STATISTICS AND DATA ANALYSIS FOR THE BEHAVIORAl SCIENCES docx

1 INTRODUCTION: STATISTICS AND DATA ANALYSIS AS TOOLS FOR RESEARCHERS 3 2 PROCESS OF RESEARCH IN PSYCHOLOGY AND RELATED FIELDS 45 3 FREQUENCY DISTRIBUTIONS, GRAPHING, AND DATA DISPLAY 85

STATISTICS AND

DATA ANALYSIS

DANA S DUNN

Moravian College

Boston Burr Ridge, IL Dubuque,IA Madison, WI New York San Francisco St Louis

Bangkok Bogota Caracas Lisbon London Madrid Mexico City Milan New Delhi Seoul Singapore Sydney Taipei Toronto

McGraw-Hill Higher Education ~

A Division of The McGraw-Hill Companies

STATISTICS AND DATA ANALYSIS FOR THE BEHAVIORAL SCIENCES

Published by McGraw-Hill, an imprint of The McGraw-Hill Companies, Inc_, 1221 Avenue of

the Americas, New York, NY 10020 Copyright © 2001 by The McGraw-Hill Companies, Inc

All rights reserved No part of this publication may be reproduced or distributed in any form or

by any means, or stored in a database or retrieval system, without the prior written consent of

The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic

storage or transmission, or broadcast for distance learning

Some ancillaries, including electronic and print components, may not be available to customers

outside the United States

This book is printed on acid-free paper

234567890VNHNNH0987654321

ISBN 0-07-234764-3

Vice president and editor-in-chief: Thalia Dorwick

Editorial director: Jane E Vaicunas

Executive editor: Joseph Terry

Marketing manager: Chris Hall

Project manager: Susan J Brusch

Senior media producer: Sean Crowley

Production supervisor: Kara Kudronowicz

Coordinator of freelance design: David W Hash

Cover/interior designer: Rebecca Lloyd Lemna

Senior photo research coordinator: Carrie K Burger

Senior supplement coordinator: Jodi K Banowetz

Compositor: York Graphic Services, Inc

!

)

r

I (

To the memory of my father and grandfather,

James L Dunn and Foster E Kennedy

"WHAT'S PAST IS PROLOGUE" - THE TEMPEST (ACT II, SC I)

DANA S DUNN

vi

ABOUT THE AUTHOR

Dana S Dunn is currently an Associate Professor and the Chair of the ment of Psychology at Moravian College, a liberal arts and sciences college in

Depart-Bethlehem, Pennsylvania Dunn received his Ph.D in experimental social psychology from the University of Virginia in 1987, having previously graduated with a BA

in psychology from Carnegie Mellon University in 1982

He has taught statistics and data analysis for over 12 years

Dunn has published numerous articles and chapters in the areas of social cognition, rehabilitation psychology, the teaching of psychology, and liberal education He is the author of a research methods book, The Practical Researcher: A Student Guide to Conducting Psychological Research (McGraw-Hill, 1999) Dunn lives in Bethlehem with his wife and two children

)

i

1 INTRODUCTION: STATISTICS AND DATA ANALYSIS AS TOOLS FOR RESEARCHERS 3

2 PROCESS OF RESEARCH IN PSYCHOLOGY AND RELATED FIELDS 45

3 FREQUENCY DISTRIBUTIONS, GRAPHING, AND DATA DISPLAY 85

4 DESCRIPTIVE STATISTICS: CENTRAL TENDENCY AND VARIABILITY 133

5 STANDARD SCORES AND THE NORMAL DISTRIBUTION 177

10 MEAN COMPARISON I: THE tTEST 365

11 MEAN COMPARISON II: ONE-VARIABLE ANALYSIS OF VARIANCE 411

12 MEAN COMPARISON III: TWO-VARIABLE ANALYSIS OF VARIANCE 459

13 MEAN COMPARISON IV: ONE-VARIABLE MEASURES ANALYSIS OF VARIANCE 499

REPEATED-14 SOME NONPARAMETRIC STATISTICS FOR CATEGORICAL AND ORDINAL DATA 523

15 CONCLUSION: STATISTICS AND DATA ANALYSIS IN CONTEXT 563

vii

Contents in Brief

Appendix A: Basic Mathematics Review and Discussion of Math Anxiety A-I

Appendix B: Statistical Tables B-1

viii

Appendix C: Writing Up Research in APA Style: Overview and Focus on Results C-l

Appendix D: Doing a Research Project Using Statistics and Data Analysis: Organization, Time Management, and Prepping Data for Analysis D-l

Appendix E: Answers to Odd-Numbered End of Chapter Problems E-l

Appendix F: Emerging Alternatives: Qualitative Research Approaches F-l

References R-l

Credits CR-l

Name Index NI-l

Subject Index SI-l

Reader Response xxviii

1 INTRODUCTION: STATISTICS AND DATA ANALYSIS AS TOOLS FOR RESEARCHERS 3

DATA BOX 1.A: What Is or Are Data? 5

Tools for Inference: David L.'s Problem 5

College Choice 6 College Choice: What Would (Did) You Do? 6

Statistics Is the Science of Data, Not Mathematics 8 Statistics, Data Analysis, and the Scientific Method 9

Inductive and Deductive Reasoning 10

Populations and Samples 12

Descriptive and Inferential Statistics 16

DATA BOX 1.B: Reactions to the David L Problem 18

Knowledge Base 19

Discontinuous and Continuous Variables 20

DATA BOX 1.C: Rounding and Continuous Variables 22

Writing About Data: Overview and Agenda 23

Scales of Measurement 24

Nominal Scales 25 Ordinal Scales 26 Interval Scales 27 Ratio Scales 28

Writing About Scales 29 Knowledge Base 31

Overview of Statistical Notation 31

What to Do When: Mathematical Rules of Priority 34

DATA BOX 1.D: The Size of Numbers is Relative 38 Mise en Place 39

ix

x Contents

About Calculators 39 Knowledge Base 40

PRO.JECT EXERCISE: Avoiding Statisticophobia 40

Looking Forward, Then Back 41 Summary 42

Key Terms 42 Problems 42

2 PROCESS OF RESEARCH IN PSYCHOLOGY AND RELATED FIELDS 45

The Research Loop of Experimentation: An Overview of the Research Process 45

Populations and Samples Revisited: The Role of Randomness 48

Distinguishing Random Assignment from Random Sampling 48 Some Other Randomizing Procedures 50

Sampling Error 52 Knowledge Base 53

DATA BOX 2.A: Recognizing Randomness, Imposing Order 54

Independent and Dependent Variables 54

Types of Dependent Measures 58 Closing or Continuing the Research Loop? 60

DATA BOX 2.B: Variable Distinctions: Simple, Sublime, and All Too Easily Forgotten 61

The Importance of Determining Causality 61

DATA BOX 2.C: The "Hot Hand in Basketball" and the Misrepresentation of Randomness 62

Operational Definitions in Behavioral Research 63

Writing Operational Definitions 64 Knowledge Base 64

Reliability and Validity 65

Reliability 66 Validity 67 Knowledge Base 69

Research Designs 70

Correlational Research 70 Experiments 72

Quasi-experiments 74

DATA BOX 2.D: Quasi-experimentation in Action: What to Do

Without Random Assignment or a Control Group 75 Knowledge Base 76

PRO.JECT EXERCISE: Using a Random Numbers Table 77

Looking Forward, Then Back 81 Summary 81

Key Terms 82 Problems 82

3 FREQUENCY DISTRIBUTIONS, GRAPHING, AND DATA DISPLAY 85

What is a Frequency Distribution? 87

Graphing Frequency Distributions 97

Bar Graphs 98 Histograms 99 Frequency Polygons 100 Misrepresenting Relationships: Biased or Misleading Graphs 102

