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Home > Generalized mean

If t is a non-zero real number, we can define the generalized mean with exponent t of the positive real numbers a₁,...,a_n as

The case t = 1 yields the arithmetic mean and the case t = −1 yields the harmonic mean. As t approaches 0, the limit of M(t) is the geometric mean of the given numbers, and so it makes sense to define M(0) to be the geometric mean. Furthermore, as t approaches ∞, M(t) approaches the maximum of the given numbers, and as t approaches −∞, M(t) approaches the minimum of the given numbers.

In general, if −∞ ≤ s < t ≤ ∞, then

and the two means are equal if and only if a₁ = a₂ = ... = a_n. Furthermore, if a is a positive real number, then the generalized mean with exponent t of the numbers aa₁,..., aa_n is equal to a times the generalized mean of the numbers a₁,..., a_n.

This could be generalized further to the generalized f-mean:

and again a suitable choice of an invertible f(x) will give the arithmetic mean with f(x) = x, the geometric mean with f(x) = log(x), the harmonic mean with f(x) = 1/x, and the generalized mean with exponent t with f(x) = x^t. But other functions could be used, such as f(x) = e^x.

See also

• average

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