Statistics In Plain English Third Edition Timothy C Urdan

Statistics In Plain English Third Edition
Timothy C Urdan download
https://ebookbell.com/product/statistics-in-plain-english-third-
edition-timothy-c-urdan-47913020
Explore and download more ebooks at ebookbell.com

Here are some recommended products that we believe you will be
interested in. You can click the link to download.
Statistics In Plain English 3rd Edition 3rd Timothy C Urdan
https://ebookbell.com/product/statistics-in-plain-english-3rd-
edition-3rd-timothy-c-urdan-2332874
Statistics In Plain English 5th Edition Timothy C Urdan
https://ebookbell.com/product/statistics-in-plain-english-5th-edition-
timothy-c-urdan-43840460
Statistics In Plain English Fifth Edition 5th Edition Timothy C Urdan
https://ebookbell.com/product/statistics-in-plain-english-fifth-
edition-5th-edition-timothy-c-urdan-43852302
Statistics In Plain English 4th Edition Timothy C Urdan
https://ebookbell.com/product/statistics-in-plain-english-4th-edition-
timothy-c-urdan-10911446

Statistics In Plain English 2nd Edition 2nd Edition Timothy C Urdan
https://ebookbell.com/product/statistics-in-plain-english-2nd-
edition-2nd-edition-timothy-c-urdan-1479498
Breaking The Law Of Averages Reallife Probability And Statistics In
Plain English William M Briggs
https://ebookbell.com/product/breaking-the-law-of-averages-reallife-
probability-and-statistics-in-plain-english-william-m-briggs-5386636
Statistics In Language Research Analysis Of Variance Reprint 2010 Toni
Rietveld Roeland Van Hout
https://ebookbell.com/product/statistics-in-language-research-
analysis-of-variance-reprint-2010-toni-rietveld-roeland-van-
hout-51032984
Statistics In Social Work An Introduction To Practical Applications
Amy Batchelor
https://ebookbell.com/product/statistics-in-social-work-an-
introduction-to-practical-applications-amy-batchelor-51837762
Statistics In Theory And Practice Robert Lupton
https://ebookbell.com/product/statistics-in-theory-and-practice-
robert-lupton-51954020

6WDWLVWLFVLQ3ODLQ(QJOLVK
7KLUG(GLWLRQ
7LPRWK8UGDQ
6DQWDODUD8QLYHUVLW

Routledge
Taylor Francis Group
270 Madison Avenue
New York, NY 10016
Routledge
Taylor Francis Group
27 Church Road
Hove, East Sussex BN3 2FA
© 2010 by Taylor and Francis Group, LLC
Routledge is an imprint of Taylor Francis Group, an Informa business
International Standard Book Number: 978-0-415-87291-1 (Paperback)
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact
the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides
licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment
has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation
without intent to infringe.
Library of Congress Cataloging‑in‑Publication Data
Urdan, Timothy C.
Statistics in plain English / Tim Urdan. ‑‑ 3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978‑0‑415‑87291‑1
1. Statistics‑‑Textbooks. I. Title.
QA276.12.U75 2010
519.5‑‑dc22 2010000438
Visit the Taylor Francis Web site at
http://www.taylorandfrancis.com
and the Psychology Press Web site at
http://www.psypress.com
This edition published in the Taylor Francis e-Library, 2011.
To purchase your own copy of this or any of Taylor Francis or Routledge’s
collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.
ISBN 0-203-85117-X Master e-book ISBN

To Ella and Nathaniel. Because you rock.

v
Contents
Preface ix
1
Chapter Introduction to Social Science Research Principles and Terminology 1
Populations and Samples, Statistics and Parameters 1
Sampling Issues 3
Types of Variables and Scales of Measurement 4
Research Designs 4
Making Sense of Distributions and Graphs 6
Wrapping Up and Looking Forward 10
Glossary of Terms for Chapter 1 10
2
Chapter Measures of Central Tendency 13
Measures of Central Tendency in Depth 14
Example: The Mean, Median, and Mode of a Skewed Distribution 15
Writing it Up 17
Glossary of Terms and Symbols for Chapter 2 18
3
Chapter Measures of Variability 19
Measures of Variability in Depth 20
Example: Examining the Range, Variance, and Standard Deviation 24
4
Chapter The Normal Distribution 29
The Normal Distribution in Depth 30
Example: Applying Normal Distribution Probabilities to a Nonnormal Distribution 33
5
Chapter Standardization and z Scores 37
Standardization and z Scores in Depth 37
Examples: Comparing Raw Scores and z Scores 45
6
Chapter Standard Errors 49
Standard Errors in Depth 49
Example: Sample Size and Standard Deviation Effects on the Standard Error 58

vi ■ Contents
7
Chapter Statistical Significance, Effect Size, and Confidence Intervals 61
Statistical Significance in Depth 62
Effect Size in Depth 68
Confidence Intervals in Depth 71
Example: Statistical Significance, Confidence Interval, and Effect Size for a
One-Sample t Test of Motivation 73
Recommended Reading 78
8
Chapter Correlation 79
Pearson Correlation Coefficients in Depth 81
A Brief Word on Other Types of Correlation Coefficients 88
Example: The Correlation between Grades and Test Scores 89
Writing It Up 90
9
Chapter t Tests 93
Independent Samples t Tests in Depth 94
Paired or Dependent Samples t Tests in Depth 98
Example: Comparing Boys’ and Girls’ Grade Point Averages 100
Example: Comparing Fifth-and Sixth-Grade GPAs 102
Writing It Up 103
1
Chapter 0 One-Way Analysis of Variance 105
One-Way ANOVA in Depth 106
Example: Comparing the Preferences of 5-, 8-, and 12-Year-Olds 113
Writing It Up 116
1
Chapter 1 Factorial Analysis of Variance 119
Factorial ANOVA in Depth 120
Example: Performance, Choice, and Public versus Private Evaluation 128
Writing It Up 129
1
Chapter 2 Repeated-Measures Analysis of Variance 131
Repeated-Measures ANOVA in Depth 133
Example: Changing Attitudes about Standardized Tests 138
Writing It Up 143

Contents ■ vii
1
Chapter 3 Regression 145
Regression in Depth 146
Multiple Regression 152
Example: Predicting the Use of Self-Handicapping Strategies 156
Writing It Up 159
1
Chapter 4 The Chi-Square Test of Independence 161
Chi-Square Test of Independence in Depth 162
Example: Generational Status and Grade Level 165
Writing It Up 166
1
Chapter 5 Factor Analysis and Reliability Analysis: Data Reduction Techniques 169
Factor Analysis in Depth 169
A More Concrete Example of Exploratory Factor Analysis 172
Reliability Analysis in Depth 178
Writing It Up 180
Wrapping Up 180
Glossary of Symbols and Terms for Chapter 15 181
Appendices 183
Appendix A : Area under the Normal Curve beyond z 185
Appendix B: Critical Values of the t Distributions 187
Appendix C: Critical Values of the F Distributions 189
Appendix D: Critical Values of the Studentized Range Statistic (for the Tukey HSD Test) 195
Appendix E: Critical Values of the χ2 Distributions 199
References 201
Glossary of Symbols 203
Index 205

ix
Preface
Why Use Statistics?
As a researcher who uses statistics frequently, and as an avid listener of talk radio, I find myself
yelling at my radio daily. Although I realize that my cries go unheard, I cannot help myself. As
radio talk show hosts, politicians making political speeches, and the general public all know,
there is nothing more powerful and persuasive than the personal story, or what statisticians
call anecdotal evidence. My favorite example of this comes from an exchange I had with a staff
member of my congressman some years ago. I called his office to complain about a pamphlet his
office had sent to me decrying the pathetic state of public education. I spoke to his staff member
in charge of education. I told her, using statistics reported in a variety of sources (e.g., Berliner
and Biddle’s The Manufactured Crisis and the annual “Condition of Education” reports in the
Phi Delta Kappan written by Gerald Bracey), that there are many signs that our system is doing
quite well, including higher graduation rates, greater numbers of students in college, rising
standardized test scores, and modest gains in SAT scores for students of all ethnicities. The staff
member told me that despite these statistics, she knew our public schools were failing because
she attended the same high school her father had, and he received a better education than she. I
hung up and yelled at my phone.
Many people have a general distrust of statistics, believing that crafty statisticians can “make
statistics say whatever they want” or “lie with statistics.” In fact, if a researcher calculates the
statistics correctly, he or she cannot make them say anything other than what they say, and sta-
tistics never lie. Rather, crafty researchers can interpret what the statistics mean in a variety of
ways, and those who do not understand statistics are forced to either accept the interpretations
that statisticians and researchers offer or reject statistics completely. I believe a better option is
to gain an understanding of how statistics work and then use that understanding to interpret the
statistics one sees and hears for oneself. The purpose of this book is to make it a little easier to
understand statistics.
Uses of Statistics
One of the potential shortfalls of anecdotal data is that they are idiosyncratic. Just as the con-
gressional staffer told me her father received a better education from the high school they both
attended than she did, I could have easily received a higher quality education than my father
did. Statistics allow researchers to collect information, or data, from a large number of people
and then summarize their typical experience. Do most people receive a better or worse educa-
tion than their parents? Statistics allow researchers to take a large batch of data and summarize
it into a couple of numbers, such as an average. Of course, when many data are summarized
into a single number, a lot of information is lost, including the fact that different people have
very different experiences. So it is important to remember that, for the most part, statistics do
not provide useful information about each individual’s experience. Rather, researchers generally
use statistics to make general statements about a population. Although personal stories are often
moving or interesting, it is often important to understand what the typical or average experience
is. For this, we need statistics.
Statistics are also used to reach conclusions about general differences between groups. For
example, suppose that in my family, there are four children, two men and two women. Suppose
that the women in my family are taller than the men. This personal experience may lead me to
the conclusion that women are generally taller than men. Of course, we know that, on average,

x ■ Preface
men are taller than women. The reason we know this is because researchers have taken large,
random samples of men and women and compared their average heights. Researchers are often
interested in making such comparisons: Do cancer patients survive longer using one drug than
another? Is one method of teaching children to read more effective than another? Do men and
women differ in their enjoyment of a certain movie? To answer these questions, we need to col-
lect data from randomly selected samples and compare these data using statistics. The results
we get from such comparisons are often more trustworthy than the simple observations people
make from nonrandom samples, such as the different heights of men and women in my family.
Statistics can also be used to see if scores on two variables are related and to make predictions.
For example, statistics can be used to see whether smoking cigarettes is related to the likelihood
of developing lung cancer. For years, tobacco companies argued that there was no relation-
ship between smoking and cancer. Sure, some people who smoked developed cancer. But the
tobacco companies argued that (a) many people who smoke never develop cancer, and (b) many
people who smoke tend to do other things that may lead to cancer development, such as eating
unhealthy foods and not exercising. With the help of statistics in a number of studies, research-
ers were finally able to produce a preponderance of evidence indicating that, in fact, there is a
relationship between cigarette smoking and cancer. Because statistics tend to focus on overall
patterns rather than individual cases, this research did not suggest that everyone who smokes
will develop cancer. Rather, the research demonstrated that, on average, people have a greater
chance of developing cancer if they smoke cigarettes than if they do not.
With a moment’s thought, you can imagine a large number of interesting and important
questions that statistics about relationships can help you answer. Is there a relationship between
self-esteem and academic achievement? Is there a relationship between the appearance of crimi-
nal defendants and their likelihood of being convicted? Is it possible to predict the violent crime
rate of a state from the amount of money the state spends on drug treatment programs? If we
know the father’s height, how accurately can we predict son’s height? These and thousands of
other questions have been examined by researchers using statistics designed to determine the
relationship between variables in a population.
How to Use This Book
This book is not intended to be used as a primary source of information for those who are
unfamiliar with statistics. Rather, it is meant to be a supplement to a more detailed statistics
textbook, such as that recommended for a statistics course in the social sciences. Or, if you have
already taken a course or two in statistics, this book may be useful as a reference book to refresh
your memory about statistical concepts you have encountered in the past. It is important to
remember that this book is much less detailed than a traditional textbook. Each of the concepts
discussed in this book is more complex than the presentation in this book would suggest, and
a thorough understanding of these concepts may be acquired only with the use of a more tradi-
tional, more detailed textbook.
With that warning firmly in mind, let me describe the potential benefits of this book, and
how to make the most of them. As a researcher and a teacher of statistics, I have found that
statistics textbooks often contain a lot of technical information that can be intimidating to non-
statisticians. Although, as I said previously, this information is important, sometimes it is useful
to have a short, simple description of a statistic, when it should be used, and how to make sense
of it. This is particularly true for students taking only their first or second statistics course, those
who do not consider themselves to be “mathematically inclined,” and those who may have taken
statistics years ago and now find themselves in need of a little refresher. My purpose in writing
this book is to provide short, simple descriptions and explanations of a number of statistics that
are easy to read and understand.

Preface ■ xi
To help you use this book in a manner that best suits your needs, I have organized each chap-
ter into three sections. In the first section, a brief (one to two pages) description of the statistic
is given, including what the statistic is used for and what information it provides. The second
section of each chapter contains a slightly longer (three to eight pages) discussion of the statistic.
In this section, I provide a bit more information about how the statistic works, an explanation of
how the formula for calculating the statistic works, the strengths and weaknesses of the statistic,
and the conditions that must exist to use the statistic. Finally, each chapter concludes with an
example in which the statistic is used and interpreted.
Before reading the book, it may be helpful to note three of its features. First, some of the
chapters discuss more than one statistic. For example, in Chapter 2, three measures of central
tendency are described: the mean, median, and mode. Second, some of the chapters cover sta-
tistical concepts rather than specific statistical techniques. For example, in Chapter 4 the normal
distribution is discussed. There are also chapters on statistical significance and on statistical
interactions. Finally, you should remember that the chapters in this book are not necessarily
designed to be read in order. The book is organized such that the more basic statistics and statis-
tical concepts are in the earlier chapters whereas the more complex concepts appear later in the
book. However, it is not necessary to read one chapter before understanding the next. Rather,
each chapter in the book was written to stand on its own. This was done so that you could
use each chapter as needed. If, for example, you had no problem understanding t tests when you
learned about them in your statistics class but find yourself struggling to understand one-way
analysis of variance, you may want to skip the t test chapter (Chapter 9) and skip directly to
the analysis of variance chapter (Chapter 10).
New Features in This Edition
There are several new and updated sections in this third edition of Statistics in Plain English.
Perhaps the biggest change is the addition of a new chapter on data reduction and organiza-
tion techniques, factor analysis and reliability analysis (Chapter 15). These are very commonly
used statistics in the social sciences, particularly among researchers who use survey methods.
In addition, the first chapter has a new section about understanding distributions of data, and
includes several new graphs to help you understand how to use and interpret graphs. I have also
added a “Writing it Up” section at the end of many of the chapters to illustrate how the statis-
tics would be presented in published articles, books, or book chapters. This will help you as you
write up your own results for publication, or when you are reading the published work of others.
The third edition also comes with a companion website at http://www.psypress.com/statistics-
in-plain-english/ that has Powerpoint summaries for each chapter, a set of interactive work
problems for most of the chapters, and links to useful websites for learning more about statistics.
Perhaps best of all, I fixed all of the mistakes that were in the last edition of the book. Of course,
I probably added some new mistakes to this edition, just to keep you on your toes.
Statistics are powerful tools that help people understand interesting phenomena. Whether
you are a student, a researcher, or just a citizen interested in understanding the world around
you, statistics can offer one method for helping you make sense of your environment. This book
was written using plain English to make it easier for non-statisticians to take advantage of the
many benefits statistics can offer. I hope you find it useful.
Acknowledgments
First, long overdue thanks to Debra Riegert at Routledge/Taylor and Francis for her helpful
ideas and the many free meals over the years. Next, my grudging but sincere thanks to the
reviewers of this third edition of the book: Gregg Bell, University of Alabama, Catherine A.

xii ■ Preface
Roster, University of New Mexico, and one anonymous reviewer. I do not take criticism well,
but I eventually recognize helpful advice when I receive it and I followed most of yours, to the
benefit of the readers. I always rely on the help of several students when producing the vari-
ous editions of this book, and for this edition I was assisted most ably by Sarah Cafasso, Stacy
Morris, and Louis Hung. Finally, thank you Jeannine for helping me find time to write and to
Ella and Nathaniel for making sure I didn’t spend too much time “doing work.”

1
1
Chapter
Introduction to Social Science Research
Principles and Terminology
When I was in graduate school, one of my statistics professors often repeated what passes,
in statistics, for a joke: “If this is all Greek to you, well that’s good.” Unfortunately, most of
the class was so lost we didn’t even get the joke. The world of statistics and research in the
social sciences, like any specialized field, has its own terminology, language, and conventions.
In this chapter, I review some of the fundamental research principles and terminology includ-
ing the distinction between samples and populations, methods of sampling, types of variables,
and the distinction between inferential and descriptive statistics. Finally, I provide a brief word
about different types of research designs.
Populations and Samples, Statistics and Parameters
A population is an individual or group that represents all the members of a certain group or
category of interest. A sample is a subset drawn from the larger population (see Figure 1.1). For
example, suppose that I wanted to know the average income of the current full-time, tenured
faculty at Harvard. There are two ways that I could find this average. First, I could get a list
of every full-time, tenured faculty member at Harvard and find out the annual income of each
member on this list. Because this list contains every member of the group that I am interested in,
it can be considered a population. If I were to collect these data and calculate the mean, I would
have generated a parameter, because a parameter is a value generated from, or applied to, a
population. Another way to generate the mean income of the tenured faculty at Harvard would
be to randomly select a subset of faculty names from my list and calculate the average income of
this subset. The subset is known as a sample (in this case it is a random sample), and the mean
that I generate from this sample is a type of statistic. Statistics are values derived from sample
data, whereas parameters are values that are either derived from or applied to population data.
It is important to keep a couple of things in mind about samples and populations. First, a
population does not need to be large to count as a population. For example, if I wanted to know
the average height of the students in my statistics class this term, then all of the members of the
class (collectively) would comprise the population. If my class only has five students in it, then
my population only has five cases. Second, populations (and samples) do not have to include
people. For example, suppose I want to know the average age of the dogs that visited a veterinary
clinic in the last year. The population in this study is made up of dogs, not people. Similarly, I
may want to know the total amount of carbon monoxide produced by Ford vehicles that were
assembled in the United States during 2005. In this example, my population is cars, but not all
cars—it is limited to Ford cars, and only those actually assembled in a single country during a
single calendar year.

