Quantitative and statistical research methods form hypothesis to results by martin brigmon

Table 1.1 Values Used to Illustrate Measures of Central Tendency and VariabilityTable 1.2 Frequency Distribution of Scores of Depressive SymptomsTable 1.3 Scores and Difference Measures

AND STATISTICAL RESEARCH METHODS From Hypothesis to Results

WILLIAM E MARTIN

KRISTA D BRIDGMON

One Montgomery Street, Suite 1200, San Francisco, CA 94104-4594 —www.josseybass.com

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Library of Congress Cataloging-in-Publication Data

Martin, William E (William Eugene),

1948-Quantitative and statistical research methods : from hypothesis to results / William E Martin, Krista D Bridgmon — First edition.

pages cm — (Research methods for the social sciences; 42)

Includes bibliographical references and index.

ISBN 978-0-470-63182-9 (pbk.); ISBN 978-1-118-22075-7 (ebk.); ISBN 978-1-118-23457-0 (ebk.); ISBN 978-1-118-25908-5 (ebk.)

1 Psychology —Methodology 2 Social sciences—Methodology 3 SPSS (Computer file)

I Bridgmon, Krista D., 1979- II Title.

BF38.5.M349 2012

150.72'7 —dc23

2012010748 Printed in the United States of America

FIRST EDITION

PB Printing 10 9 8 7 6 5 4 3 2 1

Tables and Figures ix

Review of Foundational Research Concepts 3Review of Foundational Statistical Information 6

Chapter 2 Logical Steps of Conducting Quantitative Research:

Chapter 3 Maximizing Hypothesis Decisions Using Power Analysis 39

Balance between Avoiding Type I and Type II Errors 41Chapter 4 Research and Statistical Designs 53

Formulating Experimental Conditions 54Reducing the Imprecision in Measurement 55Controlling Extraneous Experimental Influences 57Internal Validity and Experimental Designs 59Choosing a Statistic to Use for an Analysis 67

The IBM SPSS 20 Data View Screen 80Naming and Defining Variables in Variable View 80

Examples of Modifying Data Procedures 96Chapter 6 Diagnosing Study Data for Inaccuracies and

Chapter 7 Randomized Design Comparing Two Treatments

and a Control Using a One-Way Analysis of Variance 129

Hypothesis Testing Step 5: Select Sample, CollectData, Screen Data, Compute Statistic, andDetermine Probability Estimates 144Hypothesis Testing Step 6: Make Decision Regarding

the H0 and Interpret Post Hoc Effect Sizes and

Hypothesis Testing Step 1: Establish the Alternative

Statistic and Its Sampling Distribution to Test

Hypothesis Testing Step 5: Select Sample, Collect Data,Screen Data, Compute Statistic, and Determine

Probability Estimates 248Hypothesis Testing Step 6: Make Decision Regarding

the H0 and Interpret Post Hoc Effect Sizes and

Formula Calculations of the Study Results 278

Hypothesis Testing Step 5: Select Sample, CollectData, Screen Data, Compute Statistic, andDetermine Probability Estimates 307Hypothesis Testing Step 6: Make Decision Regarding

the H0 and Interpret Post Hoc Effect Sizes and

Formula ANCOVA Calculations of the Study Results 327

Chapter 11 Randomized Control Group and Repeated-Treatment

Question 349Hypothesis Testing Step 1: Establish the Alternative

Statistic and Its Sampling Distribution toTest the H0 Assuming H0Is True 354Hypothesis Testing Step 5: Select Sample, Collect

Data, Screen Data, Compute Statistic, andDetermine Probability Estimates 355Hypothesis Testing Step 6: Make Decision

Regarding the H0 and Interpret Post Hoc Effect Sizes 370

Nonparametric Research Problem Two: Friedman’sRank Test for Correlated Samples and Wilcoxon’sMatched-Pairs Signed-Ranks Test 382Chapter 12 Bivariate and Multivariate Correlation Methods

Using Multiple Regression Analysis 401

Assuming H0 Is True 407Hypothesis Testing Step 5: Select Sample, Collect Data,Screen Data, Compute Statistic, and Determine

Table 1.1 Values Used to Illustrate Measures of Central Tendency and

VariabilityTable 1.2 Frequency Distribution of Scores of Depressive SymptomsTable 1.3 Scores and Difference Measures for Dependent t AnalysisTable 3.1 Decision Balance between Type I and Type II Errors

Table 3.2 Cohen’s Strength of d Effect Sizes

Table 3.3 Cohen’s Strength of η2

Effect SizesTable 3.4 Cohen’s Strength of r Effect Sizes

Table 4.1 Threats to Internal Validity: THIS MESS DREAD

Table 5.1 Frequencies Table of Ethnicity

Table 5.2 Descriptive Statistics of Status

Table 5.3 An Independent t-Test Analysis

Table 5.4 Correlation Matrix of Age, COSE Confidence in Executing

Microskills, and COSE Dealing with Difficult Client BehaviorsTable 6.1 Mean and Standard Deviation of MAAS Scores

Table 6.2 Frequencies of MAAS Scores

Table 6.3 Missing Case for TotalMAAS

Table 6.4 Skewness and Kurtosis Values with Standard Errors of the

Dependent Variable for Both ConditionsTable 6.5 Shapiro-Wilk Statistic Results to Assess Normality

