# 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

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

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

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

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:

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

Hence

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

and:

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

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

## References

• Hoaglin, David C. (1983). Understanding Robust and Exploratory Data Analysis. John Wiley & Sons. pp. 404–414. ISBN 0-471-09777-2. Unknown parameter |coauthors= ignored (help)
• Venables, W.N. (1999). Modern Applied Statistics with S-PLUS. Springer. p. 128. ISBN 0-387-98825-4. Unknown parameter |coauthors= ignored (help)