The ‘hot hand’ theory makes a comeback

There is a strong belief among athletes and sports fans that sometimes athletes enter ‘the zone’, or have a ‘hot streak’ where it seems they can do no wrong or at least perform much better than they usually do and thus have a much greater chance of success at hitting the ball or shooting a basket than at other times. There is a kind of plausibility story built around this idea. When you achieve success, it makes you feel good and confident and that sense of assurance may lead to a greater focus and thus better performance whereas failure may lead to greater nervousness and second-guessing of oneself that could prove harmful in fast-moving actions sports.

When people used to say that certain athletes were on such a streak, I would smugly correct them with science, saying that studies had shown that when analyzed statistically, such streaks were non-existent and were merely illusions, examples of confirmation bias in action. But the victims of my condescension may have the last laugh. Joshua Miller and Adam Sanjuro write about the history of analyses of such streaks and why they may have misled us.

In the landmark 1985 paper “The hot hand in basketball: On the misperception of random sequences,” psychologists Thomas Gilovich, Robert Vallone and Amos Tversky (GVT, for short) found that when studying basketball shooting data, the sequences of makes and misses are indistinguishable from the sequences of heads and tails one would expect to see from flipping a coin repeatedly.

In GVT’s critical test of hot hand shooting conducted on the Cornell University basketball team, they examined whether players shot better when on a streak of hits than when on a streak of misses. In this intuitive test, players’ field goal percentages were not markedly greater after streaks of makes than after streaks of misses.

GVT made the implicit assumption that the pattern they observed from the Cornell shooters is what you would expect to see if each player’s sequence of 100 shot outcomes were determined by coin flips. That is, the percentage of heads should be similar for the flips that follow streaks of heads, and the flips that follow streaks of misses.

The authors then explain why that plausible implicit assumption may be wrong and it is due to what they call the principle of restricted choice.

For example, imagine flipping a coin 100 times and then collecting all the flips in which the preceding three flips are heads. While one would intuitively expect that the percentage of heads on these flips would be 50 percent, instead, it’s less.

Here’s why.

Suppose a researcher looks at the data from a sequence of 100 coin flips, collects all the flips for which the previous three flips are heads and inspects one of these flips. To visualize this, imagine the researcher taking these collected flips, putting them in a bucket and choosing one at random. The chance the chosen flip is a heads – equal to the percentage of heads in the bucket – we claim is less than 50 percent.

Since the expected value is less than 50%, when the earlier researchers found close to 50% they were not finding that there was no hot hand but that in fact there was, because the athletes were outperforming expectations. You can read the article for the reasoning behind this key point that underlies their claim. The authors then go on to apply this to the analysis of ‘hot hand’.

With this counterintuitive new finding in mind, let’s now go back to the GVT data. GVT divided shots into those that followed streaks of three (or more) makes, and streaks of three (or more) misses, and compared field goal percentages across these categories. Because of the surprising bias we discovered, their finding of only a negligibly higher field goal percentage for shots following a streak of makes (three percentage points), was, if you do the calculation, actually 11 percentage points higher than one would expect from a coin flip!

An 11 percentage point relative boost in shooting when on a hit-streak is not negligible. In fact, it is roughly equal to the difference in field goal percentage between the average and the very best 3-point shooter in the NBA. Thus, in contrast with what was originally found, GVT’s data reveal a substantial, and statistically significant, hot hand effect.

Thus, surprisingly, these recent discoveries show that the practitioners were actually right all along. It’s OK to believe in the hot hand. While perhaps you shouldn’t get too carried away, you can believe in the magic and mystery of momentum in basketball and life in general, while still maintaining your intellectual respectability.

The GVT paper’s conclusions challenged the conventional wisdom of its time and it took awhile to become generally accepted. As one can imagine, any work such as this that challenges the new conventional wisdom is also going to be challenged, with more data collected and analyses carried out to see if it holds up.


  1. Smokey says

    If I understand this correctly, which I very much doubt, the players are outperforming randomness because of weird statistical logic? The sequence of player hits and misses are as random as flipping a coin, but we expect them to do … worse?

    This is how you get Dark Lords. Monty Hall has a lot to answer for.

