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Main Essentials of Statistics for the Behavioral Science, 7th Edition
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SUMMARY OF STATISTICS FORMULAS THE MEAN SUM OF SQUARES VARIANCE STANDARD DEVIATION z-SCORE (FOR LOCATING AN X VALUE) z-SCORE (FOR LOCATING A SAMPLE MEAN) t STATISTIC (SINGLE SAMPLE) t STATISTIC (INDEPENDENT MEASURES) t STATISTIC (RELATED SAMPLES) ESTIMATION INDEPENDENT-MEASURES ANOVA REPEATED-MEASURES ANOVA TWO-FACTOR ANOVA PEARSON CORRELATION SPEARMAN CORRELATION REGRESSION CHI-SQUARE STATISTIC MEASURES OF EFFECT SIZE Essentials of Statistics for the Behavioral Sciences Essentials of Statistics for the Behavioral Sciences S E V E N T H E D I T I O N Frederick J Gravetter State University of New York College at Brockport Larry B. Wallnau State University of New York College at Brockport Essentials of Statistics for the Behavioral Sciences, 7th Edition Frederick J Gravetter Larry B. Wallnau Executive Editor: Jon-David Hague Associate Developmental Editor: Tali Beesley Assistant Editor: Rebecca Rosenberg Editorial Assistant: Alicia McLaughlin Media Editor: Mary Noel Marketing Manager: Tierra Morgan Marketing Assistant: Anna Andersen Marketing Communications Manager: Talia Wise Content Project Manager: Charlene M. Carpentier Creative Director: Rob Hugel Art Director: Vernon Boes Print Buyer: Linda Hsu Rights Acquisitions Account Manager, Text: Roberta Broyer Rights Acquisitions Account Manager, Image: Don Schlotman Production Service: Graphic World Inc. Text Designer: Lisa Henry Illustrator: Graphic World Inc. Cover Designer: Roger Knox Cover Image: Astro Retro I by Lisa Kowalski Compositor: Graphic World Inc. © 2011, 2008 Wadsworth, Cengage Learning ALL RIGHTS RESERVED. 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Library of Congress Control Number: 2009932188 ISBN-13: 978-0-495-81220-3 ISBN-10: 0-495-81220-X Wadsworth 20 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at www.cengage.com/global. Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Wadsworth, visit www.cengage.com/wadsworth Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 1 2 3 4 5 6 7 13 12 11 10 09 Contents in Brief A note about this eBook PART I INTRODUCTION AND DESCRIPTIVE STATISTICS Chapter 1 Introduction to Statistics Chapter 2 Frequency Distributions Chapter 3 Central Tendency Chapter 4 Variability PART II FOUNDATIONS OF INFERENTIAL STATISTICS Chapter 5 z-Scores: Location of Scores and Standardized Distributions Chapter 6 Probability Chapter 7 Probability and Samples: The Distribution of Sample Means Chapter 8 Introduction to Hypothesis Testing PART III USING t STATISTICS FOR INFERENCES ABOUT POPULATION MEANS AND MEAN DIFFERENCES Chapter 9 Introduction to the t Statistic Chapter 10 The t Test for Two Independent Samples Chapter 11 The t Test for Two Related Samples Chapter 12 Estimation PART IV ANALYSIS OF VARIANCE: TESTS FOR DIFFERENCES AMONG TWO OR MORE POPULATION MEANS Chapter 13 Introduction to Analysis of Variance Chapter 14 Repeated-Measures and Two-Factor Analysis of Variance PART V CORRELATIONS AND NONPARAMETRIC TESTS Chapter 15 Correlation and Regression Chapter 16 The Chi-Square Statistic: Tests for Goodness of Fit and Independence Contents PART I INTRODUCTION AND DESCRIPTIVE STATISTICS Chapter 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 Resources Focus on Problem Solving Demonstration 1.1 Problems Chapter 2 Frequency Distributions 2.1 Introduction to Frequency Distributions 2.2 Frequency Distribution Tables 2.3 Frequency Distribution Graphs 2.4 THE Shape of a Frequency Distribution Summary Resources Focus on Problem Solving Demonstration 2.1 Problems Chapter 3 Central Tendency 3.1 Overview 3.2 The Mean 3.3 The Median 3.4 The Mode 3.5 Selecting a Measure of Central Tendency 3.6 Central Tendency and the Shape of the Distribution Summary Resources Focus on Problem Solving Demonstration 3.1 Problems Chapter 4 Variability 4.1 Overview 4.2 The Range 4.3 Standard Deviation and Variance for a Population 4.4 Standard Deviation and Variance for Samples 4.5 More about Variance and Standard Deviation Summary Resources Focus on Problem Solving Demonstration 4.1 Problems PART I REVIEW PART II FOUNDATIONS OF INFERENTIAL STATISTICS Chapter 5 z-Scores: Location of Scores and Standardized Distributions 5.1 Introduction to z-scores 5.2 z-scores and Location in a Distribution 5.3 Using z-scores to Standardize a Distribution 5.4 Other Standardized Distributions Based on z-Scores 5.5 Computing z-scores for Samples 5.6 Looking Ahead to Inferential Statistics Summary Resources Focus on Problem Solving Demonstration 5.1 and Demonstration 5.2 Problems Chapter 6 Probability 6.1 Introduction to Probability 6.2 Probability and the Normal Distribution 6.3 Probabilities and Proportions for Scores from a Normal Distribution 6.4 Looking Ahead to Inferential Statistics Summary Resources Focus on Problem Solving Demonstrations 6.1 Problems Chapter 7 Probability and Samples: The Distribution of Sample Means 7.1 Samples and Populations 7.2 The Distribution of Sample Means 7.3 Probability and the Distribution of Sample Means 7.4 More about Standard Error 7.5 Looking Ahead to Inferential Statistics Summary Resources Focus on Problem Solving Demonstration 7.1 Problems Chapter 8 Introduction to Hypothesis Testing 8.1 The Logic of Hypothesis Testing 8.2 Uncertainty and Errors in Hypothesis Testing 8.3 An Example of a Hypothesis Test 8.4 Directional (One-Tailed) Hypothesis Tests 8.5 Concerns about Hypothesis Testing: Measuring Effect Size 8.6 Statistical Power Summary Resources Focus on Problem Solving Demonstrations 8.1 and 8.2 Problems PART II REVIEW PART III USING t STATISTICS FOR INFERENCES ABOUT POPULATION MEANS AND MEAN DIFFERENCES Chapter 9 Introduction to the t Statistic 9.1 The t Statistic: An Alternative to Z 9.2 Hypothesis Tests with the t Statistic 9.3 Measuring Effect Size for the t Statistic 9.4 Directional Hypotheses and One-Tailed Tests Summary Resources Focus on Problem Solving Demonstrations 9.1 and 9.2 Problems Chapter 10 The t Test for Two Independent Samples 10.1 Introduction to the Independent-Measures Design 10.2 The t Statistic for an Independent-Measures Research Design 10.3 Hypothesis Tests and Effect Size with the Independent-Measures t Statistic 10.4 Assumptions Underlying the Independent-Measures t Formula Summary Resources Focus on Problem Solving Demonstrations 10.1 and 10.2 Problems Chapter 11 The t Test for Two Related Samples 11.1 Introduction to Repeated-Measures Designs 11.2 The t Statistic for a Repeated-Measures Research Design 11.3 Hypothesis Tests and Effect Size for the Repeated-Measures Design 11.4 Uses and Assumptions for Repeated-Measures t Tests Summary Resources Focus on Problem Solving Demonstration 11.1 and Demonstration 11.2 Problems Chapter 12 Estimation 12.1 An Overview of Estimation 12.2 Estimation with the t Statistic 12.3 A Final Look at Estimation Summary Resources Focus on Problem Solving Demonstrations 12.1 and 12.2 Problems PART III REVIEW PART IV ANALYSIS OF VARIANCE: TESTS FOR DIFFERENCES AMONG TWO OR MORE POPULATION MEANS Chapter 13 Introduction to Analysis of Variance 13.1 Introduction 13.2 The Logic of Analysis of Variance 13.3 ANOVA Notation and Formulas 13.4 The Distribution of F-Ratios 13.5 Examples of Hypothesis Testing and Effect Size with ANOVA 13.6 Post Hoc Tests 13.7 The Relationship between ANOVA and t Tests Summary Resources Focus on Problem Solving Demonstrations 13.1 and 13.2 Problems Chapter 14 Repeated-Measures and Two-Factor Analysis of Variance 14.1 Overview 14.2 Repeated-Measures Analysis of Variance 14.3 Two-Factor ANOVA (Independent Measures) Summary Resources Focus on Problem Solving Demonstrations 14.1 and 14.2 Problems PART IV REVIEW PART V CORRELATIONS AND NONPARAMETRIC TESTS Chapter 15 Correlation and Regression 15.1 Introduction 15.2 The Pearson Correlation 15.3 Using and Interpreting the Pearson Correlation 15.4 Hypothesis Tests with the Pearson Correlation 15.5 Alternatives to the Pearson Correlation 15.6 Introduction to Linear Equations and Regression Summary Resources Focus on Problem Solving Demonstration 15.1 Problems Chapter 16 The Chi-Square Statistic: Tests for Goodness of Fit and Independence 16.1 Parametric and Nonparametric Statistical Tests 16.2 The Chi-Square Test for Goodness of Fit 16.3 The Chi-Square Test for Independence 16.4 Measuring Effects Size for the Chi-Square Test for Independence 16.5 Assumptions and Restrictions for Chi-Square Tests 16.6 Special Applications for the Chi-Square Tests Summary Resources Focus on Problem Solving Demonstrations 16.1 and 16.2 Problems PART V REVIEW Appendix A Basic Mathematics Review Appendix B Statistical Tables Appendix C Solutions for Odd-Numbered Problems is the Text Appendix D General Instructions for Using SPSS Statistics Organizer References Index A note about this eBook In some cases, individual items such as ancillary images or multimedia have been removed for digital delivery due to rights restrictions. This eBook does not include any extra assets or study materials that may come free with a printed textbook. If you are not sure what is included with a Cengage eBook or if you have questions about how this eBook differs from the printed text, please contact customer support at email@example.com. Web addresses listed in this book may change over time. Please contact Cengage customer support at email@example.com if you have questions about where to find information referenced in the book. Preface Many students in the behavioral sciences view the required statistics course as an intimidating obstacle that has been placed in the middle of an otherwise interesting curriculum. They want to learn about human behavior—not about math and science. As a result, the statistics course is seen as irrelevant to their education and career goals. However, as long as the behavioral sciences are founded in science, a knowledge of statistics will be necessary. Statistical procedures provide researchers with objective and systematic methods for describing and interpreting their research results. Scientific research is the system that we use to gather information, and statistics are the tools that we use to distill the information into sensible and justified conclusions. The goal of this book is not only to teach the methods of