STATISTICS AND DATA ANALYSIS FOR T~E 5E~AVIORAl SCIENCES STATISTICS AND DATA ANALYSIS FOR T~E 5E~AVIORAL SCIENCES DANA S DUNN Moravian College I' I j j \ i f I , , I " I I i Boston Burr Ridge, IL Dubuque,IA Madison, WI New York San Francisco St Louis Bangkok Bogota Caracas Lisbon London Madrid Mexico City Milan New Delhi Seoul Singapore Sydney Taipei Toronto McGraw-Hill Higher Education ~ A Division of The McGraw-Hill Companies STATISTICS AND DATA ANALYSIS FOR THE BEHAVIORAL SCIENCES Published by McGraw-Hill, an imprint of The McGraw-Hill Companies, Inc_, 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2001 by The McGraw-Hill Companies, Inc All rights reserved No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper 234567890VNHNNH0987654321 ISBN 0-07-234764-3 Vice president and editor-in-chief: Thalia Dorwick Editorial director: Jane E Vaicunas Executive editor: Joseph Terry Marketing manager: Chris Hall Project manager: Susan J Brusch Senior media producer: Sean Crowley Production supervisor: Kara Kudronowicz Coordinator of freelance design: David W Hash Cover/interior designer: Rebecca Lloyd Lemna Senior photo research coordinator: Carrie K Burger Senior supplement coordinator: Jodi K Banowetz Compositor: York Graphic Services, Inc Typeface: 10112 Minion Printer: Von Hoffmann Press, Inc Photo credit Figure 1.5; ©CorbislBettmann / I The credits section for this book begins on page C-1 and is considered an extension of the copyright page Library of Congress Cataloging-in-Publication Data Dunn, Dana Statistics and data analysis for the behavioral sciences / Dana S Dunn -1st ed p cm Includes bibliographical references and index ISBN 0-07-234764-3 Psychometrics Psychology-Research-Methodology I Title BF39.D825 2001 150' l'5195-dc21 ! ) r I ( 00-030546 CIP www.mhhe.com ; / I I / To the memory of my father and grandfather, James L Dunn and Foster E Kennedy "WHAT'S PAST IS PROLOGUE" - THE TEMPEST (ACT II, SC I) ABOUT THE AUTHOR DANA S DUNN Dana S Dunn is currently an Associate Professor and the Chair of the Department of Psychology at Moravian College, a liberal arts and sciences college in Bethlehem, Pennsylvania Dunn received his Ph.D in experimental social psychology from the University of Virginia in 1987, having previously graduated with a BA in psychology from Carnegie Mellon University in 1982 He has taught statistics and data analysis for over 12 years Dunn has published numerous articles and chapters in the areas of social cognition, rehabilitation psychology, the teaching of psychology, and liberal education He is the author of a research methods book, The Practical Researcher: A Student Guide to Conducting Psychological Research (McGraw-Hill, 1999) Dunn lives in Bethlehem with his wife and two children ) i vi J CONTrNTS IN 5Rlri Preface Acknowledgments INTRODUCTION: STATISTICS AND DATA ANALYSIS AS TOOLS FOR RESEARCHERS PROCESS OF RESEARCH IN PSYCHOLOGY AND RELATED FIELDS 45 FREQUENCY DISTRIBUTIONS, GRAPHING, AND DATA DISPLAY 85 DESCRIPTIVE STATISTICS: CENTRAL TENDENCY AND VARIABILITY 133 STANDARD SCORES AND THE NORMAL DISTRIBUTION CORRELATION LINEAR REGRESSION PROBABILITY INFERENTIAL STATISTICS: SAMPLING DISTRIBUTIONS AND HYPOTHESIS TESTING 315 /' ; 177 205 241 273 365 10 MEAN COMPARISON I: THE tTEST 11 MEAN COMPARISON II: ONE-VARIABLE ANALYSIS OF VARIANCE 411 12 MEAN COMPARISON III: TWO-VARIABLE ANALYSIS OF VARIANCE 459 13 MEAN COMPARISON IV: ONE-VARIABLE REPEATEDMEASURES ANALYSIS OF VARIANCE 499 14 SOME NONPARAMETRIC STATISTICS FOR CATEGORICAL AND ORDINAL DATA 523 15 CONCLUSION: STATISTICS AND DATA ANALYSIS IN CONTEXT 563 vii Contents in Brief viii Appendix A: Basic Mathematics Review and Discussion of Math Anxiety A-I Appendix B: Statistical Tables B-1 Appendix C: Writing Up Research in APA Style: Overview and Focus on Results C-l Appendix D: Doing a Research Project Using Statistics and Data Analysis: Organization, Time Management, and Prepping Data for Analysis D-l Appendix E: Answers to Odd-Numbered End of Chapter Problems E-l Appendix F: Emerging Alternatives: Qualitative Research Approaches F-l References R-l Credits CR-l Name Index NI-l Subject Index SI-l ! ) CONTENTS Preface xxi Acknowledgments xxvi Reader Response xxviii ) INTRODUCTION: STATISTICS AND DATA ANALYSIS AS TOOLS FOR RESEARCHERS DATA BOX 1.A: What Is or Are Data? Tools for Inference: David L.'s Problem College Choice College Choice: What Would (Did) You Do? Statistics Is the Science of Data, Not Mathematics Statistics, Data Analysis, and the Scientific Method Inductive and Deductive Reasoning 10 Populations and Samples 12 Descriptive and Inferential Statistics 16 DATA BOX 1.B: Reactions to the David L Problem Knowledge Base Discontinuous and Continuous Variables DATA BOX 1.C: 18 19 20 Rounding and Continuous Variables Writing About Data: Overview and Agenda Scales of Measurement 24 Nominal Scales 22 23 25 Ordinal Scales 26 Interval Scales 27 Ratio Scales 28 Writing About Scales Knowledge Base 29 31 Overview of