  2. Holms says

    I don’t know why there was controversy over this. Those periods of good shooting / kicking / dodging etc. might have approximately the same frequency of occurrence as streaks of coin flips, but that’s no reason to dismiss the ‘hot streak’ idea. Rather, those periods are those times that the athlete notices better than average concentration under pressure, i.e. the hot streak.

  3. Rob Grigjanis says

    For example, imagine flipping a coin 100 times and then collecting all the flips in which the preceding three flips are heads. While one would intuitively expect that the percentage of heads on these flips would be 50 percent, instead, it’s less.

    I think that’s bullshit, and intuition has nothing to do with it either way. Every flip is independent.

  4. says

    They are comparing performance on a neuromuscular task to coin flips? WTF social science.

    I have some experience with this. When you’re shooting targets you sometimes DO get in a groove -- your breathing and timing are right, you’re feeling the spot where the trigger breaks perfectly, the weather is just right (no sun glare) etc. and yeah you can bang off a streak as long as you keep the muscle memory of doing everything right and you just do it over and over again. That’s not anything like random. It’s variance in a precise learned behavior. I suppose they are going to conclude next that practice offers no benefits for motor tasks?

    I’m amazed anyone took that study seriously to begin with. Psychology’s got a lot of stupid to deal with! Get replicating!

  5. Rob Grigjanis says

    Also, what Marcus said. I experienced the same sort of thing playing soccer.

  6. Andrew G. says

    Rob G @ 3: every flip is independent, but when you take a finite sequence of flips, and sample only those flips which follow a “streak” of matching results, the independence is lost. The effect is larger for shorter sequences and longer streaks; simple simulations give me about a 54% chance of a tail after three heads when using sequences of length 100 for example, but at length 1000 this is down to 50.4%.

  7. says

    Ps -- anything where you are aiming or doing some kind of physics estimates -- you’re not measuring independent results, you’re measuring a succession of refinements.

    There was one time I was shooting a friend’s rifle in 5-shoot groups at targets at 300yd. Each target showed a pattern: the first shot was slightly off (high and left if I recall) then almost on, and then the rest were all 3 in a single hole. Simple answer: my friend’s scope was off so I had to estimate and compensate and we were alternating targets 5 minutes apart and I was not preserving the estimate. “Hot hand”? Hardly. It’s just “practice” and getting dialed in. Each shot is a measurement in a sequence -- it’s not a coinflip where each result has no bearing on the one before or the next.

    Anyone who’s done anything involving aiming has sometime or other “walked it on” or successively corrected. I despise psychologists when they publish big conclusions about stuff they clearly know nothing about. They just embarrass their field.

    Tl;dr: those practice free throws: next you’re gonna say they don’t help? It rather depends on the ball player.

  8. Rob Grigjanis says

    Andrew @6: I made a dodgy assumption; that if you find a streak of three heads at N, N+1,N+2, you store the value of N+3 and then proceed from N+4. I see that’s not what the authors were doing. They’d store the value of N+3 and proceed from N+3. So, if N+3 were heads, the value of N+4 would also be stored, whereas it couldn’t have been with my assumption.

    I stand corrected. Thanks.

  9. Rob Grigjanis says

    Marcus @4:

    They are comparing performance on a neuromuscular task to coin flips?

    No, they’re showing that just because you find 50-50 doesn’t mean there’s no real effect. In particular, if randomness actually predicted 54-46, a 50-50 result says there’s some non-random stuff going on.

  10. Andrew G. says

    Of course, that does NOT mean that when observing a sequence of coin flips, you should bet against a head after seeing three heads—that probability is still 50%.

    @ Rob G: I’ve been assuming that given a sequence like TTTTTFFTTTF the bolded elements are the ones selected.

  11. Brian English says

    For some reason this reminds me of the card trick you posted about not so long ago Mano.

  12. Eric Weatherby says

    I absolutely agree that there’s no “hot hand” in gambling (at least not if the game has been set up to be properly random) . . . but as Marcus Ranum observes, human beings DO experience periods of better or worse than average performance. The idea that athletes (human beings, just as subject as the rest of us to the thousand natural shocks that mortal flesh is heir to) MUST perform at the exact same level at ALL times is just ludicrous and I can’t imagine how the authors of the studies denying hot hands and winning streaks could have missed that.

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