By Peter Orszag
If you have a vacation coming up in August and you’re looking for a fun book to read that will also enlighten you, it would be hard to beat Jordan Ellenberg’s “How Not to Be Wrong: The Power of Mathematical Thinking.” Ellenberg, a math professor at the University of Wisconsin, shows how mathematics and statistics help us understand the world better — but does so in a way that skips the formalities and allows everyone to follow the argument. And the lessons are powerful.
Here’s an example. During World War II, the U.S. military was trying to optimize the armor plating on its airplanes. Officials noticed that the bullet holes in planes returning from combat in Europe followed certain patterns: There were more per square foot in the fuselage than in the engine section. They figured that they therefore needed to add more protection to the fuselage, but wanted help in determining how much more — to balance the extra protection against the loss of fuel efficiency and maneuverability.
The military took this problem to Abraham Wald of the Statistical Research Group. Wald, who spent most of his career as a statistics professor at Columbia University, came back with a surprising answer: Add no plating to the fuselage. Instead, add it to the engine area.
Wald's reason was that unless the enemy was for some odd reason successfully targeting the fuselages, the bullet holes on the returning planes showed where the planes could withstand attack and still survive. The paucity of bullet holes on the engine casings of the returning planes suggested that hits to that area tended to bring down the plane. Returning planes, in other words, were a biased sample of the planes that were attacked. The lesson: Think about where the bullet holes aren’t.
Another example of subtle biases involves Berkson’s fallacy, named after the medical statistician Joseph Berkson. Assume, for example, that someone has noticed that economists hardly ever drink Diet Coke and, to see if the observation is statistically correct, gathers information about everyone who drinks Diet Coke and everyone who is an economist. Berkson’s fallacy shows that the result will suggest a negative correlation between Diet Coke and being an economist, even if there isn’t one.
Why? Assume there are 1,000 people in a small town where the analyst decides to the study the subject: 250 residents drink Diet Coke, 200 of them are economists, and 50 of them are economists who drink Diet Coke. An analysis of those who either drink Diet Coke or are economists would include 50 economists who drink Diet Coke, 150 economists who don’t drink Diet Coke, and 200 non-economists who drink Diet Coke. The analysis would thus suggest that economists were much less likely to drink Diet Coke (50 out of 250) than non-economists (200 out of 200).
You can see the headlines: Drink Diet Coke if you want to avoid being an economist! But that conclusion would be wrong for two reasons. The first, as you may already know, is that correlation is not causation. Berkson’s fallacy, though, suggests the correlation itself is misleading. In reality, 25 percent of both economists and non-economists drink Diet Coke; there is no correlation between one and the other. (The problem is that the survey did not include the non-economists who don’t drink Diet Coke.)
Ellenberg shows how many real-world conclusions suffer from Berkson’s fallacy — even romantic frustrations (included in a section partially titled “why are handsome men such jerks?”). He also writes about hot hands in basketball, a statistical analysis of the Bible and how unlikely events can happen a lot, if given enough chances.
Our public discourse would be much improved if the basic lessons of “How Not to Be Wrong” were better appreciated — including how easy it is to be misled by statistical biases and simplistic extrapolations. And though the topics are weighty, Ellenberg’s writing is not. It might just be the first math book you can enjoy reading on the beach.
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