# Average Absolute Deviation

An Average Absolute Deviation is an average value of all absolute deviations

**Context:**- It can be defined as: [math]\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]X=\{x_0,x_1,...,x_N\}[/math], [math]|\quad|[/math] denotes the absolute value and [math]\langle \quad \rangle [/math] the average value

- It can be defined as: [math]\Delta X =\langle |x_i-\langle x\rangle | \rangle = \frac{1}{N}\sum{|x_i-\langle x \rangle|}[/math]
**Example(s):**- for the data set {2, 2, 3, 4, 14}:
- Average Absolute Deviation w.r.t. Mean = [math]\frac{|2 - 5| + |2 - 5| + |3 - 5| + |4 - 5| + |14 - 5|}{5} = 3.6[/math].
- Average Absolute Deviation w.r.t. Median = [math]\frac{|2 - 3| + |2 - 3| + |3 - 3| + |4 - 3| + |14 - 3|}{5} = 2.8[/math].
- Average Absolute Deviation w.r.t. Mode = [math]\frac{|2 - 2| + |2 - 2| + |3 - 2| + |4 - 2| + |14 - 2|}{5} = 3.0[/math].

- for the data set {2, 2, 3, 4, 14}:
**See:**Summary Statistics, Statistical Dispersion, Arithmetic Mean, Median, Mode (Statistics), Central Tendency.

## References

### 2016

- (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]\alpha=\frac{1}{N}\sum_{i=1}^N|x_i-\mu|=\langle |x_i-\mu|\rangle[/math].

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/average_absolute_deviation Retrieved:2015-6-9.
- The
**average absolute deviation**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 this general form, the central point can be the mean, median, mode, or the result of another measure of central tendency. Furthermore, as described in article about average, the deviation averaging operation may refer to the mean or the median. Thus the total number of combinations amounts to at least four types of average absolute deviation.

- The

### 2015

- (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.

- Several measures of statistical dispersion are defined in terms of the absolute deviation.