6-10, December 2002.
ExpertFit Software (2001), Averill M. Law & Associates Inc, Tucson, Arizona.
Faires, J. D., and Burden, R. L. (1993), Numerical Methods, PWS-Kent Publishing Company,
Boston, USA.
A-64
-------
Gilbert, R.O. (1987), Statistical Methods for Environmental Pollution Monitoring, New York:
Van Nostrand Reinhold.
Grice, J.V., and Bain, L. J. (1980), Inferences Concerning the Mean of the Gamma
Distribution. Journal of the American Statistical Association. Vol 75, Number 372, pp 929-
933.
Hall, P. (1988), Theoretical comparison of bootstrap confidence intervals; Annals of
Statistics, 16, 927-953.
Hall, P. (1992), On the Removal of Skewness by Transformation. Journal of Royal Statistical
Society, B 54, 221-228.
Hardin, J.W., and Gilbert, R.O. (1993), "Comparing Statistical Tests for Detecting Soil
Contamination Greater Than Background," Pacific Northwest Laboratory, Battelle, Technical
Report #DE 94-005498.
Hoaglin, D.C., Mosteller, F., and Tukey, J.W. (1983), Understanding Robust and
Exploratory Data Analysis. John Wiley, New York.
Hogg, R.V., and Craig, A.T. (1978), Introduction to Mathematical Statistics, New York:
Macmillan Publishing Company.
Johnson, N.J. (1978), "Modified t-Tests and Confidence Intervals for Asymmetrical
Populations," The American Statistician., Vol. 73, pp.536-544.
A-65
-------
Johnson, N.L., Kotz, S., and Balakrishnan, N. (1994), Continuous Univariate Distributions,
Volume 1. Second Edition. John Wiley.
Kleijnen, J.P.C., Kloppenburg, G.L.J., and Meeuwsen, F.L. (1986), "Testing the Mean of an
Asymmetric Population: Johnson's Modified t Test Revisited." Commun. in Statist.-Simula.,
15(3), 715-731.
Land, C. E. (1971), "Confidence Intervals for Linear Functions of the Normal Mean and
Variance," Annals of Mathematical Statistics, 42, 1187-1205.
Land, C. E. (1975), "Tables of Confidence Limits for Linear Functions of the Normal Mean
and Variance," in Selected Tables in Mathematical Statistics, Vol. Ill, American
Mathematical Society, Providence, R.I., 385-419.
Law, A.M., and Kelton, W.D. (2000), Simulation Modeling and Analysis. Third Edition.
McGrawHill.
Manly, B.F.J. (1997), Randomization, Bootstrap, and Monte Carlo Methods in Biology.
Second Edition. Chapman Hall, London.
Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. (1990). Numerical
Recipes in C, The Art of Scientific Computing. Cambridge University Press. Cambridge, MA.
Schneider, B. E. (1978), Kolmogorov-Smirnov Test Statistic for the Gamma Distribution with
Unknown Parameters, Dissertation, Department of Statistics, Temple University,
Philadelphia, Pa.
A-66
-------
Schneider, B.E., and Clickner, R.P. (1976). On the distribution of the Kolmogorov-Smirnov
Statistic for the Gamma Distribution with Unknown Parameters, Mimeo Series Number 36,
Department of Statistics, School of Business Administration, Temple University,
Philadelphia, Pa.
Schulz, T. W., and Griffin, S. (1999), Estimating Risk Assessment Exposure Point
Concentrations when Data are Not Normal or Lognormal. Risk Analysis, Vol. 19, No. 4,
1999.
Scout: A Data Analysis Program, Technology Support Project. EPA, NERL -LV, Las Vegas,
NV 89193-3478.
Singh, A. (1993), "Omnibus Robust Procedures For Assessment of Multivariate Normality
and Detection of Multivariate Outliers," Multivariate Environmental Statistics. G. P. Patil
and C.R. Rao, Editors, Elsevier Science Publishers.
Singh, A. K., Singh, A., and Engelhardt, M., "The Lognormal Distribution in Environmental
Applications," EPA/600/R-97/006, December 1997.
Singh, A. K., Singh, A., and Engelhardt, M., "Some Practical Aspects of Sample Size and
Power Computations for Estimating the Mean of Positively Skewed Distributions in
Environmental Applications," EPA/600/S-99/006, November 1999.
Singh. A., Singh, A. K., Engelhardt, M., and Nocerino, J.M. (2002a), " On the Computation
of the Upper Confidence Limit of the Mean of Contaminant Data Distributions." Under EPA
Review.
A-67
-------
Singh, A., Singh, A. K., and laci, R. J. (2002b). " Estimation of the Exposure Point
Concentration Term Using a Gamma Distribution." EPA/600/R-02/084.
Singh, A. and Singh, A.K. (2003). Estimation of the Exposure Point Concentration Term
(95% UCL) Using Bias-Corrected Accelerated (BCA) Bootstrap Method and Several other
methods for Normal, Lognormal, and Gamma Distributions. Draft EPA Internal Report.
Stephens, M. A. (1970), Use of Kolmogorov-Smirnov, Cramer-von Mises and Related
Statistics Without Extensive Tables. Journal of Royal Statistical Society, B 32, 115-122.
Sutton, C.D. (1993), "Computer -Intensive Methods for Tests About the Mean Of an
Asymmetrical Distribution," Journal Of American Statistical Society, Vol. 88, No. 423, pp
802-810.
Thorn, H.C. S. (1968), Direct and Inverse Tables of the Gamma Distribution, Silver Spring,
MD; Environmental Data Service.
Whittaker, J. (1974), Generating Gamma and Beta Random Variables with Non-integral
Shape Parameters. Applied Statistics, 23, No. 2, 210-214.
Wong, A. (1993), A Note on Inference for the Mean Parameter of the Gamma Distribution.
Statistics Probability Letters, Vol 17, 61-66.
A-68
-------
A-69
-------
APPENDIX B
CRITICAL VALUES
OF
ANDERSON-DARLING TEST STATISTIC
AND
KOLMOGOROV-SMIRNOV TEST STATISTIC
FOR
GAMMA DISTRIBUTION
WITH UNKNOWN PARAMETERS
-------
Critical Values for Anderson Darling Test - Significance Level of 0.20
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.6012 0.5867 0.5709 0.5498 0.5169 0.5017 0.4900 0.4854 0.4839 0.4819 0.4810 0.4805 0.4802 0.4803 0.4795 0.4795 0.4791 0.4790
5 0.6366 0.6085 0.5796 0.5590 0.5322 0.5166 0.5049 0.4996 0.4969 0.4949 0.4930 0.4926 0.4914 0.4919 0.4908 0.4903 0.4899 0.4901
6 0.6851 0.6362 0.5915 0.5682 0.5431 0.5264 0.5117 0.5055 0.5026 0.4996 0.4981 0.4969 0.4964 0.4960 0.4955 0.4950 0.4948 0.4951
7 0.7349 0.6671 0.6037 0.5745 0.5491 0.5318 0.5172 0.5102 0.5064 0.5036 0.5007 0.5002 0.4988 0.4991 0.4984 0.4975 0.4973 0.4973
8 0.7856 0.6966 0.6150 0.5784 0.5545 0.5372 0.5210 0.5134 0.5092 0.5058 0.5039 0.5025 0.5016 0.5017 0.4998 0.5000 0.4992 0.4990
9 0.8385 0.7291 0.6265 0.5827 0.5593 0.5407 0.5239 0.5162 0.5122 0.5082 0.5068 0.5045 0.5035 0.5024 0.5015 0.5015 0.5010 0.5014
10 0.8923 0.7600 0.6384 0.5849 0.5626 0.5436 0.5263 0.5170 0.5135 0.5097 0.5079 0.5063 0.5050 0.5041 0.5029 0.5023 0.5020 0.5020
11 0.9469 0.7926 0.6496 0.5881 0.5662 0.5463 0.5287 0.5198 0.5147 0.5110 0.5088 0.5061 0.5049 0.5048 0.5041 0.5035 0.5030 0.5024
12 1.0021 0.8247 0.6600 0.5900 0.5680 0.5485 0.5299 0.5213 0.5166 0.5113 0.5098 0.5080 0.5058 0.5050 0.5042 0.5048 0.5036 0.5041
13 1.0571 0.8571 0.6731 0.5910 0.5697 0.5499 0.5317 0.5224 0.5169 0.5134 0.5101 0.5080 0.5073 0.5064 0.5053 0.5049 0.5050 0.5047
14 1.1106 0.8897 0.6828 0.5928 0.5716 0.5508 0.5330 0.5229 0.5184 0.5131 0.5111 0.5090 0.5080 0.5072 0.5054 0.5051 0.5040 0.5045
15 1.1656 0.9221 0.6926 0.5951 0.5735 0.5525 0.5331 0.5238 0.5188 0.5134 0.5115 0.5095 0.5078 0.5073 0.5058 0.5051 0.5054 0.5051
16 1.2201 0.9542 0.7047 0.5967 0.5744 0.5535 0.5345 0.5242 0.5197 0.5143 0.5127 0.5095 0.5081 0.5082 0.5068 0.5057 0.5052 0.5054
17 1.2747 0.9856 0.7157 0.5975 0.5764 0.5553 0.5354 0.5249 0.5200 0.5152 0.5122 0.5099 0.5086 0.5085 0.5066 0.5063 0.5053 0.5055
18 1.3270 1.0187 0.7261 0.5994 0.5761 0.5556 0.5357 0.5247 0.5203 0.5151 0.5132 0.5107 0.5097 0.5090 0.5067 0.5066 0.5058 0.5063
19 1.3799 1.0502 0.7376 0.6000 0.5775 0.5563 0.5367 0.5257 0.5208 0.5155 0.5127 0.5107 0.5090 0.5080 0.5074 0.5069 0.5067 0.5057
20 1.4316 1.0812 0.7470 0.6016 0.5779 0.5567 0.5369 0.5264 0.5210 0.5159 0.5135 0.5103 0.5091 0.5090 0.5082 0.5066 0.5069 0.5069
21 1.4859 1.1119 0.7574 0.6022 0.5788 0.5569 0.5386 0.5271 0.5209 0.5160 0.5137 0.5112 0.5098 0.5092 0.5081 0.5077 0.5071 0.5071
22 1.5373 1.1433 0.7681 0.6037 0.5793 0.5584 0.5377 0.5277 0.5220 0.5160 0.5135 0.5116 0.5101 0.5093 0.5083 0.5069 0.5072 0.5064
23 1.5882 1.1774 0.7794 0.6042 0.5803 0.5589 0.5380 0.5275 0.5213 0.5166 0.5134 0.5110 0.5108 0.5097 0.5081 0.5069 0.5069 0.5070
24 1.6410 1.2064 0.7890 0.6046 0.5807 0.5595 0.5386 0.5272 0.5225 0.5173 0.5139 0.5117 0.5097 0.5093 0.5082 0.5076 0.5074 0.5072
25 1.6915 1.2376 0.8002 0.6057 0.5806 0.5601 0.5391 0.5278 0.5229 0.5169 0.5144 0.5119 0.5104 0.5095 0.5082 0.5074 0.5070 0.5071
26 1.7433 1.2691 0.8100 0.6069 0.5809 0.5601 0.5395 0.5279 0.5223 0.5170 0.5140 0.5113 0.5099 0.5098 0.5082 0.5073 0.5072 0.5076
27 1.7932 1.2981 0.8228 0.6081 0.5816 0.5608 0.5390 0.5287 0.5233 0.5171 0.5150 0.5120 0.5106 0.5097 0.5077 0.5080 0.5077 0.5073
28 1.8431 1.3284 0.8319 0.6088 0.5818 0.5610 0.5397 0.5283 0.5228 0.5170 0.5153 0.5118 0.5112 0.5100 0.5081 0.5085 0.5078 0.5073
29 1.8948 1.3600 0.8424 0.6099 0.5818 0.5613 0.5402 0.5287 0.5235 0.5175 0.5149 0.5124 0.5110 0.5097 0.5082 0.5076 0.5075 0.5074
30 1.9433 1.3895 0.8532 0.6110 0.5825 0.5617 0.5397 0.5292 0.5230 0.5176 0.5151 0.5126 0.5099 0.5097 0.5089 0.5079 0.5072 0.5081
35 2.1902 1.5371 0.9057 0.6147 0.5843 0.5626 0.5414 0.5300 0.5237 0.5178 0.5156 0.5126 0.5123 0.5105 0.5090 0.5087 0.5082 0.5074
40 2.4320 1.6829 0.9551 0.6174 0.5848 0.5630 0.5418 0.5299 0.5246 0.5183 0.5153 0.5128 0.5110 0.5108 0.5094 0.5083 0.5075 0.5083
45 2.6734 1.8275 1.0046 0.6211 0.5857 0.5646 0.5418 0.5301 0.5244 0.5191 0.5160 0.5130 0.5111 0.5110 0.5094 0.5085 0.5084 0.5083
50 2.9056 1.9669 1.0536 0.6238 0.5872 0.5651 0.5413 0.5313 0.5251 0.5192 0.5162 0.5132 0.5116 0.5111 0.5095 0.5088 0.5087 0.5089
60 3.3680 2.2458 1.1502 0.6309 0.5878 0.5655 0.5430 0.5311 0.5248 0.5189 0.5165 0.5141 0.5113 0.5112 0.5099 0.5084 0.5089 0.5089
