Revisiting a 90-year-old debate: the advantages of the mean deviation (2004)

There is a technical reason to prefer Mean Square Error derived measures (like RMSE and standard deviation) in some situations (such as machine learning and value estimation): when minimizing one of these measures you tend towards the mean and get expected values correct. Expected values are additive: so they roll up nicely (get the individuals right and you also have the group).<p>My example tends to be lottery tickets. You minimize MAD by saying they are all worth zero (which is pretty much my opinion). But then you don&#x27;t get the value of the lottery by summing up all the ticket values. You do&#x2F;should get get with mean&#x2F;expectation based estimates.<p>More of my writing on this: <a href="http:&#x2F;&#x2F;www.win-vector.com&#x2F;blog&#x2F;2014&#x2F;01&#x2F;use-standard-deviation-not-mad-about-mad&#x2F;" rel="nofollow">http:&#x2F;&#x2F;www.win-vector.com&#x2F;blog&#x2F;2014&#x2F;01&#x2F;use-standard-deviatio...</a> . Though I am also a fan of quantile regression (it just solves different problems).

There&#x27;s an epic rant on this topic here, <a href="https:&#x2F;&#x2F;www.edge.org&#x2F;response-detail&#x2F;25401" rel="nofollow">https:&#x2F;&#x2F;www.edge.org&#x2F;response-detail&#x2F;25401</a> , by Nassim Taleb, one of the 21st century&#x27;s greatest masters of epic rants.<p>This Mean Absolute Deviation is a very useful error estimator. Unfortunately it, unlike the standard deviation, requires two passes over a dataset to compute.

I actually used the Mean Absolute Deviation at work to good effect. When you&#x27;re testing variance over very large sample sizes, it&#x27;s a lot better than using something like Chi-Squared tests, which nearly always yield a significant result over a certain sample size (maybe 20,000 or so).

This may seem weird, but is this being posted because of some standardized testing?<p>My 6th grader took home a practice standardized test today. Last question on the test required a MAD computation.<p>He didn&#x27;t know what MAD was and I had to look it up myself.<p>I think understanding the <i>idea</i> behind MAD is great and potentially useful and interesting.<p>However, I&#x27;m not sure what 6th graders are going to be able to grind through two iterations over a data set to get the correct answer.

Why, oh why, didn&#x27;t they study <i>this</i> kind of stuff when I very nearly managed to drop out of the History of Science program in college?

Two thoughts:<p>1. The SD does have an interpretation: it is the (rescaled) Euclidean distance between your observed data set and a data set in which all points are replaced by the sample mean. Not terribly useful, but arguably not useless.<p>2. Are there models for which MD is in fact a sufficient statistic?