Median absolute deviation

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 X1, X2, ..., 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
As an estimate for the standard deviation &sigma;, one takes


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

where K is a constant. For normally distributed data K is taken to be 1 / &Phi;-1(3/4) (where &Phi;-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.