Statistics Question

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

  • Perform the independent samples t-test for amount spent by vegetarian status of shopper. Copy and paste the descriptive statistics and t-test output. You may need to create a screenshot.
  • Write the t-test results in APA format that includes descriptive statistics for each group, the t-test result with df and p, and the effect size (use d).
  • Make an APA-formatted bar graph on amount spent by gender. Copy and paste the graph below. You may need to create a screenshot.
  • Perform the independent samples t-test for amount spent by gender. Copy and paste the descriptive statistics and t-test output. You may need to create a screenshot.
  • Ed iti o n
    © Deborah Batt
    Statistics for the
    Behavioral Sciences
    Frederick J Gravetter
    The College at Brockport, State University of New York
    Larry B. WaLLnau
    The College at Brockport, State University of New York
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    B RiEF Co n tEn t S
    Introduction to Statistics 1
    Frequency Distributions
    Central Tendency
    z-Scores: Location of Scores and Standardized Distributions 131
    Probability and Samples: The Distribution of Sample Means 193
    Introduction to Hypothesis Testing 223
    Introduction to the t Statistic
    C H A P t E R 10
    The t Test for Two Independent Samples 299
    The t Test for Two Related Samples 335
    C H A P t E R 12
    Introduction to Analysis of Variance 365
    C H A P t E R 13
    Repeated-Measures Analysis of Variance 413
    C H A P t E R 14
    Two-Factor Analysis of Variance (Independent Measures) 447
    C H A P t E R 16
    Introduction to Regression 529
    C H A P t E R 17
    The Chi-Square Statistic: Tests for Goodness of Fit and Independence 559
    C H A P t E R 18
    The Binomial Test
    Co n tEn t S
    1 Introduction to Statistics
    1.1 Statistics, Science, and Observations 2
    1.2 Data Structures, Research Methods, and Statistics 10
    1.3 Variables and Measurement 18
    1.4 Statistical Notation 25
    Focus on Problem Solving 30
    Demonstration 1.1 30
    Problems 31
    2 Frequency Distributions
    2.1 Frequency Distributions and Frequency Distribution Tables 35
    2.2 Grouped Frequency Distribution Tables 38
    2.3 Frequency Distribution Graphs 42
    2.4 Percentiles, Percentile Ranks, and Interpolation 49
    2.5 Stem and Leaf Displays 56
    Focus on Problem Solving 59
    Demonstration 2.1 60
    Demonstration 2.2 61
    Problems 62
    3 Central Tendency
    3.1 Overview 68
    3.2 The Mean 70
    3.3 The Median 79
    3.4 The Mode 83
    3.5 Selecting a Measure of Central Tendency 86
    3.6 Central Tendency and the Shape of the Distribution 92
    Focus on Problem Solving 95
    Demonstration 3.1 96
    Problems 96
    4 Variability
    4.1 Introduction to Variability 101
    4.2 Defining Standard Deviation and Variance 103
    4.3 Measuring Variance and Standard Deviation for a Population 108
    4.4 Measuring Standard Deviation and Variance for a Sample 111
    4.5 Sample Variance as an Unbiased Statistic 117
    4.6 More about Variance and Standard Deviation 119
    Focus on Problem Solving 127
    Demonstration 4.1 128
    Problems 128
    z-Scores: Location of Scores
    C H A P t E R 5 and Standardized Distributions
    5.1 Introduction to z-Scores 133
    5.2 z-Scores and Locations in a Distribution 135
    5.3 Other Relationships Between z, X, 𝛍, and 𝛔 138
    5.4 Using z-Scores to Standardize a Distribution 141
    5.5 Other Standardized Distributions Based on z-Scores 145
    5.6 Computing z-Scores for Samples 148
    5.7 Looking Ahead to Inferential Statistics 150
    Focus on Problem Solving 154
    Demonstration 5.1 155
    Demonstration 5.2 155
    Problems 156
    6 Probability
    6.1 Introduction to Probability 160
    6.2 Probability and the Normal Distribution 165
    6.3 Probabilities and Proportions for Scores
    from a Normal Distribution
    6.4 Probability and the Binomial Distribution 179
    6.5 Looking Ahead to Inferential Statistics 184
    Focus on Problem Solving 187
    Demonstration 6.1 188
    Demonstration 6.2 188
    Problems 189
    Probability and Samples: The Distribution
    C H A P t E R 7 of Sample Means
    7.1 Samples, Populations, and the Distribution
    of Sample Means
    7.2 The Distribution of Sample Means for any Population
    and any Sample Size
    7.3 Probability and the Distribution of Sample Means 206
    7.4 More about Standard Error 210
    7.5 Looking Ahead to Inferential Statistics
    Focus on Problem Solving 219
    Demonstration 7.1 220
    Problems 221
    8 Introduction to Hypothesis Testing
    8.1 The Logic of Hypothesis Testing 225
    8.2 Uncertainty and Errors in Hypothesis Testing 236
    8.3 More about Hypothesis Tests 240
    8.4 Directional (One-Tailed) Hypothesis Tests 245
    8.5 Concerns about Hypothesis Testing: Measuring Effect Size 250
    8.6 Statistical Power 254
    Focus on Problem Solving 261
    Demonstration 8.1 262
    Demonstration 8.2 263
    Problems 263
    9 Introduction to the t Statistic
    9.1 The t Statistic: An Alternative to z 268
    9.2 Hypothesis Tests with the t Statistic 274
    9.3 Measuring Effect Size for the t Statistic 279
    9.4 Directional Hypotheses and One-Tailed Tests 288
    Focus on Problem Solving 293
    Demonstration 9.1 293
    Demonstration 9.2 294
    Problems 295
    10 The t Test for Two Independent Samples
    10.1 Introduction to the Independent-Measures Design 300
    10.2 The Null Hypothesis and the Independent-Measures t Statistic 302
    10.3 Hypothesis Tests with the Independent-Measures t Statistic 310
    10.4 Effect Size and Confidence Intervals for the
    Independent-Measures t
    10.5 The Role of Sample Variance and Sample Size in the
    Independent-Measures t Test
    Focus on Problem Solving 327
    Demonstration 10.1 328
    Demonstration 10.2 329
    Problems 329
    11 The t Test for Two Related Samples
    11.1 Introduction to Repeated-Measures Designs 336
    11.2 The t Statistic for a Repeated-Measures Research Design 339
    11.3 Hypothesis Tests for the Repeated-Measures Design 343
    11.4 Effect Size and Confidence Intervals for the Repeated-Measures t 347
    11.5 Comparing Repeated- and Independent-Measures Designs 352
    Focus on Problem Solving 358
    Demonstration 11.1 358
    Demonstration 11.2 359
    Problems 360
    12 Introduction to Analysis of Variance
    12.1 Introduction (An Overview of Analysis of Variance) 366
    12.2 The Logic of Analysis of Variance 372
    12.3 ANOVA Notation and Formulas 375
    12.4 Examples of Hypothesis Testing and Effect Size with ANOVA 383
    12.5 Post Hoc Tests 393
    12.6 More about ANOVA 397
    Focus on Problem Solving 406
    Demonstration 12.1 406
    Demonstration 12.2 408
    Problems 408
    13 Repeated-Measures Analysis of Variance
    13.1 Overview of the Repeated-Measures ANOVA 415
    13.2 Hypothesis Testing and Effect Size with the
    Repeated-Measures ANOVA
    13.3 More about the Repeated-Measures Design 429
    Focus on Problem Solving 438
    Demonstration 13.1 439
    Demonstration 13.2 440
    Problems 441
    Two-Factor Analysis of Variance
    C H A P t E R 14 (Independent Measures)
    14.1 An Overview of the Two-Factor, Independent-Measures, ANOVA: Main
    Effects and Interactions
    14.2 An Example of the Two-Factor ANOVA and Effect Size 458
    14.3 More about the Two-Factor ANOVA 467
    Focus on Problem Solving 475
    Demonstration 14.1 476
    Demonstration 14.2 478
    Problems 479
    15 Correlation
    15.1 Introduction 487
    15.2 The Pearson Correlation 489
    15.3 Using and Interpreting the Pearson Correlation 495
    15.4 Hypothesis Tests with the Pearson Correlation 506
    15.5 Alternatives to the Pearson Correlation 510
    Focus on Problem Solving 522
    Demonstration 15.1 523
    Problems 524
    16 Introduction to Regression
    16.1 Introduction to Linear Equations and Regression 530
    16.2 The Standard Error of Estimate and Analysis of Regression:
    The Significance of the Regression Equation
    16.3 Introduction to Multiple Regression with Two Predictor Variables 544
    Linear and Multiple Regression
    Focus on Problem Solving 554
    Demonstration 16.1 555
    Problems 556
    The Chi-Square Statistic: Tests for Goodness
    C H A P t E R 17 of Fit and Independence
    17.1 Introduction to Chi-Square: The Test for Goodness of Fit 561
    17.2 An Example of the Chi-Square Test for Goodness of Fit 567
    17.3 The Chi-Square Test for Independence 573
    17.4 Effect Size and Assumptions for the Chi-Square Tests
    17.5 Special Applications of the Chi-Square Tests 587
    Focus on Problem Solving 595
    Demonstration 17.1 595
    Demonstration 17.2 597
    Problems 597
    18 The Binomial Test
    18.1 Introduction to the Binomial Test 604
    18.2 An Example of the Binomial Test 608
    18.3 More about the Binomial Test: Relationship with Chi-Square
    and the Sign Test
    Focus on Problem Solving 619
    Demonstration 18.1 619
    Problems 620
    A PPE N D IX E S
    A Basic Mathematics Review 625
    Symbols and Notation 627
    Proportions: Fractions, Decimals, and Percentages 629
    Negative Numbers 635
    Basic Algebra: Solving Equations 637
    Exponents and Square Roots 640
    B Statistical Tables 647
    C Solutions for Odd-Numbered Problems in the Text 663
    D General Instructions for Using SPSS 683
    E Hypothesis Tests for Ordinal Data: Mann-Whitney,
    Wilcoxon, Kruskal-Wallis, and Friedman Tests
    Statistics Organizer: Finding the Right Statistics for Your Data
    Name Index
    Subject Index
    any students in the behavioral sciences view the required statistics course as an
    intimidating obstacle that has been placed in the middle of an otherwise interesting curriculum. They want to learn about human behavior—not about math and science.
