ProUCL Version 4.1

User Guide

(Draft)

Statistical Software for Environmental 

Applications for Data Sets with and 

without Nondetect Observations

R E S E A R C H A N D D E V E L O P M E N T
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Notice: Although this work was reviewed by EPA and approved for publication, it may not necessarily reflect official 

Agency policy. Mention of trade names and commercial products does not constitute endorsement or 

recommendation for use.

129cmb07

ProUCL Version 4.1

User Guide

(Draft)

Statistical Software for Environmental 

Applications for Data Sets with and 

without Nondetect Observations

Prepared for:

Felicia Barnett

Technology Support Center

U.S. Environmental Protection Agency

Region 4

61 Forsyth Street SW,

Atlanta, GA 30303

Prepared by:

Anita Singh, Ph.D.

Robert Maichle

Narain Armbya

Lockheed Martin Environmental Services

1050 E. Flamingo Road, Suite N240

Las Vegas, NV 89119

EPA/600/R-07/041

May 2010

www.epa.gov

U.S. Environmental Protection Agency

Office of Research and Development

Washington, DC 20460
ii 
iii

Notice

The United States Environmental Protection Agency (EPA) through its Office of Research and 

Development (ORD) funded and managed the research described here. It has been peer reviewed by the 

EPA and approved for publication. Mention of trade names and 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 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.
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v

Changes from ProUCL 4.0 (Version 4.00.00) to ProUCL 4.00.002

http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm

Although extensive changes were made in the code for ProUCL 4.0 (version 4.00.00) to produce ProUCL 

4.00.02, those changes are transparent to the users. Most of those changes were made so that

ProUCL 4.00.02 is compatible with our developing statistical software, Scout (e.g., both programs share 

the same statistical libraries). ProUCL will also reside as a separate module in Scout as a research tool.

There is a very minor correction of a displayed value in one of the hypothesis tests, the two sample t-test. 

The p-value associated with the t-test was computed in two different ways: one way is correct and the 

other way, although it produced subtle differences, is incorrect. The incorrect method has been removed 

from ProUCL 4.00.02.

Several extra warning messages have been added to ProUCL 4.00.02, mainly in regard to attempting tests 

when a data set is very small (n < 5), when the number of detected values is small (e.g., only zero, one, or 

two), or when all of the values are non-detected values. For an example, some screens depicting those 

warning messages are included in the newly added Section 2.11 (page 40) of this ProUCL 4.00.02 User 

Guide.

The only software files that were changed from ProUCL version 4.0 (4.00.00) to version 4.0.02 were 

updates in the ProUCL.exe file, and updates to the StatsLib.dll file to produce a more advanced 

ScoutLib.dll file. Very minor changes were made to this ProUCL 4.00.02 User Guide, including: changes 

to avoid inappropriate user inputs (warnings), changes to the title page, the inclusion of an 

acknowledgement page, and the inclusion of a contact information page.

Changes from ProUCL 4.00.02 to ProUCL 4.00.04

http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm

ProUCL 4.00.04 is an upgrade of ProUCL Version 4.00.02 which represents an upgrade of ProUCL 4.0 

(EPA, 2004). ProUCL 4.00.04 contains all statistical methods as available in ProUCL 4.00.02 to address 

various environmental issues for both full data sets without nondetects (NDs) and for data sets with NDs 

(also known as left-censored data sets). In addition to having all methods available in ProUCL 4.00.02, 

ProUCL 4.00.04 has extended version of Shapiro-Wilk (S-W) test that can perform normal and lognormal 

goodness-of-fit tests for data sets of sizes upto 2000. Moreover, ProUCL 4.00.04 can compute upper 

prediction and upper tolerance limits based upon gamma distribution. Some modifications have also been 

made in decision tables and recommendations made by ProUCL to estimate the EPC terms. Specifically, 

based upon recent experience, developers of ProUCL are re-iterating that the use of lognormal 

distribution to estimate EPC terms should be avoided, as the use of lognormal distribution yields 

unrealistic and highly unstable UCLs. In an effort to simplify the EPC estimation process, for highly 

skewed lognormally distributed data sets, developers are recommending the use of appropriate 

nonparametric Chebyshev (mean Sd) UCLs. These changes have been incorporated in various decision 

tables included in ProUCL 4.00.04 Technical Guide and ProUCL 4.00.04 User Guide. Recommendations 

made by ProUCL 4.00.02 have been changed accordingly in ProUCL 4.00.04. Some minor bugs as 

suggested by ProUCL 4.0 and ProUCL 4.00.02 users have also been addressed in this upgraded version of 

ProUCL software package.
vi 

Changes from ProUCL 4.00.04 to ProUCL 4.00.05

http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm

ProUCL version 4.00.05 is an upgrade of ProUCL Version 4.00.04 (EPA, 2008). ProUCL 4.00.05 

consists of all of the statistical and graphical methods that are available in previous ProUCL 4.0 versions 

to address various environmental issues for full data sets without nondetect (ND) observations and also 

for data sets with NDs and below detection limit observations. Several additions (e.g., sample size 

determination module), enhancements (File module), modifications (e.g., p-values of WRS/WMW test, 

Gehan test) have been made in ProUCL 4.00.05. Some bugs (e.g., correction in adjusted Gamma UCLs) 

as suggested and found by users of previous ProUCL 4.0 versions have been addressed in this version of 

ProUCL. With the inclusion of the sample size determination module, ProUCL 4.00.05 will serve as a 

comprehensive statistical software package equipped with statistical methods and graphical tools needed 

to address environmental sampling and statistical issues described in various CERCLA (EPA 2002a, 

2002b, 2006) and RCRA (EPA 1989b, 1992b, 2002c, 2009) guidance documents. For data sets with and 

without nondetect observations, ProUCL 4.00.05 also provides statistical methods to address reference 

area and survey unit sampling issues described in MARSSIM (EPA 2000) document. In addition to 

sample size determination methods, ProUCL 4.00.05 offers parametric and nonparametric statistical 

methods (e.g., Sign test, Wilcoxon Rank Sum test) often used to address statistical issues described in 

MARSSIM (EPA 2000) guidance document. The user friendly sample size determination module of 

ProUCL 4.00.05 has a straight forward mechanism to enter the desired/pre-specified decision parameters 

needed to compute appropriate sample size(s) for the selected statistical application. The Sample Size 

module of ProUCL 4.00.05 provides sample size determination methods for most of the parametric and 

nonparametric one-sided and two-sided hypotheses testing approaches available in the Hypothesis Testing 

module of ProUCL 4.0. Theoretical details of the Sample Size module are given in “Supplement to 

ProUCL 4.0 Technical Guide: Determining Minimum Sample Sizes for User Specified Decision 

Parameters.” Some specific changes made in ProUCL 4.00.05 are listed as follows.

 File Option: This option has been upgraded to open *.xls files by default. In the earlier versions 

of ProUCL, this option was available only for *.wst, *.ost and *.gst files and Excel files had to be 

imported. Now you can use the import option to read multiple worksheets from one Excel file. 

ProUCL 4.00.05 will import worksheets until all worksheets are read or a blank or empty 

worksheet is encountered. 

 Displaying All Menu Options: ProUCL 4.00.05 now displays all available menu options even 

before opening a valid (e.g., non empty) data file. However, the user has to open a valid data file 

before activating a menu option and using a statistical or graphical method available in ProUCL 

4.00.05. Obviously, no statistical method can be used on an empty (without any data) 

spreadsheet.

 Sample Size Module: Several parametric (assuming normal distribution) and nonparametric 

sample size determination formulae as used and described in various EPA guidance documents 

(e.g., EPA 1989a, 1989b, 1992, 2000, 2002a, 2002b, 2002c, 2006, and 2009) have been 

incorporated in ProUCL 4.00.05. Inclusion of this module will help the users to develop/design 

DQOs based sampling plans with pre-specified values of decision error rates (α = Type I and β 

=Type II) and width of the gray region, Δ around the parameter of interest (e.g., mean 

concentration, proportion of sampled observations exceeding the action level). Basic sample size 

determination formulae have been incorporated for sampling of continuous characteristics (lead, 

Ra 226) as well as for attributes (e.g., proportion exceeding a specified threshold). Additionally, 
vii

sample size formulae for acceptance sampling of discrete objects (e.g., drums) have also been 

incorporated in this module. The detailed description of the statistical methods and formulae used 

in this module are described in the supplement al document (Appendix B) for ProUCL 4.0 

Technical Guide.

 Adjusted Gamma UCL: There was a minor bug in the computation of adjusted α level of 

significance (called β level) used for calculating Adjusted Gamma UCLs. This error has been 

corrected.

 Computation of Nonparametric Percentiles: There are several ways to compute nonparametric 

percentiles; and percentiles obtained using different methods can differ slightly. For graphical 

displays, ProUCL 4.00.05 uses development software, ChartFX. Thus boxplots generated by 

ProUCL display percentiles (e.g., median and quartiles) as computed by ChartFX. In order to 

avoid confusion, the percentile algorithm used in ProUCL has been modified so that it now 

computes and displays comparable percentiles as computed by ChartFX.

 UCL based upon Winsorization Method: In the computation of Winsorized UCLs, the sample

standard deviation of the winsorized data was being used instead of the approximate unbiased 

estimate of the population standard deviation from the winsorized data as detailed in ProUCL 

4.00.04 Technical Guide. This has been corrected in ProUCL 4.00.05.

 Displaying K values used in UTLs: The tolerance factor, K, based on the number of valid 

observations, level of confidence coefficient, and coverage percentage is displayed along with 

UTL statistics and other relevant input parameters.

 Fixes in the p- values associated with WSR/WMW Test, Gehan Test, and Equality of Variances 

Test: More efficient algorithms have been incorporated in ProUCL 4.00.05 to compute p-values 

associated with the test statistics associated with these two tests.

 Additional Critical values associated with Land's H Statistic: In addition to 0.9 and 0.95 

confidence coefficients; 0.975, 0.99 and 0.995 confidence levels have been incorporated in 

ProUCL 4.00.05 to compute H-UCLs based upon a lognormal distribution. 

 Adjustment in Precision Associated with Lognormal and Gamma ROS Methods: The lower 

bound associated with Lognormal ROS and Gamma ROS extrapolated estimates have been 

extended from 1e-7 to 1e-10. ProUCL 4.00.05 issues a warning message when extrapolated ROS 

estimates lie below 1e-10. 

 Some terminology changes have been made in single sample hypotheses approaches available in 

the Hypothesis Testing module. Specifically the phrase “Compliance Limit” have been replaced 

by the phrase “Action Level”.
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Changes and Upgrades from ProUCL 4.00.05 to ProUCL 4.1.00

http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm

ProUCL version 4.1.00, a statistical software package for environmental applications for data sets with 

and without nondetect (ND) observations is an upgrade of ProUCL version 4.00.05 (EPA 

2010a). ProUCL 4.1 consists of all of the statistical and graphical methods that are available in all 

previous versions of ProUCL software package to address various environmental issues for full 

uncensored data sets (without ND observations), as well as for data sets with NDs or below detection 

limit observations. ProUCL version 4.1.00, its earlier versions (ProUCL version 3.00.02, 4.00.02, 4.00.04, 

and 4.00.05), associated Facts Sheet, User Guides and Technical Guides (e.g., EPA 2010b, 2010c) can be 

downloaded from the EPA website: http://www.epa.gov/osp/hstl/tsc/software.htm

New Modules in ProUCL 4.1 (ANOVA and Trend Tests): Two new modules, ANOVA and Trend 

Tests have been incorporated in ProUCL 4.1. ANOVA module has both classical and nonparametric 

Kruskal-Wallis Oneway ANOVA tests as described in EPA guidance documents (e.g., EPA 2006, 2009). 

Trend Tests module has linear ordinary least squares (OLS) regression method, Mann-Kendall trend test, 

Theil-Sen trend test, and time series plots as described in the Unified RCRA Guidance Document (EPA 

2009). Oneway ANOVA is used to compare means (or medians) of multiple groups such as comparing 

mean concentrations of several areas of concern; and to perform inter-well comparisons. In groundwater 

(GW) monitoring applications, OLS regression, trend tests, and time series plots (EPA, 2009) are often 

used to identify trends (e.g., upwards, downwards) in contaminant concentrations of GW monitoring 

wells over a certain period of time. 

The Number of Samples module of ProUCL 4.1 provides user friendly options to enter the desired/prespecified decision parameters (e.g., Type I and Type II error rates) and DQOs used to determine 

minimum sample sizes for the selected statistical applications including: estimation of mean, single and 

two sample hypothesis testing approaches, and acceptance sampling. Sample size determination methods 

are available for the sampling of continuous characteristics (e.g., lead or Ra 226), as well as for attributes 

(e.g., proportion of occurrences exceeding a specified threshold). Both parametric (e.g., for t-tests) and 

nonparametric (e.g., Sign test, test for proportions, WRS test) sample size determination methods as 

described in EPA (2006, 2009) and MARSIMM (2000) guidance documents are available in ProUCL 4.1. 

ProUCL 4.1 also has the sample size determination methods for acceptance sampling of lots of discrete 

objects such as a lot of drums consisting of hazardous waste (e.g., RCRA applications, EPA 2002c). 

ProUCL version 4.1 can process multiple contaminants (variables) simultaneously. ProUCL version 4.1 

also has the capability of processing data by groups (a valid group column should be included in the data 

file). ProUCL version 4.1 has a couple of simple outlier test procedures, such as the Dixon test and the 

Rosner test. ProUCL version 4.1 offers useful graphical displays for data sets with or without NDs, 

including: histograms, multiple quantile-quantile (Q-Q) plots, and side-by-side box plots. The use of 

graphical displays provides additional insight about information (such as hidden data structures) 

contained in data sets that may not be revealed by the use of estimates (e.g., 95% upper limits) or test 

statistics, such as the GOF test statistics or the t-test statistic. In addition to providing information about 

the data distributions (e.g., normal or gamma), Q-Q plots are also useful in identifying potential outliers 

or the presence of mixture samples (e.g., data from different populations) in a data set. Side-by-side box 

plots and multiple Q-Q plots are useful to visually compare two or more data sets (groups), such as: siteversus-background contaminant concentrations, surface-versus-subsurface concentrations, and 

contaminant concentrations of groundwater monitoring wells (MWs).
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As in earlier ProUCL versions, in addition to goodness-of-fit (GOF) tests for normal, lognormal and 

gamma distributions, ProUCL 4.1 has parametric and nonparametric methods including bootstrap 

methods to compute various decision making statistics such as the upper confidence limits (UCLs) of 

mean (EPA 2002a), percentiles, upper prediction limits (UPLs) for future k (≥1) observations, and upper 

tolerance limits (UTLs) (e.g., EPA 1992, EPA 2009) based upon uncensored full data sets and leftcensored data sets consisting of NDs with multiple detection limits. In addition to simple substitution 

methods (e.g., DL/2 DL), Kaplan-Meier (KM) method and Regression on Order Statistics (ROS) methods 

are also available in ProUCL. ProUCL 4.1 can also compute parametric UCLs, percentiles, UPLs for 

future k (≥1) observations, and UTLs based upon gamma distributed data sets. 

ProUCL version 4.1 has parametric and nonparametric single-sample and two-sample hypotheses testing 

approaches. Single-sample hypotheses tests (e.g., Student‟s t-test, the sign test, the Wilcoxon signed rank 

test, and the proportion test) can be used to compare site mean concentrations (or some site threshold 

value such as an upper percentile) with some average cleanup standard, Cs (or a not-to-exceed compliance 

limit, A0) to verify the attainment of cleanup levels (EPA 1989, EPA 2006) after some remediation 

activities have been performed at the impacted site areas. Several two-sample hypotheses tests as 

described in EPA and MARSSIM guidance documents (e.g., EPA 2000, 2002b, and 2006) are also 

available in ProUCL 4.1. Two sample hypotheses testing approaches in ProUCL 4.1 include: Student‟s ttest, the Wilcoxon-Mann-Whitney (WMW) test (also known as Wilcoxon Rank Sum (WRS) test), the 

quantile test, and Gehan‟s test. These tests are used for site-versus-background comparisons and 

comparisons of contaminant concentrations of two or more monitoring wells (MWs). The hypothesis 

testing approaches in ProUCL 4.1 can be used on both uncensored (without NDs) and left-censored (with 

NDs) data sets. Single sample tests (e.g., Sign test, proportion test) and upper limits such as UTLs and 

UPLs are also used to perform intra-well comparisons as described in RCRA document (EPA, 2009). 

With the inclusion of Oneway ANOVA, Regression and Trend tests, and the user-friendly DQOs based 

sample size determination modules, ProUCL version 4.1.00 represents a comprehensive statistical 

software package equipped with statistical methods and graphical tools needed to address many 

environmental sampling and statistical issues as described in various CERCLA (EPA 1989a, 2002a, 

2002b, 2006), MARSSIM (EPA 2000), and RCRA (EPA 1989b, 1992b, 2002c, 2009) guidance 

documents.

Finally, it should be noted that all known software bugs found by the various users and developers of 

ProUCL 4.00.05 (and earlier versions) and most of the suggestions made by the users have been 

addressed in ProUCL 4.1.00.

All previous and current versions of ProUCL software package can be downloaded from the following 

EPA site: http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm
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Contact Information for all Versions of ProUCL 

The ProUCL software is developed under the direction of the Technical Support Center 

(TSC). As of November 2007, the direction of the TSC is transferred from Brian Schumacher to 

Felicia Barnett. Therefore, any comments or questions concerning all versions of ProUCL 

should be addressed to:

Felicia Barnett, (HSTL) 

US EPA, Region 4

61 Forsyth Street, S.W. 

Atlanta, GA 30303-8960

barnett.felicia@epa.gov

(404) 562-8659

Fax: (404) 562-8439
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Executive Summary

Statistical inference, including both estimation and hypotheses testing approaches, is routinely used to: 

1. Estimate environmental parameters of interest, such as exposure point concentration 

(EPC) terms, not-to-exceed values, and background level threshold values (BTVs) for 

contaminants of potential concern (COPC), 

2. Identify areas of concern (AOC) at a contaminated site,

3. Compare contaminant concentrations found at two or more AOCs of a contaminated site, 

4. Compare contaminant concentrations found at an AOC with background or reference 

area contaminant concentrations, and

5. Compare site concentrations with a cleanup standard to verify the attainment of cleanup 

standards.

Several exposure and risk management and cleanup decisions in support of United States Environmental 

Protection Agency (EPA) projects are often made based upon the mean concentrations of the COPCs. A 

95% upper confidence limit (UCL95) of the unknown population (e.g., an AOC) arithmetic mean (AM), 

μ1, can be used to:

 Estimate the EPC term of the AOC under investigation, 

 Determine the attainment of cleanup standards, 

 Compare site mean concentrations with reference area mean concentrations, and

 Estimate background level mean contaminant concentrations. The background mean 

contaminant concentration level may be used to compare the mean of an area of concern. 

It should be noted that it is not appropriate to compare individual point-by-point site 

observations with the background mean concentration level.

It is important to compute a reliable and stable UCL95 of the population mean using the available data. 

The UCL95 should approximately provide the 95% coverage for the unknown population mean, μ1. Based 

upon the available background data, it is equally important to compute reliable and stable upper 

percentiles, upper prediction limits (UPLs), or upper tolerance limits (UTLs). These upper limits based 

upon background (or reference) data are used as estimates of BTVs, compliance limits (CL), or not-toexceed values. These upper limits are often used in site (point-by-point) versus background comparison 

evaluations. 

Environmental scientists often encounter trace level concentrations of COPCs when evaluating sample 

analytical results. Those low level analytical results cannot be measured accurately and, therefore, are 

typically reported as less than one or more detection limit (DL) values (also called nondetects). However, 

practitioners need to obtain reliable estimates of the population mean, μ1, and the population standard 

deviation, σ1, and upper limits including the UCL of the population mass or mean, the UPL, and the UTL

based upon data sets with nondetect (ND) observations. Additionally, they may have to use hypotheses 

testing approaches to verify the attainment of cleanup standards, and compare site and background 

concentrations of COPCs as mentioned above. 

Background evaluation studies, BTVs, and not-to-exceed values should be estimated based upon 

defensible background data sets. The estimated BTVs or not-to-exceed values are then used to identify the 

COPCs, to identify the site AOCs or hot spots, and to compare the contaminant concentrations at a site 
xii 

with background concentrations. The use of appropriate statistical methods and limits for site versus 

background comparisons is based upon the following factors: 

o Objective of the study,

o Environmental medium (e.g., soil, groundwater, sediment, air) of concern,

o Quantity and quality of the available data,

o Estimation of a not-to-exceed value or of a mean contaminant concentration,

o Pre-established or unknown cleanup standards and BTVs, and

6. Sampling distributions (parametric or nonparametric) of the concentration data sets 

collected from the site and background areas under investigation. 

In background versus site comparison evaluations, the environmental population parameters of interest 

may include: 

 Preliminary remediation goals (PRGs),

 Soil screening levels (SSLs),

 RBC standards,

 BTVs, not-to-exceed values, and 

 Compliance limit, maximum concentration limit (MCL), or alternative concentration 

limit (ACL), frequently used in groundwater applications.

When the environmental parameters listed above are not known or pre-established, appropriate upper 

statistical limits are used to estimate those parameters. The UPL, UTL, and upper percentiles are used to 

estimate the BTVs and not-to-exceed values. Depending upon the site data availability, point-by-point site 

observations are compared with the estimated (or pre-established) BTVs and not-to-exceed values. If 

enough site and background data are available, two-sample hypotheses testing approaches are used to 

compare site concentrations with background concentrations levels. These statistical methods can also be 

used to compare contaminant concentrations of two site AOCs, surface and subsurface contaminant 

concentrations, or upgradient versus monitoring well contaminant concentrations. 

ProUCL 4.00.05 is an upgrade of ProUCL Version 4.00.02 which represents an upgrade of ProUCL 4.0 

(EPA, 2007). ProUCL 4.00.05 contains all statistical methods as available in ProUCL 4.00.02 to address 

various environmental issues for both full data sets without nondetects (NDs) and for data sets with NDs 

(also known as left-censored data sets). In addition to having all methods available in ProUCL 4.0 as 

described below, ProUCL 4.00.05 has extended version of Shapiro-Wilk (S-W) test that can perform 

normal and lognormal goodness-of-fit tests for data sets of sizes upto 2000. Moreover, ProUCL 4.00.05

can compute upper prediction and upper tolerance limits based upon gamma distribution. Some 

modifications have also been made in decision tables and recommendations made by ProUCL to estimate 

the EPC terms. Specifically, based upon recent experience, developers of ProUCL are re-iterating that the 

use of lognormal distribution to estimate EPC terms should be avoided, as the use of lognormal 

distribution yields unrealistic and highly unstable UCLs. In an effort to simplify the EPC estimation 

process, for highly skewed lognormally distributed data sets, developers are recommending the use of 

appropriate nonparametric Chebyshev (mean Sd) UCLs. These changes have been incorporated in 

various decision tables included in ProUCL 4.0 Technical Guide and ProUCL 4.0 User Guide. 

Recommendations made by ProUCL 4.00.02 have been changed accordingly in ProUCL 4.00.05. Some 

minor bugs as suggested by ProUCL 4.0 users have also been addressed in this upgraded version of 

ProUCL software package.
xiii

ProUCL 4.0/ProUCL 4.00.02 contains:

o Rigorous parametric and nonparametric (including bootstrap methods) statistical methods 

(instead of simple ad hoc or substitution methods) that can be used on full data sets 

without nondetects and on data sets with below detection limit (BDL) or ND 

observations. 

2. State-of-the-art parametric and nonparametric UCL, UPL, and UTL computation 

methods. These methods can be used on full-uncensored data sets without nondetects and 

also on data sets with BDL observations. Some of the methods (e.g., Kaplan-Meier 

method, ROS methods) are applicable on left-censored data sets having multiple 

detection limits. The UCL and other upper limit computation methods cover a wide range 

of skewed data sets with and without the BDLs. 

3. Single sample (e.g., Student‟s t-test, sign test, Proportion test, Wilcoxon Singed Rank 

test) and two-sample (Student‟s t-test, Wilcoxon-Mann-Whitney test, Gehan test, quantile 

test) parametric and nonparametric hypotheses testing approaches for data sets with and 

without ND observations. These hypothesis testing approaches can be used to: verify the 

attainment of cleanup standards, perform site versus background comparisons, and 

compare two or more AOCs, monitoring wells (MWs). 

4. The single sample hypotheses testing approaches are used to compare site mean, site 

median, site proportion, or a site percentile (e.g., 95th) to a compliance limit (action level, 

regularity limit). The hypotheses testing approaches can handle both full-uncensored data 

sets without nondetects, and left-censored data sets with nondetects. Simple two-sample

hypotheses testing methods to compare two populations are available in ProUCL 4.0, 

such as two-sample t-tests, Wilcoxon-Mann-Whitney (WMW) Rank Sum test, quantile 

test, Gehan‟s test, and dispersion test. Variations of hypothesis testing methods (e.g., 

Levene‟s method to compare dispersions, generalized WRS test) are easily available in 

most commercial and freely available software packages (e.g., MINITAB, R). 

5. ProUCL 4.0 also includes graphical methods (e.g., box plots, multiple Q-Q plots, 

histogram) to compare two or more populations. ProUCL 4.0 can also be used to display 

a box plot of one population (e.g., site data) with compliance limits or upper limits (e.g., 

UPL) of other population (background area) superimposed on the same graph. This kind 

of graph provides a useful visual comparison of site data with a compliance limit or 

BTVs. Graphical displays of a data set (e.g., Q-Q plot) should be used to gain insight 

knowledge contained in a data set that may not otherwise be clear by looking at simple 

test statistics such as t-test, Dixon test statistic, or Shapiro-Wilk (S-W) test statistic.

6. ProUCL 4.0 can process multiple contaminants (variables) simultaneously and has the 

capability of processing data by groups. A valid group column should be included in the 

data file.

7. ProUCL 4.0 provides GOF test for data sets with nondetects. The user can create 

additional columns to store extrapolated (estimated) values for nondetects based upon 

normal ROS, gamma ROS, and lognormal ROS (robust ROS) methods.
xiv 

ProUCL 4.0/ProUCL 4.00.02 retains all of the capabilities of ProUCL 3.0, including goodness-of-fit 

(GOF) tests for a normal, lognormal, and a gamma distribution and computation of UCLs based upon full 

data sets without nondetects. Graphical displays and GOF tests for data sets with BDL observations have 

also been included in ProUCL 4.0. It is re-emphasized that the computation of appropriate UCLs, UPLs, 

and other limits is based upon the assumption that the data set under study represents a single a single 

population. This means that the data set used to compute the limits should represent a single statistical 

population. For example, a background data set should represent a defensible background data set free of 

outlying observations. ProUCL 4.0 includes simple and commonly used classical outlier identification 

procedures, such as the Dixon test and the Rosner test. These procedures are included as an aid to identify 

outliers. These simple classical outlier tests often suffer from masking effects in the presence of multiple 

outliers. Description and use of robust and resistant outlier procedures is beyond the scope of ProUCL 4.0

software. 

It is suggested that the classical outlier procedures should always be accompanied by graphical displays 

including box plots and Q-Q plots. The use of a Q-Q plot is useful to identify multiple or mixture samples 

that might be present in a data set. However, the decision regarding the proper disposition of outliers (e.g., 

to include or not to include outliers in statistical analyses; or to collect additional verification samples) 

should be made by members of the project team and experts familiar with site and background conditions. 

Guidance on the disposition of outliers and their accommodation in a data set by using a transformation 

(e.g., lognormal distribution) is discussed in Chapter 1 of this User Guide. 

ProUCL 4.0 has improved graphical methods, which may be used to compare the concentrations of two or 

more populations such as: 

o Site versus background populations, 

o Surface versus subsurface concentrations, 

o Concentrations of two or more AOCs, and

o Identification of mixture samples and/or potential outliers

These graphical methods include multiple quantile-quantile (Q-Q) plots, side-by-side box plots, and 

histograms. Whenever possible, it is desirable to supplement statistical results with useful visual displays 

of data sets. There is no substitute for graphical displays of a data set. For example, in addition to 

providing information about the data distribution, a normal Q-Q plot can also help identify outliers and 

multiple populations that may be present in a data set. On a Q-Q plot, observations well separated from 

the majority of the data may represent potential outliers, and jumps and breaks of significant magnitude 

may suggest the presence of observations from multiple populations in the data set. It is suggested that 

analytical outlier tests (e.g., Rosner test) and goodness-of-fit (G.O.F.) tests (e.g., SW test) should always 

be supplemented with the graphical displays such as Q-Q plot and box plot.

ProUCL 4.00.05 (and all its previous versions) serves as a companion software package for Calculating 

Upper Confidence Limits for Exposure Point Concentrations at Hazardous Waste Sites (EPA, 2002a) and 

Guidance for Comparing Background and Chemical Concentrations in Soil for CERCLA Sites (EPA, 

2002b). ProUCL 4.00.05 is also useful to verify the attainment of cleanup standards (EPA, 1989). 

ProUCL 4.00.05 can also be used to perform two-sample hypotheses tests and to compute various upper 

limits often needed in groundwater monitoring applications (EPA, 1992 and EPA, 2004).

ProUCL 4.1.00 has two new modules: ANOVA and Trend Tests. ANOVA module has both classical and 

nonparametric Kruskal-Wallis Oneway ANOVA tests as described in EPA guidance documents (e.g., 

EPA 2006, 2009). Trend Tests module has linear ordinary least squares (OLS) regression method, Mann-
xv

Kendall trend test, Theil-Sen trend test, and time series plots as described in the Unified RCRA Guidance 

Document (EPA 2009). Oneway ANOVA is used to compare means (or medians) of multiple groups such 

as comparing mean concentrations of several areas of concern; and to perform inter-well comparisons. In 

groundwater (GW) monitoring applications, OLS regression, trend tests, and time series plots (EPA, 

2009) are often used to identify trends (e.g., upwards, downwards) in contaminant concentrations of 

various GW monitoring wells over a certain period of time. 

With the inclusion of Oneway ANOVA, Regression and Trend tests, and user-friendly DQOs based 

sample size determination modules, ProUCL version 4.1.00 represents a comprehensive statistical 

software package equipped with statistical methods and graphical tools needed to address many 

environmental sampling and statistical issues as described in various CERCLA (EPA 1989a, 2002a, 

2002b, 2006), MARSSIM (2000) and RCRA (EPA 1989b, 1992b, 2002c, 2009) guidance documents.
xvi 

Acronyms and Abbreviations

% NDs Percentage of Nondetect observations

ACL Alternative concentration limit 

A-D, AD Anderson-Darling test

AM Arithmetic mean 

AOC area(s) of concern 

BC Box-Cox-type transformation

BCA bias-corrected accelerated bootstrap method 

BDL below detection limit

BTV background threshold value

BW Black and White (for printing)

CERCLA Comprehensive Environmental Response, Compensation, and 

Liability Act

CL compliance limit

CLT central limit theorem 

CMLE Cohen‟s maximum likelihood estimate

COPC contaminant(s) of potential concern 

CV Coefficient of Variation

DL detection limit 

DL/2 (t) UCL based upon DL/2 method using Student‟s t-distribution 

cutoff value

DL/2 Estimates estimates based upon data set with nondetects replaced by half 

of the respective detection limits

DQO data quality objective

EA exposure area

EDF empirical distribution function 

EM expectation maximization 

EPA Environmental Protection Agency 

EPC exposure point concentration
xvii

FP-ROS (Land) UCL based upon fully parametric ROS method using Land‟s Hstatistic 

Gamma ROS (Approx.) UCL based upon Gamma ROS method using the gamma 

approximate-UCL method

Gamma ROS (BCA) UCL based upon Gamma ROS method using the bias-corrected 

accelerated bootstrap method

GOF, G.O.F. goodness-of-fit 

H-UCL UCL based upon Land‟s H-statistic

ID identification code

IQR interquartile range 

K Next K, Other K, Future K

KM (%) UCL based upon Kaplan-Meier estimates using the percentile 

bootstrap method

KM (Chebyshev) UCL based upon Kaplan-Meier estimates using the Chebyshev 

inequality

KM (t) UCL based upon Kaplan-Meier estimates using the Student‟s tdistribution cutoff value

KM (z) UCL based upon Kaplan-Meier estimates using standard normal 

distribution cutoff value

K-M, KM Kaplan-Meier

K-S, KS Kolmogorov-Smirnov 

LN lognormal distribution

Log-ROS Estimates estimates based upon data set with extrapolated nondetect values 

obtained using robust ROS method

MAD Median Absolute Deviation

Maximum Maximum value

MCL maximum concentration limit 

Mean classical average value

Median Median value

Minimum Minimum value

MLE maximum likelihood estimate

MLE (t) UCL based upon maximum likelihood estimates using Student‟s 

t-distribution cutoff value
xviii 

MLE (Tiku) UCL based upon maximum likelihood estimates using the 

Tiku‟s method

Multi Q-Q multiple quantile-quantile plot

MVUE minimum variance unbiased estimate

ND nondetect or nondetects

NERL National Exposure Research Laboratory

NumNDs Number of Nondetects

NumObs Number of Observations

ORD Office of Research and Development

PRG preliminary remediation goals

Q-Q quantile-quantile 

RBC risk-based cleanup 

RCRA Resource Conservation and Recovery Act

ROS regression on order statistics

RU remediation unit

S substantial difference

SD, Sd, sd standard deviation

SSL soil screening levels

S-W, SW Shapiro-Wilk 

UCL upper confidence limit 

UCL95, 95% UCL 95% upper confidence limit

UPL upper prediction limit

UPL95, 95% UPL 95% upper prediction limit

USEPA United States Environmental Protection Agency 

UTL upper tolerance limit

Variance classical variance 

WMW Wilcoxon-Mann-Whitney 

WRS Wilcoxon Rank Sum 

WSR Wilcoxon Signed Rank
xix
xx 

Acknowledgements

We wish to express our gratitude and thanks to all of the many people that reviewed, tested, and 

gave some very helpful suggestions for the development of ProUCL 4.0. We wish to especially 

acknowledge: Nadine Adkins, Ray Bienert, Cheri L. Butler, Lucía Casabó, John P. Christopher,

Marion Edison, Evan Englund, Tim Ehli, Kuen Huang-Farmer, Dolores Gardner, Philip E. 

Goodrum, Scott A. Grubisich, Dennis R. Helsel, Jennifer Hubbard, Richard O. Gilbert, Sarah 

Levinson, Jayne Michaud, Jeff Myers, Yassine Nachti,Gareth Pearson, Marian Olsen, Kristina 

Rayner, Nancy Rios-Jafolla, Shahrokh Rouhani, Peter Rousseeuw, Dan Sahagun, Terry W. 

