Unlock: Variance-Stabilizing Transformations

Many distributions have variance that depends on the mean: Poisson variance equals the mean, binomial-proportion variance equals p(1-p)/n. The delta method gives Var(g(X)) ≈ [g'(μ)]^2 σ^2(μ)/n, so picking g to satisfy g'(μ) σ(μ) = constant makes the asymptotic variance independent of μ. Solving this ODE produces the canonical transformations: 2√X for Poisson, arcsin(√p̂) for binomial proportions, log for multiplicative scale data, and the Fisher z-transform for the sample correlation. Anscombe's small-count corrections and the Box-Cox family complete the toolkit.