Introductory Statistics for the Behavioral Sciences E-Book

R. Brooke Lea

• Herausgeber: John Wiley & Sons
• Kategorie: Geisteswissenschaft
• Sprache: Englisch

Seitenzahl: 1184

Veröffentlichungsjahr: 2025

Beschreibung

The accessible, hands-on statistics textbook that behavioral science students and instructors trust

Introductory Statistics for the Behavioral Sciences is a respected, practical textbook that offers carefully crafted exercises to support the teaching and learning of statistics. This revised eighth edition presents all the topics students in the behavioral sciences need in a uniquely accessible format, making statistics feel relevant and approachable. With fictitious yet realistic examples that reappear throughout the chapter, students can follow a continuous narrative that helps them engage with and internalize the content.

User-friendly integration with SPSS software enables readers to gain hands-on experience with the application of theoretical concepts. Exercises at the end of each chapter, with additional practice in the online study guide, give students the repetition they need to fully comprehend the material. After working through this textbook, students will understand, not only the what, but also the why of statistical analysis.

• Get plain-English explanations of statistical concepts and procedures important in behavioral sciences research
• Learn from relatable examples and exercises focused on psychology, sociology, and other behavioral science
• Work through well-crafted exercises designed to enhance your understanding of the material
• Get clear instructions on how to perform statistical procedures with the industry-standard SPSS software

Online resources for instructors include a test bank, chapter quizzes, and PowerPoint slides. Introductory Statistics for the Behavioral Sciences also includes a student website containing additional basic math coverage, math review exercises, a study guide, a set of additional SPSS exercises, and downloadable data sets.

Leseprobe

Introductory Statistics for the Behavioral Sciences

Eighth Edition

R. Brooke LeaBarry H. Cohen

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Library of Congress Cataloging-in-Publication Data

Names: Lea, R. Brooke author | Cohen, Barry H., 1949‐ authorTitle: Introductory statistics for the behavioral sciences / R. Brooke Lea, Barry H. Cohen.Description: Eighth edition. | Hoboken, New Jersey : Wiley, [2026] | Series: Introductory statistics for the behavioral sciences | Includes index.Identifiers: LCCN 2025030350 (print) | LCCN 2025030351 (ebook) | ISBN 9781394234738 paperback | ISBN 9781394234752 adobe pdf | ISBN 9781394234745 epubSubjects: LCSH: Social sciences–Statistical methods | Psychometrics | Sociology–Statistical methods | Educational statisticsClassification: LCC HA29 .W445 2025 (print) | LCC HA29 (ebook)LC record available at https://lccn.loc.gov/2025030350LC ebook record available at https://lccn.loc.gov/2025030351

Cover design: WileyCover image: Courtesy of Tom Navratil and Dan Navratil

Brooke dedicates this book to Emily and Jackson, the two parameters that keep him Normal,and Barry dedicates the book to Leona and the newest additions to our family: Maya, Leo, Jackson, Liv, and Roman

Preface

The eighth edition of this textbook has been a delight and honor to create. The book is one of the longest‐running statistics texts in the behavioral sciences. (This title has been in print continuously since 1971.) In creating this revision, we tried to maintain the original purpose of the text as defined by Joan Welkowitz and Jacob “Jack” Cohen, and expressed in the preface to the second edition: “to introduce and explain statistical concepts and principles clearly and in a highly readable fashion, assuming minimal mathematical sophistication, but avoiding a ‘cookbook’; approach to methodology.”

At the same time that we worked to uphold the original mission of this classic text, we have made substantial improvements to nearly every chapter. In addition, we have added a new chapter that presents an accessible introduction to the evolving and controverial role that null hypothesis testing plays in behavioral research, along with newer methods (e.g., bootstrapping and Bayesian statistics) that have gained popularity and importance since the last edition. In this new chapter—and throughout the book—we have adopted a more “comprehension‐based” approach that emphasizes how an understanding of statistical concepts informs students’ comprehension of primary sources of research. Borrowing the comprehension vs. production distinction from psycholinguistics, students of behavioral science use their statistical knowledge to comprehend (e.g., read and understand) research at least as much as they use it to produce statistical output (e.g., perform statistical tests). Our approach, especially for more advanced topics, is to ensure that readers of this book understand statistical techniques at a level that will be useful in a variety of educational and experiential contexts.

