Mean Absolute Deviation: Empirical Rule?
October 12, 2016 7:05 AM Subscribe
I'm looking for either a secondary source confirming this so-called "empirical rule" of mean absolute deviation and standard deviation -- 1.25MAD≈s -- or a quick explanation of how that may have been derived.
From: Business Analytics: Data Analysis & Decision Making (Albright, S. Christian; Winston, Wayne L): '"Still, some analysts quote the mean absolute deviation (abbreviated as MAD) as another measure of variability, particularly in time series analysis. It is defined as the average of the absolute deviations.... There is another empirical rule for MAD: For many (but not all) variables, the standard deviation is approximately 25% larger than MAD, that is, s≈1.25MAD."Thanks!
From: Business Analytics: Data Analysis & Decision Making (Albright, S. Christian; Winston, Wayne L): '"Still, some analysts quote the mean absolute deviation (abbreviated as MAD) as another measure of variability, particularly in time series analysis. It is defined as the average of the absolute deviations.... There is another empirical rule for MAD: For many (but not all) variables, the standard deviation is approximately 25% larger than MAD, that is, s≈1.25MAD."Thanks!
mark k beat me to it. The exact number, by the way, is not 1.25 but √(pi/2) ≈ 1.25331413732...
Here's the integral for the MAD using Wolfram Alpha; here's the integral for the variance. (The standard deviation is the square root of the variance, but that's still equal to 1.)
posted by Johnny Assay at 7:42 AM on October 12, 2016
Here's the integral for the MAD using Wolfram Alpha; here's the integral for the variance. (The standard deviation is the square root of the variance, but that's still equal to 1.)
posted by Johnny Assay at 7:42 AM on October 12, 2016
Here is a related fact that I have never seen anywhere, but which might be of interested to you.
Just as the mean (average) is the point that minimizes the mean squared error of a sample, the median is the point that minimizes the mean absolute error. With a symmetric distribution, this is probably not important, but with an asymmetric distribution, it gives some support to using the median as the estimate of centrality rather than the mean.
posted by SemiSalt at 2:32 PM on October 12, 2016
Just as the mean (average) is the point that minimizes the mean squared error of a sample, the median is the point that minimizes the mean absolute error. With a symmetric distribution, this is probably not important, but with an asymmetric distribution, it gives some support to using the median as the estimate of centrality rather than the mean.
posted by SemiSalt at 2:32 PM on October 12, 2016
« Older Rental househunting in College Park/Greenbelt, MD... | How do I, an American, manage the practicalities... Newer »
This thread is closed to new comments.
So I'd assume it is probably a true statement derived from a perfect, symmetrical Gaussian distribution.
posted by mark k at 7:27 AM on October 12, 2016 [1 favorite]