New Alternatives for Graphing Data: Exploratory Data Analysis 104

Stem and Leaf Diagrams 105

DATA BOX 3.B: Biased Graphical Display-Appearances Can Be Deceiving 106

Tukey's Tallies 108 Knowledge Base 109

Envisioning the Shape of Distributions III

DATA BOX 3.C: Kurtosis, or What's the Point Spread? 113

DATA BOX 3.D: Elegant Information-Napoleon's Ill-fated March

to Moscow 114

Percentiles and Percentile Ranks 115

Cumulative Frequency 116 Cumulative Percentage 117 Calculating Percentile Rank 118 Reversing the Process: Finding Scores from Percentile Ranks 119 Exploring Data: Calculating the Middle Percentiles and Quartiles 120 Writing About Percentiles 122

Knowledge Base 123

Constructing Tables and Graphs 123

Less is More: Avoiding Chart junk and Tableclutter, and Other Suggestions 124

American Psychological Association (APA) Style Guidelines for Data Display 125

PROJECT EXERCISE: Discussing the Benefits of Accurate but Persuasive Data Display 126

Looking Forward, Then Back 127

Why Represent Data By Central Tendency 134

The Mean: The Behavioral Scientist's Statistic of Choice 136

DATA BOX 4.A: How Many Are There? And Where Did They Come From? Proper Use of Nand n 138

Calculating Means from Ungrouped and Grouped Data 138 Caveat Emptor: Sensitivity to Extreme Scores 140

xi

Shapes of Distributions and Central Tendency 147 When to Use Which Measure of Central Tendency 148 Writing About Central Tendency 149

Knowledge Base 150

Understanding Variability 151 The Range 153

The Interquartile and the Semi-Interquartile Range 153

Variance and Standard Deviation 155

Sample Variance and Standard Deviation 157 Homogeneity and Heterogeneity: Understanding the Standard Deviations of Different Distributions 159

Calcuklting Variance and Standard Deviation from a Data Array 160 Population Variance and Standard Deviation 161

Looking Ahead: Biased and Unbiased Estimators of Variance and Standard Deviation 162

DATA BOX 4.C: Avoid Computation Frustration: Get to Know Your Calculator 165

Knowledge Base 165

Factors Affecting Variability 166

Writing About Range, Variance, and Standard Deviation 168

DATA BOX 4.D: Sample Size and Variability-The Hospital Problem 169

PRO.IECT EXERCISE: Proving the Least Squares Principle for the Mean 170

Looking Forward, Then Back 171 Summary 172

Key Terms 173 Problems 173

DATA BOX IIA: Social Comparison Among Behavioral and Natural Scientists: How Many Peers Review Research Before Publication? 179

DATA BOX II.B: Explaining the Decline in SAT Scores: Lay Versus Statistical Accounts 180

Why Standardize Measures? 181

The z Score: A Conceptual Introduction 182 Formulas for Calculating z Scores 185

The Standard Normal Distribution 186 Standard Deviation Revisited: The Area Under the Normal Curve 187

Application: Comparing Performance on More than One Measure 188 Knowledge Base 189

! ,

I

Working with z Scores and the Normal Distribution 190

Finding Percentile Ranks with z Scores 191 Further Examples of Using z Scores to Identify Areas Under the Normal Curve 192

DATA BOX S.C: Intelligence, Standardized IQ Scores, and the Normal Distribution 194

A Further Transformed Score: The T Score 196 Writing About Standard Scores and the Normal Distribution 197 Knowledge Base 198

Looking Ahead: Probability, z Scores, and the Normal Distribution 198

PRO.JECT EXERCISE: Understanding the Recentering of Scholastic Aptitude Test Scores 199

Looking Forward, Then Back 201 Summary 202

Key Terms 202 Problems 202

6 CORRELATION 205 Association, Causation, and Measurement 206

Galton, Pearson, and the Index of Correlation 207

A Brief But Essential Aside: Correlation Does Not Imply Causation 207

The Pearson Correlation Coefficient 209

Conceptual Definition of the Pearson r 209

DATA BOX 6.A: Mood as Misbegotten: Correlating Predictors with Mood States 213

Calculating the Pearson r 216 Interpreting Correlation 221

Magnitude of r 222

Coefficients of Determination and Nondetermination 222

Factors Influencing r 224 Writing About Correlational Relationships 226 Knowledge Base 227

Correlation as Consistency and Reliability 228

DATA BOX 6.B: Personality, Cross-Situational Consistency, and Correlation 228

Other Types of Reliability Defined 229

A Brief Word About Validity 229

DATA BOX 6.C: Examining a Correlation Matrix: A Start for Research 230

What to Do When: A Brief, Conceptual Guide to Other Measures of Association 231

DATA BOX 6.D: Perceived Importance of Scientific Topics and Evaluation Bias 232

PROJECT EXERCISE: Identifying Predictors of Your Mood 233 Looking Forward, Then Back 237

Summary 237 Key Terms 238 Problems 238

xiv Contents

7 LINEAR REGRESSION 241 Simple Linear Regression 242

The z Score Approach to Regression 242 Computational Approaches to Regression 243 The Method of Least Squares for Regression 245 Knowledge Base 249

DATA BOX 7oA: Predicting Academic Success 250

Residual Variation and the Standard Error of Estimate 251

DATA BOX 7.B The Clinical and the Statistical: Intuition Versus Prediction 253

Assumptions Underlying the Standard Error of Estimate 253

Partitioning Variance: Explained and Unexplained Variation 256

A Reprise for the Coefficients of Determination and Nondetermination 257

Proper Use of Regression: A Brief Recap 258 Knowledge Base 258

Regression to the Mean 259

DATA BOX 7.C Reinforcement, Punishment, or Regression Toward the Mean? 260

Regression as a Research Tool 261

Other Applications of Regression in the Behavioral Sciences 262 Writing About Regression Results 263

Multivariate Regression: A Conceptual Overview 263

PRo.JECT EXERCISE Perceiving Risk and Judging the Frequency of Deaths 264

Looking Forward, Then Back 268 Summary 268

Key Terms 269 Problems 269

8 PROBABILITY 273 The Gambler's Fallacy or Randomness Revisited 275 Probability: A Theory of Outcomes 277

Classical Probability Theory 277

DATA BOX 8oA: "I Once Knew a Man Who ": Beware Man- Who Statistics 278

Probability's Relationship to Proportion and Percentage 281

DATA BOX 8.B Classical Probability and Classic Probability Examples 282

Probabilities Can Be Obtained from Frequency Distributions 283 Knowledge Base 283

DATA BOX S.C A Short History of Probability 284

Calculating Probabilities Using the Rules for Probability 285

The Addition Rule for Mutually Exclusive and Nonmutually Exclusive Events 285

The Multiplication Rule for Independent and Conditional Probabilities 287

DATA BOX 8.D Conjunction Fallacies: Is Linda a Bank Teller or a Feminist Bank Teller? 288

DATA BOX 8.E: Control, Probability, and When the Stakes Are High 304

Knowledge Base 305

p Values: A Brief Introduction 305

Writing About Probability 306

PROJECT EXERCISE: Flipping Coins and the Binomial Distribution 307

Looking Forward, Then Back 310 Summary 310

Key Terms 311 Problems 311

9 INFERENTIAL STATISTICS: SAMPLING DISTRIBUTIONS AND HYPOTHESIS TESTING 315

Samples, Population, and Hypotheses: Links to Estimation and Experimentation 316

Point Estimation 317 Statistical Inference and Hypothesis Testing 318 The Distribution of Sample Means 319

Expected Value and Standard Error 320

The Central Limit Theorem 322

Law of Large Numbers Redux 322

DATA BOX 9oA: The Law of Small Numbers Revisited 323

Standard Error and Sampling Error in Depth 324

Estimating the Standard Error of the Mean 324

Standard Error of the Mean: A Concrete Example Using Population Parameters 326

Defining Confidence Intervals Using the Standard Error

of the Mean 327

DATA BOX 9.B: Standard Error as an Index of Stability and Reliability of Means 328

