Psychological Testing: Principles, Applications, and Issues

  • 86 11,981 10
  • Like this paper and download? You can publish your own PDF file online for free in a few minutes! Sign Up

Psychological Testing: Principles, Applications, and Issues

SEVENTH EDITION Psychological Testing Principles, Applications, and Issues This page intentionally left blank SEVEN

21,141 15,067 11MB

Pages 744 Page size 252 x 316.08 pts Year 2010

Report DMCA / Copyright

DOWNLOAD FILE

Recommend Papers

File loading please wait...
Citation preview

SEVENTH EDITION

Psychological Testing Principles, Applications, and Issues

This page intentionally left blank

SEVENTH EDITION

Psychological Testing Principles, Applications, and Issues

Robert M. Kaplan University of California, Los Angeles

Dennis P. Saccuzzo San Diego State University

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

Psychological Testing: Principles, Applications, and Issues, Seventh Edition

© 2009, 2005 Wadsworth, Cengage Learning

Robert M. Kaplan, Dennis P. Saccuzzo

ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or

Editor: Jaime Perkins

by any means graphic, electronic, or mechanical, including but not

Editorial Assistant: Wilson Co

limited to photocopying, recording, scanning, digitizing, taping, Web

Technology Project Manager: Amy Cohen

distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the

Marketing Manager: Kim Russell Marketing Assistant: Molly Felz

1976 United States Copyright Act, without the prior written permission of the publisher.

Marketing Communications Manager: Talia Wise Project Manager, Editorial Production: Charlene M. Carpentier

For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706.

Creative Director: Rob Hugel

For permission to use material from this text or product, submit all requests online at cengage.com/permissions. Further permissions questions can be e-mailed to [email protected]

Art Director: Vernon Boes Print Buyer: Linda Hsu Permissions Editor: Bob Kauser Production Service: Newgen–Austin Text Designer: Lisa Henry Photo Researcher: Darren Wright

Library of Congress Control Number: 2008927883 Student Edition: ISBN-13: 978-0-495-09555-2 ISBN-10: 0-495-09555-9

Copy Editor: Mary Ann Grobbel Cover Designer: Larry Didona Cover Image: “Geometric shapes below clouds” ©Pete Turner/The Image Bank/Getty Images

Wadsworth 10 Davis Drive Belmont, CA 94002-3098 USA

Compositor: Newgen Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at international.cengage.com/region. Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit academic.cengage.com. Purchase any of our products at your local college store or at our preferred online store www.ichapters.com.

Printed in the United States of America 1 2 3 4 5 6 7 12 11 10 09 08

BRIEF CONTENTS

PA R T I

1 2 3 4 5 6 7 PA R T I I

PR I NCI PLES Introduction 1 Norms and Basic Statistics for Testing 25 Correlation and Regression 65 Reliability 101 Validity 133 Writing and Evaluating Test Items 157 Test Administration 185

APPLIC ATIONS

8 Interviewing Techniques 201 9 Theories of Intelligence and the Binet Scales 229 10 The Wechsler Intelligence Scales: WAIS-III, WISC-IV, 11 12 13 14 15 16 17 18 PA R T I I I

and WPPSI-III 249 Other Individual Tests of Ability in Education and Special Education 273 Standardized Tests in Education, Civil Service, and the Military Applications in Clinical and Counseling Settings 333 Projective Personality Tests 373 Computers and Basic Psychological Science in Testing 403 Testing in Counseling Psychology 433 Testing in Health Psychology and Health Care 451 Testing in Industrial and Business Settings 483

303

ISSUES

19 Test Bias 511 20 Testing and the Law 545 21 Ethics and the Future of Psychological Testing 585 v

CONTENTS

PA R T I P R I N C I P L E S

1

Introduction 1 Basic Concepts 6 What a Test Is 6 Types of Tests 7

Overview of the Book 9 Principles of Psychological Testing 10 Applications of Psychological Testing 10 Issues of Psychological Testing 11

Historical Perspective 11 Early Antecedents 11 Charles Darwin and Individual Differences 12 Experimental Psychology and Psychophysical Measurement 12 The Evolution of Intelligence and Standardized Achievement Tests 14 Personality Tests: 1920–1940 17 The Emergence of New Approaches to Personality Testing 18 The Period of Rapid Changes in the Status of Testing 20 The Current Environment 21

Summary 22

2

Norms and Basic Statistics for Testing 25 Why We Need Statistics 26 Scales of Measurement 27

vi

CONTENTS Properties of Scales 27 Types of Scales 29 Permissible Operations 30

Frequency Distributions 31 Percentile Ranks 34 Percentiles 38 Describing Distributions 39 Mean 39 Standard Deviation 40 Z Score 42 Standard Normal Distribution 45 McCall’s T 50 Quartiles and Deciles 51

Norms

53

Age-Related Norms 54 Tracking 55 Criterion-Referenced Tests

Summary

3

60

63

Correlation and Regression 65 The Scatter Diagram 66 Correlation 68 Regression 69 The Regression Line 69 The Best-Fitting Line 71 Testing the Statistical Significance of a Correlation Coefficient 76 How to Interpret a Regression Plot 78

Other Correlation Coefficients 82 Terms and Issues in the Use of Correlation 84 Residual 84 Standard Error of Estimate 85 Coefficient of Determination 85 Coefficient of Alienation 85 Shrinkage 86 Cross Validation 86 The Correlation-Causation Problem Third Variable Explanation 88 Restricted Range 88

Multivariate Analysis (Optional)

87

88

General Approach 89 An Example Using Multiple Regression Discriminant Analysis 91 Factor Analysis 91

90

vii

viii

CONTENTS

Summary 94 Appendix 3.1: Calculation of a Regression Equation and a Correlation Coefficient 95 Calculation of a Regression Equation (Data from Table 3.5) 95 Calculation of a Correlation Coefficient (Data from Table 3.5) 98

4

Reliability 101 History and Theory of Reliability 102 Conceptualization of Error 102 Spearman’s Early Studies 103 Basics of Test Score Theory 103

The Domain Sampling Model Item Response Theory 107 Models of Reliability 108

105

Sources of Error 109 Time Sampling: The Test-Retest Method 109 Item Sampling: Parallel Forms Method 111 Split-Half Method 111 KR 20 Formula 113 Coefficient Alpha 115 Reliability of a Difference Score 116

Reliability in Behavioral Observation Studies 120 Connecting Sources of Error with Reliability Assessment Method 121 Using Reliability Information 124 Standard Errors of Measurement and the Rubber Yardstick How Reliable Is Reliable? 125 What to Do About Low Reliability 125

124

Summary 129 Appendix 4.1: Using Coefficient Alpha to Estimate Split-Half Reliability When the Variances for the Two Halves of the Test Are Unequal 130 Appendix 4.2: The Calculation of Reliability Using KR20 130

5

Validity

133

Defining Validity 135 Aspects of Validity 135 Face Validity 135 Content-Related Evidence for Validity 136 Criterion-Related Evidence for Validity 137 Construct-Related Evidence for Validity 147 Relationship Between Reliability and Validity 154

Summary

155

CONTENTS

6

Writing and Evaluating Test Items 157 Item Writing 158 Item Formats 159 Other Possibilities 168

Item Analysis 170 Item Difficulty 171 Discriminability 172 Pictures of Item Characteristics 174 Linking Uncommon Measures 180 Items for Criterion-Referenced Tests 181 Limitations of Item Analysis 182

Summary

7

183

Test Administration 185 The Examiner and the Subject

186

The Relationship Between Examiner and Test Taker The Race of the Tester 187 Language of Test Taker 188 Training of Test Administrators 189 Expectancy Effects 189 Effects of Reinforcing Responses 191 Computer-Assisted Test Administration 193 Mode of Administration 195 Subject Variables 196

Behavioral Assessment Methodology 197 Reactivity 197 Drift 198 Expectancies 198 Deception 198 Statistical Control of Rating Errors 199

Summary

PA R T I I

8

200

A P P L I C AT I O N S Interviewing Techniques 201 The Interview as a Test 204 Reciprocal Nature of Interviewing 205 Principles of Effective Interviewing 206 The Proper Attitudes 206 Responses to Avoid 207 Effective Responses 208 Responses to Keep the Interaction Flowing Measuring Understanding 213

209

186

ix

x

CONTENTS

Types of Interviews 215 Evaluation Interview 215 Structured Clinical Interviews 216 Case History Interview 220 Mental Status Examination 221 Developing Interviewing Skills 222

Sources of Error in the Interview 223 Interview Validity 223 Interview Reliability 225

Summary 226

9

Theories of Intelligence and the Binet Scales 229 The Problem of Defining Intelligence 230 Binet’s Principles of Test Construction 232 Principle 1: Age Differentiation 232 Principle 2: General Mental Ability 233

Spearman’s Model of General Mental Ability 233 Implications of General Mental Intelligence (g) The g f-gc Theory of Intelligence 234

234

The Early Binet Scales 235 The 1905 Binet-Simon Scale 235 The 1908 Scale 236

Terman’s Stanford-Binet Intelligence Scale 238 The 1916 Stanford-Binet Intelligence Scale 238 The Intelligence Quotient (IQ ) 238 The 1937 Scale 240 The 1960 Stanford-Binet Revision and Deviation IQ (SB-LM)

241

The Modern Binet Scale 242 Model for the Fourth and Fifth Editions of the Binet Scale Characteristics of the 1986 Revision 243 Characteristics of the 2003 Fifth Edition 245 Psychometric Properties of the 2003 Fifth Edition 247 Median Validity 248

Summary

10

243

248

The Wechsler Intelligence Scales: WAIS-III, WISC-IV, and WPPSI-III 249 The Wechsler Intelligence Scales 251 Point and Performance Scale Concepts

251

From the Wechsler-Bellevue Intelligence Scale to the WAIS-III 253 Scales, Subtests, and Indexes of the WAIS-III 253 The Verbal Subtests 254 Raw Scores, Scaled Scores, and the VIQ 258 The Performance Subtests 259

CONTENTS Performance IQs 261 FSIQs 261 Index Scores 261

Interpretive Features of the Wechsler Tests 262 Verbal-Performance IQ Comparisons Pattern Analysis 262 Hypothetical Case Studies 263

262

Psychometric Properties of the Wechsler Adult Scale 265 Standardization 265 Reliability 265 Validity 266

Evaluation of the Wechsler Adult Scales 266 Downward Extensions of the WAIS-III: The WISC-IV and the WPPSI-III 267 The WISC-IV 267 The WPPSI-III 270

Summary

11

271

Other Individual Tests of Ability in Education and Special Education 273 Alternative Individual Ability Tests Compared with the Binet and Wechsler Scales 274 Alternatives Compared with One Another 275 Specific Individual Ability Tests 277 Infant Scales 278 Major Tests for Young Children 285 General Individual Ability Tests for Handicapped and Special Populations 289

Testing Learning Disabilities 293 Visiographic Tests 298 Creativity: Torrance Tests of Creative Th inking (TTCT) 300 Individual Achievement Tests: Wide Range Achievement Test-3 (WRAT-3) 301

Summary

12

302

Standardized Tests in Education, Civil Service, and the Military 303 Comparison of Group and Individual Ability Tests 305 Advantages of Individual Tests 305 Advantages of Group Tests 306

Overview of Group Tests 307 Characteristics of Group Tests Selecting Group Tests 307 Using Group Tests 308

307

xi

xii

CONTENTS

Group Tests in the Schools: Kindergarten Through 12th Grade 309 Achievement Tests Versus Aptitude Tests 309 Group Achievement Tests 309 Group Tests of Mental Abilities (Intelligence) 312

College Entrance Tests 314 The SAT Reasoning Test 315 Cooperative School and College Ability Tests The American College Test 317

316

Graduate and Professional School Entrance Tests 318 Graduate Record Examination Aptitude Test Miller Analogies Test 322 The Law School Admission Test 322

318

Nonverbal Group Ability Tests 324 Raven Progressive Matrices 325 Goodenough-Harris Drawing Test (G-HDT) 328 The Culture Fair Intelligence Test 328 Standardized Tests Used in the U.S. Civil Service System 329 Standardized Tests in the U.S. Military: The Armed Services Vocational Aptitude Battery 330

Summary

13

330

Applications in Clinical and Counseling Settings 333 Strategies of Structured Personality-Test Construction 335 Deductive Strategies 336 Empirical Strategies 337 Criteria Used in Selecting Tests for Discussion

338

The Logical-Content Strategy 338 Woodworth Personal Data Sheet 339 Early Multidimensional Logical-Content Scales 339 Mooney Problem Checklist 339 Criticisms of the Logical-Content Approach 340

The Criterion-Group Strategy 340 Minnesota Multiphasic Personality Inventory 340 California Psychological Inventory (CPI)–Third Edition

351

The Factor Analytic Strategy 352 Guilford’s Pioneer Efforts 353 Cattell’s Contribution 353 Problems with the Factor Analytic Strategy 356

The Theoretical Strategy 356 Edwards Personal Preference Schedule (EPPS) 356 Personality Research Form (PRF) and Jackson Personality Inventory (JPI) 358 Self-Concept 360

CONTENTS

Combination Strategies 361 Positive Personality Measurement and the NEO Personality Inventory–Revised (NEO-PI-R) 361 The NEO Personality Inventory (NEO-PI-R) 362

Frequently Used Measures of Positive Personality Traits 365 Rosenberg Self-Esteem Scale 365 General Self-Efficacy Scale (GSE) 365 Ego Resiliency Scale 366 Dispositional Resilience Scale (DRS) 366 Hope Scale 366 Life Orientation Test–Revised (LOT-R) 367 Satisfaction with Life Scale (SWLS) 367 Positive and Negative Affect Schedule (PANAS) 368 Coping Intervention for Stressful Situations (CISS) 368 Core Self-Evaluations 368

Future of Positive Personality Research Summary 370

14

369

Projective Personality Tests 373 The Projective Hypothesis 375 The Rorschach Inkblot Test 376 Historical Antecedents 376 Stimuli, Administration, and Interpretation Psychometric Properties 383

377

An Alternative Inkblot Test: The Holtzman 391 The Thematic Apperception Test 391 Stimuli, Administration, and Interpretation Psychometric Properties 396

392

Alternative Apperception Procedures 397 Nonpictorial Projective Procedures 397 Word Association Test 397 Sentence Completion Tasks 398 Figure Drawing Tests 399

Summary

15

400

Computers and Basic Psychological Science in Testing 403 Cognitive-Behavioral Assessment Procedures 405 The Rationale for Cognitive-Behavioral Assessment Procedures Based on Operant Conditioning 407 Self-Report Techniques 409 Kanfer and Saslow’s Functional Approach 413

405

xiii

xiv

CONTENTS The Dysfunctional Attitude Scale 414 Irrational Beliefs Test 415 Cognitive Functional Analysis 415

Psychophysiological Procedures 417 Physiological Variables with Treatment Implications Evaluation of Psychophysiological Techniques 418

417

Computers and Psychological Testing 419 Computer-Assisted Interview 419 Computer-Administered Tests 420 Computer Diagnosis, Scoring, and Reporting of Results 421 Internet Usage for Psychological Testing 423 The Computerization of Cognitive-Behavioral Assessment 424 Tests Possible Only by Computer 425 Computer-Adaptive Testing 426 Psychophysical and Signal-Detection Procedures 427

Summary

16

430

Testing in Counseling Psychology 433 Measuring Interests 434 The Strong Vocational Interest Blank 435 The Strong-Campbell Interest Inventory 436 The Campbell Interest and Skill Survey 438 The Reemergence of the Strong Interest Inventory 439 The Kuder Occupational Interest Survey 440 The Career Assessment Inventory 445 The Self-Directed Search 445 Eliminating Gender Bias in Interest Measurement 446 Aptitudes and Interests 447

Measuring Personal Characteristics for Job Placement 447 Trait Factor Approach: Osipow’s Vocational Dimensions Are There Stable Personality Traits? 448

Summary

17

448

449

Testing in Health Psychology and Health Care 451 Neuropsychological Assessment 452 Clinical Neuropsychology 452 Developmental Neuropsychology 457 Adult Neuropsychology 460 California Verbal Learning Test (CVLT) 466 Automated Neuropsychological Testing 469 Anxiety and Stress Assessment 470 Stress and Anxiety 470 The State-Trait Anxiety Inventory 471

