Least Squares, Medians, and the Indy 500

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CHAPTER OBJECTIVES:

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r to show two procedures for approximating bivariate data with straight lines one of which

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uses medians to nd some surprising connections between geometry and data analysis to nd the least squares regression line without calculus to see an interesting use of an elliptic paraboloid to show how the equations of straight lines and their intersections can be used in a practical situation r to use the properties of median lines in triangles that can be used in data analysis

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r r r r

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INTRODUCTION

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We often summarize a set of data by a single number such as the mean, median, range, standard deviation, and many other measures We now turn our attention to the analysis of a data set with two variables by an equation We ask, Can a bivariate data set be described by an equation As an example, consider the following data set that represents a very small study of blood pressure and age

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Age 35 45 55 65 75 Blood pressure 114 124 143 158 166

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A Probability and Statistics Companion, John J Kinney Copyright 2009 by John Wiley & Sons, Inc

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Introduction

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We seek here an equation that approximates and summarizes the data First, let us look at a graph of the data This is shown in Figure 131

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160 150 140 130 120 40 50 Age 60 70

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It would appear that the data could be well approximated by a straight line We might guess some straight lines that might t the data well For example, we might try the straight line y = 60 + 12x, where y is blood pressure and x is age How well does this line t the data Let us consider the predictions this line makes, call them yi , and the observed values, say yi We have shown these values and the discrepancies, y y, in Table 131

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Table 131 x 35 45 55 65 75 y 114 124 143 158 166 y 102 114 126 138 150 y y 12 10 17 20 16

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The discrepancies, or what are commonly called errors or residuals, happen to be all positive in this case, but that is not always so So how are we to measure the adequacy of this straight line approximation or t Sometimes, of course, the positive residuals will offset the negative residuals, so adding up the residuals can be quite misleading To avoid this complication, it is customary to square the residuals before adding them up If we do that in this case we get 1189,but we do not know if that can be improved upon So let us try some other combinations of straight lines First, suppose the line is of the form yi = + xi Although the details of the calculations have not been shown, Table 132 shows some values for the sum of squares, SS = 5 (y y)2 , and various choices for and i=1 One could continue in this way, trying various combinations of and until a minimum is reached The minimum in this case, as we shall soon see, occurs when = 651 and = 138, producing a minimum sum of squares of 651 But trial and error is a very inef cient way to determine the minimum sum of squares and is feasible in this case because the data set consists of only ve data points It is clearly nearly impossible, even with a computer, when we consider subsequently an

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Blood pressure

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13

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Least Squares, Medians, and the Indy 500 Table 132 55 60 65 70 75 1 12 13 14 14 SS 4981 1189 13925 212 637

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example consisting of all the winning speeds at the Indianapolis 500-mile race, a data set consisting of 91 data points In our small case, it is possible to examine a surface showing the sum of squares (SS) as a function of and A graph of SS = n (yi xi )2 is shown in i=1 Figure 132 It can be shown that the surface is an elliptic paraboloid, that is, the intersections of the surface with vertical planes are parabolas and the intersections of the surface with horizontal planes are ellipses It is clear that SS does reach a minimum, although it is graphically dif cult to determine the exact values of and that produce that minimum We now show an

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16 15 14 13 12 11 1

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