Mean

In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of adiscrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving ${\displaystyle \mu =\sum xP(x)}$. An analogous formula applies to the case of acontinuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite: for example, when the probability of the value ${\displaystyle 2^{n}}$ is ${\displaystyle {\tfrac {1}{2^{n}}}}$ for n = 1, 2, 3, ....

Arithmetic mean (AM)[edit]

Main article: Arithmetic mean

The arithmetic mean (or simply "mean") of a sample ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$, usually denoted by ${\displaystyle {\bar {x}}}$, is the sum of the sampled values divided by the number of items in the sample:

${\displaystyle {\bar {x}}={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}}$

For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is

${\displaystyle {\frac {4+36+45+50+75}{5}}={\frac {210}{5}}=42.}$

Geometric mean (GM)[edit]

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.

${\displaystyle {\bar {x}}=\left(\prod _{i=1}^{n}{x_{i}}\right)^{\tfrac {1}{n}}}$

For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:

${\displaystyle (4\times 36\times 45\times 50\times 75)^{^{1}/_{5}}={\sqrt[{5}]{24\;300\;000}}=30.}$

Harmonic mean (HM)[edit]

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for examplespeed (distance per unit of time).

${\displaystyle {\bar {x}}=n\cdot \left(\sum _{i=1}^{n}{\frac {1}{x_{i}}}\right)^{-1}}$

For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is

${\displaystyle {\frac {5}{{\tfrac {1}{4}}+{\tfrac {1}{36}}+{\tfrac {1}{45}}+{\tfrac {1}{50}}+{\tfrac {1}{75}}}}={\frac {5}{\;{\tfrac {1}{3}}\;}}=15.}$