# Why is variability measured relative to a point?

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?

• 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. – whuber Apr 29 '14 at 19:14
• 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. – Nick Cox Apr 29 '14 at 19:33

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.