# Inter-quantile width

The difference between the lower and upper quantiles of the same level (cf. Quantile). Let $F(x)$ be a strictly-monotone continuous distribution function and let $p$ be an arbitrary number, $0<p<1/2$. The inter-quantile distance at level $p$ is defined as $x_{1-p}-x_p$, where $x_p$ and $x_{1-p}$ are the solutions of $F(x_p)=p$ and $F(x_{1-p})=1-p$, respectively. Inter-quantile distances at well-chosen levels $p$ are used in mathematical statistics and probability theory to characterize the dispersion (scatter) of probability distributions. E.g., the difference $x_{0.75}-x_{0.25}$, corresponding to $p=0.25$, has the name inter-quartile distance, and in the case of a normal distribution it is equal to $1.349\sigma$ (where $\sigma$ is the natural measure of dispersion, called the standard deviation); half the inter-quartile (inter-decile) is called the probable deviation (probable error or semi-inter-quartile distance). If $p=1/6$ or $p=1/10$, the inter-quantile distance is called the inter-sixtile or inter-tentile, respectively.