We also took a hard look at the examples that are used in the book and updated them to provide a more diverse and inclusive perspective on behavioral science. We have made substantial changes to the data set used for exercises by replacing variables that some students might find triggering (e.g., math phobia) with those that are more approachable.

We listened to feedback from students and adopters of the previous edition, who commented that once students have mastered the fundamentals of computing means and standard deviations by hand, the practice becomes needlessly tedious when exercises focus on more complex computations such as t tests, Pearson r, and ANOVA. Accordingly, we now supply summary statistics (means and SDs) so that students can focus their efforts on statistical concepts they are working to master. Of course, we also supply the raw data in case instructors would prefer students to continue to drill on computing descriptive statistics step by step, or use statistical calculators or spreadsheets.

As always, we emphasize the through‐lines that link newer topics with those that came before them in the text. We view this conceptual connective tissue as an important facilitator of learning content that, while divided into discrete topics and chapters, is deeply interconnected. Accordingly, we now provide specific page numbers when referring to formulas, tables, or figures presented earlier in the book to encourage students to refer back to familiar content that reinforces these connections.

Naturally, we took this opportunity to correct any mistakes we and others had noticed in previous editions, and to add some explanatory sentences for the concepts that we know our students have struggled with in the past.

As with the seventh edition of this text, the supplementary materials of this edition will overlap with those for the fourth edition of Barry Cohen’s graduate‐level statistics text, Explaining Psychological Statistics (https://www.wiley.com/en‐us/Explaining+Psychological+Statistics%2C+4th+Edition‐p‐9781118652145), also published by John Wiley & Sons.

There are several structural changes in this edition that are worth emphasizing, as described next.

1. Adding Page Numbers When Referring to Prior Material in the Text

One of our chief strategies for making statistics understandable is to point out the connections between related concepts; mastering behavioral statistics is easier when students can use prior knowledge to help scaffold new material. We have facilitated this process by providing both the name of the prior element and the page on which it appears (e.g., Formula 3.2, page 51). We hope this change will encourage students to appreciate the extent to which statistical knowledge is cumulative.

2. Reduced Emphasis on Rarely Used Procedures and Formulas

As the field advances and makes room for new techniques (see the new Chapter 12), older, rarely used procedures can be relegated to more specialized texts. Examples include linear interpolation procedures for percentiles and cumulative frequency distributions, and computational formulas that are only useful for processing a large quantity of numbers. Learning to compute the sum of squares by hand, for example, is mainly useful as a way for students to understand how examining deviations from the mean can lead to a useful descriptive measure of dispersion. The computational formulas for SS, by contrast, hold no pedagogical value and only exist to facilitate arriving at the correct value of SS with large amounts of data. The ubiquity of automated computation, however, obviates the need for such formulas and techniques.

3. Updating the Computer Exercises, SPSS Sections, and New Bridge to R

We revised Ihno’s data set to create Jackson’s data set for this edition by changing the variables so that they lead to more interesting exercises. We also took this opportunity to update our Bridge to SPSS sections to reflect changes in the most recent versions of SPSS. (The version current during the writing of this edition is 29.) In several chapters we have now included screenshots to illustrate the main SPSS dialog boxes commonly used by researchers to perform the analyses described in this text. In some cases, we have also included results boxes from the output that SPSS produces for particular analyses, in order to draw connections between the terms used by SPSS to label its output and the corresponding (sometimes different) terms used in this text. Furthermore, we now include “Bridge to R” sections as part of the online Supplementary Materials, which will help students using RStudio to successfully complete our Computer Exercises.

