Editing Average absolute deviation (section)

== Mean absolute deviation around a central point ==
{{for|arbitrary differences (not around a central point)|Mean absolute difference}}
{{for|paired differences (also known as mean absolute deviation)|Mean absolute error}}

The mean absolute deviation of a set ''X''&thinsp;=&thinsp;{''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''n''</sub>} is
<math display="block">\frac{1}{n} \sum_{i=1}^n |x_i-m(X)|.</math>

The choice of measure of central tendency, <math>m(X)</math>, has a marked effect on the value of the mean deviation. For example, for the data set {2, 2, 3, 4, 14}:

{| class="wikitable" style="margin:auto;width:100%;"
|-
!Measure of central tendency <math>m(X)</math>
!Mean absolute deviation
|-
| [[Arithmetic mean|Arithmetic Mean]] = 5
| <MATH>\frac{|2 - 5| + |2 - 5| + |3 - 5| + |4 - 5| + |14 - 5|}{5} = 3.6</MATH>
|-
| Median = 3
| <MATH>\frac{|2 - 3| + |2 - 3| + |3 - 3| + |4 - 3| + |14 - 3|}{5} = 2.8</MATH>
|-
| Mode = 2
| <MATH>\frac{|2 - 2| + |2 - 2| + |3 - 2| + |4 - 2| + |14 - 2|}{5} = 3.0</MATH>
|}

=== Mean absolute deviation around the mean ===

The ''mean absolute deviation'' (MAD), also referred to as the "mean deviation" or sometimes "average absolute deviation", is the mean of the data's absolute deviations around the data's mean: the average (absolute) distance from the mean. "Average absolute deviation" can refer to either this usage, or to the general form with respect to a specified central point (see above).

MAD has been proposed to be used in place of [[standard deviation]] since it corresponds better to real life.<ref>{{Cite web|last=Taleb|first=Nassim Nicholas |date=2014 | title=What scientific idea is ready for retirement? | url=http://www.edge.org/response-detail/25401 | url-status=bot: unknown | archive-url=https://web.archive.org/web/20140116031136/http://www.edge.org/response-detail/25401 | archive-date=2014-01-16 |access-date=2014-01-16 |website=Edge}}</ref> Because the MAD is a simpler measure of variability than the [[standard deviation]], it can be useful in school teaching.<ref name=Kader1999>{{cite journal |last=Kader|first=Gary|title=Means and MADS |journal=Mathematics Teaching in the Middle School |date=March 1999|volume=4| issue=6 | pages=398–403|doi=10.5951/MTMS.4.6.0398 | url=http://www.learner.org/courses/learningmath/data/overview/readinglist.html| access-date=20 February 2013 |archive-url=https://web.archive.org/web/20130518092027/http://learner.org/courses/learningmath/data/overview/readinglist.html|archive-date=2013-05-18| url-status=live}}</ref><ref name=GAISE>{{cite book |last=Franklin |first=Christine, Gary Kader, Denise Mewborn, Jerry Moreno, [[Roxy Peck]], Mike Perry, and Richard Scheaffer |title=Guidelines for Assessment and Instruction in Statistics Education | year=2007 | publisher=American Statistical Association | isbn=978-0-9791747-1-1| url=http://www.amstat.org/education/gaise/GAISEPreK-12_Full.pdf| access-date=2013-02-20 | archive-url=https://web.archive.org/web/20130307004604/http://www.amstat.org/education/gaise/GAISEPreK-12_Full.pdf| archive-date=2013-03-07| url-status=live}}</ref>

