From Quanta, March 16:
The central limit theorem started as a bar trick for 18th-century gamblers. Now scientists rely on it every day.
No matter where you look, a bell curve is close by.
Place a measuring cup in your backyard every time it rains and note the height of the water when it stops: Your data will conform to a bell curve. Record 100 people’s guesses at the number of jelly beans in a jar, and they’ll follow a bell curve. Measure enough women’s heights, men’s weights, SAT scores, marathon times — you’ll always get the same smooth, rounded hump that tapers at the edges.
Why does the bell curve pop up in so many datasets?
The answer boils down to the central limit theorem, a mathematical truth so powerful that it often strikes newcomers as impossible, like a magic trick of nature. “The central limit theorem is pretty amazing because it is so unintuitive and surprising,” said Daniela Witten (opens a new tab), a biostatistician at the University of Washington. Through it, the most random, unimaginable chaos can lead to striking predictability.
It’s now a pillar on which much of modern empirical science rests. Almost every time a scientist uses measurements to infer something about the world, the central limit theorem is buried somewhere in the methods. Without it, it would be hard for science to say anything, with any confidence, about anything.
“I don’t think the field of statistics would exist without the central limit theorem,” said Larry Wasserman (opens a new tab), a statistician at Carnegie Mellon University. “It’s everything.”
Purity From Vice
Perhaps it shouldn’t come as a surprise that the push to find regularity in randomness came from the study of gambling.
In the coffeehouses of early-18th-century London, Abraham de Moivre’s mathematical talents were obvious. Many of his contemporaries, including Isaac Newton and Edmond Halley, recognized his brilliance. De Moivre was a fellow of the Royal Society, but he was also a refugee, a Frenchman who had fled his home country as a young man in the face of anti-Protestant persecution. As a foreigner, he couldn’t secure the kind of steady academic post that would befit his talent. So to help pay his bills, he became a consultant to gamblers who sought a mathematical edge.
Flipping a coin, rolling a die, and drawing a card from a deck are random actions, with every outcome equally likely. What de Moivre realized is that when you combine many random actions, the result follows a reliable pattern.
Flip a coin 100 times and count how often it comes up heads. It’ll be somewhere around 50, but not very precisely. Play this game 10 times, and you may get 10 different counts.
Now imagine playing the game 1 million times. The bulk of the outcomes will be close to 50. You’ll almost never get under 10 heads or over 90. If you make a graph of how many times you see each number between zero and 100, you’ll see that classic bell shape, with 50 at the center. The more times you play the game, the smoother and clearer the bell will become.
De Moivre figured out the exact shape of this bell, which came to be called the normal distribution. It told him, without his having to actually play the game, how likely different outcomes were. For instance, the probability of getting between 45 and 55 heads is about 68%.
De Moivre marveled with religious devotion at the “steadfast order of the universe” that eventually overcame any and all deviations from the bell. “In process of time,” he wrote, “these irregularities will bear no proportion to the recurrency of that order which naturally results from original design.”
He used these insights to sustain a meager life in London, writing a book called The Doctrine of Chances that became a gambler’s bible, and holding informal office hours at the famed Old Slaughter’s Coffee House. But even de Moivre didn’t realize the full scope of his discovery. Only when Pierre-Simon Laplace ran with the idea in 1810, decades after de Moivre’s death, was its full reach uncovered.
Let’s take an example slightly more complex than coin flips: dice rolls. Every roll of a die has six equally likely outcomes. If you repeatedly roll the die and tally the results, you’ll get a chart that looks flat — you’re bound to see about as many rolls of 1 as you do 2 or 4 or 6....
....MUCH MORE