4. New Chapter on Newer, More Advanced Statistical Techniques

The new Chapter 12, titled “Beyond Null Hypothesis Testing,” addresses some important topics in behavioral statistics that have emerged since our last edition. We begin by reviewing some of the more trenchant critiques of null hypothesis testing, along with some cogent rebuttals that are sometimes omitted in treatments of this debate. A concrete analogy involving disease detection illustrates what a p value can and cannot tell you. This discussion leads to an introduction to “Robust Statistics” that includes a review of bootstrapping techniques along with the important role that nonparametric statistics can play in behavioral science. Every student should learn about the problems introduced by p Hacking, HARKing, and the file drawer problem. We explain the dangers associated with these practices, and how easily researchers can inadvertently find themselves falling into such traps. That discussion naturally leads into a treatment of the replication crisis in behavioral research, along with some proposed solutions. The last topic in this new chapter introduces alternatives to null hypothesis testing, including the “new” statistics, which emphasizes the interpretation of effect sizes and confidence intervals. Finally, we include an appendix to this chapter that serves as a primer to the use of Bayesian statistics. Overall, this chapter presents these advanced topics and debates in an accessible, friendly manner.

5. Updated Ancillaries

We have updated the Student Companion page on the Wiley website for the new edition, which can be found at: www.wiley.com/go/ISBS8e

The Student Companion Site includes the following items:

Study Guide:

A lively chapter‐by‐chapter review of the text with additional exercises and answers. Created by graduate students who recently served as teaching assistants for statistics, it provides another perspective on the material presented in this text.

PowerPoint Slides:

Expanded and updated for this new edition, these slides provide convenient summaries of the important points of each chapter, and can help instructors to organize their lectures around the key concepts for each statistical topic.

We have also completely updated the Wiley Instructor Companion Site for the eighth edition, which can also be found at: www.wiley.com/go/ISBS8e

In addition to the two items on the student site, the Instructor Website includes the following items:

Instructor’s Manual:

Step‐by‐step answers to all of the computational exercises in the text.

Test Bank:

Multiple choice questions, both conceptual and computational, that can be used to create quizzes to assess the students’ mastery of each chapter in the eighth edition. The test bank includes questions for

Chapter 12

that can be used for testing or to spur discussion of issues concerning the use of different statistical approaches.

R. Brooke LeaBarry H. Cohen

Acknowledgments

Thanks are due to our many encouraging friends and relatives, to all the colleagues and adopters who made many useful comments on previous editions, and especially to Columbia University Professor Katherine Fox‐Glassman. Professors Steve Guglielmo and Annie Pezzala at Macalester College have been inspirational co‐conspirators in convincing psychology majors that learning statistics can be a (surprisingly) exciting and rewarding voyage. Once again, we want to take this opportunity to thank our many students who, throughout the years, have provided invaluable feedback on our teaching of statistics, as well as on earlier editions of this text and its accompanying materials. Special thanks go to the recent survivors of the “RIP” sequence at Macalester College for their trenchant suggestions for improving the text.

This latest edition surely owes its very existence to Patricia Rossi (former Executive Editor), Darren Lalonde (current Acquisitions Editor), and Nathanael Jude Mcgavin (Managing Editor) at John Wiley & Sons, whose support and prodding have ensured that this text will now begin its 40th year in continuous print. This new edition has also benefited greatly from the able assistance of Christina Weyrauch (Senior Editorial Assistant), as well as Sundhar Karuthudiyan (Content Refinement Specialist). We are also grateful to those responsible for the attractive look and design of this text’s cover—especially, Tom and Daniel Navratil. We remain ever grateful to Grace Jackson and Samantha Gaies for creating the lively and engaging Study Guide for the previous edition, most of which has survived in the supplementary web material for this eighth edition.

We are more than delighted to acknowledge a substantial debt of gratitude to Jackson Pearlman for numerous invaluable contributions to this edition. In addition to revising the data set (now named after him) that we use for the computer exercises, Jackson was instrumental in helping us update many examples in the text that were of questionable cultural relevance in the mid‐2020s. He also updated and contributed importantly to the material we provide in the Instructor and Student companion sites for this text. Simply put, this edition would not have been possible without Jackson Pearlman.

Finally, we would like to express our gratitude to those who have supported us in all the small ways that matter during our work on this project. Brooke shouts out thanks and love to Emily Fields and Jackson Lea, and Barry does the same for Leona Gizzi.