This method's forecast accuracy is very closely related to the [[mean squared error]] (MSE) method which is just the average squared error of the forecasts. Although these methods are very closely related, MAD is more commonly used because it is both easier to compute (avoiding the need for squaring)<ref>{{citation | title=Production and Operations Analysis| edition=7th |first1=Steven | last1=Nahmias |first2=Tava Lennon |last2=Olsen |author2-link=Tava Olsen |publisher=Waveland Press |year=2015 |isbn=9781478628248 |page=62 |url=https://books.google.com/books?id=SIsoBgAAQBAJ&pg=PA62 | quote=MAD is often the preferred method of measuring the forecast error because it does not require squaring.}}</ref> and easier to understand.<ref>{{citation|title=Supply Chain Management and Advanced Planning: Concepts, Models, Software, and Case Studies | series=Springer Texts in Business and Economics| editor1-first=Hartmut |editor1-last=Stadtler| editor2-first=Christoph |editor2-last=Kilger |editor3-first=Herbert | editor3-last=Meyr | edition=5th | publisher=Springer |year=2014 | isbn=9783642553097 | page=143 |url=https://books.google.com/books?id=iDhpBQAAQBAJ&pg=PA143 | quote=the meaning of the MAD is easier to interpret}}.</ref>

For the [[normal distribution]], the ratio of mean absolute deviation from the mean to standard deviation is <math display="inline"> \sqrt{2/\pi} = 0.79788456\ldots</math>. Thus if ''X'' is a normally distributed random variable with expected value 0 then, see Geary (1935):<ref>Geary, R. C. (1935). The ratio of the mean deviation to the standard deviation as a test of normality. Biometrika, 27(3/4), 310–332.</ref>
<math display="block"> w=\frac{ E|X| }{ \sqrt{E(X^2)} } = \sqrt{\frac{2}{\pi}}. </math>
In other words, for a normal distribution, mean absolute deviation is about 0.8 times the standard deviation.
However, in-sample measurements deliver values of the ratio of mean average deviation / standard deviation for a given Gaussian sample ''n'' with the following bounds: <math> w_n \in [0,1] </math>, with a bias for small ''n''.<ref>See also Geary's 1936 and 1946 papers: Geary, R. C. (1936). Moments of the ratio of the mean deviation to the standard deviation for normal samples. Biometrika, 28(3/4), 295–307 and Geary, R. C. (1947). Testing for normality. Biometrika, 34(3/4), 209–242.</ref>

The mean absolute deviation from the mean is less than or equal to the [[standard deviation]]; one way of proving this relies on [[Jensen's inequality]].

{{math proof | proof = Jensen's inequality is <math>\varphi\left(\mathbb{E}[Y]\right) \leq \mathbb{E}\left[\varphi(Y)\right]</math>, where ''φ'' is a convex function, this implies for <math>Y = \vert X-\mu\vert </math> that:
<math display="block">\left(\mathbb{E} |X -\mu \right|)^{2}\leq\mathbb{E}\left(|X-\mu|^2 \right)</math>
<math display="block">\left(\mathbb{E} |X -\mu \right|)^{2}\leq \operatorname{Var}(X)</math>

Since both sides are positive, and the [[square root]] is a [[Inequality (mathematics)#Applying a function to both sides|monotonically increasing function]] in the positive domain:
<math display="block">\mathbb{E} \left(|X -\mu \right|) \leq \sqrt{\operatorname{Var}(X)}</math>

For a general case of this statement, see [[Hölder's inequality#Probability measure|Hölder's inequality]].
}}

=== Mean absolute deviation around the median ===

The [[median]] is the point about which the mean deviation is minimized. The MAD median offers a direct measure of the scale of a random variable around its median
<math display="block">D_\text{med} = E |X-\text{median}| </math>

This is the [[maximum likelihood]] estimator of the scale parameter <math>b</math> of the [[Laplace distribution]]. 

Since the median minimizes the average absolute distance, we have <math>D_\text{med} \le D_\text{mean}</math>.
The mean absolute deviation from the median is less than or equal to the mean absolute deviation from the mean. In fact, the mean absolute deviation from the median is always less than or equal to the mean absolute deviation from any other fixed number.

By using the general dispersion function, Habib (2011) defined MAD about median as
<math display="block">D_\text{med} = E |X-\text{median}| = 2\operatorname{Cov}(X,I_O) </math>
where the indicator function is
<math display="block">\mathbf{I}_O := \begin{cases}
1 &\text{if } x > \text{median}, \\
0 &\text{otherwise}.
\end{cases}
</math>

This representation allows for obtaining MAD median correlation coefficients.{{citation needed|date=November 2019}}