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ExploringStatistics
Tales of Distributions
12th Edition
Chris Spatz
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Exploring Statistics: Tales of Distributions
12th Edition
Chris Spatz
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v
Chris Spatz is at Hendrix College where he twice served as chair of
the Psychology Department. Dr. Spatz’s undergraduate education was
at Hendrix and his PhD in experimental psychology is from Tulane
University in New Orleans. He subsequently completed postdoctoral
fellowships in animal behavior at the University of California, Berkeley,
and the University of Michigan. Before returning to Hendrix to teach,
Spatz held positions at The University of the South and the University
of Arkansas at Monticello.
Spatz served as a reviewer for the journal Teaching of Psychology
for more than 20 years. He co-authored a research methods textbook,
wrote several chapters for edited books, and was a section editor for the
Encyclopedia of Statistics in Behavioral Science.
In addition to writing and publishing, Dr. Spatz enjoys the outdoors,
especially canoeing, camping, and gardening. He swims several times
a week (mode = 3). Spatz has been an opponent of high textbook prices for years, and he is
happy to be part of a new wave of authors who provide high-quality textbooks to students at
affordable prices.
About The Author
vi Dedication
With love and affection,
this textbook is dedicated to
Thea Siria Spatz, Ed.D., CHES
Introduction
CHAPTER
1
O B J E C T I V E S F O R C H A P T E R 1
After studying the text and working the problems in this chapter, you should be
able to:
1. Distinguish between descriptive and inferential statistics
2. Define population, sample, parameter, statistic, and variable as they are
used in statistics
3. Distinguish between quantitative and categorical variables
4. Distinguish between continuous and discrete variables
5. Identify the lower and upper limits of a continuous variable
6. Identify four scales of measurement and distinguish among them
7. Distinguish between statistics and experimental design
8. Define independent variable, dependent variable, and extraneous variable
and identify them in experiments
9. Describe statistics’ place in epistemology
10. List actions to take to analyze a data set
11. Identify a few events in the history of statistics
WE BEGIN OUR exploration of statistics with a trip to London. The year is 1900.
Walking into an office at University College
London, we meet a tall, well-dressed man about
40 years old. He is Karl Pearson, Professor of
Applied Mathematics and Mechanics. I ask him
to tell us a little about himself and why he is an
important person. He seems authoritative, glad
to talk about himself. As a young man, he says,
he wrote essays, a play, and a novel, and he also
worked for women’s suffrage. These days, he is
excited about this new branch of biology called
genetics. He says he supervises lots of data
gathering.
1
Karl Pearson
2 Chapter 1
Pearson, warming to our group, lectures us about the major problem in science—there
is no agreement on how to decide among competing theories. Fortunately, he just published
a new statistical method that provides an objective way to decide among competing theories,
regardless of the discipline. The method is called chi square.1 Pearson says, “Now, arguments
will be much fewer. Gather a thousand data points and calculate a chi square test. The result
gives everyone an objective way to determine whether or not the data fit the theory.”
Exploration Notes from a student: Exploration off to good start. Hit on a nice, easy-to-
remember date to start with, visited a founder of statistics, and had a statistic called chi square
described as a big deal.
Our next stop is Rothamsted Experiment Station just
north of London. Now the year is 1925. There are fields all
around the agricultural research facility, each divided into
many smaller plots. The growth in the fields seems quite
variable.
Arriving at the office, the atmosphere is congenial. The
staff is having tea. There are two topics—a new baby and
a new book. We get introduced to Ronald Fisher, the chief
statistician. Fisher is a small man with thick glasses and red
hair.
He tells us about his new child2 and then motions to
a book on the table. Sneaking a peek, we read the title:
Statistical Methods for Research Workers. Fisher becomes
focused on his book, holding forth in an authoritative way.
He says the book explains how to conduct experiments
and that an experiment is just a comparison of two or more conditions. He tells us we don’t need
a thousand data points. He says that small samples, randomly selected, are the way for science
to progress. “With an experiment and my technique of analysis of variance,” he exclaims, “you
can determine why that field out there”—here he waves toward the window—“is so variable.
We can find out what makes some plots lush and some mimsy.” Analysis of variance,3 he says,
works in any discipline, not just agriculture.
Exploration Notes: Looks like statistics had some controversy in it.4 Also looks like
progress. Statistics is used for experiments, too, and not just for testing theories. And Fisher
says experiments can be used to compare anything. If that’s right, I can use statistics no matter
what I major in.
1 Chi square, which is explained in this book in Chapter 14, has been called one of the 20 most important inventions
in the 20th century (Hacking, 1984).
2 (in what will become a family with eight children).
3 explained in Chapters 11-13
4 The slight sniping I’ve built into this story is just a hint of the strong animosity between Fisher and Pearson.
Ronald A. Fisher
3 Introduction
Next we go to Poland to visit Jerzy Neyman at his
office at the University of Warsaw. It is 1933. As we walk
in, he smiles, seems happy we’ve arrived, and makes us feel
completely welcome.
Motioning to an envelope on his desk, he tells us it holds
a manuscript that he and Egon Pearson5 wrote. “The problem
with Fisher’s analysis of variance test is that it focuses
exclusively on finding a difference between groups. Suppose
the statistical test doesn’t detect a difference. Does that prove
there is no difference? No, of course not. It may be that the test
was just not sensitive enough to detect the difference. Right?”
At his question, a few of us nod in agreement. Seeing
uncertainty, he notes, “Maybe a larger sample is needed to
find the difference, you see? Anyway, what we’ve done is
expand statistics to cover not just finding a difference, but
also what it means when the test doesn’t find a difference.
Our approach is what you people in your time will call null
hypothesis significance testing.”
Exploration Notes: Statistics seems like a work in progress. Changing. Now it is not just
about finding a difference but also about what it means not to find a difference. Also, looks like
null hypothesis significance testing is a phrase that might turn up on tests.
Our next trip is to libraries, say, anytime between 1940 and 2000. For this exploration, the task
is to examine articles in professional journals published in various disciplines. The disciplines
include anthropology, biology, chemistry, defense strategy, education, forestry, geology, health,
immunology, jurisprudence, manufacturing, medicine, neurology, ophthalmology, political
science, psychology, sociology, zoology, and others. I’m sure you get the idea—the whole range
of disciplines that use quantitative measures in their research. What this exploration produces is
the discovery that all of these disciplines rely on a data analysis technique called null hypothesis
significance testing (NHST).6 Many different statistical tests are employed. However, for all the
tests in all the disciplines, the phrase, “p < .05” turns up frequently.
Exploration Notes: It seems that all that earlier controversy has subsided and scientists
in all sorts of disciplines have agreed that NHST is the way to analyze quantitative data. All of
them seem to think that if there is a comparison to be made, applying NHST is a necessary step
to get correct conclusions. All of them use “p < .05,” so I’ll have to be sure to find out exactly
what that means.
5 Egon Pearson was Karl Pearson’s son.
6 Null hypothesis significance testing is first explained in Chapters 9 and 10.
Jerzy Neyman
4 Chapter 1
Our next excursion is a 1962 visit with Jacob Cohen at New York
University in New York City. He is holding his article about studies
published in the Journal of Abnormal and Social Psychology, a leading
psychology journal. He tells us that the NHST technique has problems.
Also, he says we should be calculating an effect size statistic, which
will show whether the differences observed in our experiments are
large or small.
Exploration Notes: The idea of an effect size index makes a lot of sense. Just knowing
there is a difference isn’t enough. How big is the difference? Wonder what “problems with
NHST” is all about.
