Average

{{Merging from|Central tendency|Mean|discuss=Talk:Average#Merge proposal|date=February 2026}}

{{About|the concept by this name|the Canadian artist with this surname|Joe Average}} {{For|the equivalent concept in geometry|centre (geometry)}} {{Short description|Number taken as representative of a list of numbers}} In mathematics, an '''average''' of a collection or group is a value that is most central, common, or typical in some sense, and represents its overall position. In mathematics, it most commonly refers to the arithmetic mean, but may also refer to other measures such as other types of mean, the median, or the mode.

==Definitions== [[File:Bivariate mean median mode.svg|thumb|class=skin-invert-image|Representation of the arithmetic mean, median and mode of a set of 250 points. The black curve represents the theoretical distribution used to generate the points, with the gray histogram depicting the actual distribution.]] [[File:MeansSemicircleChords.svg|thumb|class=skin-invert-image|Four means of two numbers, a and b, constructed as chords on a semicircle. The arithmetic, geometric and harmonic means are sometimes referred to as the "Pythagorean means"<ref>{{Cite web |last=Cantrell |first=David W. |title=Pythagorean Means |url=https://mathworld.wolfram.com/PythagoreanMeans.html |access-date=2025-11-04 |website=MathWorld |language=en}}</ref>. This type of construction highlights the ordering of the values of the different means.]]

The most commonly used definition of the average is the arithmetic mean,<ref>{{Cite journal |last=Kaplan |first=Jennifer |last2=Fisher |first2=Dianne G. |last3=Rogness |first3=Neal T. |date=July 2010 |title=Lexical Ambiguity in Statistics: How students use and define the words: association, average, confidence, random and spread |trans-title=part 2 |url=https://jse.amstat.org/v18n2/kaplan.pdf |journal=Journal of Statistics Education |volume=18 |issue=2 |doi=10.1080/10691898.2010.11889491 |eissn=1069-1898}}</ref> i.e. the sum divided by the count, so the "average" of the list of numbers [2, 3, 4, 7, 9] is generally considered to be (2+3+4+7+9)/5 = 25/5 = 5.

However, other meanings are sometimes used depending on the context, which can lead to confusion; for instance, in teaching, "average" sometimes refers to "the three Ms": mean, median, and mode.<ref>{{Cite journal |last=Stack |first=Sue |last2=Watson |first2=Jane |last3=Hindley |first3=Sue |last4=Samson |first4=Pauline |last5=Devlin |first5=Robyn |date=2010 |title=What's average? |url= |journal=Australian Mathematics Teacher (AMT) |language=en |volume=66 |issue=3 |pages=7–15 |issn=0045-0685 |id={{Academia.edu|48155690}}. {{EBSCOhost|53382428}}. Informit [https://search.informit.org/doi/10.3316/informit.354027435519820 10.3316/informit.354027435519820].}}</ref><ref>{{Cite journal |last=Quinnell |first=Lorna |date=September 2017 |title=Those muddling M's: Scaffolding understanding of averages in mathematics. |journal=The Australian Mathematics Teacher (AMT) |language=en |volume=73 |issue=3 |pages=6–12 |issn=0045-0685 |id={{EBSCOhost|125324609}}. Informit [https://search.informit.org/doi/10.3316/informit.123380293707263 10.3316/informit.123380293707263], [https://search.informit.org/doi/10.3316/aeipt.218997 10.3316/aeipt.218997].}}</ref><ref>{{Cite journal |last=Watson |first=Jane M. |last2=Moritz |first2=Jonathan B. |date=1999 |title=The Development of Concepts of Average |url= |journal=Focus on Learning Problems in Mathematics |language=en |volume=21 |issue=4 |pages=15–39 |issn=0272-8893 |id={{ERIC|EJ612005}}.}}</ref><ref>{{Cite journal |last=Watson |first=Jane M. |date=May 2007  |title=The Role of Cognitive Conflict in Developing Students’ Understanding of Average |url=https://link.springer.com/article/10.1007/s10649-006-9043-3 |journal=Educational Studies in Mathematics |language=en |volume=65 |pages=21–47 |doi=10.1007/s10649-006-9043-3 |issn=0013-1954|url-access=subscription }}</ref><ref>{{Cite journal |last=Kaplan |first=Jennifer J. |last2=Fisher |first2=Diane G. |last3=Rogness |first3=Neal T. |date=2009 |title=Lexical Ambiguity in Statistics: What do students know about the words association, average, confidence, random and spread? |trans-title=part 1 |url=https://jse.amstat.org/v17n3/kaplan.html  |journal=Journal of Statistics Education |volume=17 |issue=3 |doi=10.1080/10691898.2009.11889535 |eissn=1069-1898|doi-access=free }}</ref><ref>{{Cite book |last=Triola |first=Mario F. |url=https://api.pageplace.de/preview/DT0400.9781292055787_A24586321/preview-9781292055787_A24586321.pdf#page=58 |title=Elementary Statistics |date=2014 |publisher=Pearson Education Limited |isbn=978-1-292-03941-1 |edition=12th Pearson New International |page=53 |chapter=}}</ref><ref>{{Cite conference |last=Watson |first=Jane |last2=Chick |first2=Helen |date=July 2–6, 2012 |title=Average Revisited in Context |url=https://eric.ed.gov/?id=ED573389  |conference=35th Annual Meeting of the Mathematics Education Research Group of Australasia (MERGA), Singapore |publisher=Mathematics Education Research Group of Australasia |publication-place=Adelaide SA, Australia |pages=753–760 |id=This research was partially funded by ARC Linkage Grant No. LP0669106.}}</ref>

