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8th edition

Frederick J Gravetter

State University of New York, Brockport

Larry B. Wallnau

State University of New York, Brockport

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Essentials of Statistics for the Behavioral

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Frederick J Gravetter and Larry B. Wallnau

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Chapter 1 Introduction to Statistics 3

Chapter 2 Frequency Distributions 37

Chapter 3 Measures of Central Tendency 59

Chapter 4 Measures of Variability 89

Chapter 5 z-Scores: Location of Scores and Standardized Distributions 123

Chapter 6 Probability 149

Chapter 7 Probability and Samples: The Distribution of Sample Means 175

Chapter 8 Introduction to Hypothesis Testing 203

Using t Statistics for Inferences About Population Means and Mean DifferencesPART III

Chapter 9 Introduction to the t Statistic 249

Chapter 10 The t Test for Two Independent Samples 279

Chapter 11 The t Test for Two Related Samples 313

Analysis of Variance: Tests for Differences Among Two or More Population MeansPART IV

Chapter 12 Introduction to Analysis of Variance 345

Chapter 13 Repeated-Measures and Two-Factor Analysis of Variance 393

iii

Contents in Brief

Introduction and Descriptive StatisticsPART I

Foundations of Inferential StatisticsPART II

Correlations and Nonparametric TestsPART V

Chapter 14 Correlation 449

Chapter 15 The Chi-Square Statistic: Tests for Goodness of Fit and Independence 509

iv CONTENTS IN BRIEF

v

Contents

Introduction to Statistics 3

1.1 Statistics, Science, and Observations 4 1.2 Populations and Samples 5 1.3 Data Structures, Research Methods, and Statistics 12 1.4 Variables and Measurement 20 1.5 Statistical Notation 26 Summary 30 Focus on Problem Solving 32 Demonstration 1.1 32 Problems 33

Frequency Distributions 37

2.1 Introduction to Frequency Distributions 38 2.2 Frequency Distribution Tables 38 2.3 Frequency Distribution Graphs 44 2.4 The Shape of a Frequency Distribution 50 Summary 52 Focus on Problem Solving 54 Demonstration 2.1 55 Problems 56

Measures of Central Tendency 59

3.1 Defining Central Tendency 60 3.2 The Mean 61 3.3 The Median 69 3.4 The Mode 73 3.5 Selecting a Measure of Central Tendency 74 3.6 Central Tendency and the Shape of the Distribution 80 Summary 82 Focus on Problem Solving 84 Demonstration 3.1 84 Problems 85

Measures of Variability 89

4.1 Defining Variability 90 4.2 The Range 91

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Introduction and Descriptive StatisticsPART I

vi CONTENTS

4.3 Standard Deviation and Variance for a Population 92 4.4 Standard Deviation and Variance for a Sample 99 4.5 More About Variance and Standard Deviation 104 Summary 112 Focus on Problem Solving 114 Demonstration 4.1 115 Problems 115

Part I Review 119 Review Exercises 119

z-Scores: Location of Scores and Standardized Distributions 123

5.1 Introduction to z-Scores 124 5.2 z-Scores and Location in a Distribution 125 5.3 Using z-Scores to Standardize a Distribution 131 5.4 Other Standardized Distributions Based on z-Scores 136 5.5 Computing z-Scores for a Sample 138 5.6 Looking Ahead to Inferential Statistics 140 Summary 143 Focus on Problem Solving 144 Demonstration 5.1 145 Demonstration 5.2 145 Problems 146

Probability 149

6.1 Introduction to Probability 150 6.2 Probability and the Normal Distribution 155 6.3 Probabilities and Proportions for Scores from

a Normal Distribution 162 6.4 Looking Ahead to Inferential Statistics 169 Summary 171 Focus on Problem Solving 172 Demonstration 6.1 172 Problems 173

Probability and Samples: The Distribution of Sample Means 175

7.1 Samples and Populations 176 7.2 The Distribution of Sample Means 176 7.3 Probability and the Distribution of Sample Means 186 7.4 More About Standard Error 190

Chapter 5

Chapter 6

Chapter 7

Foundations of Inferential StatisticsPART II

CONTENTS vii

7.5 Looking Ahead to Inferential Statistics 194 Summary 198 Focus on Problem Solving 199 Demonstration 7.1 200 Problems 201

Introduction to Hypothesis Testing 203

8.1 The Logic of Hypothesis Testing 204 8.2 Uncertainty and Errors in Hypothesis Testing 213 8.3 More About Hypothesis Tests 217 8.4 Directional (One-Tailed) Hypothesis Tests 224 8.5 Concerns About Hypothesis Testing: Measuring Effect Size 227 8.6 Statistical Power 232 Summary 237 Focus on Problem Solving 239 Demonstration 8.1 240 Demonstration 8.2 240 Problems 241

Part II Review 245 Review Exercises 245

Chapter 8

Chapter 9

Chapter 10

Introduction to the t Statistic 249

9.1 The t Statistic: An Alternative to z 250 9.2 Hypothesis Tests with the t Statistic 255 9.3 Measuring Effect Size for the t Statistic 260 9.4 Directional Hypotheses and One-Tailed Tests 268 Summary 271 Focus on Problem Solving 273 Demonstration 9.1 273 Demonstration 9.2 274 Problems 275

The t Test for Two Independent Samples 279

10.1 Introduction to the Independent-Measures Design 280 10.2 The t Statistic for an Independent-Measures Research Design 281 10.3 Hypothesis Tests and Effect Size with the Independent-

Measures t Statistic 288 10.4 Assumptions Underlying the Independent-Measures

t Formula 300 Summary 303 Focus on Problem Solving 305 Demonstration 10.1 307 Demonstration 10.2 308 Problems 308

Using t Statistics for Interferences About Population Means and Mean DifferencesPART III

viii CONTENTS

The t Test for Two Related Samples 313

11.1 Introduction to Repeated-Measures Designs 314 11.2 The t Statistic for a Repeated-Measures Research Design 315 11.3 Hypothesis Tests and Effect Size for the Repeated-Measures Design 320 11.4 Uses and Assumptions for Repeated-Measures t Tests 328

Summary 331 Focus on Problem Solving 332 Demonstration 11.1 333 Demonstration 11.2 335 Problems 335

Part III Review 341 Review Exercises 341

Chapter 11

Analysis of Variance: Tests for Differences Among Two or More Population MeansPART IV

Introduction to Analysis of Variance 345

12.1 Introduction 346 12.2 The Logic of ANOVA 350 12.3 ANOVA Notation and Formulas 354 12.4 The Distribution of F-Ratios 362 12.5 Examples of Hypothesis Testing and Effect Size with ANOVA 364 12.6 Post Hoc Tests 375 12.7 The Relationships Between ANOVA and t Tests 379

Summary 381 Focus on Problem Solving 383 Demonstration 12.1 384 Demonstration 12.2 386 Problems 387

Repeated-Measures and Two-Factor Analysis of Variance 393

13.1 Overview 394 13.2 Repeated-Measures ANOVA 394 13.3 Two-Factor ANOVA (Independent Measures) 409 Summary 428 Focus on Problem Solving 432 Demonstration 13.1 434 Demonstration 13.2 435 Problems 438

Part IV Review 445 Review Exercises 445

Chapter 12

Chapter 13

CONTENTS ix

Correlations and Nonparametric TestsPART V

Correlation 449

14.1 Introduction 450 14.2 The Pearson Correlation 453 14.3 Using and Interpreting the Pearson Correlation 458 14.4 Hypothesis Tests with the Pearson Correlation 464 14.5 Alternatives to the Pearson Correlation 472 14.6 Introduction to Linear Equations and Regression 481 Summary 496 Focus on Problem Solving 500 Demonstration 14.1 502 Problems 503

The Chi-Square Statistic: Tests for Goodness of Fit and Independence 509

15.1 Parametric and Nonparametric Statistical Tests 510 15.2 The Chi-Square Test for Goodness of Fit 511 15.3 The Chi-Square Test for Independence 521 15.4 Measuring Effect Size for the Chi-Square Test for Independence 532

15.5 Assumptions and Restrictions for Chi-Square Tests 534 Summary 535 Focus on Problem Solving 539 Demonstration 15.1 539 Demonstration 15.2 541 Problems 541

Part V Review 547 Review Exercises 547

Basic Mathematics Review 549

Statistical Tables 571

Solutions for Odd-Numbered Problems in the Text 583

General Instructions for Using SPSS 601

Statistics Organizer: Finding the Right Statistics for Your Data 605

References 619

Index 625

Chapter 14

Chapter 15

Appendix A

Appendix B

Appendix C

Appendix D

xi

Many students in the behavioral sciences view the required statistics course as an intimi- dating obstacle that has been placed in the middle of an otherwise interesting curriculum. They want to learn about human behaviornot 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, a knowledge of statistics will be neces- sary. 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 informa- tion 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 Essentials of Statistics for the Behavioral Sciences will notice that some changes have been made. These changes are summarized in the section titled To the Instructor. In revising this text, our stu- dents 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 titled 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 pro- cedures 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 examplesmany based on actual research studiesalong with learning checks, a summary and list of key terms, and a set of 20 to 30 problems.