Knowledge Base 329

DATA BOX 9.C: Representing Standard Error Graphically 330

Asking and Testing Focused Questions: Conceptual Rationale for Hypotheses 331

DATA BOX 9.D: What Constitutes a Good Hypothesis? 332 Directional and Nondirectional Hypotheses 333 The Null and the Experimental Hypothesis 333

Statistical Significance: A Concrete Account 336

DATA BOX 9.E: Distinguishing Between Statistical and Practical Significance 337

xv

Single Sample Hypothesis Testing: The z Test and the Significance of r 343

What Is the Probability a Sample Is from One Population or Another? 344

Is One Sample Different from a Known Population? 345 When Is a Correlation Significant? 347

Inferential Errors Types I and II 349 Statistical Power and Effect Size 351

Effect Size 354 Writing About Hypotheses and the Results of Statistical Tests 355 Knowledge Base 357

PROJECT EXERCISE: Thinking About Statistical Significance in the Behavioral Science Literature 357

Looking Forward, Then Back 360 Summary 360

Key Terms 362 Problems 362

10 MEAN COMPARISON I: THE t TEST 365 Recapitulation: Why Compare Means? 367 The Relationship Between the t and the z Distributions 368

The t Distribution 368 Assumptions Underlying the t Test 369

DATA BOX 10.A: Some Statistical History: Who was '~Student"? 371

Hypothesis Testing with t: One-Sample Case 372

Confidence Intervals for the One-Sample t Test

DATA BOX 10.B: The Absolute Value of t 376 Power Issues and the One-Sample t Test 377

Knowledge Base 377

375

Hypothesis Testing with Two Independent Samples 378

Standard Error Revised: Estimating the Standard Error of the Difference Between Means 379

Comparing Means: A Conceptual Model and an Aside for Future Statistical Tests 383

The t Test for Independent Groups 384

DATA BOX 10.C: Language and Reporting Results, or (Too) Great Expectations 388

Effect Size and the t Test 388 Characterizing the Degree of Association Between the Independent Variable and the Dependent Measure 389

DATA BOX 10.D: Small Effects Can Be Impressive Too 390 Knowledge Base 392

Hypothesis Testing with Correlated Research Designs 393

,/

I

,.:

MEAN COMPARISON II: ONE-VARIABLE ANALYSIS

Overview of the Analysis of Variance 413

Describing the F Distribution 417 Comparing the ANOVA to the t Test: Shared Characteristics and Assumptions 418

Problematic Probabilities: Multiple t Tests and the Risk of Type I Error 420

DATA BOX 1104: R A Fischer: Statistical Genius and Vituperative Visionary 422

How is the ANOVA Distinct from Prior Statistical Tests? Some Advantages 423

Omnibus Test Comparing More than 1Wo Means Simultaneously 423

DATA BOX 11.B: Linguistically Between a Rock and Among Hard Places 424

Experimentwise Error: Protecting Against Type I Error 424

Causality and Complexity 425 Knowledge Base 426

One-Factor Analysis of Variance 426

Identifying Statistical Hypotheses for the ANOVA 427 Some Notes on Notation and the ANOVA's Steps 429

DATA BOX 11.C: Yet Another Point of View on Variance: The General Linear Model 431

One- Way ANOVA from Start to Finish: An Example with Data 431

Post Hoc Comparisons of Means: Exploring Relations in the "Big, Dumb F" 439

Tukey's Honestly Significant Difference Test 440 Effect Size for the F Ratio 442

Estimating the Degree of Association Between the Independent Variable and the Dependent Measure 443

DATA BOX 11.D: A Variance Paradox-Explaining Variance Due to Skill or Baseball is Life 444

Writing About the Results of a One-Way ANOVA 445 Knowledge Base 446

xvii

Overview of Complex Research Designs: Life Beyond Manipulating One Variable 460

Two-Factor Analysis of Variance 461

DATA BOX 12.A: Thinking Factorially 463 Reading Main Effects and the Concept of Interaction 465 Statistical Assumptions of the Two-Factor ANOVA 469 Hypotheses, Notation, and Steps for Performing for the Two-Way ANOVA 469

DATA BOX 12.B: Interpretation Qualification: Interactions Supercede Main Effects 471

The Effects of Anxiety and Ordinal Position on Affiliation: A Detailed Example of a Two-Way ANOVA 475

Knowledge Base 475

DATA BOX 12.C: The General Linear Model for the Two-Way ANOVA 476

Effect Size 486 Estimated Omega-Squared (~2) for the 1Wo-Way ANOVA 487 Writing About the Results of a 1Wo-Way ANOVA 488 Coda: Beyond 2 X 2 Designs 489

One-Factor Repeated-Measures ANOVA 501

Statistical Assumptions of the One-Way Repeated-Measures ANOVA 502

Hypothesis, Notation, and Steps for Performing the One-Variable Repeated-Measures ANOVA 503

DATA BOX 13.A: Cell Size Matters, But Keep the Cell Sizes Equat Too 508

Thkey's HSD Revisited 510 Effect Size and the Degree of Association Between the Independent Variable and Dependent Measure 511

-

How Do Nonparametric Tests Differ from Parametric Tests? 525

Advantages of Using Nonparametric Statistical Tests Over Parametric Tests 526

Choosing to Use a Nonparametric Test: A Guide for the Perplexed 527

DATA BOX 14.A: The Nonparametric Bible for the Behavioral Sciences: Siegel and Castellan (1988) 528

The Chi-Square (X 2 ) Test for Categorical Data 528

Statistical Assumptions of the Chi-Square 529 The Chi-Square Test for One- Variable: Goodness-of-Fit 529 The Chi-Square Test of Independence of Categorical Variables 534

DATA BOX 14.B: A Chi-Square Test for Independence Shortcut for

2 X 2 Tables 538 Supporting Statistics for the Chi-Square Test of Independence: Phi

(cp) and Cramer's V 538 Writing About the Result of a Chi-Square Test for Independence 539

DATA BOX 14.C: Research Using the Chi-Square Test to Analyze Data 540

Writing About the Results of the Mann- Whitney U Test 547

The Wilcoxon Matched-Pairs Signed-Ranks Test 547

DATA BOX 14.E: Even Null Results Must Be Written Up and Reported 550

Writing About the Results of the Wilcoxon ill Test 551

The Spearman Rank Order Correlation Coefficient 551

Writing About the Results of a Spearman rs Test 554 Knowledge Base 554

Research Using An Ordinal Test to Analyze Data 555

xix

The Fuss Over Null Hypothesis Significance Tests 564

Panel Recommendations: Wisdom from the APA Task Force on Statistical Inference 565

Knowledge Base 567

Statistics as Avoidable Ideology 567 Reprise: Right Answers Are Fine, but Interpretation Matters More

568 Linking Analysis to Research 569

Do Something: Collect Some Data, Run a Study, Get Involved 569 Knowing When to Say When: Seeking Statistical Help in the Future 570

DATA BOX 1S.A: Statistical Heuristics and Improving Inductive Reasoning 571

Data Analysis with Computers: The Tools Perspective Revisited 572

Appendix A: Basic Mathematics Review and Discussion of Math Anxiety A-I Appendix B: Statistical Tables B-1

Appendix C: Writing Up Research in APA Style: Overview and Focus on Results C-l Appendix D: Doing a Research Project Using Statistics and Data Analysis: Organization, Time Management, and Prepping Data for Analysis D-l

Appendix E: Answers to Odd-Numbered End of Chapter Problems E-l Appendix F: Emerging Alternatives: Qualitative Research Approaches F-l References R-l