The median, defined as the value in the center after sorting the group, is usually used as the average in situations where the data is skewed or has outliers, in order to focus on the main part of the group rather than the long tail. For example, the average personal income is usually given as the median income, so that it represents the majority of the population rather than being overly influenced by the much higher incomes of the few rich people.<ref>{{Cite web |date=4 April 2005 |title=Mean or median? 'Average' household income |url=http://www.conceptstew.co.uk/PAGES/mean_or_median.html |url-status=dead |archive-url=https://web.archive.org/web/20051205025913/http://www.conceptstew.co.uk/PAGES/mean_or_median.html |archive-date=2005-12-05 |website=Statistics for the Terrified |publisher=Concept Stew Ltd}}</ref>

The harmonic mean, defined as the reciprocal of the mean of the reciprocals, is used in a variety of situations involving rates or ratios, such as computing the average speed from multiple measurements taken ''over the same distance''<ref>{{Cite journal |last=de Mestre |first=Neville |date=2019 |title=discovery: What do you mean by an average? |url= |journal=Australian Mathematics Education Journal (AMEJ) |language=en |publisher= |volume=1 |issue=4 |pages=24–25 |issn=2652-0176 |id={{EBSCOhost|140411813}}. Informit [https://search.informit.org/doi/10.3316/informit.837688193205822 10.3316/informit.837688193205822].}}</ref>. Indeed, unlike an arithmetic mean or median of speeds, a harmonic mean of speeds will give the value of the constant speed that would cause one to travel the same distance in the same amount of time.

The mode represents the most common value found in the group. It can be used when the data is categorical rather than numeric,<ref>{{Cite web |title=Teacher Discussion - Meet the Averages |trans-title=based on an article in ''The Mercury'', Sunday, 16 August, 1998, p.9 |url=http://www.mercurynie.com.au/mathguys/discuss/1998/980816d1.htm |url-status=dead |archive-url=https://web.archive.org/web/20110301231349/http://www.mercurynie.com.au/mathguys/discuss/1998/980816d1.htm |archive-date=2011-03-01 |website=Numeracy in the News }}</ref> when the frequency of each value is relevant (such as where a histogram, bar chart, or probability density function is being referenced),<ref>{{Cite conference |last=Landtblom |first=Karin |date=July 2018 |editor-last=Sorto |editor-first=M. A. |editor2-last=White |editor2-first=A. |editor3-last=Guyot |editor3-first=L. |title=IS DATA A QUANTITATIVE THING? AN ANALYSIS OF THE CONCEPT OF THE MODE IN TEXTBOOKS FOR GRADE 4-6. |url=http://iase-web.org/icots/10/proceedings/pdfs/ICOTS10_2F1.pdf?1531364243 |conference=Tenth International Conference on Teaching Statistics (ICOTS10), Kyoto, Japan |publisher=International Statistical Institute |publication-place=Voorburg, The Netherlands |archive-url=https://web.archive.org/web/20191218144934/http://iase-web.org/icots/10/proceedings/pdfs/ICOTS10_2F1.pdf?1531364243 |archive-date=2019-12-18 |work=Looking back, looking forward |url-status=dead}}</ref> or to find a value that represents the majority of the group.<ref>{{Cite news |last=Fowler |first=Elizabeth M. |date=August 4, 1956 |title=The Three Averages: A Study of How and Why Statisticians Make Use of Mean, Median and Mode / 3 AVERAGES USED BY STATISTICIANS |url=https://www.nytimes.com/1956/08/04/archives/the-three-averages-a-study-of-how-and-why-statisticians-make-use-of.html |work=New York Times |pages=19, 25 [but digitally tagged as 28, 34] |id={{ProQuest|113794286}}.}}</ref>

Other statistics that can be used as an average include the mid-range, the quadratic mean or the geometric mean, but they are rarely referred to as "the average".