statistics but also to convey the basic principles of objectivity and logic that are essential for science and valuable for decision marking 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 entitled "To the Instructor." In revising this text, our students have been foremost in our minds. Over the years, they have provided honest and useful feedback. Their hard work and perseverance has made our writing and teaching most rewarding. We sincerely thank them. Students who are using this edition should please read the section of the preface entitled "To the Student." ANCILLARIES Ancillaries for this edition include the following: Aplia Statistics for Psychology and the Behavioral Sciences: An online interactive learning solution that ensures students stay involved with their coursework and master the basic tools and concepts of statistical analysis. Created by a research psychologist to help students excel, Aplia's content engages students with questions based on real-world scenarios that help students understand 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. Enhanced WebAssign: Enhanced WebAssign allows instructors to assign additional homework and study problems to students so they can get even more practice on the most difficult concepts. Original guided tutorials and simulations lead students through step-by-step problem-solving. Algorithmic versions of the end-of-chapter problems offer opportunities for additional practice. Study Guide: Contains chapter overviews, learning objectives, new terms and concepts, new formulas, step-by-step procedures for problem solving, study hints and cautions, self-tests, and review. The Study Guide contains answers to the self-test questions. Book Companion Website: Additional, free study resources are available online at the book companion website. Practice and reinforce statistical concepts using the Statistics Workshops, chapter quizzes, chapter objectives, interactive flash cards, Web links, and more! Visit www.cengage.com/psychology/gravetter. Instructor's 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 download or for ExamView computerized test bank software. PowerLecture with JoinIn and ExamView: This CD includes the instructor's manual, test bank, lecture slides with book figures, and more. Featuring automatic grading, ExamView™, also available within PowerLecture, allows you to create, deliver, and customize tests and study guides (both print and online) in minutes. See assessments onscreen exactly as they will print or display online. Build tests of up to 250 questions using up to 12 question types and enter an unlimited number of new questions or edit existing questions. The JoinInTM content (for use with most "clicker" systems) available within PowerLecture delivers instant classroom assessment and active learning. WebTutor on Blackboard and WebCT: Load a WebTutor cartridge into your Course Management System to access text-specific content, media assets, quizzing, Web links, and discussion topics. Easily blend, add, edit, reorganize, or delete content. ACKNOWLEDGMENTS It takes a lot of good, hard-working people to produce a book. Our friends at Wadsworth have made enormous contributions to this text book. We thank Jane Potter, Senior Sponsoring Editor; Tali Beesley, Associate Development Editor; Rebecca Rosenberg, Assistant Editor; Nicolas Albert, Editorial Assistant; Charlene Carpentier, Production Project Manager; Brenda Ginty, Production Manager; Rachel Guzman, Media Editor; and Vernon Boes, Art Director. 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 seventh edition: Kiesa Kelly, Tennessee State University; Renee V. Galliher, Utah State University; Johnathan Forbey, Ball State University; Helga Noice, Elmhurst College; Merritt Hoover, California State University, San Jose; Sister Edith Bogue, College of St. Scholastica; Michael Abrams, San Jose State University; Dr. J. T. Greco, Houston Community College; Nuria Giralt, California State University, Long Beach; and Edwin Gomez, Old Dominion University. TO THE INSTRUCTOR Those of you familiar with the previous edition of Essentials of Statistics for the Behavioral Sciences will notice a number of changes in the seventh 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 parts to emphasize the similarities among groups of statistical methods. Each part contains two to four chapters, and begins with an introduction and concludes with a review, including review exercises. Also, the Demonstration problems at the end of each chapter have been simplified to focus on the content of the chapter and minimize tedious computations. Other examples of specific and noteworthy revisions include: Chapter 1 The distinction between true experiments and other research strategies has been clarified and there is expanded discussion of continuous and discrete variables. Chapter 2 Minor editing adds variety and simplifies examples. Chapter 3 The distinction between discrete and continuous variables has been expanded, especially in the context of computing the median. An example demonstrates finding the precise median for a continuous variable. Chapter 4 New discussion relates measures of variability to the scales of measurement introduced in Chapter 1. Specifically, most measures of variability are measures of distance and require an interval or ratio scale. Alternate definitions of the range have been added and discussion of the interquartile range has been deleted. Greater emphasis is placed on the computational formula for SS because it is used for most data sets. Also, the section discussing biased and unbiased statistics has been expanded. Chapter 5 There is increased emphasis on the definition and concept of z-scores rather than formulas, encouraging students to estimate answers (based on the definition) before they begin calculations. A new section demonstrates that many z-score problems can be solved by sketching distributions rather than manipulating formulas and numbers. The section on z-scores for samples has been expanded. Chapter 6 Minor editing adds variety to the examples and simplifies the instructions for using the unit normal table. Chapter 7 The discussion of the basic definition and purpose of the standard error has been expanded. A new section emphasizes that we are now dealing with three distributions: the original population, the sample, and the theoretical distribution of sample means. Chapter 8 A new analogy compares hypothesis testing with the process of a jury trial. The section on statistical power has been expanded, including new discussion relating power to Type II errors, new examples demonstrating the calculation of power, and a new discussion concerning the relationship between power and determining an optimum sample size for a research study. Chapter 9 A new section discusses the research situations in which the t statistic can be used for hypothesis testing. Also, new text simplifies locating the critical region for a one-tailed test. Chapter 10 A new section discusses how sample variance and sample size influence the outcome of an independent-measures t hypothesis test and measures of effect size. Chapter 11 A figure has been added to illustrate the null and alternative hypotheses for the repeated-measures t test. The discussion of variability as a reflection of the consistency of the treatment effect has been expanded. Chapter 12 A new box discusses the difference between statistical significance and practical significance. Also, the comparison of hypothesis testing and estimation has been simplified. Chapter 13 A new section clarifies the advantages and disadvantages of the different methods for computing the between-treatments SS. Another new section discusses the relationship between ANOVA and t tests when evaluating the significance of a mean difference between two treatments or two populations. Chapter 14 New text clarifies the introduction to the repeated-measures F-ratio by stressing that the goal of the hypothesis test is to differentiate between the systematic differences that are caused by the treatment conditions and the random, unsystematic differences that are explained by chance or error. Chapter 15 A new section introduces the concept and formula for a partial correlation, and discusses how partial correlations are used to measures the correlation between two variables after eliminating the influence of a third variable. The section demonstrating analysis of regression has been supplemented with a note that this analysis is equivalent to testing the significance of the Pearson correlation. Chapter 16 Minor editing changes incorporates into text what was formerly a box discussing the rationale for the denominator of the chi-square formula. Matching the Text to Your Syllabus The book chapters are organized in the sequence that we use for our own statistics courses. However, different instructors may prefer different organizations and probably will choose to omit or deemphasize specific topics. We have tried to make separate chapters, and even sections of chapters, completely self-contained, so they can be deleted or reorganized to fit the syllabus for nearly any instructor. Some common examples are as follows: Many instructors omit Chapter 12, Estimation, entirely. It also is common for instructors to choose between emphasizing analysis of variance (Chapters 13 and 14) or emphasizing correlation/regression (Chapter 15). It is rare for a one-semester course to include complete coverage of both topics. Although we choose to complete all the hypothesis tests for means and mean differences before introducing correlation (Chapter 15), many instructors prefer to place correlation much earlier in the sequence of course topics. To accommodate this, Sections 15.1 and 15.2 present the calculation and interpretation of the Pearson correlation and can be introduced immediately following Chapter 4 (variability). Other sections of Chapter 15 refer to hypothesis testing and should be delayed until the process of hypothesis testing (Chapter 8) has been introduced. It is also possible for instructors to present the chi-square tests (Chapter 16) much earlier in the sequence of course topics. Chapter 16, which presents hypothesis tests for proportions, can be presented immediately after Chapter 8, which introduces the process of hypothesis testing. If this is