Median absolute deviation

In statistics, the median absolute deviation (or "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.

For a univariate data set X₁, X₂, ..., X_n, the MAD is defined as the median of the absolute deviations from the data's median:

that is, starting with the residuals (deviations) from the data's median, the MAD is the median of their absolute values.

Example

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. So the median absolute deviation for this data is 1.

Uses

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

In order to use the MAD as a consistent estimator for the estimation of the standard deviation σ, one takes

where K is a constant scale factor, which depends on the distribution.

For normally distributed data K is taken to be , 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:

In words: for a symmetric distribution, the MAD is the distance between the 1st and 2nd (equivalently, 2nd and 3rd) quartiles, so for a symmetric distribution about the mean, the MAD is the 3rd quartile (75th percentile). Thus the scale factor to use the MAD for the normal distribution is the 75th percentile of the normal distribution with σ = 1.

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.

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, 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.

The earliest known mention of the concept of the MAD occurred in 1816, in a paper by Carl Friedrich Gauss on the determination of the accuracy of numerical observations.^[1][2]

See also

• Deviation (statistics)
• Robust measures of scale

Notes

References

• 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.