On April 2, Golden State’s season reached its nadir with a 53-point loss to the Raptors in Tampa. Stephen Curry, who had recently returned from a five-game absence, sat out the second game of a back-to-back, Draymond Green was out with a finger injury, and it was the kind of night where the dynasty seemed like it was two decades ago rather than two years ago.
Golden State was 23-26 after that loss, just a game ahead of Sacramento for 10th in the Western Conference. Sure, Klay Thompson has been out all year, but the three-time champs going out like this? Unfathomable.
Then Curry got back in the lineup, and he took over. For the next 21 games, the two-time MVP averaged 36.7 points, shooting 49.7 percent from the field and 43.8 percent on three-pointers. Golden State went 14-7 over that stretch to book a place in the Western Conference Play-in tournament, which starts Wednesday.
Curry has been carrying Golden State, but the question heading into the Play-in is whether he can keep it up, whether he’ll continue to have a hot hand.
Hot hands do not exist
Mathematician Lisa Goldberg is a professor of the practice of economics at the University of California - Berkeley, and in 2018 she did a study with Alon Daks and Nishant Desai that centered around Curry, Thompson, and Golden State.
It was a follow-up to the 1985 study by Thomas Gilovich, Robert Vallone, and Amos Tversky that initially rubbished the “hot hand” theory — the idea that a player on a streak of made shots is more likely to make the next one, ie that they have a hot hand.
The difference in the study that Goldberg performed was to follow what is known as the Law of Small Numbers. As she explains in a Numberphile video about the study, if you flip a coin three times, you would think that upon seeing a heads, the chance of another heads following it is 50-50… but it turns out to be 41.6 percent, which is not intuitive, but has to do with the fact that the series is limited to three flips — a third-flip heads cannot be followed by another heads because there isn’t a fourth flip.
In the Golden State study, Goldberg and her colleagues looked at strings of shots by Curry, Thompson, and Kevin Durant. They then looked at randomized permutations of the order of makes and misses, and checked those random strings against the real performances. The idea was to see if there was a higher probability of the real players having a made shot after two straight makes than there would be if the order of makes and misses were randomized.
During the Numberphile video, Goldberg reveals that there were only a small handful of occurrences of what could be considered a hot hand effect for Curry or Thompson, and filmmaker Brady Haran exclaims, “Okay, so there’s no hot hand!”
Except, isn’t that evidence that the hot hand is real… just extremely rare compared to how often TV announcers might cite it in their commentary?
“Did we see hot hands, by definition, in some games?” Goldberg said in an interview with Deadspin. “Well, if you… let me try and ask you a question. Suppose I flip a coin five times and one coin flip is heads, heads, tails, heads, tails, and one of them is tails, tails, tails, tails, tails. Which one of those sequences is more likely for you to get?”
Being versed enough in probability and having seen the permutations of three coin flips in Goldberg’s Numberphile video, clearly they’re equally likely.
“Right,” Goldberg said. “One in two to the fifth power, because that’s how many sequences like that there are, although nobody would think twice about seeing the first one, and people would look at the second one as special. So, if you go around in data looking for something unusual, you’re going to find it if you look in enough data. If I look over a whole season of games for something unusual, I’m bound to find something unusual. It’s called data mining, and data mining is like a science these days but it’s also possibly a pejorative term. It means looking in some sea of data for patterns and you’ll always find a pattern if you’re looking for it. It’s not really good science. You’ll always find some pattern.
“If you want to find something unusual in a situation, you’ll find it even if the situation is generated totally by random processes. So what we found, given the definition of hot hand that we used — which was not one we made up but an old one — we found that it occurred just exactly as it would’ve occurred out of a random process. Five percent of the time, you found something unusual at a five percent confidence level. That’s what you’d expect in random coin flipping. Just like every once in a while, if you’re flipping a coin five times, you will see five tails in a row even if there’s nothing unusual about the coin.”
Does Curry transcend math?