Those of you familiar with the previous edition of Essentials of Statistics for the Behavioral Sciences will notice a number of changes in the eighth edition. Throughout the book, research examples have been updated, real-world examples have been added, and the end-of-chapter problems have been extensively revised. The book has been separated into five sections to emphasize the similarities among groups of statistical methods. Each section contains two to four chapters and begins with an introduction and concludes with a review, including review exercises. Major revisions for this edition include:

The former Chapter 12 on estimation has been eliminated. In its place, sec- tions on confidence intervals have been added to the three chapters presenting t statistics.

A new appendix titled Statistics Organizer: Finding the Right Statistics for Your Data, discusses the process of selecting the correct statistics to be used with different categories of data and replaces the Statistics Organizer that appeared as an appendix in earlier editions.

Preface

T O T H E I N S T R U C T O R

Other specific and noteworthy revisions include:

Chapter 1 A separate section explains how statistical methods can be classified using the same categories that are used to group data structures and research methods.

Chapter 2 The discussion of histograms has been modified to differentiate discrete and continuous variables.

Chapter 3 A modified definition of the median acknowledges that this value is not algebraically defined and that determining the median, especially for discrete variables, can be somewhat subjective.

Chapter 4 Relatively minor editing for clarity. The section on variance and inferen- tial statistics has been simplified.

Chapter 5 Relatively minor editing for clarity.

Chapter 6 The concepts of random sample and independent random sample have been clarified with separate definitions. A new figure helps demonstrate the process of using the unit normal table to find proportions for negative z-scores.

Chapter 7 Relatively minor editing for clarity.

Chapter 8 The chapter has been shortened by substantial editing that eliminated several pages of unnecessary text, particularly in the sections on errors (Type I and II) and power.

Chapter 9 The section describing how sample size and sample variance influence the outcome of a hypothesis test has been moved so that it appears immediately after the hypothesis test example. A new section introduces confidence intervals in the context of describing effect size, describes how confidence intervals are reported in the litera- ture, and discusses factors affecting the width of a confidence interval.

Chapter 10 An expanded section discusses how sample variance and sample size influ- ence the outcome of an independent-measures hypothesis test and measures of effect size. A new section introduces confidence intervals as an alternative for describing effect size. The relationship between a confidence interval and a hypothesis test is also discussed.

Chapter 11 The description of repeated-measures and matched-subjects designs has been clarified and we increased emphasis on the concept that all calculations for the related-samples test are done with the difference scores. A new section introduces con- fidence intervals as an alternative for describing effect size and discusses the relation- ship between a confidence interval and a hypothesis test.

The former Chapter 12 has been deleted. The content from this chapter discussing con- fidence intervals has been added to Chapters 9, 10, and 11.

Chapter 12 (former Chapter 13, introducing ANOVA) The discussion of testwise alpha levels versus experimentwise alpha levels has been moved from a box into the text, and definitions of the two terms have been added. To emphasize the concepts of ANOVA rather than the formulas, SS

between treatments is routinely found by subtraction

instead of being computed directly. Two alternative equations for SS between treatments

have been moved from the text into a box.

xii PREFACE

Chapter 13 (former Chapter 14, introducing repeated-measures and two-factor ANOVA) A new section demonstrates the relationship between ANOVA and the t test when a repeated-measures study is comparing only two treatments. Extensive editing has shortened the chapter and simplified the presentation.

Chapter 14 (formerly Chapter 15, introducing correlation and regression) New sec- tions present the t statistic for testing hypotheses about the Pearson correlation and demonstrate how the t test for significance of a correlation is equivalent to the F-ratio used for analysis of regression.

Chapter 15 (formerly Chapter 16, introducing chi-square tests) Relatively minor editing has shortened and clarified the chapter.

Matching the Text to Your Syllabus We have tried to make separate chapters, and even sections of chapters, completely self-contained so that 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 and 13) or emphasizing correlation/regression (Chapter 14). It is rare for a one-semester course to provide complete coverage of both topics.

Although we choose to complete all the hypothesis tests for means and mean differences before introducing correlation (Chapter 14), many instructors prefer to place correlation much earlier in the sequence of course topics. To accommodate this, sections 14.1, 14.2, and 14.3 present the calculation and interpretation of the Pearson correlation and can be introduced immediately following Chapter 4 (variability). Other sections of Chapter 14 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 15) much earlier in the sequence of course topics. Chapter 15, 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 14.1, 14.2, and 14.3) be pre- sented early to provide a foundation for the chi-square test for independence.

A primary goal of this book is to make the task of learning statistics as easy and pain- less 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 en- courage you to take advantage of these opportunities. Read the text rather than just memorize 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 un- derlying a statistical formula, you will find that learning the formula and how to use it will be much easier. In the following section, Study Hints, 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

PREFACE xiii

T O T H E S T U D E N T

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 [email protected]

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 mate- rial, 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 sev- eral 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 3}12}-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 instructorthere 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 trou- ble, 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.

Do not kid yourself! Avoid denial. Many students watch their instructor solve problems in class and think to themselves, This looks easyI 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 noth- ing 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 big- gest 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 tutor- ing can be very helpful.

xiv PREFACE

Ancillaries for this edition include the following:

Aplia Statistics for Psychology and the Behavioral Sciences: An online inter- active learning solution that ensures students stay involved with their course- work and master the basic tools and concepts of statistical analysis. Created by a research psychologist to help students excel, Aplias content engages students with questions based on real-world scenarios that help students un- derstand how statistics applies to everyday life. At the same time, all chapter assignments are automatically graded and provide students with detailed explanations, making sure they learn from and improve with every question.

Instructors Manual with Test Bank: Contains chapter outlines, annotated learning objectives, lecture suggestions, test items, and solutions to all end-of- chapter problems in the text. Test items are also available as a Word down- load or for ExamView computerized test bank software.

PowerLecture with ExamView: This CD includes the instructors manual, test bank, lecture slides with book figures, and more. Featuring automatic grading, ExamView, also available within PowerLecture, allows you to create, de- liver, and customize tests and study guides (both print and online) in minutes. Assessments appear onscreen exactly as they will print or display online; you can build tests of up to 250 questions using up to 12 question types, and you can enter an unlimited number of new questions or edit existing questions.

WebTutor on Blackboard and WebCT: Jump-start your course with customiz- able, text-specific content within your Course Management System.

Psychology CourseMate: Cengage Learnings Psychology CourseMate brings course concepts to life with interactive learning, study, and exam prepara- tion tools that support the printed textbook. Go to www.cengagebrain.com. Psychology CourseMate includes:

An interactive eBook;

Interactive teaching and learning tools, including

quizzes,

flashcards,

videos,

and more; plus

The Engagement Tracker, a first-of-its-kind tool that monitors student engage- ment in the course.

It takes a lot of good, hardworking people to produce a book. Our friends at Wadsworth/ Cengage Learning have made enormous contributions to this textbook. We thank: Jon-David Hague, Publisher; Tim Matray, Acquisitions Editor; Paige Leeds, Assistant Editor; Nicole Richards, Editorial Assistant; Charlene M. Carpentier, Content Project Manager; Jasmin Tokatlian, Media Editor; and Pam Galbreath, Art Director. Special thanks go to Liana Sarkisian and Arwen Petty, our Developmental Editors, and to Mike Ederer, who led us through production at Graphic World.

