Absolute deviation

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"Mean deviation" redirects here. For the book, see Mean Deviation (book).

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.

$D_{i}=|x_{i}-m(X)|$

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. In its general form, the average used can be the mean, median, mode, or the result of another measure of central tendency.

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 distribution, the ratio of mean absolute deviation to standard deviation is $\scriptstyle {\sqrt {2/\pi }}=0.79788456\dots$ . Thus if X is a normally distributed random variable with expected value 0 then, see Geary (1935):^[1]

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

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: $w_{n}\in [0,1]$ , with a bias for small n.^[2]

Mean absolute deviation (MAD)

The mean absolute deviation (MAD), also referred to as the mean deviation (or sometimes average absolute deviation, though see above for a distinction), is the mean of the absolute deviations of a set of data about the data's mean. In other words, it is the average distance of the data set from its mean. 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 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] as

$D_{m}=E|X-\mu |=2Cov(X,I_{O})$

where

D_m is the expected value of the absolute deviation about mean,

"Cov" is the covariance between the random variable X and the over indicator function ( $I_{{O}}$ ).

and the over indicator function is defined as

${\mathbf {I}}_{O}:={\begin{cases}1&{\text{if }}x>\mu ,\\0&{\text{else }}\end{cases}}$

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.

Average absolute deviation about median

Mean absolute deviation about median (MAD median) offers a direct measure of the scale of a random variable about its median

$D_{{med}}=E|X-median|$

For the normal distribution we have $D_{{med}}=\sigma {\sqrt (}2/\pi )$ . Since the median minimizes the average absolute distance, we have $D_{{med}}<=D_{{mean}}$ . By using the general dispersion function Habib (2011) defined MAD about median as

$D_{{med}}=E|X-median|=2Cov(X,I_{O})$

where the indicator function is

${\mathbf {I}}_{O}:={\begin{cases}1&{\text{if }}x>median,\\0&{\text{else }}\end{cases}}$

This representation allows for obtaining MAD median correlation coefficients;^[7]

Median absolute deviation (MAD)

Main article: Median absolute deviation

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.

Maximum absolute deviation

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

References

↑ Geary, R. C. (1935). The ratio of the mean deviation to the standard deviation as a test of normality. Biometrika, 27(3/4), 310-332.
↑ See also Geary's 1936 and 1946 papers: Geary, R. C. (1936). Moments of the ratio of the mean deviation to the standard deviation for normal samples. Biometrika, 28(3/4), 295-307 and Geary, R. C. (1947). Testing for normality. Biometrika, 34(3/4), 209-242.
↑ http://www.edge.org/response-detail/25401
↑ Kader, Gary (March 1999). "Means and MADS". Mathematics Teaching in the Middle School 4 (6): 398–403. Retrieved 20 February 2013.
↑ Franklin, Christine, Gary Kader, Denise Mewborn, Jerry Moreno, Roxy Peck, Mike Perry, and Richard Scheaffer (2007). Guidelines for Assessment and Instruction in Statistics Education. American Statistical Association. ISBN 978-0-9791747-1-1.
↑ Elamir, Elsayed A.H. (2012). "On uses of mean absolute deviation: decomposition, skewness and correlation coefficients". Metron: International Journal of Statistics LXX (2-3).
↑ Habib, Elsayed A.E. (2011). "Correlation coefficients based on mean absolute deviation about median". International Journal of Statistics and Systems 6 (4): pp. 413–428.

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