[David j. sheskin]_handbook_of_parametric_and_nonp

Handbook of
PARAMETRICand
NONPARAMETRIC
STATISTICAL PROCEDURES
THIRD EDITION
DavidJ.Sheskin
Western Connecticut State University
CHAPMAN & HALUCRC- -
A CRC Press Company
Boca Raton London NewYork Washington, D.C.

Library of Congress Cataloging-in-PublicationData
Sheskin, David.
Handbook of parametric and nonparametric statistical procedures I by David J.
Sheskin.-3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN 1-58488-440-1 (alk. paper)
1. Mathematical statistics-Handbook, manuals, etc. I. Title: Parametric and
nonparametric statistical procedures. 11. Title.
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Copyright 2004 by Chapman & Hal/CRC

Preface
Like the first two editions, the third edition of the Handbook of Parametric and
Nonparametric Statistical Procedures is designed to provide researchers, teachers, and
students with a comprehensivereference book in the areas of parametric and nonparametric
statistics. The addition of material not included in the first two editionsmakes the Handbook
unparalleled in terms of its coverage of the field of statistics. Rather than being directed at a
limited audience, the Handbook is intended for individuals who are involved in a broad
spectrum of academicdisciplinesencompassingthe fieldsofmathematicdstatistics,the social,
biological,andenvironmentalsciences,business, and education. Myphilosophyin writingboth
this and the previous editions was to create a reference book on parametric and nonparametric
statistical procedures that I (as well as colleagues and students I have spoken with over the
years)have alwayswanted,yet could never find. To be more specific, my primary goal was to
produceacomprehensivereferencebook onunivariateandbivariatestatisticalprocedureswhich
covers a scope of material that extends far beyond that which is covered in any singlesource.
It was essential that the book be applications oriented, yet at the same time that it address
relevant theoretical and practical issues which are of concern to the sophisticatedresearcher.
In addition, I wanted to write a book which is accessible to people who have a little or no
knowledge of statistics, as well as those who are well versed in the subject. 1 believe I have
achieved these goals, and on the basis of this I believe that the Handbook of Parametric and
Nonparametric Statistical Procedures will continue to serve as an invaluable resource for
people in multiple academicdisciplineswho conduct research,are involvedin teaching, or are
presently in the process of learning statistics.
I am not awareof any applications-orientedbook which provides in-depth coverage of as
manystatisticalproceduresasthenumberthat arecoveredintheHandbookofParametricand
NonparametricStatisticalProcedures. InspectionoftheTableofContentsandIndexshould
confirm the scope of material covered in the book. A unique feature of the Handbook, which
distinguishes it £komother reference books on statistics, is that it provides the reader with a
practical guide which emphasizes application over theory. Although the book will be of
practical valueto statisticallysophisticatedindividualswho are involved in research, it is also
accessible to those who lack the theoretical and/or mathematical background required for
understandingthe material documentedin more conventionalstatisticsreferencebooks. Since
a major goal ofthe book is to serveas a practical guide, emphasisisplaced on decision making
with respect to which test ismost appropriateto employin evaluatinga specificdesign. Within
the fiameworkofbeing user-friendly, clearcomputationalguidelines,accompaniedby easy-to-
understand examples, are provided for all procedures.
Oneshouldnot, however, getthe impression that theHandbookofParametricandNon-
parametric StatisticalProceduresis littlemore than a cookbook. In point of Edct, the design
of the Handbook is such that within the fiamework of each of the statistical procedures which
arecovered,in additiontothebasicguidelinesfordecisionmakingandcomputation, substantial
in-depth discussion is devoted to a broad spectrum of practical and theoretical issues, many of
which arenot discussed in conventional statisticsbooks. Inclusion ofthe lattermaterial ensures
that the Handbook will serve as an invaluableresource for those who are sophisticatedas well
as unsophisticatedin statistics.
It should be noted that although a major goal of this book is to provide the reader with
clear,easyto followguidelinesfor conductingstatisticalanalyses,it is essentialtokeep in mind

vi Handbook of Parametric and Nonparametric Statistical Procedures
that the statistical procedures contained within it are essentially little more than algorithms
which have been derived for evaluating a set of data under certain conditions. A statistical
procedure, in and of itself, is incapableof makingjudgements with respect to the reliability of
data. If the latter is compromised (as a result of a faulty experimental design, sloppy
methodology, or inattention with respect to whether or not certain assumptions underlying a
statistical analysismay have been violated), for all practical purposes, the result of an analysis
will be worthless. Consequently,it cannot be emphasizedtoo strongly, that a prerequisite for
the intelligent and responsibleuse ofstatisticsisthat onehas a reasonableunderstanding ofthe
conceptualbasisbehind an analysis,aswellasarealizationthattheuseofa statisticalprocedure
merelyrepresentsthefinalstagein a sequentialprocess involved in conductingresearch. Those
stages which precede the statistical analysis are comprised of whatever it is that a researcher
does within the framework of designing and executinga study. Ignorance or sloppinesson the
part of a researcher with respect to the latter can render any subsequent statistical analysis
meaningless.
Althougha number ofdifferenthypothesistestingmodelsarediscussed in this Handbook,
the primary model that is emphasized is the classical hypothesistesting model (i.e., the null
hypothesissignificancetesting model). Although the latter model (which has been subjected
to criticism by proponents of alternativeapproachesto hypothesistesting) has its limitations,
the author believes that the present scope of scientific knowledge would not be greater than it
is now ifup tothis point in timeanyof the alternative hypothesistestingmodels had been used
in its place. Throughout the book, in employing the classical hypothesis testing model, the
author emphasizesitsjudicious use, aswell asthe importanceof conductingreplication studies
when the statusof a hypothesisrequires M e r clarification..
In order to facilitate its usage, most of the procedures contained in this Handbook are
organizedwithin a standardizedformat. Specifically, for most of the procedures the following
information is provided:
I. Hypothesisevaluatedwith test and relevantbackground information Thefirst part
of this section provides a general statement of the hypothesis evaluated with the test. This is
followed by relevant background information on the test such as the following: a) Information
regardingthe experimentaldesign for which thetest isappropriate;b) Any assumptionsunder-
lying the test which, if violated, would compromiseits reliability; and c) General information
on other statisticalprocedures that are related to the test.
11.Example Thissectionpresents a descriptionofan experiment,with an accompanying
data set (or in some instances two experiments utilizing the same data set), for which the test
will be employed. All examplesemploysmallsamplesizes, as well as integerdataconsisting
of small numbers, in order to facilitate the reader's ability to follow the computational
procedures to be described in Section IV.
111. Null versus alternative hypotheses This section contains both a symbolic and
verbal descriptionofthe statisticalhypotheses evaluated with thetest (i.e., the null hypothesis
versus the alternativehypothesis). It also statesthe form the data will assumewhen the null
hypothesisis supported,as opposed to when oneor more of the possible alternativehypotheses
are supported.
IV. Test computations Thissectioncontainsa step-by-stepdescriptionof the procedure
for computingthe test statistic. Thecomputationalguidelinesare clearlyoutlined in reference
to the data for the example(s)presented in Section 11.
V. Interpretation of the test results This section describes the protocol for evaluating
the computed test statistic. Specifically: a) It provides clear guidelines for employing the
appropriate table of critical values to analyze the test statistic; b) Guidelines are provided
delineating the relationship between the tabled critical values and when a researcher should

Preface vii
retain the null hypothesis, as opposed to when the researcher can concludethat oneor more of
the possible alternative hypotheses are supported; c) The computed test statisticis interpreted
in referenceto the example(s) presented in Section 11;and d) In instances where a parametric
and nonparametrictest can be used to evaluatethe same set of data, the results obtained using
both proceduresarecomparedwith oneanother,andtherelativepower ofboth testsisdiscussed
in this section and/or in Section VI.
Vi. Additional analytical proceduresfor the test andlor related tests Sincemany of
thetestsdescribed in theHandbookhaveadditionalanalyticalproceduresassociatedwith them,
such procedures are described in this section. Many of these procedures are commonly
employed (such as comparisonsconducted followingan analysisof variance), while others are
used and/or discussed less frequently (such asthe tie correction employed for the largesample
normalapproximationofmanynonparametrictest statistics). Manyoftheanalyticalprocedures
covered in Section VI are not discussed (or if so, only briefly) in other books. Some repre-
sentative topics which are covered in Section VI are planned versus unplanned comparison
procedures, measures of association for inferential statistical tests, computation of confidence
intervals, and computation of power. In addition to the aforementioned material, for many of
the tests there is additional discussion of other statistical procedureswhich are directlyrelated
to the test under discussion. In instances where two or more tests produce equivalentresults,
examplesare provided that clearly demonstratethe equivalency of the procedures.
MI. Additional discussion of the test Section VII discussestheoretical concepts and
issues, as well as practical and procedural issues which arerelevant to a specifictest. In some
instanceswherea subjectisaccorded briefcoverage in the initialmaterial presented on thetest,
the reader is alerted to the fact that the subject is discussed in greater depth in Section VII.
Manyofthe issues discussed in this section aretopicswhich are generallynot covered in other
books, or if they are, they are only discussed briefly. Among the topicscovered in Section VII
isadditionaldiscussion oftherelationshipbetween a specifictest and otherteststhat arerelated
to it. Section VII also provides bibliographic information on less commonly employed
alternative procedures that can be used to evaluate the same design for which the test under
discussion is used.
Ma. Additional examples illustrating the use of the test This section provides
descriptions of one or more additional experiments for which a specific test is applicable.For
the most part, these examples employ the same data set as that in the original example(s)
presented in Section I1 for the test. By virtue of using standardized data for most of the
examples, the material for a test contained in Section IV (Test computations) and Section V
(Interpretation of the test results) will be applicable to most of the additional examples.
Because ofthis, the reader is ableto focuson common design elementsin variousexperiments
which indicate that a given test is appropriate for use with a specific type of design.
IX. Addendum At the conclusion of the discussion of a number of tests an Addendum
hasbeen includedwhich describesoneormorerelatedteststhat arenot discussedin SectionVI.
Asanexample,theAddendumofthebetween-subjectsfactorialanalysisofvariancecontains
an overview and computationalguidelines for the factorial analysisof variance for a mixed
design and the within-subjectsfactorial analysisofvariance.
References This section provides the reader with a listing of primary and secondary
sourcematerial on each test.
Endnotes At the conclusionofmost tests, a detailedendnotessectioncontainsadditional
usefbl informationthat further clarifiesor expands upon material discussed in the main text.
In addition to the Introduction,the third edition of the Handbook contains 32 chapters,
eachofwhichdocumentsa specificinferentialstatisticaltest (aswellasrelatedtests)ormeasure

