View source for Mean

{{Short description|Numeric quantity representing the center of a collection of numbers}}

{{hatnote group|a
{{About|quantifying the concept of "typical value"}}  {{Broader|Average}}
{{For|the state of being mean or cruel|Meanness}}
}}
A '''mean''' is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers.<ref name=":2"/> There are several kinds of '''means''' (or "measures of [[central tendency]]") in [[mathematics]], especially in [[statistics]]. Each attempts to summarize or typify a given group of [[data]], illustrating the [[magnitude (mathematics)|magnitude]] and [[sign (mathematics)|sign]] of the [[data set]]. Which of these measures is most illuminating depends on what is being measured, and on context and purpose.<ref>{{cite AV media |date=2024-08-27 |title=Why Few Math Students Actually Understand the Meaning of Means |type=YouTube video |language=en-us |url=https://www.youtube.com/watch?v=V1_4nNm8a6w |access-date=2024-09-10 |publisher=Math The World}}</ref>

The ''[[arithmetic mean]]'', also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., x''<sub>n</sub>'' is typically denoted using an [[overhead bar]], <math>\bar{x}</math>.{{refn|Pronounced "''x'' bar".|group="note"}} If the numbers are from observing a [[sampling (statistics)|sample]] of a [[ statistical population |larger group]], the arithmetic mean is termed the ''[[sample mean]]'' (<math>\bar{x}</math>) to distinguish it from the [[ population mean |group mean]] (or [[expected value]]) of the underlying distribution, denoted '''<math>\mu</math>''' or '''<math>\mu_x</math>'''.{{refn|Greek letter [[Mu (letter)|μ]], pronounced /'mjuː/.|group="note"}}<ref>Underhill, L.G.; Bradfield d. (1998) ''Introstat'', Juta and Company Ltd. {{isbn|0-7021-3838-X}} [https://books.google.com/books?id=f6TlVjrSAsgC&pg=PA181 p. 181]</ref>

Outside [[probability]] and statistics, a wide range of other notions of mean are often used in [[geometry]] and [[mathematical analysis]]; examples are given below.

==Types of means==

===Pythagorean means===
{{Main|Pythagorean means}}
In mathematics, the three classical '''Pythagorean means''' are the [[arithmetic mean]] (AM), the [[geometric mean]] (GM), and the [[harmonic mean]] (HM). These means were studied with proportions by [[Pythagoreans]] and later generations of Greek mathematicians<ref>{{cite book|first=Thomas|last=Heath|title=History of Ancient Greek Mathematics}}</ref> because of their importance in geometry and music.
==== Arithmetic mean (AM) ====
{{Main| Arithmetic mean }}
The [[arithmetic mean]] (or simply ''mean'' or ''average'') of a list of numbers, is the sum of all of the numbers divided by their count. Similarly, the mean of a sample <math>x_1,x_2,\ldots,x_n</math>, usually denoted by <math>\bar{x}</math>, is the sum of the sampled values divided by the number of items in the sample.

:<math> \bar{x} = \frac{1}{n}\sum_{i=1}^n{x_i} = \frac{x_1+x_2+\cdots +x_n}{n} </math>

For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:
:<math>\frac{4+36+45+50+75}{5} = \frac{210}{5} = 42.</math>

==== Geometric mean (GM) ====
The [[geometric mean]] is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean):<ref name=":2">{{Cite web|title=Mean {{!}} mathematics|url=https://www.britannica.com/science/mean|access-date=2020-08-21|website=Encyclopedia Britannica|language=en}}</ref>
:<math>\bar{x} = \left( \prod_{i=1}^n{x_i} \right )^\frac{1}{n} = \left(x_1 x_2 \cdots x_n \right)^\frac{1}{n}</math>  

For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:
:<math>(4 \times 36 \times 45 \times 50 \times 75)^\frac{1}{5} = \sqrt[5]{24\;300\;000} = 30.</math>

==== Harmonic mean (HM) ====
The [[harmonic mean]] is an average which is useful for sets of numbers which are defined in relation to some [[Unit of measurement|unit]], as in the case of [[speed]] (i.e., distance per unit of time):
 
:<math> \bar{x} = n \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}</math>

For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is

:<math>\frac{5}{\tfrac{1}{4}+\tfrac{1}{36}+\tfrac{1}{45} + \tfrac{1}{50} + \tfrac{1}{75}} = \frac{5}{\;\tfrac{1}{3}\;} = 15.</math>

If we have five pumps that can empty a tank of a certain size in respectively 4, 36, 45, 50, and 75 minutes, then the harmonic mean of <math>15</math>
tells us that these five different pumps working together will pump at the same rate as five pumps that can each empty the tank in <math>15</math> minutes.

