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 the value of either the median or the mean of the data set.

$|D|\; =\; |x\_i-m(X)|$

where

$|D|$ 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.

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| + |3 - 5| + |4 - 5| + |14 - 5{5} = 3.6>
Median = 3	frac + |2 - 3| + |3 - 3| + |4 - 3| + |14 - 3{5} = 2.8>
Mode = 2	frac + |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 a Gaussian distribution, where x is a random variable with a mean of 0, in expectation, the ratio of standard deviation to mean absolute deviation should satisfy the following equality

$frac\{sum|x\{sqrt\{\; sum\; x^2\}\}\; =\; sqrt\{frac\{2\}\{pi\}\}$

in other words, mean absolute deviation is about .8 times the standard deviation.

The mean absolute deviation (MAD) is the average 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 mean absolute deviation of a sample is a biased estimator of the mean absolute deviation of the population.

See also

• Deviation (statistics)
• Least absolute deviations
• Median absolute deviation

External links

• Advantages of the mean absolute deviation

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Last updated on Monday July 21, 2008 at 05:27:31 PDT (GMT -0700)
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