[falseprofit]

Median and mean are not the same distribution. Consider three uniform values. For the median to be small two need to be small, but the mean needs three.

I think order statistics are more useful than what you described, because “min” and “max” are themselves quantiles and more conceptually similar to “median” than to “mean”.

Trying to imagine how to bridge from min/max to mean, I guess you could take weighted averages with weights determined by order, but I can’t think of a canonical way to do that.

[adgjlsfhk1]

the canonical mapping is via norms. min is the 0 norm, 1 is the mean and the inf norm is maximum