done, we also recommend that the Pearson correlation (Sections 15.1 and 15.2) be presented early to provide a foundation for the chi-square test for independence. TO THE STUDENT A primary goal of this book is to make the task of learning statistics as easy and painless as possible. Among other things, you will notice that the book provides you with a number of opportunities to practice the techniques you will be learning in the form of learning checks, examples, demonstrations, and end-of-chapter problems. We encourage you to take advantage of these opportunities. Read the text rather than just 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 underlying 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 Larry Wallnau at the Department of Psychology, SUNY College at Brockport, 350 New Campus Drive, Brockport, New York 14420. You can also contact Professor Emeritus Gravetter directly at firstname.lastname@example.com. Study Hints You may find some of these tips helpful, as our own students have reported. The key to success in a statistics course is to keep up with the material. Each new topic builds on previous topics. If you have learned the previous material, then the new topic is just one small step forward. Without the proper background, however, the new topic can be a complete mystery. If you find that you are falling behind, get help immediately. You will learn (and remember) much more if you study for short periods several times per week rather than try to condense all of your studying into one long session. For example, it is far more effective to study half an hour every night than to have a single 3 1/2-hour study session once a week. We cannot even work on writing this book without frequent rest breaks. Do some work before class. Keep a little ahead of the instructor by reading the appropriate sections before they are presented in class. Although you may not fully understand what you read, you will have a general idea of the topic, which will make the lecture easier to follow. Also, you can identify material that is particularly confusing and then be sure the topic is clarified in class. Pay attention and think during class. Although this advice seems obvious, often it is not practiced. Many students spend so much time trying to write down every example presented or every word spoken by the instructor that they do not actually understand and process what is being said. Check with your instructor—there may not be a need to copy every example presented in class, especially if there are many examples like it in the text. Sometimes, we tell our students to put their pens and pencils down for a moment and just listen. Test yourself regularly. Do not wait until the end of the chapter or the end of the week to check your knowledge. After each lecture, work some of the end-of-chapter problems and do the Learning Checks. Review the Demonstration Problems, and be sure you can define the Key Terms. If you are having trouble, get your questions answered immediately (reread the section, go to your instructor, or ask questions in class). By doing so, you will be able to move ahead to new material. Do not kid yourself! Avoid denial. Many students observe their instructor solve problems in class and think to themselves, "This looks easy, I understand it." Do you really understand it? Can you really do the problem on your own without having to leaf through the pages of a chapter? Although there is nothing wrong with using examples in the text as models for solving problems, you should try working a problem with your book closed to test your level of mastery. We realize that many students are embarrassed to ask for help. It is our biggest challenge as instructors. You must find a way to overcome this aversion. Perhaps contacting the instructor directly would be a good starting point, if asking questions in class is too anxiety-provoking. You could be pleasantly surprised to find that your instructor does not yell, scold, or bite! Also, your instructor might know of another student who can offer assistance. Peer tutoring can be very helpful. Frederick J Gravetter Larry B. Wallnau 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 bachelor's degree in mathematics from M.I.T. and his Ph.D. in psychology from Duke University. In addition to publishing this textbook and several research articles, Dr. Gravetter co-authored Research Methods for the Behavioral Sciences and Statistics for the Behavioral Sciences. Fred Larry B. Wallnau is Professor Emeritus of Psychology at the State University of New York College at Brockport. At Brockport he taught course relating to the biological basis of behavior and published numerous research 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 on the seeking worldwide contacts with other radio enthusiasts. Larry PART I INTRODUCTION AND DESCRIPTIVE STATISTICS Chapter 1 Introduction to Statistics Chapter 2 Frequency Distributions Chapter 3 Central Tendency Chapter 4 Variability We have divided this book into five sections, each covering a general topic area of statistics. The first section, consisting of Chapters 1 to 4, provides a broad overview 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 techniques 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 calculating 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 integrate the elements from the separate chapters. CHAPTER 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 1.1 STATISTICS, SCIENCE, AND OBSERVATIONS PREVIEW Before we begin our discussion of statistics, we ask you to read the following paragraph taken from the philosophy of Wrong Shui (Candappa, 2000). The Journey to Enlightenment In Wrong Shui life is seen as a cosmic journey, a struggle to overcome unseen and unexpected obstacles at the end of which the traveler will find illumination and enlightenment. Replicate this quest in your home by moving light switches away from doors and over to the far side of each room.∗ Why did we begin a statistics book with a bit of twisted philosophy? 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. The objectives for this first chapter are to provide an introduction to the topic of statistics and to give you some background for the rest of the book. We discuss the role of statistics within the general field of scientific inquiry, and we introduce some of the vocabulary and notation that are necessary for the statistical methods that follow. As you read through the following chapters, keep in mind that the general topic of statistics follows a well-organized, logically developed progression that leads from basic concepts and definitions to increasingly sophisticated techniques. Thus, the material presented in the early chapters of this book will serve as a foundation for the material that follows. The content of the first nine chapters, for example, provides an essential background and context for the statistical methods presented in Chapter 10. If you turn directly to Chapter 10 without reading the first nine chapters, you will find the material confusing and incomprehensible. However, if you learn and use the background material, you will have a good frame of reference for understanding and incorporating new concepts as they are presented. DEFINITIONS OF STATISTICS By one definition, statistics consist of facts and figures such as average income, crime rate, birth rate, average snowfall, and so on. These statistics are usually informative and time-saving because they condense large quantities of information into a few simple figures. Later in this chapter we return to the notion of calculating statistics (facts and figures) but, for now, we concentrate on a much broader definition of statistics. Specifically, we use the term statistics to refer to a set of mathematical procedures. 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. Research in psychology (and other fields) involves gathering information. To determine, for example, whether violence on TV has any effect on children's behavior, you would need to gather information about children's behaviors. When researchers finish the task of gathering information, they typically find themselves with pages and pages of measurements such as IQ scores, personality scores, reaction time scores, and so on. The role of statistics is to help researchers make sense of this information. Specifically, statistics serve two general purposes: Statistics are used to organize and summarize the information so that the researcher can see what happened in the research study and can communicate the results to others. Statistics help the researcher to answer the general questions that initiated the research by determining exactly what conclusions are justified based on the results that were obtained. DEFINITION The term statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information. Statistical procedures help ensure that the information or observations are presented and interpreted in an accurate and informative way. In somewhat grandiose terms, statistics help researchers bring order out of chaos. In addition, statistics provide researchers with a set of standardized techniques that are recognized and understood throughout the scientific community. Thus, the statistical methods used by one researcher will be familiar to other researchers, who can accurately interpret the statistical analyses with a full understanding of how the analysis was done and what the results signify. 1.2 POPULATIONS AND SAMPLES WHAT ARE THEY? Scientific research typically begins with a general question about a specific group (or groups) of individuals. For example, a researcher may be interested in the effect of divorce on the self-esteem of preteen children. 