  Encyclopedia > Generalized mean

Article Content

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,...,an as

$M(t) = \left( \frac{1}{n} \sum_{i=1}^n a_{i}^t \right) ^ {1/t}$

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 = ... = an. Furthermore, if a is a positive real number, then the generalized mean with exponent t of the numbers aa1,..., aan is equal to a times the generalized mean of the numbers a1,..., an.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 French resistance ... Dericourt[?]. In January 1 1942 Jean Moulin parachuted to Arles with two other men and radio equipment and continued to Marseilles. De Gaulle had sent him coordinate ...  