Sample Variance

To estimate the population Variance from a sample of $N$ elements with a priori unknown Mean (i.e., the Mean is estimated from the sample itself), we need an unbiased Estimator for $\sigma$. This is the k-Statistic $k_2$, where

$$k_2 = {N\over N-1} m_2$$ (1)

and $m_2\equiv s^2$ is the sample variance

$$s^2\equiv {1\over N}\sum_{i=1}^N (x_i-\bar x)^2.$$ (2)

Note that some authors prefer the definition

$$s'^2\equiv {1\over N-1}\sum_{i=1}^N (x_i-\bar x)^2,$$ (3)

since this makes the sample variance an Unbiased Estimator for the population variance.

See also k-Statistic, Variance