Table 6.6 Results of Levene’s Test of Homogeneity of Variance

Table 6.7 One-Way ANOVA Results before Log10 Data TransformationTable 6.8 Skewness and Kurtosis Values after Log10 Transformation

TransformationTable 6.11 One-Way ANOVA Results for the Log10 Transformed DataTable 6.12 Data Diagnostics Study Example

Table 7.1 Descriptive Statistics of Depressive Symptoms by Condition

GroupTable 7.2 Highest 6z-Scores by Condition Group

Table 7.3 Skewness, Kurtosis, and Standard Error Values by GroupTable 7.4 Skewness z-Scores by Condition Group

Table 7.5 Kurtosis z-Scores by Condition Group

Table 7.6 Shapiro-Wilk Statistics by Condition Group

Table 7.7 Levene’s Test of Homogeneity of Variance

Table 7.8 One-Way Analysis Results

Table 7.9 HSD Post Hoc Analysis

Table 7.10 ANOVA Summary Table Specifications

Table 7.11 ANOVA Summary Table

Table 7.12 Matrix of Mean Differences

Table 7.13 One-Way Analysis of Variance Data

Table 8.1 Descriptive Statistics of Weight Loss by Condition GroupTable 8.2 Highest 6z-Scores by Condition Group

Table 8.3 Skewness, Kurtosis, and Standard Error Values by

Condition GroupTable 8.4 Skewness z-Scores by Condition Group

Table 8.5 Kurtosis z-Scores by Condition Group

Table 8.6 Shapiro-Wilk Statistics by Condition Group

Table 8.7 Mauchly’s Test of Sphericity

Table 8.8 RM-ANOVA Results for the Omnibus Null Hypothesis

Table 8.9 Post Hoc Comparisons Using the Fisher’s Protected Least

Significant Differences (PLSD) StatisticTable 8.10 Trends of Weight Loss Means Across the Condition GroupsTable 8.11 RM-ANOVA Summary Table Specifications MST/MSE

Table 8.12 Study Data with Column and Row Means by Subject and ConditionTable 8.13 RM-ANOVA Summary Table Specifications

Table 8.14 Repeated-Measures Analysis of Variance Data

Table 9.1 23 2 Factorial Design Matrix

Table 9.3 Highest 6z-Scores by Group

Table 9.4 Skewness, Kurtosis, and Standard Error Values by GroupTable 9.5 Skewness z-Scores by Condition Group

Table 9.6 Kurtosis z-Scores by Condition Group

Table 9.7 Shapiro-Wilk Statistics by Condition Group

Table 9.8 Levene’s Test Comparing Variances of the Treatment Condition

Groups (SC vs SC 1 CM)Table 9.9 Levene’s Test Comparing Variances of the Treatment Status

Groups (0–1 vs $2)Table 9.10 Descriptive Statistics by Conditions

Table 9.11 Levene’s Test of Equality of Error Variancesa

Table 9.12 Two-Way Analysis of Variance Results

Table 9.13 Treatment Condition at Each Treatment Status Level ResultsTable 9.14 Treatment Status at Each Level of Treatment Condition ResultsTable 9.15 Decisions and Conclusions Regarding Null Hypotheses of Main

Effects and Interaction EffectTable 9.16 Decisions and Conclusions Regarding Null Hypotheses of Simple

EffectsTable 9.17 CI.99for Mean Difference of Treatment Retention by Treatment

ConditionTable 9.18 CI.99for Mean Difference of Treatment Retention by Treatment

StatusTable 9.19 Two-Way ANOVA Summary Table Specifications

Table 9.20 Study Data with Column and Row Means by Subject and ConditionTable 9.21 Two-Way ANOVA Summary Table

Table 9.22 Simple Effects Summary Table

Table 9.23 Two-Way Analysis of Variance Data

Table 10.1 Highest 6z-Scores for the Covariate Age and the Dependent

Variable Longest Duration of AbstinenceTable 10.2 Skewness, Kurtosis, and Standard Error Values by GroupTable 10.3 Skewness z-Scores by Treatment Condition Group

Table 10.4 Kurtosis z-Scores by Substance Treatment Group

Table 10.5 Shapiro-Wilk Statistics by Substance Treatment ConditionTable 10.6 Levene’s Test of Homogeneity of Variance for CovAge and DVLDA

and IVTreatmentConditionTable 10.9 ANCOVA Results

Table 10.10 Estimated Marginal Means

Table 10.11 Confidence Interval (.99) for the Mean Difference between SC

and SC 1 CMTable 10.12 Data and Summary Statistics for LDALDA DV (Y )

Table 10.13 Data and Summary Statistics for Age

Table 10.14 Summary of Previous Calculations

Table 10.15 Summary of ANCOVA Results

Table 10.16 Analysis of Covariance Data

Table 11.1 K-W MWU Data

Table 11.2 Descriptive Statistics of Pain Improvement by Electric Simulation

ConditionTable 11.3 Three Highest 6z-Scores of Pain Improvement by Electric

Stimulation ConditionTable 11.4 Skewness, Kurtosis, and Standard Error Values by GroupTable 11.5 Skewness z-Scores by Condition Group

Table 11.6 Kurtosis z-Scores by Condition on Pain Improvement ScoresTable 11.7 Shapiro-Wilk Statistics by Conditions