    As a result, the statistics course is seen as irrelevant to their education and career goals.
    However, as long as the behavioral sciences are founded in science, knowledge of statistics
    will be necessary. Statistical procedures provide researchers with objective and systematic
    methods for describing and interpreting their research results. Scientific research is the
    system that we use to gather information, and statistics are the tools that we use to distill
    the information into sensible and justified conclusions. The goal of this book is not only
    to teach the methods of statistics, but also to convey the basic principles of objectivity and
    logic that are essential for science and valuable for decision making in everyday life.
    Those of you who are familiar with previous editions of Statistics for the Behavioral
    Sciences will notice that some changes have been made. These changes are summarized
    in the section entitled “To the Instructor.” In revising this text, our students have been
    foremost in our minds. Over the years, they have provided honest and useful feedback.
    Their hard work and perseverance has made our writing and teaching most rewarding. We
    sincerely thank them. Students who are using this edition should please read the section of
    the preface entitled “To the Student.”
    The book chapters are organized in the sequence that we use for our own statistics
    courses. We begin with descriptive statistics, and then examine a variety of statistical procedures focused on sample means and variance before moving on to correlational methods
    and nonparametric statistics. Information about modifying this sequence is presented in the
    To The Instructor section for individuals who prefer a different organization. Each chapter
    contains numerous examples, many based on actual research studies, learning checks, a
    summary and list of key terms, and a set of 20–30 problems.
    Ancillaries for this edition include the following.
    MindTap® Psychology: MindTap® Psychology for Gravetter/Wallnau’s Statistics
    for The Behavioral Sciences, 10th Edition is the digital learning solution that helps
    instructors engage and transform today’s students into critical thinkers. Through paths
    of dynamic assignments and applications that you can personalize, real-time course
    analytics, and an accessible reader, MindTap helps you turn cookie cutter into cutting
    edge, apathy into engagement, and memorizers into higher-level thinkers.
    As an instructor using MindTap you have at your fingertips the right content and
    unique set of tools curated specifically for your course, such as video tutorials that
    walk students through various concepts and interactive problem tutorials that provide
    students opportunities to practice what they have learned, all in an interface designed
    to improve workflow and save time when planning lessons and course structure. The
    control to build and personalize your course is all yours, focusing on the most relevant
    material while also lowering costs for your students. Stay connected and informed in
    your course through real time student tracking that provides the opportunity to adjust
    the course as needed based on analytics of interactivity in the course.
    Online Instructor’s Manual: The manual includes learning objectives, key terms,
    a detailed chapter outline, a chapter summary, lesson plans, discussion topics, student
    activities, “What If” scenarios, media tools, a sample syllabus and an expanded test
    bank. The learning objectives are correlated with the discussion topics, student
    activities, and media tools.
    Online PowerPoints: Helping you make your lectures more engaging while effectively reaching your visually oriented students, these handy Microsoft PowerPoint®
    slides outline the chapters of the main text in a classroom-ready presentation. The
    PowerPoint® slides are updated to reflect the content and organization of the new
    edition of the text.
    Cengage Learning Testing, powered by Cognero®: Cengage Learning Testing,
    Powered by Cognero®, is a flexible online system that allows you to author, edit,
    and manage test bank content. You can create multiple test versions in an instant and
    deliver tests from your LMS in your classroom.
    It takes a lot of good, hard-working people to produce a book. Our friends at Cengage
    have made enormous contributions to this textbook. We thank: Jon-David Hague, Product
    Director; Timothy Matray, Product Team Director; Jasmin Tokatlian, Content Development Manager; Kimiya Hojjat, Product Assistant; and Vernon Boes, Art Director. Special
    thanks go to Stefanie Chase, our Content Developer and to Lynn Lustberg who led us
    through production at MPS.
    Reviewers play a very important role in the development of a manuscript. Accordingly,
    we offer our appreciation to the following colleagues for their assistance: Patricia Case,
    University of Toledo; Kevin David, Northeastern State University; Adia Garrett, University of Maryland, Baltimore County; Carrie E. Hall, Miami University; Deletha Hardin,
    University of Tampa; Angela Heads, Prairie View A&M University; Roberto Heredia,
    Texas A&M International University; Alisha Janowski, University of Central Florida;
    Matthew Mulvaney, The College at Brockport (SUNY); Nicholas Von Glahn, California
    State Polytechnic University, Pomona; and Ronald Yockey, Fresno State University.
    To the Instructor
    Those of you familiar with the previous edition of Statistics for the Behavioral Sciences will
    notice a number of changes in the 10th edition. Throughout this book, research examples
    have been updated, real world examples have been added, and the end-of-chapter problems
    have been extensively revised. Major revisions for this edition include the following:
    1. Each section of every chapter begins with a list of Learning Objectives for that
    specific section.
    2. Each section ends with a Learning Check consisting of multiple-choice questions
    with at least one question for each Learning Objective.
    3. The former Chapter 19, Choosing the Right Statistics, has been eliminated and
    an abridged version is now an Appendix replacing the Statistics Organizer, which
    appeared in earlier editions.
    Other examples of specific and noteworthy revisions include the following.
    Chapter 1 The section on data structures and research methods parallels the new
    Appendix, Choosing the Right Statistics.
    Chapter 2 The chapter opens with a new Preview to introduce the concept and purpose
    of frequency distributions.
    Chapter 3
    Minor editing clarifies and simplifies the discussion the median.
    Chapter 4 The chapter opens with a new Preview to introduce the topic of Central
    Tendency. The sections on standard deviation and variance have been edited to increase
    emphasis on concepts rather than calculations.
    The section discussion relationships between z, X, μ, and σ has been
    expanded and includes a new demonstration example.
    Chapter 5
    Chapter 6
    The chapter opens with a new Preview to introduce the topic of Probability.
    The section, Looking Ahead to Inferential Statistics, has been substantially shortened and
    Chapter 7
    The former Box explaining difference between standard deviation and
    standard error was deleted and the content incorporated into Section 7.4 with editing to
    emphasize that the standard error is the primary new element introduced in the chapter.
    The final section, Looking Ahead to Inferential Statistics, was simplified and shortened to
    be consistent with the changes in Chapter 6.
    Chapter 8
    A redundant example was deleted which shortened and streamlined the
    remaining material so that most of the chapter is focused on the same research example.
    Chapter 9 The chapter opens with a new Preview to introduce the t statistic and explain
    why a new test statistic is needed. The section introducing Confidence Intervals was edited
    to clarify the origin of the confidence interval equation and to emphasize that the interval
    is constructed at the sample mean.
    Chapter 10
    The chapter opens with a new Preview introducing the independent-measures t statistic. The section presenting the estimated standard error of (M1 – M2) has been
    simplified and shortened.
    Chapter 11
    The chapter opens with a new Preview introducing the repeated-measures t
    statistic. The section discussing hypothesis testing has been separated from the section on
    effect size and confidence intervals to be consistent with the other two chapters on t tests.
    The section comparing independent- and repeated-measures designs has been expanded.
    Chapter 12 The chapter opens with a new Preview introducing ANOVA and explaining
    why a new hypothesis testing procedure is necessary. Sections in the chapter have been
    reorganized to allow flow directly from hypothesis tests and effect size to post tests.
    Chapter 13
    Substantially expanded the section discussing factors that influence the
    outcome of a repeated-measures hypothesis test and associated measures of effect size.
    Chapter 14
    The chapter opens with a new Preview presenting a two-factor research
    example and introducing the associated ANOVA. Sections have been reorganized so that
    simple main effects and the idea of using a second factor to reduce variance from individual differences are now presented as extra material related to the two-factor ANOVA.
    Chapter 15
    The chapter opens with a new Preview presenting a correlational research
    study and the concept of a correlation. A new section introduces the t statistic for evaluating the significance of a correlation and the section on partial correlations has been simplified and shortened.
    Chapter 16 The chapter opens with a new Preview introducing the concept of regression and
    its purpose. A new section demonstrates the equivalence of testing the significance of a correlation and testing the significance of a regression equation with one predictor variable. The section on residuals for the multiple-regression equation has been edited to simplify and shorten.