Back to the library for a final excursion to check out recent events. We come across a 2014
article by Geoff Cumming on the “new statistics.” We find things like, “avoid NHST and use
better techniques” (p. 26) and “we should not trust any p value” (p. 13). This seems like awfully
strong advice. Are researchers taking this advice? Looking through more of today’s research in
journals in several fields, we find that most statistical analyses use NHST and there are many
instances of “p < .05.”
Exploration Notes, Conclusion: These days, it looks like statistics is in transition again.
There’s a lot of controversy out there about how to analyze data from experiments. The NHST
approach is still very common, though, so it’s clear I must learn it. But I want to be prepared
for changes. I hope knowing NHST will be helpful for the future.7
Welcome to statistics at a time when the discipline is once again in transition. A well-
established tradition (null hypothesis significance testing) has been in place for almost a century
but is now under attack. New ways of thinking about data analysis are emerging, and along with
them, a collection of statistics that do not include the traditional NHST approach. As for the
immediate future, though, NHST remains the method most widely used by researchers in many
fields. In addition, much of the thinking required for NHST is required for other approaches.
Our exploration tour is over, so I’ll quit supplying notes; they are your responsibility
now. As your own experience probably shows, making up your own summary notes improves
retention of what you read. In addition, I have a suggestion. Adopt a mindset that thinks growth.
A student with a growth mindset expects to learn new things. When challenges arise, as they
7 Not only helpful, but necessary, I would say.
Jacob Cohen
5 Introduction
Disciplines that Use Quantitative Data
inevitably do, acknowledge them and figure out how to meet the challenge. A growth mindset
treats ability as something to be developed (see Dweck, 2016). If you engage yourself in this
course, you can expect to use what you learn for the rest of your life.
The main title of this book is “Exploring Statistics.” Exploring conveys the idea of
uncovering something that was not apparent before. An attitude of searching, wondering,
checking, and so forth is what I want to encourage. (Those who object to traditional NHST
procedures are driven by this exploration motivation.) As for this book’s subtitle, “Tales of
Distributions,” I’ll have more to say about it as we go along.
Which disciplines use quantitative data? The list is long and more variable than the list I gave
earlier. The examples and problems in this textbook, however, come from psychology, biology,
sociology, education, medicine, politics, business, economics, forestry, and everyday life.
Statistics is a powerful method for getting answers from data, and this makes it popular with
investigators in a wide variety of fields.
Statistics is used in areas that might surprise you. As examples, statistics has been used to
determine the effect of cigarette taxes on smoking among teenagers, the safety of a new surgical
anesthetic, and the memory of young school-age children for pictures (which is as good as that
of college students). Statistics show which diseases have an inheritance factor, how to improve
short-term weather forecasts, and why giving intentional walks in baseball is a poor strategy.
All these examples come from Statistics: A Guide to the Unknown, a book edited by Judith M.
Tanur and others (1989). Written for those “without special knowledge of statistics,” this book
has 29 essays on topics as varied as those above.
In American history, the authorship of 12 of The Federalist papers was disputed for a
number of years. (The Federalist papers were 85 short essays written under the pseudonym
“Publius” and published in New York City newspapers in 1787 and 1788. Written by James
Madison, Alexander Hamilton, and John Jay, the essays were designed to persuade the people
of the state of New York to ratify the Constitution of the United States.) To determine authorship
of the 12 disputed papers, each was graded with a quantitative value analysis in which the
importance of such values as national security, a comfortable life, justice, and equality was
assessed. The value analysis scores were compared with value analysis scores of papers known
to have been written by Madison and Hamilton (Rokeach, Homant, & Penner, 1970). Another
study, by Mosteller and Wallace, analyzed The Federalist papers using the frequency of words
such as by and to (reported in Tanur et al., 1989). Both studies concluded that Madison wrote
all 12 essays.
Here is an example from law. Rodrigo Partida was convicted of burglary in Hidalgo
County, a border county in southern Texas. A grand jury rejected his motion for a new trial.
Partida’s attorney filed suit, claiming that the grand jury selection process discriminated against
Mexican-Americans. In the end (Castaneda v. Partida, 430 U.S. 482 [1976]), Justice Harry
6 Chapter 1
Inferential statistics
Method that uses sample
evidence and probability
to reach conclusions about
unmeasurable populations.
Descriptive statistic
A number that conveys a
particular characteristic of a
set of data.
Mean
Arithmetic average; sum of
scores divided by number of
scores.
Blackmun of the U.S. Supreme Court wrote, regarding the number of Mexican-Americans on
grand juries, “If the difference between the expected and the observed number is greater than
two or three standard deviations, then the hypothesis that the jury drawing was random (is)
suspect.” In Partida’s case, the difference was approximately 12 standard deviations, and the
Supreme Court ruled that Partida’s attorney had presented prima facie evidence. (Prima facie
evidence is so good that one side wins the case unless the other side rebuts the evidence, which
in this case did not happen.) Statistics: A Guide to the Unknown includes two essays on the use
of statistics by lawyers.
Gigerenzer et al. (2007), in their public interest article on health statistics, point out that
lack of statistical literacy among both patients and physicians undermines the information
exchange necessary for informed consent and shared decision making. The result is anxiety,
confusion, and undue enthusiasm for testing and treatment.
Whatever your current interests or thoughts about your future as a statistician, I believe you
will benefit from this course. A successful statistics course teaches you to identify questions a
set of data can answer; determine the statistical procedures that will provide the answers; carry
out the procedures; and then, using plain English and graphs, tell the story the data reveal.
The best way for you to acquire all these skills (especially the part about telling the story)
is to engage statistics. Engaged students are easily recognized; they are prepared for exams, are
not easily distracted while studying, and generally finish assignments on time. Becoming an
engaged student may not be so easy, but many have achieved it. Here are my recommendations.
Read with the goal of understanding. Attend class. Do all the assignments (on time). Write
down questions. Ask for explanations. Expect to understand. (Disclaimer: I’m not suggesting
that you marry statistics, but just engage for this one course.)
Are you uncertain about whether your background skills are adequate for a statistics
course? For most students, this is an unfounded worry. Appendix A, Getting Started, should
help relieve your concerns.
What Do You Mean, “Statistics”?
The Oxford English Dictionary says that the word statistics came into
use almost 250 years ago. At that time, statistics referred to a country’s
quantifiable political characteristics—characteristics such as population,
taxes, and area. Statistics meant “state numbers.” Tables and charts
of those numbers turned out to be a very satisfactory way to compare
different countries and to make projections about the future. Later, tables
and charts proved useful to people studying trade (economics) and natural
phenomena (science). Statistical thinking spread because it helped. Today,
two different techniques are called statistics.
Descriptive statistics8 produce a number or a figure that summarizes
or describes a set of data. You are already familiar with some descriptive
statistics. For example, you know about the arithmetic average, called
7 Introduction
8 Boldface words and phrases are defined in the margin and also in Appendix D, Glossary of Words.
9 A summary of this study can be found in Ellis (1938). The complete reference and all others in the text are listed in
the References section at the back of the book.
the mean. You have probably known how to compute a mean since elementary school—just
add up the numbers and divide the total by the number of entries. As you already know, the
mean describes the central tendency of a set of numbers. The basic idea of descriptive statistics
is simple: They summarize a set of data with one number or graph. This book covers about a
dozen descriptive statistics.
The other statistical technique is inferential statistics. Inferential statistics use
measurements from a sample to reach conclusions about a larger, unmeasured population.
