How Not to Be Wrong

Please pardon the obnoxious title! If it matters, the author himself makes fun of it in the book's first chapter.

This one's a pop science book that tries to do for mathematical reasoning what Freakonomics did for economical reasoning. I think Ellenberg somewhat succeeds. The book doesn't feel particularly approachable by the kind of people who don't already understand some basic fundamentals of math, but then Ellenberg spends a lot of time going over some of those basic fundamentals while zipping through a few more advanced things that could use longer explanations. Here are some of my biggest takeaways, in bullet form. I'm not sure if these were necessarily meant to be the biggest takeaways, but that's the thing about pop science books - everyone can get something different out of them!

• There's a difference between being good at calculation and being good at math. Being able to divide two numbers in your head or on paper is a skill, but it's just a calculation; anyone with access to a computer can get a precise answer in no time. On the other hand, understanding when to use different mathematical operations and inferences and models to understand the real world is a different skill altogether.
• Don't use proportions or percentages when the answers might be negative. Say you're running a shop that sells coffee and pastries and sandwiches. You're making money on the coffee and the pastries but losing money on the sandwiches. If those figures sum, respectively, to +$100, +$100, and -$50 a week then you've made $150. And it might be tempting to say that the coffee, which netted you +$100 of that +$150, is responsible for 67% of your profit. Ditto the pastries. But how can the coffee and the pastries both be responsible for the majority of your profit? It's a technically accurate statement, but a misleading and useless one. You could also say that the coffee did twice as well as the sandwiches and pastries combined. Ellenberg warns us to be careful of political ads that say, for instance, "The Governor of Wisconsin was responsible for half of the jobs created in the country last year." This might be technically true, but ten or twenty other states might be able to make the same claim. Some may even be able to say they are responsible for more than 100% of the jobs created in a given timeframe, which would at least sound fishier and less intuitive. Be careful out there with negative percentages!
• Do not confuse high probability for certainty. There's an xkcd comic Ellenberg even includes in his book to illustrate this point. Say you're 95% confident that a study has shown a correlation between two things. To a layman, this sounds like a certainty. But improbable things happen all the time! This means that if there are a hundred studies conducted, each with a 95% confidence, then we'd expect five of them to be, you know, wrong. Ellenberg includes an interesting analysis of Nate Silver's famous correct prediction of all 50 states in the 2012 Presidential Election. He says that while Silver correctly predicted the outcome in every state, Silver's own uncertainties his predictions (like "Florida: 67% Obama, 33% Romney") meant Silver should have expected to get close to three states wrong. In a way, the fact that Silver nailed all fifty states suggests that his models were off; if his probabilities had been accurate, he (probably) shouldn't have had a perfect showing. It's counterintuitive but true!
• Don't confuse correlation for causation. This one's a pretty old maxim and it's easy to understand when you notice correlations between things like hand size and foot size. People with bigger hands tend to have bigger feet, but do big feet cause big hands, or do big hands cause big feet? Neither, of course; both are caused by other factors - mostly genetics. Chinese foot-binding didn't cause women to have tiny hands, after all. But it's easy to start mistaking correlation for causation when it comes to things like, say, HDL and heart disease. Years of data have shown that people with higher HDL (good cholesterol) are less likely to suffer from heart disease, and it's natural to assume that boosting your HDL will in turn boost your immunity to heart disease. Except, that's not the case. When people are given drugs or other treatments to boost their HDL levels, they don't end up any less likely to suffer from heart disease. The implication is that there's an unseen variable at play here that tends to cause both low HDL levels and a higher risk of heart disease in some people, and higher HDL and lower risk in others. But that doesn't mean boosting your HDL by artificial means will prevent heart disease anymore than it means losing your feet would stunt the growth of your hands.

There was plenty else, but like I said, these were my main takeaways. If I suddenly remember any others in the coming days and feel compelled to write about them, hey, there's always the comment section.