~~trong>Questions –~~*When the Referee Sees Red…*

1. Which data are more variable – the age of the referees or their years of experience?How do you know?

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2.What is the IV in this study?What is the DV?

3.In this study, why is a dependent t-test more appropriate than an independent t-test to analyze the data?Make sure your answer is specific with respect to what is being compared.

4.Was the number of points awarded to the competitor wearing red protective gear significantly higher than those awarded to the competitor wearing blue protective gear?How do you know, statistically speaking?

5.Was the number of points awarded to a blue competitor who was digitally transformed into a red competitor significantly higher?How do you know, statistically speaking?

6.For which comparison was the effect size the largest?How would the value be interpreted?

7.What one change or addition would you make to this study? (can’t say “none”!)

PS YC HOLOGICA L SC IENCE

Short Report

When the Referee Sees Red . . .

Norbert Hagemann, Bernd Strauss, and Jan Leiing

Westfälische Wilhelms-Universität Münster

Hill and Barton (2005) showed that wearing red sports attire has

a positive impact on one’s outcome in a combat sport (e.g., tae

kwon do or wrestling). They suggested that this effect is due to an

evolutionary or cultural association of the color red with dominance and aggression, proposing that this association triggers

a psychological effect in an athlete who wears red (or in his or

her opponent; e.g., Cuthill, Hunt, Cleary, & Clark, 1997;

Milinski & Bakker, 1990; Setchell & Wickings, 2005). Rowe,

Harris, and Roberts (2005) criticized this argument and instead

attributed the bias evident in these and other data (judo) to

differences in opponents’ visibility.

We disagree with both interpretations (see also Barton & Hill,

2005), arguing that this phenomenon is actually due to a perceptual bias in the referee. That is, we propose that the perception of colors triggers a psychological effect in referees that

can lead to bias in evaluating identical performances. Referees

and umpires exert a major influence on the outcome of sports

competitions (Plessner & Haar, 2006). Athletes frequently make

very rapid movements, and referees have to view sports competitions from a very disadvantageous perspective, so it is extremely difficult for them to make objective judgments (Oudejans et al., 2000). As a result, their judgments may show biases

like those found in other social judgments (Frank & Gilovich,

1988; Plessner & Haar, 2006; Ste-Marie & Valiquette, 1996).

Therefore, we believe that it is the referees who are the actual

cause of the advantage competitors have when they wear red.

Because the effect of red clothing on performance and on the

decisions of referees may well have been confounded in the

original data, we conducted a new experiment and found that

referees assign more points to tae kwon do competitors dressed

in red than to those dressed in blue, even when the performance

of the competitors is identical.

referee 5 8.02 years, SD 5 6.27) individually watched videotaped excerpts from sparring rounds of five different male

competitors of similar abilities. Each of two blocks contained 11

clips, with an average length of 4.4 s. The video images measured 1,024 768 pixels and were displayed on a notebook

computer with a 15.4-in. screen.

In each video, one competitor was wearing red protective gear,

and the other was wearing blue protective gear. (Underneath this

gear, each competitor wore a white tae kwon do uniform.) The two

blocks contained the same clips, but with the colors of the competitors reversed. We reversed the colors using digital graphics,

animation, and image-compositing software (Adobe After Effects

7.0).

After viewing each clip, participants indicated how many

points they would award the red and the blue competitors.

Following the rules of the World Taekwondo Federation,1 participants awarded points when permitted techniques were used

to deliver attacks to the legal scoring areas of the body: Specifically, 1 point was awarded for an attack to the trunk protector

(fist and foot techniques), and 2 points were awarded for an attack to the face (only attacks by foot technique are permitted).

Additional points could be awarded if a contestant knocked

down his opponent. Prohibited acts could be counted as a deduction of 1 point.

The video clips were presented in random order within each

block, and the order of the blocks was counterbalanced across

participants. For each referee, we calculated the total number of

points for the red and blue competitors, and these values were

subjected to separate dependent t tests. We used Cohen’s d as

our measure of effect size. We expected that changing the color

of the protective gear from blue to red would lead to an increase

in points awarded, whereas changing the color from red to blue

would have the opposite effect.

METHOD

We investigated the effect of the color of the protective gear

(trunk and head protectors) in tae kwon do on the decisions of

referees. A total of 42 experienced referees (13 female, 29 male;

mean age 5 29.31 years, SD 5 10.56; mean experience as a

Address correspondence to Norbert Hagemann, Department of Sport

Psychology, University of Münster, Horstmarer Landweg 62b, 48149

Münster, Germany, e-mail: nhageman@uni-muenster.de.

Volume 19—Number 8

RESULTS AND DISCUSSION

Figure 1 shows the effect of the color of the protective gear. The

competitor wearing red protective gear was awarded an average

1

In a regular competition, the match score is the sum of points in three 2-min

rounds. Unless there is a knockout, withdrawal, or disqualification, the winner

is usually determined by points: The winner has the higher final score, exceeds

the opponent’s score by 7 points, or reaches the maximum of 12 points.

Copyright r 2008 Association for Psychological Science

769

Seeing Red

10

9

8

Mean Points

7

6

5

4

3

Competitors A Competitors B

(red)

(blue)

Competitors A Competitors B

(blue)

(red)

Original

Color-Reversed

2

1

0

Version

Fig. 1. Mean number of points awarded to tae kwon do competitors in the original and color-reversed versions of the video

clips. Each clip depicted a sparring round with one competitor dressed in red and the other dressed in blue. Competitors A

wore red in the original clips and blue in the color-reversed clips, and Competitors B wore blue in the original clips and red

in the color-reversed clips. Error bars indicate standard errors.

