Your favorite NBA player has drilled his last four shots.
"He's heating up!" you think to yourself, eagerly awaiting the next time the ball ends up in his hands. Surely that upcoming attempt is going to be good, ripping through the bottom of the twine for more points. After all, he's catching fire and the points are going to flow in bunches.
Problem is, you're counting on a phenomenon that might not actually exist. It seems as though professional basketball players can get hot or cold, falling prey to streakiness that either aids or hinders their performance on the court, but that's only because of pure randomness. Sure, players are going to have hot and cold streaks, but there's not much statistical evidence that some players are subject to them more often than other similar players.
This—the "myth of the hot hand," as it's known in statistical circles—has been discussed quite often. It's been the subject of much study and debate over the years. We've even seen papers published and presented at the MIT Sloan Sports Analytics Conference, like this one by Harvard's Andrew Bocskocsky, John Ezekowitz and Carolyn Stein that claims the following:
For thirty years, the empirical consensus that the 'Hot Hand' in basketball is a fallacy of the human mind has been confirmed time and again. In the same way that evolutionary biologists might regard creationists as completely misguided, economists, psychologists and statisticians have viewed the persistent belief in the Hot Hand as entirely fallacious.
Nonetheless, the belief lives on.
Ask nearly any player about his performance after a run of made shots, and he'll surely tell you some variation of "I just got hot." Even Shane Battier, whom ESPN.com's Israel Gutierrez once called "the numbers guy, the statistics specialist, the percentages guru," has expressed belief in the basketball gods that cause hot streaks.
They did get hot.
There's no denying that. But even hot streaks don't mean that some players are streakier than others. To examine that, I've taken a different approach than the many researchers of the past, though this analysis is admittedly limited to data from only the 2013-14 campaign.
Sorting Shots into Buckets
With access to shot-by-shot data, we can look at this phenomenon of streakiness through this particular statistical lens. To begin, let's take a random heat-up shooter—well, a supposed heat-up shooter—and use him as an example. We'll call him John Doe.
If we put all of Doe's shots into numerical form (zero representing a miss and one representing a make), then we can form a string of numbers that represent his results throughout the season. Grouping those into sets of five shots, we can determine his shooting percentages for those five-shot stretches. So hypothetically, let's say he took 10 shots and alternated makes and misses, which would be represented as the following: 1, 0, 1, 0, 1, 0, 1, 0, 1, 0.
His field-goal percentage for Shots 1-5 would be 60 percent (three divided by five). Then for Shots 2-6, it would be 40 percent. For Shots 3-7, it would be 60 percent. So on and so forth until we've looked at the last string of five—Shots 6-10. Because of this method, there will always be four fewer intervals of shots looked at than a player has attempts.
The next step involves comparing these percentages to the player's season-long field-goal percentage so that we can see how much variance there is in each set of five. For the sake of this hypothetical, let's assume that Doe's field-goal percentage is exactly 50 percent. Now, we can see the variance for each of the six sets of five attempts:
|Attempts||Set FG%||Season-Long FG%||Variance|
Once that's done for each grouping, we can put those variances into what we call buckets. There are just six buckets, or possibilities, here—though Doe only used two—because with five shots, only six different field-goal percentages can be achieved for any stretch: 0, 20, 40, 60, 80 and 100 percent.
Then, we can form a frequency graph of the variances—a graph that shows how many times Doe's percentages fell into each bucket—which should display something that closely resembles a normal distribution. After all, that's the expectation if how much a five-shot percentage varies from a season-long percentage is perfectly random.
If we measure the kurtosis, or peakedness, of the graph, we can see how concentrated his results were around his average. Theoretically, he should have more concentration around the bucket closest to zero (littlest variance from the season-long average) than anywhere else. The less peaked the distribution is, the hotter and colder he was that season, because he's experiencing more results that deviate strongly from his typical shooting percentage.
A negative value for kurtosis indicates that a player was streakier than we'd expect, and the more negative the number is, the more hot and cold he got.
But now, let's abandon Doe and move on to Kevin Durant.
The MVP of the 2013-14 season, Durant took 1,688 shots last year, which means that we have 1,684 sets of five-shot streaks throughout the season. It's worth noting one inherent flaw of this study is that we're not using the end of games as cutoff points, but instead assuming that a streak can carry over from the final buzzer to the opening tip of an ensuing game.
Since Durant shot 50.3 percent from the field in 2013-14, that means he has six different bucket values here: 50.3 percent below average (0-of-5), 30.3 percent below average (1-of-5), 10.3 percent below average (2-of-5), 9.7 percent above average (3-of-5), 29.7 percent above average (4-of-5) and 49.7 percent above average (5-of-5).
Here's the same exercise used above with the first 15 shots of Durant's season (make, miss, make, miss, make, miss, miss, miss, miss, miss, make, make, make, make, make):
|Attempts||Set FG%||Season-Long FG%||Variance|
As you can see, all six buckets showed up within the first 15 shots. But expanding our scope to the entire season's data, how frequently did Durant's sets fit into each one? You can see that below:
That's a fairly normal-looking distribution, and I mean that in both senses of the word. It's close to being statistically normal, and with a kurtosis of minus-0.42, it's pretty typical when compared to other high-scoring jump-shooters from the 2013-14 season.
