Strength in Numbers In Search of the Warriors’ Hot Hand
Alon Daks, Nishant Desai, Lisa Goldberg*
May 30, 2017
The true believer in the law of small numbers commits his multitude of sins against the logic of statistical inference in good faith.
Amos Tversky and Daniel Kahneman, “Belief in the Law of Small Numbers”
In 1985, psychologists Thomas Gilovitch and Amos Tversky along with statistician Robert Vallone challenged conventional wisdom with the finding that the hot hand in basketball was a cognitive illusion. Their empirically grounded result predates the modern information revolution by more than a quarter of a century. An analysis of shooting records of Philadelphia 76ers, Boston Celtics and Cornell University varsity and junior varsity players indicated that the observed scoring streaks were not out of the ordinary, even if they seemed to be. They could be explained by mere chance.
But chance can be hard to pin down. In a remarkable plot twist, statisticians Josh Miller and Adam Sanjurjo found a fundamental flaw in the 1985 study thirty years after its publication. When the flaw was corrected, some of the Cornell players, at least, showed a hot hand.
Flaws in research are discovered all the time—that is the nature of research. But there is a special irony surrounding this flaw. In its characterization of “mere chance,” the 1985 hot hand study neglected a Small Sample Effect: the tendency to read too much into a few examples. Amos Tversky and his lifelong collaborator, Nobel Laureate Daniel Kahneman, are famous for their understanding of how humans make mistakes. One of their early studies documented the Small Sample Effect.
Relying on the original hot hand study for inspiration, applying Miller and Sanjurjo’s insightful small sample correction, and availing ourselves of the easy-to-use tools of modern data science, we analyzed the shooting records of Splash Brothers Steph Curry and Klay Thompson. It is a magical experience to watch these players on the court. When they are on a roll, they seem to be the essence of hot handedness. Our analysis, however, tells a different story. Among the regular season games that the Golden State Warriors played in the 2016–2017 season, Steph had a hot hand in one or two, Klay in three or four. In his phenomenal 60-point game on December 5, Klay’s hits and misses were intertwined and his streaks were explainable by mere chance, even after taking account of the Small Sample Effect. The Splash Brothers are great shooters but they are not streak shooters.
Amos Tversky was 59 years old when he died in 1996, almost two decades before the flaw in his hot hand study was found. An awesome researcher and a huge basketball fan, he would no doubt be pleased about the correction of his error if he were with us today, and he would surely be watching the Warriors play the Cavs for the 2017 championship.
*All three authors are at University of California, Berkeley. Please contact lrg [at] berkeley [dot] edu for more information.