Why are measures of dispersion calculated relative to some central point? Why wouldn't, for instance, all possible non-repeated, pairwise differences in the dataset be a valid measure of spread?
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4$\begingroup$ When $X$ and $Y$ are identically distributed random variables, then $\frac{1}{2}\mathbb{E}((X-Y)^2)$--which measures all pairwise differences--is exactly the common variance of $X$ and $Y$. This shows there isn't necessarily any difference at all between the two approaches. $\endgroup$– whuber ♦Apr 29, 2014 at 19:14
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4$\begingroup$ The same theme is developed, rather differently, in L-moments (start at en.wikipedia.org/wiki/L-moment). The second L-moment is essentially a reincarnation of an often invented measure based on comparing using absolute differences rather than squared differences. See also projecteuclid.org/download/pdf_1/euclid.ss/1028905831 for an accessible (double sense) historical perspective. $\endgroup$– Nick CoxApr 29, 2014 at 19:33
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Actually, not all measures of dispersion are calculated relative to some central point. Examples include the $Q_n$ and $S_n$ statistics. Your intuition is sharp, as indeed, the calculation of these only relies on pairwise differences.
A downside of these two estimators is that they are less efficient at the gaussian distribution than the classical variance. However, an advantage is that they are more robust.