of 13% (0.94 points) more points than the competitor wearing

blue protective gear, t(41) 5 2.85, p < .01, d 5 0.35. The
number of points awarded increased for a blue competitor who
was digitally transformed into a red competitor, t(41) 5 2.45,
p < .01 (one-tailed), d 5 0.36, and decreased for a red competitor who was digitally transformed into a blue competitor,
t(41) 5 1.66, p < .05 (one-tailed), d 5 0.25. The gender of the
referee, total number of points awarded in the two versions of the
video clip (original vs. color-reversed), and the order in which
the two versions were presented had no effect on the referees’
decisions.
Thus, competitors dressed in red are awarded more points
than competitors dressed in blue, even when their performance
is identical. The effect found in this experiment can also explain
why the effect of clothing color on the outcome is stronger when
competitors have similar abilities than when there is a large
asymmetry in their abilities (Hill & Barton, 2005). Referees’
decisions will ‘‘tip the scales’’ when athletes are relatively well
matched, but have relatively small influence when one is clearly
superior. Even though the color of athletes’ sportswear may well
exert an influence on their performance (through associations
with dominance or differences in visibility of the opponent), we
argue that the referees are responsible for the advantage conveyed to athletes who wear red. Although there is a need for
further research (including research on the effects of different
770
colors), our results suggest a need to change the rules (i.e., forbid
red sports attire) or support referees by providing electronic
decision-making aids (e.g., electronic trunk protectors) in those
sports in which this color bias may be a problem.
REFERENCES
Barton, R.A., & Hill, R.A. (2005). Sporting contests: Seeing red?
Putting sportswear in context (reply). Nature, 437, E10–E11.
Cuthill, I.C., Hunt, S., Cleary, C., & Clark, C. (1997). Colour bands,
dominance, and body mass regulation in male zebra finches
(Taeniopygia guttata). Proceedings of the Royal Society B: Biological Sciences, 264, 1093–1099.
Frank, M.G., & Gilovich, T. (1988). The dark side of self- and
social perception: Black uniforms and aggression in professional
sports. Journal of Personality and Social Psychology, 54, 74–85.
Hill, R.A., & Barton, R.A. (2005). Red enhances human performance
in contests. Nature, 435, 293.
Milinski, M., & Bakker, T.C.M. (1990). Female sticklebacks use male
coloration in mate choice and hence avoid parasitized males.
Nature, 344, 330–333.
Oudejans, R.R., Verheijen, R., Bakker, F.C., Gerrits, J.C., Steinbruckner, M., & Beek, P.J. (2000). Errors in judging ‘offside’ in
football. Nature, 404, 33.
Plessner, H., & Haar, T. (2006). Sports performance judgments from a
social cognitive perspective. Psychology of Sport and Exercise, 7,
555–575.
Volume 19—Number 8
Norbert Hagemann, Bernd Strauss, and Jan Leiing
Rowe, C., Harris, J.M., & Roberts, S.C. (2005). Sporting contests:
Seeing red? Putting sportswear in context. Nature, 437, E10.
Setchell, J.M., & Wickings, E.J. (2005). Dominance, status signals and
coloration in male mandrills (Mandrillus sphinx). Ethology, 111,
25–50.
Volume 19—Number 8
Ste-Marie, D.M., & Valiquette, S.M. (1996). Enduring memory-influenced biases in gymnastic judging. Journal of Experimental
Psychology: Learning, Memory, and Cognition, 22, 1498–1502.
(RECEIVED 12/16/07; REVISION ACCEPTED 2/13/08)
771
Ed iti o n
© Deborah Batt
10
Statistics for the
Behavioral Sciences
Frederick J Gravetter
The College at Brockport, State University of New York
Larry B. WaLLnau
The College at Brockport, State University of New York
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Printed in Canada
Print Number: 01
Print Year: 2015
B RiEF Co n tEn t S
CHAPtER
1
Introduction to Statistics 1
CHAPtER
2
Frequency Distributions
CHAPtER
3
Central Tendency
CHAPtER
4
Variability
CHAPtER
5
z-Scores: Location of Scores and Standardized Distributions 131
CHAPtER
6
Probability
CHAPtER
7
Probability and Samples: The Distribution of Sample Means 193
CHAPtER
8
Introduction to Hypothesis Testing 223
CHAPtER
9
Introduction to the t Statistic
C H A P t E R 10
CHAPtER
11
33
67
99
159
267
The t Test for Two Independent Samples 299
The t Test for Two Related Samples 335
C H A P t E R 12
Introduction to Analysis of Variance 365
C H A P t E R 13
Repeated-Measures Analysis of Variance 413
C H A P t E R 14
Two-Factor Analysis of Variance (Independent Measures) 447
CHAPtER
15
Correlation
485
C H A P t E R 16
Introduction to Regression 529
C H A P t E R 17
The Chi-Square Statistic: Tests for Goodness of Fit and Independence 559
C H A P t E R 18
The Binomial Test
603
iii
Co n tEn t S
CHAPtER
1 Introduction to Statistics
PREVIEW
1
2
1.1 Statistics, Science, and Observations 2
1.2 Data Structures, Research Methods, and Statistics 10
1.3 Variables and Measurement 18
1.4 Statistical Notation 25
Summary
29
Focus on Problem Solving 30
Demonstration 1.1 30
Problems 31
CHAPtER
2 Frequency Distributions
PREVIEW
33
34
2.1 Frequency Distributions and Frequency Distribution Tables 35
2.2 Grouped Frequency Distribution Tables 38
2.3 Frequency Distribution Graphs 42
2.4 Percentiles, Percentile Ranks, and Interpolation 49
2.5 Stem and Leaf Displays 56
Summary
58
Focus on Problem Solving 59
Demonstration 2.1 60
Demonstration 2.2 61
Problems 62
v
vi
CONTENTS
CHAPtER
3 Central Tendency
PREVIEW
67
68
3.1 Overview 68
3.2 The Mean 70
3.3 The Median 79
3.4 The Mode 83
3.5 Selecting a Measure of Central Tendency 86
3.6 Central Tendency and the Shape of the Distribution 92
Summary
94
Focus on Problem Solving 95
Demonstration 3.1 96
Problems 96
CHAPtER
4 Variability
PREVIEW
99
100
4.1 Introduction to Variability 101
4.2 Defining Standard Deviation and Variance 103
4.3 Measuring Variance and Standard Deviation for a Population 108
4.4 Measuring Standard Deviation and Variance for a Sample 111
4.5 Sample Variance as an Unbiased Statistic 117
4.6 More about Variance and Standard Deviation 119
Summary
125
Focus on Problem Solving 127
Demonstration 4.1 128
Problems 128
z-Scores: Location of Scores
C H A P t E R 5 and Standardized Distributions
PREVIEW
132
5.1 Introduction to z-Scores 133
5.2 z-Scores and Locations in a Distribution 135
5.3 Other Relationships Between z, X, 𝛍, and 𝛔 138
131
CONTENTS
vii
5.4 Using z-Scores to Standardize a Distribution 141
5.5 Other Standardized Distributions Based on z-Scores 145
5.6 Computing z-Scores for Samples 148
5.7 Looking Ahead to Inferential Statistics 150
Summary
153
Focus on Problem Solving 154
Demonstration 5.1 155
Demonstration 5.2 155
Problems 156
CHAPtER
6 Probability
PREVIEW
159
160
6.1 Introduction to Probability 160
6.2 Probability and the Normal Distribution 165
6.3 Probabilities and Proportions for Scores
from a Normal Distribution
172
6.4 Probability and the Binomial Distribution 179
6.5 Looking Ahead to Inferential Statistics 184
Summary
186
Focus on Problem Solving 187
Demonstration 6.1 188
Demonstration 6.2 188
Problems 189
Probability and Samples: The Distribution
C H A P t E R 7 of Sample Means
PREVIEW
194
7.1 Samples, Populations, and the Distribution
of Sample Means
194
7.2 The Distribution of Sample Means for any Population
and any Sample Size
199
7.3 Probability and the Distribution of Sample Means 206
7.4 More about Standard Error 210
7.5 Looking Ahead to Inferential Statistics
215
193
viii
CONTENTS
Summary
219
Focus on Problem Solving 219
Demonstration 7.1 220
Problems 221
CHAPtER
8 Introduction to Hypothesis Testing
PREVIEW
223
224
8.1 The Logic of Hypothesis Testing 225
8.2 Uncertainty and Errors in Hypothesis Testing 236
8.3 More about Hypothesis Tests 240
8.4 Directional (One-Tailed) Hypothesis Tests 245
8.5 Concerns about Hypothesis Testing: Measuring Effect Size 250
8.6 Statistical Power 254
Summary
260
Focus on Problem Solving 261
Demonstration 8.1 262
Demonstration 8.2 263
Problems 263
CHAPtER
9 Introduction to the t Statistic
PREVIEW
268
9.1 The t Statistic: An Alternative to z 268
9.2 Hypothesis Tests with the t Statistic 274
9.3 Measuring Effect Size for the t Statistic 279
9.4 Directional Hypotheses and One-Tailed Tests 288
Summary
291
Focus on Problem Solving 293
Demonstration 9.1 293
Demonstration 9.2 294
Problems 295
267
CONTENTS
CHAPtER
10 The t Test for Two Independent Samples
PREVIEW
ix
299
300
10.1 Introduction to the Independent-Measures Design 300
10.2 The Null Hypothesis and the Independent-Measures t Statistic 302
10.3 Hypothesis Tests with the Independent-Measures t Statistic 310
10.4 Effect Size and Confidence Intervals for the
Independent-Measures t
316
10.5 The Role of Sample Variance and Sample Size in the
Independent-Measures t Test
Summary
322
325
Focus on Problem Solving 327
Demonstration 10.1 328
Demonstration 10.2 329
Problems 329
CHAPtER
11 The t Test for Two Related Samples
PREVIEW
335
336
11.1 Introduction to Repeated-Measures Designs 336
11.2 The t Statistic for a Repeated-Measures Research Design 339
11.3 Hypothesis Tests for the Repeated-Measures Design 343
11.4 Effect Size and Confidence Intervals for the Repeated-Measures t 347
11.5 Comparing Repeated- and Independent-Measures Designs 352
Summary
355
Focus on Problem Solving 358
Demonstration 11.1 358
Demonstration 11.2 359
Problems 360
CHAPtER
12 Introduction to Analysis of Variance
PREVIEW
366
12.1 Introduction (An Overview of Analysis of Variance) 366
12.2 The Logic of Analysis of Variance 372
12.3 ANOVA Notation and Formulas 375
365
x
CONTENTS
12.4 Examples of Hypothesis Testing and Effect Size with ANOVA 383
12.5 Post Hoc Tests 393
12.6 More about ANOVA 397
Summary
403
Focus on Problem Solving 406
Demonstration 12.1 406
Demonstration 12.2 408
Problems 408
CHAPtER
13 Repeated-Measures Analysis of Variance
PREVIEW
413
414
13.1 Overview of the Repeated-Measures ANOVA 415
13.2 Hypothesis Testing and Effect Size with the
Repeated-Measures ANOVA
420
13.3 More about the Repeated-Measures Design 429
Summary
436
Focus on Problem Solving 438
Demonstration 13.1 439
Demonstration 13.2 440
Problems 441
Two-Factor Analysis of Variance
C H A P t E R 14 (Independent Measures)
PREVIEW
447
448
14.1 An Overview of the Two-Factor, Independent-Measures, ANOVA: Main
Effects and Interactions
448
14.2 An Example of the Two-Factor ANOVA and Effect Size 458
14.3 More about the Two-Factor ANOVA 467
Summary
473
Focus on Problem Solving 475
Demonstration 14.1 476
Demonstration 14.2 478
Problems 479
CONTENTS
CHAPtER
15 Correlation
PREVIEW
xi
485
486
15.1 Introduction 487
15.2 The Pearson Correlation 489
15.3 Using and Interpreting the Pearson Correlation 495
15.4 Hypothesis Tests with the Pearson Correlation 506
15.5 Alternatives to the Pearson Correlation 510
Summary
520
Focus on Problem Solving 522
Demonstration 15.1 523
Problems 524
CHAPtER
16 Introduction to Regression
PREVIEW
529
530
16.1 Introduction to Linear Equations and Regression 530
16.2 The Standard Error of Estimate and Analysis of Regression:
The Significance of the Regression Equation
538
16.3 Introduction to Multiple Regression with Two Predictor Variables 544
Summary
552
Linear and Multiple Regression
554
Focus on Problem Solving 554
Demonstration 16.1 555
Problems 556
The Chi-Square Statistic: Tests for Goodness
C H A P t E R 17 of Fit and Independence
PREVIEW
560
17.1 Introduction to Chi-Square: The Test for Goodness of Fit 561
17.2 An Example of the Chi-Square Test for Goodness of Fit 567
17.3 The Chi-Square Test for Independence 573
17.4 Effect Size and Assumptions for the Chi-Square Tests
17.5 Special Applications of the Chi-Square Tests 587
582
559
xii
CONTENTS
Summary
591
Focus on Problem Solving 595
Demonstration 17.1 595
Demonstration 17.2 597
Problems 597
CHAPtER
18 The Binomial Test
PREVIEW
603
604
18.1 Introduction to the Binomial Test 604
18.2 An Example of the Binomial Test 608
18.3 More about the Binomial Test: Relationship with Chi-Square
and the Sign Test
Summary
612
617
Focus on Problem Solving 619
Demonstration 18.1 619
Problems 620
A PPE N D IX E S
A Basic Mathematics Review 625
A.1
A.2
A.3
A.4
A.5
Symbols and Notation 627
Proportions: Fractions, Decimals, and Percentages 629
Negative Numbers 635
Basic Algebra: Solving Equations 637
Exponents and Square Roots 640
B Statistical Tables 647
C Solutions for Odd-Numbered Problems in the Text 663
D General Instructions for Using SPSS 683
E Hypothesis Tests for Ordinal Data: Mann-Whitney,
Wilcoxon, Kruskal-Wallis, and Friedman Tests
687
Statistics Organizer: Finding the Right Statistics for Your Data
References
717
Name Index
723
Subject Index
725
701
PREFACE
M
any students in the behavioral sciences view the required statistics course as an
intimidating obstacle that has been placed in the middle of an otherwise interesting curriculum. They want to learn about human behavior—not about math and science.
As a result, the statistics course is seen as irrelevant to their education and career goals.
