Introduction to Probability and Statistics for Engineers and Scientists

9.5The Coefficient of Determination and the Sample Correlation Coefficient 377

represents the amount of variation in the response variables that isexplainedby the different
input values; and so the quantityR^2 defined by

R^2 =

SYY−SSR

SYY

= 1 −

SSR

SYY

represents the proportion of the variation in the response variables that is explained by the
different input values.R^2 is called thecoefficient of determination.
The coefficient of determinationR^2 will have a value between 0 and 1. A value ofR^2
near 1 indicates that most of the variation of the response data is explained by the different
input values, whereas a value ofR^2 near 0 indicates that little of the variation is explained
by the different input values.

EXAMPLE 9.5a In Example 9.4c, which relates the height of a son to that of his father, the
output from Program 9.2 yielded that

SYY=38.521, SSR=1.497

Thus,

R^2 = 1 −

1.497

38.531

=.961

In other words, 96 percent of the variation of the heights of the 10 individuals is explained
by the heights of their fathers. The remaining (unexplained) 4 percent of the variation is
due to the variance of a son’s height even when the father’s height is taken into account.
(That is, it is due toσ^2 , the variance of the error random variable.) ■

The value ofR^2 is often used as an indicator of how well the regression model fits the
data, with a value near 1 indicating a good fit, and one near 0 indicating a poor fit. In
other words, if the regression model is able to explain most of the variation in the response
data, then it is considered to fit the data well.
Recall that in Section 2.6 we defined the sample correlation coefficientrof the set of
data pairs (xi,Yi),i=1,...,n,by

r=

∑n

i= 1

(xi−x)(Yi−Y)

√

∑n

i= 1

(xi−x)^2

∑n

i= 1

(Yi−Y)^2

It was noted thatrprovided a measure of the degree to which high values ofxare
paired with high values ofY and low values ofxwith low values ofY. A value ofr