Mediaanabsolute

Mediaanabsolute, commonly referred to in English as the median absolute deviation (MAD), is a robust measure of statistical dispersion. It describes how far data values typically lie from a central location, using the median rather than the mean to reduce the influence of outliers. For a univariate sample x1, x2, ..., xn, compute the median m = median(x) and the absolute deviations di = |xi − m|. The MAD is the median of these deviations: MAD(x) = median(di).

To make MAD comparable to the standard deviation for normally distributed data, it is standard to apply a scaling constant c ≈ 1.4826, obtaining the scaled MAD: MAD* = c · MAD. This scaled form is often used as a robust analogue of the standard deviation, particularly in exploratory data analysis and when detecting outliers.

Properties and interpretation: The MAD is highly robust to outliers, with a breakdown point of 50%, meaning that up to half the data can be contaminated without causing MAD* to break down. It remains unchanged under monotone transformations of the data. While it provides a resilient measure of spread, the MAD is generally less efficient than the standard deviation for estimating dispersion under clean, normally distributed data.

Applications: MAD is widely used in robust statistics, outlier detection, and as a component of robust z-scores for identifying anomalous observations. It is implemented in many statistical software packages (for example, functions that compute MAD with a scaling option). Extensions exist for multivariate settings, though the univariate MAD remains the most common form.

See also: median, absolute deviation, robust statistics.