CONTENTS Measures of Test Anxiety 472 Measures of Coping 476 Ecological Momentary Assessment 476

Quality-of-Life Assessment 477 What Is Health-Related Quality of Life? 478 Common Methods for Measuring Quality of Life 479

Summary

18

481

Testing in Industrial and Business Settings 483 Personnel Psychology—The Selection of Employees 484 Employment Interview 484

Base Rates and Hit Rates 485 Taylor-Russell Tables 489 Utility Theory and Decision Analysis Incremental Validity 496

495

Personnel Psychology from the Employee’s Perspective: Fitting People to Jobs 498 The Myers-Briggs Type Indicator 498 Tests for Use in Industry: Wonderlic Personnel Test (WPT)

499

Measuring Characteristics of the Work Setting 500 The Social-Ecology Approach 501 Classifying Environments 502

Job Analysis 504 Measuring the Person–Situation Interaction 507 Summary 509

PA R T I I I I S S U E S

19

Test Bias

511

Why Is Test Bias Controversial? 512 Test Fairness and the Law 513 The Traditional Defense of Testing 517 Content-Related Evidence for Validity 517 Criterion-Related Sources of Bias 520

Other Approaches to Testing Minority Group Members 524 Ignorance Versus Stupidity 524 The Chitling Test 526 The Black Intelligence Test of Cultural Homogeneity 527 The System of Multicultural Pluralistic Assessment 528

xv

xvi

CONTENTS

Suggestions for Solutions 532 Ethical Concerns and the Definition of Test Bias 532 Thinking Differently: Finding New Interpretations of Data Developing Different Criteria 537

535

Changing the Social Environment 540 Summary 542

20

Testing and the Law 545 Laws Governing the Use of Tests 546 Federal Authorities 546 Specific Laws 555 No Child Left Behind (NCLB) 557

Major Lawsuits That Have Affected Psychological Testing 558 Early Desegregation Cases 558 Stell v. Savannah-Chatham County Board of Education 560 Hobson v. Hansen 560 Diana v. State Board of Education 561 Larry P. v. Wilson Riles 562 Parents in Action on Special Education v. Hannon 564 Crawford et al. v. Honig et al. 564 Marchall v. Georgia 568 Debra P. v. Turlington 569 Regents of the University of California v. Bakke 571 Golden Rule Insurance Company et al. v. Washburn et al. 571 Adarand Constructors, Inc. v. Pena, Secretary of Transportation et al. 572 Affirmative Action in Higher Education 572 Grutter v. Bollinger and Gratz v. Bollinger 574 Parents v. Seattle 574 Meredith v. Jefferson County Board of Education 576 Personnel Cases 576 Cases Relevant to the Americans with Disabilities Act (ADA) 581 A Critical Look at Lawsuits 582

Summary

21

582

Ethics and the Future of Psychological Testing 585 Issues Shaping the Field of Testing 586 Professional Issues 586 Moral Issues 591 Social Issues 595

Current Trends 597 The Proliferation of New Tests 597 Higher Standards, Improved Technology, and Increasing Objectivity 598 Greater Public Awareness and Influence 599

CONTENTS The Computerization of Tests Testing on the Internet 600

xvii

600

Future Trends 601 Future Prospects for Testing Are Promising 601 The Proliferation of New and Improved Tests Will Continue 602 Controversy, Disagreement, and Change Will Continue 603 The Integration of Cognitive Science and Computer Science Will Lead to Several Innovations in Testing 603

Summary

603

APPENDIX

1

Areas of a Standard Normal Distribution 605

APPENDIX

2

Critical Values of r for 𝛂 = .05 and 𝛂 = .01 (Two-Tailed Test) 609

APPENDIX

3

Critical Values of t* 610

APPENDIX

4

Code of Fair Testing Practices in Education 612 GLOSSARY

617

REFERENCES

622

NAME INDEX

689

SUBJEC T IND E X

703

List of Sample Test Profiles FIGURE

9.7

Cover page of the Stanford-Binet Intelligence Scale 244

FIGURE

12.1

Example of a score report for the Stanford Achievement Test 311

FIGURE

12.2

A sample student profile from the ACT

FIGURE

12.3

GRE verbal ability sample items 319

FIGURE

12.4

GRE quantitative ability sample items 320

FIGURE

12.5

MAT sample items 323

FIGURE

13.2

An MMPI profile sheet 341

FIGURE

13.3

An MMPI-2 profile sheet 348

FIGURE

13.4

Jackson Personality Inventory profile sheet 359

FIGURE

13.5

NEO Personality Inventory profile sheet 363

TA B L E

14.1

FOCUSED EXAMPLE

14.2

317

Summary of Rorschach scoring 382 The danger of basing Rorschach interpretations on insufficient evidence 388–389 Sentence completion tasks 398

FIGURE TA B L E

17.4

17.4

Profile of a patient tested with the Luria-Nebraska battery 465 Some of the questions used in the Test Anxiety Questionnaire 473

FIGURE

18.2

Sample questions from the Wonderlic 500

FIGURE

19.8

Sample SOMPA profile 531

TA B L E

20.1

Examples of items from a minimum competence test 570

PR EFACE

P

sychology is a broad, exciting field. Psychologists work in settings ranging from schools and clinics to basic research laboratories, pharmaceutical fi rms, and private international companies. Despite this diversity, all psychologists have at least two things in common: They all study behavior, and they all depend to some extent on its measurement. This book concerns a particular type of measurement, psychological tests, which measure characteristics pertaining to all aspects of behavior in human beings. Psychological Testing is the result of a long-standing partnership between the authors. As active participants in the development and use of psychological tests, we became disheartened because far too many undergraduate college students view psychological testing courses as boring and unrelated to their goals or career interests. In contrast, we view psychological testing as an exciting field. It has a solid place in the history of psychology, yet it is constantly in flux because of challenges, new developments, and controversies. A book on testing should encourage, not dampen, a student’s interest. Thus, we provide an overview of the many facets of psychological tests and measurement principles in a style that will appeal to the contemporary college student. To understand the applications and issues in psychological testing, the student must learn some basic principles, which requires some knowledge of introductory statistics. Therefore, some reviewing and a careful reading of Part I will pave the way for an understanding of the applications of tests discussed in Part II. Part III examines the issues now shaping the future of testing. Such issues include test anxiety, test bias, and the interface between testing and the law. The future of applied psychology may depend on the ability of psychologists to face these challenging issues. Throughout the book, we present a series of focused discussions and focused examples. These sections illustrate the material in the book through examples or xix

xx

PREFACE provide a more detailed discussion of a particular issue. We also use box features called “Psychological Testing in Everyday Life” to demonstrate material such as statistical calculations.

INCREASED EMPHASIS ON APPLICATION Students today often favor informal discussions and personally relevant examples. Consequently, we decided to use models from various fields and to write in an informal style. However, because testing is a serious and complicated field in which major disagreements exist even among scholars and experts, we have treated the controversial aspects of testing with more formal discussion and detailed referencing. The first edition of Psychological Testing: Principles, Applications, and Issues was published in 1982. In over a quarter of a century since the text was fi rst introduced, the world has changed in many ways. For example, personal computers were new in 1982. Most students and professors had never heard of e-mail or the Internet. There were far fewer applications of psychological testing than there are today. On the other hand, principles of psychological testing have remained relatively constant. Thus, newer editions have included improvements and refinements in the Principles chapters. The later chapters on Applications and Issues have evolved considerably. Not only has the field of psychological testing changed, but so have the authors. One of us (RMK) has spent most of his career as a professor in a school of medicine and is now in a school of public health. The other (DPS) completed law school and works as both a psychology professor and an adjunct professor of law. While maintaining our central identities as psychologists, we have also had the opportunity to explore cutting-edge practice in medicine, public health, education, and law. The seventh edition goes further than any previous edition in spelling out the applications of psychological testing in a wide variety of applied fields. In developing the seventh edition, we have organized topics around the application areas. Chapter 11 considers psychological testing in education and special education. Chapter 12 looks at the use of standardized tests in education, civil service, and the military. Chapters 13 and 14 consider the use of psychological tests in clinical and counseling settings. The age of computers has completely revolutionized psychological testing. We deal with some of these issues in the Principles chapters by discussing computeradaptive testing and item response theory. In Chapter 15, we discuss applications of psychological science in the computer age. Chapter 16 discusses the use of psychological testing in the field of counseling psychology and focuses primarily on interest inventories. Chapter 17 explores the rapidly developing fields of psychological assessment in health psychology, medicine, and health care. Chapter 18 reviews psychological testing in industry and business settings. The final chapters on issues in psychological testing have been extensively updated to reflect new developments in social justice, law, and ethics. The first edition of Psychological Testing was produced on typewriters before word processors were commonly used. At the time, few professors or students had access to private computers. The early editions of the book offered instruction for

PREFACE

xxi

preparing the submission of statistical analyses to mainframe computers. As recently as the production of the third edition, the Internet was largely unused by university students. Today, nearly all students have ready access to the Internet and World Wide Web, and we now commonly provide references to Web sites. Furthermore, we provide greater discussion of computer-administered tests.

ORGANIZATION OF THE SEVENTH EDITION: A NOTE TO PROFESSORS FOR PLANNING Producing seven editions of Psychological Testing over more than 25 years has been challenging and rewarding. We are honored that hundreds of professors have adopted our text and that it is now used in hundreds of colleges and universities all over the world. However, some professors have suggested that we reorganize the book to facilitate their approach to the class. To accommodate the large variety of approaches, we have tried to keep the chapters independent enough for professors to teach them in whatever order they choose. For example, one approach to the course is to go systematically through the chapter sequence. Professors who wish to emphasize psychometric issues, however, might assign Chapters 1 through 7, followed by Chapters 19 and 20. Then, they might return to certain chapters from the Applications section. On campuses that require a strong statistics course as a prerequisite, Chapters 2 and 3 might be dropped. Professors who emphasize applications might assign Chapters 1 through 5 and then proceed directly to Part II, with some professors assigning only some of its chapters. Though Chapters 9 through 13 are the ones most likely to be used in a basic course, we have found sufficient interest in Chapters 14 through 18 to retain them. Chapters 17 and 18 represent newer areas into which psychological testing is expanding. Finally, Chapters 19 and 20 were written so that they could be assigned either at the end of the course or near the beginning. For example, some professors prefer to assign Chapters 19 and 20 after Chapter 5.

SUPPLEMENTS BEYOND COMPARE As with the previous editions, a student workbook is available. Professors have access to an instructor’s manual and a bank of electronic test items.

Book Companion Web Site The Web site contains several components that will be invaluable to instructors. First, a data set consisting of 25 examinees’ scores on several measures can be downloaded and used with accompanying reliability and validity exercises. Second, several integrative assignments—including a report on a battery of psychological tests, an evaluation of a mock test manual, and a test critique—and associated grading rubrics will be posted on the Web site. The integrative assignment fi les and grading rubrics are modifiable, allowing you to make changes so that they better fit your specific course objectives.

xxii

PREFACE

Student Workbook (ISBN 0-495-59774-0) More than a traditional study guide, the Student Workbook—written by Katherine Nicolai of Rockhurst University—truly helps students understand the connections between abstract measurement concepts and the development, evaluation, selection, and use of psychological tests in the real world. The Student Workbook contains interesting hands-on exercises and assignments, including case studies to critique, test profi les to interpret, and studies on the psychometric properties of tests to evaluate. Of course, the Student Workbook also contains traditional features such as chapter outlines and practice multiple-choice quizzes. Best of all, the workbook is presented in a three-ring binder in which students can keep other course notes and handouts. Students will discover that the Student Workbook will help them organize their study of Kaplan and Saccuzzo’s text and excel on course exams, assignments, and projects!

Instructor’s Resource Manual/Test Bank (ISBN: 0-495-50914-0) The Instructor’s Resource Manual (IRM) was written by Katherine Nicolai of Rockhurst University, and the Test Bank by Ira Bernstein and Yasushi Kyutoku of the University of Texas at Arlington. In an easy-to-use three-ring binder, the IRM contains a bevy of resources, including guides on designing your course, the use of psychological tests in the classroom, the use of student test data to teach measurement, suggested use of class time, and demonstrations, activities, and activity-based lectures. The IRM provides a description of integrative assignments found on the book companion Web site and gives the instructors unique mock projectives and much more. The test bank contains more than 750 multiple-choice questions in addition to many “thought” essay questions.

ACKNOWLEDGMENTS We are highly indebted to the many reviewers and professors who provided feedback on the sixth edition or reviewed drafts of the seventh edition. Special thanks go to reviewers of this edition, including: Virginia Allen, Idaho State University; David Bush, Utah State University; Ira Bernstein, University of Texas, Arlington; Jeff Conte, San Diego State University; Imogen Hall, University of Windsor; Maureen Hannah, Siena College; Ronald McLaughlin, Juniata College; Michael Mills, Loyola Marymount University; Philip Moberg, University of Akron; M. J. Monnot, Central Michigan University; Jennifer Neemann, University of Baltimore; Karen Obremski Brandon, University of South Florida; Frederick Oswald, Michigan State University; S. Mark Pancer, Wilfrid Laurier University; Christopher Ralston, Iowa State University; Sharon Rostosky, University of Kentucky; Stefan Schulenberg, University of Mississippi; Theresa Sparks, Clayton State University; Chockalingam Viswesvaran, Florida International University; Mark Wagner, Wagner College; and Nancy Zook, SUNY Purchase. The seven editions of this book have been developed under six different editors at Wadsworth. The earlier editions benefited from the patient and inspired

PREFACE

xxiii

supervision of Todd Lueders, C. Deborah Laughton, Phil Curson, Marianne Tafl inger, and Jim Brace-Thompson. We are most appreciative of the support we have received from our new editor, Jaime Perkins. He has been patient, helpful, and very well organized in directing the development of the current edition. Each of our editors has come to the task with a different personality and a different set of insights. We learned immensely from each of them and the seventh edition represents a collection of what we have gained from advice and consultations over many years. We also want to thank Charlene Carpentier, production project manager, Vernon Boes for coordinating the cover, and Wilson Co, editorial assistant for coordinating supplements. We want to give particular thanks to Kate Nicolai for authoring the exciting new Student Workbook and the much expanded Instructor’s Resource Manual. Special thanks go to Danielle Zuest and Nancy E. Johnson. Danielle conducted much of the research for the updating of about half the chapters and attended to numerous details. Nancy also assisted in numerous ways, including research, editing, and locating difficult-to-find sources. Without these two individuals, publication of this edition would have been much delayed. Dr. Kaplan is also thankful to the Rockefeller Foundation, who provided a quiet workspace overlooking Lake Como in Italy to allow for the completion of the final stages of this project. Robert M. Kaplan Dennis P. Saccuzzo April 2008

This page intentionally left blank

ABOUT THE AUTHORS

ROBERT M. KAPLAN is Fred W. and Pamela K. Wasserman Professor and chair of the Department of Health Services at UCLA and professor of medicine at the UCLA David Geffen School of Medicine. From 1997 to 2004, he was professor and chair of the Department of Family and Preventive Medicine, at the University of California, San Diego. He is a past president of several organizations, including the American Psychological Association Division of Health Psychology, Section J of the American Association for the Advancement of Science (Pacific), the International Society for Quality of Life Research, the Society for Behavioral Medicine, and the Academy of Behavioral Medicine Research. He is a past chair of the Behavioral Science Council of the American Thoracic Society. Dr. Kaplan is currently editor-in-chief of Health Psychology and is the former editor-in-chief of the Annals of Behavioral Medicine. He has served as associate editor of the American Psychologist, and consulting editor of four other academic journals. Selected additional honors include APA Division of Health Psychology Annual Award for Outstanding Scientific Contribution (for junior scholar 1987 and again for a senior scholar 2001), SDSU Distinguished Research Lecturer, 1988, and Health Net Distinguished Lecturer in 1991, University of California 125 Anniversary Award for Most Distinguished Alumnus, University of California, Riverside, American Psychological Association Distinguished Lecturer, Distinguished Scientific contribution award from the American Association of Medical School Psychologists, National Leadership Award from the Society of Behavioral Medicine in 2003, and President’s Award for Career Achievement from the International Society for Quality of Life Research in 2004. In 2006, he received the Distinguished Research Mentor Award from the Society of Behavioral Medicine. His public service contributions include various NIH, AHRQ , and VA grant review groups, service on the local American Lung Association (ALA) Board of Directors, and the regional research committee for the American Heart Association. He served as co-chair of the Behavioral xxv

xxvi

ABOUT THE AUTHORS Committee for the NIH Women’s Health Initiative and as a member of both the NHLBI Behavioral Medicine Task Force and the Institute of Medicine (IOM) National Academy of Sciences Committee on Health and Behavior. In addition he served on the expert advisory policy panel for the CDC-NIH Public Action Plan to Prevent Heart Disease and Stroke. Kaplan currently serves on the National Advisory Committee for the Decade of Behavior. Further, he is the chair of the Cost/ Effectiveness Committee for the NHLBI National Emphysema Treatment Trial (NETT). Dr. Kaplan is the author or co-author of more than 15 books and over 400 articles or chapters. The ISI includes him in the listing of the most cited authors in the world (defined as above the 99.5th percentile). In 2005, he was elected to the Institute of Medicine of the National Academies of Sciences. DENNIS P. SACCUZZO is professor of psychology at San Diego State University, and an adjunct professor of law at Thomas Jefferson School of Law. He has been a scholar and practitioner of psychological testing for over 32 years and has numerous peer-reviewed publications and professional presentations in the field. Dr. Saccuzzo’s research has been supported by the National Science Foundation, the National Institutes of Mental Health, the National Institutes of Health, the U.S. Department of Education, the Scottish Rite Foundation, and the U.S. armed services. He is also a California-licensed psychologist and a California-licensed attorney. He is board certified in clinical psychology by the American Board of Professional Psychology (ABPP). In addition, he is a Diplomate of the American Board of Assessment Psychology (ABAP) and the American Board of Forensic Medicine. He is a fellow of the American Psychological Association, American Psychological Society, and Western Psychological Association for outstanding and unusual contributions to the field of psychology. Dr. Saccuzzo is the author or co-author of over 300 peerreviewed papers and publications, including eight textbooks and 20 law manuals. He is the president and co-founder of Applications of Psychology to Law, Inc., an educational institution that applies psychological science to legal issues and the study of law.