Statistical Notation 31 What to Do When: Mathematical Rules of Priority The Size of Numbers is Relative Mise en Place 39 DATA BOX 1.D: 34 38 ix x Contents About Calculators 39 Knowledge Base 40 PRO.JECT EXERCISE: Avoiding Statisticophobia Looking Forward, Then Back 41 Summary 42 Key Terms 42 Problems 42 40 PROCESS OF RESEARCH IN PSYCHOLOGY AND RELATED FIELDS 45 The Research Loop of Experimentation: An Overview of the Research Process 45 Populations and Samples Revisited: The Role of Randomness 48 Distinguishing Random Assignment from Random Sampling 48 Some Other Randomizing Procedures 50 Sampling Error 52 Knowledge Base 53 DATA BOX 2.A: Recognizing Randomness, Imposing Order 54 Independent and Dependent Variables 54 Types of Dependent Measures 58 Closing or Continuing the Research Loop? 60 DATA BOX 2.B: Variable Distinctions: Simple, Sublime, and All Too Easily Forgotten 61 The Importance of Determining Causality 61 DATA BOX 2.C: The "Hot Hand in Basketball" and the Misrepresentation of Randomness 62 Operational Definitions in Behavioral Research 63 Writing Operational Definitions 64 Knowledge Base 64 Reliability and Validity 65 Reliability 66 Validity 67 Knowledge Base 69 Research Designs 70 Correlational Research 70 Experiments 72 Quasi-experiments 74 DATA BOX 2.D: Quasi-experimentation in Action: What to Do Without Random Assignment or a Control Group 75 Knowledge Base 76 PRO.JECT EXERCISE: Using a Random Numbers Table 77 Looking Forward, Then Back 81 Summary 81 Key Terms 82 Problems 82 FREQUENCY DISTRIBUTIONS, GRAPHING, AND DATA DISPLAY 85 What is a Frequency Distribution? 87 81·2 Subject Index Carryover effect, 393-394 Categorical variables, chi-square test of independence of, 534-538 Causality ANOVA for examination of, 425-426 correlation and, 71, 207-209, 221-222,232 importance of determining, 61-63 making conclusions about, 565 Cell size, for one-variable repeatedmeasures ANOVA, 508 Census sampling, 52 Central limit theorem, 322-323 Central tendency, 134-150 average and, 135-136 choosing measure to use, 132, 148, 149 mean and See Means median and, 144-145, 148, 149 mode and, 145-147, 148, 149 shapes of frequency distributions and, 147-148 utility of, 147-150 writing about, 149-150 Chartjunk, 124 Chi-square test, 528-541 for goodness-of-fit, 529-534 for independence, 534-540 statistical assumptions of, 529 supporting statistics for, 522 two-tailed critical ratios of, B-12 writing about, 533-534, 539-540 Clinical decision making, intuition versus prediction in, 253 Cluster sampling, 51, 52 Coefficient of determination, 222, 257-258 Coefficient of nondetermination, 222-224,257-258 College choice, statistical analysis for, 6-8, 18 Comparison groups, 75 Comparisons, 439 pairwise, 421 post hoc, 439-445 Computer data analysis, 572-573 Conceptual replication, 47 Conditional probabilities, 290-291, 292-293 Confidence intervals, 327 defining using standard error of the mean, 327-329 for one-sample t tests, 375-376 Confounded variables, 73-74 Conjunction fallacies, 288-289 Consistency, correlation as, 228-229 Constants, Constructs, hypothetical, 65-66 Construct validity, 67, 69 Content analysis, F-IO-I2 Continuous variables, 2, 20-21 Contrast analysis, 447-451 Contrasts, 448 as carryover effect, 394 Control, illusion of, 54-55 probability and, 304 Convenience samples, 49, 52 Convergent validity, 68, 69 Cooking data, 575 Correlated research designs, hypothesis testing with See t test for correlated (dependent) groups Correlation, 205-238 causation and, 71,207-209, 221-222,232 linear relationships and, 212-216 negative, 70-71, 211, 212-213 positive, 70, 211, 212, 213-214 as reliability, 228-229 significant, 347-349 signs of, 212 spurious, 204, 224 strength of, 212 third variable problems and, 208-209 writing about, 226-227 zero, 71,211-212,214,215 Correlational designs, 70-72 Correlation coefficients Pearson See Pearson (productmoment) correlation coefficient (Pearson r) Spearman rank order, 551-555 Correlation matrices, 230, 231 Counterbalancing, 394-395 Covariance, calculating, 217-218 Cramer's V statistic, 539 Criterion variable, 242 Critical region, 339 Critical values, rejecting null hypothesis and,338-340 Cumulative frequency distributions, 116-117 Cumulative percentages, 117-118 I!:Il Data, definition of, 2, Data analysis, computerized, 572-573 fraudulent, false, or misleading, 576 prepping data for, D-3-4 statistics versus, 4-5 Data display See Graphs; Tables Datum, 2, Deductive reasoning, 11 Definitions descriptive, 63, 64 operational, 63-64 Degrees of freedom, 341-343 Dependent events, multiplication rule for, 272, 293 Dependent groups, t test for See t test for correlated (dependent) groups Dependent variables, 44, 57-60,61 degree of association between independent variable and, 389-392,443,445,511-512 precision of, power and, 353 timing of measure of, variability and, 168 types of, 58-60 variability and, 167-168 Descriptive definitions, 63, 64 Descriptive statistics, 16-17 See also