Average absolute deviation

Mean absolute deviation around a central point

The mean absolute deviation of a set X = {x₁, x₂, …, x_n} is:$${\displaystyle \frac{1}{n}\sum _{i=1}^n|x_i-m(X)|.}$$

The choice of measure of central tendency, ${\displaystyle m(X)}$, has a marked effect on the value of the mean deviation. For example, for the data set {2, 2, 3, 4, 14}:

Mean absolute deviation around the mean

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 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 of standard deviation since it corresponds better to real life. Because the MAD is a simpler measure of variability than the standard deviation, it can be useful in school teaching.

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 because it is both easier to compute (avoiding the need for squaring) and easier to understand.

Relation to standard deviation

For the normal distribution, the ratio of mean absolute deviation from the mean to standard deviation is $\sqrt{2/\pi }=0.79788456\dots$. Thus if X is a normally distributed random variable with expected value 0 then, see Geary (1935): $${\displaystyle w=\frac{\mathrm{E}\left[|X|\right]}{\sqrt{\mathrm{E}\left[X^2\right]}}=\sqrt{\frac{2}{\pi }}.}$$ 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 sample n with the following bounds: ${\displaystyle w_n\in [0,1]}$, with a bias for small n.

The mean 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 ${\displaystyle \phi \left(\mathrm{E}[Y]\right)\le \mathrm{E}\left[\phi (Y)\right]}$, where ${\displaystyle \phi }$ is a convex function, this implies for ${\displaystyle Y=|X-\mu |}$ that: $${\displaystyle {\left(\mathrm{E}\left[|X-\mu |\right]\right)}^2\le \mathrm{E}\left[|X-\mu {|}^2\right]=\mathrm{Var}(X)}$$

Since both sides are positive, and the square root is a monotonically increasing function in the positive domain: $${\displaystyle \mathrm{E}\left[|X-\mu |\right]\le \sqrt{\mathrm{Var}(X)}}$$

For a general case of this statement, see Hölder's inequality.

Mean absolute deviation around the median

The median 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 $${\displaystyle D_{\text{med}}=\mathrm{E}\left[|X-\text{median}|\right]}$$

This is the maximum likelihood estimator of the scale parameter ${\displaystyle b}$ of the Laplace distribution.

Since the median minimizes the average absolute distance, we have ${\displaystyle D_{\text{med}}\le D_{\text{mean}}}$. 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 as $${\displaystyle D_{\text{med}}=\mathrm{E}\left[|X-\text{median}|\right]=2\mathrm{Cov}(X,I_O)}$$ where the indicator function is $${\displaystyle {\mathbf{I}}_O:=\{\begin{array}{ll}1& \text{if }x>\text{median},\\ 0& \text{otherwise}.\end{array}}$$

This representation allows for obtaining MAD median correlation coefficients.

Median absolute deviation around a central point

While in principle the mean or any other central point could be taken as the central point for the median absolute deviation, most often the median value is taken instead.

Median absolute deviation around the median

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 is 1.

For a symmetric distribution, the median absolute deviation is equal to half the interquartile range.

Maximum absolute deviation

The maximum 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 ${\displaystyle m(X)=max(X)}$, where ${\displaystyle max(X)}$ is the sample maximum.

Minimization

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:

• L² norm statistics: the mean minimizes the mean squared error
• L¹ norm statistics: the median minimizes average absolute deviation,
• L^∞ norm statistics: the mid-range minimizes the maximum absolute deviation
• trimmed L^∞ norm statistics: for example, the midhinge (average of first and third quartiles) which minimizes the median absolute deviation of the whole distribution, also minimizes the maximum absolute deviation of the distribution after the top and bottom 25% have been trimmed off.

Estimation

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 mean-unbiasedness. Each measure of location has its own form of unbiasedness (see entry on biased estimator). The relevant form of unbiasedness here is median unbiasedness.