Statistical Methods in Education

and Psychology

ALBERT K. KURTZ SAMUEL T. MAYO

Statistical Methods in Education and Psychology

SPRINGER-VERLAG New York Heidelberg Berlin

Albert K. Kurtz, Ph.D. 1810 Ivy Lane Winter Park, Florida 32792, USA

AMS Subject Classifications: 62-01

Samuel T. Mayo, Ph.D. Professor, Foundations of Education School of Education Loyola University of Chicago Chicago, lllinois 60611, USA

Library of Congress Cataloging in Publication Data

Kurtz, Albert Kenneth, 1904-Statistical methods in education and psychology. Bibliography: p. Includes index. 1. Educational statistics. 2. Educational tests and measurements. I. Mayo, Samuel

Turberville, joint author. II. Title. LB2846.K88 370'.1'82 78-15295

All rights reserved.

No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag.

Copyright © 1979 by Springer-Verlag New York Inc.

Softcover reprint of the hardcover 1st edition 1979

987654321

ISBN-13: 978-1-4612-6131-5 DOl: 10.1007/978-1-4612-6129-2

e-ISBN-13: 978-1-4612-6129-2

v

Preface

This book is intended for use in the elementary statistics course in Educa­tion or in Psychology. While it is primarily designed for use in the first semester of a two-semester course, it may also be used in a one-semester course. There are not five or ten competing texts; the number is much closer to fifty or a hundred. Why, then, should we write still another one?

A new statistics text for use in Education and Psychology is, to some slight extent, comparable to a new translation or edition of the Bible. Most of it has been said before-but this time with a difference.

The present writers realize that elementary statistics students know very little about the subject-even the meaning of I is all Greek to them. This text covers the basic course in depth, with examples using real data from the real world. It, of course, contains the usual reference tables and several new ones; it gives the appropriate formulas every time; and it accurately depicts all graphs. It is so comprehensive that if instructors can't find their own special areas of interest covered, then those interests probably don't belong in a basic text.

This book uses simple sentences written in standard English. Except for one or two sentences in this Preface, no dictionary is needed for interpreta-

VI Preface

tion. The present writers have taken their task seriously but themselves lightly. The book is crammed with facts and suggestions but it moves along at a breezy clip. The writers would even have referred to ecdysiasts if they had thought this would have aided in understanding any figures in the text. How to, why to, when to, and (even more important) how not to, are all there; poor techniques are not allowed.

In this text, the authors never utilize obfuscation in the elucidation of the untenability of doctrinaire or hypothetical tenets. They simply say that the results might well be due to chance alone.

Most of what the writers regard as basic material is in the 15 chapters; much of the other material wanted by individual instructors is in the supplements. It is expected that most instructors will wish to assign material carefully selected from several supplements; it is unlikely that many instruc­tors will wish to assign all of the supplements to their classes.

The writers have spent considerable time in locating theoretical, interpreta­tive, mechanical, and computational errors and difficulties. Then we wrote or rewrote the text in order to reduce or eliminate these troubles. The basic goal, of course, was to help the student to decide upon appropriate statisti­cal procedures and then to use them correctly. Some specific examples are:

(a) Instead of giving a rule and then violating it by using the same data for both biserial r and point biserial r, the principle differentiating the two is clearly enunciated and then some 38 variables are individually classified. Further, 9 others that are ambiguous or equivocal are discussed, showing, in each case, under what circumstances each of the two measures (biserial r or point biserial r) is appropriate.

(b) In order to let the student know more clearly why and how various statistics are to be used, two intentionally different examples are often given instead of the usual one.

(c) Because of the large number of computational errors in published research studies, special attention has been given to checking proce­dures throughout the book. A new and very efficient method for checking analysis of variance computations illustrates this change.

(d) Alternate formulas have sometimes been derived in order to get the usual results with less arithmetic. Formula (10.13) for the standard error of phi is one of them. With the two tables which accompany it, the computation is now so simple that the traditional formula no longer needs to be ignored because of computational complications.

(e) Many tables are included. A number of them have been modified to make them of maximal usefulness to the student. These tables include newly derived ones, standard ones, and several designed to illustrate relationships among various statistics.

Preface vii

(f) Where there are several uses for or interpretations of a statistic, an effort has usually been made to present them. The outstanding exam­ple of this is the correlation coefficient. While most texts give two or three, one text gives six interpretations of the meaning of Pearson r. In Chapters 8 and 9, we present fifteen!