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 preteen children. In the second example, the researcher wants to compare the group of men with the group of women. In statistical terminology, the entire group that a researcher wishes to study is called a population. DEFINITION A population is the set of all the individuals of interest in a particular study. As you can well imagine, a population can be quite large—for example, the entire set of women on the planet Earth. A researcher might be more specific, limiting the population for study to women who are registered voters in the United States. Perhaps the investigator would like to study the population consisting of women who are heads of state. Populations can obviously vary in size from extremely large to very small, depending on how the investigator defines the population. The population being studied should always be identified by the researcher. In addition, the population need not consist of people—it could be a population of rats, corporations, parts produced in a factory, or anything else an investigator wants to study. In practice, populations are typically very large, such as the population of college sophomores in the United States or the population of small businesses. Because populations tend to be very large, it usually is impossible for a researcher to examine every individual in the population of interest. Therefore, researchers typically select 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. DEFINITION 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 children in a preschool program, and another study might use a sample of more than 1000 registered voters representing the population of a major city. So far we have talked about a sample being selected from a population. However, this is actually only half of the full relationship between a sample and its population. Specifically, when a researcher finishes examining the sample, the goal is to generalize the results back to the entire population. Remember that the research started with a general question about the population. To answer the question, a researcher studies a sample and then generalizes the results from the sample to the population. The full relationship between a sample and a population is shown in Figure 1.1. VARIABLES AND DATA Typically, researchers are interested in specific characteristics of the individuals in the population (or in the sample), or they are interested in outside factors that may influence the individuals. For example, a researcher may be interested in the influence of the weather on people's moods. As the weather changes, do people's moods also change? Something that can change or have different values is called a variable. DEFINITION A variable is a characteristic or condition that changes or has different values for different individuals. Once again, variables can be characteristics that differ from one individual to another, such as height, weight, gender, or personality. Also, variables can be environmental conditions that change such as temperature, time of day, or the size of the room in which the research is being conducted. FIGURE 1.1 The relationship between a population and a sample. 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. DEFINITIONS 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 sophomores and a sample of preschool children. Be forewarned, however, that we will also refer to populations or samples of scores. Because research typically involves measuring each individual to obtain a score, every sample (or population) of individuals produces a corresponding sample (or population) of scores. PARAMETERS AND STATISTICS When describing data it is necessary to distinguish whether the data come from a population or a sample. A characteristic that describes a population—for example, the average score for the population—is called a parameter. A characteristic that describes a sample is called a statistic. Thus, the average score for a sample is an example of a statistic. Typically, the research process begins with a question about a population parameter. However, the actual data come from a sample and are used to compute sample statistics. DEFINITIONS 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. Typically, every population parameter has a corresponding sample statistic, and much of this book is concerned with the relationship between sample statistics and the corresponding population parameters. In Chapter 7, for example, we examine the relationship between the mean obtained for a sample and the mean for the population from which the sample was obtained. DESCRIPTIVE AND INFERENTIAL STATISTICAL METHODS Although researchers have developed a variety of different statistical procedures to organize and interpret data, these different procedures can be classified into two general categories. The first category, descriptive statistics, consists of statistical procedures that are used to simplify and summarize data. DEFINITION Descriptive statistics are statistical procedures used to summarize, organize, and simplify data. Descriptive statistics are techniques that take raw scores and organize or summarize them in a form that is more manageable. Often the scores are organized in a table or a graph so that it is possible to see the entire set of scores. Another common technique is to summarize a set of scores by computing an average. Note that even if the data set has hundreds of scores, the average provides a single descriptive value for the entire set. The second general category of statistical techniques is called inferential statistics. Inferential statistics are methods that use sample data to make general statements about a population. DEFINITION Inferential statistics consist of techniques that allow us to study samples and then make generalizations about the populations from which they were selected. It usually is not possible to measure everyone in the population. Because populations are typically very large, a sample is selected to represent the population. By analyzing the results from the sample, we hope to make general statements about the population. Typically, researchers use sample statistics as the basis for drawing conclusions about population parameters. One problem with using samples, however, is that a sample provides only limited information about the population. Although samples are generally representative of their populations, a sample is not expected to give a perfectly accurate picture of the whole population. There usually is some discrepancy between a sample statistic and the corresponding population parameter. This discrepancy is called sampling error, and it creates the fundamental problem inferential statistics must always address (Box 1.1). DEFINITION Sampling error is the discrepancy, or amount of 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 1000 college students and two samples, each with 5 students, that have been selected from the population. Notice that each sample contains different individuals who have different characteristics. Because the characteristics of each sample depend on the specific people in the sample, statistics will vary from one sample to another. For example, the five students in sample 1 have an average age of 19.8 years and the students in sample 2 have an average age of 20.4 years. In addition, it is very unlikely that the statistics obtained for a sample will be identical to the parameters for the entire population. In this 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. BOX 1.1 THE MARGIN OF ERROR BETWEEN STATISTICS AND PARAMETERS One common example of sampling error is the error associated with a sample proportion. For example, in newspaper articles reporting results from political polls, you frequently find statements such as this: Candidate Brown leads the poll with 51% of the vote. Candidate Jones has 42% approval, and the remaining 7% are undecided. This poll was taken from a sample of registered voters and has a margin of error of plus-or-minus 4 percentage points. The "margin of error" is the sampling error. In this case, the percentages that are reported were obtained from a sample and are being generalized to the whole population. As always, you do not expect the statistics from a sample to be perfect. There always will be some "margin of error" when sample statistics are used to represent population parameters. As a further demonstration of sampling error, imagine that your statistics class is separated into two groups by drawing a line from front to back through the middle of the room. Now imagine that you compute the average age (or height, or IQ) for each group. Will the two groups have exactly the same average? Almost certainly they will not. No matter what you chose to measure, you will probably find some difference between the two groups. However, the difference you obtain does not necessarily mean that there is a systematic difference between the two groups. For example, if the average age for students on the right-hand side of the room is higher than the average for students on the left, it is unlikely that some mysterious force has caused the older people to gravitate to the right side of the room. Instead, the difference is probably the result of random factors such as chance. The unpredictable, unsystematic differences that exist from one sample to another are an example of sampling error. STATISTICS IN THE CONTEXT OF RESEARCH The following example shows the general stages of a research study and demonstrates how descriptive statistics and inferential statistics are used to organize and interpret the data. At the end of the example, note how sampling error can affect the interpretation of experimental results, and consider why inferential statistical methods are needed to deal with this problem. FIGURE 1.2 A demonstration of sampling error. Two samples are selected from the same population. Notice that the sample statistics are different from one sample to another, and all 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. EXAMPLE 1.1 Figure 1.3 shows an overview of a general research situation and demonstrates the roles that descriptive and inferential statistics play. The purpose of the research study is to evaluate the difference between two methods for teaching reading to first-grade children. Two samples are selected from the population of first-grade children. The children in sample A are assigned to teaching method A and the children in sample B are assigned to method B. After 6 months, all of the students are given a standardized reading test. 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. FIGURE 1.3 The role of statistics in experimental research. First, descriptive statistics are used to simplify the pages of data. For example, the researcher could draw a graph showing the scores for each sample or compute the average score for each sample. Note that descriptive methods provide a simplified, organized description of the scores. In this example, the students taught by method A averaged 76 on the standardized test, and the students taught by method B averaged only 71. 