There is, of course, a problem with samples.
Samples always depend partly on the luck of the draw; chance helps determine the
particular measurements you get.
If you have the measurements for the entire population, chance doesn’t play a part—all the
variation in the numbers is “true” variation. But with samples, some of the variation is the true
variation in the population and some is just the chance ups and downs that go with a sample.
Inferential statistics was developed as a way to account for the effects of chance that come with
sampling. This book will cover about a dozen and a half inferential statistics.
Here is a textbook definition: Inferential statistics is a method that takes chance factors into
account when samples are used to reach conclusions about populations. Like most textbook
definitions, this one condenses many elements into a short sentence. Because the idea of using
samples to understand populations is perhaps the most important concept in this course, please
pay careful attention when elements of inferential statistics are explained.
Inferential statistics has proved to be a very useful method in scientific disciplines. Many
other fields use inferential statistics, too, so I selected examples and problems from a variety of
disciplines for this text and its auxiliary materials. Null hypothesis significance testing, which
had a prominent place in our exploration tour, is an inferential statistics technique.
Here is an example from psychology that uses the NHST technique. Today, there is a lot
of evidence that people remember the tasks they fail to complete better than the tasks they
complete. This is known as the Zeigarnik effect. Bluma Zeigarnik asked participants in her
experiment to do about 20 tasks, such as work a puzzle, make a clay figure, and construct a box
from cardboard.9 For each participant, half the tasks were interrupted before completion. Later,
when the participants were asked to recall the tasks they worked on, they listed more of the
interrupted tasks (average about 7) than the completed tasks (about 4).
One good question to start with is, “Did interrupting make a big difference or a small
difference?” In this case, interruption produced about three additional memory items compared
to the completion condition. This is a 75% difference, which seems like a big change, given
our experience with tests of memory. The question of “How big is the difference?” can often be
answered by calculating an effect size index.
8 Chapter 1
clue to the future
So, should you conclude that interruption improves memory? Not yet. It might be that
interruption actually has no effect but that several chance factors happened to favor the
interrupted tasks in Zeigarnik’s particular experiment. One way to meet this objection is
to conduct the experiment again. Similar results would lend support to the conclusion that
interruption improves memory. A less expensive way to meet the objection is to use inferential
statistics such as NHST.
NHST begins with the actual data from the experiment. It ends with a probability—the
probability of obtaining data like those actually obtained if it is true that interruption has no
effect on memory. If the probability is very small, you can conclude that interruption does affect
memory. For Zeigarnik’s data, the probability was tiny.
Now for the conclusion. One version might be, “After completing about 20 tasks, memory
for interrupted tasks (average about 7) was greater than memory for completed tasks (average
about 4). The approximate 75% difference cannot be attributed to chance because chance by
itself would rarely produce a difference between two samples as large as this one.” The words
chance and rarely tell you that probability is an important element of inferential statistics.
My more complete answer to what I mean by “statistics” is Chapter 6 in 21st Century
Psychology: A Reference Handbook (Spatz, 2008). This 8-page chapter summarizes in words
(no formulas) the statistical concepts usually covered in statistics courses. This chapter can orient
you as you begin your study of statistics and later provide a review after you finish your course.
clue to the future
The first part of this book is devoted to descriptive statistics (Chapters 2–6) and the
second part to inferential statistics (Chapters 7–15). Inferential statistics is the more
comprehensive of the two because it combines descriptive statistics, probability, and logic.
Calculating effect size indexes is first addressed in Chapter 5. It is also a topic in Chapters
9-14.
Statistics: A Dynamic Discipline
Many people continue to think of statistics as a collection of techniques that were developed
long ago, that have not changed, and that will be the same in the future. That view is mistaken.
Statistics is a dynamic discipline characterized by more than a little controversy. New
techniques in both descriptive and inferential statistics continue to be developed. Controversy
9 Introduction
Some Terminology
continues too, as you saw at the end of our exploration tour. To get a feel for the issues when
the controversy entered the mainstream, see Dillon (1999) or Spatz (2000) for nontechnical
summaries. For more technical explanations, see Nickerson (2000). To read about current
approaches, see Erceg-Hurn and Mirosevich (2008), Kline (2013), or Cumming (2014).
In addition to controversy over techniques, attitudes toward data analysis shifted in recent
years. The shift has been toward the idea of exploring data to see what it reveals and away
from using statistical analyses to nail down a conclusion. This shift owes much of its impetus
to John Tukey (1915–2000), who promoted Exploratory Data Analysis (Lovie, 2005). Tukey
invented techniques such as the boxplot (Chapter 5) that reveal several characteristics of a data
set simultaneously.
Today, statistics is used in a wide variety of fields. Researchers start with a phenomenon,
event, or process that they want to understand better. They make measurements that produce
numbers. The numbers are manipulated according to the rules and conventions of statistics.
Based on the outcome of the statistical analysis, researchers draw conclusions and then write
the story of their new understanding of the phenomenon, event, or process. Statistics is just one
tool that researchers use, but it is often an essential tool.
Family incomes of college students in the fall of 2017
Weights of crackers eaten by obese male students
Depression scores of Alaskans
Gestation times for human beings
Memory scores of human beings10
Population
All measurements of a
specified group.
Sample
Measurements of a subset of a
population.
Like most courses, statistics introduces you to many new words. In statistics, most of the terms
are used over and over again. Your best move, when introduced to a new term, is to stop, read
the definition carefully, and memorize it. As the term continues to be used, you will become
more and more comfortable with it. Making notes is helpful.
Populations and Samples
A population consists of all the scores of some specified group. A sample is a subset of a
population. The population is the thing of interest. It is defined by the investigator and includes
all cases. The following are some populations:
10 I didn’t pull these populations out of thin air; they are all populations that researchers have gathered data on.
Studies of these populations will be described in this book.
10 Chapter 1
Parameter
Numerical or nominal
characteristic of a population.
Statistic
Numerical or nominal
characteristic of a sample.
Variable
Something that exists in more
than one amount or in more
than one form.
Investigators are always interested in populations. However, as you can determine from
these examples, populations can be so large that not all the members can be studied. The
investigator must often resort to measuring a sample that is small enough to be manageable. A
sample taken from the population of incomes of families of college students might include only
40 students. From the last population on the list, Zeigarnik used a sample of 164.
Most authors of research articles carefully explain the characteristics of the samples they
use. Often, however, they do not identify the population, leaving that task to the reader.
The answer to the question “What is the population?” depends on the specifics of a research
area, but many researchers generalize generously. For example, for some topics it is reasonable
to generalize from the results of a study on rats to “all mammals.” In all cases, however, the
reason for gathering data from a sample is to generalize the results to a larger population even
though sampling introduces some uncertainty into the conclusions.
Parameters and Statistics
A parameter is some numerical (number) or nominal (name) characteristic
of a population. An example is the mean reading readiness score of all
first-grade pupils in the United States. A statistic is some numerical or
nominal characteristic of a sample. The mean reading readiness score of
50 first-graders is a statistic, and so is the observation that 45% are girls.
A parameter is constant; it does not change unless the population itself
changes. The mean of a population is exactly one number. Unfortunately,
the parameter often cannot be computed because the population is
unmeasurable. So, a statistic is used as an estimate of the parameter, although, as suggested
before, statistics tend to differ from one sample to another. If you have five samples from the
same population, you will probably have five different sample means. In sum, parameters are
constant; statistics are variable.
