This independence result can also be proven by Cochran's theorem.

The rank of Q can be shown to be n 1, and thus the conditions for Cochran's theorem are met.

By Cochran's theorem, for normal distributions the sample mean and the sample variance s are independent, which means there can be no gain in considering their joint distribution.

Cochran's theorem then states that Q and Q are independent, with chi-squared distributions with n 1 and 1 degree of freedom respectively.

To derive the correction, note that for normally distributed X, Cochran's theorem implies that the square of has a chi distribution with n 1 degrees of freedom.

In instances where the F-distribution is used, for instance in the analysis of variance, independence of U and U might be demonstrated by applying Cochran's theorem.

In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance.

Cochran's theorem shows that decompositions of a sum of squares of normal random variables into sums of squares of linear combinations of these variables are always independent chi-squared distributions.