Table 11.8 Levene’s Test of Homogeneity of Variance

Table 11.9 Mean Ranks of Pain Improvement by Conditions

Table 11.10 K-W Results

Table 11.11 Mean Ranks of the Low Electric Stimulation Condition

Compared to the Placebo ConditionTable 11.12 MWU Results Comparing Low Electric Stimulation to PlaceboTable 11.13 Mean Ranks of the Placebo Condition Compared to the High

Electric Stimulation ConditionTable 11.14 MWU Results Comparing Placebo to High Electric StimulationTable 11.15 Formula Kruskal-Wallis and Mann-Whitney U Calculations of

the Study ResultsTable 11.16 Low Electric Stimulation Condition Compared to Placebo

Condition on Pain ImprovementTable 11.17 High Stimulation Condition Compared to Placebo Condition on

Pain Improvement

ConditionsTable 11.20 Mean Ranks of Pain Improvement by High Electric ConditionsTable 11.21 Friedman’s Statistic of Pain Improvement by High Electric

ConditionsTable 11.22 Wilcoxon Results

Table 11.23 Friedman’s Rank Test

Table 11.24 Wilcoxon’s Matched-Pairs Signed-Ranks Test: First Treatment

Scores Compared to Removed Treatment ScoresTable 11.25 Wilcoxon’s Matched-Pairs Signed-Ranks Test: Restored

Treatment Scores Compared to Removed Treatment ScoresTable 12.1 Highest 6z-Scores for DSI, SPIPract, and SPIScient

Table 12.2 Largest (Maximum) Mahalanobis Distance Value

Table 12.3 Bivariate Correlation Coefficients between the Study VariablesTable 12.4 Multicollinearity Measures of Tolerance and Variance Inflation

Factor (VIF)Table 12.5 Model Summary of Sequential MRA

Table 12.6 Analysis of Variance of the Two Sequential MRA ModelsTable 12.7 Significance Values of Each Predictor Variable

Table 12.8 Matrix of Correlation Coefficients, Means, and Standard

DeviationsTable 12.9 Sequential MRA Data

Table 13.1 Comparisons of Effect Sizes of the Mozart Effect

FIGURES

Figure 1.1 Bar Chart of Scores of Depressive Symptoms

Figure 1.2 Histogram of Scores of Depressive Symptoms

Figure 1.3 Normal Curve Superimposed on Histogram of Scores of

Depressive SymptomsFigure 1.4 Q-Q Plot of Scores of Depressive Symptoms

Figure 1.5 The Normal Distribution and Standardized Scores

Figure 3.1 G*Power First Page

Figure 3.2 A Priori Power Analysis for the Example

Figure 4.1 Issues in Choosing a Statistic to Use for an Analysis

Figure 5.3 Variable View Screen of IBM SPSS 20

Figure 5.4 Variables Named and Defined in Variable View

Figure 5.5 Example Data in Data View

Figure 5.6 Bar Chart of Ethnicity

Figure 5.7 Scatter Plot of Age and COSE Dealing with Difficult

Client BehaviorsFigure 5.8 COSE Composite Sum Variable

Figure 5.9 COSE Composite Mean Variable

Figure 6.1 Histogram of Mindfulness Attention Awareness Scores for the

Treatment GroupFigure 6.2 Histogram of Mindfulness Attention Awareness Scores for the

Control GroupFigure 6.3 Q-Q Plot to Assess Normality of Treatment Condition ScoresFigure 6.4 Q-Q Plot to Assess Normality of Control Condition ScoresFigure 6.5 Histograms of the Dependent Variable by Condition Groups

after Log10 Data TransformationFigure 6.6 Normal Q-Q Plots after Log10 Data Transformation

Figure 7.1 A Priori Power Analysis of ANOVA Problem

Figure 7.2 Normal Q-Q Plot of Depressive Symptoms for CBT GroupFigure 7.3 Normal Q-Q Plot of Depressive Symptoms for IPT GroupFigure 7.4 Normal Q-Q Plot of Depressive Symptoms for Control GroupFigure 7.5 Matrix Scatterplot to Assess Independence

Figure 7.6 Hypothesis Testing Graph—One-Way ANOVA

Figure 8.1 Repeated-Treatment Design with One Group for

Study ExampleFigure 8.2 Same Participants Measured Repeatedly over Time

Figure 8.3 Same Participants Measured under Different Conditions

Figure 8.4 Matched Pairs of Participants Measured under Different

ConditionsFigure 8.5 Power Analysis for the RM-ANOVA Problem

Figure 8.6 Histograms of Weight Loss by Weight Loss Intervention

Figure 8.7 Normal Q-Q Plots of Weight Loss by Weight Loss Intervention

Conditions

Figure 9.1 Power Analysis for Treatment Condition of the Factorial

ANOVA ProblemFigure 9.2 Power Analysis for Treatment Status of the Factorial ANOVA

ProblemFigure 9.3 Power Analysis for Treatment Condition 3 Treatment Status

Interaction of the Factorial ANOVA ProblemFigure 9.4 Normal Q-Q Plot by Condition Groups

Figure 9.5 Matrix Scatter Plot to Assess Independence on Treatment

Retention Across the Condition GroupsFigure 9.6 Estimated Marginal Means of Treatment Retention