    Chapter 17
    A new chapter Preview presents an experimental study with data consisting
    of frequencies, which are not compatible with computing means and variances. Chi-square
    tests are introduced as a solution to this problem. A new section introduces Cohen’s w as
    a means of measuring effect size for both chi-square tests.
    Chapter 18
    Substantial editing clarifies the section explaining how the real limits for
    each score can influence the conclusion from a binomial test.
    The former Chapter 19 covering the task of matching statistical methods to specific
    types of data has been substantially shortened and converted into an Appendix.
    ■■Matching the Text to Your Syllabus
    The book chapters are organized in the sequence that we use for our own statistics courses.
    However, different instructors may prefer different organizations and probably will choose
    to omit or deemphasize specific topics. We have tried to make separate chapters, and even
    sections of chapters, completely self-contained, so they can be deleted or reorganized to fit
    the syllabus for nearly any instructor. Some common examples are as follows.
    It is common for instructors to choose between emphasizing analysis of variance
    (Chapters 12, 13, and 14) or emphasizing correlation/regression (Chapters 15 and 16).
    It is rare for a one-semester course to complete coverage of both topics.
    Although we choose to complete all the hypothesis tests for means and mean
    differences before introducing correlation (Chapter 15), many instructors prefer to
    place correlation much earlier in the sequence of course topics. To accommodate
    this, Sections 15.1, 15.2, and 15.3 present the calculation and interpretation of
    the Pearson correlation and can be introduced immediately following Chapter 4
    (variability). Other sections of Chapter 15 refer to hypothesis testing and should be
    delayed until the process of hypothesis testing (Chapter 8) has been introduced.
    It is also possible for instructors to present the chi-square tests (Chapter 17) much
    earlier in the sequence of course topics. Chapter 17, which presents hypothesis tests
    for proportions, can be presented immediately after Chapter 8, which introduces the
    process of hypothesis testing. If this is done, we also recommend that the Pearson
    correlation (Sections 15.1, 15.2, and 15.3) be presented early to provide a foundation
    for the chi-square test for independence.
    To the Student
    A primary goal of this book is to make the task of learning statistics as easy and painless
    as possible. Among other things, you will notice that the book provides you with a number
    of opportunities to practice the techniques you will be learning in the form of Learning
    Checks, Examples, Demonstrations, and end-of-chapter problems. We encourage you to
    take advantage of these opportunities. Read the text rather than just memorizing the formulas. We have taken care to present each statistical procedure in a conceptual context that
    explains why the procedure was developed and when it should be used. If you read this
    material and gain an understanding of the basic concepts underlying a statistical formula,
    you will find that learning the formula and how to use it will be much easier. In the “Study
    Hints,” that follow, we provide advice that we give our own students. Ask your instructor
    for advice as well; we are sure that other instructors will have ideas of their own.
    Over the years, the students in our classes and other students using our book have given
    us valuable feedback. If you have any suggestions or comments about this book, you can
    write to either Professor Emeritus Frederick Gravetter or Professor Emeritus Larry Wallnau
    at the Department of Psychology, SUNY College at Brockport, 350 New Campus Drive,
    Brockport, New York 14420. You can also contact Professor Emeritus Gravetter directly at
    ■■Study Hints
    You may find some of these tips helpful, as our own students have reported.
    The key to success in a statistics course is to keep up with the material. Each new
    topic builds on previous topics. If you have learned the previous material, then the
    new topic is just one small step forward. Without the proper background, however,
    the new topic can be a complete mystery. If you find that you are falling behind, get
    help immediately.
    You will learn (and remember) much more if you study for short periods several
    times per week rather than try to condense all of your studying into one long session.
    For example, it is far more effective to study half an hour every night than to have
    a single 3½-hour study session once a week. We cannot even work on writing this
    book without frequent rest breaks.
    Do some work before class. Keep a little ahead of the instructor by reading the appropriate sections before they are presented in class. Although you may not fully understand what you read, you will have a general idea of the topic, which will make the
    lecture easier to follow. Also, you can identify material that is particularly confusing
    and then be sure the topic is clarified in class.
    Pay attention and think during class. Although this advice seems obvious, often it is
    not practiced. Many students spend so much time trying to write down every example
    presented or every word spoken by the instructor that they do not actually understand
    and process what is being said. Check with your instructor—there may not be a need
    to copy every example presented in class, especially if there are many examples like
    it in the text. Sometimes, we tell our students to put their pens and pencils down for a
    moment and just listen.
    Test yourself regularly. Do not wait until the end of the chapter or the end of the
    week to check your knowledge. After each lecture, work some of the end-of-chapter
    problems and do the Learning Checks. Review the Demonstration Problems, and
    be sure you can define the Key Terms. If you are having trouble, get your questions
    answered immediately—reread the section, go to your instructor, or ask questions in
    class. By doing so, you will be able to move ahead to new material.
    Do not kid yourself! Avoid denial. Many students observe their instructor solve
    problems in class and think to themselves, “This looks easy, I understand it.” Do
    you really understand it? Can you really do the problem on your own without having
    to leaf through the pages of a chapter? Although there is nothing wrong with using
    examples in the text as models for solving problems, you should try working a problem with your book closed to test your level of mastery.
    We realize that many students are embarrassed to ask for help. It is our biggest challenge as instructors. You must find a way to overcome this aversion. Perhaps contacting the instructor directly would be a good starting point, if asking questions in class
    is too anxiety-provoking. You could be pleasantly surprised to find that your instructor does not yell, scold, or bite! Also, your instructor might know of another student
    who can offer assistance. Peer tutoring can be very helpful.
    Frederick J Gravetter
    Larry B. Wallnau
    A B o U t tH E AU tH o R S
    Frederick Gravetter
    Frederick J Gravetter is Professor Emeritus of Psychology at the
    State University of New York College at Brockport. While teaching at
    Brockport, Dr. Gravetter specialized in statistics, experimental design, and
    cognitive psychology. He received his bachelor’s degree in mathematics from
    M.I.T. and his Ph.D in psychology from Duke University. In addition to publishing this textbook and several research articles, Dr. Gravetter co-authored
    Research Methods for the Behavioral Science and Essentials of Statistics for
    the Behavioral Sciences.
    Larry B. Wallnau
    Larry B. WaLLnau is Professor Emeritus of Psychology at the State
    University of New York College at Brockport. While teaching at Brockport,
    he published numerous research articles in biopsychology. With
    Dr. Gravetter, he co-authored Essentials of Statistics for the Behavioral
    Sciences. Dr. Wallnau also has provided editorial consulting for numerous
    publishers and journals. He has taken up running and has competed in 5K
    races in New York and Connecticut. He takes great pleasure in adopting
    neglected and rescued dogs.
    CH A P T ER
    Introduction to Statistics
    © Deborah Batt
    1.1 Statistics, Science, and Observations
    1.2 Data Structures, Research Methods, and Statistics
    1.3 Variables and Measurement
    1.4 Statistical Notation
    Focus on Problem Solving
    Demonstration 1.1
    Before we begin our discussion of statistics, we ask you
    to read the following paragraph taken from the philosophy of Wrong Shui (Candappa, 2000).
    The Journey to Enlightenment
    In Wrong Shui, life is seen as a cosmic journey,
    a struggle to overcome unseen and unexpected
    obstacles at the end of which the traveler will find
    illumination and enlightenment. Replicate this quest
    in your home by moving light switches away from
    doors and over to the far side of each room.*
    Why did we begin a statistics book with a bit of twisted
    philosophy? In part, we simply wanted to lighten the
    mood with a bit of humor—starting a statistics course is
    typically not viewed as one of life’s joyous moments. In
    addition, the paragraph is an excellent counterexample for
    the purpose of this book. Specifically, our goal is to do
    everything possible to prevent you from stumbling around
    in the dark by providing lots of help and illumination as
    you journey through the world of statistics. To accomplish
    this, we begin each section of the book with clearly stated
    learning objectives and end each section with a brief quiz
    to test your mastery of the new material. We also introduce each new statistical procedure by explaining the purpose it is intended to serve. If you understand why a new
    procedure is needed, you will find it much easier to learn.
    The objectives for this first chapter are to provide
    an introduction to the topic of statistics and to give you
    some background for the rest of the book. We discuss
    the role of statistics within the general field of scientific
    inquiry, and we introduce some of the vocabulary and
    notation that are necessary for the statistical methods
    that follow.
    As you read through the following chapters, keep
    in mind that the general topic of statistics follows a
    well-organized, logically developed progression that
    leads from basic concepts and definitions to increasingly sophisticated techniques. Thus, each new topic
    serves as a foundation for the material that follows. The
    content of the first nine chapters, for example, provides
    an essential background and context for the statistical
    methods presented in Chapter 10. If you turn directly
    to Chapter 10 without reading the first nine chapters,
    you will find the material confusing and incomprehensible. However, if you learn and use the background
    material, you will have a good frame of reference for
    understanding and incorporating new concepts as they
    are presented.
    *Candappa, R. (2000). The little book of wrong shui. Kansas City:
    Andrews McMeel Publishing. Reprinted by permission.
    Statistics, Science, and Observations
    1. Define the terms population, sample, parameter, and statistic, and describe the
    relationships between them.
