Suppose I have a data set $ x=\{x_1,\dots, x_n\} $. Then does the following statistical dispersion exist?
$$ \frac 1 n \sum_{i,j\in n} |x_i-x_j| $$
If not, why? It is independent of mean values, sounds interesting. But it grows exponentially in computational effort.
I played around with it a bit and it behaves similar to the geometric mean with its GSD, but of course looks much simpler than the GSD
The statistic $$U_n=\binom{n}{2}^{-1}\sum_{1\leq i<j\leq n}\vert X_i-X_j\vert$$
is known as the absolute mean difference or Gini’s mean difference. This is a U-statistic of order 2 for the kernel $h(x_1,x_2) = \vert x_1 - x_2\vert.$