Central tendency; Means; Standard deviation; Variability Deviation IQ scores, 195 Directional hypotheses, 333 Discontinuous variables, 2, 20 Discounting, F-17 Discriminant validity, 68, 69 Discussion section of paper, C-6 Distribution of sample means, 319-322 Distributions See Frequency distributions Division, review of, A-3-4 Division symbol, 25 Dracula effect, 576 IE Economics, linear regression in, 262 Education linear regression in, 262 statistics and data analysis and, 573-574 Effect size for correlated research groups, 399-400 for F ratio, 442-443 one-variable repeated-measures ANOVA and, 511-512 power and, 354-355 ttest and, 388-389, 390-391 two-variable (two-factor) ANOVA and,486-487 S~·3 Subject Index Empiricism, Episodic narratives, F-14-15 Error experimentwise, 421-423, 424-425 inferential, 314, 349-351 See also Type I error random, power and, 354 standard See Standard error of the mean; Standard error; Standard error of estimate Error sum of squares, 256-257 Error term, 379-383 Error variance, 251, 415-416 reduction in correlated groups designs, 395-396 Estimated omegA-squared, 391-392, 444-445 for two-variable (two-factor) ANOVA, 487-488 Estimation interval, 316-317 point, 317 of population parameters, 317-318 of variance and standard deviation, 162-165 Ethical standards, statistics and data analysis and, 575-578 Ethnography, F-12-14 Evaluation bias, 232 Expected value, 320-321 Experimental designs, 73-74 Experimental hypotheses, 334 Experimentation, research loop of, 46-48,60-61 Experiments, 47 Experimentwise error, 421-423, 424-425 Explained variance, 256-258 Exploratory data analysis (EDA), 104-109 line box tally system and, 108-109 stem and leaf diagrams and, 105-108 Exponents, review of, A-4-5 External validity, 68, 69 If Face validity, 67-68, 69 Factorial designs, 462-463 Fatigue, as carryover effect, 394 F distribution, 4l3, 417-418 critical values of, B-8-10 Figures, in papers, C-6-7, 84 Focus groups, F-15-17 Forging data, 575 F ratio degree of association between independent and dependent variables and, 443, 445 effect size for, 442-443 Frequency distributions, 87-104 bimodal, 112 central tendency and, 147-148 cumulative, 116-117 graphing, 97-104 grouped, 92-95 kurtosis and, 113 normal, 111-112 obtaining probabilities from, 283 percentages and, 91-92 proportions and, 90-91 relative, 92 shapes of, 84, 147-148 skewed, 112-1l3 standard deviations of, 159-160 true limits and, 95-96 z scores and, 184-185 Frequency polygons, 100-102 G Gambler's fallacy, 275-276, 277 Gender bias, in writing, C-3 Generalizability, of results, 566 General linear model (GLM) for one-way ANOVA, 431 for two-variable (two-factor) ANOVA,476 Goodness-of-fit, chi-square test for, 529-534 Grammar, checking, C-1-2 Grand mean, 414 Graphs, 97-104 bar, 98 biased or misleading, 102-104, 105, 106 constructing, 123-124 frequency polygons, 100-102 histograms, 99-100 of standard errors of the mean, 330,331 x and y axes of, 97 Greater than or equal to symbol, 25 Greater than symbol, 25 Grouped data, calculating means from, l38-140 Grouped frequency distributions, 92-95 Groups, qualitative research in, F-15-17 IHl Habituation, as carryover effect, 394 Haphazard samples, 49, 52 Heterogeneous distributions, 159 Heteroscedasticity, 254 Heuristics, 571 of representativeness, 55-56 Histograms, 99-100 Hoaxing, 575 Homogeneous distributions, 159 Homoscedasticity, 254-256 Honestly significant difference (HSD) test, 440-442, 510-511 Hypotheses, 9-10, 317, 331-336 alternative (experimental), 334 criteria for, 332 directional and nondirectional, 333 identifying for ANOVA, 427-429 null,333-336 for one-variable repeated-measures ANOVA,503 statistical, 332, 427-429 for two-variable ANOVA, 469-472 writing about, 355 Hypothesis testing, 318-319, 343-349 See also specific statistical tests significance of rand, 347-349 steps for, 344 z test and, 344-347 Hypothetical constructs, 65-66 Ideology, statistics as, 567-568 Illusion of control, 54-55 probability and, 304 Independence chi-square test for, 534-540 probability and, 276 Independent groups t test, 366, 378-393 conceptual model for, 383-384 degree of association between independent and dependent variables and, 389-392 effect size and, 388-389, 390-391 estimating standard error of difference between means and, 379-383 writing up results of, 387-388 Independent variables, 44,56-57,61 degree of association between dependent variable and, 389-392,443,445,511-512 salience of, power and, 353-354 timing of measurement of dependent variable following, variability and, 168 variability and, 167, 168 Individual differences, 384 SI-4 Subject Index Inductive reasoning, 10-11 statistical heuristics and, 571 Inferential errors, 349-351 type I, 314,349-350,420-425 type II, 349, 350-351 Inferential statistics, 17-19 See also Hypothesis testing; Sampling distributions Interaction, 467-469, 471 Intercept, 243 Internal consistency measures of reliability, 229 Internal validity, 68, 69 Interpolation, 450 