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 5-point difference between the two samples (sample A averaged 76 and sample B averaged 71). The problem for inferential statistics is to differentiate between the following two interpretations: There is no real difference between the two teaching methods, and the 5-point difference between the samples is just an example of sampling error (like the samples in Figure 1.2). There really is a difference between the two teaching methods, and the 5-point difference between the samples was caused by the different methods of teaching. In simple English, does the 5-point difference between samples provide convincing evidence of a difference between the two teaching methods, or is the 5-point difference just chance? The purpose of inferential statistics is to answer this question. LEARNING CHECK A researcher is interested in the reading skill of fourth-grade students in the state of Texas. The average reading score for the entire group of fourth-grade students would be an example of a ________________. A researcher is interested in the effect of watching a reality television show about modeling on the eating behavior of 13-year-old girls. 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 ______________. 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-olds. The average score is an example of a __________________. Statistical techniques are classified into two general categories. What are the two categories called, and what is the general purpose for the techniques in each category? Briefly define the concept of sampling error. ANSWERS parameter sample statistic 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. Sampling error is the error or discrepancy between the value obtained for a sample statistic and the value for the corresponding population parameter. 1.3 DATA STRUCTURES, RESEARCH METHODS, AND STATISTICS RELATIONSHIPS BETWEEN VARIABLES Some research studies are conducted simply to describe individual variables as they exist naturally. For example, a college official may conduct a survey to describe the eating, sleeping, and study habits of a group of college students. Most research, however, is intended to examine the relationship between 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 academic performance for elementary school children? Is there a relationship between the number of hours of sleep and grade point average for college students? To establish the existence of a relationship, researchers must make observations—that is, measurements of the two variables. The resulting measurements can be classified into two distinct data structures that also help to classify different research methods and different statistical techniques. In the following section we identify and discuss these two data structures. I. ONE GROUP WITH TWO VARIABLES MEASURED FOR EACH INDIVIDUAL 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 measure the two variables for each individual. For example, research has demonstrated a relationship between sleep habits, especially wake-up time, and academic performance for college students (Trockel, Barnes, and Egget, 2000). The researchers used a survey to measure wake-up time and school records to measure academic performance for each student. Figure 1.4 shows an example of the kind of data obtained in the study. The researchers then look for consistent patterns in the data to provide evidence for a relationship between variables. For example, as wake-up time changes from one student to another, is there also a tendency for academic performance to change? Consistent patterns in the data are often easier to see if the scores are presented in a graph. Figure 1.4 also shows the scores for the eight students in a graph called a scatter plot. In the scatter plot, each individual is represented by a point so that the horizontal position corresponds to the student's wake-up time and the vertical position corresponds to the student's academic performance score. The scatter plot shows a clear relationship between wake-up time and academic performance: as wake-up time increases, academic performance decreases. FIGURE 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 in a graph (b). 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. The relationship between variables is usually measured and described using a statistic called a correlation. Correlations and the correlational method are discussed in detail in Chapter 15. DEFINITION In the correlational method, two different variables are observed to determine whether there is a relationship between them. Occasionally, the correlational method produces scores that are not numerical values. For example, a researcher could measure gender (male or female) and cellphone preference (talk or text) for a group of college students. Note that the researcher has two measurements for each individual but neither of the measurements is a numerical score. This type of data is typically summarized in a table showing how many individuals are classified into each of the possible categories. Table 1.1 shows an example of this kind of summary table. The relationship between variables for non-numerical data, such as the data in Table 1.1, is evaluated using a statistical technique known as a chi-square test. Chi-square tests are presented in Chapter 16. The results from a correlational study can demonstrate the existence of a relationship between two variables, but they do not provide an explanation for the relationship. In particular, a correlational study cannot demonstrate a cause-and-effect relationship. For example, the data in Figure 1.4 show a systematic relationship between wake-up time and academic performance for a group of college students; those who sleep late tend to have lower performance scores than those who wake early. However, there are many possible explanations for the relationship and we do not know exactly what factor (or factors) is responsible for late sleepers having lower grades. In particular, we cannot conclude that waking students up earlier would cause their academic performance to improve, or that studying more would cause students to wake up earlier. To demonstrate a cause-and-effect relationship between two variables, researchers must use the experimental method, which is discussed next. II. COMPARING TWO (OR MORE) GROUPS OF SCORES: EXPERIMENTAL AND NONEXPERIMENTAL METHODS The second method for examining the relationship beween 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, 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/adventure 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 will compare 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. TABLE 1.1 Correlational data consisting of nonnumerical scores. Note that there are two measurements for each individual: gender and cell phone preference. The numbers indicate how many people are in each category. For example, out of the 50 males, 30 prefer text over talk. When the measurement procedure produces numerical scores, the statistical evaluation typically involves computing the average score for each group and then comparing the averages. The process of computing averages is presented in Chapter 3, and a variety of statistical techniques for comparing averages are presented in Chapters 8-14. If the measurement process simply classifies individuals into non-numerical categories, the statistical evaluation usually consists of computing proportions for each group and then comparing proportions. Previously, in Table 1.1, we presented an example of non-numerical data examining the relationship between gender and cellphone preference. The same data can be used to compare the proportions for males with the proportions for females. For example, using text is preferred by 60% of the males compared to 50% of the females. As before, these data are evaluated using a chi-square test, which is presented in Chapter 16. THE EXPERIMENTAL METHOD One specific research method that involves comparing groups of scores is known as the experimental method or the experimental research strategy. The goal of an experimental study is to demonstrate a cause-and-effect relationship between two variables. Specifically, an experiment attempts to show that changing the value of one variable causes changes to occur in the second variable. To accomplish this goal, the experimental method has two characteristics that differentiate experiments from other types of research studies: 1. Manipulation The researcher manipulates one variable by changing its value from one level to another. A second variable is observed (measured) to determine whether the manipulation causes changes to occur. 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. 