Words for describing the quirks of the way that the "average()" operation composes

Is there some general way of expressing the fact that the operation of taking an average is sensitive not only to the set of numbers on which it is operating but also the number of instances of those numbers?

That is, for the 2-set example, (ma+nb)/(m+n) is not necessarily equal to (a+b)/2.

2.

Or, rephrased slightly: that the operation of taking an average does not compose unless the quantities of the values being averaged are equal.

So, if we imagine avg is a function that takes the average of its arguments:

avg(avg(a₁, ..., a_m), avg(b₁, ..., b_n)) = avg(a, b) which is not necessarily equal to avg(a₁, ... , a_m, b₁, ..., b_n)