The Means

As is customary, group a finite sequence of numbers a₁, a₂, ..., a_n in a vector form a = (a₁, a₂, ..., a_n). The arithmetic and harmonic means of the sequence defined as

A(a) = (∑a)/n = (∑a_i)/n and H(a) = n/(∑1/a) = n/(∑1/a_i),

where the summation is being carried from i = 1 through i = n. Observe that H(a) = 1/A(1/a). This leads to a more general definition

M_r(a) = (∑a^r/n)^1/r = (∑a_i^r/n)^1/r

where I assume that all a_i's are positive and r is a real number different from 0. For example, A(a) = M₁(a) whereas H(a) = M_-1(a).

In general, M_r(a) is the mean value of numbers a₁, a₂, ..., a_n with the exponent r. M₂(a) is known as the quadratic average.

For r = 0, the following complements the definition of M_r(a):

M₀(a) = (a₁a₂ ... a_n)^1/n

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The Means

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