viii Handbook of Parametric and Nonparametric Statistical Procedures
ofcorrelation/association.Thegenerallabeltest isemployedforall procedures describedin the
book (i.e., inferential tests as well as measures of correlation/association). New material
(encompassingapproximately200pages)which hasbeen addedtospecificchaptersftestsinthis
edition is described below.
Introduction: A largeamountofnew material has been added to the Introduction ofthis
edition in order to make the Handbook more accessible to individuals who have no prior
exposure to the field of statistics. New material on the following topics has been added:
Discussion of the geometric and harmonic means; discussion of the concepts of efficiency,
sufficiency, and consistency; a comprehensivesection on visual methods for displaying data
(which includesthe use of conventionaltables and graphsand a discussion of exploratorydata
analysis(specificallystem-and-leafdisplaysand boxplots)); expandeddiscussion ofthenormal
distribution; a history and critiqueof the classical hypothesistesting model; a comprehensive
section on experimentaldesign (which includespre-experimentaldesigns, quasi-experimental
designs, true experimental designs; single-subject designs); discussion of survey sampling
procedures; an in-depth discussion of basic principles of probability including rules and
examplesfor computingprobabilities. Eight examplesand 22 figureshave been added to the
Introduction.
The Kolmogorov-Smirnovgoodness-of-fit test for a single sample (Test 7): Further
discussion of the graphicalmethod of analysis is accompaniedby one new figure.
The binomial sign test for a single sample (Test 9): The following material has been
addedtothe Addendum ofthetest: Conditionalprobability, Bayes' theorem and itsapplication
in epidemiological studies; an in-depth discussion of Bayesian hypothesis testing. Six new
exampleshave been added to this chapter.
Thesingle-sampleruns test (and othertests of randomness)(Test 10):A fulldiscussion
of The coupon collectors test (Test 100has been added.
The t test for two independent samples(Test 11): Discussion ofthegsample analogue
for Cohen's dindex; a comprehensivediscussion ofmethodsfor dealingwith missing datahas
been added.
The chi-square test for r x c tables (Test 16): Extendeddiscussion ofheterogeneitychi-
squareanalysis; description of Cohen's kappa (Test 16k) (and computation of a confidence
interval for kappa, test ofsignificanceforCohen's kappa (Test 16k-a),andtest ofsignificance
for two independent values of kappa (Test 16k-b)). Onenew examplehas been addedtothis
chapter.
The McNemar test (Test 20): Computation of the power of the McNemar test;
description of the Gart test for order effects (Test 20s); comparison procedures for the
Bowker test for internal symmetry (Test 20b); description of the Stuart-Maxwell test of
homogeneity (Test 20c).
The single-factor between-subjects analysis of variance (Test 21): Discussion of
equivalency ofthe single-factor between-subjectsanalysis of variance and the r test for two
independent sampleswith the chi-square test for r x ctables when c = 2; discussion oftrend
analysis.
The Kruskal-Wallis one-way analysis of variance by ranks (Test 22): Description of
the JonckheereTerpstra test for ordered alternatives (Test 22a).
The single-factor within-subjects analysis of variance (Test 24): Discussion of the
intraclass correlation coefficient (Test 24i); alternative models for conceptualizing the
variance elements for the single-factor within-subjects analysis of variance.
The Friedman two-way analysis of variance by ranks (Test 25): Description of the
Page test for ordered alternatives (Test 25a).

Preface ix
Thebetween-subjectsfactorialanalysisofvariance(Test27):Discussionofemploying
thebetween-subjectsfactorialanalysisofvarianceforevaluatingarandomized-blocksdesign.
The Pearson product-moment correlation coefficient (Test 28): Discussion of the g
sampleanalogueforCoben'sdindex;discussionofanalysisofvarianceforregression analysis;
theuseoftheStoufferprocedure(Test280) forheterogeneitychi-squareanalysisisillustrated;
descriptionof the binomial effect size display(Test 28r).
Kendall's coefficientof concordance(Test 31):Discussion of Kendall's coefficient of
concordance versus the intraclass correlation coefficient (Test 219.
As noted earlier in this Prehce, in orderto insure that the Handbookof Parametricand
Nonparametric Statistical Procedures can be employed by readers who have no prior fami-
liaritywith statisticsorresearch design,the Introductionhas been expanded to includesubject
matter not covered in the first two editions. Followingthe Introduction, the reader is provided
with guidelinesand decision tables for selectingthe appropriatestatistical test for evaluatinga
specificexperimentaldesign. The Handbook of Parametricand Nonparametric Statistical
Procedurescan beused as a referencebook or it can be employed as a textbook in undergrad-
uate and graduate courseswhich are designed to cover a broad spectrum of parametric and/or
nonparametric statistical procedures.
The author would like to express his gratitude to a number of people who helped make
this book a reality. First, I would like to thank Tim Pletscher of CRC Press for his
confidence in and support of the first edition of the Handbook. Specialthanks are due to Bob
Stern, the mathematics editor at Chapman and HalVCRC, who is responsible for the subse-
quent editions of this book. I am also indebted to Glena Arne who did an excellent job
preparing the copy-ready manuscript for the first two editions of the book. Finally, I must
express my appreciation to my wife Vicki and daughter Emily, who both endured and
toleratedthe difficultiesassociated with a project of this magnitude.
David Sheskin

To Vicki and Emily

Table of Contents
with Summary of Topics
Introduction
Descriptive versus inferential statistics
Statistic versus parameter
Levels of measurement
Continuous versus discrete variables
Measures of central tendency (mode. median. mean. geometric mean. and
thehmonicmean)
Measures of variability (range; quantiles. percentiles. deciles. and quartiles;
variance and standard deviation; the coefficientof variation)
Measures of skewness and kurtosis
Visual methods for displaying data (tables and graphs. exploratory data
analysis (stem-and-leaf displays and boxplots))
The normal distribution
Hypothesistesting
A history and critique of the classical hypothesis testing model
Estimation in inferential statistics
Relevant concepts. issues. and terminology in conducting research (the case
studymethod; the experimental method; the correlational method)
Experimental design @re-experimental designs; quasi-experimental designs
true experimental designs; single-subject designs)
Sampling methodologies
Basic principles of probability
Parametric versus nonparametric inferential statistical tests
Selection of the appropriate statistical procedure
Outline of Inferential StatisticalTests and Measures of
Correlation/Association
Guidelines and Decision Tables for Selecting the Appropriate
StatisticalProcedure
Inferential Statistical Tests Employed with a Single Sample
Test 1: The Single-Sample z Test
I. HypothesisEvaluated with Test and Relevant Background Information
I1. Example
111. Null versus Alternative Hypotheses

xiv Handbook of Parametric and Nonparametric Statistical Procedures
IV. Test Computations
V. Interpretation of the Test Results
VI. Additional Analytical Procedures for the Single-Samplez Test and/or
RelatedTests
VII. Additional Discussion of the Single-Samplez Test
1. The interpretation of a negativez value
2. The standard error of the population mean and graphical
representation ofthe results ofthe single-samplez test
3. Additional examples illustratingthe interpretation of a computedz
value
4. Thez test for a populationproportio
VIII. Additional ExamplesIllustratingthe Use of the Single-Samplez Test
Test 2: The Single-Samplet Test
I. Hypothesis Evaluatedwith Test and Relevant Background Information
I1. Example
I11. Null vers
VI. Additional Analytical Procedures for the Single-Samplet Test and/or
RelatedTests
1. Determination of the power of the single-sample t test and the
single-sample z test. and the application of Test 2a: Cohen's d
index
2. Comp
lation representedby a sample
VII. Additional Discussion of the Single-Samplet Test
1. Degrees of fieedom
VIII. Additional Examples Illustratingthe Use of the Single-Samplet Test
Test 3: The Single-Sample Chi-square Test for a Population Variance
I1. Example
I11. Null versus Alternative Hypotheses
IV. Testcomputations
V. Interpretation ofthe Test Results
VI. Additional Analytical Procedures for the Single-SampleChi-square
Test for a Population Variance and/or Related Tests
1. Large samplenormal approximationof the chi-square distribution
2. Computation of a confidence interval for the variance of a
population representedby a sample
3. Sources for computing the power of the single-sample chi-square
test for a population variance
VII. Additional Discussion of the Single-SampleChi-square Test for a
Populationvariance
VIII. Additional Examples Illustrating the Use of the Single-SampleChi-
Square Test for a Population Variance

Table of Contents xv
Test 4: The Single-SampleTest for Evaluating Population Skewness
.I Hypothesis Evaluated with Test and Relevant Background Informatio
I1. Example
111. Null versusAlternative Hypothese
V. Interpretation of the Test Result
VI. Additional Analytical Proceduresfor the Single-SampleTest for
Evaluating Population Skewness and/or Related Test
1. Note on the D'Agostino-Pearson test of normality (Test 5a
VII. Additional Discussion of the Single-SampleTest for Evaluating
Population Skewnes
1. Exact tables for the single-sample test for evaluating
population skewnes
2. Note on a nonparametric test for evaluating skewness
VIII. Additional Examples Illustrating the Use of the Single-Sample Test for
Evaluating Population Skewnes
Test 5: The Single-SampleTest for Evaluating Population Kurtosi
.I1 Example
.111 Null versus Alternative Hypothese
.IV Testcomputations
.V Interpretation of the Test Result
VI. Additional Analytical Procedures for the Single-SampleTest for
Evaluating ~o~ulationKurtosis andlor Related Test
1. Test 5a: The D'Agostino-Pearson test of normality
VII. Additional Discussion of the Single-SampleTest for Evaluating
Population Kurtosi
1. Exact tables for the single-sample test for evaluating
population kurtosis
VIII. Additional Examples Illustrating the Use of the Single-SampleTest for
Evaluating Population Kurtosi
Test 6: The Wilcoxon Signed-RanksTest
I. Hypothesis Evaluated with Test and Relevant Background Informatio
I1. Example
I11. Null versus Alternative Hypothese
VI. Additional Analytical Procedures for the Wilcoxon Signed-RanksTest
and/or Related Tests
1. The normal approximation of the Wilcoxon T statistic for large
samplesizes
2. The correction for continuity for the normal approximation of the
Wilcoxon signed-ranks test
3. Tie correction for the normal approximation of the Wilcoxon test