Interpretation importance of, 568-569 main effects and interaction and, 471 Interquartile range, 153-154 Interval estimation, 316-317 Interval scales, 24, 27-28, 29 measures of association under, 233 Introduction of paper, C-5-6 IQ scores, 194-195 deviation, 195 standardization of, 181-182 Item-total reliability, 229 I Joint probability, 285-286 K Kurtosis, 113 L Large numbers, law of, 169, 322-323 Latin square designs, 514 Law of averages, 276 Law of large numbers, 169, 322-323 Learning, as carryover effect, 394 Least squares method, for regression, 245-249 Least squares principle, for mean, 170-171 Leniency bias, 232 Leptokurtic distributions, 113 Less than or equal to symbol, 25 Less than symbol, 25 Levels of significance, 314, 336-340 power and, 354, 377 Life history, F-14-15 Limits, true, 21, 23 frequency distributions and, 95-96 Linear regression, 240-269 behavioral science applications of, 262 explained and unexplained variation and, 256-258 to mean, 259-261 proper use of, 258 as research tool, 261-263 residual variance and, 251 simple (bivariate) See Simple linear regression standard error of estimate and, 251-256 writing about results of, 263 Linear relationships, 212-214 limited, 224-225 Line box tally system, 108-109 Loci, method of, 151-152 Lower true limit, 21 M MAGIC criteria, 342, 343, 388, 569 Main effects, 466, 471 Management, linear regression in, 262 Mann-Whitney Utest, 541-547 critical values of U statistic and, B-13-14 for larger samples, 546-547 writing about results of, 547 "Man-who" statistics, 278 Marginal probabilities, 292 Matched groups design, 393 See also t test for correlated (dependent) groups Math anxiety, A-9 Mathematical rules of priority, 34-38 Mathematical symbols, 24-25 Mathematics basic, references about learning, A-8-9 statistics versus, 8-9 Mathematics review, A-1-8 of absolute values, A-5 of addition and subtraction, A-I-2 of algebraic equations, A-5-8 of division, A-3-4 of exponents and square roots, A-4-5 of multiplication, A-3 of transposition, A-6-7 Mean deviation approach, for calculating Pearson r, 219 Mean differences power of t test and, 377 sampling distribution of, 379-380 standard error of, 379-383 Mean polish, 491-494 Means, 136-144, 148, 149 calculating from grouped and ungrouped data, 138-140 conceptual model for comparison of, 383-384 contrast analysis for comparison of, 447-451 grand, 414 least squares principle for, 170-171 one-variable ANOVA for comparison of See One-variable (one-way) analysis of variance; Onevariable repeated-measures analysis of variance post hoc comparisons of, 439-445 reasons to compare, 367-368 regression to, 259-261 sensitivity to extreme scores, 140-142 standard error of See Standard error of the mean t test for comparison of See t test two-variable ANOVA for comparing See Two-variable (two-factor) analysis of variance weighted, 142 z scores on either side of, area between, 192-193 Measurement, in qualitative research, F-4-9 Measurement scales, 24-31 interval, 2, 24, 27-28, 29, 233 nominal, 2, 24, 25-26, 29, 233 ordinal, 2, 24, 26-27, 29, 233 ratio, 2, 24, 28-29, 233 writing about, 29-31 Measures See also Variables standardization of, 181-186 See also z scores Median, 144-145, 148, 149 Mesokurtotic distributions, 113 Meta-analysis, 564 Method of least squares, for regression, 245-249 Method ofloci, 151-152 Method section of paper, C-6 Mise en place philosophy, 39 Misleading graphs, 102-104, 105, 106 Missing values, D-3-4 Mixed design analysis of variance, 515-516 Mixed research designs, 515-516 Mode, 145-147, 148, 149 Multifactorial designs, 489-490 Multimodal distributions, 112 SI·5 Subject Index Multiple correlation coefficients, 264 Multiple (multivariate) regression, 263-264 Multiplication, review of, A-3 Multiplication rules for probability for dependent events, 272, 293 for independent and conditional probabilities, 272, 287, 289-293 Multiplication symbol, 25 Mutually exclusive events, 286-287 N N, 32,33 Narrative interviews, F-14 Narratives, F-I4-15 Negative correlation, 70-71, 211, 212-213 Negative skew, 113 Nominal scales, 24, 25-26, 29 measures of association under, 233 nonparametric tests and, 526 Nondirectional hypotheses, 333 Nonparametric statistics, 525 Nonparametric tests, 523-559 chi-square test See Chi-square test choosing among, 522 differences from parametric tests, 525-527 guide to, 528 Mann-Whitney Utest, 541-547 Spearman rank order correlation coefficient, 551-555 steps for testing hypotheses using, 530 uses of, 527-528 Wilcoxon matched-pairs signed-ranks test, 547-551 Nonrandom sampling, 51-52 Normal distribution, 111-112, 186-199 approximating with binomial distribution, 301-305 area under curve and, B-3-5, 187-189 writing about, 197-198 z scores and, 190-191 See also z scores Not equal to symbol, 25 Null hypothesis, 333-336 acceptance of, 335 controversy over significance testing of, 564-567 establishing criteria for rejecting, 338-340 MAGIC criteria and, 342, 343 Null results, need for