These different quantities all estimate the central tendency of a group, with each having their advantages and issues. Mathematically, they can be thought as solving different variational problems.

==General properties== All averages of a collection are somewhere within its bounding box (and so for real numbers, between its maximum and minimum). Therefore, if a collection consists entirely of the same value, any average of it is that value.<ref>{{Cite journal |last=Nagumo |first=Mitio |author-link=Mitio Nagumo |date=1930 |orig-date=Eingegangen am 1. November, 1929 |title=9. Über eine Klasse der Mittelwerte. |url=https://www.jstage.jst.go.jp/article/jjm1924/7/0/7_0_71/_pdf |journal=Japanese Journal of Mathematics |publisher= |volume=VII |pages=71–79 |id=HathiTrust record [https://catalog.hathitrust.org/Record/006954247 006954247], items [https://babel.hathitrust.org/cgi/pt?id=inu.30000091433023&seq=85 inu.30000091433023], [https://babel.hathitrust.org/cgi/pt?id=uc1.31822020414991&seq=489 uc1.31822020414991], [https://babel.hathitrust.org/cgi/pt?id=rul.39030029662022&seq=483 rul.39030029662022].}}</ref>

Most averages{{Efn|One exception to this is the mode; for example, the mode of [1, 1, 2, 2, 2] is 2, but the mode of [1, 1, 2, 3, 4] is 1. Also, most kinds of averages are strictly monotone, but some, such as the median, truncated mean, and winsorized mean, are only weakly monotone, and may remain the same after some of the values are increased.}} are monotonic, i.e. moving a member of it in one direction causes the average to move in the same direction, or equivalently, if two collections of numbers ''A'' and ''B'' have the same number of elements, and they can be arranged such that each entry in ''A'' ≥ the corresponding entry in ''B'', then the average of ''A'' ≥ the average of ''B''.

All commonly-used averages are linearly homogeneous, i.e. multiplying every value by the same scale factor multiplies the average by that same scale factor.

Most averages{{Efn|Exceptions to this may include weighted averages (if the weights are assigned by position) and moving averages (if the entire resulting sequence or curve is considered). Moving averages are often done using position-weighted averages.}} remain identical when the list of items is permuted, i.e. the ordering does not matter.

==List of possible averages== {{see also|Mean#Other means|Central tendency#Solutions to variational problems}}