Schulz, Mike Schum, Tom Siard, Christopher Sibert, Bob Stewart, Rick Sugatt, J. Keith Tolson, 

Robert Tucker, John Warren, and Edward P. Wosika.
xxi
xxii 

Table of Contents

Notice............ ........................................................................................................................................iii

Changes from ProUCL 4.0 (4.00.00) to ProUCL 4.00.02.............................................................. v

Changes from ProUCL 4.00.02 to ProUCL 4.00.05....................................................................... v

Contact Information for ProUCL 4.00.02........................................................................................vi

Executive Summary............................................................................................................................vii

Acronyms and Abbreviations.........................................................................................................xvi

Acknowledgements.......................................................................................................................... xv

Introduction............................................................................................................................................ 1

The Need for ProUCL Software ................................................................................................... 1

ProUCL 4.0 Capabilities............................................................................................................... 2

ProUCL Applications.................................................................................................................... 4

ProUCL Methods.......................................................................................................................... 5

Background versus Site Comparison Evaluations........................................................................ 7

Graphical Capabilities................................................................................................................. 11

Technical Guide.......................................................................................................................... 13

Minimum Hardware Requirements............................................................................................. 13

Software Requirements............................................................................................................... 13

Installation Instructions............................................................................................................... 14

Getting Started ............................................................................................................................ 14

Chapter 1 Guidance on the Use of Statistical Methods and Associated Minimum 

Sample Size Requirements .................................................................................... 16

1.1 Background Data Sets.................................................................................................... 17

1.2 Site Data Sets................................................................................................................. 18

1.3 Discrete Samples or Composite Samples?..................................................................... 19

1.4 Upper Limits and Their Use .......................................................................................... 19

1.5 Point-by-Point Comparison of Site Observations with BTVs, Compliance Limits, 

and Other Threshold Values .......................................................................................... 22

1.6 Hypothesis Testing Approaches and Their Use............................................................. 24

1.6.1 Single Sample Hypotheses – BTVs and Not-to-Exceed Values are Known 

(Pre-established) ............................................................................................... 24

1.6.2 Two-sample Hypotheses – When BTVs and Not-to-Exceed 

Values are Unknown......................................................................................... 25

1.7 Minimum Sample Size Requirements ........................................................................... 26

1.7.1 Minimum Sample Size Requirements to Compute Decision Statistics

for Data without Non-detects............................................................................ 26

1.7.2 Minimum Sample Size Requirements for Datasets with Non-detects.............. 27

1.7.3 Minimum Sample Size for Estimation and Point-by-Point Site 

Observation Comparisons................................................................................. 27

1.7.4 Minimum Sample Size Requirements for Hypothesis Testing Approaches..... 28
xxiii

1.8 Sample Sizes for Bootstrap Methods............................................................................. 29

1.9 Statistical Analyses by a Group ID................................................................................ 29

1.10 Use of Maximum Detected Values as Estimates of Upper Limits................................. 30

1.10.1 Use of Maximum Detected Values to Estimate BTVs and 

Not-to-Exceed Values....................................................................................... 29

1.10.2 Use of Maximum Detected Values to Estimate EPC Terms ............................ 29

1.10.3 Samples with Nondetect Observations ............................................................. 31

1.10.4 Avoid the Use of DL/2 Method to Compute UCL95 ....................................... 31

1.10.5 Samples with Low Frequency of Detection...................................................... 31

1.11 Other Applications of Methods in ProUCL 4.0 ............................................................. 33

1.11.1 Identification of COPCs.................................................................................... 32

1.11.2 Identification of Non-Compliance Monitoring Wells....................................... 33

1.11.3 Verification of the Attainment of Cleanup Standards, Cs................................. 33

1.11.4 Using BTVs (Upper Limits) to identify Hot Spots........................................... 33

1.12 Some General Issues and Recommendations Made by ProUCL................................... 34

1.12.1 Multiple Detection Limits................................................................................. 34

1.12.2 Field Duplicates................................................................................................ 34

1.12.3 ProUCL Recommendations about ROS Method and

Substitution (DL/2) Method.............................................................................. 34

Chapter 2 Entering and Manipulating Data............................................................................ 36

2.1 Creating a New Data Set................................................................................................ 36

2.2 Opening an Existing Data Set........................................................................................ 36

2.3 Input File Format ........................................................................................................... 37

2.4 Number Precision .......................................................................................................... 37

2.5 Entering and Changing a Header Name......................................................................... 38

2.6 Saving Files.................................................................................................................... 39

2.7 Editing............................................................................................................................ 39

2.8 Handling Nondetect Observations................................................................................. 40

2.9 Caution........................................................................................................................... 41

2.10 Summary Statistics for Data Sets with Nondetect Observations................................... 42

2.11 Warning Messages and Recommendations for Datasets with Insufficient 

Amount of Data ............................................................................................................. 40

2.12 Handling Missing Values............................................................................................... 45

2.13 User Graphic Display Modification............................................................................... 47

2.13.1 Graphics Tool Bar............................................................................................. 44

2.13.2 Drop-Down Menu Graphics Tools................................................................... 44

Chapter 3 Select Variables Screen .......................................................................................... 50

3.1 Select Variables Screen.................................................................................................. 50

3.1.1 Graphs by Groups............................................................................................. 48

Chapter 4 Summary Statistics .................................................................................................. 54

4.1 Summary Statistics with Full Data Sets......................................................................... 54

4.2 Summary Statistics with NDs........................................................................................ 56

Chapter 5 Estimating Nondetects Using ROS Methods .................................................... 60

Chapter 6 Graphical Methods (Graph).................................................................................... 61
xxiv 

6.1 Box Plot ......................................................................................................................... 63

6.2 Histogram....................................................................................................................... 65

6.3 Multi-QQ ....................................................................................................................... 67

6.3.1 Multi-QQ (Full) ................................................................................................ 63

6.3.2 Multi-QQ (with NDs) ....................................................................................... 64

Chapter 7 Simple Classical Outlier Tests ............................................................................. 72

7.1 Outlier Test for Full Data Set......................................................................................... 74

7.2 Outlier Test for Data Set with NDs................................................................................ 74

Chapter 8 Goodness-of-Fit (G.O.F.) Tests ............................................................................. 78

8.1 ROS Estimated (Est.) NDs – Saving Extrapolated NDs................................................ 81

8.2 Goodness-of-Fit Tests with Full Data Sets.................................................................... 81

8.2.1 GOF Tests for Normal and Lognormal Distribution ........................................ 82

8.2.2 GOF Tests for Gamma Distribution ................................................................. 85

8.3 Goodness-of-Fit Tests Excluding NDs.......................................................................... 87

8.3.1 Normal and Lognormal Options....................................................................... 88

8.3.2 Gamma Distribution Option ............................................................................. 91

8.4 Goodness-of-Fit Tests with Log-ROS Estimates........................................................... 93

8.4.1 Normal or Lognormal Distribution (Log-ROS Estimates) ............................... 94

8.4.2 Gamma Distribution (Log-ROS Estimates)...................................................... 97

8.5 Goodness-of-Fit Tests with DL/2 Estimates.................................................................. 99

8.5.1 Normal or Lognormal Distribution (DL/2 Estimates) .................................... 100

8.5.2 Gamma Distribution (DL/2 Estimates)........................................................... 103

8.6 Goodness-of-Fit Tests Statistics .................................................................................. 105

Chapter 9 Single Sample and Two-Sample Hypotheses Testing Approaches .......... 108

9.1 Single Sample Hypotheses Tests................................................................................. 108

9.1.1 Single Sample Hypothesis Testing for Full Data without Nondetects............ 105

9.1.1.1 Single Sample t-Test.......................................................................... 109

9.1.1.2 Single Sample Proportion Test .......................................................... 111

9.1.1.3 Single Sample Sign Test.................................................................... 113

9.1.1.4 Single Sample Wilcoxon Signed Rank (WSR) Test.......................... 111

9.1.2 Single Sample Hypothesis Testing for Data Sets with Nondetects ................ 114

9.1.2.1 Single Proportion Test on Data Sets with NDs.................................. 114

9.1.2.2 Single Sample Sign Test with NDs.................................................... 120

9.1.2.3 Single Sample Wilcoxon Signed Rank Test with NDs...................... 122

9.2 Two-Sample Hypotheses Testing Approaches............................................................ 125

9.2.1 Two-Sample Hypothesis Tests for Full Data.................................................. 122

9.2.1.1 Two-Sample t-Test without NDs....................................................... 124

9.2.1.2 Two-Sample Wilcoxon-Mann-Whitney (WMW) Test without NDs 126

9.2.1.3 Two-Sample Quantile Test for Full Data without NDs..................... 129

9.2.2 Two-Sample Hypothesis Testing for Data Sets without Nondetects.............. 132

9.2.2.1 Two-Sample Wilcoxon-Mann-Whitney Test with Nondetects ......... 132

9.2.2.2 Two-Sample Gehan Test for Data Sets with Nondetects................... 140

9.2.2.3 Two-Sample Quantile Test for Data Sets with Nondetects............... 138
xxv

Chapter 10 Background Statistics ........................................................................................... 146

10.1 Background Statistics for Full Data Sets without Nondetects..................................... 147

10.1.1 Normal or Lognormal Distribution................................................................. 143

10.1.2 Gamma Distribution ....................................................................................... 151

10.1.3 Nonparametric Methods ................................................................................. 153

10.1.4 All Statistics Option........................................................................................ 155

10.2 Background Statistics with NDs.................................................................................. 157

10.2.1 Normal or Lognormal Distribution................................................................. 157

10.2.2 Gamma Distribution ....................................................................................... 162

10.2.3 Nonparametric Methods (with NDs) .............................................................. 164

10.2.4 All Statistics Option........................................................................................ 166

Chapter 11 Computing Upper Confidence Limits (UCLs) of Mean.................................. 169

11.1 UCLs for Full Data Sets............................................................................................... 171

11.1.1 Normal Distribution (Full Data Sets without NDs)........................................ 171

11.1.2 Gamma, Lognormal, Nonparametric, All Statistics Option 

(Full Data without NDs) ................................................................................. 174

11.2 UCL for Data Sets with NDs....................................................................................... 179

Chapter 12 Sample Sizes for User Specified Performance Measures............................ 189

12.1 Estimation of Mean...................................................................................................... 189

12.2 Sample Sizes for Single Sample Hypothesis Tests...................................................... 190

12.2.1 Sample Size for Single Sample t-Test ............................................................ 171

12.2.2 Sample Size for Single Sample Proportion Test............................................. 192

12.2.3 Sample Size for Single Sample Sign Test....................................................... 193

12.2.4 Sample Size for Single Sample Wilcoxon Signed Rank Test......................... 194

12.3 Sample Sizes for Two Sample Hypothesis Tests......................................................... 196

12.3.1 Sample Size for Two Sample t-Test ............................................................... 196

12.3.2 Sample Size for Two Sample Wilcoxon Mann-Whitney................................ 197

12.4 Sample Sizes for Acceptance Sampling ...................................................................... 199

Chapter 13 Analysis of Variance .............................................................................................. 201

13.1 One - Way ANOVA .................................................................................................... 201

13.2 Nonparametric ANOVA.............................................................................................. 202

Chapter 14 Trend Analysis......................................................................................................... 204

14.1 Simple Linear Regression............................................................................................ 204

14.2 Mann - Kendall Test .................................................................................................... 207

14.3 Theil - Sen Test............................................................................................................ 210

14.4 Time Series Plots......................................................................................................... 212

Chapter 15 Windows.................................................................................................................... 221

Chapter 16 Help............................................................................................................................. 222

Chapter 17 Handling the Output Screens and Graphs ....................................................... 224

Copying Graphs........................................................................................................... 224

Printing Graphs............................................................................................................ 225

Printing Non-graphical Outputs................................................................................... 226
xxvi 

Saving Output Screens as Excel Files.......................................................................... 227

Chapter 18 Recommendations to Compute a 95% UCL (Estimate of EPC Term) 

of the Population Mean, μ1, Using Symmetric and Positively Skewed 

Full Data Set without any Nondetects ............................................................... 230

18.1 Normally or Approximately Normally Distributed Data Sets ..................................... 230

18.2 Gamma Distributed Skewed Data Sets........................................................................ 231

18.3 Lognormally Distributed Skewed Data Sets................................................................ 232

18.4 Data Sets without a Discernable Skewed Distribution – Nonparametric Methods for 

Skewed Data Sets......................................................................................................... 234

16.5 Should the Maximum Observed Concentration be Used as an Estimate of 

the EPC Term?............................................................................................................. 239

Chapter 19 Recommendations to Compute a 95% UCL of the Population Mean, 

μ1, Using Data Sets with Nondetects with Multiple Detection Limits........ 209

19.1 General Recommendations and Suggestions............................................................... 240

19.2 Recommended UCL95 Methods for Normal (Approximate Normal) Distribution..... 242

19.3 Recommended UCL95 Methods for Gamma Distribution .......................................... 243

19.4 Recommended UCL95 Methods for Lognormal Distribution..................................... 243

19.5 Recommended Nonparametric UCL Methods............................................................. 244

Glossary .............................................................................................................................................. 248

References.......................................................................................................................................... 250
1

Introduction

The Need for ProUCL Software

Statistical inferences about the sampled populations and their parameters are made based upon defensible 

and representative data sets of appropriate sizes collected from the populations under investigation. 

Statistical inference, including both estimation and hypotheses testing approaches, is routinely used to: 

1. Estimate environmental parameters of interest such as exposure point concentration 

(EPC) terms, not-to-exceed values, and background level threshold values (BTVs) for 

contaminants of potential concern (COPC), 

2. Identify areas of concern (AOC) at a contaminated site,

3. Compare contaminant concentrations found at two or more AOCs of a contaminated site, 

4. Compare contaminant concentrations found at an AOC with background or reference 

area contaminant concentrations, 

5. Compare site concentrations with a cleanup standard to verify the attainment of cleanup 

standards.

Statistical inference about the sampled populations and their parameters are made based upon defensible 

and representative data sets of appropriate sizes collected from the populations under investigation. 

Environmental data sets originated from the Superfund and RCRA sites often consist of observations 

below one or more detection limits (DLs). In order to address the statistical issues arising in: exposure and 

risk assessment applications; background versus site comparison and evaluation studies; and various other 

environmental applications, several graphical, parametric, and nonparametric statistical methods for data 

sets with nondetects and without nondetects have been incorporated in ProUCL 4.0.

Exposure and risk management and cleanup decisions in support of United States Environmental 

Protection Agency (EPA) projects are often made based upon the mean concentrations of the COPCs. A 

95% upper confidence limit (UCL95) of the unknown population (e.g., an AOC) arithmetic mean (AM), 

μ1, can be used to:

 Estimate the EPC term of the AOC under investigation, 

 Determine the attainment of cleanup standards, 

 Compare site mean concentrations with reference area mean concentrations, and

 Estimate background level mean contaminant concentrations. The background mean contaminant 

concentration level may be used to compare the mean of an AOC. It should be noted that it is not 

appropriate to compare individual point-by-point site observations with the background mean 

concentration level.

It is important to compute a reliable and stable UCL95 of the population mean using the available data. 

The UCL95 should approximately provide the 95% coverage for the unknown population mean, μ1. Based 

upon the available background data, it is equally important to compute reliable and stable upper 

percentiles, upper prediction limits (UPLs), or upper tolerance limits (UTLs). These upper limits based 

upon background (or reference) data are used as estimates of BTVs, compliance limits (CL), or not-toexceed values. These upper limits are often used in site (point-by-point) versus background comparison 

evaluations. 

Environmental scientists often encounter trace level concentrations of COPCs when evaluating sample 
2 

analytical results. Those low level analytical results cannot be measured accurately, and therefore are 

typically reported as less than one or more detection limit (DL) values (also called nondetects). However, 

practitioners often need to obtain reliable estimates of the population mean, μ1, the population standard 

deviation, σ1, and upper limits, including the upper confidence limit (UCL) of the population mass or 

mean, the UPL, and the UTL based upon data sets with nondetect (ND) observations. Hypotheses testing 

approaches are often used to verify the attainment of cleanup standards, and compare site and background 

concentrations of COPCs. 

Background evaluation studies, BTVs, and not-to-exceed values should be estimated based upon 

defensible background data sets. The estimated BTVs or not-to-exceed values are then used to identify the 

COPCs, to identify the site AOCs or hot spots, and to compare the contaminant concentrations at a site 

with background concentrations. The use of appropriate statistical methods and limits for site versus 

background comparisons is based upon the following factors: 

1. Objective of the study,

2. Environmental medium (e.g., soil, groundwater, sediment, air) of concern,

3. Quantity and quality of the available data,

4. Estimation of a not-to-exceed value or of a mean contaminant concentration,

5. Pre-established or unknown cleanup standards and BTVs, and

6. Sampling distributions (parametric or nonparametric) of the concentration data sets collected 

from the site and background areas under investigation. 

In background versus site comparison evaluations, the environmental population parameters of interest 

may include: 

 Preliminary remediation goals (PRGs),

 Soil screening levels (SSLs),

 Risk-based cleanup (RBC) standards,

 BTVs, not-to-exceed values, and 

 Compliance limit, maximum concentration limit (MCL), or alternative concentration limit (ACL), 

frequently used in groundwater applications.

When the environmental parameters listed above are not known or have not been pre-established, 

appropriate upper statistical limits are used to estimate the parameters. The UPL, UTL, and upper 

percentiles are used to estimate the BTVs and not-to-exceed values. Depending upon the site data 

availability, point-by-point site observations are compared with the estimated (or pre-established) BTVs 

and not-to-exceed values. If enough site and background data are available, two-sample hypotheses 

testing approaches are used to compare site concentrations with background concentrations levels. These 

statistical methods can also be used to compare contaminant concentrations of two site AOCs, surface and 

subsurface contaminant concentrations, or upgradient versus monitoring well contaminant concentrations. 

ProUCL 4.00.05 Capabilities

ProUCL Version 4.00.05 is an upgrade of ProUCL Version 4.0 (EPA, 2007). ProUCL 4.00.02 contains 

statistical methods to address various environmental issues for both full data sets without nondetects and 

for data sets with NDs (also known as left-censored data sets). 
3

Some of the statistical and graphical methods available in ProUCL 4.00.02/ProUCL 4.0 and ProUCL 4.1

are listed in the following:

o Rigorous parametric and nonparametric (including bootstrap methods) statistical methods 

(instead of simple ad hoc or substitution methods) that can be used on full data sets 

without nondetects and on data sets with below detection limit (BDL) or nondetect (ND) 

observations. 

o State-of-the-art parametric and nonparametric UCL, UPL, and UTL computation 

methods. These methods can be used on full-uncensored data sets without nondetects and 

also on data sets with BDL observations. Some of the methods (e.g., Kaplan-Meier 

method, ROS methods) are applicable on left-censored data sets having multiple 

detection limits. The UCL and other upper limit computation methods cover a wide range 

of skewed data sets with and without the BDLs. 

o Single sample (e.g., Student‟s t-test, sign test, proportion test, Wilcoxon Singed Rank 

test) and two-sample (Student‟s t-test, Wilcoxon-Mann-Whitney test, Gehan test, quantile 

test) parametric and nonparametric hypotheses testing approaches for data sets with and 

without ND observations. These hypothesis testing approaches can be used to: verify the 

attainment of cleanup standards, perform site versus background comparisons, and 

compare two or more AOCs, monitoring wells (MWs). 

o The single sample hypotheses testing approaches are used to compare site mean, site 

median, site proportion, or a site percentile (e.g., 95th) to a compliance limit (action level, 

regularity limit). The hypotheses testing approaches can handle both full-uncensored data 

sets without nondetects, and left-censored data sets with nondetects. Simple two-sample

hypotheses testing methods to compare two populations are available in ProUCL 4.0, 

such as two-sample t-tests, Wilcoxon-Mann-Whitney (WMW) Rank Sum test, quantile 

test, Gehan‟s test, and dispersion test. Variations of hypothesis testing methods (e.g., 

Levene‟s method to compare dispersions, generalized WRS test) are easily available in 

most commercial and freely available software packages (e.g., MINITAB, R). 

o ProUCL 4.0 includes graphical methods (e.g., box plots, multiple Q-Q plots, histogram) 

to compare two or more populations. Additionally, ProUCL 4.0 can also be used to 

display a box plot of one population (e.g., site data) with compliance limits or upper 

limits (e.g., UPL) of other population (background area) superimposed on the same 

graph. This kind of graph provides a useful visual comparison of site data with a 

compliance limit or BTVs. Graphical displays of a data set (e.g., Q-Q plot) should be 

used to gain insight knowledge contained in a data set that may not otherwise be clear by 

looking at simple test statistics such as t-test, Dixon test statistic, or Shapiro-Wilk (S-W) 

test statistic.

6. ProUCL 4.0 can process multiple contaminants (variables) simultaneously and has the 

capability of processing data by groups. A valid group column should be included in the 

data file.

7. ProUCL 4.0 provides a GOF test for data sets with nondetects. The user can create 

additional columns to store extrapolated (estimated) values for nondetects based upon 

normal ROS, gamma ROS, and lognormal ROS (robust ROS) methods.
4 

8. The Number of Samples module of ProUCL 4.00.05 provides user friendly options to 

enter the desired/pre-specified decision parameters (e.g., Type I and Type II error rates) 

and DQOs used to determine minimum sample sizes for the selected statistical 

applications including: estimation of mean, single and two sample hypothesis testing 

approaches, and acceptance sampling. Sample size determination methods are available 

for the sampling of continuous characteristics (e.g., lead or Ra 226), as well as for 

attributes (e.g., proportion of occurrences exceeding a specified threshold). Both 

parametric (e.g., for t-tests) and nonparametric (e.g., Sign test, test for proportions, WRS 

test) sample size determination methods as described in EPA (2006, 2009) and 

MARSIMM (2000) guidance documents are available in ProUCL 4.1. ProUCL 4.1 also 

has the sample size determination methods for acceptance sampling of lots of discrete 

objects such as a lot of drums consisting of hazardous waste (e.g., RCRA applications, 

EPA 2002c). 

9. ProUCL 4.1.00 has two new modules: ANOVA and Trend Tests. ANOVA module has 

both classical and nonparametric Kruskal-Wallis Oneway ANOVA tests as described in 

EPA guidance documents (e.g., EPA 2006, 2009). Trend Tests module has linear 

ordinary least squares (OLS) regression method, Mann-Kendall trend test, Theil-Sen 

trend test, and time series plots as described in the Unified RCRA Guidance Document 

(EPA 2009). Oneway ANOVA is used to compare means (or medians) of multiple groups 

such as comparing mean concentrations of several areas of concern; and to perform interwell comparisons. In groundwater (GW) monitoring applications, OLS regression, trend 

tests, and time series plots (EPA, 2009) are often used to identify trends (e.g., upwards, 

downwards) in contaminant concentrations of various GW monitoring wells over a 

certain period of time. 

ProUCL Applications

The methods incorporated in ProUCL 4.1.0 (and in all previous versions) can be used on data 

sets with and without BDL and ND observations. Methods and recommendations as 

incorporated in ProUCL 4.0 are based upon the results and findings of the extensive 

simulation studies as summarized in Singh and Singh (2003), and Singh, Maichle, and Lee 

(EPA, 2006). 

Methods included in ProUCL 4.00.05 can be used in various other environmental 

applications including the verification of cleanup standards (EPA, 1989), and computation of 

upper limits needed in groundwater monitoring applications (EPA, 1992 and EPA, 2004).

In 2002, EPA issued guidance for calculating the UCLs of the unknown population means for 

contaminant concentrations at hazardous waste sites. The ProUCL 4.0 software package 

(EPA, 2007) has served as a companion software package for the EPA (2002a) guidance 

document for calculating UCLs of mean contaminant concentrations at hazardous waste sites. 

ProUCL 4.0 has several parametric and nonparametric statistical methods that can be used to 

compute appropriate UCLs based upon full-uncensored data sets without any ND 

observations. ProUCL 4.0 retains the capabilities of ProUCL 3.0, including goodness-of-fit 

(GOF) and the UCL computation methods for data sets without any BDL observations. 
5

However, ProUCL 4.0 has the additional capability to perform GOF tests and computing 

UCLs and other upper limits based upon data sets with BDL observations.

With the inclusion of Oneway ANOVA, Regression and Trend tests, and user-friendly DQOs

based sample size determination modules, ProUCL version 4.1.00 represents a 

comprehensive statistical software package equipped with statistical methods and graphical 

tools needed to address many environmental sampling and statistical issues as described in 

various CERCLA (EPA 1989a, 2002a, 2002b, 2006), MARSSIM (2000) and RCRA (EPA 

1989b, 1992b, 2002c, 2009) guidance documents.

ProUCL 4.0 defines log-transform (log) as the natural logarithm (ln) to the base e. ProUCL 

4.0 also computes the maximum likelihood estimates (MLEs) and the minimum variance 

unbiased estimates (MVUEs) of unknown population parameters of normal, lognormal, and 

gamma distributions. This, of course, depends upon the underlying data distribution. ProUCL 

4.0 computes the (1 – α)100% UCLs of the unknown population mean, μ1, using 5 parametric 

and 10 nonparametric methods. It should be pointed out that ProUCL 4.0 computes the 

simple summary statistics for detected raw and log-transformed data for full data sets without 

NDs, as well as for data sets with BDL observations. It is noted that estimates of mean and sd

for data sets with NDs based upon rigorous statistical methods (e.g., MLE, ROS, K-M 

methods) are note provided in the summary statistics. Those estimates and the associated 

upper limits for data sets with NDs are provided under the menu options: Background and 

UCL. 

It is emphasized that throughout this User Guide, and in the ProUCL 4.00.05 (and in all 

previous versions of 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 first separate them out (e.g., using appropriate statistical 

population partitioning techniques), and then compute appropriate respective 95% UCLs 

separately for each of the identified populations. Outliers, if any, should be identified and 

thoroughly investigated. ProUCL 4.0 provides two commonly used simple classical outlier 

identification procedures: 1) the Dixon test and 2) the Rosner test. Outliers distort most 

parametric statistics (e.g., mean, UCLs, upper prediction limits (UPLs), test statistics) of 

interest. Moreover, it should be noted that even though outliers might have minimal influence 

on hypotheses testing statistics based upon ranks (e.g., WMW test), outliers do distort those 

nonparametric statistics (including bootstrap methods), which are based upon higher order 

statistics such as UPLs and UTLs. Decisions about the disposition (exclusion or inclusion) of 

outliers in a data set used to estimate the EPC terms or BTVs should be made by all parties 

involved (e.g., project team, EPA, local agency, potentially responsible party, etc.) in the 

decision making process. 

The presence of outlying observations also distorts statistics based upon bootstrap re-samples. 

The use of higher order values (quantiles) of the distorted statistics for the computation of the 

UCLs or UPLs based upon bootstrap t and Hall‟s bootstrap methods may yield unstable and 

erratic UCL values. This is especially true for the upper limits providing higher confidence 

coefficients such as 95%, 97.5%, or 99%. Similar behavior of the bootstrap t UCL is 

observed for data sets having BDL observations. Therefore, the bootstrap t and Hall‟s 

bootstrap methods should be used with caution. It is suggested that the user should examine 

various other 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 
6 

results based upon these two bootstrap methods are much higher than the rest of methods, 

then this could be an indication of erratic behavior of those bootstrap UCL values, perhaps 

distorted by outlying observations. 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. Alternatively, one may use a 97.5% or 99% Chebyshev UCL to estimate 

the mean of a highly skewed population. It should be noted that typically, a Chebyshev UCL 

may yield conservative and higher values of the UCLs than other methods available in 

ProUCL 4.0 This is especially true when data are moderately skewed and sample size is 

large. In such cases, when the sample size is large, one may want to use a 95% Chebyshev 

UCL or a Chebyshev UCL with lower confidence coefficient such as 92.5% or 90% as 

estimate of the population mean. 

ProUCL Methods

ProUCL 4.00.05 provides 15 UCL computation methods for full data sets without any BDL observations; 

5 are parametric and 10 are nonparametric methods. The nonparametric methods do not depend upon any 

assumptions about the data distributions. The five parametric UCL computation methods are: 

o Student‟s t-UCL, 

o Approximate gamma UCL using chi-square approximation,

o Adjusted gamma UCL (adjusted for level significance), 

o Land‟s H-UCL, and 

o Chebyshev inequality-based UCL (using MVUEs of parameters of a lognormal 

distribution). 

The 10 nonparametric methods are:

1. The central limit theorem (CLT)-based UCL, 

2. Modified-t statistic (adjusted for skewness)-based UCL, 

3. Adjusted-CLT (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. 

Environmental scientists often encounter trace level concentrations of COPCs when evaluating sample 

analytical results. Those low level analytical results cannot be measured accurately, and therefore are 

typically reported as less than one or more DL values. However, the practitioners need to obtain reliable 

estimates of the population mean, μ1, and the population standard deviation, σ1, and upper limits including 

the UCL of the population mass (measure of central tendency) or mean, UPL, and UTL. Several methods 

are available and cited in the environmental literature (Helsel (2005), Singh and Nocerino (2002), Millard 

and Neerchal (2001)) that can be used to estimate the population mean and variance. However, till to date, 

no specific recommendations are available for the use of appropriate methods that can be used to compute 

upper limits (e.g., UCLs, UPLs) based upon data sets with BDL observations. Singh, Maichle, and Lee 

(EPA, 2006) extensively studied the performance of several parametric and nonparametric UCL 
7

computation methods for data sets with BDL observations. Based upon their results and findings, several 

methods to compute upper limits (UCLs, UPLs, and UTLs) needed to estimate the EPC terms and BTVs 

have been incorporated in ProUCL 4.0.

In 2002, EPA issued another Guidance for Comparing Background and Chemical Concentrations in Soil 

for CERCLA Sites (EPA, 2002b). This EPA (2002b) background guidance document is currently being 

revised to include statistical methods that can be used to estimate the BTVs and not-to-exceed values 

based upon data sets with and without the BDL observations. In background evaluation studies, BTVs, 

compliance limits, or not-to-exceed values often need to be estimated based upon defensible background 

data sets. The estimated BTVs or not-to-exceed values are then used for screening the COPCs, to identify 

the site AOCs or hot spots, and also to determine if the site concentrations (perhaps after a remediation 

activity) are comparable to background concentrations, or are approaching the background level 

concentrations. Individual point-by-point site observations (composite samples preferred) are sometimes 

compared with those not-to-exceed values or BTVs. It should be pointed out that in practice, it is 

preferred to use hypotheses testing approaches to compare site versus background concentrations 

provided enough (e.g., at least 8-10 detected observations from each of the two populations) site and 

background data are available. Chapter 1 provides practical guidance on the minimum sample size 

requirements to estimate and use the BTVs, single and two-sample hypotheses testing approaches to 

perform background evaluations and background versus site comparisons. Chapter 1 also briefly 

discusses the differences in the definitions and uses of the various upper limits as incorporated in ProUCL 

4.0. Detailed discussion of the various methods to estimate the BTVs and other not-to-exceed values for 

full-uncensored data sets (Chapter 5) without any nondetect values and for left-censored data sets 

(Chapter 6) with nondetect values are given in the revised background guidance document.

ProUCL 4.0 includes statistical methods to compute UCLs of the mean, upper limits to estimate the 

BTVs, other not-to-exceed values, and compliance limits based upon data sets with one or more detection 

limits. The use of appropriate statistical methods and limits for exposure and risk assessment, and site 

versus background comparisons, is based upon several factors: 

1. Objective of the study; 

2. Environmental medium (e.g., soil, groundwater, sediment, air) of concern; 

3. Quantity and quality of the available data; 

4. Estimation of a not-to-exceed value or of a mean contaminant concentration; 

5. Pre-established or unknown cleanup standards and BTVs; and 

6. Sampling distributions (parametric or nonparametric) of the concentration data sets 

collected from the site and background areas under investigation. 

In background versus site comparison studies, the population parameters of interest are typically 

represented by upper threshold limits (e.g., upper percentiles, upper confidence limits of an upper 

percentile, upper prediction limit) of the background data distribution. It should be noted that the upper 

threshold values are estimated and represented by upper percentiles and other values from the upper tail 

of the background data distribution. These background upper threshold values do not represent measures 

of central tendency such as the mean, the median, or their upper confidence limits. These environmental 

parameters may include: 

 Preliminary remediation goals (PRGs), Compliance Limits,

 Soil screening levels (SSLs),

 Risk-based cleanup (RBC) standards,

 BTVs, compliance limits, or not-to-exceed values, and
8 

 Maximum concentration limit (MCL) or alternative concentration limit (ACL) used in 

Groundwater applications.

When the environmental parameters listed above are not known or pre-established, appropriate upper 

statistical limits are used to estimate those parameters. The UPL, UTL, and upper percentiles are typically 

used to estimate the BTVs, not-to-exceed values, and other parameters listed above. Depending upon the 

availability of site data, point-by-point site observations are compared with the estimated (or preestablished) BTVs and not-to-exceed values. If enough site and background data are available, twosample hypotheses testing approaches (preferred method to compare two populations) are used to 

compare site concentrations with background concentrations levels. The hypotheses testing methods can 

also be used to compare contaminant concentrations of two site AOCs, surface and subsurface 

contaminant concentrations, or upgradient versus monitoring well contaminant concentrations. 

Background versus Site Comparison Evaluations

The following statistical limits have been incorporated in ProUCL 4.0 to assist in background versus site 

comparison evaluations:

Parametric Limits for Full-Uncensored Data Sets without Nondetect Observations

 UPL for a single observation (Normal, Lognormal) not belonging to the original data set

 UPL for next k (k is user specified) or k future observations (Normal, Lognormal)

 UTL, an upper confidence limit of a percentile (Normal, Lognormal)

 Upper percentiles (Normal, Lognormal, and Gamma)

Nonparametric Limits for Full-Uncensored Data Sets without Nondetect Observations

Nonparametric limits are typically based upon order statistics of a data set such as a background or a 

reference data set. Depending upon the size of the data set, higher order statistics (maximum, second 

largest, third largest, and so on) are used as these upper limits (e.g., UPLs, UTLs). The details of these 

methods with sample size requirements can be found in Chapter 5 of the revised Guidance for Comparing 

Background and Chemical Concentrations in Soil for CERCLA Sites (EPA, 2002b). It should be, noted 

that the following statistics might get distorted by the presence of outliers (if any) in the data set under 

study.

 UPL for a single observation not belonging to the original data set

 UTL, an upper confidence limit of a percentile

 Upper percentiles 

 Upper limit based upon interquartile range (IQR)

 Upper limits based upon bootstrap methods

For data sets with BDL observations, the following parametric and nonparametric methods to compute 

the upper limits were studied and evaluated by Singh, Maichle, and Lee (EPA, 2006) via Monte Carlo 

Simulation Experiments. Depending upon the performances of those methods, only some of the methods 

have been incorporated in ProUCL 4.0. Methods (e.g., Delta method, DL method, uniform (0, DL) 

generation method) not included in ProUCL 4.0 do not perform well in comparison with other methods. 
9

Note: When the percentage of NDs in a data set is high (e.g., > 40%-50%), especially when multiple 

detection limits might be present, it is hard to reliably perform GOF tests (to determine data distribution) 

on those data sets with many NDs. The uncertainty associated with those GOF tests will be high, 

especially when the data sets are of small sizes (< 10-20). It should also be noted that the parametric 

MLE methods (e.g., for normal and lognormal distributions) often yield unstable estimates of mean and 

sd. This is especially true when the number of nondetects exceeds 40%-50%. In such situations, it is 

preferable to use nonparametric (e.g., KM method) methods to compute statistics of interest such as 

UCLs, UPLs, and UTLs. Nonparametric methods do not require any distributional assumptions about the 

data sets under investigation. Singh, Maichle, and Lee (EPA, 2006) also concluded that the performance 

of the KM estimation method is better (in terms of coverage probabilities) than various other parametric 

estimation (e.g., MLE, EM, ROS) methods. 

Parametric Methods to Compute Upper Limits for Data Sets with Nondetect Observations

 Simple substitution (proxy) methods (0, DL/2, DL)

 MLE method, often known as Cohen‟s MLE method – single detection limit

 Restricted MLE method – single detection limit – not in ProUCL 4.0

 Expectation maximization (EM) method – single detection limit – not in ProUCL 4.0

 EPA Delta log method – single detection limit – not in ProUCL 4.0

 Regression method on detected data and using slope and intercept of the OLS regression 

line as estimates of standard deviation, sd, and mean (not a recommended method) 

 Robust ROS (regression on order statistics) on log-transformed data – nondetects 

extrapolated (estimated) using robust ROS; mean, sd, UCLs, and other statistics 

computed using the detected and extrapolated data in original scale – multiple detection 

limits

 Normal ROS – nondetects extrapolated (estimated) using normal distribution, mean, sd, 

UCLs, and other statistics computed using the detected and extrapolated data – multiple 

detection limits. 

 It is noted that the estimated NDs often become negative and even larger than the 

detection limits (not a recommended method)

 Gamma ROS – nondetects extrapolated (estimated) using gamma distribution, mean, sd, 

UCLs, and other statistics computed using the detected and extrapolated data – multiple 

detection limits

Nonparametric Methods to Compute Upper Limits for Data Sets with Nondetect Observations

 Bootstrap Methods 

o Percentile Bootstrap on robust ROS

o Percentile Bootstrap

o BCA Bootstrap

o Bootstrap t

 Jackknife Method 

o Jackknife on robust ROS

 Kaplan-Meier (KM) Method

o Bootstrap (percentile, BCA) using KM estimates

o Jackknife using KM estimates
10 

o Chebyshev Method using KM estimates

 Winsorization Method

For uncensored full data sets without any NDs, the performance (in terms of coverage for the mean) of 

the various UCL computation methods was evaluated by Singh and Singh (2003). The performance of the 

parametric and nonparametric UCL methods based upon data sets with nondetect observations was 

studied by Singh, Maichle, and Lee (EPA, 2006). Several of the methods listed above have been 

incorporated in ProUCL 4.0 to compute the estimates of EPC terms (95% UCL), and of BTVs (UPLs, 

UTLs, upper percentiles). Methods that did not perform well (e.g., poor coverage or unrealistically large 

values, infeasible and biased estimates) are not included in ProUCL 4.0. Methods not incorporated in 

ProUCL 4.0 are: EPA Delta Log method, Restricted MLE method, and EM method, substitution method 

(0, and DL), and Regression method. 

Note: It should be noted that for data sets with NDs, the DL/2 substitution method has been incorporated 

in ProUCL 4.0 only for historical reasons and also for its current default use. It is well known that the 

DL/2 method (with NDs replaced by DL/2) does not perform well (e.g., Singh, Maichle, and Lee (EPA,

2006)) even when the percentage of NDs is only 5%-10%. It is strongly suggested to avoid the use of 

DL/2 method for estimation and hypothesis testing approaches used in various environmental 

applications. Also, when the % of NDs becomes high (e.g., > 40%-50%), it is suggested to avoid the use 

of parametric MLE methods. For data sets with high percentage of NDs (e.g., > 40%), the distributional 

assumptions needed to use parametric methods are hard to verify; and those parametric MLE methods 

may yield unstable results.

