 Thus the "n"-th harmonic mean is related to the "n"-th geometric and arithmetic means.The general formula is

If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.For the special case of just two numbers, formula_9 and formula_10, the harmonic mean can be written

In this special case, the harmonic mean is related to the arithmetic mean formula_12 and the geometric mean formula_13 by

Since formula_15 by the inequality of arithmetic and geometric means, this shows for the "n" = 2 case that "H" ≤ "G" (a property that in fact holds for all "n").