Absolute deviation

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In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the point from which the deviation is measured is the value of either the median or the mean of the data set.

$|D| = |x_i-\overline{x}|$

where

|D| is the absolute deviation,

x_i is the data element

and $\overline{x}$ is the chosen measure of central tendency of the data set.

The average absolute deviation (or simply average deviation) of a data set is the average (or expected value) of the absolute deviations and is a summary statistic of statistical dispersion or variability.

The average absolute deviation of a set {x₁, x₂, ..., x_n} is

$\frac{1}{n}\sum_{i=1}^n |x_i-\overline{x}|$

The choice of measure of central tendency has a marked effect on the value of the average deviation. For example, for the data set {2, 2, 3, 4, 14}:

Measure of central tendency	Absolute deviation
Mean = 5	$\frac{\|2 - 5\| + \|2 - 5\| + \|3 - 5\| + \|4 - 5\| + \|14 - 5\|}{5} = 3.6$
Median = 3	$\frac{\|2 - 3\| + \|2 - 3\| + \|3 - 3\| + \|4 - 3\| + \|14 - 3\|}{5} = 2.8$
Mode = 2	$\frac{\|2 - 2\| + \|2 - 2\| + \|3 - 2\| + \|4 - 2\| + \|14 - 2\|}{5} = 3.0$

The average absolute deviation from the median is less than or equal to the average absolute deviation from the mean. In fact, the average absolute deviation from the median is always less than or equal to the average absolute deviation from any other fixed number.

The average absolute deviation from the mean is less than or equal to the standard deviation. One way of proving that relies on Jensen's inequality.

The mean absolute deviation is the average absolute deviation from the mean and is a common measure of forecast error in time series analysis. It should be noted that although the term mean deviation is used as a synonym for mean absolute deviation, to be precise it is not the same; in its strict interpretation (namely, omitting the absolute value operation), the mean deviation of any data set from its mean is always zero.

The mean absolute deviation of a sample is a biased estimator of the mean absolute deviation of the population.