2. Control The researcher must exercise control over the research situation to ensure that other, extraneous variables do not influence the relationship being examined. In more complex experiments, a researcher may systematically manipulate more than one variable and may observe more than one variable. Here we are considering the simplest case, in which only one variable is manipulated and only one variable is observed. To demonstrate these two characteristics, consider an experiment in which a researcher is examining the effect of classroom temperature on student learning. The purpose of the experiment is to determine whether changes in room temperature cause changes in learning performance. The students are shown a video lecture and then are given a test to measure what they have learned. The researcher manipulates temperature by creating two or more different treatment conditions. For example, our researcher could set the temperature at 70° for one condition and then change the temperature to 90° for a second condition. The researcher then compares the test scores for a group of participants in the 70° room with the scores for another group that is tested in the 90° room. A systematic difference between the groups of scores provides evidence that changing the temperature from 70° to 90° also caused a change in learning performance. The structure of this experiment is shown in Figure 1.6. To be able to say that the difference in learning performance is caused by temperature, the researcher must rule out any other possible explanation for the difference. That is, any other variables that might affect student learning must be controlled. There are two general categories of variables that researchers must consider: 1. Participant Variables These are characteristics such as age, gender, and intelligence that vary from one individual to another. Whenever an experiment compares different groups of participants (one group in treatment A and a different group in treatment B), researchers must ensure that participant variables do not differ from one group to another. For example, the experiment shown in Figure 1.6 compares learning performance for two different temperature conditions. Suppose, however, that the participants in the 70° room have higher IQs than those in the 90° room. In this case, there is an alternative explanation for any difference in learning performance that exists between the two groups. Specifically, it is possible that the difference was caused by temperature, but it also is possible that the difference was caused by the participants' intelligence. Whenever a research study allows more than one explanation for the results, the study is said to be confounded because it is impossible to reach an unambiguous conclusion. FIGURE 1.6 The structure of an experiment. Students are randomly assigned to one of two treatment conditions: a 70° room or a 90° room. A video lecture is presented and then the students take a test on the lecture material to measure learning performance. A systematic difference in the test scores between the two groups is attributed to the treatment (the temperature of the room). 2. Environmental Variables These are characteristics of the environment such as lighting, time of day, and weather conditions. A researcher must ensure that the individuals in treatment A are tested in the same environment as the individuals in treatment B. Using the temperature experiment (see Figure 1.6) as an example, suppose that the individuals in the 70° room were all tested in the morning and the individuals in the 90° room were all tested in the afternoon. Again, this would produce a confounded experiment because the researcher could not determine whether the differences in learning performance were caused by temperature or caused by the time of day. Researchers typically use three basic techniques to control other variables. First, the researcher could use random assignment, which means that each participant has an equal chance of being assigned to each of the treatment conditions. The goal of random assignment is to distribute the participant characteristics evenly between the two groups so that neither group is noticeably smarter (or older, or faster) than the other. Random assignment can also be used to control environmental variables. For example, participants could be randomly assigned to be tested either in the morning or in the afternoon. Second, the researcher can use matching to ensure equivalent groups or equivalent environments. For example, the researcher could measure each participant's IQ and then assign individuals to groups so that all of the groups have roughly the same average IQ. 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. DEFINITION In the experimental method, one variable is manipulated while another variable is observed and measured. To establish a cause-and-effect relationship between the two variables, an experiment attempts to control all other variables to prevent them from influencing the results. TERMINOLOGY IN THE EXPERIMENTAL METHOD Specific names are used for the two variables that are studied by the experimental method. The variable that is manipulated by the experimenter is called the independent variable. It can be identified as the treatment conditions to which participants are assigned. For the example in Figure 1.6, temperature 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, student learning (measured by their test scores) is the dependent variable. DEFINITIONS The independent variable is the variable that is manipulated by the researcher. In behavioral research, the independent variable usually consists of the two (or more) treatment conditions to which subjects are exposed. The independent variable consists of the antecedent conditions that were manipulated prior to observing the dependent variable. The dependent variable is the one that is observed to assess the effect of the treatment. Control conditions in an experiment An experimental study evaluates the relationship between two variables by manipulating one variable (the independent variable) and measuring one variable (the dependent variable). Note that in an experiment only one variable is actually measured. You should realize that this is different from a correlational study, in which both variables are measured and the data consist of two separate scores for each individual. Often an experiment will include a condition in which the subjects do not receive any treatment. The scores from these subjects are then compared with scores from subjects 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. DEFINITIONS Individuals in a control condition do not receive the experimental treatment. Instead, they either receive no treatment or they receive a neutral, placebo treatment. The purpose of a control condition is to provide a baseline for comparison with the experimental condition. Individuals in the experimental condition do receive the experimental treatment. 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 temperature experiment (see Figure 1.6), the independent variable is temperature (using values of 90° and 70°). For an experiment with an experimental group and a control group, the independent variable is treatment versus no treatment. NONEXPERIMENTAL METHODS: NONQUIVALENT GROUPS AND PRE-POST STUDIES In informal conversation, there is a tendency for people to use the term experiment to refer to any kind of research study. You should realize, however, that the term only applies to studies that satisfy the specific requirements outlined earlier. In particular, a real experiment must include manipulation of an independent variable and rigorous control of other, extraneous variables. As a result, there are a number of other research designs that are not true experiments but still examine the relationship between variables by comparing groups of scores. Two examples are shown in Figure 1.7 and are discussed in the following paragraphs. This type of research study is classified as nonexperimental. Correlational studies are also examples of nonexperimental research. In this section, however, we are discussing nonexperimental studies that compare two or more groups of scores. The top part of Figure 1.7 shows an example of a nonequivalent groups study comparing boys and girls. Notice that this study involves comparing two groups of scores (like an experiment). However, the researcher has no ability to control which participants go into which group—all the males must be in the boy group and all the females must be in the girl group. Because this type of research compares preexisting groups, the researcher cannot control the assignment of participants to groups and cannot ensure equivalent groups. Other examples of nonequivalent group studies include comparing 8-year-old children and 10-year-old children, people with an eating disorder and those with no disorder, and comparing children from a single-parent home and those from a two-parent home. Because it is impossible to use techniques like random assignment to control participant variables and ensure equivalent groups, this type of research is not a true experiment. FIGURE 1.7 Two examples of nonexperimental 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 thte two times. The bottom part of Figure 1.7 shows an example of a pre-post study comparing depression scores before therapy and after therapy. Again, the study is comparing two groups of scores (like an experiment). In this study, however, the researcher has no control over the passage of time. The "before" scores are always measured earlier than the "after" scores. Although a difference between the two groups of scores may be caused by the treatment, it is always possible that the scores simply change as time goes by. For example, the depression scores may decrease over time in the same way that the symptoms of a cold disappear over time. In a pre-post study the researcher also has no control over other variables that change with time. For example, the weather could change from dark and gloomy before therapy to bright and sunny after therapy. In this case, the depression scores could improve because of the weather and not because of the therapy. Because the researcher cannot control the passage of time or other variables related to time, this study is not a true experiment. You may have noticed that a pre-post study is similar to a correlational study in that both designs measure two scores for each individual. In a correlational study, however, the two scores correspond to two different variables, such as height and weight. In a pre-post design the two scores are obtained by measuring the same variable twice under two different conditions at two different times. In the study shown in Figure 1.7(b) for example, depression is measured twice for each participant, once before treatment and again after treatment. 