# Descriptive Statistics Project

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Mathematics

Graduate Statistics

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PSY-520 Graduate Statistics

Topic 1 – Descriptive Statistics Project

Directions: Use the following information to complete the questions below. While APA format is not required for the body of this assignment, solid academic writing is expected, and documentation of sources should be presented using APA formatting guidelines, which can be found in the APA Style Guide, located in the Student Success Center.

A researcher was interested in the anxiety present in students just prior to the midterm exam. The research used an anxiety self-quiz to gage the student’s anxiety. The score for 30 students are given here.

 69 99 91 66 70 98 70 53 40 58 61 66 94 77 70 56 88 85 86 40 84 86 54 48 86 43 78 53 79 89
• Construct a frequency table with class, frequency, relative percent, and cumulative percent that has 6 classes to describe the distribution of the data in SPSS.
• Use the frequency table created in problem to construct a histogram in SPSS.
• Use SPSS to calculate the numerical descriptive statistics mean, median, standard deviation, and variance of the anxiety scores.

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Witte11e_fm.indd 16 11/18/2016 8:18:14 PM STATISTICS Eleventh Edition Robert S. Witte Emeritus, San Jose State University John S. Witte University of California, San Francisco Witte11e_fm.indd 1 11/18/2016 8:18:13 PM VP AND EDITORIAL DIRECTOR EDITORIAL DIRECTOR EDITORIAL ASSISTANT EDITORIAL MANAGER CONTENT MANAGEMENT DIRECTOR CONTENT MANAGER SENIOR CONTENT SPECIALIST PRODUCTION EDITOR COVER PHOTO CREDIT George Hoffman Veronica Visentin Ethan Lipson Gladys Soto Lisa Wojcik Nichole Urban Nicole Repasky Abidha Sulaiman M.C. Escher’s Spirals © The M.C. Escher Company - The Netherlands This book was set in 10/11 Times LT Std by SPi Global and printed and bound by Lightning Source Inc. The cover was printed by Lightning Source Inc. Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specifications and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship. Copyright © 2017, 2010, 2007 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923 (Web site: www.copyright.com). Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008, or online at: www.wiley. com/go/permissions. Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at: www.wiley.com/go/returnlabel. If you have chosen to adopt this textbook for use in your course, please accept this book as your complimentary desk copy. Outside of the United States, please contact your local sales representative. ISBN: 978-1-119-25451-5(PBK) ISBN: 978-1-119-25445-4(EVALC) Library of Congress Cataloging-in-Publication Data Names: Witte, Robert S. | Witte, John S. Title: Statistics / Robert S. Witte, Emeritus, San Jose State University, John S. Witte, University of California, San Francisco. Description: Eleventh edition. | Hoboken, NJ: John Wiley & Sons, Inc., [2017] | Includes index. Identifiers: LCCN 2016036766 (print) | LCCN 2016038418 (ebook) | ISBN 9781119254515 (pbk.) | ISBN 9781119299165 (epub) Subjects: LCSH: Statistics. Classification: LCC QA276.12 .W57 2017 (print) | LCC QA276.12 (ebook) | DDC 519.5—dc23 LC record available at https://lccn.loc.gov/2016036766 The inside back cover will contain printing identification and country of origin if omitted from this page. In addition, if the ISBN on the back cover differs from the ISBN on this page, the one on the back cover is correct. Witte11e_fm.indd 2 11/18/2016 8:18:13 PM To Doris Witte11e_fm.indd 3 11/18/2016 8:18:13 PM Preface TO THE READER Students often approach statistics with great apprehension. For many, it is a required course to be taken only under the most favorable circumstances, such as during a quarter or semester when carrying a light course load; for others, it is as distasteful as a visit to a credit counselor—to be postponed as long as possible, with the vague hope that mounting debts might miraculously disappear. Much of this apprehension doubtless rests on the widespread fear of mathematics and mathematically related areas. This book is written to help you overcome any fear about statistics. Unnecessary quantitative considerations have been eliminated. When not obscured by mathematical treatments better reserved for more advanced books, some of the beauty of statistics, as well as its everyday usefulness, becomes more apparent. You could go through life quite successfully without ever learning statistics. Having learned some statistics, however, you will be less likely to flinch and change the topic when numbers enter a discussion; you will be more skeptical of conclusions based on loose or erroneous interpretations of sets of numbers; you might even be more inclined to initiate a statistical analysis of some problem within your special area of interest. TO THE INSTRUCTOR Largely because they panic at the prospect of any math beyond long division, many students view the introductory statistics class as cruel and unjust punishment. A halfdozen years of experimentation, first with assorted handouts and then with an extensive set of lecture notes distributed as a second text, convinced us that a book could be written for these students. Representing the culmination of this effort, the present book provides a simple overview of descriptive and inferential statistics for mathematically unsophisticated students in the behavioral sciences, social sciences, health sciences, and education. PEDAGOGICAL FEATURES • Basic concepts and procedures are explained in plain English, and a special effort has been made to clarify such perennially mystifying topics as the standard deviation, normal curve applications, hypothesis tests, degrees of freedom, and analysis of variance. For example, the standard deviation is more than a formula; it roughly reflects the average amount by which individual observations deviate from their mean. • Unnecessary math, computational busy work, and subtle technical distinctions are avoided without sacrificing either accuracy or realism. Small batches of data define most computational tasks. Single examples permeate entire chapters or even several related chapters, serving as handy frames of reference for new concepts and procedures. iv Witte11e_fm.indd 4 11/18/2016 8:18:13 PM P R E FA C E v • Each chapter begins with a preview and ends with a summary, lists of important terms and key equations, and review questions. • Key statements appear in bold type, and step-by-step summaries of important procedures, such as solving normal curve problems, appear in boxes. • Important definitions and reminders about key points appear in page margins. • Scattered throughout the book are examples of computer outputs for three of the most prevalent programs: Minitab, SPSS, and SAS. These outputs can be either ignored or expanded without disrupting the continuity of the text. • Questions are introduced within chapters, often section by section, as Progress Checks. They are designed to minimize the cumulative confusion reported by many students for some chapters and by some students for most chapters. Each chapter ends with Review Questions. • Questions have been selected to appeal to student interests: for example, probability calculations, based on design flaws, that re-create the chillingly high likelihood of the Challenger shuttle catastrophe (8.18, page 165); a t test analysis of global temperatures to evaluate a possible greenhouse effect (13.7, page 244); and a chi-square test of the survival rates of cabin and steerage passengers aboard the Titanic (19.14, page 384). • Appendix B supplies answers to questions marked with asterisks. Other appendices provide a practical math review complete with self-diagnostic tests, a glossary of important terms, and tables for important statistical distributions. INSTRUCTIONAL AIDS An electronic version of an instructor’s manual accompanies the text. The instructor’s manual supplies answers omitted in the text (for about one-third of all questions), as well as sets of multiple-choice test items for each chapter, and a chapter-by-chapter commentary that reflects the authors’ teaching experiences with this material. Instructors can access this material in the Instructor Companion Site at http://www.wiley.com/college/witte. An electronic version of a student workbook, prepared by Beverly Dretzke of the University of Minnesota, also accompanies the text. Self-paced and self-correcting, the workbook contains problems, discussions, exercises, and tests that supplement the text. Students can access this material in the Student Companion Site at http://www.wiley. com/college/witte. CHANGES IN THIS EDITION • Update discussion of polling and random digit dialing in Section 8.4 • A new Section 14.11 on the “file drawer effect,” whereby nonsignificant statistical findings are never published and the importance of replication. • Updated numerical examples. • New examples and questions throughout the book. • Computer outputs and website have been updated. Witte11e_fm.indd 5 11/18/2016 8:18:13 PM vi P R E FA C E USING THE BOOK The book contains more material than is covered in most one-quarter or one-semester courses. Various chapters can be omitted without interrupting the main development. Typically, during a one-semester course we cover the entire book except for analysis of variance (Chapters 16, 17, and 18) and tests of ranked data (Chapter 20). An instructor who wishes to emphasize inferential statistics could skim some of the earlier chapters, particularly Normal Distributions and Standard Scores (z) (Chapter 5), and Regression (Chapter 7), while an instructor who desires a more applied emphasis could omit Populations, Samples, and Probability (Chapter 8) and More about Hypothesis Testing (Chapter 11). ACKNOWLEDGMENTS The authors wish to acknowledge their immediate family: Doris, Steve, Faith, Mike, Sharon, Andrea, Phil, Katie, Keegan, Camy, Brittany, Brent, Kristen, Scott, Joe, John, Jack, Carson, Sam, Margaret, Gretchen, Carrigan, Kedrick, and Alika. The first author also wishes to acknowledge his brothers and sisters: Henry, the late Lila, J. Stuart, A. Gerhart, and Etz; deceased parents: Henry and Emma; and all friends and relatives, past and present, including Arthur, Betty, Bob, Cal, David, Dick, Ellen, George, Grace, Harold, Helen, John, Joyce, Kayo, Kit, Mary, Paul, Ralph, Ruth, Shirley, and Suzanne. Numerous helpful comments were made by those who reviewed the current and previous editions of this book: John W. Collins, Jr., Seton Hall University; Jelani Mandara, Northwestern University; L. E. Banderet, Northeastern University; S. Natasha Beretvas, University of Texas at Austin; Patricia M. Berretty, Fordham University; David Coursey, Florida State University; Shelia Kennison, Oklahoma State University; Melanie Kercher, Sam Houston State University; Jennifer H. Nolan, Loyola Marymount University; and Jonathan C. Pettibone, University of Alabama in Huntsville; Kevin Sumrall, Montgomery College; Sky Chafin, Grossmont College; Christine Ferri, Richard Stockton College of NJ; Ann Barich, Lewis University. Special thanks to Carson Witte who proofread the entire manuscript twice. Excellent editorial support was supplied by the people at John Wiley & Sons, Inc., most notably Abidha Sulaiman and Gladys Soto. Witte11e_fm.indd 6 11/18/2016 8:18:13 PM Contents PREFACE iv ACKNOWLEDGMENTS 1 INTRODUCTION vi 1 1.1 WHY STUDY STATISTICS? 2 1.2 WHAT IS STATISTICS? 2 1.3 MORE ABOUT INFERENTIAL STATISTICS 1.4 THREE TYPES OF DATA 6 1.5 LEVELS OF MEASUREMENT 7 1.6 TYPES OF VARIABLES 11 1.7 HOW TO USE THIS BOOK 15 Summary 16 Important Terms 17 Review Questions 17 3 PART 1 Descriptive Statistics: Organizing and Summarizing Data 21 2 DESCRIBING DATA WITH TABLES AND GRAPHS TABLES (FREQUENCY DISTRIBUTIONS) 2.1 2.2 2.3 2.4 2.5 2.6 2.7 22 23 FREQUENCY DISTRIBUTIONS FOR QUANTITATIVE DATA 23 GUIDELINES 24 OUTLIERS 27 RELATIVE FREQUENCY DISTRIBUTIONS 28 CUMULATIVE FREQUENCY DISTRIBUTIONS 30 FREQUENCY DISTRIBUTIONS FOR QUALITATIVE (NOMINAL) DATA INTERPRETING DISTRIBUTIONS CONSTRUCTED BY OTHERS 32 GRAPHS 31 33 2.8 GRAPHS FOR QUANTITATIVE DATA 33 2.9 TYPICAL SHAPES 37 2.10 A GRAPH FOR QUALITATIVE (NOMINAL) DATA 2.11 MISLEADING GRAPHS 40 2.12 DOING IT YOURSELF 41 Summary 42 Important Terms 43 Review Questions 43 39 vii Witte11e_fm.indd 7 11/18/2016 8:18:13 PM viii CONTENTS 3 DESCRIBING DATA WITH AVERAGES 47 3.1 MODE 48 3.2 MEDIAN 49 3.3 MEAN 51 3.4 WHICH AVERAGE? 