Reviewers play a very important role in the development of a manuscript. Accordingly, we offer our appreciation to the following colleagues for their assistance with the eighth edition: Patricia Tomich, Kent State University; Robert E. Wickham, University of Houston; Jessica Urschel, Western Michigan University; Wilson Chu, California State University, Long Beach; Melissa Platt, University of Oregon; Brian Detweiler-Bedell, Lewis and Clark College.

PREFACE xv

A N C I L L A R I E S

A C K N O W L E D G M E N T S

Larry B. Wallnau is Professor Emeritus of Psychology at the State University of New York College at Brockport. At Brockport he taught courses relating to the biological basis of behavior and published numerous re- search articles, primarily in the field of biopsychology. With Dr. Gravetter, he co-authored Statistics for the Behavioral Sciences. He also has provided editorial consulting for a number of publishers and journals. He is an FCC-licensed amateur radio operator, and in his spare time he is seeking worldwide contacts with other radio enthusiasts.

About the Authors

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 bachelors 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 Sciences and Statistics for the Behavioral Sciences.

xvi

P A R T

I Chapter 1 Introduction to Statistics 3

Chapter 2 Frequency Distributions 37

Chapter 3 Measures of Central Tendency 59

Chapter 4 Measures of Variability 89

W e have divided this book into five sections, each cover-

ing a general topic area of statistics. The first section,

consisting of Chapters 1 to 4, provides a broad over-

view of statistical methods and a more focused presentation of

those methods that are classified as descriptive statistics.

By the time you finish the four chapters in this part, you should

have a good understanding of the general goals of statistics and

you should be familiar with the basic terminology and notation

used in statistics. In addition, you should be familiar with the tech-

niques of descriptive statistics that help researchers organize and

summarize the results they obtain from their research. Specifically,

you should be able to take a set of scores and present them in a

table or in a graph that provides an overall picture of the complete

set. Also, you should be able to summarize a set of scores by cal-

culating one or two values (such as the average) that describe the

entire set.

At the end of this section, there is a brief summary and a set of

review problems that should help to integrate the elements from

the separate chapters.

Introduction and Descriptive Statistics

1

C H A P T E R

1 Introduction to Statistics

1.1 Statistics, Science, and Observations

1.2 Populations and Samples

1.3 Data Structures, Research Methods, and Statistics

1.4 Variables and Measurement

1.5 Statistical Notation

Summary

Focus on Problem Solving

Demonstration 1.1

Problems

Aplia for Essentials of Statistics for the Behavioral Sciences

After reading, go to Resources at the end of this chapter for

an introduction on how to use Aplias homework and learning

resources.

4 CHAPTER 1 INTRODUCTION TO STATISTICS

STATISTICS, SCIENCE, AND OBSERVATIONS

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? Actually, the

paragraph is an excellent (and humorous) counterexample for the purpose of this book.

Specifically, our goal is to help you avoid stumbling around in the dark by providing

lots of easily available light switches and plenty of illumination as you journey through

the world of statistics. To accomplish this, we try to present sufficient background and

a clear statement of purpose as we introduce each new statistical procedure. Remember

that all statistical procedures were developed to serve a purpose. If you understand why

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

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, the mate-

rial presented in the early chapters of this book 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, we should reassure you that the progression

from basic concepts to complex statistical techniques is a slow, step-by-step process.

As you learn the basic background material, you will develop a good frame of refer-

ence for understanding and incorporating new, more sophisticated concepts as they are

presented.

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.

Statistics are often defined as facts and figures, such as average income, crime rate,

birth rate, baseball batting averages, and so on. These statistics are usually informa-

tive 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 sta-

tistics. Specifically, we use the term statistics to refer to a set of mathematical proce-

dures. 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 for research in the

behavioral sciences.

1.1

OV E R V I E W

D E F I N I T I O N S O F STAT I ST I C S

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

by permission.

SECTION 1.2 / POPULATIONS AND SAMPLES 5

Research in the behavioral sciences (and other fields) involves gathering informa-

tion. To determine, for example, whether college students learn better by reading mate-

rial 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 IQ scores, personality scores, exam scores, and so on. In this

book, we present the statistics that researchers use to analyze and interpret the informa-

tion that they gather. Specifically, statistics serve two general purposes:

1. Statistics are used to organize and summarize the information so that the re-

searcher 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.

The term statistics refers to a set of mathematical procedures for organizing,

summarizing, and interpreting information.

Statistical procedures help to ensure that the information or observations are

presented and interpreted in an accurate and informative way. In somewhat gran-

diose terms, statistics help researchers bring order out of chaos. Statistics also

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 are 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.

A population is the entire set of the individuals of interest for a particular

research question.

As you can well imagine, a population can be quite largefor example, the entire

set of men on the planet Earth. A researcher might be more specific, limiting the popu-

lation for study to retired men who live in the United States. Perhaps the investigator

would like to study the population consisting of men who are professional basketball

players. Populations can obviously vary in size from extremely large to very small,

depending on how the researcher defines the population. The population being studied

should always be identified by the researcher. In addition, the population need not

D E F I N I T I O N

1.2

D E F I N I T I O N

6 CHAPTER 1 INTRODUCTION TO STATISTICS

consist of peopleit could be a population of rats, corporations, parts produced in a

factory, or anything else a researcher wants to study. In practice, populations are typi-

cally 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 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.

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

might examine a sample of only 10 autistic children, 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 general-

ize 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.

Typically, researchers are interested in specific characteristics of the individuals in

the population (and 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

D E F I N I T I O N

VA R I A B L E S A N D DATA

THE POPULATION All of the individuals of interest

THE SAMPLE The individuals selected to

participate in the research study

The results from the sample are generalized

to the population

The sample is selected from the population

FIGURE 1.1

The relationship between

a population and a sample.

SECTION 1.2 / POPULATIONS AND SAMPLES 7

of the weather on peoples moods. As the weather changes, do peoples moods also

change? Something that can change or have different values is called a variable.

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

different individuals.

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

other, such as height, weight, gender, or personality. Also, variables can be environmen-

tal 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.

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 college students and a sample of autistic children. Be fore-

warned, 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.

When describing data, it is necessary to distinguish whether the data come from a popula-

tion or a sample. A characteristic that describes a populationfor example, the average

score for the populationis 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.

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.

Although researchers have developed a variety of different statistical procedures to or-

ganize 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.

D E F I N I T I O N

D E F I N I T I O N S

PA R A M E T E R S A N D STAT I ST I C S

D E F I N I T I O N S

D E S C R I P T I V E A N D I N F E R E N T I A L

STAT I ST I CA L M E T H O D S

8 CHAPTER 1 INTRODUCTION TO STATISTICS

BOX

1.1 THE MARGIN OF ERROR BETWEEN STATISTICS AND PARAMETERS

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 is always some

margin of error when sample statistics are used to

represent 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.

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

simplify data.

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.

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 answer general questions 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. Thus, there typically is some discrepancy between a sample statistic and the

corresponding population parameter. This discrepancy is called sampling error, and it

creates the fundamental problem that inferential statistics must always address (Box 1.1).

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 two samples, each with 5 students, who

have been selected from the population. Notice that each sample contains differ-

ent individuals who have different characteristics. Because the characteristics of

each sample depend on the specific people in the sample, statistics vary from one

D E F I N I T I O N

D E F I N I T I O N

D E F I N I T I O N

SECTION 1.2 / POPULATIONS AND SAMPLES 9

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.

Also note that the statistics obtained for a sample are not 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.

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 stu-

dents 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.

FIGURE 1.2

A demonstration of sam-

pling error. Two samples

are selected from the

same population. Notice

that the sample statistics

are different from one

sample to another, and all

of the sample statistics

are different from the

corresponding population

parameters. The natural

differences that exist, by

chance, between a sample

statistic and a population

parameter are called

sampling error.

Population of 1000 college students

Population Parameters Average Age 21.3 years

Average IQ 112.5 65% Female, 35% Male

Sample #1

Eric Jessica Laura Karen Brian

Sample Statistics Average Age 19.8 Average IQ 104.6

60% Female, 40% Male

Sample #2

Tom Kristen Sara

Andrew John

Sample Statistics Average Age 20.4 Average IQ 114.2

40% Female, 60% Male

1 0 CHAPTER 1 INTRODUCTION TO STATISTICS

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.