2 ■ Statistics in Plain English, Third Edition
Third, the researcher generally defines the population, either explicitly or implicitly. In the
examples above, I defined my populations (of dogs and cars) explicitly. Often, however, research-
ers define their populations less clearly. For example, a researcher may say that the aim of her
study is to examine the frequency of depression among adolescents. Her sample, however, may
only include a group of 15-year-olds who visited a mental health service provider in Connecticut
in a given year. This presents a potential problem and leads directly into the fourth and final
little thing to keep in mind about samples and populations: Samples are not necessarily good
representations of the populations from which they were selected. In the example about the rates
of depression among adolescents, notice that there are two potential populations. First, there
is the population identified by the researcher and implied in her research question: adolescents.
But notice that adolescents is a very large group, including all human beings, in all countries,
between the ages of, say, 13 and 20. Second, there is the much more specific population that
was defined by the sample that was selected: 15-year-olds who visited a mental health service
provider in Connecticut during a given year.
Inferential and Descriptive Statistics
Why is it important to determine which of these two populations is of interest in this study?
Because the consumer of this research must be able to determine how well the results from the
sample generalize to the larger population. Clearly, depression rates among 15-year-olds who
visit mental health service providers in Connecticut may be different from other adolescents.
For example, adolescents who visit mental health service providers may, on average, be more
depressed than those who do not seek the services of a psychologist. Similarly, adolescents in
Connecticut may be more depressed, as a group, than adolescents in California, where the sun
shines and Mickey Mouse keeps everyone smiling. Perhaps 15-year-olds, who have to suffer the
indignities of beginning high school without yet being able to legally drive, are more depressed
than their 16-year-old, driving peers. In short, there are many reasons to suspect that the ado-
lescents who were not included in the study may differ in their depression rates than adolescents
who were in the study. When such differences exist, it is difficult to apply the results garnered
from a sample to the larger population. In research terminology, the results may not general-
ize from the sample to the population, particularly if the population is not clearly defined.
So why is generalizability important? To answer this question, I need to introduce the dis-
tinction between descriptive and inferential statistics. Descriptive statistics apply only to the
members of a sample or population from which data have been collected. In contrast, inferential
statistics refer to the use of sample data to reach some conclusions (i.e., make some inferences)
Sample (n = 3)
Population (N = 10)
Figure 1.1 A population and a sample drawn from the population.

Introduction to Social Science Research Principles and Terminology ■ 3
about the characteristics of the larger population that the sample is supposed to represent.
Although researchers are sometimes interested in simply describing the characteristics of a
sample, for the most part we are much more concerned with what our sample tells us about the
population from which the sample was drawn. In the depression study, the researcher does not
care so much about the depression levels of her sample per se. Rather, she wants to use the data
from her sample to reach some conclusions about the depression levels of adolescents in general.
But to make the leap from sample data to inferences about a population, one must be very clear
about whether the sample accurately represents the population. An important first step in this
process is to clearly define the population that the sample is alleged to represent.
Sampling Issues
There are a number of ways researchers can select samples. One of the most useful, but also the
most difficult, is random sampling. In statistics, the term random has a much more specific
meaning than the common usage of the term. It does not mean haphazard. In statistical jargon,
random means that every member of a population has an equal chance of being selected into
a sample. The major benefit of random sampling is that any differences between the sample
and the population from which the sample was selected will not be systematic. Notice that in
the depression study example, the sample differed from the population in important, systematic
(i.e., nonrandom) ways. For example, the researcher most likely systematically selected adoles-
cents who were more likely to be depressed than the average adolescent because she selected
those who had visited mental health service providers. Although randomly selected samples may
differ from the larger population in important ways (especially if the sample is small), these dif-
ferences are due to chance rather than to a systematic bias in the selection process.
Representative sampling is a second way of selecting cases for a study. With this method,
the researcher purposely selects cases so that they will match the larger population on specific
characteristics. For example, if I want to conduct a study examining the average annual income
of adults in San Francisco, by definition my population is “adults in San Francisco.” This popula-
tion includes a number of subgroups (e.g., different ethnic and racial groups, men and women,
retired adults, disabled adults, parents and single adults, etc.). These different subgroups may
be expected to have different incomes. To get an accurate picture of the incomes of the adult
population in San Francisco, I may want to select a sample that represents the population well.
Therefore, I would try to match the percentages of each group in my sample that I have in my
population. For example, if 15% of the adult population in San Francisco is retired, I would
select my sample in a manner that included 15% retired adults. Similarly, if 55% of the adult
population in San Francisco is male, 55% of my sample should be male. With random sampling,
I may get a sample that looks like my population or I may not. But with representative sam-
pling, I can ensure that my sample looks similar to my population on some important variables.
This type of sampling procedure can be costly and time-consuming, but it increases my chances
of being able to generalize the results from my sample to the population.
Another common method of selecting samples is called convenience sampling. In conve-
nience sampling, the researcher generally selects participants on the basis of proximity, ease-of-
access, and willingness to participate (i.e., convenience). For example, if I want to do a study
on the achievement levels of eighth-grade students, I may select a sample of 200 students from
the nearest middle school to my office. I might ask the parents of 300 of the eighth-grade stu-
dents in the school to participate, receive permission from the parents of 220 of the students,
and then collect data from the 200 students that show up at school on the day I hand out my
survey. This is a convenience sample. Although this method of selecting a sample is clearly less
labor-intensive than selecting a random or representative sample, that does not necessarily make
it a bad way to select a sample. If my convenience sample does not differ from my population of

interest in ways that influence the outcome of the study, then it is a perfectly acceptable method of
selecting a sample.
Types of Variables and Scales of Measurement
In social science research, a number of terms are used to describe different types of variables.
A variable is pretty much anything that can be codified and has more than a single value
(e.g., income, gender, age, height, attitudes about school, score on a meas
ure of depression). A
constant, in contrast, has only a single score. For example, if every member of a sample is male,
the “gender” category is a constant. Types of variables include quantitative (or continuous)
and qualitative (or categorical). A quantitative variable is one that is scored in such a way that
the numbers, or values, indicate some sort of amount. For example, height is a quantitative (or
continuous) variable because higher scores on this variable indicate a greater amount of height.
In contrast, qualitative variables are those for which the assigned values do not indicate more or
less of a certain quality. If I conduct a study to compare the eating habits of people from Maine,
New Mexico, and Wyoming, my “state” variable has three values (e.g., 1 = Maine, 2 = New
Mexico, 3 = Wyoming). Notice that a value of 3 on this variable is not more than a value of 1 or
2—it is simply different. The labels represent qualitative differences in location, not quantitative
differences. A commonly used qualitative variable in social science research is the dichotomous
variable. This is a variable that has two different categories (e.g., male and female).
Most statistics textbooks describe four different scales of meas
ure
ment for variables: nomi-
nal, ordinal, interval, and ratio. A nominally scaled variable is one in which the labels that
are used to identify the different levels of the variable have no weight, or numeric value. For
example, researchers often want to examine whether men and women differ on some variable
(e.g., income). To conduct statistics using most computer software, this gender variable would
need to be scored using numbers to represent each group. For example, men may be labeled “0”
and women may be labeled “1.” In this case, a value of 1 does not indicate a higher score than a
value of 0. Rather, 0 and 1 are simply names, or labels, that have been assigned to each group.
With ordinal variables, the values do have weight. If I wanted to know the 10 richest people
in America, the wealthiest American would receive a score of 1, the next richest a score of 2, and
so on through 10. Notice that while this scoring system tells me where each of the wealthiest 10
Americans stands in relation to the others (e.g., Bill Gates is 1, Oprah Winfrey is 8, etc.), it does
not tell me how much distance there is between each score. So while I know that the wealthiest
American is richer than the second wealthiest, I do not know if he has one dollar more or one
billion dollars more. Variables scored using either interval and ratio scales, in contrast, contain
information about both relative value and distance. For example, if I know that one member of
my sample is 58 inches tall, another is 60 inches tall, and a third is 66 inches tall, I know who
is tallest and how much taller or shorter each member of my sample is in relation to the others.
Because my height variable is measured using inches, and all inches are equal in length, the
height variable is measured using a scale of equal intervals and provides information about both
relative position and distance. Both interval and ratio scales use measures with equal distances
between each unit. Ratio scales also include a zero value (e.g., air temperature using the Celsius
scale of meas
ure
ment). Figure 1.2 provides an illustration of the difference between ordinal and
interval/ratio scales of meas
ure
ment.
Research Designs
There are a variety of research methods and designs employed by social scientists. Sometimes
researchers use an experimental design. In this type of research, the experimenter divides the
cases in the sample into different groups and then compares the groups on one or more variables

of interest. For example, I may want to know whether my newly developed mathematics cur-
riculum is better than the old method. I select a sample of 40 students and, using random
assignment, teach 20 students a lesson using the old curriculum and the other 20 using the new
curriculum. Then I test each group to see which group learned more mathematics concepts. By
applying students to the two groups using random assignment, I hope that any important dif-
ferences between the two groups get distributed evenly between the two groups and that any
differences in test scores between the two groups is due to differences in the effectiveness of the
two curricula used to teach them. Of course, this may not be true.
Correlational research designs are also a common method of conducting research in the
social sciences. In this type of research, participants are not usually randomly assigned to
groups. In addition, the researcher typically does not actually manipulate anything. Rather, the
researcher simply collects data on several variables and then conducts some statistical analyses
to determine how strongly different variables are related to each other. For example, I may be
interested in whether employee productivity is related to how much employees sleep (at home,
not on the job). So I select a sample of 100 adult workers, meas
ure their productivity at work,
and meas
ure how long each employee sleeps on an average night in a given week. I may find that
there is a strong relationship between sleep and productivity. Now logically, I may want to argue
that this makes sense, because a more rested employee will be able to work harder and more
efficiently. Although this conclusion makes sense, it is too strong a conclusion to reach based on
my correlational data alone. Correlational studies can only tell us whether variables are related
to each other—they cannot lead to conclusions about causality. After all, it is possible that being
more productive at work causes longer sleep at home. Getting one’s work done may relieve stress
and perhaps even allows the worker to sleep in a little longer in the morning, both of which
create longer sleep.
Experimental research designs are good because they allow the researcher to isolate specific
independent variables that may cause variation, or changes, in dependent variables. In the
example above, I manipulated the independent variable of a mathematics curriculum and was
able to reasonably conclude that the type of math curriculum used affected students’ scores on
the dependent variable, test scores. The primary drawbacks of experimental designs are that they
are often difficult to accomplish in a clean way and they often do not generalize to real-world
situations. For example, in my study above, I cannot be sure whether it was the math curricula
that influenced test scores or some other factor, such as preexisting difference in the mathemat-
ics abilities of my two groups of students or differences in the teacher styles that had nothing to
Ordinal Interval/Ratio
Finish Line
1
2
3
4
5
0.25 seconds
5 seconds
3 seconds
0.30 seconds
1
2
3
4
5
2 seconds
2 seconds
2 seconds
2 seconds
Figure 1.2 Difference between ordinal and interval/ratio scales of meas
ure
ment.

do with the curricula, but could have influenced test scores (e.g., the clarity or enthusiasm of the
teacher). The strengths of correlational research designs are that they are often easier to conduct
than experimental research, they allow for the relatively easy inclusion of many variables, and
they allow the researcher to examine many variables simultaneously. The principle drawback of
correlational research is that such research does not allow for the careful controls necessary for
drawing conclusions about causal associations between variables.
Making Sense of Distributions and Graphs
Statisticians spend a lot of time talking about distributions. A distribution is simply a collec-
tion of data, or scores, on a variable. Usually, these scores are arranged in order from smallest
to largest and then they can be presented graphically. Because distributions are so important in
statistics, I want to give them some attention early in the book and show you several examples
of different types of distributions and how they are depicted in graphs. Note that later in this
book there are whole chapters devoted to several of the most commonly used distributions in
statistics, including the normal distribution (Chapters 4 and 5), t distributions (Chapter 9 and
parts of Chapter 7), F distributions (Chapters 10, 11, and 12), and chi-square distributions
(Chapter 14).
Let’s begin with a simple example. Suppose that I am conducting a study of voter’s attitudes
and I select a random sample of 500 voters for my study. One piece of information I might
want to know is the political affiliation of the members of my sample. So I ask them if they are
Republicans, Democrats, or Independents. I find that 45% of my sample identify themselves
as Democrats, 40% report being Republicans, and 15% identify themselves as Independents.
Notice that political affiliation is a nominal, or categorical, variable. Because nominal variables
are variables with categories that have no numerical weight, I cannot arrange my scores in this
distribution from highest to lowest. The value of being a Republican is not more or less than the
value of being a Democrat or an Independent—they are simply different categories. So rather
than trying to arrange my data from the lowest to the highest value, I simply leave them as sepa-
rate categories and report the percentage of the sample that falls into each category.
There are many different ways that I could graph this distribution, including pie charts, bar
graphs, column graphs, different sized bubbles, and so on. The key to selecting the appropriate
graphic is to keep in mind that the purpose of the graph is to make the data easy to understand.
For my distribution of political affiliation, I have created two different graphs. Both are fine
choices because both of them offer very clear and concise summaries of this distribution and
are easy to understand. Figure 1.3 depicts this distribution as a column graph, and Figure 1.4
presents the data in a pie chart. Which graphic is best for these data is a matter of personal
preference. As you look at Figure 1.3, notice that the x-axis (the horizontal one) shows the party
0
5
10
15
20
25
30
35
40
45
50
Percentage
Republicans
Political Aﬃliation
Independents
Democrats
Figure 1.3 Column graph showing distribution of Republicans, Democrats, and Independents.

affiliations: Democrats, Republicans, and Independents. The y-axis (the vertical one) shows the
percentage of the sample. You can see the percentages in each group and, just by quickly glanc-
ing at the columns, you can see which political affiliation has the highest percentage of this
sample and get a quick sense of the differences between the party affiliations in terms of the per-
centage of the sample. The pie chart in Figure 1.4 shows the same information, but in a slightly
more striking and simple manner, I think.
Sometimes, researchers are interested in examining the distributions of more than one vari-
able at a time. For example, suppose I wanted to know about the association between hours
spent watching television and hours spent doing homework. I am particularly interested in how
this association looks across different countries. So I collect data from samples of high school
students in several different countries. Now I have distributions on two different variables across
5 different countries (the United States, Mexico, China, Norway, and Japan). To compare these
different countries, I decide to calculate the average, or mean (see Chapter 2) for each country on
each variable. Then I graph these means using a column graph, as shown in Figure 1.5 (note that
these data are fictional—I made them up). As this graph clearly shows, the disparity between
the average amount of television watched and the average hours of homework completed per day
is widest in the United States and Mexico and nonexistent in China. In Norway and Japan, high
school students actually spend more time on homework than they do watching TV according to
my fake data. Notice how easily this complex set of data is summarized in a single graph.
Another common method of graphing a distribution of scores is the line graph, as shown in
Figure 1.6. Suppose that I selected a random sample of 100 college freshpeople who have just
completed their first term. I asked them each to tell me the final grades they received in each
40%
45%
Republicans
Democrats
Independents
15%
Figure 1.4 Pie chart showing distribution of Republicans, Democrats, and Independents.
0
1
2
3
4
5
6
7
Japan
Hours
Country
Hours TV
Hours homework
U.S. Mexico China Norway
Figure 1.5 Average hours of television viewed and time spent on homework in five countries.

of their classes and then I calculated a grade point average (GPA) for each of them. Finally, I
divided the GPAs into 6 groups: 1 to 1.4, 1.5 to 1.9, 2.0 to 2.4, 2.5 to 2.9, 3.0 to 3.4, and 3.5 to
4.0. When I count up the number of students in each of these GPA groups and graph these data
using a line graph, I get the results presented in Figure 1.6. Notice that along the x-axis I have
displayed the 6 different GPA groups. On the y-axis I have the frequency, typically denoted by
the symbol f. So in this graph, the y-axis shows how many students are in each GPA group. A
quick glance at Figure 1.6 reveals that there were quite a few students (13) who really struggled
in their first term in college, accumulating GPAs between 1.0 and 1.4. Only 1 student was in
the next group from 1.5 to 1.9. From there, the number of students in each GPA group gener-
ally goes up with roughly 30 students in the 2.0–2.9 GPA categories and about 55 students
in the 3.0–4.0 GPA categories. A line graph like this offers a quick way to see trends in data,
either over time or across categories. In this example with GPA, we can see that the general
trend is to find more students in the higher GPA categories, plus a fairly substantial group that
is really struggling.
Column graphs are another clear way to show trends in data. In Figure 1.7, I present a
stacked-column graph. This graph allows me to show several pieces of information in a single
graph. For example, in this graph I am illustrating the occurrence of two different kinds of
crime, property and violent, across the period from 1990 to 2007. On the x-axis I have placed
the years, moving from earlier (1990) to later (2007) as we look from the left to the right.
On the y-axis I present the number of crimes committed per 100,000 people in the United
States. When presented this way, several interesting facts jump out. First, the overall trend from
0
5
10
15
20
25
30
35
1.0–1.4 1.5–1.9 2.0–2.4 2.5–2.9 3.0–3.4 3.5–4.0
Frequency
GPA
Figure 1.6 Line graph showing frequency of students in different GPA groups.
0
1000
2000
3000
4000
5000
6000
7000
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Crime
Year
Violent
Property
Figure 1.7 Stacked column graph showing crime rates from 1990 to 2007.