70 3.8261 2.5178 1.2478 0.6361 0.5882 0.5667 0.5433 0.5310 0.5252 0.5194 0.5165 0.5132 0.5122 0.5112 0.5098 0.5091 0.5090 0.5081
80 4.2729 2.7850 1.3430 0.6424 0.5889 0.5669 0.5439 0.5314 0.5258 0.5201 0.5173 0.5130 0.5131 0.5110 0.5100 0.5087 0.5089 0.5086
90 4.7189 3.0528 1.4370 0.6474 0.5883 0.5670 0.5438 0.5321 0.5256 0.5203 0.5174 0.5139 0.5124 0.5109 0.5101 0.5088 0.5087 0.5091
100 5.1658 3.3136 1.5320 0.6516 0.5886 0.5681 0.5438 0.5318 0.5260 0.5200 0.5174 0.5140 0.5117 0.5115 0.5101 0.5090 0.5092 0.5089
200 9.4620 5.8551 2.4199 0.7059 0.5910 0.5675 0.5452 0.5325 0.5264 0.5199 0.5172 0.5140 0.5122 0.5115 0.5095 0.5095 0.5090 0.5093
300 13.6454 8.3200 3.2731 0.7595 0.5915 0.5688 0.5448 0.5328 0.5260 0.5205 0.5174 0.5134 0.5126 0.5120 0.5107 0.5091 0.5092 0.5092
400 17.7759 10.7341 4.1071 0.8119 0.5902 0.5688 0.5448 0.5331 0.5266 0.5200 0.5168 0.5143 0.5127 0.5125 0.5107 0.5095 0.5090 0.5093
500 21.8687 13.1245 4.9232 0.8646 0.5910 0.5685 0.5450 0.5332 0.5267 0.5203 0.5173 0.5145 0.5129 0.5123 0.5102 0.5092 0.5094 0.5097
1000 42.0423 24.8700 8.9004 1.1234 0.5917 0.5687 0.5457 0.5327 0.5265 0.5204 0.5174 0.5143 0.5126 0.5118 0.5098 0.5098 0.5091 0.5096
2500 101.548 59.3470 20.4324 1.8628 0.5930 0.5698 0.5460 0.5336 0.5268 0.5206 0.5178 0.5143 0.5155 0.5129 0.5102 0.5093 0.5087 0.5095
B-1D
-------
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.20
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.3745 0.3681 0.3610 0.3538 0.3419 0.3360 0.3314 0.3293 0.3285 0.3275 0.3270 0.3266 0.3263 0.3263 0.3258 0.3258 0.3257 0.3256
5 0.3495 0.3407 0.3315 0.3276 0.3228 0.3179 0.3128 0.3093 0.3074 0.3055 0.3043 0.3036 0.3029 0.3026 0.3019 0.3015 0.3014 0.3014
6 0.3350 0.3220 0.3102 0.3048 0.2990 0.2942 0.2889 0.2856 0.2839 0.2822 0.2812 0.2804 0.2800 0.2795 0.2792 0.2788 0.2788 0.2787
7 0.3207 0.3062 0.2918 0.2848 0.2792 0.2745 0.2695 0.2666 0.2649 0.2631 0.2620 0.2613 0.2608 0.2606 0.2601 0.2598 0.2597 0.2596
8 0.3105 0.2932 0.2759 0.2683 0.2641 0.2598 0.2547 0.2516 0.2498 0.2480 0.2471 0.2462 0.2458 0.2456 0.2449 0.2446 0.2444 0.2444
9 0.3014 0.2831 0.2641 0.2553 0.2510 0.2468 0.2419 0.2389 0.2372 0.2354 0.2346 0.2336 0.2332 0.2327 0.2323 0.2321 0.2319 0.2319
10 0.2937 0.2738 0.2533 0.2436 0.2394 0.2352 0.2307 0.2276 0.2262 0.2244 0.2236 0.2228 0.2223 0.2220 0.2214 0.2211 0.2211 0.2209
11 0.2869 0.2660 0.2440 0.2333 0.2296 0.2255 0.2209 0.2182 0.2165 0.2149 0.2141 0.2132 0.2126 0.2124 0.2120 0.2117 0.2115 0.2115
12 0.2811 0.2592 0.2355 0.2243 0.2206 0.2168 0.2123 0.2097 0.2082 0.2064 0.2057 0.2048 0.2042 0.2040 0.2036 0.2035 0.2032 0.2033
13 0.2757 0.2531 0.2285 0.2162 0.2127 0.2091 0.2047 0.2022 0.2006 0.1991 0.1981 0.1973 0.1970 0.1967 0.1961 0.1960 0.1959 0.1958
14 0.2710 0.2478 0.2220 0.2091 0.2056 0.2020 0.1980 0.1954 0.1940 0.1922 0.1915 0.1907 0.1903 0.1900 0.1895 0.1893 0.1891 0.1891
15 0.2665 0.2427 0.2159 0.2026 0.1993 0.1958 0.1916 0.1893 0.1877 0.1862 0.1854 0.1847 0.1842 0.1840 0.1834 0.1832 0.1832 0.1831
16 0.2625 0.2383 0.2107 0.1966 0.1933 0.1900 0.1862 0.1836 0.1822 0.1807 0.1800 0.1792 0.1787 0.1785 0.1782 0.1779 0.1777 0.1777
17 0.2587 0.2341 0.2059 0.1912 0.1881 0.1850 0.1810 0.1785 0.1772 0.1756 0.1749 0.1741 0.1738 0.1736 0.1731 0.1729 0.1727 0.1727
18 0.2553 0.2304 0.2014 0.1863 0.1831 0.1799 0.1762 0.1737 0.1724 0.1710 0.1704 0.1696 0.1692 0.1690 0.1684 0.1683 0.1681 0.1681
19 0.2519 0.2267 0.1975 0.1816 0.1786 0.1754 0.1719 0.1694 0.1681 0.1668 0.1659 0.1653 0.1649 0.1646 0.1643 0.1641 0.1640 0.1639
20 0.2489 0.2236 0.1935 0.1774 0.1743 0.1713 0.1677 0.1654 0.1641 0.1628 0.1621 0.1613 0.1609 0.1608 0.1603 0.1601 0.1600 0.1600
21 0.2463 0.2205 0.1899 0.1734 0.1704 0.1673 0.1639 0.1617 0.1604 0.1590 0.1584 0.1576 0.1573 0.1571 0.1568 0.1565 0.1564 0.1564
22 0.2437 0.2176 0.1867 0.1697 0.1667 0.1639 0.1604 0.1582 0.1569 0.1555 0.1550 0.1543 0.1539 0.1537 0.1532 0.1531 0.1530 0.1529
23 0.2412 0.2151 0.1837 0.1661 0.1634 0.1604 0.1570 0.1549 0.1536 0.1524 0.1517 0.1509 0.1507 0.1505 0.1502 0.1498 0.1498 0.1498
24 0.2389 0.2124 0.1808 0.1629 0.1600 0.1573 0.1539 0.1518 0.1506 0.1494 0.1487 0.1480 0.1477 0.1475 0.1470 0.1469 0.1468 0.1467
25 0.2366 0.2101 0.1782 0.1598 0.1570 0.1542 0.1510 0.1488 0.1478 0.1465 0.1459 0.1452 0.1449 0.1446 0.1443 0.1441 0.1440 0.1439
26 0.2346 0.2080 0.1756 0.1569 0.1541 0.1513 0.1482 0.1462 0.1449 0.1437 0.1432 0.1424 0.1422 0.1419 0.1416 0.1414 0.1413 0.1412
27 0.2325 0.2058 0.1735 0.1542 0.1513 0.1487 0.1455 0.1436 0.1425 0.1412 0.1406 0.1400 0.1395 0.1394 0.1390 0.1389 0.1388 0.1388
28 0.2308 0.2038 0.1710 0.1515 0.1488 0.1462 0.1431 0.1411 0.1399 0.1388 0.1382 0.1376 0.1373 0.1371 0.1367 0.1366 0.1365 0.1364
29 0.2289 0.2018 0.1689 0.1491 0.1462 0.1439 0.1407 0.1388 0.1377 0.1365 0.1359 0.1353 0.1349 0.1347 0.1343 0.1342 0.1341 0.1341
30 0.2272 0.2000 0.1669 0.1468 0.1439 0.1414 0.1384 0.1364 0.1355 0.1343 0.1337 0.1331 0.1328 0.1325 0.1323 0.1321 0.1320 0.1320
35 0.2197 0.1921 0.1581 0.1366 0.1337 0.1314 0.1286 0.1268 0.1258 0.1248 0.1243 0.1236 0.1234 0.1231 0.1228 0.1228 0.1226 0.1226
40 0.2136 0.1857 0.1509 0.1282 0.1255 0.1232 0.1206 0.1190 0.1181 0.1170 0.1165 0.1160 0.1156 0.1155 0.1152 0.1151 0.1150 0.1150
45 0.2084 0.1803 0.1449 0.1214 0.1185 0.1166 0.1140 0.1125 0.1116 0.1106 0.1101 0.1096 0.1093 0.1091 0.1089 0.1087 0.1087 0.1086
50 0.2040 0.1756 0.1400 0.1155 0.1128 0.1108 0.1083 0.1070 0.1060 0.1051 0.1047 0.1042 0.1039 0.1038 0.1035 0.1033 0.1032 0.1033
60 0.1970 0.1682 0.1319 0.1060 0.1032 0.1014 0.0992 0.0979 0.0971 0.0962 0.0958 0.0954 0.0951 0.0950 0.0948 0.0946 0.0945 0.0945
70 0.1915 0.1623 0.1257 0.0987 0.0958 0.0942 0.0921 0.0908 0.0901 0.0893 0.0889 0.0885 0.0883 0.0882 0.0879 0.0878 0.0877 0.0877
80 0.1870 0.1576 0.1207 0.0927 0.0898 0.0882 0.0863 0.0851 0.0844 0.0837 0.0833 0.0829 0.0827 0.0826 0.0824 0.0822 0.0822 0.0822
90 0.1832 0.1538 0.1166 0.0877 0.0847 0.0833 0.0815 0.0804 0.0797 0.0790 0.0787 0.0783 0.0781 0.0780 0.0778 0.0777 0.0776 0.0776
100 0.1801 0.1504 0.1131 0.0835 0.0805 0.0792 0.0774 0.0763 0.0758 0.0751 0.0748 0.0744 0.0741 0.0741 0.0739 0.0738 0.0737 0.0737
200 0.1630 0.1325 0.0940 0.0611 0.0573 0.0563 0.0551 0.0544 0.0539 0.0534 0.0532 0.0529 0.0528 0.0527 0.0526 0.0525 0.0525 0.0525
300 0.1554 0.1247 0.0857 0.0513 0.0469 0.0461 0.0451 0.0445 0.0442 0.0438 0.0435 0.0433 0.0433 0.0432 0.0431 0.0430 0.0430 0.0430
400 0.1510 0.1200 0.0807 0.0455 0.0407 0.0400 0.0392 0.0386 0.0383 0.0379 0.0378 0.0376 0.0375 0.0375 0.0374 0.0373 0.0373 0.0373
500 0.1480 0.1169 0.0773 0.0416 0.0364 0.0358 0.0351 0.0346 0.0343 0.0340 0.0338 0.0337 0.0336 0.0336 0.0335 0.0334 0.0334 0.0334
1000 0.1407 0.1093 0.0692 0.0323 0.0258 0.0254 0.0249 0.0245 0.0243 0.0241 0.0240 0.0239 0.0238 0.0238 0.0237 0.0237 0.0237 0.0237
2500 0.1344 0.1027 0.0621 0.0242 0.0164 0.0161 0.0158 0.0156 0.0154 0.0153 0.0152 0.0151 0.0151 0.0151 0.0151 0.0150 0.0150 0.0150
B-2D
-------
Critical Values for Anderson Darling Test - Significance Level of 0.15
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.6495 0.6354 0.6212 0.5995 0.5626 0.5456 0.5321 0.5268 0.5252 0.5226 0.5217 0.5206 0.5203 0.5208 0.5202 0.5197 0.5193 0.5190
5 0.6893 0.6597 0.6317 0.6137 0.5836 0.5649 0.5505 0.5436 0.5404 0.5377 0.5357 0.5352 0.5339 0.5341 0.5328 0.5321 0.5319 0.5320
6 0.7453 0.6944 0.6484 0.6262 0.5967 0.5765 0.5591 0.5509 0.5476 0.5441 0.5419 0.5406 0.5401 0.5394 0.5391 0.5382 0.5380 0.5382
7 0.8015 0.7290 0.6625 0.6337 0.6049 0.5838 0.5667 0.5581 0.5530 0.5496 0.5460 0.5460 0.5441 0.5443 0.5436 0.5427 0.5422 0.5421
8 0.8594 0.7632 0.6757 0.6393 0.6124 0.5912 0.5711 0.5622 0.5574 0.5532 0.5504 0.5490 0.5477 0.5480 0.5460 0.5456 0.5450 0.5452
9 0.9189 0.8002 0.6896 0.6442 0.6176 0.5950 0.5754 0.5658 0.5608 0.5561 0.5542 0.5517 0.5505 0.5493 0.5478 0.5480 0.5473 0.5480
10 0.9786 0.8354 0.7026 0.6475 0.6222 0.5995 0.5786 0.5673 0.5629 0.5578 0.5559 0.5538 0.5524 0.5517 0.5498 0.5492 0.5486 0.5489
11 1.0392 0.8719 0.7163 0.6508 0.6266 0.6025 0.5813 0.5709 0.5644 0.5597 0.5574 0.5544 0.5530 0.5525 0.5515 0.5511 0.5505 0.5498
12 1.0998 0.9079 0.7288 0.6534 0.6290 0.6050 0.5826 0.5726 0.5673 0.5605 0.5587 0.5568 0.5545 0.5531 0.5523 0.5527 0.5515 0.5520
13 1.1601 0.9445 0.7437 0.6556 0.6309 0.6077 0.5850 0.5742 0.5674 0.5631 0.5590 0.5564 0.5559 0.5548 0.5534 0.5531 0.5529 0.5526
14 1.2198 0.9815 0.7543 0.6579 0.6332 0.6084 0.5870 0.5746 0.5693 0.5630 0.5602 0.5580 0.5565 0.5558 0.5535 0.5533 0.5521 0.5526
15 1.2789 1.0165 0.7658 0.6600 0.6354 0.6105 0.5870 0.5759 0.5699 0.5637 0.5609 0.5586 0.5567 0.5562 0.5547 0.5540 0.5540 0.5534
16 1.3374 1.0527 0.7796 0.6618 0.6364 0.6118 0.5892 0.5762 0.5707 0.5642 0.5625 0.5584 0.5571 0.5573 0.5557 0.5545 0.5535 0.5539
17 1.3967 1.0875 0.7922 0.6631 0.6388 0.6140 0.5898 0.5770 0.5712 0.5658 0.5621 0.5592 0.5578 0.5576 0.5554 0.5550 0.5540 0.5542
18 1.4533 1.1240 0.8037 0.6659 0.6392 0.6142 0.5905 0.5773 0.5715 0.5657 0.5635 0.5599 0.5588 0.5578 0.5555 0.5554 0.5547 0.5549
19 1.5098 1.1576 0.8169 0.6665 0.6405 0.6155 0.5919 0.5783 0.5726 0.5660 0.5626 0.5605 0.5584 0.5573 0.5567 0.5562 0.5555 0.5546
20 1.5661 1.1928 0.8279 0.6685 0.6413 0.6161 0.5921 0.5790 0.5732 0.5668 0.5635 0.5604 0.5588 0.5585 0.5570 0.5559 0.5556 0.5557
21 1.6235 1.2257 0.8396 0.6691 0.6420 0.6160 0.5937 0.5804 0.5728 0.5668 0.5641 0.5611 0.5594 0.5584 0.5573 0.5570 0.5560 0.5560
22 1.6779 1.2584 0.8514 0.6704 0.6431 0.6175 0.5932 0.5806 0.5735 0.5669 0.5646 0.5614 0.5598 0.5594 0.5577 0.5561 0.5565 0.5557
23 1.7323 1.2970 0.8644 0.6716 0.6440 0.6186 0.5935 0.5807 0.5731 0.5676 0.5637 0.5611 0.5608 0.5592 0.5575 0.5562 0.5561 0.5560
