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Concept# Average

Summary

In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, an average might be another statistic such as the median, or mode. For example, the average personal income is often given as the median—the number below which are 50% of personal incomes and above which are 50% of personal incomes—because the mean would be higher by including personal incomes from a few billionaires. For this reason, it is recommended to avoid using the word "average" when discussing measures of central tendency.
If all numbers in a list are the same number, then their average is also equal to this number. This property is shared by each of the many types of average.
Another universal property is monotonicity: if two lists of numbers A and B have the same length, and each entry of list A is at least as large as the corresponding entry on list B, then the average of list A is at least that of list B. Also, all averages satisfy linear homogeneity: if all numbers of a list are multiplied by the same positive number, then its average changes by the same factor.
In some types of average, the items in the list are assigned different weights before the average is determined. These include the weighted arithmetic mean, the weighted geometric mean and the weighted median. Also, for some types of moving average, the weight of an item depends on its position in the list. Most types of average, however, satisfy permutation-insensitivity: all items count equally in determining their average value and their positions in the list are irrelevant; the average of (1, 2, 3, 4, 6) is the same as that of (3, 2, 6, 4, 1).
Pythagorean means
Mean#Pythagorean means
The arithmetic mean, the geometric mean and the harmonic mean are known collectively as the Pythagorean means.

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Average

In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, an average might be another statistic such as the median, or mode. For example, the average personal income is often given as the median—the number below which are 50% of personal incomes and above which are 50% of personal incomes—because the mean would be higher by including personal incomes from a few billionaires.

Generalized mean

In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). If p is a non-zero real number, and are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is (See p-norm).

Interquartile mean

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