1990 to 2007 is a pretty dramatic drop in crime. From a high of nearly 6,000 crimes per 100,000
people in 1991, the crime rate dropped to well under 4,000 per 100,000 people in 2007. That is a
drop of nearly 40%. The second noteworthy piece of information that is obvious from the graph
is that violent crimes (e.g., murder, rape, assault) occur much less frequently than crimes against
property (e.g., burglary, vandalism, arson) in each year of the study.
Notice that the graph presented in Figure 1.7 makes it easy to see that there has been a drop
in crime overall from 1990 to 2007, but it is not so easy to tell whether there has been much of a
drop in the violent crime rate. That is because violent crime makes up a much smaller percent-
age of the overall crime rate than does property crime, so the scale used in the y-axis is pretty
large. This makes the tops of the columns, the part representing violent crimes, look quite small.
To get a better idea of the trend for violent crimes over time, I created a new graph, which is
presented in Figure 1.8.
In this new figure, I have presented the exact same data that was presented in Figure 1.7 as a
stacked column graph. The line graph separates violent crimes from property crimes completely,
making it easier to see the difference in the frequency of the two types of crimes. Again, this
graph clearly shows the drop in property crime over the years. But notice that it is still difficult
to tell whether there was much of a drop in violent crime over time. If you look very closely, you
0
1000
2000
3000
4000
5000
6000
Crimes
per
100,000
Year
Property
Violent
1
9
9
0
2
0
0
7
2
0
0
6
2
0
0
5
2
0
0
4
2
0
0
3
2
0
0
2
2
0
0
1
2
0
0
0
1
9
9
9
1
9
9
7
1
9
9
8
1
9
9
6
1
9
9
5
1
9
9
4
1
9
9
3
1
9
9
2
1
9
9
1
Figure 1.8 Line graph showing crime rates from 1990 to 2007.
0
100
200
300
400
500
600
700
800
Violent
Crimes
per
100,000
Year
1
9
9
0
2
0
0
7
2
0
0
6
2
0
0
5
2
0
0
4
2
0
0
3
2
0
0
2
2
0
0
1
2
0
0
0
1
9
9
9
1
9
9
7
1
9
9
8
1
9
9
6
1
9
9
5
1
9
9
4
1
9
9
3
1
9
9
2
1
9
9
1
Figure 1.9 Column graph showing violent crime rates from 1990 to 2007.

can see that the rate of violent crime dropped from about 800 per 100,000 in 1990 to about 500
per 100,000 in 2007. This is an impressive drop in the crime rate, but we had to work too hard
to see it. Remember: The purpose of the graph is to make the interesting facts in the data easy
to see. If you have to work hard to see it, the graph is not that great.
The problem with Figure 1.8, just as it was with Figure 1.7, is that the scale on the y-axis
is too large to clearly show the trends for violent crimes rates over time. To fix this problem
we need a scale that is more appropriate for the violent crime rate data. So I created one more
graph (Figure 9.1) that included the data for violent crimes only, without the property crime data.
Instead of using a scale from 0 to 6000 or 7000 on the y-axis, my new graph has a scale from 0 to
800 on the y-axis. In this new graph, a column graph, it is clear that the drop in violent crime from
1990 to 2007 was also quite dramatic.
Any collection of scores on a variable, regardless of the type of variable, forms a distribution,
and this distribution can be graphed. In this section of the chapter, several different types of
graphs have been presented, and all of them have their strengths. The key, when creating graphs,
is to select the graph that most clearly illustrates the data. When reading graphs, it is important
to pay attention to the details. Try to look beyond the most striking features of the graph to the
less obvious features, like the scales used on the x- and y-axes. As I discuss later (Chapter 12),
graphs can be quite misleading if the details are ignored.
Wrapping Up and Looking Forward
The purpose of this chapter was to provide a quick overview of many of the basic principles and
terminology employed in social science research. With a foundation in the types of variables,
experimental designs, and sampling methods used in social science research it will be easier
to understand the uses of the statistics described in the remaining chapters of this book. Now
we are ready to talk statistics. It may still all be Greek to you, but that’s not necessarily a bad
thing.
Glossary of Terms for Chapter 1
Chi-square distributions: A family of distributions associated with the chi-square (χ2)
statistic.
Constant: A construct that has only one value (e.g., if every member of a sample was 10 years
old, the “age” construct would be a constant).
Convenience sampling: Selecting a sample based on ease of access or availability.
Correlational research design: A style of research used to examine the associations among
variables. Variables are not manipulated by the researcher in this type of research
design.
Dependent variable: The values of the dependent variable are hypothesized to depend on the
values of the independent variable. For example, height depends, in part, on gender.
Descriptive statistics: Statistics used to describe the characteristics of a distribution of scores.
Dichotomous variable: A variable that has only two discrete values (e.g., a pregnancy variable
can have a value of 0 for “not pregnant” and 1 for “pregnant”).
Distribution: Any collection of scores on a variable.
Experimental research design: A type of research in which the experimenter, or researcher,
manipulates certain aspects of the research. These usually include manipulations of the
independent variable and assignment of cases to groups.
F distributions: A family of distributions associated with the F statistic, which is commonly
used in analysis of variance (ANOVA).
Frequency: How often a score occurs in a distribution.

Generalize (or Generalizability): The ability to use the results of data collected from a sample
to reach conclusions about the characteristics of the population, or any other cases not
included in the sample.
Independent variable: A variable on which the values of the dependent variable are hypoth-
esized to depend. Independent variables are often, but not always, manipulated by the
researcher.
Inferential statistics: Statistics, derived from sample data, that are used to make inferences
about the population from which the sample was drawn.
Interval or Ratio variable: Variables measured with numerical values with equal distance, or
space, between each number (e.g., 2 is twice as much as 1, 4 is twice as much as 2, the
distance between 1 and 2 is the same as the distance between 2 and 3).
Mean: The arithmetic average of a distribution of scores.
Nominally scaled variable: A variable in which the numerical values assigned to each category
are simply labels rather than meaningful numbers.
Normaldistribution: A bell-shaped frequency distribution of scores that has the mean, median,
and mode in the middle of the distribution and is symmetrical and asymptotic.
Ordinal variable: Variables measured with numerical values where the numbers are meaning-
ful (e.g., 2 is larger than 1) but the distance between the numbers is not constant.
Parameter: A value, or values, derived from population data.
Population: The collection of cases that comprise the entire set of cases with the specified
characteristics (e.g., all living adult males in the United States).
Qualitative (or categorical) variable: A variable that has discrete categories. If the categories
are given numerical values, the values have meaning as nominal references but not as
numerical values (e.g., in 1 = “male” and 2 = “female,” 1 is not more or less than 2).
Quantitative (or continuous) variable: A variable that has assigned values and the values are
ordered and meaningful, such that 1 is less than 2, 2 is less than 3, and so on.
Random assignment: Assignment members of a sample to different groups (e.g., experimental
and control) randomly, or without consideration of any of the characteristics of sample
members.
Random sample (or random sampling): Selecting cases from a population in a manner that
ensures each member of the population has an equal chance of being selected into the
sample.
Representative sampling: A method of selecting a sample in which members are purposely
selected to create a sample that represents the population on some characteristic(s) of
interest (e.g., when a sample is selected to have the same percentages of various ethnic
groups as the larger population).
Sample: A collection of cases selected from a larger population.
Statistic: A characteristic, or value, derived from sample data.
t distributions: A family of distributions associated with the t statistic, commonly used in the
comparison of sample means and tests of statistical significance for correlation coef-
ficients and regression slopes.
Variable: Any construct with more than one value that is examined in research.

13
2
Chapter
Measures of Central Tendency
Whenever you collect data, you end up with a group of scores on one or more variables. If you
take the scores on one variable and arrange them in order from lowest to highest, what you get
is a distribution of scores. Researchers often want to know about the characteristics of these
distributions of scores, such as the shape of the distribution, how spread out the scores are, what
the most common score is, and so on. One set of distribution characteristics that researchers are
usually interested in is central tendency. This set consists of the mean, median, and mode.
The mean is probably the most commonly used statistic in all social science research.
The mean is simply the arithmetic average of a distribution of scores, and researchers like it
because it provides a single, simple number that gives a rough summary of the distribution.
It is important to remember that although the mean provides a useful piece of information,
it does not tell you anything about how spread out the scores are (i.e., variance) or how many
scores in the distribution are close to the mean. It is possible for a distribution to have very
few scores at or near the mean.
The median is the score in the distribution that marks the 50th percentile. That is, 50% of
the scores in the distribution fall above the median and 50% fall below it. Researchers often use
the median when they want to divide their distribution scores into two equal groups (called a
median split). The median is also a useful statistic to examine when the scores in a distribution
are skewed or when there are a few extreme scores at the high end or the low end of the distribu-
tion. This is discussed in more detail in the following pages.
The mode is the least used of the measures of central tendency because it provides the least
amount of information. The mode simply indicates which score in the distribution occurs most
often, or has the highest frequency.
A Word abou t P opu l at ions and Sampl es
You will notice in Table 2.1 that there are two different symbols used for the mean,
–
X
and µ. Two different symbols are needed because it is important to distinguish between a
statistic that applies to a sample and a parameter that applies to a population. The sym-
bol used to represent the population mean is µ. Statistics are values derived from sample
data, whereas parameters are values that are either derived from or applied to population
data. It is important to note that all samples are representative of some population and
that all sample statistics can be used as estimates of population parameters. In the case of
the mean, the sample statistic is represented with the symbol
–
X. The distinction between
sample statistics and population parameters appears in several chapters (e.g., Chapters 1,
3, 5, and 7).

Measures of Central Tendency in Depth
The calculations for each meas
ure of central tendency are mercifully straightforward. With the
aid of a calculator or statistics software program, you will probably never need to calculate any of
these statistics by hand. But for the sake of knowledge and in the event you find yourself without
a calculator and in need of these statistics, here is the information you will need.
Because the mean is an average, calculating the mean involves adding, or summing, all of
the scores in a distribution and dividing by the number of scores. So, if you have 10 scores in a
distribution, you would add all of the scores together to find the sum and then divide the sum
by 10, which is the number of scores in the distribution. The formula for calculating the mean
is presented in Table 2.1.
The calculation of the median (P50) for a simple distribution of scores1 is even simpler than
the calculation of the mean. To find the median of a distribution, you need to first arrange all
of the scores in the distribution in order, from smallest to largest. Once this is done, you sim-
ply need to find the middle score in the distribution. If there is an odd number of scores in the
distribution, there will be a single score that marks the middle of the distribution. For example,
if there are 11 scores in the distribution arranged in descending order from smallest to largest,
the 6th score will be the median because there will be 5 scores below it and 5 scores above it.
However, if there are an even number of scores in the distribution, there is no single middle
score. In this case, the median is the average of the two scores in the middle of the distribution
(as long as the scores are arranged in order, from largest to smallest). For example, if there are
10 scores in a distribution, to find the median you will need to find the average of the 5th and
6th scores. To find this average, add the two scores together and divide by two.
To find the mode, there is no need to calculate anything. The mode is simply the category in
the distribution that has the highest number of scores, or the highest frequency. For example,
suppose you have the following distribution of IQ test scores from 10 students:
86 90 95 100 100 100 110 110 115 120
In this distribution, the score that occurs most frequently is 100, making it the mode of the
distribution. If a distribution has more than one category with the most common score, the dis-
tribution has multiple modes and is called multimodal. One common example of a multimodal
1 It is also possible to calculate the median of a grouped frequency distribution. For an excellent description of the technique for calculat-
ing a median from a grouped frequency distribution, see Spatz (2007), Basic Statistics: Tales of Distributions (9th ed.).
Table 2.1 Formula for Calculating the Mean
of a Distribution
µ =
ΣX
N
or
–
X =
ΣX
n
where
–
X is the sample mean
µ is the population mean
Σ means “the sum of”
X is an individual score in the distribution
n is the number of scores in the sample
N is the number of scores in the population

Measures of Central Tendency ■ 15
distribution is the bimodal distribution. Researchers often get bimodal distributions when they
ask people to respond to controversial questions that tend to polarize the public. For example,
if I were to ask a sample of 100 people how they feel about capital punishment, I might get the
results presented in Table 2.2. In this example, because most people either strongly oppose or
strongly support capital punishment, I end up with a bimodal distribution of scores.
On the following scale, please indicate how you feel about capital punishment.
1----------2----------3----------4----------5
Strongly Strongly
Oppose Support
Example: The Mean, Median, and Mode of a Skewed Distribution
As you will see in Chapter 4, when scores in a distribution are normally distributed, the mean,
median, and mode are all at the same point: the center of the distribution. In the messy world
of social science, however, the scores from a sample on a given variable are often not normally
distributed. When the scores in a distribution tend to bunch up at one end of the distribution
and there are a few scores at the other end, the distribution is said to be skewed. When working
with a skewed distribution, the mean, median, and mode are usually all at different points.
It is important to note that the procedures used to calculate a mean, median, and mode are
the same whether you are dealing with a skewed or a normal distribution. All that changes
are where these three measures of central tendency are in relation to each other. To illustrate,
I created a fictional distribution of scores based on a sample size of 30. Suppose that I were to
ask a sample of 30 randomly selected fifth graders whether they think it is important to do well
in school. Suppose further that I ask them to rate how important they think it is to do well in
school using a 5-point scale, with 1 = “not at all important” and 5 = “very important.” Because
most fifth graders tend to believe it is very important to do well in school, most of the scores in
this distribution are at the high end of the scale, with a few scores at the low end. I have arranged
my fictitious scores in order from smallest to largest and get the following distribution:
1 1 1 2 2 2 3 3 3 3
4 4 4 4 4 4 4 4 5 5
5 5 5 5 5 5 5 5 5 5
As you can see, there are only a few scores near the low end of the distribution (1 and 2) and
more at the high end of the distribution (4 and 5). To get a clear picture of what this skewed
distribution looks like, I have created the graph in Figure 2.1.
This graph provides a picture of what some skewed distributions look like. Notice how most
of the scores are clustered at the higher end of the distribution and there are a few scores creating
a tail toward the lower end. This is known as a negatively skewed distribution, because the tail
goes toward the lower end. If the tail of the distribution were pulled out toward the higher end,
this would have been a positively skewed distribution.
Table 2.2 Frequency of Responses
Category of Responses on the Scale
1 2 3 4 5
Frequency of Responses
in Each Category
45 3 4 3 45

A quick glance at the scores in the distribution, or at the graph, reveals that the mode is 5 because
there were more scores of 5 than any other number in the distribution.
To calculate the mean, we simply apply the formula mentioned earlier. That is, we add up all
of the scores (ΣX) and then divide this sum by the number of scores in the distribution (n). This
gives us a fraction of 113/30, which reduces to 3.7666. When we round to the second place after
the decimal, we end up with a mean of 3.77.
To find the median of this distribution, we arrange the scores in order from smallest to largest
and find the middle score. In this distribution, there are 30 scores, so there will be 2 in the middle.
When arranged in order, the 2 scores in the middle (the 15th and 16th scores) are both 4. When
we add these two scores together and divide by 2, we end up with 4, making our median 4.
As I mentioned earlier, the mean of a distribution can be affected by scores that are unusually
large or small for a distribution, sometimes called outliers, whereas the median is not affected
by such scores. In the case of a skewed distribution, the mean is usually pulled in the direction
of the tail, because the tail is where the outliers are. In a negatively skewed distribution, such as
the one presented previously, we would expect the mean to be smaller than the median, because
the mean is pulled toward the tail whereas the median is not. In our example, the mean (3.77) is
somewhat lower than the median (4). In positively skewed distributions, the mean is somewhat
higher than the median.
To provide a better sense of the effects of an outlier on the mean of a distribution, I present
two graphs showing the average life expectancy, at birth, of people in several different coun-
tries. In Figure 2.2, the life expectancy for 13 countries is presented in a line graph and the
0
2
4
6
8
10
12
14
Importance of School
1 2 3 4 5
Frequency
Figure 2.1 A skewed distribution.
50
55
60
65
70
75
80
85
Life
Expectancy
Country
J
a
p
a
n
A
u
s
t
r
a
l
i
a
C
a
n
a
d
a
F
r
a
n
c
e
G
e
r
m
a
n
y
U
n
i
t
e
d
K
i
n
g
d
o
m
U
n
i
t
e
d
S
t
a
t
e
s
C
u
b
a
S
a
u
d
i
A
r
a
b
i
a
M
e
x
i
c
o
S
e
r
b
i
a
T
u
r
k
e
y
U
g
a
n
d
a
Figure 2.2 Life expectancy at birth in several countries.

Measures of Central Tendency ■ 17
countries are arranged from the longest life expectancy (Japan) to the shortest (Uganda). As
you can see, there is a gradual decline in life expectancy from Japan through Turkey, but then
there is a dramatic drop off in life expectancy in Uganda. In this distribution of nations, Uganda
is an outlier. The average life expectancy for all of the countries except Uganda is 78.17 years,
whereas the average life expectancy for all 13 countries in Figure 2.2, including Uganda, drops to
76.21 years. The addition of a single country, Uganda, drops the average life expectancy for all of
the 13 countries combined by almost 2 full years. Two years may not sound like a lot, but when
you consider that this is about the same amount that separates the top 5 countries in Figure 2.2
from each other, you can see that 2 years can make a lot of difference in the ranking of countries
by the life expectancies of their populations.
The effects of outliers on the mean are more dramatic with smaller samples because the
mean is a statistic produced by combining all of the members of the distribution together. With
larger samples, one outlier does not produce a very dramatic effect. But with a small sample,
one outlier can produce a large change in the mean. To illustrate such an effect, I examined the
effect of Uganda’s life expectancy on the mean for a smaller subset of nations than appeared in
Figure 2.2. This new analysis is presented in Figure 2.3. Again, we see that the life expectancy
in Uganda (about 52 years) was much lower than the life expectancy in Japan, the United
States, and the United Kingdom (all near 80 years). The average life expectancy across the
three nations besides Uganda was 79.75 years, but this mean fell to 72.99 years when Uganda
was included. The addition of a single outlier pulled the mean down by nearly 7 years. In this
small dataset, the median would be between the United Kingdom and the United States, right
around 78.5 years. This example illustrates how an outlier pulls the mean in its direction. In
this case, the mean was well below the median.
Writing it Up
When you encounter descriptions of central tendency in published articles, or when you write
up such descriptions yourself, you will find such descriptions brief and simple. For the example
above, the proper write-up would be as follows: “In this distribution, the mean (–
x = 3.77) was
slightly lower than the median (P50 = 4.00), indicating a slight negative skew.”
Measures of central tendency, particularly the mean and the median, are some of the most used
and useful statistics for researchers. They each provide important information about an entire
distribution of scores in a single number. For example, we know that the average height of a
man in the United States is five feet nine inches tall. This single number is used to summarize
50
55
60
65
70
75
80
85
United Kingdom
Life
Expectancy
Country
Japan Uganda
United States
Figure 2.3 Life expectancy at birth in four countries.

information about millions of men in this country. But for the same reason that the mean and
median are useful, they can often be dangerous if we forget that a statistic such as the mean
ignores a lot of information about a distribution, including the great amount of variety that exists
in many distributions. Without considering the variety as well as the average, it becomes easy to
make sweeping generalizations, or stereotypes, based on the mean. The meas
ure of variance is
the topic of the next chapter.
Glossary of Terms and Symbols for Chapter 2
Bimodal: A distribution that has two values that have the highest frequency of scores.
Distribution: A collection, or group, of scores from a sample on a single variable. Often, but
not necessarily, these scores are arranged in order from smallest to largest.
Mean: The arithmetic average of a distribution of scores.
Median split: Dividing a distribution of scores into two equal groups by using the median
score as the divider. Those scores above the median are the “high” group whereas those
below the median are the “low” group.
Median: The score in a distribution that marks the 50th percentile. It is the score at which 50%
of the distribution falls below and 50% fall above.
Mode: The score in the distribution that occurs most frequently.
Multimodal: When a distribution of scores has two or more values that have the highest fre-
quency of scores.
Negativeskew: In a skewed distribution, when most of the scores are clustered at the higher end
of the distribution with a few scores creating a tail at the lower end of the distribution.
Outliers: Extreme scores that are more than two standard deviations above or below the
mean.
Positiveskew: In a skewed distribution, when most of the scores are clustered at the lower end of
the distribution with a few scores creating a tail at the higher end of the distribution.
Parameter: A value derived from the data collected from a population, or the value inferred to
the population from a sample statistic.
Population: The group from which data are collected or a sample is selected. The population
encompasses the entire group for which the data are alleged to apply.
Sample: An individual or group, selected from a population, from whom or which data are
collected.
Skew: When a distribution of scores has a high number of scores clustered at one end of the
distribution with relatively few scores spread out toward the other end of the distribu-
tion, forming a tail.
Statistic: A value derived from the data collected from a sample.
∑ The sum of; to sum.
X An individual score in a distribution.
∑X The sum of X; adding up all of the scores in a distribution.
–
X The mean of a sample.
µ The mean of a population.
n The number of cases, or scores, in a sample.
N The number of cases, or scores, in a population.
P50 Symbol for the median.

