The Harmonic Mean & the Mediant

The harmonic mean yields the “full mediant” of a finite sequence of values wherein the numerators of all the values in the series are made identical. The full mediant of a finite sequence of terms is obtained by dividing the sum of the all the numerators by the sum of the all the denominators. Consider:

Note that when the denominators form an arithmetic series, assuming a finite sequence of values wherein all the numerators are equal, then the harmonic mean is equal to the central term in the series (i.e., mean = median) or, in the case where there are an even number of terms, the mediant (equivalent to the harmonic mean) of the central two terms.

Here is a paper that discusses averages in detail: Averages