[[Moment (probability)]]
# Excess kurtosis

The **excess kurtosis** $\mathrm{Kurt}[X]$ of a [[real random variable]] $X : \xi \to \mathbb{R}$ with [[Expectation|mean]] $\mu$ and [[Standard deviation|variance]] $\sigma^2$ is a shifted version of the fourth [[Statistical moment|standardized moment]] of $X$ #m/def/prob 
$$
\begin{align*}
\mathrm{Kurt}[X] = \Ex\left[ \left( \frac{X-\mu}{\sigma} \right)^4 \right]-3
\end{align*}
$$
where the unshifted version is sometimes referred to as the **kurtosis** (although this term is also used for the shifted version).
The reason for the shift is that it ensures that any [[normal distribution]] has excess kurtosis of zero.


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#state/tidy | #lang/en | #SemBr