24 1.7885 1.3279 0.8745 0.6727 0.6444 0.6192 0.5944 0.5806 0.5739 0.5683 0.5643 0.5618 0.5596 0.5590 0.5582 0.5569 0.5562 0.5565
25 1.8422 1.3607 0.8871 0.6737 0.6453 0.6196 0.5948 0.5813 0.5751 0.5677 0.5652 0.5619 0.5605 0.5595 0.5581 0.5568 0.5565 0.5565
26 1.8963 1.3958 0.8982 0.6745 0.6449 0.6193 0.5950 0.5817 0.5744 0.5681 0.5649 0.5614 0.5598 0.5596 0.5575 0.5568 0.5567 0.5567
27 1.9503 1.4261 0.9129 0.6765 0.6455 0.6208 0.5944 0.5816 0.5756 0.5684 0.5656 0.5623 0.5602 0.5591 0.5575 0.5574 0.5564 0.5568
28 2.0036 1.4603 0.9224 0.6766 0.6461 0.6213 0.5955 0.5817 0.5751 0.5681 0.5663 0.5623 0.5613 0.5601 0.5579 0.5576 0.5570 0.5563
29 2.0588 1.4943 0.9338 0.6782 0.6457 0.6217 0.5965 0.5818 0.5755 0.5690 0.5653 0.5629 0.5607 0.5597 0.5575 0.5569 0.5566 0.5569
30 2.1110 1.5255 0.9463 0.6801 0.6465 0.6216 0.5955 0.5826 0.5758 0.5689 0.5661 0.5629 0.5599 0.5593 0.5584 0.5578 0.5566 0.5574
35 2.3678 1.6835 1.0038 0.6836 0.6483 0.6230 0.5974 0.5835 0.5764 0.5696 0.5667 0.5633 0.5625 0.5606 0.5591 0.5584 0.5576 0.5576
40 2.6243 1.8376 1.0582 0.6870 0.6498 0.6232 0.5979 0.5837 0.5773 0.5701 0.5662 0.5636 0.5612 0.5608 0.5593 0.5579 0.5575 0.5580
45 2.8741 1.9901 1.1118 0.6917 0.6505 0.6253 0.5979 0.5839 0.5768 0.5706 0.5669 0.5637 0.5621 0.5612 0.5599 0.5586 0.5585 0.5584
50 3.1177 2.1386 1.1654 0.6950 0.6527 0.6258 0.5976 0.5860 0.5778 0.5710 0.5676 0.5641 0.5619 0.5616 0.5599 0.5588 0.5588 0.5586
60 3.5997 2.4304 1.2695 0.7029 0.6530 0.6262 0.5995 0.5851 0.5780 0.5709 0.5673 0.5645 0.5618 0.5616 0.5605 0.5589 0.5589 0.5589
70 4.0720 2.7155 1.3751 0.7081 0.6538 0.6281 0.5996 0.5850 0.5781 0.5716 0.5684 0.5643 0.5629 0.5615 0.5600 0.5592 0.5593 0.5588
80 4.5375 2.9941 1.4768 0.7162 0.6539 0.6273 0.6005 0.5858 0.5786 0.5726 0.5690 0.5641 0.5637 0.5616 0.5602 0.5589 0.5588 0.5589
90 4.9957 3.2729 1.5758 0.7212 0.6536 0.6283 0.6001 0.5863 0.5789 0.5724 0.5691 0.5651 0.5631 0.5618 0.5602 0.5590 0.5594 0.5593
100 5.4567 3.5445 1.6772 0.7269 0.6549 0.6299 0.6005 0.5865 0.5793 0.5723 0.5693 0.5651 0.5628 0.5622 0.5608 0.5595 0.5590 0.5592
200 9.8591 6.1657 2.6088 0.7864 0.6568 0.6291 0.6020 0.5870 0.5796 0.5720 0.5690 0.5656 0.5634 0.5622 0.5600 0.5598 0.5591 0.5597
300 14.1248 8.6896 3.4931 0.8459 0.6577 0.6301 0.6019 0.5873 0.5795 0.5729 0.5685 0.5646 0.5637 0.5632 0.5616 0.5593 0.5599 0.5602
400 18.3207 11.1508 4.3546 0.9029 0.6562 0.6306 0.6017 0.5880 0.5798 0.5725 0.5684 0.5657 0.5638 0.5636 0.5615 0.5601 0.5596 0.5602
500 22.4788 13.5882 5.1945 0.9597 0.6575 0.6301 0.6021 0.5878 0.5804 0.5729 0.5698 0.5660 0.5642 0.5632 0.5611 0.5601 0.5600 0.5602
1000 42.8884 25.5062 9.2649 1.2387 0.6576 0.6303 0.6032 0.5874 0.5798 0.5726 0.5696 0.5652 0.5642 0.5631 0.5607 0.5604 0.5597 0.5603
2500 102.850 60.3279 20.9754 2.0188 0.6594 0.6314 0.6028 0.5884 0.5806 0.5726 0.5697 0.5658 0.5674 0.5643 0.5613 0.5597 0.5593 0.5605
B-3D
-------
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.15
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.3901 0.3832 0.3761 0.3698 0.3599 0.3533 0.3462 0.3417 0.3401 0.3385 0.3379 0.3373 0.3369 0.3369 0.3364 0.3363 0.3362 0.3362
5 0.3646 0.3559 0.3475 0.3445 0.3389 0.3336 0.3279 0.3240 0.3220 0.3200 0.3188 0.3180 0.3172 0.3171 0.3163 0.3159 0.3158 0.3156
6 0.3507 0.3378 0.3254 0.3199 0.3133 0.3078 0.3018 0.2983 0.2964 0.2948 0.2936 0.2928 0.2922 0.2917 0.2914 0.2910 0.2909 0.2909
7 0.3358 0.3203 0.3055 0.2988 0.2934 0.2884 0.2828 0.2794 0.2773 0.2753 0.2742 0.2733 0.2728 0.2726 0.2719 0.2716 0.2715 0.2713
8 0.3254 0.3078 0.2899 0.2825 0.2781 0.2732 0.2673 0.2639 0.2620 0.2599 0.2588 0.2579 0.2574 0.2571 0.2564 0.2560 0.2557 0.2558
9 0.3158 0.2969 0.2773 0.2686 0.2639 0.2592 0.2539 0.2504 0.2486 0.2467 0.2458 0.2447 0.2442 0.2436 0.2432 0.2429 0.2428 0.2428
10 0.3077 0.2873 0.2659 0.2561 0.2518 0.2472 0.2421 0.2386 0.2371 0.2351 0.2343 0.2334 0.2329 0.2325 0.2318 0.2315 0.2314 0.2313
11 0.3006 0.2792 0.2564 0.2453 0.2415 0.2371 0.2320 0.2290 0.2270 0.2251 0.2244 0.2234 0.2227 0.2225 0.2220 0.2218 0.2215 0.2215
12 0.2944 0.2721 0.2475 0.2360 0.2322 0.2280 0.2229 0.2201 0.2183 0.2163 0.2155 0.2147 0.2140 0.2137 0.2131 0.2132 0.2128 0.2129
13 0.2887 0.2657 0.2402 0.2276 0.2239 0.2198 0.2151 0.2121 0.2104 0.2087 0.2077 0.2067 0.2064 0.2061 0.2054 0.2052 0.2052 0.2050
14 0.2835 0.2600 0.2333 0.2200 0.2163 0.2124 0.2080 0.2050 0.2034 0.2015 0.2006 0.1998 0.1994 0.1991 0.1985 0.1983 0.1981 0.1980
15 0.2787 0.2547 0.2270 0.2132 0.2097 0.2058 0.2013 0.1987 0.1969 0.1951 0.1943 0.1936 0.1930 0.1928 0.1922 0.1919 0.1919 0.1918
16 0.2745 0.2501 0.2216 0.2070 0.2035 0.1998 0.1955 0.1926 0.1912 0.1895 0.1887 0.1877 0.1873 0.1871 0.1867 0.1863 0.1861 0.1861
17 0.2704 0.2455 0.2165 0.2012 0.1980 0.1945 0.1901 0.1874 0.1859 0.1841 0.1834 0.1825 0.1821 0.1820 0.1814 0.1811 0.1809 0.1809
18 0.2667 0.2416 0.2118 0.1961 0.1926 0.1892 0.1850 0.1824 0.1809 0.1793 0.1786 0.1778 0.1774 0.1771 0.1764 0.1763 0.1762 0.1762
19 0.2632 0.2376 0.2077 0.1912 0.1880 0.1844 0.1806 0.1778 0.1764 0.1748 0.1739 0.1733 0.1728 0.1725 0.1721 0.1719 0.1718 0.1717
20 0.2599 0.2344 0.2036 0.1868 0.1834 0.1802 0.1761 0.1736 0.1722 0.1707 0.1699 0.1691 0.1686 0.1685 0.1680 0.1678 0.1678 0.1677
21 0.2570 0.2309 0.1998 0.1825 0.1794 0.1759 0.1723 0.1698 0.1683 0.1668 0.1661 0.1653 0.1649 0.1646 0.1643 0.1640 0.1638 0.1639
22 0.2542 0.2279 0.1964 0.1786 0.1756 0.1724 0.1685 0.1661 0.1647 0.1631 0.1625 0.1618 0.1613 0.1611 0.1605 0.1604 0.1603 0.1601
23 0.2516 0.2253 0.1933 0.1749 0.1719 0.1687 0.1650 0.1626 0.1612 0.1598 0.1591 0.1583 0.1580 0.1577 0.1574 0.1570 0.1569 0.1569
24 0.2491 0.2225 0.1901 0.1715 0.1685 0.1654 0.1617 0.1593 0.1580 0.1566 0.1559 0.1552 0.1548 0.1545 0.1541 0.1539 0.1538 0.1537
25 0.2466 0.2200 0.1873 0.1683 0.1652 0.1623 0.1586 0.1563 0.1551 0.1537 0.1529 0.1522 0.1518 0.1516 0.1512 0.1510 0.1509 0.1509
26 0.2445 0.2177 0.1846 0.1652 0.1622 0.1591 0.1557 0.1534 0.1521 0.1507 0.1501 0.1493 0.1490 0.1487 0.1484 0.1482 0.1480 0.1480
27 0.2423 0.2152 0.1824 0.1624 0.1593 0.1565 0.1528 0.1507 0.1495 0.1482 0.1474 0.1467 0.1462 0.1462 0.1457 0.1455 0.1454 0.1454
28 0.2404 0.2132 0.1798 0.1596 0.1566 0.1538 0.1504 0.1481 0.1469 0.1455 0.1449 0.1442 0.1439 0.1436 0.1432 0.1431 0.1430 0.1429
29 0.2383 0.2111 0.1776 0.1570 0.1539 0.1514 0.1479 0.1456 0.1446 0.1431 0.1425 0.1418 0.1414 0.1412 0.1407 0.1407 0.1406 0.1405
30 0.2365 0.2092 0.1755 0.1545 0.1515 0.1488 0.1454 0.1433 0.1421 0.1408 0.1402 0.1396 0.1391 0.1389 0.1386 0.1384 0.1383 0.1383
35 0.2284 0.2007 0.1661 0.1438 0.1408 0.1382 0.1352 0.1332 0.1321 0.1309 0.1303 0.1295 0.1294 0.1291 0.1287 0.1286 0.1285 0.1284
40 0.2219 0.1936 0.1585 0.1350 0.1321 0.1296 0.1268 0.1249 0.1240 0.1227 0.1221 0.1216 0.1212 0.1211 0.1208 0.1206 0.1205 0.1205
45 0.2163 0.1879 0.1521 0.1278 0.1248 0.1226 0.1197 0.1180 0.1171 0.1159 0.1154 0.1148 0.1146 0.1144 0.1141 0.1139 0.1139 0.1138
50 0.2115 0.1829 0.1469 0.1216 0.1187 0.1165 0.1138 0.1123 0.1113 0.1102 0.1098 0.1092 0.1089 0.1088 0.1085 0.1083 0.1082 0.1082
60 0.2039 0.1749 0.1383 0.1117 0.1087 0.1067 0.1043 0.1027 0.1019 0.1009 0.1005 0.1000 0.0997 0.0996 0.0993 0.0991 0.0990 0.0990
70 0.1978 0.1686 0.1317 0.1039 0.1008 0.0991 0.0967 0.0953 0.0945 0.0937 0.0932 0.0927 0.0925 0.0924 0.0922 0.0920 0.0919 0.0919
80 0.1930 0.1635 0.1263 0.0977 0.0945 0.0928 0.0906 0.0893 0.0886 0.0878 0.0874 0.0869 0.0867 0.0865 0.0863 0.0862 0.0862 0.0861
90 0.1889 0.1593 0.1219 0.0924 0.0891 0.0876 0.0856 0.0844 0.0836 0.0829 0.0825 0.0821 0.0818 0.0817 0.0815 0.0814 0.0813 0.0813
100 0.1855 0.1557 0.1182 0.0880 0.0847 0.0832 0.0813 0.0801 0.0795 0.0788 0.0784 0.0780 0.0777 0.0776 0.0774 0.0773 0.0773 0.0772
200 0.1669 0.1364 0.0977 0.0643 0.0603 0.0592 0.0579 0.0571 0.0565 0.0560 0.0558 0.0555 0.0553 0.0552 0.0551 0.0550 0.0550 0.0550
300 0.1587 0.1279 0.0887 0.0540 0.0494 0.0485 0.0474 0.0467 0.0463 0.0459 0.0456 0.0454 0.0453 0.0452 0.0451 0.0450 0.0450 0.0450
400 0.1538 0.1228 0.0834 0.0479 0.0428 0.0421 0.0411 0.0405 0.0402 0.0398 0.0396 0.0394 0.0393 0.0393 0.0392 0.0391 0.0391 0.0391
500 0.1506 0.1194 0.0797 0.0438 0.0384 0.0376 0.0368 0.0363 0.0360 0.0357 0.0355 0.0353 0.0352 0.0352 0.0350 0.0350 0.0350 0.0350
1000 0.1425 0.1111 0.0709 0.0338 0.0272 0.0267 0.0261 0.0257 0.0255 0.0253 0.0252 0.0250 0.0250 0.0249 0.0249 0.0248 0.0248 0.0248
2500 0.1356 0.1039 0.0632 0.0252 0.0173 0.0169 0.0165 0.0163 0.0162 0.0160 0.0159 0.0159 0.0159 0.0158 0.0158 0.0157 0.0157 0.0157
B-4D
-------
Critical Values for Anderson Darling Test - Significance Level of 0.10
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.7088 0.6976 0.6855 0.6661 0.6259 0.6057 0.5899 0.5829 0.5809 0.5777 0.5764 0.5748 0.5747 0.5750 0.5744 0.5738 0.5733 0.5733
5 0.7611 0.7307 0.7050 0.6915 0.6540 0.6303 0.6122 0.6035 0.5988 0.5957 0.5937 0.5926 0.5907 0.5913 0.5895 0.5891 0.5883 0.5882
6 0.8243 0.7721 0.7260 0.7074 0.6719 0.6466 0.6246 0.6138 0.6099 0.6056 0.6031 0.6009 0.6001 0.5995 0.5987 0.5977 0.5979 0.5978
7 0.8907 0.8132 0.7424 0.7162 0.6838 0.6573 0.6353 0.6245 0.6180 0.6138 0.6092 0.6085 0.6065 0.6066 0.6057 0.6041 0.6045 0.6041
8 0.9573 0.8541 0.7599 0.7250 0.6944 0.6670 0.6415 0.6303 0.6236 0.6185 0.6155 0.6133 0.6120 0.6120 0.6099 0.6091 0.6081 0.6087
9 1.0261 0.8963 0.7772 0.7312 0.7009 0.6724 0.6478 0.6346 0.6289 0.6227 0.6198 0.6174 0.6158 0.6140 0.6125 0.6125 0.6118 0.6128
10 1.0941 0.9378 0.7920 0.7359 0.7070 0.6776 0.6516 0.6375 0.6315 0.6252 0.6229 0.6199 0.6186 0.6176 0.6157 0.6148 0.6138 0.6138
11 1.1631 0.9805 0.8091 0.7407 0.7121 0.6823 0.6552 0.6420 0.6339 0.6275 0.6250 0.6217 0.6195 0.6189 0.6178 0.6170 0.6164 0.6156
12 1.2308 1.0222 0.8239 0.7436 0.7156 0.6855 0.6573 0.6445 0.6378 0.6291 0.6267 0.6244 0.6214 0.6198 0.6187 0.6196 0.6178 0.6183
