Averages

In colloquial language, an average is a middle or typical number of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the sum of the numbers divided by how many numbers are being averaged; in mathematics and statistics, this would be called the arithmetic mean. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage sometimes 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 a_i (where i = 1,2, …, n), 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 = (2 + 8)/2 = 5. Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list to 2, 8, and 11, the arithmetic mean is found by solving for the value of A in the equation 2 + 8 + 11 = A + A + A. One finds that A = (2 + 8 + 11)/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, …, a_n 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 ${\displaystyle GM={\sqrt {2\cdot 8}}=4}$

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