# Average - transparent background clipart (28)

Wiki article on this topic: In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value.

The most common type of average is the arithmetic mean. If n numbers are given, each number denoted by ai , the arithmetic mean is the sum of the as divided by n or

The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. One may find that A = 3 = 7.

Along with the arithmetic mean above, the geometric mean and the harmonic mean are known collectively as the Pythagorean means.

The geometric mean of n positive numbers is obtained by multiplying them all together and then taking the nth root. In algebraic terms, the geometric mean of a1, a2, ..., an is defined as

Geometric mean can be thought of as the antilog of the arithmetic mean of the logs of the numbers.

Example: Geometric mean of 2 and 8 is G M = 2 ⋅ 8 = 4 {\displaystyle GM={\sqrt {2\cdot 8}}=4}

Harmonic mean for a non-empty collection of numbers a1, a2, ..., an, all different from 0, is defined as the reciprocal of the arithmetic mean of the reciprocals of the ai' s:

One example where the harmonic mean is useful is when examining the speed for a number of fixed-distance trips. For example, if the speed for going from point A to B was 60 kmh, then the harmonic mean speed is given by

A well known inequality concerning arithmetic, geometric, and harmonic means for any set of positive numbers is

It is easy to remember noting that the alphabetical order of the letters A, G, and H is preserved in the inequality. See Inequality of arithmetic and geometric means.

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