In [[mathematics]] and [[statistics]], the '''arithmetic [[mean]]''' (or simply the '''mean''') of a list of numbers is the sum of all the members of the list divided by the number of items in the list. If the list is a [[statistical population]], then the mean of that population is called a '''population mean'''. If the list is a [[sampling (statistics)|statistical sample]], we call the resulting [[statistic]] a '''sample mean'''.

Some are incorrectly taught that the mean is the "[[average]]". The mean is rather the most commonly used type of average. This misconception causes problems when people encounter other types of averages, such as the [[median]], and they assume it's the arithmetic mean. This behavior is sometimes deliberately exploited in an effort to mislead with statistics. For this reason, it is important to distinguish the difference between 'mean' and 'average'.

When the mean is not an accurate estimate of the [[median]], the list of numbers, or [[frequency distribution]], is said to be [[skewness|skewed]].

==Introduction==
If we denote a set of data by ''X'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>), then the sample mean is typically denoted with a horizontal bar over the variable (<math>\bar{x} \,</math>, enunciated "''x'' bar").

The symbol μ (Greek: mu) is used to denote the arithmetic mean of an entire population. Or, for a [[random number]] that has a defined mean, μ is the ''probabilistic mean'' or [[expected value]] of the random number. If the set ''X'' is a collection of random numbers with probabilistic mean of μ, then for any individual sample, ''x''<sub>''i''</sub>, from that collection, μ = E{''x''<sub>''i''</sub>} is the [[expected value]] of that sample.

In practice, the difference between μ and <math>\bar{x} \,</math> 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 <math>\bar{x} \,</math>, but not μ, as a [[random variable]], attributing a [[probability distribution]] to it (the [[sampling distribution]] of the mean).

Both are computed in the same way:
:<math>\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i = \frac{1}{n} (x_1+\cdots+x_n).</math>

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 [[generalised f-mean|generalized f-mean]], the [[harmonic mean]], the [[arithmetic-geometric mean]], and various [[weighted mean]]s.

===Examples===
*If you have 3 numbers then add them and divide them by 3: <math>\frac{x_1 + x_2 + x_3}{3}</math>
*If you have 4 numbers add them and divide by 4: <math>\frac{x_1 + x_2 + x_3 + x_4}{4}</math>

==Formulation as an optimization problem==

The arithmetic mean is the value with minimal quadratical distance from the given values:

:<math>\bar{x} = \arg \min_{y\in\R} {\sum_{k=1}^n (y-x_k)^2}. </math>

Equivalently, the mean is the best [[least squares]] fit of a constant function to the given data.

==Problems with some uses of the mean==
While the mean is often used to report [[central tendency]], it may not be appropriate for describing [[skewed distribution]]s, because it is easily misinterpreted. The arithmetic mean is greatly influenced by [[outlier]]s. These distortions can occur when the mean is different from the median. When this happens the [[median]] may be a better description of central tendency.

A classic example is [[average income]]. The arithmetic mean may be misinterpreted to imply that most people's incomes are higher than is in fact the case. When presented with an "average" one may be led to believe that ''most'' people's incomes are near this number. This "average" (arithmetic mean) income ''is'' higher than most people's incomes, because high income outliers skew the result higher (in contrast, the median income "resists" such skew). However, this "average" says nothing about the number of people near the median income (nor does it say anything about the modal income that most people are near). Nevertheless, because one might carelessly relate "average" and "most people" one might incorrectly assume that most people's incomes would be higher (nearer this inflated "average") than they are. For instance, reporting the "average" [[net worth]] in [[Medina, Washington]] as the arithmetic mean of all annual net worths would yield a surprisingly high number because of [[Bill Gates]]. Consider the scores (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five out of six scores are below this.

In certain situations, the arithmetic mean is the wrong measure of central tendency altogether. For example, if a stock fell 10 % in the first year, and rose 30 % in the second year, then it would be incorrect to report its "average" increase per year over this two year period as the arithmetic mean (&minus;10 % + 30 %)/2 = 10 %; the correct average in this case is the [[geometric mean]] which yields an average increase per year of only 8.2 %. The reason for this is that each of those percents have different starting points. If the stock starts at $30 and falls 10 %, it is now at $27. If the stock then rises 30 %, it is now $35.1. The arithmetic mean of those rises is 10 %, but since the stock rose by $5.1 in 2 years, an [[average]] of 8.2 % would result in the final $35.1 figure [$30(1-10 %)(1+30 %) = $30(1+8.2 %)(1+8.2 %) = $35.1]. If one used the arithmetic mean 10 % in the same way, one would not get the actual increase [$30(1+10 %)(1+10 %) = $36.3].

Particular care must be taken when using cyclic data such as phases or angles. Taking the arithmetic mean of 1 degree and 359 degrees yields a result of 180 degrees, whereas 1 and 359 are both adjacent to 360 degrees which may be a more correct ''average'' value. In general application such an oversight will lead to the average value artificially moving towards the middle of the numerical range. A solution to this problem is to use the optimization formulation, and redefine the difference as a modular distance.

==See also==
<div style="-moz-column-count:3; column-count:3;">
* [[Average]]
* [[Mean]]
* [[mode (statistics)|Mode]]
* [[Median]]
* [[Summary statistics]]
* [[Central tendency]]
* [[Generalized mean]]
* [[Geometric mean]]
* [[Variance]]
* [[Standard deviation]]
* [[Inequality of arithmetic and geometric means]]
* [[Muirhead's inequality]]
* [[Sample size]]
* [[Sample mean and covariance]]
* [[Empirical measure]]
</div>

==Further reading==
*[[Darrell Huff]], ''How to lie with statistics'', Victor Gollancz, 1954 (ISBN 0-393-31072-8).

==External links==
*[http://www.sengpielaudio.com/calculator-geommean.htm Calculations and comparisons between arithmetic and geometric mean between two numbers]
*[http://people.revoledu.com/kardi/tutorial/BasicMath/Average/index.html Mean or Average]
*[http://mipagina.cantv.net/arithmetic/rmdef.htm Rational Mean]

[[Category:Means]]
{{Statistics}}
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