[dahart]

Ah, fascinating, I’ve never used Order Statistics. It doesn’t look exactly like what I was thinking, but there is also a continuum from min to median to max, similar to min to mean/sum to max. I’m not sure but I might guess that for the special case of a set of independent uniform variables, the median and the mean distributions are the same? Does this mean there’s a strong or conceptual connection between the Bates distribution and the Beta distribution? (Neither Wikipedia page mentions the other.) Maybe Order Statistics are more applicable & useful than what I imagined…

[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

[btilly]

The reason that they do not look the same is that the order statistics are there presented for an exponential function, which has an unbounded upper range. When you do it on a uniform distribution with n variables, you get an n'th power and n'th root at the extremes, with varying lopsided normal-looking distributions in between.