Variables
A variable is something that exists in more than one amount or in more
than one form. Height and eye color are both variables. The notation 67
inches is a numerical way to identify a group of persons who are similar
in height. Of course, there are many other groups, each with an identifying
number. Blue and brown are common eye colors, which might be assigned
the numbers 0 and 1. All participants represented by 0 have the same eye
color. I will often refer to numbers like 67 and 0 as scores or test scores. A score is simply the
result of measuring a variable.
11 Introduction
Lower limit
Bottom of the range of possible
values that a measurement on
a continuous variable can have.
Upper limit
Top of the range of possible
values that a measurement on
a continuous variable can have.
Quantitative variable
Variable whose scores indicate
different amounts.
Quantitative Variables
Scores on quantitative variables tell you the degree or amount of the
thing being measured. At the very least, a larger score indicates more of
the variable than a smaller score does.
Continuous Variables. Continuous variables are quantitative variables whose scores
can be any value or intermediate value over the variable’s possible range. The continuous
memory scores in Zeigarnik’s experiment make up a quantitative, continuous variable. Number
of tasks recalled scores come in whole numbers such as 4 or 7, but it seems reasonable to assume
that the thing being measured, memory, is a continuous variable. Thus, of two participants who
both scored 7, one just barely got 7 and the other almost scored 8. Picture the continuous
variable, recall, as Figure 1.1.
Figure 1.1 shows that a score of 7 is used for a range of possible
recall values—the range from 6.5 to 7.5. The number 6.5 is the lower
limit and 7.5 is the upper limit of the score of 7. The idea is that recall
can be any value between 6.5 and 7.5, but that all the recall values in this
range are expressed as 7. In a similar way, a charge indicator value of 62%
on your cell phone stands for all the power values between 61.5% (the
lower limit) and 62.5% (the upper limit).
Sometimes scores are expressed in tenths, hundredths, or thousandths.
Like integers, these scores have lower and upper limits that extend
halfway to the next value on the quantitative scale.
Discrete Variables. Some quantitative variables are classified
as discrete variables because intermediate values are not possible.
The number of siblings you have, the number of times you’ve been
hospitalized, and how many pairs of shoes you have are examples.
Intermediate scores such as 2½ just don’t make sense.
Continuous variable
A quantitative variable whose
scores can be any amount.
Discrete variable
Variable for which intermediate
values between scores are not
meaningful.
F I G U R E 1 . 1 The lower and upper limits of recall scores of 6, 7, and 8
12 Chapter 1
Categorical Variables
Categorical variables (also called qualitative variables) produce scores
that differ in kind and not amount. Eye color is a categorical variable.
Scores might be expressed as blue and brown or as 0 and 1, but substituting
a number for a name does not make eye color a quantitative variable.
American political affiliation is a categorical variable with values of Democrat, Republican,
Independent, and Other. College major is another categorical variable.
Some categorical variables have the characteristic of order. College standing has ordered
measurements of senior, junior, sophomore, and freshman. Military rank is a categorical
variable with scores such as sergeant, corporal, and private. Categorical variables such as color
and gender do not have an inherent order. All categorical variables produce discrete scores, but
not all discrete scores are from a categorical variable.
Problems and Answers
Categorical variable
Variable whose scores differ in
kind, not amount.
At the beginning of this chapter, I urged you to engage statistics. Have you? For example, did
you read the footnotes? Have you looked up any words you weren’t sure of? (How near are you
to dictionary definitions when you study?) Have you read a paragraph a second time, wrinkled
your brow in concentration, made notes in the book margin, or promised yourself to ask your
instructor or another student about something you aren’t sure of? Engagement shows up as
activity. Best of all, the activity at times is a nod to yourself and a satisfied, “Now I understand.”
From time to time, I will use my best engagement tactic: I’ll give you a set of problems so
that you can practice what you have just been reading about. Working these problems correctly
is additional evidence that you have been engaged. You will find the answers at the end of the
book in Appendix G. Here are some suggestions for efficient learning.
1. Buy yourself a notebook or establish a file for statistics. Save your work there. When
you make an error, don’t remove it—note the error and rework the problem correctly.
Seeing your error later serves as a reminder of what not to do on a test. If you find that I
have made an error, write to me with a reminder of what not to do in the next edition.
2. Never, never look at an answer before you have worked the problem (or at least tried
twice to work the problem).
3. For each set of problems, work the first one and then immediately check your answer
against the answer in the book. If you make an error, find out why you made it—faulty
understanding, arithmetic error, or whatever.
4. Don’t be satisfied with just doing the math. If a problem asks for an interpretation, write
out your interpretation.
5. When you finish a chapter, go back over the problems immediately, reminding yourself
of the various techniques you have learned.
6. Use any blank spaces near the end of the book for your special notes and insights.
13 Introduction
P R O B L E M S
1.1. The history-of-statistics tour began with what easy-to-remember date?
1.2. The dominant approach to inferential statistics that is under attack is called ___________.
1.3. Identify each number below as coming from a quantitative variable or a categorical
variable.
a. 65 – seconds to work a puzzle
b. 319 – identification number for intellectual disability in the American Psychiatric
Association manual
c. 3 – group identification for small-cup daffodils
d. 4 – score on a high school advanced placement exam
e. 81 – milligrams of aspirin
1.4. Place lower and upper limits beside the continuous variables. Write discrete beside the
others.
a. _____________________ 20, seconds to work a puzzle
b. _____________________ 14, number of concerts attended
c. _____________________ 3, birth order
d. _____________________ 10, speed in miles per hour
1.5. Write a paragraph that gives the definitions of population, sample, parameter, and statistic
and the relationships among them.
1.6. Two kinds of statistics are ____________ statistics and ____________ statistics. Fill each
blank with the correct adjective.
a. To reach a conclusion about an unmeasured population, use ___________ statistics.
b. ____________ statistics take chance into account to reach a conclusion.
c. ____________ statistics are numbers or graphs that summarize a set of data.
Scales of Measurement
Now, here is an opportunity to see how actively you have been reading.
Numbers mean different things in different situations. Consider three answers that appear to be
identical but are not:
What number were you wearing in the race? “5”
What place did you finish in? “5”
How many minutes did it take you to finish? “5”
The three 5s all look the same. However, the three variables (identification number, finish
place, and time) are quite different. Because of the difference in what the variables measure,
each 5 has a different interpretation.
To illustrate this difference, consider another person whose answers to the same three
questions were 10, 10, and 10. If you take the first question by itself and know that the two
people had scores of 5 and 10, what can you say? You can say that the first runner was different
14 Chapter 1
from the second, but that is all. (Think about this until you agree.) On the second question,
with scores of 5 and 10, what can you say? You can say that the first runner was faster than the
second and, of course, that they are different.
Comparing the 5 and 10 on the third question, you can say that the first runner was twice
as fast as the second runner (and, of course, was faster and different).
The point of this discussion is to draw the distinction between the thing you are interested
in and the number that stands for the thing. Much of your experience with numbers has been
with pure numbers or quantitative measures such as time, length, and amount. Four and two
have a relationship of twice as much and half as much. And, for distance and seconds, four is
twice two; for amounts, two is half of four. But these relationships do not hold when numbers
are used to measure some things. For example, for political race finishes, twice and half are not
helpful. Second place is not half or twice anything compared to fourth place.
S. S. Stevens (1946) identified four different scales of measurement, each of which carries
a different set of information. Each scale uses numbers, but the information that can be inferred
from the numbers differs. The four scales are nominal, ordinal, interval, and ratio.
