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…