# Median absolute deviation

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In statistics, the median absolute deviation (or "MAD") is a resistant measure of the variability of a univariate sample. It is useful for describing the variability of data with outliers.

For a univariate data set X1X2, ..., Xn, the MAD is defined as

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

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

## Relation to standard deviation Edit

As an estimate for the standard deviation σ, one takes

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

where K is a constant. For normally distributed data K is taken to be 1 / Φ-1(3/4) (where Φ-1 is the inverse of the cumulative distribution function for the standard normal distribution), or 1.4826... , because the MAD is given by:

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

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 Xi is approximately equal to the standard deviation of the normal distribution.

## ReferencesEdit

• Hoaglin, David C.; Frederick Mosteller and John W. Tukey (1983). Understanding Robust and Exploratory Data Analysis, 404-414, John Wiley & Sons.
• Venables, W.N.; B.D. Ripley (1999). Modern Applied Statistics with S-PLUS, 128, Springer.