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 ad-

dress 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.

STAT I ST I C S I N T H E CO N T E X T

O F R E S E A R C H

E X A M P L E 1 . 1

Step 1

Step 2

Step 3

Experiment:

Descriptive statistics:

Inferential statistics:

Compare two studying methods

Test scores for the students in each sample

Organize and simplify

Interpret results

Sample A Read from printed

pages

25 27 30 19 29

26 21 28 23 26

28 27 24 26 22

20 23 25 22 18

22 17 28 19 24

27 23 21 22 19

Sample B Read from computer

screen

Data

Average Score = 26

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.

Population of College Students

Average Score = 22

20 25 30 20 25 30

FIGURE 1.3

The role of statistics in research.

SECTION 1.2 / POPULATIONS AND SAMPLES 1 1

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 Figure 1.3). 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 aver-

age score for each sample. Note that descriptive methods provide a simplified, organized

description of the scores. In this example, the students who studied printed pages had an aver-

age 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 two study methods, 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 two study methods, and the 4-point differ-

ence 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.

1. A researcher is interested in the texting habits of high school students in the

United States. If the researcher measures the number of text messages that each

individual sends each day and calculates the average number for the entire group of

high school students, the average number would be an example of a ___________.

2. A researcher is interested in how watching a reality television show featuring

fashion models influences the eating behavior of 13-year-old girls.

a. A group of 30 13-year-old girls is selected to participate in a research study.

The group of 30 13-year-old girls is an example of a ___________.

b. In the same study, the amount of food eaten in one day is measured for each

girl and the researcher computes the average score for the 30 13-year-old girls.

The average score is an example of a __________.

3. Statistical techniques are classified into two general categories. What are the two cat-

egories called, and what is the general purpose for the techniques in each category?

4. Briefly define the concept of sampling error.

1. parameter

2. a. sample

b. statistic

3. The two categories are descriptive statistics and inferential statistics. Descriptive techniques

are intended to organize, simplify, and summarize data. Inferential techniques use sample

data to reach general conclusions about populations.

4. Sampling error is the error, or discrepancy, between the value obtained for a sample statistic

and the value for the corresponding population parameter.

L E A R N I N G C H E C K

ANSWERS

1 2 CHAPTER 1 INTRODUCTION TO STATISTICS

DATA STRUCTURES, RESEARCH METHODS, AND STATISTICS

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 studying 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 61%

of the adults in the United States drink alcohol.

Most research, however, is intended to examine relationships between two or more

variables. For example, is there a relationship between the amount of violence that

children see on television and the amount of aggressive behavior they display? Is there a

relationship between the quality of breakfast and level of academic performance for el-

ementary 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 relation-

ship, researchers must make observationsthat 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.

Data structure I. Measuring two variables 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 mea-

sure 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?

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 students wake-up time and the vertical position corresponds to the

students academic performance score. The scatter plot shows a clear relationship be-

tween 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.

1.3

I N D I V I D UA L VA R I A B L E S

R E L AT I O N S H I P S B E T W E E N VA R I A B L E S

D E F I N I T I O N

SECTION 1.3 / DATA STRUCTURES, RESEARCH METHODS, AND STATISTICS 1 3

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 relation-

ship 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.

Data structure 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, one group of elementary school children is shown a 30-minute action/adven-

ture television program involving numerous instances of violence, and a second group is

shown a 30-minute comedy that includes no violence. Both groups are then observed on

the playground and a researcher records the number of aggressive acts committed by each

child. An example of the resulting data is shown in Figure 1.5. The researcher compares

the scores for the violence group with the scores for the no-violence group. A systematic

difference between the two groups provides evidence for a relationship between viewing

television violence and aggressive behavior for elementary school children.

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 experi-

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

T H E E X P E R I M E N TA L M E T H O D

3.8

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

7 8 9

Wake-up time

A c

a d

e m

ic p

e rf

o rm

a n

c e

10 11 12FIGURE 1.4

One of two data structures for studies 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 a graph (b).

A B C D E F G H

11 9 9

12 7

10 10

8

Student Wake-up

Time

2.4 3.6 3.2 2.2 3.8 2.2 3.0 3.0

Academic Performance

(a) (b)

1 4 CHAPTER 1 INTRODUCTION TO STATISTICS

Specifically, 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 experi-

mental 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. A second variable is observed (measured) to deter-

mine 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.

To demonstrate these two characteristics, consider an experiment in which researchers

demonstrate the pain-killing effects of handling money (Zhou & Vohs, 2009). In the

experiment, a group of college students was told that they were participating in a

manual dexterity study. The researchers then created two treatment conditions by

manipulating the kind of material that each participant would be handling. Half of

the students were given a stack of money to count and the other half got a stack of

blank pieces of paper. After the counting task, the participants were asked to dip their

hands into bowls of painfully hot water (122 F) and rate how uncomfortable it was.

Participants who had counted money rated the pain significantly lower than those who

had counted paper. The structure of the experiment is shown in Figure 1.6.

To be able to say that the difference in pain perception is caused by the money, the

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

researchers must control any other variables that might affect pain tolerance. 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.

In the money-counting experiment, for example, suppose that the participants

in the money condition were primarily females and those in the paper condition

were primarily males. In this case, there is an alternative explanation for any

difference in the pain ratings that exists between the two groups. Specifically, it

is possible that the difference was caused by the money, but it also is possible

that the difference was caused by the participants gender (females can tolerate

One variable (violence/no violence) is used to define groups

A second variable (aggressive behavior) is measured to obtain scores within each group

4 2 0 1 3 2 4 1 3

0 2 1 3 0 0 1 1 1

Violence No

Violence

Compare groups of scores

FIGURE 1.5

The second data structure

for studies evaluating

the relationship between

variables. Note that one

variable is used to define

the groups and the second

variable is measured to

obtain scores within each

group.

In more complex experi-

ments, 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.

SECTION 1.3 / DATA STRUCTURES, RESEARCH METHODS, AND STATISTICS 1 5

more pain than males can). Whenever a research study allows more than one

explanation for the results, the study is said to be confounded because it is im-

possible to reach an unambiguous conclusion.

2. Environmental Variables These are characteristics of the environment such

as lighting, time of day, and weather conditions. Using the money-counting

experiment (see Figure 1.6) as an example, suppose that the individuals in the

money condition were all tested in the morning and the individuals in the paper

condition were all tested in the evening. Again, this would produce a confounded

experiment because the researcher could not determine whether the differences

in the pain ratings were caused by the money 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, partici-

pants could be assigned randomly for testing either in the morning or in the afternoon.

Second, the researcher can use matching to ensure equivalent groups or equivalent envi-

ronments. 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, if an experiment uses only 10-year-old children

as participants (holding age constant), then the researcher can be certain that one group

is not noticeably older than another.

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

D E F I N I T I O N

Variable #1: Counting money or blank paper (the independent variable) Manipulated to create two treatment conditions.

Variable #2: Pain rating (the dependent variable) Measured in each of the treatment conditions.

7 4 5 6 6 8 6 5 5 6

8 10 8 9 8

10 7 8 8 7

Money Paper

Compare groups of scores

FIGURE 1.6

The structure of an

experiment. Participants

are randomly assigned

to one of two treatment

conditions: counting

money or counting blank

pieces of paper. Later,

each participant is tested

by placing one hand in

a bowl of hot (122 F)

water and rating the level

of pain. A difference be-

tween the ratings for the

two groups is attributed

to the treatment (paper

versus money).

1 6 CHAPTER 1 INTRODUCTION TO STATISTICS

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 1.6,

money versus blank paper 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 1.6, the level of pain is the dependent variable.

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 vari-

able consists of the antecedent conditions that are manipulated prior to observing

the dependent variable.

The dependent variable is the variable that is observed to assess the effect of the

treatment.

An experimental study evaluates the relationship between two variables by manipu-

lating one variable (the independent variable) and measuring one variable (the depen-

dent 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.

Control Conditions in an Experiment Often an experiment will include a condition

in which the participants do not receive any treatment. The scores from these individu-

als 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.

