This is an excellent question that I wish I knew the answer to. Perhaps the axioms should also cover the case of the mean of n points as well as two: Then the first axiom might be revised to say the domain of a mean is any finite subset of points of most generally a metric space. One other axiom should be that the output is a continuous function of the input. In recent unpublished research, I describe how to take a kind of mean of finite -- and many other -- subsets of any compact (Riemannian) manifold X. For example if X is the usual spherical surface S^2, the mean of any two non-antipodal points is the unique midpoint of the shortest arc between them, and the mean of two antipodal points is the equator with the uniform measure on it. (And the mean of the equator is its pair of poles.) For x,y in R+, there are of course the p-means m_p for -oo <= p <= oo: m_p(x,y) = lim as q -> p of ((x^p + y^p)/2)^(1/q) for any {x,y} \subset R+. I wonder if there is a nice set of axioms for a (symmetric) mean on R+ such that the means that satisfy these axioms are exactly the means of the form m(x,y) = g^(-1)((g(x) + g(y))/2) with g required to be continuous and increasing on R+ (and throw in the means that are limits of these means). --Dan -------------------------------------- Marc writes: << But what is the Official Definition of "mean" anyway? Looking at mathworld, it would seem that we must have x M y = y M x x min y <= x M y <= x max y ax M ay = a(x M y)