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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 deviation is reckoned from the central value, being construed as some type of average, most often the median or sometimes the mean of the data set.
where
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 from a central point and 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. See below for distinctions where average is synonymous with "mean" and the central point is also the "mean".
The average absolute deviation of a set {x_{1}, x_{2}, ..., x_{n}} is
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  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.
Proof 

Jensen's inequality is ,where φ() is a convex function, this implies that: Since both sides are positive, and the square root is a monotonically increasing function in the positive domain: For a general case of this statement, see Hölder's inequality. 
For the normal distribution, the ratio of mean absolute deviation to standard deviation is . Thus if X is a normally distributed random variable with expected value 0 then, see Geary (1935):^{[1]}
In other words, for a normal distribution, mean absolute deviation is about 0.8 times the standard deviation. However insample measurements deliver values of the ratio of mean average deviation / standard deviation for a given Gaussian sample n with the following bounds: , with a bias for small n.^{[2]}
The mean absolute deviation (MAD), also referred to as the "mean deviation" or sometimes "average absolute deviation" is the mean of the data's absolute deviations about the data's mean: the average (absolute) distance from the mean. "Average absolute deviation" can refer to either this usage, or to the general form with respect to a specified central point (see above).
MAD has been proposed to be used in place of standard deviation since it corresponds better to real life.^{[3]} Because the MAD is a simpler measure of variability than the standard deviation, it can be used as pedagogical tool to help motivate the standard deviation.^{[4]}^{[5]}
This method's forecast accuracy is very closely related to the mean squared error (MSE) method which is just the average squared error of the forecasts. Although these methods are very closely related, MAD is more commonly used^{[citation needed]} because it does not require squaring.
More recently, the mean absolute deviation about mean is expressed as a covariance between a random variable and its under/over indicator functions;^{[6]}
where
and the over indicator function is defined as
Based on this representation new correlation coefficients are derived. These correlation coefficients ensure high stability of statistical inference when we deal with distributions that are not symmetric and for which the normal distribution is not an appropriate approximation. Moreover an easy and simple way for a semi decomposition of Pietra’s index of inequality is obtained.
Mean absolute deviation about median (MAD median) offers a direct measure of the scale of a random variable about its median
For the normal distribution we have . Since the median minimizes the average absolute distance, we have . By using the general dispersion function Habib (2011) defined MAD about median as
where the indicator function is
This representation allows for obtaining MAD median correlation coefficients;^{[7]}
The median absolute deviation (also MAD) is the median of the 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} (reordered as {0, 1, 1, 1, 11}) with a median of 1, in this case unaffected by the value of the outlier 14, so the median absolute deviation (also called MAD) is 1.
The maximum absolute deviation about a point is the maximum of the absolute deviations of a sample from that point. While not strictly a measure of central tendency, the maximum absolute deviation can be found using the formula for the average absolute deviation as above with , where is the sample maximum. The maximum absolute deviation 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. Some location parameters can be compared as follows:
This section requires expansion. (March 2009) 
The mean absolute deviation of a sample is a biased estimator of the mean absolute deviation of the population. In order for the absolute deviation to be an unbiased estimator, the expected value (average) of all the sample absolute deviations must equal the population absolute deviation. However, it does not. For the population 1,2,3 both the population absolute deviation about the median and the population absolute deviation about the mean are 2/3. The average of all the sample absolute deviations about the mean of size 3 that can be drawn from the population is 44/81, while the average of all the sample absolute deviations about the median is 4/9. Therefore the absolute deviation is a biased estimator.
However, this argument is based on the notion of meanunbiasedness. Each measure of location has its own form of unbiasedness (see entry on biased estimator. The relevant form of unbiasedness here is median unbiasedness.
