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 X₁, X₂, ..., X_n, the MAD is defined as

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

where K is a constant. Typically, K is taken to be 1.4826... (1 / Φ^-1(3/4), where Φ^-1 is the inverse of the cumulative distribution function for the standard normal distribution), which for large samples of normally distributed X_i equates its expectation to the standard deviation of the distribution.