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Exploring Statistics Tales of Distributions

12th Edition

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Chris Spatz

Outcrop Publishers Conway, Arkansas

Exploring Statistics: Tales of Distributions 12th Edition Chris Spatz

Cover design: Grace Oxley Answer Key: Jill Schmidlkofer Webmaster & Ebook: Fingertek Web Design, Tina Haggard Managers: Justin Murdock, Kevin Spatz

Copyright © 2019 by Outcrop Publishers, LLC All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, contact or write to the publisher at the address below.

Outcrop Publishers 615 Davis Street Conway, AR 72034 Email: Website: Library of Congress Control Number: [Applied for]

ISBN-13 (hardcover): 978-0-9963392-2-3 ISBN-13 (ebook): 978-0-9963392-3-0 ISBN-13 (study guide): 978-0-9963392-4-7

Examination copies are provided to academics and professionals to consider for adoption as a course textbook. Examination copies may not be sold or transferred to a third party. If you adopt this textbook, please accept it as your complimentary desk copy.

Ordering information: Students and professors – visit Bookstores – email

Photo Credits – Chapter 1 Karl Pearson – Courtesy of Ronald A. Fisher – R.A. Fisher portrait, 0006973, Special Collections Research Center, North Carolina State

University Libraries, Raleigh, North Carolina Jerzy Neyman – Paul R. Halmos Photograph Collection, e_ph 0223_01, Dolph Briscoe Center for American History,

The University of Texas at Austin Jacob Cohen – New York University Archives, Records of the NYU Photo Bureau

Printed in the United States of America by Walsworth ® 2 3 4 5 6 7 24 23 22 21 20

Online study guide available at
v About The Author

Chris Spatz is at Hendrix College where he twice served as chair of the Psychology Department. Dr. Spatz’s undergraduate education was at Hendrix, and his PhD in experimental psychology is from Tulane University in New Orleans. He subsequently completed postdoctoral fellowships in animal behavior at the University of California, Berkeley, and the University of Michigan. Before returning to Hendrix to teach, Spatz held positions at The University of the South and the University of Arkansas at Monticello.

Spatz served as a reviewer for the journal Teaching of Psychology for more than 20 years. He co-authored a research methods textbook, wrote several chapters for edited books, and was a section editor for the Encyclopedia of Statistics in Behavioral Science.

In addition to writing and publishing, Dr. Spatz enjoys the outdoors, especially canoeing, camping, and gardening. He swims several times a week (mode = 3). Spatz has been an opponent of high textbook prices for years, and he is happy to be part of a new wave of authors who provide high-quality textbooks to students at affordable prices.

About The Author

vi Dedication

With love and affection,

this textbook is dedicated to

Thea Siria Spatz, Ed.D., CHES

vii Brief Contents

Brief Contents

Preface xiv 1 Introduction 1 2 Exploring Data: Frequency Distributions and Graphs 29 3 Exploring Data: Central Tendency 45 4 Exploring Data: Variability 59 5 Other Descriptive Statistics 77 6 Correlation and Regression 94 7 Theoretical Distributions Including the Normal Distribution 127 8 Samples, Sampling Distributions, and Confidence Intervals 150 9 Effect Size and NHST: One-Sample Designs 175 10 Effect Size, Confidence Intervals, and NHST:

Two-Sample Designs 200 11 Analysis of Variance: Independent Samples 231 12 Analysis of Variance: Repeated Measures 259 13 Analysis of Variance: Factorial Design 271 14 Chi Square Tests 303 15 More Nonparametric Tests 328 16 Choosing Tests and Writing Interpretations 356


A Getting Started 371 B Grouped Frequency Distributions and Central Tendency 376 C Tables 380 D Glossary of Words 401 E Glossary of Symbols 405 F Glossary of Formulas 407 G Answers to Problems 414

References 466 Index 472


Preface xiv

chapter 1 Introduction 1 Disciplines That Use Quantitative Data 5 What Do You Mean, “Statistics”? 6 Statistics: A Dynamic Discipline 8 Some Terminology 9 Problems and Answers 12 Scales of Measurement 13 Statistics and Experimental Design 16 Experimental Design Variables 17 Statistics and Philosophy 20 Statistics: Then and Now 21 How to Analyze a Data Set 22 Helpful Features of This Book 22 Computers, Calculators, and Pencils 24 Concluding Thoughts 25 Key Terms 27

