medianabsolute

Median absolute deviation, commonly abbreviated as MAD, is a robust statistic used to measure the dispersion of a univariate data set. It is defined as the median of the absolute deviations from the data’s median: MAD = median(|xi − median(x)|) for a sample x1, x2, ..., xn. In a population setting with true median m, MAD = median(|X − m|). Because it relies on medians rather than means, MAD is resistant to outliers and extreme values.

Scaling to estimate standard deviation: for data that are approximately normally distributed, MAD can be scaled by a constant to provide a consistent estimate of the standard deviation. The common scale factor is about 1.4826, giving sMAD = 1.4826 × MAD. This adjustment makes MAD comparable to the standard deviation under normality while retaining its robustness to outliers.

Properties and use cases: MAD is notable for its robustness to outliers, making it a preferred scale measure in robust statistics, in outlier detection, and as a component of resistant regression methods. It is less sensitive to extreme values than the standard deviation, though it is typically less efficient for well-behaved, normally distributed data. MAD is often used in exploratory data analysis, to describe data spread, and as a diagnostic tool alongside other robust measures such as the interquartile range.

Computation and considerations: MAD depends on the median, which makes it more computationally intensive than variance-based measures but straightforward for most data sets. It is defined for any nonempty sample; with very small samples or tied data, the interpretation of MAD can be limited.