# Basketball Players and The Hot Hand

In honor of the start of basketball season, and to commemorate the surprising victory of my Kobe-less Lakers over the Suns, I thought I’d discuss my favorite science paper on basketball. (I did a similar thing to celebrate the beginning of the football season.)

The paper is by Amos Tversky (of KahnemanandTversky fame) and Thomas Gilovich. In 1985, Tversky and Gilovich analyzed “the hot hand” among NBA players. As every basketball fan knows, players are streaky: one minute they are in the zone and can’t miss a shot, and the next minute every field goal attempt is ricocheting off the rim. But is “the hot hand” a real phenomenon, or is it just another cognitive illusion, an attempt by the brain to detect patterns inside a random process? (Kahneman and Tversky had shown a decade before that people indulge in the error of “representativeness,” which is when “people regularly exaggerate the degree to which the probability distribution in a small group will closely resemble the probability distribution in the overall population.” Think of the coin which has landed on heads a few times in a row. Most people assume that the coin is now more likely to land on tails, since “the law of averages” has to correct the imbalance. Of course, this is a false intuition. Random coin flips are statistically independent.)

After interviewing the Philadelphia 76ers, Tversky and his co-authors realized that basketball players certainly took “the hot hand” very seriously. If a teammate had made several shots in a row, they were more likely to get passed the ball. But Tversky wasn’t convinced: he thought “the hot hand” was the sporting equivalent of a coin that has landed on heads a few times in a row. In other words, it’s just random luck: players in the zone aren’t better shooters than players who aren’t in the zone. So Tversky and Gilovich began data mining, sieving through years of 76er statistics. They found absolutely no evidence of “the hot hand” phenomenon: a player’s chance of making a shot were not affected by whether or not their previous shots had gone in. Each field goal attempt was its own random event.

The 76ers were shocked by the evidence. Andrew Toney, the shooting guard, was particularly hard to convince: he was sure that he was a streaky shooter, and went through distinct “hot” and “cold” periods. But the numbers told a different story. During the regular season, Tooney made 46 percent of all of his shots. After hitting three shots in a row–a sure sign that he was now “in the zone”–Tooney’s field goal percentage dropped to 34 percent. (Ironically, when Tooney thought he was “hot” he was actually more likely to miss a shot.) The opposite phenomenon was also true: after missing three shots in a row, Tooney made 52 percent of his shots, which was significantly higher than his normal average. (Julius Erving was the only 76er player to not suffer from this inverse streakiness. When he was on a hot or cold streak, his field goal percentage remained constant.)

But maybe the 76ers were a statistical outlier. After all, 91 percent of serious NBA fans believed in “the hot hand”. Gilovich and Tversky next turned to the Boston Celtics, who had just won a world championship. But when Tversky and Gilovich analyzed the Celtics’ free throws, they found “no evidence that the outcome of free throws is affected by previous free throw attempts.” In fact, Larry Bird was like Andrew Tooney: after making several consecutive free throws, his free throw percentage actually declined.

So why do we believe in streaky shooters? According to Kahneman and Tversky, “the hot hand” is just an illusion, the standard laws of chance viewed through the prism of human cognition. That said, when Kobe is in the zone, I still want him to have the ball. Some illusions are hard to give up.

1. #1 andrew
November 1, 2006

Great post. Really interesting. But if streaks aren’t real, then why did players get worse after making three shots in a row? did they just start taking more foolish shots?

2. #2 MarkP
November 1, 2006

That seems most likely Andrew. Think about it: When are you more likely to take a 30 foot shot, when you’ve hit ten a row, or missed ten in a row? It also makes sense that you would do better after missing a few, since that would likely prompt greater concentration and/or selectivity.

But never fear, sportscasters will keep telling us that there is a zone, that people are streaky, that numbers can’t measure heart, and other ignorant nonsense usually prefaced with “I don’t know anything about math, but…”.

3. #3 Karl
November 1, 2006

I’m sorry but there is a big difference between calculating the odds of a coin coming up heads and calculating the odds of a basketball player making a shot.
The difference is inert vs living. People have emotions. People have days when they haven’t had enough sleep, or too much sugar or carb. People react to stress by producing adrenelin which affects coordination and concentration.
If you have ever played a sport under competitive conditions or even ever listened to interviews with pro, or college, athletes, you know that “nerves” are a big factor. So, an athlete, during the course of a game can go up and down in terms of energy, concentration, coordination and confidence. That is why some people are the ones that can make the crucial shot while others can’t. And that is also why athletes will get “in the zone” sometimes. It is more than statistical. Pennies do not have phyiological or emotional responses.
Sorry, it is a real phenomenon.

