In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items.

An average is a single value that is meant to typify a list of values. If all the numbers in the list are the same, then this number should be used. If the numbers are not all the same, an easy way to get a representative value from a list is to randomly pick any number from the list. However, the word 'average' is usually reserved for more sophisticated methods that are generally found to be more useful. In the latter case, the average is calculated by combining the values from the set in a specific way and computing a single number as being the average of the set.

The most common method is the arithmetic mean but there are many other types of averages, such as median (which is used most often when the distribution of the values is skewed with some small numbers of very high values, as seen with house prices or incomes).

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Arithmetic mean

main: Arithmetic mean If $n$ numbers are given, each number denoted by a_i, where $i=1,\; \backslash dots\; ,n$, the arithmetic mean is the 1 of the a_i's divided by $n$ or

$AM=\backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^na\_i$.

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 for which we want an average, we get, for example, that the arithmetic mean of 2, 8, and 11 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.

Changing the order of the three members of the list does not change the result: A = (8 + 11 + 2)/3 = 7 and that 7 is between 2 and 11. This summation method is easily generalized for lists with any number of elements. However, the mean of a list of integers is not necessarily an integer. "The average family has 1.7 children" is a jarring way of making a statement that is more appropriately expressed by "the average number of children in the collection of families examined is 1.7".

Geometric mean

main: Geometric mean

The geometric mean of n numbers is obtained by multiplying them all together and then taking the nth root. In algebraic terms, the geometric mean of $a\_1,\; a\_2,\; ...,\; a\_n$ is defined as

$GM=\backslash sqrt2\{\backslash prod\_\{i=1\}^n\; a\_i\}=\backslash sqrt3\{a\_1\; a\_2\; ...\; a\_n\}.$

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