SEVENTH EDITION

Psychological Testing Principles, Applications, and Issues

This page intentionally left blank

CHAPTER

1

Introduction

LE AR N I N G O B J EC TIVE S When you have completed this chapter, you should be able to: ■

Define the basic terms pertaining to psychological and educational tests



Distinguish between an individual test and a group test



Define the terms achievement, aptitude, and intelligence and identify a concept that can encompass all three terms



Distinguish between ability tests and personality tests



Define the term structured personality test



Explain how structured personality tests differ from projective personality tests



Explain what a normative or standardization sample is and why such a sample is important



Identify the major developments in the history of psychological testing



Explain the relevance of psychological tests in contemporary society

1

2

CHAPTER 1



Introduction

Y

ou are sitting at a table. You have just been fingerprinted and have shown a picture ID. You look around and see 40 nervous people. A stern-looking test proctor with a stopwatch passes out booklets. You are warned not to open the booklet until told to do so; you face possible disciplinary action if you disobey. Th is is not a nightmare or some futuristic fantasy—this is real. Finally, after what seems like an eternity, you are told to open your booklet to page 3 and begin working. Your mouth is dry; your palms are soaking wet. You open to page 3. You have 10 minutes to solve a five-part problem based on the following information.1 A car drives into the center ring of a circus and exactly eight clowns—Q , R, S, T, V, W, Y, and Z—get out of the car, one clown at a time. The order in which the clowns get out of the car is consistent with the following conditions: V gets out at some time before both Y and Q. Q gets out at some time after Z. T gets out at some time before V but at some time after R. S gets out at some time after V. R gets out at some time before W. Question 1. If Q is the fi fth clown to get out of the car, then each of the following could be true except: Z is the fi rst clown to get out of the car. T is the second clown to get out of the car. V is the third clown to get out of the car. W is the fourth clown to get out of the car. Y is the sixth clown to get out of the car.

Not quite sure how to proceed, you look at the next question. Question 2. If R is the second clown to get out of the car, which of the following must be true? S gets out of the car at some time before T does. T gets out of the car at some time before W does. W gets out of the car at some time before V does. Y gets out of the car at some time before Q does. Z gets out of the car at some time before W does.

Your heart beats a little faster and your mind starts to freeze up like an overloaded computer with too little working memory. You glance at your watch and notice that 2 minutes have elapsed and you still don’t have your bearings. Th e person sitting next to you looks a bit faint. Another three rows up someone storms up to the test proctor and complains frantically that he cannot do this type of problem. While the proctor struggles to calm this person down, another makes a mad dash for the restroom. Welcome to the world of competitive, “high stakes,” standardized psychological tests in the 21st century. The questions you just faced were actual problems from

1 Used by permission from the Law School Admission Test, October 2002. Answer to Question 1 is D; answer to Question 2 is E.

CHAPTER 1



Introduction

3

a past version of the LSAT—the Law School Admission Test. Whether or not a student is admitted into law school in the United States is almost entirely determined by that person’s score on the LSAT and undergraduate college grade point average. Thus, one’s future can depend to a tremendous extent on a single score from a single test given in a tension-packed morning or afternoon. Despite efforts to improve tests like the LSAT to increase diversity (Pashley, Thornton, & Duff y, 2005), standardized tests tend to disadvantage women and ethnic minorities (Sackett, Schmitt, Ellingson, & Kabin, 2001). Similar problems appear on the GRE— the Graduate Record Exam, a test that plays a major role in determining who gets to study at the graduate level in the United States. (Later in this book we discuss how to prepare for such tests and what their significance, or predictive validity, is.) Tests such as the LSAT and GRE are the most difficult modern psychological tests. The scenes we’ve described are real; some careers do ride on a single test. Perhaps you have already taken the GRE or LSAT. Or perhaps you have not graduated yet but are thinking about applying for an advanced degree or professional program and will soon be facing the GRE, LSAT, or MCAT (Medical College Admission Test). Clearly, it will help you to have a basic understanding of the multitude of psychological tests people are asked to take throughout their lives. From our birth, tests have a major influence on our lives. When the pediatrician strokes the palms of our hands and the soles of our feet, he or she is performing a test. When we enter school, tests decide whether we pass or fail classes. Testing may determine if we need special education. In the United States and many industrialized countries competence tests determine if students will graduate from high school (Carnoy, 2005; Hursh, 2005). More tests determine which college we may attend. And, of course, when we get into college we face still more tests. After graduation, those who choose to avoid tests such as the GRE may need to take tests to determine where they will work. In the modern world, a large part of everyone’s life and success depends on test results. Indeed, tests even have international significance. For example, 15-year-old children in 32 nations were given problems such as the following from the Organization for Economic Co-operation and Development (OECD) and the Programme for International Student Assessment (PISA) (Schleicher & Tamassia, 2000): A result of global warming is that ice of some glaciers is melting. Twelve years after the ice disappears, tiny plants, called lichen, start to grow on the rocks. Each lichen grows approximately in the shape of a circle. The relationship between the diameter of the circles and the age of the lichen can be approximated with the formula: d = 7.0 × the square root of (t − 12) for any t less than or equal to 12, where d represents the diameter of the lichen in millimeters, and t represents the number of years after the ice has disappeared. Calculate the diameter of the lichen 16 years after the ice disappeared. The complete and correct answer is:

d = 7.0 × the square root of (16 − 12 mm) d = 7.0 × the square root of 4 mm d = 14 mm

4

CHAPTER 1



Introduction International Mathematical Literacy Scores Brazil Mexico Luxembourg Greece Portugal Italy Latvia Poland Spain Russia Hungary Germany USA Czech Rep. Norway Ireland Sweden Liechtenstein Iceland Denmark Austria France Belgium U.K. Switzerland Canada Australia Finland New Zealand Japan Korea 300

350

400

450

500

550

600

Points

F I G U R E 1.1 Approximate average scores of 15-year-old students on the OECD mathematical literacy test. (Statistics used by permission of the OECD and PISA. Figure courtesy of W. J. Koen.)

Eighteen countries ranked above the United States in the percentage of 15-year-olds who had mastered such concepts (see Figure 1.1). The results were similar for an OECD science literacy test (see Figure 1.2), which had questions such as the following: A bus is moving along a straight stretch of road. The bus driver, named Ray, has a cup of water resting in a holder on the dashboard. Suddenly Ray has to slam on the brakes. What is most likely to happen to the water in the cup immediately after Ray slams on the brakes? A. The water will stay horizontal. B. The water will spill over side 1.

CHAPTER 1



5

Introduction

International Scientific Literacy Scores Brazil Mexico Luxembourg Portugal Latvia Russia Greece Liechtenstein Italy Denmark Poland Germany Spain Switzerland Belgium Iceland Hungary USA Norway France Czech. Rep. Sweden Ireland Austria Australia New Zealand Canada U.K. Finland Japan Korea 300

350

400

450

500

550

600

Points

F I G U R E 1.2 Approximate average scores of 15-year-old students on the OECD scientific literacy test. (Statistics used by permission of the OECD and PISA. Figure courtesy of W. J. Koen.)

C. The water will spill over side 2. D. The water will spill but you cannot tell if it will spill over side 1 or side 2.

The correct answer is C. How useful are tests such as these? Do they measure anything meaningful? How accurate are they? Such questions concern not only every U.S. citizen but also all members of the highly competitive international community. To answer them, you must understand the principles of psychological testing that you are about to learn. To answer questions about tests, you must understand the concepts presented in this book, such as reliability, validity, item analysis, and test construction. A full

6

CHAPTER 1



Introduction understanding of these concepts will require careful study and a knowledge of basic statistics, but your efforts will be richly rewarded. When you finish this book, you will be a better consumer of tests.

BASIC CONCEPTS You are probably already familiar with some of the elementary concepts of psychological testing. For the sake of clarity, however, we shall begin with definitions of the most basic terms so that you will know how they are used in this textbook.

What a Test Is Everyone has had experience with tests. A test is a measurement device or technique used to quantify behavior or aid in the understanding and prediction of behavior. A spelling test, for example, measures how well someone spells or the extent to which someone has learned to spell a specific list of words. At some time during the next few weeks, your instructor will likely want to measure how well you have learned the material in this book. To accomplish this, your instructor may give you a test. As you well know, the test your instructor gives may not measure your full understanding of the material. This is because a test measures only a sample of behavior, and error is always associated with a sampling process. Test scores are not perfect measures of a behavior or characteristic, but they do add significantly to the prediction process, as you will see. An item is a specific stimulus to which a person responds overtly; this response can be scored or evaluated (for example, classified, graded on a scale, or counted). Because psychological and educational tests are made up of items, the data they produce are explicit and hence subject to scientific inquiry. In simple terms, items are the specific questions or problems that make up a test. The problems presented at the beginning of this chapter are examples of test items. The overt response would be to fi ll in or blacken one of the spaces:

A

B

C

D

E

F

G

A psychological test or educational test is a set of items that are designed to measure characteristics of human beings that pertain to behavior. There are many types of behavior. Overt behavior is an individual’s observable activity. Some psychological tests attempt to measure the extent to which someone might engage in or “emit” a particular overt behavior. Other tests measure how much a person has previously engaged in some overt behavior. Behavior can also be covert—that is, it takes place within an individual and cannot be directly observed. For example, your feelings and thoughts are types of covert behavior. Some tests attempt to measure such behavior. Psychological and educational tests thus measure past or current behavior. Some also attempt to predict future behavior, such as success in college or in an advanced degree program. What does it mean when someone gets 75 items correct on a 100-item test? One thing it means, of course, is that 75% of the items were answered correctly.

CHAPTER 1



Introduction

7

In many situations, however, knowing the percentage of correct items a person obtained can be misleading. Consider two extreme examples. In one case, out of 100 students who took the exam, 99 had 90% correct or higher, and 1 had 75% correct. In another case, 99 of the 100 students had scores of 25% or lower, while 1 had 75% correct. The meaning of the scores can change dramatically, depending on how a well-defined sample of individuals scores on a test. In the first case, a score of 75% is poor because it is in the bottom of the distribution; in the second case, 75% is actually a top score. To deal with such problems of interpretation, psychologists make use of scales, which relate raw scores on test items to some defined theoretical or empirical distribution. Later in the book you will learn about such distributions. Scores on tests may be related to traits, which are enduring characteristics or tendencies to respond in a certain manner. “Determination,” sometimes seen as “stubbornness,” is an example of a trait; “shyness” is another. Test scores may also be related to the state, or the specific condition or status, of an individual. A determined individual after many setbacks may, for instance, be in a weakened state and therefore be less inclined than usual to manifest determination. Tests measure many types of behavior.

Types of Tests Just as there are many types of behavior, so there are many types of tests. Those that can be given to only one person at a time are known as individual tests (see Figure 1.3). The examiner or test administrator (the person giving the test) gives the test to only one person at a time, the same way that psychotherapists see only one person at a time. A group test, by contrast, can be administered to more than one person at a time by a single examiner, such as when an instructor gives everyone in the class a test at the same time. One can also categorize tests according to the type of behavior they measure. Ability tests contain items that can be scored in terms of speed, accuracy, or both. On an ability test, the faster or the more accurate your responses, the better your scores on a particular characteristic. The more algebra problems you can correctly solve in a given amount of time, the higher you score in ability to solve such problems. Historically, experts have distinguished among achievement, aptitude, and intelligence as different types of ability. Achievement refers to previous learning. A test that measures or evaluates how many words you can spell correctly is called a spelling achievement test. Aptitude, by contrast, refers to the potential for learning or acquiring a specific skill. A spelling aptitude test measures how many words you might be able to spell given a certain amount of training, education, and experience. Your musical aptitude refers in part to how well you might be able to learn to play a musical instrument given a certain number of lessons. Traditionally distinguished from achievement and aptitude, intelligence refers to a person’s general potential to solve problems, adapt to changing circumstances, think abstractly, and profit from experience. When we say a person is “smart,” we are usually referring to intelligence. When a father scolds his daughter because she has not done as well in school as she can, he most likely believes that she has not used her intelligence (general potential) to achieve (acquire new knowledge). The distinctions among achievement, aptitude, and intelligence are not always so cut-and-dried because all three are highly interrelated. Attempts to separate prior learning from potential for learning, for example, have not succeeded. In view of

8

CHAPTER 1



Introduction

FIGURE

1.3

An individual test administration.

(Ann Chwatsky/Jeroboam.)

the considerable overlap of achievement, aptitude, and intelligence tests, all three concepts are encompassed by the term human ability. There is a clear-cut distinction between ability tests and personality tests. Whereas ability tests are related to capacity or potential, personality tests are related to the overt and covert dispositions of the individual—for example, the tendency of a person to show a particular behavior or response in a given situation. Remaining isolated from others, for instance, does not require any special skill or ability, but some people typically prefer or tend to remain thus isolated. Personality tests measure typical behavior. There are several types of personality tests. In Chapter 13, you will learn about structured, or objective, personality tests. Structured personality tests provide a statement, usually of the “self-report” variety, and require the subject to choose between two or more alternative responses such as “True” or “False” (see Figure 1.4). In contrast to structured personality tests, projective personality tests are unstructured. In a projective personality test, either the stimulus (test materials) or the required response—or both—are ambiguous. For example, in the highly controversial Rorschach test, the stimulus is an inkblot. Furthermore, rather than being asked to choose among alternative responses, as in structured personality tests, the individual is asked to provide a spontaneous response. The inkblot is presented to the subject, who is asked, “What might this be?” Projective tests assume that a person’s interpretation of an ambiguous stimulus will reflect his or her unique characteristics (see Chapter 14).

CHAPTER 1



Introduction

True

9

False

1. I like heavy metal music. 2. I believe that honesty is the best policy. 3. I am in good health. 4. I am easily fatigued. 5. I sleep well at night.

FIGURE

TA B L E

1.4

Self-report test items.

1.1

Types of Tests I.