In the nominal scale, numbers are used simply as names and have no real quantitative
value. Numerals on sports uniforms are an example. Thus, 45 is different
from 32, but that is all you can say. The person represented by 45 is
not “more than” the person represented by 32, and certainly it would
be meaningless to calculate the mean of the two numbers. Examples of
nominal variables include psychological diagnoses, personality types, and
political parties. Psychological diagnoses, like other nominal variables,
consist of a set of categories. People are assessed and then classified into
one of the categories. The categories have both a name (such as posttraumatic stress disorder or
autism spectrum disorder) and a number (309.81 and 299.00, respectively). On a nominal scale,
the numbers mean only that the categories are different. In fact, for a nominal scale variable,
the numbers could be assigned to categories at random. Of course, all things that are alike must
have the same number.
The ordinal scale has the characteristic of the nominal scale (different numbers mean
different things) plus the characteristic of indicating greater than or less than. In the ordinal
scale, the object with the number 3 has less or more of something than
the object with the number 5. Finish places in a race are an example of
an ordinal scale. The runners finish in rank order, with 1 assigned to the
winner, 2 to the runner-up, and so on. Here, 1 means less time than 2.
Judgments about anxiety, quality, and recovery often correspond to an
ordinal scale. “Much improved,” “improved,” “no change,” and “worse”
are levels of an ordinal recovery variable. Ordinal scales are characterized
by rank order.
Ordinal scale
Measurement scale in which
numbers are ranks; equal
differences between numbers
do not represent equal
differences between the things
measured.
Nominal scale
Measurement scale in which
numbers serve only as labels
and do not indicate any
quantitative relationship.
15 Introduction
11 Convert 100°C and 50°C to Fahrenheit (F = 1.8C + 32) and suddenly the “twice as much” relationship disappears.
12 Convert 16 kilograms and 4 kilograms to pounds (1 kg = 2.2 lbs) and the “four times heavier” relationship is
maintained.
Interval scale
Measurement scale in which
equal differences between
numbers represent equal
differences in the thing
measured. The zero point is
arbitrarily defined.
Ratio scale
Measurement scale with
characteristics of interval scale;
also, zero means that none of
the thing measured is present.
The third kind of scale is the interval scale, which has the
properties of both the nominal and ordinal scales plus the additional
property that intervals between the numbers are equal. “Equal
interval” means that the distance between the things represented by 2
and 3 is the same as the distance between the things represented by 3
and 4. Temperature is measured on an interval scale. The difference
in temperature between 10°C and 20°C is the same as the difference
between 40°C and 50°C. The Celsius thermometer, like all interval
scales, has an arbitrary zero point. On the Celsius thermometer, this zero point is the freezing
point of water at sea level. Zero degrees on this scale does not mean the complete absence of
heat; it is simply a convenient starting point. With interval data, there is one restriction: You
may not make simple ratio statements. You may not say that 100° is twice as hot as 50° or that
a person with an IQ of 60 is half as intelligent as a person with an IQ of 120.11
The fourth kind of scale, the ratio scale, has all the characteristics
of the nominal, ordinal, and interval scales plus one other: It has
a true zero point, which indicates a complete absence of the thing
measured. On a ratio scale, zero means “none.” Height, weight, and
time are measured with ratio scales. Zero height, zero weight, and
zero time mean that no amount of these variables is present. With a
true zero point, you can make ratio statements such as 16 kilograms
is four times heavier than 4 kilograms.12 Table 1.1 summarizes the major differences among the
four scales of measurement.
T A B L E 1 . 1 Characteristics of the four scales of measurement
Nominal Yes No No No
Ordinal Yes Yes No No
Interval Yes Yes Yes No
Ratio Yes Yes Yes Yes
Scale of
measurement
Different
numbers for
different
things
Numbers convey
greater than
and less than
Equal differences
mean equal
amounts
Zero means none of
what was measured
was detected
Scale characteristics
16 Chapter 1
Knowing the distinctions among the four scales of measurement will help you in two
tasks in this course. The kind of descriptive statistics you can compute from numbers depends,
in part, on the scale of measurement the numbers represent. For example, it is senseless to
compute a mean of numbers on a nominal scale. Calculating a mean Social Security number,
a mean telephone number, or a mean psychological diagnosis is either a joke or evidence of
misunderstanding numbers.
Understanding scales of measurement is sometimes important in choosing the kind of
inferential statistic that is appropriate for a set of data. If the dependent variable (see next
section) is a nominal variable, then a chi square analysis is appropriate (Chapter 14). If the
dependent variable is a set of ranks (ordinal data), then a nonparametric statistic is required
(Chapter 15). Most of the data analyzed with the techniques described in Chapters 7–13 are
interval and ratio scale data.
The topic of scales of measurement is controversial among statisticians. Part of the
controversy involves viewpoints about the underlying thing you are interested in and the number
that represents the thing (Wuensch, 2005). In addition, it is sometimes difficult to classify some
of the variables used in the social and behavioral sciences. Often they appear to fall between the
ordinal scale and the interval scale. For example, a score may provide more information than
simply rank, but equal intervals cannot be proven. Examples include aptitude and ability tests,
personality measures, and intelligence tests. In such cases, researchers generally treat the scores
as if they were interval scale data.
Statistics and Experimental Design
Here is a story that will help you distinguish between statistics (applying straight logic) and
experimental design (observing what actually happens). This is an excerpt from a delightful
book by E. B. White, The Trumpet of the Swan (1970, pp. 63–64).
The fifth-graders were having a lesson in arithmetic, and their teacher, Miss Annie Snug, greeted Sam
with a question.
“Sam, if a man can walk three miles in one hour, how many miles can he walk in four hours?” “It
would depend on how tired he got after the first hour,” replied Sam. The other pupils roared. Miss Snug
rapped for order.
“Sam is quite right,” she said. “I never looked at the problem that way before. I always supposed that
man could walk twelve miles in four hours, but Sam may be right: that man may not feel so spunky after
the first hour. He may drag his feet. He may slow up.”
Albert Bigelow raised his hand. “My father knew a man who tried to walk twelve miles, and he died
of heart failure,” said Albert.
“Goodness!” said the teacher. “I suppose that could happen, too.”
17 Introduction
“Anything can happen in four hours,” said Sam. “A man might develop a blister on his heel. Or he
might find some berries growing along the road and stop to pick them. That would slow him up even if he
wasn’t tired or didn’t have a blister.”
“It would indeed,” agreed the teacher. “Well, children, I think we have all learned a great deal about
arithmetic this morning, thanks to Sam Beaver.”
Everyone had learned how careful you have to be when dealing with figures.
Statistics involves the manipulation of numbers and the conclusions based on those
manipulations (Miss Snug). Experimental design (also called research methods) deals with all
the things that influence the numbers you get (Sam and Albert). Figure 1.2 illustrates these
two approaches to getting an answer. This text could have been a “pure” statistics book, from
which you would learn to analyze numbers without knowing where they came from or what
they referred to. You would learn about statistics, but such a book would be dull, dull, dull. On
the other hand, to describe procedures for collecting numbers is to teach experimental design—
and this book is for a statistics course. My solution to this conflict is generally to side with
Miss Snug but to include some aspects of experimental design throughout the book. Knowing
experimental design issues is especially important when it comes time to interpret a statistical
analysis. Here’s a start on experimental design.
Experimental Design Variables
The overall task of an experimenter is to discover relationships among variables. Variables
are things that vary, and researchers have studied personality, health, gender, anger, caffeine,
memory, beliefs, age, skill…. (I’m sure you get the picture—almost anything can be a variable.)
