Cochran's Theorem

Introduces William Cochran's 1934 theorem -- developed at Rothamsted alongside Yates and Fisher -- which most applied users have never heard of by name despite using its consequences every time they read an ANOVA table or run a t-test. Derives the theorem's geometric heart in two steps: an eigenvalue-counting argument showing that a sum-of-squares decomposition's ranks adding up to the total dimension forces each piece to be an orthogonal projection, and a rotation argument showing orthogonal projections of a standard Normal vector are automatically independent chi-squares. Works two examples end to end and verifies both numerically -- the classical independence of the sample mean and sample variance (including the little-known Geary-Lukacs fact that this independence is essentially unique to the Normal distribution), and the full three-piece decomposition behind the one-way ANOVA F-test cited without proof in this site's Analysis of Variance article. Surveys where the theorem underlies practice (t-tests, ANOVA, OLS regression inference, variance confidence intervals, degrees-of-freedom bookkeeping) and closes by relating it to the Wilson-Hilferty and Fisher square-root transformations as a natural, sequential next step rather than a competing idea.