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

Instead, they either receive no treatment or they receive a neutral, placebo treat-

ment. The purpose of a control condition is to provide a baseline for comparison

with the experimental condition. The individuals in the control condition are

often called the control group.

Individuals in the experimental condition do receive the experimental treatment

and are often called the experimental group.

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

money-counting experiment (see Figure 1.6), the independent variable is money versus

plain paper. For an experiment with an experimental group and a control group, the

independent variable is treatment versus no treatment.

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 ap-

plies 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 compare groups of scores but are not true experiments. Two examples are shown

in Figure 1.7 and are discussed in the following paragraphs. This type of research study

is classified as nonexperimental.

D E F I N I T I O N S

D E F I N I T I O N S

N O N E X P E R I M E N TA L M E T H O D S :

N O N E Q U I VA L E N T G R O U P S A N D

P R E P O ST ST U D I E S

SECTION 1.3 / DATA STRUCTURES, RESEARCH METHODS, AND STATISTICS 1 7

The top part of Figure 1.7 shows an example of a nonequivalent groups study com-

paring boys and girls. Notice that this study involves comparing two groups of scores

(like an experiment). However, the researcher has no ability to control the assignment

of participants to groupsthe males automatically go in the boy group and the females

go 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 or comparing people with an eating disor-

der and those with no disorder. 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 1.7 shows an example of a prepost 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 prepost 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 prepost study, the researcher also has

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

17 19 16 12 17 18 15 16

12 10 14 15 13 12 11 13

Boys Girls

Any difference?

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

17 19 16 12 17 18 15 16

12 10 14 15 13 12 11 13

Before Therapy

After Therapy

Any difference?

(a)

(b)

FIGURE 1.7

Two examples of nonex-

perimental studies that

involve comparing

two groups of scores. In

(a), a participant variable

(gender) is used to

create groups, and then

the dependent variable

(verbal score) is measured

in each group. In

(b), time is the variable

used to define the two

groups, and the dependent

variable (depression) is

measured at each of the

two times.

Correlational studies are

also examples of nonex-

perimental research. In this

section, however, we are

discussing nonexperimental

studies that compare two or

more groups of scores.

1 8 CHAPTER 1 INTRODUCTION TO STATISTICS

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 1.7 are not true experiments, you should notice that they produce the

same kind of data that are found in an experiment (see Figure 1.6). 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 1.7, 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.

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

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

The two general data structures that we used to classify research methods can also be

used to classify statistical methods.

I. One group with two variables measured for each individual Recall that the data

from a correlational study consist of two scores, representing two different variables,

for each individual. The scores can be listed in a table or displayed in a scatter plot as

in Figure 1.5. The relationship between the two variables is usually measured and de-

scribed using a statistic called a correlation. Correlations and the correlational method

are discussed in detail in Chapter 14.

Occasionally, the measurement process used for a correlational study simply clas-

sifies individuals into categories that do not correspond to numerical values. For

example, Greitemeyer and Osswald (2010) examine the effect of prosocial video

games on prosocial behavior. One group of participants played a prosocial game and

a second group played a neutral game. After the game was finished, the experimenter

accidentally knocked a cup of pencils onto the floor and recorded whether the par-

ticipants helped to pick them up. Note that the researcher has two scores for each

individual (type of game and helping behavior) but neither of the scores is a numerical

value. This type of data is typically summarized in a table showing how many indi-

viduals are classified into each of the possible categories. Table 1.1 is an example

of this kind of summary table showing results similar to those obtained in the study.

The table shows, for example, that 12 of the 18 participants playing the prosocial

game helped to pick up pencils. This type of data can be coded with numbers (for

example, neutral 0 and prosocial 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 15.

D E F I N I T I O N

DATA ST R UC T U R E S A N D STAT I ST I CA L

M E T H O D S

SECTION 1.3 / DATA STRUCTURES, RESEARCH METHODS, AND STATISTICS 1 9

II. 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 experimental study in Figure 1.6 and the nonexperimental studies in Figure 1.7.

Specifically, we examine descriptive statistics that summarize and describe the scores

in each group, and we examine inferential statistics that allow us to use the groups, or

samples, to generalize to the entire population.

When the measurement procedure produces numerical scores, the statistical evalu-

ation typically involves computing the average score for each group and then compar-

ing 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 813.

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 type of video game and helping

behavior. The same data can be used to compare the proportions for prosocial game

players with the proportions for neutral game players. For example, 67% of those who

played the prosocial game helped the researcher compared to 33% of those who played

the neutral game. As mentioned before, these data are evaluated using a chi-square test,

which is presented in Chapter 15.

Type of Video Game

Prosocial Neutral

Helped 12 6

Did not Help 6 12

TABLE 1.1

Correlational data consisting of

non-numerical scores. Note that

there are two measurements for

each individual: type of game

played and helping behavior.

The numbers indicate how many

people are in each category. For

example, out of the 18 partici-

pants who played a prosocial

game, 12 helped the researcher.

1. 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). Is this study an

example of an experiment? Explain why or why not.

2. What two elements are necessary for a research study to be an experiment?

3. Stephens, Atkins, and Kingston (2009) conducted an experiment in which

participants were able to tolerate more pain when they shouted their favorite

swear words over and over than when they shouted neutral words. Identify the

independent and dependent variables for this study.

1. This study is nonexperimental. The researcher is simply observing, not manipulating, two

nonequivalent groups of participants.

2. First, the researcher must manipulate one of the two variables being studied. Second, all

other variables that might influence the results must be controlled.

3. The independent variable is the type of word being shouted and the dependent variable is

the amount of pain tolerated by each participant.

L E A R N I N G C H E C K

ANSWERS

2 0 CHAPTER 1 INTRODUCTION TO STATISTICS

VARIABLES AND MEASUREMENT

The scores that are obtained in a research study are the result of observing and mea-

suring 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 cannot be observed or measured

directly. However, we all assume that these variables exist and we use them 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 measures and defines a construct

in terms of external behaviors. For example, we can measure performance on an IQ test

and then use the test scores as a definition of intelligence. Or hunger can be measured

and defined by the number of hours since last eating.

Constructs, also known as hypothetical 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 an internal 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.

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 valuesfor example,

five dots and six dotsno other values can ever be observed.

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

exist between two neighboring categories.

Discrete variables are commonly restricted to whole, countable numbersfor

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,

1.4

CO N ST R UC T S A N D O P E R AT I O N A L

D E F I N I T I O N S

D E F I N I T I O N S

D I S C R E T E A N D CO N T I N U O U S

VA R I A B L E S

D E F I N I T I O N

SECTION 1.4 / VARIABLES AND MEASUREMENT 2 1

teacher, lawyer, and so on), and college students can be classified by academic major

(art, biology, chemistry, and so on). 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.

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.

Note 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,

peoples heights can be measured by simply classifying individuals into three broad

categories: tall, average, and short. Note that there is no measurement category be-

tween tall and average. Thus, it may appear that we are measuring a discrete variable.

However, the underlying variable, height, is continuous. In this example, we chose to

limit the measurement scale to three categories. We could have decided to measure

height to the nearest inch, or the nearest half inch, and so on. The key to determin-

ing 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 0.1 inch, or the nearest 0.01 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.

Measuring a continuous variable Any continuous variable, for example, weight,

can be pictured as a continuous line (Figure 1.8). Note that there are an infinite number

of possible points on 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.

D E F I N I T I O N

149

149.5

150

149.6 150.3

150.5

151 152

148.5

149

149.5

150

150.5

Real limits

151

151.5

152

152.5

FIGURE 1.8

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.

2 2 CHAPTER 1 INTRODUCTION TO STATISTICS

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 relatively 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 1.8). 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 is assigned a score of X 150.

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 measur-

ing 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.

Later in this book, real limits are used for constructing graphs and for various calcu-

lations with continuous scales. For now, however, you should realize that real limits are

a necessity whenever you make measurements of a continuous variable.

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 set of catego-

ries makes 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 peoples heights, for example,

you could measure a group of individuals by simply classifying them into three cat-

egories: 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

D E F I N I T I O N S

S CA L E S O F M E AS U R E M E N T

SECTION 1.4 / VARIABLES AND MEASUREMENT 2 3

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 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 accord-

ing 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.