==== Relationship between AM, GM, and HM ====
{{AM_GM_inequality_visual_proof.svg}}
{{Main|QM-AM-GM-HM inequalities}}

AM, GM, and HM of [[nonnegative]] [[Real number|real numbers]] satisfy these inequalities:<ref>{{Cite book |last1=Djukić |first1=Dušan |url=https://books.google.com/books?id=okx0d9jdM8oC&pg=PA7 |title=The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009 Second Edition |last2=Janković |first2=Vladimir |last3=Matić |first3=Ivan |last4=Petrović |first4=Nikola |date=2011-05-05 |publisher=Springer Science & Business Media |isbn=978-1-4419-9854-5 |language=en}}</ref>

:<math> \mathrm{AM} \ge \mathrm{GM} \ge \mathrm{HM} \, </math>

Equality holds if all the elements of the given sample are equal.

===Statistical location===
{{See also|Average#Statistical location}}
[[File:Comparison mean median mode.svg|thumb|Comparison of the [[arithmetic mean]], [[median]], and [[mode (statistics)|mode]] of two skewed ([[log-normal distribution|log-normal]]) distributions]]
[[File:visualisation mode median mean.svg|thumb|upright|Geometric visualization of the mode, median and mean of an arbitrary probability density function<ref>{{cite web|title=AP Statistics Review - Density Curves and the Normal Distributions|url=http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions?action=purge|access-date=16 March 2015|archive-url=https://web.archive.org/web/20150402183703/http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions?action=purge|archive-date=2 April 2015|url-status=dead}}</ref>]]
In [[descriptive statistics]], the mean may be confused with the [[median]], [[Mode (statistics)|mode]] or [[mid-range]], as any of these may colloquially be called an "average" (more formally, a measure of [[central tendency]]). The mean of a set of observations is the arithmetic average of the values; however, for [[skewness|skewed distributions]], the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the [[Exponential distribution|exponential]] and [[Poisson distribution|Poisson]] distributions.

====Mean of a probability distribution====
{{Main|Expected value}}
{{See also|Population mean}}
The mean of a [[probability distribution]] is the long-run arithmetic average value of a [[random variable]] having that distribution. If the random variable is denoted by <math>X</math>, then the mean is also known as the [[expected value]] of <math>X</math> (denoted <math>E(X)</math>). For a [[discrete probability distribution]], the mean is given by <math>\textstyle \sum xP(x)</math>, where the sum is taken over all possible values of the random variable and <math>P(x)</math> is the [[probability mass function]]. For a [[continuous probability distribution|continuous distribution]], the mean is <math>\textstyle \int_{-\infty}^{\infty} xf(x)\,dx</math>, where <math>f(x)</math> is the [[probability density function]].<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Population Mean|url=https://mathworld.wolfram.com/PopulationMean.html|access-date=2020-08-21|website=mathworld.wolfram.com|language=en}}</ref> In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the [[Lebesgue integration|Lebesgue integral]] of the random variable with respect to its [[probability measure]]. The mean need not exist or be finite; for some probability distributions the mean is infinite ({{math|+&infin;}} or {{math|−&infin;}}), while for others the mean is [[Undefined (mathematics)|undefined]].

===Generalized means===

====Power mean====
{{Main|Generalized mean}}
The generalized mean, also known as the power mean or Hölder mean, abstracts several other means. It is defined for positive numbers <math>x_1, \dots, x_n</math> by{{r|:2}}
:<math>M_p(x_1, \dots, x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}.</math>
This, as a function of <math>p</math>, is well defined on <math>\mathbb{R}\setminus \{0\}</math>, but can be extended continuously to <math>\mathbb{R} \cup \{-\infty, +\infty\}</math>.<ref name="Bullen">P. S. Bullen: ''Handbook of Means and Their Inequalities''. Dordrecht, Netherlands: Kluwer, 2003, pp. 176.</ref>
By choosing different values for <math>m</math>, other well known means are retrieved.