19
3
Chapter
Measures of Variability
Measures of central tendency, such as the mean and the median described in Chapter 2,
provide useful information. But it is important to recognize that these measures are limited
and, by themselves, do not provide a great deal of information. There is an old saying that
provides a caution about the mean: “If your head is in the freezer and your feet are in the
oven, on average you’re comfortable.” To illustrate, consider this example: Suppose I gave a
sample of 100 fifth-grade children a survey to assess their level of depression. Suppose fur-
ther that this sample had a mean of 10.0 on my depression survey and a median of 10.0 as
well. All we know from this information is that the mean and median are in the same place
in my distribution, and this place is 10.0. Now consider what we do not know. We do not
know if this is a high score or a low score. We do not know if all of the students in my sample
have about the same level of depression or if they differ from each other. We do not know
the highest depression score in our distribution or the lowest score. Simply put, we do not yet
know anything about the dispersion of scores in the distribution. In other words, we do not
yet know anything about the variety of the scores in the distribution.
There are three measures of dispersion that researchers typically examine: the range, the
variance, and the standard deviation. Of these, the standard deviation is perhaps the most
informative and certainly the most widely used.
Range
The range is simply the difference between the largest score (the maximum value) and the
smallest score (the minimum value) of a distribution. This statistic gives researchers a quick
sense of how spread out the scores of a distribution are, but it is not a particularly useful statistic
because it can be quite misleading. For example, in our depression survey described earlier, we
may have 1 student score a 1 and another score a 20, but the other 98 may all score 10. In this
example, the range will be 19 (20 – 1 = 19), but the scores really are not as spread out as the range
might suggest. Researchers often take a quick look at the range to see whether all or most of the
points on a scale, such as a survey, were covered in the sample.
Another common meas
ure of the range of scores in a distribution is the interquartile range
(IQR). Unlike the range, which is the difference between the largest and smallest score in the
distribution, the IQR is the difference between the score that marks the 75th percentile (the
third quartile) and the score that marks the 25th percentile (the first quartile). If the scores in
a distribution were arranged in order from largest to smallest and then divided into groups of
equal size, the IQR would contain the scores in the two middle quartiles (see Figure 3.1).
Variance
The variance provides a statistical average of the amount of dispersion in a distribution of scores.
Because of the mathematical manipulation needed to produce a variance statistic (more about
this in the next section), variance, by itself, is not often used by researchers to gain a sense of a

distribution. In general, variance is used more as a step in the calculation of other statistics (e.g.,
analysis of variance) than as a stand-alone statistic. But with a simple manipulation, the variance
can be transformed into the standard deviation, which is one of the statistician’s favorite tools.
Standard Deviation
The best way to understand a standard deviation is to consider what the two words mean.
Deviation, in this case, refers to the difference between an individual score in a distribution and
the average score for the distribution. So if the average score for a distribution is 10 (as in our
previous example), and an individual child has a score of 12, the deviation is 2. The other word
in the term standard deviation is standard. In this case, standard means typical, or average. So a
standard deviation is the typical, or average, deviation between individual scores in a distribu-
tion and the mean for the distribution.1 This is a very useful statistic because it provides a handy
meas
ure of how spread out the scores are in the distribution. When combined, the mean and
standard deviation provide a pretty good picture of what the distribution of scores is like.
In a sense, the range provides a meas
ure of the total spread in a distribution (i.e., from the
lowest to the highest scores), whereas the variance and standard deviation are measures of the
average amount of spread within the distribution. Researchers tend to look at the range when
they want a quick snapshot of a distribution, such as when they want to know whether all of
the response categories on a survey question have been used (i.e., did people use all 5 points on
the 5-point Likert scale?) or they want a sense of the overall balance of scores in the distribu-
tion. Researchers rarely look at the variance alone, because it does not use the same scale as the
original meas
ure of a variable, although the variance statistic is very useful for the calculation
of other statistics (such as analysis of variance; see Chapter 10). The standard deviation is a very
useful statistic that researchers constantly examine to provide the most easily interpretable and
meaningful meas
ure of the average dispersion of scores in a distribution.
Measures of Variability in Depth
Calculating the Variance and Standard Deviation
There are two central issues that I need to address when considering the formulas for calculat-
ing the variance and standard deviation of a distribution: (1) whether to use the formula for the
sample or the population, and (2) how to make sense of these formulas.
1 Although the standard deviation is technically not the “average deviation” for a distribution of scores, in practice this is a useful heu-
ristic for gaining a rough conceptual understanding of what this statistic is. The actual formula for the average deviation would be
Σ(|X – mean|)/N.
Interquartile range
f
75%
–
X
25%
Figure 3.1 The interquartile range.

Measures of Variability ■ 21
It is important to note that the formulas for calculating the variance and the standard devia-
tion differ depending on whether you are working with a distribution of scores taken from a
sample or from a population. The reason these two formulas are different is quite complex and
requires more space than allowed in a short book like this. I provide an overly brief explana-
tion here and then encourage you to find a more thorough explanation in a traditional statistics
textbook. Briefly, when we do not know the population mean, we must use the sample mean as
an estimate. But the sample mean will probably differ from the population mean. Whenever we
use a number other than the actual mean to calculate the variance, we will end up with a larger
variance, and therefore a larger standard deviation, than if we had used the actual mean. This
will be true regardless of whether the number we use in our formula is smaller or larger than our
actual mean. Because the sample mean usually differs from the population mean, the variance
and standard deviation that we calculate using the sample mean will probably be smaller than
it would have been had we used the population mean. Therefore, when we use the sample mean
to generate an estimate of the population variance or standard deviation, we will actually under-
estimate the size of the true variance in the population because if we had used the population
mean in place of the sample mean, we would have created a larger sum of squared deviations,
and a larger variance and standard deviation. To adjust for this underestimation, we use n – 1
in the denominator of our sample formulas. Smaller denominators produce larger overall vari-
ance and standard deviation statistics, which will be more accurate estimates of the population
parameters.
Sampl e St at ist ics As Est imat es Of Popu l at ion Paramet ers
It is important to remember that most statistics, although generated from sample data, are
used to make estimations about the population. As discussed in Chapter 1, researchers usu-
ally want to use their sample data to make some inferences about the population that the
sample represents. Therefore, sample statistics often represent estimates of the population
parameters. This point is discussed in more detail later in the book when examining infer-
ential statistics. But it is important to keep this in mind as you read about these measures
of variation. The formulas for calculating the variance and standard deviation of sample
data are actually designed to make these sample statistics better estimates of the population
parameters (i.e., the population variance and standard deviation). In later chapters (e.g., 6,
7, 8), you will see how researchers use statistics like standard errors, confidence intervals,
and probabilities to figure out how well their sample data estimate population parameters.
The formulas for calculating the variance and standard deviation of a population and the
estimates of the population variance and standard deviation based on a sample are presented in
Table 3.1. As you can see, the formulas for calculating the variance and the standard deviation
are virtually identical. Because both require that you calculate the variance first, we begin with
the formulas for calculating the variance (see the upper row of Table 3.1). This formula is known
as the deviation score formula.2
When working with a population distribution, the formulas for both the variance and the
standard deviation have a denominator of N, which is the size of the population. In the real
world of research, particularly social science research, we usually assume that we are working
with a sample that represents a larger population. For example, if I study the effectiveness of my
new reading program with a class of second graders, as a researcher I assume that these particu-
lar second graders represent a larger population of second graders, or students more generally.
2 It is also possible to calculate the variance and standard deviation using the raw score formula, which does not require that you calculate
the mean. The raw score formula is included in most standard statistics textbooks.

Because of this type of inference, researchers generally think of their research participants as
a sample rather than a population, and the formula for calculating the variance of a sample is
the formula more often used. Notice that the formula for calculating the variance of a sample
is identical to that used for the population, except the denominator for the sample formula is
n – 1.
How much of a difference does it make if we use N or n – 1 in our denominator? Well, that
depends on the size of the sample. If we have a sample of 500 people, there is virtually no differ-
ence between the variance formula for the population and for the estimate based on the sample.
After all, dividing a numerator by 500 is almost the same as dividing it by 499. But when we
have a small sample, such as a sample of 10, then there is a relatively large difference between
the results produced by the population and sample formulas.
To illustrate, suppose that I am calculating a standard deviation. After crunching the num-
bers, I find a numerator of 100. I divide this numerator by four different values depending on
the sample size and whether we divide by N or n – 1. The results of these calculations are sum-
marized in Table 3.2. With a sample size of 500, subtracting 1 from the denominator alters the
size of the standard deviation by less than one one-thousandth. With a sample size of 10, sub-
tracting 1 from the denominator increases the size of the standard deviation by nearly 2 tenths.
Note that in both the population and sample examples, given the same value in the numerator,
larger samples produce dramatically smaller standard deviations. This makes sense because the
larger the sample, the more likely each member of the sample will have a value near the mean,
thereby producing a smaller standard deviation.
The second issue to address involves making sense of the formulas for calculating the vari-
ance. In all honesty, there will be very few times that you will need to use this formula. Outside
of my teaching duties, I haven’t calculated a standard deviation by hand since my first statistics
Table 3.1 Variance and Standard Deviation Formulas
Population Estimate Based on a Sample
Variance σ
µ
2
2
=
−
Σ( )
X
N
where Σ = to sum
X = a score in the distribution
µ = the population mean
N = the number of cases in the population
s
X X
n
2
2
1
=
−
−
Σ( )
where Σ = to sum
–
X = the sample mean
n = the number of cases in the sample
Standard Deviation σ
µ
=
−
Σ( )
X
N
2
where Σ = to sum
µ = the population mean
N = the number of cases in the population
s
X X
n
=
−
−
Σ( )2
1
where Σ = to sum
–
X = the sample mean
n = the number of cases in the sample
Table 3.2 Effects of Sample Size and n – 1
on Standard Deviation
N = 500 N = 10
Population σ = =
100
500
44721
. σ = =
100
10
3 16
.
Sample s = =
100
499
44766
. s = =
100
9
3 33
.

course. Thankfully, all computer statistics and spreadsheet programs, and many calculators,
compute the variance and standard deviation for us. Nevertheless, it is mildly interesting and
quite informative to examine how these variance formulas work.
To begin this examination, let me remind you that the variance is simply an average of a
distribution. To get an average, we need to add up all of the scores in a distribution and divide
this sum by the number of scores in the distribution, which is n (remember the formula for cal-
culating the mean in Chapter 2?). With the variance, however, we need to remember that we
are not interested in the average score of the distribution. Rather, we are interested in the average
difference, or deviation, between each score in the distribution and the mean of the distribution.
To get this information, we have to calculate a deviation score for each individual score in the
distribution (see Figure 3.2). This score is calculated by taking an individual score and subtract-
ing the mean from that score. If we compute a deviation score for each individual score in the
distribution, then we can sum the deviation scores and divide by n to get the average, or stan-
dard, deviation, right? Not quite.
The problem here is that, by definition, the mean of a distribution is the mathematical middle
of the distribution. Therefore, some of the scores in the distribution will fall above the mean
(producing positive deviation scores), and some will fall below the mean (producing negative
deviation scores). When we add these positive and negative deviation scores together, the sum
will be zero. Because the mean is the mathematical middle of the distribution, we will get zero
when we add up these deviation scores no matter how big or small our sample, or how skewed
or normal our distribution. And because we cannot find an average of zero (i.e., zero divided by
n is zero, no matter what n is), we need to do something to get rid of this zero.
The solution statisticians came up with is to make each deviation score positive by squaring
it. So, for each score in a distribution, we subtract the mean of the distribution and then square
the deviation. If you look at the deviation score formulas in Table 3.1, you will see that all
that the formula is doing with (X – µ)2 is to take each score, subtract the mean, and square the
resulting deviation score. What you get when you do this is the all-important squared devia-
tion, which is used all the time in statistics. If we then put a summation sign in front, we have
Σ(X – µ)2. What this tells us is that after we produce a squared deviation score for each case in
our distribution, we then need to add up all of these squared deviations, giving us the sum of
squared deviations, or the sum of squares (SS). Once this is done, we divide by the number of
cases in our distribution, and we get an average, or mean, of the squared deviations. This is our
variance.
The final step in this process is converting the variance into a standard deviation. Remember
that to calculate the variance, we have to square each deviation score. We do this to avoid get-
ting a sum of zero in our numerator. When we square these scores, we change our statistic from
our original scale of meas
ure
ment (i.e., whatever units of meas
ure
ment were used to generate
X = 12
X = 10
–
Deviation
f
Figure 3.2 A deviation.

our distribution of scores) to a squared score. To reverse this process and give us a statistic that
is back to our original unit of meas
ure
ment, we merely need to take the square root of our vari-
ance. When we do this, we switch from the variance to the standard deviation. Therefore, the
formula for calculating the standard deviation is exactly the same as the formula for calculating
the variance, except we put a big square root symbol over the whole formula. Notice that because
of the squaring and square rooting process, the standard deviation and the variance are always
positive numbers.
Why Have Variance?
If the variance is a difficult statistic to understand, and rarely examined by researchers, why not
just eliminate this statistic and jump straight to the standard deviation? There are two reasons.
First, we need to calculate the variance before we can find the standard deviation anyway, so it
is not more work. Second, the fundamental piece of the variance formula, which is the sum of
the squared deviations, is used in a number of other statistics, most notably analysis of variance
(ANOVA). When you learn about more advanced statistics such as ANOVA (Chapter 10), fac-
torial ANOVA (Chapter 11), and even regression (Chapter 13), you will see that each of these
statistics uses the sum of squares, which is just another way of saying the sum of the squared
deviations. Because the sum of squares is such an important piece of so many statistics, the vari-
ance statistic has maintained a place in the teaching of basic statistics.
Example: Examining the Range, Variance, and Standard Deviation
I conducted a study in which I gave questionnaires to approximately 500 high school students
in the 9th and 11th grades. In the examples that follow, we examine the mean, range, variance,
and standard deviation of the distribution of responses to two of these questions. To make sense
of these (and all) statistics, you need to know the exact wording of the survey items and the
response scale used to answer the survey items. Although this may sound obvious, I mention it
here because, if you notice, much of the statistical information reported in the news (e.g., the
results of polls) does not provide the exact wording of the questions or the response choices.
Without this information, it is difficult to know exactly what the responses mean, and “lying
with statistics” becomes easier.
The first survey item we examine reads, “If I have enough time, I can do even the most dif-
ficult work in this class.” This item is designed to meas
ure students’ confidence in their abilities to
succeed in their classwork. Students were asked to respond to this question by circling a number
on a scale from 1 to 5. On this scale, circling the 1 means that the statement is “not at all true”
and the 5 means “very true.” So students were basically asked to indicate how true they felt the
statement was on a scale from 1 to 5, with higher numbers indicating a stronger belief that
the statement was true.
I received responses from 491 students on this item. The distribution of responses produced
the following statistics:
Sample Size = 491
Mean = 4.21
Standard Deviation = .98
Variance = (.98)2 = .96
Range = 5 – 1 = 4

A graph of the frequency distribution for the responses on this item appears in Figure 3.3. As
you can see in this graph, most of the students in the sample circled number 4 or number 5 on
the response scale, indicating that they felt the item was quite true (i.e., that they were confident
in their ability to do their classwork if they were given enough time). Because most students
circled a 4 or a 5, the average score on this item is quite high (4.21 out of a possible 5). This is a
negatively skewed distribution.
The graph in Figure 3.3 also provides information about the variety of scores in this distribu-
tion. Although our range statistic is 4, indicating that students in the sample circled both the
highest and the lowest number on the response scale, we can see that the range does not really
provide much useful information. For example, the range does not tell us that most of the stu-
dents in our sample scored at the high end of the scale. By combining the information from the
range statistic with the mean statistic, we can reach the following conclusion: “Although the dis-
tribution of scores on this item covers the full range, it appears that most scores are at the higher
end of the response scale.”
Now that we’ve determined that (1) the distribution of scores covers the full range of possible
scores (i.e., from 1 to 5), and (2) most of the responses are at the high end of the scale (because
the mean is 4.21 out of a possible 5), we may want a more precise meas
ure of the average amount
of variety among the scores in the distribution. For this we turn to the variance and standard
deviation statistics. In this example, the variance (.96) is almost exactly the same as the stan-
dard deviation (.98). This is something of a fluke. Do not be fooled. It is quite rare for the vari-
ance and standard deviation to be so similar. In fact, this only happens if the standard deviation
is about 1.0, because 1.0 squared is 1.0. So in this rare case, the variance and standard devia-
tion provide almost the same information. Namely, they indicate that the average difference
between an individual score in the distribution and the mean for the distribution is about 1 point
on the 5-point scale.
Taken together, these statistics tell us the same things that the graph tells us, but more pre-
cisely. Namely, we now know that (1) students in the study answered this item covering the
whole range of response choices (i.e., 1 – 5); (2) most of the students answered at or near the top
of the range, because the mean is quite high; and (3) the scores in this distribution generally pack
fairly closely together with most students having circled a number within 1 point of the mean,
because the standard deviation was .98. The variance tells us that the average squared deviation is
.96, and we scratch our heads, wonder what good it does us to know the average squared devia-
tion, and move on.
In our second example, we examine students’ responses to the item, “I would feel really good
if I were the only one who could answer the teacher’s question in class.” This item is one of
9
23
73
139
247
0
50
100
150
200
250
300
1 2 3 4 5
Frequency
Scores on Confidence Item
Figure 3.3 Frequency distribution of scores on the confidence item.

several on the survey designed to meas
ure students’ desires to demonstrate to others that they
are smart, or academically able.
We received responses from 491 students on this item, and the distribution produced the fol-
lowing statistics:
Sample Size = 491
Mean = 2.92
Standard Deviation = 1.43
Variance = (1.43)2 = 2.04
Range = 5 – 1 = 4
Figure 3.4 illustrates the distribution of students’ responses to this item across each of the
five response categories. It is obvious, when looking at this graph, how the distribution of
scores on this item differs from the distribution of scores on the confidence item presented in
Figure 3.3. But if we didn’t have this graph, how could we use the statistics to discover the
differences between the distributions of scores on these two items?
Notice that, as with the previous item, the range is 4, indicating that some students circled
the number 1 on the response scale and some circled the number 5. Because the ranges for
both the confidence and the wanting to appear able items are equal (i.e., 4), they do nothing to
indicate the differences in the distributions of the responses to these two items. That is why the
range is not a particularly useful statistic—it simply does not provide very much information.
Our first real indication that the distributions differ substantially comes from a comparison of
the means. In the previous example, the mean of 4.21 indicated that most of the students must
have circled either a 4 or a 5 on the response scale. For this second item, the mean of 2.92 is a bit
less informative. Although it provides an average score, it is impossible from just examining the
mean to determine whether most students circled a 2 or 3 on the scale, or whether roughly equal
numbers of students circled each of the five numbers on the response scale, or whether almost
half of the students circled 1 whereas the other half circled 5. All three scenarios would produce
a mean of about 2.92, because that is roughly the middle of the response scale.
To get a better picture of this distribution, we need to consider the standard deviation in
conjunction with the mean. Before discussing the actual standard deviation for this distribution
of scores, let us briefly consider what we would expect the standard deviation to be for each of
the three scenarios just described. First, if almost all of the students circled a 2 or a 3 on the
response scale, we would expect a fairly small standard deviation, as we saw in the previous
example using the confidence item. The more similar the responses are to an item, the smaller
the standard deviation. However, if half of the students circled 1 and the other half circled 5,
115
81
120
77
98
0
20
40
60
80
100
120
140
1 2 3 4 5
Frequency
Scores on Desire to Demonstrate Ability Item
Figure 3.4 Frequency distribution of scores on the desire to demonstrate ability item.

