structural:expansion, information:extraction, information:highlighting
Standard unbiased estimator

Given a sub-set of samples from a population, the sample excess kurtosis g 2 {\displaystyle g_{2}} above is a biased estimator of the population excess kurtosis. An alternative estimator of the population excess kurtosis, which is unbiased in random samples of a normal distribution, is defined as follows: G 2 = k 4 k 2 2 = n 2 (n − 1) (n − 2) (n − 3) (n − 1) 2 n 2 m 2 2 = n − 1 (n − 2) (n − 3) = n − 1 (n − 2) (n − 3) = (n + 1) n (n − 1) (n − 2) (n − 3) ∑ i = 1 n (x i − x ¯) 4 (∑ i = 1 n (x i − x ¯) 2) 2 − 3 (n − 1) 2 (n − 2) (n − 3) = (n + 1) n (n − 1) (n − 2) (n − 3) ∑ i = 1 n (x i − x ¯) 4 k 2 2 − 3 (n − 1) 2 (n − 2) (n − 3) {\displaystyle {\begin{aligned}G_{2}&={\frac {k_{4}}{k_{2}^{2}}}={\frac {n^{2}\,\left}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {(n-1)^{2}}{n^{2}\,m_{2}^{2}}}\\&={\frac {n-1}{(n-2)\,(n-3)}}\left\\&={\frac {n-1}{(n-2)\,(n-3)}}\left\\&={\frac {(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{4}}{\left(\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}\right)^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)\,(n-3)}}\\&={\frac {(n+1)\,n}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{4}}{k_{2}^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)(n-3)}}\end{aligned}}} where k is the unique symmetric unbiased estimator of the fourth cumulant, k is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance), m is the fourth sample moment about the mean, m is the second sample moment about the mean, x is the i -th value, and x ¯ {\displaystyle {\bar {x}}} is the sample mean. This adjusted Fisher–Pearson standardized moment coefficient G 2 {\displaystyle G_{2}} is the version found in Excel and several statistical packages including Minitab, SAS, and SPSS.