53 3.5 AVERAGES FOR QUALITATIVE AND RANKED DATA Summary 56 Important Terms 57 Key Equation 57 Review Questions 57 4 DESCRIBING VARIABILITY 55 60 4.1 INTUITIVE APPROACH 61 4.2 RANGE 62 4.3 VARIANCE 63 4.4 STANDARD DEVIATION 64 4.5 DETAILS: STANDARD DEVIATION 67 4.6 DEGREES OF FREEDOM (df ) 75 4.7 INTERQUARTILE RANGE (IQR) 76 4.8 MEASURES OF VARIABILITY FOR QUALITATIVE AND RANKED DATA Summary 78 Important Terms 79 Key Equations 79 Review Questions 79 5 NORMAL DISTRIBUTIONS AND STANDARD (z) SCORES 5.1 THE NORMAL CURVE 83 5.2 z SCORES 86 5.3 STANDARD NORMAL CURVE 87 5.4 SOLVING NORMAL CURVE PROBLEMS 5.5 FINDING PROPORTIONS 90 5.6 FINDING SCORES 95 5.7 MORE ABOUT z SCORES 100 Summary 103 Important Terms 103 Key Equations 103 Review Questions 103 6 Witte11e_fm.indd 8 82 89 DESCRIBING RELATIONSHIPS: CORRELATION 6.1 6.2 6.3 6.4 6.5 6.6 78 107 AN INTUITIVE APPROACH 108 SCATTERPLOTS 109 A CORRELATION COEFFICIENT FOR QUANTITATIVE DATA: r DETAILS: COMPUTATION FORMULA FOR r 117 OUTLIERS AGAIN 118 OTHER TYPES OF CORRELATION COEFFICIENTS 119 113 11/18/2016 8:18:13 PM CONTENTS ix 6.7 COMPUTER OUTPUT 120 Summary 123 Important Terms and Symbols 124 Key Equations 124 Review Questions 124 7 REGRESSION 126 7.1 TWO ROUGH PREDICTIONS 127 7.2 A REGRESSION LINE 128 7.3 LEAST SQUARES REGRESSION LINE 130 7.4 STANDARD ERROR OF ESTIMATE, sy |x 133 7.5 ASSUMPTIONS 135 7.6 INTERPRETATION OF r 2 136 7.7 MULTIPLE REGRESSION EQUATIONS 141 7.8 REGRESSION TOWARD THE MEAN 141 Summary 143 Important Terms 144 Key Equations 144 Review Questions 144 PART 2 Inferential Statistics: Generalizing Beyond Data 147 8 POPULATIONS, SAMPLES, AND PROBABILITY POPULATIONS AND SAMPLES 8.1 8.2 8.3 8.4 8.5 8.6 149 POPULATIONS 149 SAMPLES 150 RANDOM SAMPLING 151 TABLES OF RANDOM NUMBERS 151 RANDOM ASSIGNMENT OF SUBJECTS SURVEYS OR EXPERIMENTS? 154 PROBABILITY 153 155 8.7 DEFINITION 155 8.8 ADDITION RULE 156 8.9 MULTIPLICATION RULE 157 8.10 PROBABILITY AND STATISTICS Summary 162 Important Terms 163 Key Equations 163 Review Questions 163 Witte11e_fm.indd 9 148 161 11/18/2016 8:18:13 PM x CONTENTS 9 SAMPLING DISTRIBUTION OF THE MEAN 168 9.1 WHAT IS A SAMPLING DISTRIBUTION? 169 9.2 CREATING A SAMPLING DISTRIBUTION FROM SCRATCH 9.3 SOME IMPORTANT SYMBOLS 173 9.4 MEAN OF ALL SAMPLE MEANS (μ ) 173 X 9.5 STANDARD ERROR OF THE MEAN (σ ) 174 X 9.6 SHAPE OF THE SAMPLING DISTRIBUTION 176 9.7 OTHER SAMPLING DISTRIBUTIONS 178 Summary 178 Important Terms 179 Key Equations 179 Review Questions 179 10 INTRODUCTION TO HYPOTHESIS TESTING: THE z TEST 170 182 10.1 TESTING A HYPOTHESIS ABOUT SAT SCORES 183 10.2 z TEST FOR A POPULATION MEAN 185 10.3 STEP-BY-STEP PROCEDURE 186 10.4 STATEMENT OF THE RESEARCH PROBLEM 187 10.5 NULL HYPOTHESIS (H0) 188 10.6 ALTERNATIVE HYPOTHESIS (H1) 188 10.7 DECISION RULE 189 10.8 CALCULATIONS 190 10.9 DECISION 190 10.10 INTERPRETATION 191 Summary 191 Important Terms 192 Key Equations 192 Review Questions 193 11 MORE ABOUT HYPOTHESIS TESTING 195 11.1 WHY HYPOTHESIS TESTS? 196 11.2 STRONG OR WEAK DECISIONS 197 11.3 ONE-TAILED AND TWO-TAILED TESTS 199 11.4 CHOOSING A LEVEL OF SIGNIFICANCE ( ) 202 11.5 TESTING A HYPOTHESIS ABOUT VITAMIN C 203 11.6 FOUR POSSIBLE OUTCOMES 204 11.7 IF H0 REALLY IS TRUE 206 11.8 IF H0 REALLY IS FALSE BECAUSE OF A LARGE EFFECT 207 11.9 IF H0 REALLY IS FALSE BECAUSE OF A SMALL EFFECT 209 11.10 INFLUENCE OF SAMPLE SIZE 211 11.11 POWER AND SAMPLE SIZE 213 Summary 216 Important Terms 217 Review Questions 218 Witte11e_fm.indd 10 11/18/2016 8:18:13 PM CONTENTS 12 xi ESTIMATION (CONFIDENCE INTERVALS) 221 12.1 POINT ESTIMATE FOR μ 222 12.2 CONFIDENCE INTERVAL (CI) FOR μ 222 12.3 INTERPRETATION OF A CONFIDENCE INTERVAL 226 12.4 LEVEL OF CONFIDENCE 226 12.5 EFFECT OF SAMPLE SIZE 227 12.6 HYPOTHESIS TESTS OR CONFIDENCE INTERVALS? 228 12.7 CONFIDENCE INTERVAL FOR POPULATION PERCENT 228 Summary 230 Important Terms 230 Key Equation 230 Review Questions 231 13 t TEST FOR ONE SAMPLE 233 13.1 GAS MILEAGE INVESTIGATION 234 13.2 SAMPLING DISTRIBUTION OF t 234 13.3 t TEST 237 13.4 COMMON THEME OF HYPOTHESIS TESTS 238 13.5 REMINDER ABOUT DEGREES OF FREEDOM 238 13.6 DETAILS: ESTIMATING THE STANDARD ERROR (s X ) 13.7 DETAILS: CALCULATIONS FOR THE t TEST 239 13.8 CONFIDENCE INTERVALS FOR 𝜇 BASED ON t 241 13.9 ASSUMPTIONS 242 Summary 242 Important Terms 243 Key Equations 243 Review Questions 243 14 t TEST FOR TWO INDEPENDENT SAMPLES 238 245 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 EPO EXPERIMENT 246 STATISTICAL HYPOTHESES 247 SAMPLING DISTRIBUTION OF X1 – X 2 248 t TEST 250 DETAILS: CALCULATIONS FOR THE t TEST 252 p-VALUES 255 STATISTICALLY SIGNIFICANT RESULTS 258 ESTIMATING EFFECT SIZE: POINT ESTIMATES AND CONFIDENCE INTERVALS 259 14.9 ESTIMATING EFFECT SIZE: COHEN’S d 262 14.10 META-ANALYSIS 264 14.11 IMPORTANCE OF REPLICATION 264 14.12 REPORTS IN THE LITERATURE 265 Witte11e_fm.indd 11 11/18/2016 8:18:13 PM xii CONTENTS 14.13 ASSUMPTIONS 266 14.14 COMPUTER OUTPUT 267 Summary 268 Important Terms 268 Key Equations 269 Review Questions 269 15 t TEST FOR TWO RELATED SAMPLES (REPEATED MEASURES) 15.1 EPO EXPERIMENT WITH REPEATED MEASURES 274 15.2 STATISTICAL HYPOTHESES 277 15.3 SAMPLING DISTRIBUTION OF D 277 15.4 t TEST 278 15.5 DETAILS: CALCULATIONS FOR THE t TEST 279 15.6 ESTIMATING EFFECT SIZE 281 15.7 ASSUMPTIONS 283 15.8 OVERVIEW: THREE t TESTS FOR POPULATION MEANS 283 15.9 t TEST FOR THE POPULATION CORRELATION COEFFICIENT, ρ Summary 287 Important Terms 288 Key Equations 288 Review Questions 288 16 ANALYSIS OF VARIANCE (ONE FACTOR) 273 285 292 16.1 TESTING A HYPOTHESIS ABOUT SLEEP DEPRIVATION AND AGGRESSION 293 16.2 TWO SOURCES OF VARIABILITY 294 16.3 F TEST 296 16.4 DETAILS: VARIANCE ESTIMATES 299 16.5 DETAILS: MEAN SQUARES (MS ) AND THE F RATIO 304 16.6 TABLE FOR THE F DISTRIBUTION 305 16.7 ANOVA SUMMARY TABLES 307 16.8 F TEST IS NONDIRECTIONAL 308 16.9 ESTIMATING EFFECT SIZE 308 16.10 MULTIPLE COMPARISONS 311 16.11 OVERVIEW: FLOW CHART FOR ANOVA 315 16.12 REPORTS IN THE LITERATURE 315 16.13 ASSUMPTIONS 316 16.14 COMPUTER OUTPUT 316 Summary 317 Important Terms 318 Key Equations 318 Review Questions 319 17 ANALYSIS OF VARIANCE (REPEATED MEASURES) 17.1 17.2 Witte11e_fm.indd 12 322 SLEEP DEPRIVATION EXPERIMENT WITH REPEATED MEASURES F TEST 324 323 11/18/2016 8:18:13 PM CONTENTS xi i i 17.3 TWO COMPLICATIONS 325 17.4 DETAILS: VARIANCE ESTIMATES 326 17.5 DETAILS: MEAN SQUARE (MS ) AND THE F RATIO 17.6 TABLE FOR F DISTRIBUTION 331 17.7 ANOVA SUMMARY TABLES 331 17.8 ESTIMATING EFFECT SIZE 333 17.9 MULTIPLE COMPARISONS 333 17.10 REPORTS IN THE LITERATURE 335 17.11 ASSUMPTIONS 336 Summary 336 Important Terms 336 Key Equations 337 Review Questions 337 18 ANALYSIS OF VARIANCE (TWO FACTORS) 329 339 18.1 A TWO-FACTOR EXPERIMENT: RESPONSIBILITY IN CROWDS 18.2 THREE F TESTS 342 18.3 INTERACTION 344 18.4 DETAILS: VARIANCE ESTIMATES 347 18.5 DETAILS: MEAN SQUARES (MS ) AND F RATIOS 351 18.6 TABLE FOR THE F DISTRIBUTION 353 18.7 ESTIMATING EFFECT SIZE 353 18.8 MULTIPLE COMPARISONS 354 18.9 SIMPLE EFFECTS 355 18.10 OVERVIEW: FLOW CHART FOR TWO-FACTOR ANOVA 358 18.11 REPORTS IN THE LITERATURE 358 18.12 ASSUMPTIONS 360 18.13 OTHER TYPES OF ANOVA 360 Summary 360 Important Terms 361 Key Equations 361 Review Questions 361 19 CHI-SQUARE ( χ 2) TEST FOR QUALITATIVE (NOMINAL) DATA 340 365 2 ONE-VARIABLE χ TEST 366 19.1 19.2 19.3 19.4 19.5 SURVEY OF BLOOD TYPES 366 STATISTICAL HYPOTHESES 366 2 DETAILS: CALCULATING χ 367 2 TABLE FOR THE χ DISTRIBUTION 2 χ TEST 370 TWO-VARIABLE χ2 TEST 19.6 19.7 19.8 Witte11e_fm.indd 13 369 372 LOST LETTER STUDY 372 STATISTICAL HYPOTHESES 373 2 DETAILS: CALCULATING χ 373 11/18/2016 8:18:14 PM xiv CONTENTS 19.9 TABLE FOR THE χ DISTRIBUTION 376 2 19.10 χ TEST 376 19.11 ESTIMATING EFFECT SIZE 377 19.12 ODDS RATIOS 378 19.13 REPORTS IN THE LITERATURE 380 19.14 SOME PRECAUTIONS 380 19.15 COMPUTER OUTPUT 381 Summary 382 Important Terms 382 Key Equations 382 Review Questions 382 2 20 TESTS FOR RANKED (ORDINAL) DATA 386 20.1 20.2 20.3 20.4 20.5 USE ONLY WHEN APPROPRIATE 387 A NOTE ON TERMINOLOGY 387 MANN–WHITNEY U TEST (TWO INDEPENDENT SAMPLES) WILCOXON T TEST (TWO RELATED SAMPLES) 392 KRUSKAL–WALLIS H TEST (THREE OR MORE INDEPENDENT SAMPLES) 396 20.6 GENERAL COMMENT: TIES 400 Summary 400 Important Terms 400 Review Questions 400 21 POSTSCRIPT: WHICH TEST? 387 403 21.1 DESCRIPTIVE OR INFERENTIAL STATISTICS? 404 21.2 HYPOTHESIS TESTS OR CONFIDENCE INTERVALS? 404 21.3 QUANTITATIVE OR QUALITATIVE DATA? 404 21.4 DISTINGUISHING BETWEEN THE TWO TYPES OF DATA 406 21.5 ONE, TWO, OR MORE GROUPS? 407 21.6 CONCLUDING COMMENTS 408 Review Questions 408 APPENDICES 411 A MATH REVIEW 411 B ANSWERS TO SELECTED QUESTIONS C TABLES 457 D GLOSSARY 471 INDEX Witte11e_fm.indd 14 419 477 11/18/2016 8:18:14 PM STATISTICS Eleventh Edition Witte11e_fm.indd 15 11/18/2016 8:18:14 PM Witte11e_fm.indd 16 11/18/2016 8:18:14 PM C H APTER Introduction 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 WHY STUDY STATISTICS? WHAT IS STATISTICS? MORE ABOUT INFERENTIAL STATISTICS THREE TYPES OF DATA LEVELS OF MEASUREMENT TYPES OF VARIABLES HOW TO USE THIS BOOK Summary / Important Terms / Review Questions Preview Statistics deals with variability. You’re different from everybody else (and, we hope, proud of it). Today differs from both yesterday and tomorrow. In an experiment designed to detect whether psychotherapy improves self-esteem, self-esteem scores will differ among subjects in the experiment, whether or not psychotherapy improves self-esteem. Beginning with Chapter 2, descriptive statistics will provide tools, such as tables, graphs, and averages, that help you describe and organize the inevitable variability among observations. For example, self-esteem scores (on a scale of 0 to 50) for a group of college students might approximate a bell-shaped curve with an average score of 32 and a range of scores from 18 to 49. Beginning with Chapter 8, inferential statistics will supply powerful concepts that, by adjusting for the pervasive effects of variability, permit you to generalize beyond limited sets of observations. For example, inferential statistics might help us decide whether—after an adjustment has been made for background variability (or chance)— an observed improvement in self-esteem scores can be attributed to psychotherapy rather than to chance. Chapter 1 provides an overview of both descriptive and inferential statistics, and it also introduces a number of terms—some from statistics and some from math and research methods—with which you already may have some familiarity. These terms will clarify a number of important distinctions that will aid your progress through the book. 1 Witte11e_c01.indd 1 11/11/2016 11:36:59 AM 2 IN T R O D U C T ION 1 . 1 W H Y S T U D Y S TAT I S T I C S ? You’re probably taking a statistics course because it’s required, and your feelings about it may be more negative than positive. Let’s explore some of the reasons why you should study statistics. For instance, recent issues of a daily newspaper carried these items: ■ ■ ■ The annual earnings of college graduates exceed, on average, those of high school graduates by \$20,000. On the basis of existing research, there is no evidence of a relationship between family size and the scores of adolescents on a test of psychological adjustment. Heavy users of tobacco suffer significantly more respiratory ailments than do nonusers. Having learned some statistics, you’ll not stumble over the italicized phrases. Nor, as you continue reading, will you hesitate to probe for clarification by asking, “Which average shows higher annual earnings?” or “What constitutes a lack of evidence about a relationship?” or “How many more is significantly more respiratory ailments?” A statistical background is indispensable in understanding research reports within your special area of interest. Statistical references punctuate the results sections of most research reports. Often expressed with parenthetical brevity, these references provide statistical support for the researcher’s conclusions: ■ ■ ■ Subjects who engage in daily exercise score higher on tests of self-esteem than do subjects who do not exercise [p .05]. Highly anxious students are perceived by others as less attractive than nonanxious students [t (48) 3.21, p .01, d .42]. Attitudes toward extramarital sex depend on socioeconomic status [x2 (4, n 185) 11.49, p .05, 2c .03]. Having learned some statistics, you will be able to decipher the meaning of these symbols and consequently read these reports more intelligently. Sometime in the future—possibly sooner than you think—you might want to plan a statistical analysis for a research project of your own. Having learned some statistics, you’ll be able to plan the statistical analysis for modest projects involving straightforward research questions. If your project requires more advanced statistical analysis, you’ll know enough to consult someone with more training in statistics. Once you begin to understand basic statistical concepts, you will discover that, with some guidance, your own efforts often will enable you to use and interpret more advanced statistical analysis required by your research. 