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 a measure of central tendency, most often the median or sometimes the mean of the data set.
where
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Several measures of statistical dispersion are defined in terms of the absolute deviation.
The average absolute deviation, or simply average deviation of a data set is the average of the absolute deviations and is a summary statistic of statistical dispersion or variability. It is also called the mean absolute deviation, but this is easily confused with the median absolute deviation.
The average absolute deviation of a set {x_{1}, x_{2}, ..., x_{n}} is
The choice of measure of central tendency, m(X), 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 m(X)  Average absolute deviation 

Mean = 5  
Median = 3  
Mode = 2 
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 this relies on Jensen's inequality.
If x is a Gaussian random variable with a mean of 0, then, in expectation for large n, the ratio of standard deviation to mean absolute deviation should satisfy the following equality [1]
In other words, for a Gaussian, mean absolute deviation is about 0.8 times the standard deviation.
The mean absolute deviation (MAD) is the mean absolute deviation from the mean. A related quantity, the mean absolute error (MAE), is a common measure of forecast error in time series analysis, where this measures the average absolute deviation of observations from their forecasts.
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 median absolute deviation (also MAD) is the median absolute deviation from the median. It is a robust estimator of dispersion.
For the example {2, 2, 3, 4, 14}: 3 is the median, so the absolute deviations from the median are {1, 1, 0, 1, 11} (or reordered as {0, 1, 1, 1, 11}) with a median absolute deviation of 1, in this case unaffected by the value of the outlier 14.
The maximum absolute deviation about a point is the maximum of the absolute deviations of a sample from that point. It is realized by the sample maximum or sample minimum and cannot be less than half the range.
The measures of statistical dispersion derived from absolute deviation characterize various measures of central tendency as minimizing dispersion: The median is the measure of central tendency most associated with the absolute deviation, in that
The mean absolute deviation of a sample is a biased estimator of the mean absolute deviation of the population.
