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Absolute deviation

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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 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.

D_i = | x_i − m(X) |

where

D_i is the absolute deviation,

x_i is the data element

and m(X) is the chosen measure of central tendency of the data set—sometimes the mean ( $\overline{x}$ ), but most often the median.

Measures of dispersion

Several measures of statistical dispersion are defined in terms of the absolute deviation.

Average 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 {x₁, x₂, ..., x_n} is

$\frac{1}{n}\sum_{i=1}^n |x_i-m(X)|.$

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	$\frac{\|2 - 5\| + \|2 - 5\| + \|3 - 5\| + \|4 - 5\| + \|14 - 5\|}{5} = 3.6$
Median = 3	$\frac{\|2 - 3\| + \|2 - 3\| + \|3 - 3\| + \|4 - 3\| + \|14 - 3\|}{5} = 2.8$
Mode = 2	$\frac{\|2 - 2\| + \|2 - 2\| + \|3 - 2\| + \|4 - 2\| + \|14 - 2\|}{5} = 3.0$

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.

For the normal or "Gaussian" distribution, the ratio of mean absolute deviation to standard deviation is $\scriptstyle \sqrt{2/\pi}$ .[1] Thus if X is a normally distributed random variable with expected value 0 then

$\frac{ E|X| }{ \sqrt{E(X^2)} } = \sqrt{\frac{2}{\pi}}.$

In other words, for a Gaussian, mean absolute deviation is about 0.8 times the standard deviation.

Mean absolute 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.

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.

Median absolute deviation

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.

Maximum absolute deviation

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.

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, in that

L² norm statistics

just as the mean minimizes the standard deviation,

L¹ norm statistics

the median minimizes average absolute deviation,

L^∞ norm statistics

the mid-range minimizes the maximum absolute deviation, and

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 the population absolute deviation is 2/3. The average of all the sample standard deviations of size 3 that can be drawn from the population is 40/81. Therefore the absolute deviation is a biased estimator.

See also