In linear regression, mean response and predicted response are values of the dependent variable calculated from the regression parameters and a given value of the independent variable. The values of these two responses are the same, but their calculated variances are different.
Since the data in this context is defined to be (x, y) pairs for every observation, the mean response at a given value of x, say xd, is an estimate of the mean of the y values in the population at the x value of xd, that is . The variance of the mean response is given by
This expression can be simplified to
where m is the number of data points.
To demonstrate this simplification, one can make use of the identity
The confidence intervals are computed as . Thus, the confidence interval for predicted response is wider than the interval for mean response. This is expected intuitively – the variance of the population of values does not shrink when one samples from it, because the random variable εi does not decrease, but the variance of the mean of the does shrink with increased sampling, because the variance in and decrease, so the mean response (predicted response value) becomes closer to .
This is analogous to the difference between the variance of a population and the variance of the sample mean of a population: the variance of a population is a parameter and does not change, but the variance of the sample mean decreases with increased samples.