we would expect a large standard deviation (about 2.0) because each score would be about two
units away from the mean i.e., if the mean is about 3.0 and each response is either 1 or 5, each
response is about two units away from the mean. Finally, if the responses are fairly evenly spread
out across the five response categories, we would expect a moderately sized standard deviation
(about 1.50).
Now, when we look at the actual mean for this distribution (2.92) and the actual standard
deviation (1.43), we can develop a rough picture of the distribution in our minds. Because we
know that on a scale from 1 to 5, a mean of 2.92 is about in the middle, we can guess that the
distribution looks somewhat symmetrical (i.e., that there will be roughly the same number of
responses in the 4 and 5 categories as there are in the 1 and 2 categories. Furthermore, because
we’ve got a moderately sized standard deviation of 1.43, we know that the scores are pretty well
spread out, with a healthy number of students in each of the five response categories. So we
know that we didn’t get an overwhelming number of students circling 3 and we didn’t get stu-
dents circling only 1 or 5. At this point, this is about all we can say about this distribution: The
mean is near the middle of the scale, and the responses are pretty well spread out across the five
response categories. To say any more, we would need to look at the number of responses in each
category, such as that presented in Figure 3.4.
As we look at the actual distribution of scores presented in the graph in Figure 3.4, we can
see that the predictions we generated from our statistics about the shape of the distribution are
pretty accurate. Notice that we did not need to consider the variance at all, because the variance
in this example (2.04) is on a different scale of meas
ure
ment than our original 5-point response
scale, and therefore is very difficult to interpret. Variance is an important statistic for many tech-
niques (e.g., ANOVA, regression), but it does little to help us understand the shape of a distribu-
tion of scores. The mean, standard deviation, and to a lesser extent the range, when considered
together, can provide a rough picture of a distribution of scores. Often, a rough picture is all a
researcher needs or wants. Sometimes, however, researchers need to know more precisely the
characteristics of a distribution of scores. In that case, a picture, such as a graph, may be worth
a thousand words.
Another useful way to examine a distribution of scores is to create a boxplot. In Figure 3.5,
a boxplot is presented for the same variable that is represented in Figure 3.4, wanting to dem-
onstrate ability. This boxplot was produced in the SPSS statistical software program. The box
in this graph contains some very useful information. First, the thick line in the middle of the
box represents the median of this distribution of scores. The top line of the box represents the
75th percentile of the distribution and the bottom line represents the 25th percentile. Therefore,
the top and bottom lines of the box reveal the interquartile range (IQR) for this distribution.
In other words, 50% of the scores on this variable in this distribution are contained within the
upper and lower lines of this box (i.e., 50% of the scores are between just above a score of 2
6
5
4
3
2
1
0
Figure 3.5 Boxplot for the desire to appear able variable.

and just below a score of 4). The vertical lines coming out of the top and bottom of the box and
culminating in horizontal lines reveal the largest and smallest scores in the distribution, or the
range. These scores are 5 and 1, producing a range of 5 – 1 = 4. As you can see, the boxplot in
Figure 3.5 contains a lot of useful information about the spread of scores on this variable in a
single picture.
Measures of variation, such as the variance, standard deviation, and range, are important descrip-
tive statistics. They provide useful information about how spread out the scores of a distribution
are, and the shape of the distribution. Perhaps even more important than their utility as descrip-
tors of a single distribution of scores is their role in more advanced statistics such as those com-
ing in later chapters (e.g., ANOVA in Chapters 10, 11, and 12). In the next chapter, we examine
the properties of a theoretical distribution, the normal distribution, that has a specific shape and
characteristics. Using some of the concepts from Chapter 3, we can see how the normal distribu-
tion can be used to make inferences about the population based on sample data.
Glossary of Terms and Symbols for Chapter 3
Boxplot: A graphic representation of the distribution of scores on a variable that includes the
range, the median, and the interquartile range.
Interquartile range (IQR): The difference between the 75th percentile and 25th percentile
scores in a distribution.
Range: The difference between the largest score and the smallest score of a distribution.
Squared deviation: The difference between an individual score in a distribution and the
mean for the distribution, squared.
Standard deviation: The average deviation between the individual scores in the distribution
and the mean for the distribution.
Sum of squared deviations, sum of squares: The sum of each squared deviation for all of the
cases in the sample.
Variance: The sum of the squared deviations divided by the number of cases in the population,
or by the number of cases minus one in the sample.
µ The population mean.
X An individual score in a distribution.
s2 The sample variance.
s The sample standard deviation.
σ The population standard deviation.
σ2 The population variance.
SS The sum of squares, or sum of squared deviations.
n The number of cases in the sample.
N The number of cases in the population.

29
The normal distribution is a concept with which most people have some familiarity, although
they often have never heard of the term. A more familiar name for the normal distribution is
the bell curve, because a normal distribution forms the shape of a bell. The normal distribution
is extremely important to statistics and has some specific characteristics that make it so useful.
In this chapter, I briefly describe what a normal distribution is and why it is so important to
researchers. Then I discuss some of the features of the normal distribution, and of sampling, in
more depth.
Characteristics of the Normal Distribution
In Figure 4.1, I present a simple line graph that depicts a normal distribution. Recall from the
discussion of graphs in Chapter 1 that this type of graph shows the frequency, i.e., number of
cases, with particular scores on a single variable. So in this graph, the y-axis shows the frequency
of the cases and the x-axis would show the score on the variable of interest. For example, if the
variable were scores on an IQ test, the x-axis would have the scores ranging from smallest to
largest. The mean, median, and mode would be 100, and the peak of the line shows that the
frequency of cases is highest at 100 (i.e., the mode). As you move away from the mode in either
direction, the height of the line goes down, indicating fewer cases (i.e., lower frequencies) at
those other scores.
If you take a look at the normal distribution shape presented in Figure 4.1, you may notice
that the normal distribution has three fundamental characteristics. First, it is symmetrical,
meaning that the upper half and the lower half of the distribution are mirror images of each
other. Second, the mean, median, and mode are all in the same place, in the center of the distri-
bution (i.e., the top of the bell curve). Because of this second feature, the normal distribution is
highest in the middle, it is unimodal, and it curves downward toward the top and bottom of the
distribution. Finally, the normal distribution is asymptotic, meaning that the upper and lower
tails of the distribution never actually touch the baseline, also known as the x-axis.
Why Is the Normal Distribution So Important?
When researchers collect data from a sample, sometimes all they want to know about are char-
acteristics of the sample. For example, if I wanted to examine the eating habits of 100 first-year
college students, I would just select 100 students, ask them what they eat, and summarize my
data. These data might give me statistics such as the average number of calories consumed each
day by the 100 students in my sample, the most commonly eaten foods, the variety of foods
eaten, and so on. All of these statistics simply describe characteristics of my sample, and are
therefore called descriptive statistics. Descriptive statistics generally are used only to describe
a specific sample. When all we care about is describing a specific sample, it does not matter
whether the scores from the sample are normally distributed or not.
4
Chapter
The Normal Distribution

Many times, however, researchers want to do more than simply describe a sample. Sometimes,
they want to know what the exact probability is of something occurring in their sample just due
to chance. For example, if the average student in my sample consumes 2,000 calories a day, what
are the chances, or probability, of having a student in the sample who consumes 5,000 calories
a day? The three characteristics of the normal distribution are each critical in statistics because
they allow us to make good use of probability statistics.
In addition, researchers often want to be able to make inferences about the population based
on the data they collect from their sample. To determine whether some phenomenon observed
in a sample represents an actual phenomenon in the population from which the sample was
drawn, inferential statistics are used. For example, suppose I begin with an assumption that in
the population of men and women there is no difference in the average number of calories con-
sumed in a day. This assumption of no differences is known as a null hypothesis. Now suppose
that I select a sample of men and a sample of women, compare their average daily calorie con-
sumption, and find that the men eat an average of 200 calories more per day than do the women.
Given my null hypothesis of no differences, what is the probability of finding a difference this
large between my samples by chance? To calculate these probabilities, I need to rely on the normal
distribution, because the characteristics of the normal distribution allow statisticians to generate
exact probability statistics. In the next section, I will briefly explain how this works.
The Normal Distribution in Depth
It is important to note that the normal distribution is what is known in statistics as a theoretical
distribution. That is, one rarely, if ever, gets a distribution of scores from a sample that forms an
exact, normal distribution. Rather, what you get when you collect data is a distribution of scores
that may or may not approach a normal, bell-shaped curve. Because the theoretical normal dis-
tribution is what statisticians use to develop probabilities, a distribution of scores that is not nor-
mal may be at odds with these probabilities. Therefore, there are a number of statistics that begin
with the assumption that scores are normally distributed. When this assumption is violated (i.e.,
when the scores in a distribution are not normally distributed), there can be dire consequences.
The most obvious consequence of violating the assumption of a normal distribution is that
the probabilities associated with a normal distribution are not valid. For example, if you have
a normal distribution of scores on some variable (e.g., IQ test scores of adults in the United
States), you can use the probabilities based on the normal distribution to determine exactly what
percentage of the scores in the distribution will be 120 or higher on the IQ test (see Chapter 4
for a description of how to do this). But suppose the scores in our distribution do not form a nor-
mal distribution. Suppose, for some reason, we have an unusually large number of high scores
f
Mean
Median
Mode
Figure 4.1 The normal distribution.

The Normal Distribution ■ 31
(e.g., over 120) and an unusually small number of low scores (e.g., below 90) in our distribution.
If this were the case, when we use probability estimates based on the normal distribution, we
would underestimate the actual number of high scores in our distribution and overestimate the
actual number of low scores in our distribution.
The Relationship between Sampling Method and the Normal Distribution
As I discussed in Chapter 1, researchers use a variety of different ways of selecting samples.
Sometimes, samples are selected so that they represent the population in specific ways, such as
the percentage of men or the proportion of wealthy individuals (representativesampling). Other
times, samples are selected randomly with the hope that any differences between the sample and
the population are also random, rather than systematic (random sampling). Often, however,
samples are selected for their convenience rather than for how they represent the larger popula-
tion (convenience sampling). The problem of violating the assumption of normality becomes
most problematic when our sample is not an adequate representation of our population.
The relationship between the normal distribution and sampling methods is as follows. The
probabilities generated from the normal distribution depend on (1) the shape of the distribution
and (2) the idea that the sample is not somehow systematically different from the population. If
I select a sample randomly from a population, I know that this sample may not look the same as
another sample of equal size selected randomly from the same population. But any differences
between my sample and other random samples of the same size selected from the same popula-
tion would differ from each other randomly, not systematically. In other words, my sampling
method was not biased such that I would continually select a sample from one end of my popu-
lation (e.g., the more wealthy, the better educated, the higher achieving) if I continued using
the same method for selecting my sample. Contrast this with a convenience sampling method.
If I only select schools that are near my home or work, I will continually select schools with
similar characteristics. For example, if I live in the Bible Belt, my sample will probably be biased
in that my sample will more likely hold fundamentalist religious beliefs than the larger popu-
lation of schoolchildren. Now if this characteristic is not related to the variable I am studying
(e.g., achievement), then it may not matter that my sample is biased in this way. But if this bias
is related to my variable of interest (e.g., “How strongly do American schoolchildren believe in
God?”), then I may have a problem.
Suppose that I live and work in Cambridge, Massachusetts. Cambridge is in a section of
the country with an inordinate number of highly educated people because there are a number
of high-quality universities in the immediate area (Harvard, MIT, Boston College, Boston
University, etc.). If I conduct a study of student achievement using a convenience sample from
this area, and try to argue that my sample represents the larger population of students in the
United States, probabilities that are based on the normal distribution may not apply. That is
because my sample will be more likely than the national average to score at the high end of the
distribution. If, based on my sample, I try to predict the average achievement level of students
in the United States, or the percentage that score in the bottom quartile, or the score that marks
the 75th percentile, all of these predictions will be off, because the probabilities that are gen-
erated by the normal distribution assume that the sample is not biased. If this assumption is
violated, we cannot trust our results.
Skew and Kurtosis
Two characteristics used to describe a distribution of scores are skew and kurtosis. When a
sample of scores is not normally distributed (i.e., not the bell shape), there are a variety of shapes
it can assume. One way a distribution can deviate from the bell shape is if there is a bunching of
scores at one end and a few scores pulling a tail of the distribution out toward the other end. If
there are a few scores creating an elongated tail at the higher end of the distribution, it is said to
be positively skewed. If the tail is pulled out toward the lower end of the distribution, the shape

is called negatively skewed. These shapes are depicted in Figure 4.2. As you can see, the mean
in a skewed distribution is pulled in the direction of the tail. Skew does not affect the median,
however. So a positively skewed distribution will have a higher mean than median, and a nega-
tively skewed distribution will have a smaller mean than median. If you recall that the mean and
median are the same in a normal distribution, you can see how the skew affects the mean relative
to the median.
As you might have guessed, skewed distributions can distort the accuracy of the probabilities
based on the normal distribution. For example, if most of the scores in a distribution occur at
the low end with a few scores at the higher end (positively skewed distribution), the probabilities
that are based on the normal distribution will underestimate the actual number of scores at the
lower end of this skewed distribution and overestimate the number of scores at the higher end of
the distribution. In a negatively skewed distribution, the opposite pattern of errors in prediction
will occur.
Kurtosis refers to the shape of the distribution in terms of height, or flatness. When a distribu-
tion has a peak that is higher than that found in a normal, bell-shaped distribution, it is called
leptokurtic. When a distribution is flatter than a normal distribution, it is called platykurtic.
Because the normal distribution contains a certain percentage of scores in the middle area
(i.e., about 68% of the scores fall between 1 standard deviation above and 1 standard deviation
below the mean), a distribution that is either platykurtic or leptokurtic will likely have a differ-
ent percentage of scores near the mean than will a normal distribution. Specifically, a leptokurtic
distribution will probably have a greater percentage of scores closer to the mean and fewer in the
upper and lower tails of the distribution, whereas a platykurtic distribution will have more scores
at the ends and fewer in the middle than will a normal distribution.
f
Mean
f
Mean
Figure 4.2 Positively and negatively skewed distributions.

Example: Applying Normal Distribution Probabilities to a Nonnormal Distribution
To illustrate some of the difficulties that can arise when we try to apply the probabilities that
are generated from using the normal distribution to a distribution of scores that is skewed, I
present a distribution of sixth-grade students’ scores on a meas
ure of self-esteem. In these data,
677 students completed a questionnaire that included four items designed to meas
ure students’
overall sense of self-esteem. Examples of these questions include “On the whole, I am satisfied
with myself” and “I feel I have a number of good qualities.” Students responded to each of these
four questions using a 5-point rating scale with 1 = “not at all true” and 5 = “very true.” Students’
responses on these four items were then averaged, creating a single self-esteem score that ranged
from a possible low of 1 to a possible high of 5. The frequency distribution for this self-esteem
variable is presented in Figure 4.3.
As you can see, the distribution of scores presented in Figure 4.3 does not form a nice, nor-
mal, bell-shaped distribution. Rather, most of the students in this sample scored at the high end
of the distribution, and a long tail extends out toward the lower end of the scale. This is a classic,
negatively skewed distribution of scores. The happy part of this story is that most of the students
in this sample appear to feel quite good about themselves. The sad part of the story is that some
of the assumptions of the normal distribution are violated by this skewed distribution. Let’s take
a look at some specifics.
One of the qualities of a normal distribution is that it is symmetrical, with an equal percentage
of the scores between the mean and 1 standard deviation below the mean as there are between the
mean and 1 standard deviation above the mean. In other words, in a normal distribution, there
should be about 34% of the scores within 1 standard deviation above the mean and 34% within
1 standard deviation below the mean. In our distribution of self-esteem scores presented earlier,
the mean is 4.28 and the standard deviation is .72. A full 50% of the distribution falls between
the mean and 1 standard deviation above the mean in this group of scores (see Figure 4.4). So,
although I might predict that about 16% of my distribution will have scores more than 1 standard
deviation above the mean in a normal distribution, in my skewed distribution of self-esteem
scores, I can see that there are no students with scores more than 1 standard deviation above the
mean. In Chapter 5, I present a more thorough discussion of how to use the normal distribution
to calculate standard deviation units and percentile scores in a normal distribution.
As this example demonstrates, the probabilities that statisticians have generated using the
normal distribution may not apply well to skewed or otherwise nonnormal distributions of data.
This should not lead you to believe, however, that nonnormal distributions of scores are worth-
less. In fact, even if you have a nonnormal distribution of scores in your sample, these scores
0
20
40
60
80
100
120
140
160
180
200
1.4 2 2.6 3.2 3.8 4.4 5
Frequency
Self-esteem Scores
Figure 4.3 Frequency distribution for self-esteem scores.

can create normal sampling distributions for use in inferential statistics (see Chapter 6). What
is perhaps most important to keep in mind is that a nonnormal distribution of scores may be an
indication that your sample differs in important and systematic ways from the population that
it is supposed to represent. When making inferences about a population based on a sample,
be very careful to define the population precisely and to be aware of any biases you may have
introduced by your method of selecting your sample. It is also important to note, however, that
not all variables are normally distributed in the population. Therefore, non
normal sample data
may be an accurate representation of nonnormal population data, as well as an indication that
the sample does not accurately represent the population. The normal distribution can be used
to generate probabilities about the likelihood of selecting an individual or another sample with
certain characteristics (e.g., distance from the mean) from a population. If your sample is not
normal and your method of selecting the sample may be systematically biased to include those
with certain characteristics (e.g., higher than average achievers, lower than average income),
then the probabilities of the normal distribution may not apply well to your sample.
The theoretical normal distribution is a critical element of statistics primarily because many of
the probabilities that are used in inferential statistics are based on the assumption of normal
distributions. As you will see in coming chapters, statisticians use these probabilities to deter-
mine the probability of getting certain statistics and to make inferences about the population
based on the sample. Even if the data in a sample are not normally distributed, it is possible that
the data in the population from which the sample was selected may be normally distributed. In
Chapter 5, I describe how the normal distribution, through the use of z scores and standardiza-
tion, is used to determine the probability of obtaining an individual score from a sample that is
a certain distance away from the sample mean. You will also learn about other fun statistics like
percentile scores in Chapter 5.
Glossary of Terms for Chapter 4
Asymptotic: When the ends, or “tails,” of a distribution never intersect with the x-axis; they
extend indefinitely.
Bell curve: The common term for the normal distribution. It is called the bell curve because
of its bell-like shape.
Biased: When a sample is not selected randomly, it may be a biased sample. A sample is biased
when the members are selected in a way that systematically overrepresents some segment
of the population and underrepresents other segments.
0.0215 0.0215
0.1359 0.1359
0.3413 0.3413
f
–2σ
–3σ –1σ 1σ
µ 2σ 3σ
Figure 4.4 The normal distribution divided into standard deviation units.