1 . 2 W H AT I S S TAT I S T I C S ? It is difficult to imagine, even as a fantasy exercise, a world where there is no variability—where, for example, everyone has the same physical characteristics, intelligence, attitudes, etc. Knowing that one person is 70 inches tall, and has an intelligence quotient (IQ) of 125 and a favorable attitude toward capital punishment, we could immediately conclude that everyone else also has these characteristics. This mind-numbing world would have little to recommend it, other than that there would be no need for the field of statistics (and a few of us probably would be looking for work). Witte11e_c01.indd 2 11/11/2016 11:36:59 AM 1 .3 MO R E ABOUT INFERENTIAL STATISTICS 3 Descriptive Statistics Statistics exists because of the prevalence of variability in the real world. In its simplest form, known as descriptive statistics, statistics provides us with tools—tables, graphs, averages, ranges, correlations—for organizing and summarizing the inevitable variability in collections of actual observations or scores. Examples are: 1. A tabular listing, ranked from most to least, of the total number of romantic affairs during college reported anonymously by each member of your stat class 2. A graph showing the annual change in global temperature during the last 30 years 3. A report that describes the average difference in grade point average (GPA) between college students who regularly drink alcoholic beverages and those who don’t Inferential Statistics Statistics also provides tools—a variety of tests and estimates—for generalizing beyond collections of actual observations. This more advanced area is known as inferential statistics. Tools from inferential statistics permit us to use a relatively small collection of actual observations to evaluate, for example: 1. A pollster’s claim that a majority of all U.S. voters favor stronger gun control laws 2. A researcher’s hypothesis that, on average, meditators report fewer headaches than do nonmeditators 3. An assertion about the relationship between job satisfaction and overall happiness In this book, you will encounter the most essential tools of descriptive statistics (Part 1), beginning with Chapter 2, and those of inferential statistics (Part 2), beginning with Chapter 8. Progress Check *1.1 Indicate whether each of the following statements typifies descriptive statistics (because it describes sets of actual observations) or inferential statistics (because it generalizes beyond sets of actual observations). (a) Students in my statistics class are, on average, 23 years old. (b) The population of the world exceeds 7 billion (that is, 7,000,000,000 or 1 million multiplied by 7000). (c) Either four or eight years have been the most frequent terms of office actually served by U.S. presidents. (d) Sixty-four percent of all college students favor right-to-abortion laws. Answers on page 420. 1 . 3 M O R E A B O U T I N F E R E N T I A L S TAT I S T I C S Population Any complete collection of observations or potential observations. Witte11e_c01.indd 3 Populations and Samples Inferential statistics is concerned with generalizing beyond sets of actual observations, that is, with generalizing from a sample to a population. In statistics, a population 11/11/2016 11:36:59 AM 4 Sample Any smaller collection of actual observations from a population. IN T R O D U C T ION refers to any complete collection of observations or potential observations, whereas a sample refers to any smaller collection of actual observations drawn from a population. In everyday life, populations often are viewed as collections of real objects (e.g., people, whales, automobiles), whereas in statistics, populations may be viewed more abstractly as collections of properties or measurements (e.g., the ethnic backgrounds of people, life spans of whales, gas mileage of automobiles). Depending on your perspective, a given set of observations can be either a population or a sample. For instance, the weights reported by 53 male statistics students in Table 1.1 can be viewed either as a population, because you are concerned about exceeding the load-bearing capacity of an excursion boat (chartered by the 53 students to celebrate successfully completing their stat class!), or as a sample from a population because you wish to generalize to the weights of all male statistics students or all male college students. Table 1.1 QUANTITATIVE DATA: WEIGHTS (IN POUNDS) OF MALE STATISTICS STUDENTS 160 193 226 157 180 205 165 168 169 160 163 172 151 157 133 245 170 152 160 220 190 170 160 180 158 170 166 206 150 152 150 225 145 152 172 165 190 156 135 185 159 175 158 179 190 165 152 156 154 165 157 156 135 Ordinarily, populations are quite large and exist only as potential observations (e.g., the potential scores of all U.S. college students on a test that measures anxiety). On the other hand, samples are relatively small and exist as actual observations (the actual scores of 100 college students on the test for anxiety). When using a sample (100 actual scores) to generalize to a population (millions of potential scores), it is important that the sample represent the population; otherwise, any generalization might be erroneous. Although conveniently accessible, the anxiety test scores for the 100 students in stat classes at your college probably would not be representative of the scores for all students. If you think about it, these 100 stat students might tend to have either higher or lower anxiety scores than those in the target population for numerous reasons including, for instance, the fact that the 100 students are mostly psychology majors enrolled in a required stat class at your particular college. Random Sampling (Surveys) Random Sampling A procedure designed to ensure that each potential observation in the population has an equal chance of being selected in a survey. Witte11e_c01.indd 4 Whenever possible, a sample should be randomly selected from a population in order to increase the likelihood that the sample accurately represents the population. Random sampling is a procedure designed to ensure that each potential observation in the population has an equal chance of being selected in a survey. Classic examples of random samples are a state lottery where each number from 1 to 99 in the population has an equal chance of being selected as one of the five winning numbers or a nationwide opinion survey in which each telephone number has an equal chance of being selected as a result of a series of random selections, beginning with a three-digit area code and ending with a specific seven-digit telephone number. Random sampling can be very difficult when a population lacks structure (e.g., all persons currently in psychotherapy) or specific boundaries (e.g., all volunteers who could conceivably participate in an experiment). In this case, a random sample 11/11/2016 11:36:59 AM 1 .3 MO R E ABOUT INFERENTIAL STATISTICS 5 becomes an ideal that can only be approximated—always with an effort to remove obvious biases that might cause the sample to misrepresent the population. For example, lacking the resources to sample randomly the target population of all U.S. college students, you might obtain scores by randomly selecting the 100 students, not just from stat classes at your college but also from one or more college directories, possibly using some of the more elaborate techniques described in Chapter 8. Insofar as your sample only approximates a true random sample, any resulting generalizations should be qualified. For example, if the 100 students were randomly selected only from several public colleges in northern California, this fact should be noted, and any generalizations to all college students in the United States would be both provisional and open to criticism. Random Assignment (Experiments) Random Assignment A procedure designed to ensure that each person has an equal chance of being assigned to any group in an experiment. Estimating the average anxiety score for all college students probably would not generate much interest. Instead, we might be interested in determining whether relaxation training causes, on average, a reduction in anxiety scores between two groups of otherwise similar college students. Even if relaxation training has no effect on anxiety scores, we would expect average scores for the two groups to differ because of the inevitable variability between groups. The question becomes: How should we interpret the apparent difference between the treatment group and the control group? Once variability has been taken into account, should the difference be viewed as real (and attributable to relaxation training) or as transitory (and merely attributable to variability or chance)? College students in the relaxation experiment probably are not a random sample from any intact population of interest, but rather a convenience sample consisting of volunteers from a limited pool of students fulfilling a course requirement. Accordingly, our focus shifts from random sampling to the random assignment of volunteers to the two groups. Random assignment signifies that each person has an equal chance of being assigned to any group in an experiment. Using procedures described in Chapter 8, random assignment should be employed whenever possible. Because chance dictates the membership of both groups, not only does random assignment minimize any biases that might favor one group or another, it also serves as a basis for estimating the role of variability in any observed result. Random assignment allows us to evaluate any finding, such as the actual average difference between two groups, to determine whether this difference is larger than expected just by chance, once variability is taken into account. In other words, it permits us to generalize beyond mere appearances and determine whether the average difference merits further attention because it probably is real or whether it should be ignored because it can be attributed to variability or chance. Overview: Surveys and Experiments Figure 1.1 compares surveys and experiments. Based on random samples from populations, surveys permit generalizations from samples back to populations. Based on the random assignment of volunteers to groups, experiments permit decisions about whether differences between groups are real or merely transitory. PROGRESS CHECK *1.2 Indicate whether each of the following terms is associated primarily with a survey (S) or an experiment (E). (a) random assignment (b) representative (c) generalization to the population (d) control group Answers on page 420. Witte11e_c01.indd 5 11/11/2016 11:36:59 AM 6 IN T R O D U C T ION SURVEYS Random Sample Population (unknown scores) Sample (known scores) Generalize to population EXPERIMENTS Treatment Group (known scores) Volunteers (unknown scores) Random Assignment Is difference real or transitory? Control Group (known scores) FIGURE 1.1 Overview: surveys and experiments. (e) real difference (f) random selection (g) convenience sample Data A collection of actual observations (h) volunteers Answers on page 420. or scores in a survey or an experiment 1 . 