# Statistics Question

Make an APA-formatted bar graph on amount spent by vegetarian status of shopper. Copy and paste the graph below. You may need to create a screenshot.

Ed iti o n

© Deborah Batt

10

Statistics for the

Behavioral Sciences

Frederick J Gravetter

The College at Brockport, State University of New York

Larry B. WaLLnau

The College at Brockport, State University of New York

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Printed in Canada

Print Number: 01

Print Year: 2015

B RiEF Co n tEn t S

CHAPtER

1

Introduction to Statistics 1

CHAPtER

2

Frequency Distributions

CHAPtER

3

Central Tendency

CHAPtER

4

Variability

CHAPtER

5

z-Scores: Location of Scores and Standardized Distributions 131

CHAPtER

6

Probability

CHAPtER

7

Probability and Samples: The Distribution of Sample Means 193

CHAPtER

8

Introduction to Hypothesis Testing 223

CHAPtER

9

Introduction to the t Statistic

C H A P t E R 10

CHAPtER

11

33

67

99

159

267

The t Test for Two Independent Samples 299

The t Test for Two Related Samples 335

C H A P t E R 12

Introduction to Analysis of Variance 365

C H A P t E R 13

Repeated-Measures Analysis of Variance 413

C H A P t E R 14

Two-Factor Analysis of Variance (Independent Measures) 447

CHAPtER

15

Correlation

485

C H A P t E R 16

Introduction to Regression 529

C H A P t E R 17

The Chi-Square Statistic: Tests for Goodness of Fit and Independence 559

C H A P t E R 18

The Binomial Test

603

iii

Co n tEn t S

CHAPtER

1 Introduction to Statistics

PREVIEW

1

2

1.1 Statistics, Science, and Observations 2

1.2 Data Structures, Research Methods, and Statistics 10

1.3 Variables and Measurement 18

1.4 Statistical Notation 25

Summary

29

Focus on Problem Solving 30

Demonstration 1.1 30

Problems 31

CHAPtER

2 Frequency Distributions

PREVIEW

33

34

2.1 Frequency Distributions and Frequency Distribution Tables 35

2.2 Grouped Frequency Distribution Tables 38

2.3 Frequency Distribution Graphs 42

2.4 Percentiles, Percentile Ranks, and Interpolation 49

2.5 Stem and Leaf Displays 56

Summary

58

Focus on Problem Solving 59

Demonstration 2.1 60

Demonstration 2.2 61

Problems 62

v

vi

CONTENTS

CHAPtER

3 Central Tendency

PREVIEW

67

68

3.1 Overview 68

3.2 The Mean 70

3.3 The Median 79

3.4 The Mode 83

3.5 Selecting a Measure of Central Tendency 86

3.6 Central Tendency and the Shape of the Distribution 92

Summary

94

Focus on Problem Solving 95

Demonstration 3.1 96

Problems 96

CHAPtER

4 Variability

PREVIEW

99

100

4.1 Introduction to Variability 101

4.2 Defining Standard Deviation and Variance 103

4.3 Measuring Variance and Standard Deviation for a Population 108

4.4 Measuring Standard Deviation and Variance for a Sample 111

4.5 Sample Variance as an Unbiased Statistic 117

4.6 More about Variance and Standard Deviation 119

Summary

125

Focus on Problem Solving 127

Demonstration 4.1 128

Problems 128

z-Scores: Location of Scores

C H A P t E R 5 and Standardized Distributions

PREVIEW

132

5.1 Introduction to z-Scores 133

5.2 z-Scores and Locations in a Distribution 135

5.3 Other Relationships Between z, X, 𝛍, and 𝛔 138

131

CONTENTS

vii

5.4 Using z-Scores to Standardize a Distribution 141

5.5 Other Standardized Distributions Based on z-Scores 145

5.6 Computing z-Scores for Samples 148

5.7 Looking Ahead to Inferential Statistics 150

Summary

153

Focus on Problem Solving 154

Demonstration 5.1 155

Demonstration 5.2 155

Problems 156

CHAPtER

6 Probability

PREVIEW

159

160

6.1 Introduction to Probability 160

6.2 Probability and the Normal Distribution 165

6.3 Probabilities and Proportions for Scores

from a Normal Distribution

172

6.4 Probability and the Binomial Distribution 179

6.5 Looking Ahead to Inferential Statistics 184

Summary

186

Focus on Problem Solving 187

Demonstration 6.1 188

Demonstration 6.2 188

Problems 189

Probability and Samples: The Distribution

C H A P t E R 7 of Sample Means

PREVIEW

194

7.1 Samples, Populations, and the Distribution

of Sample Means

194

7.2 The Distribution of Sample Means for any Population

and any Sample Size

199

7.3 Probability and the Distribution of Sample Means 206

7.4 More about Standard Error 210

7.5 Looking Ahead to Inferential Statistics

215

193

viii

CONTENTS

Summary

219

Focus on Problem Solving 219

Demonstration 7.1 220

Problems 221

CHAPtER

8 Introduction to Hypothesis Testing

PREVIEW

223

224

8.1 The Logic of Hypothesis Testing 225

8.2 Uncertainty and Errors in Hypothesis Testing 236

8.3 More about Hypothesis Tests 240

8.4 Directional (One-Tailed) Hypothesis Tests 245

8.5 Concerns about Hypothesis Testing: Measuring Effect Size 250

8.6 Statistical Power 254

Summary

260

Focus on Problem Solving 261

Demonstration 8.1 262

Demonstration 8.2 263

Problems 263

CHAPtER

9 Introduction to the t Statistic

PREVIEW

268

9.1 The t Statistic: An Alternative to z 268

9.2 Hypothesis Tests with the t Statistic 274

9.3 Measuring Effect Size for the t Statistic 279

9.4 Directional Hypotheses and One-Tailed Tests 288

Summary

291

Focus on Problem Solving 293

Demonstration 9.1 293

Demonstration 9.2 294

Problems 295

267

CONTENTS

CHAPtER

10 The t Test for Two Independent Samples

PREVIEW

ix

299

300

10.1 Introduction to the Independent-Measures Design 300

10.2 The Null Hypothesis and the Independent-Measures t Statistic 302

10.3 Hypothesis Tests with the Independent-Measures t Statistic 310

10.4 Effect Size and Confidence Intervals for the

Independent-Measures t

316

10.5 The Role of Sample Variance and Sample Size in the

Independent-Measures t Test

Summary

322

325

Focus on Problem Solving 327

Demonstration 10.1 328

Demonstration 10.2 329

Problems 329

CHAPtER

11 The t Test for Two Related Samples

PREVIEW

335

336

11.1 Introduction to Repeated-Measures Designs 336

11.2 The t Statistic for a Repeated-Measures Research Design 339

11.3 Hypothesis Tests for the Repeated-Measures Design 343

11.4 Effect Size and Confidence Intervals for the Repeated-Measures t 347

11.5 Comparing Repeated- and Independent-Measures Designs 352

Summary

355

Focus on Problem Solving 358

Demonstration 11.1 358

Demonstration 11.2 359

Problems 360

CHAPtER

12 Introduction to Analysis of Variance

PREVIEW

366

12.1 Introduction (An Overview of Analysis of Variance) 366

12.2 The Logic of Analysis of Variance 372

12.3 ANOVA Notation and Formulas 375

365

x

CONTENTS

12.4 Examples of Hypothesis Testing and Effect Size with ANOVA 383

12.5 Post Hoc Tests 393

12.6 More about ANOVA 397

Summary

403

Focus on Problem Solving 406

Demonstration 12.1 406

Demonstration 12.2 408

Problems 408

CHAPtER

13 Repeated-Measures Analysis of Variance

PREVIEW

413

414

13.1 Overview of the Repeated-Measures ANOVA 415

13.2 Hypothesis Testing and Effect Size with the

Repeated-Measures ANOVA

420

13.3 More about the Repeated-Measures Design 429

Summary

436

Focus on Problem Solving 438

Demonstration 13.1 439

Demonstration 13.2 440

Problems 441

Two-Factor Analysis of Variance

C H A P t E R 14 (Independent Measures)

PREVIEW

447

448

14.1 An Overview of the Two-Factor, Independent-Measures, ANOVA: Main

Effects and Interactions

448

14.2 An Example of the Two-Factor ANOVA and Effect Size 458

14.3 More about the Two-Factor ANOVA 467

Summary

473

Focus on Problem Solving 475

Demonstration 14.1 476

Demonstration 14.2 478

Problems 479

CONTENTS

CHAPtER

15 Correlation

PREVIEW

xi

485

486

15.1 Introduction 487

15.2 The Pearson Correlation 489

15.3 Using and Interpreting the Pearson Correlation 495

15.4 Hypothesis Tests with the Pearson Correlation 506

15.5 Alternatives to the Pearson Correlation 510

Summary

520

Focus on Problem Solving 522

Demonstration 15.1 523

Problems 524

CHAPtER

16 Introduction to Regression

PREVIEW

529

530

16.1 Introduction to Linear Equations and Regression 530

16.2 The Standard Error of Estimate and Analysis of Regression:

The Significance of the Regression Equation

538

16.3 Introduction to Multiple Regression with Two Predictor Variables 544

Summary

552

Linear and Multiple Regression

554

Focus on Problem Solving 554

Demonstration 16.1 555

Problems 556

The Chi-Square Statistic: Tests for Goodness

C H A P t E R 17 of Fit and Independence

PREVIEW

560

17.1 Introduction to Chi-Square: The Test for Goodness of Fit 561

17.2 An Example of the Chi-Square Test for Goodness of Fit 567

17.3 The Chi-Square Test for Independence 573

17.4 Effect Size and Assumptions for the Chi-Square Tests

17.5 Special Applications of the Chi-Square Tests 587

582

559

xii

CONTENTS

Summary

591

Focus on Problem Solving 595

Demonstration 17.1 595

Demonstration 17.2 597

Problems 597

CHAPtER

18 The Binomial Test

PREVIEW

603

604

18.1 Introduction to the Binomial Test 604

18.2 An Example of the Binomial Test 608

18.3 More about the Binomial Test: Relationship with Chi-Square

and the Sign Test

Summary

612

617

Focus on Problem Solving 619

Demonstration 18.1 619

Problems 620

A PPE N D IX E S

A Basic Mathematics Review 625

A.1

A.2

A.3

A.4

A.5

Symbols and Notation 627

Proportions: Fractions, Decimals, and Percentages 629

Negative Numbers 635

Basic Algebra: Solving Equations 637

Exponents and Square Roots 640

B Statistical Tables 647

C Solutions for Odd-Numbered Problems in the Text 663

D General Instructions for Using SPSS 683

E Hypothesis Tests for Ordinal Data: Mann-Whitney,

Wilcoxon, Kruskal-Wallis, and Friedman Tests

687

Statistics Organizer: Finding the Right Statistics for Your Data

References

717

Name Index

723

Subject Index

725

701

PREFACE

M

any students in the behavioral sciences view the required statistics course as an

