Kurtosis

Kurtosis is a measure of the peakedness of the distribution of a real-valued random variable. 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.

Kurtosis is defined as μ₄ / σ⁴ − 3, where μ₄ is the fourth moment about the mean and σ is the standard deviation. 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.

For a set of N values the kurtosis can be calculated as Σ_i(x_i - μ)⁴ / Nσ⁴ − 3, where x_i is the i^th value and μ is the mean.

See also: mean, variance, skewness.