4 T H R E E T Y P E S O F D ATA Qualitative Data Any statistical analysis is performed on data, a collection of actual observations or scores in a survey or an experiment. A set of observations where any single observation is a word, letter, or numerical code that represents a class or category. Ranked Data A set of observations where any single observation is a number that indicates relative standing. Quantitative Data A set of observations where any single observation is a number that represents an amount or a count. Witte11e_c01.indd 6 The precise form of a statistical analysis often depends on whether data are qualitative, ranked, or quantitative. Generally, qualitative data consist of words (Yes or No), letters (Y or N), or numerical codes (0 or 1) that represent a class or category. Ranked data consist of numbers (1st, 2nd, . . . 40th place) that represent relative standing within a group. Quantitative data consist of numbers (weights of 238, 170, . . . 185 lbs) that represent an amount or a count. To determine the type of data, focus on a single observation in any collection of observations. For example, the weights reported by 53 male students in Table 1.1 are quantitative data, since any single observation, such as 160 lbs, represents an amount of weight. If the weights in Table 1.1 had been replaced with ranks, beginning with a rank of 1 for the lightest weight of 133 lbs and ending with a rank of 53 for the heaviest weight of 245 lbs, these numbers would have been ranked data, since any single observation represents not an amount, but only relative standing within the group of 53 students. Finally, the Y and N replies of students in Table 1.2 are qualitative data, since any single observation is a letter that represents a class of replies. 11/11/2016 11:36:59 AM 1 .5 LE V E L S OF MEASUREMENT 7 Table 1.2 QUALITATIVE DATA: “DO YOU HAVE A FACEBOOK PROFILE?” YES (Y) OR NO (N) REPLIES OF STATISTICS STUDENTS Y Y N Y Y Y N Y Y N Y Y Y Y N Y N Y Y Y Y Y Y N N Y N N Y Y N Y N N Y Y N Y N Y Y N N N N Y N N Y Y Y N Y Y Y Y Y Y Y N N N Y Y Y Y N Y Y N Y Y Y Y Y Y Y Y Y Y N Y Y Progress Check *1.3 Indicate whether each of the following terms is qualitative (because it’s a word, letter, or numerical code representing a class or category); ranked (because it’s a number representing relative standing); or quantitative (because it’s a number representing an amount or a count). (a) ethnic group (b) age (c) family size (d) academic major (e) sexual preference (f) IQ score (g) net worth (dollars) (h) third-place finish (i) gender (j) temperature Answers on page 420. Level of Measurement Specifies the extent to which a number (or word or letter) actually represents some attribute and, therefore, has implications for the appropriateness of various arithmetic operations and statistical procedures. Witte11e_c01.indd 7 1.5 LEVELS OF MEASUREMENT Learned years ago in grade school, the abstract statement that 2 + 2 4 qualifies as one of life’s everyday certainties, along with taxes and death. However, not all numbers have the same interpretation. For instance, it wouldn’t make sense to find the sum of two Social Security numbers or to claim that, when viewed as indicators of academic achievement, two GPAs of 2.0 equal a GPA of 4.0. To clarify further the differences among the three types of data, let’s introduce the notion of level of measurement. Looming behind any data, the level of measurement specifies the extent to which a number (or word or letter) actually represents some attribute and, therefore, has implications for the appropriateness of various arithmetic operations and statistical procedures. 11/11/2016 11:36:59 AM 8 IN T R O D U C T ION For our purposes, there are three levels of measurement—nominal, ordinal, and interval/ratio—and these levels are paired with qualitative, ranked, and quantitative data, respectively. The properties of these levels—and the usefulness of their associated numbers—vary from nominal, the simplest level with only one property, to interval/ ratio, the most complex level with four properties. Progressively more complex levels contain all properties of simpler levels, plus one or two new properties. More complex levels of measurement are associated with numbers that, because they better represent attributes, permit a wider variety of arithmetic operations and statistical procedures. Qualitative Data and Nominal Measurement Nominal Measurement Words, letters, or numerical codes of qualitative data that reflect differences in kind based on classification. If people are classified as either male or female (or coded as 1 or 2), the data are qualitative and measurement is nominal. The single property of nominal measurement is classification—that is, sorting observations into different classes or categories. Words, letters, or numerical codes reflect only differences in kind, not differences in amount. Examples of nominal measurement include classifying mood disorders as manic, bipolar, or depressive; sexual preferences as heterosexual, homosexual, bisexual, or nonsexual; and attitudes toward stricter pollution controls as favor, oppose, or undecided. A distinctive feature of nominal measurement is its bare-bones representation of any attribute. For instance, a student is either male or female. Even with the introduction of arbitrary numerical codes, such as 1 for male and 2 for female, it would never be appropriate to claim that, because female is 2 and male is 1, females have twice as much gender as males. Similarly, calculating an average with these numbers would be meaningless. Because of these limitations, only a few sections of this book and Chapter 19 are dedicated exclusively to an analysis of qualitative data with nominal measurement. Ranked Data and Ordinal Measurement Ordinal Measurement Relative standing of ranked data that reflects differences in degree based on order. When any single number indicates only relative standing, such as first, second, or tenth place in a horse race or in a class of graduating seniors, the data are ranked and the level of measurement is ordinal. The distinctive property of ordinal measurement is order. Comparatively speaking, a first-place finish reflects the fastest finish in a horse race or the highest GPA among graduating seniors. Although first place in a horse race indicates a faster finish than second place, we don’t know how much faster. Since ordinal measurement fails to reflect the actual distance between adjacent ranks, simple arithmetic operations with ranks are inappropriate. For example, it’s inappropriate to conclude that the arithmetic mean of ranks 1 and 3 equals rank 2, since this assumes that the actual distance between ranks 1 and 2 equals the distance between ranks 2 and 3. Instead, these distances might be very different. For example, rank 2 might be virtually tied with either rank 1 or rank 3. Only a few sections of this book and Chapter 20 are dedicated exclusively to an analysis of ranked data with ordinal measurement.* *Strictly speaking, ordinal measurement also can be associated with qualitative data whose classes are ordered. Examples of ordered qualitative data include the classification of skilled workers as master craftsman, journeyman, or apprentice; socioeconomic status as low, middle, or high; and academic grades as A, B, C, D, or F. It’s worth distinguishing between qualitative data with nominal and ordinal measurement because, as described in Chapters 3 and 4, a few extra statistical procedures are available for ordered qualitative data. Witte11e_c01.indd 8 11/11/2016 11:36:59 AM 1 .5 LE V E L S OF MEASUREMENT 9 Quantitative Data and Interval/Ratio Measurement Interval/Ratio Measurement Amounts or counts of quantitative data reflect differences in degree based on equal intervals and a true zero. Often the products of familiar measuring devices, such as rulers, clocks, or meters, the distinctive properties of interval/ratio measurement are equal intervals and a true zero. Weighing yourself on a bathroom scale qualifies as interval/ratio measurement. Equal intervals imply that hefting a 10-lb weight while on the bathroom scale always registers your actual weight plus 10 lbs. Equal intervals imply that the difference between 120 and 130 lbs represents an amount of weight equal to the difference between 130 and 140 lbs, and it’s appropriate to describe one person’s weight as a certain amount greater than another’s. A true zero signifies that the bathroom scale registers 0 when not in use—that is, when weight is completely absent. Since the bathroom scale possesses a true zero, numerical readings reflect the total amount of a person’s weight, and it’s appropriate to describe one person’s weight as a certain ratio of another’s. It can be said that the weight of a 140-lb person is twice that of a 70-lb person. In the absence of a true zero, numbers—much like the exposed tips of icebergs— fail to reflect the total amount being measured. For example, a reading of 0 on the Fahrenheit temperature scale does not reflect the complete absence of heat—that is, the absence of any molecular motion. In fact, true zero equals −459.4°F on this scale. It would be inappropriate, therefore, to claim that 80°F is twice as hot as 40°F. An appropriate claim could be salvaged by adding 459.4°F to each of these numbers: 80° becomes 539.4° and 40° becomes 499.4°. Clearly, 539.4°F is not twice as hot as 499.4°F. Interval/ratio measurement appears in the behavioral and social sciences as, for example, bar-press rates of rats in Skinner boxes; the minutes of dream-friendly rapid eye movement (REM) sleep among participants in a sleep-deprivation experiment; and the total number of eye contacts during verbal disputes between romantically involved couples. Thanks to the considerable amount of information conveyed by each observation, interval/ratio measurement permits meaningful arithmetic operations, such as calculating arithmetic means, as well as the many statistical procedures for quantitative data described in this book. Measurement of Nonphysical Characteristics When numbers represent nonphysical characteristics, such as intellectual aptitude, psychopathic tendency, or emotional maturity, the attainment of interval/ratio measurement often is questionable. For example, there is no external standard (such as the 10-lb weight) to demonstrate that the addition of a fixed amount of intellectual aptitude always produces an equal increase in IQ scores (equal intervals). There also is no instrument (such as the unoccupied bathroom scale) that registers an IQ score of 0 when intellectual aptitude is completely absent (true zero). In the absence of equal intervals, it would be inappropriate to claim that the difference between IQ scores of 120 and 130 represents the same amount of intellectual aptitude as the difference between IQ scores of 130 and 140. Likewise, in the absence of a true zero, it would be inappropriate to claim that an IQ score of 140 represents twice as much intellectual aptitude as an IQ score of 70. Other interpretations are possible. One possibility is to treat IQ scores as attaining only ordinal measurement—that is, for example, a score of 140 represents more intellectual aptitude than a score of 130—without specifying the actual size of this difference. This strict interpretation would greatly restrict the number of statistical procedures for use with behavioral and social data. A looser (and much more common) interpretation, adopted in this book, assumes that, although lacking a true zero, IQ scores provide a crude measure of corresponding differences in intellectual aptitude (equal intervals). Thus, the difference between IQ scores of 120 and 130 represents a roughly similar amount of intellectual aptitude as the difference between scores of 130 and 140. Witte11e_c01.indd 9 11/11/2016 11:36:59 AM 10 IN T R O D U C T ION Insofar as numerical measures of nonphysical characteristics approximate interval measurement, they receive the same statistical treatment as numerical measures of physical characteristics. In other words, these measures support the arithmetic operations and statistical tools appropriate for quantitative data. At this point, you might wish that a person could be injected with 10 points of intellectual aptitude (or psychopathic tendency or emotional maturity) as a first step toward an IQ scale with equal intervals