Ability tests: Measure skills in terms of speed, accuracy, or both. A. Achievement: Measures previous learning. B. Aptitude: Measures potential for acquiring a specific skill. C. Intelligence: Measures potential to solve problems, adapt to changing circumstances, and profit from experience.

II. Personality tests: Measure typical behavior—traits, temperaments, and dispositions. A. Structured (objective): Provides a self-report statement to which the person responds “True” or “False,” “Yes” or “No.” B. Projective: Provides an ambiguous test stimulus; response requirements are unclear.

See Table 1.1 for a brief overview of ability and personality tests. Psychological testing refers to all the possible uses, applications, and underlying concepts of psychological and educational tests. The main use of these tests, though, is to evaluate individual differences or variations among individuals. Such tests measure individual differences in ability and personality and assume that the differences shown on the test reflect actual differences among individuals. For instance, individuals who score high on an IQ test are assumed to have a higher degree of intelligence than those who obtain low scores. Thus, the most important purpose of testing is to differentiate among those taking the tests. We shall discuss the idea of individual differences later in this chapter.

OVERVIEW OF THE BOOK This book is divided into three parts: Principles, Applications, and Issues. Together, these parts cover psychological testing from the most basic ideas to the most complex. Basic ideas and events are introduced early and stressed throughout to reinforce

10

CHAPTER 1



Introduction what you have just learned. In covering principles, applications, and issues, we intend to provide not only the who’s of psychological testing but also the how’s and why’s of major developments in the field. We also address an important concern of many students—relevance—by examining the diverse uses of tests and the resulting data.

Principles of Psychological Testing By principles of psychological testing we mean the basic concepts and fundamental ideas that underlie all psychological and educational tests. Chapters 2 and 3 present statistical concepts that provide the foundation for understanding tests. Chapters 4 and 5 cover two of the most fundamental concepts in testing: reliability and validity. Reliability refers to the accuracy, dependability, consistency, or repeatability of test results. In more technical terms, reliability refers to the degree to which test scores are free of measurement errors. As you will learn, there are many ways a test can be reliable. For example, test results may be reliable over time, which means that when the same test is given twice within any given time interval, the results tend to be the same or highly similar. Validity refers to the meaning and usefulness of test results. More specifically, validity refers to the degree to which a certain inference or interpretation based on a test is appropriate. When one asks the question, “What does this psychological test measure?” one is essentially asking “For what inference is this test valid?” Another principle of psychological testing concerns how a test is created or constructed. In Chapter 6, we present the principles of test construction. The act of giving a test is known as test administration, which is the main topic of Chapter 7. Though some tests are easy to administer, others must be administered in a highly specific way. The final chapter of Part I covers the fundamentals of administering a psychological test.

Applications of Psychological Testing Part II, on applications, provides a detailed analysis of many of the most popular tests and how they are used or applied. It begins with an overview of the essential terms and concepts that relate to the application of tests. Chapter 8 discusses interviewing techniques. An interview is a method of gathering information through verbal interaction, such as direct questions. Not only has the interview traditionally served as a major technique of gathering psychological information in general, but also data from interviews provide an important complement to test results. Chapters 9 and 10 cover individual tests of human ability. In these chapters, you will learn not only about tests but also about the theories of intelligence that underlie them. In Chapter 11, we cover testing in education with an emphasis on special education. In Chapter 12, we present group tests of human ability. Chapter 13 covers structured personality tests, and Chapter 14 covers projective personality tests. In Chapter 15, we discuss the important role of computers in the testing field. We also consider the influence of cognitive psychology, which today is the most prominent of the various schools of thought within psychology (Kellogg, 2003; Leahy & Dowd, 2002; Weinstein & Way, 2003). These chapters not only provide descriptive information but also delve into the ideas underlying the various tests. Chapter 16 examines interest tests, which

CHAPTER 1



Introduction

11

measure behavior relevant to such factors as occupational preferences. Chapter 17 reviews the relatively new area of medical testing for brain damage and health status. It also covers important recent advancements in developmental neuropsychology. Finally, Chapter 18 covers tests for industrial and organizational psychology and business.

Issues of Psychological Testing Many social and theoretical issues, such as the controversial topic of racial differences in ability, accompany testing. Part III covers many of these issues. As a compromise between breadth and depth of coverage, we focus on a comprehensive discussion of those issues that have particular importance in the current professional, social, and political environment. Chapter 19 examines test bias, one of the most volatile issues in the field (Geisinger, 2003; Reynolds & Ramsay, 2003; Ryan & DeMark, 2002). Because psychological tests have been accused of being discriminatory or biased against certain groups, this chapter takes a careful look at both sides of the argument. Because of charges of bias and other problems, psychological testing is increasingly coming under the scrutiny of the law (Phillips, 2002; Saccuzzo, 1999). Chapter 20 examines test bias as related to legal issues and discusses testing and the law. Chapter 21 presents a general overview of other major issues currently shaping the future of psychological testing in the United States with an emphasis on ethics. From our review of the issues, we also speculate on what the future holds for psychological testing.

HISTORICAL PERSPECTIVE We now briefly provide the historical context of psychological testing. This discussion touches on some of the material presented earlier in this chapter.

Early Antecedents Most of the major developments in testing have occurred over the last century, many of them in the United States. The origins of testing, however, are neither recent nor American. Evidence suggests that the Chinese had a relatively sophisticated civil service testing program more than 4000 years ago (DuBois, 1970, 1972). Every third year in China, oral examinations were given to help determine work evaluations and promotion decisions. By the Han Dynasty (206 B.C.E. to 220 C.E.), the use of test batteries (two or more tests used in conjunction) was quite common. These early tests related to such diverse topics as civil law, military affairs, agriculture, revenue, and geography. Tests had become quite well developed by the Ming Dynasty (1368–1644 C.E.). During this period, a national multistage testing program involved local and regional testing centers equipped with special testing booths. Those who did well on the tests at the local level went on to provincial capitals for more extensive essay examinations. After this second testing, those with the highest test scores went on to the nation’s capital for a final round. Only those who passed this third set of tests were eligible for public office. The Western world most likely learned about testing programs through the Chinese. Reports by British missionaries and diplomats encouraged the English

12

CHAPTER 1



Introduction East India Company in 1832 to copy the Chinese system as a method of selecting employees for overseas duty. Because testing programs worked well for the company, the British government adopted a similar system of testing for its civil service in 1855. After the British endorsement of a civil service testing system, the French and German governments followed suit. In 1883, the U.S. government established the American Civil Service Commission, which developed and administered competitive examinations for certain government jobs. The impetus of the testing movement in the Western world grew rapidly at that time (Wiggins, 1973).

Charles Darwin and Individual Differences Perhaps the most basic concept underlying psychological and educational testing pertains to individual differences. No two snowflakes are identical, no two fingerprints the same. Similarly, no two people are exactly alike in ability and typical behavior. As we have noted, tests are specifically designed to measure these individual differences in ability and personality among people. Although human beings realized long ago that individuals differ, developing tools for measuring such differences was no easy matter. To develop a measuring device, we must understand what we want to measure. An important step toward understanding individual differences came with the publication of Charles Darwin’s highly influential book, The Origin of Species, in 1859. According to Darwin’s theory, higher forms of life evolved partially because of differences among individual forms of life within a species. Given that individual members of a species differ, some possess characteristics that are more adaptive or successful in a given environment than are those of other members. Darwin also believed that those with the best or most adaptive characteristics survive at the expense of those who are less fit and that the survivors pass their characteristics on to the next generation. Th rough this process, he argued, life has evolved to its currently complex and intelligent levels. Sir Francis Galton, a relative of Darwin’s, soon began applying Darwin’s theories to the study of human beings (see Figure 1.5). Given the concepts of survival of the fittest and individual differences, Galton set out to show that some people possessed characteristics that made them more fit than others, a theory he articulated in his book Hereditary Genius, published in 1869. Galton (1883) subsequently began a series of experimental studies to document the validity of his position. He concentrated on demonstrating that individual differences exist in human sensory and motor functioning, such as reaction time, visual acuity, and physical strength. In doing so, Galton initiated a search for knowledge concerning human individual differences, which is now one of the most important domains of scientific psychology. Galton’s work was extended by the U.S. psychologist James McKeen Cattell, who coined the term mental test (Cattell, 1890). Cattell’s doctoral dissertation was based on Galton’s work on individual differences in reaction time. As such, Cattell perpetuated and stimulated the forces that ultimately led to the development of modern tests.

Experimental Psychology and Psychophysical Measurement A second major foundation of testing can be found in experimental psychology and early attempts to unlock the mysteries of human consciousness through the scientific method. Before psychology was practiced as a science, mathematical models

CHAPTER 1

FIGURE

1.5



Introduction

13

Sir Francis Galton.

(From the National Library of Medicine.)

of the mind were developed, in particular those of J. E. Herbart. Herbart eventually used these models as the basis for educational theories that strongly influenced 19th-century educational practices. Following Herbart, E. H. Weber attempted to demonstrate the existence of a psychological threshold, the minimum stimulus necessary to activate a sensory system. Then, following Weber, G. T. Fechner devised the law that the strength of a sensation grows as the logarithm of the stimulus intensity. Wilhelm Wundt, who set up a laboratory at the University of Leipzig in 1879, is credited with founding the science of psychology, following in the tradition of Weber and Fechner (Hearst, 1979). Wundt was succeeded by E. B. Titchner, whose student, G. Whipple, recruited L. L. Thurstone. Whipple provided the basis for immense changes in the field of testing by conducting a seminar at the Carnegie Institute in 1919 attended by Thurstone, E. Strong, and other early prominent U.S. psychologists. From this seminar came the Carnegie Interest Inventory and later the Strong Vocational Interest Blank. Later in this book we discuss in greater detail the work of these pioneers and the tests they helped to develop. Thus, psychological testing developed from at least two lines of inquiry: one based on the work of Darwin, Galton, and Cattell on the measurement of individual differences, and the other (more theoretically relevant and probably stronger) based on the work of the German psychophysicists Herbart, Weber, Fechner, and Wundt. Experimental psychology developed from the latter. From this work also came the idea that testing, like an experiment, requires rigorous experimental control. Such control, as you will see, comes from administering tests under highly standardized conditions. The efforts of these researchers, however necessary, did not by themselves lead to the creation of modern psychological tests. Such tests also arose in response to important needs such as classifying and identifying the mentally and emotionally handicapped. One of the earliest tests resembling current procedures, the Seguin Form Board Test (Seguin, 1866/1907), was developed in an effort to educate and

14

CHAPTER 1



Introduction evaluate the mentally disabled. Similarly, Kraepelin (1912) devised a series of examinations for evaluating emotionally impaired people. An important breakthrough in the creation of modern tests came at the turn of the 20th century. The French minister of public instruction appointed a commission to study ways of identifying intellectually subnormal individuals in order to provide them with appropriate educational experiences. One member of that commission was Alfred Binet. Working in conjunction with the French physician T. Simon, Binet developed the first major general intelligence test. Binet’s early effort launched the first systematic attempt to evaluate individual differences in human intelligence (see Chapter 9).

The Evolution of Intelligence and Standardized Achievement Tests The history and evolution of Binet’s intelligence test are instructive. The first version of the test, known as the Binet-Simon Scale, was published in 1905. This instrument contained 30 items of increasing difficulty and was designed to identify intellectually subnormal individuals. Like all well-constructed tests, the Binet-Simon Scale of 1905 was augmented by a comparison or standardization sample. Binet’s standardization sample consisted of 50 children who had been given the test under standard conditions—that is, with precisely the same instructions and format. In obtaining this standardization sample, the authors of the Binet test had norms with which they could compare the results from any new subject. Without such norms, the meaning of scores would have been difficult, if not impossible, to evaluate. However, by knowing such things as the average number of correct responses found in the standardization sample, one could at least state whether a new subject was below or above it. It is easy to understand the importance of a standardization sample. However, the importance of obtaining a standardization sample that represents the population for which a test will be used has sometimes been ignored or overlooked by test users. For example, if a standardization sample consists of 50 white men from wealthy families, then one cannot easily or fairly evaluate the score of an African American girl from a poverty-stricken family. Nevertheless, comparisons of this kind are sometimes made. Clearly, it is not appropriate to compare an individual with a group that does not have the same characteristics as the individual. Binet was aware of the importance of a standardization sample. Further development of the Binet test involved attempts to increase the size and representativeness of the standardization sample. A representative sample is one that comprises individuals similar to those for whom the test is to be used. When the test is used for the general population, a representative sample must reflect all segments of the population in proportion to their actual numbers. By 1908, the Binet-Simon Scale had been substantially improved. It was revised to include nearly twice as many items as the 1905 scale. Even more significantly, the size of the standardization sample was increased to more than 200. The 1908 Binet-Simon Scale also determined a child’s mental age, thereby introducing a historically significant concept. In simplified terms, you might think of mental age as a measurement of a child’s performance on the test relative to other children

CHAPTER 1



Introduction

15

of that particular age group. If a child’s test performance equals that of the average 8-year-old, for example, then his or her mental age is 8. In other words, in terms of the abilities measured by the test, this child can be viewed as having a similar level of ability as the average 8-year-old. The chronological age of the child may be 4 or 12, but in terms of test performance, the child functions at the same level as the average 8-year-old. The mental age concept was one of the most important contributions of the revised 1908 Binet-Simon Scale. In 1911, the Binet-Simon Scale received a minor revision. By this time, the idea of intelligence testing had swept across the world. By 1916, L. M. Terman of Stanford University had revised the Binet test for use in the United States. Terman’s revision, known as the Stanford-Binet Intelligence Scale (Terman, 1916), was the only American version of the Binet test that flourished. It also characterizes one of the most important trends in testing—the drive toward better tests. Terman’s 1916 revision of the Binet-Simon Scale contained many improvements. The standardization sample was increased to include 1000 people, original items were revised, and many new items were added. Terman’s 1916 Stanford-Binet Intelligence Scale added respectability and momentum to the newly developing testing movement. World War I The testing movement grew enormously in the United States because of the demand for a quick, efficient way of evaluating the emotional and intellectual functioning of thousands of military recruits in World War I. The war created a demand for largescale group testing because relatively few trained personnel could evaluate the huge influx of military recruits. However, the Binet test was an individual test. Shortly after the United States became actively involved in World War I, the army requested the assistance of Robert Yerkes, who was then the president of the American Psychological Association (see Yerkes, 1921). Yerkes headed a committee of distinguished psychologists who soon developed two structured group tests of human abilities: the Army Alpha and the Army Beta. The Army Alpha required reading ability, whereas the Army Beta measured the intelligence of illiterate adults. World War I fueled the widespread development of group tests. About this time, the scope of testing also broadened to include tests of achievement, aptitude, interest, and personality. Because achievement, aptitude, and intelligence tests overlapped considerably, the distinctions proved to be more illusory than real. Even so, the 1916 Stanford-Binet Intelligence Scale had appeared at a time of strong demand and high optimism for the potential of measuring human behavior through tests. World War I and the creation of group tests had then added momentum to the testing movement. Shortly after the appearance of the 1916 Stanford-Binet Intelligence Scale and the Army Alpha test, schools, colleges, and industry began using tests. It appeared to many that this new phenomenon, the psychological test, held the key to solving the problems emerging from the rapid growth of population and technology. Achievement Tests Among the most important developments following World War I was the development of standardized achievement tests. In contrast to essay tests, standardized achievement tests provide multiple-choice questions that are standardized on a

16

CHAPTER 1



Introduction large sample to produce norms against which the results of new examinees can be compared. Standardized achievement tests caught on quickly because of the relative ease of administration and scoring and the lack of subjectivity or favoritism that can occur in essay or other written tests. In school settings, standardized achievement tests allowed one to maintain identical testing conditions and scoring standards for a large number of children. Such tests also allowed a broader coverage of content and were less expensive and more efficient than essays. In 1923, the development of standardized achievement tests culminated in the publication of the Stanford Achievement Test by T. L. Kelley, G. M. Ruch, and L. M. Terman. By the 1930s, it was widely held that the objectivity and reliability of these new standardized tests made them superior to essay tests. Their use proliferated widely. It is interesting, as we shall discuss later in the book, that teachers of today appear to have come full circle. Currently, many people favor written tests and work samples (portfolios) over standardized achievement tests as the best way to evaluate children (Boerum, 2000; Harris, 2002). Rising to the Challenge For every movement there is a countermovement, and the testing movement in the United States in the 1930s was no exception. Critics soon became vocal enough to dampen enthusiasm and to make even the most optimistic advocates of tests defensive. Researchers, who demanded nothing short of the highest standards, noted the limitations and weaknesses of existing tests. Not even the Stanford-Binet, a landmark in the testing field, was safe from criticism. Although tests were used between the two world wars and many new tests were developed, their accuracy and utility remained under heavy fi re. Near the end of the 1930s, developers began to reestablish the respectability of tests. New, improved tests reflected the knowledge and experience of the previous two decades. By 1937, the Stanford-Binet had been revised again. Among the many improvements was the inclusion of a standardization sample of more than 3000 individuals. A mere 2 years after the 1937 revision of the Stanford-Binet test, David Wechsler published the fi rst version of the Wechsler intelligence scales (see Chapter 10), the Wechsler-Bellevue Intelligence Scale (W-B) (Wechsler, 1939). The Wechsler-Bellevue scale contained several interesting innovations in intelligence testing. Unlike the Stanford-Binet test, which produced only a single score (the socalled IQ , or intelligence quotient), Wechsler’s test yielded several scores, permitting an analysis of an individual’s pattern or combination of abilities. Among the various scores produced by the Wechsler test was the performance IQ. Performance tests do not require a verbal response; one can use them to evaluate intelligence in people who have few verbal or language skills. The Stanford-Binet test had long been criticized because of its emphasis on language and verbal skills, making it inappropriate for many individuals, such as those who cannot speak or who cannot read. In addition, few people believed that language or verbal skills play an exclusive role in human intelligence. Wechsler’s inclusion of a nonverbal scale thus helped overcome some of the practical and theoretical weaknesses of the Binet test. In 1986, the Binet test was drastically revised to include performance subtests. More recently, it was overhauled again in 2003, as we shall see in Chapter 9. (Other

CHAPTER 1



Introduction

17

important concepts in intelligence testing will be formally defined in Chapter 10, which covers the various forms of the Wechsler intelligence scales.)

