Arithmetic mean

In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk ˈmiːn/, stress on third syllable of 'arithmetic'), or simply the mean or average when the context is clear, 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 of an experiment or an observational study, or frequently a set of results from a survey. The term 'arithmetic mean' is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk ˈmiːn/, stress on third syllable of 'arithmetic'), or simply the mean or average when the context is clear, 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 of an experiment or an observational study, or frequently a set of results from a survey. The term 'arithmetic mean' is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology, and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers (values that are very much larger or smaller than most of the values). Notably, for skewed distributions, such as the distribution of income for which a few people's incomes are substantially greater than most people's, the arithmetic mean may not coincide with one's notion of 'middle', and robust statistics, such as the median, may be a better description of central tendency. The arithmetic mean (or mean or average), x ¯ {displaystyle {ar {x}}} (read x {displaystyle x} bar), is the mean of the n {displaystyle n} values x 1 , x 2 , … , x n {displaystyle 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 , … , a n {displaystyle a_{1},a_{2},ldots ,a_{n}} , then the arithmetic mean A {displaystyle A} is defined by the formula: (See summation for an explanation of the summation operator).