and a true zero. Lacking this alternative, however, train yourself to look at numbers as products of measurement and to temper your numerical claims accordingly—particularly when numerical data only seem to approximate interval measurement. Overview: Types of Data and Levels of Measurement Refer to Figure 1.2 while reading this paragraph. Given some set of observations, decide whether any single observation qualifies as a word or as a number. If it is a word (or letter or numerical code), the data are qualitative and the level of measurement is nominal. Arithmetic operations are meaningless and statistical procedures are limited. On the other hand, if the observation is a number, the data are either ranked or quantitative, depending on whether numbers represent only relative standing or an amount/count. If the data are ranked, the level of measurement is ordinal and, as with qualitative data, arithmetic operations and statistical procedures are limited. If the data are quantitative, the level of measurement is interval/ratio—or approximately interval when numbers represent nonphysical characteristics—and a full range of arithmetic operations and statistical procedures are available. Progress Check *1.4 Indicate the level of measurement—nominal, ordinal, or interval/ ratio—attained by the following sets of observations or data. When appropriate, indicate that measurement is only approximately interval. DATA Words Numbers Relative Standing Amount or Count QUALITATIVE (Yes, No) RANKS (1st, 2nd,...) QUANTITATIVE (160,...193 lbs) Classification Order Equal Intervals/True Zero NOMINAL ORDINAL INTERVAL/RATIO FIGURE 1.2 Overview: types of data and levels of measurement. Witte11e_c01.indd 10 11/11/2016 11:36:59 AM 1 .6 T Y P E S OF VARIABL ES 11 NOTE: Always assign the highest permissible level of measurement to a given set of observations. For example, a list of annual incomes should be designated as interval/ratio because a \$1000 difference always signifies the same amount of income (equal intervals) and because \$0 signifies the complete absence of income. It would be wrong to describe annual income as ordinal data even though different incomes always can be ranked as more or less (order), or as nominal data even though different incomes always reflect different classes (classification). (a) height (b) religious affiliation (c) score for psychopathic tendency (d) years of education (e) military rank (f) vocational goal (g) GPA (h) marital status Answers on page 420. 1 . 6 T Y P E S O F VA R I A B L E S General Definition Variable A characteristic or property that can take on different values. Constant A characteristic or property that can take on only one value. Discrete Variable A variable that consists of isolated numbers separated by gaps. Continuous Variable A variable that consists of numbers whose values, at least in theory, have no restrictions. Another helpful distinction is based on different types of variables. A variable is a characteristic or property that can take on different values. Accordingly, the weights in Table 1.1 can be described not only as quantitative data but also as observations for a quantitative variable, since the various weights take on different numerical values. By the same token, the replies in Table 1.2 can be described as observations for a qualitative variable, since the replies to the Facebook profile question take on different values of either Yes or No. Given this perspective, any single observation in either Table 1.1 or 1.2 can be described as a constant, since it takes on only one value. Discrete and Continuous Variables Quantitative variables can be further distinguished in terms of whether they are discrete or continuous. A discrete variable consists of isolated numbers separated by gaps. Examples include most counts, such as the number of children in a family (1, 2, 3, etc., but never 11/2 in spite of how you might occasionally feel about a sibling); the number of foreign countries you have visited; and the current size of the U.S. population. A continuous variable consists of numbers whose values, at least in theory, have no restrictions. Examples include amounts, such as weights of male statistics students; durations, such as the reaction times of grade school children to a fire alarm; and standardized test scores, such as those on the Scholastic Aptitude Test (SAT). Approximate Numbers Approximate Numbers Numbers that are rounded off, as is always the case with values for continuous variables. Witte11e_c01.indd 11 In theory, values for continuous variables can be carried out infinitely far. Someone’s weight, in pounds, might be 140.01438, and so on, to infinity! Practical considerations require that values for continuous variables be rounded off. Whenever values are rounded off, as is always the case with actual values for continuous variables, the resulting numbers are approximate, never exact. For example, the weights of the 11/11/2016 11:36:59 AM 12 IN T R O D U C T ION male statistics students in Table 1.1 are approximate because they have been rounded to the nearest pound. A student whose weight is listed as 150 lbs could actually weigh between 149.5 and 150.5 lbs. In effect, any value for a continuous variable, such as 150 lbs, must be identified with a range of values from 149.5 to 150.5 rather than with a solitary value. As will be seen, this property of continuous variables has a number of repercussions, including the selection of graphs in Chapter 2 and the types of meaningful questions about normal distributions in Chapter 5. Because of rounding-off procedures, gaps appear among values for continuous variables. For example, because weights are rounded to the nearest pound, no male statistics student in Table 1.1 has a listed weight between 150 and 151 lbs. These gaps are more apparent than real; they are superimposed on a continuous variable by our need to deal with finite (and, therefore, approximate) numbers. Progress Check *1.5 Indicate whether the following quantitative observations are discrete or continuous. (a) litter of mice (b) cooking time for pasta (c) parole violations by convicted felons (d) IQ (e) age (f) population of your hometown (g) speed of a jetliner Answers on page 420. Independent and Dependent Variables Experiment A study in which the investigator decides who receives the special treatment. Independent Variable The treatment manipulated by the investigator in an experiment. Witte11e_c01.indd 12 Unlike the simple studies that produced the data in Tables 1.1 and 1.2, most studies raise questions about the presence or absence of a relationship between two (or more) variables. For example, a psychologist might wish to investigate whether couples who undergo special training in “active listening” tend to have fewer communication breakdowns than do couples who undergo no special training. To study this, the psychologist may expose couples to two different conditions by randomly assigning them either to a treatment group that receives special training in active listening or to a control group that receives no special training. Such studies are referred to as experiments. An experiment is a study in which the investigator decides who receives the special treatment. When well designed, experiments yield the most informative and unambiguous conclusions about cause-effect relationships. Independent Variable Since training is assumed to influence communication, it is an independent variable. In an experiment, an independent variable is the treatment manipulated by the investigator. The impartial creation of distinct groups, which differ only in terms of the independent variable, has a most desirable consequence. Once the data have been collected, any difference between the groups (that survives a statistical analysis, as described in Part 2 of the book) can be interpreted as being caused by the independent variable. If, for instance, a difference appears in favor of the active-listening group, the psychologist can conclude that training in active listening causes fewer communication 11/11/2016 11:36:59 AM 1 .6 T Y P E S OF VARIABL ES 13 breakdowns between couples. Having observed this relationship, the psychologist can expect that, if new couples were trained in active listening, fewer breakdowns in communication would occur. Dependent Variable Dependent Variable A variable that is believed to have been influenced by the independent variable. To test whether training influences communication, the psychologist counts the number of communication breakdowns between each couple, as revealed by inappropriate replies, aggressive comments, verbal interruptions, etc., while discussing a conflict-provoking topic, such as whether it is acceptable to be intimate with a third person. When a variable is believed to have been influenced by the independent variable, it is called a dependent variable. In an experimental setting, the dependent variable is measured, counted, or recorded by the investigator. Unlike the independent variable, the dependent variable isn’t manipulated by the investigator. Instead, it represents an outcome: the data produced by the experiment. Accordingly, the values that appear for the dependent variable cannot be specified in advance. Although the psychologist suspects that couples with special training will tend to show fewer subsequent communication breakdowns, he or she has to wait to see precisely how many breakdowns will be observed for each couple. Independent or Dependent Variable? With just a little practice, you should be able to identify these two types of variables. In an experiment, what is being manipulated by the investigator at the outset and, therefore, qualifies as the independent variable? What is measured, counted, or recorded by the investigator at the completion of the study and, therefore, qualifies as the dependent variable? Once these two variables have been identified, they can be used to describe the problem posed by the study; that is, does the independent variable cause a change in the dependent variable?* Observational Studies Observational Study A study that focuses on detecting relationships between variables not manipulated by the investigator. Instead of undertaking an experiment, an investigator might simply observe the relation between two variables. For example, a sociologist might collect paired measures of poverty level and crime rate for each individual in some group. If a statistical analysis reveals that these two variables are related or correlated, then, given some person’s poverty level, the sociologist can better predict that person’s crime rate or vice versa. Having established the existence of this relationship, however, the sociologist can only speculate about cause and effect. Poverty might cause crime or vice versa. On the other hand, both poverty and crime might be caused by one or some combination of more basic variables, such as inadequate education, racial discrimination, unstable family environment, and so on. Such studies are often referred to as observational studies. An observational study focuses on detecting relationships between variables not manipulated by the investigator, and it yields less clear-cut conclusions about causeeffect relationships than does an experiment. To detect any relationship between active listening and fewer breakdowns in communication, our psychologist could have conducted an observational study rather than an experiment. In this case, he or she would have made no effort to manipulate active-listening skills by assigning couples to special training sessions. Instead, the *For the present example, note that the independent variable (type of training) is qualitative, with nominal measurement, whereas the dependent variable (number of communication breakdowns) is quantitative. Insofar as the number of communication breakdowns is used to indicate the quality of communication between couples, its level of measurement is approximately interval. Witte11e_c01.indd 13 11/11/2016 11:36:59 AM 14 IN T R O D U C T ION psychologist might have used a preliminary interview to assign an active-listening score to each couple. Subsequently, our psychologist would have obtained a count of the number of communication breakdowns for each couple during the conflictresolution