Transition Passage to Descriptive Statistics 28

chapter 2 Exploring Data: Frequency Distributions and Graphs 29 Simple Frequency Distributions 31 Grouped Frequency Distributions 33 Graphs of Frequency Distributions 35 Describing Distributions 39




The Line Graph 41 More on Graphics 42 A Moment to Reflect 43 Key Terms 44

chapter 3 Exploring Data: Central Tendency 45 Measures of Central Tendency 46 Finding Central Tendency of Simple Frequency Distributions 49 When to Use the Mean, Median, and Mode 52 Determining Skewness From the Mean and Median 54 The Weighted Mean 55 Estimating Answers 56 Key Terms 58

chapter 4 Exploring Data: Variability 59 Range 61 Interquartile Range 61 Standard Deviation 63 Standard Deviation as a Descriptive Index of Variability 64 ŝ as an Estimate of σ 69 Variance 73 Statistical Software Programs 74 Key Terms 76

chapter 5 Other Descriptive Statistics 77 Describing Individual Scores 78 Boxplots 82 Effect Size Index 86 The Descriptive Statistics Report 89 Key Terms 92

Transition Passage to Bivariate Statistics 93

chapter 6 Correlation and Regression 94 Bivariate Distributions 96 Positive Correlation 96 Negative Correlation 99 Zero Correlation 101 Correlation Coefficient 102 Scatterplots 106



Interpretations of r 106 Uses of r 110 Strong Relationships but Low Correlation Coefficients 112 Other Kinds of Correlation Coefficients 115 Linear Regression 116 The Regression Equation 117 Key Terms 124 What Would You Recommend? Chapters 2-6 125

Transition Passage to Inferential Statistics 126

chapter 7 Theoretical Distributions Including the Normal Distribution 127 Probability 128 A Rectangular Distribution 129 A Binomial Distribution 130 Comparison of Theoretical and Empirical Distributions 131 The Normal Distribution 132 Comparison of Theoretical and Empirical Answers 146 Other Theoretical Distributions 146 Key Terms 147

Transition Passage to the Analysis of Data From Experiments 149

chapter 8 Samples, Sampling Distributions, and Confidence Intervals 150 Random Samples 152 Biased Samples 155 Research Samples 156 Sampling Distributions 157 Sampling Distribution of the Mean 157 Central Limit Theorem 159 Constructing a Sampling Distribution When σ Is Not Available 164 The t Distribution 165 Confidence Interval About a Population Mean 168 Categories of Inferential Statistics 172 Key Terms 173



Transition Passage to Null Hypothesis Significance Testing 174

chapter 9 Effect Size and NHST: One-Sample Designs 175 Effect Size Index 176 The Logic of Null Hypothesis Significance Testing (NHST) 179 Using the t Distribution for Null Hypothesis Significance Testing 182 A Problem and the Accepted Solution 184 The One-Sample t Test 186 An Analysis of Possible Mistakes 188 The Meaning of p in p < .05 191 One-Tailed and Two-Tailed Tests 192 Other Sampling Distributions 195 Using the t Distribution to Test the Significance of a Correlation Coefficient 195 t Distribution Background 197 Why .05? 198 Key Terms 199

chapter 10 Effect Size, Confidence Intervals, and NHST: Two-Sample Designs 200 A Short Lesson on How to Design an Experiment 201 Two Designs: Paired Samples and Independent Samples 202 Degrees of Freedom 206 Paired-Samples Design 208 Independent-Samples Design 212 The NHST Approach 217 Statistical Significance and Importance 222 Reaching Correct Conclusions 222 Statistical Power 225 Key Terms 228 What Would You Recommend? Chapters 7-10 229