R. Brooke LeaBarry H. Cohen

Postscript

We cannot close this Acknowledgments section without sadly paying homage to the two departed professors who, along with a colleague now living in Florida, wrote and edited the original text and kept it fresh and alive over the course of several decades. BHC had the pleasure of working closely with the senior author, Dr. Joan Welkowitz, on the previous edition of this text, and it is not possible to overestimate the continued influence of both her and her former co‐author and friend, Jacob Cohen, on this text and on our approach to the teaching of statistics. The one living author of the original text, Robert Ewen, wrote nearly all of the exercises and thought questions you see in this edition. Though many of the exercises and examples in the text have been updated, Dr. Ewen’s careful craftsmanship and editorial skills are still visible in this edition.

R. Brooke LeaBarry H. Cohen

Glossary of Symbols

Numbers in parentheses indicate the chapter in which the symbol first appears.

a

YX

Y

‐intercept of linear regression line for predicting

Y

from

X

(10)

α

criterion (or level) of significance; probability of Type I error (5)

α

EW

experiment‐wise alpha (14)

α

FW

family‐wise alpha (15)

α

pc

alpha per comparison (14)

b

YX

slope of the linear regression line for predicting

Y

from

X

(10)

β

probability of Type II error (5)

1 − β

power (11)

Cf

cumulative frequency (2)

χ

2

statistic following the chi square distribution (13)

D

difference between two scores (7)

mean of the

D

s (7)

d

effect size involving two samples (7)

d

effect size involving two populations (11)

Df

degrees of freedom (6)

df

Bet

degrees of freedom between groups (13)

df

W

degrees of freedom within groups (13)

df

1

degrees of freedom for factor 1 (15)

df

2

degrees of freedom for factor 2 (15)

df

1×2

degrees of freedom for interaction (15)

δ

delta (11)

η

2

eta squared; effect size in multiple samples (13)

f

effect size involving multiple populations (13)

f

frequency (2)

f

e

expected frequency (18)

f

o

observed frequency (18)

F

statistic following the

F

distribution (13)

g

Hedge’s adjusted effect size involving two samples (7)

H

statistic for the Kruskal–Wallis test (13)

H

0

null hypothesis (5)

H

A

alternative hypothesis (5)

HSD

Tukey’s Honestly Significant Difference (14)

i

case number (1)

k

a constant (1)

k

number of groups (13)

LSD

Fisher’s Least Significant Difference (14)

Mdn

median (3)

MS

mean square (13)

MS

Bet

mean square between groups (13)

MS

W

mean square within groups (13)

MS

1

mean square for factor 1 (15)

MS

2

mean square for factor 2 (15)

MS

1×2

mean square for interaction (15)

μ

population mean (3)

n

number of observations in one of two or more equal‐sized samples (7)

N

T

total number of subjects or observations (1)

N

i

number of observations or subjects in group

i

(13)

ω

2

omega squared; proportion of variance accounted for in a population (10)

π

hypothetical population proportion (6)

p

probability of attaining results as extreme as yours when the null hypothesis is true (5)

P

observed sample proportion (6)

P(A)

probability of event

A

(17)

PR

percentile rank (2)

ϕ

phi coefficient (18)

ϕ

C

Cramér’s ϕ (18)

q

studentized range statistic (14)

r

C

matched pairs rank biserial correlation coefficient (8)

r

G

Glass rank biserial correlation coefficient (8)

r

pb

point‐biserial correlation coefficient (10)

r

s

Spearman rank‐order correlation coefficient (9)

r

XY

sample Pearson correlation coefficient between

X

and

Y

(9)

mean of a set of ranks (8)

ρ

XY

population correlation coefficient between

X

and

Y

(9)

s

sample standard deviation (5)

s

2

population variance estimate (5)

variance of the

D

s (7)

pooled variance (7)

standard error of the mean (6)

standard error of the difference (7)

s

est

estimate of σ

y

′ obtained from a sample (10)

SIQR

semi‐interquartile range (3)