It should also be noted that even though the lognormal distribution and some statistics based upon 

lognormal assumption (e.g., Robust ROS, DL/2 method) are available in ProUCL 4.0, ProUCL 4.0 does 

not compute MLEs of mean and sd based upon a lognormal distribution. The main reason is that the 

estimates need to be computed in the original scale via back-transformation (Shaarawi, 1989, and Singh, 

Maichle, and Lee (EPA, 2006)). Those back-transformed estimates often suffer from an unknown amount 

of significant bias. Hence, it is also suggested to avoid the use of a lognormal distribution to compute 

MLEs of mean and sd, and associated upper limits, especially UCLs based upon those MLEs obtained 

using a lognormal distribution.

ProUCL 4.0 recommends the use of an appropriate UCL to estimate the EPC terms. It is desirable that the 

user consults with the project team and experts familiar with the site before using those recommendations. 

Furthermore, there does not seem to be a general agreement about the use of an upper limit (e.g., UPL, 

percentile, or UTL) to estimate not-to-exceed values or BTVs to be used for screening of the COPCs and

in site versus background comparison studies. ProUCL 4.0 can compute both parametric and 

nonparametric upper percentiles, UPLs, and UTLs for uncensored and censored data sets. However, no 

specific recommendations have been made regarding the use of UPLs, UTLs, or upper percentiles to 

estimate the BTVs, compliance limits, and other related background or reference parameters. However, 

the developers of ProUCL 4.0 prefer the use of UPLs or upper percentiles to estimate the background 

population parameters (e.g., BTVs, not-to-exceed values) that may be needed to perform point-by-point 

site versus background comparisons.

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 other bootstrap methods. However, it has been noted (e.g., 
11

Efron and Tibshirani (1993), and Singh, Singh, and Iaci (2002b)) that these two bootstrap methods 

sometimes yield erratic and inflated UCL values (orders of magnitude higher than the other UCLs). This 

may occur when outliers are present in a data set. Similar behavior of the bootstrap t UCL is observed 

based upon data sets with NDs. Therefore, whenever applicable, ProUCL 4.0 provides cautionary 

statements regarding the use of bootstrap methods. 

ProUCL 4.0 provides several state-of-the-art parametric and nonparametric UCL, UPL, and UTL 

computation methods that can be used on uncensored data sets (full data sets) and on data sets with BDL 

observations. Some of the methods (e.g., Kaplan-Meier method, ROS methods) incorporated in ProUCL 

4.0 are applicable on left-censored data sets having multiple detection limits. The UCLs and other upper 

limits computation methods in ProUCL 4.0 cover a wide range of skewed data distributions with and 

without the BDLs arising from the environmental applications. 

ProUCL 4.0 also has parametric and nonparametric single and two-sample hypotheses testing approaches 

required to: compare site location (e.g., mean, median) to a specified cleanup standard; perform site 

versus background comparisons; or compare of two or more AOCs. These hypotheses testing methods 

can handle both full (uncensored data sets without NDs) and left-censored (with nondetects) data sets. 

Specifically, two-sample tests such as t-test, Wilcoxon Mann-Whitney (WMW) Rank Sum test, quantile 

test, and Gehan‟s test are available in ProUCL 4.0 to compare concentrations of two populations.

Single sample parametric (Student‟s t-test) and nonparametric (sign test, Wilcoxon Signed Rank (WSR) 

test, tests for proportions and percentiles) hypotheses testing approaches are also available in ProUCL 4.0. 

The single sample hypotheses tests are useful when the environmental parameters such as the clean 

standard, action level, or compliance limits are known, and the objective is to compare site concentrations 

with those known threshold values. Specifically, a t-test (or a sign test) may be used to verify the 

attainment of cleanup levels at an AOC after a remediation activity; and a test for proportion may be used 

to verify if the proportion of exceedances of an action level (or a compliance limit) by sample 

concentrations collected from the AOC (or a MW) exceeds a certain specified proportion (e.g., 1%, 5%, 

10%). As mentioned before, ProUCL 4.0 can perform these hypotheses on data sets with and without 

nondetect observations. 

Note: It should be noted that as cited in the literature, some of the hypotheses testing approaches (e.g., 

nonparametric two-sample WMW) deal with the single detection limit scenario. If multiple detection 

limits are present, all NDs below the largest detection limit need to be considered as NDs (Gilbert, 1987, 

and Helsel, 2005). This in turn may reduce the power and increase uncertainty associated with test. As 

mentioned before, it is always desirable to supplement the test statistics and test conclusions with 

graphical displays such as the multiple Q-Q plots and side-by-side box plots. ProUCL 4.0 can graph box 

plots and Q-Q plots for data sets with nondetect observations. Gehan test as available in ProUCL 4.0 

should be used in case multiple detection limits are present. ProUCL 4.0 can draw Q-Q plots and box 

plots for data sets with and without nondetect observations.

It should be pointed out that when using two-sample hypotheses approaches (WMW test, Gehan test, and 

quantile test) on data sets with NDs, both samples and variables (e.g., site-As, Back-As) should be 

specified as having nondetects. This means, a ND column (0 = ND, and 1 = detect) should be provided 

for each variable (here D_site-As, and D_Back-As) to be used in this comparison. If a variable (e.g., siteAs) does not have any nondetects, still a column with label D_site-As should be included in the data set 

with all entries = 1 (detected values).
12 

Moreover, in single sample hypotheses tests (e.g., sign test, proportion test) used to compare site 

mean/median concentration level with a cleanup standard, Cs, or compliance limit (e.g., proportion test), 

all NDs (if any) should lie below the cleanup standard, Cs.

The differences between these tests should be noted and understood. Specifically, a t-test or a Wilcoxon 

Signed Rank (WSR) test are used to compare the measures of location and central tendencies (e.g., mean, 

median) of a site area (e.g., AOC) to a cleanup standard, Cs, or action level also representing a measure of 

central tendency (e.g., mean, median); whereas, a proportion test compares if the proportion of site 

observations from an AOC exceeding a compliance limit (CL) exceeds a specified proportion, P0 (e.g., 

5%, 10%). The percentile test compares a specified percentile (e.g., 95th) of the site data to a prespecified upper threshold (e.g., reporting limit, action level). All of these tests have been incorporated in 

ProUCL 4.0. Most of the single sample and two-sample hypotheses tests also report associated p-values. 

For some of the hypotheses tests (e.g., WMW test, WSR test, proportion test), large sample approximate 

p-values are computed using continuity correction factors. 

Determining sample sizes for various statistical applications: The Number of Samples module of ProUCL 

4.00.05 provides user friendly options to enter the desired/pre-specified decision parameters (e.g., Type I 

and Type II error rates) and DQOs used to determine minimum sample sizes for the selected statistical 

applications including: estimation of mean, single and two sample hypothesis testing approaches, and 

acceptance sampling. Sample size determination methods are available for the sampling of continuous 

characteristics (e.g., lead or Ra 226), as well as for attributes (e.g., proportion of occurrences exceeding a 

specified threshold). Both parametric (e.g., for t-tests) and nonparametric (e.g., Sign test, test for 

proportions, WRS test) sample size determination methods as described in EPA (2006, 2009) and 

MARSIMM (2000) guidance documents are available in ProUCL 4.1. ProUCL 4.1 also has the sample 

size determination methods for acceptance sampling of lots of discrete objects such as a lot of drums 

consisting of hazardous waste (e.g., RCRA applications, EPA 2002c). 

ANOVA and Trend Tests: ProUCL 4.1.00 has Oneway ANOVA and Trend Tests. ANOVA module has 

both classical and nonparametric Kruskal-Wallis Oneway ANOVA tests as described in EPA guidance 

documents (e.g., EPA 2006, 2009). Trend Tests module has linear ordinary least squares (OLS) 

regression method, Mann-Kendall trend test, Theil-Sen trend test, and time series plots as described in the 

Unified RCRA Guidance Document (EPA 2009). Oneway ANOVA is used to compare means (or 

medians) of multiple groups such as comparing mean concentrations of several areas of concern; and to 

perform inter-well comparisons. In groundwater (GW) monitoring applications, OLS regression, trend 

tests, and time series plots (EPA, 2009) are often used to identify trends (e.g., upwards, downwards) in 

contaminant concentrations of various GW monitoring wells over a certain period of time. 

Graphical Capabilities

ProUCL 4.0 has useful exploratory graphical methods that may be used to visually compare the 

concentrations of: 

1. A site area of concern (AOC) with an action level. This can be done using a box plot of 

site data with action level superimposed on that graph,

2. Two or more populations, including site versus background populations, surface versus 

subsurface concentrations, and

3. Two or more AOCs. 
13

The graphical methods include double and multiple quantile-quantile (Q-Q) plots, side-by-side box plots, 

and histograms. Whenever possible, it is desirable to supplement statistical test results and statistics with 

visual graphical displays of data sets. There is no substitute for graphical displays of a data set as the 

visual displays often provide useful information about a data set, which cannot be revealed by simple test 

statistics such as t-test, SW test, Rosner test, WMW test. For example, in addition to providing 

information about the data distribution, a normal Q-Q plot can also help identify outliers and multiple 

populations that might be present in a data set. This kind of information cannot be revealed by simple test 

statistics such as a Shapiro-Wilk (SW) test or Rosner‟s outlier test statistic. Specifically, the SW test may 

lead to the conclusion that a mixture data set (representing two or more populations) can be modeled by a 

normal (or lognormal) distribution, whereas the occurrence of obvious breaks and jumps in the associated 

Q-Q plot may suggest the presence of multiple populations in the mixture data set. It is suggested that the 

user should use exploratory tools to gain necessary insight into a data set and the underlying assumptions 

(e.g., distributional, single population) that may not be revealed by simple test statistics.

Note: On a Q-Q plot, observations well separated from the majority of the data may represent potential 

outliers, and obvious jumps and breaks of significant magnitude may suggest the presence of observations 

from multiple populations in the data set.

The analyses of data categorized by a group ID variable such as: 1) Surface vs. Subsurface;

2) AOC1 vs. AOC2; 3) Site vs. Background; and 4) Upgradient vs. Downgradient monitoring wells are 

quite common in many environmental applications. ProUCL 4.0 offers this option for data sets with and 

without nondetects. The Group Option provides a powerful tool to perform various statistical tests and 

methods (including graphical displays) separately for each of the group (samples from different 

populations) that may be present in a data set. For an example, the same data set may consist of samples 

from the various groups or populations representing site, background, two or more AOCs, surface, 

subsurface, monitoring wells. The graphical displays (e.g., box plots, Q-Q plots) and statistics 

(computations of background statistics, UCLs, hypotheses testing approaches) of interest can be 

computed separately for each group by using this option. 

Technical Guide

In addition to this User Guide, a Technical document also accompanies ProUCL 4.0, providing useful 

technical details of the graphical and statistical methods as incorporated in ProUCL 4.0. Most of the 

mathematical algorithms and formulas (with references) used in the development of ProUCL 4.0 are 

summarized in the Technical Guide.

Minimum Hardware Requirements

 Intel Pentium 1.0 GHz

 45 MB of hard drive space

 512 MB of memory (RAM)

 CD-ROM drive

 Windows 98 or newer. ProUCL was thoroughly tested on NT-4, Windows 2000, and 

Windows XP Operating Systems (limited testing on Windows ME).

Software Requirements

ProUCL 4.00.05 has been developed in the Microsoft .NET Framework using the C# programming 

language. As such, to properly run ProUCL 4.00.05, the computer using the program must have the .NET 
14 

Framework pre-installed. The downloadable .NET files can be found at one of the following two Web

sites:

 http://msdn.microsoft.com/netframework/downloads/updates/default.aspx

Note: Download .Net version 1.1

 http://www.microsoft.com/downloads/details.aspx?FamilyId=262D25E3-F589-4842-

8157-034D1E7CF3A3&displaylang=en

The first Web site lists all of the downloadable .NET Framework files, while the second Web site 

provides information about the specific file (s) needed to run ProUCL 4.0. Download times are estimated 

at 57 minutes for a dialup connection (56K), and 13 minutes on a DSL/Cable connection (256K). 

Installation Instructions

 Download the file SETUP.EXE from the EPA Web site and save to a temporary location. 

Note: This download is pending release – beta-testers: See text file on CD. 

 Run the SETUP.EXE program. This will create a ProUCL directory and two folders: 

1) the USER GUIDE (this document), and 2) DATA (example data sets).

 To 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 the 

ProUCL.exe file, and run ProUCL.exe.

 To uninstall the program, use Windows Explorer to locate and delete the ProUCL folder.

Caution: If you have previous versions of the ProUCL, which were installed on your computer, you 

should remove or rename the directory in which earlier ProUCL versions are currently located.

Getting Started

The functionality and the use of the methods and options available in ProUCL 4.0 have been illustrated 

using Screen Shots of output screen generated by ProUCL 4.0. ProUCL 4.0 uses a pull-down menu 

structure, similar to a typical Windows program. 

The screen below appears when the program is executed. 
15

The screen consists of three main window panels:

 The MAIN WINDOW displays data sheets and outputs from the procedure used.

 The NAVIGATION PANEL displays the name of data sets and all generated outputs. 

o At present, the navigation panel can hold at most 20 outputs. In order to see more 

files (data files or generated output files), one can click on Widow Option.

 The LOG PANEL displays transactions in green, warnings in orange, and errors in red. 

For an example, when one attempts to run a procedure meant for censored data sets on a 

full-uncensored data set, ProUCL 4.0 will print out a warning message in orange in this 

panel. 

o Should both panels be unnecessary, you can click or choose Configure ►

Panel ON/OFF. 

The use of this option will give extra space to see and print out the statistics of interest. For an example, 

one may want to turn off these panels when multiple variables (e.g., multiple Q-Q plots) are analyzed and 

GOF statistics and other statistics may need to be captured for all of the variables.

← Main 

 Window Navigation

Panel 

 ↓

← Log Panel
16 

Chapter 1

Guidance on the Use of Statistical Methods and Associated 

Minimum Sample Size Requirements 

This chapter briefly describes the differences between the various statistical limits (e.g., UCLs, UPLs, 

UTLs) often used to estimate the environmental parameters of interest including exposure point 

concentration (EPC) terms and background threshold values (BTVs). Suggestions are provided about the 

minimum sample size requirements needed to use statistical inferential methods to estimate the 

environmental parameters: EPC terms, BTVs and not-to-exceed values, and to compare site data with 

background data or with some pre-established reference limits (e.g., preliminary remediation goals 

(PRGs), action levels, compliance limits). It is noted that several EPA guidance documents (e.g., EPA 

1997, 2002a, 2006) discuss in details about data quality objectives (DQOs) and sample size 

determinations based upon those DQOs needed for the various statistical methods used in environmental 

applications. 

Also, appropriate sample collection methods (e.g., instruments, sample weights, discrete or composite, 

analytical methods) depend upon the medium (e.g., soil, sediment, water) under consideration. For an

example, Gerlach and Nocerino (EPA, 2003) describe optimal soil sample (based upon Gy theory) 

collection methods. Therefore, the topics of sample size determination based upon DQOs, data validation, 

and appropriate sample collection methods for the various environmental media are not considered in 

ProUCL 4.0 and its associated Technical Guide. It is assumed that data sets to be used in ProUCL are of 

good quality, and whenever possible have been obtained using the guidance provided in various EPA 

(2003, 2006) documents. It is the users‟ responsibility to assure that adequate amount of data have been 

collected, and the collected data are of good quality. 

Note: In ProUCL 4.0 and its associated guidance documents, emphasis is given on the practical 

applicability and appropriate use of statistical methods needed to address statistical issues arising in risk 

management, background versus site evaluation studies, and various other environmental applications. 

Specifically, guidance on minimum sample size requirements as provided in this chapter is useful when 

data have already been collected, or it is not possible (e.g., due to resource limitations) to collect the 

number of samples obtained using DQO processes as described in EPA (2006). 

Decisions based upon statistics obtained using data sets of small sizes (e.g., 4 to 6 detected observations) 

cannot be considered reliable enough to make a remediation decision that affects human health and the 

environment. For an example, a background data set of size 4 to 6 is not large enough to characterize 

background population, to compute BTV values, or to perform background versus site comparisons. In 

order to perform reliable and meaningful statistical inference (estimation and hypothesis testing), one 

should determine the sample sizes that need to be collected from the populations under investigation 

using appropriate DQO processes and decision error rates (EPA, 2006). However, in some cases, it may 

not be possible (e.g., resource constraints) to collect the same number of samples recommended by the 

DQO process. In order to address such cases, minimum sample size requirements for background and site 

data sets are described.

The use of an appropriate statistical method depends upon the environmental parameter(s) being 

estimated or compared. The measures of central tendency (e.g., means, medians, or their upper confidence 

limits (UCLs)) are often used to compare site mean concentrations (e.g., after remediation activity) with a 
17

cleanup standard, Cs, representing some central tendency measure of a reference area or other known 

threshold representing a measure of central tendency. The upper threshold values, such as the compliance 

limits (e.g., alternative concentration limit (ACL), maximum concentration limit (MCL)), or not-toexceed values, are used when individual point-by-point observations are compared with those not-toexceed values or other compliance limit. It should be noted that depending upon whether the 

environmental parameters (e.g., BTVs, not-to-exceed value, EPC term, cleanup standards) are known or 

unknown, different statistical methods with different data requirements are needed to compare site 

concentrations with pre-established (known) or estimated (unknown) cleanup standards and BTVs. 

ProUCL 4.0 has been developed to address issues arising in exposure assessment, risk assessment, and 

background versus site comparison applications. Several upper limits, and single- and two-sample 

hypotheses testing approaches, for both full uncensored and left-censored data sets, are available in 

ProUCL 4.0. The details of the statistical and graphical methods included in ProUCL 4.0 can be found in 

the ProUCL Technical Guidance. In order to make sure that the methods in ProUCL 4.0 are properly 

used, this chapter provides guidance on: 

1. analysis of site and background areas and data sets, 

2. collection of discrete or composite samples, 

3. appropriate use of the various upper limits, 

4. guidance regarding minimum sample sizes,

5. point-by-point comparison of site observations with BTVs,

6. use of hypotheses testing approaches, 

7. using small data sample sets, 

8. use of maximum detected value, and

9. discussion of ProUCL usage for special cases. 

1.1 Background Data Sets

The project team familiar with the site should identify and chose a background area. Depending upon the 

site activities and the pollutants, the background area can be site-specific or a general reference area. An 

appropriate random sample of independent observations should be collected from the background area. A 

defensible background data set should represent a “single” background population (e.g., representing 

pristine site conditions before any of the industrial site activities) free of contaminating observations such 

as outliers. In a background data set, outliers may represent potentially contaminated observations from 

impacted site areas under study or possibly from other polluted site(s). This scenario is common when 

background samples are obtained from the various onsite areas (e.g., large federal facilities). Outlying 

observations should not be included in the estimation (or hypotheses testing procedures) of the BTVs. 

The presence of outliers in the background data set will yield distorted estimates of the BTVs and 

hypothesis testing statistics. The proper disposition of outliers to include or not include them in the data 

set should be decided by the project team. 

Decisions based upon distorted statistics can be incorrect, misleading, and expensive. It should be noted 

that the objective is to compute background statistics based upon the majority of the data set representing 

the dominant background population, and not to accommodate a few low probability outliers that may 

also be present in the background data set. A couple of simple classical outlier tests (Dixon and Rosner 

tests) are available in ProUCL 4.0. Since these classical tests suffer from masking effects (e.g., extreme 

outliers may mask the occurrence of other intermediate outliers), it is suggested that these classical outlier 

tests should always be supplemented with graphical displays such as a box plot or a Q-Q plot. The use of 
18 

robust and resistant outlier identification procedures (Singh and Nocerino, 1995, Rousseeuw and Leroy, 

1987) is recommended when multiple outliers may be present in a data set. Those methods are beyond the 

scope of ProUCL 4.0. 

An appropriate background data set of a reasonable size (preferably computed using DQO processes) is 

needed to characterize a background area including computation of upper limits (e.g., estimates of BTVs, 

not-to-exceed values) based upon background data sets and also to compare site and background data sets 

using hypotheses testing approaches. As mentioned before, a small background data set of size 4 to 6 is 

not large enough to compute BTVs or to perform background versus site comparisons. At the minimum, a 

background sample should have at least 8 to 10 (more observations are preferable) detected observations 

to estimate BTVs or to use hypotheses testing approaches.

1.2 Site Data Sets

A defensible data set from a site population (e.g., AOC, EA, RU, group of monitoring wells) should be 

representative of the site area under investigation. Depending upon the site areas under investigation, 

different soil depths and soil types may be considered as representing different statistical populations. In 

such cases, background-versus-site comparisons may have to be conducted separately for each of those 

site sub-populations (e.g., surface and sub-surface layers of an AOC, clay and sandy site areas). These 

issues, such as comparing depths and soil types, should also be considered in a planning and sampling 

design before starting to collect samples from the various site areas under investigation. Specifically, the 

availability of an adequate amount of representative site data is required from each of those site subpopulations defined by sample depths, soil types, and the various other characteristics. For detailed 

guidance on soil sample collections, the reader is referred to Gerlach and Nocerino (EPA (2003)). 

The site data collection requirements depend upon the objective of the study. Specifically, in backgroundversus-site comparisons, site data are needed to perform:

 Individual point-by-point site observation comparisons with pre-established or estimated 

BTVs, PRGs, cleanup standards, and not-to-exceed-values. Typically, this approach is 

used when only a small number (e.g., < 4 to 6) of detected site observations (preferably 

based upon composite samples) are available which need to be compared with BTVs and 

not-to-exceed values. 

 Single sample hypotheses tests to compare site data with pre-established cleanup 

standards, Cs (e.g., representing a measure of central tendency); or with BTVs and not-toexceed values (used for tests for proportions and percentiles). The hypotheses testing 

approaches are used when enough site data are available. Specifically, when at least 8 to 

10 detected (more are desirable) site observations are available, it is preferable to use 

hypotheses testing approaches to compare site observations with specified threshold 

values. The use of hypotheses testing approaches can control the two types (Type 1 and 

Type 2) of error rates more efficiently than the point-by-point individual observation 

comparisons. This is especially true as the number of point-by-point comparisons 

increases. This issue is illustrated by the following table summarizing the probabilities of 

exceedances (false positive error rate) of the background threshold value (e.g., 95th

percentile) by site observations, even when the site and background populations have 

comparable distributions. The probabilities of these chance exceedances increase as the 

sample size increases.
19

Sample Size Probability of Exceedance

1 0.05

2 0.10

5 0.23

8 0.34

10 0.40

12 0.46

64 0.96

 Two-sample hypotheses testing to compare site data distribution with background data 

distribution to determine if the site concentrations are comparable to background 

concentrations. Adequate amount of data need to be made available from the site as well 

as the background populations. It is preferable to collect at least 8 to 10 detected 

observations from each of the population under comparison.

1.3 Discrete Samples or Composite Samples?

In a data set (background or site), collected samples should be either all discrete or all composite. In 

general, both discrete and composite site samples may be used for individual point-by-point site 

comparisons with a threshold value, and for single and two-sample hypotheses testing applications. 

 If possible, the use of composite site samples is preferred when comparing individual 

point-by-point site observations from an area (e.g., area of concern (AOC), remediation 

unit (RU), exposure area (EA)) with pre-established or estimated BTV, compliance limit 

(CL), or other not-to-exceed value. This comparison approach is useful when few (< 4 to 

6) detected site observations are compared with a pre-established or estimated BTV or 

other not-to-exceed threshold.

 When using a single sample hypothesis testing approach, site data can be obtained by 

collecting all discrete or all composite samples. The hypothesis testing approach is used 

when many (e.g., exceeding 8 to 10) detected site observations are available. Details of 

the single sample hypothesis approaches are widely available in EPA documents (1989, 

1997, and 2006). Selected single sample hypotheses testing procedures are available in 

ProUCL 4.0.

 If a two-sample hypotheses testing approach is used to perform site versus background 

comparisons, then samples from both of the populations should be either all discrete or 

all composite samples. The two-sample hypothesis testing approach is used when many 

(e.g., exceeding 8 to 10) site, as well as background, observations are available. For better 

and more accurate results with higher statistical power, the availability of more 

observations (e.g., exceeding 10-15) from each of the two populations is desirable, 

perhaps based upon an appropriate DQO process, as described in an EPA guidance 

document (2006).

1.4 Upper Limits and Their Use

The appropriate computation and use of statistical limits depend upon their applications and the 

parameters (e.g., EPC term, not-to-exceed value) they are supposed to be estimating. Depending upon the 

objective of the study, a pre-specified cleanup standard, Cs, or a risk-based cleanup (RBC) can be viewed 
20 

as to represent: 1) as average contaminant concentration; or 2) a not-to-exceed upper threshold value. 

These two threshold values, an average value, 0, and a not-to-exceed value, A0, represent two 

significantly different parameters, and different statistical methods and limits are used to compare the site 

data with these two different parameters or threshold values. Statistical limits, such as an upper 

confidence limit (UCL) of the population mean, an upper prediction limit (UPL) for an independently 

obtained “single” observation, or independently obtained k observations (also called future k 

observations, next k observations, or k different observations), upper percentiles, and upper tolerance 

limits (UTLs), are often used to estimate the environmental parameters, including the EPC terms, 

compliance limits (e.g., ACL, MLC), BTVs, and other not-to-exceed values. Here, UTL95%-95% 

represents a 95% confidence limit of the 95th percentile of the distribution of the contaminant under study.

It is important to understand and note the differences between the uses and numerical values of these 

statistical limits so that they can be properly used. Specifically, the differences between UCLs and UPLs 

(or upper percentiles), and UCLs and UTLs should be clearly understood and acknowledged. A UCL with 

a 95% confidence limit (UCL95) of the mean represents an estimate of the population mean (measure of 

the central tendency of a data distribution), whereas a UPL95, a UTL95%-95%, and an upper 95th

percentile represent estimates of a threshold value in the upper tail of the data distribution. Therefore, a 

UCL95 should represent a smaller number than an upper percentile or an upper prediction limit. Also, 

since a UTL 95%-95% represents a 95% UCL of the upper 95th percentile, a UTL should be  the 

corresponding UPL95 and the 95th upper percentile. Typically, it is expected that the numerical values of 

these limits should follow the order given as follows:

Sample Mean  UCL95 of Mean  Upper 95th Percentile  UPL95 of a Single Observation  UTL95%-

95% 

It should also be pointed out that as the sample size increases, a UCL95 of the mean approaches 

(converges to) the population mean, and a UPL95 approaches the 95th percentile. The differences among 

the various upper limits are further illustrated in Example 1-1 below. It should be noted that, in some 

cases, these limits might not follow the natural order described above. This is especially true when the 

upper limits are computed based upon a lognormal distribution (Singh, Singh, and Engelhardt, 1997). It is 

well known that a lognormal distribution-based H-UCL95 (Land‟s UCL95) often yields unstable and 

impractically large UCL values. An H-UCL95 often becomes larger than UPL95 and even larger than a 

UTL 95%-95%. This is especially true when dealing with skewed data sets of smaller sizes. Moreover, it 

should also be noted that in some cases, a H-UCL95 becomes smaller than the sample mean, especially 

when the data are mildly skewed to moderately skewed and the sample size is large (e.g., > 50, 100). 

Example 1-1: Consider a simple site-specific background data set associated with a Superfund site. The 

data set (given in Appendix 5 of the revised Guidance for Comparing Background and Chemical 

Concentrations in Soil for CERCLA Sites (EPA, 2002b)) has several inorganic contaminants of potential 

concern, including aluminum, arsenic, chromium, iron, and lead. It is noted that iron concentrations 

follow a normal distribution. Upper limits for the iron data set are summarized in Table 1-1. It is noted 

that the upper limits do follow the order as described above.

Table 1-1. Computation of Upper Limits for Iron (Normally Distributed)

Mean Median Min Max UCL95

UPL95 for a 

Single 

Observation

UPL95 for 4 

Observations UTL95/95 95% Upper Percentile

9618 9615 3060 18700 11478 18145 21618 21149 17534
21

A 95% UCL (UCL95) of the mean is the most commonly used limit in environmental applications. For an 

example, a 95% UCL of mean is used as an estimate of the EPC. A UCL95 should not be used to estimate 

a background threshold value (a value in the upper tail of the background data distribution) to be 

compared with individual site observations. There are many instances in background evaluations and 

background versus site comparison studies, when it is not appropriate to use a 95% UCL. Specifically, 

when point-by-point site observations are to be compared with a BTV, then that BTV should be estimated 

(or represented) by a limit from the upper tail of the reference set (background) data distribution. 

A brief discussion about the differences between the applications and uses of the various statistical limits 

is provided below. This will assist a typical user in determining which upper limit (e.g., UCL95 or 

UPL95) to use to estimate the parameter of interest (e.g., EPC or BTV).

 A UCL represents an average value that should be compared with a threshold value also 

representing an average value (pre-established or estimated), such as a mean cleanup 

standard, Cs. For an example, a site 95% UCL exceeding a cleanup value, Cs, may lead to 

the conclusion that the cleanup level, Cs, has not been attained by the site area under 

investigation. It should be noted that UCLs of means are typically computed based upon 

the site data set.

 When site averages (and not individual site observations) are compared with a threshold 

value (pre-determined or estimated), such as a PRG or a RBC, or with some other 

cleanup standard, Cs, then that threshold should represent an average value, and not a notto-exceed threshold value for individual observation comparisons. 

 A UCL represents a “collective” measure of central tendency, and it is not appropriate to 

compare individual site observations with a UCL. Depending upon data availability,

single or two-sample hypotheses testing approaches are used to compare site averages: 

with a specified or pre-established cleanup standard (single sample hypothesis), or with 

the background population averages (two-sample hypothesis).

 A UPL, an upper percentile, or an UTL represents an upper limit to be used for point-bypoint individual site observation comparisons. UPLs and UTLs are computed based upon 

background data sets, and individual site observations are compared with those limits. A 

site observation for a contaminant exceeding a background UTL or UPL may lead to the 

conclusion that the contaminant is a contaminant of potential concern (COPC) to be 

included in further risk evaluation and risk management studies. 

 When individual point-by-point site observations are compared with a threshold value 

(pre-determined or estimated) of a background population or some other threshold and 

compliance limit value, such as a PRG, MLC, or ACL, then that threshold value should 

represent a not-to-exceed value. Such BTVs or not-to-exceed values are often estimated 

by a 95% UPL, UTL 95%-95%, or by an upper percentile. ProUCL 4.0 can be used to 

compute any of these upper limits based upon uncensored data sets as well as data sets 

with nondetect values. 

 As the sample size increases, a UCL approaches the sample mean, and a UPL95 

approaches the corresponding 95th upper percentile.
22 

 It is pointed out that the developers of ProUCL 4.0 prefer the use of a 95% UPL (UPL95) 

as an estimate of BTV or a not-to-exceed value. As mentioned before, the option of 

comparing individual site observations with a BTV (specified or estimated) should be 

used when few (< 4 to 6) detected site observations (preferably composite values) are to 

be compared with a BTV.

 When enough (e.g., > 8 to 10) detected site observations are available, it is preferred to 

use hypotheses testing approaches. Specifically, single sample hypotheses testing 

(comparing site to a specified threshold) approaches should be used to perform site 

versus a known threshold comparison; and two-sample hypotheses testing (provided 

enough background data are also available) approaches should be used to perform site 

versus background comparison. Several parametric and nonparametric single and twosample hypotheses testing approaches are available in ProUCL 4.0.

It is re-emphasized that only averages should be compared with the averages or UCLs, and individual site 

observations should be compared with UPLs, upper percentiles, or UTLs. For an example, the comparison 

of a 95% UCL of one population (e.g., site) with a 90% or 95% upper percentile of another population 

(e.g., background) cannot be considered fair and reasonable as these limits (e.g., UCL and UPL) estimate 

and represent different parameters. It is hard to justify comparing a UCL of one population with a UPL of 

the other population. Conclusions (e.g., site dirty or site clean) derived by comparing UCLs and UPLs, or 

UCLs and upper percentiles as suggested in Wyoming DEQ, Fact Sheet #24 (2005), cannot be considered 

fair and reliable. Specifically, the decision error rates associated with such comparisons can be 

significantly different from the specified (e.g., Type I error = 0.1, Type II error = 0.1) decision errors. 

1.5 Point-by-Point Comparison of Site Observations with BTVs, 

Compliance Limits, and Other Threshold Values

Point-by-point observation comparison method is used when a small number (e.g., 4 to 6 locations) of 

detected site observations are compared with pre-established or estimated BTVs, screening levels, or 

preliminary remediation goals (PRGs). In this case, individual point-by-point site observations (preferably 

based upon composite samples from various site locations) are compared with estimated or preestablished background (e.g., USGS values) values, PRGs, or some other not-to-exceed value. Typically, 

a single exceedance of the BTV, PRG, or of a not-to-exceed value by a site (or from a monitoring well) 

observation may be considered as an indication of contamination at the site area under investigation. The 

conclusion of an exceedance by a site value is some times confirmed by re-sampling (taking a few more 

collocated samples) that site location (or a monitoring well) exhibiting contaminant concentration in 

excess of the BTV or PRG. If all collocated (or collected during the same time period) sample 

observations collected from the same site location (or well) exceed the PRG (or MLC) or a not-to-exceed 

value, then it may be concluded that the location (well) requires further investigation (e.g., continuing 

treatment and monitoring) and cleanup. 

When BTV contaminant concentrations are not known or pre-established, one has to collect, obtain, or 

extract a data set of an appropriate size that can be considered as representative of the site related 

background. Statistical upper limits are computed using the data set thus obtained, which are used as 

estimates of BTVs and not-to-exceed values. It should be noted that in order to compute reasonably 

reliable and accurate estimates of BTVs and not-to-exceed values based upon a background (or reference) 

data set, enough background observations (minimum of 8 to 10) should be collected, perhaps using an 

appropriate DQO process as described in EPA (2006). Typically, background samples are collected from 

a comparable general reference area or site-specific areas that are known to be free of contamination due 
23

to any of the site related activities. Several statistical limits can be used to estimate the BTVs based upon 

a defensible data set of an adequate size. A detailed description of the computation and estimation of 

BTVs is given in Chapter 5 (for uncensored data sets) and in Chapter 6 for data sets with nondetects of 

the revised background guidance document. Once again, the use of this point-by-point comparison 

method is recommended when not many (e.g., < 4 to 6) site observations are to be compared with 

estimated BTVs or PRGs. An exceedance of the estimated BTV by a site value may be considered as an 

indication of the existing or continuing contamination at the site. 

Note: When BTVs are not known, it is suggested that at least 8 to 10 (more are preferable) detected 

representative background observations be made available to compute reasonably reliable estimates of 

BTVs and other not-to-exceed values. 

The point-by-point comparison method is also useful when quick turnaround comparisons are required. 

Specifically, when the decisions have to be made in real time by a sampling or screening crew, or when 

few detected site samples are available, then individual point-by-point site concentrations are compared 

either with pre-established PRGs, cleanup goals and standards, or with estimated BTVs and not-to-exceed 

values. The crew can use these comparisons to make the following informative decisions: 

1. Screen and identify the COPCs, 

2. Identify the polluted site AOCs,

3. Continue or stop remediation or excavation at a site AOC or a RU, or 

4. Move the cleanup apparatus and crew to the next AOC or RU.

During the screening phase, an exceedance of a compliance limit, action level, a BTV, or a PRG by site 

values for a contaminant may declare that contaminant as a COPC. Those COPCs are then included in 

future site remediation and risk management studies. During the remediation phase, an exceedance of the 

threshold value such as a compliance limit (CL) or a BTV by sample values collected from a site area (or 

a monitoring well (MW)) may declare that site area as a polluted AOC, or a hot spot requiring further 

sampling and cleanup. This comparison method can also be used to verify if the site concentrations (e.g., 

from the base or side walls of an excavated site area) are approaching or meeting PRG, BTV, or a cleanup 

standard after excavation has been conducted at that site area.

If a larger number of detected samples (e.g., greater than 8 to10) are available from the site locations 

representing the site area under investigation (e.g., RU, AOC, EA), then the use of hypotheses testing 

approaches (both single sample and two-sample) is preferred. The use of a hypothesis testing approach 

will control the error rates more tightly and efficiently than the individual point-by-point site observations 

versus BTV comparisons, especially when many site observations are compared with a BTV or a not-toexceed value.

Note: In background versus site comparison evaluations, scientists usually prefer the use of hypotheses 

testing approaches to point-by-point site observation comparisons with BTVs or not-to-exceed values. 

Hypotheses testing approaches require the availability of larger data sets from the populations under 

investigation. Both single sample (used when BTVs, not-to-exceed values, compliance limits, or cleanup 

standards are known and pre-established) and two-sample (used when BTVs and compliance limits are 

unknown) hypotheses testing approaches are available in ProUCL 4.0. 
24 

1.6 Hypothesis Testing Approaches and Their Use

Both single sample and two-sample hypotheses testing approaches are used to make cleanup decisions at 

polluted sites, and also to compare contaminant concentrations of two (e.g., site versus background) or 

more (several monitoring wells (MWs)) populations. The uses of hypotheses testing approaches in those 

environmental applications are described as follows.

