Take the 2-minute tour ×
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It's 100% free, no registration required.

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?

share|improve this question
4  
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 at 19:14
    
@whuber awesome, that's neat insight thank you! –  trichman22 Apr 29 at 19:27
4  
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 at 19:33

1 Answer 1

up vote 7 down vote accepted

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.

share|improve this answer
    
Thanks for taking the time! –  trichman22 Apr 29 at 19:04
    
My pleasure, kiddo. –  Deathkill14 Apr 29 at 19:05

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.