{| class="wikitable" |- ! Name !! Equation or description !! As solution to optimization problem |- | Arithmetic mean || <math>\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i = \frac{1}{n} (x_1 + \cdots + x_n)</math> || <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n (x - x_i)^2</math> |- | Median || A middle value that separates the higher half from the lower half of the data set; may not be unique if the data set contains an even number of points || <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n |x - x_i|</math> |- | Geometric median || A rotation invariant extension of the median for points in <math>\mathbb{R}^d</math> || <math>\underset{\vec{x} \in \mathbb{R}^d}{\operatorname{argmin}}\, \sum_{i=1}^n ||\vec{x} - \vec{x}_i||_2</math> |- | Tukey median || Another rotation invariant extension of the median for points in <math>\mathbb{R}^d</math>—a point that maximizes the Tukey depth || <math>\underset{\vec{x} \in \mathbb{R}^d}{\operatorname{argmax}}\, \underset{\vec{u} \in \mathbb{R}^d}{\operatorname{min}} \, \sum_{i=1}^n \left(\begin{cases}1, \text{ if }(\vec{x}_i-\vec{x})\cdot\vec{u} \geq 0 \\ 0, \text{ otherwise}\end{cases}\right)</math> |- | Mode || The most frequent value in the data set || <math>\underset{x \in \mathbb{R}}{\operatorname{argmax}}\, \sum_{i=1}^n \left(\begin{cases}1, \text{ if }x = x_i \\ 0, \text{ if }x \neq x_i\end{cases}\right)</math> |- | Geometric mean || <math>\sqrt[n]{\prod_{i=1}^n x_i} = \sqrt[n]{x_1 \cdot x_2 \dotsb x_n}</math> || <math>\underset{x \in \mathbb{R}_{> 0}}{\operatorname{argmin}}\, \sum_{i=1}^n (\ln(x) - \ln(x_i))^2,\qquad \text{if }x_i > 0\,\forall\, i \in \{1,\dots,n\}</math> |- | Harmonic mean || <math>\frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}</math> || <math>\underset{x \in \mathbb{R}_{\neq 0}}{\operatorname{argmin}}\, \sum_{i=1}^n \left(\frac{1}{x} - \frac{1}{x_i}\right)^2</math> |- |Contraharmonic mean |<math>\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{{x_1} +{x_2} + \cdots + {x_n}}</math> |<math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n x_i(x - x_i)^2</math> |- | Lehmer mean|| <math>\frac{\sum_{i=1}^n x_i^p}{\sum_{i=1}^n x_i^{p-1}}</math>|| <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n x_i^{p-1}(x - x_i)^2</math> |- | Quadratic mean (or RMS) || <math>\sqrt{\frac{1}{n} \sum_{i=1}^{n} x_i^2} = \sqrt{\frac{1}{n}\left(x_1^2 + x_2^2 + \cdots + x_n^2\right)}</math>|| <math>\underset{x \in \mathbb{R}_{\geq 0}}{\operatorname{argmin}}\, \sum_{i=1}^n (x^2 - x_i^2)^2</math> |- | Cubic mean|| <math>\sqrt[3]{\frac{1}{n} \sum_{i=1}^{n} x_i^3} = \sqrt[3]{\frac{1}{n}\left(x_1^3 + x_2^3 + \cdots + x_n^3\right)}</math>|| <math>\underset{x \in \mathbb{R}_{\geq 0}}{\operatorname{argmin}}\, \sum_{i=1}^n (x^3 - x_i^3)^2,\qquad \text{if }x_i \geq 0\,\forall\, i \in \{1,\dots,n\}</math> |- | Generalized mean|| <math>\sqrt[p]{\frac{1}{n} \cdot \sum_{i=1}^n x_{i}^p}</math>|| <math>\underset{x \in \mathbb{R}_{\geq 0}}{\operatorname{argmin}}\, \sum_{i=1}^n (x^p - x_i^p)^2,\qquad \text{if }x_i \geq 0\,\forall\, i \in \{1,\dots,n\}</math> |- | Quasi-arithmetic mean|| <math> f^{-1}\left(\frac{1}{n} \sum_{k=1}^{n}f(x_k) \right)</math>|| <math>\underset{x \in \operatorname{dom}(f)}{\operatorname{argmin}}\, \sum_{i=1}^n (f(x) - f(x_i))^2,\qquad \text{if } f</math> is monotonic |- | Weighted mean || <math>\frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}</math> || <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n w_i(x - x_i)^2</math> |- | Truncated mean || The arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded |- | Interquartile mean || A special case of the truncated mean, using the interquartile range. A special case of the inter-quantile truncated mean, which operates on quantiles (often deciles or percentiles) that are equidistant but on opposite sides of the median. |- | Midrange || <math>\frac{1}{2}\left(\max x + \min x\right)</math> || <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \underset{i \in \{1,\dots,n\}}{\operatorname{max}}\, |x - x_i|</math> |- | Winsorized mean || Similar to the truncated mean, but, rather than deleting the extreme values, they are set equal to the largest and smallest values that remain |- | Medoid || A representative object of a set <math>\mathcal X</math> of objects with minimal sum of dissimilarities to all the objects in the set, according to some dissimilarity function <math>d</math>. || <math>\underset{y \in \mathcal X}{\operatorname{argmin}} \sum_{i=1}^n d(y, x_i)</math> |}

Even though perhaps not an average, the <math>\tau</math>th quantile (another summary statistic that generalizes the median) can similarly be expressed as a solution to the optimization problem

:<math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n \max\big((1-\tau)(x_i - x),\, \tau(x - x_i)\big) = \underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n \big(|x - x_i| + (1 - 2\tau)\,x\big)</math>,

which aims to minimize the total tilted absolute value loss (or ''quantile'' loss or ''pinball'' loss).

Other more sophisticated averages are: trimmore sophistiean, trimedian, and normalized mean, with their generalizations.<ref>{{cite journal |last1=Merigo |first1=Jose M. |last2=Cananovas |first2=Montserrat |title=The Generalized Hybrid Averaging Operator and its Application in Decision Making |date=2009 |journal=Journal of Quantitative Methods for Economics and Business Administration |volume=9 |pages=69–84 |issn=1886-516X }}</ref>

In a more general fashion, one can create their own average metric using the generalized ''f''-mean:

: <math>y = f^{-1}\left(\frac{1}{n}\left[f(x_1) + f(x_2) + \cdots + f(x_n)\right]\right)</math>

where ''f'' is any invertible function. The harmonic mean is an example of this using ''f''(''x'') = 1/''x'', and the geometric mean is another, using ''f''(''x'') = log ''x''.