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 manipulating the independent variable and the 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. DEFINITION In a nonexperimental study, the "independent variable" that is used to create the different groups of scores is often called the quasi-independent variable. Most of the statistical procedures presented in this book are designed for research studies that compare sets 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. LEARNING CHECK Researchers have observed that high school students who watched educational television programs as young children tend to have higher grades than their peers who did not watch educational television. Is this study an example of an experiment? Explain why or why not. What two elements are necessary for a research study to be an experiment? Loftus and Palmer (1974) conducted an experiment in which participants were shown a video of an automobile accident. After the video, some participants were asked to estimate the speed of the cars when they "smashed into" each other. Others were asked to estimate the speed when the cars "hit" each other. The "smashed into" group produced significantly higher estimates than the "hit" group. Identify the independent and dependent variables for this study. ANSWERS This study could be correlational or nonexperimental, but it is definitely not an example of a true experiment. The researcher is simply observing, not manipulating, the amount of educational television. First, the researcher must manipulate one of the two variables being studied. Second, all other variables that might influence the results must be controlled. The independent variable is the phrasing of the question and the dependent variable is the speed estimated by each participant. 1.4 VARIABLES AND MEASUREMENT The scores that make up the data from a research study are the result of observing and measuring variables. For example, a researcher may finish a study with a set of IQ scores, personality scores, or reaction-time scores. In this section, we take a closer look at the variables that are being measured and the process of measurement. CONSTRUCTS AND OPERATIONAL DEFINITIONS Some variables, such as height, weight, and eye color are well-defined, concrete entities that can be observed and measured directly. On the other hand, many variables studied by behavioral scientists are internal characteristics that people use to help understand and explain behavior. For example, we say that a child 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 representive of the construct. For example, we cannot "see" intelligence but we can see examples of intelligent behavior. The external behaviors can then be used to create an operational definition for the construct. An operational definition defines a construct in terms of external behaviors that can be observed and measured. For example, your intelligence is measured and defined by your performance on an IQ test, or hunger can be measured and defined by the number of hours since last eating. DEFINITIONS Constructs are internal attributes or characteristics that cannot be directly observed but are useful for describing and explaining behavior. An operational definition identifies a measurement procedure (a set of operations) for measuring an external behavior and uses the resulting measurements as a definition and a measurement of a hypothetical construct. Note that an operational definition has two components: First, it describes a set of operations for measuring a construct. Second, it defines the construct in terms of the resulting measurements. DISCRETE AND CONTINUOUS VARIABLES The variables in a study can be characterized by the type of values that can be assigned to them. A discrete variable consists of separate, indivisible categories. For this type of variable, there are no intermediate values between two adjacent categories. Consider the values displayed when dice are rolled. Between neighboring values—for example, seven dots and eight dots—no other values can ever be observed. DEFINITION A discrete variable consists of separate, indivisible categories. No values can exist between two neighboring categories. Discrete variables are commonly restricted to whole, countable numbers—for example, the number of children in a family or the number of students attending class. If you observe class attendance from day to day, you may count 18 students one day and 19 students the next day. However, it is impossible ever to observe a value between 18 and 19. A discrete variable may also consist of observations that differ qualitatively. For example, people can be classified by gender (male or female), by occupation (nurse, teacher, lawyer, etc.), and college students can by classified by academic major (art, biology, chemistry, etc.). In each case, the variable is discrete because it consists of separate, indivisible categories. On the other hand, many variables are not discrete. Variables such as time, height, and weight are not limited to a fixed set of separate, indivisible categories. You can measure time, for example, in hours, minutes, seconds, or fractions of seconds. These variables are called continuous because they can be divided into an infinite number of fractional parts. DEFINITION For a continuous variable, there are an infinite number of possible values that fall between any two observed values. A continuous variable is divisible into an infinite number of fractional parts. Suppose, for example, that a researcher is measuring weights for a group of individuals participating in a diet study. Because weight is a continuous variable, it can be pictured as a continuous line (Figure 1.8). 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. Two other factors apply to continuous variables: 1. When measuring a continuous variable, it should be very rare to obtain identical measurements for two different individuals. Because a continuous variable has an infinite number of possible values, it should be almost impossible for two people to have exactly the same score. If the data show a substantial number of tied scores, then you should suspect that the measurement procedure is very crude or that the variable is not really continuous. 2. When measuring a continuous variable, each measurement category is actually an interval that must be defined by boundaries. For example, two people who both claim to weigh 150 pounds are probably not exactly the same weight. 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. 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 will be assigned a score of X = 150. DEFINITION Real limits are the boundaries of intervals for scores that are represented on a continuous number line. The real limit separating two adjacent scores is located exactly halfway between the scores. Each score has two real limits. The upper real limit is at the top of the interval, and the lower real limit is at the bottom. The concept of real limits applies to any measurement of a continuous variable, even when the score categories are not whole numbers. For example, if you were measuring time to the nearest tenth of a second, the measurement categories would be 31.0, 31.1, 31.2, and so on. Each of these categories represents an interval on the scale that is bounded by real limits. For example, a score of X = 31.1 seconds indicates that the actual measurement is in an interval bounded by a lower real limit of 31.05 and an upper real limit of 31.15. Remember that the real limits are always halfway between adjacent categories. Later in this book, real limits are used for constructing graphs and for various calculations with continuous scales. For now, however, you should realize that real limits are a necessity whenever you make measurements of a continuous variable. Finally, we should warn you that the term continuous applies to the variable that is being measured and not to the scores that are obtained from the measurement. For example, measuring people's heights to the nearest inch produces scores of 60, 61, 62, and so on. Although the scores may appear to be discrete numbers, the underlying variable is continuous. One key to determining whether a variable is continuous or discrete is that a continuous variable can be divided into any number of fractional parts. Height can be measured to the nearest inch, the nearest 0.5 inch, or the nearest 0.1 inch. Similarly, a professor evaluating students' knowledge could use a pass/fail system that classifies students into two broad categories. However, the professor could choose to use a 10-point quiz that divides student knowledge into 11 categories corresponding to quiz scores from 0 to 10. Or the professor could use a 100-point exam that potentially divides student knowledge into 101 categories from 0 to 100. Whenever you are free to choose the degree of precision or the number of categories for measuring a variable, the variable must be continuous. SCALES OF MEASUREMENT It should be obvious by now that data collection requires that we make measurements of our observations. Measurement involves assigning individuals or events to categories. The categories can simply be names such as male/female or employed/unemployed, or they can be numerical values such as 68 inches or 175 pounds. The complete set of categories 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 people's heights, for example, you could measure a group of individuals by simply classifying them into three categories: tall, medium, and short. However, this simple classification would not tell you much about the actual heights of the individuals, and these measurements would not give you enough information to calculate an average height for the group. Although the simple classification would be adequate for some purposes, you would need more sophisticated measurements before you could answer more detailed questions. In this section, we examine four different scales of measurement, beginning with the simplest and moving to the most sophisticated. THE NOMINAL SCALE The word nominal means "having to do with names." Measurement on a nominal scale involves classifying individuals into categories that have different names but are not related to each other in any systematic way. For example, if you were measuring the academic majors for a group of college students, the categories would be art, biology, business, chemistry, and so on. Each student would be classified in one category according to his or her major. The measurements from a nominal scale allow us to determine whether two individuals are different, but they do not identify either the direction or the size of the difference. If one student is an art major and another is a biology major we can say that they are different, but we cannot say that art is "more than" or "less than" biology and we cannot specify how much difference there is between art and biology. Other examples of nominal scales include classifying people by race, gender, or occupation. DEFINITION A nominal scale consists of a set of categories that have different names. Measurements on a nominal scale label and categorize observations, but do not make any quantitative distinctions