Understanding Statistics 106656_FM_R1.indd 1 11/9/17 4:37 PM 106656_FM_R1.indd 2 11/9/17 4:37 PM Understanding Statistics AN INTRODUCTION Antony Davies CATO INSTITUTE WASHINGTON, D.C. 106656_FM_R1.indd 3 11/9/17 4:37 PM Copyright ©2017 by the Cato Institute. All rights reserved. Paperback ISBN: 978-1-944424-35-0 eBook ISBN: 978-1-944424-36-7 Library of Congress Cataloging-in-Publication Data available. Printed in the United States of America. CATO INSTITUTE 1000 Massachusetts Avenue, N.W. Washington, D.C. 20001 www.cato.org 106656_FM_R1.indd 4 11/9/17 4:37 PM C O N T E N T S Introduction: Why We Think Statistics Is Difficult � � � � � 1 1� Common Errors � � � � � � � � � � � � � � � � � � � � � �11 2� What Are Probabilities?� � � � � � � � � � � � � � � � � �41 3� How Do We Know When Two Things Are Different? � � � � � � � � � � � � � � � � � � � � � � 63 4� How Do We Know When One Thing Affects Another? � � � � � � � � � � � � � � � � � � � � � 77 5� What Do Economic Measures Tell Us? � � � � � � � � � 107 Conclusion � � � � � � � � � � � � � � � � � � � � � � � � � � 121 Endnotes� � � � � � � � � � � � � � � � � � � � � � � � � � �125 Recommended Readings� � � � � � � � � � � � � � � � � � � 131 About the Author � � � � � � � � � � � � � � � � � � � � � � 133 Index� � � � � � � � � � � � � � � � � � � � � � � � � � � � � 135 106656_FM_R1.indd 5 11/9/17 4:37 PM 106656_FM_R1.indd 6 11/9/17 4:37 PM Introduction: Why We Think Statistics Is Difficult This book is an introduction to statistics, not an introduction to statistical analysis. Statistical analysis is about performing statistical calculations and estimating probabilities. This book is for someone who wants to learn how to interpret the results of statistical analyses. Think of it like the difference between being able to read a book and being able to write a book. One doesn’t have to be an author to enjoy reading. Sim- ilarly, one does not need to know how to perform statistical analysis to understand statistics. From a practical perspective, 1 106656_Introduction_R2.indd 1 11/9/17 5:18 PM UNDERSTANDING STATISTICS: AN INTRODUCTION understanding statistics can be as important as being able to perform statistical analysis. Statistical analysis is a mathematical endeavor. Therefore, the formulation of the questions, the tools used in performing analyses, and the results those analyses yield are all expressed in mathematical language. Translating the mathematical results into English is fraught with difficulty. Understanding statistics helps to mitigate the errors that arise from inexact translations. For example, an analyst might discover a statis- tically positive relationship between the minimum wage and unemployment. But that finding doesn’t necessarily mean that a noticeable increase in the unemployment rate will follow an increase in the minimum wage. Whether the relation- ship is noticeable depends on several factors, such as: (1) the magnitude of the minimum wage increase; (2) the difference between the proposed minimum wage and the current aver- age hourly wage rate; (3) which sectors of the economy are growing and contracting; and (4) whether by “unemploy- ment rate” we mean the unemployment rate in general, or the unemployment rate for hourly workers, or the unemployment rate for hourly entry-level workers, or the unemployment rate for hourly, entry-level, unskilled workers. Whereas measur- ing the effect of a minimum wage hike on the unemployment rate requires knowledge of statistical analysis, understanding 2 106656_Introduction_R2.indd 2 11/9/17 5:18 PM IntroductIon: Why We thInk StatIStIcS IS dIffIcult the nuances of the results that the analysis yields requires knowledge of statistics. Many children learn simple addition and subtraction by locating numbers on a number line. For example, to add three and two, you first find three on the number line and then move two places to the right. That puts you at five—three plus two is five. This is a useful way for children to begin to understand numbers and arithmetic. Unfortunately, most of us keep this model in our heads as adults—and this model is antithetical to understanding statistics. In statistics, things aren’t as nicely arranged and well-behaved as they are on a number line. In statistics, things move around and vibrate in random ways, leading to all sorts of very real but counterintuitive results. For example, suppose your car can travel 30 miles (on average) on a gallon of regular gas and 36 miles (on average) on a gallon of premium gas. If you put one gallon of regular gas and one gallon of premium gas in your tank, how many miles will you travel? If you answered “66,” you are suffering from the erro- neous “number line” view of the world. The correct answer is “I don’t know.” You do know that you will travel 66 miles on average, but that’s very different from saying you will travel 66 miles. The miles you can travel on a gallon of gas is a ran- dom variable. On average, it is 30 miles for regular gas. And, on average, it is 36 miles for premium gas. But those numbers 3 106656_Introduction_R2.indd 3 11/9/17 5:18 PM UNDERSTANDING STATISTICS: AN INTRODUCTION Figure 1 How We Imagine Measurements Work You are here. 25 30 35 are only averages. On any given trip, you may get better or worse gas mileage depending on how much weight is in the car, whether the engine is well tuned, how much air pressure is in the tires, and many other factors. The mileage you get from a gallon of gas is not a fixed point on a number line (as in Figure 1). It is a cloud of points that vibrate around a com- mon center (as in Figure 2). That common center is the mean (which is the technical name for “average”). The degree to which the cloud of points is spread out is called the standard deviation (an alternate measure is the variance). Figure 2 How Measurements Actually Work Sometimes you are here... ...or here. ...or here. ...or here. ...or here. 25 30 35 4 106656_Introduction_R2.indd 4 11/9/17 5:18 PM