xvi Handbook of Parametric and Nonparametric Statistical Procedures
statisti
VII. Additional Discussion of the Wilcoxon Signed-RanksTest
1. Power-efficiency of the Wilcoxon signed-ranks test and the
concept of asymptotic relative efficienc
2. Note on symmetric population concerning hypotheses regarding
medianandmea
3. Confidenceinterval for the median difference
VIII. Additional Examples Illustratingthe Use of the Wilcoxon Signed-
RanksTest
Test 7: The Kolmogorov-SmirnovGoodness-of-FitTest for a Single
Sampl
IT. Example
111. Null versus Alternative Hypothese
VI. Additional Analytical Proceduresfor the Kolmogorov-Smirnov
Goodness-of-Fit Test for a SingleSampleandlor Related Test
1. Computing a confidence interval for the Kolmogorov-
Smirnovgoodness-of-fit test for a single sampl
2. The power of the Kolmogorov-Smirnov goodness-of-fit test
for a single sample
3. Test 7a: The Lillieforstest for normality
VII. Additional Discussion of the Kolmogorov-Smirnov Goodness-of-Fit
Test for a SingleSample
1. Effect of sample size on the result of a goodness-of-fittest
2. The Kolmogorov-Smirnov goodness-of-fit test for a single
sample versus the chi-square goodness-of-fit test and altema-
tive goodness-of-fit tests
VIII. Additional Examples Illustrating the Use of the Kolmogorov-Smirnov
Goodness-of-Fit Test for a SingleSample
Test 8: The Chi-square Goodness-of-Fit Test
I1. Examples
IV. TestComputations
VI. Additional Analytical Procedures for the Chi-square Goodness-of-Fit
Test and/or Related Tests
1. Comparisonsinvolvingindividual cells when k > 2
2. The analysis of standardizedresiduals
3. Computation of a confidence interval for the chi-square goodness-
of-fit test (confidenceinterval for a population proportion
4. Brief discussion of the z test for a population proportion
(Test 9a) and the single-sampletest for the median (Test 9b)
5. The correction for continuity for the chi-squaregoodness-of-fit test

Tableof Contents xvii
6. Application of the chi-square goodness-of-fit test for assessing
goodness-of-fit for a theoretical population distribution
7. Sources for computing of the power of the chi-square goodness-of-
fittes
8. Heterogeneitychi-square analysi
VII. Additional Discussion of the Chi-square Goodness-of-Fit Test
1. Directionalityof the chi-squaregoodness-of-fittest
2. Additional goodness-of-fittests
VIII. AdditionalExamplesIllustratingthe Use of the Chi-square Goodness-
of-Fit Test
Test 9: The Binomial SignTest for a Single Sample
.I HypothesisEvaluated with Test and Relevant Background Informatio
.I1 Examples
.I11 Null vers
.V Interpretation ofthe Test Result
VI. Additional Analytical Procedures for the Binomial Sign Test for a
Single Sampleand/or Related Tests
1. Test 9a: The z test for a population proportion (with discussi
of correction for continuity; computation of a confidence interval;
extension of the z test for a population proportion to evaluate the
performance of m subjects on n trials on a binomially distributed
variable
2. Test 9b: The single-sampletest for the median
3. Computingthe power ofthe binomial sign test for a singlesampl
VII. Additional Discussion of the Binomial Sign Test for a Single Sampl
1. Evaluating goodness-of-fit for a binomial distribution
VIII. AdditionalExample Illustrating the Use of the Binomial Sign Test for
a SingleSample
IX. Addendum
1. Discussion of additional discrete probability distributions
(multinomial distribution; negative binomial distribution; hyperge-
ometric distribution; Poisson distribution (and evaluating goodness-
of-fit for a Poisson distribution);matching distribution)
2. Conditional probability, Bayes' theorem, Bayesian statistics and
hypothesistesting
Test 10: The Single-SampleRuns Test (and Other Testsof Randomness)
I1. Example
I11. Null versus AlternativeHypotheses
V. Interpretation ofthe Test Results
VI. AdditionalAnalytical Procedures for the Single-Sample Runs Test
1. The normal approximation of the singlesample runs test for large
samplesizes

xviii Handbook of Parametric and Nonparametric Statistical Procedures
single-sampleruns test
3. Extension of the runs test to data with more than two categorie
4. Test 10a: The runs test for serial randomnes
VII. Additional Discussion of the Single-SampleRuns Test
1. Additional discussion of the concept of randomness
VII. Additional Examples Illustrating the Use of the Single-Sample
RunsTest
IX. Addendum
1. The generation of pseudorandom numbers (The midsquare
method; the midproduct method; the linear congruential method)
2. Alternative tests of randomness (Test lob: The frequency test;
Test 10c: The gap test; Test 10d: The poker test; Test 10e: The
maximum test; Test 10f: The coupon collector's test; Test log:
The mean square successive difference test (for serial random-
ness); Additional tests of randomness (Autocorrelation;The serial
test; The d2 square test of random numbers; Tests of trend
analysidtime series analysis)
Inferential StatisticalTests Employed with Two Independent
Samples (and Related Measures of Association/Correlation)
Test 11: The t Test for Two IndependentSamples
I. HypothesisEvaluated with Test and Relevant Background Informatio
.I1 Example
VI. Additional Analytical Procedures for the t Test for Two Independent
Samples andlor Related Tests
1. The equation for the t test for two independent samples when a
value for a difference other than zero is stated in the null
hypothesis
2. Test lla: Hartley's F,, test for homogeneity of variance1Ftest
for two population variances: Evaluation of the homogeneity of
variance assumption of the t test for two independentsamples
3. Computation of the power of the t test for two independent samples
and the application of Test llb: Cohen's d index
4. Measures of magnitude of treatment effect for the t test for two
independent samples: Omega squared (Test Ilc) and Eta
squared(Testl1d)
5. Computation of a
independentsamples
6. Test lle: The z test
VII. Additional Discussion of the t Test for Two Independent Sample
1. Unequal sample sizes

Handbook of Parametric and Nonparametric Statistical Procedures
2. Computing sample confidence intervals for the Kolmogorov-
Smirnovtest for two independentsamples
3. Large sample chi-square approximation for a one-tailed analysis of
the Kolrnogorov-Smirnov test for two independentsamples
VII. Additional Discussion of the Kolmogorov-Smirnov Test for Two
Independent Samples
1. Additional comments on the Kolmogorov-Smirnov test for two
independent samples
VIII. Additional Examples Illustratingthe Use of the Kolmogorov-Smirnov
Test for Two Independent Samples
Test 14: The SiegeLTukeyTest for Equal Variability
I. Hypothesis Evaluated with Test and Relevant Background Information
I1. Example
VI. Additional AnalyticalProceduresfor the Siegel-Tukey Test for Equal
Variability and/or Related Tests
1. The normal approximation of the Siegel-Tukey test statistic for
large sample sizes
Siegel-Tukey test for equal variability
3. Tie correction for the normal approximation of the Siegel-Tukey
test statistic
4. Adjustment of scores for the Siegel-Tukey test for equal variability
when 0, + 0,
VII. Additional Discussion of the Siegel-Tukey Test for Equal Variability
1. Analysis of the homogeneity of variance hypothesis for the same
set of data with both a parametric and nonparametric test. and the
power-efficiencyof the Siegel-Tukey Test for Equal Variability
2. Alternative nonparametrictests of dispersion
VIII. Additional Examples Illustrating the Use of the Siegel-Tukey Test for
Equalvariability
Test 15: The Moses Test for Equal Variability
I1. Example
VI. Additional Analytical Procedure r the Moses Test for Equal
Variabilityand/or Related Tests
I. The normal approximationofthe Moses test statisticfor large
samplesizes
VII. Additional Discussion of the Moses Test for Equal Variability
1. Power-efficiencyof the Moses Test for equal variability
2. Issue of repetitiveresampling

Tableof Contents xxi
3. Alternative nonparametric tests of dispersio
VIII. Additional Examples Illustrating the Use of the Moses Test for Equal
Variabilit
Test 16: The Chi-square Test for r x c Tables (Test 16a: The Chi-Square
Test for Homogeneity; Test 16b: The Chi-Square Test of Indepen-
dence (employedwith a single sample)
I1. Examples
.V Interpretation of the Test Result
VI. Additional Analytical Procedures for the Chi-Square Test for r x c
Tables and/or Related Test
1. Yates' correction for continuity
2. Quick computational equation for a 2 x 2 tabl
3. Evaluation of a directional alternative hypothesis in the case of a 2
x 2 contingencytable
4. Test 16c: The Fisher exact test
5. Test 16d: The z test for two independent proportions
6. Computation of a confidence interval for a differencebetween two
proportions
7. Test 16e: The median test for independent samples
8. Extension of the chi-square test for r x c tables to contingency
tables involving more than two rows and/or columns, and asso-
ciated comparisonprocedure
9. The analysis of standardized residuals
10. Sources for computing the power of the chi-square test for r x c
tables
11. Heterogeneity chi-square analysis for a 2 x 2 contingencytable
12. Measures of association for r x c contingency tables (Test 16f:
The contingencycoefficient; Test 16g: The phi coefficient; Test
16h: Cramdr's phi coefficient; Test 16i: Yule's Q; Test 16j: The
odds ratio (and the concept of relative risk); Test 16j-a: Test of
significance for an odds ratio and computation of a confidence
interval for an odds ratio); Test 16k; Cohen's kappa (and
computation of a confidence interval for kappa, Test 16k-a: Test
of significance for Cohen's kappa, and Test 16k-b: Test of
significancefor two independent values of Cohen's kappa)
VII. Additional Discussion of the Chi-square Test for r x c Tables
1. Equivalency of the chi-square test for r x c tables when c = ith
the t test for two independent samples (when r = 2) and the single-
factor between-subjects analysis of variance (when r > 2)
2. Simpson's Paradox
3. Analysis of multidimensional contingencytables
VIII. Additional Examples Illustrating the Use of the Chi-Square Test for
rxcTables

xxii Handbook of Parametric and Nonparametric Statistical Procedures
Inferential StatisticalTests Employed with Two Dependent
Samples (and Related Measures of Association/Correlation)
Test 17: The t Test for Two Dependent Samples
11. Exampl
IV. TestComputation
VI. Additional Analytical Procedures for the t Test for Two Dependent
Samples andor Related Tests
1. Alternative equation for the t test for two dependent sample
2. The equation for the t test for two dependent samples when a value
for a differenceother than zero is stated in the null hypothesi
3. Test 17a: The t test for homogeneity of variance for two depen-
dent samples:Evaluation of the homogeneity of variance assurnp-
tion of the t test for two dependent samples
4. Computation of the power of the t test for two dependent samples
and the application of Test 17b: Cohen's d index
5. Measure of magnitude of treatment effect for the t test for two
dependent samples: Omega squared (Test l7c
6. Computation of a confidence interval for the t test for two
dependentsample
7. Test l7d: Sandler's A test
8. Test 17e: The z test for two dependentsamples
VII. Additional Discussion ofthe t Test for Two Dependent Sample
1. The use of matched subjects in a dependent samples design
2. Relative power of the t test for two dependent samples and the t test
for two independent samples
3. Counterbalancing and order effects
4. Analysis of a one-group pretest-posttest design with the t test for
two dependent sample
VIII. Additional Example Illustrating the Use of the t Test for Two
Dependent Sample
Test 18: The Wilcoxon Matched-PairsSigned-RanksTest
I. Hypothesis Evaluatedwith Test and Relevant Background Informatio
I1. Example
I11. Null versusAlternative Hypotheses
VI. Additional Analytical Procedures for the Wilcoxon Matched-Pairs
Signed-RanksTest andor Related Tests
1. The normal approximation of the
sample sizes
2. The correcti
Wilcoxon matched-pairs signed-ranks test