intimidating obstacle that has been placed in the middle of an otherwise interesting curriculum. They want to learn about human behavior—not about math and science.

As a result, the statistics course is seen as irrelevant to their education and career goals.

However, as long as the behavioral sciences are founded in science, knowledge of statistics

will be necessary. Statistical procedures provide researchers with objective and systematic

methods for describing and interpreting their research results. Scientific research is the

system that we use to gather information, and statistics are the tools that we use to distill

the information into sensible and justified conclusions. The goal of this book is not only

to teach the methods of statistics, but also to convey the basic principles of objectivity and

logic that are essential for science and valuable for decision making in everyday life.

Those of you who are familiar with previous editions of Statistics for the Behavioral

Sciences will notice that some changes have been made. These changes are summarized

in the section entitled “To the Instructor.” In revising this text, our students have been

foremost in our minds. Over the years, they have provided honest and useful feedback.

Their hard work and perseverance has made our writing and teaching most rewarding. We

sincerely thank them. Students who are using this edition should please read the section of

the preface entitled “To the Student.”

The book chapters are organized in the sequence that we use for our own statistics

courses. We begin with descriptive statistics, and then examine a variety of statistical procedures focused on sample means and variance before moving on to correlational methods

and nonparametric statistics. Information about modifying this sequence is presented in the

To The Instructor section for individuals who prefer a different organization. Each chapter

contains numerous examples, many based on actual research studies, learning checks, a

summary and list of key terms, and a set of 20–30 problems.

Ancillaries

Ancillaries for this edition include the following.

■■

MindTap® Psychology: MindTap® Psychology for Gravetter/Wallnau’s Statistics

for The Behavioral Sciences, 10th Edition is the digital learning solution that helps

instructors engage and transform today’s students into critical thinkers. Through paths

of dynamic assignments and applications that you can personalize, real-time course

analytics, and an accessible reader, MindTap helps you turn cookie cutter into cutting

edge, apathy into engagement, and memorizers into higher-level thinkers.

As an instructor using MindTap you have at your fingertips the right content and

unique set of tools curated specifically for your course, such as video tutorials that

walk students through various concepts and interactive problem tutorials that provide

students opportunities to practice what they have learned, all in an interface designed

to improve workflow and save time when planning lessons and course structure. The

control to build and personalize your course is all yours, focusing on the most relevant

xiii

xiv

PREFACE

■■

■■

■■

material while also lowering costs for your students. Stay connected and informed in

your course through real time student tracking that provides the opportunity to adjust

the course as needed based on analytics of interactivity in the course.

Online Instructor’s Manual: The manual includes learning objectives, key terms,

a detailed chapter outline, a chapter summary, lesson plans, discussion topics, student

activities, “What If” scenarios, media tools, a sample syllabus and an expanded test

bank. The learning objectives are correlated with the discussion topics, student

activities, and media tools.

Online PowerPoints: Helping you make your lectures more engaging while effectively reaching your visually oriented students, these handy Microsoft PowerPoint®

slides outline the chapters of the main text in a classroom-ready presentation. The

PowerPoint® slides are updated to reflect the content and organization of the new

edition of the text.

Cengage Learning Testing, powered by Cognero®: Cengage Learning Testing,

Powered by Cognero®, is a flexible online system that allows you to author, edit,

and manage test bank content. You can create multiple test versions in an instant and

deliver tests from your LMS in your classroom.

Acknowledgments

It takes a lot of good, hard-working people to produce a book. Our friends at Cengage

have made enormous contributions to this textbook. We thank: Jon-David Hague, Product

Director; Timothy Matray, Product Team Director; Jasmin Tokatlian, Content Development Manager; Kimiya Hojjat, Product Assistant; and Vernon Boes, Art Director. Special

thanks go to Stefanie Chase, our Content Developer and to Lynn Lustberg who led us

through production at MPS.

Reviewers play a very important role in the development of a manuscript. Accordingly,

we offer our appreciation to the following colleagues for their assistance: Patricia Case,

University of Toledo; Kevin David, Northeastern State University; Adia Garrett, University of Maryland, Baltimore County; Carrie E. Hall, Miami University; Deletha Hardin,

University of Tampa; Angela Heads, Prairie View A&M University; Roberto Heredia,

Texas A&M International University; Alisha Janowski, University of Central Florida;

Matthew Mulvaney, The College at Brockport (SUNY); Nicholas Von Glahn, California

State Polytechnic University, Pomona; and Ronald Yockey, Fresno State University.

To the Instructor

Those of you familiar with the previous edition of Statistics for the Behavioral Sciences will

notice a number of changes in the 10th edition. Throughout this book, research examples

have been updated, real world examples have been added, and the end-of-chapter problems

have been extensively revised. Major revisions for this edition include the following:

1. Each section of every chapter begins with a list of Learning Objectives for that

specific section.

2. Each section ends with a Learning Check consisting of multiple-choice questions

with at least one question for each Learning Objective.

PREFACE

xv

3. The former Chapter 19, Choosing the Right Statistics, has been eliminated and

an abridged version is now an Appendix replacing the Statistics Organizer, which

appeared in earlier editions.

Other examples of specific and noteworthy revisions include the following.

Chapter 1 The section on data structures and research methods parallels the new

Appendix, Choosing the Right Statistics.

Chapter 2 The chapter opens with a new Preview to introduce the concept and purpose

of frequency distributions.

Chapter 3

Minor editing clarifies and simplifies the discussion the median.

Chapter 4 The chapter opens with a new Preview to introduce the topic of Central

Tendency. The sections on standard deviation and variance have been edited to increase

emphasis on concepts rather than calculations.

The section discussion relationships between z, X, μ, and σ has been

expanded and includes a new demonstration example.

Chapter 5

Chapter 6

The chapter opens with a new Preview to introduce the topic of Probability.

The section, Looking Ahead to Inferential Statistics, has been substantially shortened and

simplified.

Chapter 7

The former Box explaining difference between standard deviation and

standard error was deleted and the content incorporated into Section 7.4 with editing to

emphasize that the standard error is the primary new element introduced in the chapter.

The final section, Looking Ahead to Inferential Statistics, was simplified and shortened to

be consistent with the changes in Chapter 6.

Chapter 8

A redundant example was deleted which shortened and streamlined the

remaining material so that most of the chapter is focused on the same research example.

Chapter 9 The chapter opens with a new Preview to introduce the t statistic and explain

why a new test statistic is needed. The section introducing Confidence Intervals was edited

to clarify the origin of the confidence interval equation and to emphasize that the interval

is constructed at the sample mean.

Chapter 10

The chapter opens with a new Preview introducing the independent-measures t statistic. The section presenting the estimated standard error of (M1 – M2) has been

simplified and shortened.

Chapter 11

The chapter opens with a new Preview introducing the repeated-measures t

statistic. The section discussing hypothesis testing has been separated from the section on

effect size and confidence intervals to be consistent with the other two chapters on t tests.

The section comparing independent- and repeated-measures designs has been expanded.

Chapter 12 The chapter opens with a new Preview introducing ANOVA and explaining

why a new hypothesis testing procedure is necessary. Sections in the chapter have been

reorganized to allow flow directly from hypothesis tests and effect size to post tests.

xvi

PREFACE

Chapter 13

Substantially expanded the section discussing factors that influence the

outcome of a repeated-measures hypothesis test and associated measures of effect size.

Chapter 14

The chapter opens with a new Preview presenting a two-factor research

example and introducing the associated ANOVA. Sections have been reorganized so that

simple main effects and the idea of using a second factor to reduce variance from individual differences are now presented as extra material related to the two-factor ANOVA.

Chapter 15

The chapter opens with a new Preview presenting a correlational research

study and the concept of a correlation. A new section introduces the t statistic for evaluating the significance of a correlation and the section on partial correlations has been simplified and shortened.

Chapter 16 The chapter opens with a new Preview introducing the concept of regression and

its purpose. A new section demonstrates the equivalence of testing the significance of a correlation and testing the significance of a regression equation with one predictor variable. The section on residuals for the multiple-regression equation has been edited to simplify and shorten.

Chapter 17

A new chapter Preview presents an experimental study with data consisting

of frequencies, which are not compatible with computing means and variances. Chi-square

tests are introduced as a solution to this problem. A new section introduces Cohen’s w as

a means of measuring effect size for both chi-square tests.