    2. Define descriptive and inferential statistics and describe how these two general
    categories of statistics are used in a typical research study.
    3. Describe the concept of sampling error and explain how this concept creates the
    fundamental problem that inferential statistics must address.
    ■■Definitions of Statistics
    By one definition, statistics consist of facts and figures such as the average annual snowfall
    in Denver or Derrick Jeter’s lifetime batting average. These statistics are usually informative
    and time-saving because they condense large quantities of information into a few simple figures. Later in this chapter we return to the notion of calculating statistics (facts and figures)
    but, for now, we concentrate on a much broader definition of statistics. Specifically, we use
    the term statistics to refer to a general field of mathematics. In this case, we are using the
    term statistics as a shortened version of statistical procedures. For example, you are probably using this book for a statistics course in which you will learn about the statistical techniques that are used to summarize and evaluate research results in the behavioral sciences.
    SEctIon 1.1 | Statistics, Science, and Observations
    Research in the behavioral sciences (and other fields) involves gathering information.
    To determine, for example, whether college students learn better by reading material on
    printed pages or on a computer screen, you would need to gather information about students’ study habits and their academic performance. When researchers finish the task of
    gathering information, they typically find themselves with pages and pages of measurements such as preferences, personality scores, opinions, and so on. In this book, we present
    the statistics that researchers use to analyze and interpret the information that they gather.
    Specifically, statistics serve two general purposes:
    1. Statistics are used to organize and summarize the information so that the researcher can
    see what happened in the research study and can communicate the results to others.
    2. Statistics help the researcher to answer the questions that initiated the research by
    determining exactly what general conclusions are justified based on the specific
    results that were obtained.
    The term statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information.
    Statistical procedures help ensure that the information or observations are presented
    and interpreted in an accurate and informative way. In somewhat grandiose terms, statistics
    help researchers bring order out of chaos. In addition, statistics provide researchers with a
    set of standardized techniques that are recognized and understood throughout the scientific
    community. Thus, the statistical methods used by one researcher will be familiar to other
    researchers, who can accurately interpret the statistical analyses with a full understanding
    of how the analysis was done and what the results signify.
    ■■Populations and Samples
    Research in the behavioral sciences typically begins with a general question about a specific
    group (or groups) of individuals. For example, a researcher may want to know what factors
    are associated with academic dishonesty among college students. Or a researcher may want
    to examine the amount of time spent in the bathroom for men compared to women. In the
    first example, the researcher is interested in the group of college students. In the second
    example, the researcher wants to compare the group of men with the group of women. In statistical terminology, the entire group that a researcher wishes to study is called a population.
    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
    chaPtER 1 | Introduction to Statistics
    a smaller, more manageable group from the population and limit their studies to the individuals in the selected group. In statistical terms, a set of individuals selected from a population
    is called a sample. A sample is intended to be representative of its population, and a sample
    should always be identified in terms of the population from which it was selected.
    A sample is a set of individuals selected from a population, usually intended to
    represent the population in a research study.
    Just as we saw with populations, samples can vary in size. For example, one study might
    examine a sample of only 10 students in a graduate program and another study might use a
    sample of more than 10,000 people who take a specific cholesterol medication.
    So far we have talked about a sample being selected from a population. However, this is
    actually only half of the full relationship between a sample and its population. Specifically,
    when a researcher finishes examining the sample, the goal is to generalize the results back
    to the entire population. Remember that the research started with a general question about
    the population. To answer the question, a researcher studies a sample and then generalizes
    the results from the sample to the population. The full relationship between a sample and a
    population is shown in Figure 1.1.
    ■■Variables and Data
    Typically, researchers are interested in specific characteristics of the individuals in the population (or in the sample), or they are interested in outside factors that may influence the
    individuals. For example, a researcher may be interested in the influence of the weather on
    people’s moods. As the weather changes, do people’s moods also change? Something that
    can change or have different values is called a variable.
    A variable is a characteristic or condition that changes or has different values for
    different individuals.
    All of the individuals of interest
    The results
    from the sample
    are generalized
    to the population
    F I G U R E 1.1
    The relationship between a
    population and a sample.
    The sample
    is selected from
    the population
    The individuals selected to
    participate in the research study
    SEctIon 1.1 | Statistics, Science, and Observations
    Once again, variables can be characteristics that differ from one individual to another,
    such as height, weight, gender, or personality. Also, variables can be environmental conditions that change such as temperature, time of day, or the size of the room in which the
    research is being conducted.
    To demonstrate changes in variables, it is necessary to make measurements of the variables
    being examined. The measurement obtained for each individual is called a datum, or more commonly, a score or raw score. The complete set of scores is called the data set or simply the data.
    Data (plural) are measurements or observations. A data set is a collection of measurements or observations. A datum (singular) is a single measurement or observation and is commonly called a score or raw score.
    Before we move on, we should make one more point about samples, populations, and
    data. Earlier, we defined populations and samples in terms of individuals. For example,
    we discussed a population of graduate students and a sample of cholesterol patients. Be
    forewarned, however, that we will also refer to populations or samples of scores. Because
    research typically involves measuring each individual to obtain a score, every sample (or
    population) of individuals produces a corresponding sample (or population) of scores.
    ■■Parameters and Statistics
    When describing data it is necessary to distinguish whether the data come from a population or a sample. A characteristic that describes a population—for example, the average
    score for the population—is called a parameter. A characteristic that describes a sample is
    called a statistic. Thus, the average score for a sample is an example of a statistic. Typically,
    the research process begins with a question about a population parameter. However, the
    actual data come from a sample and are used to compute sample statistics.
    A parameter is a value, usually a numerical value, that describes a population. A
    parameter is usually derived from measurements of the individuals in the population.
    A statistic is a value, usually a numerical value, that describes a sample. A statistic
    is usually derived from measurements of the individuals in the sample.
    Every population parameter has a corresponding sample statistic, and most research
    studies involve using statistics from samples as the basis for answering questions about
    population parameters. As a result, much of this book is concerned with the relationship
    between sample statistics and the corresponding population parameters. In Chapter 7, for
    example, we examine the relationship between the mean obtained for a sample and the
    mean for the population from which the sample was obtained.
    ■■Descriptive and Inferential Statistical Methods
    Although researchers have developed a variety of different statistical procedures to organize and interpret data, these different procedures can be classified into two general categories. The first category, descriptive statistics, consists of statistical procedures that are used
    to simplify and summarize data.
    Descriptive statistics are statistical procedures used to summarize, organize, and
    simplify data.
    chaPtER 1 | Introduction to Statistics
    Descriptive statistics are techniques that take raw scores and organize or summarize
    them in a form that is more manageable. Often the scores are organized in a table or a graph
    so that it is possible to see the entire set of scores. Another common technique is to summarize a set of scores by computing an average. Note that even if the data set has hundreds
    of scores, the average provides a single descriptive value for the entire set.
    The second general category of statistical techniques is called inferential statistics.
    Inferential statistics are methods that use sample data to make general statements about a
    Inferential statistics consist of techniques that allow us to study samples and then
    make generalizations about the populations from which they were selected.
    Because populations are typically very large, it usually is not possible to measure
    everyone in the population. Therefore, a sample is selected to represent the population.
    By analyzing the results from the sample, we hope to make general statements about the
    population. Typically, researchers use sample statistics as the basis for drawing conclusions
    about population parameters. One problem with using samples, however, is that a sample
    provides only limited information about the population. Although samples are generally
    representative of their populations, a sample is not expected to give a perfectly accurate
    picture of the whole population. There usually is some discrepancy between a sample statistic and the corresponding population parameter. This discrepancy is called sampling
    error, and it creates the fundamental problem inferential statistics must always address.
    Sampling error is the naturally occurring discrepancy, or error, that exists between
    a sample statistic and the corresponding population parameter.
    The concept of sampling error is illustrated in Figure 1.2. The figure shows a population of 1,000 college students and 2 samples, each with 5 students who were selected from
    the population. Notice that each sample contains different individuals who have different
    characteristics. Because the characteristics of each sample depend on the specific people in
    the sample, statistics will vary from one sample to another. For example, the five students
    in sample 1 have an average age of 19.8 years and the students in sample 2 have an average
    age of 20.4 years.
    It is also very unlikely that the statistics obtained for a sample will be identical to the
    parameters for the entire population. In Figure 1.2, for example, neither sample has statistics that are exactly the same as the population parameters. You should also realize that
    Figure 1.2 shows only two of the hundreds of possible samples. Each sample would contain
    different individuals and would produce different statistics. This is the basic concept of
    sampling error: sample statistics vary from one sample to another and typically are different from the corresponding population parameters.
    One common example of sampling error is the error associated with a sample proportion. For example, in newspaper articles reporting results from political polls, you frequently find statements such as this:
    Candidate Brown leads the poll with 51% of the vote. Candidate Jones has 42%
    approval, and the remaining 7% are undecided. This poll was taken from a sample of registered voters and has a margin of error of plus-or-minus 4 percentage points.
    The “margin of error” is the sampling error. In this case, the percentages that are reported
    were obtained from a sample and are being generalized to the whole population. As always,
    you do not expect the statistics from a sample to be perfect. There always will be some
    “margin of error” when sample statistics are used to represent population parameters.
