Editing Kurtosis (section)

== Sample kurtosis ==

=== Definitions ===
==== A natural but biased estimator ====

For a [[sample (statistics)|sample]] of ''n'' values, a [[Method of moments (statistics)|method of moments]] estimator of the population excess kurtosis can be defined as
<math display="block"> g_2 = \frac{m_4}{m_2^2} -3
= \frac{\tfrac{1}{n} \sum_{i=1}^n \left(x_i - \overline{x}\right)^4}{\left[\tfrac{1}{n} \sum_{i=1}^n \left(x_i - \overline{x}\right)^2\right]^2} - 3 </math>
where {{math|''m''<sub>4</sub>}} is the fourth sample [[moment about the mean]], ''m''<sub>2</sub> is the second sample moment about the mean (that is, the [[sample variance]]), {{math|''x''<sub>''i''</sub>}} is the {{mvar|i}}-th value, and <math>\overline{x}</math> is the [[sample mean]].

This formula has the simpler representation,
<math display="block"> g_2 = \frac{1}{n} \sum_{i=1}^n z_i^4 - 3 </math>
where the <math> z_i </math> values are the standardized data values using the standard deviation defined using {{mvar|n}} rather than {{math|''n'' − 1}} in the denominator.

For example, suppose the data values are 0, 3, 4, 1, 2, 3, 0, 2, 1, 3, 2, 0, 2, 2, 3, 2, 5, 2, 3, 999.

Then the {{math|''z''<sub>i</sub>}} values are −0.239, −0.225, −0.221, −0.234, −0.230, −0.225, −0.239, −0.230, −0.234, −0.225, −0.230, −0.239, −0.230, −0.230, −0.225, −0.230, −0.216, −0.230, −0.225, 4.359

and the {{math|''z''<sub>''i''</sub><sup>4</sup>}} values are  0.003, 0.003, 0.002, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.002, 0.003, 0.003, 360.976.

The average of these values is 18.05 and the excess kurtosis is thus {{nowrap|1=18.05 − 3 = 15.05}}.  This example makes it clear that data near the "middle" or "peak" of the distribution do not contribute to the kurtosis statistic, hence kurtosis does not measure "peakedness". It is simply a measure of the outlier, 999 in this example.

==== Standard unbiased estimator ====
Given a sub-set of samples from a population, the sample excess kurtosis <math>g_2</math> 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:{{r|Joanes1998}}
<math display="block">
\begin{align}
G_2 & = \frac{k_4}{k_2^2}
= \frac{n^2\,\left[(n+1)\,m_4 - 3\,(n-1)\,m_2^2\right]}{(n-1)\,(n-2)\,(n-3)} \; \frac{(n-1)^2}{n^2\,m_2^2} \\[6pt]
& = \frac{n-1}{(n-2)\,(n-3)} \left[(n+1)\,\frac{m_4}{m_2^2} - 3\,(n-1) \right] \\[6pt]
& = \frac{n-1}{(n-2)\,(n-3)} \left[(n+1)\,g_2 + 6 \right] \\[6pt]
& = \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)} \\[6pt]
& = \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{align}
</math>
where {{math|''k''<sub>4</sub>}} is the unique symmetric [[bias of an estimator|unbiased]] estimator of the fourth [[cumulant]], {{math|''k''<sub>2</sub>}} is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance), {{math|''m''<sub>4</sub>}} is the fourth sample moment about the mean, {{math|''m''<sub>2</sub>}} is the second sample moment about the mean, {{math|''x''<sub>''i''</sub>}} is the {{mvar|i}}-th value, and <math>\bar{x}</math> is the sample mean. This adjusted Fisher–Pearson standardized moment coefficient <math> G_2 </math> is the version found in [[Microsoft Excel|Excel]] and several statistical packages including [[Minitab]], [[SAS (software)|SAS]], and [[SPSS]].<ref name=Doane2011>Doane DP, Seward LE (2011) J Stat Educ 19 (2)</ref>

Unfortunately, in nonnormal samples <math>G_2</math> is itself generally biased.

=== Upper bound ===
An upper bound for the sample kurtosis of {{mvar|n}} ({{math|''n'' > 2}}) real numbers is{{r|Sharma2015}}
<math display="block"> g_2 \le \frac{1}{2} \frac{n-3}{n-2} g_1^2 + \frac{n}{2} - 3.</math>
where <math>g_1 = m_3/m_2^{3/2}</math> is the corresponding sample skewness.

=== Variance under normality ===
The variance of the sample kurtosis of a sample of size {{mvar|n}} from the [[normal distribution]] is{{r|Fisher1930}}
<math display="block"> \operatorname{var}(g_2) = \frac{24n(n-1)^2}{(n-3)(n-2)(n+3)(n+5)} </math>

Stated differently, under the assumption that the underlying random variable <math>X</math> is normally distributed, it can be shown that <math> \sqrt{n} g_2 \,\xrightarrow{d}\, \mathcal{N}(0, 24)</math>.{{r|Kendall1969|p=Page number needed}}