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STATISTICS
Eleventh Edition
Robert S. Witte
Emeritus, San Jose State University
John S. Witte
University of California, San Francisco
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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.
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To Doris
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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322
SLEEP DEPRIVATION EXPERIMENT WITH REPEATED MEASURES
F TEST 324
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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
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369
372
LOST LETTER STUDY 372
STATISTICAL HYPOTHESES 373
2
DETAILS: CALCULATING χ 373
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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
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STATISTICS
Eleventh Edition
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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
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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).
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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.
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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
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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
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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.
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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.
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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.
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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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