Just noticed a new study on a very old debate: the “hot hand” in basketball. There was a very famous study back in 1985 that concluded that our belief that players have hot streaks and cold streaks is simply an example of the “clustering illusion.” Think of it this way: if you flip a coin over and over, you’ll occasionally have streaks of six or seven heads in a row — but the probability of the next toss is still 50-50. What Gilovich, Vallone and Tversky found when they analyzed NBA game data was the same pattern: the probability of hitting a given shot was independent of whether the player had hit or missed his previous attempts.
Of course, most people simply refuse to acknowledge this sort of result that conflicts with what seems “obvious.” As Tversky noted:
I’ve been in a thousand arguments over this topic, won them all, but convinced no one.
After all, we can all remember, say, Michael Jordan taking over a game in the fourth quarter and pouring in basket after basket. But part of this may be because, with the game on the line, he starts taking more shots. And we tend to forget all the nights when he didn’t manage to take over the game, despite presumably making the same effort. This is one of those issues when we can really only trust the cold, hard data.
Anyway, the new study (full text freely available here; press release here) takes advantage of the Moneyball era of statistical abundance to revisit this question with a larger data set. The researchers, from Yale, looked at every free throw taken in the five seasons between 2005 and 2010 — a staggering total of 308,862 free throws — and tried to determine whether the patterns in the data could truly be explained by considering each shot as an independent event. And indeed, they found some evidence that — according to the press release, at least — supports the existence of the hot hand.
The key result they found is that, when players were taking two foul shots, they had a slightly greater chance of hitting the second shot when they hit the first (~76%) compared to when they missed the first (~73%). There are two possible ways to explain this:
- Players have periods when they’re “hot” and “cold.” The success of the first free throw is an indicator of which of those zones (if any) they’re in for the second throw.
- The outcome of the first throw causally influences the outcome of the second throw. For example, if you hit the first, you relax, feel confident, and drain the second; if you miss the first, you tense up, feel the pressure, and (become infinitesimally more likely to) miss.
The researchers argue against that second explanation, for the following reason. When they analyzed the individual data, some players shot better after hitting the first throw, while others shot better after missing the first throw. This is to be expected: since the supposed effect is psychological, different players will react differently to hitting/missing the first throw. But when they drilled deeper and broke the data down into individual seasons, they found that players who shot better after hitting the first shot in one season had a 50-50 chance of showing the opposite pattern the next season. That suggests that the connection between the first and second shots isn’t actually causal.
So what does this all mean? Well, in a sense it makes a fairly obvious point. It would be ludicrous to imagine that pairs of free throws are totally uncorrelated — consider, say, a pair of throws taken late in an insignificant game where the outcome is already decided, just after returning from a prolonged injury and having sustained a hard foul that hurts your shooting hand, the night after a coast-to-coast flight that was delayed by weather for seven hours, during which you got hammered because one of your teammates was celebrating his 21st birthday. The probability of both those throws will be slightly lower than a pair of throws under optimal conditions — and that leaves the kind of statistical footprint detected in this paper.
In a sense, of course, that’s precisely the point: under those conditions, you might say the player has a “cold hand.” But that’s not usually what we think of when we talk about hot and cold hands — we’re usually referring to time frames that are longer than two back-to-back free throws (it usually takes more than two shots before announcers start pulling out the “hot hand” trope), but far shorter than game-to-game variations. So in the end, I’m going to keep believing that the hot hand doesn’t exist until better evidence emerges.