    SEctIon 1.1 | Statistics, Science, and Observations
    F I G U R E 1. 2
    A demonstration of sampling error. Two
    samples are selected from the same population.
    Notice that the sample statistics are different
    from one sample to another and all the sample
    statistics are different from the corresponding
    population parameters. The natural differences that exist, by chance, between a sample
    statistic and population parameter are called
    sampling error.
    of 1000 college students
    Population Parameters
    Average Age 5 21.3 years
    Average IQ 5 112.5
    65% Female, 35% Male
    Sample #1
    Sample #2
    Sample Statistics
    Average Age 5 19.8
    Average IQ 5 104.6
    60% Female, 40% Male
    Sample Statistics
    Average Age 5 20.4
    Average IQ 5 114.2
    40% Female, 60% Male
    As a further demonstration of sampling error, imagine that your statistics class is separated into two groups by drawing a line from front to back through the middle of the room.
    Now imagine that you compute the average age (or height, or IQ) for each group. Will the
    two groups have exactly the same average? Almost certainly they will not. No matter what
    you chose to measure, you will probably find some difference between the two groups.
    However, the difference you obtain does not necessarily mean that there is a systematic
    difference between the two groups. For example, if the average age for students on the
    right-hand side of the room is higher than the average for students on the left, it is unlikely
    that some mysterious force has caused the older people to gravitate to the right side of
    the room. Instead, the difference is probably the result of random factors such as chance.
    The unpredictable, unsystematic differences that exist from one sample to another are an
    example of sampling error.
    ■■Statistics in the Context of Research
    The following example shows the general stages of a research study and demonstrates
    how descriptive statistics and inferential statistics are used to organize and interpret the
    data. At the end of the example, note how sampling error can affect the interpretation of
    experimental results, and consider why inferential statistical methods are needed to deal
    with this problem.
    chaPtER 1 | Introduction to Statistics
    ExamplE 1.1
    Figure 1.3 shows an overview of a general research situation and demonstrates the roles that
    descriptive and inferential statistics play. The purpose of the research study is to address a
    question that we posed earlier: Do college students learn better by studying text on printed
    pages or on a computer screen? Two samples are selected from the population of college
    students. The students in sample A are given printed pages of text to study for 30 minutes
    and the students in sample B study the same text on a computer screen. Next, all of the
    students are given a multiple-choice test to evaluate their knowledge of the material. At this
    point, the researcher has two sets of data: the scores for sample A and the scores for sample
    B (see the figure). Now is the time to begin using statistics.
    First, descriptive statistics are used to simplify the pages of data. For example, the
    researcher could draw a graph showing the scores for each sample or compute the average score for each sample. Note that descriptive methods provide a simplified, organized
    Step 1
    Compare two
    studying methods
    Test scores for the
    students in each
    Step 2
    Descriptive statistics:
    Organize and simplify
    Population of
    Sample A
    Read from printed
    Score = 26
    Step 3
    Inferential statistics:
    Interpret results
    F i g u r E 1. 3
    The role of statistics in experimental
    Sample B
    Read from computer
    Score = 22
    The sample data show a 4-point difference
    between the two methods of studying. However,
    there are two ways to interpret the results.
    1. There actually is no difference between
    the two studying methods, and the sample
    difference is due to chance (sampling error).
    2. There really is a difference between
    the two methods, and the sample data
    accurately reflect this difference.
    The goal of inferential statistics is to help researchers
    decide between the two interpretations.
    SEctIon 1.1 | Statistics, Science, and Observations
    description of the scores. In this example, the students who studied printed pages had an average score of 26 on the test, and the students who studied text on the computer averaged 22.
    Once the researcher has described the results, the next step is to interpret the outcome.
    This is the role of inferential statistics. In this example, the researcher has found a difference
    of 4 points between the two samples (sample A averaged 26 and sample B averaged 22). The
    problem for inferential statistics is to differentiate between the following two interpretations:
    1. There is no real difference between the printed page and a computer screen, and
    the 4-point difference between the samples is just an example of sampling error
    (like the samples in Figure 1.2).
    2. There really is a difference between the printed page and a computer screen, and
    the 4-point difference between the samples was caused by the different methods
    of studying.
    In simple English, does the 4-point difference between samples provide convincing
    evidence of a difference between the two studying methods, or is the 4-point difference just
    chance? The purpose of inferential statistics is to answer this question.

    lE arn in g Ch ECk
    1. A researcher is interested in the sleeping habits of American college students.
    A group of 50 students is interviewed and the researcher finds that these students
    sleep an average of 6.7 hours per day. For this study, the average of 6.7 hours is an
    example of a(n)
    a. parameter
    b. statistic
    c. population
    d. sample
    2. A researcher is curious about the average IQ of registered voters in the state of Florida.
    The entire group of registered voters in the state is an example of a
    a. sample
    b. statistic
    c. population
    d. parameter
    3. Statistical techniques that summarize, organize, and simplify data are classified
    a. population statistics
    b. sample statistics
    c. descriptive statistics
    d. inferential statistics
    4. In general,
    statistical techniques are used to summarize the data from
    a research study and
    statistical techniques are used to determine what
    conclusions are justified by the results.
    a. inferential, descriptive
    b. descriptive, inferential
    c. sample, population
    d. population, sample
    chaPtER 1 | Introduction to Statistics
    5. IQ tests are standardized so that the average score is 100 for the entire group of
    people who take the test each year. However, if you selected a group of 20 people
    who took the test and computed their average IQ score you probably would not get
    100. What statistical concept explains the difference between your mean and the
    mean for the entire group?
    a. statistical error
    b. inferential error
    c. descriptive error
    d. sampling error
    an s wE r s
    1. B, 2. C, 3. C, 4. B, 5. D
    1.2 Data Structures, Research Methods, and Statistics
    4. Differentiate correlational, experimental, and nonexperimental research and describe
    the data structures associated with each.
    5. Define independent, dependent, and quasi-independent variables and recognize
    examples of each.
    ■■Individual Variables: Descriptive Research
    Some research studies are conducted simply to describe individual variables as they exist
    naturally. For example, a college official may conduct a survey to describe the eating, sleeping, and study habits of a group of college students. When the results consist of numerical
    scores, such as the number of hours spent studying each day, they are typically described
    by the statistical techniques that are presented in Chapters 3 and 4. Non-numerical scores
    are typically described by computing the proportion or percentage in each category. For
    example, a recent newspaper article reported that 34.9% of Americans are obese, which is
    roughly 35 pounds over a healthy weight.
    ■■Relationships Between Variables
    Most research, however, is intended to examine relationships between two or more variables. For example, is there a relationship between the amount of violence in the video
    games played by children and the amount of aggressive behavior they display? Is there a
    relationship between the quality of breakfast and academic performance for elementary
    school children? Is there a relationship between the number of hours of sleep and grade
    point average for college students? To establish the existence of a relationship, researchers must make observations—that is, measurements of the two variables. The resulting
    measurements can be classified into two distinct data structures that also help to classify
    different research methods and different statistical techniques. In the following section we
    identify and discuss these two data structures.
    I. One Group with Two Variables Measured for Each Individual: The Correlational Method One method for examining the relationship between variables is to
    observe the two variables as they exist naturally for a set of individuals. That is, simply
    SEctIon 1.2 | Data Structures, Research Methods, and Statistics
    measure the two variables for each individual. For example, research has demonstrated a
    relationship between sleep habits, especially wake-up time, and academic performance
    for college students (Trockel, Barnes, and Egget, 2000). The researchers used a survey to
    measure wake-up time and school records to measure academic performance for each student. Figure 1.4 shows an example of the kind of data obtained in the study. The researchers then look for consistent patterns in the data to provide evidence for a relationship
    between variables. For example, as wake-up time changes from one student to another, is
    there also a tendency for academic performance to change?
    Consistent patterns in the data are often easier to see if the scores are presented in a
    graph. Figure 1.4 also shows the scores for the eight students in a graph called a scatter
    plot. In the scatter plot, each individual is represented by a point so that the horizontal
    position corresponds to the student’s wake-up time and the vertical position corresponds
    to the student’s academic performance score. The scatter plot shows a clear relationship
    between wake-up time and academic performance: as wake-up time increases, academic
    performance decreases.
    A research study that simply measures two different variables for each individual and
    produces the kind of data shown in Figure 1.4 is an example of the correlational method,
    or the correlational research strategy.
    In the correlational method, two different variables are observed to determine
    whether there is a relationship between them.
    ■■Statistics for the Correlational Method
    When the data from a correlational study consist of numerical scores, the relationship
    between the two variables is usually measured and described using a statistic called a
    correlation. Correlations and the correlational method are discussed in detail in Chapters 15 and 16. Occasionally, the measurement process used for a correlational study
    simply classifies individuals into categories that do not correspond to numerical values.
    For example, a researcher could classify a group of college students by gender (male
    Academic performance
    F i g u r E 1. 4
    Wake-up time
    One of two data structures for evaluating the relationship between variables. Note that there are two separate measurements for each individual (wake-up time and academic performance). The same scores are shown in a table (a) and in
    a graph (b).
    chaPtER 1 | Introduction to Statistics
    or female) and by cell-phone preference (talk or text). Note that the researcher has two
    scores for each individual but neither of the scores is a numerical value. This type of data
    is typically summarized in a table showing how many individuals are classified into each
    of the possible categories. Table 1.1 shows an example of this kind of summary table. The
    table shows for example, that 30 of the males in the sample preferred texting to talking.
