in colloquial language average usually refers to the sum of a list of numbers divided by the size of the list, in other words the arithmetic mean. however, the word "average" can be used to refer to the median, the mode, or some other central or typical value. in statistics, these are all known as measures of central tendency. thus the concept of an average can be extended in various ways in mathematics, but in those contexts it is usually referred to as a mean (for example the mean of a function). ==calculation== ===arithmetic mean===  the most common type of average is the arithmetic mean. if n numbers are given, each number denoted by ai, where i&nbsp;=&nbsp;1,&nbsp;â¦,&nbsp;n, the arithmetic mean is the [sum] of the ai's divided by n or :am = \frac{1}{n}\sum_{i=1}^na_i = \frac{1}{n}\left(a_1 + a_2 + \cdots + a_n\right) 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 =&nbsp;(2&nbsp;+&nbsp;8)/2 =&nbsp;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&nbsp;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&nbsp;+&nbsp;8&nbsp;+&nbsp;11 =&nbsp;a&nbsp;+&nbsp;a&nbsp;+&nbsp;a. one finds that a =&nbsp;(2&nbsp;+&nbsp;8&nbsp;+&nbsp;11)/3 =&nbsp;7. ===pythagorean means===   along with the arithmetic mean above, the geometric mean and the harmonic mean are known collectively as the pythagorean means. ====geometric mean==== the geometric mean of n non-negative numbers is obtained by multiplying them all together and then taking the nth root. in algebraic terms, the geometric mean of a1,&nbsp;a2,&nbsp;â¦,&nbsp;a'n is defined as : gm= \sqrt[n]{\prod_{i=1}^n a_i} = \sqrt[n]{a_1 a_2 \cdots a_n} 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 sm = \sqrt{2 \cdot 8} = 4 ====harmonic mean==== harmonic mean for a non-empty collection of numbers a1,&nbsp;a2,&nbsp;â¦,&nbsp;a'n, all different from 0, is defined as the reciprocal of the arithmetic mean of the reciprocals of the a'is: : hm = \frac{1}{\frac{1}{n}\sum_{i=1}^n \frac{1}{a_i}} = \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}} 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
Arithmetic_functions Means Statistical_terminology Summary_statistics