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Is there any known approach or method to measure the dispersion of a set depending on the distances between its points (i.e.: without calculating the average or the mean) ?

thanks.

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Suppose you have a finite set $X$, and some system of numbers $d(x,y)$ that can reasonably be interpreted as the distances between the points in $X$. You can then use $d$ to regard $X$ as a finite metric space. There is an interesting measure of dispersion for finite metric spaces, called the magnitude. If I understand correctly, it was largely developed by Tom Leinster and Simon Willerton. They say that it has useful applications in a number of areas, such as quantifying biodiversity.

You could start by looking at these talks: http://www.maths.ed.ac.uk/~tl/barcelona_ig/

Or at this preprint: http://arxiv.org/pdf/1012.5857v3.pdf

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