However, this method for generating means is not general enough to capture all averages. A more general method<ref name="Bibby"/>{{failed verification|date=May 2022}} for defining an average takes any function ''g''(''x''1, ''x''2, ..., ''x''''n'') of a list of arguments that is continuous, strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average ''y'' is then the value that, when replacing each member of the list, results in the same function value: {{nowrap|1=''g''(''y'', ''y'', ..., ''y'') =}} {{nowrap|''g''(''x''1, ''x''2, ..., ''x''''n'')}}. This most general definition still captures the important property of all averages that the average of a list of identical elements is that element itself. The function {{nowrap|1=''g''(''x''1, ''x''2, ..., ''x''''n'') =}} {{nowrap|''x''1+''x''2+ ··· + ''x''''n''}} provides the arithmetic mean. The function {{nowrap|1 = ''g''(''x''1, ''x''2, ..., ''x''''n'') =}} {{nowrap|''x''1''x''2···''x''''n''}} (where the list elements are positive numbers) provides the geometric mean. The function {{nowrap|1 = ''g''(''x''1, ''x''2, ..., ''x''''n'') =}} {{nowrap|(''x''1−1+''x''2−1+ ··· + ''x''''n''−1)−1)}} (where the list elements are positive numbers) provides the harmonic mean.<ref name=Bibby>{{cite journal |last1=Bibby |first1=John |date= 1974 |title=Axiomatisations of the average and a further generalisation of monotonic sequences |journal=Glasgow Mathematical Journal |volume=15 |pages=63–65 |doi=10.1017/s0017089500002135 |doi-access=free}}</ref>

==Moving average== {{main|Moving average}}

Given a time series, such as daily stock market prices or yearly temperatures, people often want to create a smoother series.<ref>{{cite book | first1=George E.P. | last1= Box |first2=Gwilym M.| last2= Jenkins| title= Time Series Analysis: Forecasting and Control | edition= revised| publisher=Holden-Day |date=1976 | isbn=0816211043}}</ref> This helps to show underlying trends or perhaps periodic behavior. An easy way to do this is the ''moving average'': one chooses a number ''n'' and creates a new series by taking the arithmetic mean of the first ''n'' values, then moving forward one place by dropping the oldest value and introducing a new value at the other end of the list, and so on. This is the simplest form of moving average. More complicated forms involve using a weighted average. The weighting can be used to enhance or suppress various periodic behaviors and there is extensive analysis of what weightings to use in the literature on filtering. In digital signal processing the term "moving average" is used even when the sum of the weights is not 1.0 (so the output series is a scaled version of the averages).<ref>{{cite book | first=Simon | last= Haykin | title= Adaptive Filter Theory | publisher=Prentice-Hall |date=1986 | isbn=0130040525}}</ref> The reason for this is that the analyst is usually interested only in the trend or the periodic behavior.

==History==

===Origin=== The first recorded time that the arithmetic mean was extended from 2 to n cases for the use of estimation was in the sixteenth century. From the late sixteenth century onwards, it gradually became a common method to use for reducing errors of measurement in various areas.<ref name="ReferenceA">{{cite journal |doi=10.2307/2333051 | volume=45 | issue=1/2 | title=Studies in the History of Probability and Statistics: VII. The Principle of the Arithmetic Mean | journal=Biometrika | pages=130–135| jstor=2333051 |date=1958 | last1=Plackett | first1=R. L. }}</ref><ref name="york.ac.uk">[http://www.york.ac.uk/depts/maths/histstat/eisenhart.pdf Eisenhart, Churchill. "The development of the concept of the best mean of a set of measurements from antiquity to the present day." Unpublished presidential address, American Statistical Association, 131st Annual Meeting, Fort Collins, Colorado. 1971.]</ref> At the time, astronomers wanted to know a real value from noisy measurement, such as the position of a planet or the diameter of the moon. Using the mean of several measured values, scientists assumed that the errors add up to a relatively small number when compared to the total of all measured values. The method of taking the mean for reducing observation errors was mainly developed in astronomy.<ref name="ReferenceA"/><ref name="amstatbakker">{{Cite web |url=http://www.amstat.org/publications/jse/v11n1/bakker.html |title=Bakker, Arthur. "The early history of average values and implications for education." Journal of Statistics Education 11.1 (2003): 17-26. |access-date=2015-10-22 |archive-date=2015-12-04 |archive-url=https://web.archive.org/web/20151204181338/http://www.amstat.org/publications/jse/v11n1/bakker.html |url-status=live }}</ref> A possible precursor to the arithmetic mean is the mid-range (the mean of the two extreme values), used for example in Arabian astronomy of the ninth to eleventh centuries, but also in metallurgy and navigation.<ref name="york.ac.uk"/>