Personality Tests: 1920–1940 Just before and after World War II, personality tests began to blossom. Whereas intelligence tests measured ability or potential, personality tests measured presumably stable characteristics or traits that theoretically underlie behavior. Traits are relatively enduring dispositions (tendencies to act, think, or feel in a certain manner in any given circumstance) that distinguish one individual from another. For example, we say that some people are optimistic and some pessimistic. Optimistic people tend to remain so regardless of whether or not things are going well. A pessimist, by contrast, tends to look at the negative side of things. Optimism and pessimism can thus be viewed as traits. One of the basic goals of traditional personality tests is to measure traits. As you will learn, however, the notion of traits has important limitations. The earliest personality tests were structured paper-and-pencil group tests. These tests provided multiple-choice and true-false questions that could be administered to a large group. Because it provides a high degree of structure—that is, a definite stimulus and specific alternative responses that can be unequivocally scored—this sort of test is a type of structured personality test. The first structured personality test, the Woodworth Personal Data Sheet, was developed during World War I and was published in final form just after the war (see Figure 1.6). As indicated earlier, the motivation underlying the development of the first personality test was the need to screen military recruits. History indicates that tests such as the Binet and the Woodworth were created by necessity to meet unique challenges. Like the early ability tests, however, the first structured personality test was simple by today’s standards. Interpretation of the Woodworth test depended on the now-discredited assumption that the content of an item could be accepted at face value. If the person marked “False” for the statement “I wet the bed,” then it was assumed that he or she did not “wet the bed.” As logical as this assumption

Yes

No

1. I wet the bed. 2. I drink a quart of whiskey each day. 3. I am afraid of closed spaces. 4. I believe I am being followed. 5. People are out to get me. 6. Sometimes I see or hear things that other people do not hear or see.

F I G U R E 1.6 The Woodworth Personal Data Sheet represented an attempt to standardize the psychiatric interview. It contains questions such as those shown here.

18

CHAPTER 1



Introduction seems, experience has shown that it is often false. In addition to being dishonest, the person responding to the question may not interpret the meaning of “wet the bed” the same way as the test administrator does. (Other problems with tests such as the Woodworth are discussed in Chapter 13.) The introduction of the Woodworth test was enthusiastically followed by the creation of a variety of structured personality tests, all of which assumed that a subject’s response could be taken at face value. However, researchers scrutinized, analyzed, and criticized the early structured personality tests, just as they had done with the ability tests. Indeed, the criticism of tests that relied on face value alone became so intense that structured personality tests were nearly driven out of existence. The development of new tests based on more modern concepts followed, revitalizing the use of structured personality tests. Thus, after an initial surge of interest and optimism during most of the 1920s, structured personality tests declined by the late 1930s and early 1940s. Following World War II, however, personality tests based on fewer or different assumptions were introduced, thereby rescuing the structured personality test. During the brief but dramatic rise and fall of the fi rst structured personality tests, interest in projective tests began to grow. In contrast to structured personality tests, which in general provide a relatively unambiguous test stimulus and specific alternative responses, projective personality tests provide an ambiguous stimulus and unclear response requirements. Furthermore, the scoring of projective tests is often subjective. Unlike the early structured personality tests, interest in the projective Rorschach inkblot test grew slowly (see Figure 1.7). The Rorschach test was fi rst published by Herman Rorschach of Switzerland in 1921. However, several years passed before the Rorschach came to the United States, where David Levy introduced it. The first Rorschach doctoral dissertation written in a U.S. university was not completed until 1932, when Sam Beck, Levy’s student, decided to investigate the properties of the Rorschach test scientifically. Although initial interest in the Rorschach test was lukewarm at best, its popularity grew rapidly after Beck’s work, despite suspicion, doubt, and criticism from the scientific community. Today, however, the Rorschach is under a dark cloud (see Chapter 14). Adding to the momentum for the acceptance and use of projective tests was the development of the Thematic Apperception Test (TAT) by Henry Murray and Christina Morgan in 1935. Whereas the Rorschach test contained completely ambiguous inkblot stimuli, the TAT was more structured. Its stimuli consisted of ambiguous pictures depicting a variety of scenes and situations, such as a boy sitting in front of a table with a violin on it. Unlike the Rorschach test, which asked the subject to explain what the inkblot might be, the TAT required the subject to make up a story about the ambiguous scene. The TAT purported to measure human needs and thus to ascertain individual differences in motivation.

The Emergence of New Approaches to Personality Testing The popularity of the two most important projective personality tests, the Rorschach and TAT, grew rapidly by the late 1930s and early 1940s, perhaps because of disillusionment with structured personality tests (Dahlstrom, 1969a). However, as we shall see in Chapter 14, projective tests, particularly the Rorschach, have not

CHAPTER 1



Introduction

19

F I G U R E 1.7 Card 1 of the Rorschach inkblot test, a projective personality test. Such tests provide an ambiguous stimulus to which a subject is asked to make some response.

withstood a vigorous examination of their psychometric properties (Wood, Nezworski, Lilienfeld, & Garb, 2003). In 1943, the Minnesota Multiphasic Personality Inventory (MMPI) began a new era for structured personality tests. The idea behind the MMPI—to use empirical methods to determine the meaning of a test response—helped revolutionize structured personality tests. The problem with early structured personality tests such as the Woodworth was that they made far too many assumptions that subsequent scientific investigations failed to substantiate. The authors of the MMPI, by contrast, argued that the meaning of a test response could be determined only by empirical research. The MMPI, along with its updated companion the MMPI-2 (Butcher, 1989, 1990), is currently the most widely used and referenced personality test. Its emphasis on the need for empirical data has stimulated the development of tens of thousands of studies. Just about the time the MMPI appeared, personality tests based on the statistical procedure called factor analysis began to emerge. Factor analysis is a method of finding the minimum number of dimensions (characteristics, attributes), called factors, to account for a large number of variables. We may say a person is outgoing, is gregarious, seeks company, is talkative, and enjoys relating to others. However, these descriptions contain a certain amount of redundancy. A factor analysis can identify how much they overlap and whether they can all be accounted for or subsumed under a single dimension (or factor) such as extroversion. In the early 1940s, J. R Guilford made the first serious attempt to use factor analytic techniques in the development of a structured personality test. By the end of that decade, R. B. Cattell had introduced the Sixteen Personality Factor Questionnaire (16PF); despite its declining popularity, it remains one of the most wellconstructed structured personality tests and an important example of a test developed with the aid of factor analysis. Today, factor analysis is a tool used in the design

20

CHAPTER 1



Introduction TA B L E

1.2

Summary of Personality Tests Woodworth Personal Data Sheet: An early structured personality test that assumed that a test response can be taken at face value. The Rorschach Inkblot Test: A highly controversial projective test that provided an ambiguous stimulus (an inkblot) and asked the subject what it might be. The Thematic Apperception Test (TAT): A projective test that provided ambiguous pictures and asked subjects to make up a story. The Minnesota Multiphasic Personality Inventory (MMPI): A structured personality test that made no assumptions about the meaning of a test response. Such meaning was to be determined by empirical research. The California Psychological Inventory (CPI): A structured personality test developed according to the same principles as the MMPI. The Sixteen Personality Factor Questionnaire (16PF): A structured personality test based on the statistical procedure of factor analysis.

or validation of just about all major tests. (Factor analytic personality tests will be discussed in Chapter 13.) See Table 1.2 for a brief overview of personality tests.

The Period of Rapid Changes in the Status of Testing The 1940s saw not only the emergence of a whole new technology in psychological testing but also the growth of applied aspects of psychology. The role and significance of tests used in World War I were reaffirmed in World War II. By this time, the U.S. government had begun to encourage the continued development of applied psychological technology. As a result, considerable federal funding provided paid, supervised training for clinically oriented psychologists. By 1949, formal university training standards had been developed and accepted, and clinical psychology was born. Other applied branches of psychology—such as industrial, counseling, educational, and school psychology—soon began to blossom. One of the major functions of the applied psychologist was providing psychological testing. The Shakow, Hilgard, Kelly, Sanford, and Shaffer (1947) report, which was the foundation of the formal training standards in clinical psychology, specified that psychological testing was a unique function of the clinical psychologist and recommended that testing methods be taught only to doctoral psychology students. A position paper of the American Psychological Association published 7 years later (APA, 1954) affi rmed that the domain of the clinical psychologist included testing. It formally declared, however, that the psychologist would conduct psychotherapy only in “true” collaboration with physicians. Thus, psychologists could conduct testing independently, but not psychotherapy. Indeed, as long as psychologists assumed the role of testers, they played a complementary but often secondary role vis-à-vis medical practitioners. Though the medical profession could have hindered the emergence of clinical psychology, it did not, because as tester the psychologist aided the physician. Therefore, in the late 1940s and early 1950s, testing was the major function of the clinical psychologist (Shaffer, 1953).

CHAPTER 1



Introduction

21

For better or worse, depending on one’s perspective, the government’s efforts to stimulate the development of applied aspects of psychology, especially clinical psychology, were extremely successful. Hundreds of highly talented and creative young people were attracted to clinical and other applied areas of psychology. These individuals, who would use tests and other psychological techniques to solve practical human problems, were uniquely trained as practitioners of the principles, empirical foundations, and applications of the science of psychology. Armed with powerful knowledge from scientific psychology, many of these early clinical practitioners must have felt frustrated by their relationship to physicians (see Saccuzzo & Kaplan, 1984). Unable to engage independently in the practice of psychotherapy, some psychologists felt like technicians serving the medical profession. The highly talented group of post-World War II psychologists quickly began to reject this secondary role. Further, because many psychologists associated tests with this secondary relationship, they rejected testing (Lewandowski & Saccuzzo, 1976). At the same time, the potentially intrusive nature of tests and fears of misuse began to create public suspicion, distrust, and contempt for tests. Attacks on testing came from within and without the profession. These attacks intensified and multiplied so fast that many psychologists jettisoned all ties to the traditional tests developed during the first half of the 20th century. Testing therefore underwent another sharp decline in status in the late 1950s that persisted into the 1970s (see Holt, 1967).

The Current Environment During the 1980s, 1990s, and 2000s several major branches of applied psychology emerged and flourished: neuropsychology, health psychology, forensic psychology, and child psychology. Because each of these important areas of psychology makes extensive use of psychological tests, psychological testing again grew in status and use. Neuropsychologists use tests in hospitals and other clinical settings to assess brain injury. Health psychologists use tests and surveys in a variety of medical settings. Forensic psychologists use tests in the legal system to assess mental state as it relates to an insanity defense, competency to stand trial or to be executed, and emotional damages. Child psychologists use tests to assess childhood disorders. Tests are presently in use in developed countries throughout the world (Marsh, Hau, Artelt, Baumet, & Peschar, 2006; Black & William, 2007). As in the past, psychological testing remains one of the most important yet controversial topics in psychology. As a student, no matter what your occupational or professional goals, you will find the material in this text invaluable. If you are among those who are interested in using psychological techniques in an applied setting, then this information will be particularly significant. From the roots of psychology to the present, psychological tests have remained among the most important instruments of the psychologist in general and of those who apply psychology in particular. Testing is indeed one of the essential elements of psychology. Though not all psychologists use tests and some psychologists are opposed to them, all areas of psychology depend on knowledge gained in research studies that rely on measurements. The meaning and dependability of these measurements are essential to psychological research. To study any area of human behavior effectively, one must understand the basic principles of measurement.

22

CHAPTER 1



Introduction In today’s complex society, the relevance of the principles, applications, and issues of psychological testing extends far beyond the field of psychology. Even if you do not plan to become a psychologist, you will likely encounter psychological tests. Attorneys, physicians, social workers, business managers, educators, and many other professionals must frequently deal with reports based on such tests. Even as a parent, you are likely to encounter tests (taken by your children). To interpret such information adequately, you need the information presented in this book. The more you know about psychological tests, the more confident you can be in your encounters with them. Given the attacks on tests and threats to prohibit or greatly limit their use, you have a responsibility to yourself and to society to know as much as you can about psychological tests. The future of testing may well depend on you and people like you. A thorough knowledge of testing will allow you to base your decisions on facts and to ensure that tests are used for the most beneficial and constructive purposes. Tests have probably never been as important as they are today. For example, consider just one type of testing—academic aptitude. Every year more than 2.5 million students take tests that are designed to measure academic progress or suitability, and the testing process begins early in students’ lives. Some presecondary schools require certain tests, and thousands of children take them each year. When these students become adolescents and want to get into college preparatory schools, tens of thousands will take a screening examination. Few students who want to go to a 4-year college can avoid taking a college entrance test. The SAT Reasoning Test alone is given to some 2 million high-school students each year. Another 100,000 high-school seniors take other tests in order to gain advanced placement in college. These figures do not include the 75,000 people who take a special test for admission to business school or the 148,000 who take a Law School Admission Test—or tests for graduate school, medical school, dental school, the military, professional licenses, and others. In fact, the Educational Testing Service alone administers more than 11 million tests annually in 181 countries (Gonzalez, 2001). Nor do they include the millions of tests given around the world for research and evaluation purposes (Black & William, 2007; Marsh et al., 2006). As sources of information about human characteristics, the results of these tests affect critical life decisions.

SUMMARY The history of psychological testing in the United States has been brief but intense. Although these sorts of tests have long been available, psychological testing is very much a product of modern society with its unprecedented technology and population growth and unique problems. Conversely, by helping to solve the challenges posed by modern developments, tests have played an important role in recent U.S. and world history. You should realize, however, that despite advances in the theory and technique of psychological testing, many unsolved technical problems and hotly debated social, political, and economic issues remain. Nevertheless, the prevalence of tests despite strong opposition indicates that, although they are far from perfect, psychological tests must fulfi ll some important need in the decision-making processes permeating all facets of society. Because decisions must be made, such tests will probably flourish until a better or more objective way of making decisions emerges.

CHAPTER 1



Introduction

23

Modern history shows that psychological tests have evolved in a complicated environment in which hostile and friendly forces have produced a balance characterized by innovation and a continuous quest for better methods. One interesting thing about tests is that people never seem to remain neutral about them. If you are not in favor of tests, then we ask that you maintain a flexible, open mind while studying them. Our goal is to give you enough information to assess psychological tests intelligently throughout your life.

