For large samples the fraction (N 2 1) (N 2 2) is essentially 1, and we can thus write

the equation as it is often found in statistics texts:

or

Keep in mind, however, that for small samples these equations are only an approxima-

tion and will underestimate the error variance by the fraction (N 2 1) (N 2 2). For

samples of any size, however,. This particular formula is going to

play a role throughout the rest of the book, especially in Chapters 15 and 16.

Errors of Prediction as a Function of r

Now that we have obtained an expression for the standard error of estimate in terms of r, it

is instructive to consider how this error decreases as rincreases. In Table 9.4, we see the

magnitude of the standard error relative to the standard deviation of Y(the error to be ex-

pected when Xis unknown) for selected values of r.

The values in Table 9.4 are somewhat sobering in their implications. With a correlation

of .20, the standard error of our estimate is fully 98% of what it would be if Xwere un-

known. This means that if the correlation is .20, using as our prediction rather than

(i.e., taking Xinto account) reduces the standard error by only 2%. Even more discourag-

ing is that if ris .50, as it is in our example, the standard error of estimate is still 87% of

the standard deviation. To reduce our error to one-half of what it would be without knowl-

edge of Xrequires a correlation of .866, and even a correlation of .95 reduces the error by

only about two-thirds. All of this is not to say that there is nothing to be gained by using a

regression equation as the basis of prediction, only that the predictions should be inter-

preted with a certain degree of caution. All is not lost, however, because it is often the kinds

of relationships we see, rather than their absolute magnitudes, that are of interest to us.

r^2 as a Measure of Predictable Variability

From the preceding equation expressing residual error in terms of , it is possible to derive

an extremely important interpretation of the correlation coefficient. We have already seen that

Expanding and rearranging, we have

r^2 =

SSY 2 SSresidual

SSY

SSresidual=SSY 2 SSY(r^2 )

SSresidual=SSY(1 2 r^2 )

r^2

YN Y

SSresidual=SSY(1 2 r^2 )

s^2 Y#X >

sY#X=sY 3 (1 2 r^2 )

s^2 Y#X=s^2 Y(1 2 r^2 )

>

Section 9.7 The Accuracy of Prediction 261

Table 9.4 The standard error of estimate as a function of r

rsYX rsYX

.00 sY .60 0.800sY

.10 0.995sY .70 0.714sY

.20 0.980sY .80 0.600sY

.30 0.954sY .866 0.500sY

.40 0.917sY .90 0.436sY

.50 0.866sY .95 0.312sY