Chapter 18

Substantial editing clarifies the section explaining how the real limits for

each score can influence the conclusion from a binomial test.

The former Chapter 19 covering the task of matching statistical methods to specific

types of data has been substantially shortened and converted into an Appendix.

■■Matching the Text to Your Syllabus

The book chapters are organized in the sequence that we use for our own statistics courses.

However, different instructors may prefer different organizations and probably will choose

to omit or deemphasize specific topics. We have tried to make separate chapters, and even

sections of chapters, completely self-contained, so they can be deleted or reorganized to fit

the syllabus for nearly any instructor. Some common examples are as follows.

■■

■■

■■

It is common for instructors to choose between emphasizing analysis of variance

(Chapters 12, 13, and 14) or emphasizing correlation/regression (Chapters 15 and 16).

It is rare for a one-semester course to complete coverage of both topics.

Although we choose to complete all the hypothesis tests for means and mean

differences before introducing correlation (Chapter 15), many instructors prefer to

place correlation much earlier in the sequence of course topics. To accommodate

this, Sections 15.1, 15.2, and 15.3 present the calculation and interpretation of

the Pearson correlation and can be introduced immediately following Chapter 4

(variability). Other sections of Chapter 15 refer to hypothesis testing and should be

delayed until the process of hypothesis testing (Chapter 8) has been introduced.

It is also possible for instructors to present the chi-square tests (Chapter 17) much

earlier in the sequence of course topics. Chapter 17, which presents hypothesis tests

for proportions, can be presented immediately after Chapter 8, which introduces the

process of hypothesis testing. If this is done, we also recommend that the Pearson

correlation (Sections 15.1, 15.2, and 15.3) be presented early to provide a foundation

for the chi-square test for independence.

PREFACE

xvii

To the Student

A primary goal of this book is to make the task of learning statistics as easy and painless

as possible. Among other things, you will notice that the book provides you with a number

of opportunities to practice the techniques you will be learning in the form of Learning

Checks, Examples, Demonstrations, and end-of-chapter problems. We encourage you to

take advantage of these opportunities. Read the text rather than just memorizing the formulas. We have taken care to present each statistical procedure in a conceptual context that

explains why the procedure was developed and when it should be used. If you read this

material and gain an understanding of the basic concepts underlying a statistical formula,

you will find that learning the formula and how to use it will be much easier. In the “Study

Hints,” that follow, we provide advice that we give our own students. Ask your instructor

for advice as well; we are sure that other instructors will have ideas of their own.

Over the years, the students in our classes and other students using our book have given

us valuable feedback. If you have any suggestions or comments about this book, you can

write to either Professor Emeritus Frederick Gravetter or Professor Emeritus Larry Wallnau

at the Department of Psychology, SUNY College at Brockport, 350 New Campus Drive,

Brockport, New York 14420. You can also contact Professor Emeritus Gravetter directly at

fgravett@brockport.edu.

■■Study Hints

You may find some of these tips helpful, as our own students have reported.

■■

■■

■■

■■

■■

The key to success in a statistics course is to keep up with the material. Each new

topic builds on previous topics. If you have learned the previous material, then the

new topic is just one small step forward. Without the proper background, however,

the new topic can be a complete mystery. If you find that you are falling behind, get

help immediately.

You will learn (and remember) much more if you study for short periods several

times per week rather than try to condense all of your studying into one long session.

For example, it is far more effective to study half an hour every night than to have

a single 3½-hour study session once a week. We cannot even work on writing this

book without frequent rest breaks.

Do some work before class. Keep a little ahead of the instructor by reading the appropriate sections before they are presented in class. Although you may not fully understand what you read, you will have a general idea of the topic, which will make the

lecture easier to follow. Also, you can identify material that is particularly confusing

and then be sure the topic is clarified in class.

Pay attention and think during class. Although this advice seems obvious, often it is

not practiced. Many students spend so much time trying to write down every example

presented or every word spoken by the instructor that they do not actually understand

and process what is being said. Check with your instructor—there may not be a need

to copy every example presented in class, especially if there are many examples like

it in the text. Sometimes, we tell our students to put their pens and pencils down for a

moment and just listen.

Test yourself regularly. Do not wait until the end of the chapter or the end of the

week to check your knowledge. After each lecture, work some of the end-of-chapter

problems and do the Learning Checks. Review the Demonstration Problems, and

be sure you can define the Key Terms. If you are having trouble, get your questions

answered immediately—reread the section, go to your instructor, or ask questions in

class. By doing so, you will be able to move ahead to new material.

xviii

PREFACE

■■

■■

Do not kid yourself! Avoid denial. Many students observe their instructor solve

problems in class and think to themselves, “This looks easy, I understand it.” Do

you really understand it? Can you really do the problem on your own without having

to leaf through the pages of a chapter? Although there is nothing wrong with using

examples in the text as models for solving problems, you should try working a problem with your book closed to test your level of mastery.

We realize that many students are embarrassed to ask for help. It is our biggest challenge as instructors. You must find a way to overcome this aversion. Perhaps contacting the instructor directly would be a good starting point, if asking questions in class

is too anxiety-provoking. You could be pleasantly surprised to find that your instructor does not yell, scold, or bite! Also, your instructor might know of another student

who can offer assistance. Peer tutoring can be very helpful.

Frederick J Gravetter

Larry B. Wallnau

A B o U t tH E AU tH o R S

Frederick Gravetter

Frederick J Gravetter is Professor Emeritus of Psychology at the

State University of New York College at Brockport. While teaching at

Brockport, Dr. Gravetter specialized in statistics, experimental design, and

cognitive psychology. He received his bachelor’s degree in mathematics from

M.I.T. and his Ph.D in psychology from Duke University. In addition to publishing this textbook and several research articles, Dr. Gravetter co-authored

Research Methods for the Behavioral Science and Essentials of Statistics for

the Behavioral Sciences.

Larry B. Wallnau

Larry B. WaLLnau is Professor Emeritus of Psychology at the State

University of New York College at Brockport. While teaching at Brockport,

he published numerous research articles in biopsychology. With

Dr. Gravetter, he co-authored Essentials of Statistics for the Behavioral

Sciences. Dr. Wallnau also has provided editorial consulting for numerous

publishers and journals. He has taken up running and has competed in 5K

races in New York and Connecticut. He takes great pleasure in adopting

neglected and rescued dogs.

xix

CH A P T ER

Introduction to Statistics

1

© Deborah Batt

PREVIEW

1.1 Statistics, Science, and Observations

1.2 Data Structures, Research Methods, and Statistics

1.3 Variables and Measurement

1.4 Statistical Notation

Summary

Focus on Problem Solving

Demonstration 1.1

Problems

1

PREVIEW

Before we begin our discussion of statistics, we ask you

to read the following paragraph taken from the philosophy of Wrong Shui (Candappa, 2000).

The Journey to Enlightenment

In Wrong Shui, life is seen as a cosmic journey,

a struggle to overcome unseen and unexpected

obstacles at the end of which the traveler will find

illumination and enlightenment. Replicate this quest

in your home by moving light switches away from

doors and over to the far side of each room.*

Why did we begin a statistics book with a bit of twisted

philosophy? In part, we simply wanted to lighten the

mood with a bit of humor—starting a statistics course is

typically not viewed as one of life’s joyous moments. In

addition, the paragraph is an excellent counterexample for

the purpose of this book. Specifically, our goal is to do

everything possible to prevent you from stumbling around

in the dark by providing lots of help and illumination as

you journey through the world of statistics. To accomplish

this, we begin each section of the book with clearly stated

learning objectives and end each section with a brief quiz

to test your mastery of the new material. We also introduce each new statistical procedure by explaining the purpose it is intended to serve. If you understand why a new

procedure is needed, you will find it much easier to learn.

1.1

The objectives for this first chapter are to provide

an introduction to the topic of statistics and to give you

some background for the rest of the book. We discuss

the role of statistics within the general field of scientific

inquiry, and we introduce some of the vocabulary and

notation that are necessary for the statistical methods

that follow.

As you read through the following chapters, keep

in mind that the general topic of statistics follows a

well-organized, logically developed progression that

leads from basic concepts and definitions to increasingly sophisticated techniques. Thus, each new topic

serves as a foundation for the material that follows. The

content of the first nine chapters, for example, provides

an essential background and context for the statistical

methods presented in Chapter 10. If you turn directly

to Chapter 10 without reading the first nine chapters,

you will find the material confusing and incomprehensible. However, if you learn and use the background

material, you will have a good frame of reference for

understanding and incorporating new concepts as they

are presented.

*Candappa, R. (2000). The little book of wrong shui. Kansas City:

Andrews McMeel Publishing. Reprinted by permission.

Statistics, Science, and Observations

LEARNING OBJECTIVEs

1. Define the terms population, sample, parameter, and statistic, and describe the

relationships between them.

2. Define descriptive and inferential statistics and describe how these two general

categories of statistics are used in a typical research study.

3. Describe the concept of sampling error and explain how this concept creates the

fundamental problem that inferential statistics must address.

■■Definitions of Statistics

By one definition, statistics consist of facts and figures such as the average annual snowfall

in Denver or Derrick Jeter’s lifetime batting average. These statistics are usually informative

and time-saving because they condense large quantities of information into a few simple figures. Later in this chapter we return to the notion of calculating statistics (facts and figures)

but, for now, we concentrate on a much broader definition of statistics. Specifically, we use

the term statistics to refer to a general field of mathematics. In this case, we are using the

term statistics as a shortened version of statistical procedures. For example, you are probably using this book for a statistics course in which you will learn about the statistical techniques that are used to summarize and evaluate research results in the behavioral sciences.

