= Kurtosis
{title2=$\kappa=\mathbb E[(X-\mathbb EX)^4]/\operatorname{Var}(X)^2$}
For a real <random variable> $X$ with <mean> $\mu=\mathbb EX$, finite positive <variance> $0<\sigma^2=\operatorname{Var}(X)<\infty$, and finite <fourth moment>, <kurtosis> is the fourth <central moment> divided by the square of the <variance>:
$$
\kappa(X)=\frac{\mathbb E[(X-\mu)^4]}{\sigma^4}=\mathbb E[Z^4],\qquad Z=\frac{X-\mu}{\sigma}.
$$
The <standard score> $Z$ measures distance from the <mean> in <standard deviations>. Its fourth power gives a value ten <standard deviations> away a weight $10^4=10000$ inside the <expected value>. Thus a small <probability mass> far from the <mean> can strongly affect <kurtosis>. It is useful for describing the contribution of extreme <standard scores>, but does not by itself describe the height or shape of a <probability density function> near its peak.<Kurtosis as Peakedness, 1905 – 2014. R.I.P. by Peter H. Westfall>[§2, Terminology, definition of κ; §4, counterexamples with different peaks; §5, central and tail contributions]{ref}
The <normal distribution> has <kurtosis> $3$. A random scale can increase <kurtosis> even when the <conditional distribution> is normal. A constant <random variable> has zero <variance>, so this ratio is undefined. If the <variance> is finite and positive but the fourth <central moment> is infinite, the ratio can be written $+\infty$ in an extended sense; it is not a finite <kurtosis>. If the <variance> itself is infinite or the <mean> is undefined, the displayed normalization is unavailable.
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