averaged

TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
and
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical ...

, the arithmetic mean (, stress on first and third syllables of "arithmetic"), or simply the
mean There are several kinds of mean in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
or the 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 An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated. Experiments vary ...

or an
observational study In fields such as epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics), ...
, or frequently a set of results from a
survey Survey may refer to: Statistics and human research * Statistical survey Survey methodology is "the study of survey methods". As a field of applied statistics concentrating on Survey (human research), human-research surveys, survey methodology s ...
. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other
mean There are several kinds of mean in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...

s, such as the
geometric mean In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean In mathematics and statistics, the arit ...

and the
harmonic mean In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. Sometimes it is appropriate for situations when the average rate (mathematics), rate is desired. The harmonic mean can be express ...
. In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as
economics Economics () is a social science Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those societies. The term was fo ...

,
anthropology Anthropology is the of ity, concerned with , , , and , in both the present and past, including . studies patterns of behaviour, while studies cultural meaning, including norms and values. studies how language influences social life. studi ...
and
history History (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 millio ...

, and it is used in almost every academic field to some extent. For example,
per capita income Per capita income (PCI) or total income measures the average income earned per person in a given area (city, region, country, etc.) in a specified year. It is calculated by dividing the area's total income by its total population. Per capita in ...
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 statisticRobust statistics are statistic A statistic (singular) or sample statistic is any quantity computed from values in a Sample (statistics), sample that is used for a statistical purpose. Statistical purposes include estimating a population parameter ...
, meaning that it is greatly influenced by
outlier Figure 1. Box plot of data from the Michelson–Morley experiment displaying four outliers in the middle column, as well as one outlier in the first column. In statistics, an outlier is a data point that differs significantly from other observa ...

s (values that are very much larger or smaller than most of the values). For
skewed distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by e ...
s, such as the
distribution of incomeIn economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and ...
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 In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...

, may provide better description of central tendency.

# Definition

Given a
data set A data set (or dataset) is a collection of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sense, data are a set of values of qualitative property, qualitative or quantity, quantit ...
$X = \$, the arithmetic mean (or mean or average), denoted $\bar$ (read $x$ ''bar''), is the mean of the $n$ values $x_1,x_2,\ldots,x_n$. The arithmetic mean is the most commonly used and readily understood measure of central tendency in a data set. In statistics, the term
average In colloquial, ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2 ...

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 $a_1, a_2, \ldots, a_n$, then the arithmetic mean $A$ is defined by the formula: :$A=\frac\sum_^n a_i=\frac$ (for an explanation of the
summation operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, see
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: function (mathematics), fun ...

.) For example, consider the monthly salary of 10 employees of a firm: 2500, 2700, 2400, 2300, 2550, 2650, 2750, 2450, 2600, 2400. The arithmetic mean is : $\frac=2530.$ If the data set is a
statistical population In statistics, a population is a Set (mathematics), set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way gal ...
(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 The Greek alphabet has been used to write the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is an independent branch of the Indo-European languages, Indo-European family of languages, nat ...

$\mu$. If the data set is a
statistical sample In statistics and quantitative research methodology, a sample is a set of individuals or objects collected or selected from a statistical population by a defined procedure. The elements of a sample are known as Elementary event, sample points, St ...
(a subset of the population), then we call the statistic resulting from this calculation a sample mean (which for a data set $X$ is denoted as $\overline$). The arithmetic mean can be similarly defined for
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
in multiple dimension, not only
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
values; this is often referred to as a
centroid In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

. More generally, because the arithmetic mean is a
convex combinationImage:Convex combination illustration.svg, Given three points x_1, x_2, x_3 in a plane as shown in the figure, the point P ''is'' a convex combination of the three points, while Q is ''not.'' (Q is however an affine combination of the three points, a ...
(coefficients sum to 1), it can be defined on a
convex space Illustration of a non-convex set. Since the (red) part of the (black and red) line-segment joining the points x and y lies ''outside'' of the (green) set, the set is non-convex. In geometry Geometry (from the grc, γεωμετρία; ''wi ...
, not only a vector space.

# Motivating properties

The arithmetic mean has several properties that make it useful, especially as a measure of central tendency. These include: * If numbers $x_1,\dotsc,x_n$ have mean $\bar$, then $\left(x_1-\bar\right) + \dotsb + \left(x_n-\bar\right) = 0$. Since $x_i-\bar$ is the distance from a given number to the mean, one way to interpret this property is as saying that the numbers to the left of the mean are balanced by the numbers to the right of the mean. The mean is the only single number for which the residuals (deviations from the estimate) sum to zero. * If it is required to use a single number as a "typical" value for a set of known numbers $x_1,\dotsc,x_n$, then the arithmetic mean of the numbers does this best, in the sense of minimizing the sum of squared deviations from the typical value: the sum of $\left(x_i-\bar\right)^2$. (It follows that the sample mean is also the best single predictor in the sense of having the lowest
root mean squared error The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents th ...
.) If the arithmetic mean of a population of numbers is desired, then the estimate of it that is
unbiased Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
is the arithmetic mean of a sample drawn from the population.