This page intentionally left blank

CHAPTER

2

Norms and Basic Statistics for Testing LE AR N I N G O B J EC TIVE S When you have completed this chapter, you should be able to: ■

Discuss three properties of scales of measurement



Determine why properties of scales are important in the field of measurement



Identify methods for displaying distributions of scores



Calculate the mean and the standard deviation for a set of scores



Define a Z score and explain how it is used



Relate the concepts of mean, standard deviation, and Z score to the concept of a standard normal distribution



Define quartiles, deciles, and stanines and explain how they are used



Tell how norms are created



Relate the notion of tracking to the establishment of norms

25

26

CHAPTER 2



Norms and Basic Statistics for Testing

W

e all use numbers as a basic way of communicating: Our money system requires us to understand and manipulate numbers, we estimate how long it will take to do things, we count, we express evaluations on scales, and so on. Think about how many times you use numbers in an average day. There is no way to avoid them. One advantage of number systems is that they allow us to manipulate information. Through sets of well-defined rules, we can use numbers to learn more about the world. Tests are devices used to translate observations into numbers. Because the outcome of a test is almost always represented as a score, much of this book is about what scores mean. Th is chapter reviews some of the basic rules used to evaluate number systems. These rules and number systems are the psychologist’s partners in learning about human behavior. If you have had a course in psychological statistics, then this chapter will reinforce the basic concepts you have already learned. If you need additional review, reread your introductory statistics book. Most such books cover the information in this chapter. If you have not had a course in statistics, then this chapter will provide some of the information needed for understanding other chapters in this book.

WHY WE NEED STATISTICS Through its commitment to the scientific method, modern psychology has advanced beyond centuries of speculation about human nature. Scientific study requires systematic observations and an estimation of the extent to which observations could have been influenced by chance alone (Salkind, 2007). Statistical methods serve two important purposes in the quest for scientific understanding. First, statistics are used for purposes of description. Numbers provide convenient summaries and allow us to evaluate some observations relative to others (Cohen & Lea, 2004; Pagano, 2004; Thompson, 2006). For example, if you get a score of 54 on a psychology examination, you probably want to know what the 54 means. Is it lower than the average score, or is it about the same? Knowing the answer can make the feedback you get from your examination more meaningful. If you discover that the 54 puts you in the top 5% of the class, then you might assume you have a good chance for an A. If it puts you in the bottom 5%, then you will feel differently. Second, we can use statistics to make inferences, which are logical deductions about events that cannot be observed directly. For example, you do not know how many people watched a particular television movie unless you ask everyone. However, by using scientific sample surveys, you can infer the percentage of people who saw the fi lm. Data gathering and analysis might be considered analogous to criminal investigation and prosecution (Cox, 2006; Regenwetter, 2006; Tukey, 1977). First comes the detective work of gathering and displaying clues, or what the statistician John Tukey calls exploratory data analysis. Then comes a period of confirmatory data analysis, when the clues are evaluated against rigid statistical rules. This latter phase is like the work done by judges and juries. Some students have an aversion to numbers and anything mathematical. If you find yourself among them, you are not alone. Not only students but also professional psychologists can feel uneasy about statistics. However, statistics and the basic

CHAPTER 2



Norms and Basic Statistics for Testing

27

principles of measurement lie at the center of the modern science of psychology. Scientific statements are usually based on careful study, and such systematic study requires some numerical analysis. This chapter reviews both descriptive and inferential statistics. Descriptive statistics are methods used to provide a concise description of a collection of quantitative information. Inferential statistics are methods used to make inferences from observations of a small group of people known as a sample to a larger group of individuals known as a population. Typically, the psychologist wants to make statements about the larger group but cannot possibly make all the necessary observations. Instead, he or she observes a relatively small group of subjects (sample) and uses inferential statistics to estimate the characteristics of the larger group (Salkind, 2007).

SCALES OF MEASUREMENT One may define measurement as the application of rules for assigning numbers to objects. The rules are the specific procedures used to transform qualities of attributes into numbers (Camilli, Cizek, & Lugg, 2001; Nunnally & Bernstein, 1994; Yanai, 2003). For example, to rate the quality of wines, wine tasters must use a specifi c set of rules. They might rate the wine on a 10-point scale where 1 means extremely bad and 10 means extremely good. For a taster to assign the numbers, the system of rules must be clearly defined. The basic feature of these types of systems is the scale of measurement. For example, to measure the height of your classmates, you might use the scale of inches; to measure their weight, you might use the scale of pounds. There are numerous systems by which we assign numbers in psychology. Indeed, the study of measurement systems is what this book is about. Before we consider any specific scale of measurement, however, we should consider the general properties of measurement scales.

Properties of Scales Three important properties make scales of measurement different from one another: magnitude, equal intervals, and an absolute 0. Magnitude Magnitude is the property of “moreness.” A scale has the property of magnitude if we can say that a particular instance of the attribute represents more, less, or equal amounts of the given quantity than does another instance (Aron & Aron, 2003; Hurlburt, 2003; McCall, 2001; Howell, 2008). On a scale of height, for example, if we can say that John is taller than Fred, then the scale has the property of magnitude. A scale that does not have this property arises, for example, when a gym coach assigns identification numbers to teams in a league (team 1, team 2, and so forth). Because the numbers only label the teams, they do not have the property of magnitude. If the coach were to rank the teams by the number of games they have won, then the new numbering system (games won) would have the property of magnitude.

CHAPTER 2



Norms and Basic Statistics for Testing 10 Ratings of Artwork

28

8 6 4 2 0

0

2 4 6 8 Manual Dexterity Test

10

F I G U R E 2.1 Hypothetical relationship between ratings of artwork and manual dexterity. In some ranges of the scale, the relationship is more direct than it is in others.

Equal Intervals The concept of equal intervals is a little more complex than that of magnitude. A scale has the property of equal intervals if the difference between two points at any place on the scale has the same meaning as the difference between two other points that differ by the same number of scale units. For example, the difference between inch 2 and inch 4 on a ruler represents the same quantity as the difference between inch 10 and inch 12: exactly 2 inches. As simple as this concept seems, a psychological test rarely has the property of equal intervals. For example, the difference between Intelligence Quotients (IQs) of 45 and 50 does not mean the same thing as the difference between IQs of 105 and 110. Although each of these differences is 5 points (50 ⫺ 45 ⫽ 5 and 110 ⫺ 105 ⫽ 5), the 5 points at the first level do not mean the same thing as 5 points at the second. We know that IQ predicts classroom performance. However, the difference in classroom performance associated with differences between IQ scores of 45 and 50 is not the same as the differences in classroom performance associated with IQ score differences of 105 and 110. In later chapters we will discuss this problem in more detail. When a scale has the property of equal intervals, the relationship between the measured units and some outcome can be described by a straight line or a linear equation in the form Y ⫽ a ⫹ bX. This equation shows that an increase in equal units on a given scale reflects equal increases in the meaningful correlates of units. For example, Figure 2.1 shows the hypothetical relationship between scores on a test of manual dexterity and ratings of artwork. Notice that the relationship is not a straight line. By examining the points on the figure, you can see that at first the relationship is nearly linear: Increases in manual dexterity are associated with increases in ratings of artwork. Then the relationship becomes nonlinear. The figure shows that after a manual dexterity score of approximately 5, increases in dexterity produce relatively smaller increases in quality of artwork.

CHAPTER 2 TA B L E



Norms and Basic Statistics for Testing

29

2.1

Scales of Measurement and Their Properties Property Type of scale

Magnitude

Equal intervals

Absolute 0

Nominal

No

No

No

Ordinal

Yes

No

No

Interval

Yes

Yes

No

Ratio

Yes

Yes

Yes

Absolute 0 An absolute 0 is obtained when nothing of the property being measured exists. For example, if you are measuring heart rate and observe that your patient has a rate of 0 and has died, then you would conclude that there is no heart rate at all. For many psychological qualities, it is extremely difficult, if not impossible, to define an absolute 0 point. For example, if one measures shyness on a scale from 0 through 10, then it is hard to define what it means for a person to have absolutely no shyness (McCall, 2001).

Types of Scales Table 2.1 defines four scales of measurement based on the properties we have just discussed. You can see that a nominal scale does not have the property of magnitude, equal intervals, or an absolute 0. Nominal scales are really not scales at all; their only purpose is to name objects. For example, the numbers on the backs of football players’ uniforms are nominal. Nominal scales are used when the information is qualitative rather than quantitative. Social science researchers commonly label groups in sample surveys with numbers (such as 1 ⫽ African American, 2 ⫽ white, and 3 ⫽ Mexican American). When these numbers have been attached to categories, most statistical procedures are not meaningful. On the scale for ethnic groups, for instance, what would a mean of 1.87 signify? Th is is not to say that the sophisticated statistical analysis of nominal data is impossible. Indeed, several new and exciting developments in data analysis allow extensive and detailed use of nominal data (Chen, 2002; Miller, Scurfield, Drga, Galvin, & Whitmore, 2002; Stout, 2002). A scale with the property of magnitude but not equal intervals or an absolute 0 is an ordinal scale. This scale allows you to rank individuals or objects but not to say anything about the meaning of the differences between the ranks. If you were to rank the members of your class by height, then you would have an ordinal scale. For example, if Fred was the tallest, Susan the second tallest, and George the third tallest, you would assign them the ranks 1, 2, and 3, respectively. You would not give any consideration to the fact that Fred is 8 inches taller than Susan, but Susan is only 2 inches taller than George. For most problems in psychology, the precision to measure the exact differences between intervals does not exist. So, most often one must use ordinal scales of

30

CHAPTER 2



Norms and Basic Statistics for Testing measurement. For example, IQ tests do not have the property of equal intervals or an absolute 0, but they do have the property of magnitude. If they had the property of equal intervals, then the difference between an IQ of 70 and one of 90 should have the same meaning as the difference between an IQ of 125 and one of 145. Because it does not, the scale can only be considered ordinal. Furthermore, there is no point on the scale that represents no intelligence at all—that is, the scale does not have an absolute 0. When a scale has the properties of magnitude and equal intervals but not absolute 0, we refer to it as an interval scale. The most common example of an interval scale is the measurement of temperature in degrees Fahrenheit. This temperature scale clearly has the property of magnitude, because 35°F is warmer than 32°F, 65°F is warmer than 64°F, and so on. Also, the difference between 90°F and 80°F is equal to a similar difference of 10° at any point on the scale. However, on the Fahrenheit scale, temperature does not have the property of absolute 0. If it did, then the 0 point would be more meaningful. As it is, 0 on the Fahrenheit scale does not have a particular meaning. Water freezes at 32°F and boils at 212°F. Because the scale does not have an absolute 0, we cannot make statements in terms of ratios. A temperature of 22°F is not twice as hot as 11°F, and 70°F is not twice as hot as 35°F. The Celsius scale of temperature is also an interval rather than a ratio scale. Although 0 represents freezing on the Celsius scale, it is not an absolute 0. Remember that an absolute 0 is a point at which nothing of the property being measured exists. Even on the Celsius scale of temperature, there is still plenty of room on the thermometer below 0. When the temperature goes below freezing, some aspect of heat is still being measured. A scale that has all three properties (magnitude, equal intervals, and an absolute 0) is called a ratio scale. To continue our example, a ratio scale of temperature would have the properties of the Fahrenheit and Celsius scales but also include a meaningful 0 point. There is a point at which all molecular activity ceases, a point of absolute 0 on a temperature scale. Because the Kelvin scale is based on the absolute 0 point, it is a ratio scale: 22°K is twice as cold as 44°K. Examples of ratio scales also appear in the numbers we see on a regular basis. For example, consider the number of yards gained by running backs on football teams. Zero yards actually means that the player has gained no yards at all. If one player has gained 1000 yards and another has gained only 500, then we can say that the fi rst athlete has gained twice as many yards as the second. Another example is the speed of travel. For instance, 0 miles per hour (mph) is the point at which there is no speed at all. If you are driving onto a highway at 30 mph and increase your speed to 60 when you merge, then you have doubled your speed.

Permissible Operations Level of measurement is important because it defines which mathematical operations we can apply to numerical data. For nominal data, each observation can be placed in only one mutually exclusive category. For example, you are a member of only one gender. One can use nominal data to create frequency distributions (see the next section), but no mathematical manipulations of the data are permissible. Ordinal measurements can be manipulated using arithmetic; however, the result is

CHAPTER 2



31

Norms and Basic Statistics for Testing

often difficult to interpret because it reflects neither the magnitudes of the manipulated observations nor the true amounts of the property that have been measured. For example, if the heights of 15 children are rank ordered, knowing a given child’s rank does not reveal how tall he or she stands. Averages of these ranks are equally uninformative about height. With interval data, one can apply any arithmetic operation to the differences between scores. The results can be interpreted in relation to the magnitudes of the underlying property. However, interval data cannot be used to make statements about ratios. For example, if IQ is measured on an interval scale, one cannot say that an IQ of 160 is twice as high as an IQ of 80. Th is mathematical operation is reserved for ratio scales, for which any mathematical operation is permissible.

FREQUENCY DISTRIBUTIONS A single test score means more if one relates it to other test scores. A distribution of scores summarizes the scores for a group of individuals. In testing, there are many ways to record a distribution of scores. The frequency distribution displays scores on a variable or a measure to reflect how frequently each value was obtained. With a frequency distribution, one defines all the possible scores and determines how many people obtained each of those scores. Usually, scores are arranged on the horizontal axis from the lowest to the highest value. The vertical axis reflects how many times each of the values on the horizontal axis was observed. For most distributions of test scores, the frequency distribution is bell-shaped, with the greatest frequency of scores toward the center of the distribution and decreasing scores as the values become greater or less than the value in the center of the distribution. Figure 2.2 shows a frequency distribution of 1000 observations that takes on values between 61 and 90. Notice that the most frequent observations fall toward

Frequency of Occurrence

100 80 60 40 20 0

62

64

66

68

70

72

74

76 Score

78

80

82

84

86

F I G U R E 2.2 Frequency distribution approximating a normal distribution of 1000 observations.

88

90

32

CHAPTER 2



Norms and Basic Statistics for Testing the center of the distribution, around 75 and 76. As you look toward the extremes of the distribution, you will find a systematic decline in the frequency with which the scores occur. For example, the score of 71 is observed less frequently than 72, which is observed less frequently than 73, and so on. Similarly, 78 is observed more frequently than 79, which is noted more often than 80, and so forth. Though this neat symmetric relationship does not characterize all sets of scores, it occurs frequently enough in practice for us to devote special attention to it. In the section on the normal distribution, we explain this concept in greater detail. Table 2.2 lists the rainfall amounts in San Diego, California, between 1964 and 2007. Figure 2.3 is a histogram based on the observations. The distribution is

TA B L E

2.2

Inches of Rainfall in San Diego, 1964–2007 Year

Rainfall (Inches)

Year

Rainfall (Inches)

1964

5.15

1988

12.44

1965

8.81

1989

5.88

1966

14.76

1990

7.62

1967

10.86

1991

12.31

1968

7.86

1992

12.48

1969

11.48

1993

18.26

1970

6.23

1994

9.93

1971

8.03

1995

17.13

1972

6.12

1996

5.18

1973

10.99

1997

8.74

1974

6.59

1998

20.89

1975

10.64

1999

6.51

1976

10.14

2000

5.77

1977

9.18

2001

8.82

2002

3.44

14.93

2003

10.24

15.62

2004

5.31

1981

8.13

2005

22.81

1982

11.85

2006

5.35

1983

18.49

2007

1984

5.37

Sum

1985

9.6

Mean

1986

14.64

1987

9.3

1978

17.3

1979 1980

Data from http://cdec.water.ca.gov.

Standard Deviation

3.62 454.8 10.34 4.71

CHAPTER 2



Norms and Basic Statistics for Testing

33

12

Frequency

10 8 6 4 2 0

0–3

FIGURE

2.3

3.1–6

6.1–9

9.1–12 12.1–15 15.1–18 18.1–21 21.1–24 24.1–27 Inches of Rainfall

Histogram for San Diego rainfall, 1964–2007.

