On Proving That and S 2 are Independent

The need to establish the independence of the sample mean and the sample variance in sampling from a normal population arises early in a course in statistics. For the result is an essential ingredient in the derivation of the Student-t distribution for statistical inference. Often this need arises before the tools, notably multivariate methods, for a rigorous proof are available. Occasionally one will find attempts to derive this result using only bivariate assumptions. A recent article in this journal, as well as some current textbooks, offer such a proof. In all cases there are serious questions about the validity of the proofs.