 Independence of the normal sample mean and sample variance

For an independent normal sample with common mean $\mu$ and positive variance ${\sigma }^2$, its sample mean and sample variance are independent. Indeed, the mean and residual vector $(X_i-\overline{X}{)}_i$ are jointly multivariate normal, and $\mathrm{Cov}(\overline{X},X_i-\overline{X})={\sigma }^2/n-{\sigma }^2/n=0$. Vanishing cross-covariance between jointly Gaussian blocks proves independence of the mean from the entire residual vector. Since the sample variance is a measurable function of that vector, independence follows. Independence from each coordinate separately would not suffice without the joint Gaussian argument.