Figure 9.7 Hypothesis Testing Graph Factorial ANOVA

Figure 10.1 G*Power Screen Shots for ANCOVA Problem

Figure 10.2 Normal Q-Q Plots of CovAge by Groups

Figure 10.3 Normal Q-Q Plots of DVLDA by Groups

Figure 10.4 Matrix Scatter Plot to Assess Independence

Figure 10.5 Profile Plot

Figure 10.6 Hypothesis Testing Graph ANCOVA

Figure 11.1 Randomized Pretest-Posttest Control Group Design

Figure 11.2 A Priori Power Analysis Results for Low Electric Stimulation

Versus PlaceboFigure 11.3 A Priori Power Analysis Results for High Electric Stimulation

Versus PlaceboFigure 11.4 Histogram of the Low Electric Stimulation Condition on Pain

ImprovementFigure 11.5 Histogram of the Placebo Condition on Pain ImprovementFigure 11.6 Histogram of the High Electric Stimulation Condition on Pain

ImprovementFigure 11.7 Normal Q-Q Plot of Pain Improvement Scores for the Low

Electric Stimulation ConditionFigure 11.8 Normal Q-Q Plot of Pain Improvement Scores for the Placebo

ConditionFigure 11.9 Normal Q-Q Plot of Pain Improvement Scores for the High

Electric Stimulation Condition

Figure 11.11 Post Hoc Effect Size and Power Analysis for High Electric

Stimulation versus PlaceboFigure 11.12 A Priori Power Analysis Results for High Electric Stimulation

at First Treatment (or Restored Treatment) versus RemovedTreatment

Figure 11.13 Post Hoc Effect Size and Power Analysis for High Electric

Stimulation at First Treatment versus Removed TreatmentFigure 11.14 Post Hoc Effect Size and Power Analyses Using G*Power 3.1 for

High Electric Stimulation at First Treatment versus RestoredTreatment

Figure 11.15 Post Hoc Effect Size and Power Analysis for High Electric

Stimulation at Removed Treatment versus Restored TreatmentFigure 12.1 A Priori Power Analysis of MRA Problem

Figure 12.2 Histogram of Residuals of DSI Predicted by SPIScient and

SPIPractFigure 12.3 Normal P-P Plot of Residuals of DSI Predicted by SPIScient

and SPIPractFigure 12.4 Scatter Plot of Residuals of DSI Predicted by SPIScient and

SPIPract

Working through a solution to a research problem is a stimulating process Thefocus of this book is learning statistics while progressing through the steps of thehypothesis-testing process from hypothesis to results The hypothesis-testingprocess is the most commonly used tool in science and entails following a logicalsequence of actions, judgments, decisions, and interpretations as statistics areapplied to research problems Statistics emerged as a discipline with the purpose

of developing and applying mathematical theory and scientific operations toenhance human understanding of phenomena experienced in life For example,William Gossett developed the t-statistic while working at the Guinness Brewery

in the late 1800s He worked to explain the factors that contribute to Guinnessbeer remaining suitable for drinking and what fertilizers produce the best yield

of barley used in brewing Analysis of variance is the most widely used family ofstatistics in the world, and Sir Ronald A Fisher developed the procedure in 1921while researching the factors contributing to better yields of wheat and potatoes.The research problems used in the book reflect statistical applications related tointeresting and important topics For example, research problems for students

to work through include findings on the efficacy of using cognitive-behavioraltherapy to treat depression among adolescents and evaluating if support partnersadded to weight loss treatment can improve weight loss among persons who areoverweight It is hoped that students will find the problems that they work through

to be interesting and relevant to their field of study The research problems sented are consistent with findings in the field

pre-The format for each chapter on a major statistic is to cover the research problem

by taking the student through identifying research questions and hypotheses;identifying, classifying, and operationally defining the study variables; choosing

(IBM SPSS); interpreting the statistics; and writing the results related to theproblem.

It is the intent of the authors to provide a user-friendly guide to students tounderstand and apply procedural steps in completing quantitative studies Stu-dents will know how to plan research and conduct statistical analyses usingseveral common statistical and research designs after completion of the book Thequantitative methodological tools learned by students can actually be applied totheir own research with less oversight by faculty

Students will develop competencies in using IBM SPSS for statistical analyses.Computer-generated statistical analysis is the primary method used by quantitativeresearchers Students will have the opportunity to also calculate statistics by handfor a fuller understanding of mathematics used in computations

Moreover, the curriculum includes having students analyze research articles

in psychology using a research analysis and interpretation guide These learningexperiences allow students to enhance their understanding of consuming researchusing the information they have learned about statistical and research methods

ACKNOWLEDGMENTS

The authors would like to gratefully acknowledge the outstanding editorialleadership and support provided by Andrew Pasternack, Senior Editor; SethSchwartz, Associate Editor; and Kelsey McGee, Senior Production Editor, all ofJossey-Bass We also wish to thank the following reviewers for their thoughtfuland valuable feedback in the early stages of the manuscript: Joel Nadler, KathrynOleson, Richard Osbaldiston, and Joseph Taylor

William E Martin Jr is a professor of educational psychology and senior scholar

in the College of Education at Northern Arizona University His areas ofteaching include intermediate, computer, and multivariate statistics; researchmethods; and psychodiagnostics His research relates to person-environmentpsychology and psychosocial adaptation

Krista D Bridgmon received a PhD in educational psychology from NorthernArizona University with emphasis in counseling She is an assistant professor ofpsychology at Colorado State University Pueblo She has taught undergraduatecourses in abnormal psychology, child psychology, clinical psychology, statistics,tests and measurements, and theories of personality, and has taught graduatecourses in appraisal and assessment, clinical counseling, ethics, and schoolcounseling Her doctoral dissertation examined the stress factors that all-but-dissertation (ABD) students encounter in the disciplines of counselor education andsupervision, counseling psychology, and clinical psychology The study created aninstrument using multivariate correlational methods to measure stress factorsassociated with being ABD, named the BASS (Bridgmon All-But-DissertationStress Survey)

To my grandchildren: Grace, Adriana, Hudson, Lillee,

Uriah, Naaman, and Isaac

—W.E.M Jr

To Jerrad and Coltin: Thank you for always making me laugh!