13 1.2985 1.0649 0.8408 0.7479 0.7175 0.6887 0.6603 0.6466 0.6376 0.6330 0.6276 0.6237 0.6230 0.6222 0.6208 0.6197 0.6195 0.6194
14 1.3637 1.1060 0.8544 0.7493 0.7210 0.6903 0.6627 0.6473 0.6399 0.6326 0.6291 0.6264 0.6243 0.6237 0.6212 0.6203 0.6189 0.6197
15 1.4305 1.1465 0.8679 0.7521 0.7238 0.6927 0.6629 0.6497 0.6416 0.6342 0.6299 0.6277 0.6242 0.6237 0.6222 0.6215 0.6210 0.6204
16 1.4956 1.1879 0.8847 0.7551 0.7246 0.6943 0.6655 0.6493 0.6424 0.6343 0.6319 0.6272 0.6259 0.6255 0.6236 0.6220 0.6209 0.6218
17 1.5594 1.2263 0.8989 0.7571 0.7272 0.6971 0.6664 0.6506 0.6438 0.6362 0.6318 0.6281 0.6265 0.6264 0.6234 0.6226 0.6217 0.6217
18 1.6219 1.2670 0.9131 0.7597 0.7286 0.6976 0.6675 0.6512 0.6433 0.6361 0.6337 0.6292 0.6279 0.6264 0.6234 0.6237 0.6229 0.6229
19 1.6831 1.3041 0.9283 0.7613 0.7302 0.6989 0.6699 0.6516 0.6453 0.6372 0.6333 0.6300 0.6271 0.6259 0.6251 0.6244 0.6239 0.6230
20 1.7445 1.3440 0.9407 0.7629 0.7316 0.7004 0.6688 0.6527 0.6451 0.6377 0.6333 0.6299 0.6282 0.6274 0.6258 0.6244 0.6244 0.6243
21 1.8079 1.3798 0.9536 0.7648 0.7323 0.6994 0.6713 0.6549 0.6458 0.6374 0.6341 0.6308 0.6288 0.6278 0.6264 0.6257 0.6243 0.6240
22 1.8649 1.4173 0.9679 0.7666 0.7337 0.7020 0.6705 0.6552 0.6462 0.6377 0.6355 0.6315 0.6295 0.6284 0.6266 0.6249 0.6254 0.6243
23 1.9250 1.4589 0.9827 0.7679 0.7349 0.7032 0.6711 0.6551 0.6451 0.6388 0.6346 0.6307 0.6306 0.6287 0.6269 0.6249 0.6247 0.6244
24 1.9846 1.4938 0.9951 0.7700 0.7356 0.7039 0.6719 0.6543 0.6468 0.6397 0.6347 0.6318 0.6298 0.6281 0.6271 0.6256 0.6254 0.6254
25 2.0426 1.5281 1.0090 0.7703 0.7364 0.7044 0.6727 0.6560 0.6484 0.6393 0.6355 0.6322 0.6299 0.6294 0.6274 0.6257 0.6251 0.6248
26 2.1022 1.5681 1.0213 0.7705 0.7353 0.7044 0.6729 0.6564 0.6484 0.6397 0.6355 0.6314 0.6294 0.6291 0.6268 0.6265 0.6260 0.6256
27 2.1572 1.6005 1.0378 0.7732 0.7370 0.7063 0.6730 0.6562 0.6486 0.6404 0.6360 0.6323 0.6297 0.6288 0.6271 0.6268 0.6260 0.6262
28 2.2173 1.6381 1.0486 0.7741 0.7372 0.7061 0.6742 0.6563 0.6486 0.6401 0.6374 0.6330 0.6311 0.6299 0.6272 0.6274 0.6261 0.6255
29 2.2750 1.6749 1.0620 0.7755 0.7369 0.7073 0.6753 0.6561 0.6488 0.6407 0.6371 0.6330 0.6311 0.6296 0.6270 0.6262 0.6260 0.6255
30 2.3305 1.7089 1.0763 0.7781 0.7378 0.7064 0.6744 0.6581 0.6496 0.6408 0.6374 0.6334 0.6307 0.6289 0.6279 0.6269 0.6261 0.6267
35 2.6059 1.8806 1.1413 0.7832 0.7395 0.7082 0.6765 0.6588 0.6502 0.6424 0.6382 0.6334 0.6332 0.6306 0.6293 0.6280 0.6270 0.6273
40 2.8792 2.0456 1.2022 0.7872 0.7421 0.7090 0.6768 0.6597 0.6514 0.6426 0.6370 0.6343 0.6320 0.6309 0.6287 0.6277 0.6276 0.6279
45 3.1396 2.2085 1.2601 0.7927 0.7430 0.7115 0.6768 0.6605 0.6507 0.6433 0.6387 0.6342 0.6321 0.6319 0.6304 0.6281 0.6289 0.6286
50 3.3978 2.3668 1.3201 0.7966 0.7457 0.7124 0.6772 0.6622 0.6518 0.6429 0.6398 0.6354 0.6326 0.6322 0.6302 0.6286 0.6285 0.6287
60 3.9028 2.6768 1.4351 0.8062 0.7470 0.7129 0.6785 0.6609 0.6518 0.6441 0.6395 0.6360 0.6333 0.6324 0.6313 0.6290 0.6291 0.6292
70 4.3942 2.9790 1.5495 0.8114 0.7472 0.7152 0.6789 0.6615 0.6531 0.6454 0.6408 0.6357 0.6339 0.6327 0.6302 0.6296 0.6295 0.6293
80 4.8817 3.2693 1.6602 0.8202 0.7471 0.7141 0.6801 0.6625 0.6535 0.6458 0.6412 0.6362 0.6352 0.6326 0.6312 0.6299 0.6291 0.6288
90 5.3579 3.5630 1.7658 0.8266 0.7481 0.7155 0.6801 0.6633 0.6537 0.6457 0.6417 0.6366 0.6339 0.6337 0.6310 0.6289 0.6303 0.6298
100 5.8377 3.8476 1.8740 0.8334 0.7487 0.7170 0.6807 0.6634 0.6538 0.6457 0.6422 0.6370 0.6341 0.6335 0.6315 0.6300 0.6296 0.6291
200 10.3750 6.5699 2.8606 0.9021 0.7513 0.7163 0.6820 0.6641 0.6545 0.6458 0.6418 0.6380 0.6351 0.6339 0.6311 0.6305 0.6302 0.6307
300 14.7424 9.1683 3.7864 0.9692 0.7517 0.7178 0.6822 0.6641 0.6549 0.6466 0.6411 0.6363 0.6359 0.6341 0.6334 0.6305 0.6302 0.6311
400 19.0253 11.6939 4.6787 1.0324 0.7510 0.7180 0.6826 0.6649 0.6554 0.6460 0.6413 0.6372 0.6358 0.6349 0.6329 0.6312 0.6306 0.6313
500 23.2588 14.1892 5.5513 1.0949 0.7520 0.7179 0.6825 0.6644 0.6554 0.6465 0.6426 0.6382 0.6356 0.6349 0.6320 0.6308 0.6305 0.6315
1000 43.9612 26.3237 9.7366 1.3989 0.7522 0.7183 0.6844 0.6645 0.6549 0.6460 0.6429 0.6373 0.6363 0.6348 0.6321 0.6310 0.6300 0.6311
2500 104.511 61.5654 21.6770 2.2303 0.7537 0.7193 0.6835 0.6645 0.6563 0.6462 0.6424 0.6375 0.6403 0.6357 0.6322 0.6302 0.6307 0.6316
B-5D
-------
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.10
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.4102 0.4026 0.3956 0.3925 0.3872 0.3817 0.3746 0.3699 0.3677 0.3647 0.3633 0.3622 0.3615 0.3616 0.3605 0.3601 0.3599 0.3596
5 0.3856 0.3761 0.3680 0.3655 0.3586 0.3524 0.3459 0.3416 0.3395 0.3373 0.3361 0.3351 0.3344 0.3343 0.3334 0.3331 0.3330 0.3328
6 0.3694 0.3570 0.3448 0.3393 0.3324 0.3261 0.3192 0.3151 0.3130 0.3110 0.3096 0.3087 0.3080 0.3075 0.3073 0.3068 0.3066 0.3066
7 0.3558 0.3394 0.3234 0.3175 0.3128 0.3071 0.3006 0.2968 0.2941 0.2919 0.2903 0.2893 0.2886 0.2885 0.2877 0.2872 0.2871 0.2870
8 0.3441 0.3265 0.3083 0.3010 0.2959 0.2903 0.2837 0.2799 0.2776 0.2754 0.2741 0.2731 0.2725 0.2722 0.2714 0.2710 0.2707 0.2709
9 0.3343 0.3148 0.2947 0.2856 0.2808 0.2754 0.2695 0.2656 0.2635 0.2612 0.2602 0.2590 0.2585 0.2578 0.2574 0.2571 0.2569 0.2570
10 0.3255 0.3048 0.2824 0.2726 0.2682 0.2630 0.2572 0.2533 0.2514 0.2492 0.2483 0.2471 0.2465 0.2463 0.2454 0.2451 0.2448 0.2448
11 0.3182 0.2962 0.2726 0.2613 0.2574 0.2523 0.2465 0.2430 0.2408 0.2386 0.2378 0.2365 0.2360 0.2356 0.2351 0.2348 0.2346 0.2345
12 0.3113 0.2887 0.2633 0.2515 0.2473 0.2425 0.2368 0.2337 0.2316 0.2293 0.2284 0.2274 0.2267 0.2263 0.2256 0.2257 0.2255 0.2254
13 0.3051 0.2817 0.2554 0.2425 0.2385 0.2340 0.2287 0.2253 0.2232 0.2214 0.2202 0.2191 0.2187 0.2183 0.2177 0.2174 0.2173 0.2172
14 0.2995 0.2758 0.2482 0.2344 0.2306 0.2262 0.2211 0.2177 0.2158 0.2137 0.2127 0.2119 0.2113 0.2111 0.2104 0.2101 0.2098 0.2096
15 0.2943 0.2702 0.2414 0.2271 0.2235 0.2192 0.2139 0.2110 0.2091 0.2070 0.2061 0.2052 0.2046 0.2043 0.2037 0.2034 0.2034 0.2032
16 0.2898 0.2651 0.2358 0.2207 0.2168 0.2127 0.2078 0.2047 0.2028 0.2009 0.2001 0.1990 0.1985 0.1983 0.1978 0.1974 0.1972 0.1973
17 0.2854 0.2602 0.2304 0.2145 0.2110 0.2071 0.2021 0.1990 0.1973 0.1954 0.1944 0.1935 0.1930 0.1928 0.1921 0.1919 0.1917 0.1917
18 0.2814 0.2559 0.2253 0.2091 0.2054 0.2014 0.1967 0.1938 0.1920 0.1903 0.1894 0.1884 0.1880 0.1876 0.1870 0.1868 0.1867 0.1867
19 0.2776 0.2518 0.2210 0.2038 0.2004 0.1965 0.1920 0.1888 0.1873 0.1854 0.1845 0.1838 0.1832 0.1828 0.1824 0.1822 0.1820 0.1820
20 0.2740 0.2480 0.2166 0.1991 0.1956 0.1919 0.1873 0.1844 0.1828 0.1812 0.1801 0.1793 0.1788 0.1786 0.1781 0.1779 0.1778 0.1777
21 0.2708 0.2445 0.2125 0.1945 0.1912 0.1873 0.1832 0.1804 0.1787 0.1770 0.1761 0.1753 0.1747 0.1745 0.1741 0.1738 0.1737 0.1736
22 0.2677 0.2412 0.2090 0.1904 0.1872 0.1835 0.1792 0.1764 0.1749 0.1730 0.1724 0.1715 0.1710 0.1708 0.1701 0.1700 0.1699 0.1698
23 0.2648 0.2383 0.2056 0.1865 0.1834 0.1797 0.1755 0.1728 0.1712 0.1696 0.1686 0.1678 0.1674 0.1673 0.1668 0.1663 0.1663 0.1662
24 0.2620 0.2354 0.2023 0.1829 0.1797 0.1762 0.1720 0.1693 0.1678 0.1662 0.1654 0.1646 0.1641 0.1639 0.1633 0.1631 0.1630 0.1629
25 0.2595 0.2325 0.1993 0.1794 0.1762 0.1729 0.1686 0.1660 0.1647 0.1630 0.1622 0.1614 0.1611 0.1608 0.1603 0.1601 0.1599 0.1599
26 0.2570 0.2299 0.1964 0.1762 0.1729 0.1695 0.1657 0.1630 0.1616 0.1599 0.1592 0.1583 0.1579 0.1576 0.1573 0.1570 0.1569 0.1569
27 0.2547 0.2274 0.1941 0.1731 0.1699 0.1667 0.1626 0.1602 0.1588 0.1572 0.1564 0.1556 0.1551 0.1549 0.1544 0.1543 0.1541 0.1541
28 0.2526 0.2253 0.1912 0.1701 0.1669 0.1638 0.1599 0.1573 0.1560 0.1544 0.1537 0.1529 0.1526 0.1523 0.1518 0.1517 0.1515 0.1515
29 0.2503 0.2230 0.1888 0.1674 0.1641 0.1612 0.1573 0.1547 0.1535 0.1518 0.1512 0.1505 0.1501 0.1497 0.1492 0.1491 0.1490 0.1489
30 0.2484 0.2208 0.1866 0.1648 0.1615 0.1585 0.1547 0.1523 0.1509 0.1495 0.1488 0.1480 0.1475 0.1473 0.1469 0.1467 0.1466 0.1466
35 0.2395 0.2115 0.1766 0.1534 0.1500 0.1472 0.1437 0.1415 0.1402 0.1389 0.1383 0.1374 0.1372 0.1368 0.1365 0.1363 0.1361 0.1361
40 0.2324 0.2039 0.1684 0.1441 0.1407 0.1381 0.1348 0.1327 0.1316 0.1302 0.1295 0.1290 0.1286 0.1284 0.1280 0.1278 0.1277 0.1278
45 0.2262 0.1976 0.1614 0.1363 0.1331 0.1306 0.1273 0.1255 0.1243 0.1230 0.1224 0.1218 0.1215 0.1213 0.1210 0.1208 0.1207 0.1206
50 0.2210 0.1922 0.1558 0.1297 0.1265 0.1241 0.1210 0.1193 0.1182 0.1170 0.1165 0.1158 0.1155 0.1153 0.1150 0.1147 0.1147 0.1147
60 0.2126 0.1835 0.1466 0.1191 0.1159 0.1136 0.1109 0.1091 0.1082 0.1071 0.1066 0.1061 0.1057 0.1056 0.1053 0.1051 0.1049 0.1050
70 0.2060 0.1766 0.1394 0.1108 0.1075 0.1056 0.1028 0.1013 0.1004 0.0994 0.0989 0.0983 0.0981 0.0979 0.0977 0.0975 0.0974 0.0974
80 0.2007 0.1710 0.1336 0.1042 0.1008 0.0988 0.0964 0.0949 0.0941 0.0932 0.0927 0.0922 0.0920 0.0918 0.0916 0.0913 0.0913 0.0913
90 0.1962 0.1665 0.1288 0.0985 0.0950 0.0933 0.0910 0.0896 0.0888 0.0880 0.0876 0.0870 0.0868 0.0866 0.0864 0.0863 0.0862 0.0861
100 0.1925 0.1625 0.1248 0.0938 0.0902 0.0886 0.0865 0.0851 0.0844 0.0835 0.0831 0.0826 0.0824 0.0823 0.0820 0.0819 0.0819 0.0819
200 0.1719 0.1413 0.1024 0.0686 0.0643 0.0631 0.0616 0.0606 0.0600 0.0594 0.0592 0.0588 0.0587 0.0586 0.0583 0.0583 0.0583 0.0583
300 0.1629 0.1320 0.0926 0.0576 0.0526 0.0516 0.0504 0.0496 0.0492 0.0487 0.0484 0.0482 0.0481 0.0480 0.0478 0.0477 0.0477 0.0477
400 0.1575 0.1263 0.0868 0.0510 0.0456 0.0448 0.0437 0.0430 0.0426 0.0422 0.0420 0.0418 0.0417 0.0416 0.0415 0.0414 0.0414 0.0414
500 0.1538 0.1226 0.0828 0.0466 0.0409 0.0401 0.0391 0.0385 0.0382 0.0378 0.0376 0.0374 0.0373 0.0373 0.0372 0.0371 0.0371 0.0371