1.6.1 Single Sample Hypotheses – BTVs and Not-to-Exceed Values are Known 

(Pre-established)

When pre-established BTVs and not-to-exceed values are used, such as the USGS background values 

(Shacklette and Boerngen (1984)), thresholds obtained from similar sites, or pre-established not-to-exceed 

values, PRGs, or RBCs, there is no need to extract, establish, or collect a background or reference data 

set. When the BTVs and cleanup standards are known, one-sample hypotheses are used to compare site 

data (provided enough site data are available) with known and pre-established threshold values. It is 

suggested that the project team determine (e.g., using DQO) or decide (depending upon resources) about 

the number of site observations that should be collected and compared with the “pre-established” 

standards before coming to a conclusion about the status (clean or polluted) of the site area (e.g., RU, 

AOC) under investigation. When the number of available detected site samples is less than 4 to 6, one 

might perform point-by-point site observation comparisons with a BTV; and when enough detected site 

observations (> 8 to 10, more are preferable) are available, it is desirable to use single sample hypothesis 

testing approaches. 

Depending upon the parameter (e.g., the average value, 0, or a not-to-exceed value, A0), represented by 

the known threshold value, one can use single sample hypothesis tests for population mean (t-test, sign 

test) or single sample tests for proportions and percentiles. The details of the single sample hypotheses 

testing approaches can be found in EPA (2006) and the Technical Guide for ProUCL 4.0. Several single 

sample tests listed as follows are available in ProUCL 4.0. 

One-Sample t-Test: This test is used to compare the site mean, , with a specified cleanup standard, Cs,

where the cleanup standard, Cs, represents an average threshold value, 0. The Student‟s t-test (or a UCL 

of mean) is often used (assuming normality of site data or when site sample size is large such as larger 

than 30, 50) to determine the attainment of cleanup levels at a polluted site after some remediation 

activities.

One-Sample Sign Test or Wilcoxon Signed Rank (WSR) Test: These tests are nonparametric tests and can 

also handle nondetect observations provided all nondetects (e.g., associated detection limits) fall below 

the specified threshold value, Cs. These tests are used to compare the site location (e.g., median, mean) 

with a specified cleanup standard, Cs, representing a similar location measure.

One-Sample Proportion Test or Percentile Test: When a specified cleanup standard, A0, such as a 

PRG or a BTV represents an upper threshold value of a contaminant concentration distribution 

(e.g., not-to-exceed value, compliance limit) rather than the mean threshold value, 0, of the 

contaminant concentration distribution, then a test for proportion or a test for percentile (or 

equivalently a UTL 95%-95%) can be used to compare site proportion or site percentile with the 

specified threshold or action level, A0. This test can also handle ND observations provided all 

NDs are below the compliance limit.
25

In order to obtain reasonably reliable estimates and test statistics, an adequate amount of representative 

site data (8 to 10 detected observations) is needed to perform the hypotheses tests. As mentioned before, 

in case only a few (e.g., < 4 to 6) detected site observations are available, then point-by-point site 

concentrations may be compared with the specified action level, A0.

1.6.2 Two-Sample Hypotheses – When BTVs and Not-to-Exceed Values are Unknown

When BTVs, not-to-exceed values, and other cleanup standards are not available, then site data are 

compared directly with the background data. In such cases, a two-sample hypothesis testing approach can 

be used to perform site versus background comparisons. Note that this approach can be used to compare 

concentrations of any two populations including two different site areas or two different monitoring wells 

(MWs). In order to use and perform a two-sample hypothesis testing approach, enough data should be 

available (collected) from each of the two populations under investigation. Site and background data 

requirements (e.g., based upon DQOs) to perform two-sample hypothesis test approaches are described in 

EPA (1989b, 2006), Breckenridge and Crockett (1995), and the VSP (2005) software package. While 

collecting site and background data, for better representation of populations under investigation, one may 

also want to account for the size of the background area (and site area for site samples) into sample size 

determination. That is, a larger number (> 10 to 15) of representative background (or site) samples should 

be collected from larger background (or site) areas. As mentioned before, every effort should be made to 

collect as many samples as determined using DQO processes as described in EPA documents (2006).

The two-sample (or more) hypotheses approaches are used when the site parameters (e.g., mean, shape, 

distribution) are being compared with the background parameters (e.g., mean, shape, distribution). The 

two-sample hypotheses testing approach is also used when the cleanup standards or screening levels are 

not known a priori, and they need to be estimated based upon a data set from a background or reference 

population. Specifically, two-sample hypotheses testing approaches are used to compare 1) the average 

contaminant concentrations of two or more populations such as the background population and the 

potentially contaminated site areas, or 2) the proportions of site and background observations exceeding a 

pre-established compliance limit, A0. In order to derive reliable conclusions with higher statistical power 

based upon hypothesis testing approaches, enough data (e.g., minimum of 8 to 10 samples) should be 

available from all of the populations under investigation. It is also desirable to supplement statistical 

methods with graphical displays, such as the double Q-Q plots, or side-by-side multiple box plots, as 

available in ProUCL 4.0. Several parametric and nonparametric two-sample hypotheses testing 

approaches, including Student‟s t-test, the Wilcoxon-Mann-Whitney (WMW) test, Gehan‟s test, and 

quantile test are included in ProUCL 4.0. Details of those methods are described in the ProUCL 4.0 

Technical Guide. It should be noted that the WMW, Gehan, and quantile tests are also available for data 

sets with NDs. Gehan‟s test is specifically meant to be used on data sets with multiple detection limits. It 

is also suggested that for best and reliable conclusions, both the WMW and quantile tests should be used 

on the same data set. The details of these two tests with examples are given in EPA (1994, 2006).

The samples collected from the two (or more) populations should all be of the same type obtained using 

similar analytical methods and apparatus. In other words, the collected site and background samples 

should be all discrete or all composite (obtained using the same design and pattern), and be collected from 

the same medium (soil) at similar depths (e.g., all surface samples or all subsurface samples) and time 

(e.g., during the same quarter in groundwater applications) using comparable (preferably same) analytical 

methods. Good sample collection methods and sampling strategies are given in EPA (1996, 2003) 

guidance documents.
26 

1.7 Minimum Sample Size Requirements 

Due to resource limitations, it may not be possible (nor needed) to sample the entire population (e.g., 

background area, site area, areas of concern, exposure areas) under study. Statistics is used to draw 

inference(s) about the populations (clean, dirty) and their known or unknown parameters (e.g., 

comparability of population means, not-to-exceed values, upper percentiles, and spreads) based upon 

much smaller data sets (samples) collected from those populations under study. In order to determine and 

establish BTVs, not-to-exceed values, or site-specific screening levels, defensible data set(s) of 

appropriate size(s) needs to be collected from background areas (e.g., site-specific, general reference or 

pristine area, or historical data). The project team and site experts should decide what represents a site 

population and what represents a background population. The project team should determine the 

population size and boundaries based upon all current and future objectives for the data collection. The 

size and area of the population (e.g., a remediation unit, area of concern, or an exposure unit) may be 

determined based upon the potential land use, and other exposure and risk management objectives and 

decisions. Moreover, appropriate effort should be made to properly collect soil samples (e.g., methods 

based upon Gy sampling theory), as described in Gerlach and Nocerino (2003).

Using the collected site and background data sets, statistical methods supplemented with graphical 

displays are used to perform site versus background comparisons. The test results and statistics obtained 

by performing such site versus background comparisons are used to determine if the site and background 

level contaminant concentration are comparable; or if the site concentrations exceed the background 

threshold concentration level; or if an adequate amount of cleanup and remediation approaching the BTV 

or a cleanup level have been performed at polluted areas (e.g., AOC, RU) of the site under study. 

In order to perform statistical inference (estimation and hypothesis testing), one needs to determine the 

sample sizes that need to be collected from the populations (e.g., site and background) under investigation 

using appropriate DQO processes (EPA 2006). However, in some cases, it may not be possible to collect 

the same number of samples as determined by using a DQO process. For example, the data might have 

already been collected (often is the case in practice) without using a DQO process, or due to resource 

constraints, it may not be possible to collect as many samples as determined by using a DQO-based 

sample size formula. It is observed that, in practice, the project team and the decision makers may not 

collect enough background samples, perhaps due to various resource constraints. However, every effort 

should be made to collect at least 8 to 10 (more are desirable) background observations before using 

methods as incorporated in ProUCL 4.0. The minimum sample size recommendations as described here 

are useful when resources are limited (as often is the case), and it may not be possible to collect as many 

background and site (e.g., AOC, EU) samples as computed using DQOs and the sample size 

determination formulae given in the EPA (2006). Some minimum sample size requirements are also given 

in Fact Sheet #24, prepared by Wyoming Department of Environmental Quality (June 2005).

As mentioned before, the topics of DQO processes and the sample size determination are described in 

detail in the EPA (2006) guidance document. Therefore, the sample size determination formulae based 

upon DQO processes are not included in ProUCL 4.0 and its Technical Guide. It should be noted that 

DQO-based sample size determination routines are available in DataQUEST (EPA, 1997) and VSP 

(2005) software packages. Guidance and suggestions are provided on the minimum number of 

background and site samples needed to be able to use statistical methods for the computation of upper 

limits, and to perform single sample tests, two-sample tests such as t-test, and the Wilcoxon-MannWhitney (WMW) test. The minimum sample size recommendations (requirements) as described here are 
27

made so that reasonably reliable estimates of EPC terms and BTVs, and defensible values of test statistics 

for single or two-sample hypotheses tests (e.g., t-test, WMW test), can be computed. 

Finally, it is also important to note and understand the differences between the following two minimum 

sample size concepts:

1. Minimum sample needed to compute a statistic based upon theoretical formulae; and

2. Minimum sample size needed to compute a reliable and usable decision statistics. 

Even though, most of the decision statistics such as upper confidence limits (UCLs), upper prediction 

limits (UPLs), and upper tolerance limits (UTLs) can be computed based upon very small samples of 

sizes 2, 3, and 4, those decision statistics are not reliable and representative enough to make defensible 

and correct cleanup and remediation decisions. Use of such statistics should be avoided.

Specific recommendations regarding the minimum sample size requirement (when data sets of DQOs 

based sample sizes cannot be collected) needed to compute reliable and usable decision statistics have 

also been described in this chapter. It should be pointed out that those minimum sample size 

recommendation (at least 8-10 observations) should be used only when samples of size determined by a 

DQO process (EPA, 2006) cannot be collected (e.g., due to resource constraints). The intention of the 

developers of ProUCL 4.0 is to provide statistically rigorous and defensible methods and decision 

statistics. Success of the applicability of a statistical method depends upon the quality and quantity of the 

available data set. It is always desirable to collect appropriate number of samples based upon data quality 

objectives (DQOs) so that reliable decision statistics (e.g., UPLs, UCLs, and hypotheses test statistics) 

can be computed to make appropriate decisions. 

1.7.1 Minimum Sample Size Requirements to Compute Decision Statistics for Data without 

NDs 

It was noted by the developers of ProUCL software that some users of earlier versions of ProUCL (e.g., 

ProUCL 3.0 and ProUCL 4.0) were computing decisions statistics (e.g., UCLs, UPLs) based upon small 

data sets of sizes 2, 3 etc. As a result, in later versions of ProUCL such as ProUCL 4.00.02, the 

developers restricted the use of ProUCL for samples of size at least 5. ProUCL 4.00.02 and higher 

versions will not compute decision statistics (e.g., UCLs, UPLs, UTLs) based upon samples of sizes less 

than 5. Some users did complain about being not able to compute decision statistics based upon samples 

of size 3 or 4; but that is fine as one should not be computing decision statistics based upon such small 

samples. It is desirable that the ProUCL users follow the sample size requirements as described in this 

chapter. 

At present, ProUCL 4.00.02 and higher versions will not compute any decision statistics such as UCLs 

and UPLs, UTLs for data sets of size less than 5 (without NDs). It may compute other summary statistics 

and graphs but will not compute decision making statistics. For such small data sets of size less than 5, 

ProUCL 4.00.02 provides warning messages informing the user about the potential deficiencies present in 

his data set submitted to ProUCL.
28 

1.7.2 Minimum Sample Size Requirements for Data Sets with NDs

ProUCL 4.00.02 and higher versions will not compute any decision statistics based upon data sets of sizes 

less than 5 consisting of NDs. Moreover, for data sets of size at least 5, no decision statistics will be 

computed when not more than one detected observation is present in the data set. For such extreme data 

situations, ProUCL 4.00.02 provides some warning messages regarding the lack of appropriate amount of 

data. For data sets of size 5 or larger with only two detected values, ProUCL 4.00.02 will produce only 

Kaplan –Meier method based decision statistics (UCLs, UPLs, UTLs); and for data sets of size 5 or larger 

with 3 detected values, most nonparametric and parametric (except for gamma distribution based ) 

decision statistics will be computed and printed. For data sets of size 5 or higher with 4 or more detected 

observations, ProUCL 4.00.02 will produce values for all parametric and nonparametric decision 

statistics.

For all small data sets (e.g., size <8-10), ProUCL 4.00.02 informs the user by providing appropriate 

warning messages about the potential deficiencies present in his data set submitted to ProUCL. Some 

recommendations about how to determine a value of a decision statistic based upon data sets consisting of 

all or only a few (1 or 2) detected values are also provided in this chapter. It is suggested that the project 

team and experts associated with the site should come to an agreement about an appropriate value that 

may be used for the decision statistic under consideration. 

1.7.3 Minimum Sample Size for Estimation and Point-by-Point Site Observation Comparisons

 Point-by-point observation comparison method is used when a small number (e.g., 4 to 6 

locations) of detected site observations are compared with pre-established or estimated 

BTVs, screening levels, or PRGs. In this case, individual point-by-point site observations 

(preferably based upon composite samples from various site locations) are compared with 

estimated or pre-established background (e.g., USGS values) values, PRGs, or some 

other not-to-exceed value. 

 When BTV contaminant concentrations are not known or pre-established, one has to 

collect, obtain, or extract a data set of an appropriate size that can be considered as 

representative of the site related background. Statistical upper limits are computed using 

the data set thus obtained; which are used as estimates of BTVs and not-to-exceed values. 

It should be noted that in order to compute reasonably reliable and accurate estimates of 

BTVs and not-to-exceed values based upon a background (or reference) data set, enough 

background observations (minimum of 8 to 10) should be collected perhaps using an 

appropriate DQO process as described in EPA (2006). Typically, background samples are 

collected from a comparable general reference area or a site-specific area.

 When enough (e.g., > 8 to 10) detected site observations are available, it is preferred to 

use hypotheses testing approaches. Specifically, single sample hypotheses testing 

(comparing site to a specified threshold) approaches should be used to perform site 

versus a known threshold comparison and two-sample hypotheses testing (provided 

enough background data are also available) approaches should be used to perform site 

versus background comparison. 
29

1.7.4 Minimum Sample Size Requirements for Hypothesis Testing Approaches

Statistical methods (as in ProUCL 4.0) used to estimate EPC terms, BTVs, PRGs, or to compare the site 

contaminant concentration data distribution with the background data distribution can be computed based 

upon small site and background data sets (e.g., of sizes 3, 4, 5, or 6). However, those statistics cannot be 

considered representative and reliable enough to make important cleanup and remediation decisions. It is 

recommended not to use those statistics to draw cleanup and remediation decisions potentially impacting 

the human health and the environment. It is suggested that the estimation and hypothesis testing methods 

as incorporated in ProUCL 4.0 may not be used on background data sets with fewer than 8 to 10 detected 

observations. Also, when using hypotheses testing approaches, it is suggested that the site and 

background data be obtained using an appropriate DQO process as described in EPA (2006). In case that 

is not possible, it is suggested that the project team at least collect 8 to 10 observations from each of the 

populations (e.g., site area, MWs, background area) under investigation.

Site versus background comparisons and computation of the BTVs depend upon many factors, some of 

which cannot be controlled. These factors include the site conditions, lack of historical information, site 

medium, lack of adequate resources, measurement and analytical errors, and accessibility of the site areas. 

Therefore, whenever possible, it is desirable to use more than one statistical method to perform site versus 

background comparison. The use of statistical methods should always be supplemented with appropriate 

graphical displays.

1.8 Sample Sizes for Bootstrap Methods

Several parametric and nonparametric (including bootstrap methods) UCL, UPL, and other limits 

computation methods for both full-uncensored data sets (without nondetects) and left-censored data sets 

with nondetects are available in ProUCL 4.0. It should be noted that bootstrap resampling methods are 

useful when not too few (e.g., < 10-15) and not too many (e.g., > 500-1000) detected observations are 

available. For bootstrap methods (e.g., percentile method, BCA bootstrap method, bootstrap t method), a 

large number (e.g., 1000, 2000) of bootstrap resamples (with replacement) are drawn from the same data 

set. Therefore, in order to obtain bootstrap resamples with some distinct values (so that statistics can be 

computed from each resample), it is suggested that a bootstrap method should not be used when dealing 

with small data sets of sizes less than 10-15. Also, it is not required to bootstrap a large data set of size 

greater than 500 or 1000; that is when a data set of a large size (e.g., > 1000) is available, there is no need 

to obtain bootstrap resamples to compute statistics of interest (e.g., UCLs). One can simply use a 

statistical method on the original large data set. Moreover, bootstrapping a large data set of size greater 

than 500 or 1000 will be time consuming.

1.9 Statistical Analyses by a Group ID 

The analyses of data categorized by a group ID variable such as: 1) Surface vs. Subsurface;

2) AOC1 vs. AOC2; 3) Site vs. Background; and 4) Upgradient vs Downgradient monitoring wells are 

quite common in many environmental applications. ProUCL 4.0 offers this option for data sets with and 

without nondetects. The Group Option provides a powerful tool to perform various statistical tests and 

methods (including graphical displays) separately for each of the group (samples from different 

populations) that may be present in a data set. For an example, the same data set may consist of samples 

from the various groups or populations representing site, background, two or more AOCs, surface, 

subsurface, monitoring wells. The graphical displays (e.g., box plots, Q-Q plots) and statistics 
30 

(computations of background statistics, UCLs, hypotheses testing approaches) of interest can be 

computed separately for each group by using this option. 

It should be pointed out that it is the users‟ responsibility to provide adequate amount of detected data to 

perform the group operations. For an example, if the user desires to produce a graphical Q-Q plot (using 

only detected data) with regression lines displayed, then there should be at least two detected points (to 

compute slope, intercept, sd) in the data set. Similarly if the graphs are desired for each of the group 

specified by the group ID variable, there should be at least two detected observations in each group 

specified by the group variable. ProUCL 4.0 generates a warning message (in orange color) in the lower 

panel of the ProUCL 4.0 screen. Specifically, the user should make sure that a variable with nondetects 

and categorized by a group variable should have enough detected data in each group to perform the 

various methods (e.g., GOF tests, Q-Q plots with regression lines) as incorporated in ProUCL 4.0.

1.10 Use of Maximum Detected Values as Estimates of Upper Limits

Some practitioners tend to use the maximum detected value as an estimate of the EPC term. This is 

especially true when the sample size is small such as  5, or when a UCL95 exceeds the maximum 

detected values (EPA, 1992b). Also, many times in practice, the BTVs and not-to-exceed values are 

estimated by the maximum detected value. This section discusses the appropriateness of using the 

maximum detected value as estimates of the EPC term, BTVs, or other nor-to-exceed values. 

1.10.1 Use of Maximum Detected Values to Estimate BTVs and Not-to-Exceed Values

It is noted that BTVs and not-to-exceed values represent upper threshold values in the upper tail of a data 

distribution; therefore, depending upon the data distribution and sample size, the BTVs and other not-toexceed values may be estimated by the maximum detected value. As described earlier, upper limits, such 

as UPLs, UTLs, and upper percentiles, are used to estimate the BTVs and not-to-exceed values. It is noted 

that a nonparametric UPL or UTL is often estimated by higher order statistics such as the maximum value 

or the second largest value (EPA 1992a, RCRA Guidance Addendum). The use of higher order statistics 

to estimate the UTLs depends upon the sample size. For an example: 1) 59 to 92 samples, a 

nonparametric UTL95%-95 is given by the maximum detected value; 2) 93 to 123 samples, a 

nonparametric UTL95%-95 is given by the second largest maximum detected value; and 3) 124 to 152 

samples, a UTL95%-95 is given by the third largest detected value in the sample. 

Note: Therefore, when a data set does not follow a discernable distribution, the maximum observed value 

(or other high order statistics) may be used as an estimate of BTV or a not-to-exceed value, provided the 

maximum value does not represent an outlier or a contaminating observation perhaps representing a hot 

location.

1.10.2 Use of Maximum Detected Values to Estimate EPC Terms

Some practitioners tend to use the maximum detected value as an estimate of the EPC term. This is 

especially true when the sample size is small such as  5, or when a UCL95 exceeds the maximum 

detected values (EPA, 1992b). Specifically, a RAGS document (EPA, 1992) suggests the use of the 

maximum detected value as a default value to estimate the EPC term when a 95% UCL (e.g., the H-UCL) 

exceeded the maximum value. ProUCL 4.0 can compute a 95% UCL of mean using several methods 

based upon normal, Gamma, lognormal, and non-discernable distributions. In past (e.g., EPA, 1992b), 
31

only two methods were used to estimate the EPC term based upon: 1) Student‟s t-statistic and a normal 

distribution, and 2) Land‟s H-statistic (1975) and a lognormal model. The use of H-statistic often yields 

unstable and impractically large UCL95 of the mean (Singh, Singh, and Iaci, 2002). For skewed data sets 

of smaller sizes (e.g., < 30, < 50), H-UCL often exceeds the maximum detected value. This is especially 

true when extreme high outliers may be present in the data set. Since the use of a lognormal distribution 

has been quite common (e.g., suggested as a default model in a RAGS document (EPA, 1992)), the 

exceedance of the maximum detected value by H-UCL95 is frequent for many skewed data sets of 

smaller sizes (e.g., < 30, < 50). It is 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. 

All of these occurrences result in the possibility of using the maximum detected value as an estimate of 

the EPC term. It should be pointed out that in some cases, the maximum observed value actually might 

represent a highly polluted outlying observation. Obviously, it is not desirable to use a highly polluted 

value as an estimate of average exposure (EPC term) for an exposure area. 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 that cannot be modeled by a gamma distribution, a 95% UCL of the mean should be 

computed using an appropriate distribution-free nonparametric method. 

It should be pointed out that 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. One needs to compute an average exposure and not the maximum exposure. It is unlikely 

that an individual will visit the location (e.g., in an EA) of the maximum detected value all of the time. 

One can argue that the use of this practice results in a conservative (higher) estimate of the EPC term. The 

objective is to compute an accurate estimate of the EPC term. Several other methods (instead of H-UCL) 

as described in EPA (2002), and included in ProUCL 4.0 (EPA 2007), are available to estimate the EPC 

terms. It is unlikely (but possible with outliers) that the UCLs based upon those methods will exceed the 

maximum detected value, unless some outliers are present in the data set. ProUCL 4.0 displays a warning 

message when the recommended 95% UCL (e.g., Hall‟s or bootstrap t UCL with outliers) of the mean 

exceeds the observed maximum concentration. When a 95% UCL does exceed the maximum observed 

value, ProUCL4.0 recommends the use of an alternative UCL computation method based upon the 

Chebyshev inequality. The detailed recommendations (as functions of sample size and skewness) for the 

use of those UCLs are summarized in ProUCL 3.0 User Guide (EPA, 2004). 

Singh and Singh (2003) studied the performance of the max test (using the maximum observed value as 

an estimate of the EPC term) via Monte Carlo simulation experiments. They 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. The use of the maximum value as an estimate of the EPC term also ignores most 

(except for maximum value) of the information contained in the data set. With the availability of so many 

UCL computation methods (15 of them), the developers of ProUCL 4.0 do not recommend using the 

maximum observed value as an estimate of the EPC term representing an average exposure by an 

individual over an EA. Also, for the distributions considered, the maximum value is not a sufficient 

statistic for the unknown population mean. 

Note: It is recommended that the maximum observed value NOT be used as an estimate of the EPC term 

representing average exposure contracted by an individual over an EA. For the sake of interested users, 

ProUCL displays a warning message when the recommended 95% UCL (e.g., Hall’s bootstrap UCL etc.) 
32 

of the mean exceeds the observed maximum concentration. For such scenarios (when a 95% UCL does 

exceed the maximum observed value), an alternative 95% UCL computation method is recommended by 

ProUCL 4.0.

1.10.3 Samples with Nondetect Observations

Nondetect observations (or less than obvious values) are inevitable in most environmental data sets. 

Singh, Maichle, and Lee (EPA, 2006) studied the performances (in terms of coverages) of the various 

UCL95 computation methods including the simple substitution methods (such as the DL/2 and DL 

methods) for data sets with nondetect observations. They concluded that the UCLs obtained using the 

substitution methods, including the replacement of nondetects by respective DL/2, do not perform well 

even when the percentage of nondetect observations is low, such as 5%-10%. They recommended 

avoiding the use of substitution methods to compute UCL95 based upon data sets with nondetect 

observations.

1.10.4 Avoid the Use of DL/2 Method to Compute UCL95

Based upon the results of the report by Singh, Maichle, and Lee (EPA, 2006), it is strongly recommended 

to avoid the use of the DL/2 method to perform GOF test, and to compute the summary statistics and 

various other limits (e.g., UCL, UPL) often used to estimate the EPC terms and BTVs. Until recently, the 

DL/2 method has been the most commonly used method to compute the various statistics of interest for 

data sets with BDL observations. The main reason of its common use has been the lack of the availability 

of other defensible methods and associated programs that can be used to estimate the various 

environmental parameters of interest. Today, several other methods (e.g., KM method, bootstrap 

methods) with better performances are available that can be used to compute the various upper limits of 

interest. Some of those parametric and nonparametric methods are available in ProUCL 4.0. Even though 

the DL/2 method (to compute UCLs, UPLs, and for goodness-of-fit test) has also been incorporated in 

ProUCL 4.0, its use is not recommended due to its poor performance. The DL/2 method is included in 

ProUCL 4.0 only for historical reasons as it had been the most commonly used and recommended method 

until recently (EPA, 2006). Some of the reviewers of ProUCL 4.0 suggested and requested the inclusion 

of DL/2 method in ProUCL for comparison purposes.

Note: The DL/2 method has been incorporated in ProUCL 4.0 for historical reasons only. NERL-EPA, 

Las Vegas strongly recommends avoiding the use of DL/2 method even when the percentage (%) of NDs 

is as low as 5%-10%. There are other methods available in ProUCL 4.0 that should be used to compute 

the various summary statistics and upper limits based upon data sets with multiple detection limits. 

1.10.5 Samples with Low Frequency of Detection

When all of the sampled data values are reported as nondetects, the EPC term should also be reported as a 

nondetect value, perhaps by the maximum reporting limit (RL) or maximum RL/2. Statistics (e.g., 

UCL95) computed based upon only a few detected values (e.g., < 4 to 6) cannot be considered reliable 

enough to estimate the EPC terms having potential impact on the human heath and the environment. 

When the number of detected data is small, it is preferable to use simple ad hoc methods rather than using 

statistical methods to compute the EPC terms and other upper limits. Specifically, it is suggested that in 

cases when the detection frequency is low (e.g., < 4%-5%) and the number of detected observations is 

low, the project team and the decision makers together should make a decision on site-specific basis on 

how to estimate the average exposure (EPC term) for the contaminant and area under consideration. For 

such data sets with low detection frequencies, other measures such as the median or mode represent better 

estimates (with lesser uncertainty) of the population measure of central tendency. 
33

Additionally, it is also suggested that when most (e.g., > %95) of the observations for a contaminant lie 

below the detection limit(s) or reporting limits (RLs), the sample median or the sample mode (rather than 

the sample average which cannot be computed accurately) may be used as an estimate the EPC term. Note 

that when the majority of the data are nondetects, the median and the mode will also be a nondetect. The 

uncertainty associated with such estimates will be high. It is noted that the statistical properties, such as 

the bias, accuracy, and precision of such estimates, would remain unknown. In order to be able to 

compute defensible estimates, it is always desirable to collect more samples. 

Note: In case the number of available detected samples is small (< 5), it is suggested that the project 

team decide about the estimation of the EPC term on site-specific basis. For such small data sets with 

very few detected values (< 5), the final decision (“policy decision”) on how to estimate the EPC term 

should be determined by the project team and decision makers.

1.11 Other Applications of Methods in ProUCL 4.0 

In addition to performing background versus site comparisons for CERCLA and RCRA sites, and 

estimating the EPC terms in exposure and risk evaluation studies, the statistical methods as incorporated 

in ProUCL 4.0 can be used to address other issues dealing with environmental investigations that are 

conducted at Superfund or RCRA sites. 

1.11.1 Identification of COPCs

Risk assessors and RPMs often use screening levels or BTVs to identify the COPCs during the screening 

phase of a cleanup project to be conducted at a contaminated site. The screening for the COPCs is 

performed prior to any characterization and remediation activities that may have to be conducted at the 

site under investigation. This comparison is performed to screen out those contaminants that may be 

present in the site medium of interest at low levels (e.g., at or below the background levels or preestablished screening levels) and may not pose any threat and concern to human health and the 

environment. Those contaminants may be eliminated from all future site investigations, and risk 

assessment and risk management studies. 

In order to identify the COPCs, point-by-point site observations (preferably composite samples) are

compared with pre-established screening levels, SSL, or estimated BTVs. This is especially true when the 

comparisons of site concentrations with screening levels or BTVs are conducted in real time by the 

sampling or cleanup crew right there in the site field. The project team should decide about the type of 

site samples (discrete or composite) and the number of detected site observations (not more than 4 to 6) 

that should be collected and compared with the screening levels or the BTVs. In case BTVs, screening 

levels, or not-to-exceed values are not known, the availability of a defensible background or reference 

data set of reasonable size (e.g., > 8 to 10, more are preferable) is required to obtain reliable estimates of 

BTVs and screening levels. When a reasonable number of detected site observations are available, it is 

preferable to use hypotheses testing approaches. The contaminants with concentrations exceeding the 

respective screening values or BTVs may be considered as COPCs, whereas contaminants with 

concentrations (in all collected samples) lower than the screening value, PRG, or an estimated BTV may 

be omitted from all future evaluations including the risk assessment and risk management investigations.
34 

1.11.2 Identification of Non-Compliance Monitoring Wells 

In monitoring well (MW) compliance assessment applications, individual (often discrete) contaminant 

concentrations from a MW are compared with pre-established ACL, MCL, or an estimated compliance 

limit (CL) based upon a group of upgradient wells representing the background population. An 

exceedance of the MCL or the BTV by a MW concentration may be considered as an indication of 

contamination in that MW. In such individual concentration comparisons, the presence of contamination 

(determined by an exceedance) may have to be confirmed by re-sampling from that MW. If 

concentrations of contaminants in both the original sample and the re-sample(s) exceed the MCL or BTV, 

then that MW may require closer scrutiny, perhaps triggering the remediation remedies as determined by 

the project team. If the concentration data from a MW for about 4 to 5 continuous quarters (or another 

designated time period determined by the project team) are below the MCL or BTV level, then that MW 

may be considered as complying with (achieving) the pre-established or estimated standards. Statistical 

methods as described in Chapters 5 and 6 of the revised background guidance document (EPA, 2002b) 

can be used to estimate the not-to-exceed values or BTVs based upon background or upgradient wells in 

case the ACLs or MCLs are not pre-determined.

1.11.3 Verification of the Attainment of Cleanup Standards, Cs

Hypothesis testing approaches may be used to verify the attainment of the cleanup standard, Cs, at 

polluted site areas of concern after conducting remediation and cleanup at the site AOC (EPA, 2006). In 

order to properly address this scenario, a site data set of adequate size (minimum of 8 to 10 detected site 

observations) needs to be made available from the remediated or excavated areas of the site under 

investigation. The sample size should also account for the size of the remediated site area; meaning that 

larger site areas should be sampled more (with more observations) to obtain a representative sample of the 

site under investigation. 

Typically, the null hypothesis of interest is H0: Site Mean, s >=Cs versus the alternative hypothesis, H1: 

Site Mean, s < Cs, where the cleanup standard, Cs, is known a priori. The sample size needed to 

perform such single sample hypotheses tests can be obtained using the DQO process-based sample size 

formula as given in the EPA (2006) documents. In any case, in order to use this test, a minimum of 8 to 

10 detected site samples should be collected. The details of statistical methods used to perform single 

sample hypothesis as described above can be found in EPA (2006).

1.11.4 Using BTVs (Upper Limits) to Identify Hot Spots

The use of upper limits (e.g., UTLs) to identify hot spot(s) has also been mentioned in the Guidance for 

Comparing Background and Chemical Concentrations in Soil for CERCLA Sites (EPA, 2002b). Point-bypoint site observations (preferably using composite samples representing a site location) are compared 

with a pre-established or estimated BTV. Exceedances of the BTV by site observations may be 

considered as representing locations with elevated concentrations (hot spots). Chapters 5 and 6 of the 

revised background guidance document (EPA, 2002b) describe several methods to estimate the BTVs 

based upon full data sets without nondetects and left-censored data sets with nondetect observations.

The rest of the chapters of this User Guide illustrate the use of the various procedures as incorporated in 

ProUCL 4.0. Those methods are useful to analyze environmental data sets with and without the nondetect 

observations. It is noted that ProUCL 4.0 is the first software package equipped with single sample and 

two-sample hypotheses testing approaches that can be used on data sets with nondetect observations. 
35

1.12 Some General Issues and Recommendations made by ProUCL

Some general issues regarding the handling of multiple detection limits and field duplicates by ProUCL 

and recommendations made about various substitution and ROS methods for data sets with NDs are 

described in the following sections.

1.12.1 Multiple Detection Limits

ProUCL 4.0 and its later versions do not make distinctions between method detection limits (MDLs), 

adjusted MDLs, sample quantitation limits (SQLs), or detection limits (DLs). Multiple detection limits in 

ProUCL mean different values of the detection limit. All these nondetect (ND) observations in ProUCL 

are indentified by the value „0‟ of the indicator variable used in ProUCL 4.0 to distinguish between 

detected and nondetected observations. It is users‟ responsibility to supply correct numerical values 

(should be entered as the reported detection limit value) for ND observations in the data set, and to create 

an indicator variable column associated with each variable/analyte consisting of ND values. It should be 

noted that some of the methods (e.g., Kaplan-Meier - KM Method) included in ProUCL 4.0 can handle 

data sets with detection limits greater than the detected values. 

1.12.2 Field Duplicates

Since, collection of analytical data are not cheap, field duplicates collected from the same site area under 

investigation may be used in the statistical analysis to compute various decision statistics of interest. 

Duplicates should be considered just like any other discrete samples collected from the site area under 

study. Alternatively, the project team may come to an agreement regarding the use of duplicates in 

statistical analyses (e.g., computing decision statistics) of data collected from the site.

1.12.3 ProUCL Recommendation about ROS Method and Substitution (DL/2) Method

In order to estimate EPC (computing UCL95) for data sets with NDs, ProUCL 4.0 and higher versions 

have several methods including substitution methods (e.g., DL/2, DL), ROS methods (normal, lognormal, 

and gamma), and Kaplan-Meier Method. Extensive simulation study conducted by Singh, Maichle and 

Lee (2006) demonstrated that statistically rigorous K-M method yields more precise and accurate 

estimates of EPC terms than those based upon substitution and ROS methods. Even though several of the 

substitution and ROS methods have been incorporated in ProUCL (for historical reasons and comparison 

purposes), those methods are not recommended by ProUCL to estimate the EPC terms or to compute 

other decision statistics. 
36 

Chapter 2

Entering and Manipulating Data

2.1 Creating a New Data Set

By simply executing the ProUCL 4.0, a new worksheet is generated and displayed for the user to enter 

data. 

To create a new worksheet: click or choose File ► New

2.2 Opening an Existing Data Set

If your data sets are stored in the ProUCL data format (*.wst), then click or choose File ► Open

 If your data sets are stored in the Microsoft Excel format (*.xls), then choose File ►

Other Files... ►Import Excel... OR File ► Load Excel Data

 Possible Error Messages:

o When you import an Excel file, make sure that you have an empty worksheet. If there 

is no empty worksheet, then you must create a new worksheet before importing an 

Excel file. Otherwise, there will be an error message in the Log Panel: “[Error] 

Worksheet must be empty.” 

o First open a new worksheet and then import the Excel file. 

o Make sure that the file you trying to import is not currently open. Otherwise, there 

will be the following warning message in the Log panel:

o “[Information] Unable to open C:\***.xls.” Check the validity of this file.
37

2.3 Input File Format 

 Note that ProUCL 4.0 does not require that data in each column must end with a nonzero 

value (ProUCL 3.0 requires this). Therefore, all zero values (in the beginning, middle, or 

end of data columns) are treated as valid zero values as part of the data set.