However, as long as the behavioral sciences are founded in science, knowledge of statistics
will be necessary. Statistical procedures provide researchers with objective and systematic
methods for describing and interpreting their research results. Scientific research is the
system that we use to gather information, and statistics are the tools that we use to distill
the information into sensible and justified conclusions. The goal of this book is not only
to teach the methods of statistics, but also to convey the basic principles of objectivity and
logic that are essential for science and valuable for decision making in everyday life.
Those of you who are familiar with previous editions of Statistics for the Behavioral
Sciences will notice that some changes have been made. These changes are summarized
in the section entitled “To the Instructor.” In revising this text, our students have been
foremost in our minds. Over the years, they have provided honest and useful feedback.
Their hard work and perseverance has made our writing and teaching most rewarding. We
sincerely thank them. Students who are using this edition should please read the section of
the preface entitled “To the Student.”
The book chapters are organized in the sequence that we use for our own statistics
courses. We begin with descriptive statistics, and then examine a variety of statistical procedures focused on sample means and variance before moving on to correlational methods
and nonparametric statistics. Information about modifying this sequence is presented in the
To The Instructor section for individuals who prefer a different organization. Each chapter
contains numerous examples, many based on actual research studies, learning checks, a
summary and list of key terms, and a set of 20–30 problems.
Ancillaries
Ancillaries for this edition include the following.
■■
MindTap® Psychology: MindTap® Psychology for Gravetter/Wallnau’s Statistics
for The Behavioral Sciences, 10th Edition is the digital learning solution that helps
instructors engage and transform today’s students into critical thinkers. Through paths
of dynamic assignments and applications that you can personalize, real-time course
analytics, and an accessible reader, MindTap helps you turn cookie cutter into cutting
edge, apathy into engagement, and memorizers into higher-level thinkers.
As an instructor using MindTap you have at your fingertips the right content and
unique set of tools curated specifically for your course, such as video tutorials that
walk students through various concepts and interactive problem tutorials that provide
students opportunities to practice what they have learned, all in an interface designed
to improve workflow and save time when planning lessons and course structure. The
control to build and personalize your course is all yours, focusing on the most relevant
xiii
xiv
PREFACE
■■
■■
■■
material while also lowering costs for your students. Stay connected and informed in
your course through real time student tracking that provides the opportunity to adjust
the course as needed based on analytics of interactivity in the course.
Online Instructor’s Manual: The manual includes learning objectives, key terms,
a detailed chapter outline, a chapter summary, lesson plans, discussion topics, student
activities, “What If” scenarios, media tools, a sample syllabus and an expanded test
bank. The learning objectives are correlated with the discussion topics, student
activities, and media tools.
Online PowerPoints: Helping you make your lectures more engaging while effectively reaching your visually oriented students, these handy Microsoft PowerPoint®
slides outline the chapters of the main text in a classroom-ready presentation. The
PowerPoint® slides are updated to reflect the content and organization of the new
edition of the text.
Cengage Learning Testing, powered by Cognero®: Cengage Learning Testing,
Powered by Cognero®, is a flexible online system that allows you to author, edit,
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Acknowledgments
It takes a lot of good, hard-working people to produce a book. Our friends at Cengage
have made enormous contributions to this textbook. We thank: Jon-David Hague, Product
Director; Timothy Matray, Product Team Director; Jasmin Tokatlian, Content Development Manager; Kimiya Hojjat, Product Assistant; and Vernon Boes, Art Director. Special
thanks go to Stefanie Chase, our Content Developer and to Lynn Lustberg who led us
through production at MPS.
Reviewers play a very important role in the development of a manuscript. Accordingly,
we offer our appreciation to the following colleagues for their assistance: Patricia Case,
University of Toledo; Kevin David, Northeastern State University; Adia Garrett, University of Maryland, Baltimore County; Carrie E. Hall, Miami University; Deletha Hardin,
University of Tampa; Angela Heads, Prairie View A&M University; Roberto Heredia,
Texas A&M International University; Alisha Janowski, University of Central Florida;
Matthew Mulvaney, The College at Brockport (SUNY); Nicholas Von Glahn, California
State Polytechnic University, Pomona; and Ronald Yockey, Fresno State University.
To the Instructor
Those of you familiar with the previous edition of Statistics for the Behavioral Sciences will
notice a number of changes in the 10th edition. Throughout this book, research examples
have been updated, real world examples have been added, and the end-of-chapter problems
have been extensively revised. Major revisions for this edition include the following:
1. Each section of every chapter begins with a list of Learning Objectives for that
specific section.
2. Each section ends with a Learning Check consisting of multiple-choice questions
with at least one question for each Learning Objective.
PREFACE
xv
3. The former Chapter 19, Choosing the Right Statistics, has been eliminated and
an abridged version is now an Appendix replacing the Statistics Organizer, which
appeared in earlier editions.
Other examples of specific and noteworthy revisions include the following.
Chapter 1 The section on data structures and research methods parallels the new
Appendix, Choosing the Right Statistics.
Chapter 2 The chapter opens with a new Preview to introduce the concept and purpose
of frequency distributions.
Chapter 3
Minor editing clarifies and simplifies the discussion the median.
Chapter 4 The chapter opens with a new Preview to introduce the topic of Central
Tendency. The sections on standard deviation and variance have been edited to increase
emphasis on concepts rather than calculations.
The section discussion relationships between z, X, μ, and σ has been
expanded and includes a new demonstration example.
Chapter 5
Chapter 6
The chapter opens with a new Preview to introduce the topic of Probability.
The section, Looking Ahead to Inferential Statistics, has been substantially shortened and
simplified.
Chapter 7
The former Box explaining difference between standard deviation and
standard error was deleted and the content incorporated into Section 7.4 with editing to
emphasize that the standard error is the primary new element introduced in the chapter.
The final section, Looking Ahead to Inferential Statistics, was simplified and shortened to
be consistent with the changes in Chapter 6.
Chapter 8
A redundant example was deleted which shortened and streamlined the
remaining material so that most of the chapter is focused on the same research example.
Chapter 9 The chapter opens with a new Preview to introduce the t statistic and explain
why a new test statistic is needed. The section introducing Confidence Intervals was edited
to clarify the origin of the confidence interval equation and to emphasize that the interval
is constructed at the sample mean.
Chapter 10
The chapter opens with a new Preview introducing the independent-measures t statistic. The section presenting the estimated standard error of (M1 – M2) has been
simplified and shortened.
Chapter 11
The chapter opens with a new Preview introducing the repeated-measures t
statistic. The section discussing hypothesis testing has been separated from the section on
effect size and confidence intervals to be consistent with the other two chapters on t tests.
The section comparing independent- and repeated-measures designs has been expanded.
Chapter 12 The chapter opens with a new Preview introducing ANOVA and explaining
why a new hypothesis testing procedure is necessary. Sections in the chapter have been
reorganized to allow flow directly from hypothesis tests and effect size to post tests.
xvi
PREFACE
Chapter 13
Substantially expanded the section discussing factors that influence the
outcome of a repeated-measures hypothesis test and associated measures of effect size.
Chapter 14
The chapter opens with a new Preview presenting a two-factor research
example and introducing the associated ANOVA. Sections have been reorganized so that
simple main effects and the idea of using a second factor to reduce variance from individual differences are now presented as extra material related to the two-factor ANOVA.
Chapter 15
The chapter opens with a new Preview presenting a correlational research
study and the concept of a correlation. A new section introduces the t statistic for evaluating the significance of a correlation and the section on partial correlations has been simplified and shortened.
Chapter 16 The chapter opens with a new Preview introducing the concept of regression and
its purpose. A new section demonstrates the equivalence of testing the significance of a correlation and testing the significance of a regression equation with one predictor variable. The section on residuals for the multiple-regression equation has been edited to simplify and shorten.
Chapter 17
A new chapter Preview presents an experimental study with data consisting
of frequencies, which are not compatible with computing means and variances. Chi-square
tests are introduced as a solution to this problem. A new section introduces Cohen’s w as
a means of measuring effect size for both chi-square tests.
Chapter 18
Substantial editing clarifies the section explaining how the real limits for
each score can influence the conclusion from a binomial test.
The former Chapter 19 covering the task of matching statistical methods to specific
types of data has been substantially shortened and converted into an Appendix.
■■Matching the Text to Your Syllabus
The book chapters are organized in the sequence that we use for our own statistics courses.
However, different instructors may prefer different organizations and probably will choose
to omit or deemphasize specific topics. We have tried to make separate chapters, and even
sections of chapters, completely self-contained, so they can be deleted or reorganized to fit
the syllabus for nearly any instructor. Some common examples are as follows.
■■
■■
■■
It is common for instructors to choose between emphasizing analysis of variance
(Chapters 12, 13, and 14) or emphasizing correlation/regression (Chapters 15 and 16).
It is rare for a one-semester course to complete coverage of both topics.
Although we choose to complete all the hypothesis tests for means and mean
differences before introducing correlation (Chapter 15), many instructors prefer to
place correlation much earlier in the sequence of course topics. To accommodate
this, Sections 15.1, 15.2, and 15.3 present the calculation and interpretation of
the Pearson correlation and can be introduced immediately following Chapter 4
(variability). Other sections of Chapter 15 refer to hypothesis testing and should be
delayed until the process of hypothesis testing (Chapter 8) has been introduced.
It is also possible for instructors to present the chi-square tests (Chapter 17) much
earlier in the sequence of course topics. Chapter 17, which presents hypothesis tests
for proportions, can be presented immediately after Chapter 8, which introduces the
process of hypothesis testing. If this is done, we also recommend that the Pearson
correlation (Sections 15.1, 15.2, and 15.3) be presented early to provide a foundation
for the chi-square test for independence.
PREFACE
xvii
To the Student
A primary goal of this book is to make the task of learning statistics as easy and painless
as possible. Among other things, you will notice that the book provides you with a number
of opportunities to practice the techniques you will be learning in the form of Learning
Checks, Examples, Demonstrations, and end-of-chapter problems. We encourage you to
take advantage of these opportunities. Read the text rather than just memorizing the formulas. We have taken care to present each statistical procedure in a conceptual context that
explains why the procedure was developed and when it should be used. If you read this
material and gain an understanding of the basic concepts underlying a statistical formula,
