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“Heating Up in NBA Free Throw Shooting”

Paul Pudaite writes:

I demonstrate that repetition heats players up, while interruption cools players down in NBA free throw shooting. My analysis also suggests that fatigue and stress come into play. If, as seems likely, all four of these effects have comparable impact on field goal shooting, they would justify strategic choices throughout a basketball game that take into account the hot hand. More generally my analysis motivates approaching causal investigation of the variation in the quality of all types of human performance by seeking to operationalize and measure these effects. Viewing the hot hand as a dynamic, causal process motivates an alternative application of the concept of the hot hand: instead of trying to detect which player happens to be hot at the moment, promote that which heats up you and your allies.

Pudaite says his paper is related to this post (and also, of course, this).


  1. Dave Meyer says:

    If you’re dead, you can’t shoot free throws very well; percentage is very low. Ergo, definitely helps to warm up at least a bit; just be sure to keep body temp below 101F…

  2. Anoneuoid says:

    Has anyone managed to construct a machine that makes 100% of free throws?

    • What’s the time-scale?

      From Churchland, Afshar and Shenoy (2006):

      In 1990, Larry Bird of America’s Boston Celtics basketball team made 71 consecutive free throws, or foul shots, across almost two month’s worth of games. While this is a remarkable feat, one cannot help but wonder: why did he miss the 72nd?

    • Toequemada in Training says:

      Junior year of high school (previous century) I and my gang of fellow delinquents thought it would be cool to build a catapult as an extracurricular project. We had visions of hurling pumpkins into the clouds, but the administration (back then, The Man) let us know that launching anything bigger than a ping pong ball would have serious repercussions on our rear ends (so you know this was long, long ago). But you can’t cultivate interesting amounts of mischief without ingenuity so we countered with a launcher of basketballs, said project woven with appropriate buzzwords from athletics and physics so as to establish sober intent. They bought it, so we slapped together something resembling (if memory serves) some sort of catapult/ballista hybrid (research, we don’t need no stinkin’ research). Coach was curious enough to supply us with the balls and a freshman retriever. It took us two hours of tinkering and fiddling and positioning, but then, from the foul line, we dropped something like fifty in a row. Then we called it a day for two reasons: 1. To get those consecutive fifty we threw over 200 times. It got boring as hell. And 2. setting up the launcher over and over turned out to be hard work. The whole thing excessively challenged both our attention spans and our “work ethic”, and if we were never going to get to fling a VW Beetle over the administration building, then to heck with the whole thing. It never occurred to anybody that this might be pioneering or record setting, and I had forgotten all about it until now.

  3. Interesting approach. Quick comments here.

    I see Paul Pudaite is aware of Jeremy Arkes’ free throw paper, that is good. There are some omissions though.

    The Pudaite says:

    Hot hand research typically employs some type of statistic that is conditioned by past outcomes. For pairs of free throws, the condition is limited to the binary outcome of the 1st shot. Gelman (2015) shows that even for a nontrivial effect size, conditioning on a single shot has a very large variance in comparison because past outcomes only weakly identify the shooter’s state.

    Fortuitously, free throw shooting exhibits a large causal effect that does not depend on past outcomes: the act of shooting the first free throw, regardless of whether hit or missed causes a typical player to hit 5‐ to 6‐percentage‐points higher on his 2nd shot than on his 1st. This effect is large enough (about twice the size of the conditional effect Arkes estimated) to emerge clearly in the following analysis performed on GVT’s relatively small sample of free throw data

    1. On the weak identification point: In the 2014 version of this paper ] this point is made (appendix); weak identification was first discussed on this blog in 2014, point #3 here