{| class="wikitable"
|-
! Name
! Exponent
! Value
|-
| [[Minimum]]
| <math>p = -\infty</math>
| <math>\min \{x_1, \dots, x_n\}</math>
|-
| [[Harmonic mean]]
| <math>p = -1</math>
| <math>\frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}}</math>
|-
| [[Geometric mean]]
| <math>p = 0</math>
| <math>\sqrt[n]{x_1\dots x_n}</math>
|-
| [[Arithmetic mean]]
| <math>p = 1</math>
| <math>\frac{x_1 + \dots + x_n}{n}</math>
|-
| [[Root mean square]]
| <math>p = 2</math>
| <math>\sqrt{\frac{x_1^2 + \dots + x_n^2}{n}}</math>
|-
| [[Cubic mean]]
| <math>p = 3</math>
| <math>\sqrt[3]{\frac{x_1^3 + \dots + x_n^3}{n}}</math>
|-
| [[Maximum]]
| <math>p = +\infty</math>
| <math>\max\{x_1, \dots, x_n\}</math>
|}

====Quasi-arithmetic mean====
{{Main|Quasi-arithmetic mean}}
A similar approach to the power mean is the <math>f</math>-mean, also known as the quasi-arithmetic mean.
For an [[injective]] function <math>f \colon I \rightarrow \mathbb{R}</math> on an interval <math>I \subset \mathbb{R}</math> and real numbers <math>x_1, \dots, x_n \in I</math> we define their <math>f</math>-mean as
: <math> M_f(x_1, \dots, x_n) = f^{-1}\left({\frac{1}{n} \sum_{i=1}^n{f\left(x_i\right)}}\right). </math>
By choosing different functions <math>f</math>, other well known means are retrieved. 
{| class="wikitable"
|-
! Mean
! <math>I</math>
! Function{{refn|For this column we will use the "mapping arrow" to denote a function. Under this notation, the function <math>f</math> is denoted by <math>x \mapsto f(x)</math>.|group="note"}}
|-
| [[Arithmetic mean]]
| <math>\mathbb{R}</math>
| <math>x \mapsto x</math>
|-
| [[Geometric mean]]
| <math>]0, +\infty[</math>{{refn|The geometric mean is well defined on <math>[0, +\infty[</math>, but this is not captured by this approach.|group="note"}}
| <math>x \mapsto \ln(x)</math>
|-
| [[Harmonic mean]]
| <math>\mathbb{R} \setminus \{0\}</math>
| <math>x \mapsto x^{-1}</math>
|-
| [[Power mean]]
| <math>\mathbb{R} \setminus \{0\}</math>{{refn|For <math>m \neq 0</math> the domain can be <math>\mathbb{R}</math>.|group="note"}}
| <math>x \mapsto x^m</math>
|}

===Weighted arithmetic mean===
The [[weighted mean|weighted arithmetic mean]] (or weighted average) is used if one wants to combine average values from different sized samples of the same population, and is define by<ref name=":2" />

:<math>\bar{x} = \frac{\sum_{i=1}^n {w_i x_i}}{\sum_{i=1}^n w_i}, </math> 

where <math>x_i</math> and <math>w_i</math> are the mean and size of sample <math>i</math> respectively. In other applications, they represent a measure for the reliability of the influence upon the mean by the respective values.

===Truncated mean===
Sometimes, a set of numbers might contain outliers (i.e., data values which are much lower or much higher than the others). Often, outliers are erroneous data caused by [[artifact (observational)|artifacts]]. In this case, one can use a [[truncated mean]]. It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of the total number of values.

===Interquartile mean===
The [[interquartile mean]] is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.
: <math>\bar{x} = \frac{2}{n} \;\sum_{i = \frac{n}{4} + 1}^{\frac{3}{4}n}\!\! x_i</math>

assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights.