—K.B

INTRODUCTION AND OVERVIEW

Review measures of central tendency and variability.

Review visual representations of data, including the normal distribution.

Review descriptive and inferential statistical applications of the normal distribution.

Tstudents to understand and apply procedural steps in completing

quantitative studies The book emphasizes a step-by-step guide usingresearch examples for students to move through the hypothesis-testing processfor commonly used statistical procedures and research methods Statisticaland research designs are integrated as they are applied to the examples.The structure of each chapter covers the following nine quantitative researchprocedural steps:

1 A description of a research problem, taking the student through identifyingresearch questions and hypotheses

2 A method of identifying, classifying, and operationally defining the studyvariables

3 A discussion of appropriate research designs

4 A procedure for conducting an a priori power analysis

5 A discussion of choosing an appropriate statistic for the problem

6 A statistical analysis of a data set

7 A process for conducting data screening and analyses (IBM SPSS) to test nullhypotheses

8 A discussion of interpretation of the statistics

9 A method of writing the results related to the problem

The underlying philosophy of the book is to view the quantitative researchprocess from a more holistic and sequential perspective Concepts are discussed asthey are applied during the procedural steps It is hoped that after completion ofthe book readers will be better able to plan research and conduct statisticalanalyses using several commonly used statistical and research designs Thequantitative methodological tools learned by students can actually be applied totheir own research, hopefully with less oversight by faculty

The use of statistical software is an essential tool of researchers Psychological,educational, social, and behavioral areas of research typically have multifactor

or multivariate explanations Statistical software provides a researcher withsophisticated techniques to analyze the effects and relationships among

software, which has been developed over many decades and is one of the mostwidely used statistics programs in the world.

Statistical techniques may have more meaning, understandability, and vance when learned within the context of research One needs to have anunderstanding of statistical analyses to consume and construct professionalresearch competently Knowledge of quantitative research methods is especiallyimportant today because of the emphasis on evidence-based practice in psy-chology (EBPP) to improve clinical work with clients EBPP refers to using thebest available research with clinical expertise in the context of patient char-acteristics, culture, and preferences (American Psychological Association, 2006).Ideally, the goal is to help a student achieve self-efficacy in understanding,planning, and conducting actual independent research Information and skillsgrow, leading to advanced understanding We next present a review of founda-tional information related to research and statistics that will be useful to reviewprior to completing the chapters that follow

rele-REVIEW OF FOUNDATIONAL RESEARCH CONCEPTS

A review of foundational concepts related to research and statistics is presentednext Quantitative research involves the interplay among variables after they havebeen operationalized, allowing a researcher to measure study outcomes Essentialstatistical methods used to assess scores of variables include central tendency,variability, and the characteristics of the normal distribution

Independent, Dependent, and Extraneous Variables

At the core of quantitative research is studying and measuring how variableschange Kerlinger and Pedhazur (1973) stated, “It can be asserted that all thescientist has to work with is variance If variables do not vary, if they do not havevariance, the scientist cannot do his work” (p 3) Even the father of modernstatistics, Sir Ronald Fisher (1973), said, “Yet, from the modern point of view,the study of the causes of variation of any variable phenomenon, from the yield ofwheat to the intellect of man, should be begun by the examination and mea-surement of the variation which presents itself” (p 3)

variable It is an antecedent condition to an observed resultant behavior Changes

in the independent variable produce changes in the dependent variable

All variables need to be able to vary Kerlinger and Lee (2000) identified twotypes of independent variables: active and attribute An active independent variable

is one that is manipulated by the researcher For example, a researcher designs astudy with an IV that has a researcher-specified treatment condition compared to

a no-treatment control condition Other terms used for an active IV are stimulusvariable, treatment variable, experimental variable, intervention variable, and

X variable

A second type of IV is called an attribute independent variable, which is notmanipulated but is ready-made or has preexisting values such as gender, age, orethnocultural grouping Other terms used are organismic or personological variables.The terms classification variable and categorical variable are often used as an IVlabel They can be used as either active or attribute types For example, amanipulated IV that has a treatment condition and a control condition could becalled a classification variable Also, an attribute variable such as gender (male orfemale) may be referred to as a classification or categorical variable

A dependent variable (DV) is the presumed resulting outcome in research It isusually observed and measured in response to an IV We look for changes in a DVcaused by an IV A DV is also referred to as a response variable or a Y variable

An extraneous variable (EV) is an unwanted and contaminating variable An

EV acts on a dependent variable like an independent variable does but in aconfounding way that confuses an understanding of how the IV is changing the

DV An extraneous variable is undesired noise in a research study A researcherwants to control an extraneous variable to neutralize its effects

