Harmonic Mean - Relationship With Other Means

Relationship With Other Means

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.

Let r be a non zero real number and let the rth power mean ( Mr ) of a series of real variables ( a₁, a₂, a₃, ... ) be defined as

For r = -1, 1 and 2 we have the harmonic, the arithmetic and the quadratic means respectively. Define r = 0, -∞ and +∞ to be the geometric mean, the minimum of the variates and the maximum of the variates respectively. Then for any two real numbers s and t such that s < t we have

with equality only if all the a_i are equal.

Let R be the quadratic mean (or root mean square). Then