We have also done several things designed to make life just a little easier both for the student and for the instructor.

(a) There are many cross-references and most of them are to specific pages rather than to topics or chapters.

(b) Formulas are written so they will be easy to use-not easy to set in type. Thus, we never enclose material in parentheses which is then raised to the one-half power; we simply use a square root sign. Another example is illustrated by the derivation and use of our new formula (4.2) instead of the usual formula (4.1) for the normal proba­bility curve.

(c) This is minor, but we even introduced a new symbol, "P , for the ordinate of the normal probability curve.

(d) A Workbook has been prepared for any instructors who may care to use it.

It is trite for authors of elementary statistics texts to deny that previous mathematics is a prerequisite. The same claim is made for this text: most of the book can be understood by an average student who has had either one year of high school algebra or none, but, of course, much of the book will be more readily comprehensible to the student with a more extensive background in mathematics.

Despite the comprehensive coverage of elementary statistics referred to earlier in this Preface, the aim throughout has been to place primary emphasis on those principles, methods, and techniques which are, and of right ought to be, most useful in research studies in Education and Psychology.

The instructor will note a greater than usual emphasis on standard errors (or other tests of significance) and on large numbers of observations. This, briefly and bluntly, is because the writers are completely fed up with conclusions which start out with, "While not significant, the results are suggestive and indicate ... " Let's get results that are significant and let's interpret them properly.

The writers were aided by many different persons in the preparation of this manuscript. We wish to acknowledge our indebtedness and express our gratitude to a few of the most important of them.

viii Preface

First, we owe a great deal to our statistics professors who taught us much of what we know.

Next, we are grateful to our students, our colleagues, and our competi­tors-from all of whom we have learned much about how to present our material and, on occasion, how not to present it.

Most of all, our greatest thanks are due to:

Elaine Weimer, a former editor and statistician, who added tremendously to the clarity of the text by revising or rewriting most, we hope, of the obscure portions of it.

Melba Mayo, who provided stimulation and encouragement to both the writers, assisting us in many ways in bringing the book to fruition.

Nathan Jaspen, who provided us with valuable suggestions for improve­ment of earlier drafts of the manuscript.

We are, of course, grateful to the many authors and publishers who have graciously permitted us to use their copyrighted materials. More specific acknowledgements are made in the text near the cited materials.

We shall end this preface with the hope that this text will provide you with "everything you always wanted to know and teach about statistics but were afraid to ask." What more can we say?

October, 1978 ALBERT K. KURTZ

SAMUEL T. MAYO

Contents

PREFACE

CHAPTER 1 THE NATURE OF STATISTICAL METHODS

General Interest in Numbers The Purposes of Statistical Methods Preview of This Text Rounding, Significant Figures, and Decimals

Supplement 1 Rounding

CHAPTER 2 AVERAGES

Raw Data The Mean Computed from Raw Data Grouping of Data The Mean Computed from Grouped Data The Median Choice of Mean or Median The Histogram and the Frequency Polygon

ix

v

1

1 3 3 8

8 8

15

15 15 17 22 27 31 33

x

Summary

Supplement 2 Other Averages Proof that Error Due to Grouping Is Small Cumulative Graphs Bar Diagrams

CHAPTER 3 THE STANDARD DEVIATION

Need for a Measure of Variability Formula for the Standard Deviation Computing the Standard Deviation with Grouped Data Standard Deviation of a Finite Population Standard Scores Other Measures of Dispersion

Summary

Supplement 3 How to Square Numbers Methods of Extracting Square Roots How to Check Square Roots How to Compute and Check the Standard Deviation Sheppard's Correction for Coarseness of Grouping Some Other Measures of Dispersion Types of Standard Scores

CHAPTER 4 NORMAL PROBABILITY CURVE

The Nature of the Normal Probability Curve The Ordinates of the Normal Probability Curve Binomial Coefficients and the Normal Probability Curve Applications of the Binomial Coefficients The Area under the Normal Probability Curve

Summary

Contents

36

36 36 42 42 44

46

46 48 52 58 61 64

64

65 65 67 73 75 76 78 80

82

82 84 89 92 97

108

Supplement 4 108 Simplifying the Equation for the Normal Curve 108 Fitting a Normal Probability Curve to Any Frequency Distribution 109

