Sample Standard Deviation - Estimation - Corrected Sample Standard Deviation

Corrected Sample Standard Deviation

When discussing the bias, to be more precise, the corresponding estimator for the variance, the biased sample variance:

equivalently the second central moment of the sample (as the mean is the first moment), is a biased estimator of the variance (it underestimates the population variance). Taking the square root to pass to the standard deviation introduces further downward bias, by Jensen's inequality, due to the square root 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 apply Bessel's correction, using N − 1 instead of N to yield the unbiased sample variance, denoted s2:

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 residuals,

Taking square roots reintroduces bias, and yields the corrected sample standard deviation, denoted by s:

While s2 is an unbiased estimator for the population variance, s is a biased estimator for the population standard deviation, though markedly less biased than the uncorrected sample standard deviation. The bias is still significant for small samples (n less than 10), and also drops off as 1/n as sample size increases. This estimator is commonly used, and generally known simply as the "sample standard deviation".