    This type of data can be coded with numbers (for example, male = 0 and female = 1)
    so that it is possible to compute a correlation. However, the relationship between variables for non-numerical data, such as the data in Table 1.1, is usually evaluated using
    a statistical technique known as a chi-square test. Chi-square tests are presented in
    Chapter 17.
    Ta b lE 1.1
    Correlational data consisting of non-numerical scores. Note that there are two measurements for
    each individual: gender and cell phone preference. The numbers indicate how many people are in
    each category. For example, out of the 50 males, 30 prefer text over talk.
    Cell Phone Preference
    ■■Limitations of the Correlational Method
    The results from a correlational study can demonstrate the existence of a relationship
    between two variables, but they do not provide an explanation for the relationship. In
    particular, a correlational study cannot demonstrate a cause-and-effect relationship. For
    example, the data in Figure 1.4 show a systematic relationship between wake-up time and
    academic performance for a group of college students; those who sleep late tend to have
    lower performance scores than those who wake early. However, there are many possible
    explanations for the relationship and we do not know exactly what factor (or factors) is
    responsible for late sleepers having lower grades. In particular, we cannot conclude that
    waking students up earlier would cause their academic performance to improve, or that
    studying more would cause students to wake up earlier. To demonstrate a cause-and-effect
    relationship between two variables, researchers must use the experimental method, which
    is discussed next.
    II. Comparing Two (or More) Groups of Scores: Experimental and Nonexperimental Methods The second method for examining the relationship between two
    variables involves the comparison of two or more groups of scores. In this situation, the
    relationship between variables is examined by using one of the variables to define the
    groups, and then measuring the second variable to obtain scores for each group. For example, Polman, de Castro, and van Aken (2008) randomly divided a sample of 10-year-old
    boys into two groups. One group then played a violent video game and the second played
    a nonviolent game. After the game-playing session, the children went to a free play period
    and were monitored for aggressive behaviors (hitting, kicking, pushing, frightening, name
    calling, fighting, quarreling, or teasing another child). An example of the resulting data is
    shown in Figure 1.5. The researchers then compare the scores for the violent-video group
    with the scores for the nonviolent-video group. A systematic difference between the two
    groups provides evidence for a relationship between playing violent video games and
    aggressive behavior for 10-year-old boys.
    SEctIon 1.2 | Data Structures, Research Methods, and Statistics
    F i g u r E 1. 5
    Evaluating the relationship between
    variables by comparing groups of scores.
    Note that the values of
    one variable are used
    to define the groups
    and the second variable is measured to
    obtain scores within
    each group.
    One variable (type of video game)
    is used to define groups
    A second variable (aggressive behavior)
    is measured to obtain scores within each group
    Compare groups
    of scores
    ■■Statistics for Comparing Two (or More) Groups of Scores
    Most of the statistical procedures presented in this book are designed for research studies that compare groups of scores like the study in Figure 1.5. Specifically, we examine
    descriptive statistics that summarize and describe the scores in each group and we use
    inferential statistics to determine whether the differences between the groups can be generalized to the entire population.
    When the measurement procedure produces numerical scores, the statistical evaluation typically involves computing the average score for each group and then comparing
    the averages. The process of computing averages is presented in Chapter 3, and a variety
    of statistical techniques for comparing averages are presented in Chapters 8–14. If the
    measurement process simply classifies individuals into non-numerical categories, the statistical evaluation usually consists of computing proportions for each group and then comparing proportions. Previously, in Table 1.1, we presented an example of non-numerical
    data examining the relationship between gender and cell-phone preference. The same data
    can be used to compare the proportions for males with the proportions for females. For
    example, using text is preferred by 60% of the males compared to 50% of the females. As
    before, these data are evaluated using a chi-square test, which is presented in Chapter 17.
    ■■Experimental and Nonexperimental Methods
    There are two distinct research methods that both produce groups of scores to be compared:
    the experimental and the nonexperimental strategies. These two research methods use
    exactly the same statistics and they both demonstrate a relationship between two variables.
    The distinction between the two research strategies is how the relationship is interpreted.
    The results from an experiment allow a cause-and-effect explanation. For example, we can
    conclude that changes in one variable are responsible for causing differences in a second
    variable. A nonexperimental study does not permit a cause-and effect explanation. We can
    say that changes in one variable are accompanied by changes in a second variable, but we
    cannot say why. Each of the two research methods is discussed in the following sections.
    ■■The Experimental Method
    One specific research method that involves comparing groups of scores is known as the
    experimental method or the experimental research strategy. The goal of an experimental
    study is to demonstrate a cause-and-effect relationship between two variables. Specifically,
    chaPtER 1 | Introduction to Statistics
    an experiment attempts to show that changing the value of one variable causes changes to
    occur in the second variable. To accomplish this goal, the experimental method has two
    characteristics that differentiate experiments from other types of research studies:
    1. Manipulation The researcher manipulates one variable by changing its value from
    one level to another. In the Polman et al. (2008) experiment examining the effect
    of violence in video games (Figure 1.5), the researchers manipulate the amount of
    violence by giving one group of boys a violent game to play and giving the other
    group a nonviolent game. A second variable is observed (measured) to determine
    whether the manipulation causes changes to occur.
    2. Control The researcher must exercise control over the research situation to ensure
    that other, extraneous variables do not influence the relationship being examined.
    In more complex experiments, a researcher
    may systematically
    manipulate more than
    one variable and may
    observe more than one
    variable. Here we are
    considering the simplest
    case, in which only one
    variable is manipulated
    and only one variable is
    To demonstrate these two characteristics, consider the Polman et al. (2008) study examining the effect of violence in video games (see Figure 1.5). To be able to say that the difference in aggressive behavior is caused by the amount of violence in the game, the researcher
    must rule out any other possible explanation for the difference. That is, any other variables
    that might affect aggressive behavior must be controlled. There are two general categories
    of variables that researchers must consider:
    1. Participant Variables These are characteristics such as age, gender, and intelligence that vary from one individual to another. Whenever an experiment compares
    different groups of participants (one group in treatment A and a different group
    in treatment B), researchers must ensure that participant variables do not differ
    from one group to another. For the experiment shown in Figure 1.5, for example,
    the researchers would like to conclude that the violence in the video game causes
    a change in the participants’ aggressive behavior. In the study, the participants in
    both conditions were 10-year-old boys. Suppose, however, that the participants in
    the nonviolent condition were primarily female and those in the violent condition
    were primarily male. In this case, there is an alternative explanation for the difference in aggression that exists between the two groups. Specifically, the difference
    between groups may have been caused by the amount of violence in the game,
    but it also is possible that the difference was caused by the participants’ gender
    (females are less aggressive than males). Whenever a research study allows more
    than one explanation for the results, the study is said to be confounded because it is
    impossible to reach an unambiguous conclusion.
    2. Environmental Variables These are characteristics of the environment such as
    lighting, time of day, and weather conditions. A researcher must ensure that the
    individuals in treatment A are tested in the same environment as the individuals
    in treatment B. Using the video game violence experiment (see Figure 1.5) as an
    example, suppose that the individuals in the nonviolent condition were all tested in
    the morning and the individuals in the violent condition were all tested in the evening. Again, this would produce a confounded experiment because the researcher
    could not determine whether the differences in aggressive behavior were caused by
    the amount of violence or caused by the time of day.
    Researchers typically use three basic techniques to control other variables. First, the
    researcher could use random assignment, which means that each participant has an equal
    chance of being assigned to each of the treatment conditions. The goal of random assignment is to distribute the participant characteristics evenly between the two groups so that
    neither group is noticeably smarter (or older, or faster) than the other. Random assignment
    can also be used to control environmental variables. For example, participants could be
    assigned randomly for testing either in the morning or in the afternoon. A second technique
    SEctIon 1.2 | Data Structures, Research Methods, and Statistics
    for controlling variables is to use matching to ensure equivalent groups or equivalent environments. For example, the researcher could match groups by ensuring that every group
    has exactly 60% females and 40% males. Finally, the researcher can control variables by
    holding them constant. For example, in the video game violence study discussed earlier
    (Polman et al., 2008), the researchers used only 10-year-old boys as participants (holding
    age and gender constant). In this case the researchers can be certain that one group is not
    noticeably older or has a larger proportion of females than the other.
    In the experimental method, one variable is manipulated while another variable
    is observed and measured. To establish a cause-and-effect relationship between the
    two variables, an experiment attempts to control all other variables to prevent them
    from influencing the results.
    ■■Terminology in the Experimental Method
    Specific names are used for the two variables that are studied by the experimental method. The
    variable that is manipulated by the experimenter is called the independent variable. It can be
    identified as the treatment conditions to which participants are assigned. For the example in
    Figure 1.5, the amount of violence in the video game is the independent variable. The variable
    that is observed and measured to obtain scores within each condition is the dependent variable. For the example in Figure 1.5, the level of aggressive behavior is the dependent variable.