you will find that learning the formula and how to use it will be much easier. In the “Study
Hints,” that follow, we provide advice that we give our own students. Ask your instructor
for advice as well; we are sure that other instructors will have ideas of their own.
Over the years, the students in our classes and other students using our book have given
us valuable feedback. If you have any suggestions or comments about this book, you can
write to either Professor Emeritus Frederick Gravetter or Professor Emeritus Larry Wallnau
at the Department of Psychology, SUNY College at Brockport, 350 New Campus Drive,
Brockport, New York 14420. You can also contact Professor Emeritus Gravetter directly at
fgravett@brockport.edu.
■■Study Hints
You may find some of these tips helpful, as our own students have reported.
■■
■■
■■
■■
■■
The key to success in a statistics course is to keep up with the material. Each new
topic builds on previous topics. If you have learned the previous material, then the
new topic is just one small step forward. Without the proper background, however,
the new topic can be a complete mystery. If you find that you are falling behind, get
help immediately.
You will learn (and remember) much more if you study for short periods several
times per week rather than try to condense all of your studying into one long session.
For example, it is far more effective to study half an hour every night than to have
a single 3½-hour study session once a week. We cannot even work on writing this
book without frequent rest breaks.
Do some work before class. Keep a little ahead of the instructor by reading the appropriate sections before they are presented in class. Although you may not fully understand what you read, you will have a general idea of the topic, which will make the
lecture easier to follow. Also, you can identify material that is particularly confusing
and then be sure the topic is clarified in class.
Pay attention and think during class. Although this advice seems obvious, often it is
not practiced. Many students spend so much time trying to write down every example
presented or every word spoken by the instructor that they do not actually understand
and process what is being said. Check with your instructor—there may not be a need
to copy every example presented in class, especially if there are many examples like
it in the text. Sometimes, we tell our students to put their pens and pencils down for a
moment and just listen.
Test yourself regularly. Do not wait until the end of the chapter or the end of the
week to check your knowledge. After each lecture, work some of the end-of-chapter
problems and do the Learning Checks. Review the Demonstration Problems, and
be sure you can define the Key Terms. If you are having trouble, get your questions
answered immediately—reread the section, go to your instructor, or ask questions in
class. By doing so, you will be able to move ahead to new material.
xviii
PREFACE
■■
■■
Do not kid yourself! Avoid denial. Many students observe their instructor solve
problems in class and think to themselves, “This looks easy, I understand it.” Do
you really understand it? Can you really do the problem on your own without having
to leaf through the pages of a chapter? Although there is nothing wrong with using
examples in the text as models for solving problems, you should try working a problem with your book closed to test your level of mastery.
We realize that many students are embarrassed to ask for help. It is our biggest challenge as instructors. You must find a way to overcome this aversion. Perhaps contacting the instructor directly would be a good starting point, if asking questions in class
is too anxiety-provoking. You could be pleasantly surprised to find that your instructor does not yell, scold, or bite! Also, your instructor might know of another student
who can offer assistance. Peer tutoring can be very helpful.
Frederick J Gravetter
Larry B. Wallnau
A B o U t tH E AU tH o R S
Frederick Gravetter
Frederick J Gravetter is Professor Emeritus of Psychology at the
State University of New York College at Brockport. While teaching at
Brockport, Dr. Gravetter specialized in statistics, experimental design, and
cognitive psychology. He received his bachelor’s degree in mathematics from
M.I.T. and his Ph.D in psychology from Duke University. In addition to publishing this textbook and several research articles, Dr. Gravetter co-authored
Research Methods for the Behavioral Science and Essentials of Statistics for
the Behavioral Sciences.
Larry B. Wallnau
Larry B. WaLLnau is Professor Emeritus of Psychology at the State
University of New York College at Brockport. While teaching at Brockport,
he published numerous research articles in biopsychology. With
Dr. Gravetter, he co-authored Essentials of Statistics for the Behavioral
Sciences. Dr. Wallnau also has provided editorial consulting for numerous
publishers and journals. He has taken up running and has competed in 5K
races in New York and Connecticut. He takes great pleasure in adopting
neglected and rescued dogs.
xix
CH A P T ER
Introduction to Statistics
1
© Deborah Batt
PREVIEW
1.1 Statistics, Science, and Observations
1.2 Data Structures, Research Methods, and Statistics
1.3 Variables and Measurement
1.4 Statistical Notation
Summary
Focus on Problem Solving
Demonstration 1.1
Problems
1
PREVIEW
Before we begin our discussion of statistics, we ask you
to read the following paragraph taken from the philosophy of Wrong Shui (Candappa, 2000).
The Journey to Enlightenment
In Wrong Shui, life is seen as a cosmic journey,
a struggle to overcome unseen and unexpected
obstacles at the end of which the traveler will find
illumination and enlightenment. Replicate this quest
in your home by moving light switches away from
doors and over to the far side of each room.*
Why did we begin a statistics book with a bit of twisted
philosophy? In part, we simply wanted to lighten the
mood with a bit of humor—starting a statistics course is
typically not viewed as one of life’s joyous moments. In
addition, the paragraph is an excellent counterexample for
the purpose of this book. Specifically, our goal is to do
everything possible to prevent you from stumbling around
in the dark by providing lots of help and illumination as
you journey through the world of statistics. To accomplish
this, we begin each section of the book with clearly stated
learning objectives and end each section with a brief quiz
to test your mastery of the new material. We also introduce each new statistical procedure by explaining the purpose it is intended to serve. If you understand why a new
procedure is needed, you will find it much easier to learn.
1.1
The objectives for this first chapter are to provide
an introduction to the topic of statistics and to give you
some background for the rest of the book. We discuss
the role of statistics within the general field of scientific
inquiry, and we introduce some of the vocabulary and
notation that are necessary for the statistical methods
that follow.
As you read through the following chapters, keep
in mind that the general topic of statistics follows a
well-organized, logically developed progression that
leads from basic concepts and definitions to increasingly sophisticated techniques. Thus, each new topic
serves as a foundation for the material that follows. The
content of the first nine chapters, for example, provides
an essential background and context for the statistical
methods presented in Chapter 10. If you turn directly
to Chapter 10 without reading the first nine chapters,
you will find the material confusing and incomprehensible. However, if you learn and use the background
material, you will have a good frame of reference for
understanding and incorporating new concepts as they
are presented.
*Candappa, R. (2000). The little book of wrong shui. Kansas City:
Andrews McMeel Publishing. Reprinted by permission.
Statistics, Science, and Observations
LEARNING OBJECTIVEs
1. Define the terms population, sample, parameter, and statistic, and describe the
relationships between them.
2. Define descriptive and inferential statistics and describe how these two general
categories of statistics are used in a typical research study.
3. Describe the concept of sampling error and explain how this concept creates the
fundamental problem that inferential statistics must address.
■■Definitions of Statistics
By one definition, statistics consist of facts and figures such as the average annual snowfall
in Denver or Derrick Jeter’s lifetime batting average. These statistics are usually informative
and time-saving because they condense large quantities of information into a few simple figures. Later in this chapter we return to the notion of calculating statistics (facts and figures)
but, for now, we concentrate on a much broader definition of statistics. Specifically, we use
the term statistics to refer to a general field of mathematics. In this case, we are using the
term statistics as a shortened version of statistical procedures. For example, you are probably using this book for a statistics course in which you will learn about the statistical techniques that are used to summarize and evaluate research results in the behavioral sciences.
2
SEctIon 1.1 | Statistics, Science, and Observations
3
Research in the behavioral sciences (and other fields) involves gathering information.
To determine, for example, whether college students learn better by reading material on
printed pages or on a computer screen, you would need to gather information about students’ study habits and their academic performance. When researchers finish the task of
gathering information, they typically find themselves with pages and pages of measurements such as preferences, personality scores, opinions, and so on. In this book, we present
the statistics that researchers use to analyze and interpret the information that they gather.
Specifically, statistics serve two general purposes:
1. Statistics are used to organize and summarize the information so that the researcher can
see what happened in the research study and can communicate the results to others.
2. Statistics help the researcher to answer the questions that initiated the research by
determining exactly what general conclusions are justified based on the specific
results that were obtained.
DEFInItIon
The term statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information.
Statistical procedures help ensure that the information or observations are presented
and interpreted in an accurate and informative way. In somewhat grandiose terms, statistics
help researchers bring order out of chaos. In addition, statistics provide researchers with a
set of standardized techniques that are recognized and understood throughout the scientific
community. Thus, the statistical methods used by one researcher will be familiar to other
researchers, who can accurately interpret the statistical analyses with a full understanding
of how the analysis was done and what the results signify.
■■Populations and Samples
Research in the behavioral sciences typically begins with a general question about a specific
group (or groups) of individuals. For example, a researcher may want to know what factors
are associated with academic dishonesty among college students. Or a researcher may want
to examine the amount of time spent in the bathroom for men compared to women. In the
first example, the researcher is interested in the group of college students. In the second
example, the researcher wants to compare the group of men with the group of women. In statistical terminology, the entire group that a researcher wishes to study is called a population.
DEFInItIon
A population is the set of all the individuals of interest in a particular study.
As you can well imagine, a population can be quite large—for example, the entire set
of women on the planet Earth. A researcher might be more specific, limiting the population
for study to women who are registered voters in the United States. Perhaps the investigator
would like to study the population consisting of women who are heads of state. Populations
can obviously vary in size from extremely large to very small, depending on how the investigator defines the population. The population being studied should always be identified by
the researcher. In addition, the population need not consist of people—it could be a population of rats, corporations, parts produced in a factory, or anything else an investigator wants
to study. In practice, populations are typically very large, such as the population of college
sophomores in the United States or the population of small businesses.
Because populations tend to be very large, it usually is impossible for a researcher to