2

SEctIon 1.1 | Statistics, Science, and Observations

3

Research in the behavioral sciences (and other fields) involves gathering information.

To determine, for example, whether college students learn better by reading material on

printed pages or on a computer screen, you would need to gather information about students’ study habits and their academic performance. When researchers finish the task of

gathering information, they typically find themselves with pages and pages of measurements such as preferences, personality scores, opinions, and so on. In this book, we present

the statistics that researchers use to analyze and interpret the information that they gather.

Specifically, statistics serve two general purposes:

1. Statistics are used to organize and summarize the information so that the researcher can

see what happened in the research study and can communicate the results to others.

2. Statistics help the researcher to answer the questions that initiated the research by

determining exactly what general conclusions are justified based on the specific

results that were obtained.

DEFInItIon

The term statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information.

Statistical procedures help ensure that the information or observations are presented

and interpreted in an accurate and informative way. In somewhat grandiose terms, statistics

help researchers bring order out of chaos. In addition, statistics provide researchers with a

set of standardized techniques that are recognized and understood throughout the scientific

community. Thus, the statistical methods used by one researcher will be familiar to other

researchers, who can accurately interpret the statistical analyses with a full understanding

of how the analysis was done and what the results signify.

■■Populations and Samples

Research in the behavioral sciences typically begins with a general question about a specific

group (or groups) of individuals. For example, a researcher may want to know what factors

are associated with academic dishonesty among college students. Or a researcher may want

to examine the amount of time spent in the bathroom for men compared to women. In the

first example, the researcher is interested in the group of college students. In the second

example, the researcher wants to compare the group of men with the group of women. In statistical terminology, the entire group that a researcher wishes to study is called a population.

DEFInItIon

A population is the set of all the individuals of interest in a particular study.

As you can well imagine, a population can be quite large—for example, the entire set

of women on the planet Earth. A researcher might be more specific, limiting the population

for study to women who are registered voters in the United States. Perhaps the investigator

would like to study the population consisting of women who are heads of state. Populations

can obviously vary in size from extremely large to very small, depending on how the investigator defines the population. The population being studied should always be identified by

the researcher. In addition, the population need not consist of people—it could be a population of rats, corporations, parts produced in a factory, or anything else an investigator wants

to study. In practice, populations are typically very large, such as the population of college

sophomores in the United States or the population of small businesses.

Because populations tend to be very large, it usually is impossible for a researcher to

examine every individual in the population of interest. Therefore, researchers typically select

4

chaPtER 1 | Introduction to Statistics

a smaller, more manageable group from the population and limit their studies to the individuals in the selected group. In statistical terms, a set of individuals selected from a population

is called a sample. A sample is intended to be representative of its population, and a sample

should always be identified in terms of the population from which it was selected.

A sample is a set of individuals selected from a population, usually intended to

represent the population in a research study.

DEFInItIon

Just as we saw with populations, samples can vary in size. For example, one study might

examine a sample of only 10 students in a graduate program and another study might use a

sample of more than 10,000 people who take a specific cholesterol medication.

So far we have talked about a sample being selected from a population. However, this is

actually only half of the full relationship between a sample and its population. Specifically,

when a researcher finishes examining the sample, the goal is to generalize the results back

to the entire population. Remember that the research started with a general question about

the population. To answer the question, a researcher studies a sample and then generalizes

the results from the sample to the population. The full relationship between a sample and a

population is shown in Figure 1.1.

■■Variables and Data

Typically, researchers are interested in specific characteristics of the individuals in the population (or in the sample), or they are interested in outside factors that may influence the

individuals. For example, a researcher may be interested in the influence of the weather on

people’s moods. As the weather changes, do people’s moods also change? Something that

can change or have different values is called a variable.

DEFInItIon

A variable is a characteristic or condition that changes or has different values for

different individuals.

THE POPULATION

All of the individuals of interest

The results

from the sample

are generalized

to the population

F I G U R E 1.1

The relationship between a

population and a sample.

The sample

is selected from

the population

THE SAMPLE

The individuals selected to

participate in the research study

SEctIon 1.1 | Statistics, Science, and Observations

5

Once again, variables can be characteristics that differ from one individual to another,

such as height, weight, gender, or personality. Also, variables can be environmental conditions that change such as temperature, time of day, or the size of the room in which the

research is being conducted.

To demonstrate changes in variables, it is necessary to make measurements of the variables

being examined. The measurement obtained for each individual is called a datum, or more commonly, a score or raw score. The complete set of scores is called the data set or simply the data.

DEFInItIon

Data (plural) are measurements or observations. A data set is a collection of measurements or observations. A datum (singular) is a single measurement or observation and is commonly called a score or raw score.

Before we move on, we should make one more point about samples, populations, and

data. Earlier, we defined populations and samples in terms of individuals. For example,

we discussed a population of graduate students and a sample of cholesterol patients. Be

forewarned, however, that we will also refer to populations or samples of scores. Because

research typically involves measuring each individual to obtain a score, every sample (or

population) of individuals produces a corresponding sample (or population) of scores.

■■Parameters and Statistics

When describing data it is necessary to distinguish whether the data come from a population or a sample. A characteristic that describes a population—for example, the average

score for the population—is called a parameter. A characteristic that describes a sample is

called a statistic. Thus, the average score for a sample is an example of a statistic. Typically,

the research process begins with a question about a population parameter. However, the

actual data come from a sample and are used to compute sample statistics.

DEFInItIon

A parameter is a value, usually a numerical value, that describes a population. A

parameter is usually derived from measurements of the individuals in the population.

A statistic is a value, usually a numerical value, that describes a sample. A statistic

is usually derived from measurements of the individuals in the sample.

Every population parameter has a corresponding sample statistic, and most research

studies involve using statistics from samples as the basis for answering questions about

population parameters. As a result, much of this book is concerned with the relationship

between sample statistics and the corresponding population parameters. In Chapter 7, for

example, we examine the relationship between the mean obtained for a sample and the

mean for the population from which the sample was obtained.

■■Descriptive and Inferential Statistical Methods

Although researchers have developed a variety of different statistical procedures to organize and interpret data, these different procedures can be classified into two general categories. The first category, descriptive statistics, consists of statistical procedures that are used

to simplify and summarize data.

DEFInItIon

Descriptive statistics are statistical procedures used to summarize, organize, and

simplify data.

6

chaPtER 1 | Introduction to Statistics

Descriptive statistics are techniques that take raw scores and organize or summarize

them in a form that is more manageable. Often the scores are organized in a table or a graph

so that it is possible to see the entire set of scores. Another common technique is to summarize a set of scores by computing an average. Note that even if the data set has hundreds

of scores, the average provides a single descriptive value for the entire set.

The second general category of statistical techniques is called inferential statistics.

Inferential statistics are methods that use sample data to make general statements about a

population.

DEFInItIon

Inferential statistics consist of techniques that allow us to study samples and then

make generalizations about the populations from which they were selected.

Because populations are typically very large, it usually is not possible to measure

everyone in the population. Therefore, a sample is selected to represent the population.

By analyzing the results from the sample, we hope to make general statements about the

population. Typically, researchers use sample statistics as the basis for drawing conclusions

about population parameters. One problem with using samples, however, is that a sample

provides only limited information about the population. Although samples are generally

representative of their populations, a sample is not expected to give a perfectly accurate

picture of the whole population. There usually is some discrepancy between a sample statistic and the corresponding population parameter. This discrepancy is called sampling

error, and it creates the fundamental problem inferential statistics must always address.

DEFInItIon

Sampling error is the naturally occurring discrepancy, or error, that exists between

a sample statistic and the corresponding population parameter.

The concept of sampling error is illustrated in Figure 1.2. The figure shows a population of 1,000 college students and 2 samples, each with 5 students who were selected from

the population. Notice that each sample contains different individuals who have different

characteristics. Because the characteristics of each sample depend on the specific people in

the sample, statistics will vary from one sample to another. For example, the five students

in sample 1 have an average age of 19.8 years and the students in sample 2 have an average

age of 20.4 years.

It is also very unlikely that the statistics obtained for a sample will be identical to the

parameters for the entire population. In Figure 1.2, for example, neither sample has statistics that are exactly the same as the population parameters. You should also realize that

Figure 1.2 shows only two of the hundreds of possible samples. Each sample would contain

different individuals and would produce different statistics. This is the basic concept of

sampling error: sample statistics vary from one sample to another and typically are different from the corresponding population parameters.

One common example of sampling error is the error associated with a sample proportion. For example, in newspaper articles reporting results from political polls, you frequently find statements such as this:

Candidate Brown leads the poll with 51% of the vote. Candidate Jones has 42%

approval, and the remaining 7% are undecided. This poll was taken from a sample of registered voters and has a margin of error of plus-or-minus 4 percentage points.

The “margin of error” is the sampling error. In this case, the percentages that are reported

were obtained from a sample and are being generalized to the whole population. As always,

you do not expect the statistics from a sample to be perfect. There always will be some

“margin of error” when sample statistics are used to represent population parameters.

SEctIon 1.1 | Statistics, Science, and Observations

7

F I G U R E 1. 2

A demonstration of sampling error. Two

samples are selected from the same population.

Notice that the sample statistics are different

from one sample to another and all the sample

statistics are different from the corresponding

population parameters. The natural differences that exist, by chance, between a sample

statistic and population parameter are called

sampling error.

Population

of 1000 college students

Population Parameters

Average Age 5 21.3 years

Average IQ 5 112.5

65% Female, 35% Male

Sample #1

Sample #2

Eric

Jessica

Laura

Karen

Brian

Tom

Kristen

Sara

Andrew

John

Sample Statistics

Average Age 5 19.8

Average IQ 5 104.6

60% Female, 40% Male

Sample Statistics

Average Age 5 20.4

Average IQ 5 114.2

40% Female, 60% Male

As a further demonstration of sampling error, imagine that your statistics class is separated into two groups by drawing a line from front to back through the middle of the room.

