I came across Jordan Ellenberg’s book How Not to Be Wrong: The Power of Mathematical Thinking (Penguin, 2014) when I was looking for another title. It was a serendipitous find, one of the best books I’ve read this year.
The book’s ideal reader is numerate and politically liberal. Ellenberg’s examples often display an openly liberal bias, which would not go down well with the Fox News crowd. A case in point: Ellenberg’s explanation of why, contrary to Republican dogma, the Reagan tax cut resulted in less tax revenue, not more. Even if one believes in the power of the Laffer curve (which is overly simplistic in and of itself because, for instance, it ignores spending as a variable), the question is where we are on the curve. Assume the x-axis represents the tax rate, from 0% to 100%, and the y-axis revenue. The Laffer curve slopes up from 0%, peaks at some point, and then slopes down to 100%. If we’re to the right of the peak, a government that adopts Laffer-curve thinking should lower the tax rate to increase revenue; if we’re to the left, however, it should raise the tax rate. Most likely, Ellenberg suggests, we were already to the left of the Laffer peak when Reagan lowered taxes—and saw a significant decrease in revenue from personal income taxes.
Ellenberg uses another political example to show why you shouldn’t talk about percentages of numbers when you’re dealing with a combination of positive and negative numbers. In June of 2011 Wisconsin’s Republican Party issued a news release touting the job-creating record of its governor, Scott Walker. That month the U.S. economy had added only 18,000 jobs. Wisconsin, by contrast, added 9,500 jobs. “Today,” the statement read, “we learned that over 50% of U.S. job growth in June came from our state.” The problem with that claim is that Minnesota added 13,000 jobs (as Ellenberg writes, “70% of all jobs created—by now the arithmetical problem should be evident”), and four other states also outpaced Wisconsin’s job gains. Job losses in other states came close to balancing out job gains in states like Wisconsin and Minnesota.
Ellenberg, to my mind, is at his best when he makes tough mathematical concepts, such as a ten-dimensional vector, comprehensible. And he does just that when explaining the correlation between average January 2011 and January 2012 temperatures in ten California cities. The two vectors point in roughly the same direction. “The correlation between the two variables is determined by the angle between the two vectors.” (p. 277) When the angle is acute, the two variables are positively correlated; when it is obtuse, they are negative correlated; when the angle is a right angle, the vectors are orthogonal. (If you want to know the meaning of the word “orthogonal,” just ask Chief Justice John Roberts and Justice Antonin Scalia. Ellenberg includes an amusing exchange from a recent Supreme Court oral argument.)
Establishing the efficacy of drugs or other medical treatment is notoriously difficult, in part because correlation is not transitive. For instance, niacin increases HDL, and a higher HDL is associated with a lower risk of cardiovascular events. But patients who got niacin had just as many heart attacks and strokes as the rest of the population. That is, niacin is correlated with high HDL and high HDL is correlated with a low risk of heart attack, but niacin isn’t ipso facto correlated with a low risk of heart attack.
As books on mathematical (primarily statistical) thinking go, Ellenberg’s is a keeper. It’s informative, witty, and a page-turner. Who could ask for anything more?