writing up and reporting, 550 Numbers, relative size of, 38 o Observations, number of, 33, 138 Observed scores, 66 Odd-even reliability, 229 Omnibus statistical tests, ANOVA as, 423-424 One-factor analysis of variance, 426-439 example of, 431-439 factors, 427 general linear model for, 431 identifying statistical hypotheses for, 427-429 notation for, 429 repeated-measures See One-variable repeated-measures analysis of variance steps in, 429-430 One-sample ttest, 314, 366, 372-377 confidence intervals for, 375-376 power of, 377 writing about results of, 374-375 One-tailed significance tests, 340-341 power and, 354 One-variable (one-way) analysis of variance, 412-426 comparison with other statistical tests, 423-426 decision to use, 410 error variance and, 415-416 F distribution and, 417-418 risk of type I error with multiple t tests and, 420-423 total variance and, 414-415 treatment variance and, 415-417 t test compared with, 418-420 yielding omnibus F ratio, procedure following, 410 One-variable repeated-measures analysis of variance, 499-514, 516-519 effect size and, 511-512 hypotheses for, 503 procedure following, 498 statistical assumptions of, 502-503 steps in, 503-510 Tukey's HSD test and, 510-511 writing about results of, 512-513 Operational definitions, 63-64 writing, 64 Order of operations, 34-38 Ordinal data, 541 Mann-Whitney Utest for, 541-547 Spearman rank order correlation coefficient for, 551-555 Wilcoxon matched-pairs signed-ranks test for, 547-551 Ordinal scales, 24, 26-27, 29 measures of association under, 233 non parametric tests and, 526 tied ranks and, 544 Ordinate, 97 Organismic variables, 378-379 Outlying scores, correlation and, 225 Oversight bias, 232 p Paired t test See t test for correlated (dependent) groups Pairwise comparison, 421 Papers, writing See Writing Parametric statistics, 525 Parametric tests, differences from nonparametric tests, 525-527 Participant observation, F-12 Pearson (product-moment) correlation coefficient (Pearson r), 207, 209-221 calculating, 216-221 conceptual defmition of, 209-210 critical values for, B-6 factors influencing, 224-226 magnitude of, 222 mean deviation approach to, 219-220 pitfalls in calculating, 220-221 relation to z scores, 216-217 sum of the squares and, 217-221 Peer review, of manuscripts, 179 Percentages (percents), 91-92 cumulative, 117-118 probability related to, 281-282 Percentile ranks, 115 calculating, 118-119 finding scores from, 119-120 finding with z scores, 191-192 of z scores, 176 Percentiles, 115-116 middle, calculating, 120-122 writing about, 122-123 Personnel decisions, linear regression in, 262 Phi coefficient, 539 Physiological measures, 59, 60 Platycurtic distributions, 113 Point estimation, 317 Population parameters, 13, 132 estimation of, 132,317-318 Populations, 12,316 51-6 Subject Index Population standard deviation, 162 Population variance, 155, 161-162 Positive correlation, 70, 211, 212, 213-214 Positive skew, 113 Post hoc tests, 424, 439-445 Power, 351-355 effect size and, 354-355 factors affecting, 314, 351-354 one-sample ttest and, 377 Power analysis, 225-226, 362, 400-402 Practical significance, 337 Predictor variable, 242 Priority, mathematical rules of, 34-38 Probability, 273-311 addition rule for, 272, 285-287 binomial distribution to determine, 299-305,307-310 calculating, 272 classical theory of, 277-281, 282 conditional, 290-291, 292-293 gambler's fallacy and, 275-276, 277 history of, 284 joint, 285-286 marginal, 292 multiplication rule for dependent events and, 272, 293 multiplication rule for independent and conditional probabilities and,272,287,289-293 obtaining from frequency distributions, 283 proportion and percentage related to, 281-282 p values and, 272, 305-307, 377 subjective, 274 writing about, 306-307 z scores and, 198-199,294-299 Probability tests, one-and two-tailed, 340-341 Probability values See p values Problems, in research procedure, accounting for, 566 Proportion, probability related to, 281-282 Proportions, 90-91 Psychology, as science, F-6 Psychometric information, 566 Punctuation, checking, C-1-2 p values, 272, 305-307 power of ttest and, 377 Q Q test, 440-442 Qualitative research, F-1-20 content analysis and, F-10-12 in groups, F-15-17 history of, F-5, F-7 measurement in, F-4-9 narratives and life history and, F-14-15 participant observation and ethnography and, F-12-14 readings on, F-18 rules of thumb for, F-lO skills needed for, F-16 writing about, F-17-19 Quartiles, calculating, 121-122 Quasi-experiments, 74-76 Quota sampling, 51-52 R Random assignment, 49, 52 performing, 80-81 random sampling versus, 44, 48-50 Random errors, power and, 354 Randomization documentation of procedures for, 566 variability and, 167 Randomness, 62 illusion of control and, 54-55 probability and, 275-276, 277 Random numbers tables, B-2, 77-81 random assignment using, 80-81 random selection using, 79-80 sample of, 78-79 Random sampling performing, 79-80 procedures for, 50-51 random assignment