Copyright © Cengage Learning. All rights reserved. 1 STRAIGHT LINES AND LINEAR FUNCTIONS

Copyright © Cengage Learning. All rights reserved. 1.5 The Method of Least Squares

3 We saw how a linear equation may be used to approximate the sales trend for a local sporting goods store. The trend line, as we saw, may be used to predict the store’s future sales. Here, we describe a general method known as the method of least squares for determining a straight line that, in some sense, best fits a set of data points when the points are scattered about a straight line.

4 The Method of Least Squares To illustrate the principle behind the method of least squares, suppose, for simplicity, that we are given five data points, P 1 (x 1, y 1 ), P 2 (x 2, y 2 ), P 3 (x 3, y 3 ), P 4 (x 4, y 4 ), P 5 (x 5, y 5 ) describing the relationship between the two variables x and y.

5 The Method of Least Squares By plotting these data points, we obtain a graph called a scatter diagram (Figure 41). A scatter diagram Figure 41

6 The Method of Least Squares If we try to fit a straight line to these data points, the line will miss the first, second, third, fourth, and fifth data points by the amounts d 1, d 2, d 3, d 4, and d 5, respectively (Figure 42). d i is the vertical distance between the straight line and a given data point. Figure 42

7 The Method of Least Squares We can think of the amounts d 1, d 2,..., d 5 as the errors made when the values y 1, y 2,..., y 5 are approximated by the corresponding values of y lying on the straight line L. The principle of least squares states that the straight line L that fits the data points best is the one chosen by requiring that the sum of the squares of d 1, d 2,..., d 5 — that is, be made as small as possible.

8 Finding an Equation of Least-Squares Line INSERT PODCAST

9 Practice p. 62 #1, 3