View source for Arithmetic mean

{{Short description|Type of average of a collection of numbers}}
{{redirect|X̄|the character|macron (diacritic)}}
{{broader|Mean}}{{Use dmy dates|date=December 2020}}

In [[mathematics]] and [[statistics]], the '''arithmetic mean''' ({{IPAc-en|pron|ˌ|æ|r|ɪ|θ|ˈ|m|ɛ|t|ɪ|k}} {{respell|arr|ith|MET|ik}}), '''arithmetic average''', or just 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.<ref>{{cite book|last=Jacobs|first=Harold R.|title=Mathematics: A Human Endeavor|edition=Third|year=1994|publisher=[[W. H. Freeman]]|page=547|isbn=0-7167-2426-X}}</ref> The collection is often a set of results from an [[experiment]], an [[observational study]], or a [[Survey (statistics)|survey]]. The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as [[geometric mean|geometric]] and [[harmonic mean|harmonic]].

In addition to mathematics and statistics, the arithmetic mean is frequently used in [[economics]], [[anthropology]], [[history]], and 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 tendency|central tendencies]], it is not a [[robust statistic]]: it is greatly influenced by [[outlier]]s (values much larger or smaller than most others). For [[skewed distribution]]s, 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.

==Definition==
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 <math>x_1,\dots,x_n</math>, the arithmetic mean is defined by the formula:

:<math>\bar{x}=\frac{1}{n}\left (\sum_{i=1}^n{x_i}\right)
=\frac{x_1+x_2+\dots+x_n}{n}</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Arithmetic Mean|url=https://mathworld.wolfram.com/ArithmeticMean.html|access-date=2020-08-21|website=mathworld.wolfram.com|language=en}}</ref>

(For an explanation of the summation operator, see [[summation]].)

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

<math>\frac{\text{Total of all numbers within the data}}{\text{Amount of total numbers within the data}}
</math>

For example, if the monthly salaries of <math>10</math> employees are <math>\{2500,2700,2400,2300,2550,2650,2750,2450,2600,2400\}</math>, then the arithmetic mean is:

:<math>\frac{2500+2700+2400+2300+2550+2650+2750+2450+2600+2400}{10}=2530</math>

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 alphabet|Greek letter]] <math>\mu</math>. If the data set is a [[Sampling (statistics)|statistical sample]] (a subset of the population), it is called the ''[[sample mean]]'' (which for a data set <math>X</math> is denoted as <math>\overline{X}</math>).

The arithmetic mean can be similarly defined for [[Vector (mathematics and physics)|vectors]] in multiple dimensions, not only [[Scalar (mathematics)|scalar]] values; this is often referred to as a [[centroid]]. More generally, because the arithmetic mean is a [[convex combination]] (meaning its coefficients sum to <math>1</math>), it can be defined on a [[convex space]], not only a vector space.

==History==
The statistician [[Churchill Eisenhart]], senior researcher fellow at the [[National Institute of Standards and Technology|U. S. National Bureau of Standards]], traced the history of the arithmetic mean in detail. In the modern age it started to be used as a way of combining various observations that should be identical, but were not such as estimates of the direction of magnetic north.
In 1635 the mathematician [[Henry Gellibrand]] described as “meane” the midpoint of a lowest and highest number, not quite the arithmetic mean. In 1668, a person known as “DB” was quoted in the [[Transactions of the Royal Society]] describing “taking the mean” of five values:<ref>{{cite web |last1=Eisenhart |first1=Churchill |title=The Development of the Concept of the Best Mean of a Set of Measurements from Antiquity to the Present Day |url=https://www.york.ac.uk/depts/maths/histstat/eisenhart.pdf |publisher=Presidential Address, 131st Annual Meeting of the American Statistical Association, Colorado State University |pages=68–69 |date=24 August 1971}}</ref>
{{quote|In this Table, he [Capt. Sturmy] notes the greatest difference to be 14 minutes; and so taking the mean for the true Variation, he concludes it then and there to be just 1. deg. 27. min.|D.B. p. 726}}

==Motivating properties==
The arithmetic mean has several properties that make it interesting, especially as a measure of central tendency. These include:

*If numbers <math>x_1,\dotsc,x_n</math> have mean <math>\bar{x}</math>, then <math>(x_1-\bar{x})+\dotsb+(x_n-\bar{x})=0</math>. Since <math>x_i-\bar{x}</math> is the distance from a given number to the mean, one way to interpret this property is by saying that the numbers to the left of the mean are balanced by the numbers to the right. The mean is the only number for which the [[Errors and residuals in statistics|residuals]] (deviations from the estimate) sum to zero.  This can also be interpreted as saying that the mean is [[Translational symmetry|translationally invariant]] in the sense that for any real number <math>a</math>, <math>\overline{x + a} = \bar{x} + a</math>.
*If it is required to use a single number as a "typical" value for a set of known numbers <math>x_1,\dotsc,x_n</math>, then the arithmetic mean of the numbers does this best since it minimizes the sum of squared deviations from the typical value: the sum of <math>(x_i-\bar{x})^2</math>. The sample mean is also the best single predictor because it has the lowest [[root mean squared error]].<ref name="JM">{{cite book|last=Medhi|first=Jyotiprasad|title=Statistical Methods: An Introductory Text|url=https://books.google.com/books?id=bRUwgf_q5RsC|year=1992|publisher=New Age International|isbn=9788122404197|pages=53–58}}</ref> If the arithmetic mean of a population of numbers is desired, then the estimate of it that is [[Unbiased estimate|unbiased]] is the arithmetic mean of a sample drawn from the population.

*The arithmetic mean is independent of scale of the units of measurement, in the sense that <math>\text{avg}(ca_{1},\cdots,ca_{n})=c\cdot\text{avg}(a_{1},\cdots,a_{n}).</math>  So, for example, calculating a mean of liters and then converting to gallons is the same as converting to gallons first and then calculating the mean.  This is also called [[Homogeneous function|first order homogeneity]].

===Additional properties===
* The arithmetic mean of a sample is always between the largest and smallest values in that sample.
*The arithmetic mean of any amount of equal-sized number groups together is the arithmetic mean of the arithmetic means of each group.

==Contrast with median==
{{main|Median}}

The arithmetic mean may be contrasted with the [[median]]. The median is defined such that no more than half the values are larger, and no more than half are smaller than it. If elements in the data [[arithmetic progression|increase arithmetically]] when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample <math>\{1,2,3,4\}</math>. The mean is <math>2.5</math>, as is the median. However, when we consider a sample that cannot be arranged to increase arithmetically, such as <math>\{1,2,4,8,16\}</math>, the median and arithmetic average can differ significantly. In this case, the arithmetic average is <math>6.2</math>, while the median is <math>4</math>. The average value can vary considerably from most values in the sample and can be larger or smaller than most.

There are applications of this phenomenon in many fields. For example, since the 1980s, the median income in the United States has increased more slowly than the arithmetic average of income.<ref>{{cite magazine|first=Paul|last=Krugman|url=http://prospect.org/article/rich-right-and-facts-deconstructing-inequality-debate|title=The Rich, the Right, and the Facts: Deconstructing the Income Distribution Debate|magazine=The American Prospect|date=4 June 2014|orig-year=Fall 1992}}</ref>

==Generalizations==

===Weighted average===
{{main|Weighted average}}

A weighted average, or weighted mean, is an average in which some data points count more heavily than others in that they are given more weight in the calculation.<ref>{{Cite web|title=Mean {{!}} mathematics|url=https://www.britannica.com/science/mean|access-date=2020-08-21|website=Encyclopedia Britannica|language=en}}</ref> For example, the arithmetic mean of <math>3</math> and <math>5</math> is <math>\frac{3+5}{2}=4</math>, or equivalently <math>3 \cdot \frac{1}{2}+5 \cdot \frac{1}{2}=4</math>. In contrast, a ''weighted'' mean in which the first number receives, for example, twice as much weight as the second (perhaps because it is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as <math>3 \cdot \frac{2}{3}+5 \cdot \frac{1}{3}=\frac{11}{3}</math>. Here the weights, which necessarily sum to one, are <math>\frac{2}{3}</math> and <math>\frac{1}{3}</math>, the former being twice the latter. The arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all weights are equal to the same number (<math>\frac{1}{2}</math> in the above example and <math>\frac{1}{n}</math> in a situation with <math>n</math> numbers being averaged).