Credits CR-l Name Index NI-l Subject Index SI-l

/

; /

,

(

r

/

discover new ways to think about the world around them, uncover previously ognized relationships among disparate variables, and make better judgments about how and why people behave the way they do

statistics and the analysis of data through a practical, hands-on approach Students will learn the "how to" side of statistics: how to select an appropriate test, how to collect data for research, how to perform statistical calculations in a step-by-step manner, how

to be intelligent consumers of statistical information, and how to write up analyses and results in American Psychological Association (APA) style Linking theory with prac-tice will help students retain what they learn for use in future behavioral science courses, research projects, graduate school, or any career where problem solving is used Com-bining statistics with data analysis leads to a practical pedagogical goal-helping stu-dents to see that both are tools for intellectual discovery that examine the world and events in it in new ways

• To the Student

Two events spurred me to write this book, and I want you to know that I wrote it with students foremost in my mind First, I have taught statistics for over 12 years In that time, I've come to believe that some students struggle with statistics and quantitative material simply because it is not well presented by existing textbooks Few authors, for example, adequately translate abstract ideas into concrete terms and examples that can

be easily understood Consequently, as I wrote this book, I consciously tried to make even the most complex material as accessible as possible I also worked to develop ap-plications and asides that bring the material to life, helping readers to make connec-tions between abstract statistical ideas and their concrete application in daily life

xxi

xxii Preface

Second, the first statistics course that I took as an undergraduate was an gated disaster, really, a nightmare-it was dull, difficult, and daunting I literally had no idea what the professor was talking about, nor did I know how to use statistics for any purpose I lost that battle but later won the war by consciously trying to think about how statistics and the properties of data reveal themselves in everyday life I came to appreciate the utility and even dare I say it-the beauty of statistics In doing so, I also vowed that when I became a professor, no student of mine would suffer the pain and intellectual doubt that I did as a first-time statistics student Thus, I wrote this book with my unfortunate "growing" experience in mind I never want anyone in my classes

unmiti-or using my book to feel the anxiety that I did and, though it is a cliche, I think that the book is better because of my trying first experience

How can you ensure that you will do well in your statistics class? Simple: Attend classes, do the reading, do the homework, and review what you learn regularly Indeed,

it is a very good idea to reserve some meaningful period of time each day for studying

statistics and data analysis (yes, I am quite serious) When you do not understand thing mentioned in this book or during class, ask the instructor for clarification im- mediately, not later, when your uncertainty has had time to blossom into full-blown confusion (remember my first experience in a statistics class-I know whereof I speak)

some-Remember, too, the importance of reminding yourself that statistics is for something

You should be able to stop at any given point in the course of performing a statistical test in order to identify what you are doing, why, and what you hope to find out by us-ing it If you cannot do so, then you must backtrack to the point where you last un-derstood what you were doing and why; to proceed without such understanding is not only a waste of time, it is perilous, even foolhardy, and will not help you to compre-hend the material By the way, if you feel that you need a review of basic mathematics, Appendix A provides one, including some helpful ideas on dealing with math anxiety

Beyond these straightforward steps, you should also take advantage of the gogical tools I created for this book They are reviewed in detail in the To the Instruc- tor section, and I suggest you take a look at their descriptions below I do, however, take

peda-the time to explain peda-these tools and peda-their use as they appear in the first few chapters of

the book I urge you to take these devices seriously, to see them as complementary to and not replacements for your usual study habits I promise you that your diligence will have a favorable payoff in the end-actual understanding, reduced anxiety, and prob-ably a higher grade than you expected when you first began the class

This book was written for use in a basic, first, non-calculus-based statistics course for undergraduate students in psychology, education, sociology, or one of the other be-havioral sciences I assume little mathematical sophistication, as any statistical proce-dure is presented conceptually first, followed by calculations demonstrated in a step-by-step manner Indeed, it is important for both students and instructors to remember that statistics is not mathematics, nor is it a subfield of mathematics (Moore, 1992)

This book has a variety of pedagogical features designed to make it appeal to structors of statistics (as well as students) including the following:

in-Decision Trees Appearing on the opening page of each chapter, these very simple

flow charts identify the main characteristics of the descriptive or inferential procedures reviewed therein, guiding readers through what a given test does (e.g., mean compari-son), when to use it (i.e., to what research designs does it apply), and what sort of data

it analyzes (e.g., continuous) At the close of each chapter, readers are reminded to rely

Key Terms and Concepts Key terms (e.g., mean, variance) and concepts (e.g., dom sampling, central limit theorem) are highlighted throughout the text to gain read-ers' attention and to promote retention An alphabetical list of key terms (including the page number where each is first cited) appears at the end of every chapter

ran-Marginal Notes The reader's attention will occasionally be drawn by marginal notes-key concepts, tips, suggestions, important points, and the like-appearing in the margins of the text An icon III drawn from the book's cover design identifies these brief marginal notes

Straightforward Calculation of Descriptive and Inferential Statistics by Hand tistical symbols and notation are explained early in the book (chapter 1) All of the de-scriptive and inferential statistics in the book are presented conceptually in the context

Sta-of an example, and then explained in a step-by-step manner Each step in any tion is numbered for ease of reference (example: [2.2.3] refers to chapter 2, formula 2, step 3) Readers who have access to a basic calculator can do any statistical procedure presented in the book Naturally, step-by-step advice also teaches students to read, un-derstand, and use statistical notation as well as the statistical tables presented in Ap-pendix B Appendix A reviews basic mathematics and algebraic manipulation for those students who need a self-paced refresher course The second half of Appendix A dis-cusses math anxiety, providing suggestions and references to alleviate it

calcula-Data Boxes Specific examples of published research or methodological issues using germane statistical procedures or concepts appear in Data Boxes throughout the text

By reading Data Boxes, students learn ways in which statistics and data analysis are tools

to aid the problem solver To quote Box, they are tools for "learning and discovery."

Focus on Interpretation of Results and Presenting Them in Written Form All tistical procedures conclude with a discussion of how to interpret what a result actu- ally means These discussions have two points: what the test literally concludes about

sta-some statistical relationship in the data and what it means descriptively-how did ticipants behave in a study, what did they do? The focus then turns to clearly commu-nicating results in prose form Students will learn how to put these results into words for inclusion in American Psychological Association (APA) style reports or draft arti-cles I used this approach successfully in a previous book (Dunn, 1999) Appendix C, which provides a brief overview of writing APA style reports, gives special emphasis to properly presenting research results and statistical information

par-Statistical Power, Effect Size, and Planned and Post Hoc Comparisons Increasingly, consideration of statistical power and effect size estimates is becoming more common-place in psychology textbooks as well as journals I follow this good precedent by at-taching discussion of the strength of association of independent to dependent variables along with specific inferential tests (e.g., estimated omega-squared-c;)2 -is presented with the F ratio) In the same way, review of planned or post hoc comparisons of means are attached to discussions of particular tests I focus on conceptually straightforward approaches for doing mean comparisons (e.g., Tukey's Honestly Significant Difference

Project Exercises Each chapter contains a "Project Exercise," an activity that applies

or extends issues presented therein Project Exercises are designed to give students the opportunity to think about how statistical concepts can actually be employed in re-search or to identify particular issues that can render data analysis useful for the design

of experiments or the interpretation of behavior On occasion, a chapter's Project ercise might be linked to a Data Box

Ex-End-of-Chapter Problems Each chapter in the text concludes with a series of lems Most problems require traditional numerical answers, but many are designed to help students think coherently and write cogently about the properties of statistics and data Answers to the odd-numbered problems are provided in the back of the textbook

prob-in Appendix E

Special Appendixes Beyond the traditional appendixes devoted a review of basic math (with suggestions about combating math anxiety; Appendix A), statistical tables (Appendix B), and answers to odd-numbered end-of-chapter problems (Appendix E),

I also include three more specialized offerings Appendix C presents guidance on ing up research in APA style, highlighting specific ways to write and cogently present statistical results Advice on organizing a research project using statistics and data analy-sis is presented in Appendix D I emphasize the importance of being organized, how to manage time, and-most importantly-how to prepare raw data for analysis in this ap-pendix Finally, Appendix F introduces qualitative research approaches as emerging al-ternatives-not foils-for the statistical analysis of data Though by no means com-monplace, such approaches are gradually being accepted as new options-really, opportunities-for researchers

