MSE (\(\bar{X}\)) = \(\frac{s^2}{n}\), where \(s^2\) is the sample variance for the sample of \(X\)s.

S.E. (\(\bar{X}\)) = \(\sqrt\frac{s^2}{n}\) = \(\frac{s}{\sqrt{n}}\).

However, in OLS regressions, the standard error of the regression is defined as

\(\frac{\text{SSR}}{n - K}\) where SSR = Sum of squared residuals.

My question is does this not need to be corrected again by dividing by the sample size n? Surely \(\frac{\text{SSR}}{n - K}\) is just the sample estimate of the population variance and, according to the formula above, it needs to be corrected. I understand that in both cases the standard error is the square root of the MSE but in the first case the MSE is the sample variance that the estimator came from divided by n and in the second example the MSE is just the sample variance. Does anyone have an explanation?

Thanks in advance,

This has been bugging me for a while.