Median absolute deviation
In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.
Consider the data (1, 1, 2, 2, 4, 6, 9). It has a median value of 2. The absolute deviations about 2 are (1, 1, 0, 0, 2, 4, 7) which in turn have a median value of 1 (because the sorted absolute deviations are (0, 0, 1, 1, 2, 4, 7)). So the median absolute deviation for this data is 1.
The median absolute deviation is a measure of statistical dispersion. It is a more robust estimator of scale than the sample variance or standard deviation. It thus behaves better with distributions without a mean or variance, such as the Cauchy distribution.
For instance, the MAD is a robust statistic, being more resilient to outliers in a data set than the standard deviation. In the standard deviation, the distances from the mean are squared, so on average, large deviations are weighted more heavily, and thus outliers can heavily influence it. In the MAD, the magnitude of the distances of a small number of outliers is irrelevant.
Relation to standard deviation
where K is a constant scale factor, which depends on the distribution.
For normally distributed data K is taken to be 1/Φ−1(3/4) 1.4826, where Φ−1 is the inverse of the cumulative distribution function for the standard normal distribution, i.e., the quantile function. This is because the MAD is given by:
Therefore we must have that Φ(MAD/σ) − Φ(−MAD/σ) = 1/2. Since Φ(−MAD/σ) = 1 − Φ(MAD/σ) we have that MAD/σ = Φ−1(3/4) from which we obtain the scale factor K = 1/Φ−1(3/4).
In other words, the expectation of 1.4826 times the MAD for large samples of normally distributed Xi is approximately equal to the population standard deviation. Other distributions behave differently: for example for large samples from a uniform continuous distribution, this factor is about 1.1547 (the square root of 4/3).
The population MAD
The population MAD is defined analogously to the sample MAD, but is based on the complete distribution rather than on a sample. For a symmetric distribution with zero mean, the population MAD is the 75th percentile of the distribution.
Unlike the variance, which may be infinite or undefined, the population MAD is always a finite number. For example, the standard Cauchy distribution has undefined variance, but its MAD is 1.
- Deviation (statistics)
- Robust measures of scale
- Hoaglin, David C.; Frederick Mosteller and John W. Tukey (1983). Understanding Robust and Exploratory Data Analysis. John Wiley & Sons. pp. 404–414. ISBN 0-471-09777-2.
- Russell, Roberta S.; Bernard W. Taylor III. (2006). Operations Management. John Wiley & Sons. pp. 497–498. ISBN 0-471-69209-3.
- Venables, W.N.; B.D. Ripley (1999). Modern Applied Statistics with S-PLUS. Springer. pp. 128. ISBN 0-387-98825-4.
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