However, there are various older vague references to the use of the arithmetic mean (which are not as clear, but might reasonably have to do with our modern definition of the mean). In a text from the 4th century, it was written that (text in square brackets is a possible missing text that might clarify the meaning):<ref>{{Cite web |url=https://arcaneknowledgeofthedeep.files.wordpress.com/2014/02/theologyarithmetic.pdf |title=Waterfield, Robin. "The theology of arithmetic." On the Mystical, mathematical and Cosmological Symbolism of the First Ten Number (1988). page 70. |access-date=2018-11-27 |archive-url=https://web.archive.org/web/20160304212935/https://arcaneknowledgeofthedeep.files.wordpress.com/2014/02/theologyarithmetic.pdf |archive-date=2016-03-04 |url-status=dead }}</ref> : In the first place, we must set out in a row the sequence of numbers from the monad up to nine: 1, 2, 3, 4, 5, 6, 7, 8, 9. Then we must add up the amount of all of them together, and since the row contains nine terms, we must look for the ninth part of the total to see if it is already naturally present among the numbers in the row; and we will find that the property of being [one] ninth [of the sum] only belongs to the [arithmetic] mean itself...

Even older potential references exist. There are records that from about 700 BC, merchants and shippers agreed that damage to the cargo and ship (their "contribution" in case of damage by the sea) should be shared equally among themselves.<ref name="amstatbakker"/> This might have been calculated using the average, although there seem to be no direct record of the calculation.

===Etymology=== The root is found in Arabic as عوار ''ʿawār'', a defect, or anything defective or damaged, including partially spoiled merchandise; and عواري ''ʿawārī'' (also عوارة ''ʿawāra'') = "of or relating to ''ʿawār'', a state of partial damage".{{Efn|1=Medieval Arabic had عور ''ʿawr'' meaning "blind in one eye" and عوار ''ʿawār'' meant "any defect, or anything defective or damaged". Some medieval Arabic dictionaries are at [http://www.baheth.info/ Baheth.info] {{Webarchive |url=https://web.archive.org/web/20131029192325/http://www.baheth.info/ |date=2013-10-29 }}, and some translation to English of what's in the medieval Arabic dictionaries is in [http://dict.yulghun.com/lane/ Lane's ''Arabic-English Lexicon'', pages 2193 and 2195]. The medieval dictionaries do not list the word-form عوارية ''ʿawārīa''. ''ʿAwārīa'' can be naturally formed in Arabic grammar to refer to things that have ''ʿawār'', but in practice in medieval Arabic texts ''ʿawārīa'' is a rarity or non-existent, while the forms عواري ''ʿawārī'' and عوارة ''ʿawāra'' are frequently used when referring to things that have ''ʿawār'' or damage – this can be seen in the searchable collection of medieval texts at [http://www.alwaraq.net/Core/SearchServlet/searchall?book=-1&option=1&offset=1&searchtext=2LnZiNin2LHZig==&WordForm=1&RangeOp=-1 AlWaraq.net] (book links are clickable on righthand side).}} Within the Western languages the word's history begins in medieval sea-commerce on the Mediterranean. 12th and 13th century Genoa Latin ''avaria'' meant "damage, loss and non-normal expenses arising in connection with a merchant sea voyage"; and the same meaning for ''avaria'' is in Marseille in 1210, Barcelona in 1258 and Florence in the late 13th.{{Efn|name=Avaria|1=The Arabic origin of ''avaria'' was first reported by Reinhart Dozy in the 19th century. Dozy's original summary is in his 1869 book [https://archive.org/stream/glossairedesmot00englgoog#page/n235/mode/1up ''Glossaire'']. Summary information about the word's early records in Italian-Latin, Italian, Catalan, and French is at [http://www.cnrtl.fr/definition/avarie ''avarie'' @ CNRTL.fr] {{Webarchive |url=https://web.archive.org/web/20190106172304/http://www.cnrtl.fr/definition/avarie |date=2019-01-06 }}. The seaport of Genoa is the location of the earliest-known record in European languages, year 1157. A set of medieval Latin records of ''avaria'' at Genoa is in the downloadable lexicon [http://www.storiapatriasavona.it/vocabolario-ligure/ ''Vocabolario Ligure''], by Sergio Aprosio, year 2001, ''avaria'' in