# Kurtosis

## kurtosis

*β*

_{2}or

*α*

_{4}.)

A measure of the heaviness (remoteness and mass) of the tails of a probability distribution. It is defined as

where *μ*_{4} is the fourth (statistical) moment about the mean and *σ*^{2} the variance. For the normal distribution, *β*_{2} = 3; cases for which *β*_{2} > 3 indicate distributions that are more outlier-prone (i.e., have heavier tails) than the normal (Gaussian) distribution, while those for which *β*_{2} < 3 indicate distributions that are less outlier-prone than the normal. In particular, the rectangular distribution *f*(*x*) = 1 (0 < *x* < 1) has *β*_{2} = 1.8. The terms leptokurtic, mesokurtic, and platykurtic refer to curves for which the values of *β*_{2} are, respectively, greater than 3, equal to 3, and less than 3. Excess is a relative expression for kurtosis, and the coefficient of excess *γ*_{2} is defined as *β*_{2} − 3. For more information about correct and incorrect interpretations of kurtosis, see Westfall (2014).

*Amer. Stat.*,

**68**, 191–195, doi:10.1080/00031305.2014.917055.

*Term edited 15 March 2019.*