SS

sum of squares (3)

SS

T

total sum of squares (13)

SS

Bet

sum of squares between groups (13)

SS

W

sum of squares within groups (13)

SS

1

sum of squares for factor 1 (15)

SS

2

sum of squares for factor 2 (15)

SS

1×2

sum of squares for interaction (15)

∑

summation sign (1)

σ

population standard deviation (3)

σ

2

population variance (3)

σ

p

standard error of a sample proportion (6)

σ

T

standard error of the ranks of independent samples (8)

standard error of the ranks of matched samples (8)

standard error of the mean when σ is known (5)

σ

est

standard error of estimate for predicting

Y

(10)

t

statistic following the

t

distribution (6)

T

T score (4)

T

E

expected sum of the ranks (8)

T

i

sum of ranks in group

i

(8)

X

′

predicted

X

score (10)

sample mean (3)

i

mean of group

i

(13)

G

grand mean (13)

Y

′

predicted

Y

score (10)

z

standard score (4)

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/ISBS8e 

The website includes:

Power Point

Study Guide

Test Bank

Instructor Manual

Part IDescriptive Statistics

Chapter 1

Introduction

Chapter 2

Frequency Distributions and Graphs

Chapter 3

Measures of Central Tendency and Variability

Chapter 4

Standardized Scores and the NormalDistribution

Chapter 1Introduction

PREVIEW

Why Study Statistics?

What are four important reasons why knowledge of statistics is essential for anyone majoring in psychology, sociology, or education?

Descriptive and Inferential Statistics

What is the difference between descriptive and inferential statistics?

Why must behavioral science researchers use inferential statistics?

Populations, Samples, Parameters, and Statistics

What is the difference between a population and a sample?

Why is it important to specify clearly the population from which a sample is drawn?

What is the difference between a parameter and a statistic?

Measurement Scales

What types of scales are used to measure variables in the behavioral sciences?

Independent and Dependent Variables

What is the difference between observational and experimental studies?

Summation Notation

Why is summation notation used by statisticians?

What are the eight rules involving summation notation?

Jackson’s Study

An example that provides a common thread tying together all of the subsequent chapters.

Summary

Exercises

Thought Questions

Computer Exercises

Bridge to SPSS

Why Study Statistics?

This book is written primarily for undergraduates majoring in psychology, or one of the other behavioral sciences. There are four reasons why a knowledge of statistics is essential for those who wish to conduct or consume behavioral science research:

To understand the professional literature

.

Most professional literature in the behavioral sciences includes results that are based on statistical analyses. Therefore, you will be unable to understand important articles in scientific journals and books unless you understand statistics. It is possible to seek out secondhand reports that are designed for the statistically uninformed, but those who prefer this alternative to obtain firsthand information perhaps should not be majoring in the field of behavioral science.

To understand and evaluate statistical claims made in the popular media

.

“This has been an important reason to acquire quantitative reasoning skills for decades, as advances in technology have led to an explosion of quantitative claims in a variety of media outlets.” Unfortunately, the difference between claims that are statistically sound and those that merely appear that way can be difficult to detect without some formal training. You will make many important life decisions over the next decade that will require weighing probabilities under conditions of uncertainty. A competence with basic statistics will empower you to make maximal use of the available information and help protect you from those who may wish to mislead you with pretty graphs and numbers.

To understand the rationale underlying research in the behavioral sciences

.

Statistics is not just a catalog of procedures and formulas. It offers the rationale on which much of behavioral science research is based—namely drawing inferences about a population based on data obtained from a sample. Those familiar with statistics understand that research consists of a series of educated guesses and fallible decisions, not right or wrong answers. Those without knowledge of statistics, by contrast, cannot understand the strengths and weaknesses of the techniques used by behavioral scientists to collect information and draw conclusions.

To carry out behavioral science research

.

In order to contribute competent research to the behavioral sciences, it is necessary to design the statistical analysis

before

the data are collected. Otherwise, the research procedures may be so poorly planned that not even an expert statistician can make any sense out of the results. To be sure, it is possible (and often advisable) to consult someone more experienced in statistics for assistance. Without some statistical knowledge of your own, however, you will find it difficult or impossible to convey your needs to someone else and to understand the replies.

