The Means

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

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

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

Mr(a) = (∑ar/n)1/r = (∑air/n)1/r

where I assume that all ai'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, Mr(a) is the mean value of numbers a1, a2, ..., an with the exponent r. M2(a) is known as the quadratic average.

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

M0(a) = (a1a2 ... an)1/n


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The Means


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