 The program can read Excel files. The user can perform typical Cut, Paste, and Copy 

operations.

 The first row in all input data files consist of alphanumeric (strings of numbers and 

characters) names representing the header row. Those header names may represent 

meaningful variable names such as Arsenic, Chromium, Lead, Group-ID, and so on. 

o The Group-ID column has the labels for the groups (e.g., Background, AOC1, 

AOC2, 1, 2, 3, a, b, c, Site 1, Site 2, and so on) that might be present in the data set. 

The alphanumeric strings (e.g., Surface, Sub-surface) can be used to label the various 

groups. 

o The data file can have multiple variables (columns) with unequal number of 

observations.

o Except for the header row and columns representing the group labels, only numerical 

values should appear in all other columns. 

o All alphanumeric strings and characters (e.g., blank, other characters, and strings), 

and all other values (that do not meet the requirements above) in the data file are 

treated as missing values.

o Also, a large value denoted by 1E31 (= 1x1031) can be used to represent missing data 

values. All entries with this value are ignored from the computations. These values 

are counted under number of missing values.

2.4 Number Precision 

 You may turn “Full Precision” on or off by choosing Configure ► Full Precision 

On/OFF

 By leaving “Full Precision” turned off, ProUCL will display numerical values using an 

appropriate (default) decimal digit option. However, by turning “Full Precision” off, all 

decimal values will be rounded to the nearest thousandths place.

 “Full Precision” on option is specifically useful when one is dealing with data sets 

consisting of small numerical values (e.g., < 1) resulting in small values of the various 

estimates and test statistics. These values may become so small with several leading zeros 
38 

(e.g., 0.00007332) after the decimal. In such situations, on may want to use “Full 

Precision” option to see nonzero values after the decimal. 

Note: For the purpose of this User Guide, unless noted otherwise, all examples have been described 

using the “Full Precision” off option. This option prints out results up to 3 significant digits after the 

decimal.

2.5 Entering and Changing a Header Name 

1. Highlight the column whose header name (variable name) you want to change by clicking either 

the column number or the header as shown below.

2. Right-click and then click “Header Name”

3. Change the Header Name.

4. Click the OK button to get the following output with the changed variable name.
39

2.6 Saving Files

 Save option allows the user to save the active window. 

 Save As option allows the user to save the active window. This option follows typical 

Windows standards, and saves the active window to a file in Excel 95 (or higher) format. 

All modified/edited data files, and output screens (excluding graphical displays) 

generated by the software can be saved as Excel 95 (or higher) spreadsheet.

2.7 Editing

Click on the Edit menu item to reveal the following drop-down options.

The following Edit drop-down menu options are available:

 Cut option: similar to a standard Windows Edit option, such as in Excel. It performs 

standard edit functions on selected highlighted data (similar to a buffer).

 Copy option: similar to a standard Windows Edit option, such as in Excel. It performs 
40 

typical edit functions on selected highlighted data (similar to a buffer).

 Paste option: similar to a standard Windows Edit option, such as in Excel. It performs 

typical edit functions of pasting the selected (highlighted) data to the designated 

spreadsheet cells or area. 

 It should be noted that the Edit option could also be used to Copy Graphs. This topic 

(copying and pasting graphs) is illustrated in detail in Chapter 13.

2.8 Handling Nondetect Observations

 ProUCL 4.0 can handle data sets with single and multiple detection limits.

 For a variable with nondetect observations (e.g., arsenic), the detected values, and the 

numerical values of the associated detection limits (for less than values) are entered in the 

appropriate column associated with that variable. 

 Specifically, the data for variables with nondetect values are provided in two columns. 

One column consists of the detected numerical values with less than (< DLi) values 

entered as the corresponding detection limits (or reporting limits), and the second column 

represents their detection status consisting of only 0 (for less than values) and 1 (for 

detected values) values. The name of the corresponding variable representing the 

detection status should start with d_, or D_ (not case sensitive) and the variable name. 

The detection status column with variable name starting with a D_ (or a d_) should have 

only two values: 0 for nondetect values, and 1 for detected observations. 

 For an example, the header name, D_Arsenic is used for the variable, Arsenic having 

nondetect observations. The variable D_Arsenic contains a 1 if the corresponding Arsenic 

value represents a detected entry, and contains a 0 if the corresponding entry for variable, 

Arsenic, represents a nondetect. An example data set illustrating these points is given as 

follows.
41

2.9 Caution

 Care should be taken to avoid any misrepresentation of detected and nondetected values. 

Specifically, it is advised not to have any missing values (blanks, characters) in the 

D_column (detection status column). If a missing value is located in the D_ column (and 

not in the associated variable column), the corresponding value in the variable column is 

treated as a nondetect, even if this might not have been the intention of the user. 

 It is mandatory that the user makes sure that only a 1 or a 0 are entered in the detection 

status D_column. If a value other than a 0 or a 1 is entered in the D_ column (the 

detection column), results may become unreliable, as the software defaults to any number 

other than 0 or 1 as a nondetect value. 

 When computing statistics for full data sets without any nondetect values, the user should 

select only those variables (from the list of available variables) that contain no nondetect 

observations. Specifically, nondetect values found in a column chosen for the summary 

statistics (full-uncensored data set) will be treated as a detected value; whatever value 

(e.g., detection limit) is entered in that column will be used to compute summary 

statistics for a full-uncensored data set without any nondetect values. 

 Two-Sample Hypotheses: It should be noted that at present, when using two-sample

hypotheses approaches (WMW test, Gehan test, and quantile test) on data sets with NDs, 
42 

both samples or variables (e.g., site-As, Back-As) should be specified as having 

nondetects, even though one of the variables may not have any ND observations. This 

means, a ND column (with 0 = ND, and 1 = detect) should be provided for each variable 

(here D_site-As, and D_Back-As) to be used in this comparison. If a variable (e.g., siteAs) does not have any nondetects, still a column with label D_site-As should be included 

in the data set with all entries = 1 (detected values).

 The following sample (not from a Superfund site) data set given on the next page 

illustrates points related to this option and issues listed above. The data set considered 

contains some nondetect measurements for Arsenic and Mercury. It should also be noted 

that the Mercury concentrations are used to illustrate the points related to nondetect 

observations only. Arsenic and Zinc concentrations are used to illustrate the use of the 

group variable, Group (Surface, Subsurface).

 If for mercury, one computes summary statistics (assuming no nondetect values) using 

“Full” data set option, then all nondetect values (with “0” entries in D_Mercury column) 

will be treated as detected values, and summary statistics will be computed accordingly.

2.10 Summary Statistics for Data Sets with Nondetect Observations

 In order to compute simple summary statistics or to compute other statistics of interest 

(e.g., background statistics, GOF test, UCLs, outliers) for variables with nondetect 

values, one should choose the nondetect option, “With NDs” from the various available 

menu options such as Outliers, Background Statistics, UCLs, Goodness-of-Fit test, Q-Q 

plot, Box Plot. 

 It should be noted that “summary” statistics for a data set with nondetect observations 

represent (at least in ProUCL 4.0) simple summary statistics based upon the data set 

without using nondetect observations. All other parametric and nonparametric statistics 

and estimates of population mean, variance, percentiles (e.g., MLEs, KM, and ROS 

estimates) for variables with nondetect observations are given in other menu options such 

as background statistics and UCL. The simple “Summary Statistics/With NDs” option 

only provides simple statistics (e.g., % NDs, max ND, Min ND, and Mean of detected 

values) based upon detected values. These statistics (e.g., sd of log-transformed detected 

values) may help a user to determine the degree of skewness (e.g., mild, moderate, high) 

of the data set consisting of detected values. These statistics may also help the user to 

choose the most appropriate method (e.g., KM (BCA) UCL or KM (t) UCL) to compute 

UCLs, UPLs, and other limits.

 As mentioned before, various other statistics and estimates of interest (e.g., mean, sd, 

UCLs, UTLs, MLEs, and KM estimates) for data sets with nondetect observations are 

computed in other the menu options (UCLs, Outliers, Background Statistics, GOF tests) 

available in ProUCL 4.0. 
43

2.11 Warning Messages and Recommendations for Datasets with 

Insufficient Amount of Data 

 ProUCL 4.0 provides some warning messages and recommendations for datasets with 

insufficient amount of data to calculate meaningful estimates and statistics of interest. For 

an example, it is not desirable to compute an estimate the EPC term based upon a data set 

of size less than 5, especially when nondetects may be present in the data set. In such 

cases, it is suggested to use site specific values (perhaps determined by the Project Team) 

to estimate environmental parameters (e.g., EPC term, not-to-exceed background value) 

of interest.

 Some examples of datasets with insufficient amount of data include datasets with less 

than 4 distinct observations, datasets with only one detected observation, and datasets

consisting of all nondetects.

 Some of the warning messages given by ProUCL 4.0 are shown below.
44 
45

2.12 Handling Missing Values

 ProUCL 4.0 can handle missing values within a data set.

 All blanks, alphanumeric strings (except for group variables), or the specific large 

number value 1e31 are considered as missing values. 

 A group variable (representing two or more groups, populations, AOCs, MWs) can have 

alphanumeric values (e.g., MW1, MW2, ...). 

 ProUCL 4.0 ignores all missing values in all mathematical operations it performs. 

Missing values are therefore not treated as being part of a data set. 

 Number of Valid Samples or Number of Valid Observations represents the Total Number 

of Observations less the Number of Missing Values. If there are missing values, then 

number of valid samples = total number of observations.

Valid Samples = Total Number of Observations – Missing Values.
46 

 It is important to note, however, that if a missing value not meant (e.g., a blank, or 1e31) 

to represent a group category is present in a “Group” variable, ProUCL 4.0 will treat that 

blank value (or 1e31 value) as a new group. Any variable that corresponds to this missing 

value will be treated as part of a new group and not with any existing groups. It is 

therefore very important to check the consistency and validity of all data sets before 

performing complex mathematical operations. 

 ProUCL 4.0 prints out the number of missing values (if any) associated with each 

variable in the data sheet. This information is provided in several output sheets (e.g., 

summary statistics, background statistics, UCLs) generated by ProUCL 4.0.

For further clarification of labeling of missing values, the following example illustrates the terminology 

used for the number of valid samples, number of unique and distinct samples on the various output sheets 

generated by ProUCL 4.0. 

Example: The following example illustrates the notion of Valid Samples, Unique or Distinct Samples, 

and Missing Values. The data set also has nondetect values. ProUCL 4.0 computes these numbers and 

prints them on the UCLs and background statistics output.

x D_x

2 1

4 1

2.3 1

1.2 0

w34 0

1.0E+031 0

0

anm 0

34 1

23 1

0.5 0

0.5 0

2.3 1

2.3 1

2.3 1

34 1

73 1

Valid Samples: Represents the total number of observations (censored and uncensored) excluding the 

missing values. If a data set has no missing value, then the total number of data points equals number of 

valid samples.

Missing Values: All values not representing a real numerical number are considered as missing values. 

Specifically, all alphanumeric values including blanks are considered as missing values. Also unrealistic 

big numbers such as 1.0e31 are also considered as missing values and are considered as not valid 

observations.

Unique or Distinct Samples: The number of unique samples or number of distinct samples represents all 

unique (or distinct) detected values. Number of unique or distinct values is computed for detected values 
47

only. This number is especially useful when using bootstrap methods. As well known, it is not desirable 

and advisable to use bootstrap methods, when the number of unique samples is less than 4-5. 

2.13 User Graphic Display Modification 

Advanced users are provided two sets of tools to modify graphics displays. A graphics tool bar is 

available above the graphics display and the user can right-click on the desired object within the graphics 

display, and a drop-down menu will appear. The user can select an item from the drop-down menu list by 

clicking on that item. This will allow the user to make desired modifications as available for the selected 

menu item. An illustration is given as follows.

2.13.1 Graphics Tool Bar

The user can change fonts, font sizes, vertical and horizontal axis‟s, select new colors for the various 

features and text. All these actions are generally used to modify the appearance of the graphic display. 

The user is cautioned that these tools can be unforgiving and may put the user in a situation where the 

user cannot go back to the original display. Users are on their own in exploring the robustness of these 

tools. Therefore, less experienced users may want to stay away from using these drop-down menu graphic 

tools. 

2.13.2 Drop-Down Menu Graphics Tools

These tools allow the user to move the mouse to a specific graphic item like an Axis Label or a display 

feature. The user then right-clicks their mouse and a drop-down menu will appear. This menu presents the 

user with available options for that particular control or graphic object. There is less of chance of making 

an unrecoverable error but that risk is always present. As a cautionary note, the user can always delete the 

graphics window and redraw the graphical displays by repeating their operations from the datasheet and 

menu options available in ProUCL 4.0. An example of a drop-down menu obtained by right-clicking the 

mouse on the background area of the graphics display is given as follows.
48 
49
50 

Chapter 3

Select Variables Screen

3.1 Select Variables Screen

 Variables need to be selected to perform statistical analyses.

 When the user clicks on a drop-down menu for a statistical procedure, the following 

window will appear. 

 The Options button is available in certain menus. The use of this option leads to a 

different pop-up window. 

 Multiple variables can be processed simultaneously in ProUCL 4.0. Note that this option 

was not available in ProUCL 3.0. ProUCL 4.0 can generate graphs, compute UCLs, and 

background statistics simultaneously for all selected variables.

 Moreover, if the user wants to perform statistical analysis on a variable (e.g., 

contaminant) by a Group variable, click the arrow below the Group by variable to get a 

drop-down list of available variables to select an appropriate group variable. For an 

example, a group variable (e.g., Site Area) can have alphanumeric values such as AOC1, 
51

AOC2, AOC3, and Background. Thus in this example, the group variable name, Site 

Area, takes four values such as AOC1, AOC2, AOC3, and Background.

 The Group variable is particularly useful when data from two or more samples need to be 

compared.

 Any variable can be a group variable. However, for meaningful results, only a variable, 

that really represents a group variable (categories) should be selected as a group variable.

 The number of observations in the group variable and the number observations in the 

selected variables (to be used in a statistical procedure) should be the same. In the 

example below, the variable “Mercury” is not selected because the number of 

observations for Mercury is 30; in other words mercury values have not been grouped. 

The group variable and each of the selected variables have 20 data values. 

 It is recommended not to assign any missing value such as a “Blank” for the group 

variable. If there is a missing value (represented by blanks, strings or 1E31) for a group 

variable, ProUCL 4.0 will treat those missing values as a new group. As such, data values 

corresponding to the missing Group will be assigned to a new group. 

Caution: Once again, care should be taken to avoid misrepresentation and improper use of group 

variables. It is recommended not to assign any missing value for the group variable.
52 

More on Group Option 

 The Group Option provides a powerful tool to perform various statistical tests and 

methods (including graphical displays) separately for each of the group (samples from 

different populations) that may be present in a data set. For an example, the same data set 

may consist of samples from the various groups (populations). The graphical displays 

(e.g., box plots, Q-Q plots) and statistics of interest can be computed separately for each 

group by using this option.

 In order to use this option, at least one group variable (with alphanumeric values) should 

be included in the data set. The various values of the group variable represent different 

group categories that may be present in the data set. This can be seen in the example.wst 

data set used earlier in Chapter 2.

 At this time, the number of values (representing group membership) in a Group variable 

should equal to the number of values in the variable (e.g., Arsenic) of interest that needs 

to be partitioned into various groups (e.g., monitoring wells).

 Typically, the data for the various groups (categorized by the group variable) represent 

data from the various site areas (e.g., background, AOC1, AOC2, ...), or from monitoring 

wells (e.g., MW1, MW2, ...). 

3.1.1 Graphs by Groups

 Individual or multiple graphs (Q-Q plots, box plots, and histograms) can be displayed on 

a graph by selecting the “Graphs by Groups” option.

 Individual graph for each group (specified by the selected group variable) is produced by 

selecting the “Individual Graph” option.

 Multiple graphs (e.g., side-by-side box plots, multiple Q-Q plots on the same graph) are 

produced by selecting the “Group Graph” option for a variable categorized by a group 

variable. Using this “Group Graph” option, multiple graphs can be displayed for all subgroups included in the Group variable. This option is useful when data to be compared 

are given in the same column and are classified by the group variable. 

 Multiple graphs (e.g., side-by-side box plots, multiple Q-Q plots) for selected variables 

are produced by selecting the “Group Graph” option. Using the “Group Graph” option, 

multiple graphs can be displayed for all selected variables. This option is useful when 

data (e.g., lead) to be compared are given in different columns, perhaps representing 

different populations. 

Note: It should be noted that it is the users’ responsibility to provide adequate amount of detected data to 

perform the group operations. For an example, if the user desires to produce a graphical Q-Q plot (using 

only detected data) with regression lines displayed, then there should be at least two detected points (to 

compute slope, intercept, sd) in the data set. Similarly if the graphs are desired for each of the group 

specified by the group ID variable, there should be at least two detected observations in each group 

specified by the group variable. ProUCL 4.0 generates a warning message (in orange color) in the lower 

panel of the ProUCL 4.0 screen. Specifically, the user should make sure that a variable with nondetects 
53

and categorized by a group variable should have enough detected data in each group to perform the 

various methods (e.g., GOF tests, Q-Q plots with regression lines) as incorporated in ProUCL 4.0.

As mentioned before, the analyses of data categorized by a group ID variable such as: 

1) Surface vs. Subsurface; 2) AOC1 vs. AOC2; 3) Site vs. Background; and 4) Upgradient vs. 

Downgradient monitoring wells are quite common in many environmental applications. 

The usefulness of the group option is illustrated throughout the User Guide using various methods as 

incorporated in ProUCL 4.0.
54 

Chapter 4

Summary Statistics

This option is used to compute general summary statistics for all variables in the data file. Summary 

statistics can be generated for full data sets without nondetect observations, and for data sets with 

nondetect observations. Two Menu options: Full and With NDs are available.

 Full – This option computes summary statistics for all variables in a data set without any 

nondetect values.

 With NDs – This option computes simple summary statistics for all variables in a data set 

that have nondetect (ND) observations. For variables with ND observations, only simple

summary statistics are computed based upon detected observations only. 

o For this option, no attempt is made to compute estimates of population parameters 

(e.g., mean, sd, SE) using parametric (e.g., MLE) or nonparametric (e.g., KM, 

bootstrap) estimation methods. Those statistics are generated in other estimation 

modules (e.g., Background and UCL) of ProUCL 4.0.

Each menu option (Full and With NDs) has two sub-menu options:

 Raw Statistics 

 Log-Transformed 

 In ProUCL, log-transformation means natural logarithm (ln)

 When computing summary statistics for raw data, a message will be displayed for each 

variable that contains non-numeric values.

 The Summary Statistics option computes log-transformed data only if all of the data 

values for the selected variable(s) 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. 

4.1 Summary Statistics with Full Data Sets

1. Click Summary Statistics ► Full

2. Select either Log-Transformed or Raw Statistics option.
55

3. The Select Variables Screen (see Chapter 3) will appear. 

 Select one or more variables from the Select Variables screen.

 If summary statistics are to be computed by a Group variable, then select a group variable 

by clicking the arrow below the Group by variable button. This will result in drop-down

list of available variables, and select the proper group variable. 

 Click on the OK button to continue or on the Cancel button to cancel the Summary 

Statistic option.

Raw Statistics

Log-Transformed Statistics

4. The resulting Summary Statistics screen as shown above can be saved as an Excel file. Click 

Save from the file menu. 
56 

5. On the output screen, the following summary statistics are displayed for each selected variable in 

the data file. 

NumObs = Number of Observations

Minimum = Minimum value

Maximum = Maximum value

Mean = Sample average value

Median = Median value

Variance = Classical sample variance 

SD = Classical sample standard deviation

MAD = Median absolute deviation

MAD/0.675 = Robust estimate of variability, population standard deviation, 

Skewness = Skewness statistic

Kurtosis = Kurtosis statistic

CV = Coefficient of Variation

The details of these summary statistics are described in an EPA (2006) guidance document and also in the 

ProUCL Technical Guide (A. Singh and A.K. Singh (EPA, 2007).

4.2 Summary Statistics with NDs

1. Click Summary Statistics ► With NDs

2. Select either Log-Transformed or Raw Statistics option.

3. The Select Variables Screen (Chapter 3) will appear. 

 Select variable(s) from the list of variables.

 Only those variables that have nondetect values will be shown. 

 If summary statistics are to be computed by a Group variable, then select a group variable 

by clicking the arrow below the Group by variable button. This will result in a dropdown list of available variables; then select the proper group variable. 

 Click on the OK button to continue or on the Cancel button to cancel the summary 

statistics operations.

Note: It should be noted that in ProUCL 4.0, “Summary Statistics” for a data set with nondetect 

observations represent simple summary statistics based upon the data set without using nondetect 

observations. All other parametric and nonparametric statistics and estimates of population mean, 

variance, percentiles (e.g., MLEs, KM, and ROS estimates) for variables with nondetect observations are 

given in other estimation menu options such as background statistics and UCL. The simple “Summary 
57

Statistics/With NDs” option only provides simple statistics (e.g., % NDs, max ND, Min ND, Mean of 

detected values) based upon detected values. These statistics (e.g., sd of log-transformed detected values) 

may help a user to determine the degree of skewness (e.g., mild, moderate, high) of the data set consisting 

of detected values. These statistics may also help the user to choose the most appropriate method (e.g., 

KM (BCA) UCL or KM (t) UCL) to compute UCLs, UPLs, and other limits.

Raw Statistics – Data Set with NDs

Log-Transformed Statistics – Data Set with NDs

 The Summary Statistics screen shown above can be saved as an Excel file. Click the save 

from the file menu. 

 On the results screen, the following summary statistics are displayed for each selected 

variable from the data file. 

Num Obs = Number of Observations

NumNDs = Number of Nondetects
58 

% NDs = Percentage of Nondetect observations

Minimum = Minimum value

Maximum = Maximum value

Mean = Sample average value

Median = Median value

SD = Classical sample standard deviation

MAD = Median Absolute Deviation

MAD/0.675 = Robust estimate of variability (standard deviation)

CV = Coefficient of Variation
59
60 

Chapter 5

Estimating Nondetects Using ROS Methods

Regression on order statistics (ROS) can be used to extrapolate nondetect observations using a normal, 

lognormal, or gamma model. ProUCL 4.0 has three ROS estimation methods that can be used to estimate 

or extrapolate nondetect observations. The use of this option generates additional columns consisting of 

all extrapolated nondetects and detected observations. These columns are appended to the existing open 

spreadsheet. The user should save the updated file if they want to use the generated data for their other 

application(s).

1. Click ROS Est. NDs ► Gamma ROS

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen.

 Click on the OK button to continue or on the Cancel button to cancel the option.

Output Screen for ROS Est. NDs (Gamma) Option
61

Note: Columns with similar naming convention are generated for each selected variable and distribution 

using this ROS option.
62 

Chapter 6

Graphical Methods (Graph)

Three commonly used graphical displays are available under the Graphs option:

o Box Plot

o Histogram 

o Multi-QQ

 The box plots and multiple Q-Q plots can be used for Full data sets without nondetects 

and also for data sets with nondetect values. 

 Three options are available to draw Q-Q plots with nondetect (ND) observations.

Specifically, Q-Q plots are displayed only for detected values, or with NDs replaced by 1⁄2 

DL values, or with NDs replaced by the respective detection limits. The statistics 

displayed on a Q-Q plot (mean, sd, slope, intercept) are computed according to the

method used. The NDs are displayed using the smaller font and in red color. 

 ProUCL 4.0 can display box plots for data sets with NDs. This kind of graph may not be 

very useful when many NDs may be present in a data set. 

o A few choices are available to construct box plots for data sets with NDs. For an 

example, some texts (e.g., Helsel) display box plots only for the detected 

observations. Specifically, all nondetects below the largest detection limit (DL), and 

portion of the box plot (if any) below the largest DL are not shown on the box plot. A 

horizontal line is displayed at the largest detection limit level. 

o ProUCL 4.0 constructs a box plot using all detected and nondetect (using DL values) 

values. ProUCL 4.0 shows the full box plot. However, a horizontal line is displayed 

at the largest detection limit.

 When multiple variables are selected, one can choose to: 1) produce a multiple graphs on 

the same display by choosing the Graph by group variable option, or 2) produce separate 

graphs for each selected variable. 

 The Graph by group variable option produces side-by-side box plots, or multiple Q-Q 

plots, or histograms for the groups of the selected variable representing samples obtained 

from multiple populations (groups). These multiple graphs are particularly useful to 

perform two (background versus site) or more sample visual comparisons.

o Additionally, Box Plot has an optional feature, which can be used to draw lines at 

statistical (e.g., upper limits of background data set) limits computed from one 

population on the box plot obtained using the data from another population (a site 

area of concern). This type of box plot represents a useful visual comparison of site 

data with background threshold values (background upper limits). 

o Up to four (4) statistics can be added (drawn) on a box plot. If the user inputs a value 

in the value column, the check box in that row will get activated. For example, the 
63

user may want to draw horizontal lines at 80th percentile, 90th percentile, 95th

percentile, or a 95% UPL) on a box plot. 

6.1 Box Plot

1. Click Graphs ► Box Plot

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select an appropriate variable representing a 

group variable. 

 When the Group by variable button is clicked, the following window is shown. 
64 

 The default option for Graph by Groups is Individual Graphs. This option will 

produce one graph for each selected variable. If you want to put all the selected variables 

into a single graph, then select the Group Graphs option. This Group Graphs option is 

used when multiple graphs categorized by a Group variable have to be produced on the 

same graph.

 The default option for Graphical Display Options is Color Gradient. If you want to use 

and import graphs in black and white into a document or report, then check the radio 

button next to For Export (BW Printers).

 Click on the OK button to continue or on the Cancel button to cancel the Box Plot (or 

other selected graphical) option.

Box Plot Output Screen (Single Graph)

Selected options: Label (Background UPL), Value (103.85), Individual Graphs, and Color Gradient.
65

Box Plot Output Screen (Group Graphs)

Selected options: Group Graphs and Color Gradient.

6.2 Histogram

1. Click Graphs ► Histogram

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable. 

 When that option button is clicked, the following window will be shown. 
66 

 The default selection for Graph by Groups is Individual Graphs. This option produces 

a histogram (or other graphs) separately for each selected variable. If multiple graphs or 

graphs by groups are desired, then check the radio button next to Group Graphs.

 The default option for Graphical Display Options is Color Gradient. If you want to use 

and import graphs in black and white into a document or report, then check the radio 

button next to For Export (BW Printers).

 Click on the OK button to continue or on the Cancel button to cancel the Histogram (or 

other selected graphical) option.

Histogram Output Screen

Selected options: Group Graphs and Color Gradient.
67

6.3 Multi-QQ 

6.3.1 Multi-QQ (Full)

1. Click Graphs ► Multi-QQ

2. Multi-QQ can be obtained for data sets with (With NDs) and without NDs (Full).

 When that option button is clicked, the following window will be shown. 

3. Select either Full or With NDs.

4. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable. 

 When the Group by variable option button is clicked, the following window will appear. 
68 

 The default option for Display Regression Lines is Do Not Display. If you want to see 

regression lines on graphs, then check the radio button next to Display Regression 

Lines.

 The default option for Graphical Display Options is Color Gradient. If you want to see 

the graphs in black and white, then check the radio button next to For Export (BW 

Printers).

 Click on the OK button to continue or on the Cancel button to cancel the selected MultiQQ option.

Note: For Multi-QQ plot option, for both “Full” as well as for data sets “With NDs,” the values along 

the horizontal axis represent quantiles of a standardized normal distribution (Normal distribution with 

mean 0 and standard deviation 1). Quantiles for other distributions (e.g., Gamma distribution) are used 

when using Goodness-of-Fit (GOF) test option.

Output Screen for Multi-QQ (Full) 

Selected options: Group Graph, Do Not Display Regression Lines, and Color Gradient.

6.3.2 Multi-QQ (with NDs)

1. Click Graphs ► Multi-QQ
69

2. Select With NDs option by clicking on it.

3. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable. 

 When the Group by variable option button is clicked, the following screen appears. 

 The default option for Display Regression Lines is Do Not Display. If you want to see 

regression lines, then check the radio button next to Display Regression Lines.

 The default option for Display Nondetects is Display Nondetect Values. 

o Do not Display Nondetects: Selection of this option excludes the NDs detects and 

plots only detected values on the associated Q-Q plot. The statistics are computed 

using only detected data.
70 

o Display Nondetect Values: Selection of this option treats detection limits as detected 

values and plots those detection limits and detected values on the Q-Q plot. The 

statistics are computed accordingly.

o Display 1⁄2 Nondetect Values: Selection of this option replaces the detection limits 

with their half values, and plots half detection limits and detected values on the Q-Q 

plot. The statistics are computed accordingly.

 The default option for Graphical Display Options is Color Gradient. If you want to see 

the graphs in black and white, then check the radio button next to For Export (BW 

Printers).

 Click on the OK button to continue or on the Cancel button to cancel the Multi-QQ 

option. 

Output Screen for Multi-QQ (without NDs)

Options: Do Not Display Regression Lines, Do not display Nondetects, and Color Gradient.
71

Output Screen for Multi-QQ (with NDs)

Options: Do not Display Regression Line, Display Nondetect Values, and Color Gradient.

Note: The legend size of nondetect values is smaller than that of the detected values and is shown in red. 

The legend size is made smaller for BW printers.

Output Screen for Multi-QQ (with NDs)

Selected options: Do not Display Regression Lines, Display 1⁄2 Nondetect Values, and Color Gradient.

Note: The legend size of nondetect values is smaller than that of the detected values and is shown in red.
72 

Chapter 7

Simple Classical Outlier Tests

Outliers are inevitable in data sets originating from environmental applications. There are many graphical 

(Q-Q plots, Box plots), classical (Dixon, Rosner, Welch), and robust methods (biweight, Huber, PROP) 

available to identify outliers. It is well known that the classical outlier tests (e.g., Dixon test, Rosen test, 

EPA, 2006) suffer from masking (e.g., extreme outliers may mask intermediate outliers) effects. The use 

of robust outlier identification procedures is recommended to identify multiple outliers, especially when 

dealing with multivariate (having multiple contaminants) data sets. However, those preferred and more 

effective robust outlier identification methods are beyond the scope of ProUCL 4.0. Several robust outlier 

identification methods (e.g., based upon biweight, Huber, and PROP influence functions) are available in 

the Scout software package (EPA, 1999). 

The two simple classical outlier tests (often cited in environmental literature): Dixon and Rosner tests are 

available in ProUCL 4.0. These tests can be used on data sets with and without nondetect observations. 

These tests also require the assumption of normality of the data set without the outliers. It should be noted 

that in environmental applications, one of the objectives is to identify high outlying observations that 

might be present in the right tail of a data distribution as those observations often represent contaminated 

locations of a polluted site. Therefore, for data sets with nondetects, two options are available in ProUCL 

4.0 to deal with data sets with outliers. These options are: 1) exclude nondetects and 2) replace NDs by 

DL/2 values. These options are used only to identify outliers and not to compute any estimates and limits 

used in decision-making process. 

It is suggested that these two classical outlier identification procedures be supplemented with graphical 

displays such as Q-Q plots, Box and Whisker plot (called box plot), and IQR (= upper quartile, Q3 -

lower quartile, Q1). These graphical displays are available in ProUCL 4.0. Box plots with whiskers are 

often used to identify outliers (e.g., EPA, 2006). Typically, a box plot gives a good indication of extreme 

(outliers) observations that may present in a data set. The statistics (lower quartile, median, upper quartile, 

and IQR) used in the construction of a box plot do not get distorted by outliers. On a box plot, 

observations beyond the two whiskers may be considered as candidates for potential outliers. 

Q-Q plots are also quite useful to identify outliers in a data set. For an example, on a normal Q-Q plot, 

observations that are well separated from the bulk (central part) of the data typically represent potential 

outliers needing further investigation. Also, significant and obvious jumps and breaks in a Q-Q plot (for 

any distribution) are indications of the presence of more than one population. Data sets exhibiting such 

behavior of Q-Q plots should be partitioned out in component sub-populations before estimating an EPC 

Term or a background threshold value (BTV). It is strongly recommended that both graphical and formal 

outlier identification tests should be used on the same data set to identify potential outliers that may be 

present in a data set under study. More details about the construction of graphical displays and outliers 

test can be found in the Technical Guide for ProUCL 4.0.

Dixon‟s Test (Extreme Value Test)

 This test is used to identify statistical outliers when the sample size is less than or equal 

to 25. 
73

 This test can be used to identify outliers or extreme values in both the left tail (Case 1) 

and the right tail (Case 2) of a data distribution. In environmental data sets, extremes 

found in the right tail may represent potentially contaminated site areas needing further 

investigation or remediation. The extremes in the left tail may represent ND values. 

 This test assumes that the data without the suspected outlier are normally distributed; 

therefore, it is necessary to perform a test for normality on the data without the suspected 

outlier before applying this test.

 This test may suffer from masking in the presence of multiple outliers. This means that if 

more than one outlier is suspected, this test may fail to identify all of the outliers. 

Therefore, if you decide to use the Dixon‟s test for multiple outliers, apply the test to the 

least extreme value first. Alternatively, use more effective robust outlier identification 

procedures. Those outlier identification procedures will be available in Scout (EPA,

1999) software.

Rosner‟s Test 

 This test can be used to identify and detect up to 10 outliers in data sets of sizes 25 and 

higher.

 This test also assumes that the data are normally distributed; therefore, it is necessary to 

perform a test for normality before applying this test.

Depending upon the selected variable(s) and the number of observations associated with them, either the 

Dixon‟s test or the Rosner‟s test will be performed. 

NOTE: Throughout this User Guide, and in ProUCL 4.0, it is assumed that the user is dealing with a 

single population. If multiple populations are present in a data set, it is recommended to separate them 

out using appropriate population partitioning methods and techniques. Appropriate tests and statistics 

(e.g., goodness-of-fit tests, 95% UCLs, 95% UPLs) should be computed separately for each of the 

identified populations. Also, outliers if any should be identified and thoroughly investigated. The presence 

of outliers distorts all statistics including the all of the upper limits (UCLs, UPLs, upper percentiles). The 

use of distorted statistics and limits may lead to incorrect conclusions having potential adverse effects on 

the human health and the environment. Decisions about the disposition of outliers: inclusion or exclusion 

in the data set to be used to compute the UCLs, UPLs, and other statistics should be made by all parties 

involved. Statistical methods supplemented with graphical displays (e.g., Q-Q plot and box plots) can 

only help identify statistical outliers that may be present in a data set. The project team and experts 

familiar with the site should interpret and assign physical meaning and significance to those identified 

outliers. The entire project team should be involved in taking decisions about the appropriate disposition 

(include or not include) of outliers.
74 

7.1 Outlier Test for Full Data Set

1. Click Outlier Tests ► Full ►Compute

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable. 

 If at least one of the selected variables has 25 or more observations, then click the option 

button for the Rosner Test. 

 The default option for the number of suspected outliers is 1. In order to use this test, the 

user has to obtain an initial guess about the number of outliers that may be present in the 

data set. This can be done by using graphical displays such as a Q-Q plot. On this 

graphical Q-Q plot, higher observations that are well separated from the rest of the data 

may be considered as potential or suspected outliers.

 Click on the OK button to continue or on the Cancel button to cancel the Outlier Tests.

7.2 Outlier Test for Data Set with NDs

Typically, in environmental applications, one is interested in identifying high outliers (perhaps 

representing contaminated parts of a site area, hot spots) that might be present in the right tail of the data 

distribution. Therefore, one may want to use the same outlier identification procedures (e.g., Dixon test, 
75

Rosner test) that are used on full-uncensored data sets (without any NDs). The processes to perform such 

tests using ProUCL 4.0 are described as follows.

1. Click Outlier Tests ► With NDs ► Exclude NDs

Output Screen for Dixon’s Outlier Test
76 

Output Screen for Rosner’s Outlier Test

There are many observations in this data set that may represent potential outliers. The potential outliers 

can also be seen in the graphical displays associated with this data set.

 Selected Options: Number of Suspected Outliers for the Rosner Test = 4

Box plot of the Aroclor1254 Data Set
77

Q-Q Plot of the Aroclor1254 Data Set 
78 

Chapter 8

Goodness-of-Fit (G.O.F.) Tests

Several goodness-of-fit (G.O.F.) tests for full data sets (without nondetects) and for data sets with NDs 

are available in ProUCL 4.0. Details of those tests are described in the ProUCL 4.0 Technical Guide. In 

this User Guide, those tests and available options have been illustrated using screen shots generated by 

ProUCL 4.0.

Two choices are available for Goodness-of-Fit menu: Full and With NDs.

 Full 

o This option is used to analyze full data sets without any nondetect observations. 

Throughout this User Guide and in ProUCL 4.0, “Full” represents data sets without 

nondetect observations.

o This option tests for normal, gamma, or lognormal distribution of the variable(s) 

selected using the Select Variables option.

o G.O.F. Statistics: This option is available for both full data sets and for data sets with 

NDs. This option simply generates output log of GOF test statistics and derived 

conclusions about the data distributions of all selected variables. This option is also 

available for variables categorized by a group variable. 