Save for these introductory remarks, we do not see it as our task to persuade you that statistics is important in psychology and other behavioral sciences. If you are seriously interested in any of these fields, you will find this out for yourself. Accordingly, this book does not rely on documented examples selected from the professional literature to prove to you that statistics really is used in these fields. Instead, we have devised several artificial, but realistic, examples with numerical values that reveal the processes and issues involved in statistical analyses as clearly as possible.

One example we use throughout this book is based on a hypothetical study performed by a new student named Mia on the relative friendliness of four dormitory halls on her campus. We present the data for this study in the first exercise at the end of this chapter and return to it in many of the subsequent chapters.

We have tried to avoid a “cookbook” approach that places excessive emphasis on computational recipes. Instead, the various statistical procedures and the essential underlying concepts have been explained at length, and insofar as possible in standard English, so that you will know not only what to do but why you are doing it. Do not, however, expect to learn the material in this book from a single reading; the concepts involved in statistics, especially inferential statistics, are sufficiently challenging that it is often said that the only way to completely understand the material is to teach it (or write a book about it). Having said that, however, there is no reason to approach statistics with fear and trembling. You certainly do not need any advanced understanding of mathematics to obtain a good working knowledge of basic statistics. What is needed is mathematical comprehension sufficient to cope with beginning high school algebra and a willingness to work at new concepts until they are understood, which requires, in turn, a willingness to spend some time working through at least half of the exercises at the end of each chapter (and, perhaps, some additional exercises from the online Study Guide).

Descriptive and Inferential Statistics

One purpose of statistics is to summarize or describe the characteristics of a set of data in a clear and convenient fashion. This is accomplished by what is called descriptive statistics. For example, your grade point average (GPA) serves as a convenient summary of all of the grades that you have received in college. Part I of this book is devoted to descriptive statistics.

A second function of statistics is to make possible the solution of an extremely important problem. Behavioral scientists can never measure all of the cases in which they are interested. For example, a clinical psychologist studying the effects of various kinds of therapies cannot obtain data on every single mental health patient in the world; a developmental psychologist studying age differences in attitudes cannot measure all of the millions of children and adults in the United States; a cognitive psychologist cannot observe the reading behavior of all literate adults. Behavioral scientists want to know what is happening in a given population—a large group (theoretically an infinitely large group) of people, animals, objects, or responses that are alike in at least one respect (e.g., all college students in the United States). They cannot measure the entire population, however, because it is so large that it would be too time‐consuming and expensive to do so. What to do?

Turns out there’s a reasonably simple solution: Measure just a relatively small number of cases drawn from the population (i.e., a sample), and use inferential statistics to make educated guesses about the population. Inferential statistics makes it possible to draw inferences about what is happening in the population based on what is observed in a sample from that population. (This point is discussed at greater length in Chapter 5.) The subsequent parts of this book are devoted to inferential statistics, which makes frequent use of some of the descriptive statistics discussed in Part I.

Populations, Samples, Parameters, and Statistics

As the above discussion indicates, the term population as used in statistics does not necessarily refer to people. For example, the population of interest may be that of all white rats of a given genetic strain or all responses of a single participant’s eyelid in a conditioning experiment.

Whereas the population consists of all of the cases of interest, a sample consists of any subgroup drawn from the specified population. It is important that the population be clearly specified. For example, a group of 100 Macalester College freshmen might be a well‐drawn sample from the population of all Macalester freshmen or a poorly drawn sample from the population of all undergraduates in the United States (poorly drawn because it probably will not be representative of all US undergraduates). It is strictly proper to apply (i.e., generalize) the research results only to the specified population from which the sample was drawn. (A researcher may justifiably argue that her results are more widely generalizable, but she is on her own if she does so because the rules of statistical inference do not justify this.)