As a basketball fan, Goldberg acknowledges that the scientific “hot hand” definition is quite different from the way the term is used colloquially. By the way we talk about it in sports, Curry, for the last month, has been hot. He broke Kobe Bryant’s NBA record for consecutive 30-point games by a player age 33 or older, and raised his career scoring average by 0.4 points per game in a little more than a month.
The thing is, Curry’s increased scoring has come in large part because he’s been shooting a lot more, due to Golden State’s need for him to do so to get to the postseason. He’s shot well since getting back into the lineup — two percent better than his career average — but Curry’s stretch run isn’t the best form of his career, or even this season.
From January 9 through February 15, Curry shot 52.1 percent from the field over 20 games, his best vein of shooting since a 52.7 percent stretch early in the 2017-18 season. And that 20-game split this season included the best 10-game vein of shooting in Curry’s career, 58 percent from January 28 through February 15, including his 57-point game in Dallas on February 6.
“If you look in rich enough data, you see lots of streaks,” Goldberg said. “If you look at sufficiently many hypotheses, you’ll see something unusual relative to any pure chance norm, just because you’re looking at lots of different things and it’s hard to keep all that sorted out when you’re trying to run experiments. … When you look over the whole season, by pure chance, you’re going to — even if there’s nothing hot-handed about Mr. Curry — by pure chance you’re going to see some unusual streaks. When you watch him play, it’s like watching a dancer. Watching him play is like going to a museum. He’s just a joy to watch. I don’t know how you quantify that with a statistical test.”
If you walk through a museum, you’ll see galleries full of wonderful, amazing, awe-inspiring art… and then a room where, sure, it’s fine art, but it just doesn’t connect. Why do Curry’s 10-game and 20-game hot streaks from this winter end at the same time? Curry’s next 10 games represented his worst shooting in two years, 41.7 percent from February 15 through March 15.
Curry’s career rolling 10-game shooting percentages come out as a series of peaks and valleys, all clustered between 35 percent and 60 percent, with his career shooting percentage steady for years at just under 48 percent. Not surprisingly, Curry’s 40-game rolling averages are even more tightly clustered.
As good as he’s been for the past six weeks, Curry’s career shooting mark has gone up just 0.027 percentage points.
“Here’s another way maybe we could think about a hot hand,” Goldberg said. “[A player] has his season average and every game he has an average and we think that his game averages are going to be somewhere around his season average. Makes sense. But you can ask whether they’re tightly clustered around that season average or whether they’re very spread out. I could get an average of 50 by averaging 49 and 51 or I could get an average of 50 by averaging zero and 100, right? One of them is way more spread out than the other.
“So, one thing you might look at if you were looking at hot-handed players is to see whether their game averages are very dispersed around their season average and whether there’s some big cluster of unusually high ones and maybe that’s what hotness means to a fan. But this is very different in a scientific sense from asking the question. … So, it does kind of raise the question, what is it fans are reacting to? What is it that makes your heart go faster when you’re watching these games and somebody seems to be on a streak? Is it the streak or is it just a higher percentage than usual, or what is it that we’re doing?”
The key thing here going into the Play-In is that there’s no telling whatsoever whether Curry, at any moment, is rising or falling. The oscillations aren’t constant, and none of it is statistically significant to Curry being a career 47.7 percent shooter. In fact, Curry’s game-by-game shooting, as you’d imagine — and importantly heading into the Play-In — varies wildly, but even so, that coke fiend EKG pattern doesn’t even result in big changes to his 10-game or 40-game rolling averages.
Each shot — and yes, there are variables for how open he is, how he’s feeling physically, where he is on the floor, on and on — still has about a 47.7 percent chance of getting to the bottom of the net.
The Heat Check
None of this is to say that Curry isn’t special. He’s a future inner-circle Hall of Famer who next season stands to break Ray Allen’s career record for three-pointers made, while also ranking seventh all-time in three-point accuracy. Curry can make shots that nobody else on the planet can.
So, even though his flamethrowing runs might not be statistically defined as a hot hand that’s predictive of what he’ll do on his next shot or in his next game, Curry is such an accomplished marksman that he makes the impossible seem real.