Convenience sampling: When a sample is selected because it is convenient rather than
random.
Descriptive statistics: Statistics that describe the characteristics of a given sample or popula-
tion. These statistics are only meant to describe the characteristics of those from whom
data were collected.
Inferential statistics: Statistics generated from sample data that are used to make inferences about
the characteristics of the population the sample is alleged to represent.
Kurtosis: The shape of a distribution of scores in terms of its flatness or peakedness.
L eptokurtic: A term regarding the shape of a distribution. A leptokurtic distribution is one
with a higher peak and thinner tails.
Negatively skewed: When a tail of a distribution of scores extends toward the lower end of the
distribution.
Normaldistribution: A bell-shaped frequency distribution of scores that has the mean, median,
and mode in the middle of the distribution and is symmetrical and asymptotic.
Null hypothesis: A hypothesis that there is no effect.
Platykurtic: A term regarding the shape of a distribution. A platykurtic distribution is one
with a lower peak and thicker tails.
Population: The group from which data are collected or a sample is selected. The population
encompasses the entire group for which the data are alleged to apply.
Positively skewed: When a tail of a distribution of scores extends toward the upper end of the
distribution.
Probability: The likelihood of an event occurring.
Random sampling: A method of selecting a sample in which every member of the population
has an equal chance of being selected.
Representative sampling: A method of selecting a sample in which members are purposely
selected to create a sample that represents the population on some characteristic(s)
of interest (e.g., when a sample is selected to have the same percentages of various
ethnic groups as the larger population).
Sample: An individual or group, selected from a population, from whom data are collected.
Skew: The degree to which a distribution of scores deviates from normal in terms of asym-
metrical extension of the tails.
Symmetrical: When a distribution has the same shape on either side of the median.
Theoretical distribution: A distribution based on statistical probabilities rather than empirical
data.
U nimodal: A distribution that has a single mode.

37
If you know the mean and standard deviation of a distribution of scores, you have enough
information to develop a picture of the distribution. Sometimes researchers are interested in
describing individual scores within a distribution. Using the mean and the standard deviation,
researchers are able to generate a standard score, also called a z score, to help them understand
where an individual score falls in relation to other scores in the distribution. Through a process
of standardization, researchers are also better able to compare individual scores in the distribu-
tions of two separate variables. Standardization is simply a process of converting each score in a
distribution to a z score. A z score is a number that indicates how far above or below the mean
a given score in the distribution is in standard deviation units. So standardization is simply the
process of converting individual raw scores in the distribution into standard deviation units.
Suppose that you are a college student taking final exams. In your biology class, you take your
final exam and get a score of 65 out of a possible 100. In your statistics final, you get a score of
42 out of 200. On which exam did you get a “better” score? The answer to this question may be
more complicated than it appears. First, we must determine what we mean by “better.” If better
means percentage of correct answers on the exam, clearly you did better on the biology exam.
But if your statistics exam was much more difficult than your biology exam, is it fair to judge
your performance solely on the basis of percentage of correct responses? A more fair alternative
may be to see how well you did compared to other students in your classes. To make such a
comparison, we need to know the mean and standard deviation of each distribution. With these
statistics, we can generate z scores.
Suppose the mean on the biology exam was 60 with a standard deviation of 10. That means
you scored 5 points above the mean, which is half of a standard deviation above the mean
(higher than the average for the class). Suppose further that the average on the statistics test was
37 with a standard deviation of 5. Again, you scored 5 points above the mean, but this represents
a full standard deviation over the average. Using these statistics, on which test would you say you
performed better? To fully understand the answer to this question, let’s examine standardization
and z scores in more depth.
Standardization and z Scores in Depth
As you can see in the previous example, it is often difficult to compare two scores on two vari-
ables when the variables are measured using different scales. The biology test in the example was
measured on a scale from 1 to 100, whereas the statistics exam used a scale from 1 to 200. When
variables have such different scales of meas
ure
ment, it is almost meaningless to compare the raw
scores (i.e., 65 and 42 on these exams). Instead, we need some way to put these two exams on
the same scale, or to standardize them. One of the most common methods of standardization
used in statistics is to convert raw scores into standard deviation units, or z scores. The formula
for doing this is very simple and is presented in Table 5.1.
5
Chapter
Standardization and z Scores

Random documents with unrelated
content Scribd suggests to you:

sua fedeltà. Non così fu al palazzo di Guido dalla Torre. Quivi erano
molti armati, quivi si cominciò un tumulto, e si venne alle mani coi
tedeschi. Trassero colà i parziali de' Torriani, e dall'altro canto
s'andarono ingrossando le truppe del re, il quale fu in gran pena per
questo, massimamente dappoichè gli fu riferito che anche Matteo
Visconte e Galeazzo suo figliuolo erano uniti coi Torriani. Ma eccoti
comparir Matteo col mantello alla corte; ecco da lì un pezzo un
messo, che assicurò Arrigo, come Galeazzo Visconte combatteva
insieme coi Tedeschi contra de' Torriani: il che tranquillò l'animo di
sua maestà. La conclusione fu, che i serragli e palagi dei Torriani
furono superati, dato il sacco alle lor ricche suppellettili, spogliate
anche tutte le case innocenti del vicinato. Guido dalla Torre e gli altri
suoi parenti, chi qua chi là fuggendo, si sottrassero al furor dei
Tedeschi, e se ne andarono in esilio, nè mai più ritornarono in
Milano. Non si seppe mai bene la verità di questo fatto. Fu detto che
i Torriani veramente aveano congiurato, e che nel dì seguente dovea
scoppiar la mina [Johann. de Cermenate, cap. 22, tom. 9 Rer. Ital. Giovanni
Villani, lib. 9, cap. 11. Ferretus Vicentinus, lib. 4, tom. 9 Rer. Ital.]. Ma i più
credettero, e con fondamento, che questa fosse una sottile orditura
dello scaltro Matteo Visconte per atterrare i Torriani, siccome gli
venne fatto, con fingersi prima unito ad essi, e con poscia
abbandonarli nel bisogno. Nulladimeno, con tutto che egli si facesse
conoscer fedele in tal congiuntura ad Arrigo, da lì ad alquanti dì
l'invidia di molti grandi milanesi, ed il timore che Matteo tornasse al
principato, e si vendicasse di chi l'avea tradito nell'anno 1302,
cotanto poterono presso Arrigo, che Matteo fu mandato a' confini ad
Asti, e Galeazzo suo figliuolo a Trivigi. Poco nondimeno stette Matteo
in esilio. Il suo fedele amico Francesco da Garbagnate, fatto
conoscere al re che per fini torti aveano gl'invidiosi allontanato da lui
un sì savio consigliere [Annal. Mediol., tom. 16 Rer. Ital.], cagion fu che
Arrigo nel dì 7 d'aprile il richiamò e rimise in sua grazia.
Gran terrore diede alle città guelfe di Lombardia la caduta de'
Torriani guelfi. Lodi, Cremona e Brescia per questo alzarono le
bandiere contra d'Arrigo. Per confessione di Giovanni Villani, i
Fiorentini e Bolognesi con loro maneggi e danari soffiarono in questo

fuoco. Antonio da Fissiraga signore di Lodi corse colà; ma, ritrovata
quivi dell'impotenza a sostenersi per la poca provvision di
vettovaglia, tornò a Milano ad implorar la misericordia del re, e, per
mezzo della regina e di Amedeo conte di Savoia, l'ottenne. Mandò
Arrigo a prendere il possesso di quella città, e v'introdusse tutti i
fuorusciti; poscia nel dì 17 d'aprile coll'armata s'inviò alla volta della
ribellata Cremona. S'era imbarcato quel popolo senza biscotto; e ciò
per la prepotenza di Guglielmo Cavalcabò capo della fazione guelfa,
il quale avea fatto sconsigliatamente un trattato col fallito Guido
dalla Torre. Sicchè, all'udire che il re veniva in persona con tutte le
sue forze e con quelle de' Milanesi contra di Cremona, se ne fuggì.
Sopramonte degli Amati, altro capo de' Ghibellini, uomo savio e
amante della patria, allora consigliò di gittarsi alla misericordia del
re. Venne egli coi principali della nobiltà e del popolo sino a Paderno,
dieci miglia lungi da Cremona; e tutti colle corde al collo,
inginocchiati sulla strada, allorchè arrivò Arrigo, con pietose voci e
lagrime implorarono il perdono. Era la clemenza una delle virtù di
questo re; ma se ne dimenticò egli questa volta, ed ebbe bene a
pentirsene col tempo. Comandò che ognun di loro fosse imprigionato
e mandato in varii luoghi, dove quasi tutti nelle carceri miseramente
terminarono dipoi i lor giorni. Fu questo un nulla. Arrivato a
Cremona, non volle entrarvi sotto il baldacchino preparato da'
cittadini, fece smantellar le mura, spianar le fosse, abbassar le torri
della città. Da lì ancora a qualche giorno impose una gravissima
contribuzione di cento mila fiorini d'oro, e fu dato il sacco all'infelice
città [Chron. Placent., tom. 16 Rer. Ital.], che restò anche priva di tutti i
suoi privilegii e diritti. Da qualsivoglia saggio fu creduto che questi
atti di crudeltà, sconvenevoli ad un re fornito di tante virtù, pel
terrore che diedero a tutti, rompessero affatto il corso alla pace
d'Italia ed alla fortuna d'Arrigo, addosso a cui vennero poi le dure
traversie che andremo accennando. Dacchè per benignità e favore
d'esso re rientrò in Brescia Tebaldo Brusato cogli altri fuorusciti
guelfi, andò costui pensando come esaltar la sua fazione [Ferretus
Vicentinus, lib. 4, tom. 9 Rer. Italic.]. Nel dì 24 di febbraio, levato rumore,
prese Matteo Maggi, capo de' Ghibellini, con altri grandi di quella
città, e si fece proclamar signore, o almen capo della fazion guelfa,

che restò sola al dominio. Albertino Mussato [Albertinus Mussat., Hist.
Aug., tom. 8 Rer. Ital.] scrive che i Maggi furono i primi a rompere la
concordia, e che poi rimasero al disotto. Jacopo Malvezzo [Malvecius,
Chronic. Brixian., tom. 14 Rer. Ital.] ed altri scrittori bresciani non la
finiscono di esaltar con lodi la persona di Tebaldo Brusato. Ma gli
autori contemporanei ed il fatto stesso ci vengono dicendo che egli
fu un ingrato ai benefizii ricevuti dal re Arrigo, e un traditore, avendo
egli scacciato il di lui vicario, e fatta ribellare contra di lui quella città,
in cui la real clemenza, di bandito e ramingo ch'egli era, l'avea
rimesso. Dopo avere il re tentato, col mandare innanzi Valerano suo
fratello, se i Bresciani si voleano umiliare, e trovato che no [Dino
Compagni. Chron., tom. 9 Rer. Ital.], tutto sdegno nel mese di maggio
mosse l'armata contra di quella città, e n'intraprese l'assedio. Fu
parere del Villani, che s'egli, dopo la presa di Cremona, continuava il
viaggio, Bologna, Firenze e la Toscana tutta veniva facilmente
all'ubbidienza sua. A quell'assedio furono chiamate le milizie delle
città lombarde. Spezialmente vi comparve la cavalleria e fanteria
milanese. Giberto da Correggio, oltre all'aver condotto colà la milizia
di Parma, donò ad Arrigo la corona di Federigo II Augusto, presa
allorchè quell'imperadore fu rotto sotto Parma. Per questo egli, se
crediamo al Corio [Corio, Istor. di Milano.], ottenne il vicariato di quella
città. Albertino Mussato scrive che quivi fu messo per vicario un
Malaspina. Nulla mi fermerò io a descrivere gli avvenimenti del
famoso assedio di Brescia. Basterammi di dire che la città era forte
per mura e per torri, ma più per la bravura de' cittadini, i quali per
più di quattro mesi renderono inutili tutti gli assalti e le macchine
dell'esercito nemico. Circa la metà di giugno, in una sortita restò
prigion de' Tedeschi l'indefesso Tebaldo Brusato, e coll'essere
strascinato e squartato pagò la pena dei suoi misfatti. Infierirono
perciò i Bresciani contra dei prigioni tedeschi, e si accesero
maggiormente ad un'ostinata difesa. In un incontro anche Valerano
fratello del re, mortalmente ferito, cessò di vivere.
Per tali successi era forte scontento il re Arrigo. L'onor suo non gli
permettea di ritirarsi; ed intanto maniera non si vedea di vincere la
nemica città. Mancava il danaro per la sussistenza dell'armata; e il

peggio fu, che in essa entrò una fiera epidemia, ossia la peste vera,
che facea grande strage [Johannes de Cermenat., tom. 9 Rer. Italic.]. Dio
portò al campo tre cardinali legati spediti dal papa per coronare in
Roma, e sollecitar per questo il re Arrigo, cioè i vescovi d'Ostia e
d'Albano, e Luca dal Fiesco. Questi mossero parola di perdono e di
pace. Entrò il Fiesco col patriarca d'Aquileia in Brescia, e trovò delle
durezze. Vi ritornò, e finalmente conchiuse l'accordo. Fu in salvo la
vita e la roba dei cittadini, e si scaricò sopra le mura della città il
gastigo della ribellione, le quali furono smantellate, e per esse entrò
Arrigo nella città nel dì 24 di settembre, seco menando i fuorusciti.
Oltre a ciò, settanta mila fiorini d'oro volle da quel popolo, con altri
aggravii, per quanto scrive il Malvezzi, e lo conferma Ferreto
Vicentino, contro le promesse fatte al cardinale dal Fiesco. Da
Brescia passò a Cremona, indi a Piacenza, dove lasciò un vicario
[Albertinus Mussat., lib. 4, tom. 8 Rer. Ital.], rimanendo deluso Alberto
Scotto, il quale poco dopo ricominciò le ostilità contro la patria.
Trasferitosi a Pavia, quivi si trovarono per la peste calata a tal segno
le sue soldatesche, che Filippone da Langusco, non più signore di
quella città, avrebbe potuto assassinarlo, se il mal talento gliene
fosse venuto. E ne corse anche il sospetto; perlochè portossi colà
Matteo Visconte con possente corpo di Milanesi; ma Filippone gli
chiuse le porte in faccia. Matteo, dico, il quale, stando Arrigo sotto
Brescia, non tralasciò ossequio e diligenza veruna per assisterlo con
gente, danari e vettovaglie; laonde meritò d'essere creato vicario di
Milano, e di poter accudire da lì innanzi all'esaltazione della propria
casa. In Pavia mancò di vita, per le malattie contratte all'assedio di
Brescia, il valoroso Guido conte di Fiandra. E quivi, a persuasione di
Amedeo conte di Savoia, Arrigo dichiarò vicario di Pavia, Vercelli,
Novara e Piemonte Filippo di Savoia, principe allora solamente di
titolo della Morea. Scrive Giovanni da Cermenate [Johannes de Cermen.,
tom. 9 Rer. Ital.], e con lui va d'accordo Galvano Fiamma [Gualv. Flamma,
Manipul. Flor.] col Malvezzi [Malvec., Chron. Brix., tom. 14 Rer. Ital.], che
questo principe, unitosi dipoi con Filippone di Langusco e cogli altri
Guelfi, fece ribellar quelle città, ed altre ancora al re suo benefattore.
Nel dì 21 d'ottobre arrivò Arrigo a Genova, accolto da quel popolo
con sommo onore; ed avuta che ebbe la signoria della città, si studiò

di metter pace fra que' di lor natura alteri, ed allora troppo
discordanti, cittadini, e rimise in città Obizzino Spinola con tutti i
fuorusciti [Georg. Stella, Annal. Genuens., tom. 17 Rer. Ital. Giovanni Villani.
Albertinus Mussatus, et alii.]. Ma quivi nel dì 13 di dicembre da immatura
morte fu rapita la regal sua moglie Margherita di Brabante,
principessa per le sue rare virtù degna di più lunga vita. Intanto si
scoprirono suoi palesi nemici i Fiorentini, Lucchesi, Perugini, Sanesi
ed altri popoli di Toscana, i quali, sommossi ed assistiti dal re
Roberto, fatto grande armamento, presero i passi della Lunigiana,
per impedirgli il viaggio per terra. Erano all'incontro per lui gli Aretini
e Pisani; i quali ultimi mandarono a Genova una solenne ambasceria
ad invitarlo, con fargli il dono di una sì magnifica tenda militare, che
sotto vi poteano stare dieci mila persone. Lo scrive Albertino
Mussato; e chi non vuol credere sì smisurata cosa dazio non
pagherà. Per più di due mesi si fermò in Genova il re Arrigo, nè si
può negare che tendeva il suo buon volere a ricuperare bensì i diritti
molto scaduti del romano imperio; ma insieme, se avesse potuto, a
rimettere la quiete in ogni città, e ad abolir le matte e sanguinarie
fazioni de' Guelfi e Ghibellini. Tutto il contrario avvenne. La venuta
sua mise in maggior moto gli animi alterati e divisi de' popoli.
Giberto da Correggio, guadagnato e soccorso da' Fiorentini e
Bolognesi, mosse a ribellione Parma e Reggio. In Cremona fu una
sedizione non picciola, e ne fu cacciato il ministro del re. Filippone da
Langusco insorse in Pavia contra dei Beccheria ed altri Ghibellini, e,
col favore di Filippo di Savoia, li scacciò. Lo stesso accadde ai
Ghibellini d'Asti, Novara e Vercelli. Anche in Brescia ed in altre città
furono tumulti e sedizioni. In Romagna altresì il vicario del re
Roberto mise le mani addosso ai capi dei Ghibellini di Imola, Faenza,
Forlì e d'altri luoghi, e sbandì la loro fazione [Giovanni Villani, lib. 9, cap.
18.]. Pesaro e Fano, città ribellate al papa, furono ricuperate dal
marchese d'Ancona [Ferretus Vicentinus, tom. 9 Rer. Ital.]. In Mantova
volle il re Arrigo che tornassero gli sbanditi guelfi, e quivi pose per
vicario Lappo Farinata degli liberti. Ma Passerino e Butirone de'
Bonacossi, dianzi padroni della città, presero un giorno l'armi col
popolo, e costrinsero que' miseri a tornarsene in esilio, senza