examine every individual in the population of interest. Therefore, researchers typically select
4
chaPtER 1 | Introduction to Statistics
a smaller, more manageable group from the population and limit their studies to the individuals in the selected group. In statistical terms, a set of individuals selected from a population
is called a sample. A sample is intended to be representative of its population, and a sample
should always be identified in terms of the population from which it was selected.
A sample is a set of individuals selected from a population, usually intended to
represent the population in a research study.
DEFInItIon
Just as we saw with populations, samples can vary in size. For example, one study might
examine a sample of only 10 students in a graduate program and another study might use a
sample of more than 10,000 people who take a specific cholesterol medication.
So far we have talked about a sample being selected from a population. However, this is
actually only half of the full relationship between a sample and its population. Specifically,
when a researcher finishes examining the sample, the goal is to generalize the results back
to the entire population. Remember that the research started with a general question about
the population. To answer the question, a researcher studies a sample and then generalizes
the results from the sample to the population. The full relationship between a sample and a
population is shown in Figure 1.1.
■■Variables and Data
Typically, researchers are interested in specific characteristics of the individuals in the population (or in the sample), or they are interested in outside factors that may influence the
individuals. For example, a researcher may be interested in the influence of the weather on
people’s moods. As the weather changes, do people’s moods also change? Something that
can change or have different values is called a variable.
DEFInItIon
A variable is a characteristic or condition that changes or has different values for
different individuals.
THE POPULATION
All of the individuals of interest
The results
from the sample
are generalized
to the population
F I G U R E 1.1
The relationship between a
population and a sample.
The sample
is selected from
the population
THE SAMPLE
The individuals selected to
participate in the research study
SEctIon 1.1 | Statistics, Science, and Observations
5
Once again, variables can be characteristics that differ from one individual to another,
such as height, weight, gender, or personality. Also, variables can be environmental conditions that change such as temperature, time of day, or the size of the room in which the
research is being conducted.
To demonstrate changes in variables, it is necessary to make measurements of the variables
being examined. The measurement obtained for each individual is called a datum, or more commonly, a score or raw score. The complete set of scores is called the data set or simply the data.
DEFInItIon
Data (plural) are measurements or observations. A data set is a collection of measurements or observations. A datum (singular) is a single measurement or observation and is commonly called a score or raw score.
Before we move on, we should make one more point about samples, populations, and
data. Earlier, we defined populations and samples in terms of individuals. For example,
we discussed a population of graduate students and a sample of cholesterol patients. Be
forewarned, however, that we will also refer to populations or samples of scores. Because
research typically involves measuring each individual to obtain a score, every sample (or
population) of individuals produces a corresponding sample (or population) of scores.
■■Parameters and Statistics
When describing data it is necessary to distinguish whether the data come from a population or a sample. A characteristic that describes a population—for example, the average
score for the population—is called a parameter. A characteristic that describes a sample is
called a statistic. Thus, the average score for a sample is an example of a statistic. Typically,
the research process begins with a question about a population parameter. However, the
actual data come from a sample and are used to compute sample statistics.
DEFInItIon
A parameter is a value, usually a numerical value, that describes a population. A
parameter is usually derived from measurements of the individuals in the population.
A statistic is a value, usually a numerical value, that describes a sample. A statistic
is usually derived from measurements of the individuals in the sample.
Every population parameter has a corresponding sample statistic, and most research
studies involve using statistics from samples as the basis for answering questions about
population parameters. As a result, much of this book is concerned with the relationship
between sample statistics and the corresponding population parameters. In Chapter 7, for
example, we examine the relationship between the mean obtained for a sample and the
mean for the population from which the sample was obtained.
■■Descriptive and Inferential Statistical Methods
Although researchers have developed a variety of different statistical procedures to organize and interpret data, these different procedures can be classified into two general categories. The first category, descriptive statistics, consists of statistical procedures that are used
to simplify and summarize data.
DEFInItIon
Descriptive statistics are statistical procedures used to summarize, organize, and
simplify data.
6
chaPtER 1 | Introduction to Statistics
Descriptive statistics are techniques that take raw scores and organize or summarize
them in a form that is more manageable. Often the scores are organized in a table or a graph
so that it is possible to see the entire set of scores. Another common technique is to summarize a set of scores by computing an average. Note that even if the data set has hundreds
of scores, the average provides a single descriptive value for the entire set.
The second general category of statistical techniques is called inferential statistics.
Inferential statistics are methods that use sample data to make general statements about a
population.
DEFInItIon
Inferential statistics consist of techniques that allow us to study samples and then
make generalizations about the populations from which they were selected.
Because populations are typically very large, it usually is not possible to measure
everyone in the population. Therefore, a sample is selected to represent the population.
By analyzing the results from the sample, we hope to make general statements about the
population. Typically, researchers use sample statistics as the basis for drawing conclusions
about population parameters. One problem with using samples, however, is that a sample
provides only limited information about the population. Although samples are generally
representative of their populations, a sample is not expected to give a perfectly accurate
picture of the whole population. There usually is some discrepancy between a sample statistic and the corresponding population parameter. This discrepancy is called sampling
error, and it creates the fundamental problem inferential statistics must always address.
DEFInItIon
Sampling error is the naturally occurring discrepancy, or error, that exists between
a sample statistic and the corresponding population parameter.
The concept of sampling error is illustrated in Figure 1.2. The figure shows a population of 1,000 college students and 2 samples, each with 5 students who were selected from
the population. Notice that each sample contains different individuals who have different
characteristics. Because the characteristics of each sample depend on the specific people in
the sample, statistics will vary from one sample to another. For example, the five students
in sample 1 have an average age of 19.8 years and the students in sample 2 have an average
age of 20.4 years.
It is also very unlikely that the statistics obtained for a sample will be identical to the
parameters for the entire population. In Figure 1.2, for example, neither sample has statistics that are exactly the same as the population parameters. You should also realize that
Figure 1.2 shows only two of the hundreds of possible samples. Each sample would contain
different individuals and would produce different statistics. This is the basic concept of
sampling error: sample statistics vary from one sample to another and typically are different from the corresponding population parameters.
One common example of sampling error is the error associated with a sample proportion. For example, in newspaper articles reporting results from political polls, you frequently find statements such as this:
Candidate Brown leads the poll with 51% of the vote. Candidate Jones has 42%
approval, and the remaining 7% are undecided. This poll was taken from a sample of registered voters and has a margin of error of plus-or-minus 4 percentage points.
The “margin of error” is the sampling error. In this case, the percentages that are reported
were obtained from a sample and are being generalized to the whole population. As always,
you do not expect the statistics from a sample to be perfect. There always will be some
“margin of error” when sample statistics are used to represent population parameters.
SEctIon 1.1 | Statistics, Science, and Observations
7
F I G U R E 1. 2
A demonstration of sampling error. Two
samples are selected from the same population.
Notice that the sample statistics are different
from one sample to another and all the sample
statistics are different from the corresponding
population parameters. The natural differences that exist, by chance, between a sample
statistic and population parameter are called
sampling error.
Population
of 1000 college students
Population Parameters
Average Age 5 21.3 years
Average IQ 5 112.5
65% Female, 35% Male
Sample #1
Sample #2
Eric
Jessica
Laura
Karen
Brian
Tom
Kristen
Sara
Andrew
John
Sample Statistics
Average Age 5 19.8
Average IQ 5 104.6
60% Female, 40% Male
Sample Statistics
Average Age 5 20.4
Average IQ 5 114.2
40% Female, 60% Male
As a further demonstration of sampling error, imagine that your statistics class is separated into two groups by drawing a line from front to back through the middle of the room.
Now imagine that you compute the average age (or height, or IQ) for each group. Will the
two groups have exactly the same average? Almost certainly they will not. No matter what
you chose to measure, you will probably find some difference between the two groups.
However, the difference you obtain does not necessarily mean that there is a systematic
difference between the two groups. For example, if the average age for students on the
right-hand side of the room is higher than the average for students on the left, it is unlikely
that some mysterious force has caused the older people to gravitate to the right side of
the room. Instead, the difference is probably the result of random factors such as chance.
The unpredictable, unsystematic differences that exist from one sample to another are an
example of sampling error.
■■Statistics in the Context of Research
The following example shows the general stages of a research study and demonstrates
how descriptive statistics and inferential statistics are used to organize and interpret the
data. At the end of the example, note how sampling error can affect the interpretation of
experimental results, and consider why inferential statistical methods are needed to deal
with this problem.
8
chaPtER 1 | Introduction to Statistics
ExamplE 1.1
Figure 1.3 shows an overview of a general research situation and demonstrates the roles that
descriptive and inferential statistics play. The purpose of the research study is to address a
question that we posed earlier: Do college students learn better by studying text on printed
pages or on a computer screen? Two samples are selected from the population of college
students. The students in sample A are given printed pages of text to study for 30 minutes
and the students in sample B study the same text on a computer screen. Next, all of the
students are given a multiple-choice test to evaluate their knowledge of the material. At this
point, the researcher has two sets of data: the scores for sample A and the scores for sample
B (see the figure). Now is the time to begin using statistics.
First, descriptive statistics are used to simplify the pages of data. For example, the
researcher could draw a graph showing the scores for each sample or compute the average score for each sample. Note that descriptive methods provide a simplified, organized
Step 1
Experiment:
Compare two
studying methods
Data
Test scores for the
students in each
sample
Step 2
Descriptive statistics:
Organize and simplify
Population of
College
Students
Sample A
Read from printed