    2. On players shooting better on the second shot: Bob Wardrop’s 1995 study of free throw data finds evidence of this “heating up” or calibration. Here is the study on Fermat’s library. This was replicated in the by Yaari & Eisenmann in this 2011 paper, whom also replicated Arkes’ work. In Bob Wardrop’s paper he has a nice conjecture how to justify people’s hot hand beliefs: When the average is fan is asked if a player is more likely to hit shot after hitting a few shots in a row, the fan says yes, and is giving a reasonable answer given the way the question was asked. If one were to pool together all the instances in which a shooter hit a few shots in a row, this is a true statement due to the aggregation bias commonly known as Simpson’s paradox. I fan watching the game, might feel the same. [of course this doesn’t explain why fans believe certain players are streaky, but that can be justified in this paper ]

    3. There is more evidence on heating up. Across all controlled and semi-controlled studies, we find players shoot worse in their first few shots, even after they have an opportunity to warm-up, this includes the NBA’s Three Point Contest.

    4. On free throws there is more somewhat related work. For example players shoot better in the second half of the season this paper (page 10). Goldman and Rao find a home vs. away effect in this paper.

    • anon says:


      I asked the foll. in comments to an older post, but I don’t think you saw it…

      Where exactly in your MS paper are the results for computing the bias when k>1?
      I gather there’s no neat formula except when k=1, but you can easily calculate
      it for any specific k,n,p. Is that right?

      • anon:

        yeah, didn’t see it. Too bad this wordpress doesn’t send notifications.

        In this paper Appendix E (p. 47) we outline the approach. The basic idea is in the first paragraph, and the gory combinatorics follow that (though Lemma 2 is pretty cool and simple).

        To answer your question, yes, for k>1 there is a formula, but it’s complicated and slow to calculate when n gets large. We use the formula in the paper, happy to share the matlab code if you email me privately. It’s been cross-validated with simulations for n=100, and both have been cross-validated with direct enumeration for n=10.

      • anon says:

        Thanks, Joshua!

        I’ll add that to my reading todo list and perhaps email you in the (maybe far) future.

  4. Jordan Anaya says:

    Riddle me this. If the hot hand doesn’t exist, why is there something called a heat check?

    • Whelp, apparently practitioners must be biased.

      From How We Know What Isn’t So:

      The reaction of the professional basketball world to our research on the hot hand is instructive in this regard. Do those close to the game give up their belief in the hot hand when confronted with the relevant data? Hardly. Red Auerbach, the brains behind what is arguably the most successful franchise in American sports history, the Boston Celtics, had this to say upon hearing about our results: “Who is this guy? So he makes a study. I couldn’t care less.” Another prominent coach, Bobby Knight of the 1987 NCAA champion Indiana Hoosiers, responded by saying “… there are so many variables involved in shooting the basketball that a paper like this really doesn’t mean anything.” These comments are not terribly surprising. Because a truly random arrangement of hits and misses contains a number of streaks of various lengths, the belief in the hot hand should be held most strongly by those closest to the game. Furthermore, simply hearing that the hot hand does not exist, or merely taking another look at the game is not sufficient to disabuse oneself of this belief. It is only through the kind of objective assessment we performed that the illusion can be overcome.

      More specifically, on free throws, here is a doozy from 2 weeks ago on Barry Ritholz’s podcast

      RITHOLTZ: So some of the pushback has been you know player gets hot and the defense collapses on them and they have less good looks at the basket and so naturally after a certain streak, they are going to start missing but you don’t have that same defensive pressure with foul shooting, do you?

      GILOVICH: That is right.

      RITHOLTZ: And what does the data show with that?

      GILOVICH: The data show that the outcome of the second free throws completely independent of the outcome of the first.

      RITHOLTZ: Totally so if you see hit or miss the first one, the outcome of the second one is statistically no different?

      GILOVICH: Right. And a lot of basketball players and fans will say that I’m not so impressed by that because you can’t really be streaky when it comes to free throws how they say that after the fact but —

      RITHOLTZ: I mean you are standing there nobody’s guarding you, there are certain, think of Reggie Miller of the Indiana Pacers, used to shoot like 93 percent free throws, he was an outstanding free-throw shooter, why would people not assume that if there is a streak when you’re on the — on the court why would you not assume there is a streak at the foul line?

