User:Sphilbrick/MathML example 1

Definition

Given a data set $X=\{x_{1},\ldots ,x_{n}\}$, the arithmetic mean (or mean or average), denoted ${\bar {x}}$ (read $x$ bar), is the mean of the $n$ values $x_{1},x_{2},\ldots ,x_{n}$.

The arithmetic mean is the most commonly used and readily understood measure of central tendency in a data set. In statistics, the term average refers to any of the measures of central tendency. The arithmetic mean of a set of observed data is defined as being equal to the sum of the numerical values of each and every observation, divided by the total number of observations. Symbolically, if we have a data set consisting of the values $a_{1},a_{2},\ldots ,a_{n}$, then the arithmetic mean $A$ is defined by the formula:

$A={\frac {1}{n}}\sum _{i=1}^{n}a_{i}={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}$

(for an explanation of the summation operator, see summation.)

For example, consider the monthly salary of 10 employees of a firm: 2500, 2700, 2400, 2300, 2550, 2650, 2750, 2450, 2600, 2400. The arithmetic mean is

${\frac {2500+2700+2400+2300+2550+2650+2750+2450+2600+2400}{10}}=2530.$

If the data set is a statistical population (i.e., consists of every possible observation and not just a subset of them), then the mean of that population is called the population mean, and denoted by the Greek letter $\mu$. If the data set is a statistical sample (a subset of the population), then we call the statistic resulting from this calculation a sample mean (which for a data set $X$ is denoted as ${\overline {X}}$).

The arithmetic mean can be similarly defined for vectors in multiple dimension, not only scalar values; this is often referred to as a centroid. More generally, because the arithmetic mean is a convex combination (coefficients sum to 1), it can be defined on a convex space, not only a vector space.