Statistics and Data Analysis for the Behavioral Sciences has several supplements designed

to help both instructors and students These supplements include:

Elementary Data Analysis Using Microsoft Excel by Mehan and Warner (2000) This easy to use workbook introduces students to Microsoft Excel speadsheets as a tool to

be used in introductory statistics courses By utilizing a familiar program such as cel, students can concentrate more on statistical concepts and outcomes and less on the mechanics of software

Instructor's Manual and Test Bank The book has a detailed Instructor's Manual

(1M) and Test Bank (TB) The 1M includes syllabus outlines for one- or two-semester statistics courses, detailed chapter outlines, key terms, lecture suggestions, sugges-tions for classroom activities and discussions, film recommendations (where avail-able and appropriate), and suggested readings for the instructor (i.e., articles and books containing teaching tips, exercises) The TB contains test items (i.e., multiple choice items, short essays, problems), and is also available on computer diskette for

PC and Macintosh

Dedicated Website The book has a dedicated website (www.mhhe.com.dunn) so that potential instructors can examine a synopsis of the book, its table of contents, descrip-tions of the available supplements, and ordering information Links to other sites on the Web related to statistics, data analysis, and psychology (including links to other parts

of the McGraw-Hill site) are available In addition, portions of the Instructor's Manual

and Test Bank appear on the website and are "password" accessible to instructors who have selected the text and their students The website also has an online SPSS guide, which is an alternative to the expensive printed guides Beginning with computing a correlation between two variables and a continuing with t tests, ANOVAs, and chi-square, this site will help your students understand the basics of the SPSS program

Study Guide for Statistics and Data Analysis for the Behavioral Sciences

Instruc-tors (or students) can order a study guide to accompany Statistics and Data Analysis for the Behavioral Sciences The Study Guide contains a review of key terms, concepts, and

practice problems designed to highlight statistical issues Answers to any problems will

be provided in the back of the Study Guide

xxvi

ACKNOWlrDCJf'\rNTS

riters of statistics books require willing, even charitable, readers of rough drafts

My colleagues and friends, Stacey Zaremba, Matthew Schulz, and Robert Brill, read and commented on most of the chapters in this book Peter von Allmen and Jeanine S Stewart provided valuable suggestions regarding specific issues and chapters Dennis Glew and Clif Kussmaul improved the clarity of some examples During spring

1999, several students in my Statistics and Research Methods class took the time to read initial drafts of the first half of the book, and their subsequent suggestions refined the material The Reference Librarians and the Interlibrary Loan Department of Reeves Library helped me to track down sometimes obscure materials or references Ever patient, Jackie Giaquinto shepherded the manuscript and me through our appropriate paces, and reminded me of my other responsibilities Sarah Hoffman helped to pull bits and pieces of the manuscript together at the end of the revision process I want to express my gratitude to the Moravian College Faculty Development and Research Committee for the summer grant that enabled me to finish the book on time My friend, Steve Gordy, studiously avoided reading anything this time round, but his support was there, and welcome, nonetheless

Beyond my campus, I am very grateful to the constructive comments and criticism offered by an excellent group of peer reviewers, including:

Charles Ansorge

University of Nebraska-Lincoln

Phillip J Best Miami University, Ohio

recommenda-I remain convinced that the professionals who work there are rare and true recommenda-I hope our relationship is a long one My editor and friend, Joe Terry, established the project's vision, and then developmental editor, Susan Kunchandy, helped to move it forward Editorial director Jane Vaicunas continued to show confidence in my work Barbara Santoro-a tireless and dedicated individual-answered all my queries, organized end-less details, and provided help at every turn Marketing manager Chris Hall provided sage advice about the book's development in its later stages Project manager Susan Brusch steered the book (and me) skillfully through the production schedule Wayne Harms created the book's elegant and clear design I am grateful to copy editor, Pat Steele, for consistently improving my prose

Finally, my family-past and present-enabled me to write this book Daily, my wife, Sarah, and my children, Jake and Hannah, reminded me of the link between love and work I am grateful for their patience and good humor Dah K Dunn's faith in my writing was as steadfast as ever I dedicate this book to two fine men from my family

re-to the publisher, who will share it with me You may also contact me directly at the partment of Psychology, Moravian College, 1200 Main Street, Bethlehem, PA 18018-6650; via e-mail: dunn@moravian.edu.lsincerely look forward to hearing from you

made about tlit population

Which Scale of Measurement Is Being Used?

Are differences between num-bers based on

an equal unit of measurement?

If yes,then

go to 2

If'NE;'lhen this is an

C H A P T E R 1

INTRODUCTION: STATISTICS AND

DATA ANALYSIS AS TOOLS

n my view statistics has no reason for existence except as a catalyst for learning and

discovery." This quotation from George Box serves as the guiding rationale for the

book you are now reading Here at the outset, it is essential for you to understand

that statistics are aids to improving inference, guides that help us to make sense out of

events in the world in particular ways As an undergraduate student in psychology or a

related behavioral science discipline, you should know that statistics and data analysis

can help you to answer focused questions about cause and effect, to simplify complexity,

to uncover heretofore unrecognized relationships among observations, and to make

more precise judgments about how and why people behave the way they do This book

will teach you about some of the theory behind statistics and the analysis of data

through a practical, hands-on approach As you read, you will learn the"how to" side of

statistics and data analysis, including:

• How to select an appropriate statistical test

• How to collect the right kinds of information for analysis

• How to perform statistical calculations in a straightforward, step-by-step manner

• How to accurately interpret and present statistical results

• How to be an intelligent consumer of statistical information

• How to write up analyses and results in American Psychological Association (APA) style

Linking theory with practice will help you to retain what you learn so that you can use it

in future courses in psychology or the other behavioral sciences, research projects,

grad-uate or professional school, or any career where problem solving, statistics, and data

analysis are used

But we are getting ahead of ourselves First, we need to define some terms, terms

that have been used as if you already understood them! What is a statistic, anyway? What

is data analysis? Why are these terms important?

• Statistics Is the Science of

Not Mathematics

• Statistics, Data Analysis, and Scientific Method Inductive and Deductive Reasoning

Populations and Samples Descriptive and Inferential Statistics

Data Box I.B: Reactions to David L Problem Knowledge Base

• Discontinuous and COlltil1lUOI

Knowledge Base

• Overview of Statistical What To Do When: Rules of Priority

Data Box I.D: The Size of Numbers Is Relative

Mise En Place About Calculators

Knowledge Base Project Exercise: Avoiding Statisticophobia

• Looking Forward, Then Back

• Summary

• KeyTerms

4 Chapter 1 Introduction: Statistics and Data Analysis as Tools for Researchers

KEY T E R M A statistic is some piece of information that is presented in numerical form For example, a nation's

5% unemployment rate is a statistic, and so is the average number of words per minute read by a group of second-graders or the reported high temperature on a July day in Juneau, Alaska

appro-

., - - '.:":', '

"Tonight, were going to let the statistics speakfor themselves "

Source: The Cartoon Bank: Ed Koren, The New Yorker

Data analysis refers to the systematic examination of a collection of observations The examination

can answer a question, search for a pattern, or otherwise make some sense out of the observations These observations are either numerical (i.e., quantitative) or not based on numbers (i.e., qualitative) If the observations are quantitative-for example, number of puzzles solved in an experiment-statistics are frequently used in the data analysis What was the highest number solved? The lowest? In the case of qualitative information, some organiz-ing principle-identifying the emotional content of words exchanged between husbands and wives, for instance-can draw meaning from the observations Do women use words that establish relationships, whereas men rely on words that express their individuality? Despite popular opinion, as terms, statistics and data analysis are neither synony-mous nor redundant with one another For our purposes, the first term emphasizes the importance of working through necessary calculations in order to identify or discover relationships within the quantitative results of research The second term, however, ac-knowledges the interpretive, methodological, or analytic side of working with informa-tion-knowing, for instance, what information to collect and what to do with it once it

is collected Unlike the term statistics, data analysis also allows for the possibility that not