# Contrast with median

The arithmetic mean may be contrasted with the
median In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...

. The median is defined such that no more than half the values are larger than, and no more than half are smaller than, the median. If elements in the data , when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample . The average is $2.5$, as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as , the median and arithmetic average can differ significantly. In this case, the arithmetic average is 6.2, while the median is 4. In general, the average value can vary significantly from most values in the sample, and can be larger or smaller than most of them. 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.

# Generalizations

## 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. For example, the arithmetic mean of $3$ and $5$ is $\frac\left\{\left(3+5\right)\right\}\left\{2\right\} = 4$, or equivalently $\left\left( \frac\left\{1\right\}\left\{2\right\} \cdot 3\right\right) + \left\left( \frac\left\{1\right\}\left\{2\right\} \cdot 5\right\right) = 4$. 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 $\left\left( \frac\left\{2\right\}\left\{3\right\} \cdot 3\right\right) + \left\left(\frac\left\{1\right\}\left\{3\right\} \cdot 5\right\right) = \frac\left\{11\right\}\left\{3\right\}$. Here the weights, which necessarily sum to the value one, are $\left(2/3\right)$ and $\left(1/3\right)$, 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 the weights are equal to each other (equal to $\frac\left\{1\right\}\left\{2\right\}$ in the above example, and equal to $\frac\left\{1\right\}\left\{n\right\}$ in a situation with $n$ numbers being averaged).

## Continuous probability distributions

If a numerical property, and any sample of data from it, could take on any value from a continuous range, instead of, for example, just integers, then the
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their c ...

of a number falling into some range of possible values can be described by integrating a
continuous probability distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
across this range, even when the naive probability for a sample number taking one certain value from infinitely many is zero. The analog of a weighted average in this context, in which there are an infinite number of possibilities for the precise value of the variable in each range, is called the ''mean of the probability distribution''. A most widely encountered probability distribution is called the
normal distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

; it has the property that all measures of its central tendency, including not just the mean but also the aforementioned median and the
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections that expresses an attitude of mind ** Imperative mood ** Subjunctive mo ...
(the three M's{{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), are equal to each other. This equality does not hold for other probability distributions, as illustrated for the
log-normal distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
here.

## Angles

{{Main, Mean of circular quantities Particular care must be taken when using cyclic data, such as phases or
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two rays lie in the plane (ge ...

s. Naively taking the arithmetic mean of 1° and 359° yields a result of 180°. This is incorrect for two reasons: * Firstly, angle measurements are only defined up to an additive constant of 360° (or 2π, if measuring in
radian The radian, denoted by the symbol \text, is the SI unit for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''verte ...

s). Thus one could as easily call these 1° and −1°, or 361° and 719°, since each one of them gives a different average. * Secondly, in this situation, 0° (equivalently, 360°) is geometrically a better ''average'' value: there is lower
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variation ...
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 (
viz. The abbreviation ''viz.'' (or ''viz'' without a full stop) is short for the Latin , which itself is a contraction of the Latin phrase ''videre licet'', meaning "it is permitted to see". It is used as a synonym for "namely", "that is to say", "to ...
, 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, for example as in $\bar\left\{x\right\}$ (read $x$ ''bar''). Some software ( text processors,
web browser A web browser (commonly referred to as a browser) is application software Application software (app for short) is computing software designed to carry out a specific task other than one relating to the operation of the computer itself, typical ...

s) may not display the x̄ symbol properly. For example, the x̄ symbol in
HTML The HyperText Markup Language, or HTML is the standard markup language In computer text processing, a markup language is a system for annotation, annotating a document in a way that is Syntax (logic), syntactically distinguishable from the ...

is actually a combination of two codes - the base letter x plus a code for the line above (&#772; or ¯).{{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 In some texts, such as pdfs, the x̄ symbol may be replaced by a
cent Cent may refer to: Currency * Cent (currency) The cent is a monetary unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete ...

(¢) symbol (
Unicode Unicode, formally the Unicode Standard, is an information technology standard Standard may refer to: Flags * Colours, standards and guidons * Standard (flag), a type of flag used for personal identification Norm, convention or requireme ...

¢), when copied to text processor such as
Microsoft Word Microsoft Word is a word processing software developed by Microsoft Microsoft Corporation is an American multinational corporation, multinational technology company with headquarters in Redmond, Washington. It develops, manufactures, li ...
.

{{QM_AM_GM_HM_inequality_visual_proof.svg * Fréchet mean *
Generalized mean In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
* Geometric mean * Harmonic mean * Inequality of arithmetic and geometric means * Mode (statistics), Mode * Sample mean and covariance * Standard deviation * Standard error of the mean * Summary statistics

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