In fact, of the 28 qualified scorers who averaged at least 18 points per game last campaign, Durant's distribution showed that he was more streaky—or more prone to getting hot and cold—than everyone but Arron Afflalo and James Harden. Yes, that means we looked at a staggering 34,568 field-goal attempts while researching this question.
But here's the more interesting part.
The streakiest player among that group was Harden, and his kurtosis value was minus-0.46, which indicates that his distribution was the least normal but still by a fairly insignificant margin. Meanwhile, Chris Paul, Goran Dragic and Dwight Howard were the steadiest, but their kurtoses were minus-0.18, still indicating that they were streakier than a normal distribution would show.
Most players—and by that I mean 21 of the 28 studied—were clumped between minus-0.26 and minus-0.42, which is hardly enough to make any type of meaningful differentiation. That's simply because shot percentages, even in samples of just five attempts, tend to cluster around the season-long average. Hot streaks are cancelled out by cold ones for the most part, and neither occurs with too much frequency.
So, why do these outlier streaks stand out? Why do people continue to think the hot hand is a valid explanation for players apparently heating up?
That answer isn't too complicated.
It's psychological for both players and observers. NBA standouts and those watching them aren't going to notice when two or three of five attempts are made, but the five-in-a-row streaks or string of five consecutive misses tend to stand out more. They're much rarer, after all.
Durant, for example, had these stand out as his longest streaks of consecutive makes from the 2013-14 season:
- Nine in a row (three times)
- Eight in a row (six times)
- Seven in a row (twice)
- Six in a row (five times)
- Five in a row (20 times)
The NBA's scoring champion, a guy who's one of the best shooters the sport has ever paid witness to, made at least five shots in a row only 36 times throughout the entire season. That's once every nine quarters, based on the 81 games he suited up in for the Oklahoma City Thunder.
Meanwhile, it took him until Nov. 13 to have 36 streaks in which he made two or three of his five shots. On Nov. 13, he was suiting up against the Dallas Mavericks in his fourth outing of the season.
And that's legitimately surprising—to me, at least. Surely, Durant made at least five consecutive buckets more than once every nine quarters, right? Those stretches in which he heats up seem to happen much more frequently, but only because they're burned into our brains by virtue of being seemingly unique. Even when highlights are shown, it seems as though the makes are spliced together and the misses are just forgotten.
It's not streakiness that leads some players to seem like they're heating up more often. It's randomness, though there's one confounding factor helping some small differentiations fit into a pattern.
Shot Distance Affects Streakiness
There are certainly exceptions to the rule. But for the most part, those who tend to take shots closer to the basket are steadier shooters, while those who take long-range shots are more prone to hot and cold stretches.
It's really that simple in some cases, and it stands to reason. More can go wrong on longer shots, as they're naturally harder to connect than five-foot bunnies in the lane. So when players exhibit more kurtosis in this analysis and are shooting from farther away, it doesn't necessarily mean they're actually streakier.
Below, you can see how all 28 analyzed players fared in both kurtosis and average shot distance, via NBAMiner.com:
There is a correlation.
It's not necessarily a strong one, but enough that it can be identified. In fact, when the two biggest outliers are taken out from each side, we can perform a linear regression and come up with expected values of kurtosis based on a player's average shot distance. The regression has an r-squared value of just 0.32391 (by no means a large enough number to satisfy us), so it's still worth remembering that this is not an airtight correlation.
Now, we can say that a player like Durant, whose average shot came from 13.3 feet, should have produced a kurtosis of minus-0.35. His actual value was minus-0.42, so he was slightly streakier than expected.
By navigating the interactive graphic below, you can see each of the 28 players' kurtoses, average shot distance, expected kurtoses and true levels of steadiness, determined just as we did with Durant:
Again, Paul, Dragic and James were as steady as they come, while Davis and Lee were the streakiest of the studied players.
But beyond those five players, the remaining 23 were all within one-tenth of their expected kurtosis. Think about that. By essentially taking this potentially confounding factor out of the equation, players really don't get remotely hot or cold.
Isn't that proof enough that streakiness is more random than anything else?
Of course, there will always be the strong contingent that swears by the hot hand. As Bocskocsky and Co. wrote for the 2014 Sloan conference, "Amos Tversky, co-author of the canonical paper on the subject, typifies this view when he says, 'I've been in a thousand arguments over this topic, won them all, but convinced no one.'"
It just seems nonsensical to conclude that hotness and coldness are determined randomly and balance out over the course of the season, especially when shot distance is taken into account. Yet, based on the data, that's the correct conclusion, just as it always has been no matter who's looking at the numbers while using any number of methods.
Next time your favorite player seems to be heating up, hope and pray that it continues. Just know that there's no real advantage in play, even if he's made each of his last four attempts.
He may well make the next look. He may indeed go on a streak of consecutive makes, like Klay Thompson did during his 37-point third quarter on Friday, and you can still enjoy that all the same.
You don't need a hot hand for that.