Quantitative relationships are

numerical Qualitative relationships

are based on descriptions or

organizing themes, not numbers

all the information you encounter or are interested in will necessarily be quantitative in nature As we will see later in this book (Appendix F), qualitative-that is, non-numerical, often descriptive or narrative-relationships within data can be equally re-vealing and, increasingly, social scientists are taking an active interest in them

Key terms like these will be highlighted throughout the book Whenever you come across a new term, plan to take a few minutes to study it and to make sure that you un-derstand it thoroughly Why? Because learning the vocabulary and conceptual back-ground of statistics is akin to learning a foreign language; before you can have an actual conversation in, say, French or German, you need to know how to conjugate verbs, use pronouns, recognize nouns, and so on Like learning the parts of speech in another lan-guage, it takes a bit of time and a little effort to learn the language of statistics As you will see, it can be done-you will learn to use this new language, understand, and even ben-efit from it One important point, though: the more you work with the statistical terms and their meanings, the more quickly they will become second nature to you-but you

must make the effort starting here and now Similar to studying a foreign language, tistical concepts build upon one another; learning the rudimentary parts of speech, as it were, is essential to participating in the more complex dialog that comes later

sta-What Is or Are Data?

Inexplicably, the word data has developed a certain cachet in contemporary society Although it is usually associated with science, use of the word is now common in everyday speech As a"buzz" word, it seems to lend an air of credibility to people's pronouncements on any number of topics But what does the word data actually mean? "Data" refer to a body of information, usually a col-lection of facts, items, observations, or even statistics The word is Latin in origin, meaning "a thing given." Thus, medical information from one patient, such as heart rate, blood pressure, and weight, constitute data, as does the same information from everyone admitted to a hospital in the course of a year Conceptually, then, the term is flexible

The grammatical usage of data, however, is proscribed How so? The word "data" is the word datum, which means a piece of information, is singular So, all the medical entries on a patient's chart are data, whereas the patient's weight when admitted to the ward-say, 1651b-is a datum Why does this distinction matter? When writing about or describing data, you will want to

plural-be both correct and precise Data are, datum is:

"These data are flawed." (correct)

"This data is flawed." (incorrect)

"The datum is flawed." (correct)

"These data were helpful." (correct)

"The data was helpful." (incorrect)

"The datum helped." (correct)

I urge you to listen carefully to how your friends, faculty, and family members use the term data-usually incorrectly, I'll wager-not to mention newscasters and some newspaper colum-nists, all professionals who should know better Do your best to turn the tide by resolving to use the terms data and datum correctly from now on

Statistics and data analysis are tools that behavioral scientists use to understand the sults of the research they conduct As a tool, a statistical analysis is simply the means to accomplish some task-it is by no means as important as that task (Dunn, 1999) Stu-dents often see statistics as a hindrance, not a help, as something much more involved

re-6 Chapter 1 Introduction: Statistics and Data Analysis as Tools for Researchers

than the question they are trying to answer or the topic they are exploring In fact, time statistics students can sometimes feel overwhelmed by the trappings of statistical analysis-the formulas, the math, the tables and graphs-so that they lose sight of what statistics are supposed to offer as a way of looking at things Remember the message from George Box that appeared earlier-to paraphrase him, statistics are for something, they are supposed to enlighten us, to help us discover things For most people, teachers like

first-me and students like you, they are not ends in themselves

Let's consider an example of how statistics can shed some light on a decision Read the following "problem;' as it deals with a situation you probably know firsthand After you read the problem and think about it, take out a piece of paper and answer the ques-tion that appears below

College Choice

David 1 was a senior in high school on the East Coast who was planning to go to college

He had completed an excellent record in high school and had been admitted to his two top choices: a small liberal arts college and an Ivy League university The two schools were about equal in prestige and were equally costly Both were located in attractive East Coast cities, about equally distant from his home town David had several older friends who were attending the liberal arts college and several who were attending the Ivy league university They were all excellent students like himself and had interests that were sim-ilar to his His friends at the liberal arts college all reported that they liked the place very much and that they found it very stimulating The friends at the Ivy League university re-ported that they had many complaints on both personal and social grounds and on ed-ucational grounds David thought that he would initially go to the liberal arts college However, he decided to visit both schools for a day He did not like what he saw at the private liberal arts college: Several people whom he met seemed cold and unpleasant; a professor he met with briefly seemed abrupt and uninterested in him; and he did not like the "feel" of the campus He did like what he saw at the Ivy League university: Several of the people he met seemed like vital, enthusiastic, pleasant people; he met with two dif-ferent professors who took a personal interest in him; and he came away with a very pleasant feeling about the campus

Question Which school should David 1 choose, and why? Try to analyze the ments on both sides, and explain which side is stronger (Nisbett, Krantz, Jepson, & Fong, 1982,pp.457-458)

argu-College Choice: What Would (Did) You Do?

Where should David 1 go to school? More than one waggish student has remarked that there is really no decision here: He should just go to the Ivy League university because his diploma will carry weight in the world four years hence! Other students compare the virtues of small versus large campuses-that is, intimate settings (i.e., you're noticed) are more desirable than sprawling ones (i.e., you're just a number) Let's try to approach the problem more critically-that is, statistically-and with fewer preconceptions about the two types of schools

The situation faced by the fictional David 1 is by no means unique: Many, perhaps most, college applicants must eventually select one school over another or possible oth-ers As you may well know, such decisions are rarely easy to make, and any number of factors-what are typically called variables-can be influential

KEY T E R M A variable is any factor that can be measured or have a different value Such factors can vary from

person to person, place to place, or experimental situation to experimental situation Hair color can

!'

I

be a variable (Le., blonde, brunette, redhead), as can a score on a personality test, the day of the week, or your weight

As we will see later in the chapter, statistics usually rely on variables X and Y to represent numerical values in formulas or statistics about data

Several variables stand out in the David L problem First and foremost, David's friends at the liberal arts institution were generally satisfied, as they told him they liked

it and even found it to be a stimulating place His pals at the Ivy League school, however, reported just the opposite, voicing complaints and qualifications on personal, social, and educational grounds You will recall that David planned to go to the smaller school until visits at both places called his initial decision into question-he liked what he saw at the university but had an unpleasant time at the college In short, his experiences were the opposite of his friends' experiences

What else do we know? Well, a few factors appear to be what are called constants, not variables

KEY T E R M A constant is usually a number whose value does not change, such as 11" (pronounced"pie"), which

equals 3.1416 A constant can also refer to a characteristic pertaining to a person or environment that does not change

Variables take on different values,

constants do not change

We know, for example, that David's friends attending both schools were strong students (like himself, apparently) and shared outlooks like his own In other words, intellectual ability appears to be a constant, as David is not very different from his friends Yet we know he had decidedly different experiences than they at the two schools We also know that both schools are equally prestigious, cost about the same, are metropolitan, and are equidistant from his home These normally important factors do not appear to be very influential in David's decision making because they, too, are constants and, in any case,

he is more focused on his experiences at the schools than on money issues, location, or distance

Nonetheless, these constants do tell us something-and perhaps they should tell David something, as well In a sense, his friends are constants and because they are so similar to him, he might do well to pay close attention to their experiences and to won-der rather critically why his experiences when he visited were so different from their own What would a statistician say about this situation? In other words, could a rudi-mentary understanding of statistics and the properties of data help David choose be-tween the small liberal arts college and the Ivy League university?

