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. Moreover, 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 large deviations are weighted more heavily, and thus outliers can heavily influence it. In the MAD, the deviations of a small number of outliers are irrelevant.
Relation to standard deviation
For normally distributed data K is taken to be , where 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 . Since we have that from which we obtain the scale factor .
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.
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.
- Rousseeuw, P. J.; Croux, C. (1993). "Alternatives to the median absolute deviation". Journal of the American Statistical Association 88 (424): 1273–1283. doi:10.1080/01621459.1993.10476408.
- Ruppert, D. (2010). Statistics and Data Analysis for Financial Engineering. Springer. p. 118. ISBN 9781441977878. Retrieved 2015-08-27.
- Leys, C.; et al. (2013). "Detecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median". Journal of Experimental Social Psychology 49: 764–766. doi:10.1016/j.jesp.2013.03.013.
- Gauss, Carl Friedrich (1816). "Bestimmung der Genauigkeit der Beobachtungen". Zeitschrift für Astronomie und verwandte Wissenschaften 1: 187–197.
- Walker, Helen (1931). Studies in the History of the Statistical Method. Baltimore, MD: Williams & Wilkins Co. pp. 24–25.
- Hoaglin, David C.; Frederick Mosteller; 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. p. 128. ISBN 0-387-98825-4.