pages
25
27
30
19
29
26
21
28
23
26
20
25
28
27
24
26
22
30
Average
Score = 26
Step 3
Inferential statistics:
Interpret results
F i g u r E 1. 3
The role of statistics in experimental
research.
Sample B
Read from computer
screen
20
20
23
25
22
18
22
17
28
19
24
25
30
27
23
21
22
19
Average
Score = 22
The sample data show a 4-point difference
between the two methods of studying. However,
there are two ways to interpret the results.
1. There actually is no difference between
the two studying methods, and the sample
difference is due to chance (sampling error).
2. There really is a difference between
the two methods, and the sample data
accurately reflect this difference.
The goal of inferential statistics is to help researchers
decide between the two interpretations.
SEctIon 1.1 | Statistics, Science, and Observations
9
description of the scores. In this example, the students who studied printed pages had an average score of 26 on the test, and the students who studied text on the computer averaged 22.
Once the researcher has described the results, the next step is to interpret the outcome.
This is the role of inferential statistics. In this example, the researcher has found a difference
of 4 points between the two samples (sample A averaged 26 and sample B averaged 22). The
problem for inferential statistics is to differentiate between the following two interpretations:
1. There is no real difference between the printed page and a computer screen, and
the 4-point difference between the samples is just an example of sampling error
(like the samples in Figure 1.2).
2. There really is a difference between the printed page and a computer screen, and
the 4-point difference between the samples was caused by the different methods
of studying.
In simple English, does the 4-point difference between samples provide convincing
evidence of a difference between the two studying methods, or is the 4-point difference just
chance? The purpose of inferential statistics is to answer this question.
■
lE arn in g Ch ECk
1. A researcher is interested in the sleeping habits of American college students.
A group of 50 students is interviewed and the researcher finds that these students
sleep an average of 6.7 hours per day. For this study, the average of 6.7 hours is an
example of a(n)
.
a. parameter
b. statistic
c. population
d. sample
2. A researcher is curious about the average IQ of registered voters in the state of Florida.
The entire group of registered voters in the state is an example of a
.
a. sample
b. statistic
c. population
d. parameter
3. Statistical techniques that summarize, organize, and simplify data are classified
as
.
a. population statistics
b. sample statistics
c. descriptive statistics
d. inferential statistics
4. In general,
statistical techniques are used to summarize the data from
a research study and
statistical techniques are used to determine what
conclusions are justified by the results.
a. inferential, descriptive
b. descriptive, inferential
c. sample, population
d. population, sample
10
chaPtER 1 | Introduction to Statistics
5. IQ tests are standardized so that the average score is 100 for the entire group of
people who take the test each year. However, if you selected a group of 20 people
who took the test and computed their average IQ score you probably would not get
100. What statistical concept explains the difference between your mean and the
mean for the entire group?
a. statistical error
b. inferential error
c. descriptive error
d. sampling error
an s wE r s
1. B, 2. C, 3. C, 4. B, 5. D
1.2 Data Structures, Research Methods, and Statistics
LEARNING OBJECTIVEs
4. Differentiate correlational, experimental, and nonexperimental research and describe
the data structures associated with each.
5. Define independent, dependent, and quasi-independent variables and recognize
examples of each.
■■Individual Variables: Descriptive Research
Some research studies are conducted simply to describe individual variables as they exist
naturally. For example, a college official may conduct a survey to describe the eating, sleeping, and study habits of a group of college students. When the results consist of numerical
scores, such as the number of hours spent studying each day, they are typically described
by the statistical techniques that are presented in Chapters 3 and 4. Non-numerical scores
are typically described by computing the proportion or percentage in each category. For
example, a recent newspaper article reported that 34.9% of Americans are obese, which is
roughly 35 pounds over a healthy weight.
■■Relationships Between Variables
Most research, however, is intended to examine relationships between two or more variables. For example, is there a relationship between the amount of violence in the video
games played by children and the amount of aggressive behavior they display? Is there a
relationship between the quality of breakfast and academic performance for elementary
school children? Is there a relationship between the number of hours of sleep and grade
point average for college students? To establish the existence of a relationship, researchers must make observations—that is, measurements of the two variables. The resulting
measurements can be classified into two distinct data structures that also help to classify
different research methods and different statistical techniques. In the following section we
identify and discuss these two data structures.
I. One Group with Two Variables Measured for Each Individual: The Correlational Method One method for examining the relationship between variables is to
observe the two variables as they exist naturally for a set of individuals. That is, simply
SEctIon 1.2 | Data Structures, Research Methods, and Statistics
11
measure the two variables for each individual. For example, research has demonstrated a
relationship between sleep habits, especially wake-up time, and academic performance
for college students (Trockel, Barnes, and Egget, 2000). The researchers used a survey to
measure wake-up time and school records to measure academic performance for each student. Figure 1.4 shows an example of the kind of data obtained in the study. The researchers then look for consistent patterns in the data to provide evidence for a relationship
between variables. For example, as wake-up time changes from one student to another, is
there also a tendency for academic performance to change?
Consistent patterns in the data are often easier to see if the scores are presented in a
graph. Figure 1.4 also shows the scores for the eight students in a graph called a scatter
plot. In the scatter plot, each individual is represented by a point so that the horizontal
position corresponds to the student’s wake-up time and the vertical position corresponds
to the student’s academic performance score. The scatter plot shows a clear relationship
between wake-up time and academic performance: as wake-up time increases, academic
performance decreases.
A research study that simply measures two different variables for each individual and
produces the kind of data shown in Figure 1.4 is an example of the correlational method,
or the correlational research strategy.
In the correlational method, two different variables are observed to determine
whether there is a relationship between them.
DEFInItIon
■■Statistics for the Correlational Method
When the data from a correlational study consist of numerical scores, the relationship
between the two variables is usually measured and described using a statistic called a
correlation. Correlations and the correlational method are discussed in detail in Chapters 15 and 16. Occasionally, the measurement process used for a correlational study
simply classifies individuals into categories that do not correspond to numerical values.
For example, a researcher could classify a group of college students by gender (male
Student
Wake-up
Time
Academic
Performance
A
B
C
D
E
F
G
H
11
9
9
12
7
10
10
8
2.4
3.6
3.2
2.2
3.8
2.2
3.0
3.0
(b)
Academic performance
(a)
3.8
3.6
3.4
3.2
3.0
2.8
2.6
2.4
2.2
2.0
7
F i g u r E 1. 4
8
9
10
11
12
Wake-up time
One of two data structures for evaluating the relationship between variables. Note that there are two separate measurements for each individual (wake-up time and academic performance). The same scores are shown in a table (a) and in
a graph (b).
12
chaPtER 1 | Introduction to Statistics
or female) and by cell-phone preference (talk or text). Note that the researcher has two
scores for each individual but neither of the scores is a numerical value. This type of data
is typically summarized in a table showing how many individuals are classified into each
of the possible categories. Table 1.1 shows an example of this kind of summary table. The
table shows for example, that 30 of the males in the sample preferred texting to talking.
This type of data can be coded with numbers (for example, male = 0 and female = 1)
so that it is possible to compute a correlation. However, the relationship between variables for non-numerical data, such as the data in Table 1.1, is usually evaluated using
a statistical technique known as a chi-square test. Chi-square tests are presented in
Chapter 17.
Ta b lE 1.1
Correlational data consisting of non-numerical scores. Note that there are two measurements for
each individual: gender and cell phone preference. The numbers indicate how many people are in
each category. For example, out of the 50 males, 30 prefer text over talk.
Cell Phone Preference
Text
Talk
Males
30
20
50
Females
25
25
50
■■Limitations of the Correlational Method
The results from a correlational study can demonstrate the existence of a relationship
between two variables, but they do not provide an explanation for the relationship. In
particular, a correlational study cannot demonstrate a cause-and-effect relationship. For
example, the data in Figure 1.4 show a systematic relationship between wake-up time and
academic performance for a group of college students; those who sleep late tend to have
lower performance scores than those who wake early. However, there are many possible
explanations for the relationship and we do not know exactly what factor (or factors) is
responsible for late sleepers having lower grades. In particular, we cannot conclude that
waking students up earlier would cause their academic performance to improve, or that
studying more would cause students to wake up earlier. To demonstrate a cause-and-effect
relationship between two variables, researchers must use the experimental method, which
is discussed next.
II. Comparing Two (or More) Groups of Scores: Experimental and Nonexperimental Methods The second method for examining the relationship between two
variables involves the comparison of two or more groups of scores. In this situation, the
relationship between variables is examined by using one of the variables to define the
groups, and then measuring the second variable to obtain scores for each group. For example, Polman, de Castro, and van Aken (2008) randomly divided a sample of 10-year-old
boys into two groups. One group then played a violent video game and the second played
a nonviolent game. After the game-playing session, the children went to a free play period
and were monitored for aggressive behaviors (hitting, kicking, pushing, frightening, name
calling, fighting, quarreling, or teasing another child). An example of the resulting data is
shown in Figure 1.5. The researchers then compare the scores for the violent-video group
with the scores for the nonviolent-video group. A systematic difference between the two
groups provides evidence for a relationship between playing violent video games and
aggressive behavior for 10-year-old boys.
SEctIon 1.2 | Data Structures, Research Methods, and Statistics
F i g u r E 1. 5
Evaluating the relationship between
variables by comparing groups of scores.
Note that the values of
one variable are used
to define the groups
and the second variable is measured to
obtain scores within
each group.
One variable (type of video game)
is used to define groups
A second variable (aggressive behavior)
is measured to obtain scores within each group
Violent
Nonviolent
7
8
10
7
9
8
6
10
9
6
8
4
8
3
6
5
3
4
4
5
13
Compare groups
of scores
■■Statistics for Comparing Two (or More) Groups of Scores
Most of the statistical procedures presented in this book are designed for research studies that compare groups of scores like the study in Figure 1.5. Specifically, we examine
descriptive statistics that summarize and describe the scores in each group and we use
inferential statistics to determine whether the differences between the groups can be generalized to the entire population.
When the measurement procedure produces numerical scores, the statistical evaluation typically involves computing the average score for each group and then comparing
