Mean Absolute Deviation Measure

Context:
- It can be defined as: [math]\displaystyle{ \Delta X =\langle |x_i-\langle x\rangle | \rangle = \frac{1}{N}\sum{|x_i-\langle x \rangle|} }[/math]
  where X is dataset or random variable with N elements, i.e. [math]\displaystyle{ X=\{x_0,x_1,...,x_N\} }[/math], [math]\displaystyle{ |\quad| }[/math] denotes the absolute value and [math]\displaystyle{ \langle \quad \rangle }[/math] the average value.
Example(s):
- for the data set {2, 2, 3, 4, 14}: [math]\displaystyle{ \frac{|2 - 5| + |2 - 5| + |3 - 5| + |4 - 5| + |14 - 5|}{5} = 3.6 }[/math].
- …
Counter-Example(s):
- Mean Absolute Error.
- Mean Squared Error.
- Median Absolute Deviation.
- Average Absolute Deviation w.r.t. Mode = [math]\displaystyle{ \frac{|2 - 2| + |2 - 2| + |3 - 2| + |4 - 2| + |14 - 2|}{5} = 3.0 }[/math].
See: Absolute Deviation; Mean Statistic; Population Statistic; Summary Statistics, Statistical Dispersion.

References

(Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/average_absolute_deviation Retrieved:2020-7-24.
- The average absolute deviation, or mean absolute deviation (MAD), of a data set is the average of the absolute deviations from a central point. It is a summary statistic of statistical dispersion or variability. In the general form, the central point can be a mean, median, mode, or the result of any other measure of central tendency or any random data point related to the given data set. The absolute values of the differences between the data points and their central tendency are totaled and divided by the number of data points.

(Eric W. Weisstein, 2016) ⇒ Weisstein, Eric W. (1999-2016) "Average Absolute Deviation." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/AverageAbsoluteDeviation.html Retrieved 2016-07-10
- [math]\displaystyle{ \alpha=\frac{1}{N}\sum_{i=1}^N|x_i-\mu|=\langle |x_i-\mu|\rangle }[/math].

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/average_absolute_deviation#Measures_of_dispersion Retrieved:2015-6-9.
- Several measures of statistical dispersion are defined in terms of the absolute deviation.
  The term "average absolute deviation" does not uniquely identify a measure of statistical dispersion, as there are several measures that can be used to measure absolute deviations, and there are several measures of central tendency that can be used as well. Thus, to uniquely identify the absolute deviation it is necessary to specify both the measure of deviation and the measure of central tendency. Unfortunately, the statistical literature has not yet adopted a standard notation, as both the #Mean absolute deviation around the mean and the #Median absolute deviation around the median have been denoted by their initials "MAD" in the literature, which may lead to confusion, since in general, they may have values considerably different from each other.

(Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
- QUOTE: A measure of spread. For observations [math]\displaystyle{ x_l, x_g, , x }[/math] with mean [math]\displaystyle{ i_t }[/math] and median [math]\displaystyle{ m }[/math], the mean absolute deviation about the mean [math]\displaystyle{ i_s }[/math] and the mean absolute deviation about the median [math]\displaystyle{ i_s }[/math]

(Torgo, 1999) ⇒ Luis Torgo. (1999). “Inductive Learning of Tree-based Regression Models." Ph.D. Thesis, Thesis, Faculty of Sciences, University of Porto
- QUOTE: … we describe in detail two different methods of growing a regression tree: minimising the mean squared error and minimising the mean absolute deviation. Our study is particularly focussed on the computational efficiency of these tasks.