STATISTICS IN CONTEXT BAR BAR A BLATCHLEY Agnes Scott College NEW YORK OXFORD OXFORD UNIVERSITY PRESS bla78953_fm_i-1 i 11/27/17 08:34 PM Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and certain other countries. Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America. © 2019 by Oxford University Press For titles covered by Section 112 of the US Higher Education Opportunity Act, please visit www.oup.com/us/he for the latest information about pricing and alternate formats. 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, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by license, or under terms agreed with the appropriate reproduction rights organization. Inquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not circulate this work in any other form and you must impose this same condition on any acquirer. Library of Congress Cataloging-in-Publication Data Names: Blatchley, Barbara, author. Title: Statistics in context / Barbara Blatchley. Description: First Edition. | New York : Oxford University Press, 2018. Identifiers: LCCN 2017036139 (print) | LCCN 2017052156 (ebook) | ISBN 9780190278991 (ebook) | ISBN 9780190278953 (hardback) | ISBN 9780190864682 (looseleaf) Subjects: LCSH: Statistics—Textbooks. | BISAC: PSYCHOLOGY / Assessment, Testing & Measurement. | PSYCHOLOGY / Statistics. Classification: LCC QA276 (ebook) | LCC QA276 .B614 2018 (print) | DDC 519.5—dc23 LC record available at https://lccn.loc.gov/2017036139 9 8 7 6 5 4 3 2 1 Printed by Sheridan Books, Inc., United States of America Printed in the United Stated of America bla78953_fm_i-1 ii 11/27/17 08:34 PM CONTENTS IN BRIEF CHAPTER 1 INTRODUCTION: STATISTICS—WHO NEEDS THEM? 2 CHAPTER 2 TYPES OF DATA 32 CHAPTER 3 A PICTURE IS WORTH A THOUSAND WORDS: CREATING AND INTERPRETING GRAPHICS 100 CHAPTER 4 MEASURES OF CENTRAL TENDENCY: WHAT’S SO AVERAGE ABOUT THE MEAN? 150 CHAPTER 5 VARIABILITY: THE “LAW OF LIFE” 190 CHAPTER 6 WHERE AM I? NORMAL DISTRIBUTIONS AND STANDARD SCORES 252 CHAPTER 7 BASIC PROBABILITY THEORY 294 CHAPTER 8 THE CENTRAL LIMIT THEOREM AND HYPOTHESIS TESTING 322 CHAPTER 9 THE z-TEST 354 CHAPTER 10 t-TESTS 388 CHAPTER 11 ANALYSIS OF VARIANCE 442 CHAPTER 12 CONFIDENCE INTERVALS AND EFFECT SIZE: BUILDING A BETTER MOUSETRAP 512 CHAPTER 13 CORRELATION AND REGRESSION: ARE WE RELATED? 550 CHAPTER 14 THE CHI-SQUARE TEST 598 CHAPTER 15 NONPARAMETRIC TESTS 638 CHAPTER 16 WHICH TEST SHOULD I USE, AND WHY? 686 APPENDIX A THE PROPORTIONS UNDER THE STANDARD NORMAL CURVE 697 APPENDIX B THE STUDENT’S TABLE OF CRITICAL t-VALUES 702 APPENDIX C CRITICAL F-VALUES 705 APPENDIX D CRITICAL TUKEY HSD VALUES 707 APPENDIX E CRITICAL VALUES OF CHI SQUARE 709 APPENDIX F THE PEARSON CORRELATION COEFFICIENT: CRITICAL r-VALUES 712 APPENDIX G CRITICAL r VALUES FOR THE SPEARMAN CORRELATION S COEFFICIENT 713 APPENDIX H MANN–WHITNEY CRITICAL U-VALUES 716 APPENDIX I CRITICAL VALUES FOR THE WILCOXON SIGNED-RANK, MATCHED-PAIRS t-TEST 719 iii bla78953_fm_i-1 iii 11/27/17 08:34 PM CONTENTS Figures, Tables, and Boxes xi Acknowledgments xxii Preface xix Contents Overview xxiii Introducing . . . Statistics in Context xx CHAPTER 1 INTRODUCTION: STATISTICS—WHO NEEDS THEM? 2 Overview 3 Chance Error 11 Learning Objectives 3 Using Statistics 12 Everyday Statistics 3 Some Cautionary Notes About Statistics 13 What Are Statistics? 4 Statistics in Context 15 Types of Statistics 4 Summary 16 Terms You Should Know 17 THE HISTORICAL CONTEXT Roll Them Bones 4 Writing Assignment 17 Descriptive Statistics 5 Practice Problems 17 Inferential Statistics 6 Think About It . . . 26 Variables 7 References 27 Independent and Dependent Variables 8 INTRODUCING SPSS: THE STATISTICAL PACKAGE THINK ABOUT IT . . .  