I responded to Rich,
rcs> avg( avg(a,b), avg(c,d) ) = same thing with b and c swapped
rwg>Bingo. "Association" was a poor choice of words. Dyadic symmetry?
rcs>This *is* true for ordinary averages, and geometric, harmonic, etc.
means.
And the resulting symmetric expressions make it obvious how to define avg(a,b,c). A nonobvious def in terms of dyadic avg is the root of
avg(avg(a,b),avg(c,x)) = x .
This does *not* work with elliptic mean. One plausible def is to iterate [a,b,c] <- [em(b,c),em(c,a),em(a,b)].
The skewed?/biased?/slanted?/weighted? mean avg(a,b) := t*max(a,b)+(1-t)*min(a,b) is an unscrupulous (dyadically asymmetrical) mean for which the [a,b,c] iteration gives (t^2 max(a,b,c) + (1-t) t median(a,b,c) + (1-t)^2 min(a,b,c) avg(a,b,c) := ------------------------------------------------------------ , 1 - t + t^2 (likewise the fixed point equation if a>=b>=c or a<=b<=c), with the obvious generalization to n (sorted) meanands. Yet avg(a,b,c,d) /= avg(avg(a,b),avg(c,d)) except for t= 0 (min), 1/2 (am), and 1 (max). Various iteration schemes for the avg of n meanands in terms of fewer can make inefficient sorting algorithms. Any efficient ones? --rwg