12

Frequency

10 8 6 4 2 0

0–3

FIGURE

2.4

3.1–6

6.1–9

9.1–12 12.1–15 15.1–18 18.1–21 21.1–24 24.1–27 Inches of Rainfall

Frequency polygon for rainfall in San Diego, 1964–2007.

slightly skewed, or asymmetrical. We say that Figure 2.3 has a positive skew because the tail goes off toward the higher or positive side of the X axis. There is a slight skew in Figures 2.3 and 2.4, but the asymmetry in these figures is relatively hard to detect. Figure 2.5 gives an example of a distribution that is clearly skewed. The figure summarizes annual household income in the United States in 2007. Very few people make high incomes, while the great bulk of the population is bunched toward the low end of the income distribution. Of particular interest is that this figure only includes household incomes less than $100,000. For household incomes greater than $100,000, the government only reports incomes using class intervals of $50,000. In 2007, about 16% of the U.S. households had incomes greater than $100,000. Because some households have extremely high incomes, you can imagine that the tail of this distribution would go very far to the right. Thus, income is an example of a variable that has positive skew. One can also present this same set of data as a frequency polygon (see Figure 2.4). Here the amount of rainfall is placed on the graph as a point that represents the

34

CHAPTER 2



Norms and Basic Statistics for Testing 4.0% 3.5%

Percent

3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 50 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 2, $7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97, $ $ $ $ $ $ $ $ $ $ $ $ $ r o $ $ $ $ $ $ de 0 t to to to to to to to to to to to to to to to to to to Un ,00 000 000 000 ,000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 $5 10, 15, 20, 25 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

Personal Income (US$)

F I G U R E 2.5 Household income up to $100,000 in the United States for 2007. This is an example of positive skew. (Data from the United States Department of Labor Statistics and the Bureau the Census. http://ferret.bls. census .gov/macro/ 032003/hhinc/ new06_000.htm.)

frequencies with which each interval occurs. Lines are then drawn to connect these points. Whenever you draw a frequency distribution or a frequency polygon, you must decide on the width of the class interval. The class interval for inches of rainfall is the unit on the horizontal axis. For example, in Figures 2.3 and 2.4, the class interval is 3 inches—that is, the demarcations along the X axis increase in 3-inch intervals. This interval is used here for convenience; the choice of 3 inches is otherwise arbitrary.

PERCENTILE RANKS Percentile ranks replace simple ranks when we want to adjust for the number of scores in a group. A percentile rank answers the question “What percent of the scores fall below a particular score (Xi)?” To calculate a percentile rank, you need only follow these simple steps: (1) determine how many cases fall below the score of interest, (2) determine how many cases are in the group, (3) divide the number of cases below the score of interest (Step 1) by the total number of cases in the group (Step 2), and (4) multiply the result of Step 3 by 100.

CHAPTER 2



Norms and Basic Statistics for Testing

35

The formula is Pr ⫽

B ⫻100 ⫽ percentile rank of X i N

where Pr = percentile rank Xi = the score of interest B = the number of scores below Xi N = the total number of scores This means that you form a ratio of the number of cases below the score of interest and the total number of scores. Because there will always be either the same or fewer cases in the numerator (top half) of the equation than there are in the denominator (bottom half), this ratio will always be less than or equal to 1. To get rid of the decimal points, you multiply by 100. As an example, consider the runner who finishes 62nd out of 63 racers in a gym class. To obtain the percentile rank, divide 1 (the number of people who finish behind the person of interest) by 63 (the number of scores in the group). This gives you 1/63, or .016. Then multiply this result by 100 to obtain the percentile rank, which is 1.6. Th is rank tells you the runner is below the 2nd percentile. Now consider the Bay to Breakers race, which attracts 50,000 runners to San Francisco. If you had finished 62nd out of 50,000, then the number of people who were behind you would be 49,938. Dividing this by the number of entrants gives .9988. When you multiply by 100, you get a percentile rank of 99.88. This tells you that finishing 62nd in the Bay to Breakers race is exceptionally good because it places you in the 99.88th percentile. Psychological Testing in Everyday Life 2.1 presents the calculation of percentile ranks of the infant mortality rates of selected countries as reported by the World Health Organization in 2007. Infant mortality is defined as the number of babies out of 1000 who are born alive but die before their fi rst birthday. Before proceeding, we should point out that the meaning of this calculation depends on which countries are used in the comparison. In this example, the calculation of the percentile rank is broken into five steps and uses the raw data in the table. In Step 1, we arrange the data points in ascending order. Singapore has the lowest infant mortality rate (2.3), Japan is next (2.8), and Afghanistan has the highest rate (157.0). In Step 2, we determine the number of cases with worse rates than that of the case of interest. In this example, the case of interest is the United States. Therefore, we count the number of cases with a worse rate than that of the United States. Eleven countries—Israel, Saudi Arabia, Colombia, China, Turkey, Morocco, Bolivia, Laos, Ethiopia, Mozambique, and Afghanistan—have infant mortality rates greater than 6.4.

36

CHAPTER 2



Norms and Basic Statistics for Testing

PSYCHOLOGICAL TESTING IN E V ERYDAY LIFE 2.1

Infant Mortality in Selected Countries, 2007 Infant Mortality per 1000 Live Births

Country Afghanistan Australia

157.0 4.6

Bolivia

45.6

China

23.0

Colombia

19.1

Ethiopia

86.9

France

3.4

Israel

6.8

Italy

5.7

Japan

2.8

Laos

51.4

Morocco

30.6

Mozambique

95.9

Saudi Arabia

18.8

Singapore

2.3

Spain

4.3

Turkey

27.5

United States

6.4

Mean

32.9

SD

41.9

To calculate the percentile rank of infant mortality in the United States in comparison to that in selected countries, use the following formula: Pr ⫽ where

B ⫻100 N

Pr ⫽ the percentile rank B ⫽ the number of cases with worse rates than the case of interest N ⫽ the total number of cases

CHAPTER 2

Country



Norms and Basic Statistics for Testing

Infant Mortality per 1000 Live Births

Singapore

2.3

Japan

2.8

France

3.4

Spain

4.3

Australia

4.6

Italy

5.7

United States

6.4

Israel

6.8

Saudi Arabia

18.8

Colombia

19.1

China

23.0

Turkey

27.5

Morocco

30.6

Bolivia

45.6

Laos

51.4

Ethiopia

86.9

Mozambique

95.9

Afghanistan

157.0

STEPS 1. Arrange data in ascending order—that is, the lowest score first, the second lowest score second, and so on. N = 18, mean = 32.9, standard deviation = 41.9 2. Determine the number of cases with worse rates than the score of interest. There are 11 countries in this sample with infant mortality rates greater than that in the United States. 3. Determine the number of cases in the sample (18). 4. Divide the number of scores worse than the score of interest (Step 2) by the total number of scores (Step 3): 11 ⫽ .61 18 5. Multiply by 100: .61 × 100 = 61st percentile rank

37

38

CHAPTER 2



Norms and Basic Statistics for Testing In Step 3, we determine the total number of cases (18). In Step 4, we divide the number of scores worse than the score of interest by the total number of scores: 11 ⫽ .61 18 Technically, the percentile rank is a percentage. Step 4 gives a proportion. Therefore, in Step 5 you transform this into a whole number by multiplying by 100: .61 × 100 = 61 Thus, the United States is in the 61st percentile. The percentile rank depends absolutely on the cases used for comparison. In this example, you calculated that the United States is in the 61st percentile for infant mortality within this group of countries. If all countries in the world had been included, then the ranking of the United States might have been different. Using this procedure, try to calculate the percentile rank for Bolivia. The calculation is the same except that there are four countries with worse rates than Bolivia (as opposed to 11 worse than the United States). Thus, the percentile rank for Bolivia is 4 ⫽ .22 ⫻ 100 ⫽ 22 18 or the 22nd percentile. Now try France. You should get a percentile rank of 83.

PERCENTILES Percentiles are the specific scores or points within a distribution. Percentiles divide the total frequency for a set of observations into hundredths. Instead of indicating what percentage of scores fall below a particular score, as percentile ranks do, percentiles indicate the particular score, below which a defined percentage of scores falls. Try to calculate the percentile and percentile rank for some of the data in Psychological Testing in Everyday Life 2.1. As an example, look at Italy. The infant mortality rate in Italy is 5.72/1000. When calculating the percentile rank, you exclude the score of interest and count those below (in other words, Italy is not included in the count). There are 12 countries in this sample with infant mortality rates worse than Italy’s. To calculate the percentile rank, divide this number of countries by the total number of cases and multiply by 100: Pr ⫽

B 12 ⫻ 100 ⫽ ⫻ 100 ⫽ .67 ⫻ 100 ⫽ 67 N 18

Thus, Italy is in the 67th percentile rank, or the 67th percentile in this example is 5.72/1000 or 5.72 deaths per 1000 live births. Now take the example of Israel. The calculation of percentile rank requires looking at the number of cases below the case of interest. In this example, 10 countries

CHAPTER 2



Norms and Basic Statistics for Testing

39

in this group have infant mortality rates worse than Israel’s. Thus, the percentile rank for Israel is 10/18 × 100 = 56. The 56th percentile corresponds with the point or score of 6.75 (6.75/1000 live births). In summary, the percentile and the percentile rank are similar. The percentile gives the point in a distribution below which a specified percentage of cases fall (6.75/1000 for Israel). The percentile is in raw score units. The percentile rank gives the percentage of cases below the percentile; in this example, the percentile rank is 56. When reporting percentiles and percentile ranks, you must carefully specify the population you are working with. Remember that a percentile rank is a measure of relative performance. When interpreting a percentile rank, you should always ask the question “Relative to what?” Suppose, for instance, that you finished in the 17th percentile in a swimming race (or fifth in a heat of six competitors). Does this mean that you are a slow swimmer? Not necessarily. It may be that this was a heat in the Olympic games, and the participants were the fastest swimmers in the world. An Olympic swimmer competing against a random sample of all people in the world would probably finish in the 99.99th percentile. The example for infant mortality rates depends on which countries in the world were selected for comparison. The United States actually does quite poorly when compared with European countries and the advanced economies in Asia (Singapore and Japan). However, the U.S. infant mortality rate looks much better compared with countries in the developing world.1

DESCRIBING DISTRIBUTIONS Mean Statistics are used to summarize data. If you consider a set of scores, the mass of information may be too much to interpret all at once. That is why we need numerical conveniences to help summarize the information. An example of a set of scores that can be summarized is shown in Table 2.2 (see page 32), amounts of rainfall in San Diego. We signify the variable as X. A variable is a score that can have different values. The amount of rain is a variable because different amounts of rain fell in different years. The arithmetic average score in a distribution is called the mean. To calculate the mean, we total the scores and divide the sum by the number of cases, or N. The capital Greek letter sigma (Σ) means summation. Thus, the formula for the mean, which we signify as X , is X⫽

1

⌺X N

We used a similar example in the last edition based on data from 2003. By 2007, there were significant improvements in the infant mortality rates in developing countries. Th e rate for Mozambique declined from 148.6 down to 95.9 per thousand live births. Ethiopia reduced its infant mortality rate from 142.6 to 86.9. However, the rates worsened slightly for several developed countries, including Israel, Italy, and Spain.

40

CHAPTER 2



Norms and Basic Statistics for Testing In words, this formula says to total the scores and divide the sum by the number of cases. Using the information in Table 2.2, we find the mean by following these steps: 1. 2.

Obtain ΣX, or the sum of the scores: 5.15 + 8.81 + 14.76 + 10.86 + 7.86 + ... + 3.62 = 454.80 Find N, or the number of scores: N = 44

3.

Divide ΣX by N: 454.80/44 = 10.34

Psychological Testing in Everyday Life 2.2 summarizes common symbols used in basic statistics.

Standard Deviation The standard deviation is an approximation of the average deviation around the mean. The standard deviation for the amount of rainfall in San Diego is 4.71. To understand rainfall in San Diego, you need to consider at least two dimensions: fi rst, the amount of rain that falls in a particular year; second, the degree of variation from year to year in the amount of rain that falls. The calculation suggests that, on the average, the variation around the mean is approximately 4.71 inches. However informative, knowing the mean of a group of scores does not give you that much information. As an illustration, look at the following sets of numbers. Set 1 4 4 4 4 4 4

Set 2 5 5 4 4 3 3

Set 3 8 8 6 2 0 0

PSYCHOLOGICAL TESTING IN E V ERYDAY LIFE 2. 2

Common Symbols You need to understand and recognize the symbols used throughout this book. X is the mean; it is pronounced “X bar.” Σ is the summation sign. It means sum, or add, scores together and is the capital Greek letter sigma. X is a variable that takes on different values. Each value of Xi represents a raw score, also called an obtained score.

CHAPTER 2



Norms and Basic Statistics for Testing

41

Calculate the mean of the fi rst set. You should get 4. What is the mean of the second set? If you calculate correctly, you should get 4 again. Next find the mean for Set 3. It is also 4. The three distributions of scores appear quite different but have the same mean, so it is important to consider other characteristics of the distribution of scores besides the mean. The difference between the three sets lies in variability. There is no variability in Set 1, a small amount in Set 2, and a lot in Set 3. Measuring this variation is similar to finding the average deviation around the mean. One way to measure variability is to subtract the mean from each score (X − X ) and then total the deviations. Statisticians often signify this with a lowercase x, as in x = (X − X ). Try this for the data in Table 2.2. Did you get 0? You should have, and this is not an unusual example. In fact, the sum of the deviations around the mean will always equal 0. However, you do have an alternative: You can square all the deviations around the mean in order to get rid of any negative signs. Then you can obtain the average squared deviation around the mean, known as the variance. The formula for the variance is

␴2 ⫽

⌺( X ⫺ X )2 N

where (X − X ) is the deviation of a score from the mean. The symbol s is the lowercase Greek sigma; s 2 is used as a standard description of the variance. Though the variance is a useful statistic commonly used in data analysis, it shows the variable in squared deviations around the mean rather than in deviations around the mean. In other words, the variance is the average squared deviation around the mean. To get it back into the units that will make sense to us, we need to take the square root of the variance. The square root of the variance is the standard deviation (s), and it is represented by the following formula

␴⫽

⌺( X ⫺ X )2 N

The standard deviation is thus the square root of the average squared deviation around the mean. Although the standard deviation is not an average deviation, it gives a useful approximation of how much a typical score is above or below the average score. Because of their mathematical properties, the variance and the standard deviation have many advantages. For example, knowing the standard deviation of a normally distributed batch of data allows us to make precise statements about the distribution. The formulas just presented are for computing the variance and the standard deviation of a population. That is why we use the lowercase Greek sigma (s and s2). Psychological Testing in Everyday Life 2.3 summarizes when you should use Greek and Roman letters. Most often we use the standard deviation for a sample to estimate the standard deviation for a population. When we talk about a sample, we replace the Greek s with a Roman letter S. Also, we divide by

42

CHAPTER 2



Norms and Basic Statistics for Testing N − 1 rather than N to recognize that S of a sample is only an estimate of the variance of the population.

S⫽

⌺( X ⫺ X )2 N ⫺1

In calculating the standard deviation, it is often easier to use the raw score equivalent formula, which is

S⫽

( ⌺X )2 N N ⫺1

⌺X 2 ⫺

This calculation can also be done automatically by some minicalculators. In reading the formula, you may be confused by a few points. In particular, be careful not to confuse ΣX 2 and (ΣX)2. To get ΣX 2, each individual score is squared and the values are summed. For the scores 3, 5, 7, and 8, ΣX 2 would be 32 + 52 + 72 + 82 = 9 + 25 + 49 + 64 = 147. To obtain (ΣX)2, the scores are first summed and the total is squared. Using the example, (ΣX)2 = (3 + 5 + 7 + 8)2 = 232 = 529.

