Absolute deviation

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.

The average absolute deviation of a data set is the average of the absolute deviations and is a summary statistic of statistical dispersion or variability.

The average absolute deviation of a set {x₀, x₁, ..., x_n−1} is:

$\frac{\sum_{i=0}^{n-1} |x_i-\hat{x}|}{n}$

where $\hat{x}$ is the selected value of central tendency of the set about which the average absolute deviation is being measured.

The median is the point which minimises the average absolute deviation of a data set. For example, for the set {1,2,2,4,6}, the median is 2 while the mean is 3. The average absolute deviation from the median is (1+0+0+2+4)/5=1.4 while the average absolute deviation from the mean (sometimes called the mean deviation) is (2+1+1+1+3)/5=1.6.

In general, the average absolute deviation from the mean is between one and two times the average absolute deviation from the median; it is also less than or equal to the standard deviation.

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