Table Of ContentQUICK ANSWERS
TO
Q U A N T I T A T I VE
P R O B L E MS
A Pocket Primer
G. William Page Carl V. Patton
Dean, College of Urban and Vice President for
Public Affairs Academic Affairs
Florida Atlantic University University of Toledo
Ft. Lauderdale, Florida Toledo, Ohio
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
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London Sydney Tokyo Toronto
This book is printed on acid-free paper.
Copyright © 1991 by Academic Press, Inc.
All rights reserved.
No part of this publication may be reproduced or
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS LIMITED
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Cover and Interior Design by Elizabeth E. Tustian
Library of Congress Cataloging-in-Publication Data:
Page, G. William (George William), date.
Quick answers to quantitative problems : a pocket primer / G.
William Page, Carl V. Patton.
p. cm.
Includes bibliographical references and index.
ISBN 0-12-543570-3 (alk. paper)
1. Statistics. I. Patton, Carl V. II. Title.
QA276.12.P33 1991
519.5^dc20 90-46488
CIP
Printed in the United States of America
91 92 93 94 9 8 7 6 5 4 3 2 1
INTRODUCTION
This primer presents a wide variety of techniques useful in analyzing
quantitative data. The techniques are all relatively simple and can be
completed quickly with only a pencil and paper in most instances and,
in some cases, with basic drawing equipment. These techniques also
lend themselves to easy computation using electronic spreadsheets and
other common microcomputer software. The techniques are presented
with instructions for their use, and examples with contexts and prob-
lem solutions are also provided. Although the techniques are not highly
complicated, they are extremely useful in a wide variety of public and
private sector situations, where conclusions from quantitative data are
needed quickly to help make decisions.
Often busy professionals have only limited time to prepare for meet-
ings. Most of the techniques presented here can be completed in a few
minutes and many may even be done on the back of an envelope
during a meeting. As portable microcomputers of increasing capabili-
ties and decreasing cost are rapidly coming on the market, these quick
quantitative techniques will become even more important. Unlike the
use of many computer programs, however, these quick techniques are
not "black boxes" where the process of quantitative analysis is hidden.
Rather, these techniques are open and provide the analyst with impor-
tant insights into the data and any relationships imbedded in them. The
ix
χ Introduction
process of using these techniques and their results is easily communi-
cated because they are not complex and are widely used.
These quick methods are also useful as tests to ensure that the
results of a complex computer program are reasonable. They can also
be used as a first step to decide which of a wide variety of more
complex techniques would be most appropriate for a given set of data.
There are other situations in which these quick methods can be used to
check the reasonableness of someone else's work. Reviewing the work
of others often raises questions that the analyst did not address or
questions about the appropriateness of the analyst's conclusions. In
many circumstances, these quick techniques can be applied to data
from a table to explore the potential of a new idea or to gain insights
into the conclusions. In summary, these quick quantitative methods can
be used as a first approximation to a more complex analysis, as a
complete analysis by themselves, or as a check on the reasonableness
of a more thorough analysis.
The quick methods presented in this primer are largely statistical in
nature. An understanding of statistical theories would be helpful in
mastering the methods, but is not essential. This primer does not pre-
sent statistical theory, nor does it attempt to derive or prove any formu-
lae. The essential components of the methods are presented with clear,
step-by-step instructions and examples. Citations to more complete
discussions of each technique are provided. This book is small so that
it can be carried in a briefcase and be available when the need for
quick analysis arises.
No matter the field, today's professionals need to be able to respond
quickly to problems and to develop a quick understanding of what the
data mean. Whether the person is an aide to a city council member
trying to figure out what a citizen opinion poll really means, a private
consultant to the health department trying to estimate the number of
pregnant teenagers in a particular neighborhood, or the executive
director of a small agency who wants to present its budget facts pre-
cisely and clearly, the techniques in this book can be helpful.
This primer is divided into five parts. Part 1 presents basic methods
for describing and displaying data, including descriptive statistics, tab-
ular analysis, and graphic techniques. Using these methods, the analyst
can describe and present the basic facts in a set of data.
Part 2 includes three chapters about ways to analyze data, including
scatterplots, correlation analysis, and statistical significance. With
Introduction xi
these methods, the analyst can explain the basic relationships among
variables, that is, how two or more variables are associated and the
extent to which relationships between or among variables are or are
not due to chance.
Part 3 contains methods for examining data over time, including
projection techniques, computing rates of change, analyzing economic
change by region, and using multipliers. These methods allow the ana-
lyst to estimate future conditions based on assumptions about trends
and relationships.
Part 4 addresses the question of how to obtain data and assess their
validity. Topics include determining the optimum size for a sample,
procedures for selecting a sample and obtaining other data, and deter-
mining the accuracy of sample estimates. This information allows the
analyst to determine the extent to which the data can be relied upon as
a basis for decision making, that is, to estimate how close values
derived from the sample are to values in the population from which the
sample was taken.
Part 5 presents several methods that allow the analyst to compare
options, including the location quotient, indices, and evaluation meth-
ods. The location quotient allows us to compare the concentration of a
given economic activity between regions, while indices are used to
summarize several measurements into a single value that simplifies
comparisons among areas, groups, or even countries. The evaluation
methods we present focus on economic benefit-cost comparisons, and
include the concept of net present value or net benefit as a decision
criterion. Each of these methods provides a quantitative way to com-
pare competing options.
While the focus of this book is on analysis, it is essential to remem-
ber that good analysis depends on good data and careful data collection
techniques as well as on the clear specification of problems, accurate
identification of independent and dependent variables, and the applica-
tion of the proper statistical tests. Quick analysis is not intended to
replace other methods, but rather to be used as a first approximation
that can be followed by more sophisticated techniques if time permits.