1000 0.1449 0.1134 0.0731 0.0359 0.0290 0.0284 0.0278 0.0273 0.0271 0.0268 0.0267 0.0265 0.0265 0.0264 0.0263 0.0263 0.0263 0.0263
2500 0.1371 0.1054 0.0647 0.0266 0.0184 0.0180 0.0176 0.0173 0.0172 0.0170 0.0169 0.0168 0.0168 0.0168 0.0167 0.0167 0.0167 0.0167
B-6D
-------
Critical Values for Anderson Darling Test - Significance Level of 0.05
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.7933 0.7883 0.7863 0.7785 0.7325 0.7041 0.6809 0.6703 0.6666 0.6624 0.6605 0.6594 0.6589 0.6590 0.6571 0.6571 0.6559 0.6565
5 0.8730 0.8462 0.8304 0.8264 0.7753 0.7392 0.7110 0.6983 0.6913 0.6864 0.6845 0.6826 0.6812 0.6807 0.6789 0.6787 0.6783 0.6781
6 0.9490 0.8965 0.8535 0.8446 0.8025 0.7667 0.7359 0.7210 0.7151 0.7078 0.7042 0.7013 0.7000 0.6980 0.6983 0.6971 0.6969 0.6962
7 1.0305 0.9476 0.8762 0.8598 0.8211 0.7841 0.7515 0.7361 0.7275 0.7212 0.7149 0.7122 0.7097 0.7099 0.7085 0.7067 0.7077 0.7077
8 1.1136 1.0006 0.8986 0.8720 0.8359 0.7973 0.7624 0.7451 0.7355 0.7284 0.7240 0.7215 0.7186 0.7191 0.7150 0.7162 0.7148 0.7147
9 1.1971 1.0535 0.9227 0.8810 0.8451 0.8073 0.7707 0.7515 0.7435 0.7344 0.7298 0.7268 0.7250 0.7228 0.7218 0.7210 0.7205 0.7200
10 1.2792 1.1063 0.9420 0.8881 0.8539 0.8136 0.7770 0.7570 0.7483 0.7392 0.7356 0.7322 0.7294 0.7295 0.7251 0.7246 0.7244 0.7238
11 1.3623 1.1586 0.9637 0.8950 0.8601 0.8201 0.7811 0.7625 0.7516 0.7422 0.7389 0.7335 0.7326 0.7314 0.7294 0.7287 0.7284 0.7257
12 1.4414 1.2089 0.9834 0.8989 0.8656 0.8239 0.7849 0.7656 0.7567 0.7455 0.7415 0.7390 0.7358 0.7320 0.7303 0.7319 0.7296 0.7312
13 1.5200 1.2605 1.0039 0.9049 0.8682 0.8298 0.7900 0.7703 0.7574 0.7503 0.7431 0.7392 0.7372 0.7359 0.7337 0.7333 0.7325 0.7323
14 1.5958 1.3106 1.0234 0.9072 0.8735 0.8320 0.7933 0.7706 0.7600 0.7507 0.7459 0.7425 0.7401 0.7380 0.7345 0.7340 0.7330 0.7334
15 1.6732 1.3605 1.0405 0.9113 0.8768 0.8355 0.7930 0.7751 0.7630 0.7538 0.7468 0.7445 0.7400 0.7386 0.7374 0.7347 0.7345 0.7341
16 1.7482 1.4088 1.0618 0.9164 0.8783 0.8378 0.7963 0.7745 0.7633 0.7547 0.7503 0.7443 0.7419 0.7413 0.7390 0.7365 0.7354 0.7360
17 1.8194 1.4552 1.0796 0.9205 0.8827 0.8418 0.7979 0.7764 0.7660 0.7557 0.7494 0.7454 0.7428 0.7416 0.7388 0.7378 0.7367 0.7362
18 1.8905 1.4995 1.0965 0.9229 0.8842 0.8421 0.8001 0.7780 0.7666 0.7562 0.7526 0.7458 0.7426 0.7430 0.7392 0.7395 0.7383 0.7372
19 1.9614 1.5452 1.1162 0.9250 0.8877 0.8428 0.8028 0.7788 0.7692 0.7569 0.7518 0.7481 0.7451 0.7424 0.7412 0.7403 0.7404 0.7379
20 2.0284 1.5917 1.1322 0.9289 0.8880 0.8447 0.8025 0.7795 0.7679 0.7578 0.7524 0.7475 0.7455 0.7452 0.7418 0.7407 0.7394 0.7405
21 2.0984 1.6336 1.1480 0.9288 0.8903 0.8458 0.8053 0.7828 0.7696 0.7582 0.7538 0.7494 0.7473 0.7453 0.7426 0.7425 0.7412 0.7395
22 2.1639 1.6751 1.1669 0.9334 0.8918 0.8476 0.8043 0.7830 0.7708 0.7587 0.7561 0.7494 0.7466 0.7464 0.7436 0.7403 0.7429 0.7406
23 2.2329 1.7214 1.1839 0.9338 0.8939 0.8488 0.8051 0.7822 0.7693 0.7601 0.7547 0.7503 0.7491 0.7465 0.7441 0.7419 0.7414 0.7404
24 2.2974 1.7630 1.2009 0.9377 0.8938 0.8512 0.8063 0.7825 0.7719 0.7615 0.7551 0.7511 0.7487 0.7462 0.7443 0.7423 0.7421 0.7418
25 2.3601 1.8028 1.2161 0.9394 0.8955 0.8518 0.8069 0.7835 0.7731 0.7615 0.7565 0.7513 0.7487 0.7470 0.7448 0.7432 0.7423 0.7418
26 2.4252 1.8483 1.2315 0.9393 0.8936 0.8516 0.8085 0.7842 0.7734 0.7616 0.7573 0.7505 0.7478 0.7463 0.7441 0.7438 0.7428 0.7422
27 2.4909 1.8820 1.2531 0.9437 0.8957 0.8531 0.8074 0.7839 0.7733 0.7630 0.7566 0.7517 0.7490 0.7474 0.7436 0.7440 0.7433 0.7439
28 2.5562 1.9280 1.2634 0.9432 0.8971 0.8543 0.8103 0.7847 0.7738 0.7627 0.7580 0.7537 0.7497 0.7485 0.7453 0.7446 0.7431 0.7431
29 2.6160 1.9685 1.2809 0.9478 0.8976 0.8562 0.8115 0.7837 0.7742 0.7627 0.7573 0.7527 0.7503 0.7475 0.7457 0.7442 0.7439 0.7423
30 2.6778 2.0063 1.2983 0.9482 0.8976 0.8538 0.8092 0.7878 0.7755 0.7630 0.7585 0.7524 0.7495 0.7465 0.7455 0.7443 0.7441 0.7451
35 2.9819 2.1959 1.3736 0.9546 0.8995 0.8565 0.8122 0.7877 0.7757 0.7666 0.7597 0.7537 0.7526 0.7497 0.7483 0.7467 0.7447 0.7459
40 3.2742 2.3805 1.4435 0.9625 0.9028 0.8577 0.8131 0.7891 0.7787 0.7658 0.7590 0.7547 0.7525 0.7515 0.7480 0.7467 0.7458 0.7468
45 3.5595 2.5587 1.5106 0.9690 0.9054 0.8619 0.8134 0.7897 0.7769 0.7679 0.7605 0.7556 0.7529 0.7532 0.7484 0.7482 0.7473 0.7471
50 3.8334 2.7329 1.5791 0.9737 0.9074 0.8624 0.8143 0.7934 0.7800 0.7672 0.7629 0.7569 0.7535 0.7537 0.7496 0.7482 0.7476 0.7481
60 4.3789 3.0659 1.7118 0.9844 0.9099 0.8633 0.8160 0.7921 0.7791 0.7689 0.7629 0.7580 0.7536 0.7533 0.7512 0.7489 0.7476 0.7484
70 4.9012 3.3923 1.8398 0.9917 0.9096 0.8657 0.8167 0.7926 0.7805 0.7689 0.7634 0.7575 0.7557 0.7542 0.7507 0.7492 0.7486 0.7490
80 5.4154 3.7091 1.9620 1.0021 0.9104 0.8649 0.8189 0.7931 0.7820 0.7703 0.7631 0.7589 0.7563 0.7545 0.7505 0.7508 0.7484 0.7488
90 5.9167 4.0188 2.0787 1.0111 0.9113 0.8679 0.8184 0.7936 0.7828 0.7715 0.7651 0.7592 0.7554 0.7548 0.7521 0.7495 0.7513 0.7502
100 6.4255 4.3222 2.1954 1.0194 0.9123 0.8676 0.8184 0.7950 0.7830 0.7698 0.7650 0.7585 0.7564 0.7543 0.7522 0.7495 0.7504 0.7500
200 11.1598 7.1943 3.2677 1.1031 0.9142 0.8692 0.8209 0.7962 0.7839 0.7713 0.7664 0.7604 0.7564 0.7561 0.7512 0.7513 0.7503 0.7506
300 15.6877 9.9089 4.2544 1.1798 0.9166 0.8707 0.8217 0.7969 0.7840 0.7720 0.7659 0.7594 0.7588 0.7568 0.7552 0.7509 0.7516 0.7523
400 20.0982 12.5299 5.1940 1.2564 0.9170 0.8713 0.8230 0.7977 0.7846 0.7728 0.7660 0.7602 0.7586 0.7572 0.7542 0.7515 0.7523 0.7511
500 24.4270 15.1069 6.1097 1.3280 0.9178 0.8716 0.8223 0.7971 0.7851 0.7721 0.7671 0.7615 0.7590 0.7563 0.7534 0.7522 0.7515 0.7530
1000 45.5811 27.5755 10.4679 1.6707 0.9188 0.8707 0.8244 0.7966 0.7846 0.7713 0.7679 0.7603 0.7597 0.7573 0.7527 0.7519 0.7497 0.7520
2500 107.018 63.4597 22.7439 2.5739 0.9200 0.8732 0.8223 0.7969 0.7860 0.7722 0.7666 0.7603 0.7641 0.7572 0.7527 0.7513 0.7522 0.7525
B-7D
-------
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.05
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.4371 0.4323 0.4289 0.4296 0.4244 0.4181 0.4103 0.4050 0.4024 0.3995 0.3979 0.3966 0.3962 0.3959 0.3949 0.3942 0.3938 0.3940
5 0.4191 0.4093 0.4009 0.3982 0.3885 0.3799 0.3716 0.3667 0.3644 0.3617 0.3605 0.3594 0.3584 0.3583 0.3576 0.3572 0.3569 0.3568
6 0.3971 0.3844 0.3726 0.3688 0.3637 0.3568 0.3486 0.3434 0.3408 0.3375 0.3358 0.3346 0.3335 0.3328 0.3325 0.3317 0.3318 0.3315
7 0.3849 0.3688 0.3528 0.3478 0.3419 0.3351 0.3272 0.3227 0.3196 0.3170 0.3151 0.3137 0.3129 0.3130 0.3119 0.3115 0.3114 0.3113
8 0.3724 0.3541 0.3356 0.3287 0.3233 0.3170 0.3090 0.3041 0.3015 0.2989 0.2975 0.2963 0.2953 0.2952 0.2943 0.2939 0.2934 0.2936
9 0.3617 0.3418 0.3208 0.3123 0.3075 0.3015 0.2941 0.2893 0.2869 0.2840 0.2825 0.2813 0.2804 0.2798 0.2793 0.2788 0.2787 0.2788
10 0.3523 0.3314 0.3081 0.2984 0.2941 0.2878 0.2808 0.2760 0.2737 0.2711 0.2698 0.2685 0.2678 0.2674 0.2666 0.2662 0.2659 0.2658
11 0.3440 0.3221 0.2976 0.2862 0.2819 0.2759 0.2691 0.2649 0.2624 0.2597 0.2585 0.2570 0.2565 0.2560 0.2554 0.2550 0.2546 0.2544
12 0.3368 0.3137 0.2873 0.2753 0.2710 0.2653 0.2587 0.2548 0.2524 0.2495 0.2485 0.2474 0.2464 0.2459 0.2454 0.2452 0.2450 0.2450
13 0.3297 0.3064 0.2789 0.2659 0.2613 0.2563 0.2498 0.2459 0.2430 0.2409 0.2394 0.2382 0.2377 0.2374 0.2367 0.2361 0.2362 0.2360
14 0.3234 0.2996 0.2711 0.2568 0.2530 0.2478 0.2416 0.2376 0.2351 0.2327 0.2316 0.2305 0.2298 0.2293 0.2287 0.2283 0.2280 0.2279
15 0.3178 0.2934 0.2638 0.2489 0.2451 0.2400 0.2338 0.2302 0.2279 0.2255 0.2244 0.2233 0.2226 0.2221 0.2215 0.2212 0.2210 0.2209
16 0.3127 0.2878 0.2577 0.2419 0.2376 0.2329 0.2272 0.2232 0.2212 0.2189 0.2179 0.2167 0.2161 0.2158 0.2152 0.2146 0.2144 0.2144
17 0.3078 0.2826 0.2519 0.2352 0.2315 0.2268 0.2209 0.2173 0.2151 0.2129 0.2117 0.2106 0.2100 0.2097 0.2089 0.2087 0.2085 0.2084
18 0.3033 0.2776 0.2464 0.2292 0.2253 0.2208 0.2151 0.2114 0.2094 0.2072 0.2063 0.2050 0.2046 0.2041 0.2034 0.2033 0.2031 0.2031
19 0.2990 0.2731 0.2415 0.2236 0.2198 0.2151 0.2100 0.2061 0.2043 0.2021 0.2008 0.2000 0.1992 0.1990 0.1986 0.1981 0.1981 0.1979
20 0.2949 0.2691 0.2366 0.2185 0.2145 0.2101 0.2049 0.2014 0.1994 0.1974 0.1961 0.1950 0.1946 0.1945 0.1938 0.1935 0.1934 0.1934
21 0.2915 0.2649 0.2323 0.2132 0.2097 0.2054 0.2004 0.1969 0.1949 0.1928 0.1918 0.1909 0.1903 0.1900 0.1894 0.1892 0.1889 0.1889
22 0.2879 0.2612 0.2283 0.2090 0.2055 0.2011 0.1959 0.1927 0.1908 0.1885 0.1879 0.1867 0.1862 0.1859 0.1853 0.1849 0.1851 0.1848
23 0.2847 0.2580 0.2247 0.2046 0.2013 0.1969 0.1919 0.1886 0.1867 0.1849 0.1838 0.1827 0.1824 0.1821 0.1815 0.1810 0.1809 0.1809
24 0.2813 0.2546 0.2211 0.2007 0.1971 0.1932 0.1881 0.1849 0.1830 0.1812 0.1802 0.1792 0.1787 0.1783 0.1777 0.1775 0.1774 0.1772
25 0.2786 0.2516 0.2179 0.1969 0.1933 0.1895 0.1845 0.1813 0.1796 0.1777 0.1767 0.1759 0.1753 0.1749 0.1745 0.1742 0.1739 0.1739
26 0.2759 0.2486 0.2146 0.1933 0.1896 0.1858 0.1812 0.1780 0.1764 0.1742 0.1734 0.1724 0.1719 0.1716 0.1712 0.1708 0.1707 0.1707
27 0.2732 0.2459 0.2118 0.1899 0.1863 0.1827 0.1779 0.1750 0.1730 0.1714 0.1705 0.1694 0.1689 0.1686 0.1681 0.1678 0.1676 0.1677
28 0.2709 0.2434 0.2088 0.1867 0.1832 0.1795 0.1749 0.1719 0.1702 0.1684 0.1676 0.1666 0.1661 0.1659 0.1652 0.1652 0.1649 0.1648
29 0.2683 0.2409 0.2062 0.1837 0.1802 0.1767 0.1721 0.1690 0.1675 0.1655 0.1647 0.1639 0.1634 0.1630 0.1625 0.1623 0.1622 0.1620
30 0.2663 0.2386 0.2037 0.1809 0.1772 0.1736 0.1692 0.1663 0.1648 0.1629 0.1621 0.1611 0.1607 0.1603 0.1600 0.1597 0.1595 0.1596
35 0.2561 0.2281 0.1927 0.1683 0.1647 0.1613 0.1571 0.1545 0.1530 0.1515 0.1507 0.1497 0.1494 0.1490 0.1486 0.1484 0.1482 0.1481