Variables need to be assigned meaning by specifying activities or operationsnecessary to measure the variable, which is known as an operational definition(OD) A comprehensive operational definition entails all of the activities andoperations that define the variable For example, an active IV psychotherapyapproach might have two conditions (Gestalt therapy and control condition) Wecan say there are two operational definitions for the IV psychotherapy approachfor the sake of brevity However, each condition has a detailed, comprehensiveoperational definition that is clearly and fully specified A brief operational def-inition of a dependent variable of depressive symptomatology may be scores on

In correlations research, an independent variable is often called a predictorvariable (PV), and a dependent variable is called a criterion variable (CV).Scales of Measurement of Variables

Variables can be assigned scales of measurement A variable does not have anabsolute scale of measurement The scale of measurement of a variable can changedepending on how the variable is being used in different studies and even withinthe same study Therefore, there is a research contextual consideration thathelps determine the scale of measurement of a variable The process of thinkingthrough the connection between scales of measurement and variables helpsthe researcher more clearly see how variables can be measured in a study Also, thescales of measurement assigned to a variable can be useful in selecting appropriatestatistics to use in research

There are two general classifications of scales of measurement, each havingtwo subcategories; they are discrete scale (nominal and ordinal) and continuousscale (interval and ratio)

A variable using a discrete-nominal scale of measurement has mutuallyexclusive categories For example, gender has mutually exclusive categories ofmale or female, and political affiliation has categories of Republicans, Democrats,

or independents A discrete-ordinal scale of measurement variable has orderingalong some continuum It is rank scaled For example, the order (first, second,third, etc.) in which runners complete a race reflects an ordinal scale

A continuous-interval scale of measurement variable has numerical distances

on a scale that are considered approximately equal numerical distances of theattribute being measured There is no true zero point on the scale; it is consideredarbitrary For example, scores on the Wechsler IQ test are considered intervalscaled, and there is an arbitrary zero, but the test does not measure a total absence(true zero) of intelligence A continuous-ratio scale of measurement variable has atrue zero point, and the numerical distances on the scale are equal to the attributebeing measured Weight is an example of a ratio-scaled variable A zero number ofpounds is meaningful, and 100 pounds is one-third as heavy as 300 pounds.Other examples of ratio-scaled variables include height, length, and time.There are times when ordinal-scaled variables such as Likert-type scales arestatistically analyzed as a continuous-interval variable (Tabachnick & Fidell,

only the numbers we obtain and our faith in the relationship between thosenumbers and the underlying objects or events” (p 8) A more important gauge ofunderstanding the meaning of scores on dependent variables in a study has to dowith their distributions Measures of central tendency and variability of scores ofdistributions are discussed next.

REVIEW OF FOUNDATIONAL STATISTICAL

INFORMATION

One of the most important tasks of quantitative researchers is to understand thedata they are working with Researchers need to assess their data for issuesincluding dishonest data, cases with atypical scores, and noncompliance withappropriate use of statistical requirements Also, it is important for researchers tounderstand the uniqueness of their data sets by examining typical scores, vari-ability among scores, and characteristics and shapes of distributions of scoresrelated to variables in a data set

Measures of Central Tendency

Measures of central tendency are values that represent typical scores in a bution or set of scores We will be using the data in Table 1.1 to demonstratehow to calculate the three most common measures of central tendency: mode,median, and mean

distri-TABLE 1.1 Values Used to Illustrate Measures of Central

Tendency and Variability

the highest point on a graph such as a frequency distribution or a histogram and

is referred to as unimodal If there are two scores in a sample that are equally themost frequently occurring, then the distribution is called bimodal The columnheaded by X (individual score) and N¼ 6 represents six individual scores in thedistribution of scores example The only score that is represented more than once

is 48 Thus, the Mo¼ 48 and it is a unimodal distribution

Median (M dn )

The median (Mdn) is a value in the set of which 50 percent of cases fall below and

50 percent above If the number of a set of ordered scores from low to high isodd, then the score that has half of the other scores below it and half above

it is the median For example, in the set of numbers 4, 5, 7, 8, and 9, the number

7 is the median

In the set in Table 1.1, the number of scores is even The six scores orderedare 43, 48, 48, 50, 52, 53 To obtain the median requires calculating the averagevalue between the score at N/2 and the score at (N/2)1 1 So, N/2 ¼ 6/2 ¼ 3(i.e., the third score) and (N/2)1 1 ¼ 3 1 1 ¼ 4 (i.e., the fourth score) The thirdscore in the set is 48 and the fourth score is 50 The average of 48 and 50 is(481 50)/2 ¼ 98/2 ¼ 49 So, the median of the data set in Table 1.1 is 49

Mean ( X or M)

The mean (X or M ) is the sum of individual scores (ΣX ) in a data set divided bythe number of scores (N ) The mean is typically a more precise measure of centraltendency than the mode or the median, because the specific value of each score isused to calculate the mean Also, the mean has the properties of being contin-uously scaled as an interval or a ratio The mean is more stable than the mode ormedian when it is used as a sample measure of central tendency drawn from apopulation On the downside, when a sample has extreme scores (skewed), themean is drawn way from the clustered scores toward the extreme scores In thesesituations, the mean may not be the most typical score in a data set For example,

an analysis of salaries of company employees that includes the high salaries of

executives can produce a company employee mean salary that the vast majority ofemployees are well below The median might be a better indication of the typicalemployee salary in the company Presented next is the calculation of the mean ofour sample data in Table 1.1.