The quick methods presented in this primer are reliable within cer-
tain parameters, and any analyst with a knowledge of statistics can
easily take the methods to a higher level. For example, data presented
in a simple tabular analysis could be further analyzed with inferential
statistics. We present a measure of correlation for ordinal data that can
xii Introduction
also be used to analyze interval data. A more experienced analyst
would, however, want to use an interval level correlation measure in
this case if time permits.
The tests of significance we present are known as parametric tests,
meaning that the statistics make certain assumptions about the parame-
ters that describe the population from which the sample is taken. These
assumptions are often violated in practice. For example, the popula-
tions are seldom normally distributed, the data may not be interval or
ratio scale, samples are seldom simple random samples with replace-
ment, and beyond this, nonsampling errors are seldom considered.
While there are non-parametric or distribution-free tests that do not
require the knowledge of the precise form or distribution of the popu-
lation, these tests sometimes require a deeper knowledge of statistics
than we assume the reader of this book has, as well as more involved
mathematical calculations. When under severe time constraints, we
believe that the solution to this dilemma is exactly the one used in
practice: relax the assumption of normality, apply parametric tests cau-
tiously, and interpret the results conservatively.
Use the methods in this book to help you find quick answers to
quantitative questions, but remember that often you cannot base your
conclusions and proposed policies on these statistical tests alone. There
must be an underlying logic to the analysis, the conclusions must make
sense intellectually, and they must be important as well as statistically
significant.
G. William Page
Carl V. Patton
ACKNOWLEDGMENTS
This primer was developed over a number of years during which the
authors taught quantitative analysis in several universities. A debt of
gratitude is due our colleagues at these and other institutions who
reviewed earlier versions of this work, and to our students who sug-
gested improvements over the years. We especially appreciate the criti-
cal comments and suggestions received from Curtis Roseman, Michael
Romanos, Barry Checkoway, David Forkenbrock, David Lindsley,
Jane Patton, John Swift, Catherine Dadlez, and several anonymous
reviewers. Elizabeth Tustian and Charles Glaser of Academic Press
provided assistance throughout the production of the book.
xiii
Chapter Ί
DESCRIPTIVE
STATISTICS
Definition
Descriptive statistics are used to summarize and communicate what we
find in quantitative data. We often need simple, quick ways to convey
the essential information present in tens or even thousands of individ-
ual observations about the subject of interest. We present two types of
descriptive statistics: measures of central tendency and measures of
dispersion. Measures of central tendency say something about the
"average" characteristic of our subject and are one of the most useful
descriptions we can provide. The statistics for this purpose are mean,
median, and mode. Measures of dispersion tell us how much the data
deviate from the measures of central tendency. They tell us if most of
the observations in the distribution (data set) are close in value to the
mean or median, or if there is a wide variation in the values. Three
common measures of dispersion are the range, variance, and standard
deviation.
Mean
The mean, or arithmetic average, equals the sum of the values of the
observations divided by the number of observations.
3
4 How to Describe and Display Data
η
where
X = the mean
Σ = the addition of what follows from the first observation (/ = 1)
to the n* observation,
χ = the individual observations (from 1 to «), and
η = the number of observations.
The symbol for the mean is the variable symbol (x in this example)
with a bar above it, pronounced: "JC bar." This is the symbol for the
sample mean. If all of the possible observations of the variable, called
the population or universe of observations, are collected, then you have
the population mean. The symbol of the population mean is: μ. This
Greek letter is pronounced: "mu" or "mew."
Example 1
The sample data are: 10, 12, 14, 17, 27, 36. X = 116/6 = 19.33. •
Sometimes, one must calculate statistics for data that have been con-
verted from directly measured values into categories. This is usually
called grouped data. The formulas used to calculate descriptive statis-
tics for grouped data are modified because one doesn't know where the
original measurement belongs within the range of each category of the
data available for analysis (Blalock, 1979).
The mean is the most commonly used measure of central tendency.
It is particularly valuable because everyone understands it. The mean,
however, is sensitive to extreme values. Consequently, it can be a poor
measure of central tendency if the data contain a few values that are
much larger or smaller than the rest of the data. Data on incomes,
where one person with a huge income can distort the average (mean)
income statistic, are classic examples of the potential problem.
Example 2
The sample data are: 10, 12, 14, 17, 27, 245. X = 325/6 = 54.167. Note
that this can be a misleading measure of the central tendency of the
Descriptive Statistics 5
data. See the discussion of the median for use when extreme values are
present. •
Median
The median is the measure of central tendency that identifies the mid-
most value. The median is like the median strip in a highway: it sepa-
rates the observations into two equal groups, one lower in value than
the median value and one greater in value than the median.
X = the symbol for the median. If χ is the variable, it is pronounced
"JC tilde."
To calculate the median:
1. Order the values from the smallest to the largest;
2. In an odd number of observations, the median is the mid-most
value. Ex.: 9, 10, 12, 14, 17, 27, 36. X = 14;
3. In an even number of observations, take the average of the two
mid-most values. Ex.: 10, 12, 14, 17, 27, 245. X = 14+17/2
= 15.5.
Mode
The mode is the most frequently occurring value or category of the
variable in the data.
The mode is often most easily identified by constructing a frequency
table, which is an ordering of the data indicating how often each value
or category of the data occurs (see Table 1).
Table 1
Frequency Table of Grades on Mid-term Examination
grade (the variable) frequency (F)
30 2
40 3
50 18
60 26
70 22
80 12
90 6
yv = 89
Source: Data developed for example.