Now imagine that you compute the average age (or height, or IQ) for each group. Will the

two groups have exactly the same average? Almost certainly they will not. No matter what

you chose to measure, you will probably find some difference between the two groups.

However, the difference you obtain does not necessarily mean that there is a systematic

difference between the two groups. For example, if the average age for students on the

right-hand side of the room is higher than the average for students on the left, it is unlikely

that some mysterious force has caused the older people to gravitate to the right side of

the room. Instead, the difference is probably the result of random factors such as chance.

The unpredictable, unsystematic differences that exist from one sample to another are an

example of sampling error.

■■Statistics in the Context of Research

The following example shows the general stages of a research study and demonstrates

how descriptive statistics and inferential statistics are used to organize and interpret the

data. At the end of the example, note how sampling error can affect the interpretation of

experimental results, and consider why inferential statistical methods are needed to deal

with this problem.

8

chaPtER 1 | Introduction to Statistics

ExamplE 1.1

Figure 1.3 shows an overview of a general research situation and demonstrates the roles that

descriptive and inferential statistics play. The purpose of the research study is to address a

question that we posed earlier: Do college students learn better by studying text on printed

pages or on a computer screen? Two samples are selected from the population of college

students. The students in sample A are given printed pages of text to study for 30 minutes

and the students in sample B study the same text on a computer screen. Next, all of the

students are given a multiple-choice test to evaluate their knowledge of the material. At this

point, the researcher has two sets of data: the scores for sample A and the scores for sample

B (see the figure). Now is the time to begin using statistics.

First, descriptive statistics are used to simplify the pages of data. For example, the

researcher could draw a graph showing the scores for each sample or compute the average score for each sample. Note that descriptive methods provide a simplified, organized

Step 1

Experiment:

Compare two

studying methods

Data

Test scores for the

students in each

sample

Step 2

Descriptive statistics:

Organize and simplify

Population of

College

Students

Sample A

Read from printed

pages

25

27

30

19

29

26

21

28

23

26

20

25

28

27

24

26

22

30

Average

Score = 26

Step 3

Inferential statistics:

Interpret results

F i g u r E 1. 3

The role of statistics in experimental

research.

Sample B

Read from computer

screen

20

20

23

25

22

18

22

17

28

19

24

25

30

27

23

21

22

19

Average

Score = 22

The sample data show a 4-point difference

between the two methods of studying. However,

there are two ways to interpret the results.

1. There actually is no difference between

the two studying methods, and the sample

difference is due to chance (sampling error).

2. There really is a difference between

the two methods, and the sample data

accurately reflect this difference.

The goal of inferential statistics is to help researchers

decide between the two interpretations.

SEctIon 1.1 | Statistics, Science, and Observations

9

description of the scores. In this example, the students who studied printed pages had an average score of 26 on the test, and the students who studied text on the computer averaged 22.

Once the researcher has described the results, the next step is to interpret the outcome.

This is the role of inferential statistics. In this example, the researcher has found a difference

of 4 points between the two samples (sample A averaged 26 and sample B averaged 22). The

problem for inferential statistics is to differentiate between the following two interpretations:

1. There is no real difference between the printed page and a computer screen, and

the 4-point difference between the samples is just an example of sampling error

(like the samples in Figure 1.2).

2. There really is a difference between the printed page and a computer screen, and

the 4-point difference between the samples was caused by the different methods

of studying.

In simple English, does the 4-point difference between samples provide convincing

evidence of a difference between the two studying methods, or is the 4-point difference just

chance? The purpose of inferential statistics is to answer this question.

■

lE arn in g Ch ECk

1. A researcher is interested in the sleeping habits of American college students.

A group of 50 students is interviewed and the researcher finds that these students

sleep an average of 6.7 hours per day. For this study, the average of 6.7 hours is an

example of a(n)

.

a. parameter

b. statistic

c. population

d. sample

2. A researcher is curious about the average IQ of registered voters in the state of Florida.

The entire group of registered voters in the state is an example of a

.

a. sample

b. statistic

c. population

d. parameter

3. Statistical techniques that summarize, organize, and simplify data are classified

as

.

a. population statistics

b. sample statistics

c. descriptive statistics

d. inferential statistics

4. In general,

statistical techniques are used to summarize the data from

a research study and

statistical techniques are used to determine what

conclusions are justified by the results.

a. inferential, descriptive

b. descriptive, inferential

c. sample, population

d. population, sample

10

chaPtER 1 | Introduction to Statistics

5. IQ tests are standardized so that the average score is 100 for the entire group of

people who take the test each year. However, if you selected a group of 20 people

who took the test and computed their average IQ score you probably would not get

100. What statistical concept explains the difference between your mean and the

mean for the entire group?

a. statistical error

b. inferential error

c. descriptive error

d. sampling error

an s wE r s

1. B, 2. C, 3. C, 4. B, 5. D

1.2 Data Structures, Research Methods, and Statistics

LEARNING OBJECTIVEs

4. Differentiate correlational, experimental, and nonexperimental research and describe

the data structures associated with each.

5. Define independent, dependent, and quasi-independent variables and recognize

examples of each.

■■Individual Variables: Descriptive Research

Some research studies are conducted simply to describe individual variables as they exist

naturally. For example, a college official may conduct a survey to describe the eating, sleeping, and study habits of a group of college students. When the results consist of numerical

scores, such as the number of hours spent studying each day, they are typically described

by the statistical techniques that are presented in Chapters 3 and 4. Non-numerical scores

are typically described by computing the proportion or percentage in each category. For

example, a recent newspaper article reported that 34.9% of Americans are obese, which is

roughly 35 pounds over a healthy weight.

■■Relationships Between Variables

Most research, however, is intended to examine relationships between two or more variables. For example, is there a relationship between the amount of violence in the video

games played by children and the amount of aggressive behavior they display? Is there a

relationship between the quality of breakfast and academic performance for elementary

school children? Is there a relationship between the number of hours of sleep and grade

point average for college students? To establish the existence of a relationship, researchers must make observations—that is, measurements of the two variables. The resulting

measurements can be classified into two distinct data structures that also help to classify

different research methods and different statistical techniques. In the following section we

identify and discuss these two data structures.

I. One Group with Two Variables Measured for Each Individual: The Correlational Method One method for examining the relationship between variables is to

observe the two variables as they exist naturally for a set of individuals. That is, simply

SEctIon 1.2 | Data Structures, Research Methods, and Statistics

11

measure the two variables for each individual. For example, research has demonstrated a

relationship between sleep habits, especially wake-up time, and academic performance

for college students (Trockel, Barnes, and Egget, 2000). The researchers used a survey to

measure wake-up time and school records to measure academic performance for each student. Figure 1.4 shows an example of the kind of data obtained in the study. The researchers then look for consistent patterns in the data to provide evidence for a relationship

between variables. For example, as wake-up time changes from one student to another, is

there also a tendency for academic performance to change?

Consistent patterns in the data are often easier to see if the scores are presented in a

graph. Figure 1.4 also shows the scores for the eight students in a graph called a scatter

plot. In the scatter plot, each individual is represented by a point so that the horizontal

position corresponds to the student’s wake-up time and the vertical position corresponds

to the student’s academic performance score. The scatter plot shows a clear relationship

between wake-up time and academic performance: as wake-up time increases, academic

performance decreases.

A research study that simply measures two different variables for each individual and

produces the kind of data shown in Figure 1.4 is an example of the correlational method,

or the correlational research strategy.

In the correlational method, two different variables are observed to determine

whether there is a relationship between them.

DEFInItIon

■■Statistics for the Correlational Method

When the data from a correlational study consist of numerical scores, the relationship

between the two variables is usually measured and described using a statistic called a

correlation. Correlations and the correlational method are discussed in detail in Chapters 15 and 16. Occasionally, the measurement process used for a correlational study

simply classifies individuals into categories that do not correspond to numerical values.

For example, a researcher could classify a group of college students by gender (male

Student

Wake-up

Time

Academic

Performance

A

B

C

D

E

F

G

H

11

9

9

12

7

10

10

8

2.4

3.6

3.2

2.2

3.8

2.2

3.0

3.0

(b)

Academic performance

(a)

3.8

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

7

F i g u r E 1. 4

8

9

10

11

12

Wake-up time

One of two data structures for evaluating the relationship between variables. Note that there are two separate measurements for each individual (wake-up time and academic performance). The same scores are shown in a table (a) and in

a graph (b).

12

chaPtER 1 | Introduction to Statistics

or female) and by cell-phone preference (talk or text). Note that the researcher has two

scores for each individual but neither of the scores is a numerical value. This type of data

is typically summarized in a table showing how many individuals are classified into each

of the possible categories. Table 1.1 shows an example of this kind of summary table. The

table shows for example, that 30 of the males in the sample preferred texting to talking.

This type of data can be coded with numbers (for example, male = 0 and female = 1)

so that it is possible to compute a correlation. However, the relationship between variables for non-numerical data, such as the data in Table 1.1, is usually evaluated using

a statistical technique known as a chi-square test. Chi-square tests are presented in

Chapter 17.

Ta b lE 1.1

Correlational data consisting of non-numerical scores. Note that there are two measurements for

each individual: gender and cell phone preference. The numbers indicate how many people are in

each category. For example, out of the 50 males, 30 prefer text over talk.