Z Score One problem with means and standard deviations is that they do not convey enough information for us to make meaningful assessments or accurate interpretations of data. Other metrics are designed for more exact interpretations. The Z score

PSYCHOLOGICAL TESTING IN E V ERYDAY LIFE 2. 3

Terms and Symbols Used to Describe Populations and Samples Population

Sample

Definition

All elements with the same definition

A subset of the population, usually drawn to represent it in an unbiased fashion

Descriptive characteristics

Parameters

Statistics

Symbols used to describe

Greek

Roman

Symbol for mean

m

X

Symbol for standard deviation

s

S

CHAPTER 2



Norms and Basic Statistics for Testing

43

transforms data into standardized units that are easier to interpret. A Z score is the difference between a score and the mean, divided by the standard deviation: Z⫽

Xi ⫺ X S

In other words, a Z score is the deviation of a score Xi from the mean X in standard deviation units. If a score is equal to the mean, then its Z score is 0. For example, suppose the score and the mean are both 6; then 6 − 6 = 0. Zero divided by anything is still 0. If the score is greater than the mean, then the Z score is positive; if the score is less than the mean, then the Z score is negative. Let’s try an example. Suppose that Xi = 6, the mean X = 3, and the standard deviation S = 3. Plugging these values into the formula, we get Z⫽

6⫺3 3 ⫽ ⫽1 3 3

Let’s try another example. Suppose Xi = 4, X = 5.75, and S = 2.11. What is the Z score? It is −.83: Z⫽

4 ⫺ 5.74 ⫺1.74 ⫽ ⫽⫺.83 2.11 2.11

This means that the score we observed (4) is .83 standard deviation below the average score, or that the score is below the mean but its difference from the mean is slightly less than the average deviation. Example of Depression in Medical Students: Center for Epidemiologic Studies Depression Scale (CES-D) The CES-D is a general measure of depression that has been used extensively in epidemiological studies. The scale includes 20 items and taps dimensions of depressed mood, hopelessness, appetite loss, sleep disturbance, and energy level. Each year, students at the University of California, San Diego, School of Medicine are asked to report how often they experienced a particular symptom during the fi rst week of school on a 4-point scale ranging from rarely or none of the time [0 to 1 days (0)] to most or all of the time [5 to 7 days (3)]. Items 4, 8, 12, and 16 on the CES-D are reverse scored. For these items, 0 is scored as 3, 1 is scored as 2, 2 as 1, and 3 as 0. The CES-D score is obtained by summing the circled numbers. Scores on the CES-D range from 0 to 60, with scores greater than 16 indicating clinically significant levels of depressive symptomatology in adults. Feel free to take the CES-D measure yourself. Calculate your score by summing the numbers you have circled. However, you must fi rst reverse the scores on items 4, 8, 12, and 16. As you will see in Chapter 5, the CES-D does not have high validity for determining clinical depression. If your score is less than 16, the evidence suggests that you are not clinically depressed. If your score is high, it raises suspicions about depression—though this does not mean you have a problem. (Of course, you may want to talk with your college counselor if you are feeling depressed.)

44

CHAPTER 2



Norms and Basic Statistics for Testing

Center for Epidemiologic Studies Depression Scale (CES-D) Instructions: Circle the number for each statement that best describes how often you felt or behaved this way DURING THE PAST WEEK.

R

R

Rarely or none of the time (less than 1 day)

Some or a little of the time (1–2 days)

Occasionally or a moderate amount of the time (3–4 days)

Most or all of the time (5–7 days)

1. I was bothered by things that usually don’t bother me . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

2. I did not feel like eating . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

3. I felt that I could not shake off the blues even with help from my family or friends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

4. I felt that I was just as good as other people. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

5. I had trouble keeping my mind on what I was doing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

6. I felt depressed . . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

7. I felt that everything I did was an effort.

0

.......1

.........2

.......3

8. I felt hopeful about the future . . . . . . . . . .

0

.......1

.........2

.......3

9. I thought my life had been a failure. . . . .

0

.......1

.........2

.......3

10. I felt fearful. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

11. My sleep was restless . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

R 12. I was happy . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

13. I talked less than usual. . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

14. I felt lonely. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

15. People were unfriendly. . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

R 16. I enjoyed life . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

17. I had crying spells. . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

18. I felt sad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

19. I felt that people disliked me . . . . . . . . . . .

0

.......1

.........2

.......3

20. I could not get “going.” . . . . . . . . . . . . . . . .

0

.......1

.........2

.......3

Table 2.3 shows CES-D scores for a selected sample of medical students. You can use these data to practice calculating means, standard deviations, and Z scores. In creating the frequency distribution for the CES-D scores of medical students we used an arbitrary class interval of 5.

CHAPTER 2 TA B L E



Norms and Basic Statistics for Testing

45

2.3

The Calculation of Mean, Standard Deviation, and Z Scores for CES-D Scores Name

Test score (X)

John Carla Fred Monica Eng Fritz Mary Susan Debbie Elizabeth Sarah Marcel Robin Mike Carl Phyllis Jennie Richard Tyler Frank

14 10 8 8 26 0 14 3 9 10 7 12 10 25 9 12 23 7 13 1 ΣX = 221

X⫽

S⫽

X2

Z score

196 100 64 64 676 0 196 9 81 100 49 144 100 625 81 144 529 49 169 1 ΣX 2 = 3377

.42 ⫺.15 ⫺.44 ⫺.44 2.13 ⫺1.58 .42 ⫺1.15 ⫺.29 ⫺.15 ⫺.58 .14 ⫺.15 1.99 .29 .14 1.70 ⫺.58 .28 ⫺1.43

⌺X 221 ⫽ ⫽11.05 N 20 ( ⌺X )2 (221)2 3377 ⫺ 20 ⫽ 7.01 N ⫽ 20 ⫺1 N ⫺1

⌺X 2 ⫺

Monica's Z score ⫽

X ⫺ X 8 ⫺11.05 ⫽ ⫽⫺.44 S 7.01

Marcel’s Z score⫽

X ⫺ X 12 ⫺11.05 ⫽ ⫽ .14 S 7.01

Jennie’s Z score⫽

X ⫺ X 23⫺11.05 ⫽ ⫽1.70 S 7.01

Standard Normal Distribution Now we consider the standard normal distribution because it is central to statistics and psychological testing. First, though, you should participate in a short exercise. Take any coin and fl ip it 10 times. Now repeat this exercise of 10 coin fl ips 25 times. Record the number of heads you observe in each group of 10 flips. When you are



Norms and Basic Statistics for Testing 6 Frequency of Occurrence

CHAPTER 2

5 4 3 2 1 0

FIGURE

0

2.6

Frequency of Occurrence

46

1

2

3

4 5 6 Number of Heads

7

8

9

10

Frequency distribution of the number of heads in 25 sets of 10 flips.

.3413

.00135

.3413

.00135

.1359 –3

.0214 –2

.1359 –1

0 Z Scores

1

.0214 2 3

F I G U R E 2.7 The theoretical distribution of the number of heads in an infinite number of coin flips.

done, make a frequency distribution showing how many times you observed 1 head in your 10 fl ips, 2 heads, 3 heads, and so on. Your frequency distribution might look like the example shown in Figure 2.6. The most frequently observed events are approximately equal numbers of heads and tails. Toward the extremes of 10 heads and 0 tails or 10 tails and 0 heads, events are observed with decreasing frequency. For example, there were no occasions in which fewer than 2 heads were observed and only one occasion in which more than 8 heads were observed. Th is is what we would expect from the laws of probability. On the average, we would expect half of the fl ips to show heads and half to show tails if heads and tails are equally probable events. Although observing a long string of heads or tails is possible, it is improbable. In other words, we sometimes see the coin come up heads in 9 out of 10 fl ips. The likelihood that this will happen, however, is quite small. Figure 2.7 shows the theoretical distribution of heads in an infinite number of fl ips of the coin. This figure might look a little like the distribution from your coinfl ipping exercise or the distribution shown in Figure 2.6. Actually, this is a normal distribution, or what is known as a symmetrical binomial probability distribution.

CHAPTER 2



Norms and Basic Statistics for Testing

47

On most occasions, we refer to units on the X (or horizontal) axis of the normal distribution in Z score units. Any variable transformed into Z score units takes on special properties. First, Z scores have a mean of 0 and a standard deviation of 1.0. If you think about this for a minute, you should be able to figure out why this is true. Recall that the sum of the deviations around the mean is always equal to 0. The numerator of the Z score equation is the deviation around the mean, while the denominator is a constant. Thus, the mean of Z scores can be expressed as ⌺( X i

X)/S N

or

⌺Z N

Because Σ(Xi − X ) will always equal 0, the mean of Z scores will always be 0. In Figure 2.7, the standardized, or Z score, units are marked on the X axis. The numbers under the curve are the proportions of cases (in decimal form) that we would expect to observe in each area. Multiplying these proportions by 100 yields percentages. For example, we see that 34.13% or .3413 of the cases fall between the mean and one standard deviation above the mean. Do not forget that 50% of the cases fall below the mean. Putting these two bits of information together, we can conclude that if a score is one standard deviation above the mean, then it is at about the 84th percentile rank (50 + 34.13 = 84.13 to be exact). A score that is one standard deviation below the mean would be about the 16th percentile rank (50 − 34.13 = 15.87). Thus, you can use what you have learned about means, standard deviations, Z scores, and the normal curve to transform raw scores, which have little meaning, into percentile scores, which are easier to interpret. These methods can be used only when the distribution of scores is normal or approximately normal. Methods for nonnormal distributions are discussed in most statistics books under “nonparametric statistics.” Percentiles and Z Scores These percentile ranks are the percentage of scores that fall below the observed Z score. For example, the Z score −1.6 is associated with the percentile rank of 5.48. The Z score 1.0 (third column) is associated with the percentile rank of 84.13. Part I of Appendix 1 is a simplified version of Part II, which you need for more advanced use of Z scores. Part II gives the areas between the mean and various Z scores. Standard scored values are listed in the “Z” column. To find the proportion of the distribution between the mean of the distribution and a given Z score, you must locate the entry indicated by a specific Z score. Z scores are carried to a second decimal place in the columns that go across the table. First, consider the second column of the table because it is similar to Part I of Appendix 1. Take the Z score of 1.0. The second column is labeled .00, which means that the second decimal place is also 0. The number listed in the table is .3413. Because this is a positive number, it is above the mean. Because the area below the mean is .5, the total area below a Z score of 1.0 is .5 + .3413 = .8413. To make this into a percentile (as shown in Part I of the appendix), multiply by 100 to get 84.13. Now try the example of a Z score of 1.64. To locate this value, find 1.6 in the first column. Then move your hand across the row until you get to the number below the heading .04. The number is .4495. Again, this is a positive Z score, so you must add the observed proportion to the .5

48

CHAPTER 2



Norms and Basic Statistics for Testing that falls below the mean. The proportion below 1.64 is .9495. Stated another way, 94.95% of the cases fall below a Z score of 1.64. Now try to find the percentile rank of cases that fall below a Z score of 1.10. If you are using the table correctly, you should obtain 86.43. Now try −.75. Because this is a negative Z score, the percentage of cases falling below the mean should be less than 50. But there are no negative values in Part II of Appendix 1. For a negative Z score, there are several ways to obtain the appropriate area under the curve. The tables in Appendix 1 give the area from the mean to the Z score. For a Z score of −.75, the area between the mean and the Z score is .2734. You can find this by entering the table in the row labeled .7 and then moving across the row until you get to the figure in that row below the heading .05. There you should find the number .2734. We know that .5 of the cases fall below the mean. Thus, for a negative Z score, we can obtain the proportion of cases falling below the score by subtracting .2734, the tabled value listed in the appendix, from .5. In this case, the result is .5 − .2734 = .2266 Because finding the percentile ranks associated with negative Z scores can be tricky, you might want to use Part I of Appendix 1 to see if you are in the right ballpark. This table gives both negative and positive Z scores but does not give the detail associated with the second decimal place. Look up −.7 in Part I. The percentile rank is 24.20. Now consider a Z score of −.8. That percentile rank is 21.19. Thus, you know that a Z score of −.75 should be associated with a percentile rank between 21.19 and 24.20. In fact, we have calculated that the actual percentile rank is 22.66. Practice with Appendix 1 until you are confident you understand how it works. Do not hesitate to ask your professor or teaching assistant if you are confused. Th is is an important concept that you will need throughout the rest of the book. After you have mastered using the Tables in Appendix 1, you might try a nifty website (http:// davidmlane.com/hyperstat/z_table.html) that can fi nd the probabilities for you. Look at one more example from Table 2.2 (rainfall in San Diego, page 32). California had a dry year in 1999 and in 2007. In both years, the newscasters frequently commented that this was highly unusual. They described it as the “La Nina” effect, and some even claimed that it signaled global warming. The question is whether or not the amount of rainfall received in 1999 and 2007 was unusual given what we know about rainfall in general. To evaluate this, calculate the Z score for rainfall. According to Table 2.2, there were 6.51 inches of rainfall in 1999 and 3.62 inches in 2007. The mean for rainfall is 10.33 inches and the standard deviation is 4.71. Thus, the Z score for 1999 is 6.51 ⫺10.33 ⫽⫺.81 4.71 Next determine where a Z score of −.81 falls within the Z distribution. According to Appendix 1, a Z score of −.81 is equal to the 20.9th percentile (50 − 29.1). Thus, the low rainfall year in 1999 was unusual—given all years, it was in about the 21st percentile. However, it was not that unusual. You can estimate that there would be less

CHAPTER 2



Norms and Basic Statistics for Testing

49

rainfall in approximately 17% of all years. 2007 was a different case. The Z score for 2007 was −1.43. Rainfall in 2007 was in the 7.64th percentile. (Using Appendix 1, you can look up Z score of −1.43 and find an area below the mean of 0.4236. Then you can estimate the percentile as 50 − 42.36 = 7.64.) You can also turn the process around. Instead of using Z scores to find the percentile ranks, you can use the percentile ranks to find the corresponding Z scores. To do this, look in Part II of Appendix 1 under percentiles and find the corresponding Z score. For example, suppose you wish to find the Z score associated with the 90th percentile. When you enter Part II of Appendix 1, look for the value closest to the 90th percentile. This can be a little tricky because of the way the table is structured. Because the 90th percentile is associated with a positive Z score, you are actually looking for the area above the 50th percentile. So you should look for the entry closest to .4000 (.5000 + .4000 = .9000). The closest value to .4000 is .3997, which is found in the row labeled 1.2 and the column labeled .08. This tells you that a person who obtains a Z score of 1.28 is at approximately the 90th percentile in the distribution. Now return to the example of CES-D scores for medical students (Table 2.3). Monica had a Z score on the CES-D of −.44. Using Appendix 1, you can see that she was in the 33rd percentile (obtained as .50 − .1700 = .33 × 100 = 33). Marcel, with his Z score of .14, was in the 56th percentile; and Jennie, with a Z score of 1.70, was in the 96th percentile. You might have few worries about Monica and Marcel. However, it appears that Jennie is more depressed than 96% of her classmates and may need to talk to someone. An Example Close to Home One of the difficulties in grading students is that performance is usually rated in terms of raw scores, such as the number of items a person correctly answers on an examination. You are probably familiar with the experience of having a test returned to you with some number that makes little sense to you. For instance, the professor comes into class and hands you your test with a 72 on it. You must then wait patiently while he or she draws the distribution on the board and tries to put your 72 into some category that you understand, such as B+. An alternative way of doing things would be to give you a Z score as feedback on your performance. To do this, your professor would subtract the average score (mean) from your score and divide by the standard deviation. If your Z score was positive, you would immediately know that your score was above average; if it was negative, you would know your performance was below average. Suppose your professor tells you in advance that you will be graded on a curve according to the following rigid criteria. If you are in the top 15% of the class, you will get an A (85th percentile or above); between the 60th and the 84th percentiles, a B; between the 20th and the 59th percentiles, a C; between the 6th and the 19th percentiles, a D; and in the 5th percentile or below, an F. Using Appendix 1, you should be able to find the Z scores associated with each of these cutoff points for normal distributions of scores. Try it on your own and then consult Table 2.4 to see if you are correct. Looking at Table 2.4, you should be able to determine what your grade would be in this class on the basis of your Z score. If your Z score is 1.04 or greater, you would receive an A; if it were greater than .25 but less than 1.04,

50

CHAPTER 2



Norms and Basic Statistics for Testing TA B L E

2.4

Z Score Cutoffs for a Grading System Grade

Percentiles

A

85–100

Z score cutoff 1.04

B

60–84

.25

C

20–59

⫺.84

D

6–19

⫺1.56

F

0–5