Tell me the method, I'll give you the mean

Different procedures underlying the determination of the mean of a set of values impart different weights to these values. Thus, different methods result in different weighted means. In particular, when the weights are directly proportional, or in fact equal, to the (weighted) values, the self-weighted mean is obtained, and when they are inversely proportional to the values, the harmonic mean results. Generally, the former is greater and the latter is smaller than the arithmetic (uniformly weighted) mean. We illustrate cases of sampling procedures and of geometric requirements that give rise to the self-weighted and harmonic means. We call attention to some psychological difficulties and to the didactic value of dealing with cases of differential weighting.