How a Tan Affects Attractiveness 10 FOR THE SOCIAL SCIENCES 28 CHAPTER 2 TYPES OF DATA 32 Overview 33 Grouped Frequency Distributions 51 Learning Objectives 33 THINK ABOUT IT . . . Frequency Distributions (Part 2) 58 Everyday Statistics 33 Scales of Measurement in Context 59 Data, Data Everywhere 34 . . . And Frequency Distributions 62 Scales of Measurement 34 Summary 62 Qualitative Data 34 Terms You Should Know 62 THE HISTORICAL CONTEXT S. S. Stevens and His Power Glossary of Equations 63 Law 35 Writing Assignment 63 Quantitative Data 39 Practice Problems 63 THINK ABOUT IT . . . Scales of Measurement 40 Think About It . . .  88 Organizing Data 42 References 91 Frequency Distributions 42 GETTING DATA INTO YOUR STATS PROGRAM 92 Ungrouped Frequency Distributions 45 Reading in Data with SPSS 92 THINK ABOUT IT . . . Frequency Distributions (Part 1) 50 Reading in Data with R 97 iv bla78953_fm_i-1 iv 11/27/17 08:34 PM CHAPTER 3 A PICTURE IS WORTH A THOUSAND WORDS: CREATING AND INTERPRETING GRAPHICS 100 Overview 101 Graphing Relationships 112 Learning Objectives 101 Graphing Time 113 Everyday Statistics 101 Graphics in Context: Rules for Creating Good Visualizing Patterns in Data 102 Graphs 114 THE HISTORICAL CONTEXT William Playfair and the Use Rules for Creating a Good Graph 114 of Graphics in Publishing 102 Summary 115 Terms You Should Know 115 Bar Charts and Histograms 103 Writing Assignment 115 Discrete Data 104 A Note on Graphing with Statistical Software 116 Continuous Data 105 Practice Problems 116 Stem-and-Leaf Graphs 106 Think About It . . .  129 Frequency Polygons 107 References 129 Pie Charts 108 THINK ABOUT IT . . . Interpreting Graphics 110 GRAPHING WITH SPSS AND R 130 Other Graphics 111 Graphing with SPSS 130 Graphing Means 111 Graphing with R 143 CHAPTER 4 MEASURES OF CENTRAL TENDENCY: WHAT’S SO AVERAGE ABOUT THE MEAN? 150 Overview 151 THINK ABOUT IT . . . Estimating Measures of Center 160 Learning Objectives 151 Shapes of Distributions 161 Everyday Statistics 151 Normal Distributions 162 Measures of Center 152 Finding Center with Grouped Data 163 Measures of Center: What Is Typical? 152 THINK ABOUT IT . . . Shapes of Distributions 165 THE HISTORICAL CONTEXT Adolphe Quetelet and the Measures of Center in Context 166 “Average Man” 153 Summary 168 The Mode and the Median 154 Terms You Should Know 168 Finding the Position of the Mode 154 Glossary of Equations 168 Finding the Position of the Median 155 Writing Assignment 168 The Mean 156 Practice Problems 169 Mode, Median, and Mean: Which Is the “Best” Think About It . . .  188 Measure of Center? 158 References 189 CHAPTER 5 VARIABILITY: THE “LAW OF LIFE” 190 Overview 191 Consistency and Inconsistency in Data 192 Learning Objectives 191 THE HISTORICAL CONTEXT What Is the Shape of the Everyday Statistics 191 Earth? 193 Measuring Variability 192 Measures of Variability 194 CONTENTS v bla78953_fm_i-1 v 11/27/17 08:34 PM The Range 194 Descriptive Statistics in Context 210 The Interquartile Range 195 Summary 212 Graphing the IQR 197 Terms You Should Know 212 The Variance 198 Glossary of Equations 213 Average Deviation from the Mean 198 Writing Assignment 213 The Standard Deviation 200 Practice Problems 215 Finding the Variance in a Population 200 Think About It . . .  231 Finding the Standard Deviation in a Population 202 References 231 Finding Standard Deviation in a Population versus a Sample 204 DESCRIPTIVE STATISTICS WITH SPSS AND R 232 Finding Variance and Standard Deviation: An Example 204 Descriptive Statistics with SPSS 232 Descriptive Statistics with R 243 THINK ABOUT IT . . . The Range Rule 206 Standard Deviation in Context 207 CHAPTER 6 W HERE AM I? NORMAL DISTRIBUTIONS AND STANDARD SCORES 252 Overview 253 Converting a Percentile Rank Learning Objectives 253 into a Raw Score 273 Everyday Statistics 253 THINK ABOUT IT . . . The Range Rule Statistics So Far 254 Revisited 276 Standard Scores 254 Standard Scores in Context 277 THE HISTORICAL CONTEXT Alfred Binet and Intelligence Summary 277 Testing 255 Terms You Should Know 278 The z-Score 256 Glossary of Equations 278 The “3-Sigma Rule” 259 Writing Assignment 278 Proportions in the Standard Normal Curve 262 Practice Problems 279 The Benefits of Standard Scores 270 Think About It . . .  