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Average

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From Wikipedia, the free encyclopedia

This article is about the concept by this name. For the Canadian artist with this surname, seeJoe Average.

For the equivalent concept in geometry, seecentre (geometry).

Number taken as representative of a list of numbers

Fourmeans of two numbers, a and b, constructed as chords on a semicircle

Anaverage of a collection or group is a single actual or hypothetical member of it that is mostcentral or most common in some sense, and represents its overallposition.

Inmathematics, especially incolloquial usage, it most commonly refers to thearithmetic mean, 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. In situations where the data isskewed or hasoutliers, and it is desired to focus on the main part of the group rather than thelong tail, "average" often instead refers to themedian; for example, the averagepersonal income is usually given as themedian income, so that it represents themajority of thepopulation rather than being overly influenced by the much higher incomes of the few rich people. In certainreal-world scenarios, such computing theaverage speed from multiple measurements takenover the same distance,the average used is theharmonic mean. In situations where ahistogram orprobability density function is being referenced, the "average" could instead refer to themode. Otherstatistics that can be used as an average include themid-range andgeometric mean, but they would rarely, if ever, be colloquially referred to as "the average".

Type	Description	Example	Result
Arithmetic mean	Sum of values of a data set divided by number of values: $\scriptstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}$	(1+2+2+3+4+7+9) / 7	4
Median	Middle value separating the greater and lesser halves of a data set	1, 2, 2,3, 4, 7, 9	3
Mode	Most frequent value in a data set	1,2,2, 3, 4, 7, 9	2
Mid-range	The arithmetic mean of the highest and lowest values of a set	(1+9) / 2	5

Name	Equation or description	As solution to optimization problem
Arithmetic mean	${\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {1}{n}}(x_{1}+\cdots +x_{n})$	${\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x-x_{i})^{2}$
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	${\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}\|x-x_{i}\|$
Geometric median	Arotation invariant extension of themedian for points in $\mathbb {R} ^{d}$	${\underset {{\vec {x}}\in \mathbb {R} ^{d}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}\|\|{\vec {x}}-{\vec {x}}_{i}\|\|_{2}$
Tukey median	Another rotation invariant extension of the median for points in $\mathbb {R} ^{d}$ —a point that maximizes theTukey depth	${\underset {{\vec {x}}\in \mathbb {R} ^{d}}{\operatorname {argmax} }}\,{\underset {{\vec {u}}\in \mathbb {R} ^{d}}{\operatorname {min} }}\,\sum _{i=1}^{n}\left({\begin{cases}1,{\text{ if }}({\vec {x}}_{i}-{\vec {x}})\cdot {\vec {u}}\geq 0\\0,{\text{ otherwise}}\end{cases}}\right)$
Mode	The most frequent value in the data set	${\underset {x\in \mathbb {R} }{\operatorname {argmax} }}\,\sum _{i=1}^{n}\left({\begin{cases}1,{\text{ if }}x=x_{i}\\0,{\text{ if }}x\neq x_{i}\end{cases}}\right)$
Geometric mean	${\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}={\sqrt[{n}]{x_{1}\cdot x_{2}\dotsb x_{n}}}$	${\underset {x\in \mathbb {R} _{>0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(\ln(x)-\ln(x_{i}))^{2},\qquad {\text{if }}x_{i}>0\,\forall \,i\in \{1,\dots ,n\}$
Harmonic mean	${\frac {n}{{\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}}}$	${\underset {x\in \mathbb {R} _{\neq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}\left({\frac {1}{x}}-{\frac {1}{x_{i}}}\right)^{2}$
Contraharmonic mean	${\frac {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{{x_{1}}+{x_{2}}+\cdots +{x_{n}}}}$	${\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}x_{i}(x-x_{i})^{2}$
Lehmer mean	${\frac {\sum _{i=1}^{n}x_{i}^{p}}{\sum _{i=1}^{n}x_{i}^{p-1}}}$	${\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}x_{i}^{p-1}(x-x_{i})^{2}$
Quadratic mean (or RMS)	${\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}$	${\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x^{2}-x_{i}^{2})^{2}$
Cubic mean	${\sqrt[{3}]{{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{3}}}={\sqrt[{3}]{{\frac {1}{n}}\left(x_{1}^{3}+x_{2}^{3}+\cdots +x_{n}^{3}\right)}}$	${\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x^{3}-x_{i}^{3})^{2},\qquad {\text{if }}x_{i}\geq 0\,\forall \,i\in \{1,\dots ,n\}$
Generalized mean	${\sqrt[{p}]{{\frac {1}{n}}\cdot \sum _{i=1}^{n}x_{i}^{p}}}$	${\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x^{p}-x_{i}^{p})^{2},\qquad {\text{if }}x_{i}\geq 0\,\forall \,i\in \{1,\dots ,n\}$
Quasi-arithmetic mean	$f^{-1}\left({\frac {1}{n}}\sum _{k=1}^{n}f(x_{k})\right)$	${\underset {x\in \operatorname {dom} (f)}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(f(x)-f(x_{i}))^{2},\qquad {\text{if }}f$ ismonotonic
Weighted mean	${\frac {\sum _{i=1}^{n}w_{i}x_{i}}{\sum _{i=1}^{n}w_{i}}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}$	${\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}w_{i}(x-x_{i})^{2}$
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 theinterquartile range. A special case of the inter-quantile truncated mean, which operates on quantiles (oftendeciles orpercentiles) that are equidistant but on opposite sides of the median.
Midrange	${\frac {1}{2}}\left(\max x+\min x\right)$	${\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,{\underset {i\in \{1,\dots ,n\}}{\operatorname {max} }}\,\|x-x_{i}\|$
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 ${\mathcal {X}}$ of objects with minimal sum of dissimilarities to all the objects in the set, according to some dissimilarity function $d {\displaystyle d}$ .	${\underset {y\in {\mathcal {X}}}{\operatorname {argmin} }}\sum _{i=1}^{n}d(y,x_{i})$

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General properties

Pythagorean means

Statistical location

Mode

Median

Mid-range

Summary of types

Miscellaneous types

Average percentage return and CAGR

Moving average

History

Origin

Etymology

Averages as a rhetorical tool

See also

Notes

References

External links