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Median absolute deviation: Definition from Answers.com

In statistics, the median absolute deviation (or "MAD") is a robust measure of the variability of a univariate sample.

For a univariate data set X₁, X₂, ..., X_n, the MAD is defined as the median absolute deviation from the median:

$\operatorname{MAD} = \operatorname{median}_{i}\left(\ \left| X_{i} - \operatorname{median}_{j} (X_{j}) \right|\ \right), \,$

that is, starting with the residuals (deviations) from the data's median, the MAD is the median of their absolute values.

Example

Consider the data (1, 1, 2, 2, 4, 6, 9). It has a median value of 2. The absolute deviations about 2 are (1, 1, 0, 0, 2, 4, 7) which in turn have a median value of 1. So the median absolute deviation for this data is 1.

Uses

The median absolute deviation is a measure of statistical dispersion. It is a more robust estimator of scale than the sample variance or standard deviation. It thus behaves better with distributions without a mean or variance, such as the Cauchy distribution.

For instance, the MAD is a robust statistic, being more resilient to outliers in a data set than the standard deviation. In the standard deviation, the distances from the mean are squared, so on average, large deviations are weighted more heavily, and thus outliers can heavily influence it. In the MAD, the magnitude of the distances of a small number of outliers is irrelevant.

Relation to standard deviation

In order to use the MAD as a consistent estimator for the estimation of the standard deviation σ, one takes

$\hat{\sigma}=K\cdot \operatorname{MAD},$

where K is a constant scale factor, which depends on the distribution.

For normally distributed data K is taken to be $1/\Phi^{-1}(3/4) \approx 1.4826$ , where Φ^-1 is the inverse of the cumulative distribution function for the standard normal distribution, i.e., the quantile function. This is because the MAD is given by:

$\frac 12 =P(|X-\mu|\le \operatorname{MAD})=P\left(\left|\frac{X-\mu}{\sigma}\right|\le \frac {\operatorname{MAD}}\sigma\right)=P\left(|Z|\le \frac {\operatorname{MAD}}\sigma\right).$

In words: for a symmetric distribution, the MAD is the distance between the 1st and 2nd (equivalently, 2nd and 3rd) quartiles, so for a symmetric distribution about the mean, the MAD is the 3rd quartile (75th percentile). Thus the scale factor to use the MAD for the normal distribution is the 75th percentile of the normal distribution with σ = 1.

Hence

$\frac {\operatorname{MAD}}\sigma=\Phi^{-1}(3/4) \approx 0.6745$

and:

$\sigma \approx 1.4826\ \operatorname{MAD}.$

In this case, its expectation for large samples of normally distributed X_i is approximately equal to the standard deviation of the normal distribution.

Alternatives and extensions

Rousseeuw and Croux^[1] propose alternatives to the MAD, pointing out two drawbacks of it:

It is inefficient (37% efficiency) at Gaussian distributions.
it computes a symmetric statistic about a location estimate, thus not dealing with skewness.

They propose two alternatives, based on pairwise differences: S_n and Q_n, defined as:

$\begin{align} S_n &:= 1.1926 \, \operatorname{med}_i \left( \operatorname{med}_j (\,\left| x_i - x_j \right|\,) \right) \\ Q_n & := \text{first quartile of} \left( \left| x_i - x_j \right| : i < j \right) \end{align}$

These can be computed in O(n log n) time and O(n) space.

Neither of these requires location estimation, as they are based only on differences between samples. They are both more efficient than MAD at Gaussians: S_n is 58% efficient, while Q_n is 82% efficient.

Notes

^ Rousseeuw, Peter J.; Croux, Christophe (December 1993), "Alternatives to the Median Absolute Deviation", Journal of the American Statistical Association 88 (424): 1273–1283, http://www.jstor.org/stable/2291267

References

Hoaglin, David C.; Frederick Mosteller and John W. Tukey (1983). Understanding Robust and Exploratory Data Analysis. John Wiley & Sons. pp. 404-414. ISBN 0-471-09777-2.
Russell, Roberta S.; Bernard W. Taylor III. (2006). Operations Management. John Wiley & Sons. pp. 497-498. ISBN 0-471-69209-3.
Venables, W.N.; B.D. Ripley (1999). Modern Applied Statistics with S-PLUS. Springer. pp. 128. ISBN 0-387-98825-4.

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