292 Comparing Scores from Different Distributions 270 References 293 Converting a z-Score into a Raw Score 272 CHAPTER 7 BASIC PROBABILITY THEORY 294 Overview 295 THINK ABOUT IT . . . Probability Theory and Card Learning Objectives 295 Games 309 Everyday Statistics 295 Probability in Context 310 Probability 296 Summary 312 Probability and Frequency 296 Terms You Should Know 313 Basic Set Theory 296 Glossary of Equations 313 THE HISTORICAL CONTEXT The Gambler’s Fallacy 297 Practice Problems 313 Conditional Probability 300 Think About It . . .  321 Combining Probabilities 301 References 321 Using Probability 306 vi CONTENTS bla78953_fm_i-1 vi 11/27/17 08:34 PM CHAPTER 8 THE CENTRAL LIMIT THEOREM AND HYPOTHESIS TESTING 322 Overview 323 The Sampling Distribution of the Means 329 Learning Objectives 323 The Three Statements That Make Up the Central Limit Everyday Statistics 323 Theorem 330 Introduction: Error in Statistics 324 Random Sampling 333 Inferential Statistics 324 Using the Central Limit Theorem 337 The Scientific Method 324 Estimating Parameters and Hypothesis Testing 338 THE HISTORICAL CONTEXT Kinnebrook’s Error and The Null and Alternative Hypotheses 338 Statistics 325 Directional Hypotheses 339 The Central Limit Theorem 325 Hypothesis Testing in Context 342 Measuring the Distribution of Large Sets of Events 325 Summary 344 The Law of Large Numbers and the Central Limit Terms You Should Know 345 Theorem 327 Glossary of Equations 345 THINK ABOUT IT . . . The Law of Large Numbers and Practice Problems 345 Dice Games 328 Think About It . . . 352 Drawing Samples from Populations 329 References 352 CHAPTER 9 THE z-TEST 354 Overview 355 Statistics as Estimates 370 Learning Objectives 355 On Being Right: Type I and Type II Errors 371 Everyday Statistics 355 An Example 371 Error Revisited 356 p-Values 373 THE HISTORICAL CONTEXT The Trial of the Pyx 358 Inferential Statistics in Context: Galton and the Quincunx 373 How Different Is Different Enough? Critical Summary 375 Values and p 359 Terms You Should Know 376 Assumptions in Hypothesis Testing 359 Glossary of Equations 376 The Outer 5%: The Rejection Region 360 Writing Assignment 376 THINK ABOUT IT . . . p-Values and Alpha Levels 362 Practice Problems 376 Finding the z-Value in a Nondirectional Hypothesis 363 Think About It . . .  387 The z-Test 364 References 387 Another Example 369 CHAPTER 10 t-TESTS 388 Overview 389 THE HISTORICAL CONTEXT Statistics and Beer 393 Learning Objectives 389 The Single-Sample t-Test 394 Everyday Statistics 389 Degrees of Freedom 396 Inferential Testing So Far 390 When Both σ and µ Are Unknown 400 William Gosset and the Development of the Independent-Samples t-Test 402 t-Test 390 The Standard Error of the Difference 403 “Student’s” Famous Test 392 Finding the Difference Between Two Means: An Example 404 CONTENTS vii bla78953_fm_i-1 vii 11/27/17 08:34 PM Finding the Difference Between Means with Unequal Terms You Should Know 422 Samples 406 Glossary of Equations 422 Assumptions 411 Writing Assignment 423 THINK ABOUT IT . . . t-Tests and Sample Size 411 Practice Problems 423 Using the Formula for Unequal n’s with Equal n’s 433 Dependent-Samples t-Tests 413 Introduction 413 Think About It . . . 