Sample Standard Deviation - Estimation - Unbiased Sample Standard Deviation

Unbiased Sample Standard Deviation

For unbiased estimation of standard deviation, there is no formula that works across all distributions, unlike for mean and variance. Instead, s is used as a basis, and is scaled by a correction factor to produce an unbiased estimate. For the normal distribution, an unbiased estimator is given by s/c4, where the correction factor (which depends on N) is given in terms of the Gamma function, and equals:

This arises because the sampling distribution of the sample standard deviation follows a (scaled) chi distribution, and the correction factor is the mean of the chi distribution.

An approximation is given by replacing N − 1 with N − 1.5, yielding:

\hat\sigma = \sqrt{ \frac{1}{N - 1.5} \sum_{i=1}^n (x_i - \bar{x})^2 },

The error in this approximation decays quadratically (as 1/N2), and it is suited for all but the smallest samples or highest precision: for n = 3 the bias is equal to 1.3%, and for n = 9 the bias is already less than 0.1%.

For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation:

\hat\sigma = \sqrt{ \frac{1}{n - 1.5 - \tfrac14 \gamma_2} \sum_{i=1}^n (x_i - \bar{x})^2 },

where γ2 denotes the population excess kurtosis. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data.

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