We all remember NBA Jam, the basketball video game where after three consecutive field goals made a player is “on fire” and practically cannot miss. Some of us have infused this concept into our beer pong games to give players even more opportunities to boast and trash talk, but this hot hand phenomenon is referenced well beyond the beer pong table. There are many sayings in the world of basketball that refer to the dependence of any one shot on the shots that came before it. These sayings include, but are not limited to, “He is a streaky shooter,” “He’s on fire,” “He’s hot right now,” “En fuego, baby!” etc. What commentators and basketball fans are implying when they say these things is that, at certain times in games, a player will tend to somehow temporarily increase his ability to make shots because he is having a good shooting game up to that point. There have been numerous studies attempting to find this aspect of “streakiness,” or the “hot hand” phenomenon, that have all come to a similar conclusion: the data are unable to support the “heating up” theory.
I focused my version of this popular study solely on the dependence of three-point shots on prior three-point shots taken by a player in a game. All the players in my study were selected from the NBA during various seasons between 2004 and 2009, and each player’s career three-point percentage is between 38.5% and 41.5% in an attempt to homogenize the sample players. I took 35 200-shot samples, and I tested for the hot hand using two distinct strategies: one for measuring the effect of only the most recent shot on the result of the next shot, and one for measuring the effect of all previous shots in the game on the result of the next shot. Basically I want to see if there is an obvious increase in three-point percentage when a player is shooting a high percentage already during that game, or if a player made his most recent shot.
The actual data from my sample showed that the sample players shot just 38.2% after makes while knocking down 40.3% of 3-pointers after misses. Wow…players actually shoot slightly worse after a make than after a miss, according this data. Other studies have come to the same conclusion, but why?
There are a few theories as to the discrepancy between the expected probabilities and the sample probabilities. According to a study done by Sportmetricians Consulting and recounted by Henry Abbot of ESPN.com, players who made their most recent attempt were more likely to take their next shot from a distance farther away from the hoop, implying that they took a harder shot. Basically players get cocky and think they are on fire, so they pull up from deeper. Also the defender that is guarding the player who just made a shot may play better defense the next time, forcing a more difficult shot[i]. Either of these theories can help to explain why players still shoot lower percentages after a make that our expected probabilities predict.
I have a complementary theory, as well, which may have biased the results slightly. I believe that, assuming complete independence from shot to another, there is a more accurate way to predict the shooting percentages after makes and misses. Let’s use coin flipping as an example because it assures us of streaks of independent events. Say I were to take 35 coins and flip them 10 times each, recording the sequences of heads and tails. In ten flips, say a certain coin flipped heads 6 times in the following pattern:
H T T H H T H H H T
In my analysis of the probability of flipping heads–after heads has just been flipped–I would observe that in the six flips following heads flips, there were 3 tails and 3 heads, or 3/6 flipped heads. Notice that this percentage is less than the total percentage of heads. If you think about it, picking a random head flip in the string leaves us with just 5 possible heads that could be flipped. So our maximum possible percentage of heads after head is 5/6, or 83%. This indicates that in our study, lower percentages can be expected following makes.
Now take a look at this hypothetical coin:
T H H T H T H T H H
While there are, again, 6 heads flips, one of them occurs at the end of the sequence, analogous to a made three pointer occurring at the end of the game when there is nothing to follow it. The following shot will occur a few days later during the player’s next game, and should not be taken into consideration.
Taking these observations into account, we can apply the concepts to our three-point shooters.
Here is the pertinent information about the whole sample: The average makes per 200-shot sample was 79.43, The average number of makes on the first shot of a game was 17.74, and players averaged 43.66 games per 200 shots. Employing our expected probability strategies we see that, after a make, the shooter is expected to knock down the proportion of makes remaining in his sample that aren’t the first shot of a game. This proportion would be: (79.43-17.74-1)/(200-43.66-1), or about 39.1%. For misses it would simply be (79.43-17.74)/(200-43.66-1), the only difference being that we don’t subtract 1 from the numerator because the most recent shot was not a make.
These figures are closer to the actual data observed–38.2% and 40.3%, you may recall–and in combination with the above-mentioned theories can help to explain the discrepancy between percentage after makes and misses.
A Different Look
My second method aimed to take into account not only the most recent three pointer taken, but the shooter’s percentage on all previous shots during the game. For instance, if a player had made three shots in five attempts, then he was shooting 60% going into his sixth shot. The result of sixth shot, make or miss, could then be statted. I took the percentage going into the shot (60% in this case) as the explanatory variable, and the result of the next shot as the response variable. Only games in which a player took more than three shots were recorded in this part of the study in an attempt to get slightly more significant data, and to differentiate this method from the first method. In the results listed below, the left column represents the percent-range players were shooting (during a given game) going into the next shot, and the right column shows the collective percentage players shot on all of those “next shots.”
Game % Next Shot %
(0 <= % <= 40)——->38.7%
(40 < % <=100)——>39.5%
(0 <= % < 20)———>38.7%
(20 <= % < 40)——–>39.0%
(40 <= % < 60)——–>40.3%
(60 <= % <80)——–>37.9%
(80 <= % < 100)——>39.2%
Regardless of how well or poorly players were shooting during a given game, it didn’t seem to have any effect on their next shot. The response variable results only range from 37.9% to 40.3%, and they do not seem to have any correlation to the left-column explanatory percentages.
Both these methods failed to find any evidence of a hot hand phenomenon. I don’t mean to say that the heating up theory does not exist, only that the data cannot find it. It could be that players really do improve their chances of making the next shot by recreating the muscle movements that put the last shot in the hole. Then the “cockiness” factor–where players take more difficult shots–cancels out the player heating up, and we observe shooting percentages that appear to be independent of how well that player was shooting to that point.
In either case, regardless of the causation, the fact that players don’t seem to increase their shooting percentages after makes is useful information for a coach drawing up a last-second play. I say go to the career 45% three-point shooter as long as he’s healthy…even if he’s 1/10.
[i] Abbot, Henry, “Hot and Heavy: About NBA shooting.”
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Reading into the Numbers 