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# Kurtosis

In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable.

Note: The fourth standardized moment is defined as μ4 / σ4, where μ4 is the fourth moment about the mean and σ is the standard deviation. This is sometimes used as the definition of kurtosis in older works, but is not the definition used here.

Kurtosis is more commonly defined as μ4 / σ4 − 3. The minus 3 at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero. Another reason can be seen by looking at the formula for the kurtosis of the sum of random variables. If Y is the sum of n independent random variables, all with the same distribution as X, then Kurt[Y] = Kurt[X] / n, while the formula would be more complicated if kurtosis were defined as μ4 / σ4.

A normal distribution has a kurtosis of zero (distributions with zero kurtosis are called mesokurtic). A distribution with positive kurtosis is called leptokurtic, and one with negative kurtosis platykurtic.

For a set of N values the kurtosis can be calculated as Σi(xi - μ)4 / Nσ4 − 3, where xi is the ith value and μ is the mean.

Given a sub-set of samples from a population, the equation for population kurtosis above is a biased estimator of the population kurtosis. An unbiased estimator of the population kurtosis is

$\mbox{Kurt} = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^N \left( \frac{x_i - \bar{x}}{\sigma} \right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)}$

where σ is the sample standard deviation and μ is the sample mean.