Average absolute deviation

Several measures of statistical dispersion are defined in terms of the absolute deviation. The term "average absolute deviation" does not uniquely identify a measure of statistical dispersion, as there are several measures that can be used to measure absolute deviations, and there are several measures of central 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. Unfortunately, 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 in general, they may have values considerably different from each other.

The mean absolute deviation of a set {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}:

Measure of central tendency ${\displaystyle m(X)}$	Mean absolute deviation
Mean = 5	${\displaystyle {\frac {|2-5|+|2-5|+|3-5|+|4-5|+|14-5|}{5}}=3.6}$
Median = 3	${\displaystyle {\frac {|2-3|+|2-3|+|3-3|+|4-3|+|14-3|}{5}}=2.8}$
Mode = 2	${\displaystyle {\frac {|2-2|+|2-2|+|3-2|+|4-2|+|14-2|}{5}}=3.0}$

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.

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 ${\displaystyle \varphi \left(\mathbb {E} [Y]\right)\leq \mathbb {E} \left[\varphi (Y)\right]}$,where φ is a convex function, this implies for ${\displaystyle Y=\vert X-\mu \vert }$ that: ${\displaystyle \mathbb {E} \left(|X-\mu \right|)^{2}\leq \mathbb {E} \left(|X-\mu |^{2}\right)}$ ${\displaystyle \mathbb {E} \left(|X-\mu \right|)^{2}\leq \operatorname {Var} (X)}$ Since both sides are positive, and the square root is a monotonically increasing function in the positive domain: ${\displaystyle \mathbb {E} \left(|X-\mu \right|)\leq {\sqrt {\operatorname {Var} (X)}}}$ For a general case of this statement, see Hölder's inequality.

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

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

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

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.

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.

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.

• Deviation (statistics)
• 
  Mean absolute error
• Errors and residuals in statistics
• Least absolute deviations
• Loss function
• Mean absolute percentage error
• Mean difference
• Mean squared error
• Median absolute deviation
• Squared deviations

• Advantages of the mean absolute deviation