Rick,
This is related to Mallows Distance (avg(x_i - y_i)^p)^(1/p), which has been used (p=2) to measure the distance between an observed sample and a reference sample (e.g. a normal distribution) with the same descriptors (median, mean, ...) I first came across it in Diaconis' article on "theories of data analysis". It is also related to Wasserstein or Earth Mover's distance in multiple dimensions.

Bill