Finding the hot (and cold) hand at a local gym
Sorry about all the sports talk lately, but the NBA playoffs are afoot and I have basketball fever. I’ll get back to physics soon, I promise.
There’s been a lot of talk lately in the “geeky basketball fans” community about whether the “hot hand” exists in basketball. If a player has been making most of his recent shots, is he more likely to make the next ones? Or is it safe to assume that his recent success is just a matter of statistical fluctuation? In other words, should players/coaches try to “feed the guy with the hot hand”, or is it always better to go with the player with a higher career shooting percentage?
I, personally, believe in the existence of the hot hand. The main reason is this: I once made 25 consecutive free throws. At the time I was only about a 50 percent shooter (I’m only marginally better today), and I had just finished a rough game of church-league basketball. Before going home I decided to take fifty free throws, and somewhere in the middle I made 25 in a row. If you think that those 50 shots were all statistically independent of each other, then the odds of me, a 50% shooter, making 25 consecutive free throws somewhere in the middle of a set of 50 are literally one in a million (about 0.75 in a million, actually).
The hot hand, in my experience, is better described as a “hot brain”. When I’m “hot”, I feel like making a shot is just a matter of concentration. It’s like I remember, perfectly, the sequence of muscle firings required to make a shot. So if I just concentrate hard enough on the shot, I’ll make it. Conversely, when I’m “cold”, it’s like I can’t remember how to shoot correctly anymore. Some shots go in, but I can tell that it was luck. My mind does not have the right memory loaded up.
So I believe in the hot hand. And if it can happen for me, I was fairly confident that it must happen for professional basketball players as well. I was pretty sure that I had seen it, too. After all, I was watching when Tracy McGrady scored 13 points in 35 seconds. In the interview afterwards, he said “the basket felt as big as the ocean to me.”
This weekend, I searched for the hot hand in two places. The first was in shot-by-shot data from the NBA. The second was at the local gym, where my wife and I took two hundred shots and recorded the sequence of outcomes in an attempt to create that statistically improbable streak that would provide evidence for the hot hand.
Ryan at basketballgeek.com has compiled wonderful data sets for every play from every game in the last three NBA seasons. I decided to test the data from last season, 2007-2008, in a few different ways. The first was to define a correlation function, the standard technique in physics, of a given player’s shots. What I hoped to find was some player for whom a shot was influenced by the one that preceded it. In other words, I wanted a player whose misses and makes tended to group together in a way that was statistically unlikely. I most certainly did not find this. Every player I could think to check (Kobe Bryant, Ray Allen, Dwight Howard, Ben Gordon, Kyle Korver, Dwyane Wade, and a few more ) showed absolutely no significant correlation between the outcome of one shot and the outcome of the subsequent ones. Previous studies have shown that there is a slight negative correlation between a shot and the one that immediately follows it, which has been interpreted as an “overconfidence” problem. i.e. when a player makes a shot he is more likely to take a bad shot the next time he has the chance.
My second method was to look for correlations between a player’s shooting percentage at the beginning of the game and their shooting percentage at the end of the game. For example: if Kobe Bryant opens the game shooting 8/10, how many of the next 10 shots is he likely to make? Should you expect him to shoot close to 80%, or close to his season average of 45.6%? The answer, resoundingly, is that his season field goal percentage is a much better predictor than his shooting percentage in the first 10 shots. But is there any correlation at all between early-game and late-game shooting percentage? Maybe a small one… that depends on whether you trust that these data points really do have an upward slope:
Finally, I searched through data from dozens of shooters for streaks of consecutive shots in order to find a highly improbable sequence. Was there some equivalent among the NBA ranks of my 25-in-a-row performance? The answer, surprisingly, was no. Of course there were some unlikely sequences, but none so unlikely that you shouldn’t have expected them to happen over the course of a season. For example, Kobe had a streak of 7 consecutive makes on Nov. 4, 2007 against the Utah Jazz. The odds that any seven of Kobe’s shots would all go in is about 0.4%, and I’m sure that any observer in the building would have said that Kobe had the hot hand at the time. But Kobe Bryant took 1,602 shots that season. The odds that there would be a streak of 7 consecutive makes somewhere during the season is almost 70%. So the fact that Kobe had this streak of 7 is not at all inconsistent with the hypothesis that every shot is uncorrelated from the ones before it. Every NBA shooter that I examined gave me similar results. Sure, there were some improbable streaks, like Ben Gordon making 8 in a row, but nothing so unlikely as to be inconsistent with random variation.
So, in the end, I find no real evidence for the hot hand in the NBA. That doesn’t mean it doesn’t exist, just that people’s shot patterns would look the same if it didn’t. The results are perhaps best summarized by a quote I found in this marvelous little document on sports streaks and the stock market: “Long streaks are, and must be, a matter of extraordinary luck imposed on great skill.” It’s well worth reading. Apparently a Nobel laureate in physics once examined data in baseball and found that the only streak which could not be explained by random variation was Joe DiMaggio’s “one in a million” 56-game hitting streak.
With my NBA analysis rebuked, I went to the gym with my wife to take some shots and collect some data. The goal was to do what no NBA player had been able to do: provide irrefutable evidence for the hot hand. We decided to take 200 shots each: 100 free throws and 100 randomly-distributed shots from about 15-foot range. Each time we had one person shooting and the other rebounding/recording the results. For the randomly-distributed shots, the rebounder also “randomly” chose the location of the next shot. We were careful to spend a roughly equal amount of time on each shot. These came fairly rapid-fire, with no breaks in between. The results are as follows, where 1 represents a make and 0 a miss:
I’ll save you from counting and tell you that my wife beat me 65 – 61 in free throws and 57 – 54 in the random shots. Highlighted in yellow are three statistically unlikely streaks. The first is a streak of 13 consecutive during my free throw shooting. And I have to say, I definitely felt “hot” during that streak, like my mind and body were on the same page. Statistically, though, the odds of a 61% shooter making at least 13 consecutive somewhere during a 100-shot sequence is 23%. So a statistician might say that my streak of 13 was unlikely, but hardly irrefutable evidence. Similarly, my streak of 11 makes during the random shots had a probability of about 19.7% of happening, given my 54% rate of making those shots. Taken together, you might conclude that I’m a little bit of a streaky shooter. This data is hardly going to rock the basketball world, though.
The really strong evidence is in Mrs. Gravity and Levity’s streak of 10 misses during the random shots. She shot 57% in that set, so the probability of her missing 10 straight was very small. In fact, the probability of a 57% shooter missing 10 consecutive somewhere during a sequence of 100 is only 5.8%. So for her to miss 10 in a row is pretty strong evidence of something going wrong — she had a “cold hand” during that stretch.
So what do we take away from this? The shooting patterns of my wife and I show some evidence for the hot hand, and stronger evidence for the “cold hand”. But why didn’t this show up in the NBA data?
My conclusion is that really good shooting is a matter of concentration. When you’re shooting free throws alone in a gym, it’s relatively easy to get into “the zone”, where your concentration level is exactly where it needs to be to make long streaks of shots. But in a real game, where defenses are doing everything they can to break your concentration, “the zone” is a much harder place to reach and maintain. I will probably continue to believe that Kobe Bryant “had the hot hand” last night when he torched my beloved Utah Jazz with ridiculous, hand-in-the-face, turnaround jumpshots, because being a basketball fan is largely about belief. But now I will do so with the awareness that his shot patterns would look the same even if there were no such thing.