rispetto alcuno al vicario regio. Era l'Augusto Arrigo in gran bisogno
di moneta. Una buona offerta gli fu fatta da essi Bonacossi, ed
ottennero con ciò il privilegio di vicarii imperiali di Mantova. Di
questo potente strumento seppe ben valersi anche Ricciardo da
Camino per impetrare il vicariato di Trivigi. E per la stessa via
parimente giunsero Alboino e Cane dalla Scala fratelli ad ottener
quello di Verona. Nè qui si fermò l'industria loro. In questi tempi la
città di Padova per la goduta lunga pace [Albertinus Mussatus, lib. 2 et 3,
rub. 3, tom. 8 Rer. Ital.], e perchè dominava anche in Vicenza, si trovava
in un invidiabile stato per le ricchezze e per la cresciuta popolazione.
Questa grassezza, secondo il solito, serviva di eccitamento e fomento
all'alterigia de' cittadini, in guisa che, avendo il re Arrigo fatto lor
sapere di voler inviare colà un vicario, e richiesti sessanta mila fiorini
d'oro per la sua coronazione, quel popolo se ne irritò forte; e, a
suggestione ancora de' Bolognesi e Fiorentini, negò di ubbidire, e
proruppe inoltre in parole di ribellione. Cane dalla Scala, siccome
quegli che già aspirava a gran cose, conosciuta anche la disposizion
de' Vicentini, che pretendeano d'essere maltrattati dagli uffiziali
padovani, e s'erano invogliati di mettersi in libertà, prese il tempo, e
consigliò ad Arrigo di gastigar l'arroganza di Padova con levarle
Vicenza. Ebbe effetto la mina. Cane accompagnato da Aimone
vescovo di Genevra, e colle milizie di Verona e Mantova [Cortus, Histor.,
lib. 1, tom. 12 Rer. Ital.], nel dì 15 d'aprile (e non già di marzo, come ha
lo scorretto testo di Ferreto Vicentino) entrò in quella città, e ne
cacciò il presidio padovano. I Vicentini, che si credeano di ricoverar
la libertà, non solamente caddero sotto un più pesante giogo, ma
piansero il saccheggio della loro città per iniquità di Cane, che non
attenne i patti. Calò allora l'albagia del popolo padovano; cercò poi
accordo, e l'ottenne, ma con suo notabile svantaggio; perchè, oltre
all'avere ricevuto per vicario imperiale Gherardo da Enzola da Parma,
in vece di sessanta, dovette pagare cento mila fiorini d'oro alla cassa
del re.
Morì in quest'anno Alboino dalla Scala, e restò solo Can Grande
suo fratello nella signoria di Verona, con tener anche il piede in
Vicenza. Tale era allora lo stato, ma fluttuante, della Lombardia e

dell'Italia. I soli Veneziani si stavano in pace, osservando senza
muoversi le commozioni altrui. Aveano spediti ad Arrigo, subito
ch'egli fu giunto in Italia, i loro ambasciatori con regali, a titolo non
già di suggezione, ma d'amicizia, e con ordine di non baciargli il
piede [Albertinus Mussat., lib. 3, rub. 8, tom. 8 Rer. Ital.]. Venne poscia in
quest'anno a Venezia il vescovo di Genevra ambasciatore d'Arrigo;
ma non dimandò a quel popolo nè fedeltà nè ubbidienza. Terminò i
suoi giorni in quest'anno appunto [Continuator Danduli, tom. 12 Rer. Ital.]
Pietro Gradenigo doge di Venezia, e nel dì 22 d'agosto (il Sanuto
[Marino Sanuto, tom. 21 Rer. Ital.] scrive nel dì 13) fu surrogato in suo
luogo Marino Giorgi, assai vecchio, che poco più di dieci mesi tenne
quel governo. Sotto Brescia, siccome accennammo, cominciò ad
infierir la peste nell'armata regale, e si diffuse poi per varie città. Ne
restò spopolala Piacenza, Brescia, Pavia, ed altri popoli empierono i
lor cimiterii. Portò il re Arrigo colle sue genti a Genova questo
malore, e però quivi fu gran mortalità. Diede principio papa
Clemente V [Raynaldus, Annal. Eccles. Baluzius, in Vita Pontific.] nell'ottobre
di quest'anno al concilio generale in Vienna del Delfinato, al quale
intervennero circa trecento vescovi. Era riuscito alla saggia destrezza
d'esso pontefice e de' cardinali il far desistere Filippo il Bello re di
Francia dal proseguir le calunniose accuse contro la memoria di papa
Bonifazio VIII. Nel concilio si avea da trattare, ma poco si trattò de'
tanti abusi che allora si osservavano nel clero e nella stessa corte
pontificia, massimamente in riguardo alla collazion de' benefizii e alla
simonia: intorno a che restano varie memorie e scritture di quei
tempi, che io tralascio, rimettendo i lettori alla storia ecclesiastica,
dove se ne parla ex professo.

Anno di
Cristo mcccxii. Indizione x.
Clemente V papa 8.
Arrigo VII re 5, imperad. 1.
I lamenti de' Genovesi, e il non poter più l'Augusto Arrigo ricavar
da essi alcun sussidio di moneta, di cui troppo egli scarseggiava, gli
fecero prendere la risoluzion di passare durante il verno a Pisa. Per
terra non si potea, essendo serrati i passi dalla lega di Toscana.
Trenta galee adunque de' Genovesi e Pisani furono allestite affine di
condurre per mare lui, e la corte e gente sua [Giovanni Villani, lib. 9, cap.
36.]. Nel dì 16 di febbraio imbarcatosi fu forzato dal mare grosso a
fermarsi parecchi dì in Porto Venere. Finalmente nel dì 6 di marzo
sbarcò a Porto Pisano, accolto con indicibil festa ed onore dal popolo
di Pisa. Colà concorsero a furia i Ghibellini fuorusciti di Toscana e di
Romagna, ed egli nella stessa città aspettò il rinforzo di gente che gli
dovea venir di Germania. Intanto recò qualche molestia ai Lucchesi
ribelli, con tor loro alcune castella. Ma quel che dava a lui più da
pensare, era che il re Roberto, fingendo prima di volere amicizia con
lui, gli avea anche spediti ambasciatori a Genova per intavolar seco
un trattato di concordia e di matrimonio; ma furono sì alte ed
ingorde le pretensioni di Roberto, che Arrigo non potè consentirvi.
Dipoi mandò esso re Roberto a Roma Giovanni suo fratello con più di
mille cavalli, il quale prese possesso della Basilica Vaticana e di altre
fortezze di quella insigne non sua città. Volle intendere Arrigo le di
lui intenzioni. Gli fu risposto (credo io per beffarsi di lui) esser egli

venuto per onorar la coronazione d'Arrigo, e non per fine cattivo. Ma
intanto s'andò esso Giovanni sempre più ingrossando di gente, e,
fatto venire a Roma un rinforzo di soldati fiorentini, si unì cogli Orsini
ed altri Guelfi di Roma, e cominciò la guerra contra de' Colonnesi
ghibellini e fautori del futuro novello imperadore. Allora si accertò
Arrigo che l'invidia ed ambizione del re Roberto, non offeso finora,
nè minacciato da Arrigo, aveano mosse quelle armi contra di lui per
impedirgli il conseguimento della imperial corona. Tuttavia, preso
consiglio dal suo valore, ed, animato dai Colonnesi e da altri Romani
suoi fedeli che teneano il Laterano, il Coliseo ed altre fortezze di
Roma, nel dì 23 d'aprile s'inviò con due mila cavalieri e grosse
brigate di fanteria a quella volta. Arrivò a Viterbo, e per più giorni
quivi si fermò, perchè le genti del re Roberto aveano preso e
fortificato Ponte Molle. Nel qual tempo avendo tentato i Ghibellini
d'Orvieto di cacciare i Monaldeschi e gli altri Guelfi di quella città,
senza voler aspettare il soccorso di Arrigo, ebbero essi la peggio, e
furono spinti fuori di quella città. Finalmente rimessosi in viaggio e
superati gli oppositori a Ponte Molle, nel dì 7 di maggio entrò in
Roma con sue genti [Ferretus Vicentinus, lib. 5, tom. 9 Rer. Ital.], e
cominciò la guerra contro le milizie del re Roberto con varii incontri
ora prosperosi ed ora funesti de' suoi. In uno d'essi lasciarono la vita
Teobaldo vescovo di Liegi e Pietro di Savoia fratello di Lodovico
senatore di Roma. Conoscendo poi l'impossibilità di snidare dalla
città leonina e dal Vaticano gli armati spediti colà dal re Roberto,
quasi per violenza a lui fatta dal popolo romano, determinò di farsi
coronare imperadore nella basilica lateranense: funzione che fu
solennemente eseguita nella festa de' santi Apostoli Pietro e Paolo
[Albertus Mussatus. Ptolom. Lucens., in Vita Clementis V.], cioè nel dì 29 di
giugno, e non già nella festa di san Pietro in Vincola al primo giorno
d'agosto, come ha Giovanni Villani [Giovanni Villani, lib. 9, cap. 42.]. Nel
qual giorno ancora si contrassero gli sponsali fra una figliuola del
novello imperadore e Pietro figliuolo di Federigo re di Sicilia, con cui
Arrigo, dacchè vide il mal animo del re Roberto, avea stabilita lega.
Seguitò poi la guerra in Roma. E qui può chiedere taluno: come mai
si attribuì il re Roberto tanta autorità di spedir le sue armi a Roma,
con fare il padrone dove niun diritto egli avea, e con chiara offesa ed

obbrobrio del papa, signore d'essa città? Non v'erano eglino più
scomuniche per reprimere una si fatta violenza? In altri tempi che
strepito non si sarebbe udito? Eppure niun risentimento non ne fu
fatto, in maniera che avrebbe potuto talun credere delle segrete
intelligenze fra il pontefice e il re Roberto. Ma il papa troppo s'era
legate le mani, dappoichè antepose il soggiorno della Provenza e di
stare fra i ceppi, per così dire, del re Roberto e del re di Francia,
piuttosto che di portarsi alla sedia di Roma, destinata dalla
provvidenza di Dio alla libertà dei papi. Non potea egli ciò che volea,
nè ciò che esigeva il debito suo. Ce ne avvedremo all'anno seguente.
Intanto cominciava a rincrescere di troppo questa musica al
popolo romano. Era sminuita non poco l'armata cesarea; quella di
Giovanni fratello di Roberto ogni di più s'andava rinforzando
[Albertinus Mussatus, lib. 8, cap. 8.]. Però l'Augusto Arrigo nel dì 20 di
luglio si ritirò a Tivoli; poscia perchè i fuorusciti toscani continue
istanze gli faceano di volgere le sue armi contro la Toscana, si inviò a
quella volta nel seguente agosto. Diede dei gravi danni ai Perugini, in
passando pel loro distretto, ed arrivò ad Arezzo, dove si vide ben
accolto. Straordinarii preparamenti fecero di armati e di viveri i
Fiorentini [Giovanni Villani, lib. 9, cap. 44.], nè poco fu il loro terrore,
dacchè, entrato l'imperadore nel territorio loro, prese Monte Varchi,
San Giovanni, e Feghine, e fece fuggire dall'Ancisa l'esercito di essi
Fiorentini, con dar loro una spelazzata, e poi si accampò intorno alla
medesima città di Firenze nel dì 19 di settembre. Mandarono le città
collegate gagliardi soccorsi di gente armata ai Fiorentini, i quali certo
ne aveano almeno il doppio più che l'esercito imperiale; pure non
osarono mai di uscire a battaglia. A sacco e fuoco era messo intanto
il loro contado. Immenso fu il bottino che fecero i Tedeschi e i
fuorusciti di Toscana. Veggendo poscia l'imperadore che perdeva il
tempo intorno a Firenze, si ritirò a San Casciano, ed ivi celebrò la
festa del santo Natale. Ma se la Toscana si trovava in gran moto,
minor non era quello della Lombardia. I Padovani, siccome quelli che
non poteano digerire la perdita di Vicenza, loro tolta da Cane dalla
Scala, ribellatisi espressamente all'imperadore, diedero principio alla
guerra contra di quella città, che divenne, e per lungo tempo fu, il

teatro delle miserie. Saccheggiarono le ville del Veronese sino a
Legnago e Tiene, Marostica ed altri luoghi del Vicentino. Ma non
istette colle mani alla cintola lo Scaligero. Anch'egli entrò nel
Padovano, distrusse colle fiamme varie terre, e fra l'altre quella di
Montagnana, senza potere impadronirsi del castello. Avea
l'imperadore Arrigo, all'udire gli sconcerti della Lombardia, inviato
per suo vicario generale il conte Guarnieri di Oemburg [Bonincontrus
Morigia, Chronic., tom. 12 Rer. Ital.], da altri appellato di Ottomburg,
cavaliere tedesco. In una sua lettera al comune di Monza è scritto de
Humbergh. Questi fu chiamato in suo aiuto da Cane dalla Scala; ma
per poco tempo stette ai danni de' Padovani. Essi, rinforzati da
Francesco marchese d'Este e dai Trivisani, fecero dipoi nuove
scorrerie sul Vicentino e Veronese. In quest'anno Ricciardo da
Camino, signore di Trivigi, Feltre e Belluno, fu ucciso con una ronca
da un contadino [Cortus, Hist., lib. 1, tom. 12 Rer. Ital.], il quale fu subito
messo in pezzi dalle guardie, senza sapersi chi fosse, nè da chi
mandato. In quella signoria succedette Guecelo suo fratello. Anche il
suddetto Francesco marchese d'Este [Chron. Estense, tom. 15 Rer. Ital.]
venuto a Ferrara, mentre tornava dalla caccia del falcone in città, alla
porta del Lione fu assalito dai soldati catalani, e per ordine di
Dalmasio, governatore di quella città pel re Roberto, fu
barbaramente ucciso: cosa che fece orrore a tutta la Lombardia.
Guglielmo Cavalcabò, gran fazionario della parte guelfa (e che avea
poc'anzi nel mese di marzo fatto ribellare Cremona [Albertinus
Mussatus, lib. 6, rubr. 2. Johannes de Cermenat., cap. 46, tom. 9 Rer. Ital.], con
farne fuggire Galeazzo Visconte, che era ivi vicario imperiale),
mentre, unito con Passerino dalla Torre, dopo essersi impadronito
della ricca terra di Soncino, era intento ad espugnar quel castello,
trovò anch'egli ciò che non s'aspettava. Veniva il conte Guarnieri
vicario generale da Brescia per dar soccorso al castello suddetto; ed
accoppiatesi con lui le soldatesche milanesi, inviategli da Matteo
Visconte, prima sconfisse lo sforzo de' Cremonesi che andava in
aiuto del Cavalcabò, poscia, entrato in Soncino, mise in fuga quegli
assedianti. Condotto a lui preso Guglielmo Cavalcabò, gli disse: Io
non vo' che da qui innanzi tu abbi a cavalcare nè bue nè cavallo; e
con un colpo di mazza lo stese morto a terra. Per questa perdita

saltò un gran terrore addosso ai Cremonesi, presso i quali in questi
giorni diede fine alla sua vita Guido dalla Torre, già signor di Milano.
In Lodi la fazion guelfa de' Vistarini, coll'aiuto di Giberto da
Correggio e degli altri Guelfi, cacciò fuori della città il vicario
imperiale; ed, oppressa e dispersa la fazione de' Sommariva, si fece
padrona di quella città. In Pavia Filippone conte di Langusco, e gran
caporale de' Guelfi, pose in prigione Manfredi da Beccaria, e cacciò
dalla città i grandi della fazion ghibellina: al che parve che
consentisse Filippo di Savoia principe della Morea, vicario allora di
quella città, e di Vercelli e Novara. La pendenza di questo principe
verso i Guelfi rendè dubbiosa la sua fede all'imperadore. Ma l'astuto
Matteo Visconte seppe indurlo ad inimicarsi con esso Filippone e con
Simone da Colobiano, capo de' Guelfi in Vercelli. E in effetto quel
principe con frode ritenne prigioniere Ricciardino primogenito di
Filippone e il suddetto Simone con molti altri de' maggiori di Pavia:
per la quale azione si screditò non poco in Lombardia. Allora il
Visconte, chiamati a sè i marchesi di Monferrato e di Saluzzo, spinse
Galeazzo suo figliuolo nella Lomellina a' danni de' Pavesi, con
rovinare i raccolti, saccheggiar le castella, e prendere Mortara e
Garlasco. Prima di questo fatto si suscitò anche in Vercelli una fiera
ed impetuosa guerra tra le fazioni degli Avvocati e de' Tizzoni [Chron.
Placentin., tom. 16 Rer. Ital.]: guerra che dicono durata entro quella città
circa quarantanove giorni. Fu essa cagione di aperta rottura fra il
suddetto Filippo di Savoia e il conte Guarnieri vicario generale
dell'imperadore. Accorsero amendue a Vercelli colle lor milizie, e si
venne ad una zuffa fra loro, in cui restarono tutti e due feriti. Il
principe dipoi, sentendo che veniva lo sforzo de' Milanesi, se ne
tornò a Torino. Abbiamo da Giovanni da Cermenate [Johannes de
Cermenat., cap. 50, tom. 9 Rer. Italic.], che essendo restato questo Filippo,
appellato principe della Morea, in età pupillare sotto la tutela di
Amedeo di Savoia suo zio, gli fu da lui usurpata la contea di Savoia,
e che il conte Amedeo, per compensazione, gli cedette infine, oltre
ad alcune castella del Piemonte, la città di Torino, ch'egli
probabilmente avea conseguito dall'Augusto Arrigo in ricompensa del
suo fedele attaccamento. Il bello fu che, essendo restata indecisa la

question di Vercelli, perchè n'era stato fatto compromesso nella
contessa di Savoia e nel marchese di Monferrato: Filippone da
Langusco coi Pavesi ed altri amici guelfi corse colà nel mese di luglio
[Albertinus Mussatus, lib. 7, rubr. 9, tom. 8 Rer. Ital.], ben ricevuto da Oberto
da Colobiano vescovo della città, chiamato con errore Simone dal
Mussato; ed abbattuta affatto la parte dei Tizzoni ghibellini, ridusse
in poter suo e degli Avvocati guelfi quella città. Nella Cronica di
Piacenza [Chron. Placentin., tom. 16 Rer. Ital.] è distintamente narrato
questo fatto; e come Filippone, dopo avere sconfitto un corpo di
Milanesi inviato da Matteo Visconte a Vercelli, si portò colà col
pennone d'esso Matteo, fingendosi Marco di lui figliuolo; e con
questo avendo ingannato Teodoro marchese di Monferrato, ch'era
rimasto alla guardia della città, con facilità se ne impadronì. Di molte
novità furono ancora in Piacenza. Nel dì 18 di febbraio fu in armi
quel popolo, e i Guelfi ne scacciarono il vicario imperiale e i
Ghibellini. Unitisi questi fuorusciti con Alberto Scotto, ebbero
maniera nel dì 18 di marzo di rientrare in Piacenza, e di dar la fuga
ai Guelfi: con che tornò ivi a signoreggiar l'imperadore, che vi pose
per vicario Lodrisio Visconte. Poscia nel dì 20 di settembre lo stesso
Alberto Scotto, levato rumore, spinse fuori della città Ubertino Lando
co' suoi seguaci ghibellini, e per la terza volta si fece proclamar
signor di Piacenza.
Peggiori e più strepitosi furono in quest'anno gli avvenimenti di
Modena [Chron. Mutinens., tom. 11 Rer. Ital. Mussatus, lib. 7, rubr. 7.]. Qui era
per vicario dell'imperadore Francesco Pico della Mirandola. I
Rangoni, Boschetti, Guidoni e da Rodeglia, cogli altri di fazione
guelfa, segretamente tessevano un trattato coi Bolognesi. Non fu
esso sì occulto che non traspirasse; e però queste famiglie,
conosciuto il periglio, fuggendo dalla città, e ridottesi alle loro
castella, cominciarono la guerra contro la patria, assistite da un buon
nerbo di cavalleria e fanteria bolognese, e da quei di Sassuolo.
Essendo essi Guelfi venuti a dare il sacco e il fuoco alla villa di
Bazovara, Francesco dalla Mirandola coi Modenesi arditamente diede
loro battaglia nel dì 9 di luglio, ma ne andò sconfitto. Restarono sul
campo uccisi de' principali Prendiparte suo figliuolo, Tommasino da