F I G U R E 1 . 2 Travel time from an experimental design viewpoint and a statistical viewpoint
18 Chapter 1
Independent variable
Variable controlled by the
researcher; changes in this
variable may produce changes
in the dependent variable.
Dependent variable
Observed variable that is
expected to change as a result
of changes in the independent
variable in an experiment.
Level
One value of the independent
variable.
Treatment
One value (or level) of the
independent variable.
Extraneous variable
Variable other than the
independent variable that may
affect the dependent variable.
Independent and Dependent Variables
A simple experiment has two major variables, the independent variable
and the dependent variable. In the simplest experiment, the researcher
selects two values of the independent variable for investigation. Values of
the independent variable are usually called levels and sometimes called
treatments.
The basic idea is that the researcher finds or creates two groups of
participants that are similar except for the independent variable. These
individuals are measured on the dependent variable. The question is
whether the data will allow the experimenter to claim that the values on
the dependent variable depend on the level of the independent variable.
The values of the dependent variable are found by measuring or
observing participants in the investigation. The dependent variable might
be scores on a personality test, number of items remembered, or whether
or not a passerby offered assistance. For the independent variable, the two
groups might have been selected because they were already different—in
age, gender, personality, and so forth. Alternatively, the experimenter might have produced
the difference in the two groups by an experimental manipulation such as creating different
amounts of anxiety or providing different levels of practice.
An example might help. Suppose for a moment that as a budding gourmet cook you want to
improve your spaghetti sauce. One of your buddies suggests adding marjoram. To investigate,
you serve spaghetti sauce at two different gatherings. For one group of guests, the sauce is
spiced with marjoram; for the other it is not. At both gatherings, you count the number of
favorable comments about the spaghetti sauce. Stop reading; identify the independent and the
dependent variables.
The dependent variable is the number of favorable comments, which is a measure of the
taste of the sauce. The independent variable is marjoram, which has two levels: present and
absent.
Extraneous Variables
One of the pitfalls of experiments is that every situation has other variables
besides the independent variable that might possibly be responsible
for the changes in the dependent variable. These other variables are
called extraneous variables. In the story, Sam and Albert noted several
extraneous variables that could influence the time to walk 12 miles.
19 Introduction
13 Try for answers. Then, if need be, here’s a hint: First, identify the dependent variable; for the dependent variable,
you don’t know values until data are gathered. Next, identify the independent variable; you can tell what the values of
the independent variable are just from the description of the design.
Are there any extraneous variables in the spaghetti sauce example? Oh yes, there are many,
and just one is enough to raise suspicion about a conclusion that relates the taste of spaghetti
sauce to marjoram. Extraneous variables include the amount and quality of the other ingredients
in the sauce, the spaghetti itself, the “party moods” of the two groups, and how hungry everyone
was. If any of these extraneous variables was actually operating, it weakens the claim that a
difference in the comments about the sauce is the result of the presence or absence of marjoram.
The simplest way to remove an extraneous variable is to be sure that all participants are
equal on that variable. For example, you can ensure that the sauces are the same except for
marjoram by mixing up the ingredients, dividing it into two batches, adding marjoram to
one batch but not the other, and then cooking. The “party moods” variable can be controlled
(equalized) by conducting the taste test in a laboratory. Controlling extraneous variables is a
complex topic covered in courses that focus on research methods and experimental design.
In many experiments, it is impossible or impractical to control all the extraneous variables.
Sometimes researchers think they have controlled them all, only to find that they did not. The
effect of an uncontrolled extraneous variable is to prevent a simple cause-and-effect conclusion.
Even so, if the dependent variable changes when the independent variable changes, something
is going on. In this case, researchers can say that the two variables are related, but that other
variables may play a part, too.
At this point, you can test your understanding by engaging yourself with these questions:
What were the independent and dependent variables in the Zeigarnik experiment? How many
levels of the independent variable were there?13
How well did Zeigarnik control extraneous variables? For one thing, each participant was
tested at both levels of the independent variable. That is, the recall of each participant was
measured for interrupted tasks and for completed tasks. One advantage of this technique is
that it naturally controls many extraneous variables. Thus, extraneous variables such as age
and motivation were exactly the same for tasks that were interrupted as for tasks that were not
because the same people contributed scores to both levels.
At various places in the following chapters, I will explain experiments and the statistical
analyses using the terms independent and dependent variables. These explanations usually
assume that all extraneous variables were controlled; that is, you may assume that the
experimenter knew how to design the experiment so that changes in the dependent variable
could be attributed correctly to changes in the independent variable. However, I present a few
investigations (like the spaghetti sauce example) that I hope you recognize as being so poorly
designed that conclusions cannot be drawn about the relationship between the independent
variable and dependent variable. Be alert.
20 Chapter 1
Statistics and Philosophy
14 In philosophy, those who emphasize reason are rationalists and those who emphasize experience are empiricists.
Here’s my summary of the relationship between statistics and experimental design.
Researchers suspect that there is a relationship between two variables. They design and conduct
an experiment; that is, they choose the levels of the independent variable (treatments), control
the extraneous variables, and then measure the participants on the dependent variable. The
measurements (data) are analyzed using statistical procedures. Finally, the researcher tells a
story that is consistent with the results obtained and the procedures used.
The two previous sections directed your attention to the relationship between statistics and
experimental design; this section will direct your thoughts to the place of statistics in the grand
scheme of things.
Explaining the grand scheme of things is the task of philosophy
and, over the years, many schemes have been advanced. For a scheme
to be considered a grand one, it has to address epistemology—that is, to
propose answers to the question: How do we acquire knowledge?
Both reason and experience have been popular answers among
philosophers.14 For those who emphasize the importance of reason, mathematics has been a
continuing source of inspiration. Classical mathematics starts with axioms that are assumed
to be true. Theorems are thought up and are then proved by giving axioms as reasons. Once a
theorem is proved, it can be used as a reason in a proof of other theorems.
Statistics has its foundations in mathematics and thus, a statistical analysis is based on
reason. As you go about the task of calculating 𝑋 or ŝ, finding confidence intervals, and telling
the story of what they mean, know deep down that you are engaged in logical reasoning.
(Experimental design is more complex; it includes experience and observation as well as
reasoning.)
In the 19th century, science concentrated on observation and description. The variation
that always accompanied a set of observations was thought to be due to imprecise observing,
imprecise instruments, or failure of nature to “hit the mark.” During the 20th century, however,
statistical methods such as NHST revolutionized the philosophy of science by focusing on the
variation that was always present in data (Salsburg, 2001; Gould, 1996). Focusing on variation
allowed changes in the data to be associated with particular causes.
As the 21st century approached, flaws in the logic of NHST statistics began to be recognized.
In addition to logical flaws, the practices of researchers and journal editors (such as requiring
a statistical analysis to show that p < .05) came under scrutiny. This concern with how science
is conducted has led to changes in how data are analyzed and how information is shared. The
practice of statistics is in transition.
Epistemology
The study or theory of the
nature of knowledge.
21 Introduction
Statistics: Then and Now
Let’s move from formal descriptions of philosophy to a more informal one. A very common
task of most human beings can be described as trying to understand. Statistics has helped many in
their search for better understanding, and it is such people who have recommended (or demanded)
that statistics be taught in school. A reasonable expectation is that you, too, will find statistics
useful in your future efforts to understand and persuade.
