It is officially winter in New England. A season that has me braving frigid temperatures and dreaming of hot streaks in basketball. When I’m on the court, nothing beats the feeling of heating up, the feeling that I can’t miss. On those days, that’s all the warmth I need. After a recent game, I was sitting with a teammate discussing that feeling and a thought came up: Is it real? We both paused a moment before shrugging and moving on.
And yet, I find myself wondering about it again. Are streaky shooters a thing?
In 1985, psychologists Thomas Gilovich of Cornell and Robert Vallone & Amos Tversky of Stanford published a study called “The Hot Hand in Basketball: On the Misperception of Random Sequences” about exactly that topic. I’ll spoil the ending: the study concluded that, in their research sample, there was no statistical basis for a hot streak and that any series of makes or misses generally fell within the expected range of outcomes as with any issue of probability.
I read the whole paper. And I disagree.
I’m not the only person who has looked into this. Groups have looked at the potential biases within the original study. The gist of their argument is that the original study has flaws in how it picked when to start counting, which affected the data. I read that too. It made my head hurt analyzing streams of make make make miss vs make make miss make vs miss miss miss make make make. I want to acknowledge them, but I cannot spend the word count trying to understand it. Thus, I will offer my own refutation along with the statistical data.
First and foremost, as a young guard who was forced to play point at times, I was raised on the idea that it is my job to get the other players involved. Basketball was played a different way 20 years ago. Non-dynamic non-scorers like me had to do something, so in my early teens – when I started playing with the older kids – I totally bought into the idea. If one of my teammates had hit their last three shots, I had a responsibility to get them the ball so they could go hit another one.
The flip side was true too. If one of my teammates missed a few shots, we would try to get him a nice and easy one to get him going, as he would be more likely to make the next one.
The Houston Rockets missed 27 straight three-pointers in a playoff game in 2018. 27! I’m going to say that that defies any and all probability models. I imagine that the folks behind the 1985 study would agree. The season this study was conducted, the Rockets averaged 2.3 three-point attempts per game! Statistics aside, I think there are mountains of anecdotal evidence in support of streak shooting.
“Sequences of hits and misses in a basketball game offer an interesting context for investigating the perception of randomness outside the psychological laboratory. Consider a professional basketball player who makes 50% of his shots. This player will occasionally hit four or more shots in a row. Such runs can be properly called streak shooting, however, only if their length or frequency exceeds what is expected on the basis of chance alone. The player’s performance, then, can be compared to a sequence of hits and misses generated by tossing a coin. A player who produces longer sequences of hits than those produced by tossing a coin can be said to have a “hot hand” or be described as a “streak shooter.” Similarly, these terms can be applied to a player who has a better chance of hitting a basket after one or more successful shots than after one or more misses.”
The study examined the performance of actual NBA players to advance the findings. I will let them explain their intentions:
“Field goal records of individual players were obtained for 48 home games of the Philadelphia 76ers and their opponents during the 1980-1981 season. …Our analysis of these data divides into three parts. First we examine the probability of a hit conditioned on players’ recent histories of hits and misses, second we investigate the frequency of different sequences of hits and misses in players’ shooting records, and third we analyze the stability of players’ performance records across games.”
The authors explain in great detail that, scientifically, there are expected sequences of chance. The overall takeaway from the study is that despite a couple of outliers, the data overwhelmingly supported the idea that within streaks, within individual games, and within the course of several games, the probability of shooting makes was not affected by the previous shots. In the larger sample size, the amount of “hot” and “cold” games as a whole were also within the expected ratio, as dictated by the scientific understanding of chance. This is the densest paragraph I have ever written.
Essentially, it boils down to our understanding of what “chance” actually means. The study relays all the various numbers and percentages based on flipping a coin, which they then compare to shooting – with a caveat about the difference between flipping a coin and shooting a basketball. Still, the laws of chance remain. Those series of makes or misses, what they call runs, are to be expected based on the mathematical understanding of chance and probability.
