# Kurtosis

From AMS Glossary

## kurtosis

(Symbol β

_{2}or α_{4}.) A descriptive measure of a random variable in terms of the flatness of its probability distribution.It is defined as follows: 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 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.*Term edited 16 October 2017.*