      GILOVICH: I think they say that after having seen the data and they want to maintain that belief, but we went a step further and we conducted and other people have done this too — where you have people shoot in the gym for you and the feeling for all of us who have played basketball is — you can feel it in warm up sometimes, it doesn’t have to be in the heat of the game that you can feel hot and we have people take a series of shots along an arc, an equidistance from the basket, and before each shot they place a bet on themselves, take a risky bet if they are feeling hot or more conservative that if they are not feeling so hot.

      And turns out we can predict the best they are going to make — that is if they’ve made several shots in a row, that is a very strong predictor of whether they’re going to choose the risky bet or not; however, the bets that they choose are not very good predictors of what’s going to happen next, again, this three link chain, no problem of during the first and the second.

      How you’ve done influences how you feel. But surprisingly, how you feel has either no or very little impact on your likelihood of making the next shot. And again, I want to stress, it has to do with your ability to as they now say score the basketball to get the ball in the cylinder, it may affect you in other ways, it may make you a better defender, better passer or whatever.

      • Anoneuoid says:

        Because a truly random arrangement of hits and misses contains a number of streaks of various lengths, the belief in the hot hand should be held most strongly by those closest to the game.

        Premise: a truly random arrangement of hits and misses contains a number of streaks of various lengths
        Conclusion: belief in the hot hand should be held most strongly by those closest to the game

        I don’t see why???

        • I guess he means that the practitioner is: (i) pre-disposed to believe in the hot hand, and (ii) paying attention to everything. If this is true then the practitioner will be around when all the streaks happen, and attribute them to the hot hand. A casual observer may miss those rare occasions when the streaks happen. By analogy, if I watch 100 coin flips, it’s unlikely I’ll witness a long streak, but if I watch 10,000 flips, it’s a lot more likely. Those rare events become a part of basketball lore, and contribute to the myth. That’s the story anyway, and it makes sense if you buy the coin flip analogy.

          • Allan C says:

            The literature and debate surrounding the hot-hand always has been interesting to me. As a former Junior/University hockey goaltender you would be unable to convince me that confidence doesn’t affect performance; and surely my good/bad play and wins/losses (not necessarily the same thing) both had a significant impact on my confidence. Consequently, I could not have been and can never be convinced a hot-hand effect is not real in sport. It just doesn’t compute at any fundamental level; my prior is strictly 0 for the hot hand not being a real phenomenon. What I always took the original result to mean is that there is probably less of an effect than that perceived by the player because of their closeness to the streaks you describe (slightly different setup in hockey but surely comparable).

            • Allan:

              Your take on the original result was not how the authors took it, or how it was taken by the subsequent literature and popular culture.

              Your take seems reasonable based on casual observation, but even this take is not supported by the evidence the original study. In fact, there is no way to know whether people exaggerate the hot hand or not. The lay and professional belief in the hot hand appears to correspond to the subjective probability that a player will make a shot conditional on the player being “hot”/”in rhythm”/”in the zone”; it is impossible to test whether this belief is exaggerated or not due to a form of measurement error we first discussed in this email to Andrew Gelman in 2014, and elaborated upon in our “Surprised” paper in the Appendix titled “Appendix: Size of the bias when the DGP is hot hand/streak shooting”.

              The most one can potentially show is that basketball practitioners exaggerate the diagnostic value of streaks; this is likely true as they sort of admit this themselves, without any need of academic researcher telling them that they are doing it wrong, and that they suffer from a cognitive illusion.

      • Andrew says:


        Ironic that a book called “How We Know What Isn’t So” has something the author thinks he and his colleagues know, but actually isn’t so.

    • MJT says:

      i think applied bayesians call it ‘burn-in’ or the hipper… ‘warm up phase’

  5. Josh Chang says:

    In some (more limited) analysis on a player-by-player basis I also found similar effects – in particular it appears that many players perform markedly worse on their first free throw: Some reader(s) of this blog probably reviewed my manuscript at some point so thanks if you did.

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