session. Now data for both variables would have been collected (or observed) by the psychologist—and the cause-effect basis of any relationship would be speculative. For example, couples already possessing high active-listening scores might also tend to be more seriously committed to each other, and this more serious commitment itself might cause both the higher active-listening score and fewer breakdowns in communication. In this case, any special training in active listening, without regard to the existing degree of a couple’s commitment, would not reduce the number of breakdowns in communication. Confounding Variable Confounding variable An uncontrolled variable that compromises the interpretation of a study. Whenever groups differ not just because of the independent variable but also because some uncontrolled variable co-varies with the independent variable, any conclusion about a cause-effect relationship is suspect. If, instead of random assignment, each couple in an experiment is free to choose whether to undergo special training in active listening or to be in the less demanding control group, any conclusion must be qualified. A difference between groups might be due not to the independent variable but to a confounding variable. For instance, couples willing to devote extra effort to special training might already possess a deeper commitment that co-varies with more active-listening skills. An uncontrolled variable that compromises the interpretation of a study is known as a confounding variable. You can avoid confounding variables, as in the present case, by assigning subjects randomly to the various groups in the experiment and also by standardizing all experimental conditions, other than the independent variable, for subjects in both groups. Sometimes a confounding variable occurs because it’s impossible to assign subjects randomly to different conditions. For instance, if we’re interested in possible differences in active-listening skills between males and females, we can’t assign the subject’s gender randomly. Consequently, any difference between these two preexisting groups must be interpreted cautiously. For example, if females, on average, are better listeners than males, this difference could be caused by confounding variables that co-vary with gender, such as preexisting disparities in active-listening skills attributable not merely to gender, but also to cultural stereotypes, social training, vocational interests, academic majors, and so on. Overview: Two Active-Listening Studies Figure 1.3 summarizes the active-listening study when viewed as an experiment and as an observational study. An experiment permits a decision about whether or not the average difference between treatment and control groups is real. An observational study permits a decision about whether or not the variables are related or correlated. Progress Check *1.6 For each of the listed studies, indicate whether it is an experiment or an observational study. If it is an experiment, identify the independent variable and note any possible confounding variables. (a) years of education and annual income (b) prescribed hours of sleep deprivation and subsequent amount of REM (dream) sleep (c) weight loss among obese males who choose to participate either in a weight-loss program or a self-esteem enhancement program (d) estimated study hours and subsequent test score Witte11e_c01.indd 14 11/11/2016 11:36:59 AM 1 .7 H O W TO USE THIS BOOK 15 EXPERIMENT Treatment Group Control Group INDEPENDENT VARIABLE Active-Listening Training No Active-Listening Training DEPENDENT VARIABLE Number of Communication Breakdowns Number of Communication Breakdowns Is difference real or transitory? OBSERVATIONAL STUDY FIRST VARIABLE Pre-existing Score for Active Listening Are the two variables related? SECOND VARIABLE Number of Communication Breakdowns FIGURE 1.3 Overview: two active-listening studies. (e) recidivism among substance abusers assigned randomly to different rehabilitation programs (f) subsequent GPAs of college applicants who, as the result of a housing lottery, live either on campus or off campus Answers on page 420. 1.7 HOW TO USE THIS BOOK This book contains a number of features that will help your study of statistics. Each chapter begins with a preview and ends with a summary, a list of important terms, and, whenever appropriate, a list of key equations. Use these aids to orient yourself before reading a new chapter and to facilitate your review of previous chapters. Frequent reviews are desirable, since statistics is cumulative, with earlier topics forming the basis for later topics. For easy reference, important terms are defined in the margins. Progress checks appear within chapters, and review questions appear at the end of each chapter. Do not shy away from the progress checks or review questions; they will clarify and expand your understanding as well as improve your ability to work with statistics. Appendix B supplies answers to all questions marked with asterisks, including all progress checks and selected review questions. The math review in Appendix A summarizes most of the basic math symbols and operations used throughout this book. If you are anxious about your math background—and almost everyone is—check Appendix A. Be assured that no special math background is required. If you can add, subtract, multiply, and divide, you can learn (or relearn) the simple math described in Appendix A. If this material looks unfamiliar, it would be a good idea to study Appendix A within the next few weeks. Witte11e_c01.indd 15 11/11/2016 11:36:59 AM 16 IN T R O D U C T ION An electronic version of a student workbook, prepared by Beverly Dretzke of the Center for Applied Research and Educational Improvement, University of Minnesota, Minneapolis, also accompanies the text. Self-paced and self-correcting, it supplies additional problems, questions, and tests that supplement the text. You can access this material by clicking on the Student Study Guide in the Student Companion website for the text at http://www.wiley.com/college/witte. We cannot resist ending this chapter with a personal note, as well as a few suggestions based on findings from the learning laboratory. A dear relative lent this book to an elderly neighbor, who not only praised it, saying that he wished he had had such a stat text many years ago while he was a student at the University of Pittsburgh, but subsequently died with the book still open next to his bed. Upon being informed of this, the first author’s wife commented, “I wonder which chapter killed him.” In all good conscience, therefore, we cannot recommend this book for casual bedside reading if you are more than 85 years old. Otherwise, read it anywhere or anytime. Seriously, not only read assigned material before class, but also reread it as soon as possible after class to maximize the retention of newly learned material. In the same vein, end reading sessions with active rehearsal: Close the book and attempt to re-create mentally, in an orderly fashion and with little or no peeking, the material that you have just read. With this effort, you should find the remaining chapters accessible and statistics to be both understandable and useful. Summary Statistics exists because of the prevalence of variability in the real world. It consists of two main subdivisions: descriptive statistics, which is concerned with organizing and summarizing information for sets of actual observations, and inferential statistics, which is concerned with generalizing beyond sets of actual observations—that is, generalizing from a sample to a population. Ordinarily, populations are quite large and exist only as potential observations, while samples are relatively small and exist as actual observations. Random samples increase the likelihood that the sample accurately represents the population because all potential observations in the population have an equal chance of being in the random sample. When populations consist of only limited pools of volunteers, as in many investigations, the focus shifts from random samples to random assignment. Random assignment ensures that each volunteer has an equal chance of occupying any group in the investigation. Not only does random assignment minimize any initial biases that might favor one group over another, but it also allows us to determine whether an observed difference between groups probably is real or merely due to chance variability. There are three types of data—qualitative, ranked, and quantitative—which are paired with three levels of measurement—nominal, ordinal, and interval/ratio, respectively. Qualitative data consist of words, letters, or codes that represent only classes with nominal measurement. Ranked data consist of numbers that represent relative standing with ordinal measurement. Quantitative data consist of numbers that represent an amount or a count with interval/ratio measurement. Distinctive properties of the three levels of measurement are classification (nominal), order (ordinal), and equal intervals and true zero (interval/ratio). Shifts to more complex levels of measurement permit a wider variety of arithmetic operations and statistical procedures. Even though the numerical measurement of various nonphysical characteristics fails to attain an interval/ratio level, the resulting data usually are treated as approximating interval measurement. The limitations of these data should not, however, be ignored completely when making numerical claims. Witte11e_c01.indd 16 11/11/2016 11:36:59 AM R E V IE W Q UESTIONS 17 It is helpful to distinguish between discrete and continuous variables. Discrete variables consist of isolated numbers separated by gaps, whereas continuous variables consist of numbers whose values, at least in theory, have no restrictions. In practice, values of continuous variables always are rounded off and, therefore, are approximate numbers. It is also helpful to distinguish between independent and dependent variables. In experiments, independent variables are manipulated by the investigator; dependent variables are outcomes measured, counted, or recorded by the investigator. If well designed, experiments yield the most clear-cut information about cause-effect relationships. Investigators may also undertake observational studies in which variables are observed without intervention. Observational studies yield less clear-cut information about cause-effect relationships. Both types of studies can be weakened by confounding variables. Important Terms Descriptive statistics Population Random sampling Data Ranked data Level of measurement Ordinal measurement Variable Discrete variable Independent variable Experiment Confounding variable Inferential statistics Sample Random assignment Qualitative data Quantitative data Nominal measurement Interval/ratio measurement Constant Continuous variable Approximate numbers Dependent variable Observational study REVIEW QUESTIONS 1.7 Indicate whether each of the following statements typifies descriptive statistics (because it describes sets of actual observations) or inferential statistics (because it generalizes beyond sets of actual observations). (a) On the basis of a survey conducted by the Bureau of Labor Statistics, it is estimated that 5.1 percent of the entire workforce was unemployed during the last month. (b) During a recent semester, the ages of students at my college ranged from 16 to 75 years. (c) Research suggests that an aspirin every other day reduces the chance of heart attacks (by almost 50 percent) in middle-age men. (d) Joe’s GPA has hovered near 3.5 throughout college. (e) There is some evidence that any form of frustration—whether physical, social, economic, or political—always leads to some form of aggression by the frustrated person. Witte11e_c01.indd 17 11/11/2016 11:36:59 AM 18 IN T R O D U C T ION (f) According to tests conducted by the Environmental Protection Agency, the 2016 Toyota Prius should average approximately 52 miles per gallon for combined city/ highway travel. (g) On average, Babe Ruth hit 32 homeruns during each season of his major league baseball career. (h) Research on learning suggests that active rehearsal increases the retention of newly read material; therefore, immediately after reading a chapter in this book, you should close the book and try to organize the new material. (i) Children with no siblings tend to be more adult-oriented than children with one or more siblings. 