= Probable error
{title2=$\mathbb P(|X-\mu|\leq r)=\tfrac12$}
{wiki}
The <probable error> is the half-width of an interval around a specified central value containing <probability> $1/2$. For a <normal distribution> $X\sim N(\mu,\sigma^2)$ with $\sigma>0$, its <quantile function> gives
$$
r=\sigma\Phi^{-1}(3/4)\approx0.67448975\,\sigma,
$$
where $\Phi$ is the <standard normal cumulative distribution function> and $\sigma$ is the <standard deviation>. The factor is specific to the <normal distribution>. The centered <Cauchy distribution> with parameter $\gamma>0$ has <probable error> $\gamma$, despite having no finite <variance>. Thus a central probability width need not require a finite <second moment>.
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