 With NDs

o Analyzes data sets that have both nondetected and detected values.

o Six sub-menu items listed and shown below are available for this option.

1. Exclude NDs: tests for normal, gamma, or lognormal distribution of the selected 

variable(s) using only the detected values.

2. ROS Estimates: tests for normal, gamma, or lognormal distribution of the selected 

variable(s) using the detected values and the extrapolated values for the nondetects. 

o Three ROS methods for normal, lognormal, and gamma distributions are 

available. This option is used to estimate or extrapolate the NDs based upon the 

specified distribution. 

o By using the menu item ROS Est. NDs, ProUCL 4.0 actually generates additional 

column(s) of ROS estimated NDs based upon the selected distribution. This 
79

option should be used for variables with NDs. This is further illustrated by a 

screen shot given in the following

3. DL/2 Estimates: tests for normal, gamma, or lognormal distribution of the selected 

variable(s) using the detected values and the ND values replaced by their respective 

DL/2 values. This option is included for historical reasons and also for curious users. 

Note: The use of fabricated data obtained using DL/2 (DL or 0 values) values is not 

a recommended method. At best, the user may use the fabricated data (e.g., DL/2 

option) for exploratory reasons. It is suggested that these substitution methods should 

not be used for estimation and for hypotheses testing approaches.

4. G.O.F. Statistics: As for the full data sets, this option simply generates output log of 

GOF test statistics and other relevant statistics for data sets with nondetects. The 

conclusions about the data distributions for all selected variables are also listed on the 

generated output file (Excel-type spreadsheet). This option is also available for 

variables categorized by a group variable.

 When multiple variables are selected from the Select Variables screen, you can choose 

either:

o Group Graph option to produce multiple Q-Q plots for all selected variables in a 

single graph. The relevant statistics (e.g., slope, intercept, correlation, test statistic 

and critical value) associated with the selected variables are shown on the right panel 

(in tan color; see page 77). In order to capture all the graphs and results shown on the

window screen, it is preferable to print the graph using the Landscape option. The use 

of this option is recommended when a selected variable has data coming from two or 

more groups or populations.

o Group Graph option is particularly useful to generate multiple Q-Q plots for the 

groups associated with a selected variable. In order to capture all the graphs and 

results shown on the window screen, it is preferable to print the graph using the 

Landscape option. The user may also want to turn off the Navigation Panel and Log 

Panel. 

 Individual Graph option is used to generate individual Q-Q plots and the associated 

statistics separately for each of the selected variable, one variable at a time. 

o The linear pattern displayed by a Q-Q plot suggests an approximate goodness-of-fit 

for the selected distribution.
80 

o The 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 a good fit for that distribution. This high correlation 

should exhibit a definite linear pattern in the Q-Q plot. Specifically, when data are 

sparse and correlation is high, the use of correlation statistic to determine data 

distribution is not desirable. Note that these statistics are displayed on the Q-Q plot.

o On a Q-Q plot, observations that are well separated from the bulk (central part) of the 

data typically represent potential outliers needing further investigation.

o Significant and obvious jumps and breaks in a Q-Q plot (for any distribution) are 

indications of the presence of more than one population. Data sets exhibiting such 

behavior of Q-Q plots should be partitioned out in component sub-populations before 

estimating an EPC term or a background threshold value (BTV). 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.

 Normality or Lognormality Tests: In addition to informal graphical normal and 

lognormal Q-Q plots, a formal Goodness-of-Fit (GOF) test is also available to test the 

normality or lognormality of the data set.

o Lilliefors Test: a test typically used for samples of size larger than 50 (> 50). 

However, the Lilliefors test (generalized Kolmogorov Smirnov test) is available for 

samples of all sizes. There is no applicable upper limit for sample size for the 

Lilliefors test.

o Shapiro and Wilk (SW) Test: a test used for samples of size smaller than or equal to 

2000 (<= 2000). In ProUCL 4.0, the SW test uses the SW Critical values for samples 

of size 50 or less. For samples of size, greater than 50, the SW Test statistic is 

displayed along with the P-value of the test (Royston, 1982). These values are not as 

yet available in ProUCL 4.0. This extension of SW test will be available in Scout 

(EPA, 1999) software package. 

o It should be noted that sometimes these two tests might lead to different conclusions. 

Therefore, the user should exercise caution interpreting the results. Specifically, the 

user should the pattern exhibited by the associated Q-Q plot.

 GOF test for Gamma Distribution: In addition to the graphical gamma Q-Q plot, two 

formal empirical distribution function (EDF) procedures are also available to test the 

gamma distribution of a data set. These tests are the Anderson-Darling test and the 

Kolmogorov-Smirnov test.

o It is noted that these two tests might lead to different conclusions. Therefore, the user 

should exercise caution interpreting the results.

o These two tests may be used for samples of sizes in the range of 4-2500. Also, for 

these two tests, the value of the shape parameter, k (k hat) should lie in the interval 

[0.01, 100.0]. Consult the ProUCL 4.0 Technical Guide (A. Singh and A.K. Singh 

(EPA, 2007)) for a detailed description of gamma distribution and its parameters,

including k. Extrapolation beyond these sample sizes and values of k is not 

recommended.
81

 ProUCL 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.

 Even though, the G.O.F. Statistics option prints out all GOF test statistics for all selected 

variables, it is suggested that the user should look at the graphical Q-Q plot displays to 

gain extra insight (e.g., outliers, multiple population) into the data set. 

Note: It is highly recommended not to skip the use of a graphical Q-Q plot to determine the data 

distribution as a Q-Q plot also provides a useful information about the presence of multiple populations 

or outliers.

8.1 ROS Estimated (Est.) NDs – Saving Extrapolated NDs

 As mentioned before, for a variable with NDs, ProUCL 4.0 can generate additional 

column(s) consisting of detected data and the estimated (extrapolated) values of NDs 

using the ROS method assuming a normal, lognormal, or a gamma distribution. 

 The user may want to use the resulting full data set (detected and estimated NDs) thus 

obtained to compute the statistics of interest such as a bootstrap BCA UCL95 or a gamma 

95% upper percentile.

 This option of saving estimated NDs is provided only for experienced users and 

researchers. It is expected that the user knows and understands the theory behind these 

methods. Therefore, it is suggested that this option be used with care. For an example, 

often, the use of a ROS method yields infeasible (e.g., negative, exceeding the DLs) 

estimates of NDs, and therefore, the associated estimates of EPC terms and of BTVs may 

be biased and not reliable. This is especially true when the data set contains potential 

outlier(s).

 

8.2 Goodness-of-Fit Tests with Full Data Sets

1. Click Goodness-of-Fit ► Full
82 

2. Select the distribution to be tested: Normal, Lognormal, or Gamma

 To test your variable for normality, click on Normal from the drop-down menu list.

 To test variable for lognormality, click on Lognormal from the drop-down menu list.

 To test your variable for gamma distribution, click on Gamma from the drop-down menu 

list.

8.2.1 GOF Tests for Normal and Lognormal Distribution 

1. Click Goodness-of-Fit ► Full ► Normal or Lognormal

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable. 

 When the option button is clicked, the following window will be shown. 
83

o The default option for the Confidence Level is 95%. 

o The default GOF Method is Shapiro Wilk. If the sample size is greater than 50, the 

program automatically uses the Lilliefors test. 

o The default method for Display Regression Lines is Do Not Display. If you want to 

see regression lines on a Q-Q plot, then check the radio button next to Display 

Regression Lines.

o The default option for Graphs by Group is Individual Graphs. If you want to see 

the plots for all selected variables on a single graph, then check the radio button next 

to Group Graphs.

Note: This option for Graphs by Group is specifically provided when the user wants to display multiple 

graphs for a variable by a group variable (e.g., site AOC1, site AOC2, background). This kind of display 

represents a useful visual comparison of the values of a variable (e.g., concentrations of COPC-Arsenic) 

collected from two or more groups (e.g., upgradient wells, monitoring wells, residential wells). 

o The default option for Graphical Display Options is Color Gradient. If you want 

to see the graphs in black and white to be included in reports for later use, then check 

the radio button next to For Export (BW Printers).
84 

 Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests. 

Output Screen for Normal Distribution (Full)

Selected Options: Shapiro Wilk, Display Regression Line, and For Export (BW Printers).

Output Screen for Normal Distribution (Full)

Selected Options: Shapiro Wilk (n > 50), Display Regression Line, and For Export (BW Printers).
85

Output Screen for Lognormal Distribution (Full)

Selected options: Shapiro Wilk, Display Regression Lines, and Color Gradient.

8.2.2 GOF Tests for Gamma Distribution 

1. Click Goodness-of-Fit ► Full ► Gamma

2. The Select Variables Screen (described in Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable. 

 When the option button is clicked, the following window will be shown. 
86 

o The default option for the Confidence Level is 95%. 

o The default GOF method is Anderson Darling. 

o The default option for Display Regression Lines is Do Not Display. If you want to 

see regression lines on the Gamma Q-Q plot, then check the radio button next to 

Display Regression Lines.

o The default option for Graph by Groups is Individual Graphs. If you want to see 

the graphs for all the selected variables into a single graph, then check the radio 

button next to Group Graphs.

o The default option for Graphical Display Options is Color Gradient. If you want 

to see the graphs in black and white, check the radio button next to For Export (BW 

Printers).

 Click the OK button to continue or the Cancel button to cancel the option.

 Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
87

Output Screen for Gamma Distribution (Full)

Selected options: Anderson Darling, Display Regression Lines, Individual Graphs, and Color Gradient.

8.3 Goodness-of-Fit Tests Excluding NDs

1. Click Goodness-of-Fit ► With NDs ► Exclude NDs

2. Select distribution to be tested: Normal, Gamma, or Lognormal.

 To test for normality, click on Normal from the drop-down menu list.

 To test for lognormality, click on Lognormal from the drop-down menu list.

 To test for gamma distribution, click on Gamma from the drop-down menu list.
88 

8.3.1 Normal and Lognormal Options 

1. Click Goodness-of-Fit ► With NDs ►Excluded NDs ► Normal or Lognormal

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable.

 When the option button (Normal or Lognormal) is clicked, the following window will be 

shown. 
89

o The default option for the Confidence Level is 95%. 

o The default GOF Method is Shapiro Wilk. If the sample size is greater than 50, the 

program defaults to Lilliefors test.

o The default for Display Regression Lines is Do Not Display. If you want to see 

regression lines on the associated Q-Q plot, check the radio button next to Display 

Regression Lines.

o The default option for Graphs by Group is Individual Graphs. If you want to see 

the plots for all selected variables on a single graph, check the radio button next to 

Group Graphs.

Note: This option for Graphs by Group is specifically useful when the user wants to display multiple 

graphs for a variable by a group variable (e.g., site AOC1, Site AOC2, background). This kind of display 

represents a useful visual comparison of the values of a variable (e.g., concentrations of COPC-Arsenic) 

collected from two or more groups (e.g., upgradient wells, monitoring wells, and residential wells). 
90 

o The default option for Graphical Display Option is Color Gradient. If you want to 

see the graphs in black and white, check the radio button next to For Export (BW 

Printers).

 Click the OK button to continue or the Cancel button to cancel the option.

 Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.

Output Screen for Normal Distribution (Exclude NDs)

Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and For Export (BW Printers).
91

Output Result for Lognormal Distribution (Exclude NDs)

Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and Color Gradient.

8.3.2 Gamma Distribution Option

1. Click Goodness-of-Fit ► With NDs ►Excluded NDs ► Gamma

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable.

 When the option button (Gamma) is clicked, the following window is shown. 
92 

o The default option for the Confidence Level is 95%.

o The default GOF test Method is Anderson Darling. 

o The default method for Display Regression Lines is Do Not Display. If you 

want to see regression lines on the normal Q-Q plot, check the radio button next 

to Display Regression Lines.

o The default option for Graph by Groups is Individual Graphs. If you want to 

display all selected variables on a single graph, check the radio button next to 

Group Graphs.

o The default option for Graphical Display Options is Color Gradient. If you 

want to see the graphs in black and white, check the radio button next to For 

Export (BW Printers).

 Click the OK button to continue or the Cancel button to cancel the option.

 Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
93

Output Screen for Gamma Distribution (Exclude NDs)

Selected options: Anderson Darling, Do Not Display, Individual Graphs, and For Export (BW Printers).

8.4 Goodness-of-Fit Tests with Log-ROS Estimates

1. Click Goodness-of-Fit ► With NDs ► Log-ROS Estimates

2. Select the distribution to be tested: Normal, Lognormal, or Gamma

 To test your variable for normality, click on Normal from the drop-down menu list.

 To test a variable for gamma distribution, click Gamma from drop-down menu list.

 To test your variable for lognormality, click on Lognormal from drop-down menu. 
94 

8.4.1 Normal or Lognormal Distribution (Log-ROS Estimates)

1. Click Goodness-of-Fit ► With NDs ► Log-ROS Estimates ► Normal, Lognormal

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable.

 When the option button (Normal or Lognormal) is clicked, the following window will be 

shown. 
95

o The default option for the Confidence Level is 95%. 

o The default GOF test Method is Shapiro Wilk. If the sample size is greater than 50, 

the program defaults to use the Lilliefors test.

o The default method for Display Regression Lines is Do Not Display. If you want to 

see regression lines on the normal Q-Q plot, check the radio button next to Display 

Regression Lines.

o The default option for Graphs by Group is Individual Graphs. If you want to 

display all selected variables into a single graph, check the radio button next to 

Group Graphs.

o The default option for Graphical Display Options is Color Gradient. If you want 

to see the graphs in black and white, check the radio button next to For Export (BW 

Printers).
96 

 Click the OK button to continue or the Cancel button to cancel the option.

 Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.

Output Screen for Normal Distribution (Log-ROS Estimates)

Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and For Export (BW Printers).

Note: The legend size of nondetect values is smaller than that of the detected values.
97

Output Screen for Lognormal Distribution (Log-ROS Estimates)

Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and Color Gradient.

Note: The legend size of nondetect values is smaller than that of the detected values and is shown in red.

8.4.2 Gamma Distribution (Log-ROS Estimates)

1. Click Goodness-of-Fit ► With NDs ► Log-ROS Estimates ► Gamma

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable.

 When the option button (Gamma) is clicked, the following window will be shown. 
98 

o The default option for the Confidence Level is 95%. 

o The default GOF test Method is Anderson Darling. 

o The default method for Display Regression Lines is Do Not Display. If you want to 

see regression lines on the normal Q-Q plot, check the radio button next to Display 

Regression Lines.

o The default option for Graph by Groups is Individual Graphs. If you want to put 

all of the selected variables into a single graph, check the radio button next to Group 

Graphs.

o The default option for Graphical Display Options is Color Gradient. If you want 

to see the graphs in black and white, check the radio button next to For Export (BW 

Printers).

 Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.

Output Screen for Gamma Distribution (Log-ROS Estimates)
99

Selected options: Anderson Darling, Display Regression Lines, Individual Graphs, and Color Gradient.

Note: The legend size of nondetect values is smaller than that of detected values and is shown in red.

8.5 Goodness-of-Fit Tests with DL/2 Estimates

1. Click Goodness-of-Fit ► With NDs ► DL/2 Estimates

2. Select the distribution to be tested: Normal, Gamma, or Lognormal

 To test the variable for normality, click on Normal from the drop-down menu list.

 To test the variable for lognormality, click on Lognormal from the drop-down menu list.

 To test your variable for gamma distribution, click on Gamma from the drop-down menu 

list.
100 

8.5.1 Normal or Lognormal Distribution (DL/2 Estimates)

1. Click Goodness-of-Fit ► With NDs ► DL/2 Estimates ► Normal or Lognormal

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable.

 When Normal or Lognormal button is clicked, following window is displayed. 
101

o The default option for the Confidence Level is 95%. 

o The default Method is Shapiro Wilk. If the sample size is greater than 50, the 

program defaults to the Lilliefors test.

o The default method for Display Regression Lines is Do Not Display. If you want to 

see regression lines on the normal Q-Q plot, check the radio button next to Display 

Regression Lines.

o The default option for Graphs by Group is Individual Graphs. If you want to put 

all of the selected variables into a single graph, check the radio button next to Group 

Graphs.

o The default option for Graphical Display Options is Color Gradient. If you want 

to see the graphs in black and white, check the radio button next to For Export (BW 

Printers)
102 

 Click the OK button to continue or the Cancel button to cancel the option.

 Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.

Output Screen for Normal Distribution (DL/2 Estimates)

Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and Color Gradient.

Note: The legend size of nondetect values is smaller than that of the detected values and is shown in red.
103

Output Screen for Lognormal Distribution (DL/2 Estimates)

Selected options: Shapiro Wilk, Display Regression Lines, Individual Graphs, and For Export (BW Printers).

Note: The legend size of nondetect values is smaller than that of the detected values. The color is not 

shown on this graph as this graph is for BW printers.

8.5.2 Gamma Distribution (DL/2 Estimates)

1. Click Goodness-of-Fit ► With NDs ►DL/2 Estimates ► Gamma

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen. 

 If graphs have to be produced by using a Group variable, then select a group variable by 

clicking the arrow below the Group by variable button. This will result in a drop-down

list of available variables. The user should select and click on an appropriate variable 

representing a group variable.
104 

 When the Gamma option button is clicked, the following window will be shown. 

o The default option for the Confidence Level is 95%. 

o The default Method is Anderson Darling. 

o The default method for Display Regression Lines is Do Not Display. If you want to 

see regression lines on the normal Q-Q plot, check the radio button next to Display 

Regression Lines.

o The default option for Graph by Groups is Individual Graphs. If you want to put 

all of the selected variables into a single graph, check the radio button next to Group 

Graphs.

o The default option for Graphical Display Options is Color Gradient. If you want 

to see the graphs in black and white, check the radio button next to For Export (BW 

Printers).

 Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
105

Output Screen for Gamma Distribution (DL/2 Estimates)

Selected options: Anderson Darling, Display Regression Lines, Individual Graphs, and Color Gradient. 

Note: The legend size of nondetect values is smaller than that of the detected values and is shown in red.

8.6 Goodness-of-Fit Tests Statistics

1. Click Goodness-of-Fit ► With NDs ► G.O.F. Statistics

2. The Select Variables Screen (Chapter 3) will appear. 

 Select one or more variable(s) from the Select Variables screen.

 When the option button is clicked, the following window will be shown.
106 

 The default confidence level is 95%.

 Click the OK button to continue or the Cancel button to cancel the option.

Sample Output Screen for G.O.F. Test Statistics
107

Output Screen for the G.O.F. Test Statistics – (continued)
108 

Chapter 9

Single Sample and Two-Sample Hypotheses 

Testing Approaches

This chapter illustrates single sample and two-sample parametric and nonparametric hypotheses testing 

approaches as incorporated in ProUCL 4.0. ProUCL 4.0 can perform these hypotheses tests on data sets 

with and without nondetect observations. It should be pointed out that, when one wants to use two-sample

hypotheses tests on data sets with NDs, ProUCL 4.0 assumes that samples from both of the groups have 

nondetect observations. All this means that, a ND column (with 0 or 1 entries only) needs to be provided 

for the variable in each of the two groups. This has to be done even if one of the groups has all detected 

entries; in this case the associated ND column will have all entries equal to “1.” This will allow the user 

to compare two groups (e.g., arsenic in background vs. site samples) with one of the groups having some 

NDs and the other group having all detected data. 

9.1 Single Sample Hypotheses Tests

In many environmental applications, single sample hypotheses tests are used to compare site data 

(provided enough site data are available) with pre-specified cleanup standards or compliance limits. 

ProUCL 4.0 contains single sample parametric and nonparametric tests including Student‟s t-test, sign 

test, Wilcoxon Signed Rank (WSR) test, and test for proportion. The single sample hypotheses tests are 

useful when the environmental parameters such as the clean standard, action level, or compliance limits 

(CLs) are known, and the objective is to compare site concentrations with those known threshold values. 

Specifically, a t-test (or a sign test) may be used to verify the attainment of cleanup levels at an AOC after 

remediation activity; and a test for proportion may be used to verify if the proportion of exceedances of an 

action level (or a compliance limit) by sample concentrations collected from an AOC (or a MW) exceeds 

a certain specified proportion (e.g., 1%, 5%, 10%). 

ProUCL 4.0 can perform these hypotheses on data sets with and without nondetect observations. 

However, it should be noted that for single sample hypotheses tests (e.g., sign test, proportion test) used 

to compare site mean/median concentration level with a cleanup standard, Cs, or a compliance limit (e.g., 

proportion test), all NDs (if any) should lie below the cleanup standard, Cs. For proper use of these 

hypotheses testing approaches, the differences between these tests should be noted and understood. 

Specifically, a t-test or a WSR test are used to compare the measures of location and central tendencies 

(e.g., mean, median) of a site area (e.g., AOC) to a cleanup standard, Cs, or action level also representing 

a measure of central tendency (e.g., mean, median); whereas, a proportion test compares if the proportion 

of site observations from an AOC exceeding a compliance limit (CL) exceeds a specified proportion, P0

(e.g., 5%, 10%). ProUCL 4.0 has useful graphical methods that may be used to visually compare the 

concentrations of a site area of concern (AOC) with an action level. This can be done using a box plot of 

site data with action level superimposed on that graph. The details of the various single sample 

hypotheses testing approaches can be found in EPA guidance documents (1989, 2006). A brief discussion 

of these methods is also given in the ProUCL 4.0 Technical Guide.
109

9.1.1 Single Sample Hypothesis Testing for Full Data without Nondetects

1. Click Hypothesis Testing► Single Sample

2. Select Full (w/o NDs) – This option is used for full data sets without nondetects.

 To perform a t-test, click on t-Test from the drop-down menu as shown above.

 To perform a proportion test, click on Proportion from the drop-down menu.

 To run a sign test, click on Sign test from the drop-down menu.

 To run a Wilcoxon Signed Rank test, click on Wilcoxon Signed Rank from the dropdown menu.

9.1.1.1 Single Sample t-Test

1. Click Hypothesis Testing ► Single Sample ► Full (w/o NDs) ► t-Test

2. The Select Variables Screen (see page 130) will appear.

 Select variable (variables) from the Select Variables screen.

 When the Options button is clicked, the following window will be shown.
110 

o Specify the Confidence Level; default is 0.95.

o Specify meaningful values for Substantial Difference, S and the Action Level. The 

default choice for S is “0.”

o Select the form of Null Hypothesis; default is Mean <= Action Level (Form 1).

o Click on OK button to continue or on Cancel button to cancel the test.
111

Output for Single Sample t-Test (Full Data without NDs)

9.1.1.2 Single Sample Proportion Test

1. Click Hypothesis Testing ► Single Sample ► Full (w/o NDs) ► Proportion 

2. The Select Variables Screen will appear.

 Select variable (variables) from the Select Variables screen.

 When the Options button is clicked, the following window will be shown.
112 

o Specify the Confidence level; default is 0.95.

o Specify the Proportion level and a meaningful Action Level.

o Select the form of Null Hypothesis; default is P <= Proportion (Form 1).

o Click on OK button to continue or on Cancel button to cancel the test.
113

Output for Single Sample Proportion Test (Full Data without NDs)

9.1.1.3 Single Sample Sign Test

1. Click Hypothesis Testing ► Single Sample ► Full (w/o NDs) ► Sign test

2. The Select Variables Screen will appear.

 Select variable (variables) from the Select Variables screen.

 When the Options button is clicked, the following window will be shown.
114 

o Specify the Confidence Level; default choice is 0.95.

o Specify meaningful values for Substantial Difference, S and Action Level.

o Select the form of Null Hypothesis; default is Median <= Action Level (Form 1).

o Click on OK button to continue or on Cancel button to cancel the test.
115

Output for Single Sample Proportion Test (Full Data without NDs)

9.1.1.4 Single Sample Wilcoxon Signed Rank (WSR) Test

1. Click Hypothesis Testing ► Single Sample ► Full (w/o NDs) ► Wilcoxon Signed Rank

2. The Select Variables Screen will appear.

 Select variable (variables) from the Select Variables screen.

 When the Options button is clicked, the following window will be shown.
116 

o Specify the Confidence Level; default is 0.95.

o Specify meaningful values for Substantial Difference, S, and Action Level.

o Select the form of Null Hypothesis; default is Mean/Median <= Action Level (Form 

1).

o Click on OK button to continue or on Cancel button to cancel the test.
117

Output for Single Sample Proportion Test (Full Data without NDs)
118 

9.1.2 Single Sample Hypothesis Testing for Data Sets with Nondetects 

Most of the one-sample tests such as the Proportion test and the Sign test on data sets with nondetect 

values assume that all nondetect observations lie below the compliance limit (CL) or an action level, A0. 

The single sample tests cannot be performed if ND observations exceed the CL or action levels.

1. Click on Hypothesis Testing► Single Sample

2. Select the With NDs option

 To perform a proportion test, click on Proportion from the drop-down menu.

 To perform a sign test, click on Sign test from the drop-down menu.

 To perform a Wilcoxon Signed Rank test, click on Wilcoxon Signed Rank from the dropdown menu list.

9.1.2.1 Single Proportion Test on Data Sets with NDs

1. Click Hypothesis Testing ► Single Sample ► With NDs ► Proportion 

2. The Select Variables screen will appear.

 Select variable (variables) from the Select Variables screen.

 When the Options button is clicked, the following window will be shown.
119

o Specify the Confidence Level; default is 0.95.

o Specify meaningful values for Proportion and the Action Level.

o Select the form of Null Hypothesis; default is P <= Proportion (Form 1).

o Click on OK button to continue or on Cancel button to cancel the test.
120 

Output for Single Sample Proportion Test (with NDs)

9.1.2.2 Single Sample Sign Test with NDs

1. Click Hypothesis Testing ► Single Sample ► With NDs ► Sign test

2. The Select Variables screen will appear.

 Select variable (variables) from the Select Variables screen.

 When the Options button is clicked, the following window will be shown.
121

o Specify the Confidence Level; default is 0.95.

o Specify meaningful values for Substantial Difference, S and Action Level.

o Select the form of Null Hypothesis; default is Median <= Action Level (Form 1).

o Click on OK button to continue or on Cancel button to cancel the test.
122 

Output for Single Sample Sign Test (Data with Nondetects)

9.1.2.3 Single Sample Wilcoxon Signed Rank Test with NDs

1. Click Hypothesis Testing ► Single Sample ► With NDs ► Wilcoxon Signed Rank 

2. The Select Variables Screen will appear.
123

 Select variable (variables) from the Select Variables screen.

 When the Options button is clicked, the following window will be shown.

o Specify the Confidence Level; default is 0.95.

o Specify meaningful values for Substantial Difference, S and Action Level.

o Select the form of Null Hypothesis; default is Mean/Median <= Action Level (Form 

1).

o Click on OK button to continue or on Cancel button to cancel the test.
124 

Output for Single Sample Wilcoxon Signed Rank Test (Data with Nondetects)
125

9.2 Two-Sample Hypotheses Testing Approaches 

In this section, the two-sample hypotheses testing approaches as incorporated in ProUCL 4.0 have been 

illustrated. These approaches are used to compare the parameters and distributions of the two populations 

(e.g., Background vs. AOC) based upon data sets collected from those populations. Both forms (Form 1 

and Form 2, Form 2 with Substantial Difference, S) of two-sample hypothesis testing approaches have 

been included in ProUCL 4.0. The methods are available for full data sets as well as for data sets with 

below detection limit (BDL) values.

 Full – analyzes data sets consisting of all detected values. The following parametric and 

nonparametric tests are available:

o Student‟s t and Satterthwaite tests to compare the means of two populations (e.g. 

Background versus AOC). 

o F-test to the check the equality of dispersions of two populations. 

o Two-sample nonparametric Wilcoxon-Mann-Whitney (WMW) test. This test is 

equivalent to Wilcoxon Rank Sum (WRS) test. 

o Quantile test is often used to compare upper tails of two data distributions. This test 

is normally performed in parallel with WMW test.

 With NDs – analyzes data sets consisting of both nondetected and detected values. The 

following tests are available: 

o Wilcoxon-Mann-Whitney test. All observations (including detected values) below the 

highest detection limit are treated as ND (less than the highest DL) values.

o Quantile test is used to compare upper tails of two data distributions. This test is 

performed in parallel with WMW test.

o Gehan‟s test, useful when multiple detection limits may be present.

The details of these methods can be found in the ProUCL 4.0 Technical Guide and are also available in 

EPA (1997, 2006). It is re-stated that the use of informal graphical displays (e.g., side-by-side box plots, 

multiple Q-Q plots) should always accompany the formal hypothesis testing approaches listed above. 

This is especially warranted when the data sets may consist of observations from multiple populations 

(e.g., mixture samples collected from various onsite locations) and outliers. 

Note: As mentioned before, it is pointed out that, when one wants to use two-sample hypotheses tests on 

data sets with NDs, ProUCL 4.0 assumes that samples from both of the groups have nondetect 

observations. This may not be the case, as data from a polluted site may not have any ND observations. 

ProUCL can handle such data sets. However, the user will have to provide a ND column (with 0 or 1 

entries only) for the selected variable of each of the two groups. Thus when one of the groups (e.g., site 

arsenic) has no ND value, the user supplies an associated ND column with all entries equal to “1.” This 

will allow the user to compare two groups (e.g., arsenic in background vs. site samples) with one of the 

group having some NDs and the other group having all detected data. 
126 

9.2.1 Two-Sample Hypothesis Tests for Full Data

Full – This option is used to analyze data sets consisting of all detected values. The following two-sample

tests are available in ProUCL 4.0.

 Student‟s t and Satterthwaite tests to compare the means of two populations (e.g.,

Background versus AOC). 

 F-test is also available to test the equality of dispersions of two populations. 

 Two-sample nonparametric Wilcoxon-Mann-Whitney (WMW) test. 

 Two-sample quantile test.

 Student‟s t-Test

o This test can be used to compare the site mean concentration of a COPC with that of 

the background mean concentration provided the populations are normally 

distributed. The data sets are given by independent random observations, X1, X2, . . . , 

Xn collected from a site, and independent random observations, Y1, Y2, . . . , Ym

collected from a background population. The same terminology is used for all other 

two-sample tests in ProUCL 4.0.

o Student‟s t-test also assumes that the spread (variance) of the two populations are 

approximately equal. 

o The F-test can be used to the check the equality of dispersions of two populations. 

 Satterthwaite t-Test

o This test is used to compare the population means of two populations when the 

variances or Spreads of those populations may not be equal. As mentioned before, the 

F-distribution based test can be used to verify the equality of dispersions of two 

populations.

 Test for Equality of two Dispersions (F-test)

o This test is used to determine whether the true underlying variances of two 

populations are equal. Usually the F-test is employed as a preliminary test, before 

conducting the two-sample t-test for testing the equality of means of two populations. 

o The assumptions underlying the F-test are that the two-samples represent 

independent random samples from two normal populations. The F-test for equality of 

variances is highly sensitive to departures from normality.

 Two-Sample Nonparametric WMW Test

o This test is used to determine the approximate equality of the two continuous data 

distributions. This test also assumes that the shapes (e.g., as determined by spread, 

skewness, and graphical displays) of the two populations are roughly equal. The test 

is often used to determine if the measures of central locations of the two populations 
127

are significantly different. Specifically, the test can be used to determine if the site 

concentrations exceed the background concentrations.

o The Wilcoxon-Mann-Whitney test does not assume that the data are normally or lognormally distributed. For large samples (e.g.,  20), the distribution of the WMW test 

statistic can be approximated by a normal distribution. 

o This test is used to determine if measurements from one population consistently tend 

to be larger (or smaller) than those from the other population. 

 Two-Sample Quantile Test

o The nonparametric quantile test does not assume that the data are normally or lognormally distributed. For large samples (e.g.,  20), the distribution of the quantile 

test statistic can be approximated by a normal distribution. 

o This test is used in parallel with the WMW test. This test is often used in Background 

Test Form 1 to determine if the concentrations from the upper tail of site data 

distribution are comparable to (lower than or equal to) that of the background data 

distribution. The critical values for this Form 1 test are available in EPA, 1994. The 

details of the test are given in EPA (1994, 2006). 

Note: The use of the tests listed above is not recommended on log-transformed data sets, especially when 

the parameters of interests are the population means. In practice, the cleanup and remediation decisions 

have to be made in original scale based upon statistics and estimates computed in the original scale. The 

equality of means in log-scale does not necessarily imply the equality of means in the original scale. This 

topic is discussed in detail in Chapter 3 of the revised background document (EPA, 2002) for CERCLA 

sites (currently under revision).

1. Click on Hypothesis Testing ► Two Sample

2. Select the Full (w/o NDs) option

 To perform a t-test, click on t Test from the drop-down menu.

 To perform a Wilcoxon-Mann-Whitney, click on Wilcoxon-Mann-Whitney from the 

drop-down menu list.

 To perform a quantile test, click on Quantile test from the drop-down menu.
128 

9.2.1.1 Two-Sample t-Test without NDs

1. Click Hypothesis Testing ► Two Sample ► Full (w/o NDs) ► t Test

2. The Select Variables screen will appear.

 Select variable (variables) from the Select Variables screen.

 Without Group Variable: This option is used when the data values of the variable 

(COPC) for the site and the background are given in separate columns.

 With Group Variable: This option is used when data values of the variable (COPC) for 

the site and the background are given in the same column. The values are separated into 

different populations (groups) by the values of an associated Group Variable. The group 

variable may represent several populations (e.g., several AOCs, MWs). The user can 

compare two groups at a time by using this option. 

 When using this option, the user should select a group variable by clicking the arrow next 

to the Group Var option for a drop-down list of available variable. The user selects an 

appropriate (meaningful) variable representing groups such as Background and AOC. 

The user is allowed to use letters, numbers, or alphanumeric labels for the group names.

o When the options button is clicked, the following window will be shown.
129

o Specify a useful Substantial Difference, S value. The default choice is 0. 

o Choose the Confidence level. The default choice is 95%. 

o Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).

o Click on OK button to continue or on Cancel button to cancel the option.

 Click on the OK button to continue or on the Cancel button to cancel the Site versus 

Background Comparison.
130 

Output for Two-Sample t-Test (Full Data without NDs)

9.2.1.2 Two-Sample Wilcoxon-Mann-Whitney (WMW) Test without NDs

1. Click Hypothesis Testing ► Two Sample ► Full (w/o NDs) ► Wilcoxon-Mann-Whitney 

Test
131

2. The Select Variables Screen will appear.

 Select variable (variables) from the Select Variables screen.

 Without Group Variable: This option is used when the data values of the variable 

(COPC) for the site and the background are given in separate columns.

 With Group Variable: This option is used when data values of the variable (COPC) for 

the site and the background are given in the same column. The values are separated into 

different populations (groups) by the values of an associated Group Variable. When using 

this option, the user should select a group variable by clicking the arrow next to the 

Group Var option for a drop-down list of available variables. 

 ProUCL 4.0 has been written using environmental terminology such as performing 

background versus site comparisons. However, all the tests and procedures in ProUCL 

4.0 can be used for any other application if used properly. The user selects an appropriate 

group variable representing groups such as Background and AOC. For other applications 

such as comparing a new treatment drug versus older treatment drug, the group variable 

may represent the two groups: old drug group and new drug group. The user is allowed to 

use letters, numbers, or alphanumeric labels for the group names.

 When the Options button is clicked, the following window is shown.
132 

o Specify a Substantial Difference, S value. The default choice is 0. 

o Choose the Confidence level. The default choice is 95%. 

o Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).

o Click on OK button to continue or on Cancel button to cancel the selected options.

 Click on the OK button to continue or on the Cancel button to cancel the Site versus 

Background Comparison.
133

Output for Two-Sample Wilcoxon-Mann-Whitney Test (Full Data)

9.2.1.3 Two-Sample Quantile Test for Full Data without NDs

As mentioned before, the quantile test is often used in parallel with the WMW test. Typically, both tests 

are performed on the same data set before coming to the conclusion about comparability (or noncomparability) of the data distributions of the two populations.

1. Click Hypothesis Testing ► Two Sample ► Full (w/o NDs) ► Quantile test
134 

2. The Select Variables Screen shown below will appear.

 Select variable (variables) from the Select Variables screen.

 Without Group Variable: This option is used when the data values of the variable 

(COPC) for the site and the background are given in different columns.

 With Group Variable: This option is used when data values of the variable (COPC) for 

the site and the background are given in the same column. The values are separated into 

different groups by using the values of the associated Group Variable. When using this 

option, the user should select a group variable by clicking the arrow next to the Group 

Var option for a drop-down list of available variables. The user selects an appropriate 

group variable representing groups such as Background and AOC. The user is allowed to 

use letters, numbers, or alphanumeric labels for the group names.

 When the Options button is clicked, the following window will be shown.
135

o Choose the Confidence level; the default choice is 95%.

o Click on OK button to continue or on Cancel button to cancel the option.