Gorzano, Uberto da Fredo, Niccolò degli Adelardi, con circa cento
cinquanta altri de' migliori cittadini, e presi circa cento. Per questa
rotta fu in somma costernazione Modena, e il popolo ricorse tosto
per aiuto a Can Grande dalla Scala signor di Verona, a Rinaldo,
appellato Passerino de' Bonacossi, signor di Mantova, e a Matteo
Visconte signor di Milano; ben prevedendo che i Bolognesi nel caldo
di questa vittoria sarebbono corsi con grande sforzo per
impossessarsi della loro città, siccome infatti fu da essi tentato. Ma
accorsi in persona Cane e Passerino con gente assai, frastornarono
tutti i disegni dell'armata di Bologna, la quale, frettolosamente
venuta, era fin giunta alle fosse della città, ed avea già dato principio
all'assedio e agli assalti. Allora fu che Passerino seppe profittare del
tempo propizio; perchè, trovandosi i Modenesi in tanto bisogno, si
fece nel quarto, oppur quinto giorno d'ottobre, eleggere signor di
Mantova, e governolla dipoi per anni parecchi da tiranno. Fiera
eziandio continuò in questo anno la guerra fra i Padovani e Can
Grande dalla Scala. Distrussero i primi una gran quantità di ville del
Vicentino ne' mesi d'agosto e di settembre, e pervennero
saccheggiando fin quasi alle porte di Vicenza, mancando allo
Scaligero forze da poter loro resistere. Non finì quest'anno, che
Guecelo da Camino, partendosi dalla lega de' Padovani, trattò di
unirsi con Cane dalla Scala, col conte di Gorizia e coi Ghibellini.
Essendosi ciò scoperto, e venendo riprovato dal popolo di Trivigi
[Cortus, Hist., lib. 1, tom. 12 Rer. Ital.], congiurarono contra di lui
Castellano vescovo della città, Rambaldo conte di Collalto, Biachino
da Camino ed altri Guelfi; e poscia nel dì 15 di dicembre, gridato
all'armi, per forza il privarono del dominio. Cacciato egli dalla città, si
ritirò al suo castello di Serravalle; e Trivigi tornò all'essere di
repubblica. Nella città d'Asti [Chron. Astense, cap. 69, tom. 11 Rer. Ital.]
regnava il partito de' Gottuari, ossia di quei da Castello ghibellini, e
v'era per vicario dell'imperadore Tommasino da Enzola. I Solari cogli
altri Guelfi fuorusciti si raccomandarono ad Ugo del Balzo Provenzale
siniscalco del re Roberto, che diede loro assistenza colle sue genti.
Nel dì 4 di aprile fu aspra battaglia fra loro e gli Astigiani, ed,
essendo rimasti perditori gli ultimi, e fatti ben mille prigioni d'essi, i
fuorusciti entrarono in Asti, e giurarono poi fedeltà al re Roberto

nella maniera che aveano praticato gli Alessandrini. Il medesimo Ugo
del Balzo, nel mentre che Teodoro marchese di Monferrato era nel
mese di giugno al guasto delle ville del Pavese, entrò per forza in
Casale di Monferrato, bandì molti di quei cittadini, ed obbligò gli altri
a riconoscere per lor signore il suddetto re Roberto. Aggiugne il
Ventura, da cui abbiam tali notizie, autore contemporaneo, che
anche la città di Pavia prestò al medesimo re un simile giuramento,
con iscusarsi Filippone conte di Langusco di essere stato tradito da
Filippo di Savoia, principe della Morea, che avea sotto la buona fede
fatto prigione, e tuttavia ritenea nelle carceri, Riccardino, ossia
Ricciardino suo figliuolo, e dieci de' primarii cittadini di Pavia; con
allegar eziandio d'essere stato troppo maltrattato dal conte
Guarnieri, da Matteo Visconte e dai Milanesi, che aveano distrutte e
prese tante ville e castella del Pavese. Dopo aver Marino Giorgi per
poco più di dieci mesi tenuto il governo di Venezia, sbrigossi da
questa vita, e in suo luogo fu eletto doge di quella repubblica
Giovanni Soranzo nel dì 13 di giugno, secondo il Continuator del
Dandolo [Contin. Danduli, tom. 12 Rer. Ital.]; ma, secondo il Sanuto [Marino
Sanuto, Istor. Venet., tom. 22 Rer. Ital.] (e forse più fondatamente), nel dì
15 di luglio. Diede fine in quest'anno papa Clemente V al concilio
generale di Vienna, in cui fu abolito l'ordine de' Templari, e posto
fine alle ingiuriose procedure contro la memoria di papa Bonifazio
VIII, la cui credenza fu dichiarata cattolica ed incorrotta [Giovanni
Villani, lib. 9, cap. 22.]. Due cavalieri catalani si esibirono pronti a
provarla in duello: il che confuse chiunque gli volea male. Fece
anche il papa una promozione di nove cardinali tutti franzesi in grave
danno della Sedia di san Pietro, che sempre più veniva a restare in
mano degli oltramontani [Raynald., in Annal. Ecclesiast.]. Allorchè
l'Augusto Arrigo si partì dalla vinta città di Brescia, seco menò per
ostaggi settanta de' migliori cittadini d'essa città sino a Genova
[Malvec., Chron. Brix., tom. 14 Rer. Ital.]. Siccome erano tenuti senza
guardia, di là se ne fuggirono tutti, e, tornati alla patria, fecero
commozione nel popolo, e fu battaglia civile fra i Guelfi e Ghibellini.
Gli ultimi ne furono cacciati, e contra l'imperadore si ribellò la città.
Aiutarono parimente essi Bresciani guelfi i Guelfi di Cremona a
rientrar nella loro città. Ma perciocchè i fuorusciti ghibellini bresciani

occupavano di molte castella, e faceano gran guerra alla patria, fu
mossa parola di concordia fra loro; e andò sì innanzi il trattato, che,
per mezzo di Federigo vescovo di quella città, nel dì 13 di ottobre si
conchiuse pace fra loro, ed ognuno potè ritornare alle proprie case:
pace maggiormente poi fortificata da molti maritaggi che seguirono
fra quelle fazioni. E tale fu l'anno presente, fecondo di tanti
avvenimenti, funesto per tante rivoluzioni, e per uno quasi universale
sconcerto di tutta quanta l'Italia, di modo che a voler minutamente
riferire i fatti d'allora, moltissimi fogli non basterebbono. L'assunto
mio, inclinato alla brevità, non mi permette di più. Il che dico ancora
per quello che resta della presente storia, in cui piuttosto accennerò
le avventure dell'Italia, lasciando, a chi più ne desidera, il ricorrere ai
fonti, cioè agli scrittori che cominciano ad abbondare in questo
secolo, e diffusamente trattano di questi affari.

Anno di
Cristo mcccxiii. Indizione xi.
Clemente V papa 9.
Arrigo VII re 6, imperad. 2.
Da San Casciano nel dì 6 di gennaio si ritirò l'Augusto Arrigo a
Poggibonzi, dove fece fare un castello sul Poggio, dandogli il nome di
castello imperiale [Giovanni Villani, lib. 9, cap. 47.]. Stette ivi sino al dì 6 di
marzo; e perciocchè cominciò a patir difetto di vettovaglia, e per le
infermità si assottigliò forte la sua armata, se ne tornò a Pisa. A
Poggibonzi furono a trovarlo gli ambasciatori di Federigo re di Sicilia,
che, oltre all'avergli portato un sussidio di venti mila doble d'oro
(regalo opportuno al suo estremo bisogno), concertarono seco di
portar la guerra contra del re Roberto nel regno di Napoli.
Quantunque l'imperadore si vedesse in mal arnese per l'esercito
tanto sminuito, e che maggiormente calò per la partenza di Roberto
conte di Fiandra colle sue genti; pure, siccome principe di rara virtù,
che per niuna avversità si turbava, per niuna prosperità si gonfiava,
attese a rimettersi in buono stato, già risoluto di far pentire Roberto
re di Napoli delle offese indebitamente a lui fatte finora. E,
dimorando egli in Pisa, Arrigo di Fiandra suo maliscalco, ossia
maresciallo, con ottocento cavalieri ed otto mila pedoni passò in
Versiglia e Lunigiana a' danni de' Lucchesi. Fra le altre terre, prese
per forza la ricca di Pietrasanta. Degna è di memoria la fondazione
d'essa, fatta dopo la metà del secolo precedente da Guiscardo nobile
milanese della famiglia Pietrasanta, allora podestà di Lucca, il quale

dal suo cognome la nominò. Odasi Giovanni da Cermenate, autore di
questi tempi, che così ne parla [Johann. de Cermenate, cap. 62, tom. 6 Rer.
Ital.]: Henricum de Flandria expugnare Petram-Sanctam mittit
oppidum, licet dives, novum. Ipsum namque construxerat quondam.
Guiscardus de Petra-Sancta, nobilis civis Mediolani, urbe sua exulans,
prima Turrianorum regnante tyrannide, in districtu aut prope confinia
lucanae urbis, cujus rector erat, oppido sui cognominis imponens
nomen. Aggiungasi Tolomeo da Lucca, istorico anche esso di questi
tempi, che mette all'anno 1255 [Ptolom. Lucens., Annal. brev., tom. 11 Rer.
Ital.] Guiscardo da Pietra Santa per podestà di Lucca, qui de Versilia
duos burgos, unum ex suo nomine nominavit, alterum vero Campum
majorem. Non ho voluto tacer questa notizia, affinchè si tocchi con
mano la falsità del decantato editto di Desiderio re de' Longobardi,
inciso in marmo in Viterbo, creduto vero dal Sigonio e da tanti
eruditi, anche ultimamente spacciato per tale da un avvocato de'
Viterbiesi. Quivi il re Desiderio dice d'aver fabbricato la terra di
Pietra-santa. Ci vuol egli di più a conoscere l'impostura? Anche i
marchesi Malaspina tolsero in tal occasione Sarzana, ch'era allora de'
Lucchesi. In Pisa Arrigo Augusto, valendosi de' consigli e della penna
de' suoi legati, fece i più strani ed orridi processi contra del re
Roberto, dichiarandolo nemico pubblico, traditore ed usurpator delle
terre del romano imperio, privandolo di tutti gli Stati, e d'ogni onore
e privilegio, e proferendo la sentenza di morte contra di lui [Albertinus
Mussatus, lib. 13. rubr. 5, tom. 8 Rer. Ital.]. Altri processi e terribili
condanne fece contra di Giberto da Correggio signor di Parma, e di
Filippone da Langusco signor di Pavia, e contro le città di Firenze,
Brescia, Cremona, Padova ed altre, che s'erano ribellate all'imperio
[Giovanni Villani, lib. 9, cap. 48.]. Ma, siccome osserva il Cermenate,
questi fulmini, benchè solo di carte, produssero piuttosto contrario
effetto, perchè più s'indurò nella nemicizia chi già era nemico.
Fece inoltre delle vive istanze a papa Clemente, acciocchè,
secondo l'uso d'altri suoi predecessori, scomunicasse i ribelli
dell'imperio in Italia, e procedesse ancora contra del re Roberto per
gli attentati da lui fatti in Roma in disprezzo della giurisdizione e
degli ordini del papa, e insieme dell'imperador de' Romani. E il

pontefice dovea aver preparato delle bolle in favor d'Arrigo, quando
avvenne un fatto, la cui memoria ci è stata conservata dal suddetto
Giovanni da Cermenate [Johann. de Cermen., cap. 62, tom. 9 Rer. Ital.], ed
è importante per la storia. Albertino Mussato differentemente ne
parla. Filippo il Bello re di Francia, informato di questi affari dal re
Roberto suo parente, e pregato d'aiuto, mandò alla corte pontificia
que' medesimi sgherri che aveano fatta in Anagni la detestabile
insolenza a papa Bonifazio VIII. Al vederseli comparire davanti con
volto burbero, Clemente si tenne perduto. Interrogati che
cercassero, risposero di voler vedere la cancelleria; e, senz'altre
cerimonie andati colà, vi trovarono un converso dell'ordine
cisterciense, che non sapea leggere, tenuto apposta per mettere il
sigillo di piombo alle bolle papali, ed incapace per la sua ignoranza di
lasciarsi corrompere coll'anteporre l'ultime alle prime. Presero
costoro tutti que' brevi e bolle, e le portarono sotto gli occhi del
papa, e senza rispetto alcuno il capo loro gli disse con orrida voce:
Se conveniva ad un papa il provveder d'armi i nemici della casa di
Francia, che tanto avea fatto e speso in servigio della Chiesa
romana; e perchè non avesse egli per anche profittato di ciò che era
accaduto a papa Bonifazio VIII. Che se egli non avea imparato
dall'esempio altrui, insegnerebbe agli altri col propio. Poi se ne
andarono. Oh da lì innanzi non si parlò più di prestar favore
all'Augusto Arrigo; anzi contra di lui si fece quanto volle dipoi la corte
di Francia. Ed ecco i deplorabili effetti della schiavitù, in cui si era
messo il pontefice, col preferire il soggiorno della Provenza a quello
d'Italia. Intanto i Fiorentini [Giovanni Villani, lib. 9, cap. 35.], parendo loro
d'essere in cattivo stano, diedero la signoria della lor città al re
Roberto per cinque anni. Ma l'imperadore Arrigo non la voleva più
contra di loro. Tutti i suoi pensieri erano volti contra d'esso re
Roberto per iscacciarlo, se gli veniva fatto, dal regno di Napoli. A
questo fine chiamò dalla Germania quanta gente potè; molta ne
raccolse dall'Italia; e collegatosi con Federigo re di Sicilia, ed assistito
dai Genovesi, preparò anche una possente armata marittima per
passare colà. Settanta galee si armarono in Genova e Pisa; il
Mussato dice molto meno. Il re di Sicilia ne mise cinquanta in mare,
e, trasportata in Calabria la sua cavalleria, diede principio alla guerra

colla presa di Reggio. Comune credenza fu, che se andava innanzi
questa impresa, era spedito il re Roberto; anzi fu detto ch'egli avea
preparato delle navi per fuggirsene in Provenza. Ma l'uomo propone,
e Dio dispone. Tutto in un momento andò per terra questo sì
strepitoso apparato di guerra.
Nel dì quinto d'agosto si mosse l'imperadore da Pisa con più di
quattro mila cavalieri, i più tedeschi, e con un fiorito esercito di
fanteria; il concorso era stato grande, perchè grande era la speranza
di far buon bottino. Passò nel territorio di Siena fino alle porte di
quella città, la quale ben fornita dagli aiuti della lega, non tremò
punto alla di lui comparsa. Vi era nondimeno trattato con alcuni di
que' cittadini di rendersi; ma questo, per l'avvedutezza di quel
governo, andò in fumo. Accampatosi a Monte Aperto, quivi fu
sorpreso da alcune terzane, delle quali non fece conto sulle prime.
S'inoltrò dodici miglia di là da Siena, ed, aggravatosi il male, si fece
portare a Buonconvento, dove nel dì festivo di san Bartolommeo 24
d'agosto [Albertinus Mussat. Johannes de Cermenat. Giovanni Villani. Ptolom.
Lucens. et alii.] con esemplare rassegnazione ai voleri di Dio spirò
l'anima sua: principe, in cui anche i nemici guelfi riconobbero un
complesso di tante virtù e di sì belle doti, che potè paragonarsi ai più
gloriosi che abbiano retto il romano imperio. Io non mi fermerò
punto ne' suoi elogi, e solamente dirò, che se i mali straordinarii
dell'Italia erano allora capaci di rimedio, non si potea scegliere
medico più a proposito di questo. Ma l'improvvisa sua morte guastò
tutte le misure, e peggiorò sempre più da lì innanzi la malattia
degl'Italiani. Sparsesi voce ch'egli fosse morto di veleno, e che un
frate dell'ordine dei Predicatori, suo confessore, l'avesse attossicato
nel dargli alcuni dì prima la sacra comunione; e tal voce, secondo il
solito, si dilatò per tutta Europa, credendola chiunque è più disposto
a persuadersi del male che del bene. Molti sono gli autori che ne
parlano. Ma non ha essa punto del verisimile. Albertino Mussato,
Guglielmo Ventura [Ventur., Chron. Astense, cap. 64, tom. 11 Rer. Ital.],
Ferreto Vicentino [Ferretus Vicentinus, lib. 5, tom. 9 Rer. Italic.], Giovanni da
Cermenate e Tolomeo da Lucca, autori tutti contemporanei, scrissero
che egli era mancato di morte naturale e di febbre, oppure di peste:

Welcome to our website – the perfect destination for book lovers and
knowledge seekers. We believe that every book holds a new world,
offering opportunities for learning, discovery, and personal growth.
That’s why we are dedicated to bringing you a diverse collection of
books, ranging from classic literature and specialized publications to
self-development guides and children's books.
More than just a book-buying platform, we strive to be a bridge
connecting you with timeless cultural and intellectual values. With an
elegant, user-friendly interface and a smart search system, you can
quickly find the books that best suit your interests. Additionally,
our special promotions and home delivery services help you save time
and fully enjoy the joy of reading.
Join us on a journey of knowledge exploration, passion nurturing, and
personal growth every day!
ebookbell.com

Statistics In Plain English Third Edition Timothy C Urdan

More Related Content

Similar to Statistics In Plain English Third Edition Timothy C Urdan (20)

Recently uploaded (20)

Statistics In Plain English Third Edition Timothy C Urdan