4. #4 Caledonian
November 2, 2006

5. #5 Jonah
November 2, 2006

Thanks for your comment Karl. I know this is a deeply counterintuitive study, and it’s possible that basketball players have gotten streaky since 1985. It’s also important to note that Tversky never denied the importance of emotions. In fact, he and Kahneman continually stressed the large role emotions play during decision making. But Tversky’s point in this study was simple: being “in the zone” doesn’t make you a better shooter. You might feel like a better shooter, you might be convinced that you are suddenly able to make more shots, but this is just a cognitive illusion. It’s like the gambler who’s convinced that he’s on a hot streak. (Tversky and Gilovich didn’t rule out that oppposite claim, which is that players who feel “off” or “cold” might make fewer shots.)

6. #6 Caledonian
November 2, 2006

(Tversky and Gilovich didn’t rule out that oppposite claim, which is that players who feel “off” or “cold” might make fewer shots.)

Indeed not. But the study did show that overall each shot is an independent event. Given that evidence, Karl’s assertion that streaks are real phenomena requires some serious contrary evidence to be credible.

7. #7 Karl
November 2, 2006

Caledonian:
I assume you mean that I need evidence to the contrary of the statement that each shot is an independent event.
Short of attaching an eeg to a player that would be hard to do. But, there is anecdotal evidence from the testimony of many (hundreds of) players in all sports that the mind can get in the way of physical performance. The ability to get the mind out of the loop when performing a physical task is extremely important, especially after a missed attempt. Many players have said that it is important to forget the last bad shot, not to let it affect the next shot. And for each shot to NOT THINK but just let the body do what it has been trained to do. That is the difference between coins and people.
There is a great book on this topic: The Inner Game of Tennis” by W. Timothy Gallwey.

8. #8 Caledonian
November 3, 2006

You don’t seem to grasp the point, Karl. A statistical analysis demonstrated that players do not act in the way your assertion implies that they do. The analysis wouldn’t reveal what would be responsible for causing the throws to be linked if they were – it would merely show that they were. They weren’t.

Mechanisms are irrelevant – it doesn’t matter that “people aren’t coins”. The results speak for themselves. Some people are hearing impaired, it seems.

9. #9 Alvaro
November 6, 2006

Hi Jonah,

Regards

10. #10 Dave
December 6, 2009

Probably nobody will see this post since it’s so long after the original blog, but I’m really getting annoyed by these types of sports analyzes, which are poorly thought out and usually invalid. This is similar to the invalid statistical analyzes done to prove that there is no such thing as clutch hitting or clutch hitters. Having played basketball myself I can tell you that there is such a thing as “being in the zone”. My guess is that it relates to some sort of brain activity that would be difficult to measure, but that could be measured if anybody decided to spend the time and effort to do it. Since there is ample scientific evidence that a persons “state of mind” can have a profound effect on performance, and since players are stating that they are in fact in a high performance state of mind when they are “in the zone”, it’s up to the researchers to prove the players wrong, and the Tversky study doesn’t to it. To do it:
1) there would have to be some other way of determining when a player was “in the zone” than starting after they hit 3 shots in a row(actually 1, 2, or 3 in the study), unless they are going to include the already made shots in the percentage calculations(which the study doesn’t do). I did a quick and dirty adjustment of their statistical analysis and found out that if you do that there is a pronounced statistically significant “in the zone” effect – particularly after 2 shots have been made,
2) there would have to be some way of controlling for changes in defense of the opposing team,
3) there would have to be some way of adjusting for the fact that a player generally has a much easier shot after they get their own rebound from a missed shot than other types of shots,
4) there would have to be some way of adjusting for the types of shot taken when the player is “in the zone” – a team to be successful needs to be taking and making a mix of shots and it’s likely that the player with the “hot hand” would be taking a higher percentage of more difficult shots.

I hope Tversky isn’t using this type of statistical analysis in his other types of research. Isn’t he a “behavioral economist”? Maybe that’s why the economy is in such bad shape

11. #11 Sharon
December 12, 2009

Amos Tversky died June 1996 at age 59 from melanoma. There is a nice obituary written by the NY Times that is still available online.

12. #12 Ash
January 17, 2011

Tversky and Gilovich are wrong and here’s why; unlike flipping a coin, the success of making a shot in basketball does not depend on random chance. It depends on, among other things, muscle memory and good shooting habits. Even NBA players fall into bad habits when it comes to shooting a basketball (ie., they may not follow through on their shot, keep their elbow in etc.) This can be the case even game to game or week to week. A little adjustment in shooting mechanics, consciously or not, will affect a player’s success shooting the ball. Also when you have immediately made a shot, the reason it is often easier to make another shot soon after is that it is easier to shoot the ball the same way (muscle memory). The longer time you have in between shots the greater chance you have of “forgetting” how you shot the ball. There is also other things like bad defence to consider. If you make a shot chances are that the player guarding you wasn’t doing their job properly. It might be a good idea to attack that same defender again. Also there’s simple probability to consider as well. For example, I make approximately 60% of my free throws. One time when I was shooting free throws I made 20 in a row. If what Tversky and Gilovich claim is true, this kind of success would be next to impossible because there was approximately a .0036562 % chance of this happening.