3. Tie correction for the normal approximation of the Wilcoxon test
statistic
4. Sources for computing a confidence interval for the Wilcoxon
matched-paus signed-ranks test
VII. Additional Discussionof the Wilcoxon Matched-PairsSigned-Ranks
Test
1. Power-efficiencyof the Wilcoxon matched-pairssigned-ranks test
2. Alternative nonparametric procedures for evaluating a design
involving twodependent samples
VIII. Additional Examples Illustrating the Useof the Wilcoxon Matched-
Pairs Signed-RanksTest
Test 19: The Binomial Sign Test for Two DependentSamples
11. Example
VI. Additional Analytical Procedures for the Binomial Sign Test for Two
Dependent Samples and/or Related Tests
1. The normal approximation of the binomial sign test for two
dependent samples with and without a correction for continuity
2. Computation of a confidence interval for the binomial sign test for
two dependent samples
3. Sources for computing the power of the binomial sign test for two
dependent samples, and comments on asymptotic relative
efficiencyof the test
VII. Additional Discussionof the Binomial Sign Test for Two Dependent
Samples
1. The problem of an excessivenumber of zero difference scores
2. Equivalency of the binomial sign test for two dependent samples
and the Friedman tw*way analysis variance by ranks when k = 2
VIII. Additional Examples Illustrating the Useof the Binomial Sign Test for
Two Dependent Samples
Test 20: The McNemar Test
11. Examples
VI. Additional Analytical Procedures for the McNemar Test andlor Related
Tests .
1. Alternative equation for the McNernar test statistic based on the
normal distribution
2. The correction for continuity for the McNemartest
3. Computation of the exact binomial probability fo
test model with a small sample size

xxiv Handbook of Parametric and Nonparametric Statistical Procedures
4. Computation of the power of the McNemar test
5. Additional analyticalprocedures for the McNemar tes
6. Test 20a: The Cart test for order effects
VII. Additional Discussion of the McNemar Test
1. Alternative format for the McNernar test summary table and
modified test equation
2. Alternative nonparametric procedures for evaluatinga design with
two dependentsamples involving categorical data
VIII. Additional Examples Illustrating the Use of the McNemar Test
IX. Addendum
Extension of the McNemar test model beyond 2 x 2 contingency
tables
1. Test 2Ob:The Bowker test of internal symmetry
2. Test 20c:The Stuart-Maxwell test of marginal homogeneity
Inferential StatisticalTests Employed with Two or More
Independent Samples (and Related Measures of
Association/Correlation)
Test 21: The Single-FactorBetween-SubjectsAnalysis of Variance
11. Example
111. Null vers
1V. Testcomputations
VI. AdditionalAnalytical Procedures for the Single-Factor Between-
SubjectsAnalysis of Variance andtor Related Test
I. Comparisonsfollowing computation of the omnibus Fvalue for the
single-hctor between-subjectsanalysis of variance (Planned versus
unplanned comparisons (including simple versus complex com-
parisons); Linear contrasts; Orthogonal comparisons; Test 21a:
Multiple t testsIFisher's LSD test; Test 21b: The Bonferroni-
Dunn test; Test 21c: Tukey's HSD test; Test 21d: The
Newman-Keuls test; Test 21e: The Scheffk test; Test 21f: The
Dunnett test; Additional discussion of comparison procedures and
final recommendations; The computation of a confidence interval
for a comparison
2. Comparingthe means of three or more groupswhen k 2
3. Evaluation of the homogeneity of variance assumption of the
single-hctor between-subjectsanalysis of variance
4. Computation of the power of the single-factor between-subjects
analysis of variance
5. Measures of magnitude of treatment effect for the single-fkctor
between-subjectsanalysis of variance: Omega squared (Test 21g),
Eta squared (Test 21h), and Cohen'sf index (Test 21i)
6. Computation of a confidence interval for the mean of a treatment
population

Table of Contents xxv
VII. Additional Discussion of the Single-Factor Between-SubjectsAnalysis
ofvariance
1. Theoretical rationale underlying the single-factorbetween-subjects
2. Definitional equations for the single-factor between-subjects
3. Equivalency of the single-factor between-subjects analysis of
variance and the t test for two independent samples when k =
4. Robustness of the single-factorbetween-subjects analysis of
varianc
5. Equivalency of the single-fador between-subjects analysis of
variance and the t test for two independent samples with the chi-
squaretestforrx ctableswhenc=2
6. Fixed-effects versus random-effects models for the single-factor
between-subjects analysis of variance
7. Multivariate analysis of variance (MANOVA)
VIII. Additional Examples Illustrating the Use of the Single-FactorBetween-
SubjectsAnalysis of Variance
IX. Addendum
1. Test 21j: The Single-Factor Between-Subjects Analysis of
Covariance
Test 22: The Kruskal-Wallis One-way Analysis of Variance by Ranks
1. Hypothesis Evaluated with Test and Relevant Background Informatio
.I1 Example
I11. Null vers
1V. Testcomputations
VI. Additional Analytical Procedures for the Kruskal-Wallis One-Way
Analysis of Variance by Ranks and/or Related Tests
1. Tie correction for the Kruskal-Wallis one-way analysis of variance
byranks
2. Pairwise comparisons following computation of the test statistic for
the Kruskal-Wallis one-way analysis of variance by ranks
VII. Additional Discussion of the Kruskal-Wallis One-Way Analysis of
VariancebyRanks
I. Exact tables of the Kruskal-Wallis distribution
2. Equivalencyof the Kruskal-Wallis one-way an
variance by ranks and the Mann-Whitney U test when k = 2
3. Power-efficiency of the Kruskal-Wallis one-way analysis of
variancebyranks
4. Alternative nonp
design involving k independent sample
VIII. Additional Examples Illustrating the Use of
Way Analysis of Variance by Ranks
IX. Addendum
1. Test 22a: The Jonckheere-TerpstraTest for Ordered
Alternatives

xxvi Handbook of Parametric and Nonparametric Statistical Procedures
Test 23: The Van der Waerden Normal-ScoresTest for R Independent
Samples
I1. Example
.V Interpretation of the Test Results
VI. Additional Analytical Proceduresfor the van der Waerden Normal-
Scores Test for k Independent Samplesandfor Related Tests
1. Pairwise comparisonsfollowing computation of the test statisticfor
the van der Waerden normal-scorestest for k independentsamples
VII. Additional Discussion of the van der Waerden Normal-Scores Test for
k Independent Samples
.1 Alternativenormal-scorestests
VIII. Additional ExamplesIllustratingthe Use of the van der Waerden
Normal-Scores Test for k Independent Samples
Inferential StatisticalTests Employed with Two or More
Dependent Samples (and Related Measures of
Association/Correlation)
Test 24: The Single-FactorWithin-SubjectsAnalysis of Variance
.I Hypothesis Evaluated with Test and Relevant Background Information
I1. Example
.I11 Null versus Alternative Hypotheses
VI. Additional Analytical Procedures for the Single-Factor Within-Subjects
Analysis of Variance andlor Related Tests
1. Comparisonsfollowing computation of the omnibusFvalue for the
single-factor within-subjects analysis of variance (Test 24a:
Multiple t tests/Fisher's LSD test; Test 24b: The Bonferroni-
Dunn test; Test 24c: Tukey's HSD test; Test 24d: The
NewmawKeuls test; Test 24e: The Scheff6 test; Test 24fi The
Dunnett test; The computation of a confidence interval for a
comparison; Alternative methodology for computing MS, for a
comparison
.2 Comparingthe means of three or more conditionswhen k 2 4
3. Evaluation of the sphericity assumption underlying the single-
factor within-subjects analysis of variance
4. Computation of the power of the single-factor within-subjects
analysisof variance
5. Measures of magni
within-subjects analysis of variance: Omega squared (Test 24g)
and Cohen'sfindex (Test 24h)