40 0.2482 0.2196 0.1835 0.1581 0.1544 0.1514 0.1476 0.1449 0.1437 0.1420 0.1412 0.1404 0.1401 0.1399 0.1394 0.1391 0.1390 0.1390
45 0.2412 0.2124 0.1759 0.1496 0.1461 0.1432 0.1393 0.1370 0.1356 0.1342 0.1334 0.1327 0.1323 0.1322 0.1317 0.1315 0.1314 0.1313
50 0.2353 0.2063 0.1695 0.1425 0.1389 0.1361 0.1324 0.1304 0.1289 0.1275 0.1269 0.1262 0.1257 0.1256 0.1252 0.1249 0.1249 0.1249
60 0.2258 0.1963 0.1592 0.1308 0.1272 0.1245 0.1214 0.1192 0.1180 0.1168 0.1161 0.1156 0.1151 0.1150 0.1147 0.1144 0.1143 0.1143
70 0.2183 0.1886 0.1513 0.1216 0.1179 0.1157 0.1126 0.1107 0.1095 0.1084 0.1078 0.1071 0.1069 0.1067 0.1064 0.1061 0.1061 0.1061
80 0.2122 0.1823 0.1447 0.1143 0.1105 0.1083 0.1055 0.1037 0.1027 0.1016 0.1011 0.1005 0.1002 0.0999 0.0997 0.0995 0.0993 0.0994
90 0.2071 0.1771 0.1392 0.1082 0.1044 0.1023 0.0996 0.0979 0.0970 0.0959 0.0954 0.0949 0.0945 0.0944 0.0941 0.0939 0.0939 0.0938
100 0.2029 0.1727 0.1347 0.1031 0.0991 0.0971 0.0946 0.0929 0.0921 0.0911 0.0906 0.0901 0.0898 0.0896 0.0894 0.0892 0.0892 0.0892
200 0.1794 0.1487 0.1096 0.0753 0.0705 0.0691 0.0673 0.0662 0.0655 0.0648 0.0645 0.0641 0.0639 0.0638 0.0635 0.0635 0.0635 0.0634
300 0.1691 0.1380 0.0985 0.0631 0.0577 0.0566 0.0551 0.0542 0.0537 0.0531 0.0528 0.0524 0.0523 0.0522 0.0521 0.0520 0.0519 0.0520
400 0.1629 0.1316 0.0919 0.0559 0.0501 0.0491 0.0478 0.0470 0.0465 0.0460 0.0457 0.0455 0.0454 0.0453 0.0452 0.0451 0.0451 0.0451
500 0.1587 0.1274 0.0874 0.0510 0.0448 0.0439 0.0428 0.0421 0.0417 0.0412 0.0410 0.0408 0.0406 0.0406 0.0404 0.0404 0.0403 0.0404
1000 0.1484 0.1168 0.0764 0.0390 0.0318 0.0311 0.0303 0.0298 0.0295 0.0292 0.0291 0.0289 0.0288 0.0288 0.0286 0.0286 0.0286 0.0286
2500 0.1394 0.1076 0.0668 0.0286 0.0202 0.0197 0.0192 0.0189 0.0187 0.0185 0.0184 0.0183 0.0183 0.0182 0.0182 0.0181 0.0181 0.0181
B-8D
-------
Critical Values for Anderson Darling Test - Significance Level of 0.025
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.8615 0.8677 0.8863 0.9039 0.8378 0.7984 0.7665 0.7511 0.7460 0.7399 0.7369 0.7355 0.7346 0.7346 0.7325 0.7317 0.7305 0.7311
5 0.9689 0.9509 0.9503 0.9609 0.8987 0.8497 0.8119 0.7925 0.7854 0.7759 0.7730 0.7718 0.7693 0.7693 0.7678 0.7662 0.7659 0.7643
6 1.0630 1.0130 0.9765 0.9863 0.9370 0.8879 0.8446 0.8255 0.8160 0.8059 0.8026 0.7993 0.7961 0.7948 0.7943 0.7932 0.7922 0.7918
7 1.1594 1.0729 1.0070 1.0066 0.9620 0.9114 0.8676 0.8467 0.8348 0.8258 0.8170 0.8144 0.8106 0.8106 0.8089 0.8067 0.8072 0.8070
8 1.2574 1.1392 1.0345 1.0216 0.9817 0.9290 0.8842 0.8577 0.8457 0.8353 0.8304 0.8278 0.8236 0.8235 0.8187 0.8195 0.8182 0.8185
9 1.3550 1.2024 1.0646 1.0334 0.9943 0.9451 0.8930 0.8689 0.8564 0.8441 0.8394 0.8360 0.8325 0.8298 0.8294 0.8278 0.8270 0.8273
10 1.4495 1.2659 1.0902 1.0433 1.0063 0.9536 0.9025 0.8758 0.8651 0.8524 0.8468 0.8419 0.8389 0.8396 0.8344 0.8330 0.8324 0.8317
11 1.5453 1.3293 1.1162 1.0521 1.0135 0.9598 0.9087 0.8832 0.8700 0.8571 0.8514 0.8445 0.8443 0.8416 0.8400 0.8385 0.8389 0.8349
12 1.6379 1.3875 1.1414 1.0568 1.0220 0.9648 0.9143 0.8900 0.8752 0.8611 0.8557 0.8507 0.8475 0.8447 0.8405 0.8420 0.8406 0.8430
13 1.7256 1.4490 1.1649 1.0652 1.0232 0.9720 0.9209 0.8944 0.8772 0.8665 0.8592 0.8539 0.8501 0.8486 0.8458 0.8445 0.8426 0.8433
14 1.8117 1.5045 1.1897 1.0698 1.0315 0.9777 0.9247 0.8944 0.8811 0.8688 0.8634 0.8573 0.8538 0.8515 0.8473 0.8463 0.8456 0.8454
15 1.8988 1.5637 1.2102 1.0738 1.0342 0.9816 0.9236 0.8986 0.8847 0.8708 0.8649 0.8601 0.8556 0.8527 0.8497 0.8478 0.8481 0.8481
16 1.9814 1.6174 1.2393 1.0815 1.0363 0.9840 0.9318 0.9015 0.8852 0.8745 0.8678 0.8610 0.8572 0.8564 0.8521 0.8506 0.8489 0.8507
17 2.0598 1.6722 1.2611 1.0865 1.0428 0.9892 0.9315 0.9020 0.8879 0.8750 0.8685 0.8619 0.8596 0.8565 0.8536 0.8520 0.8505 0.8503
18 2.1409 1.7231 1.2808 1.0905 1.0435 0.9902 0.9336 0.9056 0.8892 0.8751 0.8701 0.8626 0.8581 0.8589 0.8551 0.8555 0.8532 0.8519
19 2.2162 1.7764 1.3033 1.0927 1.0480 0.9919 0.9357 0.9069 0.8932 0.8776 0.8713 0.8662 0.8630 0.8583 0.8566 0.8552 0.8555 0.8534
20 2.2915 1.8258 1.3244 1.0978 1.0493 0.9941 0.9375 0.9060 0.8917 0.8789 0.8712 0.8637 0.8626 0.8621 0.8580 0.8561 0.8538 0.8559
21 2.3715 1.8738 1.3396 1.0962 1.0515 0.9951 0.9410 0.9105 0.8939 0.8793 0.8727 0.8681 0.8644 0.8614 0.8589 0.8591 0.8572 0.8552
22 2.4415 1.9185 1.3632 1.1035 1.0557 0.9968 0.9391 0.9107 0.8948 0.8791 0.8763 0.8668 0.8644 0.8644 0.8612 0.8572 0.8609 0.8568
23 2.5164 1.9742 1.3839 1.1023 1.0576 0.9990 0.9418 0.9103 0.8927 0.8811 0.8757 0.8706 0.8680 0.8627 0.8620 0.8566 0.8586 0.8569
24 2.5831 2.0197 1.4035 1.1105 1.0566 1.0019 0.9417 0.9126 0.8978 0.8822 0.8761 0.8710 0.8658 0.8640 0.8615 0.8595 0.8569 0.8577
25 2.6565 2.0644 1.4219 1.1114 1.0614 1.0024 0.9430 0.9131 0.8991 0.8836 0.8768 0.8705 0.8671 0.8651 0.8643 0.8602 0.8585 0.8579
26 2.7258 2.1088 1.4384 1.1131 1.0567 1.0018 0.9462 0.9143 0.8994 0.8838 0.8765 0.8708 0.8666 0.8637 0.8619 0.8617 0.8609 0.8594
27 2.7952 2.1511 1.4646 1.1156 1.0600 1.0060 0.9440 0.9154 0.8986 0.8861 0.8779 0.8717 0.8674 0.8649 0.8604 0.8612 0.8603 0.8599
28 2.8692 2.1998 1.4766 1.1174 1.0627 1.0042 0.9485 0.9158 0.8990 0.8850 0.8797 0.8739 0.8680 0.8672 0.8644 0.8627 0.8613 0.8602
29 2.9301 2.2438 1.4940 1.1235 1.0606 1.0077 0.9508 0.9147 0.9008 0.8857 0.8789 0.8743 0.8698 0.8665 0.8637 0.8625 0.8609 0.8588
30 3.0015 2.2866 1.5156 1.1243 1.0611 1.0049 0.9456 0.9170 0.9021 0.8860 0.8795 0.8727 0.8688 0.8650 0.8647 0.8625 0.8611 0.8624
35 3.3266 2.4946 1.6032 1.1307 1.0635 1.0088 0.9494 0.9187 0.9017 0.8905 0.8828 0.8730 0.8718 0.8686 0.8671 0.8647 0.8611 0.8643
40 3.6396 2.6943 1.6803 1.1409 1.0679 1.0105 0.9513 0.9198 0.9068 0.8895 0.8814 0.8759 0.8730 0.8689 0.8677 0.8658 0.8635 0.8665
45 3.9425 2.8877 1.7545 1.1477 1.0716 1.0157 0.9507 0.9215 0.9049 0.8916 0.8834 0.8760 0.8734 0.8741 0.8687 0.8689 0.8664 0.8653
50 4.2378 3.0745 1.8299 1.1563 1.0756 1.0154 0.9541 0.9252 0.9081 0.8919 0.8869 0.8786 0.8753 0.8741 0.8690 0.8668 0.8677 0.8687
60 4.8100 3.4320 1.9809 1.1694 1.0776 1.0157 0.9548 0.9248 0.9065 0.8932 0.8870 0.8805 0.8760 0.8744 0.8718 0.8699 0.8685 0.8670
70 5.3566 3.7754 2.1204 1.1781 1.0761 1.0204 0.9572 0.9257 0.9083 0.8935 0.8881 0.8801 0.8774 0.8766 0.8704 0.8702 0.8686 0.8694
80 5.9031 4.1163 2.2521 1.1918 1.0787 1.0207 0.9602 0.9257 0.9116 0.8949 0.8880 0.8805 0.8771 0.8753 0.8713 0.8702 0.8689 0.8687
90 6.4293 4.4397 2.3772 1.2011 1.0823 1.0220 0.9579 0.9271 0.9135 0.8981 0.8894 0.8815 0.8766 0.8763 0.8731 0.8688 0.8728 0.8708
100 6.9592 4.7648 2.5055 1.2102 1.0801 1.0215 0.9593 0.9267 0.9118 0.8948 0.8885 0.8808 0.8782 0.8778 0.8727 0.8702 0.8719 0.8705
200 11.8779 7.7654 3.6516 1.3093 1.0827 1.0257 0.9641 0.9305 0.9142 0.8985 0.8923 0.8845 0.8789 0.8789 0.8719 0.8727 0.8714 0.8716
300 16.5240 10.5806 4.6843 1.3997 1.0856 1.0260 0.9631 0.9310 0.9155 0.8988 0.8914 0.8824 0.8824 0.8801 0.8772 0.8727 0.8721 0.8732
400 21.0493 13.2831 5.6639 1.4844 1.0867 1.0286 0.9643 0.9331 0.9148 0.8989 0.8918 0.8843 0.8815 0.8801 0.8761 0.8729 0.8733 0.8725
500 25.4819 15.9306 6.6237 1.5670 1.0880 1.0277 0.9645 0.9305 0.9155 0.8992 0.8921 0.8843 0.8820 0.8790 0.8774 0.8738 0.8721 0.8751
1000 47.0169 28.6841 11.1336 1.9413 1.0900 1.0281 0.9665 0.9295 0.9142 0.8976 0.8946 0.8845 0.8834 0.8791 0.8744 0.8742 0.8703 0.8719
2500 109.217 65.1297 23.6988 2.9078 1.0895 1.0304 0.9647 0.9295 0.9171 0.8991 0.8932 0.8854 0.8877 0.8812 0.8755 0.8720 0.8748 0.8732
B-9D
-------
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.025
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.4542 0.4526 0.4535 0.4577 0.4505 0.4430 0.4343 0.4287 0.4258 0.4229 0.4211 0.4198 0.4192 0.4190 0.4176 0.4174 0.4167 0.4169
5 0.4429 0.4346 0.4277 0.4258 0.4171 0.4075 0.3974 0.3908 0.3877 0.3838 0.3823 0.3810 0.3798 0.3796 0.3786 0.3783 0.3779 0.3777
6 0.4240 0.4103 0.3981 0.3971 0.3912 0.3830 0.3737 0.3681 0.3651 0.3612 0.3596 0.3579 0.3570 0.3563 0.3556 0.3550 0.3550 0.3549
7 0.4086 0.3933 0.3784 0.3743 0.3680 0.3600 0.3509 0.3454 0.3419 0.3389 0.3365 0.3348 0.3339 0.3342 0.3329 0.3325 0.3323 0.3324
8 0.3968 0.3786 0.3595 0.3537 0.3484 0.3414 0.3324 0.3267 0.3233 0.3204 0.3185 0.3173 0.3161 0.3160 0.3146 0.3144 0.3138 0.3140
9 0.3852 0.3655 0.3447 0.3366 0.3315 0.3251 0.3160 0.3108 0.3080 0.3044 0.3027 0.3014 0.3006 0.2998 0.2993 0.2986 0.2986 0.2985
10 0.3752 0.3547 0.3310 0.3217 0.3172 0.3103 0.3020 0.2964 0.2941 0.2910 0.2893 0.2878 0.2868 0.2866 0.2854 0.2852 0.2848 0.2848
11 0.3665 0.3448 0.3199 0.3088 0.3043 0.2977 0.2897 0.2847 0.2819 0.2787 0.2773 0.2755 0.2749 0.2744 0.2737 0.2733 0.2730 0.2726
12 0.3588 0.3358 0.3090 0.2972 0.2926 0.2861 0.2785 0.2740 0.2712 0.2678 0.2666 0.2655 0.2642 0.2637 0.2630 0.2629 0.2626 0.2626
13 0.3513 0.3278 0.2999 0.2871 0.2820 0.2763 0.2692 0.2645 0.2613 0.2588 0.2570 0.2558 0.2550 0.2547 0.2537 0.2532 0.2532 0.2530
14 0.3441 0.3205 0.2915 0.2773 0.2732 0.2674 0.2604 0.2553 0.2528 0.2500 0.2488 0.2473 0.2466 0.2460 0.2452 0.2448 0.2445 0.2445
15 0.3383 0.3138 0.2839 0.2686 0.2647 0.2591 0.2519 0.2478 0.2451 0.2423 0.2408 0.2397 0.2389 0.2383 0.2375 0.2372 0.2370 0.2371
16 0.3324 0.3077 0.2774 0.2614 0.2567 0.2514 0.2448 0.2403 0.2376 0.2351 0.2341 0.2327 0.2319 0.2317 0.2309 0.2303 0.2302 0.2301
17 0.3272 0.3022 0.2712 0.2539 0.2502 0.2449 0.2381 0.2339 0.2312 0.2289 0.2275 0.2261 0.2256 0.2251 0.2242 0.2238 0.2238 0.2236
18 0.3227 0.2968 0.2653 0.2475 0.2435 0.2384 0.2319 0.2278 0.2253 0.2228 0.2217 0.2203 0.2196 0.2191 0.2182 0.2183 0.2178 0.2179
19 0.3180 0.2919 0.2599 0.2415 0.2377 0.2325 0.2265 0.2220 0.2198 0.2171 0.2158 0.2147 0.2142 0.2136 0.2132 0.2128 0.2126 0.2125
