Standard deviation
Corrected sample standard deviation

If the biased sample variance (the second central moment of the sample, which is a downward-biased estimate of the population variance) is used to compute an estimate of the population's standard deviation, the result is s N = 1 N ∑ i = 1 N (x i − x ¯) 2. {\displaystyle s_{N}={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}.}

Here taking the square root introduces further downward bias, by Jensen's inequality, due to the square root's being a concave function. The bias in the variance is easily corrected, but the bias from the square root is more difficult to correct, and depends on the distribution in question.

An unbiased estimator for the variance is given by applying Bessel's correction, using N − 1 instead of N to yield the unbiased sample variance, denoted s: s 2 = 1 N − 1 ∑ i = 1 N (x i − x ¯) 2. {\displaystyle s^{2}={\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}.}

This estimator is unbiased if the variance exists and the sample values are drawn independently with replacement. N − 1 corresponds to the number of degrees of freedom in the vector of deviations from the mean, (x 1 − x ¯, …, x n − x ¯). {\displaystyle \textstyle (x_{1}-{\bar {x}},\;\dots,\;x_{n}-{\bar {x}}).}

Taking square roots reintroduces bias (because the square root is a nonlinear function which does not commute with the expectation, i.e. often E ≠ E {\textstyle E\neq {\sqrt {E}}}), yielding the corrected sample standard deviation, denoted by s: s = 1 N − 1 ∑ i = 1 N (x i − x ¯) 2. {\displaystyle s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}.}

As explained above, while s is an unbiased estimator for the population variance, s is still a biased estimator for the population standard deviation, though markedly less biased than the uncorrected sample standard deviation. This estimator is commonly used and generally known simply as the "sample standard deviation". The bias may still be large for small samples (N less than 10). As sample size increases, the amount of bias decreases. We obtain more information and the difference between 1 N {\displaystyle {\frac {1}{N}}} and 1 N − 1 {\displaystyle {\frac {1}{N-1}}} becomes smaller.
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