Tableof Contents xxvii
6. Computation of a confidence interval for the mean of a treatment
population
7. Test 24i: The intraclass correlationcoeff~cien
VII. Additional Discussion ofthe SingleFactor Within-Subjects Analysisof
Variance
1. Theoretical rationale underlying the singlefactor within-subjects
2. Definitional equatio
ofvariance
3. Relative power of the singlefactor within-subjects analysisof var-
iance and the singlehctor between-subjectsanalysisof variance
4. Equivalency of the single-factor within-subjects analysis of vari-
ance and the t test for two dependent sampleswhen k = 2
5. The Latin square design
VIII. Additional ExamplesIllustratingthe Use of the Single-FactorWithin-
SubjectsAnalysis of Variance
Test 25: The Friedman Two-way Analysis of Variance by Ranks
I1. Example
I11. Null vers
IV. Test computations
V. Interpretation of th
VI. Additional Analytical Procedures f
Varianceby Ranks and/or Related Tests
1. Tie correction for the Friedman tweway analysis variance
byranks
2. Painvise
the Friedman two-way analysisof variance by ranks
VII. Additional Discussion of the Friedman TweWay Analysis Variance
byRanks
1. Exac
2. Equivalency of the Friedman two-way analysis variance by ranks
and the binomial sign test for two dependent sampleswhen k = 2
3. Power-efficiency of the Friedman two-way analysis variance
byranks
4. Alternative nonparametric rank-order procedures for evaluating a
design involvingk dependentsamples
5. Relationship between the Friedman t
by ranks and Kendall's coefficient of concordance
VIII. Additional Examples Illustrating the Use of the Friedman Two-way
Analysis of Variance by Ranks
IX.Addendum
1.Test25a: The Page Test for OrderedAlternatives
Test 26: The Cochran Q Test
I1. Example

xxviii Handbook of Parametric and Nonparametric Statistical Procedures
VI. Additional Analytical Procedur r the Cochran Q Test andor
RelatedTest
1. Pairwise comparisonsfollowing computation of the test statisticfor
theCochranQtest
VII. Additional Discussion of the Cochran Q Test
1. Issues relating to subjects who obtain the same score under all of
the experimentalconditions
2. Equivalencyof the Cochran Qtest and the McNemar test when
k =
3. Alternative nonparametric procedures with categorical data for
evaluating a design involvingk dependent samples
VIII. Additional Examples Illustrating the Use of the Cochran Q Test
InferentialStatisticalTest Employedwith Factorial Design
(and Related Measures of Association/Correlation)
Test 27: The Between-SubjectsFactorial Analysis of Variance
11. Example
111. Null vers
VI. Additional Analytical Procedures for the Between-SubjectsFactorial
Analysis of Variance andlor Related Test
1. Comparisons followingcomputation of the Fvalues for the betwe
subjects factorial analysis of variance (Test 27a: Multiple t
testsmisher's LSD test; Test 27b: The Bonferroni-Dunn test;
Test 27c: Tukey's HSD test; Test 27d: The Newman-Keuls test;
Test 27e: The Scheffk test; Test 27f: The Dunnett test;
Comparisons between the marginal means; Evaluation of an
omnibus hypothesis involving more than two marginal means;
Comparisons between specificgroups that are a combination of both
factors; The computation of a confidence interval for a comparison;
Analysis of simpleeffects)
2. Evaluation of the homogeneity of variance assumption of the
between-subjectsfactorialanalysis of variance
3. Computation of the power of the between-subjects factorial analysis
ofvariance
4. Measures of magnitude of treatment effect for the between-subjects
factorial analysis of variance: Omega squared (Test 27g) and
Cohen'sf index (Test 27h)
5. Computation of a confidence interval for the mean of a population
represented by a group
6. Additional analysisofvariance procedures for factorial designs

Tableof Contents xxix
VII. Additional Discussion of the Between-SubjectsFactorial Analysis of
Variance
1. Theoretical rationale underlying the between-subjects factorial
2. Definitional equations for the between-subjects factorial analysis of
variance
3. Unequal sample size
4. The randomized-blocksdesign
5. Final comments on the between-subjects factorial analysis of
variance (Fixed-effects versus random-effects versus mixed-effects
models; Nested factorshierarchical designs and designs involving
more than two factors)
VIII. Additional Examples Illustrating the Use of the Between-Subjects
Factorial Analysis of Variance
IX. Addendum
1. Discussion of and computational procedures for additional analysis
of variance procedures for factorial designs: Test 27i: The factorial
analysis of variance for a mixed design; Test 27j: The within-
subjects factorial analysis of variance
Measures of Association/Correlation
Test 28: The Pearson Product-Moment Correlation Coefficient
11. Example
V. Interpretation of the Test Results (Test 28a: Test of significancefor
a Pearson product-moment correlation coefficient;The coefficient
ofdetermination)
VI. Additional Analytical Proceduresfor the Pearson Product-Moment
Correlation Coefficient andlor Related Tests
1. Derivation of a regression line
2. The standard error of estimate
3. Computation of a confidence interval f
variable
4. Computation of a confidence interval for a Pearson product-moment
correlation coefficient
5. Test 28b: Test for evaluating the hypothesis that the true
population correlation is a specificvalue other than zero
6. Computation of power for the Pearson product-moment correlation
coefficient
7. Test 28c: Test for evaluating a hypothesis on whether there is a
significant difference between two independent correlations
8. Test 28d: Test for evaluating a hypothesis on whether k
independent correlations are homogeneous
9. Test 28e: Test for evaluating the null hypothesisH, pxZ= p,
10. Tests for evaluating a hypothesis regarding one or more regression

coefficients (Test 28f: Test for evaluating the null hypothesis H,:
$ = 0; Test 28g: Test for evaluating the null hypothesis H,: $, =
pr)
11. Additional correlationalprocedures
VII. AdditionalDiscussion of the Pearson Product-Moment Correlation
Coefficient
1. The definitional equation for the Pearson product-moment
correlation coefficien
2. Residuals and analysi
3. Covariance
4. The homoscedasticity assumption of the Pearson product-moment
correlation coefficien
5. The phi coefficient as a special case of the Pearson product-moment
correlationcoefficien
6. Autocorrelation/serialcorrelation
VIII. Additional Examples Illustrating the Use of the Pearson Product-
Moment CorrelationCoefficient
IX.Addendum
1. Bivariate measures of correlation that are related to the Pearson
product-moment correlation coefficient (Test 28h: The point-
biserial correlation coefficient (and Test 28h-a: Test of
significance for a point-biserial correlation coefficient); Test 28i:
The biserial correlation coefficient (and Test 28i-a: Test of
significance for a biserial correlation coefficient); Test 28j: The
tetrachorie correlation coefficient (and Test 28j-a: Test of
significance for a tetrachoric correlation coefficient))
2. Multiple regression analysis (General introduction to multiple
regression analysis; Computational procedures for multiple
regression analysis involving three variables: Test 28k:
The multiple correlation coefficient; The coefficient of multiple
determination; Test 28k-a: Test of significance for a multiple
correlation coefficient; The multiple regression equation; The
standard error of multiple estimate; Computation of a confidence
interval for Y'; Evaluation of the relative importance of the
predictor variables; Evaluating the significance of a regression
coefficient; Computation of a confidence interval for a regression
coefficient; Analysis of variance for multiple regression; Partial
and semipartial correlation (Test 281: The partial correlation
coefficient and Test 281-a: Test of significance for a partial
correlation coefficient; Test 28m: The semipartial correlation
coefficient and Test 28m-a: Test of significance for a semipartial
correlation coefficient); Final comments on multiple regression
analysis
3. Additional multivariate procedures involving correlational analysis
(Factor analysis; Canonical correlation; Discriminant analysis and
logisticregression)
4. Meta-analysis and related topics (Measures of effect size; Meta-
analytic procedures (Test 28n: Procedure for comparing k studies
with respect to significance level; Test 280: The Stouffer

Table of Contents xxxi
procedure for obtaining a combined significance level (p value)
for k studies; The file drawer problem; Test 28p: Procedure for
comparing k studies with respect to effect size; Test 28q:
Procedure for obtaining a combined effect size for k studies);
Practical implications of magnitude of effect size value; Test 28r:
Binomial effect size display; The significancetest controversy; The
minimum-effect hypothesis testing model)
Test 29: Spearman's Rank-Order Correlation Coefficient
I1. Example
I11. Null versus AlternativeHypotheses
V. Interpretation of the Test Results (Test 29a: Test of significance for
Spearman's rank-order correlation coefficient)
VI. Additional Analytical Proceduresfor Spearman's Rank-Order
Correlation Coefficient and/or Related Tests
1. Tie correction for Spearman's rank-order correlation coefficient
2. Spearman's rank-order correlation coefficient as a specialcase of the
Pearson product-moment correlationcoefficient
3. Regression analysis and Spearman's rank-order correlation
coefficient
4. Partial rank correlation
5. Use of Fisher's z, transformation with Spearman's rank-
order correlationcoefficient
VII. Additional Discussion of Spearman's Rank-Order Correlation
Coefficient
1. The relationship between Spearman's rank-order correlation
coefficient. Kendall's coefficient of concordance. and the Friedman
tweway analysisof variance by ranks
2. Power efficiencyof Spearman's rank-o
3. Brief discussion of Kendall's tau: An alternative measure of
association for two sets of ranks
4. Weighted rankhopdown correlation
VIII. Additional Examples Illustrating the Use of the Spearman's Rank-
Order Correlation Coefficient
Test 30: Kendall's Tau
I1. Example
I11. Null vers
Kendall'stau)
VI. Additional Analytical Procedures for Kendall's Tau andlor Related
Tests
1. Tie correction for Kendall's tau
2. Regression analysisand Kendall's tau

xxxii Handbook of Parametric and Nonparametric Statistical Procedures
3. Partial rank correlatio
4. Sourcesfor computing a confidenceinterval for Kendall's tau
VII. Additional Discussion of Kendall's Tau
1. Power efficiencyof Kendall's tau
2. Kendall's coefficient of agreement
VIII. Additional Examples Illustrating the Use of Kendall's Tau
Test 31: Kendall's Coefficient of Concordance
I1. Example
I11. Null versus Alternative Hypothese
V. Interpretation of th est 31a: Test of significance for
Kendall's coefficient of concordance
VI. Additional Analytical Procedures for Kendall's Coefficient of
Concordanceand/or Related Test
1. Tie correction for Kendall's c
VII. Additional Discussion of Kendall's Coefficient of Concordanc
1. Relationship between Kendall's coefficient of concordance and
Spearman's rank-order correlation coefficient
2. Relationship between Kendall's coefficient of concordance and the
Friedman two-way analysisof variance by ranks
3. Weighted rankftop-down concordance
4. Kendall's coefficient of concordanceversus the intraclasscorrelation
coefficient
VIII. Additional Examples Illustrating the Use of Kendall's Coefficient of
Concordanc
Test 32: Goodman and Kruskal's Gamma
I1. Example
I11.Null vers
Goodman and Kruskal's gamma)
VI. Additional Analytical Procedures f
1. The computation of a confidence interval for the value of Goodman
and Kruskal's gamma
2. Test 32b: Test for ev
3. Sources for computing a partial correlation coefficient for Goodman
and Kruskal's gamma
VII. Additional Discussion of Goodman and Kruskal's Gamma
1. Relationshipbetween Goodman and Kruskal's gamma and Yule's Q
2. Somers' delta as an alternativemeasure of association for an ordered
contingency table
VIII. Additional Examples
Gamma