 Click on the OK button to continue or on the Cancel button to cancel the Site versus 

Background Comparison.

Output for Two-Sample Quantile Test (Full Data)
136 

9.2.2 Two-Sample Hypothesis Testing for Data Sets with Nondetects 

1. Click Hypothesis Testing► Two Sample

2. Select the With NDs option. A list of available tests will appear (shown above).

 To perform a Wilcoxon-Mann-Whitney test, click on Wilcoxon-Mann-Whitney from the 

drop-down menu list.

 To perform a Gehan test, click on Gehan from the drop-down menu.

 To perform a quantile test, click on Quantile Test from the drop-down menu.

9.2.2.1 Two-Sample Wilcoxon-Mann-Whitney Test with Nondetects

1. Click Hypothesis Testing ► Two Sample ► With NDs ► Wilcoxon-Mann-Whitney 
137

2. The Select Variables Screen shown below will appear.

 Select variable (variables) from the Select Variables screen.

 Without Group Variable: This option is used when the data values of the variable 

(COPC) for the site and the background are given in separate columns.

 With Group Variable: This option is used when data values of the variable (COPC) for 

the site and the background are given in the same column. The values are separated into 

different populations (groups) by the values of an associated Group Variable. When using 

this option, the user should select a group variable by clicking the arrow next to the 

Group Var option for a drop-down list of available variables. The user selects an 

appropriate variable representing groups such as Background and AOC. The user is 

allowed to use letters, numbers, or alphanumeric labels for the group names.

 When the Options button is clicked, the following window will be shown.
138 

o Specify a meaningful Substantial Difference, S value. The default choice is 0. 

o Choose the Confidence level. The default choice is 95%. 

o Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).

o Click on the OK button to continue or on the Cancel button to cancel the selected 

options.

 Click on OK button to continue or on Cancel button to cancel the Site versus Background 

Comparison.
139

Output for Two-Sample Wilcoxon-Mann-Whitney Test (with Nondetects)

Note: In the WMW test, all observations below the largest detection limit are considered as NDs 

(potentially including some detected values) and hence they all receive the same average rank. This 

action may reduce the associated power of the WMW test considerably. This in turn may lead to incorrect 

conclusion. As mentioned before, all hypotheses testing approaches should be supplemented with 

graphical displays such as Q-Q plots and box plots. When multiple detection limits are present, the use of 

the Gehan test is preferable.
140 

9.2.2.2 Two-Sample Gehan Test for Data Sets with Nondetects

1. Click Hypothesis Testing ► Two Sample ► With NDs ► Gehan 

2. The Select Variables Screen will appear.

 Select variable (variables) from the Select Variables screen.

 Without Group Variable: This option is used when the data values of the variable 

(COPC) for the site and the background are given in separate columns.

 With Group Variable: This option is used when data values of the variable (COPC) for 

the site and the background are given in the same column. The values are separated into 

different populations (groups) by the values of an associated Group Variable. When using 

this option, the user should select a group variable by clicking the arrow next to the 

Group Var option for a drop-down list of available variables. The user selects a group 

variable representing groups such as Background and AOC. 

 When the Options button is clicked, the following window will be shown.
141

o Specify a Substantial Difference, S value. The default choice is 0. 

o Choose the Confidence level. The default choice is 95%. 

o Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).

o Click on OK button to continue or on Cancel button to cancel selected options.

 Click on the OK button to continue or on the Cancel button to cancel the Site versus 

Background Comparison.
142 

Output for Two-Sample Gehan Test (with Nondetects)

9.2.2.3 Two-Sample Quantile Test for Data Sets with Nondetects 

Quantile test as described in EPA (1994) has been included in ProUCL 4.0. The detailed power of the test 

with many ND values is not well studied. The conclusion of this test should also be supplemented with 

graphical displays. The use of the Gehan test is preferred when the data set may consist of many NDs 

with multiple detection limits.
143

1. Click Hypothesis Testing ► Two Sample ► With NDs ► Quantile Test

2. The Select Variables Screen will appear.

 Select variable (variables) from the Select Variables screen.

 Without Group Variable: This option is used when the data values of the variable 

(COPC) for the site and the background are given in separate columns.

 With Group Variable: This option is used when data values of the variable (COPC) for 

the site and the background are given in the same column. The values are separated into 

different populations (groups) by the values of an associated Group Variable. When using 

this option, the user should select a group variable by clicking the arrow next to the 

Group Var option for a drop-down list of available variables. The user selects an 

appropriate group variable representing groups such as Background and AOC. 

 When the Options button is clicked, the following window will be shown.
144 

o Choose the Confidence level; the default choice is 95%.

o Click on OK button to continue or on Cancel button to cancel the option.

 Click on the OK button to continue or on the Cancel button to cancel the Site versus 

Background Comparison.

Output for Two-Sample Quantile Test (with Nondetects)
145
146 

Chapter 10

Background Statistics

This chapter illustrates the computations of various parametric and nonparametric statistics and upper 

limits that can be used as estimates of background threshold values (BTVs) and other not-to-exceed 

values. The BTV estimation methods are available for all data sets with and without nondetect (ND) 

observations. The details of those methods are given in Chapter 5 (full data sets without NDs) and 

Chapter 6 (data sets with NDs) of the revised background document for CERCLA sites (EPA, 2002). 

Technical details can also be found in the Technical Guide associated with ProUCL 4.0. For each selected 

variable, this option computes various upper limits such as UPLs, UTLs, and upper percentiles to estimate 

the background threshold values (BTVs) and other compliance limits that are used in site versus 

background evaluations. 

As before, two choices for data sets are available to compute background statistics:

 Full – computes background statistics for a Full data set without any NDs.

 With NDs – computes background statistics for a data set with nondetected as wells as 

detected values. Multiple detection limits are allowed.

The user specifies the confidence level (probability) associated with each interval estimate. The 

reasonable confidence level as incorporated in ProUCL 4.0 represents a number in the interval [0.5, 1), 

0.5 inclusive. The default choice is 0.95. 

For data sets with and without NDs, ProUCL 4.0 can compute the following statistics that can be used as 

estimates of BTVs and not-to-exceed values.

 Parametric and nonparametric upper percentiles.

 Parametric and nonparametric upper prediction limits (UPLs) for a single observation, 

future or next k ( 1) observations, mean of next k observations. Here future k, or next k 

observations may also represent k observations from another population (e.g., site) 

different from the sampled (background) population (used to compute UPLs, UTLs).

 Parametric and nonparametric upper tolerance Limits (UTLs). 

 Nonparametric IQR-based upper limits. 
147

10.1 Background Statistics for Full Data Sets without Nondetects

1. Click Background ► Full (w/o NDs) Background Statistics

2. Select Full Background Statistics.

 To compute the background statistics assuming the normal distribution, click on Normal 

from the drop-down menu list.

 To compute the background statistics assuming the gamma distribution, click on Gamma 

from the drop-down menu list.

 To compute the background statistics assuming the lognormal distribution, click on 

Lognormal from the drop-down menu list.

 To compute the background statistics using distribution-free nonparametric methods, 

click on Non-Parametric from the drop-down menu list.

 To compute and see all background statistics available in ProUCL 4.0, click on the All 

option from the drop-down menu list.

10.1.1 Normal or Lognormal Distribution 

1. Click Background ► Full (w/o NDs) Background Statistics ► Normal or Lognormal

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If needed, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables and select an appropriate group variable. 
148 

 When the option button is clicked, the following window will be shown. 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Specify the Coverage coefficient (for a percentile) needed to compute UTLs. 

Coverage represents a number in the interval (0.0, 1). The default choice is 0.9. 

Remember, a UTL is an upper confidence limit (e.g., with confidence level = 0.95) 

for a 90% (e.g., with coverage = 0.90) percentile.

o Specify the Different or Future K Values. The default choice is 1. It is noted that 

when K = 1, the resulting interval will be a UPL for a single future (or site) 

observations. In the example shown above, a value of K = 1 has been used. 

o Specify the Number of Bootstrap Operations (resamples). The default choice is

2000.

o Click on OK button to continue or on Cancel button to cancel this option.

 Click on OK button to continue or on Cancel button to cancel the Background Statistics 

Options.
149

Output Screen for Normal Distribution (Full)
150 

Output Screen for Lognormal Distribution (Full)
151

10.1.2 Gamma Distribution 

1. Click Background ► Full (w/o NDs) Background Statistics ► Gamma

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If needed, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables, and select a proper group variable. 

 When the option button is clicked, the following window will be shown. 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Specify the Coverage level; a number in interval (0.0, 1). Default choice is 0.9. 

o Specify the next K. The default choice is 1.

o Specify the Number of Bootstrap Operations. The default choice is 2000.

o Click on OK button to continue or on Cancel button to cancel the option.

 Click on OK button to continue or on Cancel button to cancel the Background Statistics 

Options.
152 

Output Screen for Gamma Distribution (Full)
153

10.1.3 Nonparametric Methods 

1. Click Background ► Full (w/o NDs) Background Statistics ► Non-Parametric

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If needed, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables, and select a proper group variable. 

 When the option button is clicked, the following window will be shown. 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9. 

o Specify the next K. The default choice is 1.

o Specify the Number of Bootstrap Operations. The default choice is 2000.

o Click on the OK button to continue or on the Cancel button to cancel the option.

 Click on OK button to continue or on Cancel button to cancel the Background Statistics

Options.
154 

Output Screen for Nonparametric Option (Full)
155

10.1.4 All Statistics Option

1. Click Background ► Full Background Statistics ► All

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If needed, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables, and select a proper group variable. 

 When the option button is clicked, the following window will be shown. 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Specify the Coverage level; a number in the interval (0.0, 1). Default is 0.9. 

o Specify the next K. The default choice is 1. 

o Specify the Number of Bootstrap Operations. The default choice is 2000.

o Click on OK button to continue or on Cancel button to cancel the option.

 Click on OK button to continue or on Cancel button to cancel the Background Statistics

Options.
156 

Output Screen for All Statistics Option (Full)
157

10.2 Background Statistics with NDs

1. Click Background ► With NDs Background Statistics

2. Select the With NDs Background Statistics option.

 To compute the background statistics assuming the normal distribution, click on Normal 

from the drop-down menu list.

 To compute the background statistics assuming the gamma distribution, click on Gamma 

from the drop-down menu list.

 To compute the background statistics assuming the lognormal distribution, click on 

Lognormal from the drop-down menu list.

 To compute the background statistics using distribution-free methods, click on NonParametric from the drop-down menu list.

 To compute all available background statistics in ProUCL 4.0, click on the All option

from the drop-down menu list.

10.2.1 Normal or Lognormal Distribution 

1. Click Background ► With NDs Background Statistics ► Normal or Lognormal

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If needed, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables, and select a proper group variable. 

 When the option button is clicked, the following window will be shown. 
158 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9. 

o Specify the next K. The default choice is 1. 

o Specify the Number of Boostrap Operations. The default choice is 2000.

o Click on the OK button to continue or on the Cancel button to cancel the option.

 Click on OK button to continue or on Cancel button to cancel the Background Statistics

Options.
159

Output Screen for Normal Distribution (with NDs)
160 
161

Output Screen for Lognormal Distribution (with NDs)
162 

10.2.2 Gamma Distribution 

1. Click Background ► With NDs Background Statistics ► Gamma

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If needed, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables and select a proper group variable. 

 When the option button is clicked, the following window will be shown. 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The

default choice is 0.95. 

o Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9. 

o Click on the OK button to continue or on the Cancel button to cancel option.

 Click on OK button to continue or on Cancel button to cancel the Background Statistics.
163

Output Screen for Gamma Distribution (with NDs)
164 

10.2.3 Nonparametric Methods (with NDs)

1. Click Background ► With NDs Background Statistics ► Non-Parametric

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If needed, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables and select a proper group variable. 

 When the option button is clicked, the following window will be shown. 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Specify the Coverage level; a number in interval (0.0, 1). Default choice is 0.9. 

o Click on the OK button to continue or on the Cancel button to cancel the option.

 Click on OK button to continue or on Cancel button to cancel the Background Statistics.
165

Output Screen for Nonparametric Methods (with NDs)
166 

10.2.4 All Statistics Option 

1. Click Background ► With NDs Background Statistics ► All

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If needed, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables, and select a proper group variable. 

 When the option button is clicked, the following window will be shown. 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9. 

o Specify the Next K. The default choice is 1. 

o Click on the OK button to continue or on the Cancel button to cancel the option.

 Click on OK button to continue or on Cancel button to cancel the Background Statistics.
167

Output Screen for All Statistics Option (with NDs)
168 
169
170 

Chapter 11

Computing Upper Confidence Limits (UCLs) of Mean

The UCL computation module of ProUCL 4.0 represents an update of the UCL module of ProUCL 3.0. 

The detailed theory and formulae used to compute 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). 

Several parametric and nonparametric UCL computation methods for data sets with NDs have been 

incorporated in ProUCL 4.0. Methods such as the Kaplan-Meier (KM) and regression on order statistics 

(ROS) methods as incorporated in ProUCL 4.0 can handle multiple detection limits. For details regarding 

the distributions and methods available in ProUCL 4.0, refer to the ProUCL 4.0 Technical Guide and 

Singh, Maichle, and Lee (USEPA, 2006). Recommendations for the computations of UCLs for data sets 

with NDs have been made based upon the findings of the simulation experiments performed by Singh, 

Maichle, and Lee (USEPA, 2006). 

In ProUCL 4.0, two choices are available to compute UCL statistics:

 Full – Computes UCLs for full data sets without any nondetected values.

 With NDs - Computes UCLs for data sets that have detected as well as BDL 

observations. It is pointed out that it is not desirable to use statistical methods as 

incorporated in ProUCL 4.0 on data sets consisting of all nondetect values. Discussion 

about the detection sampling frequency is provided in Chapter 1 of this User Guide. 

Some of the available methods can handle multiple detection limits. The program 

provides a message to the user about the use of an appropriate method when multiple 

detection limits may be present. 

 For full data sets without NDs and also for data sets with NDs, the following options and 

choices are available to compute UCLs of the population mean.

o The user specifies the confidence level; a number in the interval [0.5, 1), 0.5 

inclusive. The default choice is 0.95. 

o The program computes several nonparametric UCLs using the central limit theorem

(CLT), Chebyshev inequality, jackknife, and bootstrap re-sampling methods.

o For the bootstrap method, the user can select the number of bootstrap runs (resamples). The default choice for the number of bootstrap runs is 2000. 

o The user is responsible for selecting an appropriate choice for the data distribution: 

normal, gamma, lognormal, or nonparametric. It is desirable that user determines data 

distribution using the Goodness-of-Fit test option prior to using the UCL option. The 

UCL option informs the user if data are normal, gamma, lognormal, or a nondiscernable distribution. Program computes statistics depending on the user selection. 

o For data sets, which are not normal, one may try the gamma UCL next. The program 

will offer you advice if you chose the wrong UCL option.

o For data sets, which are neither normal nor gamma, one may try the lognormal UCL. 

The program will offer you advice if you chose the wrong UCL option.
171

o Data sets that are not normal, gamma, or lognormal are classified as distribution-free 

nonparametric data sets. The user may use nonparametric UCL option for such data 

sets. The program will offer you advice if you chose the wrong UCL option.

o The program also provides the All option. By selecting this option, the UCLs are 

computed using most of the relevant methods available in ProUCL 4.0. The program 

informs the user about the distribution of the underlying data set, and offers advice 

regarding the use of an appropriate UCL.

o For lognormal data sets, ProUCL can compute only a 90% or a 95% Land‟s statisticbased 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.

o If you have selected a distribution, then ProUCL will provide a recommended UCL 

computation method for 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; as EPC term is estimated by a 

95% UCL of the mean. 

Note: It is recommended that the user identify a few low probability outlying observations that may be 

present in the data set. Outliers distort many statistics of interest including summary statistics, data 

distributions, test statistics, UCLs, and estimates of BTVs. Decisions based upon distorted statistics may 

be misleading and incorrect. The objective is to compute relevant statistics and estimates based upon the 

majority of the data set(s) representing the dominant population(s). Those few low probability outlying 

observations require separate attention and investigation. The project team should decide about the 

proper disposition (to include or not to include) of outliers before computing the statistics to estimate the 

EPC terms and BTVs. In order to determine and compare the improper and unbalanced influence of 

outliers on UCLs and background statistics, the project team may want to compute statistics using data 

sets with outliers and without outliers.

11.1 UCLs for Full Data Sets

11.1.1 Normal Distribution (Full Data Sets without NDs)

1. Click UCL ► Full ► Normal

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If needed, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables, and select a proper group variable. 

 When the option button is clicked, the following window will be shown. 
172 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Click on OK button to continue or on Cancel button to cancel the option.

 Click on OK button to continue or on Cancel button to cancel the UCL computation 

option.
173

Output Screen for Normal Distribution (Full Data without NDs)
174 

11.1.2 Gamma, Lognormal, Nonparametric, All Statistics Option (Full Data without NDs)

1. Click UCL ► Full ► Gamma, Lognormal, Non-Parametric, or All

2. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If desired, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables, and select a proper group variable. 

 When the option button is clicked, the following window will be shown. 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Specify the Number of Bootstrap Operations (runs). Default choice is 2000. 

o Click on OK button to continue or on Cancel button to cancel the UCLs option.

 Click on OK button to continue or on Cancel button to cancel the selected UCL 

computation option.
175

Output Screen for Gamma Distribution (Full)
176 

Output Screen for Lognormal Distribution (Full)
177

Output for Nonparametric Methods (Full Data without NDs)
178 

Output Screen for All Statistics Option (Full Data without NDs)
179

Note: Once again, it should be noted that the number of valid samples represents the total number of 

samples minus (-) the missing values (if any). The number of unique or distinct samples simply represents 

number of distinct observations. The information about the number of distinct samples is useful when 

using bootstrap methods. Specifically, it is not desirable to use bootstrap methods on data sets with only a 

few (< 4-5) distinct values.

11.2 UCL for Data Sets with NDs

1. Click UCL ► With NDs 

2. Choose the Normal, Gamma, Lognormal, Non-Parametric, or All option.

3. The Select Variables Screen (Chapter 3) will appear. 

 Select a variable (variables) from the Select Variables screen. 

 If desired, select a group variable by clicking the arrow below the Group by variable to 

obtain a drop-down list of available variables, and select a proper group variable. The

selection of this option will compute the relevant statistics separately for each group that 

may be present in the data set.

 When the option button is clicked, the following window will be shown. 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Specify the Number of Bootstrap Operations. The default choice is 2000. 

o Click on OK button to continue or on Cancel button to cancel the UCLs option.
180 

 Click on OK button to continue or on Cancel button to cancel the selected UCL 

computation option.

Output Screen for Normal Distribution (with NDs)
181

Output Screen for Normal Distribution (with NDs) – Continued
182 

Output Screen for Gamma Distribution (with NDs)
183

Output Screen for Gamma Distribution (with NDs) – Continued
184 

Output Screen for Lognormal Distribution (with NDs)

Note: Once again, it should be noted that the number of valid samples represents total number of samples 

minus (-) the missing values (if any). The number of unique or distinct samples simply represents number 

of distinct observations. The information about the number of distinct samples is useful when using 

bootstrap methods. Specifically, it is not desirable to use bootstrap methods on data sets with only a few 

(< 4-5) distinct values.
185

Output Screen for Lognormal Distribution (with NDs) – Continued
186 

Output Screen for Nonparametric Methods (with NDs)
187

Output for Nonparametric Methods (with NDs) – Continued
188 

Output Screen for All Statistics Option (with NDs)

Note: Once again, it should be noted that the number of valid samples represents the total number of 

samples minus (-) the missing values (if any). The number of unique or distinct samples simply represents 

number of distinct observations. The information about the number of distinct samples is useful when 

using bootstrap methods. Specifically, it is not desirable to use bootstrap methods on data sets with only a 

few (< 4-5) distinct values.
189

Output Screen for All Statistics Option (with NDs) – Continued
190 

Chapter 12

Sample Sizes for User Specified Performance Measures

One of the most frequent problems in the application of statistical theory to practical applications 

including environmental projects is the determination of appropriate sample sizes for sampling of 

reference/background areas and survey units (e.g., potentially impacted site areas, remediation 

units, areas of concern, decision units) needed to make defensible and cost effective decisions

based upon those sampled data. The sample size determination formulae for estimation of the 

population mean (or some other parameters) depend upon certain decision parameters including 

the confidence coefficient, (1-α) and the specified error margin, Δ from the unknown true 

population mean, μ. Similarly, for hypotheses testing approaches, sample size determination 

formulae depend upon pre-specified values of the decision parameters selected while defining

and describing the data quality objectives (DQOs) associated with an environmental project. The 

decision parameters (DQOs) associated with hypotheses testing approaches include Type I (false 

positive error rate, α) and Type II (false negative error rate, β=1-power) error rates; and the 

allowable width, Δ of the gray region. For values of the parameter of interest (e.g., mean, 

proportion) lying in the gray region, the consequences of committing the two types of errors 

described above are not significant from both human health and cost effectiveness point of view.

Both parametric (assuming normality) and nonparametric (distribution free) sample size 

determination formulae as described in EPA guidance documents (EPA 2000, 2002c, 2006, 

2009) have been incorporated in ProUCL 4.00.05. The details can be found in “Supplement to 

ProUCL 4.0 Technical Guide – Determining Minimum Sample Sizes for User Specified 

Performance Measures.”

12.1 Estimation of Mean

1. Click Number of Samples ► Estimate Mean

2. The following options window is shown.
191

 Specify the Confidence Coefficient. Default is 0.95.

 Specify the Estimate of standard deviation. Default is 3

 Specify the Allowable Error Margin in Mean Estimate. Default is 10.

 Click on OK button to continue or on Cancel button to cancel the options.

Output Screen for Sample sizes for estimation of mean (CC = 95%, sd = 25, error margin = 10)

12.2 Sample Sizes for Single Sample Hypothesis Tests

12.2.1 Sample Size for Single Sample t-Test

1. Click Number of Samples ► Hypothesis Tests ► Single Sample Tests ► t-Test
192 

2. The following options window is shown.

 Specify the False Rejection Rate (Alpha). Default is 0.05.

 Specify the False Acceptance Rate (Beta). Default is 0.1.

 Specify the Estimate of standard deviation. Default is 3

 Specify the Width of the Gray Region (Delta). Default is 2.

 Click on OK button to continue or on Cancel button to cancel the options.
193

Output Screen for sample sizes for single sample t-test (α = 0.05, β = 0.2, s = 10.41, Δ = 10 – Example from EPA 2006 

(page 49))

12.2.2 Sample Size for Single Sample Proportion Test

1. Click Number of Samples ► Hypothesis Tests ► Single Sample Tests ► Proportion

2. The following options window is shown.

 Specify the False Rejection Rate (Alpha). Default is 0.05.
194 

 Specify the False Acceptance Rate (Beta). Default is 0.1.

 Specify the Desirable Proportion (P0). Default is 0.3.

 Specify the Width of the Gray Region (Delta). Default is 0.15.

 Click on OK button to continue or on Cancel button to cancel the options.

Output Screen for sample sizes for single sample proportion test (α = 0.05, β = 0.2, P0 = 0.2, Δ = 0.05 - Example from EPA 

2006 (page 59))

12.2.3 Sample Size for Single Sample Sign Test

1. Click Number of Samples ► Hypothesis Tests ► Single Sample Tests ► Sign Test

2. The following options window is shown.
195

 Specify the False Rejection Rate (Alpha). Default is 0.05.

 Specify the False Acceptance Rate (Beta). Default is 0.1.

 Specify the Estimate of standard deviation. Default is 3

 Specify the Width of the Gray Region (Delta). Default is 2.

 Click on OK button to continue or on Cancel button to cancel the options.

Output Screen for sample sizes for single sample sign test (default options)

12.2.4 Sample Size for Single Sample Wilcoxon Signed Rank Test

1. Click Number of Samples ► Hypothesis Tests ► Single Sample Tests ► Wilcoxon Signed 

Rank
196 

2. The following options window is shown.

 Specify the False Rejection Rate (Alpha). Default is 0.05.

 Specify the False Acceptance Rate (Beta). Default is 0.1.

 Specify the Estimate of standard deviation of WSR Test Statistic. Default is 3

 Specify the Width of the Gray Region (Delta). Default is 2.

 Click on OK button to continue or on Cancel button to cancel the options.
197

Output Screen for sample sizes for single sample WSR test (α = 0.1, β = 0.2, sd = 130, Δ = 100 - Example from EPA 2006 

(page 65))

12.3 Sample Sizes for Two Sample Hypothesis Tests

12.3.1 Sample Size for Two Sample t-Test

1. Click Number of Samples ► Hypothesis Tests ► Two Sample Tests ► t-Test

2. The following options window is shown.

 Specify the False Rejection Rate (Alpha). Default is 0.05.
198 

 Specify the False Acceptance Rate (Beta). Default is 0.1.

 Specify the Estimate of standard deviation. Default is 3

 Specify the Width of the Gray Region (Delta). Default is 2.

 Click on OK button to continue or on Cancel button to cancel the options.

Output Screen for sample sizes for two sample t-test (α = 0.05, β = 0.2, sp = 1.467, Δ = 2.5 - Example from EPA 2006 (page 

68))

12.3.2 Sample Size for Two Sample Wilcoxon Mann-Whitney Test

1. Click Number of Samples ► Hypothesis Tests ► Two Sample Tests ► Wilcoxon MannWhitney Test

2. The following options window is shown.
199

 Specify the False Rejection Rate (Alpha). Default is 0.05.

 Specify the False Acceptance Rate (Beta). Default is 0.1.

 Specify the Estimate of standard deviation of WMW Test Statistic. Default is 3

 Specify the Width of the Gray Region (Delta). Default is 2.

 Click on OK button to continue or on Cancel button to cancel the options.

Output Screen for sample sizes for single sample WMW test (default options)
200 

12.4 Sample Sizes for Acceptance Sampling

1. Click Number of Samples ► Acceptance Sampling

2. The following options window is shown.

 Specify the Confidence Coefficient. Default is 0.95.

 Specify the Pre-specified proportion of non-conforming items. Default is 0.05.

 Specify the Number of allowable non-conforming items. Default is 0.

 Click on OK button to continue or on Cancel button to cancel the options.
201

Output Screen for sample sizes for acceptance sampling (default options)
202 

Chapter 13

Analysis Of Variance

Oneway ANOVA is a statistical technique that is used to compare the measures of central 

tendencies: means or medians of more than two populations/groups. Oneway ANOVA is often 

used to perform inter-well comparisons in groundwater monitoring projects. Classical Oneway 

ANOVA is a generalization of two sample t-test (Hogg and Craig, 1995); and nonparametric 

ANOVA, Kruskal-Wallis test (Hollander and Wolfe, 1999) is a generalization of two- sample 

Wilcoxon Mann Whitney test. Some details of these tests are also described in various USEPA 

guidance documents (2006, 2009).

It is a significance test which begins with the null hypothesis stating that all the groups have 

equal means and that any differences in the sample means are due to chance in, for example, the 

way the samples were taken. Then on the basis of this hypothesis, ANOVA computes the value 

of a variable F; the larger the value of F, the more unlikely it is to have occurred by chance, and 

hence the more likely that at least one of the populations has an average different from the 

others. That is, the F statistic will be large if the between-group variability is large relative to the 

within-group variability. Otherwise, the conclusion is that all the populations have equal 

averages.

13.1 Classical ANOVA

1. Click ANOVA ► Classical.

2. The “Select Variables” screen will appear. 

 Select the variables for testing. 

 Select the “Group” variable.
203

 Click “OK” to continue or “Cancel” to cancel the test.
204 

Output for a Classical ANOVA.

13.2 Nonparametric ANOVA

Nonparametric One–Way ANOVA or the Kruskal–Wallis Test is a generalization of the MannWhitney test that tests for the equality of medians of two or more groups. This test is used when 

data from various are not normally distributed.

1. Click ANOVA ► Nonparametric.

2. The “Select Variables” screen will appear. 

 Select the variables for testing. 

 Select the “Group” variable.
205

 Click “OK” to continue or “Cancel” to cancel the test.
206 

Output for a Nonparametric ANOVA.
207

Chapter 14

Trend Analyses

In groundwater (GW) monitoring applications, it is desired to identify trends in contaminant 

concentrations. Ordinary least squares (OLS) regression and two trend tests: Mann-Kendall test 

and Theil-Sen test have been incorporated in ProUCL version 4.1.00. The details of these tests 

can be found in Hollander and Wolfe (1999), USEPA (2006, 2009).Some time series plots useful 

to compare contaminant concentrations of multiple groups such as GW monitoring wells are also 

available in ProUCL 4.1.

14.1 Simple Linear Regression

1. Click Trend Tests ► Regression.

2. The “Select Variables” screen will appear. 
208 

 Select the “Dependent Variable” and the “Independent Variable” for the 

regression analysis. 

 Select the “Group” variable if any. The analysis will be done separately for each 

group.

 When the “Options” button is clicked, the following window will be shown. 

o Select “Display Intervals” for the confidence limits and the prediction limits of each 

observation to be displayed at the specified “Confidence Coefficient”. 

o Select “Display Regression Table” to display Y-hat, Residuals and the standardized 

residuals.

o Select “XY Plot” for the graph displaying the regression line.

o Select “Confidence Interval” and “Prediction Interval” to display the confidence 

and the prediction bands for the regression line.

o Click on OK button to continue or on Cancel button to cancel the option.

 Click “OK” to continue or “Cancel” to cancel the test.
209

OLS Regression Graph.

Partial output of OLS Regression Analysis
210 
211

14.2 Mann Kendall Test

1. Click Trend Tests ► Mann-Kendall.

2. The “Select Variables” screen will appear. 

 Select the “Event/Time” variable. For graphical display, this variable has to be 

selected.

 Select the “Values/Measured Data” variable for the test. 

 Select the “Group” variable if any.

 When the “Options” button is clicked, the following window will be shown. 
212 

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Select the trend lines to be displayed. “OLS Regression Line” and/or “Theil-Sen 

Trend Line.”

o Click on OK button to continue or on Cancel button to cancel the option.

 Click “OK” to continue or “Cancel” to cancel the test.

Output for a Mann Kendall Test.
213
214 

14.3 Theil – Sen Test

1. Click Trend Tests ► Theil - Sen.

2. The “Select Variables” screen will appear. 

 Select the “Event/Time” variable.

 Select the “Values/Measured Data” variable for the test. 

 Select the “Group” variable if any.

 When the “Options” button is clicked, the following window will be shown. 
215

o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The 

default choice is 0.95. 

o Select the trend lines to be displayed. “OLS Regression Line” and/or “Theil-Sen 

Trend Line.” 

o Click on OK button to continue or on Cancel button to cancel the option.

 Click “OK” to continue or “Cancel” to cancel the test.

Output for a Theil – Sen Test.
216 

14.4 Time Series Plots

1. Click Trend Tests ► Time Series Plots.

2. The “Select Variables” screen will appear. 
217

 Select the variables for testing. 

o Data only (Values/Measured Data Variable) option. This option takes in only the 

Measured Data. The event or the time variable is specified using the “Options.”

o Event/Data (Time vs. Measured Data Variable) option. This option takes in the 

Measured Data and the Event/Time variable.”

 Select the “Group” variable if any.

 When the “Options” button is clicked, the following window will be shown.

o Data only (Values/Measured Data Variable) option. 
218 

o Input a starting value for the index of the plot using the “Select Initial Start Value”.

o Input the increment steps for the index of the plot using the “Select Increment for 

Index”.

o Specify the lines (Regression or Theil-Sen) to be displayed on the time series graph.

o Select “Plot Groups Together” option for comparing the time series trends for more 

than one group on the same graph. 

Note: All the groups must be of the same size and the event values for each of the 

groups must be the same for this option to work.

o If this option is not selected but a “Group Variable” is selected, different graphs will 

be plotted for each group.

o Click on OK button to continue or on Cancel button to cancel the options.
219

o Event/Data (Time vs. Measured Data Variable) option. 

o Specify the lines (Regression or Theil-Sen) to be displayed on the time series graph.

o Select “Plot Groups Together” option for comparing the time series trends for more 

than one group on the same graph. 

o

Note: All the groups must be of the same size and the event values for each of the 

groups must be the same for this option to work.

o If this option is not selected but a “Group Variable” is selected, different graphs will 

be plotted for each group.

o Click on OK button to continue or on Cancel button to cancel the options.

 Click “OK” to continue or “Cancel” to cancel the graphs.

Output for a Time Series Plot – Event/Data option.
220 

Output for a Time Series Plot – Data Only option.
221

Chapter 15

Windows

Click on the Window menu to reveal the drop-down options shown above.

The following Window drop-down menu options are available:

 Cascade option: arranges windows in a cascade format. This is similar to a typical 

Windows program option.

 Tile option: resizes each window vertically or horizontally and then displays all open 

windows. This is similar to a typical Windows program option.

 The drop-down options list also includes 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. This is especially useful if you have more than 20 windows open, as the 

navigation panel only holds the first 20 windows. 
222 

Chapter 16

Help

When the Help menu is clicked, the following window will appear.

Three options are available under Help menu:

 About ProUCL: This option provides a brief description of ProUCL 4.0, and all 

improvements made compared to ProUCL 3.0.

 Statistical Help: This option executes an online help directory. This option provides 

information about the various algorithms and formulae (with references) used in the 

development of ProUCL 4.0. More information on the various topics covered under 

Statistical Help is provided below.

 Technical Support: This option will provide contact information for primary technical 

support via phone and e-mail.

Statistical Help provides online help notes for the methods and options available in ProUCL 4.0. A screen 

(shown below), listing the topics containing help notes, appears after clicking on the Help menu.

The online help directory is divided into the following submenus:
223

 Getting Started: This chapter provides basic information on the software, including 

software installation to various menu displays.

 Summary Statistics: This chapter provides information and examples on procedures for 

simple classical summary statistics for data sets with and without nondetects.

 ROS Estimates of NDs: This chapter briefly describes the estimation (extrapolation) of 

nondetects using regression on order statistics (ROS) for normal, lognormal, and gamma 

distribution.

 Graphs: This chapter provides information and examples on the graphical displays that 

ProUCL 4.0 can produce: box plots, histograms, and Multi Q-Q plots.

 Outlier Tests: This chapter provides information and examples for two classical outlier 

tests available in ProUCL 4.0: Dixon‟s and Rosner‟s tests for data sets with and without 

outliers. 

 Goodness-of-Fit: This chapter provides information and examples for several goodnessof-fit tests available in ProUCL 4.0 for data sets with and without NDs.

 Background Statistics: This chapter provides information and examples for the 

computation of Background Statistics needed to estimate the BTVs and not-to-exceed 

values. These statistics are sometimes used to compare point-by-point site data (not more 

than 4 to 5 site samples) with the BTVs.

 Hypotheses Testing: This chapter provides brief descriptions (with examples) of the 

various single sample and two-sample hypotheses testing approaches as incorporated in 

ProUCL 4.0. 

o Single Sample Hypotheses Testing: This chapter provides brief description and 

examples of single sample hypotheses testing approaches that are useful to compare 

site concentrations with cleanup standards, compliance limits, or not-to-exceed 

limits. The minimum sample size requirements for site data are briefly discussed in 

Chapter 1 of this User Guide.

o Two-sample hypotheses are used to compare site data with background data (or 

upgradient and downgradient wells) provided enough site and background data are 

available. The minimum sample size requirements for site and background data are 

briefly discussed in Chapter 1 of this User Guide. 

 Upper Confidence Limits (UCLs): This option provides information and examples for the 

various UCL computation methods as incorporated in ProUCL 4.0.
224 

Chapter 17

Handling the Output Screens and Graphs

Copying Graphs

1. Click the graph you want to copy or save in the Navigation Panel. The graphs can be saved using 

the copy option.

2. Click Edit ► Copy Graph.

3. Once the user has clicked “Copy Graph,” the graph is ready to be imported (pasted) into most 

Microsoft Office applications (Word, Excel, and PowerPoint have been tested) by clicking 

Edit ► Paste in these Microsoft applications. 
225

4. It is important to note that the graph cannot be saved as its own file and must be imported into an 

application to be saved. This will save the graph, but the overall file attribute and properties will 

be that of the application in which the graph was saved in. For example, if the graph was saved 

within Microsoft Word, the graph will be saved in a document with a .doc extension.

Printing Graphs

1. Click the graph you want to print in the Navigation Panel.

2. Click File ► Page Setup. 

3. Check the radio button next to Portrait or Landscape, and click OK. In some cases, with larger 

headings and captions, it may be desirable to use the Landscape printing option.