Transition Passage to More Complex Designs 230


xii Contents

chapter 11 Analysis of Variance: Independent Samples 231 Rationale of ANOVA 233 More New Terms 240 Sums of Squares 240 Mean Squares and Degrees of Freedom 245 Calculation and Interpretation of F Values Using the F Distribution 246 Schedules of Reinforcement—A Lesson in Persistence 248 Comparisons Among Means 250 Assumptions of the Analysis of Variance 254 Random Assignment 254 Effect Size Indexes and Power 255 Key Terms 258

chapter 12 Analysis of Variance: Repeated Measures 259 A Data Set 260 Repeated-Measures ANOVA: The Rationale 261 An Example Problem 262 Tukey HSD Tests 265 Type I and Type II Errors 266 Some Behind-the-Scenes Information About Repeated-Measures ANOVA 267 Key Terms 270

chapter 13 Analysis of Variance: Factorial Design 271 Factorial Design 272 Main Effects and Interaction 276 A Simple Example of a Factorial Design 282 Analysis of a 2 × 3 Design 291 Comparing Levels Within a Factor—Tukey HSD Tests 297 Effect Size Indexes for Factorial ANOVA 299 Restrictions and Limitations 299 Key Terms 301

Transition Passage to Nonparametric Statistics 302

chapter 14 Chi Square Tests 303 The Chi Square Distribution and the Chi Square Test 305 Chi Square as a Test of Independence 307 Shortcut for Any 2 × 2 Table 310 Effect Size Indexes for 2 × 2 Tables 310 Chi Square as a Test for Goodness of Fit 314

xiii Contents

Chi Square With More Than One Degree of Freedom 316 Small Expected Frequencies 321 When You May Use Chi Square 324 Key Terms 327

chapter 15 More Nonparametric Tests 328 The Rationale of Nonparametric Tests 329 Comparison of Nonparametric to Parametric Tests 330 Mann-Whitney U Test 332 Wilcoxon Signed-Rank T Test 339 Wilcoxon-Wilcox Multiple-Comparisons Test 344 Correlation of Ranked Data 348 Key Terms 353 What Would You Recommend? Chapters 11-15 353

chapter 16 Choosing Tests and Writing Interpretations 356 A Review 356 My (Almost) Final Word 357 Future Steps 358 Choosing Tests and Writing Interpretations 359 Key Term 368

Appendixes A Getting Started 371 B Grouped Frequency Distributions and Central

Tendency 376 C Tables 380 D Glossary of Words 401 E Glossary of Symbols 405 F Glossary of Formulas 407 G Answers to Problems 414

References 466 Index 472

xiv Preface

Exploring Statistics: Tales of Distributions (12th edition) is a textbook for a one-term statistics course in the social or behavioral sciences, education, or an allied health/nursing field. Its focus is conceptualization, understanding, and interpretation, rather than computation. Designed to be comprehensible and complete for students who take only one statistics course, it also includes elements that prepare students for additional statistics courses. For example, basic experimental design terms such as independent and dependent variables are explained so students can be expected to write fairly complete interpretations of their analyses. In many places, the student is invited to stop and think or do a thought exercise. Some problems ask the student to decide which statistical technique is appropriate. In sum, this book’s approach is in tune with instructors who emphasize critical thinking in their course.

This textbook has been remarkably successful for more than 40 years. Students, professors, and reviewers have praised it. A common refrain is that the book has a conversational, narrative style that is engaging, especially for a statistics text. Other features that distinguish this textbook from others include the following:

• Data sets are approached with an attitude of exploration. • Changes in statistical practice over the years are acknowledged, especially the recent

emphasis on effect sizes and confidence intervals. • Criticism of null hypothesis significance testing (NHST) is explained. • Examples and problems represent a variety of disciplines and everyday life. • Most problems are based on actual studies rather than fabricated scenarios. • Interpretation is emphasized throughout. • Problems are interspersed within a chapter, not grouped at the end. • Answers to all problems are included. • Answers are comprehensively explained—over 50 pages of detail. • A final chapter, Choosing Tests and Writing Interpretations, requires active responses to

comprehensive questions.


Even if our statistical appetite is far from keen, we all of us should like to know enough to understand, or to withstand, the statistics that are constantly being thrown at us in print or conversation—much of it pretty bad statistics. The only cure for bad statistics is apparently more and better statistics. All in all, it certainly appears that the rudiments of sound statistical sense are coming to be an essential of a liberal education.

– Robert Sessions Woodworth

xv Preface

• Effect size indexes are treated as important descriptive statistics, not add-ons to NHST. • Important words and phrases are defined in the margin when they first occur. • Objectives, which open each chapter, serve first for orientation and later as review

items. • Key Terms are identified for each chapter. • Clues to the Future alert students to concepts that come up again. • Error Detection boxes tell ways to detect mistakes or prevent them. • Transition Passages alert students to a change in focus in chapters that follow. • Comprehensive Problems encompass all (or most) of the techniques in a chapter. • What Would You Recommend? problems require choices from among techniques in

several chapters.