X ¼ΣXN

¼2946

X ¼ 49:00where

ΣX ¼ sum of the individual scores

N ¼ number of scores

Measures of Variability (Dispersion) of Scores

Foundational to quantitative research is the study of the measures of variability(dispersion) of scores in a sample data set Here we review common measures ofrange, mean deviation scores, sum of squares, variance, and standard deviationusing the data set from Table 1.1

The range of scores in a data set is simply the difference between the highestand lowest scores (scorehighest2 scorelowest) In the example data in Table 1.1, thehighest score is 53 and the lowest score is 43; 532 43 ¼ 10, which is the range

A measure of variability that is used as a component in many statisticalformulas that are used in fundamental statistics is called the total sum of squares,which is the sum of squared differences of all scores in the data set from theiroverall mean,ΣðX  X Þ2

.Variance of the Sample (s2)

The variance of the sample (s2) is the total sum of squares divided by the number ofscores,ΣðX  X ÞN 2 The symbol of the variance of the population is sigma squared(σ2

) If all the cases are the same value, the variance will equal zero The larger thevariance value, the more the values are spread out in the distribution

more accurate estimate (unbiased estimate) of a population parameter We do notknow what the population mean is, so the sample values have less variability.

We use the sample mean instead and with N2 1 as a compensation for notknowing the population mean This use of N2 1 also is referred to as degrees offreedom (df ), a practice used in most statistical analyses As Hays (1963) states,

“Thus we say that there are N 2 1 degrees of freedom for a sample variance,reflecting the fact that only N2 1 deviations are ‘free’ to be any number, but thatgiven these free values, the last deviation is completely determined” (p 311) If thegoal of studying a sample is to describe the sample and not to estimate a popu-lation, then N may be used and not N2 1 The variance is calculated next usingthe data from Table 1.1

s2 ¼ΣðX  X Þ2

N  1

¼645

s2 ¼ 12:80Standard Deviation of the Sample (s)

The standard deviation of the sample (s) is the square root of the variance Thesymbol for the standard deviation of the population isσ The standard deviation

is a more useful explanatory measure of variability when compared to the variancebecause it is in the same units as the original data For example, when presentingthe mean and the standard deviation together, they are both in the same metric

s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΣðX  X Þ2

N  1s

¼

ffiffiffiffiffi645s

¼pffiffiffiffiffiffiffiffiffiffiffi12:80

s ¼ 3:58

mean The coefficient of variation can be used to compare the variability ofdifferent variables as well as the means of the variables Using the s¼ 3.58 and

X ¼ 49 from the data in Table 1.1, the coefficient of variation is calculated here

as a ratio and as a percentage:

C ¼ sX

¼3:5849

C ¼ :073

C 3 100 ¼ 7:30%

The coefficient of variation for the DV is 0.073, and converted to a percentage it is7.30 percent The standard deviation is approximately 7.30 percent of the mean

Visual Representations of a Data Set

There are many charts available to assist researchers in more fully understanding theirdata Illustrations of a bar chart, histogram, and normal Q-Q plot are presented

A data set of scores on a dependent variable of depressive symptoms is sented in a frequency distribution in Table 1.2 There are scores from 145 par-ticipants and there are no missing data Hence, the data set has complete scoresand the values in the Percent and Valid Percent columns are the same The values

pre-in the first column are the scores representpre-ing depressive symptoms For example,the score of 17 has a frequency of nine scores (9/145¼ 6.2%) in the data set Thevalue 17 represents a cumulative percentage of 49.7 percent, which is the closestvalue to the 50th percentile rank of the distribution of scores

This data can be shown as a bar chart (see Figure 1.1) The horizontal line ofthe bar chart is called the abscissa or x-axis, and in this example each numberrepresents a value for depressive symptoms in the data set The vertical line, alsocalled the ordinate or y-axis, represents the frequency of scores by participants ateach value for depressive symptoms in the data set For example, you can see themost frequently scored value (16) and the least frequently scored values (5, 7, 9,35) by participants in the sample

A histogram of the data set is presented in Figure 1.2 In a histogram the x-axisdepicts intervals of scores and the y-axis still represents frequencies of scores.However, the scores on the x-axis are reported as falling within intervals In thisgraph, the intervals are 10-point intervals, so all scores in the data set between 0and 9 are represented by bars in this interval Using Table 1.1 for assistance,the scores by frequencies in the data set interval 09 are score 5 (frequency ¼ 1score), score 7 (frequency¼ 1 score), score 8 (frequency ¼ 4 scores), and score 9(frequency¼ 1 score) Therefore, a total of seven scores in the data set are withinthe range of 09 The total numbers of scores within the intervals are interval09 (seven scores), interval 1019 (80 scores), interval 2029 (46 scores), andinterval 3039 (12 scores).

A histogram of a data set provides the researcher with a quick visualinspection of the shape of the distribution of scores For example, we can see

that scores cluster around the mean and there is reasonable symmetry (balance) ofscores on either side of the mean Also, there appear to be no extreme scores tothe negative side (left) or the positive side (right) of the distribution.