4. Click File ► Print to print the graph, and File ► Print Preview to preview the graph before 

printing.
226 

Printing Non-graphical Outputs

1. Click the output you want to copy or print in the Navigation Panel.
227

2. Click File ► Print.

Saving Output Screens as Excel Files

1. Click on the output you want to save in the Navigation Panel List.

2. Click File ► Other Files... ►Export Excel (preserve formatting).
228 

3. Enter the desired file name you want to use, and click Save, and save the file in the desired folder 

using your browser.
229
230 

Chapter 18

Recommendations to Compute a 95% UCL (Estimate of EPC 

Term) of the Population Mean, μ1, Using Symmetric and 

Positively Skewed Full Data Sets without any Nondetects

This chapter describes the recommendations for the computation of a 95% UCL of the unknown 

population arithmetic mean, μ1, of a contaminant data distribution based upon full data sets without any 

nondetect observations. These recommendations are based upon the findings of Singh, Singh, and 

Engelhardt (1997, 1999); Singh et al. (2002a); Singh, Singh, and Iaci (2002b); and Singh and Singh 

(2003). These recommendations are applicable to full data sets without censoring and nondetect (ND) 

observations. Recommendations to compute UCL95 based upon data sets with NDs are summarized in 

Chapter 16.

For skewed parametric as well as nonparametric data sets, there is no simple solution to compute a 95% 

UCL of the population mean, μ1. Contrary to the general conjecture, Singh et al. (2002a); Singh, Singh, 

and Iaci (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-CLT do not provide the specified 

coverage (e.g., 95 %) to the population mean even for mildly to moderately skewed (e.g., 

σ

ˆ

the sd of logtransformed data in interval [0.5, 1.0)) data sets for samples of size as large as 100. The coverage of the 

population mean by the skewness-adjusted UCLs becomes much smaller than the specified coverage of 

0.95 for highly skewed data sets, where skewness is defined as a function of σ or 

σ

ˆ

(sd of logtransformed data). 

It is noted that even though, the simulation results for highly skewed data sets of small sizes suggest that 

the bootstrap t and Hall‟s bootstrap methods do approximately provide the adequate coverage to the 

population mean, sometimes in practice these two bootstrap methods yield erratic inflated values (orders 

of magnitude higher than the other UCL values) when dealing with individual highly skewed data sets of 

small sizes. This is especially true when potential outliers may be present in the data set. ProUCL 4.0 

provides warning messages whenever the recommendations are made regarding the use the bootstrap t

method or Hall‟s bootstrap method.

18.1 Normally or Approximately Normally Distributed Data Sets

 For normally distributed data sets, a UCL based upon the Student‟s t-statistic 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 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. Typically, a data set is approximately normal when the 

normal Q-Q plot displays a linear pattern (without outliers and jumps of significant 

magnitude) and the resulting correlation coefficient is high (e.g., 0.95 or higher). The 

jumps and breaks in a Q-Q plot (even with a high correlation coefficient) suggest the 

presence of multiple populations in the data set under study.
231

 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 

3

ˆ

k . A value of 

3

ˆ

k

close 

to zero (e.g., if the absolute value of the 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.

18.2 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 behaves in an erratic manner. Specifically, 95% H-UCL

often results in an unjustifiably large and impractical 95% UCL value when the sample size is small (e.g., 

n <20, 50, ..) and skewness is high. Moreover, it is also observed that a 95% UCL based upon Land Hstatistic becomes even smaller than the sample arithmetic mean. This is especially true for mildly skewed 

to moderately skewed data sets of large sizes (e.g., > 50-70). In such cases, a gamma model, G(k, θ), may 

be used to compute a reliable 95% UCL of the unknown population mean, μ1. 

 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 could differ significantly. A 

lognormal model based upon a highly skewed (e.g.,

σ

ˆ

≥ 2.5) data set will have an 

unjustifiably large and impractical population mean, μ1, 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., 

σ

ˆ

≥ 2.5-3.0) lognormal distributions should be avoided. For such highly skewed 

lognormally distributed data sets that cannot be modeled by a gamma or an approximate 

gamma distribution, nonparametric 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 the 

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 population mean closer to the nominal level of 0.95). This is especially 

true when the skewness is quite high (e.g., 

k

ˆ

< 0.1) and the sample size is small (e.g., n < 

10-15). Whenever the use of Hall‟s UCL or bootstrap t UCL is recommended, an 

informative warning message about their use is also provided.

 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, μ1, 

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

ˆ

< 1) and the sample size is small (e.g., 

Singh and Singh (2003), and Singh, Singh, and Iaci (2002b)).
232 

 From the results presented in Singh, Singh, and Iaci (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: 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, μ1. 

 For highly skewed gamma distributed data sets with k < 0.1, the 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 the sample size starts 

approaching or exceeding 15. The small sample size requirement increases as the 

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 these 

methods may 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 4.0 prints out a warning message associated with the 

recommended use of the UCLs based upon the bootstrap t method or Hall‟s bootstrap 

method. The recommendations for gamma distribution are summarized in Table 18-1.

Table 18-1. Computation of a UCL95 of the Unknown Mean, μ1, of a Gamma Distribution

k

ˆ

Sample Size, n Recommendation

k

ˆ

≥ 0.5 For all n Approximate gamma 95% UCL

0.1 ≤ 

k

ˆ

< 0.5 For all n Adjusted gamma 95% UCL

k

ˆ

< 0.1 n < 15 95% UCL based upon bootstrap t

or Hall’s bootstrap method*

k

ˆ

< 0.1 n ≥ 15 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 the adjusted gamma 

UCL.

Note: Suggestions regarding the selection of a 95% UCL are provided to help the user to select 

the most appropriate 95% UCL. These recommendations are based upon the results of the 

simulation studies summarized in Singh, Singh, and Iaci (2002) and Singh and Singh (2003). 

For additional insight, the user may want to consult a statistician.

18.3 Lognormally Distributed Skewed Data Sets

For lognormally, LN (μ, σ

2

), distributed data sets, the H-statistic-based H-UCL provides specified 0.95, 

coverage for the population mean for all values of σ. However, the H-statistic often results in unjustifiably 

large UCL values that do not occur in practice. This is especially true when the skewness is high (e.g., σ > 
233

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, μ1, of a lognormal model becomes impractically large for larger values of σ, which in 

turn results in an inflated H-UCL of the population mean, μ1. Since the population mean of a lognormal 

model becomes too large, none of the other methods, except for the H-UCL, provides the specified 95% 

coverage for that inflated population mean, μ1. This is especially true when the sample size is small and 

the skewness is high. For extremely highly skewed data sets (with σ > 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)

and 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 nonparametric bootstrap methods or the Chebyshev 

inequality-based methods, are desirable. All skewed data sets should first be tested for a gamma 

distribution. For lognormally distributed data (that cannot be modeled by gamma distribution), methods 

summarized in Table 15-2 may be used to compute a 95% UCL of mean. 

ProUCL can compute an H-UCL for samples of sizes up to 1000. For highly skewed lognormally 

distributed data sets of smaller sizes, alternative methods to compute a 95% UCL of the population mean, 

μ1, are summarized in Table 15-2. Since skewness is a function of σ (or 

σ

ˆ

), the recommendations for the 

computation of the UCL of the population mean are also summarized in terms of 

σ

ˆ

and the sample size, 

n. Here, 

σ

ˆ

is an MLE of σ, and is given by the Sd of log-transformed data. Note that Table 18-2 is 

applicable only to the computation of a 95% UCL of the population mean based upon lognormally 

distributed data sets without nondetect observations. 

Due to unstable and erratic behavior of H-UCL, the users are discouraged to use lognormal 

distribution to estimate the EPC terms. ProUCL computes and outputs H-statistic based UCLs 

for historical reasons only. H-statistic often results in unstable (both high and low) values of 

UCL95 as shown in examples 1 and 2 below. It is therefore recommended to avoid the use of Hstatistic based 95% UCLs. Use of nonparametric methods is preferred to compute UCL95 for 

skewed data sets which do not follow a gamma distribution.

Table 18-2. Computation of a 95% UCL of Mean, μ1 of a Lognormal Population

σ

ˆ

Sample Size, n Recommendation

σ

ˆ

< 0.5 For all n Student’s t, modified t, or H-UCL

0.5 ≤ 

σ

ˆ

< 1.0 For all n H-UCL

1.0 ≤ 

σ

ˆ

< 1.5

n < 25 95% Chebyshev (Mean, Sd) UCL

n ≥ 25 H-UCL

1.5 ≤ 

σ

ˆ

< 2.0

n < 20 99% Chebyshev (Mean, Sd) UCL

20 ≤ n < 50 95% Chebyshev (Mean, Sd) UCL

n ≥ 50 H-UCL

1.5 ≤ 

σ

ˆ

< 2.0

n < 20 99% Chebyshev (Mean, Sd) UCL

20 ≤ n < 50 97.5% Chebyshev (Mean, Sd) UCL

50 ≤ n < 70 95% Chebyshev (Mean, Sd) UCL

n ≥ 70 H-UCL

2.5 ≤ 

σ

ˆ

< 3.0**

n < 30 99% Chebyshev (Mean, Sd)

30 ≤ n < 70 97.5% Chebyshev (Mean, Sd) UCL
234 

70 ≤ n < 100 95% Chebyshev (Mean, Sd) UCL

n ≥ 100 H-UCL

3.0 ≤ 

σ

ˆ

≤ 3.5**

n < 15 Hall’s bootstrap method*

15 ≤ n < 50 99% Chebyshev (Mean, Sd)

50 ≤ n < 100 97.5% Chebyshev (Mean, Sd) UCL

100 ≤ n < 150 95% Chebyshev (Mean, Sd) UCL

n ≥ 150 H-UCL

σ

ˆ

> 3.5** For all n Use nonparametric methods*

*If Hall‟s bootstrap method yields an erratic or unrealistically large UCL value, then the UCL of the mean 

may be computed based upon the Chebyshev inequality.

** For highly skewed data sets with 

σ

ˆ

exceeding 3.0, 3.5, it is suggested to pre-process the data. 

It is very likely that the data consist of outliers and/or come from multiple populations. The 

population partitioning methods may be used to identify mixture populations present in the data 

set. For defensible conclusions, the decision statistics such as UCL95 should be computed 

separately for each of the identified sub-population. 

Note: Suggestions regarding the selection of a 95% UCL are provided to help the user to select 

the most appropriate 95% UCL. These recommendations are based upon the results of the 

simulation studies summarized in Singh, Singh, and Iaci (2002) and Singh and Singh (2003). 

For additional insight, the user may want to consult a statistician.

Based upon the results of the research conducted to evaluate the appropriateness of applicability of 

lognormal distribution based estimates of EPC term (Singh, et. al., 1997, 1999, 2002, 2003), the 

developers of ProUCL 4.0 had suggested to avoid the use of lognormal distribution to estimate the EPC 

term. However, computations of various lognormal distribution based statistics were included in ProUCL 

4.00.05 for historical reasons and for the sake of comparison for interested practitioners. For Lognormally 

distributed data set, the developers again suggest and recommend the use of nonparametric distribution 

free methods (Efron, 1981, 1982 and Efron and Tibshirani, 1993) to estimate environmental parameters of 

interest (e.g., EPC, BTV), especially when skewness is high, that is when Sd of log-transformed data 

starts exceeding 1.5. As a result, the developers have revised recommendations and decision tables given 

in ProUCL Technical Guide and User Guide. Those revisions have been incorporated in this updated 

ProUCL 4.00.05 version.

18.4 Data Sets without a Discernable Skewed Distribution – Nonparametric 

Methods for Skewed Data Sets

The use of gamma and lognormal distributions as discussed here cover a wide range of skewed data 

distributions. For skewed data sets which are neither gamma nor lognormal, one can use a nonparametric 

Chebyshev UCL or Hall‟s bootstrap UCL (for small samples) of the mean to estimate the EPC term. 

 For skewed nonparametric data sets with negative and zero values, use a 95% Chebyshev 

(Mean, Sd) UCL for the population mean, μ1.
235

For all other nonparametric data sets with only positive values, the following method may be used to 

estimate the EPC term.

 For mildly skewed data sets with 

σ

ˆ

≤ 0.5, one can use the Student‟s t-statistic or 

modified t-statistic to compute a 95% UCL of mean, μ1.

 For nonparametric moderately skewed data sets (e.g., σ or its estimate, 

σ

ˆ , in the interval 

(0.5, 1]), one may use a 95% Chebyshev (Mean, Sd) UCL of the population mean, μ1.

 For nonparametric moderately to highly skewed data sets (e.g., 

σ

ˆ

in the interval (1.0, 

2.0]), one may use a 99% Chebyshev (Mean, Sd) UCL or a 97.5% Chebyshev (Mean, Sd)

UCL of the population mean, μ1, to obtain an estimate of the EPC term.

 For highly skewed to extremely highly skewed data sets with 

σ

ˆ

in the interval (2.0, 3.0], 

one may use Hall‟s UCL or a 99% Chebyshev (Mean, Sd) UCL to compute the EPC term. 

 Extremely skewed nonparametric data sets with σ exceeding 3 provide poor coverage. 

For such highly skewed data distributions, none of the methods considered provide the 

specified 95% coverage for the population mean, μ1. The coverage provided by the 

methods decrease as σ 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 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 the 99% Chebyshev (Mean, Sd) UCL

method also increases. Therefore, for larger samples, a UCL should be computed based 

upon the 99% Chebyshev (Mean, Sd) method. This large sample size requirement 

increases as

σ

ˆ

increases. These recommendations are summarized in Table 18-3.

Table 18-3. Computation of a 95% UCL of the Unknown Mean, μ1, Based Upon a Skewed Data Set (with all Positive 

Values) without a Discernable Distribution, Where 

σ

ˆ

is the sd of Log-transformed Data

σ

ˆ

Sample Size, n Recommendation

σ

ˆ

< 0.5 For all n Student’s t, modified t, or H-UCL

0.5 ≤ 

σ

ˆ

< 1.0 For all n 95% Chebyshev (Mean, Sd) UCL

1.0 ≤ 

σ

ˆ

< 1.5 For all n 95% Chebyshev (Mean, Sd) UCL

1.5 ≤ 

σ

ˆ

< 2.0

n < 20 99% Chebyshev (Mean, Sd) UCL

20 ≤ n 95% Chebyshev (Mean, Sd) UCL

2.0 ≤ 

σ

ˆ

< 2.5

n < 10 Hall’s bootstrap method

10 ≤ n < 20 99% Chebyshev (Mean, Sd) UCL

20 ≤ n < 50 97.5% Chebyshev (Mean, Sd) UCL

50 ≤ n 95% Chebyshev (Mean, Sd) UCL

2.5 ≤ 

σ

ˆ

< 3.0

n < 10 Hall’s bootstrap method

10 ≤ n < 30 99% Chebyshev (Mean, Sd)

30 ≤ n < 70 97.5% Chebyshev (Mean, Sd) UCL
236 

70 ≤ n 95% Chebyshev (Mean, Sd) UCL

3.0 ≤ 

σ

ˆ

≤ 3.5**

n < 15 Hall’s bootstrap method*

15 ≤ n < 50 99% Chebyshev(Mean, Sd)

50 ≤ n < 100 97.5% Chebyshev (Mean, Sd) UCL

100 ≤ n 95% Chebyshev (Mean, Sd) UCL

σ

ˆ

> 3.5** For all n Use nonparametric methods*

*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. 

** For highly skewed data sets with 

σ

ˆ

exceeding 3.0, 3.5, it is suggested to pre-process the data. 

It is very likely that the data consist of outliers and/or come from multiple populations. The 

population partitioning methods may be used to identify mixture populations present in the data

set. For defensible conclusions, the decision statistics such as UCL95 should be computed 

separately for each of the identified sub-population. 

Note: Suggestions regarding the selection of a 95% UCL are provided to help the user to select 

the most appropriate 95% UCL. These recommendations are based upon the results of the 

simulation studies summarized in Singh, Singh, and Iaci (2002) and Singh and Singh (2003). 

For additional insight, the user may want to consult a statistician.
237

Table 18-4. Recommended UCL95 Computation Methods for Full-Uncensored Data Sets without Nondetect Observations

Skewness Sample Size

95% Student t95% modified t95% H-UCL95% Hall’s Bootstrap95% Chebyshev (MVUE)97.5% Chebyshev (MVUE)99% Chebyshev (MVUE)95% Chebyshev (Mean, Sd)97.5% Chebyshev (Mean, Sd)99% Chebyshev (Mean, Sd)95% Approx. Gamma 95% Bootstrap t95% Adjusted Gamma

Normal or Approximate Normal (with 

σ

ˆ

< 0.5) Distribution

All n  

Gamma Distribution

k

ˆ

< 0.1

n < 15  

n > 15  

0.1 ≤ 

k

ˆ

< 0.5 All n 

k

ˆ

≥ 0.5 All n 

Lognormal Distribution

σ

ˆ

< 0.5 All n   

0.5 ≤ 

σ

ˆ

< 1.0 All n 

1.0 ≤ 

σ

ˆ

< 1.5

n < 25 

n ≥ 25 

1.5 ≤ 

σ

ˆ

< 2.0

n < 20 

20 ≤ n < 50 

n ≥ 50 

2.0 ≤ 

σ

ˆ

< 2.5

n < 20 

20 ≤ n < 50 

50 ≤ n < 70 

n ≥ 70 

2.5 ≤ 

σ

ˆ

< 3.0

n < 30 

30 ≤ n < 70 

70 ≤ n < 100 

n ≥ 100 

3.0 ≤ 

σ

ˆ

≤ 3.5

n < 15 

15 ≤ n < 50 

50 ≤ n < 100 

100 ≤ n < 150 

n ≥ 150 
238 

Table 18-4. Recommended UCL95 Computation Methods for Full-Uncensored Data Sets without Nondetect 

Observations-Continued

Sample Size

95% Student t95% modified t95% H-UCL95% Hall's Bootstrap95% Chebyshev (MVUE)97.5% Chebyshev (MVUE)99% Chebyshev (MVUE)95% Chebyshev (Mean, Sd)97.5% Chebyshev (Mean, Sd)99% Chebyshev (Mean, Sd)95% Approx. Gamma 95% Bootstrap t95% Adjusted Gamma

Nonparametric Distribution Free Methods

σ

ˆ

≤ 0.5 All n  

0.5 < 

σ

ˆ

≤ 1.0 All n 

1.0 < 

σ

ˆ

≤ 1.5 All n 

1.0 < 

σ

ˆ

≤ 2.0

n < 20 

n ≥ 20 

2.0 < 

σ

ˆ

≤ 2.5

n < 10 

10 ≤ n < 20 

20 ≤ n < 50 

50 ≤ n 

2.5 < 

σ

ˆ

≤ 3.0

n < 10 

10 ≤ n < 30 

30 ≤ n < 70 

70 ≤ n 

3.0 < 

σ

ˆ

≤ 3.5

n < 15 

15 ≤ n < 50 

50 ≤ n < 100 

100 ≤ n 

σ

ˆ

> 3.5

n < 100 

n ≥ 100 

Note: Suggestions regarding the selection of a 95% UCL are provided to help the user to select the most 

appropriate 95% UCL. These recommendations are based upon the results of the simulation studies 

summarized in Singh, Singh, and Iaci (2002) and Singh and Singh (2003). For additional insight, the 

user may want to consult a statistician.
239

18.5 Should the Maximum Observed Concentration be Used as an 

Estimate of the EPC Term?

This topic has been discussed earlier in Chapter 1. It is included here only for the convenience of the user. 

In practice, a typical user tends to use the maximum sample value as an estimate of the EPC term. This is 

especially true when the sample size is small or the data are highly skewed. The discussion and 

suggestions as described in Chapter 1 apply to both Chapters 15 and 16. Singh and Singh (2003) studied 

the max test (using the maximum observed value as an estimate of the EPC term) in their simulation 

study. Previous (e.g., RAGS document (EPA, 1992)) use of the maximum observed value has been 

recommended as a default option 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 t-UCL and Land‟s 

H-UCL, were used previously to estimate the EPC term (e.g., EPA, 1992). ProUCL 4.0 can compute a 

95% UCL of the mean using several methods based upon the normal, gamma, lognormal, and 

“nonparametric” distributions. Furthermore, since the EPC term represents the average exposure 

contracted by an individual over an exposure area (EA) during a long period of time, 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. 

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. 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. 

For the sake of interested users, ProUCL displays a warning message when the recommended 95% UCL

(e.g., Hall‟s bootstrap UCL) of the mean exceeds the observed maximum concentration. When a 95% 

UCL exceeds the maximum observed value, ProUCL recommends the use of an alternative UCL method 

based upon a 97.5% or 99% Chebyshev UCL.

It should also be noted that for highly skewed data sets, the sample mean indeed can even exceed the 

upper 90%, or higher, 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 lognormally 

distributed data sets of small sizes. For such highly skewed data sets which cannot be modeled by a 

gamma distribution, a 95% UCL of the mean should be computed using an appropriate nonparametric 

method as summarized in Table 18-4. 
240 

Chapter 19

Recommendations to Compute a 95% UCL of the Population 

Mean, μ1, Using Data Sets with Nondetects with Multiple 

Detection Limits

This chapter summarizes the recommendations based on the simulation experiments conducted by Singh, 

Maichle, and Lee (USEPA, 2006) to compare the performances of the UCL computation methods based 

upon data sets with BDLs and multiple detection limits (DLs). ProUCL 4.0 suggests the use of 

appropriate UCLs based upon the findings of Singh, Maichle, and Lee (USEPA, 2006). For convenience, 

the recommended UCL95 computation methods have been tabulated in Table 16 as functions of the 

sample size, skewness, and censoring intensity. General observations and recommendations regarding the 

difficulties associated with data sets with NDs are described first.

19.1 General Recommendations and Suggestions

 In practice, it is not easy to verify (perform goodness-of-fit) the distribution of a leftcensored data set with NDs. Therefore, emphasis is given on the use of nonparametric 

UCL95 computation methods, which can also be used to handle multiple detection limits. 

 This is specifically true when the percentage (%) of nondetects exceeds 40%-50%.

 Most of the parametric MLE methods assume that there is only one detection limit. It 

should also be noted that the MLEs behave in an unstable manner when the % of NDs 

exceeds 40%-50%. Moreover, as mentioned before, it is hard to verify and justify the 

conclusion of a GOF test for data sets with nondetects in excess of 40%-50%. 

 Therefore, for data sets with many nondetects (> 40%-50%), it is suggested to use 

nonparametric methods to estimate the various environmental parameters (BTVs, EPC 

terms) of interest and to perform site versus background comparisons.

 In practice, a left-censored data set often has multiple detection limits. For such methods, 

the KM method can be used. ProUCL 4.0 provides UCL computation methods that can 

be used on data sets with multiple detection limits including the DL/2 method, KM 

method, and robust ROS method. 

 As mentioned earlier, for reliable and accurate results, it is suggested that the user make 

sure that the data set under study represents a single statistical population (e.g., 

background reference area, or an AOC) and not a mixture population (e.g., clean and 

contaminated site areas).

 It is recommended to identify all of the potential outliers and study them separately. The 

computation of the statistics such as UCL95 and background statistics should be based 

upon the majority of the data set representing a single dominant population. Decisions 

about the appropriate disposition (include or not include) of outliers should be made by 

all interested members of the project team. When in doubt, it is suggested to compute and 
241

compare all relevant statistics and estimates based upon data sets with and without the 

outliers. This step will exhibit the undue influence of outliers on the statistics of interest 

such as UCL95 and UPL95.

 Simple classical outlier identification methods (Dixon test and Rosner test) are also 

available in ProUCL 4.0. More effective robust outlier procedures (e.g., Rousseeuw and 

Leroy (1987) and Singh and Nocerino (1995)) are available in Scout (1999). 

 In case a data set represents a mixture sample (from two or more populations), one should 

partition the mixture sample into component sub-samples (e.g., Singh, Singh, and 

Flatman (1994)). 

 Avoid the use of transformations (to achieve symmetry) while computing the upper limits 

for various environmental applications, as all remediation, cleanup, background 

evaluation decisions, and risk assessment decisions have to be made using statistics in the 

original scale. Also, it is more accurate and easier to interpret the results computed in the 

original scale. The results and statistics computed in the original scale do not suffer from 

an unknown amount of transformations bias.

 Specifically, avoid the use of a lognormal model even when the data appear to be 

lognormally distributed. Its use often results in incorrect and unrealistic statistics of no 

practical purpose or importance or significance. Several variations of estimation methods 

(e.g., robust ROS and FP-ROS on log-transformed data, delta lognormal method) on logtransformed data have been developed and used by the practitioners. This has caused 

some confusion among the users of the statistical methods dealing with environmental 

data sets. The proper use of a lognormal distribution (e.g., how to properly backtransform UCL of mean in the log-scale to obtain a UCL of mean in original scale) is not 

clear to many users, which in turn may result in the incorrect use and computation of an 

estimate (= UCL95) of the population mean.

 The parameter in the transformed space may not be of interest to make cleanup decisions. 

The cleanup and remediation decisions are often made in the original raw scale; 

therefore, the statistics (e.g., UCL95) computed in transformed space need to be backtransformed in the original scale. It is not clear to a typical user how to back-transform 

results in the log-scale or any other scale obtained using a Box-Cox (BC)-type 

transformation to original raw scale. The transformed results often suffer from significant 

amount of transformation bias. 

 The question now arises-how one should back-transform results from a log-space (or any 

other transformed space) to the original space? Unfortunately, no defensible guidance is 

available in the environmental literature to address this question. Moreover, the backtransformation formula will change from transformation to transformation (BC-type 

transformations), and the bias introduced by such transformations will remain unknown.

Therefore, in cases when a data set in the “raw” scale cannot be modeled by a parametric 

distribution, it is desirable to use nonparametric methods (many available in ProUCL 4.0) 

rather than testing or estimating a parameter in the transformed space. 

 On page (78) of Helsel (2005), the use of the robust ROS MLE method (Kroll, C.N. and 

J.R. Stedinger (1996)) has been suggested to compute summary statistics. In this hybrid 
242 

method, MLEs are computed using log-transformed data. Using the regression model as 

given by equation (3-21) of Section 3, the MLEs of the mean (used as intercept) and sd

(used as slope) in the log-scale are used to extrapolate the NDs in the log-scale. Just like 

in the robust ROS method, all of the NDs are transformed back in the original scale by 

exponentiation. This results in a full data set in the original scale. One may then compute 

the mean and sd using the full data set. The estimates thus obtained are called robust ROS 

ML estimates (Helsel (2005), and Kroll and Stedinger (1996)). However, the 

performance of such a hybrid estimation method is not well known. Moreover, for higher 

censoring levels, the MLE methods sometimes behave in an unstable manner, especially 

when dealing with moderately skewed to highly skewed data sets (e.g., with σ >1.0).

o It should be noted that the performance of this hybrid method is unknown. 

o It is not clear why this method is called a robust method. 

o The stability of the MLEs obtained using the log-transformed data is doubtful, 

especially for higher censoring levels. 

o The BCA and (% bootstrap) UCLs based upon this method will fail to provide the 

adequate coverage for the population mean for moderately skewed to highly skewed 

data sets.

 The DL/2 (t) UCL method does not provide adequate coverage (for any distribution and 

sample size) for the population mean, even for censoring levels as low as 5%, 10%, 15%. 

This is contrary to the conjecture and assertion (e.g., EPA (2000)) often made that the 

DL/2 method can be used for lower (≤ 20%) censoring levels. The coverage provided by 

the DL/2 (t) method deteriorates fast as the censoring intensity increases.

This DL/2 (t) UCL method is not recommended by the authors and developers of ProUCL 4.0; it is 

included only for comparison or research purposes.

 The KM method is a preferred method as it can handle multiple detection limits. 

Moreover, the nonparametric UCL95 methods (KM (BCA), KM (z), KM (%), KM (t)) 

based upon the KM estimates provide good coverages for the population mean (e.g., 

Helsel (2005) and Singh et al. (2006)).

 For a symmetric distribution (approximate normality), several UCL95 methods provide 

good coverage (~95%) for the population mean, including the Winsorization mean, 

Cohen‟s MLE (t), Cohen‟s MLE (Tiku), KM (z), KM (t), KM (%) and KM (BCA) (e.g., 

Helsel (2005) and Singh et al. (2006)).

19.2 Recommended UCL95 Methods for Normal (Approximate Normal) 

Distribution

 For normal and approximately normal (e.g., symmetric or with sd, 

σ

ˆ

< 0.5) distribution: 

The most appropriate UCL95 computation methods for normal or approximately normal 

distributions are the KM (t) or KM (%) methods. For symmetric distributions, both of 

these methods perform equally well on left-censored data sets for all censoring levels (% 

nondetects) and sample sizes.
243

19.3 Recommended UCL95 Methods for Gamma Distribution

 Highly skewed gamma distributions, G(k, ), with shape parameter, k  1:

o Use the nonparametric KM (Chebyshev) UCL95 method for censoring levels < 30%.

o Use the nonparametric KM (BCA) UCL95 method for censoring levels in the interval 

[30%, 50%).

o Use the nonparametric KM (t) UCL95 method for censoring levels  50%. 

 Moderately skewed gamma distributions, G(k, ), with shape parameter, 1< k  2:

o For censoring level < 10%, use the KM (Chebyshev) UCL95 method.

o For higher censoring levels [10%, 25%), use the KM (BCA) UCL95 method.

o For censoring levels in [25%, 40%), use the KM (%) UCL95 method.

o For censoring levels  40%, use the KM (t) UCL95 method.

 Mildly skewed gamma distributions, G(k, ), with k > 2:

o Use the KM (BCA) UCL95 method for lower censoring levels (< 20%).

o For censoring levels in the interval [20%, 40%), use the KM (%) UCL95.

o For censoring  40%, use the KM (t) UCL95 computation method.

19.4 Recommended UCL95 Methods for Lognormal Distribution

 Mildly skewed data sets with 

σ

ˆ

 1:

o For censoring levels (< 20%) and sample of sizes less than 50-70, use the KM 

(Chebyshev) UCL95.

o For censoring levels (< 20%) and samples of sizes greater than 50-70, use the KM 

(BCA) UCL95.

o For censoring levels in the interval [20%, 40%) and all sample sizes, use the KM 

(BCA) UCL95.

o For censoring level  40%, use the KM (%) or KM (t) UCL95 method.

 Data sets with

σ

ˆ

in the interval (1, 1.5]:

o For censoring levels < 50% and samples of sizes < 40, use the 97.5% KM 

(Chebyshev) UCL.

o For censoring levels < 50% , samples of sizes  40, use 95% KM (Chebyshev) UCL.

o For censoring levels  50%, use the KM (BCA) UCL95 for samples of all sizes.

 Highly skewed data sets with 

σ

ˆ

in the interval (1.5, 2]:

o For sample sizes < 40, censoring levels <50%, use 99% KM (Chebyshev) UCL.

o For sample sizes  40, censoring levels < 50%, use 97.5% KM (Chebyshev) UCL.

o For samples of sizes < 40-50 and censoring levels  50%, use the 97.5% KM 

(Chebyshev) UCL.

o For samples of sizes  40-50, and censoring levels  50%, use the 95% KM 

(Chebyshev) UCL. 

 Use a similar pattern for more highly skewed data sets with 

σ

ˆ

> 2.0, 3.0:

o For extremely highly skewed data sets, an appropriate estimate of the EPC term (in 

terms of adequate coverage) is given by a UCL based upon the Chebyshev inequality 
244 

and KM estimates. The confidence coefficient to be used will depend upon the 

skewness. For highly skewed data sets, a higher (e.g., > 95%) confidence coefficient 

may have to be used to estimate the EPC. 

o As the skewness increases, the confidence coefficient also increases. 

o For such highly skewed distributions (with 

σ

ˆ

> 2.0, 3.0), for lower sample sizes (e.g., 

< 50-60), one may simply use 99% KM (Chebyshev) UCL to estimate the population 

mean, EPC term, and other relevant threshold (e.g., UPL, percentiles) values. 

o For sample sizes greater than 60, one may use a 97.5% KM (Chebyshev) UCL as an 

estimate of the population mean or mass. 

19.5 Recommended Nonparametric UCL Methods 

 For symmetric or approximately symmetric distribution-free, nonparametric data sets 

with 

σ

ˆ

< 0.5: Use the same UCL computation methods as for the data sets coming from 

a normal or an approximate normal (symmetric) population. These methods are 

summarized above in the normal UCL computation section.

 For skewed distribution-free, nonparametric data sets with 

σ

ˆ

 0.5: Most of the 

recommended UCL computation methods for a lognormal distribution, as described 

above in the lognormal UCL section, do not assume the lognormality of the data set. 

Therefore, the UCL computation methods, as described in the lognormal UCL 

computation section, can be used on skewed nonparametric data sets that do not follow 

any of the well-known parametric distributions. 

The suggested parametric and nonparametric UCL95 computation methods for data sets with nondetect 

observations are summarized in Table 16, shown on the next page.
245

Table 19. Recommended UCL95 Computation Methods for Left-Censored Data Sets with Nondetect Observations

Skewness Sample Size % ND

95% KM (t)95% KM (%)95% KM (Chebyshev)97.5% KM (Chebyshev)99% KM (Chebyshev)95% KM (BCA)

Normal or Approximate Normal (with 

σ

ˆ

< 0.5) Distribution 

σ

ˆ

< 0.5 All n > 0%  

Gamma Distribution

k

ˆ

≤ 1

All n < 30% 

All n [30%, 50%) 

All n ≥ 50% 

1 < 

k

ˆ

≤ 2

All n < 10% 

All n [10%, 25%) 

All n [25%, 40%) 

All n ≥ 40% 

k

ˆ

> 2

All n < 20% 

All n [20%, 40%) 

All n ≥ 40% 

Lognormal Distribution

σ

ˆ

≤ 1.0

n ≤ 50-70 < 20% 

n > 50-70 

All n [20%, 40%) 

All n ≥ 40%  

1 < 

σ

ˆ

≤ 1.5

n < 40 < 50% 

n ≥ 40 

All n  50% 

1.5 < 

σ

ˆ

≤ 2.0

n < 40 < 50% 

n ≥ 40 

n < 40-50 ≥ 50% 

n ≥ 40-50 

σ

ˆ

> 2.0,3.0

n < 50-60 > 0% 

n  60 
246 

Table 19. Recommended UCL95 Computation Methods for Left-Censored Data Sets with Nondetect Observations –

Continued

Skewness Sample Size % ND

95% KM (t)95% KM (%)95% KM (Chebyshev)97.5% KM (Chebyshev)99% KM (Chebyshev)95% KM (BCA)

Symmetric or Approximate Symmetric Non-Discernable Distribution

σ

ˆ

< 0.5 All n > 0%  

Moderately Skewed to Highly Skewed Non-Discernable Distribution

0.5 ≤ 

σ

ˆ

≤ 1.0

n ≤ 50-70 < 20% 

n > 50-70 

All n [20%, 40%) 

All n ≥ 40%  

1 < 

σ

ˆ

≤ 1.5

n < 40 < 50% 

n ≥ 40 

All n  50% 

1.5 < 

σ

ˆ

≤ 2.0

n < 40 < 50% 

n ≥ 40 

n < 40-50 ≥ 50% 

n ≥ 40-50 

σ

ˆ

> 2.0,3.0 n < 50-60 > 0% 

n  60 

Note: In Table 16, phrase “All n” represents only valid (e.g., n > 3) and recommended (n > 8 to 10) 

values of the sample size, n. As mentioned throughout the report, it is not desirable to use statistical 

methods on data sets of small sizes (e.g., with n < 8 to 10). However, it should be noted that the sample 

size requirements and recommendations (n > 8 to 10) as described in this report are not the limitations of 

the methods considered in this report. One of the main reasons for the recommendation that the sample 

size should be at least 8 to 10 is that the estimates and UCLs based upon small data sets, especially with 

many below detection limit observations (e.g., 30%, 40%, 50%, and more), may not be reliable and 

accurate enough to draw conclusions for environmental applications. It should be noted that in order to 

be able to use bootstrap re-sampling methods, it is desirable to have a minimum of 10-15 observations 

(e.g., n > 10-15). Therefore, the phrase “All n” in Table 16, should be interpreted as that the sample size, 

n, is least 8 to 10. The software, ProUCL 4.0, will provide appropriate warning messages when a user 

tries to use a method on data sets of small sizes. 

Also, Hall‟s bootstrap and bootstrap t methods to compute a UCL based upon a full data set (without 

nondetects) should be used with caution. These two bootstrap methods may yield erratic and unstable 

UCL results, especially, when outliers are present. In such cases, it is desirable to use alternative UCL 
247

methods based upon Chebyshev inequality. ProUCL software provides a warning message for erratic 

UCL results based upon Hall‟s bootstrap t methods.

Note: Suggestions regarding the selection of a 95% UCL of mean are provided to help the user to select 

the most appropriate 95% UCL. These recommendations are based upon the results of the simulation 

studies summarized in Singh, Maichle, and Lee (2006). For additional insight, the user may want to 

consult a statistician.
248 

Glossary

This glossary defines selected words in this User Guide to describe impractically large UCL values of the 

unknown population mean, m1. In practice, the UCLs based upon Land‟s H-statistic (H-UCL), and 

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” are 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 a 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 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. 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. 
249
250 

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