Volume 1 pages 115-116. Many more records in medieval Latin at Genoa are at [http://StoriaPatriaGenova.it StoriaPatriaGenova.it], usually in the plurals ''avariis'' and ''avarias''. At the port of Marseille in the 1st half of the 13th century notarized commercial contracts have dozens of instances of Latin ''avariis'' (ablative plural of ''avaria''), as published in [https://archive.org/details/documentsindit01blan Blancard year 1884]. Some information about the English word over the centuries is at [https://archive.org/stream/oed01arch#page/582/mode/1up NED (year 1888)]. See also the definition of English "average" in English dictionaries published in the early 18th century, i.e., in the time period just before the big transformation of the meaning: [https://books.google.com/books?id=PHBUAAAAYAAJ&pg=PT71&dq=dammage%20cargo%20goods Kersey-Phillips' dictionary (1706)], [https://archive.org/stream/glossographiaan00blougoog#page/n64/mode/1up Blount's dictionary (1707 edition)], [https://archive.org/stream/merchantsmagazi00hattgoog#page/n275/mode/1up Hatton's dictionary (1712)], [https://archive.org/stream/universaletymolo00bailuoft#page/n89/mode/1up Bailey's dictionary (1726)], [https://books.google.com/books?id=e-U_AQAAMAAJ&pg=PA120 Martin's dictionary (1749)]. Some complexities surrounding the English word's history are discussed in [https://archive.org/stream/contestedetymolo00wedgiala#page/10/mode/2up Hensleigh Wedgwood year 1882 page 11] and [https://archive.org/stream/etymologicaldict00skeauoft#page/781/mode/1up Walter Skeat year 1888 page 781]. Today there is consensus that: (#1) today's English "average" descends from medieval Italian ''avaria'', Catalan ''avaria'', and (#2) among the Latins the word ''avaria'' started in the 12th century and it started as a term of Mediterranean sea-commerce, and (#3) there is no root for ''avaria'' to be found in Latin, and (#4) a substantial number of Arabic words entered Italian, Catalan and Provençal in the 12th and 13th centuries starting as terms of Mediterranean sea-commerce, and (#5) the Arabic ''ʿawār {{!}} ʿawārī'' is phonetically a good match for ''avaria'', as conversion of w to v was regular in Latin and Italian, and ''-ia'' is a suffix in Italian, and the Western word's earliest records are in Italian-speaking locales (writing in Latin). And most commentators agree that (#6) the Arabic ''ʿawār {{!}} ʿawārī'' = "damage {{!}} relating to damage" is semantically a good match for ''avaria'' = "damage or damage expenses". A minority of commentators have been dubious about this on the grounds that the early records of Italian-Latin ''avaria'' have, in some cases, a meaning of "an expense" in a more general sense – [http://tlio.ovi.cnr.it/voci/005020.htm see TLIO (in Italian)]. The majority view is that the meaning of "an expense" was an expansion from "damage and damage expense", and the chronological order of the meanings in the records supports this view, and the broad meaning "an expense" was never the most commonly used meaning. On the basis of the above points, the inferential step is made that the Latinate word came or probably came from the Arabic word.}} 15th-century French ''avarie'' had the same meaning, and it begot English "averay" (1491) and English "average" (1502) with the same meaning. Today, Italian ''avaria'', Catalan ''avaria'' and French ''avarie'' still have the primary meaning of "damage". The transformation of the meaning in English began in later medieval and early modern Western merchant-marine law contracts under which if the ship met a bad storm and some of the goods had to be thrown overboard to make the ship lighter and safer, then all merchants whose goods were on the ship were to suffer proportionately (and not whoever's goods were thrown overboard); and more generally there was to be proportionate distribution of any ''avaria''{{Citation needed|date=December 2025}}. From there the word was adopted by British insurers, creditors, and merchants for talking about their losses as being spread across their whole portfolio of assets and having a mean proportion.{{Efn|name=Avaria}}<ref>{{Cite Dictionary.com |average |access-date=2023-05-25 }}</ref>