435 Using a Dependent-Samples t-Test: An Example 413 References 436 Calculations and Results 414 CONDUCTING t-TESTS WITH SPSS AND R 437 THINK ABOUT IT . . . t-Tests and Variability 417 t-Tests with SPSS 437 t-Tests in Context: “Garbage In, Garbage Out” 419 t-Tests with R 440 Summary 421 CHAPTER 11 ANALYSIS OF VARIANCE 442 Overview 443 Factorial Designs or Two-Way ANOVAs 471 Learning Objectives 443 An Example: Albert Bandura’s Study of Imitating Violence 472 Everyday Statistics 443 Graphing the Main Effects 474 Comparing More Than Two Groups 444 The Logic of the Two-Way ANOVA 476 Analysis of Variance: What Does It Mean? 444 Using the Two-Way ANOVA Source Table 477 Interpreting the Results 479 THE HISTORICAL CONTEXT Fertilizer, Potatoes, and ANOVA in Context: Interpretation and Fisher’s Analysis of Variance 445 Misinterpretation 480 A Hypothetical Study of Blood Doping 446 Summary 482 Assessing Between-Group and Within-Group Variability in Terms You Should Know 482 Our Hypothetical Results 447 Glossary of Equations 483 ANOVA Terminology 448 Writing Assignment 484 The One-Way ANOVA Procedure 449 Sums of Squares 449 Practice Problems 485 Think About It . . . 495 THINK ABOUT IT . . . t for Two and F for Many 456 References 496 THINK ABOUT IT . . . The F-Statistic 460 USING SPSS AND R FOR ANOVA 496 Post-hoc Testing 460 One-Way ANOVA in SPSS 497 The Tukey HSD Test with Equal n’s 462 Two-Way ANOVA in SPSS 501 The Tukey HSD Test with Unequal n’s 466 One-Way ANOVA in R 504 Models of F 470 Two-Way ANOVA in R 508 One-Way ANOVA Assumptions 471 CHAPTER 12 CONFIDENCE INTERVALS AND EFFECT SIZE: BUILDING A BETTER MOUSETRAP 512 Overview 513 CIs and the z-Test 518 Learning Objectives 513 THINK ABOUT IT . . . What Does a Confidence Interval Everyday Statistics 513 Really Mean? (Part 1) 520 Using Estimations 514 CIs and the Single-Sample t-Test 522 Estimates and Confidence Intervals 515 CIs and Independent- and Dependent-Samples t-Tests 524 THE HISTORICAL CONTEXT Jerzy Neyman: What’s a Lemma? 516 viii CONTENTS bla78953_fm_i-1 viii 11/27/17 08:34 PM THINK ABOUT IT . . . What Does a Confidence Interval Terms You Should Know 537 Really Mean? (Part 2) 528 Glossary of Equations 537 Writing Assignment 538 Effect Size: How Different Are These Means, Really? 529 Practice Problems 538 Effect Size and ANOVA 533 Think About It . . . 547 Statistics in Context: The CI versus the Inferential Test 535 References 548 Summary 537 CHAPTER 13 CORRELATION AND REGRESSION: ARE WE RELATED? 550 Overview 551 Coefficient of Determination 569 Learning Objectives 551 THINK ABOUT IT . . . Shared Variability and Everyday Statistics 551 Restricted Range 570 The Correlation Coefficient 552 Statistics in Context: Correlations and Causation 571 THE HISTORICAL CONTEXT Statistics and Sweet Peas 552 Summary 573 Positive and Negative Correlations 554 Terms You Should Know 573 Weak and Strong Correlations 555 Glossary of Equations 573 Calculating r 557 Writing Assignment 574 Testing Hypotheses About r 560 Practice Problems 575 The Least Squares Regression Line, a.k.a. the Line of Think About It . . .  587 Best Fit 561 References 588 Connecting the Dots 561 USING SPSS AND R FOR CORRELATION AND Finding the Regression Line 562 REGRESSION 589 THINK ABOUT IT . . . What Does Perfect Mean? 565 Pearson Correlation Coefficient in SPSS 589 Some Cautionary Notes 566 Conducting a Regression in SPSS 591 Linear versus Curvilinear Relationships 566 Pearson Correlation Coefficient in R 595 Truncated Range 567 Conducting a Regression in R 596 CHAPTER 14 THE CHI-SQUARE TEST 598 Overview 599 THINK ABOUT IT . . . The Risk Ratio 617 Learning Objectives 599 Nonparametric Tests in Context: Types of Data Everyday Statistics 599 Revisited 619 The Chi-Square Test and Why We Need It 600 Summary 620 Parametric versus Nonparametric Testing 600 Terms You Should Know 621 THE HISTORICAL CONTEXT The Questionnaire 601 Glossary of Equations 621 Making Assumptions 602 Practice Problems 621 The One-Way Chi-Square Test for Goodness Think About It . . .  629 of Fit 604 References 629 The Two-Way Chi-Square Test of Independence 609 CONDUCTING CHI-SQUARE TESTS WITH An Example of the Two-Way Test: Americans’ Belief in SPSS AND R 629 Ghosts by Region 609 A Shortcut 611 Chi-Square Test for Goodness-of-Fit in SPSS 629 A Special Case for Chi Square: The “2 by 2” Design 615 CONTENTS ix bla78953_fm_i-1 ix 11/27/17 08:34 PM