QUICK 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 Boston San Diego New York 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 transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101 United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW1 7DX 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.