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.

Although the categories on a nominal scale are not quantitative values, they are oc-

casionally 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 dif-

ference. The scales that follow do reflect an attempt to make quantitative distinctions.

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.

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 dont 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

T H E N O M I N A L S CA L E

D E F I N I T I O N

T H E O R D I N A L S CA L E

D E F I N I T I O N

2 4 CHAPTER 1 INTRODUCTION TO STATISTICS

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.

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 mea-

sured. For example, a temperature of 0 degrees 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 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 meaning-

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

existence of an absolute, nonarbitrary zero point means that we can measure the abso-

lute 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. 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.

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

the same size. Equal differences between numbers on the 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.

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

the researcher simply records each childs 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.

T H E I N T E R VA L A N D R AT I O S CA L E S

D E F I N I T I O N S

E X A M P L E 1 . 2

SECTION 1.4 / VARIABLES AND MEASUREMENT 2 5

Now suppose that the researcher converts the initial measurement into a new scale by

calculating the difference between each childs actual height and the average height for

this age group. A child who is 1 inch taller than average now gets a score of 11; a child who is 4 inches taller than average gets a score of 14. Similarly, a child who is 2 inches shorter than average gets a score of 2. On this scale, a score of zero is a convenient

reference point corresponding to the average height. Because zero no longer indicates a

complete absence of height, the new scores constitute an interval scale of measurement.

Notice that original scores and the converted scores both involve measurement in

inches, and you can compute differences, or distances, on either scale. For example,

there is a 6-inch difference in height between two boys who measure 57 and 51 inches

tall on the first scale. Likewise, there is a 6-inch difference between two boys who

measure 19 and 13 on the second scale. However, you should also notice that ratio comparisons are not possible on the second scale. For example, a boy who measures

19 is not three times taller than a boy who measures 13.

For our purposes, scales of measurement are important because they influence the kind

of statistics that can and cannot be used. For example, if you measure IQ scores for a

group of students, it is possible to add the scores together and calculate a mean score for

the group. On the other hand, if you measure the academic major for each student, you

STAT I ST I C S A N D S CA L E S

O F M E AS U R E M E N T

1. A tax form asks people to identify their annual income, number of dependents, and

social security number. For each of these three variables, identify the scale of measure-

ment that probably is used and identify whether the variable is continuous or discrete.

2. An English professor uses letter grades (A, B, C, D, and F) to evaluate a set of student

essays. What kind of scale is being used to measure the quality of the essays?

3. The teacher in a communications class asks students to identify their favorite

reality television show. The different television shows make up a ______ scale

of measurement.

4. A researcher studies the factors that determine the number of children that couples

decide to have. The variable, number of children, is a ______________ (discrete/

continuous) variable.

5. a. When measuring height to the nearest inch, what are the real limits for a score

of 68 inches?

b. When measuring height to the nearest half inch, what are the real limits for a

score of 68 inches?

1. Annual income and number of dependents are measured on ratio scales, and income is a

continuous variable. Social security number is measured on a nominal scale and is a discrete

variable. The number of dependents is also discrete.

2. ordinal

3. nominal

4. discrete

5. a. 67.5 and 68.5

b. 67.75 and 68.25

L E A R N I N G C H E C K

ANSWERS

2 6 CHAPTER 1 INTRODUCTION TO STATISTICS

cannot compute the mean. (What is the mean of three psychology majors, an English

major, and two chemistry majors?) The vast majority of the statistical techniques pre-

sented in this book are designed for numerical scores from an interval or a ratio scale.

For most statistical applications, the distinction between an interval scale and a ratio

scale is not important because both scales produce numerical values that permit us to

compute differences between scores, to add scores, and to calculate mean scores. On

the other hand, measurements from nominal or ordinal scales are typically not numeri-

cal values and are not compatible with many basic arithmetic operations. Therefore,

alternative statistical techniques are necessary for data from nominal or ordinal scales

of measurement (for example, the median and the mode in Chapter 3, the Spearman

correlation in Chapter 14, and the chi-square tests in Chapter 15).

STATISTICAL NOTATION

The measurements obtained in research studies provide the data for statistical analysis.

Most statistical techniques use the same general mathematical operations, notation,

and basic arithmetic that you have learned during previous years of school. In case you

are unsure of your mathematical skills, there is a mathematics review section in

Appendix A at the back of this book. The appendix also includes a skills-assessment

exam (p. 550) to help you determine whether you need the basic mathematics review. In

this section, we introduce some of the specialized notation that is used for statistical cal-

culations. In later chapters, additional statistical notation is introduced as it is needed.

Measuring a variable in a research study typically yields a value or a score for each

individual. Raw scores are the original, unchanged scores obtained in the study. Scores

for a particular variable are represented by the letter X. For example, if performance in

your statistics course is measured by tests and you obtain a 35 on the first test, then we

could state that X 35. A set of scores can be presented in a column that is headed by

X. For example, a list of quiz scores from your class might be presented as shown in

the margin (the single column on the left).

When two variables are measured for each individual, the data can be presented as two

lists labeled X and Y. For example, measurements of peoples height in inches (variable X)

and weight in pounds (variable Y) can be presented as shown in the double column in the

margin. Each pair X, Y represents the observations made of a single participant.

The letter N is used to specify how many scores are in a set. An uppercase letter N

identifies the number of scores in a population and a lowercase letter n identifies the

number of scores in a sample. Throughout the remainder of the book you will notice

that we often use notational differences to distinguish between samples and populations.

For the height and weight data in the preceding table, n 7 for both variables. Note that

by using a lowercase letter n, we are indicating that these scores come from a sample.

Many of the computations required in statistics involve adding a set of scores. Because

this procedure is used so frequently, a special notation is used to refer to the sum of a set

of scores. The Greek letter sigma, or o, is used to stand for summation. The expression

oX means to add all the scores for variable X. The summation sign, o, can be read as the

sum of. Thus, oX is read the sum of the scores. For the following set of quiz scores,

10, 6, 7, 4

oX 27 and N 4.

1.5

S CO R E S

S U M M AT I O N N OTAT I O N

Score

X X Y

37 72 165

35 68 151

35 67 160

30 67 160

25 68 146

17 70 160

16 66 133

SECTION 1.5 / STATISTICAL NOTATION 2 7

To use summation notation correctly, keep in mind the following two points:

1. The summation sign, o, is always followed by a symbol or mathematical ex-

pression. The symbol or expression identifies exactly which values are to be

added. To compute oX, for example, the symbol following the summation sign

is X, and the task is to find the sum of the X values. On the other hand, to com-

pute o(X 1)2, the summation sign is followed by a relatively complex math-

ematical expression, so your first task is to calculate all of the (X 1)2 values

and then add the results.

2. The summation process is often included with several other mathematical

operations, such as multiplication or squaring. To obtain the correct answer,

it is essential that the different operations be done in the correct sequence.

Following is a list showing the correct order of operations for performing

mathematical operations. Most of this list should be familiar, but you should

note that we have inserted the summation process as the fourth operation in

the list.

Order of Mathematical Operations

1. Any calculation contained within parentheses is done first.

2. Squaring (or raising to other exponents) is done second.

3. Multiplying and/or dividing is done third. A series of multiplication and/or

division operations should be done in order from left to right.

4. Summation using the o notation is done next.

5. Finally, any other addition and/or subtraction is done.

The following examples demonstrate how summation notation is used in most of the

calculations and formulas we present in this book.

A set of four scores consists of values 3, 1, 7, and 4. We will compute oX, oX2, and

(oX)2 for these scores. To help demonstrate the calculations, we will use a computa-

tional table showing the original scores (the X values) in the first column. Additional

columns can then be added to show additional steps in the series of operations.

You should notice that the first three operations in the list (parentheses, squaring, and

multiplying) all create a new column of values. The last two operations, however,

produce a single value corresponding to the sum.

The table to the left shows the original scores (the X values) and the squared scores

(the X2 values) that are needed to compute oX2.