Cell Phone Preference

Text

Talk

Males

30

20

50

Females

25

25

50

■■Limitations of the Correlational Method

The results from a correlational study can demonstrate the existence of a relationship

between two variables, but they do not provide an explanation for the relationship. In

particular, a correlational study cannot demonstrate a cause-and-effect relationship. For

example, the data in Figure 1.4 show a systematic relationship between wake-up time and

academic performance for a group of college students; those who sleep late tend to have

lower performance scores than those who wake early. However, there are many possible

explanations for the relationship and we do not know exactly what factor (or factors) is

responsible for late sleepers having lower grades. In particular, we cannot conclude that

waking students up earlier would cause their academic performance to improve, or that

studying more would cause students to wake up earlier. To demonstrate a cause-and-effect

relationship between two variables, researchers must use the experimental method, which

is discussed next.

II. Comparing Two (or More) Groups of Scores: Experimental and Nonexperimental Methods The second method for examining the relationship between two

variables involves the comparison of two or more groups of scores. In this situation, the

relationship between variables is examined by using one of the variables to define the

groups, and then measuring the second variable to obtain scores for each group. For example, Polman, de Castro, and van Aken (2008) randomly divided a sample of 10-year-old

boys into two groups. One group then played a violent video game and the second played

a nonviolent game. After the game-playing session, the children went to a free play period

and were monitored for aggressive behaviors (hitting, kicking, pushing, frightening, name

calling, fighting, quarreling, or teasing another child). An example of the resulting data is

shown in Figure 1.5. The researchers then compare the scores for the violent-video group

with the scores for the nonviolent-video group. A systematic difference between the two

groups provides evidence for a relationship between playing violent video games and

aggressive behavior for 10-year-old boys.

SEctIon 1.2 | Data Structures, Research Methods, and Statistics

F i g u r E 1. 5

Evaluating the relationship between

variables by comparing groups of scores.

Note that the values of

one variable are used

to define the groups

and the second variable is measured to

obtain scores within

each group.

One variable (type of video game)

is used to define groups

A second variable (aggressive behavior)

is measured to obtain scores within each group

Violent

Nonviolent

7

8

10

7

9

8

6

10

9

6

8

4

8

3

6

5

3

4

4

5

13

Compare groups

of scores

■■Statistics for Comparing Two (or More) Groups of Scores

Most of the statistical procedures presented in this book are designed for research studies that compare groups of scores like the study in Figure 1.5. Specifically, we examine

descriptive statistics that summarize and describe the scores in each group and we use

inferential statistics to determine whether the differences between the groups can be generalized to the entire population.

When the measurement procedure produces numerical scores, the statistical evaluation typically involves computing the average score for each group and then comparing

the averages. The process of computing averages is presented in Chapter 3, and a variety

of statistical techniques for comparing averages are presented in Chapters 8–14. If the

measurement process simply classifies individuals into non-numerical categories, the statistical evaluation usually consists of computing proportions for each group and then comparing proportions. Previously, in Table 1.1, we presented an example of non-numerical

data examining the relationship between gender and cell-phone preference. The same data

can be used to compare the proportions for males with the proportions for females. For

example, using text is preferred by 60% of the males compared to 50% of the females. As

before, these data are evaluated using a chi-square test, which is presented in Chapter 17.

■■Experimental and Nonexperimental Methods

There are two distinct research methods that both produce groups of scores to be compared:

the experimental and the nonexperimental strategies. These two research methods use

exactly the same statistics and they both demonstrate a relationship between two variables.

The distinction between the two research strategies is how the relationship is interpreted.

The results from an experiment allow a cause-and-effect explanation. For example, we can

conclude that changes in one variable are responsible for causing differences in a second

variable. A nonexperimental study does not permit a cause-and effect explanation. We can

say that changes in one variable are accompanied by changes in a second variable, but we

cannot say why. Each of the two research methods is discussed in the following sections.

■■The Experimental Method

One specific research method that involves comparing groups of scores is known as the

experimental method or the experimental research strategy. The goal of an experimental

study is to demonstrate a cause-and-effect relationship between two variables. Specifically,

14

chaPtER 1 | Introduction to Statistics

an experiment attempts to show that changing the value of one variable causes changes to

occur in the second variable. To accomplish this goal, the experimental method has two

characteristics that differentiate experiments from other types of research studies:

1. Manipulation The researcher manipulates one variable by changing its value from

one level to another. In the Polman et al. (2008) experiment examining the effect

of violence in video games (Figure 1.5), the researchers manipulate the amount of

violence by giving one group of boys a violent game to play and giving the other

group a nonviolent game. A second variable is observed (measured) to determine

whether the manipulation causes changes to occur.

2. Control The researcher must exercise control over the research situation to ensure

that other, extraneous variables do not influence the relationship being examined.

In more complex experiments, a researcher

may systematically

manipulate more than

one variable and may

observe more than one

variable. Here we are

considering the simplest

case, in which only one

variable is manipulated

and only one variable is

observed.

To demonstrate these two characteristics, consider the Polman et al. (2008) study examining the effect of violence in video games (see Figure 1.5). To be able to say that the difference in aggressive behavior is caused by the amount of violence in the game, the researcher

must rule out any other possible explanation for the difference. That is, any other variables

that might affect aggressive behavior must be controlled. There are two general categories

of variables that researchers must consider:

1. Participant Variables These are characteristics such as age, gender, and intelligence that vary from one individual to another. Whenever an experiment compares

different groups of participants (one group in treatment A and a different group

in treatment B), researchers must ensure that participant variables do not differ

from one group to another. For the experiment shown in Figure 1.5, for example,

the researchers would like to conclude that the violence in the video game causes

a change in the participants’ aggressive behavior. In the study, the participants in

both conditions were 10-year-old boys. Suppose, however, that the participants in

the nonviolent condition were primarily female and those in the violent condition

were primarily male. In this case, there is an alternative explanation for the difference in aggression that exists between the two groups. Specifically, the difference

between groups may have been caused by the amount of violence in the game,

but it also is possible that the difference was caused by the participants’ gender

(females are less aggressive than males). Whenever a research study allows more

than one explanation for the results, the study is said to be confounded because it is

impossible to reach an unambiguous conclusion.

2. Environmental Variables These are characteristics of the environment such as

lighting, time of day, and weather conditions. A researcher must ensure that the

individuals in treatment A are tested in the same environment as the individuals

in treatment B. Using the video game violence experiment (see Figure 1.5) as an

example, suppose that the individuals in the nonviolent condition were all tested in

the morning and the individuals in the violent condition were all tested in the evening. Again, this would produce a confounded experiment because the researcher

could not determine whether the differences in aggressive behavior were caused by

the amount of violence or caused by the time of day.

Researchers typically use three basic techniques to control other variables. First, the

researcher could use random assignment, which means that each participant has an equal

chance of being assigned to each of the treatment conditions. The goal of random assignment is to distribute the participant characteristics evenly between the two groups so that

neither group is noticeably smarter (or older, or faster) than the other. Random assignment

can also be used to control environmental variables. For example, participants could be

assigned randomly for testing either in the morning or in the afternoon. A second technique

SEctIon 1.2 | Data Structures, Research Methods, and Statistics

15

for controlling variables is to use matching to ensure equivalent groups or equivalent environments. For example, the researcher could match groups by ensuring that every group

has exactly 60% females and 40% males. Finally, the researcher can control variables by

holding them constant. For example, in the video game violence study discussed earlier

(Polman et al., 2008), the researchers used only 10-year-old boys as participants (holding

age and gender constant). In this case the researchers can be certain that one group is not

noticeably older or has a larger proportion of females than the other.

DEFInItIon

In the experimental method, one variable is manipulated while another variable

is observed and measured. To establish a cause-and-effect relationship between the

two variables, an experiment attempts to control all other variables to prevent them

from influencing the results.

■■Terminology in the Experimental Method

Specific names are used for the two variables that are studied by the experimental method. The

variable that is manipulated by the experimenter is called the independent variable. It can be

identified as the treatment conditions to which participants are assigned. For the example in

Figure 1.5, the amount of violence in the video game is the independent variable. The variable

that is observed and measured to obtain scores within each condition is the dependent variable. For the example in Figure 1.5, the level of aggressive behavior is the dependent variable.

DEFInItIon

The independent variable is the variable that is manipulated by the researcher. In

behavioral research, the independent variable usually consists of the two (or more) treatment conditions to which subjects are exposed. The independent variable consists of the

antecedent conditions that were manipulated prior to observing the dependent variable.

The dependent variable is the one that is observed to assess the effect of the treatment.

Control Conditions in an Experiment

An experimental study evaluates the relationship between two variables by manipulating one variable (the independent variable) and

measuring one variable (the dependent variable). Note that in an experiment only one

variable is actually measured. You should realize that this is different from a correlational

study, in which both variables are measured and the data consist of two separate scores

for each individual.

Often an experiment will include a condition in which the participants do not receive

any treatment. The scores from these individuals are then compared with scores from participants who do receive the treatment. The goal of this type of study is to demonstrate that

the treatment has an effect by showing that the scores in the treatment condition are substantially different from the scores in the no-treatment condition. In this kind of research,

the no-treatment condition is called the control condition, and the treatment condition is

called the experimental condition.

DEFInItIon

Individuals in a control condition do not receive the experimental treatment.

Instead, they either receive no treatment or they receive a neutral, placebo treatment. The purpose of a control condition is to provide a baseline for comparison

with the experimental condition.

Individuals in the experimental condition do receive the experimental treatment.

16

chaPtER 1 | Introduction to Statistics

Note that the independent variable always consists of at least two values. (Something

must have at least two different values before you can say that it is “variable.”) For the

video game violence experiment (see Figure 1.5), the independent variable is the amount

of violence in the video game. For an experiment with an experimental group and a control

group, the independent variable is treatment versus no treatment.

