# 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 *X*_{1}, *X*_{2}, ..., *X _{n}*, the MAD is defined as

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

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:

Hence

and:

In this case, its expectation for large samples of normally distributed *X _{i}* 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)