versus, 44, 48-50 simple, 14-15,52 Range, 153-154 interquartile, 153-154 restricted, correlation and, 224 semi-interquartile, 153 writing about, 168 Ratio IQs, 194-195 Ratio scales, 24, 28-29 measures of association under, 233 Raw scores, 182 converting to z scores, 176 Reasoning deductive, 11 inductive, 10-11 Record keeping, D-4 Redundancy, in charts and tables, 125 References section of paper, C-6 Region of rejection, 339 Region of retention, 340 Regression linear See Linear regression; Simple linear regression multiple (multivariate), 263-264 Regression analysis, 242 Regression lines, 244-245 Regression sum of squares, 256 Rejection, region of, 339 Relative frequency distributions, 92 Reliability, 66-67 alternate form, 229 correlation as, 228-229 internal consistency measures of,229 item-total, 229 odd-even, 229 split-half, 229 standard error of mean as measure of,328 test-retest, 229 Reliability coefficients, 66-67, 229 Repeated-measures analysis of variance, one-variable See One-variable repeated-measures analysis of variance Repeated measures research designs, 393 See also t test for correlated (dependent) groups Repeated-measures research designs, 500-501 See also One-variable repeated-measures analysis of variance Replacement sampling with, 279-280 sampling without, 280-281 Replication, 47-48 conceptual (systematic), 47 Reports, writing See Writing Representativeness heuristic, 55-56 Research designs, 44,70-76 between-subjects (between-groups), 378 See also Independent groups t test complex, 460-461 correlated (matched) groups (beforeafter; repeated-measures; withinsubjects), 393 See also t test for correlated (dependent) groups correlational,70-72 experimental,72-74 factorial, 462-463 Latin square, 514 matching statistical tests to, 527-528,562 mixed (between-within), 515-516 multifactorial, 489-490 Subject Index quasi-experimental,74-76 repeated measures, 393, 500-50l See also One-variable repeatedmeasures analysis of variance; ttest for correlated (dependent) groups repeated-measures (within-subjects), 500-501 See also One-variable repeated-measures analysis of variance Research loop of experimentation, 46 48,60-61 Research process, 45-82 causality and, 61-63 linking analysis to, 569-570 operational definitions in behavioral research and, 63-65 overview of, 45-48 random assignment versus random sampling in, 48-50 reliability and validity in, 65-69 research designs and, 70-76 sampling error and, 52-53 sampling techniques in, 50-52 variables in, 54-60 Research projects, D-l-4 organizing, D-1-2 prepping data for analysis and, D-3-4 Research reports, writing See Writing Residual values, 491 Residual variance, 251 Results section of paper, C-6, C-7-10 Retention, region of, 340 Revision, in writing, C-2-3 Risk, perception of, 264-265 Robustness of results, 566 of statistical tests, 370 S Sample means, distribution of, 319-322 Samples, 12-16,316-317 census, 52 characteristics of, variability and, 167 cluster, 51, 52 convenience (haphazard), 49, 52 quota, 51-52 random, stratified, 51, 52 reasoning from, systematic, 50-51, 52 Sample size correlation and, 225-226 Mann-Whitney Utest and, 546-547 for nonparametric tests, 526 for one-variable repeated-measures ANOVA,508 powerand,352-353,377 sampling error and, 52-53 variability and, 166-167, 169 Sample space, 277-281 Sample statistics, 13-14, 132 Sample variance, 155, 157-158, 157-168 Sampling distribution critical region of, 339 of difference between means, 379-380 Sampling distributions, 320 Sampling error, 52-53 Sampling without replacement, 280-281 Sampling with replacement, 279-280 Scales of measurement See Measurement scales Scatterplots, correlation and, 212-216 Scholastic Aptitude Test (SAT), 178 decline in scores on, 180 recentering of scores on, 199-201 Scientific method, 9-19 descriptive and inferential statistics and, 16-19 inductive and deductive reasoning and,lO-11 populations and samples and, 12-16 Scores extreme, sensitivity of mean to, 140-142 finding from percentile ranks, 119-120 IQ, 181-182, 194-195 observed, 66 outlying, correlation and, 225 raw, 176, 182 on SAT, 178, 180, 199-201 standard, 182, 197-198 standardization of, 181-186 See also z scores Tscores, 196-197 z scores See z scores Self-report measures, 58-59, 60 Semi-interquartile range, 153 Sensitization, as carryover effect, 394 Sexist language, in writing, C-3 Sigma, 33-34 Sign, of Pearson r, 212 Significance practical,337 statistical See Statistical significance 81·7 Significance levels, 314, 336-340 power and, 354, 377 Significance testing, 336 Significant differences, 336 Significant digits, 22 Simple linear regression, 242-249 computational approaches to, 243-245 method of least squares for, 245-249 predicting X from Yand, 248-249 z score approach to, 242-243 Simple random sampling, 14-15,52 Single-sample t test See One-sample t test Skew, 112-113 Slope, 243 Social welfare, statistics and data analysis and,574-575 Source tables, ANOVA, 433 Spearman rank order