■■Nonexperimental Methods: Nonequivalent Groups

and Pre-Post Studies

In informal conversation, there is a tendency for people to use the term experiment to refer

to any kind of research study. You should realize, however, that the term only applies to

studies that satisfy the specific requirements outlined earlier. In particular, a real experiment must include manipulation of an independent variable and rigorous control of other,

extraneous variables. As a result, there are a number of other research designs that are not

true experiments but still examine the relationship between variables by comparing groups

of scores. Two examples are shown in Figure 1.6 and are discussed in the following paragraphs. This type of research study is classified as nonexperimental.

The top part of Figure 1.6 shows an example of a nonequivalent groups study comparing boys and girls. Notice that this study involves comparing two groups of scores (like an

experiment). However, the researcher has no ability to control which participants go into

F i g u r E 1.6

(a)

Two examples of nonexperimental

studies that involve comparing two

groups of scores. In (a) the study

uses two preexisting groups (boys/

girls) and measures a dependent

variable (verbal scores) in each

group. In (b) the study uses time

(before/after) to define the two

groups and measures a dependent

variable (depression) in each group.

Variable #1: Subject gender

(the quasi-independent variable)

Not manipulated, but used

to create two groups of subjects

Variable #2: Verbal test scores

(the dependent variable)

Measured in each of the

two groups

Boys

Girls

17

19

16

12

17

18

15

16

12

10

14

15

13

12

11

13

Any

difference?

(b)

Variable #1: Time

(the quasi-independent variable)

Not manipulated, but used

to create two groups of scores

Variable #2: Depression scores

(the dependent variable)

Measured at each of the two

different times

Before

Therapy

After

Therapy

17

19

16

12

17

18

15

16

12

10

14

15

13

12

11

13

Any

difference?

SEctIon 1.2 | Data Structures, Research Methods, and Statistics

Correlational studies are

also examples of nonexperimental research. In

this section, however, we

are discussing nonexperimental studies that

compare two or more

groups of scores.

17

which group—all the males must be in the boy group and all the females must be in the

girl group. Because this type of research compares preexisting groups, the researcher cannot control the assignment of participants to groups and cannot ensure equivalent groups.

Other examples of nonequivalent group studies include comparing 8-year-old children and

10-year-old children, people with an eating disorder and those with no disorder, and comparing children from a single-parent home and those from a two-parent home. Because it

is impossible to use techniques like random assignment to control participant variables and

ensure equivalent groups, this type of research is not a true experiment.

The bottom part of Figure 1.6 shows an example of a pre–post study comparing depression scores before therapy and after therapy. The two groups of scores are obtained by

measuring the same variable (depression) twice for each participant; once before therapy

and again after therapy. In a pre-post study, however, the researcher has no control over

the passage of time. The “before” scores are always measured earlier than the “after”

scores. Although a difference between the two groups of scores may be caused by the

treatment, it is always possible that the scores simply change as time goes by. For example, the depression scores may decrease over time in the same way that the symptoms of

a cold disappear over time. In a pre–post study the researcher also has no control over

other variables that change with time. For example, the weather could change from dark

and gloomy before therapy to bright and sunny after therapy. In this case, the depression

scores could improve because of the weather and not because of the therapy. Because the

researcher cannot control the passage of time or other variables related to time, this study

is not a true experiment.

Terminology in Nonexperimental Research

Although the two research studies

shown in Figure 1.6 are not true experiments, you should notice that they produce the

same kind of data that are found in an experiment (see Figure 1.5). In each case, one variable is used to create groups, and a second variable is measured to obtain scores within

each group. In an experiment, the groups are created by manipulation of the independent

variable, and the participants’ scores are the dependent variable. The same terminology is

often used to identify the two variables in nonexperimental studies. That is, the variable

that is used to create groups is the independent variable and the scores are the dependent

variable. For example, the top part of Figure 1.6, gender (boy/girl), is the independent

variable and the verbal test scores are the dependent variable. However, you should realize that gender (boy/girl) is not a true independent variable because it is not manipulated.

For this reason, the “independent variable” in a nonexperimental study is often called a

quasi-independent variable.

DEFInItIon

lE arn in g Ch ECk

In a nonexperimental study, the “independent variable” that is used to create the

different groups of scores is often called the quasi-independent variable.

1. In a correlational study, how many variables are measured for each individual and

how many groups of scores are obtained?

a. 1 variable and 1 group

b. 1 variable and 2 groups

c. 2 variables and 1 group

d. 2 variables and 2 groups

18

chaPtER 1 | Introduction to Statistics

2. A research study comparing alcohol use for college students in the United States

and Canada reports that more Canadian students drink but American students drink

more (Kuo, Adlaf, Lee, Gliksman, Demers, and Wechsler, 2002). What research

design did this study use?

a. correlational

b. experimental

c. nonexperimental

d. noncorrelational

3. Stephens, Atkins, and Kingston (2009) found that participants were able to tolerate

more pain when they shouted their favorite swear words over and over than when

they shouted neutral words. For this study, what is the independent variable?

a. the amount of pain tolerated

b. the participants who shouted swear words

c. the participants who shouted neutral words

d. the kind of word shouted by the participants

an s wE r s

1.3

1. C, 2. C, 3. D

Variables and Measurement

LEARNING OBJECTIVEs

6. Explain why operational definitions are developed for constructs and identify the two

components of an operational definition.

7. Describe discrete and continuous variables and identify examples of each.

8. Differentiate nominal, ordinal, interval, and ratio scales of measurement.

■■Constructs and Operational Definitions

The scores that make up the data from a research study are the result of observing and

measuring variables. For example, a researcher may finish a study with a set of IQ scores,

personality scores, or reaction-time scores. In this section, we take a closer look at the variables that are being measured and the process of measurement.

Some variables, such as height, weight, and eye color are well-defined, concrete entities that can be observed and measured directly. On the other hand, many variables studied

by behavioral scientists are internal characteristics that people use to help describe and

explain behavior. For example, we say that a student does well in school because he or

she is intelligent. Or we say that someone is anxious in social situations, or that someone

seems to be hungry. Variables like intelligence, anxiety, and hunger are called constructs,

and because they are intangible and cannot be directly observed, they are often called

hypothetical constructs.

Although constructs such as intelligence are internal characteristics that cannot be

directly observed, it is possible to observe and measure behaviors that are representative

of the construct. For example, we cannot “see” intelligence but we can see examples of

intelligent behavior. The external behaviors can then be used to create an operational definition for the construct. An operational definition defines a construct in terms of external

SEctIon 1.3 | Variables and Measurement

19

behaviors that can be observed and measured. For example, your intelligence is measured

and defined by your performance on an IQ test, or hunger can be measured and defined by

the number of hours since last eating.

DEFInItIon

Constructs are internal attributes or characteristics that cannot be directly

observed but are useful for describing and explaining behavior.

An operational definition identifies a measurement procedure (a set of operations) for measuring an external behavior and uses the resulting measurements as

a definition and a measurement of a hypothetical construct. Note that an operational definition has two components. First, it describes a set of operations for

measuring a construct. Second, it defines the construct in terms of the resulting

measurements.

■■Discrete and Continuous Variables

The variables in a study can be characterized by the type of values that can be assigned to

them. A discrete variable consists of separate, indivisible categories. For this type of variable, there are no intermediate values between two adjacent categories. Consider the values

displayed when dice are rolled. Between neighboring values—for example, seven dots and

eight dots—no other values can ever be observed.

DEFInItIon

A discrete variable consists of separate, indivisible categories. No values can exist

between two neighboring categories.

Discrete variables are commonly restricted to whole, countable numbers—for

example, the number of children in a family or the number of students attending class.

If you observe class attendance from day to day, you may count 18 students one day

and 19 students the next day. However, it is impossible ever to observe a value between

18 and 19. A discrete variable may also consist of observations that differ qualitatively.

For example, people can be classified by gender (male or female), by occupation

(nurse, teacher, lawyer, etc.), and college students can by classified by academic major

(art, biology, chemistry, etc.). In each case, the variable is discrete because it consists

of separate, indivisible categories.

On the other hand, many variables are not discrete. Variables such as time, height, and

weight are not limited to a fixed set of separate, indivisible categories. You can measure

time, for example, in hours, minutes, seconds, or fractions of seconds. These variables

are called continuous because they can be divided into an infinite number of fractional

parts.

DEFInItIon

For a continuous variable, there are an infinite number of possible values that fall

between any two observed values. A continuous variable is divisible into an infinite

number of fractional parts.

Suppose, for example, that a researcher is measuring weights for a group of individuals

participating in a diet study. Because weight is a continuous variable, it can be pictured as

a continuous line (Figure 1.7). Note that there are an infinite number of possible points on

20

chaPtER 1 | Introduction to Statistics

F i g u r E 1.7

149.6

When measuring weight to

the nearest whole pound,

149.6 and 150.3 are assigned

the value of 150 (top). Any

value in the interval between

149.5 and 150.5 is given the

value of 150.

150.3

149

151

152

151

152

150

149.5

149

148.5

150.5

150

149.5

150.5

151.5

152.5

Real limits

the line without any gaps or separations between neighboring points. For any two different

points on the line, it is always possible to find a third value that is between the two points.

Two other factors apply to continuous variables:

1. When measuring a continuous variable, it should be very rare to obtain identical

measurements for two different individuals. Because a continuous variable has an

infinite number of possible values, it should be almost impossible for two people to

have exactly the same score. If the data show a substantial number of tied scores,

then you should suspect that the measurement procedure is very crude or that the

variable is not really continuous.