    The independent variable is the variable that is manipulated by the researcher. In
    behavioral research, the independent variable usually consists of the two (or more) treatment conditions to which subjects are exposed. The independent variable consists of the
    antecedent conditions that were manipulated prior to observing the dependent variable.
    The dependent variable is the one that is observed to assess the effect of the treatment.
    Control Conditions in an Experiment
    An experimental study evaluates the relationship between two variables by manipulating one variable (the independent variable) and
    measuring one variable (the dependent variable). Note that in an experiment only one
    variable is actually measured. You should realize that this is different from a correlational
    study, in which both variables are measured and the data consist of two separate scores
    for each individual.
    Often an experiment will include a condition in which the participants do not receive
    any treatment. The scores from these individuals are then compared with scores from participants who do receive the treatment. The goal of this type of study is to demonstrate that
    the treatment has an effect by showing that the scores in the treatment condition are substantially different from the scores in the no-treatment condition. In this kind of research,
    the no-treatment condition is called the control condition, and the treatment condition is
    called the experimental condition.
    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.
    chaPtER 1 | Introduction to Statistics
    Note that the independent variable always consists of at least two values. (Something
    must have at least two different values before you can say that it is “variable.”) For the
    video game violence experiment (see Figure 1.5), the independent variable is the amount
    of violence in the video game. For an experiment with an experimental group and a control
    group, the independent variable is treatment versus no treatment.
    ■■Nonexperimental Methods: Nonequivalent Groups
    and Pre-Post Studies
    In informal conversation, there is a tendency for people to use the term experiment to refer
    to any kind of research study. You should realize, however, that the term only applies to
    studies that satisfy the specific requirements outlined earlier. In particular, a real experiment must include manipulation of an independent variable and rigorous control of other,
    extraneous variables. As a result, there are a number of other research designs that are not
    true experiments but still examine the relationship between variables by comparing groups
    of scores. Two examples are shown in Figure 1.6 and are discussed in the following paragraphs. This type of research study is classified as nonexperimental.
    The top part of Figure 1.6 shows an example of a nonequivalent groups study comparing boys and girls. Notice that this study involves comparing two groups of scores (like an
    experiment). However, the researcher has no ability to control which participants go into
    F i g u r E 1.6
    Two examples of nonexperimental
    studies that involve comparing two
    groups of scores. In (a) the study
    uses two preexisting groups (boys/
    girls) and measures a dependent
    variable (verbal scores) in each
    group. In (b) the study uses time
    (before/after) to define the two
    groups and measures a dependent
    variable (depression) in each group.
    Variable #1: Subject gender
    (the quasi-independent variable)
    Not manipulated, but used
    to create two groups of subjects
    Variable #2: Verbal test scores
    (the dependent variable)
    Measured in each of the
    two groups
    Variable #1: Time
    (the quasi-independent variable)
    Not manipulated, but used
    to create two groups of scores
    Variable #2: Depression scores
    (the dependent variable)
    Measured at each of the two
    different times
    SEctIon 1.2 | Data Structures, Research Methods, and Statistics
    Correlational studies are
    also examples of nonexperimental research. In
    this section, however, we
    are discussing nonexperimental studies that
    compare two or more
    groups of scores.
    which group—all the males must be in the boy group and all the females must be in the
    girl group. Because this type of research compares preexisting groups, the researcher cannot control the assignment of participants to groups and cannot ensure equivalent groups.
    Other examples of nonequivalent group studies include comparing 8-year-old children and
    10-year-old children, people with an eating disorder and those with no disorder, and comparing children from a single-parent home and those from a two-parent home. Because it
    is impossible to use techniques like random assignment to control participant variables and
    ensure equivalent groups, this type of research is not a true experiment.
    The bottom part of Figure 1.6 shows an example of a pre–post study comparing depression scores before therapy and after therapy. The two groups of scores are obtained by
    measuring the same variable (depression) twice for each participant; once before therapy
    and again after therapy. In a pre-post study, however, the researcher has no control over
    the passage of time. The “before” scores are always measured earlier than the “after”
    scores. Although a difference between the two groups of scores may be caused by the
    treatment, it is always possible that the scores simply change as time goes by. For example, the depression scores may decrease over time in the same way that the symptoms of
    a cold disappear over time. In a pre–post study the researcher also has no control over
    other variables that change with time. For example, the weather could change from dark
    and gloomy before therapy to bright and sunny after therapy. In this case, the depression
    scores could improve because of the weather and not because of the therapy. Because the
    researcher cannot control the passage of time or other variables related to time, this study
    is not a true experiment.
    Terminology in Nonexperimental Research
    Although the two research studies
    shown in Figure 1.6 are not true experiments, you should notice that they produce the
    same kind of data that are found in an experiment (see Figure 1.5). In each case, one variable is used to create groups, and a second variable is measured to obtain scores within
    each group. In an experiment, the groups are created by manipulation of the independent
    variable, and the participants’ scores are the dependent variable. The same terminology is
    often used to identify the two variables in nonexperimental studies. That is, the variable
    that is used to create groups is the independent variable and the scores are the dependent
    variable. For example, the top part of Figure 1.6, gender (boy/girl), is the independent
    variable and the verbal test scores are the dependent variable. However, you should realize that gender (boy/girl) is not a true independent variable because it is not manipulated.
    For this reason, the “independent variable” in a nonexperimental study is often called a
    quasi-independent variable.
    lE arn in g Ch ECk
    In a nonexperimental study, the “independent variable” that is used to create the
    different groups of scores is often called the quasi-independent variable.
    1. In a correlational study, how many variables are measured for each individual and
    how many groups of scores are obtained?
    a. 1 variable and 1 group
    b. 1 variable and 2 groups
    c. 2 variables and 1 group
    d. 2 variables and 2 groups
    chaPtER 1 | Introduction to Statistics
    2. A research study comparing alcohol use for college students in the United States
    and Canada reports that more Canadian students drink but American students drink
    more (Kuo, Adlaf, Lee, Gliksman, Demers, and Wechsler, 2002). What research
    design did this study use?
    a. correlational
    b. experimental
    c. nonexperimental
    d. noncorrelational
    3. Stephens, Atkins, and Kingston (2009) found that participants were able to tolerate
    more pain when they shouted their favorite swear words over and over than when
    they shouted neutral words. For this study, what is the independent variable?
    a. the amount of pain tolerated
    b. the participants who shouted swear words
    c. the participants who shouted neutral words
    d. the kind of word shouted by the participants
    an s wE r s
    1. C, 2. C, 3. D
    Variables and Measurement
    6. Explain why operational definitions are developed for constructs and identify the two
    components of an operational definition.
    7. Describe discrete and continuous variables and identify examples of each.
    8. Differentiate nominal, ordinal, interval, and ratio scales of measurement.
    ■■Constructs and Operational Definitions
    The scores that make up the data from a research study are the result of observing and
    measuring variables. For example, a researcher may finish a study with a set of IQ scores,
    personality scores, or reaction-time scores. In this section, we take a closer look at the variables that are being measured and the process of measurement.
    Some variables, such as height, weight, and eye color are well-defined, concrete entities that can be observed and measured directly. On the other hand, many variables studied
    by behavioral scientists are internal characteristics that people use to help describe and
    explain behavior. For example, we say that a student does well in school because he or
    she is intelligent. Or we say that someone is anxious in social situations, or that someone
    seems to be hungry. Variables like intelligence, anxiety, and hunger are called constructs,
    and because they are intangible and cannot be directly observed, they are often called
    hypothetical constructs.
    Although constructs such as intelligence are internal characteristics that cannot be
    directly observed, it is possible to observe and measure behaviors that are representative
    of the construct. For example, we cannot “see” intelligence but we can see examples of
    intelligent behavior. The external behaviors can then be used to create an operational definition for the construct. An operational definition defines a construct in terms of external
    SEctIon 1.3 | Variables and Measurement
    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.
    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
    ■■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.
    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
    For a continuous variable, there are an infinite number of possible values that fall
    between any two observed values. A continuous variable is divisible into an infinite
    number of fractional parts.
    Suppose, for example, that a researcher is measuring weights for a group of individuals
    participating in a diet study. Because weight is a continuous variable, it can be pictured as
    a continuous line (Figure 1.7). Note that there are an infinite number of possible points on
    chaPtER 1 | Introduction to Statistics
    F i g u r E 1.7
    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.
    Real limits
    the line without any gaps or separations between neighboring points. For any two different
    points on the line, it is always possible to find a third value that is between the two points.
    Two other factors apply to continuous variables:
    1. When measuring a continuous variable, it should be very rare to obtain identical
    measurements for two different individuals. Because a continuous variable has an
    infinite number of possible values, it should be almost impossible for two people to
    have exactly the same score. If the data show a substantial number of tied scores,
    then you should suspect that the measurement procedure is very crude or that the
    variable is not really continuous.
    2. When measuring a continuous variable, each measurement category is actually an
    interval that must be defined by boundaries. For example, two people who both
    claim to weigh 150 pounds are probably not exactly the same weight. However,
    they are both around 150 pounds. One person may actually weigh 149.6 and the
    other 150.3. Thus, a score of 150 is not a specific point on the scale but instead is
    an interval (see Figure 1.7). To differentiate a score of 150 from a score of 149 or
    151, we must set up boundaries on the scale of measurement. These boundaries are
    called real limits and are positioned exactly halfway between adjacent scores. Thus,
    a score of X = 150 pounds is actually an interval bounded by a lower real limit
    of 149.5 at the bottom and an upper real limit of 150.5 at the top. Any individual
    whose weight falls between these real limits will be assigned a score of X = 150.