the averages. The process of computing averages is presented in Chapter 3, and a variety
of statistical techniques for comparing averages are presented in Chapters 8–14. If the
measurement process simply classifies individuals into non-numerical categories, the statistical evaluation usually consists of computing proportions for each group and then comparing proportions. Previously, in Table 1.1, we presented an example of non-numerical
data examining the relationship between gender and cell-phone preference. The same data
can be used to compare the proportions for males with the proportions for females. For
example, using text is preferred by 60% of the males compared to 50% of the females. As
before, these data are evaluated using a chi-square test, which is presented in Chapter 17.
■■Experimental and Nonexperimental Methods
There are two distinct research methods that both produce groups of scores to be compared:
the experimental and the nonexperimental strategies. These two research methods use
exactly the same statistics and they both demonstrate a relationship between two variables.
The distinction between the two research strategies is how the relationship is interpreted.
The results from an experiment allow a cause-and-effect explanation. For example, we can
conclude that changes in one variable are responsible for causing differences in a second
variable. A nonexperimental study does not permit a cause-and effect explanation. We can
say that changes in one variable are accompanied by changes in a second variable, but we
cannot say why. Each of the two research methods is discussed in the following sections.
■■The Experimental Method
One specific research method that involves comparing groups of scores is known as the
experimental method or the experimental research strategy. The goal of an experimental
study is to demonstrate a cause-and-effect relationship between two variables. Specifically,
14
chaPtER 1 | Introduction to Statistics
an experiment attempts to show that changing the value of one variable causes changes to
occur in the second variable. To accomplish this goal, the experimental method has two
characteristics that differentiate experiments from other types of research studies:
1. Manipulation The researcher manipulates one variable by changing its value from
one level to another. In the Polman et al. (2008) experiment examining the effect
of violence in video games (Figure 1.5), the researchers manipulate the amount of
violence by giving one group of boys a violent game to play and giving the other
group a nonviolent game. A second variable is observed (measured) to determine
whether the manipulation causes changes to occur.
2. Control The researcher must exercise control over the research situation to ensure
that other, extraneous variables do not influence the relationship being examined.
In more complex experiments, a researcher
may systematically
manipulate more than
one variable and may
observe more than one
variable. Here we are
considering the simplest
case, in which only one
variable is manipulated
and only one variable is
observed.
To demonstrate these two characteristics, consider the Polman et al. (2008) study examining the effect of violence in video games (see Figure 1.5). To be able to say that the difference in aggressive behavior is caused by the amount of violence in the game, the researcher
must rule out any other possible explanation for the difference. That is, any other variables
that might affect aggressive behavior must be controlled. There are two general categories
of variables that researchers must consider:
1. Participant Variables These are characteristics such as age, gender, and intelligence that vary from one individual to another. Whenever an experiment compares
different groups of participants (one group in treatment A and a different group
in treatment B), researchers must ensure that participant variables do not differ
from one group to another. For the experiment shown in Figure 1.5, for example,
the researchers would like to conclude that the violence in the video game causes
a change in the participants’ aggressive behavior. In the study, the participants in
both conditions were 10-year-old boys. Suppose, however, that the participants in
the nonviolent condition were primarily female and those in the violent condition
were primarily male. In this case, there is an alternative explanation for the difference in aggression that exists between the two groups. Specifically, the difference
between groups may have been caused by the amount of violence in the game,
but it also is possible that the difference was caused by the participants’ gender
(females are less aggressive than males). Whenever a research study allows more
than one explanation for the results, the study is said to be confounded because it is
impossible to reach an unambiguous conclusion.
2. Environmental Variables These are characteristics of the environment such as
lighting, time of day, and weather conditions. A researcher must ensure that the
individuals in treatment A are tested in the same environment as the individuals
in treatment B. Using the video game violence experiment (see Figure 1.5) as an
example, suppose that the individuals in the nonviolent condition were all tested in
the morning and the individuals in the violent condition were all tested in the evening. Again, this would produce a confounded experiment because the researcher
could not determine whether the differences in aggressive behavior were caused by
the amount of violence or caused by the time of day.
Researchers typically use three basic techniques to control other variables. First, the
researcher could use random assignment, which means that each participant has an equal
chance of being assigned to each of the treatment conditions. The goal of random assignment is to distribute the participant characteristics evenly between the two groups so that
neither group is noticeably smarter (or older, or faster) than the other. Random assignment
can also be used to control environmental variables. For example, participants could be
assigned randomly for testing either in the morning or in the afternoon. A second technique
SEctIon 1.2 | Data Structures, Research Methods, and Statistics
15
for controlling variables is to use matching to ensure equivalent groups or equivalent environments. For example, the researcher could match groups by ensuring that every group
has exactly 60% females and 40% males. Finally, the researcher can control variables by
holding them constant. For example, in the video game violence study discussed earlier
(Polman et al., 2008), the researchers used only 10-year-old boys as participants (holding
age and gender constant). In this case the researchers can be certain that one group is not
noticeably older or has a larger proportion of females than the other.
DEFInItIon
In the experimental method, one variable is manipulated while another variable
is observed and measured. To establish a cause-and-effect relationship between the
two variables, an experiment attempts to control all other variables to prevent them
from influencing the results.
■■Terminology in the Experimental Method
Specific names are used for the two variables that are studied by the experimental method. The
variable that is manipulated by the experimenter is called the independent variable. It can be
identified as the treatment conditions to which participants are assigned. For the example in
Figure 1.5, the amount of violence in the video game is the independent variable. The variable
that is observed and measured to obtain scores within each condition is the dependent variable. For the example in Figure 1.5, the level of aggressive behavior is the dependent variable.
DEFInItIon
The independent variable is the variable that is manipulated by the researcher. In
behavioral research, the independent variable usually consists of the two (or more) treatment conditions to which subjects are exposed. The independent variable consists of the
antecedent conditions that were manipulated prior to observing the dependent variable.
The dependent variable is the one that is observed to assess the effect of the treatment.
Control Conditions in an Experiment
An experimental study evaluates the relationship between two variables by manipulating one variable (the independent variable) and
measuring one variable (the dependent variable). Note that in an experiment only one
variable is actually measured. You should realize that this is different from a correlational
study, in which both variables are measured and the data consist of two separate scores
for each individual.
Often an experiment will include a condition in which the participants do not receive
any treatment. The scores from these individuals are then compared with scores from participants who do receive the treatment. The goal of this type of study is to demonstrate that
the treatment has an effect by showing that the scores in the treatment condition are substantially different from the scores in the no-treatment condition. In this kind of research,
the no-treatment condition is called the control condition, and the treatment condition is
called the experimental condition.
DEFInItIon
Individuals in a control condition do not receive the experimental treatment.
Instead, they either receive no treatment or they receive a neutral, placebo treatment. The purpose of a control condition is to provide a baseline for comparison
with the experimental condition.
Individuals in the experimental condition do receive the experimental treatment.
16
chaPtER 1 | Introduction to Statistics
Note that the independent variable always consists of at least two values. (Something
must have at least two different values before you can say that it is “variable.”) For the
video game violence experiment (see Figure 1.5), the independent variable is the amount
of violence in the video game. For an experiment with an experimental group and a control
group, the independent variable is treatment versus no treatment.
■■Nonexperimental Methods: Nonequivalent Groups
and Pre-Post Studies
In informal conversation, there is a tendency for people to use the term experiment to refer
to any kind of research study. You should realize, however, that the term only applies to
studies that satisfy the specific requirements outlined earlier. In particular, a real experiment must include manipulation of an independent variable and rigorous control of other,
extraneous variables. As a result, there are a number of other research designs that are not
true experiments but still examine the relationship between variables by comparing groups
of scores. Two examples are shown in Figure 1.6 and are discussed in the following paragraphs. This type of research study is classified as nonexperimental.
The top part of Figure 1.6 shows an example of a nonequivalent groups study comparing boys and girls. Notice that this study involves comparing two groups of scores (like an
experiment). However, the researcher has no ability to control which participants go into
F i g u r E 1.6
(a)
Two examples of nonexperimental
studies that involve comparing two
groups of scores. In (a) the study
uses two preexisting groups (boys/
girls) and measures a dependent
variable (verbal scores) in each
group. In (b) the study uses time
(before/after) to define the two
groups and measures a dependent
variable (depression) in each group.
Variable #1: Subject gender
(the quasi-independent variable)
Not manipulated, but used
to create two groups of subjects
Variable #2: Verbal test scores
(the dependent variable)
Measured in each of the
two groups
Boys
Girls
17
19
16
12
17
18
15
16
12
10
14
15
13
12
11
13
Any
difference?
(b)
Variable #1: Time
(the quasi-independent variable)
Not manipulated, but used
to create two groups of scores
Variable #2: Depression scores
(the dependent variable)
Measured at each of the two
different times
Before
Therapy
After
Therapy
17
19
16
12
17
18
15
16
12
10
14
15
13
12
11
13
Any
difference?
SEctIon 1.2 | Data Structures, Research Methods, and Statistics
Correlational studies are
also examples of nonexperimental research. In
this section, however, we
are discussing nonexperimental studies that
compare two or more
groups of scores.
17
which group—all the males must be in the boy group and all the females must be in the
girl group. Because this type of research compares preexisting groups, the researcher cannot control the assignment of participants to groups and cannot ensure equivalent groups.
Other examples of nonequivalent group studies include comparing 8-year-old children and
10-year-old children, people with an eating disorder and those with no disorder, and comparing children from a single-parent home and those from a two-parent home. Because it
is impossible to use techniques like random assignment to control participant variables and