2. When measuring a continuous variable, each measurement category is actually an

interval that must be defined by boundaries. For example, two people who both

claim to weigh 150 pounds are probably not exactly the same weight. However,

they are both around 150 pounds. One person may actually weigh 149.6 and the

other 150.3. Thus, a score of 150 is not a specific point on the scale but instead is

an interval (see Figure 1.7). To differentiate a score of 150 from a score of 149 or

151, we must set up boundaries on the scale of measurement. These boundaries are

called real limits and are positioned exactly halfway between adjacent scores. Thus,

a score of X = 150 pounds is actually an interval bounded by a lower real limit

of 149.5 at the bottom and an upper real limit of 150.5 at the top. Any individual

whose weight falls between these real limits will be assigned a score of X = 150.

DEFInItIon

Real limits are the boundaries of intervals for scores that are represented on a continuous number line. The real limit separating two adjacent scores is located exactly

halfway between the scores. Each score has two real limits. The upper real limit is

at the top of the interval, and the lower real limit is at the bottom.

The concept of real limits applies to any measurement of a continuous variable, even

when the score categories are not whole numbers. For example, if you were measuring time

to the nearest tenth of a second, the measurement categories would be 31.0, 31.1, 31.2, and

so on. Each of these categories represents an interval on the scale that is bounded by real

limits. For example, a score of X = 31.1 seconds indicates that the actual measurement

is in an interval bounded by a lower real limit of 31.05 and an upper real limit of 31.15.

Remember that the real limits are always halfway between adjacent categories.

SEctIon 1.3 | Variables and Measurement

Students often ask

whether a value of

exactly 150.5 should

be assigned to the

X = 150 interval or the

X = 151 interval. The

answer is that 150.5 is

the boundary between

the two intervals and is

not necessarily in one

or the other. Instead,

the placement of 150.5

depends on the rule that

you are using for rounding numbers. If you

are rounding up, then

150.5 goes in the higher

interval (X = 151) but if

you are rounding down,

then it goes in the lower

interval (X = 150).

21

Later in this book, real limits are used for constructing graphs and for various calculations with continuous scales. For now, however, you should realize that real limits are a

necessity whenever you make measurements of a continuous variable.

Finally, we should warn you that the terms continuous and discrete apply to the variables that are being measured and not to the scores that are obtained from the measurement.

For example, measuring people’s heights to the nearest inch produces scores of 60, 61, 62,

and so on. Although the scores may appear to be discrete numbers, the underlying variable

is continuous. One key to determining whether a variable is continuous or discrete is that

a continuous variable can be divided into any number of fractional parts. Height can be

measured to the nearest inch, the nearest 0.5 inch, or the nearest 0.1 inch. Similarly, a professor evaluating students’ knowledge could use a pass/fail system that classifies students

into two broad categories. However, the professor could choose to use a 10-point quiz that

divides student knowledge into 11 categories corresponding to quiz scores from 0 to 10. Or

the professor could use a 100-point exam that potentially divides student knowledge into

101 categories from 0 to 100. Whenever you are free to choose the degree of precision or

the number of categories for measuring a variable, the variable must be continuous.

■■Scales of Measurement

It should be obvious by now that data collection requires that we make measurements of

our observations. Measurement involves assigning individuals or events to categories. The

categories can simply be names such as male/female or employed/unemployed, or they

can be numerical values such as 68 inches or 175 pounds. The categories used to measure

a variable make up a scale of measurement, and the relationships between the categories determine different types of scales. The distinctions among the scales are important

because they identify the limitations of certain types of measurements and because certain

statistical procedures are appropriate for scores that have been measured on some scales

but not on others. If you were interested in people’s heights, for example, you could measure a group of individuals by simply classifying them into three categories: tall, medium,

and short. However, this simple classification would not tell you much about the actual

heights of the individuals, and these measurements would not give you enough information to calculate an average height for the group. Although the simple classification would

be adequate for some purposes, you would need more sophisticated measurements before

you could answer more detailed questions. In this section, we examine four different

scales of measurement, beginning with the simplest and moving to the most sophisticated.

■■The Nominal Scale

The word nominal means “having to do with names.” Measurement on a nominal scale

involves classifying individuals into categories that have different names but are not related

to each other in any systematic way. For example, if you were measuring the academic

majors for a group of college students, the categories would be art, biology, business,

chemistry, and so on. Each student would be classified in one category according to his

or her major. The measurements from a nominal scale allow us to determine whether two

individuals are different, but they do not identify either the direction or the size of the difference. If one student is an art major and another is a biology major we can say that they

are different, but we cannot say that art is “more than” or “less than” biology and we cannot

specify how much difference there is between art and biology. Other examples of nominal

scales include classifying people by race, gender, or occupation.

DEFInItIon

A nominal scale consists of a set of categories that have different names. Measurements on a nominal scale label and categorize observations, but do not make any

quantitative distinctions between observations.

22

chaPtER 1 | Introduction to Statistics

Although the categories on a nominal scale are not quantitative values, they are occasionally represented by numbers. For example, the rooms or offices in a building may be

identified by numbers. You should realize that the room numbers are simply names and do

not reflect any quantitative information. Room 109 is not necessarily bigger than Room

100 and certainly not 9 points bigger. It also is fairly common to use numerical values as a

code for nominal categories when data are entered into computer programs. For example,

the data from a survey may code males with a 0 and females with a 1. Again, the numerical

values are simply names and do not represent any quantitative difference. The scales that

follow do reflect an attempt to make quantitative distinctions.

■■The Ordinal Scale

The categories that make up an ordinal scale not only have different names (as in a nominal

scale) but also are organized in a fixed order corresponding to differences of magnitude.

DEFInItIon

An ordinal scale consists of a set of categories that are organized in an ordered

sequence. Measurements on an ordinal scale rank observations in terms of size or

magnitude.

Often, an ordinal scale consists of a series of ranks (first, second, third, and so on) like

the order of finish in a horse race. Occasionally, the categories are identified by verbal

labels like small, medium, and large drink sizes at a fast-food restaurant. In either case, the

fact that the categories form an ordered sequence means that there is a directional relationship between categories. With measurements from an ordinal scale, you can determine

whether two individuals are different and you can determine the direction of difference.

However, ordinal measurements do not allow you to determine the size of the difference

between two individuals. In a NASCAR race, for example, the first-place car finished faster

than the second-place car, but the ranks don’t tell you how much faster. Other examples of

ordinal scales include socioeconomic class (upper, middle, lower) and T-shirt sizes (small,

medium, large). In addition, ordinal scales are often used to measure variables for which it

is difficult to assign numerical scores. For example, people can rank their food preferences

but might have trouble explaining “how much” they prefer chocolate ice cream to steak.

■■The Interval and Ratio Scales

Both an interval scale and a ratio scale consist of a series of ordered categories (like an

ordinal scale) with the additional requirement that the categories form a series of intervals

that are all exactly the same size. Thus, the scale of measurement consists of a series of

equal intervals, such as inches on a ruler. Other examples of interval and ratio scales are the

measurement of time in seconds, weight in pounds, and temperature in degrees Fahrenheit.

Note that, in each case, one interval (1 inch, 1 second, 1 pound, 1 degree) is the same size,

no matter where it is located on the scale. The fact that the intervals are all the same size

makes it possible to determine both the size and the direction of the difference between two

measurements. For example, you know that a measurement of 80° Fahrenheit is higher than

a measure of 60°, and you know that it is exactly 20° higher.

The factor that differentiates an interval scale from a ratio scale is the nature of the zero

point. An interval scale has an arbitrary zero point. That is, the value 0 is assigned to a particular location on the scale simply as a matter of convenience or reference. In particular, a

value of zero does not indicate a total absence of the variable being measured. For example

a temperature of 0º Fahrenheit does not mean that there is no temperature, and it does

not prohibit the temperature from going even lower. Interval scales with an arbitrary zero

SEctIon 1.3 | Variables and Measurement

23

point are relatively rare. The two most common examples are the Fahrenheit and Celsius

temperature scales. Other examples include golf scores (above and below par) and relative

measures such as above and below average rainfall.

A ratio scale is anchored by a zero point that is not arbitrary but rather is a meaningful

value representing none (a complete absence) of the variable being measured. The existence

of an absolute, non-arbitrary zero point means that we can measure the absolute amount of

the variable; that is, we can measure the distance from 0. This makes it possible to compare

measurements in terms of ratios. For example, a gas tank with 10 gallons (10 more than 0) has

twice as much gas as a tank with only 5 gallons (5 more than 0). Also note that a completely

empty tank has 0 gallons. To recap, with a ratio scale, we can measure the direction and the

size of the difference between two measurements and we can describe the difference in terms

of a ratio. Ratio scales are quite common and include physical measures such as height and

weight, as well as variables such as reaction time or the number of errors on a test. The distinction between an interval scale and a ratio scale is demonstrated in Example 1.2.

DEFInItIon

An interval scale consists of ordered categories that are all intervals of exactly the

same size. Equal differences between numbers on scale reflect equal differences in

magnitude. However, the zero point on an interval scale is arbitrary and does not

indicate a zero amount of the variable being measured.

A ratio scale is an interval scale with the additional feature of an absolute zero

point. With a ratio scale, ratios of numbers do reflect ratios of magnitude.

ExamplE 1.2

A researcher obtains measurements of height for a group of 8-year-old boys. Initially, the

researcher simply records each child’s height in inches, obtaining values such as 44, 51, 49,

and so on. These initial measurements constitute a ratio scale. A value of zero represents no

height (absolute zero). Also, it is possible to use these measurements to form ratios. For example, a child who is 60 inches tall is one and a half times taller than a child who is 40 inches tall.

Now suppose that the researcher converts the…

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