Sample variance ($s_n^2$)

For observations $X_1,\dots ,X_n$, the unbiased sample variance isFor independent identically distributed observations with finite variance ${\sigma }^2$, it converges in probability to ${\sigma }^2$.

The sample standard deviation is the nonnegative square root of the usual sample variance for $n\ge 2$. The displayed convention uses divisor $n-1$. Although the sample variance is an unbiased estimator of the population variance under independent identically distributed sampling with finite variance, taking its square root does not in general give an unbiased estimator of the population standard deviation.

For a normal sample of variance ${\sigma }^2>0$, define $S^2=(n-1{)}^{-1}{\sum }_i(X_i-\overline{X}{)}^2$. The exact chi-squared distribution identity $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$ and the central limit theorem for independent squared standard normal random variables implyThe delta method for $g(v)=\sqrt{v}$, with derivative $g^{\prime }({\sigma }^2)=1/(2\sigma )$, therefore gives the displayed limit for $S$. Exact independence of the normal sample mean and sample variance also makes $S$ independent of $\overline{X}$. The resulting joint limit supports combining location and scale estimates when a model relates their parameters.

Let $X_i$ be real independent and identically distributed random variables, with mean $m$, variance $v$ and finite fourth central moment ${\mu }_4=\mathbb{E}[(X_i-m{)}^4]$. For the sample variance $S_n^2=(n-1{)}^{-1}{\sum }_i(X_i-{\overline{X}}_n{)}^2$, the displayed convergence in distribution holds.

Indeed $n^{-1}{\sum }_i(X_i-{\overline{X}}_n{)}^2=n^{-1}{\sum }_i(X_i-m{)}^2-({\overline{X}}_n-m{)}^2$. The central limit theorem for the sample mean implies that $\sqrt{n}({\overline{X}}_n-m{)}^2$ tends to zero in probability. Changing the divisor from $n$ to $n-1$ also has negligible effect at this scale, by the law of large numbers. Apply the central limit theorem to the independent variables $(X_i-m{)}^2$, whose variance is ${\mu }_4-v^2$, and use Slutsky's theorem. Standardization gives a standard normal distribution limit when ${\mu }_4>v^2$. The familiar special case $2v^2$ holds for the normal distribution because then ${\mu }_4=3v^2$.

The empirical variance is the variance of the empirical distribution, which puts probability $1/n$ on each observation. It uses divisor $n$, whereas the usual sample variance uses divisor $n-1$. For independent and identically distributed random variables with finite second moment, jackknife bias correction converts the empirical variance into the unbiased estimator $s_n^2$ exactly.

For two independent normal samples of sizes $m,n$ with a common unknown variance, pooling their within-sample sums of squares gives the unbiased estimator $S_p^2=(S_{XX}+S_{YY})/(m+n-2)$. The numerator divided by the common population variance has a chi-squared distribution with $m+n-2$ degrees of freedom and is independent of both sample means. Therefore $(\bar{X}-\bar{Y}-({\mu }_X-{\mu }_Y))/(S_p\sqrt{1/m+1/n})$ has a Student t-distribution. This exact construction requires a positive residual number of degrees of freedom and the common-variance assumption.