Why You Should Be MAD at Standard Deviation

There are three common ways to measure how spread out a set of numbers is, and they all begin the same way: find the average, then look at how far each number sits from it. The only real disagreement is what you do with those distances next.

Mean absolute deviation (MAD) simply drops the minus signs and averages the distances:

Variance squares each distance first (which also gets rid of the minus signs), then averages them:

Standard deviation is just variance with the squaring undone at the very end, to drag the number back into normal units:

where x̄ is the average and n is how many numbers you have. So standard deviation is the scenic route: square everything, average it, then take a square root to reverse the exact thing you just did. MAD is the direct flight.

Here are ten numbers, 1 through 10, run through both:

Average the middle column: MAD = 25 / 10 = 2.5. Average the right column and you get the variance, 82.5 / 10 = 8.25, then square-root it for a standard deviation of about 2.87. Same data, and standard deviation always comes out heavier.

So why does basically the whole world take the scenic route? Not because it is more correct. Because it is easy and predictable. And there is exactly one place where that is gloriously true: variances add. Combine two independent sources of randomness and the variance of the total is just one variance plus the other, with no need to relist every possible combined outcome. MAD has no such trick. That clean, predictable “just add them” behavior is the real engine under standard deviation’s popularity. It is a perfectly good reason to keep it around. It is not the same as a reason to believe it is right.

Now the story, because there is one.

In 1914 the astrophysicist Arthur Eddington looked at real observations and said the plain average distance, MAD, was the better tool. A few years later Ronald Fisher, the towering statistician, disagreed and came armed with a proof. Using an idea he was busy inventing called efficiency, he showed that on perfectly normal, bell-curved data, standard deviation is the more reliable measure, roughly 12% more efficient than MAD. That single result won the argument and steered most of twentieth-century statistics behind it.

But read the fine print, because Fisher wrote it himself. He let Eddington attach a footnote admitting that in practice, with real observations, minimizing absolute error often worked better, precisely because real data comes speckled with outliers. The winning verdict shipped with its own disclaimer: it holds only if your data is genuinely normal.

And here is the part everyone conveniently forgot. Real data is almost never genuinely normal. In 1960 Tukey showed that contaminating a normal distribution with even a pinch of weirdness flips the result, and MAD pulls ahead, which launched the whole field of robust statistics. Decades later Stephen Gorard (2005) revived MAD outright: it is simpler, it means something a human can actually feel, and the old excuse that squaring was easier to compute died the day pocket calculators arrived. Nassim Taleb (2020) went harder still, arguing that for the fat-tailed data of the actual world, reaching for standard deviation was not merely quaint but actively harmful. For the record, Geary pinned down the exchange rate back in 1935: on a perfect bell curve, MAD is about 80% of the standard deviation, the same information wearing different units.

So squaring buys standard deviation two things MAD genuinely cannot match: variances add cleanly, and it is the native language of the normal curve. Those are real, and worth keeping. Everything else is inheritance. Fisher won at exactly one distribution, the flawless bell curve, and the messy, outlier-ridden real world, which is to say everywhere you will ever actually stand, has been quietly voting for Eddington ever since. So the next time you square a distance only to square-root it right back, and someone assures you this is simply the correct way, you have every right to be MAD.