Theaverage absolute deviation (AAD) of a data set is theaverage of theabsolutedeviations from acentral point. It is asummary statistic ofstatistical dispersion or variability. In the general form, the central point can be amean,median,mode, or the result of any other measure of central tendency or any reference value related to the given data set. AAD includes themean absolute deviation and themedian absolute deviation (both abbreviated asMAD).
Several measures ofstatistical dispersion are defined in terms of the absolute deviation.The term "average absolute deviation" does not uniquely identify a measure ofstatistical dispersion, as there are several measures that can be used to measure absolute deviations, and there are several measures ofcentral 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. 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 they generally have values considerably different from each other.
The mean absolute deviation of a setX = {x1,x2, …,xn} is:
The choice of measure of central tendency,, has a marked effect on the value of the mean deviation. For example, for the data set {2, 2, 3, 4, 14}:
| Measure of central tendency | Mean absolute deviation |
|---|---|
| Arithmetic Mean = 5 | |
| Median = 3 | |
| Mode = 2 |
Themean absolute deviation (MAD), also referred to as the "mean deviation" or sometimes "average absolute deviation", is the mean of the data's absolute deviations around 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 ofstandard deviation since it corresponds better to real life.[1] Because the MAD is a simpler measure of variability than thestandard deviation, it can be useful in school teaching.[2][3]
This method's forecast accuracy is very closely related to themean 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 because it is both easier to compute (avoiding the need for squaring)[4] and easier to understand.[5]
For thenormal distribution, the ratio of mean absolute deviation from the mean to standard deviation is. Thus ifX is a normally distributed random variable with expected value 0 then, see Geary (1935):[6]In other words, for a normal distribution, mean absolute deviation is about 0.8 times the standard deviation.However, in-sample measurements deliver values of the ratio of mean average deviation / standard deviation for a given Gaussian samplen with the following bounds:, with a bias for smalln.[7]
The mean absolute deviation from the mean is less than or equal to thestandard deviation; one way of proving this relies onJensen's inequality.
Jensen's inequality is, where is a convex function, this implies for that:
Since both sides are positive, and thesquare root is amonotonically increasing function in the positive domain:
For a general case of this statement, seeHölder's inequality.
Themedian is the point about which the mean deviation is minimized. The MAD median offers a direct measure of the scale of a random variable around its median
This is themaximum likelihood estimator of the scale parameter of theLaplace distribution.
Since the median minimizes the average absolute distance, we have.The mean absolute deviation from the median is less than or equal to the mean absolute deviation from the mean. In fact, the mean absolute deviation from the median is always less than or equal to the mean absolute deviation from any other fixed number.
By using the general dispersion function, Habib (2011) defined MAD about median aswhere the indicator function is
This representation allows for obtaining MAD median correlation coefficients.[citation needed]
While in principle the mean or any other central point could be taken as the central point for the median absolute deviation, most often themedian value is taken instead.
Themedian absolute deviation (also MAD) is themedian of the absolute deviation from themedian. It is arobust 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 is 1.
For a symmetric distribution, the median absolute deviation is equal to half theinterquartile range.
Themaximum absolute deviation around an arbitrary 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 thesample maximum.
The measures of statistical dispersion derived from absolute deviation characterize various measures of central tendency asminimizing dispersion:The median is the measure of central tendency most associated with the absolute deviation. Some location parameters can be compared as follows:
The mean absolute deviation of a sample is abiased 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 mean-unbiasedness. Each measure of location has its own form of unbiasedness (see entry onbiased estimator). The relevant form of unbiasedness here is median unbiasedness.
{{cite web}}: CS1 maint: bot: original URL status unknown (link){{cite book}}: CS1 maint: multiple names: authors list (link)MAD is often the preferred method of measuring the forecast error because it does not require squaring.
the meaning of the MAD is easier to interpret.
