This is because r is an estimate of and varies quite a bit when the sample size is small. Nevertheless, the values of r and r 2 in the illustration should help calibrate your notion of what they mean. The practical use of r 2 is that when r 2 is close to 1, we know that a straight line is a good fit to the data. Source Parameter Estimate Std. Recall from HW 9 that the plot of words versus age has all the points nearly on the least squares line.
Most data sets are in between.
Here, for a given X value, the average of all the Y's for that X lies close to a straight line. But any one of the individual Y's can be fairly far from the line. Soon you will learn how to generalize the linear regression situation to the case where there are two or more possible explanatory X variables. For example, in the earthquake data set just discussed there is an X variable distance in addition to magnitude. Hence distance is not nearly as good a variable as magnitude for use with frequency.