The first calculation, oX, does not include any parentheses, squaring, or multiplica-

tion, so we go directly to the summation operation. The X values are listed in the first

column of the table, and we simply add the values in this column:

oX 3 1 1 1 7 1 4 15

To compute oX2, the correct order of operations is to square each score and then

find the sum of the squared values. The computational table shows the original scores

and the results obtained from squaring (the first step in the calculation). The second

step is to find the sum of the squared values, so we simply add the numbers in the

X2 column.

oX2 9 1 1 1 49 1 16 75

E X A M P L E 1 . 3

More information on the

order of operations for

mathematics is available in

the Math Review appendix,

page 551.

X X2

3 9

1 1

7 49

4 16

2 8 CHAPTER 1 INTRODUCTION TO STATISTICS

The final calculation, (oX)2, includes parentheses, so the first step is to perform the

calculation inside the parentheses. Thus, we first find oX and then square this sum.

Earlier, we computed oX 15, so

(oX)2 (15)2 225

Next, we use the same set of four scores from Example 1.3 and compute o(X 1) and

o(X 1)2. The following computational table will help demonstrate the calculations.

E X A M P L E 1 . 4

X (X 2 1) (X 2 1)2 The first column lists the

original scores. A second

column lists the (X 1)

values, and a third column

shows the (X 1)2 values.

3 2 4

1 0 0

7 6 36

4 3 9

To compute o(X 1), the first step is to perform the operation inside the parentheses.

Thus, we begin by subtracting one point from each of the X values. The resulting values

are listed in the middle column of the table. The next step is to add the (X 1) values.

o(X 1) 2 1 0 1 6 1 3 1 11

The calculation of o(X 1)2 requires three steps. The first step (inside parentheses)

is to subtract 1 point from each X value. The results from this step are shown in the

middle column of the computational table. The second step is to square each of the

(X 1) values. The results from this step are shown in the third column of the table.

The final step is to add the (X 1)2 values to obtain

o(X 1)2 4 1 0 1 36 1 9 49

Notice that this calculation requires squaring before adding. A common mistake is

to add the (X 1) values and then square the total. Be careful!

In both of the preceding examples, and in many other situations, the summation opera-

tion is the last step in the calculation. According to the order of operations, parentheses,

exponents, and multiplication all come before summation. However, there are situations

in which extra addition and subtraction are completed after the summation. For this

example, use the same scores that appeared in the previous two examples, and compute

oX 1.

With no parentheses, exponents, or multiplication, the first step is the summation.

Thus, we begin by computing oX. Earlier we found oX 15. The next step is to sub-

tract one point from the total. For these data,

oX 1 15 1 14

E X A M P L E 1 . 5

SECTION 1.5 / STATISTICAL NOTATION 2 9

For this example, each individual has two scores. The first score is identified as X, and

the second score is Y. With the help of the following computational table, we compute

oX, oY, and oXY.

To find oX, simply add the values in the X column.

oX 3 1 1 1 7 1 4 15

Similarly, oY is the sum of the Y values.

oY 5 1 3 1 4 1 2 14

To compute oXY, the first step is to multiply X by Y for each individual. The result-

ing products (XY values) are listed in the third column of the table. Finally, we add the

products to obtain

oXY 15 1 3 1 28 1 8 54

E X A M P L E 1 . 6

Person X Y XY

A 3 5 15

B 1 3 3

C 7 4 28

D 4 2 8

1. Calculate each value requested for the following scores: 4, 3, 7, 1.

a. oX d. oX 1

b. oX2 e. o(X 1)

c. (oX)2 f. o(X 1)2

2. Identify the first step in each of the following calculations.

a. oX2 c. o(X 2)2

b. (oX)2

3. Use summation notation to express each of the following.

a. Subtract 2 points from each score and then add the resulting values.

b. Subtract 2 points from each score, square the resulting values, and then add the

squared numbers.

c. Add the scores and then square the total.

1. a. 15 d. 14

b. 75 e. 11

c. 225 f. 49

2. a. Square each score.

b. Add the scores.

c. Subtract 2 points from each score.

3. a. o(X 2) c. (oX)2

b. o(X 2)2

L E A R N I N G C H E C K

ANSWERS

3 0 CHAPTER 1 INTRODUCTION TO STATISTICS

SUMMARY

1. The term statistics is used to refer to methods for organizing, summarizing, and interpreting data.

2. Scientific questions usually concern a population, which is the entire set of individuals one wishes to study. Usually, populations are so large that it is impossible to examine every individual, so most research is conducted with samples. A sample is a group selected from a population, usually for purposes of a research study.

3. A characteristic that describes a sample is called a statistic, and a characteristic that describes a popula- tion is called a parameter. Although sample statistics are usually representative of corresponding population parameters, there is typically some discrepancy between a statistic and a parameter. The naturally occurring difference between a statistic and a parameter is called sampling error.

4. Statistical methods can be classified into two broad categories: descriptive statistics, which organize and summarize data, and inferential statistics, which use sample data to draw inferences about populations.

5. The correlational method examines relationships between variables by measuring two different variables for each individual. This method allows researchers to measure and describe relationships, but cannot produce a cause-and-effect explanation for the relationship.

6. The experimental method examines relationships between variables by manipulating an independent variable to create different treatment conditions and then measuring a dependent variable to obtain a group of scores in each condition. The groups of scores are then compared. A systematic difference between groups provides evidence that changing the independent variable from one condition to another also caused a change in the dependent variable. All other variables are controlled to prevent them from influencing the relationship. The intent of the experimental method is to demonstrate a cause-and-effect relationship between variables.

7. Nonexperimental studies also examine relationships between variables by comparing groups of scores,

but they do not have the rigor of true experiments and cannot produce cause-and-effect explanations. Instead of manipulating a variable to create different groups, a nonexperimental study uses a preexisting participant characteristic (such as male/female) or the passage of time (before/after) to create the groups being compared.

8. A discrete variable consists of indivisible categories, often whole numbers that vary in countable steps. A continuous variable consists of categories that are infinitely divisible and each score corresponds to an interval on the scale. The boundaries that separate intervals are called real limits and are located exactly halfway between adjacent scores.

9. A measurement scale consists of a set of categories that are used to classify individuals. A nominal scale consists of categories that differ only in name and are not differentiated in terms of magnitude or direction. In an ordinal scale, the categories are differentiated in terms of direction, forming an ordered series. An interval scale consists of an ordered series of categories that are all equal-sized intervals. With an interval scale, it is possible to differentiate direc- tion and magnitude (or distance) between categories. Finally, a ratio scale is an interval scale for which the zero point indicates none of the variable being measured. With a ratio scale, ratios of measurements reflect ratios of magnitude.

10. The letter X is used to represent scores for a variable. If a second variable is used, Y represents its scores. The letter N is used as the symbol for the number of scores in a population; n is the symbol for the number of scores in a sample.

11. The Greek letter sigma (o) is used to stand for summation. Therefore, the expression oX is read the sum of the scores. Summation is a mathematical operation (like addition or multiplication) and must be performed in its proper place in the order of operations; summation occurs after operations in parentheses, exponents, and multiplication/division have been completed.

RESOURCES 31

KEY TERMS

statistics (5)

population (5)

sample (6)

variable (7)

datum (7)

raw score (7)

data (7)

data set (7)

parameter (7)

statistic (7)

descriptive statistics (8)

inferential statistics (8)

sampling error (8)

correlational method (12)

correlational research strategy (12)

experimental method (15)

confounded (15)

random assignment (15)

matching (15)

independent variable (16)

dependent variable (16)

control condition or control

group (16)

experimental condition or

experimental group (16)

nonequivalent groups study (17)

prepost study (17)

quasi-independent variable (18)

construct or hypothetical

construct (20)

operational definition (20)

discrete variable (20)

continuous variable (21)

real limits (22)

lower real limit (22)

upper real limit (22)

nominal scale (23)

ordinal scale (23)

interval scale (24)

ratio scale (24)

sigma (26)

order of operations (27)

RESOURCES

Go to CengageBrain.com to access Psychology CourseMate, where you will find an

interactive eBook, glossaries, flashcards, quizzes, statistics workshops, and more.

If your professor has assigned Aplia:

1. Sign in to your account.

2. Complete the corresponding exercises as required by your professor.

3. When finished, click Grade It Now to see which areas you have mastered, which

areas need more work, and detailed explanations of every answer.

