On meaningfulness of means

We are concerned with the definition of means or averaging operators, i.e., aggregators M(x(1),..., x(i),..., x(m)) where x(i) is an element of R present measures on the reals corresponding to a given type of scale (ordinal, interval or ratio scale). The adequate mean which varies between min(x(1),...,x(m)) should correspond to some comparison meaningfulness property or some functional equation that induces invariance or stability. We characterize the averaging operators which present some ''natural'' properties (continuity, monotonicity, neutrality, unanimity, etc.) and comparison meaningfulness or invariance and we show that operators like order statistics and ordered weighted averaging operators can be used to rank elements defined by vectors (x(1),...,x(m)) in a meaningful way in the spirit of measurement theory.