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

M(s) <= M(t)

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 as

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