Tableof Contents xxxiii
Appendk Tables
.Table A1 Table of the Normal Distributio
.Table A2 Table of Student's t Distribution
.Table A3 Power Curves for Student's t Distribution
.Table A4 Table of the Chi-square Distribution
Table A5.Table of Critical TValues for Wilcoxon's Signed-Ranks and
Matched-Pairs Signed-Ranks Test
Table A6.Table of the BinomialDistribution. Individual Probabilities
Table A7.Table of the Binomial Distribution. Cumulative Probabiliti
Table AS.Table of Critical Values for the Single-Sample Runs Test
.Table A9 Table of the F,, Distribution
.Table A10 Table of the FDistributio
Table A l l.Table of Critical Values for Mann-Whitney U Statisti
Table A12.Table of Sandler's A Statisti
Table A13.Table of the Studentized Range Statisti
Table A14.Table of Dunnett's Modified t Statistic for a Control Group
Compariso
Table A15.Graphs of t
Table A16.Table of Critical Values for Pearson
Table A17.Table of Fisher's & Transformatio
Table A18.Table of Critical Values for Spearman's Rho
Table A19.Table of Critical Values for Kendall's Ta
Table A20.Table of Critical Values for Kendall's Coefficientof
Concordance
Table 14.21. Table of Critical Values for the KolmogorovSmirnov
Goodness-of-Fit Test for a SingleSample
Table A22.Table of Critical Values for the Lilliefors test for Normality
Table A23.Table of Critical Values for the KolmogorovSmirnov Test
Two Independent Samples
Table A24.Table of Critical Values for the JonckheereTerpstra
Teststatistic
Table A25.Table of Crit

Introduction
The intent of this Introduction is to provide the reader with a general overview of basic
terminology, concepts, and methods employed within the areas of descriptive statistics and
experimental design. To be more specific, the following topics will be covered: a)
Computationalprocedures formeasuresofcentraltendency,variability,skewness,andkurtosis;
b) Visual methods for displaying data; c) The normal distribution; d) Hypothesis testing; e)
Experimental design. Within the contextofthe latter discussions, thereader ispresented with
thenecessaryinformation forboth understandingand usingthe statisticalprocedureswhich are
described in this book. Following the Introduction is an outline of all the procedures that are
covered, as well as decision tables to aid the reader in selecting the appropriate statistical
procedure.
Descriptiveversus Inferential Statistics
Theterm statisticsisderivedfromLatin andItaliantermswhichrespectivelymean "status" and
"state arithmetic" (i.e., the present conditionswithin a stateor nation). In a more formalsense,
statisticsis a field within mathematics that involves the summaryand analysis of data. The
field of statistics can be divided intotwo general areas, descriptive statisticsand inferential
statistics.
Descriptive statisticsis a branch of statistics in which data are only used for descriptive
purposes and are not employed to make predictions. Thus, descriptive statistics consists of
methodsandprocedures forpresentingandsummarizingdata. Theproceduresmostcommonly
employed in descriptive statistics are the use of tables and graphs, and the computation of
measures of central tendency and variability. Measuresof associationor correlation,which
are covered in this book, are also categorized by most sources as descriptive statistical
procedures, insofarasthey serveto describethe relationship between two or more variables. A
variable is any property of an object or an organism with respect to which there is variation -
i.e., not every object or organism is the same with respect to that property. Examples of
variables are color, weight, height, gender, intelligence, etc.
Inferential statisticsemploys data in order to draw inferences(i.e., derive conclusions)
or make predictions. Typically, in inferential statistics sample data are employed to draw
inferencesaboutoneor morepopulationsfrom which the sampleshavebeen derived. Whereas
a population consists of the sum total of subjects or objects that share something in common
with one another, a sample is a set of subjects or objects which have been derived from a
population. For a sampleto be useful in drawing inferencesabout the larger population from
which it was drawn, itmust berepresentativeofthepopulation. Thus,typically(althoughthere
are exceptions), the ideal sampletoemployin research isa randomsample. A random sample
must adhereto the following criteria: a) Each subject or object in the population has an equal
likelihood ofbeing selected as a member of the sample;b) The selection of each subjectfobject
is independent of the selection of all other subjectslobjects in the population; and c) For a
specified sample size, every possible sample that can be derived from the population has an
equal likelihoodof occurring.
In point of fact, it would be highly unusual to find an experiment that employed a truly
random sample. Pragmaticandlor ethical factorsmake it literallyimpossiblein most instances

2 Handbook of Parametric andNonparametric Statistical Procedures
to obtain random samples for research. Insofar as a sample is not random, it will limit the
degreeto which a researcher will be able to generalizeone's results. Put simply, one can only
generalize to objects or subjects that are similar to the sample employed. (A more detailed
discussion of the general subject of sampling is provided later in this Introduction.)
Statistic versus Parameter
A statistic refers to a characteristicof a sample, such as the average score (also known as the
mean). A parameter, on the other hand, refersto a characteristicof a population (such asthe
averageofa wholepopulation). A statisticcan be employedfor either descriptiveor inferential
purposes. An example of using a statisticfor descriptivepurposes is obtaining the mean of a
group(which representsa sample)in orderto summarizethe averageperformanceofthegroup.
On the other hand, ifweusethemean ofa groupto estimatethemean ofa larger populationthe
group is supposed to represent, the statistic (i.e., the group mean) is being employed for
inferential purposes. The most basic statistics that are employed for both descriptive and
inferential purposesare measures of central tendency (ofwhich the mean is an example)and
measures of variability.
In inferential statisticsthe computedvalue ofa statistic(e.g., a samplemean) is employed
to make inferences about a parameter in the population from which the sample was derived
(e.g., the population mean). The inferential statisticalprocedures described in this book all
employdata derivedfrom oneor more samplesin orderto drawinferencesor makepredictions
with respect to the larger population(s) from which the sample(s) waslwere drawn.
Sampling error is the discrepancybetween the value of a statistic and the parameter it
estimates. Due to sampling error, the value of a statistic will usually not be identical to the
parameter it is employedto estimate. The largerthe sample size employedin a study, the less
the influence of sampling error, and consequently the closer one can expect the value of a
statisticto be to the actual value of a parameter.
When data from a sampleare employedto estimate a population parameter, any statistic
derived from the sample shouldbe unbiased. Although samplingerrorwill be associatedwith
an unbiased statistic, an unbiasedstatisticprovidesthemost accurateestimateofa population
parameter. A biased statistic, on the other hand, does not provide as accurate an estimate of
that parameter as an unbiased statistic, and consequently a biased statisticwill be associated
with a largerdegreeofsamplingerror. Statedin amore formalway, an unbiased statistic (also
referredto as an unbiased estimator) is onewhoseexpected value isequalto the parameter it
is employed to estimate. The expected value of a statistic is based on the premise that an
infinitenumber of samplesof equal sizeare derivedfrom the relevantpopulation, and for each
samplethe value of the statistic is computed. The average of all the values computed for the
statistic will represent the expected value of that statistic. The latter distribution of average
values for the statistic is more formally referred to as a sampling distribution (which is a
concept discussed in greater depth later in thebook). The subjectofbias in statisticswill be dis-
cussedlaterin referencetothe mean (whichisthemostcommonlyemployedmeasureofcentral
tendency), and the variance (which is the most commonlyemployedmeasure of variability).
Levels of Measurement
Typically, information which is quantified in research for purposes of analysis is categorized
with respect to the level of measurement the data represent. Different levels of measurement
contain different amounts of information with respect to whatever the data are measuring. A

Introduction 3
data classification system developedby Stevens(1946), which is commonly employed within
the frameworkof many scientificdisciplines, will be presented in this section.
Statisticians generally conceptualize data as fitting within one of the following four
measurement categories: nominal data (alsoknown as categorical data), ordinal data (also
know as rank-order data), interval data, and ratiodata. As onemoves from the lowest level
of measurement, nominal data, to the highest level, ratio data, the amount of information
provided by the numbersincreases,aswell themeaningfulmathematicaloperationsthat can be
performedon thosenumbers. Each ofthe levels ofmeasurementwill now be discussedin more
detail.
a) NominaUcategorical level measurement In nominal/categorical measurement
numbers are employed merely to identify mutually exclusive categories, but cannot be
manipulatedin amathematicallymeaningfulmanner. Asan example,aperson's socialsecurity
number represents nominal measurement since it is used purely for purposes of identification
and cannotbemeaningfullymanipulatedin a mathematicalsense(i.e., adding, subtracting,etc.
the social securitynumbers of people does not yield anything of tangible value).
b) OrdinaUrank-order level measurement In an ordinal scale, the numbers represent
rank-orders, and donot giveanyinformation regardingthedifferencesbetween adjacentranks.
Thus, the order of finish in a horse race represents an ordinal scale. If in a race HorseA beats
HorseB in a photo finish,and HorseB beats HorseC by twenty lengths, the respectiveorder of
finishofthe three horses reveals nothing about the fact that the distance between the first and
second place horses was minimal, while the differencebetween second and third place horses
was substantial.
c) Interval level measurement An interval scalenot onlyconsiderstherelativeorderof
the measuresinvolved(as isthe casewith an ordinal scale)but, in addition, is characterized by
the fact that throughout the length of the scale equal differences between measurementscor-
respond toequal differencesin the amountoftheattributebeingmeasured. Whatthis translates
intoisthat ifIQisconceptualizedasan intervalscale,the onepoint differencebetween aperson
who has an IQ of 100and someonewho has an IQof 101shouldbe equivalentto the onepoint
differencebetween a person who has an IQof 140and someonewith an IQof 141. In actuality
somepsychologistsmight arguethis point, suggestingthat a greater increase in intelligenceis
required tojump from an IQ of 140to 141than tojurnp from an IQof 100to 101. In fact, ifthe
latter istrue,a onepoint differencedoesnot reflectthe samemagnitudeofdifferenceacrossthe
full range of the IQ scale. Although in practice IQ and most other human characteristics
measured by psychological tests (such as anxiety, introversion, self esteem, etc.) are treated as
interval scales, many researchers would arguethat they are more appropriatelycategorizedas
ordinal scales. Such an argument would be based on the fact that such measures do not really
meet the requirements of an interval scale, because it cannot be demonstrated that equal
numerical differences at different points of the scale are comparable.
It shouldalsobenotedthat unlikeratio scales,which will bediscussednext,intervalscales
do not have a true zero point. If interval scales have a zero score that can be assigned to a
person or object, it is assumed to be arbitrary. Thus, in the case of IQ we can ask the question
of whether or not there is truly an IQ which is so low that it literally represents zero IQ. In
reality, you probably can only saya person who is deadhas a zeroIQ! In point of fact, someone
who has obtained an IQ of zero on an IQ test has been assigned that score because his
performanceonthetest was extremelypoor. ThezeroIQdesignationdoesnot necessarilymean
the person could not answer any of the test questions (or, to go further, that the individual
possessesnone ofthe requisite skillsor knowledge for intelligence). Thedevelopersofthetest
just decidedto select a certain minimum scoreon the test and designate it asthe zeroIQpoint.