correlation coefficient, 551-555 critical values of, B-16 writing about results of, 554 Spelling, checking, C-1-2 Split-half reliability, 229 Spurious correlation, 224 preventing, 204 Square roots, review of, A-4-5 Stability, standard error of mean as measure of, 328 Standard deviation, 158-161, 162-165 area under normal curve and, 187-189 around regression line, 251-256 biases and unbiased estimators of, 162-165 calculating from a data array, 160-161 homogeneity and heterogeneity and, 159-160 population, 162 writing about, 168-170 Standard error, 321-322 of difference between means, 379-383 Standard error of estimate, 251-256 assumptions underlying, 253-256 Standard error of the mean, 321 concrete example of, 326-327 defining confidence intervals using, 327-329 estimating, 324-325 graphical representation of, 330, 331 as measure of stability and reliability of means, 328 SI-8 Subject Index Standardization of measures, 181-186 See also z scores reasons for, 181-182 Standard scores, 182 writing about, 197-198 Statistical analysis, linking to research, 569-572 Statistical functions, programmed into calculators, 165 Statistical heuristics, 571 Statistical hypotheses, 332 Statistical notation, 31-34 Statistical significance, 336-343 critical values and, 338-340 degrees of freedom and, 341-343 one- and two-tailed tests and, 340-341 practical significance versus, 337 Statistical symbols, 137 Statistical tests matching to research designs, 527-528,562 robustness of, 370 writing about results of, 355-357 Statistics, as avoidable ideology, 567-568 data analysis versus, 4-5 example of use of, 5-8, 18 mathematics versus, 8-9 Stem and leaf diagrams, 105-108 Stimuli, 24 Stratified random samples, 51, 52 Studentized range, percentage points of, B-11 Student's t See Independent groups t test; One-sample t test; t test; ttest for correlated (dependent) groups Subjective probability, 274 Subject variables, 378-379 Subpopulations, preexisting, correlation and, 226 Subtraction, review of, A-1-2 Subtraction symbol, 25 Summary tables, ANOVA, 433 Summation rules, 33-34 Sum of squares (SS), 155-157 Pearson rand, 217-221 Supporting statistics, 522, 566 Symbolic interactionism, F-8-9 Symbols mathematical, 24-25 statistical, 31-34, 137 Systematic replication, 47 Systematic sampling, 50-51, 52 T Tableclutter, 125 Tables constructing, 123-125 in papers, C-6-7, 84 t distributions, z distribution related to, 368 Test-retest reliability, 229 Test scores See Scores Theories, 10 Third variable problem, 208-209 Title of paper, C-5 Total variance, 414-415 Transposition, review of, A-6-7 Treatment variance, 415-417 Trimming data, 576 True limits, 21, 23 frequency distributions and, 95-96 T scores, 196-197 See also Wilcoxon matched-pairs signed-ranks test (Wilcoxon 1) t statistic, absolute value of, 376 t test, 365-406 ANOVA compared with, 418-420 assumptions underlying, 369-371 choosing, 364 with correlated research designs See ttest for correlated (dependent) groups critical values of t and, B-7 multiple, risk of type I error with, 420-423 one-sample See One-sample t test relationship between t and z distributions and, 368-369 with two independent samples See Independent groups t test ttest for correlated (dependent) groups, 364,366,393-400 calculating effect size for correlated groups and, 399-400 statistical advantage of correlated groups designs and, 395-396 writing up results of, 399 Tukey's honestly significant difference (HSD) test, 440-442, 510-511 Two-tailed significance tests, 340-341 Two-variable (two-factor) analysis of variance, 459-495 assumptions of, 469 effect size and, 486-487 estimated omegA-squared for, 487-488 example of, 475-488 factorial design and, 462-463 general linear model for, 476 hypotheses for, 469-472 interaction and, 467-469 main effects and, 466 mean polish and, 491-494 procedure following, 456 steps in, 472-475 writing about results of, 488-489 Type I error, 314, 349-350 ANOVA as protection against, 424-425 with multiple t tests, 420-423 Type II error, 349, 350-351 U Unbiased estimates (estimators), 132, 163-165 Unexplained variance, 256-258 Ungrouped data, calculating means from, 138-140 Upper true limit, 21 V Validity, 67-69, 229, 231 construct, 67, 69 convergent, 68, 69 discriminant, 68, 69 external, 68, 69 face, 67-68, 69 internal, 68, 69 Variability, 151-170 calculating using calculators, 165 choosing measure of, 132 factors affecting, 166-168 method of loci and, 151-152 power of ttest and, 377 range and, 153-154 standard deviation and See Standard deviation variance, 155, 157-158 writing about, 168-170 Variables, 6-7, 54-60 categorical, chi-square test of independence of, 534-538 confounded, 73-74 continuous, 2, 20-21 correlation between See Correlation; Pearson (product-moment) correlation coefficient (Pearson r) criterion, 242 defining and explaining relevance of, 566 dependent See Dependent variables discontinuous, 2, 20 SI-g Subject Index independent See Independent variables predictor, 