ensure equivalent groups, this type of research is not a true experiment.
The bottom part of Figure 1.6 shows an example of a pre–post study comparing depression scores before therapy and after therapy. The two groups of scores are obtained by
measuring the same variable (depression) twice for each participant; once before therapy
and again after therapy. In a pre-post study, however, the researcher has no control over
the passage of time. The “before” scores are always measured earlier than the “after”
scores. Although a difference between the two groups of scores may be caused by the
treatment, it is always possible that the scores simply change as time goes by. For example, the depression scores may decrease over time in the same way that the symptoms of
a cold disappear over time. In a pre–post study the researcher also has no control over
other variables that change with time. For example, the weather could change from dark
and gloomy before therapy to bright and sunny after therapy. In this case, the depression
scores could improve because of the weather and not because of the therapy. Because the
researcher cannot control the passage of time or other variables related to time, this study
is not a true experiment.
Terminology in Nonexperimental Research
Although the two research studies
shown in Figure 1.6 are not true experiments, you should notice that they produce the
same kind of data that are found in an experiment (see Figure 1.5). In each case, one variable is used to create groups, and a second variable is measured to obtain scores within
each group. In an experiment, the groups are created by manipulation of the independent
variable, and the participants’ scores are the dependent variable. The same terminology is
often used to identify the two variables in nonexperimental studies. That is, the variable
that is used to create groups is the independent variable and the scores are the dependent
variable. For example, the top part of Figure 1.6, gender (boy/girl), is the independent
variable and the verbal test scores are the dependent variable. However, you should realize that gender (boy/girl) is not a true independent variable because it is not manipulated.
For this reason, the “independent variable” in a nonexperimental study is often called a
quasi-independent variable.
DEFInItIon
lE arn in g Ch ECk
In a nonexperimental study, the “independent variable” that is used to create the
different groups of scores is often called the quasi-independent variable.
1. In a correlational study, how many variables are measured for each individual and
how many groups of scores are obtained?
a. 1 variable and 1 group
b. 1 variable and 2 groups
c. 2 variables and 1 group
d. 2 variables and 2 groups
18
chaPtER 1 | Introduction to Statistics
2. A research study comparing alcohol use for college students in the United States
and Canada reports that more Canadian students drink but American students drink
more (Kuo, Adlaf, Lee, Gliksman, Demers, and Wechsler, 2002). What research
design did this study use?
a. correlational
b. experimental
c. nonexperimental
d. noncorrelational
3. Stephens, Atkins, and Kingston (2009) found that participants were able to tolerate
more pain when they shouted their favorite swear words over and over than when
they shouted neutral words. For this study, what is the independent variable?
a. the amount of pain tolerated
b. the participants who shouted swear words
c. the participants who shouted neutral words
d. the kind of word shouted by the participants
an s wE r s
1.3
1. C, 2. C, 3. D
Variables and Measurement
LEARNING OBJECTIVEs
6. Explain why operational definitions are developed for constructs and identify the two
components of an operational definition.
7. Describe discrete and continuous variables and identify examples of each.
8. Differentiate nominal, ordinal, interval, and ratio scales of measurement.
■■Constructs and Operational Definitions
The scores that make up the data from a research study are the result of observing and
measuring variables. For example, a researcher may finish a study with a set of IQ scores,
personality scores, or reaction-time scores. In this section, we take a closer look at the variables that are being measured and the process of measurement.
Some variables, such as height, weight, and eye color are well-defined, concrete entities that can be observed and measured directly. On the other hand, many variables studied
by behavioral scientists are internal characteristics that people use to help describe and
explain behavior. For example, we say that a student does well in school because he or
she is intelligent. Or we say that someone is anxious in social situations, or that someone
seems to be hungry. Variables like intelligence, anxiety, and hunger are called constructs,
and because they are intangible and cannot be directly observed, they are often called
hypothetical constructs.
Although constructs such as intelligence are internal characteristics that cannot be
directly observed, it is possible to observe and measure behaviors that are representative
of the construct. For example, we cannot “see” intelligence but we can see examples of
intelligent behavior. The external behaviors can then be used to create an operational definition for the construct. An operational definition defines a construct in terms of external
SEctIon 1.3 | Variables and Measurement
19
behaviors that can be observed and measured. For example, your intelligence is measured
and defined by your performance on an IQ test, or hunger can be measured and defined by
the number of hours since last eating.
DEFInItIon
Constructs are internal attributes or characteristics that cannot be directly
observed but are useful for describing and explaining behavior.
An operational definition identifies a measurement procedure (a set of operations) for measuring an external behavior and uses the resulting measurements as
a definition and a measurement of a hypothetical construct. Note that an operational definition has two components. First, it describes a set of operations for
measuring a construct. Second, it defines the construct in terms of the resulting
measurements.
■■Discrete and Continuous Variables
The variables in a study can be characterized by the type of values that can be assigned to
them. A discrete variable consists of separate, indivisible categories. For this type of variable, there are no intermediate values between two adjacent categories. Consider the values
displayed when dice are rolled. Between neighboring values—for example, seven dots and
eight dots—no other values can ever be observed.
DEFInItIon
A discrete variable consists of separate, indivisible categories. No values can exist
between two neighboring categories.
Discrete variables are commonly restricted to whole, countable numbers—for
example, the number of children in a family or the number of students attending class.
If you observe class attendance from day to day, you may count 18 students one day
and 19 students the next day. However, it is impossible ever to observe a value between
18 and 19. A discrete variable may also consist of observations that differ qualitatively.
For example, people can be classified by gender (male or female), by occupation
(nurse, teacher, lawyer, etc.), and college students can by classified by academic major
(art, biology, chemistry, etc.). In each case, the variable is discrete because it consists
of separate, indivisible categories.
On the other hand, many variables are not discrete. Variables such as time, height, and
weight are not limited to a fixed set of separate, indivisible categories. You can measure
time, for example, in hours, minutes, seconds, or fractions of seconds. These variables
are called continuous because they can be divided into an infinite number of fractional
parts.
DEFInItIon
For a continuous variable, there are an infinite number of possible values that fall
between any two observed values. A continuous variable is divisible into an infinite
number of fractional parts.
Suppose, for example, that a researcher is measuring weights for a group of individuals
participating in a diet study. Because weight is a continuous variable, it can be pictured as
a continuous line (Figure 1.7). Note that there are an infinite number of possible points on
20
chaPtER 1 | Introduction to Statistics
F i g u r E 1.7
149.6
When measuring weight to
the nearest whole pound,
149.6 and 150.3 are assigned
the value of 150 (top). Any
value in the interval between
149.5 and 150.5 is given the
value of 150.
150.3
149
151
152
151
152
150
149.5
149
148.5
150.5
150
149.5
150.5
151.5
152.5
Real limits
the line without any gaps or separations between neighboring points. For any two different
points on the line, it is always possible to find a third value that is between the two points.
Two other factors apply to continuous variables:
1. When measuring a continuous variable, it should be very rare to obtain identical
measurements for two different individuals. Because a continuous variable has an
infinite number of possible values, it should be almost impossible for two people to
have exactly the same score. If the data show a substantial number of tied scores,
then you should suspect that the measurement procedure is very crude or that the
variable is not really continuous.
2. When measuring a continuous variable, each measurement category is actually an
interval that must be defined by boundaries. For example, two people who both
claim to weigh 150 pounds are probably not exactly the same weight. However,
they are both around 150 pounds. One person may actually weigh 149.6 and the
other 150.3. Thus, a score of 150 is not a specific point on the scale but instead is
an interval (see Figure 1.7). To differentiate a score of 150 from a score of 149 or
151, we must set up boundaries on the scale of measurement. These boundaries are
called real limits and are positioned exactly halfway between adjacent scores. Thus,
a score of X = 150 pounds is actually an interval bounded by a lower real limit
of 149.5 at the bottom and an upper real limit of 150.5 at the top. Any individual
whose weight falls between these real limits will be assigned a score of X = 150.
DEFInItIon
Real limits are the boundaries of intervals for scores that are represented on a continuous number line. The real limit separating two adjacent scores is located exactly
halfway between the scores. Each score has two real limits. The upper real limit is
at the top of the interval, and the lower real limit is at the bottom.
The concept of real limits applies to any measurement of a continuous variable, even
when the score categories are not whole numbers. For example, if you were measuring time
to the nearest tenth of a second, the measurement categories would be 31.0, 31.1, 31.2, and
so on. Each of these categories represents an interval on the scale that is bounded by real
limits. For example, a score of X = 31.1 seconds indicates that the actual measurement
is in an interval bounded by a lower real limit of 31.05 and an upper real limit of 31.15.
Remember that the real limits are always halfway between adjacent categories.
SEctIon 1.3 | Variables and Measurement
Students often ask
whether a value of
exactly 150.5 should
be assigned to the
X = 150 interval or the
X = 151 interval. The
answer is that 150.5 is
the boundary between
the two intervals and is
not necessarily in one
or the other. Instead,
the placement of 150.5
depends on the rule that
you are using for rounding numbers. If you
are rounding up, then
150.5 goes in the higher
interval (X = 151) but if
you are rounding down,
then it goes in the lower
interval (X = 150).
21
Later in this book, real limits are used for constructing graphs and for various calculations with continuous scales. For now, however, you should realize that real limits are a
necessity whenever you make measurements of a continuous variable.
Finally, we should warn you that the terms continuous and discrete apply to the variables that are being measured and not to the scores that are obtained from the measurement.
For example, measuring people’s heights to the nearest inch produces scores of 60, 61, 62,
and so on. Although the scores may appear to be discrete numbers, the underlying variable
is continuous. One key to determining whether a variable is continuous or discrete is that
a continuous variable can be divided into any number of fractional parts. Height can be
measured to the nearest inch, the nearest 0.5 inch, or the nearest 0.1 inch. Similarly, a professor evaluating students’ knowledge could use a pass/fail system that classifies students
into two broad categories. However, the professor could choose to use a 10-point quiz that
divides student knowledge into 11 categories corre...

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