Speaking of persuasion, you have probably heard it said, “You can prove anything with
statistics.” The implied message is that a conclusion based on statistics is suspect because statistical
methods are unreliable. Well, it just isn’t true that statistical methods are unreliable, but it is true
that people can misuse statistics (just as any tool can be misused). One of the great advantages of
studying statistics is that you get better at recognizing statistics that are used improperly.
Statistics began with counting, which, of course, was prehistory. The origin of the mean is almost as
obscure. It was in use by the early 1700s, but no one is credited with its discovery. Graphs, however,
began when J. H. Lambert, a Swiss-German scientist and mathematician, and William Playfair, an
English political economist, invented and improved graphs in the period 1765 to 1800 (Tufte, 2001).
The Royal Statistical Society was established in 1834 by a group of Englishmen in London. Just
5 years later, on November 27, 1839, at 15 Cornhill in Boston, a group of Americans founded the
American Statistical Society. Less than 3 months later, for a reason that you can probably figure out,
the group changed its name to the American Statistical Association, which continues today (www.
amstat.org).
According to Walker (1929), the first university course in statistics in the United States was
probably “Social Science and Statistics,” taught at Columbia University in 1880. The professor was a
political scientist, and the course was offered in the economics department. In 1887, at the University
of Pennsylvania, separate courses in statistics were offered by the departments of psychology and
economics. By 1891, Clark University, the University of Michigan, and Yale had been added to
the list of schools that taught statistics, and anthropology had been added to the list of departments.
Biology was added in 1899 (Harvard) and education in 1900 (Columbia).
You might be interested in when statistics was first taught at your school and in what department.
College catalogs are probably your most accessible source of information.
This course provides you with the opportunity to improve your ability to understand and use
statistics. Kirk (2008) identifies four levels of statistical sophistication:
Category 1—those who understand statistical presentations
Category 2—those who understand, select, and apply statistical procedures
Category 3—applied statisticians who help others use statistics
Category 4—mathematical statisticians who develop new statistical techniques and
discover new characteristics of old techniques
I hope that by the end of your statistics course, you will be well along the path to becoming a
Category 2 statistician.
22 Chapter 1
How to Analyze a Data Set
Helpful Features of This Book
15 You are reading the footnotes, aren’t you? Your answer — “Well, yes, it seems I am.”
The end point of analyzing a data set is a story that explains the relationships among the
variables in the data set. I recommend that you analyze a data set in three steps. The first step is
exploratory. Read all the information and examine the data. Calculate descriptive statistics and
focus on the differences that are revealed. In this textbook, descriptive statistics are emphasized
in Chapters 2 through 6 and include graphs, means, and effect size indexes. Calculating
descriptive statistics helps you develop preliminary ideas for your story (Step 3). The second
step is to answer the question, What are the effects that chance could have on the descriptive
statistics I calculated? An answer requires inferential statistics (Chapter 7 through Chapter 15).
The third step is to write the story the data reveal. Incorporate the descriptive and inferential
statistics to support the conclusions in the story. Of course, the skills you’ve learned and taught
yourself about composition will be helpful as you compose and write your story. Don’t worry
about length; most good statistical stories about simple data sets can be told in one paragraph.
Write your story using journal style, which is quite different from textbook style. Textbook
style, at least this textbook, is chatty, redundant, and laced with footnotes.15 Journal style, on
the other hand, is terse, formal, and devoid of footnotes. Paragraphs labeled Interpretation in
Appendix G, this textbook’s answer section, are examples of journal style. And for guidance in
writing up an entire study, see Appelbaum et al. (2018).
At various points in this chapter, I encouraged your engagement in statistics. Your active
participation is necessary if you are to learn statistics. For my part, I worked to organize this
book and write it in a way that encourages active participation. Here are some of the features
you should find helpful.
Objectives
Each chapter begins with a list of skills the chapter is designed to help you acquire. Read this
list of objectives first to find out what you are to learn to do. Then thumb through the chapter
and read the headings. Next, study the chapter, working all the problems as you come to them.
Finally, reread the objectives. Can you meet each one? If so, put a check mark beside that
objective.
23 Introduction
Often a concept is presented that will be used again in later chapters. These ideas are
separated from the rest of the text in a box labeled “Clue to the Future.” You have already
seen two of these “Clues” in this chapter. Attention to these concepts will pay dividends
later in the course.
I have boxed in, at various points in the book, ways to detect errors. Some of these
“Error Detection” tips will also help you better understand the concept. Because many of
these checks can be made early, they can prevent the frustrating experience of getting an
impossible answer when the error could have been caught in Step 2.
Problems and Answers
The problems in this text are in small groups within the chapter rather than clumped together at
the end. This encourages you to read a little and work problems, followed by more reading and
problems. Psychologists call this pattern spaced practice. Spaced practice patterns lead to better
performance than massed practice patterns. The problems come from a variety of disciplines;
the answers are in Appendix G.
Some problems are conceptual and do not require any arithmetic. Think these through and
write your answers. Being able to calculate a statistic is almost worthless if you cannot explain
in English what it means. Writing reveals how thoroughly you understand. To emphasize the
importance of explanations, I highlited Interpretation in the answers in Appendix G. On
occasion, problems or data sets are used again, either later in that chapter or in another. If you do
not work the problem when it is first presented, you are likely to be frustrated when it appears
again. To alert you, I have put an asterisk (*) beside problems that are used again.
At the end of many chapters, comprehensive problems are marked with a Working
these problems requires knowing most of the material in the chapter. For most students, it is
best to work all the problems, but be sure you can work those marked with a
clue to the future
error detection
Figure and Table References
Sometimes the words Figure and Table are in boldface print. This means that you should
examine the figure or table at that point. Afterward, it will be easy for you to return to your
place in the text—just find the boldface type.
24 Chapter 1
Computers, Calculators, and Pencils
16 The original name of the program was Statistical Package for the Social Sciences.
Transition Passages
At six places in this book, there are major differences between the material you just finished and
the material in the next section. “Transition Passages,” which describe the differences, separate
these sections.
Glossaries
This book has three separate glossaries of words, symbols, and formulas.
1. Words. The first time an important word is used in the text, it appears in boldface type
accompanied by a definition in the margin. In later chapters, the word may be boldfaced
again, but margin definitions are not repeated. Appendix D is a complete glossary of
words (p. 401). I suggest you mark this appendix.
2. Symbols. Statistical symbols are defined in Appendix E (p. 405). Mark it too.
3. Formulas. Formulas for all the statistical techniques used in the text are printed
in Appendix F (p. 407), in alphabetical order according to the name of the technique.
Computer software, calculators, and pencils with erasers are all tools used at one time or another
by statisticians. Any or all of these devices may be part of the course you are taking. Regardless
of the calculating aids that you use, however, your task is the same:
• Read a problem.
• Decide what statistical procedure to use.
• Apply that procedure using the tools available to you.
• Write an interpretation of the results.
Pencils, calculators, and software represent, in ascending order, tools that are more and
more error-free. People who routinely use statistics routinely use computers. You may or may
not use one at this point. Remember, though, whether you are using a software program or not,
your principal task is to understand and describe.
For many of the worked examples in this book, I included the output of a popular statistical
software program, IBM SPSS.16 If your course includes IBM SPSS, these tables should help
familiarize you with the program.
25 Introduction
P R O B L E M S
1.7. Name the four scales of measurement identified by S. S. Stevens.
1.8. Give the properties of each of the scales of measurement.
1.9. Identify the scale of measurement in each of the following cases.
a. Geologists have a “hardness scale” for identifying different rocks, called Mohs’ scale.