    Real limits are the boundaries of intervals for scores that are represented on a continuous number line. The real limit separating two adjacent scores is located exactly
    halfway between the scores. Each score has two real limits. The upper real limit is
    at the top of the interval, and the lower real limit is at the bottom.
    The concept of real limits applies to any measurement of a continuous variable, even
    when the score categories are not whole numbers. For example, if you were measuring time
    to the nearest tenth of a second, the measurement categories would be 31.0, 31.1, 31.2, and
    so on. Each of these categories represents an interval on the scale that is bounded by real
    limits. For example, a score of X = 31.1 seconds indicates that the actual measurement
    is in an interval bounded by a lower real limit of 31.05 and an upper real limit of 31.15.
    Remember that the real limits are always halfway between adjacent categories.
    SEctIon 1.3 | Variables and Measurement
    Students often ask
    whether a value of
    exactly 150.5 should
    be assigned to the
    X = 150 interval or the
    X = 151 interval. The
    answer is that 150.5 is
    the boundary between
    the two intervals and is
    not necessarily in one
    or the other. Instead,
    the placement of 150.5
    depends on the rule that
    you are using for rounding numbers. If you
    are rounding up, then
    150.5 goes in the higher
    interval (X = 151) but if
    you are rounding down,
    then it goes in the lower
    interval (X = 150).
    Later in this book, real limits are used for constructing graphs and for various calculations with continuous scales. For now, however, you should realize that real limits are a
    necessity whenever you make measurements of a continuous variable.
    Finally, we should warn you that the terms continuous and discrete apply to the variables that are being measured and not to the scores that are obtained from the measurement.
    For example, measuring people’s heights to the nearest inch produces scores of 60, 61, 62,
    and so on. Although the scores may appear to be discrete numbers, the underlying variable
    is continuous. One key to determining whether a variable is continuous or discrete is that
    a continuous variable can be divided into any number of fractional parts. Height can be
    measured to the nearest inch, the nearest 0.5 inch, or the nearest 0.1 inch. Similarly, a professor evaluating students’ knowledge could use a pass/fail system that classifies students
    into two broad categories. However, the professor could choose to use a 10-point quiz that
    divides student knowledge into 11 categories corresponding to quiz scores from 0 to 10. Or
    the professor could use a 100-point exam that potentially divides student knowledge into
    101 categories from 0 to 100. Whenever you are free to choose the degree of precision or
    the number of categories for measuring a variable, the variable must be continuous.
    ■■Scales of Measurement
    It should be obvious by now that data collection requires that we make measurements of
    our observations. Measurement involves assigning individuals or events to categories. The
    categories can simply be names such as male/female or employed/unemployed, or they
    can be numerical values such as 68 inches or 175 pounds. The categories used to measure
    a variable make up a scale of measurement, and the relationships between the categories determine different types of scales. The distinctions among the scales are important
    because they identify the limitations of certain types of measurements and because certain
    statistical procedures are appropriate for scores that have been measured on some scales
    but not on others. If you were interested in people’s heights, for example, you could measure a group of individuals by simply classifying them into three categories: tall, medium,
    and short. However, this simple classification would not tell you much about the actual
    heights of the individuals, and these measurements would not give you enough information to calculate an average height for the group. Although the simple classification would
    be adequate for some purposes, you would need more sophisticated measurements before
    you could answer more detailed questions. In this section, we examine four different
    scales of measurement, beginning with the simplest and moving to the most sophisticated.
    ■■The Nominal Scale
    The word nominal means “having to do with names.” Measurement on a nominal scale
    involves classifying individuals into categories that have different names but are not related
    to each other in any systematic way. For example, if you were measuring the academic
    majors for a group of college students, the categories would be art, biology, business,
    chemistry, and so on. Each student would be classified in one category according to his
    or her major. The measurements from a nominal scale allow us to determine whether two
    individuals are different, but they do not identify either the direction or the size of the difference. If one student is an art major and another is a biology major we can say that they
    are different, but we cannot say that art is “more than” or “less than” biology and we cannot
    specify how much difference there is between art and biology. Other examples of nominal
    scales include classifying people by race, gender, or occupation.
    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.
    chaPtER 1 | Introduction to Statistics
    Although the categories on a nominal scale are not quantitative values, they are occasionally represented by numbers. For example, the rooms or offices in a building may be
    identified by numbers. You should realize that the room numbers are simply names and do
    not reflect any quantitative information. Room 109 is not necessarily bigger than Room
    100 and certainly not 9 points bigger. It also is fairly common to use numerical values as a
    code for nominal categories when data are entered into computer programs. For example,
    the data from a survey may code males with a 0 and females with a 1. Again, the numerical
    values are simply names and do not represent any quantitative difference. The scales that
    follow do reflect an attempt to make quantitative distinctions.
    ■■The Ordinal Scale
    The categories that make up an ordinal scale not only have different names (as in a nominal
    scale) but also are organized in a fixed order corresponding to differences of magnitude.
    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
    Often, an ordinal scale consists of a series of ranks (first, second, third, and so on) like
    the order of finish in a horse race. Occasionally, the categories are identified by verbal
    labels like small, medium, and large drink sizes at a fast-food restaurant. In either case, the
    fact that the categories form an ordered sequence means that there is a directional relationship between categories. With measurements from an ordinal scale, you can determine
    whether two individuals are different and you can determine the direction of difference.
    However, ordinal measurements do not allow you to determine the size of the difference
    between two individuals. In a NASCAR race, for example, the first-place car finished faster
    than the second-place car, but the ranks don’t tell you how much faster. Other examples of
    ordinal scales include socioeconomic class (upper, middle, lower) and T-shirt sizes (small,
    medium, large). In addition, ordinal scales are often used to measure variables for which it
    is difficult to assign numerical scores. For example, people can rank their food preferences
    but might have trouble explaining “how much” they prefer chocolate ice cream to steak.
    ■■The Interval and Ratio Scales
    Both an interval scale and a ratio scale consist of a series of ordered categories (like an
    ordinal scale) with the additional requirement that the categories form a series of intervals
    that are all exactly the same size. Thus, the scale of measurement consists of a series of
    equal intervals, such as inches on a ruler. Other examples of interval and ratio scales are the
    measurement of time in seconds, weight in pounds, and temperature in degrees Fahrenheit.
    Note that, in each case, one interval (1 inch, 1 second, 1 pound, 1 degree) is the same size,
    no matter where it is located on the scale. The fact that the intervals are all the same size
    makes it possible to determine both the size and the direction of the difference between two
    measurements. For example, you know that a measurement of 80° Fahrenheit is higher than
    a measure of 60°, and you know that it is exactly 20° higher.
    The factor that differentiates an interval scale from a ratio scale is the nature of the zero
    point. An interval scale has an arbitrary zero point. That is, the value 0 is assigned to a particular location on the scale simply as a matter of convenience or reference. In particular, a
    value of zero does not indicate a total absence of the variable being measured. For example
    a temperature of 0º Fahrenheit does not mean that there is no temperature, and it does
    not prohibit the temperature from going even lower. Interval scales with an arbitrary zero
    SEctIon 1.3 | Variables and Measurement
    point are relatively rare. The two most common examples are the Fahrenheit and Celsius
    temperature scales. Other examples include golf scores (above and below par) and relative
    measures such as above and below average rainfall.
    A ratio scale is anchored by a zero point that is not arbitrary but rather is a meaningful
    value representing none (a complete absence) of the variable being measured. The existence
    of an absolute, non-arbitrary zero point means that we can measure the absolute amount of
    the variable; that is, we can measure the distance from 0. This makes it possible to compare
    measurements in terms of ratios. For example, a gas tank with 10 gallons (10 more than 0) has
    twice as much gas as a tank with only 5 gallons (5 more than 0). Also note that a completely
    empty tank has 0 gallons. To recap, with a ratio scale, we can measure the direction and the
    size of the difference between two measurements and we can describe the difference in terms
    of a ratio. Ratio scales are quite common and include physical measures such as height and
    weight, as well as variables such as reaction time or the number of errors on a test. The distinction between an interval scale and a ratio scale is demonstrated in Example 1.2.
    An interval scale consists of ordered categories that are all intervals of exactly the
    same size. Equal differences between numbers on scale reflect equal differences in
    magnitude. However, the zero point on an interval scale is arbitrary and does not
    indicate a zero amount of the variable being measured.
    A ratio scale is an interval scale with the additional feature of an absolute zero
    point. With a ratio scale, ratios of numbers do reflect ratios of magnitude.
    ExamplE 1.2
    A researcher obtains measurements of height for a group of 8-year-old boys. Initially, the
    researcher simply records each child’s height in inches, obtaining values such as 44, 51, 49,
    and so on. These initial measurements constitute a ratio scale. A value of zero represents no
    height (absolute zero). Also, it is possible to use these measurements to form ratios. For example, a child who is 60 inches tall is one and a half times taller than a child who is 40 inches tall.
    Now suppose that the researcher converts the…

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