between observations. Although the categories on a nominal scale are not quantitative values, they are occasionally represented by numbers. For example, the rooms or offices in a building may be identified by numbers. You should realize that the room numbers are simply names and do not reflect any quantitative information. Room 109 is not necessarily bigger than Room 100 and certainly not 9 points bigger. It also is fairly common to use numerical values as a code for nominal categories when data are entered into computer programs. For example, the data from a survey may code males with a 0 and females with a 1. Again, the numerical values are simply names and do not represent any quantitative difference. The scales that follow do reflect an attempt to make quantitative distinctions. THE ORDINAL SCALE The categories that make up an ordinal scale not only have different names (as in a nominal scale) but also are organized in a fixed order corresponding to differences of magnitude. DEFINITION An ordinal scale consists of a set of categories that are organized in an ordered sequence. Measurements on an ordinal scale rank observations in terms of size or magnitude. Often, an ordinal scale consists of a series of ranks (first, second, third, and so on) like the order of finish in a horse race. Occasionally, the categories are identified by verbal labels like small, medium, and large drink sizes at a fast-food restaurant. In either case, the fact that the categories form an ordered sequence means that there is a directional relationship between categories. With measurements from an ordinal scale, you can determine whether two individuals are different and you can determine the direction of difference. However, ordinal measurements do not allow you to determine the magnitude of the difference between two individuals. For example, if Billy is placed in the low-reading group and Tim is placed in the high-reading group, you know that Tim is a better reader, but you do not know how much better. Other examples of ordinal scales include socioeconomic class (upper, middle, lower) and T-shirt sizes (small, medium, large). In addition, ordinal scales are often used to measure variables for which it is difficult to assign numerical scores. For example, people can rank their food preferences but might have trouble explaining "how much" they prefer chocolate ice cream to steak. THE INTERVAL AND RATIO SCALES Both an interval scale and a ratio scale consist of a series of ordered categories (like an ordinal scale) with the additional requirement that the categories form a series of intervals that are all exactly the same size. Thus, the scale of measurement consists of a series of equal intervals, such as inches on a ruler. Other examples of interval and ratio scales are the measurement of time in seconds, weight in pounds, and temperature in degrees Fahrenheit. Note that, in each case, one interval (1 inch, 1 second, 1 pound, 1 degree) is the same size, no matter where it is located on the scale. The fact that the intervals are all the same size makes it possible to determine both the size and the direction of the difference between two measurements. For example, you know that a measurement of 80° Fahrenheit is higher than a measure of 60°, and you know that it is exactly 20° higher. The factor that differentiates an interval scale from a ratio scale is the nature of the zero point. An interval scale has an arbitrary zero point. That is, the value 0 is assigned to a particular location on the scale simply as a matter of convenience or reference. In particular, a value of zero does not indicate a total absence of the variable being measured. For example a temperature of 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 meaningful value representing none (a complete absence) of the variable being measured. The existence of an absolute, nonabitrary zero point means that we can measure the absolute amount of the variable; that is, we can measure the distance from 0. This makes it possible to compare measurements in terms of ratios. For example, an individual who requires 10 seconds to solve a problem (10 more than 0) has taken twice as much time as an individual who finishes in only 5 seconds (5 more than 0). 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. DEFINITIONS An interval scale consists of ordered categories that are all intervals of exactly the same size. Equal differences between numbers on scale reflect equal differences in magnitude. However, the zero point on an interval scale is arbitrary and does not indicate a zero amount of the variable being measured. A ratio scale is an interval scale with the additional feature of an absolute zero point. With a ratio scale, ratios of numbers do reflect ratios of magnitude. EXAMPLE 1.2 A researcher obtains measurements of height for a group of 8-year-old boys. Initially, the researcher simply records each child's height in inches, obtaining values such as 44, 51, 49, and so on. These initial measurements constitute a ratio scale. A value of zero represents no height (absolute zero). Also, it is possible to use these measurements to form ratios. For example, a child who is 60 inches tall is one and a half times taller than a child who is 40 inches tall. Now suppose that the researcher converts the initial measurement into a new scale by calculating the difference between each child's actual height and the average height for this age group. A child who is 1 inch taller than average now gets a score of +1; a child 4 inches taller than average gets a score of +4. Similarly, a child who is 2 inches shorter than average gets a score of -2. The new scores constitute an interval scale of measurement. A score of zero no longer indicates an absence of height; now it simply means average height. Notice that both sets of scores 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 + 9 and + 3 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 + 9 is not three times taller than a boy who measures + 3. 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 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 presented 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 numerical 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 15, and the chi-square tests in Chapter 16). LEARNING CHECK A survey asks people to identify their age, annual income, and marital status (single, married, divorced, etc.). For each of these three variables, identify the scale of measurement that probably is used. An educational psychologist classifies students as high, medium, and low intelligence. What kind of scale is being used? In a study on perception of facial expressions, participants must classify the emotions displayed in photographs of people as anger, sadness, joy, disgust, fear, or surprise. Emotional expression is measured on a(n) ___________________ scale. A researcher studies the factors that determine how many children couples decide to have. The variable, number of children, is a _________________ (discrete/continuous) variable. An investigator studies how concept-formation ability changes with age. Age is a _________________ (discrete/continuous) variable. 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? ANSWERS Age and annual income are measured on ratio scales. Marital status is measured on a nominal scale. ordinal nominal discrete continuous a. 67.5 and 68.5 b. 67.75 and 68.25 1.5 STATISTICAL NOTATION The measurements obtained in research studies provide the data for statistical analysis. Most of the statistical analyses 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. 563) 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 calculations. In later chapters, additional statistical notation is introduced as it is needed. SCORES X X Y 37 72 165 35 68 151 35 67 160 30 67 160 25 68 146 17 70 160 16 66 133 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 observations are made for two variables, there will be two scores for each individual. The data can be presented as two lists labeled X and Y for the two variables. For example, observations for people's height in inches (variable X) and weight in pounds (variable Y) can be presented as shown in the margin (the double column on the right). Each pair X, Y represents the observations made of a single participant. It is also useful to specify how many scores are in a set. We will use an uppercase letter N to represent the number of scores in a population and a lowercase letter n to represent 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 implying that these data are a sample. SUMMATION NOTATION 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 2, is used to stand for summation. The expression ΣX means to add all the scores for variable X. The summation sign Σ can be read as "the sum of." Thus, ΣX is read "the sum of the scores." For the following set of quiz scores, 10, 6, 7, 4 ∑X = 27 and N = 4. To use summation notation correctly, keep in mind the following two points: The summation sign, ∑, is always followed by a symbol or mathematical expression. The symbol or expression identifies exactly which values are to be added. To compute ∑X, 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 compute ∑(X - 1)2, the summation sign is followed by a relatively complex mathematical expression, so your first task is to calculate all of the (X - 1)2 values and then add the results. 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 Any calculation contained within parentheses is done first. Squaring (or raising to other exponents) is done second. Multiplying and/or dividing is done third. A series of multiplication and/or division operations should be done in order from left to right. Summation using the Σ notation is done next. Finally, any other addition and/or subtraction is done. More information on the order of operations for mathematics is available in the Math Review appendix, page 562. The following examples demonstrate how summation notation is used in most of the calculations and formulas we present in this book. EXAMPLE 1.3 A set of four scores consists of values 3, 1, 7, and 4. We will compute ΣX, Σ X2, and (ΣX)2 for these scores. To help demonstrate the calculations, we will use a computational 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. X X2 3 9 1 1 7 49 4 16 The table to th