= Excess kurtosis
{title2=$\gamma_2=\kappa-3$}
<Excess kurtosis> subtracts $3$ from <kurtosis>, making the <normal distribution> baseline zero. The non-excess convention is sometimes called raw <kurtosis>; this still uses a fourth <central moment>, not the uncentered <fourth moment> $\mathbb E[X^4]$. Software may report either <kurtosis> or <excess kurtosis>, so the convention must be stated.<NIST and SEMATECH e-Handbook of Statistical Methods/1.3.5.11. Measures of Skewness and Kurtosis>[§1.3.5.11, Alternative Definition of Kurtosis, normal baseline and software convention warning]{ref}
For a <standard normal distribution>, the <moment-generating function of a standard normal variable> gives $\mathbb E[Z^4]=3$, hence <excess kurtosis> zero. For a <continuous uniform distribution> on $[-\sqrt3,\sqrt3]$, the <mean> is zero, the <variance> is one, and direct <integration> gives
$$
\kappa=\frac1{2\sqrt3}\int_{-\sqrt3}^{\sqrt3}z^4\,dz=\frac95,\qquad \gamma_2=-\frac65.
$$
The <rare-outlier three-point distribution> can instead have <kurtosis> $100$ and <excess kurtosis> $97$ with <mean> zero and <variance> one. These are calculated properties of specified <probability distributions>, rather than a measured <statistic>.
\Image[/kurtosis-normal-uniform-and-rare-support.png]
{title=<Kurtosis> comparison for a <standard normal distribution>, a <continuous uniform distribution>, and a <rare-outlier three-point distribution>}
{description=The upper panels show <probability density functions> or a <probability mass function>; the lower panels show their contributions to the fourth <central moment>. All three <probability distributions> have <mean> zero and <variance> one. Areas in the continuous panels and sums in the discrete panels give <kurtosis>. The <standard normal distribution> continues beyond the displayed window; density and mass heights have different meanings.}
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