1.8 Indicate whether each of the following studies is an experiment or an observational study. If it is an experiment, identify the independent variable and note any possible confounding variables. (a) A psychologist uses chimpanzees to test the notion that more crowded living conditions trigger aggressive behavior. Chimps are placed, according to an impartial assignment rule, in cages with either one, several, or many other chimps. Subsequently, during a standard observation period, each chimp is assigned a score based on its aggressive behavior toward a chimplike stuffed doll. (b) An investigator wishes to test whether, when compared with recognized scientists, recognized artists tend to be born under different astrological signs. (c) To determine whether there is a relationship between the sexual codes of primitive tribes and their behavior toward neighboring tribes, an anthropologist consults available records, classifying each tribe on the basis of its sexual codes (permissive or repressive) and its behavior toward neighboring tribes (friendly or hostile). (d) In a study of group problem solving, an investigator assigns college students to groups of two, three, or four students and measures the amount of time required by each group to solve a complex puzzle. (e) A school psychologist wishes to determine whether reading comprehension scores are related to the number of months of formal education, as reported on school transcripts, for a group of 12-year-old migrant children. (f) To determine whether Graduate Record Exam (GRE) scores can be increased by cramming, an investigator allows college students to choose to participate in either a GRE test-taking workshop or a control (non-test-taking) workshop and then compares the GRE scores earned subsequently by the two groups of students. (g) A social scientist wishes to determine whether there is a relationship between the attractiveness scores (on a 100-point scale) assigned to college students by a panel of peers and their scores on a paper-and-pencil test of anxiety. (h) A political scientist wishes to determine whether males and females differ with respect to their attitudes toward defense spending by the federal government. She asks each person if he or she thinks that the current level of defense spending should be increased, remain the same, or be decreased. (i) Investigators found that four year-old children who delayed eating one marshmallow in order to eat two marshmallows later, scored higher than non-delayers on the Scholastic Aptitude Test (SAT) taken over a decade later. Witte11e_c01.indd 18 11/11/2016 11:36:59 AM R E V IE W Q UESTIONS 19 1.9 Recent studies, as summarized, for example, in E. Mortensen et al. (2002). The association between duration of breastfeeding and adult intelligence. Journal of the American Medical Association, 287, 2365–2371, suggest that breastfeeding of infants may increase their subsequent cognitive (IQ) development. Both experiments and observational studies are cited. (a) What determines whether some of these studies are experiments? (b) Name at least two potential confounding variables controlled by breastfeeding experiments. 1.10 If you have not done so already, familiarize yourself with the various appendices in this book. (a) Particularly note the location of Appendix B (Answers to Selected Questions) and Appendix D (Glossary). (b) Browse through Appendix A (Math Review). If this material looks unfamiliar, study Appendix A, using the self-diagnostic tests as guides. Witte11e_c01.indd 19 11/11/2016 11:36:59 AM Witte11e_c01.indd 20 11/11/2016 11:36:59 AM PA R T 1 Descriptive Statistics Organizing and Summarizing Data 2 Describing Data with Tables and Graphs 3 Describing Data with Averages 4 Describing Variability 5 Normal Distributions and Standard (z) Scores 6 Describing Relationships: Correlation 7 Regression Preview You probably associate statistics with sets of numbers. Numerical sets—or, more generally, sets of data—usually represent the point of departure for a statistical analysis. While focusing on descriptive statistics in the next six chapters, we’ll avoid extensive sets of numbers (and the discomfort they trigger in some of us) without, however, shortchanging your exposure to key statistical tools and concepts. As will become apparent, these tools will help us make sense out of data, with its inevitable variability, and communicate information about data to others. Witte11e_c02.indd 21 11/18/2016 9:02:52 PM C H APTER Describing Data 2 with Tables and Graphs TABLES (FREQUENCY DISTRIBUTIONS) 2.1 2.2 2.3 2.4 2.5 2.6 2.7 FREQUENCY DISTRIBUTIONS FOR QUANTITATIVE DATA GUIDELINES OUTLIERS RELATIVE FREQUENCY DISTRIBUTIONS CUMULATIVE FREQUENCY DISTRIBUTIONS FREQUENCY DISTRIBUTIONS FOR QUALITATIVE (NOMINAL) DATA INTERPRETING DISTRIBUTIONS CONSTRUCTED BY OTHERS GRAPHS 2.8 2.9 2.10 2.11 2.12 GRAPHS FOR QUANTITATIVE DATA TYPICAL SHAPES A GRAPH FOR QUALITATIVE (NOMINAL) DATA MISLEADING GRAPHS DOING IT YOURSELF Summary / Important Terms / Review Questions Preview A frequency distribution helps us to detect any pattern in the data (assuming a pattern exists) by superimposing some order on the inevitable variability among observations. For example, the appearance of a familiar bell-shaped pattern in the frequency distribution of reaction times of airline pilots to a cockpit alarm suggests the presence of many small chance factors whose collective effect must be considered in pilot retraining or cockpit redesign. Frequency distributions will appear in their various forms throughout the remainder of the book. Graphs of frequency distributions further aid our effort to detect data patterns and make sense out of the data. For example, knowing that the silhouette of a graph is balanced, as is the distribution of IQs for the general population, or that the silhouette is lopsided, as is the distribution of wealth for U.S. citizens, might supply important clues for understanding the data. Because they vividly summarize information, graphs sometimes serve as the final products of simple statistical analyses. Given some data, as in Table 1.1 on page 4, how do you make sense out of them—both for yourself and for others? Hidden among all those observations, is there an important message, possibly one that either supports or fails to support one of your ideas? 22 Witte11e_c02.indd 22 11/18/2016 9:02:52 PM 2 .1 F R E Q U ENCY DISTRIBUTIONS FOR QUANTITATIVE DATA Table 2.1 FREQUENCY DISTRIBUTION (UNGROUPED DATA) WEIGHT 245 244 243 242 * * * 161 160 159 158 157 * * * 136 135 134 133 Total f 1 0 0 0 0 4 1 2 3 0 2 0 1 53 23 (Or, more interestingly, is there a difference between two or more sets of data—for instance, between the GRE scores of students who do or do not attend a test-taking workshop; or between the survival rates of coronary bypass patients who do or do not own a dog; or between the starting salaries of male and female executives?) At this point, especially if you are facing a fresh set of data in which you have a special interest, statistics can be exciting as well as challenging. Your initial responsibility is to describe the data as clearly, completely, and concisely as possible. Statistics supplies some tools, including tables and graphs, and some guidelines. Beyond that, it is just the data and you. There is no single right way to describe data. Equally valid descriptions of the same data might appear in tables or graphs with different formats. By following just a few guidelines, your reward will be a well-summarized set of data. TABLES (FREQUENCY DISTRIBUTIONS) 2 . 1 F R E Q U E N C Y D I S T R I B U T I O N S F O R Q U A N T I TAT I V E D ATA Table 2.1 shows one way to organize the weights of the male statistics students listed in Table 1.1. First, arrange a column of consecutive numbers, beginning with the lightest weight (133) at the bottom and ending with the heaviest weight (245) at the top. (Because of the extreme length of this column, many intermediate numbers have been omitted in Table 2.1, a procedure never followed in practice.) Then place a short vertical stroke or tally next to a number each time its value appears in the original set of data; once this process has been completed, substitute for each tally count (not shown in Table 2.1) a number indicating the frequency ( f ) of occurrence of each weight. A frequency distribution is a collection of observations produced by sorting observations into classes and showing their frequency (f ) of occurrence in each class. Frequency Distribution A collection of observations produced by sorting observations into classes and showing their frequency (f) of occurrence in each class. Frequency Distribution When observations are sorted into classes of single values, as in Table 2.1, the result is referred to as a frequency distribution for ungrouped data. Not Always Appropriate The frequency distribution shown in Table 2.1 is only partially displayed because there are more than 100 possible values between the largest and smallest observations. Frequency distributions for ungrouped data are much more informative when the number of possible values is less than about 20. Under these circumstances, they are a straightforward method for organizing data. Otherwise, if there are 20 or more possible values, consider using a frequency distribution for grouped data. for Ungrouped Data A frequency distribution produced Progress Check *2.1 Students in a theater arts appreciation class rated the classic film whenever observations are sorted The Wizard of Oz on a 10-point scale, ranging from 1 (poor) to 10 (excellent), as follows: into classes of single values. 3 3 2 9 8 7 1 5 7 9 2 4 3 6 7 7 10 5 3 3 8 3 8 7 6 Since the number of possible values is relatively small—only 10—it’s appropriate to construct a frequency distribution for ungrouped data. Do this. Answer on page 420. Witte11e_c02.indd 23 11/18/2016 9:02:52 PM 24 D E S C R IB ING DATA WITH TABL ES AND GRAPHS Grouped Data Frequency Distribution for Grouped Data A frequency distribution produced whenever observations are sorted into classes of more than one value. Table 2.2 FREQUENCY DISTRIBUTION (GROUPED DATA) WEIGHT 240–249 230–239 220–229 210–219 200–209 190–199 180–189 170–179 160–169 150–159 140–149 130–139 Total f 1 0 3 0 2 4 3 7 12 17 1 3 53 Table 2.2 shows another way to organize the weights in Table 1.1 according to their frequency of occurrence. When observations are sorted into classes of more than one value, as in Table 2.2, the result is referred to as a frequency distribution for grouped data. Let’s look at the general structure of this frequency distribution. Data are grouped into class intervals with 10 possible values each. The bottom class includes the smallest observation (133), and the top class includes the largest observation (245). The distance between bottom and top is occupied by an orderly series of classes. The frequency ( f ) column shows the frequency of observations in each class and, at the bottom, the total number of observations in all classes. Let’s summarize the more important properties of the distribution of weights in Table 2.2. Although ranging from the 130s to the 240s, the weights peak in the 150s, with a progressively decreasing but relatively heavy concentration in the 160s and 170s. Furthermore, the distribution of weights is not balanced about its peak, but tilted in the direction of the heavier weights. 