A statistic is a numerical quantity (such as an average) that summarizes some characteristic of a sample. A parameter is the corresponding value of that characteristic in the population. For example, if the average studying time of a sample of 100 New York University sophomores is 7.4 hours per week, then 7.4 is a statistic. If the average studying time of the population of all NYU freshmen is 9.6 hours per week, then 9.6 is the corresponding population parameter. Usually, the values of population parameters are unknown because the population is too large to measure in its entirety, and appropriate techniques of inferential statistics are therefore used to estimate the values of population parameters from sample statistics. If the sample is properly selected, the sample statistics will often give good estimates of the parameters of the population from which the sample was drawn; if the sample is poorly chosen, erroneous conclusions are likely to occur. Whether you are doing your own research or reading about that produced by someone else, you should always check to be sure that the population to which the results are generalized is proper in light of the sample from which the results were obtained.

Measurement Scales

You may have noticed that we have used the term data several times without talking about where the data come from. It should come as no surprise that in the behavioral sciences, the data generally come from measuring some aspect of the behavior of a human or animal. Unlike physics, in which there are quite a few important constants (values that are always the same, such as the speed of light or the mass of an electron), the behavioral sciences deal mainly with the measurement of variables, which can take on a range of different values. An additional complication faced by the behavioral scientist is that some of the variables of most interest can be difficult to measure (e.g., self‐esteem). In this section, we discuss the measurement scales most commonly used in the behavioral sciences. It is important to know which scale you are using because that choice often determines which statistical technique is appropriate. The scales differ with respect to how finely they can distinguish differences among instances. For example, the nominal scale can only distinguish whether an item is in one category or another, but the categories have no inherent order (e.g., the color of your iPhone). The ordinal scale involves measurements that can distinguish order on a set of values (e.g., bigger, smaller). Interval scales add the capability to measure quantities on a scale that has equal intervals between units (e.g., inches; temperature), and ratio scales are interval scales for which a value of zero means that absolutely none of the variables being measured is present. We describe them below in order of complexity.

Nominal Scales

The crudest form of measurement is to classify items by assigning names to them (categorization), which does not involve any numerical precision at all. Such a scale is called a nominal scale. For example, a person’s occupation can only be “measured” on a nominal scale (e.g., accountant, lawyer, politician, computer programmer). We can count the number of people who fall into each category, but (unlike ordinal, interval, or ratio scales) there is no obvious order to the categories, and certainly no regular intervals between them. We refer to such categorical data as being qualitative, as distinguished from data measured on one of the quantitative scales described next.

Ordinal Scales

Sometimes it is possible to order your categories, even though the intervals are not precise. The most common example of this in psychological research is called a Likert scale (after its creator, Rensis Likert, pronounced “Like‐ert”), on which respondents rate their agreement with some statement by choosing, for instance, among “strongly agree,” “agree,” “uncertain,” “disagree,” and “strongly disagree.” The order of the categories is clear, but because there is no way to be sure that they are equally spaced (is the psychological distance between “strongly agree” and “agree” the same as between “agree” and “uncertain?”), this type of scale lacks the interval property and is therefore called an ordinal scale. Although it is a somewhat controversial practice, many behavioral researchers simply assign numbers to the categories (e.g., strongly agree is 1, agree is 2, etc.) and then treat the data as though they came from an interval scale.

Another, less common, way that an ordinal scale can be created is by rank ordering. It may not be possible to measure, in a precise way, the creativity of paintings produced by students in an art class, but a panel of judges could rank them from most to least creative, with perhaps a few paintings tied at the same rank.

It is important to distinguish between the characteristics of the variable we are measuring, on one hand, and the particular scale with which we choose to measure that variable, on the other. For example, suppose a teacher wishes to measure how often middle school students raise their hands in class. Hand‐raising frequency is a variable that can be measured quantitatively by just counting how often each student raises a hand. However, when the teacher uses these data to understand hand‐raising behavior in his class, his purposes may be best served by organizing the students into three broad, but ordinal, groups: (1) those who never raised their hand (most of the group); (2) those who did so just a few times; and (3) those who did so many times. In that case, he will want to create an ordinal scale to present data that was originally measured on an interval scale (described next). In other words, just because you are using a variable that has the potential