Marine damage is either ''particular average'', which is borne only by the owner of the damaged property, or general average, where the owner can claim a proportional contribution from all the parties to the marine venture. The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean".

A second English usage, documented as early as 1674 and sometimes spelled "averish", is as the residue and second growth of field crops, which were considered suited to consumption by draught animals ("avers").<ref>{{cite book |last=Ray |first=John |author-link=John Ray |title=A Collection of English Words not Generally Used |date=1674 |publisher=H. Bruges |location=London |url=http://babel.hathitrust.org/cgi/pt?id=njp.32101037601729;view=1up;seq=19 |access-date=18 May 2015}}</ref>

There is earlier (from at least the 11th century), unrelated use of the word. It appears to be an old legal term for a tenant's day labour obligation to a sheriff, probably anglicised from "avera" found in the English Domesday Book (1085).

The Oxford English Dictionary, however, says that derivations from German ''hafen'' haven, and Arabic ''ʿawâr'' loss, damage, have been "quite disposed of" and the word has a Romance origin.<ref>{{cite OED |term=average, n.2 |date=September 2019 |id=13681 |access-date=September 5, 2019 }}</ref>

== Averages as a rhetorical tool == Due to the aforementioned colloquial nature of the term "average", the term can be used to obfuscate the true meaning of data and suggest varying answers to questions based on the averaging method (most frequently arithmetic mean, median, or mode) used. In his article "Framed for Lying: Statistics as In/Artistic Proof", University of Pittsburgh faculty member Daniel Libertz comments that statistical information is frequently dismissed from rhetorical arguments for this reason.<ref name="Libertz 2018">{{Cite journal |last=Libertz |first=Daniel |date=2018-12-31 |title=Framed for Lying: Statistics as In/Artistic Proof |url=https://resrhetorica.com/index.php/RR/article/view/289 |journal=Res Rhetorica |volume=5 |issue=4 |doi=10.29107/rr2018.4.1 |issn=2392-3113 |doi-access=free }}</ref> However, due to their persuasive power, averages and other statistical values should not be discarded completely, but instead used and interpreted with caution. Libertz invites us to engage critically not only with statistical information such as averages, but also with the language used to describe the data and its uses, saying: "If statistics rely on interpretation, rhetors should invite their audience to interpret rather than insist on an interpretation."<ref name="Libertz 2018" /> In many cases, data and specific calculations are provided to help facilitate this audience-based interpretation.

==See also== {{Portal|Mathematics}} * Average absolute deviation * Central limit theorem * Expected value * Law of averages * Population mean * Sample mean

==Notes== {{Notelist}}

==References== {{Reflist}}

== Further reading ==

* {{Cite journal |last=Bakker |first=Arthur |author-link=Arthur Bakker |date=March 2003 |title=The Early History of Average Values and Implications for Education |url=https://jse.amstat.org/v11n1/bakker.html  |journal=Journal of Statistics Education |volume=11 |issue=1 |pages= |doi=10.1080/10691898.2003.11910694|doi-access=free }} * {{Cite journal |last=Edgeworth |first=F. Y. |author-link=Francis Ysidro Edgeworth |date=1889 |orig-date=Read May 25, 1885. |others= |title=VII. Observations and Statistics. An Essay on the Theory of Errors of Observation and the First Principles of Statistics. |journal=Transactions of the Cambridge Philosophical Society |volume=XIV |issue=II  |pages=138–169 |id=Google Books [https://www.google.com/books/edition/Proceedings/TdIsAAAAYAAJ?gbpv=1&pg=PA101 TdIsAAAAYAAJ]. HathiTrust record [https://catalog.hathitrust.org/Record/000526741 000526741], item [https://babel.hathitrust.org/cgi/pt?id=mdp.39015008919659&seq=158 mdp.39015008919659]; record [https://catalog.hathitrust.org/Record/100324784 100324784], item [https://babel.hathitrust.org/cgi/pt?id=hvd.32044092879402&seq=164 hvd.32044092879402].}} ** {{Cite journal |last=Edgeworth |first=F. Y. |date=May 25, 1885  |title=(4) Observations and statistics : Abstract. |journal=Proceedings of the Cambridge Philosophical Society |volume=V |issue=IV  |pages=310–312 |id=Google Books [https://www.google.com/books/edition/Proceedings/KdIsAAAAYAAJ?gbpv=1&pg=PA310 KdIsAAAAYAAJ], [https://www.google.com/books/edition/Proceedings_of_the_Cambridge_Philosophic/zpw1AAAAIAAJ?gbpv=1&pg=PA310 zpw1AAAAIAAJ]. HathiTrust record [https://catalog.hathitrust.org/Record/100528415 100528415], item [https://babel.hathitrust.org/cgi/pt?id=hvd.hwhqbe&seq=350 hvd.hwhqbe]. Internet Archive [https://archive.org/details/proceedingscambr05camb/page/310 proceedingscambr05camb], [https://archive.org/details/proceedingsofcam5188386camb/page/310 proceedingsofcam5188386camb], [https://archive.org/details/proceedingscamb09socigoog proceedingscamb09socigoog].}} ** {{Cite journal |date=May 30, 1887 |title=Corrigendum on Mr F. Y. Edgeworth's paper. |journal=Proceedings of the Cambridge Philosophical Society |volume=VI |issue=II  |pages=101–102 |id=Google Books [https://www.google.com/books/edition/Proceedings_of_the_Cambridge_Philosophic/E501AAAAIAAJ?gbpv=1&pg=PA101 E501AAAAIAAJ]. Internet Archive [https://archive.org/details/proceedingscamb12socigoog/page/n126 proceedingscamb12socigoog].}} * {{Cite journal |last=Kennedy |first=Christopher |last2=Stanley |first2=Jason |date=July 2009  |title=On 'Average' |url=https://academic.oup.com/mind/article/118/471/583/985199 |journal=Mind |language=en |volume=118 |issue=471 |pages=583–646 |doi=10.1093/mind/fzp094 |issn=0026-4423 |eissn=1460-2113|url-access=subscription }}

==External links== {{Wiktionary}} * [http://www.sengpielaudio.com/calculator-geommean.htm Calculations and comparison between arithmetic and geometric mean of two values]

Category:Arithmetic functions Category:Means Category:Summary statistics