20 0.3135 0.2875 0.2548 0.2360 0.2318 0.2270 0.2207 0.2167 0.2144 0.2123 0.2107 0.2094 0.2090 0.2088 0.2081 0.2077 0.2076 0.2077
21 0.3096 0.2829 0.2500 0.2303 0.2266 0.2219 0.2162 0.2120 0.2099 0.2072 0.2061 0.2053 0.2044 0.2042 0.2033 0.2031 0.2027 0.2028
22 0.3055 0.2789 0.2457 0.2260 0.2221 0.2172 0.2113 0.2075 0.2054 0.2026 0.2020 0.2008 0.1998 0.1997 0.1989 0.1986 0.1987 0.1984
23 0.3022 0.2754 0.2417 0.2211 0.2175 0.2126 0.2069 0.2032 0.2008 0.1988 0.1977 0.1964 0.1960 0.1956 0.1949 0.1944 0.1942 0.1942
24 0.2984 0.2718 0.2378 0.2169 0.2131 0.2087 0.2029 0.1991 0.1971 0.1948 0.1937 0.1926 0.1919 0.1916 0.1909 0.1906 0.1904 0.1904
25 0.2954 0.2684 0.2345 0.2128 0.2091 0.2046 0.1988 0.1954 0.1932 0.1910 0.1901 0.1890 0.1883 0.1880 0.1874 0.1871 0.1867 0.1867
26 0.2927 0.2654 0.2308 0.2089 0.2050 0.2005 0.1954 0.1918 0.1900 0.1874 0.1862 0.1853 0.1847 0.1844 0.1839 0.1835 0.1834 0.1834
27 0.2895 0.2622 0.2278 0.2054 0.2015 0.1974 0.1919 0.1884 0.1863 0.1842 0.1832 0.1820 0.1816 0.1811 0.1805 0.1802 0.1802 0.1801
28 0.2870 0.2593 0.2246 0.2018 0.1980 0.1940 0.1888 0.1851 0.1832 0.1811 0.1803 0.1790 0.1785 0.1782 0.1774 0.1775 0.1772 0.1769
29 0.2841 0.2566 0.2218 0.1986 0.1948 0.1908 0.1857 0.1820 0.1801 0.1781 0.1771 0.1761 0.1756 0.1752 0.1746 0.1742 0.1742 0.1741
30 0.2819 0.2540 0.2191 0.1955 0.1916 0.1875 0.1826 0.1791 0.1775 0.1752 0.1743 0.1733 0.1726 0.1722 0.1719 0.1715 0.1714 0.1714
35 0.2708 0.2426 0.2072 0.1819 0.1779 0.1743 0.1696 0.1666 0.1647 0.1629 0.1620 0.1609 0.1605 0.1602 0.1596 0.1594 0.1592 0.1591
40 0.2618 0.2333 0.1969 0.1710 0.1669 0.1635 0.1591 0.1563 0.1548 0.1528 0.1519 0.1510 0.1505 0.1503 0.1497 0.1495 0.1493 0.1495
45 0.2543 0.2254 0.1887 0.1619 0.1579 0.1548 0.1503 0.1476 0.1459 0.1444 0.1435 0.1426 0.1421 0.1420 0.1416 0.1413 0.1412 0.1411
50 0.2479 0.2187 0.1818 0.1542 0.1502 0.1470 0.1428 0.1405 0.1388 0.1372 0.1365 0.1357 0.1352 0.1350 0.1346 0.1343 0.1341 0.1343
60 0.2373 0.2078 0.1706 0.1414 0.1374 0.1345 0.1309 0.1286 0.1270 0.1257 0.1249 0.1242 0.1237 0.1236 0.1232 0.1229 0.1228 0.1229
70 0.2290 0.1992 0.1618 0.1316 0.1276 0.1250 0.1214 0.1193 0.1178 0.1167 0.1159 0.1152 0.1148 0.1147 0.1143 0.1141 0.1140 0.1140
80 0.2223 0.1924 0.1546 0.1237 0.1195 0.1170 0.1138 0.1118 0.1107 0.1093 0.1087 0.1080 0.1076 0.1074 0.1071 0.1069 0.1068 0.1068
90 0.2165 0.1866 0.1486 0.1170 0.1129 0.1105 0.1075 0.1055 0.1045 0.1033 0.1027 0.1020 0.1015 0.1015 0.1011 0.1009 0.1009 0.1008
100 0.2120 0.1817 0.1436 0.1114 0.1072 0.1048 0.1021 0.1002 0.0992 0.0980 0.0974 0.0968 0.0966 0.0964 0.0960 0.0959 0.0958 0.0958
200 0.1860 0.1552 0.1159 0.0814 0.0762 0.0747 0.0726 0.0714 0.0705 0.0698 0.0694 0.0689 0.0687 0.0686 0.0682 0.0682 0.0682 0.0682
300 0.1745 0.1434 0.1037 0.0682 0.0624 0.0611 0.0595 0.0584 0.0578 0.0571 0.0567 0.0564 0.0563 0.0561 0.0560 0.0558 0.0557 0.0558
400 0.1676 0.1363 0.0964 0.0603 0.0541 0.0531 0.0515 0.0507 0.0501 0.0495 0.0492 0.0489 0.0488 0.0487 0.0486 0.0485 0.0484 0.0484
500 0.1630 0.1316 0.0915 0.0550 0.0485 0.0474 0.0462 0.0453 0.0449 0.0444 0.0441 0.0438 0.0437 0.0436 0.0434 0.0433 0.0433 0.0434
1000 0.1514 0.1198 0.0792 0.0419 0.0343 0.0336 0.0327 0.0321 0.0318 0.0314 0.0313 0.0310 0.0310 0.0309 0.0308 0.0308 0.0307 0.0307
2500 0.1413 0.1095 0.0686 0.0304 0.0218 0.0213 0.0207 0.0203 0.0202 0.0199 0.0198 0.0197 0.0197 0.0196 0.0195 0.0195 0.0195 0.0195
B-10D
-------
Critical Values for Anderson Darling Test - Significance Level of 0.01
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.9603 1.0073 1.0852 1.1154 0.9876 0.9047 0.8544 0.8345 0.8262 0.8197 0.8164 0.8153 0.8135 0.8137 0.8112 0.8106 0.8099 0.8097
5 1.0754 1.0772 1.1053 1.1446 1.0682 1.0000 0.9453 0.9167 0.9054 0.8930 0.8897 0.8878 0.8822 0.8831 0.8817 0.8791 0.8791 0.8767
6 1.1951 1.1556 1.1420 1.1831 1.1214 1.0533 0.9896 0.9580 0.9461 0.9314 0.9275 0.9211 0.9173 0.9182 0.9159 0.9113 0.9119 0.9153
7 1.3145 1.2298 1.1755 1.2066 1.1562 1.0841 1.0186 0.9924 0.9785 0.9630 0.9510 0.9460 0.9414 0.9438 0.9383 0.9354 0.9377 0.9389
8 1.4299 1.3111 1.2089 1.2256 1.1807 1.1087 1.0439 1.0086 0.9905 0.9787 0.9699 0.9646 0.9602 0.9612 0.9553 0.9558 0.9527 0.9550
9 1.5478 1.3855 1.2502 1.2415 1.1959 1.1293 1.0584 1.0232 1.0075 0.9908 0.9844 0.9778 0.9739 0.9674 0.9685 0.9685 0.9669 0.9655
10 1.6581 1.4643 1.2809 1.2490 1.2139 1.1420 1.0707 1.0339 1.0181 1.0012 0.9944 0.9873 0.9829 0.9812 0.9769 0.9755 0.9732 0.9721
11 1.7667 1.5444 1.3164 1.2615 1.2233 1.1525 1.0789 1.0445 1.0271 1.0098 1.0031 0.9895 0.9878 0.9864 0.9840 0.9827 0.9812 0.9762
12 1.8736 1.6117 1.3464 1.2700 1.2310 1.1546 1.0876 1.0515 1.0327 1.0140 1.0061 1.0013 0.9955 0.9940 0.9883 0.9894 0.9860 0.9879
13 1.9804 1.6847 1.3763 1.2796 1.2370 1.1672 1.0983 1.0603 1.0358 1.0233 1.0115 1.0065 0.9999 0.9984 0.9939 0.9917 0.9891 0.9902
14 2.0727 1.7510 1.4057 1.2913 1.2452 1.1756 1.1010 1.0607 1.0403 1.0245 1.0157 1.0084 1.0054 1.0010 0.9959 0.9939 0.9918 0.9932
15 2.1736 1.8182 1.4338 1.2899 1.2476 1.1814 1.0996 1.0655 1.0480 1.0281 1.0184 1.0119 1.0065 1.0024 0.9990 0.9967 0.9986 0.9984
16 2.2603 1.8791 1.4693 1.3039 1.2534 1.1827 1.1117 1.0683 1.0470 1.0316 1.0186 1.0171 1.0104 1.0070 1.0043 0.9996 0.9986 1.0004
17 2.3532 1.9419 1.4945 1.3074 1.2604 1.1911 1.1104 1.0710 1.0526 1.0331 1.0230 1.0158 1.0109 1.0078 1.0045 1.0032 1.0005 1.0001
18 2.4406 2.0044 1.5210 1.3165 1.2604 1.1921 1.1158 1.0729 1.0536 1.0341 1.0270 1.0146 1.0150 1.0152 1.0061 1.0050 1.0033 1.0057
19 2.5322 2.0686 1.5482 1.3187 1.2670 1.1909 1.1153 1.0762 1.0592 1.0379 1.0259 1.0216 1.0180 1.0130 1.0096 1.0064 1.0080 1.0063
20 2.6114 2.1235 1.5704 1.3288 1.2680 1.1960 1.1178 1.0770 1.0563 1.0416 1.0311 1.0181 1.0193 1.0163 1.0121 1.0049 1.0091 1.0113
21 2.7041 2.1723 1.5955 1.3235 1.2697 1.1993 1.1264 1.0811 1.0572 1.0410 1.0310 1.0271 1.0200 1.0166 1.0127 1.0130 1.0081 1.0100
22 2.7796 2.2262 1.6194 1.3388 1.2791 1.2002 1.1191 1.0822 1.0621 1.0404 1.0359 1.0244 1.0199 1.0227 1.0141 1.0111 1.0125 1.0107
23 2.8617 2.2902 1.6460 1.3343 1.2809 1.2001 1.1249 1.0831 1.0589 1.0430 1.0342 1.0286 1.0243 1.0171 1.0196 1.0117 1.0127 1.0100
24 2.9336 2.3402 1.6668 1.3413 1.2767 1.2057 1.1257 1.0856 1.0648 1.0453 1.0346 1.0291 1.0194 1.0200 1.0155 1.0122 1.0127 1.0132
25 3.0189 2.3833 1.6899 1.3424 1.2808 1.2078 1.1272 1.0862 1.0643 1.0451 1.0376 1.0294 1.0256 1.0212 1.0171 1.0140 1.0127 1.0118
26 3.1002 2.4393 1.7106 1.3444 1.2777 1.2050 1.1302 1.0887 1.0669 1.0460 1.0369 1.0306 1.0248 1.0196 1.0152 1.0168 1.0158 1.0129
27 3.1672 2.4875 1.7364 1.3482 1.2853 1.2104 1.1299 1.0878 1.0661 1.0475 1.0393 1.0310 1.0257 1.0214 1.0160 1.0169 1.0125 1.0153
28 3.2487 2.5418 1.7524 1.3541 1.2839 1.2107 1.1342 1.0886 1.0667 1.0512 1.0406 1.0345 1.0266 1.0251 1.0200 1.0208 1.0194 1.0149
29 3.3187 2.5865 1.7783 1.3557 1.2859 1.2176 1.1353 1.0878 1.0670 1.0498 1.0410 1.0336 1.0267 1.0268 1.0194 1.0180 1.0171 1.0171
30 3.3926 2.6336 1.8001 1.3651 1.2861 1.2121 1.1326 1.0908 1.0724 1.0500 1.0438 1.0336 1.0261 1.0227 1.0231 1.0192 1.0179 1.0183
35 3.7444 2.8654 1.9035 1.3711 1.2858 1.2174 1.1360 1.0951 1.0715 1.0529 1.0450 1.0302 1.0290 1.0267 1.0246 1.0211 1.0180 1.0199
40 4.0854 3.0880 1.9882 1.3819 1.2938 1.2179 1.1375 1.0964 1.0760 1.0551 1.0457 1.0352 1.0324 1.0295 1.0272 1.0229 1.0221 1.0241
45 4.4084 3.2993 2.0772 1.3883 1.2976 1.2210 1.1413 1.0994 1.0744 1.0590 1.0476 1.0368 1.0340 1.0361 1.0300 1.0263 1.0245 1.0212
50 4.7335 3.4996 2.1620 1.4067 1.3038 1.2232 1.1419 1.1011 1.0786 1.0595 1.0526 1.0403 1.0381 1.0338 1.0295 1.0277 1.0248 1.0252
60 5.3343 3.8894 2.3298 1.4187 1.3079 1.2229 1.1435 1.1034 1.0790 1.0619 1.0540 1.0426 1.0381 1.0323 1.0317 1.0291 1.0302 1.0244
70 5.9151 4.2582 2.4835 1.4298 1.3071 1.2296 1.1446 1.1041 1.0785 1.0604 1.0549 1.0438 1.0382 1.0376 1.0311 1.0311 1.0280 1.0276
80 6.5032 4.6205 2.6216 1.4453 1.3016 1.2303 1.1499 1.1044 1.0849 1.0641 1.0550 1.0452 1.0372 1.0360 1.0328 1.0316 1.0294 1.0287
90 7.0504 4.9602 2.7593 1.4575 1.3124 1.2311 1.1486 1.1077 1.0861 1.0660 1.0558 1.0455 1.0374 1.0381 1.0345 1.0311 1.0326 1.0310
100 7.6095 5.3024 2.8954 1.4713 1.3083 1.2288 1.1492 1.1072 1.0851 1.0648 1.0542 1.0464 1.0417 1.0421 1.0353 1.0330 1.0325 1.0316
200 12.7383 8.4639 4.1296 1.5840 1.3097 1.2364 1.1565 1.1103 1.0885 1.0668 1.0587 1.0507 1.0450 1.0412 1.0314 1.0322 1.0329 1.0323
300 17.5414 11.3900 5.2242 1.6967 1.3138 1.2409 1.1535 1.1113 1.0898 1.0679 1.0582 1.0489 1.0468 1.0431 1.0376 1.0328 1.0313 1.0352
400 22.1764 14.1813 6.2523 1.7932 1.3205 1.2404 1.1579 1.1151 1.0928 1.0677 1.0573 1.0480 1.0424 1.0433 1.0390 1.0350 1.0336 1.0329
500 26.7379 16.9148 7.2528 1.8846 1.3188 1.2395 1.1552 1.1139 1.0886 1.0695 1.0570 1.0498 1.0484 1.0466 1.0402 1.0339 1.0337 1.0382
1000 48.7347 30.0039 11.9358 2.2962 1.3248 1.2438 1.1570 1.1103 1.0923 1.0676 1.0601 1.0480 1.0496 1.0430 1.0347 1.0358 1.0309 1.0329
2500 111.798 67.1014 24.8571 3.3313 1.3247 1.2420 1.1559 1.1102 1.0900 1.0683 1.0606 1.0495 1.0552 1.0476 1.0351 1.0345 1.0383 1.0364
B-11D
-------
Critical Values for Kolmogorov Smirnov Test - Significance Level of 0.01
n\k 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.750 1.000 1.500 2.000 3.000 4.000 5.000 10.000 20.000 50.000 100.000
4 0.4698 0.4724 0.4853 0.4961 0.4783 0.4662 0.4552 0.4491 0.4458 0.4426 0.4409 0.4394 0.4387 0.4384 0.4373 0.4368 0.4365 0.4365
5 0.4641 0.4581 0.4536 0.4559 0.4509 0.4415 0.4314 0.4244 0.4207 0.4157 0.4139 0.4121 0.4103 0.4104 0.4097 0.4083 0.4080 0.4077
6 0.4528 0.4411 0.4306 0.4314 0.4234 0.4137 0.4022 0.3947 0.3912 0.3873 0.3852 0.3833 0.3821 0.3819 0.3808 0.3801 0.3797 0.3805