Approaching the problem from a statistical perspective would highlight two cepts relevant to David L.'s data-base rate and sampling Base rate, in this case, refers to the common or shared reactions of his friends to their respective schools; that is, those attending the liberal arts college liked the place, while those at the Ivy League school did not If we know that David L is highly similar to his friends, shouldn't we assume that across time he will have reactions similar to theirs, that he will like the college but not the university? Thus, the base rate experiences of his friends could reasonably be weighed more heavily in the college choice deliberations than his own opinion

con-Similarity of reaction leads to the second issue, that of sampling, which may explain why David's reactions were different from those of his peers Put simply, is one day's ex-posure to any campus sufficient to really know what it is like? Probably not, especially

when you consider that his friends have repeatedly sampled what the respective schools offer for at least a year, possibly longer (Consider these thought questions: Did you know what your present school was really like before you started? What do you know

now that you did not know then?) In other words, David really doesn't have enough formation-enough data-to make a sound choice His short visits to each campus were

in-8

Statistics and data analysis highlight

possible solutions or answers to

questions, not absolute or definitive

conclusions

Iml

Statistics ;t mathematics

Chapter 1 Introduction: Statistics and Data Analysis as Tools for Researchers

characterized by biased or distorted samples of information: he met few people and, on short visits, could not have learned all there was to know about either school Sampling issues will be explored in greater detail shortly (but see Data Box l.B later in the chapter for more detail on how people typically answer the David L problem and factors that can influence such answers)

What did you do? That is, how did you select your present college or university? Were you unduly (or appropriately) influenced by the respective variables and constants found on your own campus? Did the weather matter-was a crisp, bright autumn day in October more pleasant (and influential) than a cold, rainy March visit? Were people cold and aloof or warm and friendly? What factors unique to you and your school helped you decide to enroll there? Finally, was David's situation similar to or dissimilar from your own?

Here is the important point: there is no right answer, rather, there are possible swers, the choice of which depends on the decision-maker's perspective and goal An un-derstanding of statistics and data analysis provides only one account-though admit-tedlya good one-of how one might decide between two similar scholastic alternatives

an-If David L wanted to make a statistically sound inference, then he would assume his friends' experiences at the small liberal arts college or the Ivy League university were use-ful because they have repeatedly sampled what the respective institutions have to offer

He would also realize that his short-term experiences did not provide enough tion to make a decisive choice

informa-On the other hand, he could avoid taking any statistical considerations into account and rely on his own opinion, which would be a different but acceptable way to proceed Remember, statistics serve as inferential tools, as guidelines for a particular way to make a choice; the individual using them must interpret the result and decide how to proceed The statistical analysis itself has no opinion, nor will it enroll at the college

or university As a data analyst, you must decide how to use the results and to determine what they mean for you This interpretive choice is critical-and it is common to every statistical analysis you will do Statistics and data analyses are about thinking first, calcu- lation second

Mathematics is an ancient discipline, but statistical reasoning did not really occur until late in the 17th century (Cowles, 1989), and the bulk of statistical advances took place in the 19th century (Moore, 1992) Mathematics and statistics are actually separate disci-plines with very different agendas Many students and more than a few behavioral scientists are surprised to learn that statistics is not a part of mathematics Mathematics

is the science of quantity and space and the analysis of relationships and patterns, whereas statistics is the science of data, data production, and inference (e.g., Moore, 1992) Both disciplines, however, rely on symbols to show relationships between quanti-ties and differences among quantities

These definitions can seem to be similar, but let me illustrate the distinction tween the two disciplines If we take the simple average of the five numbers 80,85, 90, 91, and 94, which is 88 (i.e., 80 + 85 + 90 + 91 + 94 = 440, and 440/5 = 88), we are per-forming a mathematical operation If, however, those five numbers are test scores of gifted students in an advanced chemistry class, then the average becomes a statistical op-eration This is particularly true if we try to determine why some students performed better than others, or if we compare the students' performance with those of another gifted group, and so on

The field of statistics is concerned with making sense out of empirical data,

particu-larly when those data contain some element of uncertainty so that we do not know the true state of affairs, how, say, a set of variables affect one another

Empirical refers to anything derived from experience or experiment

Empiricism is a philosophical stance arguing that all knowledge is developed from sory experience Indeed, one can verify what the world is like by experiencing it I know the floor is solid because I am presently standing on it If I had doubts about its struc-tural integrity, I could do an experiment by testing how much weight it would support before buckling The philosophical doctrine seen as the traditional foil to empiricism is called rationalism Rationalism posits that reason is the source of all knowledge, and that such knowledge is completely independent of sensory experience, which is deemed faulty (Leahey, 1997)

sen-Certainly it is the case that statistics relies on mathematical operations, but these operations are secondary to the process of reasoning behind statistics In fact, I always tell students in my statistics classes that the meaning behind the data, the inferences we make about the data, are more important than the math Please don't miss the subtle message here: Understanding how to do a mathematical procedure is very useful, as is getting the"right" answer, but these facts will not do you much good if you cannot inter-pret the statistical result Thus, the ideal you should strive for is the ability to select an ap-propriate statistic, perform the calculation, and to know what the result means

Some students who take a first statistics course are concerned about whether their math background is sufficient to do well in the class Other students are actually fearful

of taking statistics precisely because they believe that they will do poorly because of the math involved If you fall into either group, let me offer some solace First, I firmly be-lieve that if you can balance your checkbook, then you will be able to follow the formu-las and procedures presented in this book Second, Appendix A contains a review of sim-ple mathematics and algebraic manipulation of symbols if you feel that your math skills are a little rusty If so, consult Appendix A after you finish reading this chapter Third, you may actually be experiencing what is commonly called math anxiety To help with the latter possibility, a discussion of this common-and readily curable-form of anxiety, as well as references, can also be found in Appendix A Finally, a project exercise presented at the end of this chapter will help you to overcome the normal trepidation students feel when studying statistics for the first time

1Il1 Statistics, Data Analysis, and the Scientific Method

Researchers in the behavioral sciences use statistics to analyze data collected within the framework of the scientific method There are numerous definitions for this method, but most of them entail similar elements

KEY T E R M The scientific method guides research by identifying a problem, formulating a hypothesis, and

col-lecting empirical data to test the hypothesis,

KEY TERM

The only new term here is hypothesis, and it may already be a familiar concept to you

A hypothesis is a testable question or prediction, one usually designed to explain some

phenomenon

A developmental psychologist who is interested in how infants learn, for example, would rely on the scientific method to test her hypotheses One perceptual hypothesis is that infants are most interested in information that is moderately novel Infants like to

10 Chapter 1 Introduction: Statistics and Data Analysis as Tools for Researchers

look at objects that are not too familiar or that are not too novel; the former are boring and the latter can be confusing (McCall, Kennedy, & Appelbaum, 1977) By presenting a group of young children with different groupings of objects (e.g., blocks with patterns) representing different degrees of familiarity, the researcher could measure their interest-how long they look at each object, for instance Different hypotheses examin-ing the same issue are then combined to form what is called a theory

KEY T E R M A theory is a collection of related facts, often derived from hypotheses and the scientific method,

forming a coherent explanation for a larger phenomenon

KEY TERM

Induction: data lead to theory

One theory is that infants' interests in novelty also reveal their innate preferences for learning Some researchers have suggested that preferences for novelty, in turn, are linked with intelligence Another theory suggests that intelligent infants are drawn to novel information more readily than less intelligent infants (e.g., Bornstein & Sigman, 1986) Note that these theories were developed by examining a variety of hypotheses and the results of many studies about how infants perceive the objects they encounter A re-searcher using any theory would be aware of the existing data and would make certain to use the scientific method and careful reasoning before executing research aimed at test-ing any hypotheses consistent with it

Inductive and Deductive Reasoning

What sort of reasoning underlies the scientific method? The scientific community uses two types of reasoning, inductive and deductive Depending on what is already known about a research problem or theory, each form of reasoning serves a different function