CHAPTER 5 STATISTICAL INFERENCE 115

Dependability of Figures 115 Speculation 117

Samples and Populations 118 Sampling Distributions and Standard Error of the Mean 119 The t Test for Means 121 Levels of Significance 124 The z Test for Means 125 Point and Interval Estimates 127 Statistical Inference 129 Sampling Distribution and Standard Error of the Median 130 Sampling Distribution and Standard Error of the Standard Deviation 132 Hypothesis Testing 134

Summary 140

Supplement 5 140 Stability of the Median 140 Standard Error of a Proportion 143

CHAPTER 6 PERCENTILES AND PERCENTILE 145 RANKS

Percentiles 146 Percentile Ranks 146 Computation of Percentiles 147 Percentiles and Percentile Ranks Compared 150 Deciles 152 Quartiles 153 Standard Error of a Percentile 153

Summary 156

Supplement 6 156 Method of Obtaining Percentile Ranks for Grouped Data 156 Measures of Variability Based upon Percentiles 158 The Range 160 Relative Value of Measures of Variability 161

CHAPTER 7 SKEWNESS AND TRANSFORMED 164 SCORES

Skewness 164 Kurtosis 171 Transformed Scores 179 The Description of Frequency Distributions 183

Summary 185

Supplement 7 185 Additional Measures of Skewness and Kurtosis 185

xii Contents

CHAPTER 8 PEARSON PRODUCT MOMENT 192 COEFFICIENT OF CORRELATION

Definition of Pearson r 193 Plotting a Scatter Diagram 195 Illustrations of Pearson r' s of Various Sizes 197 Published Correlation Coefficients 211 Some Characteristics of Pearson r 229 Computing r Without Plotting the Data 232 Plotting the Data and Computing r from the Scatter Diagram 243 The z' Transformation and Its Standard Error 248 Assumptions upon Which Pearson r Is Based 252 Interpretation of Pearson r 254

Summary 260

Supplement 8 262 Other Formulas for Pearson r 262 Alternate Ways to Test the Significance of an Obtained Pearson r 264 Reliability and Validity 264

CHAPTER 9 REGRESSION EQUATIONS 278

The Purpose of a Regression Equation 278 Formulas for Regression Equations 279 The Use of Regression Equations 282 The Graphic Representation of Prediction 290 A Second Illustration of Regression Equations 292 Further Interpretations of r 298

Summ~ ~8

Supplement 9 308 Making a Large Number of Predictions ~8

CHAPTER 10 MORE MEASURES OF 311 CORRELATION

Why Other Correlations 311 Biserial r 312 Multiserial Correlation 328 Point Biserial r 328 Classification of Dichotomous Variables 335 Tetrachoric r 340

Contents xiii

Phi 346 Interrelations among rbis, r pb, r t. and cp 353

Summary 355

Supplement 10 356 Cosine Pi Correlation Coefficient 356 Rank Correlation Coefficient 358

CHAPTER 11 CHI SQUARE 362

Nature of Chi Square 362 Illustration of Correct Use of Chi Square 364 Sources of Error in Chi Square 366 Chi Square in the General Contingency Table 369 The Exact Test of Significance in 2 x 2 Tables 374 Use of Chi Square in Curve Fitting 378 Advantages and Disadvantages of Chi Square 383

Summary 384

Supplement 11 384 Unique Characteristics of Chi Square 384

CHAPTER 12 NONPARAMETRIC STATISTICS 392 OTHER THAN CHI SQUARE

The Purposes of Nonparametric Statistics 392 The Sign Test 394 The Runs Test 396 The Median Test 399 The Mann-Whitney U Test 402 Which Nonparametric Statistic Should Be Used? 405 Other Nonparametric Statistics 406

Summary 407

CHAPTER 13 SIMPLE ANALYSIS OF VARIANCE 408

Why Not the t Test 408 The Basis of Analysis of Variance 409 A First Example 412 Assumptions 416 Checking 417

xiv Contents

Technicalities 419 A Second Example 421

Summary 426

Supplement 13 428

CHAPTER 14 STANDARD ERRORS OF 432 DIFFERENCES

Standard Error of Any Difference 433 Standard Error of the Difference between Means 434 Standard Error of the Difference between Standard Deviations 440 The Standard Error of Other Differences 444

Summary 444

Supplement 14 446

CHAPTER 15 REORIENTATION 447

Moments 447 Correlation 449 Popular Concepts 450 Future Courses 450 U sing Statistics 453

Statistical Methods in Education

and Psychology