===Mean of a function===
{{Main|Mean of a function}}
In some circumstances, mathematicians may calculate a mean of an infinite (or even an [[uncountable]]) set of values. This can happen when calculating the mean value <math>y_\text{avg}</math> of a function <math>f(x)</math>. Intuitively, a mean of a function can be thought of as calculating the area under a section of a curve, and then dividing by the length of that section. This can be done crudely by counting squares on graph paper, or more precisely by [[integral|integration]]. The integration formula is written as:

: <math>y_\text{avg}(a, b) = \frac{1}{b - a} \int_a^b f(x)\,dx.</math>

In this case, care must be taken to make sure that the integral converges. But the mean may be finite even if the function itself tends to infinity at some points.

===Mean of angles and cyclical quantities===
[[Angle]]s, times of day, and other cyclical quantities require [[modular arithmetic]] to add and otherwise combine numbers. These quantities can be averaged using the [[circular mean]]. In all these situations, it is possible that no mean exists, for example if all points being averaged are equidistant. Consider a [[color wheel]]—there is no mean to the set of all colors. Additionally, there may not be a ''unique'' mean for a set of values: for example, when averaging points on a clock, the mean of the locations of 11:00 and 13:00 is 12:00, but this location is equivalent to that of 00:00.

=== Fréchet mean ===
The [[Fréchet mean]] gives a manner for determining the "center" of a mass distribution on a [[Surface (mathematics)|surface]] or, more generally, [[Riemannian manifold]]. Unlike many other means, the Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars.
It is sometimes also known as the ''Karcher mean'' (named after Hermann Karcher).

=== Triangular sets ===

In geometry, there are thousands of different
definitions for [[Triangle center|the center of a triangle]] that can all be interpreted as the mean of a triangular set of points in the plane.<ref>{{cite journal
 | last1 = Narboux | first1 = Julien
 | last2 = Braun | first2 = David
 | doi = 10.1007/s11786-016-0254-4
 | issue = 1
 | journal = Mathematics in Computer Science
 | mr = 3483261
 | pages = 57–73
 | title = Towards a certified version of the encyclopedia of triangle centers
 | volume = 10
 | year = 2016
 | url = https://hal.inria.fr/hal-01174131/file/certified-etc.pdf
 | quote = under the guidance of Clark Kimberling, an electronic encyclopedia of triangle centers (ETC) has been developed, it contains more than 7000 centers and many properties of these points}}</ref>

===Swanson's rule===

This is an approximation to the mean for a moderately skewed distribution.<ref name=Hurst2000>Hurst A, Brown GC, Swanson  RI (2000) Swanson's 30-40-30 Rule.  American Association of Petroleum Geologists Bulletin 84(12) 1883-1891</ref> It is used in [[hydrocarbon exploration]] and is defined as:

: <math> m = 0.3P_{10} + 0.4P_{50} + 0.3P_{90} </math>

where <math display="inline">P_{10}</math>, <math display="inline">P_{50}</math> and <math display="inline">P_{90}</math> are the 10th, 50th and 90th percentiles of the distribution, respectively.

===Other means===
{{main cat|Means}}
{{div col|colwidth=22em}}
*[[Arithmetic-geometric mean]]
*[[Arithmetic-harmonic mean]]
*[[Cesàro mean]]
*[[Chisini mean]]
*[[Contraharmonic mean]]
*[[Elementary symmetric mean]]
*[[Geometric-harmonic mean]]
*[[Grand mean]]
*[[Heinz mean]]
*[[Heronian mean]]
*[[Identric mean]]
*[[Lehmer mean]]
*[[Logarithmic mean]]
*[[Moving average]]
*[[Neuman–Sándor mean]]
*[[Quasi-arithmetic mean]]
*[[Root mean square]] (quadratic mean)
*[[Rényi's entropy]] (a [[generalized f-mean]])
*[[Spherical mean]]
*[[Stolarsky mean]]
*[[Weighted geometric mean]]
*[[Weighted harmonic mean]]
{{div col end}}

==See also==
{{Portal|Mathematics}}
*[[Statistical dispersion]]
*[[Central tendency]]
**[[Median]]
**[[Mode (statistics)|Mode]]
*[[Descriptive statistics]]
*[[Kurtosis]]
*[[Law of averages]]
*[[Mean value theorem]]
*[[Moment (mathematics)]]
*[[Summary statistics]]
*[[Taylor's law]]

==Notes==
{{Reflist|group=note}}

==References==
{{Reflist}}

{{Statistics|descriptive}}
{{Authority control}}

[[Category:Means| ]]
[[Category:Moments (mathematics)]]