2.2 GUIDELINES The “Guidelines for Frequency Distributions” box lists seven rules for producing a well-constructed frequency distribution. The first three rules are essential and should not be violated. The last four rules are optional and can be modified or ignored as circumstances warrant. Satisfy yourself that the frequency distribution in Table 2.2 actually complies with these seven rules. How Many Classes? The seventh guideline requires a few more comments. The use of too many classes—as in Table 2.3, in which the weights are grouped into 24 classes, each with an interval of 5—tends to defeat the purpose of a frequency distribution, namely, to provide a reasonably concise description of data. On the other hand, the use of too few classes—as in Table 2.4, in which the weights are grouped into three classes, each with an interval of 50—can mask important data patterns such as the high density of weights in the 150s and 160s. When There Are Either Many or Few Observations There is nothing sacred about 10, the recommended number of classes. When describing large sets of data, you might aim for considerably more than 10 classes in order to portray some of the more fine-grained data patterns that otherwise could vanish. On the other hand, when describing small batches of data, you might aim for fewer than 10 classes in order to spotlight data regularities that otherwise could be blurred. It is best, therefore, to think of 10, the recommended number of classes, as a rough rule of thumb to be applied with discretion. Gaps between Classes Unit of Measurement The smallest possible difference between scores. Witte11e_c02.indd 24 In well-constructed frequency tables, the gaps between classes, such as between 149 and 150 in Table 2.2, show clearly that each observation or score has been assigned to one, and only one, class. The size of the gap should always equal one unit of measurement; that is, it should always equal the smallest possible difference between scores within a particular set of data. Since the gap is never bigger than one unit of measurement, no score can fall into the gap. In the present case, in which the weights are reported to the nearest pound, one pound is the unit of measurement, and therefore, the gap between classes equals one pound. These gaps would not be appropriate if the weights had been reported to the nearest tenth of a pound. In this case, one-tenth of a pound is the unit of measurement, and therefore, the gap should equal one-tenth of a pound. The smallest class interval would be 130.0–139.9 (not 130–139), and the next class interval would be 11/18/2016 9:02:52 PM 2 .2 G U ID EL INES Table 2.3 FREQUENCY DISTRIBUTION WITH TOO MANY INTERVALS WEIGHT 245–249 240–244 235–239 230–234 225–229 220–224 215–219 210–214 205–209 200–204 195–199 190–194 185–189 180–184 175–179 170–174 165–169 160–164 155–159 150–154 145–149 140–144 135–139 130–134 Total f 1 0 0 0 2 1 0 0 2 0 0 4 1 2 2 5 7 5 9 8 1 0 2 1 53 Table 2.4 FREQUENCY DISTRIBUTION WITH TOO FEW INTERVALS WEIGHT 200–249 150–199 100–149 Total f 6 43 4 53 25 GUIDELINES FOR FREQUENCY DISTRIBUTIONS Essential 1. Each observation should be included in one, and only one, class. Example: 130–139, 140–149, 150–159, etc. It would be incorrect to use 130–140, 140–150, 150–160, etc., in which, because the boundaries of classes overlap, an observation of 140 (or 150) could be assigned to either of two classes. 2. List all classes, even those with zero frequencies. Example: Listed in Table 2.2 is the class 210–219 and its frequency of zero. It would be incorrect to skip this class because of its zero frequency. 3. All classes should have equal intervals. Example: 130–139, 140–149, 150–159, etc. It would be incorrect to use 130–139, 140–159, etc., in which the second class interval (140–159) is twice as wide as the first class interval (130–139). Optional 4. All classes should have both an upper boundary and a lower boundary. Example: 240–249. Less preferred would be 240–above, in which no maximum value can be assigned to observations in this class. (Nevertheless, this type of open-ended class is employed as a space-saving device when many different tables must be listed, as in the Statistical Abstract of the United States. An open-ended class appears in the table “Two Age Distributions” in Review Question 2.17 at the end of this chapter.) 5. Select the class interval from convenient numbers, such as 1, 2, 3, . . . 10, particularly 5 and 10 or multiples of 5 and 10. Example: 130–139, 140–149, in which the class interval of 10 is a convenient number. Less preferred would be 130–142, 143–155, etc., in which the class interval of 13 is not a convenient number. 6. The lower boundary of each class interval should be a multiple of the class interval. Example: 130–139, 140–149, in which the lower boundaries of 130, 140, are multiples of 10, the class interval. Less preferred would be 135–144, 145–154, etc., in which the lower boundaries of 135 and 145 are not multiples of 10, the class interval. 7. Aim for a total of approximately 10 classes. Example: The distribution in Table 2.2 uses 12 classes. Less preferred would be the distributions in Tables 2.3 and 2.4. The distribution in Table 2.3 has too many classes (24), whereas the distribution in Table 2.4 has too few classes (3). 140.0–149.9 (not 140–149), and so on. These new boundaries would guarantee that any observation, such as 139.6, would be assigned to one, and only one, class. Gaps between classes do not signify any disruption in the essentially continuous nature of the data. It would be erroneous to conclude that, because of the gap between 149 and 150 for the frequency distribution in Table 2.2, nobody can weigh between 149 and 150 lbs. As noted in Section 1.6, a man who reports his weight as 150 lbs actually could weigh anywhere between 149.5 and 150.5 lbs, just as a man who reports his weight as 149 lbs actually could weigh anywhere between 148.5 and 149.5 lbs. Witte11e_c02.indd 25 11/18/2016 9:02:52 PM 26 D E S C R IB ING DATA WITH TABL ES AND GRAPHS Real Limits of Class Intervals Real Limits Located at the midpoint of the gap between adjacent tabled boundaries. Gaps cannot be ignored when you are determining the actual width of any class interval. The real limits are located at the midpoint of the gap between adjacent tabled boundaries; that is, one-half of one unit of measurement below the lower tabled boundary and one-half of one unit of measurement above the upper tabled boundary. For example, the real limits for 140–149 in Table 2.2 are 139.5 (140 minus one-half of the unit of measurement of 1) and 149.5 (149 plus one-half of the unit of measurement of 1), and the actual width of the class interval would be 10 (from 149.5 139.5 = 10). If weights had been reported to the nearest tenth of a pound, the real limits for 140.0–149.9 would be 139.95 (140.0 minus one-half of the unit of measurement of .1) and 149.95 (149.9 plus one-half of one unit of measurement of .1), and the actual width of the class interval still would be 10 (from 149.95 139.95 = 10). Constructing Frequency Distributions Now that you know the properties of well-constructed frequency distributions, study the step-by-step procedure listed in the “Constructing Frequency Distributions” box, which shows precisely how the distribution in Table 2.2 was constructed from the weight data in Table 1.1. You might want to refer back to this box when you need to construct a frequency distribution for grouped data. Progress Check *2.2 The IQ scores for a group of 35 high school dropouts are as follows: 91 87 95 123 98 110 112 85 96 71 80 69 109 90 84 75 105 100 99 94 90 79 86 90 93 95 100 98 80 104 77 108 90 103 89 (a) Construct a frequency distribution for grouped data. (b) Specify the real limits for the lowest class interval in this frequency distribution. Answers on pages 420 and 421. Progress Check *2.3 What are some possible poor features of the following frequency distribution? ESTIMATED WEEKLY TV VIEWING TIME (HRS) FOR 250 SIXTH GRADERS VIEWING TIME 35–above 30–34 25–30 20–22 15–19 10–14 5–9 0–4 Total f 2 5 29 60 60 34 31 29 250 Answers on page 421. Witte11e_c02.indd 26 11/18/2016 9:02:53 PM 2 .3 O U T LIERS 27 CONSTRUCTING FREQUENCY DISTRIBUTIONS 1. Find the range, that is, the difference between the largest and smallest observations. The range of weights in Table 1.1 is 245 133 = 112. 2. Find the class interval required to span the range by dividing the range by the desired number of classes (ordinarily 10). In the present example, Class inteeerval = range 112 = = 11.2 desired number of classes 10 3. Round off to the nearest convenient interval (such as 1, 2, 3, . . . 10, particularly 5 or 10 or multiples of 5 or 10). In the present example, the nearest convenient interval is 10. 4. Determine where the lowest class should begin. (Ordinarily, this number should be a multiple of the class interval.) In the present example, the smallest score is 133, and therefore the lowest class should begin at 130, since 130 is a multiple of 10 (the class interval). 5. Determine where the lowest class should end by adding the class interval to the lower boundary and then subtracting one unit of measurement. In the present example, add 10 to 130 and then subtract 1, the unit of measurement, to obtain 139—the number at which the lowest class should end. 6. Working upward, list as many equivalent classes as are required to include the largest observation. In the present example, list 130–139, 140–149, . . . , 240–249, so that the last class includes 245, the largest score. 7. Indicate with a tally the class in which each observation falls. For example, the first score in Table 1.1, 160, produces a tally next to 160–169; the next score, 193, produces a tally next to 190–199; and so on. 8. Replace the tally count for each class with a number—the frequency (f )—and show the total of all frequencies. (Tally marks are not usually shown in the final frequency distribution.) 9. Supply headings for both columns and a title for the table. 2.3 OUTLIERS Outlier A very extreme score. Be prepared to deal occasionally with the appearance of one or more very extreme scores, or outliers. A GPA of 0.06, an IQ of 170, summer wages of \$62,000—each requires special attention because of its potential impact on a summary of the data. Check for Accuracy Whenever you encounter an outrageously extreme value, such as a GPA of 0.06, attempt to verify its accuracy. For instance, was a respectable GPA of 3.06 recorded erroneously as 0.06? If the outlier survives an accuracy check, it should be treated as a legitimate score. Witte11e_c02.indd 27 11/18/2016 9:02:53 PM 28 D E S C R IB ING DATA WITH TABL ES AND GRAPHS Might Exclude from Summaries You might choose to segregate (but not to suppress!) an outlier from any summary of the data. For example, you might relegate it to a footnote instead of using excessively wide class intervals in order to include it in a frequency distribution. Or you might use various numerical summaries, such as the median and interquartile range, to be discussed in Chapters 3 and 4, that ignore extreme scores, including outliers. Might Enhance Understanding Insofar as a valid outlier can be viewed as the product of special circumstances, it might help you to understand the data. For example, you might understand better why crime rates differ among communities by studying the special circumstances that produce a community with an extremely low (or high) crime rate, or why learning rates differ among third graders by studying a third grader who learns very rapidly (or very slowly). Progress Check *2.4 Identify any outliers in each of the following sets of data collected from nine college students. SUMMER INCOME AGE FAMILY SIZE GPA \$6,450 \$4,820 \$5,650 \$1,720 \$600 \$0 \$3,482 \$25,700 \$8,548 20 19 61 32 19 22 23 27 21 2 4 3 6 18 2 6 3 4 2.30 4.00 3.56 2.89 2.15 3.01 3.09 3.50 3.20 Answers on page 421. 2 . 4 R E L AT I V E F R E Q U E N C Y D I S T R I B U T I O N S An important variation of the frequency distribution is the relative frequency distribution. Relative Frequency Distribution A frequency distribution showing the frequency of each class as a fraction of the total frequency for the entire distribution. Relative frequency distributions show the frequency of each class as a part or fraction of the total frequency for the entire distribution. This type of distribution allows us to focus on the relative concentration of observations among different classes within the same distribution. In the case of the weight data in Table 2.2, it permits us to see that the 160s account for about one-fourth (12/53 = 23, or 23%) of all observations. This type of distribution is especially helpful when you must compare two or more distributions based on different total numbers of observations. For instance, as in Review Question 2.17, you might want to compare the distribution of ages for 500 residen...
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