# 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 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

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:

${\frac {1}{2}}=P(|X-\mu |\leq \operatorname {MAD} )=P\left(\left|{\frac {X-\mu }{\sigma }}\right|\leq {\frac {\operatorname {MAD} }{\sigma }}\right)=P\left(|Z|\leq {\frac {\operatorname {MAD} }{\sigma }}\right).$ 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. 