242 representativeness heuristic and, 55-56 subject (organismic), 378-379 symbols for, 32-33 Variance, 155 calculating from a data array, 160-161 error, 251, 415-416 estimators of, 162-165 explained and unexplained, 256-258 population, 155, 161-162 residual, 251 sample, 155, 157-168 total,414-415 treatment, 415-417 writing about, 168 Variance key, on calculators, 165 Variation between-group (between-sample), 383 within-group (within-sample), 383-384 W Weighted means, 142 WIlcoxon matched-pairs signed-ranks test (Wilcoxon 1'),547-551 critical values of T statistic and, B-15 writing about results of, 547-551 Within-group (within-sample) variation, 383-384 Within-subjects design, 393 See also t test for correlated (dependent) groups Within-subjects designs, 500-501 See also One-variable repeatedmeasures analysis of variance Writing, 23-24, C-I-I0 active voice for, C-2 APA style and See American Psychological Association (APA) style audience for, C-2 avoiding gender bias and sexist language in, C-3 basic advice and techniques for, C-I-3 about central tendency, 149-150 charts and tables in, 123-126 about chi-square test results, 533-534, 539-540 about correlated groups t test results, 399 about correlation, 226-227 about correlational relationships, 226-227 about hypotheses, 355 about independent groups t test results, 387-388 about Mann-Whitney Utest results, 546-547 about normal distribution, 197-198 about null results, 550 about one-variable repeated-measures ANOVA results, 512-513 about one-way ANOVA results, 445-446 of operational definitions, 64 peer review of manuscripts and, 179 about percentiles, 122-123 about probability, 306-307 about qualitative research, F-17-19 about regression results, 263 about results of one-sample t tests, 374-375 revising and, C-2-3 about scales of measurement, 29-31 about Spearman rank order correlation coefficient, 554 spelling, grammar, and punctuation and, C-I-2 about standard scores, 197-198 about statistical test results, 355-357 about two-variable (two-factor) ANOVA results, 488-489 about variability, 168-170 about Wilcoxon matched-pairs signedranks test results, 547-551 x X,32-33 x axis, 97 y y, 32-33 yaxis, 97 z z distribution, t distributions related to, 368 Zero correlation, 71, 211-212, 214, 215 z scores, 182-185, 190-199 converting raw scores to, 176 distributions of, 184-185 on either side of mean, area between, 192-193 finding percentile ranks with, 191-192 formulas for calculating, 185-186 linear regression and, 242-243 normal distribution and, 190-191 Pearson ts in relation to, 216-217 percentile rank of, 176 probability and, 198-199,294-299 on same side of distribution, area between, 193-196 z test, 344-347 Subject Index Multiple correlation coefficients, 264 Multiple (multivariate) regression, 263-264 Multiplication, review of, A-3 Multiplication rules for probability for dependent events, 272, 293 for independent and conditional probabilities, 272, 287, 289-293 Multiplication symbol, 25 Mutually exclusive events, 286-287 Null results, need for writing up and reporting, 550 Numbers, relative size of, 38 o S\·5 Spearman rank order correlation coefficient for, 551-555 Wilcoxon matched-pairs signed-ranks test for, 547-551 Ordinal scales, 24, 26-27, 29 measures of association under, 233 nonparametric tests and, 526 tied ranks and, 544 Ordinate, 97 Organismic variables, 378-379 Outlying scores, correlation and, 225 Oversight bias, 232 Observations, number of, 33, 138 Observed scores, 66 Odd-even reliability, 229 Omnibus statistical tests, ANOVA as, 423-424 One-factor analysis of variance, 426-439 example of, 431-439 factors, 427 N p general linear model for, 431 N, 32, 33 Paired t test See t test for correlated identifying statistical hypotheses for, Narrative interviews, F-14 (dependent) groups 427-429 Narratives, F-I4-15 notation for, 429 Pairwise comparison, 421 Negative correlation, 70-71, 211, repeated-measures, See One-variable Papers, writing, See Writing 212-213 repeated-measures analysis of Negative skew, 113 Parametric statistics, 525 Nominal scales, 24, 25-26, 29 variance Parametric tests, differences from measures of association under, 233 nonparametric tests, 525-527 steps in, 429-430 nonparametric tests and, 526 Participant observation, F-12 One-sample ttest, 314, 366, 372-377 Nondirectional hypotheses, 333 Pearson (product-moment) correlation confidence intervals for, 375-376 Nonparametric statistics, 525 coefficient (Pearson r), 207, power of, 377 209-221 writing about results of, 374-375 Nonparametric tests, 523-559 calculating, 216-221 One-tailed significance tests, 340-341 chi-square test, See Chi-square test conceptual definition of, 209-210 power and, 354 choosing among, 522 critical values for, B-6 One-variable (one-way) analysis of differences from parametric tests, factors influencing, 224-226 525-527 variance, 412-426 magnitude of, 222 guide to, 528 comparison with other statistical tests, mean deviation approach to, 219-220 M.