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In mathematics and statistics, the **arithmetic mean** of a set of numbers is the sum of all the members of the set divided by the number of items in the set (cardinality). (The word *set* is used perhaps somewhat loosely; for example, the number 3.8 could occur more than once in such a "set".) If one particular number occurs more times than others in the set, it is called a mode. The arithmetic mean is what pupils are taught very early to call the "average." If the set is a statistical population, then we speak of the **population mean**. If the set is a statistical sample, we call the resulting statistic a **sample mean**.

When the mean is not an accurate estimate of the median, the set of numbers, or frequency distribution, is said to be skewed.

The symbol μ (Greek: mu) is used to denote the arithmetic mean of a population.

If we denote a set of data by *X* = { *x*_{1}, *x*_{2}, ..., *x*_{n}}, then the sample mean is typically denoted with a horizontal bar over the variable (*x̅*, generally enunciated "*x* bar").

In practice, the difference between μ and *x̅* is that μ is typically unobservable because one observes only a sample rather than the whole population, and if the sample is drawn randomly, then one may treat *x̅*, but not μ, as a random variable, attributing a probability distribution to it.

Both are computed in the same way:

- $ \mathrm{mean} = (x_1+\cdots+x_n)/n. $

The arithmetic mean is greatly influenced by outliers. For instance, reporting the "average" net worth in Redmond, Washington as the arithmetic mean of all annual net worths would yield a surprisingly high number because of Bill Gates. These distortions occur when the mean is different from the median, and the median is a superior alternative when that happens.

In certain situations, the arithmetic mean is the wrong concept of "average" altogether. For example, if a stock rose 10% in the first year, 30% in the second year and fell 10% in the third year, then it would be incorrect to report its "average" increase per year over this three year period as the arithmetic mean (10% + 30% + (−10%))/3 = 10%; the correct average in this case is the geometric mean which yields an average increase per year of only 8.8%.

If *X* is a random variable, then the expected value of *X* can be seen as the long-term arithmetic mean that occurs on repeated measurements of *X*. This is the content of the law of large numbers. As a result, the sample mean is used to estimate unknown expected values.

Note that several other "means" have been defined, including the generalized mean, the generalized f-mean, the harmonic mean, the arithmetic-geometric mean, and the weighted mean.

## Alternate notationsEdit

The arithmetic mean may also be expressed using the sum notation:

- $ \bar{x} = \frac1n\sum_{i=1}^n x_i. $

## See alsoEdit

mean, average, summary statistics, variance, central tendency, standard deviation, inequality of arithmetic and geometric means, Muirhead's inequality

## External linksEdit

- Calculations and comparisons between arithmetic and geometric mean between two numbers
- Arithmetic and geometric means - cut-the-knotar:متوسط حسابي

cs:Aritmetický průměr de:Mittelwert es:Media aritmética fr:Moyenne arithmétique hr:Aritmetička_sredina nl:Rekenkundig gemiddeldeno:Gjennomsnittpt:Média aritmética fi:Aritmeettinen keskiarvo zh:算术平均数

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