7 0.4367 0.4207 0.4065 0.4041 0.3989 0.3902 0.3800 0.3738 0.3694 0.3651 0.3624 0.3604 0.3593 0.3599 0.3576 0.3568 0.3571 0.3575
8 0.4235 0.4066 0.3879 0.3843 0.3789 0.3700 0.3604 0.3535 0.3493 0.3456 0.3436 0.3420 0.3408 0.3403 0.3393 0.3388 0.3383 0.3385
9 0.4122 0.3928 0.3726 0.3655 0.3603 0.3530 0.3427 0.3362 0.3330 0.3289 0.3269 0.3253 0.3240 0.3233 0.3227 0.3222 0.3219 0.3218
10 0.4019 0.3816 0.3580 0.3497 0.3450 0.3378 0.3279 0.3212 0.3184 0.3143 0.3125 0.3107 0.3101 0.3095 0.3081 0.3076 0.3074 0.3071
11 0.3925 0.3713 0.3461 0.3355 0.3314 0.3238 0.3144 0.3085 0.3053 0.3018 0.2999 0.2976 0.2971 0.2965 0.2954 0.2949 0.2953 0.2942
12 0.3844 0.3613 0.3348 0.3231 0.3186 0.3110 0.3024 0.2974 0.2936 0.2898 0.2880 0.2871 0.2857 0.2853 0.2843 0.2841 0.2837 0.2835
13 0.3762 0.3530 0.3248 0.3121 0.3071 0.3008 0.2927 0.2868 0.2833 0.2803 0.2783 0.2764 0.2758 0.2755 0.2740 0.2738 0.2737 0.2736
14 0.3685 0.3447 0.3160 0.3019 0.2976 0.2910 0.2833 0.2768 0.2741 0.2709 0.2694 0.2674 0.2670 0.2662 0.2651 0.2645 0.2643 0.2646
15 0.3622 0.3379 0.3076 0.2921 0.2884 0.2820 0.2736 0.2689 0.2657 0.2626 0.2606 0.2596 0.2585 0.2577 0.2572 0.2566 0.2563 0.2564
16 0.3556 0.3310 0.3009 0.2845 0.2798 0.2738 0.2663 0.2609 0.2578 0.2547 0.2535 0.2520 0.2510 0.2507 0.2499 0.2491 0.2486 0.2489
17 0.3502 0.3250 0.2939 0.2767 0.2725 0.2669 0.2592 0.2538 0.2508 0.2480 0.2463 0.2448 0.2442 0.2436 0.2428 0.2424 0.2422 0.2419
18 0.3448 0.3192 0.2879 0.2696 0.2655 0.2597 0.2524 0.2472 0.2445 0.2415 0.2403 0.2383 0.2376 0.2374 0.2363 0.2362 0.2359 0.2357
19 0.3399 0.3139 0.2819 0.2632 0.2592 0.2534 0.2461 0.2410 0.2383 0.2353 0.2337 0.2325 0.2322 0.2315 0.2307 0.2302 0.2301 0.2299
20 0.3350 0.3093 0.2764 0.2572 0.2529 0.2475 0.2403 0.2356 0.2328 0.2301 0.2285 0.2267 0.2265 0.2262 0.2254 0.2247 0.2248 0.2247
21 0.3308 0.3041 0.2709 0.2510 0.2474 0.2416 0.2352 0.2303 0.2277 0.2248 0.2235 0.2223 0.2214 0.2211 0.2204 0.2199 0.2195 0.2195
22 0.3265 0.2998 0.2666 0.2460 0.2423 0.2363 0.2297 0.2256 0.2229 0.2198 0.2190 0.2175 0.2163 0.2161 0.2156 0.2150 0.2152 0.2148
23 0.3226 0.2960 0.2621 0.2411 0.2372 0.2313 0.2250 0.2208 0.2180 0.2155 0.2145 0.2130 0.2124 0.2117 0.2112 0.2105 0.2103 0.2103
24 0.3183 0.2923 0.2580 0.2365 0.2323 0.2271 0.2208 0.2161 0.2140 0.2114 0.2098 0.2087 0.2077 0.2076 0.2067 0.2065 0.2062 0.2064
25 0.3153 0.2880 0.2540 0.2317 0.2284 0.2229 0.2164 0.2121 0.2099 0.2073 0.2059 0.2047 0.2039 0.2035 0.2031 0.2027 0.2025 0.2023
26 0.3120 0.2848 0.2501 0.2279 0.2235 0.2188 0.2126 0.2085 0.2061 0.2033 0.2022 0.2009 0.2002 0.1997 0.1990 0.1988 0.1987 0.1986
27 0.3087 0.2813 0.2471 0.2241 0.2199 0.2150 0.2088 0.2048 0.2022 0.1997 0.1986 0.1972 0.1967 0.1964 0.1955 0.1952 0.1952 0.1950
28 0.3058 0.2783 0.2434 0.2203 0.2158 0.2115 0.2055 0.2012 0.1989 0.1966 0.1955 0.1941 0.1934 0.1930 0.1924 0.1925 0.1921 0.1917
29 0.3027 0.2749 0.2404 0.2166 0.2125 0.2082 0.2021 0.1976 0.1955 0.1931 0.1923 0.1909 0.1904 0.1899 0.1892 0.1887 0.1889 0.1886
30 0.3000 0.2723 0.2374 0.2132 0.2092 0.2047 0.1987 0.1946 0.1926 0.1902 0.1890 0.1878 0.1870 0.1865 0.1862 0.1860 0.1854 0.1856
35 0.2878 0.2597 0.2242 0.1984 0.1941 0.1901 0.1847 0.1812 0.1788 0.1769 0.1757 0.1742 0.1741 0.1737 0.1730 0.1728 0.1724 0.1724
40 0.2780 0.2495 0.2128 0.1865 0.1822 0.1782 0.1733 0.1699 0.1682 0.1661 0.1649 0.1638 0.1635 0.1628 0.1622 0.1620 0.1620 0.1620
45 0.2695 0.2408 0.2041 0.1765 0.1721 0.1688 0.1637 0.1605 0.1584 0.1570 0.1559 0.1547 0.1542 0.1541 0.1536 0.1533 0.1531 0.1529
50 0.2626 0.2332 0.1964 0.1683 0.1641 0.1604 0.1557 0.1528 0.1511 0.1490 0.1483 0.1471 0.1470 0.1463 0.1460 0.1456 0.1455 0.1455
60 0.2509 0.2213 0.1840 0.1544 0.1501 0.1466 0.1425 0.1399 0.1380 0.1364 0.1357 0.1349 0.1343 0.1340 0.1336 0.1333 0.1333 0.1331
70 0.2416 0.2118 0.1743 0.1435 0.1395 0.1362 0.1322 0.1298 0.1281 0.1268 0.1259 0.1250 0.1248 0.1244 0.1240 0.1238 0.1236 0.1235
80 0.2343 0.2043 0.1662 0.1350 0.1303 0.1276 0.1240 0.1216 0.1203 0.1189 0.1180 0.1172 0.1167 0.1166 0.1162 0.1161 0.1157 0.1158
90 0.2277 0.1978 0.1594 0.1278 0.1232 0.1207 0.1170 0.1148 0.1135 0.1122 0.1114 0.1107 0.1102 0.1101 0.1098 0.1094 0.1096 0.1093
100 0.2228 0.1923 0.1541 0.1216 0.1169 0.1143 0.1112 0.1090 0.1078 0.1065 0.1058 0.1052 0.1049 0.1046 0.1043 0.1041 0.1038 0.1039
200 0.1938 0.1628 0.1235 0.0888 0.0831 0.0815 0.0791 0.0776 0.0767 0.0758 0.0753 0.0748 0.0746 0.0744 0.0741 0.0740 0.0739 0.0739
300 0.1808 0.1496 0.1101 0.0742 0.0680 0.0667 0.0648 0.0635 0.0628 0.0621 0.0616 0.0612 0.0611 0.0609 0.0607 0.0606 0.0604 0.0606
400 0.1731 0.1418 0.1019 0.0657 0.0591 0.0579 0.0562 0.0551 0.0545 0.0537 0.0534 0.0531 0.0529 0.0528 0.0526 0.0526 0.0525 0.0525
500 0.1680 0.1365 0.0963 0.0598 0.0529 0.0517 0.0503 0.0493 0.0487 0.0482 0.0478 0.0476 0.0474 0.0473 0.0471 0.0470 0.0470 0.0471
1000 0.1549 0.1234 0.0827 0.0452 0.0375 0.0367 0.0356 0.0349 0.0345 0.0341 0.0340 0.0337 0.0336 0.0336 0.0333 0.0333 0.0333 0.0333
2500 0.1436 0.1118 0.0708 0.0325 0.0238 0.0233 0.0226 0.0221 0.0219 0.0216 0.0215 0.0213 0.0213 0.0213 0.0211 0.0211 0.0211 0.0211
B-12D
-------
APPENDIX C
GRAPHS
OF
COVERAGE COMPARISONS
FOR THE VARIOUS METHODS
FOR
NORMAL, GAMMA, AND LOGNORMAL
DISTRIBUTIONS
-------
Figure 1. Graphs of Coverage Probabilities by 95% UCLs of the Mean of N(|i=50,a=20)
96 -r - - -- - - -- - - -- - - -- - - -- - - -- - - -- - -
o
D
£ 94 H
en
01
ta
92
n>
o_
o>
a)
a
90 -
O
O
88
1D
20
4D 50 BO
Sample Size
70
Student's-t
Modified-t
Bootstrap-t
Hall's Bootstrap
Bootstrap BCA
80
90
100
Figure 2. Graphs of Coverage Probabilities by 95% UCLs of Mean of G(k=0.05,0=50)
tn
_j
O
O)
TO
^-f
0>
a.
a>
CD
a
o
O
100 -[-
94 -
Be
82
7B
70 -
64 -
5B -
52
- Max Test
- Modified-t
- 95% Chebyshev
- Bootstrap-t
- Bootstrap BCA
-Approximate Gamma
-Adjusted Gamma
10 20 30 40 50 BO
Sample Size
70
80
90
100
C-1
-------
Figure 3. Graphs of Coverage Probabilities by 95% UCLs of the Mean of G(k=0.10,0=50)
co 9B
_i
O
^ 9D
o>
O)
73
Q_
0>
CT
CO
O
O
72 -
54
Max Test
- Modified-t
- 95% Chebyshev
- Bootstrap-t
- Bootstrap BCA
-Approximate Gamma
-Adjusted Gamma
10 20 3D 40 50 60
Sample Size
70
80
90
100
Figure 4. Graphs of Coverage Probabilities by 95% UCLs of Mean of G(k=0.15,0=50)
100 -""" '" """"" '" """"" '" " "
o
D
10
OT
95 -
90 -
85 -
O)
co
c GD H
n>
u
a! 75 -
O)
fe 70 H
o
o
65 -
60
- Max Test
- Modified-t
- 95% Chebyshev
- Bootstrap-t
- Bootstrap BCA
-Approximate Gamma
- Adjusted Gamma
10 2Q 30 40 5Q BD
Sample Size
70
80
BO
100
C-2
-------
Graphs of Coverage Probabilities by 95% UCLs of the Mean of G(k=0.20,0=50)
- Max Test
- Modified-t
- 95% Chebyshev
- Bootstrap-!
- Bootstrap BCA
-Approximate Gamma
- Adjusted Gamma
10 20 30 40 50 60
Sample Size
70
80
90
100
Figure 6. Graphs of Coverage Probabilities by 95% UCLs of Mean of G(k=0.50,0=50)
CO
_l
O
as
in
o>
ni
n)
n>
01
CO
O
90 -
B6 -
B2 -
7B -
74
Max Test
- Modified-t
- 95% Chebyshev
- Bootstrap-t
- Bootstrap BCA
- Approximate Gamma
- Adjusted Gamma
0 10 20
30 40 50 60
Sample Size
70
80
90
100
C-3
-------
Figure 7. Graphs of Coverage Probabilities by 95% UCLs of the Mean of G(k= 1.00,0=50)
M
_i
O
D
£ 04
go -
Hi
O)
c
o>
u
0>
O)
TO
0>
o B2 H
O
78
- Max Test
- Modified-!
-95% Chebyshev
- Boatstrap-t
- Bootstrap BCA
-Approximate Gamma
- Adjusted Gamma
10 20 30 40 50 60
Sample Size
70
80
90
100
Figure 8. Graphs of Coverage Probabilities by 95% UCLs of the Mean of G(k=2.00,0=50)
in
O 98
D
£
o>
D)
E
o
o
90 -
ae -
- Max Test
- Modified-t
- 95% Chebyshev
- Bootstrap-t
- Bootstrap BCA
-Approximate Gamma
- Adjusted Gamma
10 20 30 40 50 BO
Sample Size
70
80
90
100
C-4
-------
Graphs of Coverage Probabilities by 95% UCLs of the Mean of G(k=5.00,0=50)
_ -3£ £ J£~
&C
Max Test
Modifiecl-t
95% Chebyshev
Bootstrap-t
Bootstrap BCA
Approximate Gamma
Adjusted Gamma
10 20 30 40 50 BO
Sample Size
70
BO
90
100
Figure 10. Graphs of Coverage Probabilities by UCLs of the Mean of LN(|j,=5,a=0.5)
100 -
_
o
95 -
O>
a
90 -
a>
O)
u>
o 65
GO
Max Test
Modified-t
95% MVUE Chebyshev
Hall's Bootstrap
Bootstrap BCA
H-Statistic UCL
10 20 30 40 50 60
Sample Size
70
80
90
100
C-5
-------
Figure 11. Graphs of Coverage Probabilities by UCLs of the Mean of LN(|i=5,a=1.0)
1DO -
V)
0 95
a>
0)
to
CL
n>
o
O
65 -
80 -
75 -
70
Max Test
Madified-t
95% MVUE Chebyshev
97.5%MVUEChebyshev
- Hall's Bootstrap
Bootstrap BCA
- H-Statistic UCL
10 20 30 40 50 60
Sample Size
70
eo
90
100
Figure 12. Graphs of Coverage Probabilities by UCLs of the Mean of LN(|a=5,a=1.5)
ta
_i
O
a>
O)
nt
4-f
a>
a>
CL
o
O
1DD -
94 -
BB -
B2 -
76 -
7D -
64 -
56
- Max Test
- 95% MVUE Chebyshev
-97.5% MVUE Chebyshev
99% MVUE Chebyshev
Hall's Bootstrap
Bootstrap BCA
- H-Statistic UCL
0 10 20 30 40 50 60
Sample Size
70
80
90
100
C-6
-------
Figure 13. Graphs of Coverage Probabilities by UCLs of the Mean of LN(|j,=5,a=2.0)
100 -
95 -
at
0 90 -
=>
> B5 -
S> ao -
£ 75 -
o
I 70 -
n>
gi 05 -
O
O
60 -
55 -
50 -
45
Max Test
- 95% MVUE Chebshev
-97.5%MVUEChebshev
99% MVUE Chebyshev
- Hall's Bootstrap
- Bootstrap BCA
- H-Stall Stic UCL
0 10 20 30 40 50 BB
Sample Size
70
BO
90
1QO
Figure 14. Graphs of Coverage Probabilities by UCLs of the Mean of LN(|a=5,a=2.5)
100 H
O
90 -
60 -
o>
I 70
a>
BO -
£ 50 -i
o
O
40 -
30
- Max Test
- 95% MVUE Chebyshev
-97.5% MVUE Chebyshev
- 99% MVUE Chebyshev
- Hall's Bootstrap
- Bootstrap BCA
- H-Statistic UCL
10 20 30 40 50 60
Sample Size
70
80
90
100
C-7
-------
Figure 15. Graphs of Coverage Probabilities by UCLs of the Mean of LN(|i=5,a=3.0)
v>
_i
O
a>
aa
a>
Z
a>
CL
a
(0
a>
o
O
100 -
BD -
BO -
70 -
BD -
50 -
40 -
30 -
2D -
Max Test
95% MVUE Chebyshev
97.5%tvlVUEChebyshev
99% MVUE Chebyshev
Hall's Bootstrap
Bootstrap BCA
H-Statistic UCL
D 10 2D 30 40 50 BO
Sample Size
70
eo
BO
1DO
C-8
------- |