The Statistical Package for the Social Sciences, known as SPSS, is a computer program

that performs most of the statistical calculations that are presented in this book, and is

commonly available on college and university computer systems. Appendix D contains

a general introduction to SPSS. In the Resources section at the end of each chapter for

which SPSS is applicable, there are step-by-step instructions for using SPSS to perform

the statistical operations presented in the chapter.

3 2 CHAPTER 1 INTRODUCTION TO STATISTICS

FOCUS ON PROBLEM SOLVING

It may help to simplify summation notation if you observe that the summation sign is

always followed by a symbol or symbolic expressionfor example, oX or o(X 1 3). This symbol specifies which values you are to add. If you use the symbol as a column

heading and list all the appropriate values in the column, your task is simply to add up

the numbers in the column. To find o(X 1 3) for example, start a column headed with (X 1 3) next to the column of Xs. List all the (X 1 3) values; then find the total for the column.

Often, summation notation is part of a relatively complex mathematical expres-

sion that requires several steps of calculation. The series of steps must be performed

according to the order of mathematical operations (see page 27). The best procedure

is to use a computational table that begins with the original X values listed in the first

column. Except for summation, each step in the calculation creates a new column

of values. For example, computing o(X 1 1)2 involves three steps and produces a computational table with three columns. The final step is to add the values in the

third column (see Example 1.4).

DEMONSTRATION 1.1

SUMMATION NOTATION

A set of scores consists of the following values:

7 3 9 5 4

For these scores, compute each of the following:

oX

(oX)2

oX2

oX 1 5 o(X 2)

Compute oX To compute oX, we simply add all of the scores in the group.

oX 7 1 3 1 9 1 5 1 4 28

Compute (oX)2 The first step, inside the parentheses, is to compute oX. The second step

is to square the value for oX.

oX 28 and (oX)2 (28)2 784

Compute oX2 The first step is to square each score. The second step is to add the squared

scores. The computational table shows the scores and squared scores. To compute oX2 we

add the values in the X2 column.

oX2 49 1 9 1 81 1 25 1 16 180

X X2

7 49

3 9

9 81

5 25

4 16

PROBLEMS 33

Compute oX 1 5 The first step is to compute oX. The second step is to add 5 points to the total.

oX 28 and oX 1 5 28 1 5 33

Compute o(X 2) The first step, inside parentheses, is to subtract 2 points from each

score. The second step is to add the resulting values. The computational table shows the

scores and the (X 2) values. To compute o(X 2), add the values in the (X 2) column

o(X 2) 5 1 1 1 7 1 3 1 2 18

X X 2

7 5

3 1

9 7

5 3

4 2

PROBLEMS

*1. A researcher is investigating the effectiveness of a treatment for adolescent boys who are taking medi- cation for depression. A group of 30 boys is selected and half receive the new treatment in addition to their medication and the other half continue to take their medication without any treatment. For this study,

a. Identify the population. b. Identify the sample.

2. Define the terms population, sample, parameter, and statistic.

3. Statistical methods are classified into two major categories: descriptive and inferential. Describe the general purpose for the statistical methods in each category.

4. Define the concept of sampling error and explain why this phenomenon creates a problem to be ad- dressed by inferential statistics.

5. Describe the data for a correlational research study. Explain how these data are different from the data obtained in experimental and nonexperimental studies, which also evaluate relationships between two variables.

6. What is the goal for an experimental research study? Identify the two elements that are necessary for an experiment to achieve its goal.

7. Knight and Haslam (2010) found that office workers who had some input into the design of their office space were more productive and had higher well- being compared to workers for whom the office design was completely controlled by an office manager. For this study, identify the independent variable and the dependent variable.

8. Judge and Cable (2010) found that thin women had higher incomes than heavier women. Is this an exam- ple of an experimental or a nonexperimental study?

9. Two researchers are both interested in determining whether large doses of vitamin C can help prevent

the common cold. Each obtains a sample of n 20 college students.

a. The first researcher interviews each student to determine whether they routinely take a vitamin C supplement. The researcher then records the number of colds each individual gets during the winter. Is this an experimental or a nonexperimental study? Explain your answer.

b. The second researcher separates the students into two roughly equivalent groups. The students in one group are given a daily multivitamin contain- ing a large amount of vitamin C, and the other group gets a multivitamin with no vitamin C. The researcher then records the number of colds each individual gets during the winter. Is this an experi- mental or a nonexperimental study? Explain your answer.

10. Weinstein, McDermott, and Roediger (2010) con- ducted an experiment to evaluate the effectiveness of different study strategies. One part of the study asked students to prepare for a test by reading a passage. In one condition, students generated and answered questions after reading the passage. In a second condition, students simply read the passage a second time. All students were then given a test on the passage material and the researchers recorded the number of correct answers.

a. Identify the dependent variable for this study. b. Is the dependent variable discrete or continuous? c. What scale of measurement (nominal, ordinal,

interval, or ratio) is used to measure the dependent variable?

11. A research study reports that alcohol consumption is significantly higher for students at a state university than for students at a religious college (Wells, 2010). Is this study an example of an experiment? Explain why or why not.

12. Oxytocin is a naturally occurring brain chemical that is nicknamed the love hormone because it seems to play a role in the formation of social relationships such as mating pairs and parentchild bonding. A recent study demonstrated that oxytocin appears to

*Solutions for odd-numbered problems are provided in Appendix C.

3 4 CHAPTER 1 INTRODUCTION TO STATISTICS

increase peoples tendency to trust others (Kosfeld, Heinrichs, Zak, Fischbacher, and Fehr, 2005). Using an investment game, the study demonstrated that people who inhaled oxytocin were more likely to give their money to a trustee compared to people who inhaled an inactive placebo. For this experimen- tal study, identify the independent variable and the dependent variable.

13. For each of the following, determine whether the variable being measured is discrete or continuous and explain your answer.

a. Social networking (number of daily minutes on Facebook)

b. Family size (number of siblings) c. Preference between digital or analog watch d. Number of correct answers on a statistics quiz

14. Four scales of measurement were introduced in this chapter: nominal, ordinal, interval, and ratio.

a. What additional information is obtained from measurements on an ordinal scale compared to measurements on a nominal scale?

b. What additional information is obtained from measurements on an interval scale compared to measurements on an ordinal scale?

c. What additional information is obtained from measurements on a ratio scale compared to measurements on an interval scale?

15. In an experiment examining the effects Tai Chi on arthritis pain, Callahan (2009) selected a large sample of individuals with doctor-diagnosed arthritis. Half of the participants immediately began a Tai Chi course and the other half (the control group) waited 8 weeks before beginning the program. At the end of 8 weeks, the individuals who had experienced Tai Chi had less arthritis pain that those who had not participated in the course.

a. Identify the independent variable for this study. b. What scale of measurement is used for the inde-

pendent variable? c. Identify the dependent variable for this study. d. What scale of measurement is used for the

dependent variable?

16. Explain why shyness is a hypothetical construct instead of a concrete variable. Describe how shyness might be measured and defined using an operational definition.

17. Ford and Torok (2008) found that motivational signs were effective in increasing physical activity on a college campus. Signs such as Step up to a healthier lifestyle and An average person burns 10 calories a minute walking up the stairs were posted by the elevators and stairs in a college building. Students and faculty increased their use of the stairs during

times that the signs were posted compared to times when there were no signs.

a. Identify the independent and dependent variables for this study.

b. What scale of measurement is used for the inde- pendent variable?

18. For the following scores, find the value of each expression:

a. oX

b. oX2

c. oX 1 1 d. o(X 1 1)

19. For the following set of scores, find the value of each expression:

a. oX2

b. (oX)2

c. o(X 1) d. o(X 1)2

20. For the following set of scores, find the value of each expression:

a. oX

b. oX2

c. o(X 1 3)

21. Two scores, X and Y, are recorded for each of n 4 subjects. For these scores, find the value of each expression.

a. oX

b. oY

c. oXY

Subject X Y

A 3 4

B 0 7

C 1 5

D 2 2

22. Use summation notation to express each of the following calculations:

a. Add 1 point to each score, and then add the result- ing values.

X

3

2

5

1

3

X

6

22

0

23

21

X

3

5

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