Arithmetic mean

In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk/ ^ⓘ arr-ith-MET-ik), arithmetic average, or just the mean or average is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps to distinguish it from other types of means, such as geometric and harmonic.

Arithmetic means are also frequently used in economics, anthropology, history, and almost every other academic field to some extent. For example, per capita income is the arithmetic average of the income of a nation's population.

While the arithmetic mean is often used to report central tendencies, it is not a robust statistic: it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions, such as the distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median, may provide a better description of central tendency.

The arithmetic mean of a set of observed data is equal to the sum of the numerical values of each observation, divided by the total number of observations. Symbolically, for a data set consisting of the values ${\displaystyle x_{1},\dots ,x_{n}}$, the arithmetic mean is defined by the formula:

In simpler terms, the formula for the arithmetic mean is:

${\displaystyle {\frac {\text{Total of all numbers within the data}}{\text{Amount of total numbers within the data}}}}$

For example, if the monthly salaries of ${\displaystyle 5}$ employees are ${\displaystyle \{2500,2700,2300,2650,2450\}}$, then the arithmetic mean is:

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 ${\displaystyle \mu }$. If the data set is a statistical sample (a subset of the population), it is called the sample mean (which for a data set ${\displaystyle X}$ is denoted as ${\displaystyle {\overline {X}}}$).