===Functions===
{{excerpt|Arithmetic mean of a function}}

===Continuous probability distributions===
[[File:Comparison mean median mode.svg|thumb|300px|Comparison of two [[log-normal distribution]]s with equal median, but different [[skewness]], resulting in various means and [[mode (statistics)|mode]]s]]

If a numerical property, and any sample of data from it, can take on any value from a continuous range instead of, for example, just integers, then the [[probability]] of a number falling into some range of possible values can be described by integrating a [[continuous probability distribution]] across this range, even when the naive probability for a sample number taking one certain value from infinitely many is zero. In this context, the analog of a weighted average, in which there are infinitely many possibilities for the precise value of the variable in each range, is called the ''mean of the [[probability distribution]]''. The most widely encountered probability distribution is called the [[normal distribution]]; it has the property that all measures of its central tendency, including not just the mean but also the median mentioned above and the mode (the three Ms<ref name=ThreeMs>{{cite web|url=https://www.visualthesaurus.com/cm/lessons/the-three-ms-of-statistics-mode-median-mean/|title=The Three M's of Statistics: Mode, Median, Mean June 30, 2010|website=www.visualthesaurus.com|author=Thinkmap Visual Thesaurus|date=2010-06-30|access-date=2018-12-03}}</ref>), are equal. This equality does not hold for other probability distributions, as illustrated for the log-normal distribution here.

===Angles===
{{main|Mean of circular quantities}}

Particular care is needed when using cyclic data, such as phases or [[angle]]s. Taking the arithmetic mean of 1° and 359° yields a result of 180[[degree (angle)|°]].
This is incorrect for two reasons:
*Firstly, angle measurements are only defined up to an additive constant of 360° (<math>2\pi</math> or <math>\tau</math>, if measuring in [[radian]]s). Thus, these could easily be called 1° and -1°, or 361° and 719°, since each one of them produces a different average.
*Secondly, in this situation, 0° (or 360°) is geometrically a better ''average'' value: there is lower [[Statistical dispersion|dispersion]] about it (the points are both 1° from it and 179° from 180°, the putative average).

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 (that is, define the mean as the central point: the point about which one has the lowest dispersion) and redefine the difference as a modular distance (i.e., the distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°).

{{AM_GM_inequality_visual_proof.svg}}

==Symbols and encoding==
The arithmetic mean is often denoted by a bar ([[Vinculum (symbol)|vinculum]] or [[Macron (diacritic)|macron]]), as in <math>\bar{x}</math>.<ref name="JM"/>

Some software ([[Text processing|text processors]], [[web browser]]s) may not display the "x̄" symbol correctly. For example, the [[HTML]] symbol "x̄" combines two codes — the base letter "x" plus a code for the line above ( &#772; or ¯).<ref>{{Cite web|url=http://www.personal.psu.edu/ejp10/psu/gotunicode/statsym.html|title=Notes on Unicode for Stat Symbols|website=www.personal.psu.edu|access-date=2018-10-14|archive-date=31 March 2022|archive-url=https://web.archive.org/web/20220331114343/http://www.personal.psu.edu/ejp10/psu/gotunicode/statsym.html|url-status=dead}}</ref>

In some document formats (such as [[PDF]]), the symbol may be replaced by a "¢" ([[Euro coins|cent]]) symbol when copied to a text processor such as [[Microsoft Word]].

==See also==
{{QM AM GM HM inequality visual proof.svg}}
* [[Fréchet mean]]
*[[Generalized mean]]
*[[Inequality of arithmetic and geometric means]]
* [[Sample mean and covariance]]
* [[Standard deviation]]
* [[Standard error of the mean]]
* [[Summary statistics]]

== Notes ==
{{notelist}}
{{reflist|group=note}}

==References==
{{Reflist}}

==Further reading==
*{{cite book|last=Huff|first=Darrell|title=How to Lie with Statistics|year=1993|publisher=W. W. Norton|isbn=978-0-393-31072-6|url-access=registration|url=https://archive.org/details/howtoliewithstat00huff}}

==External links==
*[http://www.sengpielaudio.com/calculator-geommean.htm Calculations and comparisons between arithmetic mean and geometric mean of two numbers]
*[http://www.fxsolver.com/browse/formulas/Arithmetic+Mean Calculate the arithmetic mean of a series of numbers on fxSolver]

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[[Category:Means]]