In addition to studying the shooting statistics data, they presented sequences of makes and misses to both players and observers. When the subjects were presented with the sequences, they labeled these chance outcomes as predictable because of the shooter’s previous outcomes. The authors concluded that observers view positive correlations as streak shooting no matter the data presented. In fact, they saw that the subjects would see negative sequences as purely random because of biased retrieval and selective coding. That’s a lot of science-speak for “hoopers are stubborn.” Here’s an excerpt of some of the numbers they got from basketball fans:
“Their responses revealed considerable agreement: 91% of the fans believed that a player has “a better chance of making a shot after having just made his last two or three shots than he does after having just missed his last two or three shots”; …84% of the fans believed that “it is important to pass the ball to someone who has just made several (two, three, or four) shots in a row.” The belief in a positive dependence between successive shots was reflected in numerical estimates as well. The fans were asked to consider a hypothetical player who shoots 50% from the field. Their average estimate of his field goal percentage was 61% “after having just made a shot,” and 42% “after having just missed a shot.” …Thus, our survey revealed that basketball fans believe in “streak shooting.”
That data remains consistent almost 40 years later. The more I play basketball, the more opportunities I have to witness moments that confirm my own already-held biases. Gilovich, Vallone, and Tversky point out in their conclusion that they are not stating that basketball is a game of chance, rather than a game of skill that nerds like them could never comprehend. Okay, maybe I added that last part. They do, however, state that the outcome of a shot is affected by many factors, like position on the floor, the level of defense, and the actual skill of the shooter. The one factor that does not influence the outcome, in their findings, is the outcome of the previous shots. And yet, we believe.
A part of me does find this intriguing, in the way that it governs the conduct of the game. The idea that a team needs to get the ball to the hot hand is perhaps a statistical and probability-based fallacy. The study does a good job of trying to allow for all the variables of in-game shooting. I particularly appreciate their acknowledgment that after I have hit a few shots in a row, I will have the confidence to take a shot that I probably shouldn’t, resulting in a miss on my stat sheet. I do think that there is an expansion of the hot hand idea that uses this data as an effective coaching strategy. Teams need to get the best shot possible on every possession. It is a simple idea. The player may still miss it, but if enough of the variables are eliminated and it truly doesn’t matter if that individual has hit the previous shot, then this is the strategy that should rule all. Drawing up plays to get a specific player involved in the offense only works if the play is creating the best shot.
In a recent game against the Denver Nuggets, the Boston Celtics were zipping the ball all around the court, constantly finding the most open shot, regardless of who took it. Part of the benefit of their team construction is that they have many players who can shoot well. Therefore, when the ball is flying from player to player, the defense scrambles, and eventually there is someone open. The Denver Nuggets stayed in the game, and eventually won it, by doing the opposite strategy. They had two players who scored over ⅔ of their points.
They kept giving the ball to Nikola Jokić and Jamal Murray, not because they had hit the last one, but because they were generating the highest probability of a made shot. Jokić was eating the Celtics alive in the post. Murray was losing defenders and getting open looks. I will have to do an entirely separate piece on the biomechanics of shooting, but in this case, Jokić and Murray were removing most of the other variables involved in the chance of shooting. Two different approaches by the two different teams, yet the result was largely the same. When the players’ confidence grew, yes, they were hitting hard shots. Were they hot? Maybe the paper and I are both saying the same thing: Yes, BUT within statistical expectations.
There is a reason this study was conducted by psychologists and not mathematicians. The part that fascinates me most is that I will continue to believe in the hot hand because of the innate feeling that the rim feels two sizes bigger when I’m on. Conversely, I know I will hang my head after I start a game 0 for 3, and I will hesitate on my next shot, trying to find a teammate who will have a better chance of making it.
Perhaps I will be able to use this information as fuel as I’m trying to warm up before the next game. More likely, I will continue to imagine the NBA Jam flaming basketball in my hands as I slide on the icy path from my car to the gym.