The first, inductive reasoning, is also referred to as induction

Generalizing from one or more observations in the course of developing a more general explanation

is called inductive reasoning Observations are used to generate theories

No doubt more than one developmental psychologist noticed that infants look for longer periods of time at moderately novel objects than with very familiar or completely unfamiliar displays Once the psychologist noticed this modest behavioral discrepancy and then generated an explanation, which required induction-the data prompted the development of a theory Such induction is ideal for developing preliminary hypotheses, which in turn can be refined into a coherent theory

Formulating related concepts, generalizing from one instance to another, or making some prediction based on what is already known, are all examples of inductive reason-ing (Holland, Holyoak, Nisbett, & Thagard, 1986) Induction is a powerful tool, but it can miss influential factors that could be responsible for whatever interesting phenome-non we are studying It would be a mistake, for example, to assume that all the learning infants do is based on their abilities to visually discriminate the new from the old How, after all, do children with visual impairments learn? The other sensory processes, espe-cially hearing, must also playa prominent role in the acquisition of knowledge Separate but complementary theories must be developed to explain auditory and other forms of learning, as well

Psychology and the other behavioral sciences-economics, education, sociology, and anthropology-often rely on inductive reasoning, which is the hallmark of newer sciences Why is inductive reasoning associated with these newer areas of empirical in-quiry? In contrast to the natural sciences, the behavioral sciences usually lack unified theories and research methodologies Psychology, for example, does not have a single, dominant approach that purports to explain the cognitive, emotional, physiological, and behavioral aspects of human behavior Rather, there are many distinct approaches that

I

i

Deduction: theory leads to data

seek to explain individual aspects of human behavior-just look at the table of contents

of any introductory textbook in the field-suggesting that we are far from having a fied position that ties them all together (Kuhn, 1970, 1977; Watson, 1967)

uni-Much older areas of science, especially physics, have unified theories that enable them to employ the second type of reasoning, which is called deductive

Deductive reasoning is characterized by the use of existing theories to develop conclusions, called

deductions, about how some unexamined phenomenon is likely to operate Theory is used to search for confirming observations

Deduction promotes a particular type of prediction: whenever event X occurs, event Y

usually follows Deductive reasoning is essentially fact-based reasoning, so that what we already know points us in the direction of what we can also expect to be true In physics, for example, Albert Einstein created the theory of relativity, which, among other things, posited that time was not absolute and that it depended on an object's state of motion Initially, there was no experimental evidence to support his radical contentions, but across the 75 or so years since it first appeared, Einstein's theory has been overwhelm-ingly supported by experimental data From theory, then, Einstein deduced how time and motion should behave

As I am sure you recognize, our understanding of how infants learn is not as finely honed as what we know about time, motion, or the speed oflight Numerous theories in the behavioral sciences are still being developed, tested, revised, or discarded in accor-dance with empirical data and inductive reasoning Until a generally accepted theory of human behavior based on facts arrives (if ever!), we will need to be content with induc-tive reasoning, and the statistical analyses which allow us to verify our induction Figure 1.1 illustrates the direction of inference inherent in inductive and deductive reasoning As you can see, when observations lead a researcher to create a theory to ex-plain some event, the process is inductive Incidentally, the inferences David L (and you) made about college choice were largely inductive When an investigator relies on some existing theory to posit the existence of particular observations, the process is deductive Let's turn now to consider how our ability to use these two types of reasoning can help

us to determine when an observation is representative of the whole

12 Chapter 1 Introduction: Statistics and Data Analysis as Tools for Researchers

Populations and Samples

The idea of an adequate sample of data was informally introduced in the context of the David L problem earlier in this chapter In David's case, we were working with a more or less intuitive sense of what constitutes a good sample: How much of a campus do you have to see to know something about it? Are others' views useful as a sample of opinion about a campus, even when those views contradict your own experience? We now want

to consider samples in a somewhat more formal, statistical sense

Researchers usually talk about samples when they are trying to determine if some data properly characterize the population of interest

KEY T E R M A population is a complete set of data possessing some observable characteristic, or a theoretical

set of potential observations

Perhaps because the word is commonly associated with groups of people-the tion of a city or country, for instance-students are quick to assume that the term is ex-clusively demographic Keep in mind that a population is any complete set of data, and

popula-these data can be animal, vegetable, or mineral Test scores can comprise a population, as can birthrates of Monarch butterflies in Nebraska, sales figures for the East Coast fishing industry, or all the words printed in a given book like this one Typically, of course, some numerical characteristic of the population will be used in statistical calculations When psychologists study people's behavior, they typically want to describe and un-derstand the behavior of some population of people The behavior studied can be an ac-tion, an attitude, or some other measurable response Psychologists may talk about pop-ulations of people, but keep in mind that they are usually focused on a population of some characteristic displayed by the population of people A developmental psychologist studying preschool social relations might work with a population of children who are 5 years of age or younger, observing the content of comments made by one peer to an-other A gerontologist interested in memory decline and age could examine information processing speed and efficiency in recalling words among persons 80 years of age or older Again, note that the term population does not literally mean"every person"; rather, it means the numerical responses for observations-here, comments or words-

of every person within some identified group (e.g., children under age 5 years or persons

80 years or older)

When any psychologist examines the behavior of interest in a population, he or she cannot examine the responses of all members of that population A given clinical psy-chologist who does therapy with people who have agoraphobia (i.e., fear of crowds or public spaces) does not literally work with every person who has the disorder The group

of people receiving research attention constitutes a sample from the population of ple who have agoraphobia

peo-KEY T E R M A sample is a smaller unit or subset bearing the same characteristic or characteristics of the

pop-ulation of interest

When researchers collect data from a sample, they hope to be able to demonstrate that the sample is representative of-is highly similar to-the population from which it was drawn Why is it necessary to rely on a sample? Practically speaking, most popula-tions are simply too large and unwieldy to allow a researcher to gauge every observation within them Such undertakings would be too expensive, too time consuming, not feasible, and given good samples-the observations within them reflect the characteris-tics of their populations of origin-completely unnecessary anyway Thus, the reactions

of the group of agoraphobics to a new therapy are assumed to hold true in general for

Figure 1.2 Samples Are Drawn from Some Population

Note: A sample is a subset of members of some larger population Samples are used to learn what populations are like

the population of all extant (or potential) agoraphobics Similarly, a good sample of year-old men and women should be sufficient to illustrate how aging affects processing speed and memory In both cases, what we learn from a sample should enable us to ac-curately describe the population at large Figure 1.2 illustrates this process: Samples are drawn from a population in order to discern what the population is like

80-Wait a moment-what constitutes a good sample? How do we know if a sample approximates the characteristics of the population from which it was drawn? These and related questions are actually the foundation of statistics and data analysis, as all we will really do throughout this book is variations on this same theme: Does our sample of be-havior accurately reflect its population of origin? Will an observed change in behavior in

a sample reflect a similar change in behavior within the population? To answer these questions, researchers rely on statistics and what are called popUlation parameters

A population parameter is a value that summarizes some important, measurable characteristic of

a population Although population parameters are estimated from statistics, they are constants

For all intents and purposes, we will probably never know the true parameters of any population unless it is reasonably small or extensive research funds are available When you hear advertisers say, "four out of five dentists recommend" a mouthwash or toothpaste, for example, not all practicing dentists were asked to give an opinion! Gen-erally, then, researchers must content themselves with estimating what the parameters are apt to be like Many populations have parameters that could never be measured be-cause their observations are constantly changing Consider the number of people who are born and die in the United States each minute of every day-the American popula-tion is theoretically the same from moment to moment, but in practical terms it is ever changing Despite this apparent change, we can still estimate the average height and weight of most Americans, as well as their projected life spans from sample statistics That is, sample statistics enable us to approximate the population parameters

When we discuss sample statistics and their relations to population parameters, the former term takes on added meaning

KEY T E R M A sample statistic is a summary value based upon some measurable characteristic of a sample

The values of sample statistics can vary from sample to sample