4 Handbook of Parametric and Nonparametric Statistical Procedures
d) Ratio level measurement As is the case with interval level measurement,ratio level
measurement is alsocharacterizedby the fact that throughout the length ofthe scale,equal dif-
ferencesbetween measurements correspondto equal differencesin the amount ofthe attribute
beingmeasured. However, ratio level measurementis alsocharacterizedby the fact that it has
a true zeropoint. Becauseofthelatter, with ratiomeasurementoneis ableto make meaningful
ratio statementswith regardtothe attributelvariablebeing measured. Toillustratethesepoints,
most physical measures such as weight, height, blood glucose level, as well as measures of
certain behaviors such as the number of times a person coughs or the number of times a child
cries, represent ratio scales. For all of the aforementionedmeasures there is a true zero point
(i.e., zeroweight, zeroheight,zeroblood glucose,zerocoughs, zeroepisodesofcrying),and for
each of these measures one is able to make meaningful ratio statements (such as Ann weighs
twice as much as Joan, Bill is one-half the height of Steve, Phil's blood glucose is 100times
Sam's, Marycoughsfivetimesasoften asPete, and Billycriesthreetimes asmuch asHeather).
Continuousversus Discrete Variables
When measures are obtained on people or objects, in most instances we assumethat there will
be variability. Since we assume variability, if we are quantifying whatever it is that is being
measured not everyone or everything will produce the same score. For this reason, when
somethingismeasured it is commonlyreferredto as a variable. As noted above, variablescan
be categorizedwith respect tothelevelofmeasurementtheyrepresent. In contrasttoa variable,
a constant is a number which never exhibits variation. Examples of constants are the
mathematical constantspi and e (whicharerespectively3.14159...and 2.71828...),thenumber
ofdaysin a week (which will alwaysbe 7),thenumber ofdaysin themonthofApril (whichwill
always be 30), etc.
A variable can be categorized with respect to whether it is continuous or discrete. A
continuous variable can assumeany value within the range of scoresthat definethe limits of
that variable. A discrete variable, on the other hand, can only assume a limited number of
values. Toillustrate,temperature(whichcan assumeboth integerand fractional/decimalvalues
within a given range) is a continuous variable. Theoretically,there are an infinitenumber of
possibletemperature values, and the number of temperature values we can measure is limited
onlybythe precision oftheinstrument we are employingto obtain the measurements. The face
value of a die, on the other hand, is a discrete variable, since it can only assumethe integer
values 1through 6.
Measures of Central Tendency
Earlier in the Introduction it was noted that the most commonly employed statistics are
measures of central tendency and measures of variability. This section will describe five
measures of centraltendency: the mode, the median, the mean, the geometric mean, and the
harmonic mean.
The mode The mode is the most frequentlyoccurring score in a distribution of scores.
A mode that is derived for a sample is a statistic, whereas the mode of a population is a
parameter. In the following distribution of scores the mode is 5, since it occurs two times,
whereasall other scoresoccuronlyonce: 0, 1,2,5,5,8, 10. Ifmore than onescoreoccurswith
the highest frequency, it is possible to have two or more modes in a distribution. Thus, in the
distribution 0, 1,2,5,6,8, 10,all ofthe scoresrepresent the mode, since each scoreoccurs one
time. A distributionwith more than onemode is referred to as a multimodal distribution (as
opposed to a unimodal distribution which has one mode). If it happens that two scores both

Introduction 5
occurwiththe highest frequency,the distributionwould be describedasa bimodal distribution,
which represents one type of multimodal distribution. The distribution 0, 5, 5, 8, 9, 9, 12 is
bimodal, since the scores 5 and 9 both occur two times and all other scores appear once.
The most common situation in which the mode is employed as a descriptivemeasure is
within the context of a frequency distribution. A frequency distribution is a table which
summarizesa set of data in a tabular format,listing the frequencyofeach scoreadjacentto that
score. Table1.1isa frequencydistributionfor Distribution Anoted below, which iscomprised
ofn = 20 scores. (Amore detaileddiscussion ofDistribution A can be found in the discussion
ofvisual methods for displaying data.) It shouldbe noted that Column 1ofTable1.1(i.e., the
column attheleft with thenotationxat thetop)onlylists those scoresin Distribution A which
fall within the range of values 22-96 that have a frequency of occurrence greater than zero.
Although all ofthe scoreswithin the range of values 22 -96 couldhavebeen listedin Column
1(i.e., including all ofthe scoreswith a frequencyof zero), the latter would increasethe size of
the table substantially,and in the process make it more difficultto interpret. Consequently, it
is more efficient tojust list those scoreswhich occur at least onetime, since it is more effective
in providing a succinct summaryofthe data-the latter being a major reason why a frequency
distribution is employed.
Distribution A: 22, 55. 60, 61, 61, 62, 62, 63, 63, 67, 71, 71, 72, 72. 72, 74. 74, 76, 82, 96
Table 1.1 Frequency Distribution
of Distribution A
In addition to presenting data in a tabular format, a researcher can also summarizedata
within the format of graph. Indeed, it is recommended that researchers obtain a plot of their
data prior to conductinganysortofformal statisticalanalysis. Thereason for this is that a body
of data can have certain characteristics which may be important in determining the most
appropriatemethod ofanalysis. Often such characteristicswill not be apparent to a researcher
purely on the basis of cursory visual inspection -especially if there is a large amount of data
andlor the researcher is relativelyinexperiencedin dealing with data. A commonlyemployed
method for visually presenting data is to construct a frequency polygon, which is a graph of a
frequencydistribution. Figure I.1 is a frequencypolygon of Distribution A.
Note that a frequencypolygon is comprised of two axes, a horizontal axis and a vertical
axis. TheX-axis or horizontal axis (which isreferredto asthe abscissa) is employedto record

Figure L l FrequencyPolygon of Distribution A
therange ofpossible scores on a variable. (Theelement-4I- on the left sideof theX-axis of
Figure 1.1 is employed when a researcher only wants to begin recording scores on theX-axis
which fallatsomepoint above0,andnot listanyscoresin between0andthat point.) The Y-axis
or vertical axis (which isreferred toasthe ordinate)isemployedtorepresent the frequency(f)
with which eachofthescoresnoted on theX-axis occursin the sampleor population. In order
toprovide somedegreeof standardizationin graphingdata,manysourcesrecommendthat the
length of the Y-axis be approximatelythree-quarters the length of theX-axis.
InspectionofFigure1.1revealsthat a frequencypolygon isa seriesof lineswhich connect
a set of points. One point is employedfar eachofthe scoresthat comprisetherange of scores
in the distribution. Thepoint which representsanyscorein the distributionthat occursoneor
moretimeswill falldirectlyabovethat scoreat aheight correspondingtothe frequencyforthat
scorerecorded onthe Y-axis.When the frequencypolygondescendstoand/or moves alongthe
X-axis, it indicatesa frequency of zero for those scoreson theX-axis. The highest point on a
frequencypolygon will always fall directly abovethe scorewhich correspondsto the mode of
the distribution (which in the case of Distribution A is 72). (hi the caseof a multimodal
distributionthe frequencypolygon will havemultiplehigh points.) A more detaileddiscussion
ofthe use oftables and graphs for descriptivepurposes, aswell asa discussionofexploratory
dataanalysis(which isanalternativemethodologyforscrutinizingdata),willbepresentedlater
in the Introductionin the section on the visual display of data.
The median Themedian isthemiddlescorein adistribution. Ifthereisan odd number
of scores in a distribution,in order to determinethe median the following protocol shouldbe
employed: Dividethe total number of scoresby 2 and add .5 to the result of the division. The
obtained value indicatesthe ordinal position of the scorewhich represents the median of the
distribution(note, however, that this value doesnot represent the median). Thus, if we have a
distributionconsistingoffivescores(e.g., 6,8,9, 13, 16),we dividethenumber ofscoresin the
distributionbytwo, and add .5 to theresult ofthedivision. Thus, (5/2) +.5=3. The obtained
value of 3 indicatesthat if the fivescoresare arranged ordinally(i.e., from lowest to highest),

Introduction 7
the median is the 3rd highest (or 3rd lowest) score in the distribution. With respect to the
distribution 6,8,9, 13, 16,the value ofthe median will equal 9, since9 isthe scorein the third
ordinal position.
Ifthere isan even number ofscoresin a distribution,therewill betwomiddlescores. The
median is the averageofthetwo middle scores. To determinethe ordinal positionsofthe two
middle scores, divide the total number of scores in the distribution by 2. The number value
obtained by that division and the number value that is one above it represent the ordinal
positions of the two middle scores. To illustrate,assumewe have a distribution consisting of
the following six scores: 6, 8, 9, 12, 13, 16. To determinethe median, we initiallydivide6 by
2 which equals 3. Thus, if we arrangethe scoresordinally,the 3rd and 4th scores (since 3 + 1
=4)arethemiddlescores. Theaverageofthese scores,which are, respectively, 9and 12,isthe
median (which will be representedby the notation M). Thus, M =(9+ 12)12= 10.5. Note once
again that in this example, as was the case in the previous one, the initial values computed (3
and 4) do not themselves represent the median, but instead represent the ordinal positions of
the scoresused to computethe median. As wasthe casewith the mode, a median value derived
for a sampleis a statistic, whereas the median of a whole population is a parameter.
The mean The mean (also referred to as the arithmetic mean), which is the most
commonly employed measure of central tendency, is the average score in a distribution.
Typically, when the mean isused asa measure ofcentraltendency, it is employedwith interval
or ratio level data. Within the framework of the discussion to follow, the notation n will
represent the number of subjects or objects in a sample, and the notation N will represent the
total number of subjects or objects in the population from which the sampleis derived.
Equation I.I is employed to compute the mean of a sample. E, which is the upper case
Greek letter sigma, is a summation sign. Thenotation EYindicatesthat the set of n scoresin
the sampleldistributionshould be summed.
- Exx = - (Equation 1.1)
n
Sometimes Equation 1.1 is written in the following more complex but equivalent form
containingsubscripts: X = Xi In. In the latter equation,thenotation T=,x,indicatesthat
beginning with the first score, scores 1 through n (i.e., all the scores) are to be summed. XI
represents the scoreof the i th subject or object.
Equation 1.1 will now be applied to the following distribution of five scores: 6, 8, 9, 13,
16. Sincen= 5 and EY" = 52,X = m n = 5215 = 10.4.
WhereasEquation 1.1 describeshow one can computethemean ofa sample, Equation 1.2
describeshow onecan computethe mean ofa population. The simplifiedversion without sub-
scriptsistotheright ofthefirst= sign, andthe subscriptedversion oftheequation istotheright
ofthe second = sign. The mean of a population is representedby the notation p, which is the
lower caseGreeklettermu. In practice, it would behighlyunusualto have occasionto compute
the mean of a population. Indeed, a great deal of analysis in inferential statistics is concerned
with trying to estimate the mean of a population from the mean of a sample.
(Equation 1.2)
Where: n = The number of scores in the distribution
Xi = The 1"' score in a distribution comprised of n scores