In his career, Curry has had 131 games in which he attempted at least 10 three-pointers and made at least half of them. That’s not just the most such games in NBA history, it’s more than the next two players on the all-time list, Damian Lillard and James Harden, combined. LeBron James has done it 19 times in his career. Curry has done it 19 times this season.
And when it comes to being a really hot shooter, by the non-mathematical definition, Curry is in a league by himself. The NBA record streak of 10+ three-point attempts, 50 percent accuracy games is six, by Lillard last year, but that’s the only time anyone has had more than three straight such games. Of the 14 three-game streaks that there have been, six belong to Curry, including April 10-14 of this season. Nobody else has done it more than once.
While Curry might not be mathematically more likely to make any given shot based on having been hot over a recent stretch, he does know when he’s in a zone, feeling it, whatever you want to call it. And that gives rise to the heat check, firing up what appears to be a low-percentage shot, and seeing if your form is so good at that moment, the ball can’t help but go in.
“There’s certainly, for anyone who practices a craft or an art or a sport or something that’s got an intellectual and an emotional and a physical component to it, there are times when you feel better about it and it seems to go,” Goldberg said. “You feel you can do it, and other times it’s the opposite. I think we all have that. The [Nobel laureate Daniel] Kahneman and Tversky questions are, given some underlying randomness model, is this just kind of normal or are there unusual streaks in it relative to some norm you set up?”
In this case, Curry’s normal is so far removed from everyone else’s, it skews our perception.
“What they find, I can’t ever emphasize this to people enough, are two things. One of them is that we underestimate the role that chance plays in basically everything. We’re always looking for reasons for things. ‘He had this talent, he had this inspiration.’ Luck plays a role. That’s one of their points. Another one of their points is that what randomness really means is very foreign to human intuition.”
As an example of this, Goldberg described a Berkeley statistics class in which one of her colleagues divides students into two groups. One group gets real coins to flip 100 times, while the other is asked to make up a random sequence. When the students share their sequences of 100 flips, the professor “can always tell which group any answer came from, because the kids who were asked to make up the sequence put in too many reversals and not enough long streaks. A perfectly random coin flip will have more streaks in it than people expect. It’s just not in our intuition.”
Given 100 random coin flips, there’s a 25 percent chance that there will be a sequence of seven consecutive heads at some point, but the statistics students never put in a streak that long because to the human mind, it doesn’t feel possible.
That’s what we see with Curry. He might not defy math, but time in and time out, he stretches the bounds of our imagination to do things that just don’t seem possible on a basketball court. Nobody knows when or how it starts and stops, but Golden State will go into the Play-In tomorrow night hoping that this above-average run from an all-time great continues just a little bit longer.
The idea of the statistical study is to see if there’s, as Goldberg puts it, any conclusion that is “actionable.”
“Should I, the coach, try and get the ball to the guy who seems to have the hot hand?” Goldberg said. “And here you get this army of statisticians who say you should never run the kind of experiment that Gilovich and Vallone and Tversky ran, or that we ran. Why is that? Because shots are different difficulties, because there’s all kinds of subtleties of the game, because when someone starts to shoot well, the defense starts to adjust and your analysis, in terms of making any decisions based on a perceived hot hand, should take account of all of that.
“So, let’s suppose we want to make a model of the hot hand that a coach could use and then we would start to worry about whether Curry is being guarded closely or not being guarded closely, whether he’s off-balance, how many defenders are around him and making adjustments for stuff like that. We would be complicating our analysis greatly and making even more assumptions than we’re aware of. … I don’t know how actionable any of it is even if you had analysis that showed that there was some elevated likelihood taking into account all of those other things, that a player was more likely to make a next shot given that she or he had just made a bunch of them. It would be hard for me to have too much confidence in it because in a model, a lot of things are left in and a lot of things are left out.”
But when it comes down to it, the actionable thing for Golden State would be Steve Kerr telling his team, “get the ball to Steph,” which, well, he’s already going to do.
“I agree,” Goldberg said. “He’s a great shooter. Why not bet on him?”