The hardest rock (diamond) has a value of 10 and will scratch all others. The second
hardest will scratch all but the diamond, and so on. Talc, with a value of 1, can be
scratched by every other rock. (A fingernail, a truly handy field-test instrument, has
a value between 2 and 3.)
b. The volumes of three different cubes are 40, 64, and 65 cubic inches.
c. Three different highways are identified by their numbers: 40, 64, and 65.
d. Republicans, Democrats, Independents, and Others are identified on the voters’ list
with the numbers 1, 2, 3, and 4, respectively.
e. The winner of the Miss America contest was Miss New York; the two runners-up
were Miss Ohio and Miss California.18
f. The prices of the three items are $3.00, $10.00, and $12.00.
g. She earned three degrees: B.A., M.S., and Ph.D.
Concluding Thoughts
17 This book’s index is unusually extensive. If you make margin notes, they will help too.
18 Contest winners have come most frequently from these states, which have had six winners each.
The first leg of your exploration of statistics is complete. Some of the ideas along the path are
familiar and perhaps a few are new or newly engaging. As your course progresses, you will
come to understand what is going on in many statistical analyses, and you will learn paths to
follow if you analyze data that you collect yourself.
This book is a fairly complete introduction to elementary statistics. Of course, there is
lots more to statistics, but there is a limit to what you can do in one term. Even so, exploration
of paths not covered in a textbook can be fun. Encyclopedias, both general and specialized,
often reward such exploration. Try the Encyclopedia of Statistics in Behavioral Sciences or
the International Encyclopedia of the Social and Behavioral Sciences. Also, when you finish
this course (but before any final examination), I recommend Chapter 16, the last chapter in this
book. It is an overview/integrative chapter.
Most students find that this book works well as a textbook in their statistics course. I
recommend that you keep the book after the course is over to use as a reference book. In
courses that follow statistics and even after leaving school, many find themselves looking up a
definition or reviewing a procedure.17 Thus, a familiar textbook becomes a valued guidebook
that serves for years into the future. For me, exploring statistics and using them to understand
the world is quite satisfying. I hope you have a similar experience.
26 Chapter 1
1.10. Undergraduate students conducted the three studies that follow. For each study, identify
the dependent variable, the independent variable, the number of levels of the independent
variable, and the names of the levels of the independent variable.
a. Becca had students in a statistics class rate a résumé, telling them that the person had
applied for a position that included teaching statistics at their college. The students
rated the résumé on a scale of 1 (not qualified) to 10 (extremely qualified). All the
students received identical résumés, except that the candidate’s first name was Jane
on half the résumés and John on the other half.
b. Michael’s participants filled out the Selfism scale, which measures narcissism.
(Narcissism is neurotic self-love.) In addition, students were classified as first-born,
second-born, and later-born.
c. Johanna had participants read a description of a crime and “Mr. Anderson,” the
person convicted of the crime. For some participants, Mr. Anderson was described
as a janitor. For others, he was described as a vice president of a large corporation.
For still others, no occupation was given. After reading the description, participants
recommended a jail sentence (in months) for Mr. Anderson.
1.11. Researchers who are now well known conducted the three classic studies that follow. For
each study, identify the dependent variable, the independent variable, and the number and
names of the levels of the independent variable. Complete items i and ii.
a. Theodore Barber hypnotized 25 people, giving each a series of suggestions. The
suggestions included arm rigidity, hallucinations, color blindness, and enhanced
memory. Barber counted the number of suggestions the participants complied with
(the mean was 4.8). For another 25 people, he simply asked them to achieve the best
score they could (but no hypnosis was used). This second group was given the same
suggestions, and the number complied with was counted (the mean was 5.1). (See
Barber, 1976.)
i. Identify a nominal variable and a statistic.
ii. In a sentence, describe what Barber’s study shows.
b. Elizabeth Loftus had participants view a film clip of a car accident. Afterward, some
were asked, “How fast was the car going?” and others were asked, “How fast was
the car going when it passed the barn?” (There was no barn in the film.) A week later,
Loftus asked the participants, “Did you see a barn?” If the barn had been mentioned
earlier, 17% said yes; if it had not been mentioned, 3% said yes. (See Loftus, 1979.)
i. Identify a population and a parameter.
ii. In a sentence, describe what Loftus’s study shows.
c. Stanley Schachter and Larry Gross gathered data from obese male students for about
an hour in the afternoon. At the end of this time, a clock on the wall was correct (5:30
p.m.) for 20 participants, slow (5:00 p.m.) for 20 others, and fast (6:00 p.m.) for
20 more. The actual time, 5:30, was the usual dinnertime for these students. While
participants filled out a final questionnaire, Wheat Thins® were freely available.
The weight of the crackers each student consumed was measured. The means were
27 Introduction
as follows: 5:00 group—20 grams; 5:30 group—30 grams; 6:00 group—40 grams.
(See Schachter and Gross, 1968.)
i. Identify a ratio scale variable.
ii. In a sentence, describe what this study shows.
1.12. There are uncontrolled extraneous variables in the study described here. Name as many as
you can. Begin by identifying the dependent and independent variables.
An investigator concluded that Textbook A was better than Textbook B after comparing
the exam scores of two statistics classes. One class met MWF at 10:00 a.m. for 50
minutes, used Textbook A, and was taught by Dr. X. The other class met for 2.5 hours
on Wednesday evening, used Textbook B, and was taught by Dr. Y. All students took the
same comprehensive test at the end of the term. The mean score for Textbook A students
was higher than the mean score for Textbook B students.
1.13. In philosophy, the study of the nature of knowledge is called .
1.14. a. The two approaches to epistemology identified in the text are and .
b. Statistics has its roots in .
1.15. Your textbook recommends a three-step approach to analyzing a data set. Summarize the
steps.
1.16. Read the objectives at the beginning of this chapter. Responding to them will help you
consolidate what you have learned.
KEY TERMS
Categorical variable (p. 12)
Continuous variable (p. 11)
Dependent variable (p. 18)
Descriptive statistics (p. 6)
Discrete variable (p. 11)
Epistemology (p. 20)
Extraneous variable (p. 18)
Independent variable (p. 18)
Inferential statistics (p. 6)
Interval scale (p. 15)
Level (p. 18)
Lower limit (p. 11)
Mean (p. 7)
Nominal scale (p. 14)
Ordinal scale (p. 14)
Parameter (p. 10)
Population (p. 9)
Qualitative variable (p. 12)
Quantitative variable (p. 11)
Ratio scale (p. 15)
Sample (p. 9)
Statistic (p. 10)
Treatment (p. 18)
Upper limit (p. 11)
Variable (p. 10)
The online Study Guide for Exploring Statistics (12th ed.)
is available for sale at exploringstatistics.com
http://exploringstatistics.com
28
To Descriptive Statistics
STATISTICAL TECHNIQUES ARE often categorized as descriptive statistics and
inferential statistics. The next five chapters are about descriptive statistics. You are already
familiar with some of these descriptive statistics, such as the mean, range, and bar graphs.
Others may be less familiar—the correlation coefficient, effect size index, and boxplot. All of
these and others that you study will be helpful in your efforts to understand data.
The phrase Exploring Data appears in three of the chapter titles that follow. This phrase
is a reminder to approach a data set with the attitude of an explorer, an attitude of What can I
find here? Descriptive statistics are especially valuable in the early stages of an analysis as you
explore what the data have to say. Later, descriptive statistics are essential when you convey
your story of the data to others. In addition, many descriptive statistics have important roles in
the inferential statistical techniques that are covered in later chapters. Let’s get started.
Transition Passage
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