Note that in the numerator of Equation 1.2all N scoresin the population are summed, as
opposed to just summing n scores when the value of 2 is computed. The sample meany
provides an unbiased estimate of the population mean p, which indicates that if one has a
distributionofnscores,i providesthebest possibleestimateofthetrue value of p. Later in the
book (specifically,underthe discussionofthesingle-samplez test (Test1))itwill benoted that
the mean of the sampling distribution of means (which represents the expected value of the
statisticrepresentedby the mean) will equal to the value ofthe populationmean. (A sampling
distribution of means is a frequency distribution of sample means derived from the same
population, in which the samenumber of scores is employedfor each sample.) Recollectthat
earlierin the Introductionit was noted that an unbiasedstatisticis onewhose expectedvalue
is equal to the parameter it is employedto estimate. This applies to the samplemean, since its
expected value is equal to the population mean.
The geometric mean The geometric mean is a measure of central tendency which is
primarilyemployed within the context of certain types of analysis in business and economics.
It is most commonly used as an average of index numbers, ratios, and percent changes over
time. (An index number is a metricofthe degreeto which a variable changes over time. It is
calculatedby determiningthe ratio ofthe current valueofthevariableto a previous value. The
most commonlyemployed indexnumber is a price index,which is used to contrastprices from
oneperiod oftimeto another.) Thegeometricmean (GM)ofa distribution isthe n* root ofthe
product ofthe n scoresin the distribution. Equation 1.3 is employedto computethe geometric
mean of a distribution.
GM = "/x= (Equation 1.3)
To illustratethe aboveequation,the geometricmean ofthe five values 2,5, 15,20and 30
is GM = ,/(2)(5)(15)(20)(30) = 6.18.
Only positivenumbers should be employed in computingthe geometricmean, since one
or more zero values will render GM = 0, and negative numbers will render the equation
insoluble(when there are an oddnumberofnegativevalues) ormeaningless(when thereare an
even number ofnegativevalues). Specifically,let us assumewe wish to computethe geometric
mean for the four values -2, -2, -2, -2. Employing the equation noted above,
4
GM = ,/(-2)(-2)(-2)(-2) = 2. Obviouslylattervaluedoesnotmakesense,sincelogicallythe
geometricmean shouldhave a minussign -specifically, GMshouldbe equal to-2. When all
ofthe values in a distribution are equivalent,the geometricmean and arithmetic meanwill be
equal to one another. In all other instances, the value of the geometricmean will be less than
thevalueofthearithmeticmean. Notethat in theaboveexamplein which GM= 6.18,the value
computed for the arithmeticmean is i = 14.4, which is larger than the geometricmean.
Before the introduction of hand calculators, a computationally simpler method for
computing the geometric mean utilized logarithms (which are discussed in Endnote 13).
Specifically, the following equation can also be employed to compute the geometric mean:
log (GM) = (ElogX)/n. The latter equation indicates that the logarithm of the geometric
mean is equivalent to the arithmeticmean of the logarithm of the values of the scores in the
distribution. The antilogarithm of log (GM) will represent the geometricmean.
Chou (1989, pp. 107-1 10))notes that when a distribution ofnumbers takes the form of a
geometric series or a logarithmically distributed series (which is positively skewed), the
geometricmean is a more suitablemeasureofcentral tendencythan the arithmeticmean. (The
concept of skewness is discussedin detail later in the Introduction.) A geometric series is a
sequenceofnumbersin which the ratio ofanyterm to the precedingterm isthe samevalue. As

Introduction 9
an example, the series 2, 4, 8, 16, 32, 64, .... represents a geometric series in which each
subsequent term is twice the value of the preceding term. A logarithmic series (also referred
to as a power series) is onein which successiveterms aretheresult ofa constant(a)multiplied
by successive integer powers of a variable (to be designated as x) (e.g., a x , ax2, ax?, ..., ax").
Thus, if x =3,the seriesa3, ~ 3 ~ ,a34,....,a3"representsan exampleofa logarithmicseries.
A major consequence of employing the geometricmean in lieu of the arithmeticmean is
that the presence of extreme values (which is often the casewith a geometricseries) will have
less of an impact on the value of the geometric mean. Although it would not generally be
employedas a measure of centraltendency for a symmetrical distribution,the geometricmean
can reduce the impact of skewness when it is employed as a measure of centraltendency for a
nonsyrnmetrical distribution.
The harmonic mean Another measure of centraltendency is the harmonic mean. The
harmonic mean is determined by computing the reciprocal of each of the n scores in a
distribution. (The reciprocal of a number is the value computed when the number is divided
into 1-i.e., the reciprocal ofX= IIX.) Themeanlaveragevalue of the n reciprocals is then
computed. Thereciprocal ofthe latter mean representsthe harmonicmean (representedby the
notation K )which is computed with Equation 1.4.
- n
Xh = - (Equation 1.4)
Where: n = The number of scores in the distribution
X = The I* scorein a distribution comprised of n scores
To illustratethe aboveequation, theharmonicmean ofthe fivevalues 2,5, 15,20 and 30
is computed below.
Chou (1989,p. 111)notesthat for any distributionin which there is variabilityamongthe
scoresand in which no score is equal to zero, the harmonicmean will always be smaller than
both themean (which in the caseofthe abovedistribution is 2= 14.4)and the geometricmean
(whichisGM=6.18). This isthe case, sincethe harmonicmean is least influenced by extreme
scores in a distribution. Chou (1989) provides a good discussion of the circumstanceswhen it
is prudent to employ the harmonic mean as a measure of central tendency. He notesthat the
harmonicmean isrecommended when scoresare expressed inverselyto what isrequired in the
desired measure of central tendency. Examples of such circumstancesare certain conditions
where a measure of central tendencyis desired for time ratesandlor prices. Further discussion
of the harmonic mean can be found in Section VI of the t test for two independent samples
(Test 11)and in Chou (1989, pp. 110-113).
Measures of Variability
In this section a number of measures of variability will be discussed. Primary emphasis,
however, will be given to the standard deviation and the variance, which are the most
commonly employedmeasures of variability.

a) The range The range is the difference between the highest and lowest scores in a
distribution. Thusin thedistribution2,3,5,6,7,12, therangeisthedifferencebetween 12(the
highest score)and 2 (the lowest score). Thus: Range = 12-2 = 10. Some sources add oneto
the obtainedvalue, and would thus say that the Range = 11. Although the range is employed
on occasion for descriptivepurposes, it is of little use in inferential statistics.
b) Quantiles, percentiles, deciles, and quartiles A quantile is a measure that divides
a distribution into equidistant percentage points. Examples of quantiles are percentiles,
quartiles, and deciles. Percentiles divide a distribution into blocks comprised of one
percentage point (or blocks that comprise a proportion equal to .O1 of the distribution).' A
specific percentile value correspondsto the point in a distribution at which a given percentage
of scoresfalls at or below. Thus, if an IQ test score of 115 falls at the 84"' percentile, it means
84% ofthe populationhas an IQ of 115or less. Theterm percentile rank is also employed to
mean the samething as a percentile-in other words, we cansay that an IQ scoreof 115has
a percentile rank of 84%.
Decilesdividea distributionintoblockscomprisedoften percentagepoints(or blocks that
comprise a proportion equal to .10 ofthe distribution). A distribution canbe divided into ten
deciles, the upper limits of which are defined by the 10"' percentile, 20'" percentile, ...,90"'
percentile, and 100'"percentile. Thus, a scorethat correspondstothe 10"'percentile fallsat the
upper limit ofthe first decileofthe distribution.A scorethat correspondsto the 20thpercentile
fallsat theupper limit oftheseconddecileofthedistribution,and soon. Theinterdecile range
isthedifferencebetweenthe scoresat the 90"' percentile(theupper limit ofthe ninth decile)and
the 10'"percentile.
Quartiles divide a distribution into blocks comprised of 25 percentage points (or blocks
that comprisea proportion equal to .25 of the distribution). A distribution canbe divided into
fourquartiles, theupper limitsofwhicharedefinedbythe25"' percentile,50"' percentile(which
correspondsto the median of the distribution), 75thpercentile, and 100'" percentile. Thus, a
score that corresponds to the 25'" percentile falls at the upper limit of the first quartile of
the distribution. A scorethat correspondsto the 50"' percentile falls at the upper limit of the
secondquartileofthedistribution,andsoon. Theinterquartile range isthedifferencebetween
the scores at the 75thpercentile (which is the upper limit of the third quartile) and the 25"'
percentile.
Infrequently, the interdecile or interquartile ranges may be employed to represent
variability. An exampleofa situation where a researchermight elect to employeitherofthese
measures to represent variability would be when the researcher wishes to omit a few extreme
scoresin a distribution. Suchextremescoresarereferredtoasoutliers. Specifically,an outlier
is a scorein a set ofdata which is soextremethat, by all appearances,it isnot representativeof
the population fromwhich the sampleis ostensiblyderived. Sincethe presenceof outliers can
dramaticallyaffectvariability(aswell asthe valueofthesamplemean),their presencemay lead
a researcher to believe that the variabilityof a distributionmight best be expressedthrough use
of the interdecile or interquartilerange (as well asthe fact that when outliers are present, the
samplemedianismore likelythan themean tobearepresentativemeasureofcentraltendency).
Further discussion of outliers can be found in the latter part of the Introduction, as well as in
SectionVI ofthe t test for two independent samples.
c) The variance and the standard deviation The most commonlyemployed measures
of variability in both inferential and descriptivestatistics are the variance and the standard
deviation. Thesetwomeasuresaredirectlyrelatedto oneanother, sincethe standard deviation
is the squareroot ofthe variance(and thusthe varianceisthe squareofthe standard deviation).
As is the case with the mean, the standard deviation and the variance are generally only
employedwith interval or ratio level data.

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