Kurtosis

Definition

Let $X$ be a real random variable with $E[X^4] < \infty$, mean $\mu$ and standard deviation $\sigma > 0$. The kurtosis of $X$ is $$\operatorname{Kurt}(X) = E\left[\left(\frac{X - \mu}{\sigma}\right)^4\right] = \frac{\mu_4}{\sigma^4},$$ where $\mu_4$ is the fourth central moment. The excess kurtosis is $\operatorname{Kurt}(X) - 3$.

Remarks

Every normal distribution has kurtosis $3$, so its excess kurtosis is $0$. For every $X$, $\operatorname{Kurt}(X) \ge 1$. The Bernoulli distribution with $p = 1/2$ reaches this bound. Distributions with positive excess kurtosis are called leptokurtic. Those with negative excess kurtosis are called platykurtic. The kurtosis does not change under $X \mapsto aX + b$ with $a \ne 0$.

Explanations

L1 human

Kurtosis compares the tails of a distribution with those of the normal distribution. A high kurtosis means that rare extreme values are more frequent than for a normal variable with the same variance.