Gur Yaari, a researcher at Yale School of Medicine, recently notified me of an online article on free-throw shooting and the hot hand that he co-authored with Shmuel Eisenmann. The study focused on whether basketball players have a higher probability of making the second of two free throws after making the first than after missing the first.

The classic Gilovich, Vallone, and Tversky (1985) article failed to find evidence of streakiness in free-throw shooting, but Yaari and Eisenmann noted that they were using a much larger sample than that of Gilovich and colleagues -- 308,862 free-throws taken from 2005-06 through 2009-10 in the National Basketball Association (situations in which shooters were awarded three shots after being fouled behind the arc were studied, but I don't discuss them).

Some of the statistical concepts cited were beyond my expertise, such as the hypergeometric distribution. The basic finding was clear, however. Players made around 72-75% of their free throws (depending on the season) after missing a free throw, whereas they hit on around 76-80% after making the first. The paper is full of technicality and subtlety. The authors precisely characterize their findings as: "essentially that the results are unlikely to emerge from a collection of uncorrelated sequences each with a constant probability of success and no auto correlation."

Yaari and Eisenmann also raised two possible explanations for their findings: what I would consider a traditional hot-hand scenario ("success breeds success and failure breeds failure"); and fluctuations between "better and worse periods." The authors found that players' hotness levels were uncorrelated from season to season, thus going against the idea that some shooters possess inherent streakiness.

The website "Sweat Science" also reviewed the Yaari and Eisenmann study. The review ended with the following call for caution, with which I concur:

...we’re usually referring to time frames that are longer than two back-to-back free throws [for a sequence to be considered a hot or cold hand] ... 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.