Statistical sample ($(X_1,\dots ,X_n)$)

A collection of observed random variables used to infer a statistical model or its parameters. A common assumption is that the observations are independent and identically distributed random variables; dependence must otherwise be included in the model. Its randomness induces the sampling distribution of a statistic.

A collection of independent identically distributed normal random variables. Orthogonal coordinates in the standardized Gaussian vector separate the sample mean from the residual sum of squares, proving their independence and giving the normal and chi-squared distributions used for exact inference.

In the normal mean proportional to standard deviation model, both $\overline{X}$ and $S$ estimate the positive statistical parameter $\mu$. Their independent limiting normal distributions have rescaled variances ${\mu }^2$ and ${\mu }^2/2$, respectively, by asymptotic normality of a normal sample standard deviation. Thus for fixed $w\in \mathbb{R}$,The bracket is $3(w-1/3{)}^2/2+1/3$, minimized uniquely at $w=1/3$. Hence $(\overline{X}+2S)/3$ has leading sampling variance ${\mu }^2/(3n)$, the inverse expected Fisher information. The result is first-order optimality within these fixed-weight combinations and does not assert that $S$ is an exactly unbiased estimator.

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.