Editing Standard deviation (section)

===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, {{mvar|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 {{math|{{sfrac|{{var|s}}|{{var|c}}{{sub|4}}}}}}, where the correction factor (which depends on {{mvar|N}}) is given in terms of the [[Gamma function]], and equals:
<math display="block">c_4(N)\,=\,\sqrt{\frac{2}{N-1}}\,\,\,\frac{\Gamma\left(\frac{N}{2}\right)}{\Gamma\left(\frac{N-1}{2}\right)}.</math>

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 can be given by replacing {{math|{{var|N}}&nbsp;−&nbsp;1}} with {{math|{{var|N}}&nbsp;−&nbsp;1.5}}, yielding:
<math display="block">\hat\sigma = \sqrt{\frac{1}{N - 1.5} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2},</math>

The error in this approximation decays quadratically (as {{math|{{sfrac|1|{{var|N}}{{sup|2}}}}}}), and it is suited for all but the smallest samples or highest precision: for {{math|1={{var|N}} = 3}} the bias is equal to 1.3%, and for {{math|1={{var|N}} = 9}} the bias is already less than 0.1%.

A more accurate approximation is to replace {{math|{{var|N}} − 1.5}} above with {{math|{{var|N}} − 1.5 + {{sfrac|1|8({{var|N}} − 1)}}}}.<ref>{{Citation|first1=John |last1=Gurland |first2=Ram C. |last2=Tripathi|title=A Simple Approximation for Unbiased Estimation of the Standard Deviation|journal=The American Statistician|volume=25|issue=4|year=1971|pages=30–32|doi=10.2307/2682923|jstor=2682923 }}</ref>

For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation:
<math display="block">\hat\sigma = \sqrt{\frac{1}{N - 1.5 - \frac{1}{4}\gamma_2} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2},</math>

where {{math|{{var|γ}}{{sub|2}}}} denotes the population [[excess kurtosis]]. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data.<ref>{{Cite web|date=2021-07-11|title=Standard Deviation Calculator|url=https://purecalculators.com/standard-deviation-calculator|access-date=2021-09-14|website=PureCalculators|language=en}}</ref>