For this 12th edition, I increased the emphasis on effect sizes and confidence intervals, moving them to the front of Chapter 9 and Chapter 10. The controversy over NHST is addressed more thoroughly. Power gets additional attention. Of course, examples and problems based on contemporary data are updated, and there are a few new problems. In addition, a helpful Study Guide to Accompany Exploring Statistics (12th edition) was written by Lindsay Kennedy, Jennifer Peszka, and Leslie Zorwick, all of Hendrix College. The study guide is available online at

Students who engage in this book and their course can expect to:

• Solve statistical problems • Understand and explain statistical reasoning • Choose appropriate statistical techniques for common research designs • Write explanations that are congruent with statistical analyses

After many editions with a conventional publisher, Exploring Statistics: Tales of Distributions is now published by Outcrop Publishers. As a result, the price of the print edition is about one-fourth that of the 10th edition. Nevertheless, the authorship and quality of earlier editions continue as before.

xvi Preface


The person I acknowledge first is the person who most deserves acknowledgment. And for the 11th and 12th editions, she is especially deserving. This book and its accompanying publishing company, Outcrop Publishers, would not exist except for Thea Siria Spatz, encourager, supporter, proofreader, and cheer captain. This edition, like all its predecessors, is dedicated to her.

Kevin Spatz, manager of Outcrop Publishers, directed the distribution of the 11th edition, advised, week by week, and suggested the cover design for the 12th edition. Justin Murdock now serves as manager, continuing the tradition that Kevin started. Tina Haggard of Fingertek Web Design created the book’s website, the text’s ebook, and the online study guide. She provided advice and solutions for many problems. Thanks to Jill Schmidlkofer, who edited the extensive answer section again for this edition. Emily Jones Spatz created new drawings for the text. I’m particularly grateful to Grace Oxley for a cover design that conveys exploration, and to Liann Lech, who copyedited for clarity and consistency. Walsworth® turned a messy collection of files into a handsome book—thank you Nathan Stufflebean and Dennis Paalhar. Others who were instrumental in this edition or its predecessors include Jon Arms, Ellen Bruce, Mary Kay Dunaway, Bob Eslinger, James O. Johnston, Roger E. Kirk, Rob Nichols, Jennifer Peszka, Mark Spatz, and Selene Spatz. I am especially grateful to Hendrix College and my Hendrix colleagues for their support over many years, and in particular, to Lindsay Kennedy, Jennifer Peszka, and Leslie Zorwick, who wrote the study guide that accompanies the text.

This textbook has benefited from perceptive reviews and significant suggestions by some 90 statistics teachers over the years. For this 12th edition, I particularly thank

Jessica Alexander, Centenary College Lindsay Kennedy, Hendrix College Se-Kang Kim, Fordham University Roger E. Kirk, Baylor University Kristi Lekies, The Ohio State University Jennifer Peszka, Hendrix College Robert Rosenthal, University of California, Riverside

I’ve always had a touch of the teacher in me—as an older sibling, a parent, a professor, and now a grandfather. Education is a first-class task, in my opinion. I hope this book conveys my enthusiasm for it. (By the way, if you are a student who is so thorough as to read even the acknowledgments, you should know that I included phrases and examples in a number of places that reward your kind of diligence.)

If you find errors in this book, please report them to me at I will post corrections at the book’s website:

Introduction CHAPTER


O B J E C T I V E S F O R C H A P T E R 1

After studying the text and working the problems in this chapter, you should be able to:

1. Distinguish between descriptive and inferential statistics 2. Define population, sample, parameter, statistic, and variable as they are

used in statistics 3. Distinguish between quantitative and categorical variables 4. Distinguish between continuous and discrete variables 5. Identify the lower and upper limits of a continuous variable 6. Identify four scales of measurement and distinguish among them 7. Distinguish between statistics and experimental design 8. Define independent variable, dependent variable, and extraneous variable

and identify them in experiments 9. Describe statistics’ place in epistemology 10. List actions to take to analyze a data set 11. Identify a few events in the history of statistics

WE BEGIN OUR exploration of statistics with a trip to London. The year is 1900. Walking into an office at University College.To continue with the answer check on

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