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''inter-quantile distance, inter-quantile range''

The difference between the lower and upper quantiles of the same level (cf. [[Quantile|Quantile]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i0517901.png" /> be a strictly-monotone continuous distribution function and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i0517902.png" /> be an arbitrary number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i0517903.png" />. The inter-quantile distance at level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i0517904.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i0517905.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i0517906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i0517907.png" /> are the solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i0517908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i0517909.png" />, respectively. Inter-quantile distances at well-chosen levels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i05179010.png" /> are used in mathematical statistics and probability theory to characterize the dispersion (scatter) of probability distributions. E.g., the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i05179011.png" />, corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i05179012.png" />, has the name inter-quartile distance, and in the case of a normal distribution it is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i05179013.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i05179014.png" /> is the natural measure of dispersion, called the [[Standard deviation|standard deviation]]); half the inter-quartile (inter-decile) is called the [[Probable deviation|probable deviation]] (probable error or semi-inter-quartile distance). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i05179015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051790/i05179016.png" />, the inter-quantile distance is called the inter-sixtile or inter-tentile, respectively.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.U. Yale,   "An introduction to the theory of statistics" , Griffin  (1916)</TD></TR></table>