The same histogram of the data set is presented in Figure 1.3 with a normalcurve superimposed on the graph This provides additional information abouthow well the sample distribution of scores fits a normal curve If the data setscores were a perfect fit to the normal curve, then the bars would fit fully withinthe superimposed normal curve

An example of a useful plot to assess normality of a data set is called a Q-Q(quantile-quantile) plot (see Figure 1.4) The Q-Q plot is derived by first sub-tracting each observed score from the group mean Then, these residuals

DV

Histogram

20

Mean ⫽ 18.73 Std Dev ⫽ 6.793

(differences) are plotted against the expected observed scores if the data are from anormal distribution (Norusis, 2003) Normality exists in the sample distribution ofscores if the points on the Q-Q plot fall on or near the straight line In this example,

it appears that the data set shows reasonable normality among the scores

THE NORMAL DISTRIBUTION

The normal distribution (bell curve) is the most studied and widely used curve in thefield of probability (Tabak, 2005) Many measurements of human activities havebeen shown to be normally distributed A great deal has been discovered about thenormal curve over the past 300 years since Abraham de Moivre formulated a

DV

Histogram

20

Mean ⫽ 18.73 Std Dev ⫽ 6.793

mathematical proof of the normal distribution The normal distribution has usefulproperties If two random variables have a normal distribution, their sum has anormal distribution In general, all kinds of sums and differences of normal variableshave normal distributions So, many statistics derived from normal variates arethemselves normally“distributed” (Salsburg, 2001) The normal distribution hastwo parameters (constants): the population mean (μ) and the population standarddeviation (σ) There are many different normal curves that are based on these twoparameters (Snedecor & Cochran, 1967).

Characteristics of the Normal Distribution

The normal distribution has a peak (highest point on the curve), tails (the extremeleft and right points of the curve) and shoulders (the left and right sections ofthe curve between the peak and the tails) (see Figure 1.5) The right side of the

The base of the curve is called the abscissa (horizontal axis, x-axis) and tions off measurements in standard deviation units of constant percentages such

sec-as percentile ranks, z-scores, and T-scores The height of the normal curve iscalled the ordinate (vertical axis, y-axis), which represents the percentage of casesunder portions of the normal curve

The area within the normal curve is referred to as a density of 100 percent or

a unit of 1.0 for using probability While the tails on both sides of the normalcurve extend to infinity (N) and never touch the abscissa, 991 percent ofthe curve falls within 63 standard deviations of the curve Most of the area is

in the middle of the curve at the highest point where 68.26 percent is between

61 standard deviation of the normal curve The percentage of the area under thecurve decreases as the shape of the curve moves toward the tails

The normal distribution is symmetrical, and each half of the curve is exactly

50 percent density The mean, median, and mode of the normal curve are thesame, as represented by 0 at the midpoint of the curve Illustrations of usingthe normal curve in descriptive statistical analyses are discussed next

Descriptive Statistical Applications of the Normal Distribution

We will assess an individual’s measured IQ score compared to the IQ scores ofothers who are part of a normative sample of individuals whose scores reflect anormal curve Bob has a measured IQ score on a standardized IQ test that is 80.The population mean of the normative sample is μ ¼ 100, and the standarddeviation is σ ¼ 15

A z-score can be used with this information to compare Bob’s score with thenormative sample A z-score is a standard score that shows the relative standing of

a raw score in a normal distribution The formula for a z-score is z¼ X 2 μ/σ, where

X is an individual score, μ is the population mean, and σ is the population standarddeviation The z-score of Bob’s individual IQ score is z ¼ 80 2 100/15 ¼ 1.33.One can visualize where z¼ 1.33 is placed on the normal curve in Figure 1.5.Next we will find the percentile rank of Bob’s z-score ¼ 1.33 (raw score of80) A percentile rank is the score that indicates what percentage of persons beingmeasured fall equal to or below the particular score The exact percentages in thenormal curve associated with z-scores are found using an online statistics calculator

on the Normal Distribution button click on the Cumulative Area Under theNormal Curve Calculator.

3 Type in Bob’s z-score of 1.33 click on the Calculate button theCumulative area: is 09175914

If you take 09175914 times 100 or move the decimal two places to the right,you obtain 9.175914, which is approximately 9.18% So a z-score¼ 1.33indicates that equal to or less than 9.18 percent of the norm group obtain an IQscore of 80, when the mean is 100 and the standard deviation is 15 Bob’spercentile rank is 9.18

In another example, Jean scored 105 on the same IQ test Jean’s z-score ¼

1052 100/15 ¼ 1.33 A z-score ¼ 1.33 is to the right of the center point mean

1 Go to www.danielsoper.com, where there are several free statistics calculators

2 On the home website click on Statistics Calculator scroll down and click

on the Normal Distribution button click on the Cumulative Area Under theNormal Curve Calculator

3 Type in Jean’s z-score of 33 click on the Calculate button the lative area: is 62930002

Cumu-If you take 62930002 times 100 or move the decimal place two places tothe right, you obtain 62.930002, which is approximately 62.93 percent So az-score ¼ 33 indicates that equal to or less than 62.93 percent of the norm groupobtain an IQ score of 105, when the mean is 100 and the standard deviation

is 15 Jean’s IQ score of 105 represents a percentile rank of 62.93

Inferential Statistical Applications of the Normal Distribution

We just showed how the normal curve can be used in a descriptive statisticalanalysis The normal curve also plays a key role in inferential statistics, whichinvolves inferring information about samples to generalize to populations