Latrinsorm
08-24-2014, 05:16 PM
A recent paper (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2481494) found that "yeah-huh there is so a hot hand", and while it does a very nice job of making its case it still has one major flaw: do players make more in game shots in a row because they are hot, or by dint of making the first one are they more likely to be facing an inferior defense? The paper does control for relative position and angle of defender, which is very good, but does not (and cannot) control for quality of defender.
If there is a hot hand, it must exist for free throws. I acquired a set of free throw data sometime somewhere. All told, players shot 116,706 free throws when taking 2 in a row in this sample: 73.8% ± 0.3% on the first, 77.9% ± 0.2% on the second. "Aha!" you might cry, "the hot hand!" But the real picture is far more sinister.
There are two states.
"Hot" is defined as making the first free throw.
"Cold" is defined as missing it.
For various reasons it's not a good idea to simply combine everyone's data from hot and cold. Put most simply: Dwight Howard misses a ton of his first free throws, then misses a ton of his second free throws. Steve Nash makes a ton of his first, then makes a ton of his second. This isn't because either are cold/hot, it's because they're terrible/awesome free throw shooters. What we can look at instead is for every given player if their hot and cold shooting %s deviate significantly from the expected random noise. It turns out that there were 355 player-entries in this sample to take at least 100 free throws and have a less than 100% percentage for both the first and second (because 100% success does weird things to statistical noise). If we take the standard deviation of every one of their percentages and divide it into the difference between overall and hot/cold, we obtain a Z score. Generally speaking a Z score of 2 or higher indicates statistically significant deviation, but in a sample of 355 we would expect non-zero player-entries to display those scores. What we can do instead of that is sum all the Z scores: a sample with nothing but Gaussian noise should sum to zero (or nearly so, about 4). The results are:
Hot: 168
Cold: 25(!!!)
In a stunning twist, even missing the first free throw has a positive impact on making the second one! How can this be?
1. Free throws are always taken from the same distance, but that doesn't mean that distance is inherently known by players. Especially in the flow of a game, where players are liable to take shots of many kinds in between free throws, it's very plausible that the first free throw can act as a range finder. "That one hit the front iron, I'll shoot this one a little harder."
2. Free throws like most tasks are a question of focus. "Oh shoot, I missed the first one! I'll bear down on the next one."
3. Fluke. It's a very significant effect, but as is always the case in empiricism there's no guarantee it's not just noise.
Alright, so we have two significant effects, let's look at the graphs...
http://img.photobucket.com/albums/v456/johnnyoldschool/NBAHotHandFreeThrows_zpsc330b8d7.png
BOGGLE?!? There should be 48 apparent outliers each, hot has only 11 and cold has only 6, both on the positive side. My first thought was that free throw percentage happens to be too high for three standard deviations to be possible (because it can only go up to 100%), and that is the case for some player entries, but that still leaves 226 cases where it should be possible for hot and 95 for cold (because cold shots are less frequent their standard deviation is higher). That works out to 1 apiece in +3z, so that's pretty close at 1 and 0, but our +2z is still a shit show: we should have 20 and 12.5, we do have 10 and 6. How can there be both a positive effect and dramatically reduced positive outliers?
1. Overcorrection. "That one grazed the front iron on the way in, I'll shoot this one a little harder OH GEEZ TOO HARD!"
2. Focus. "Heck yeah, I made the first one! Who cares if I make the second..."
3. Fluke.
.
Bottom line:
Taking the first of two free throws (make or miss) has a positive effect on making the second.
Making the first of two free throws has an additional positive effect.
But in all cases the second of two free throws shows self-correcting forces on the distribution, pushing all shooters towards the mean.
If there is a hot hand, it must exist for free throws. I acquired a set of free throw data sometime somewhere. All told, players shot 116,706 free throws when taking 2 in a row in this sample: 73.8% ± 0.3% on the first, 77.9% ± 0.2% on the second. "Aha!" you might cry, "the hot hand!" But the real picture is far more sinister.
There are two states.
"Hot" is defined as making the first free throw.
"Cold" is defined as missing it.
For various reasons it's not a good idea to simply combine everyone's data from hot and cold. Put most simply: Dwight Howard misses a ton of his first free throws, then misses a ton of his second free throws. Steve Nash makes a ton of his first, then makes a ton of his second. This isn't because either are cold/hot, it's because they're terrible/awesome free throw shooters. What we can look at instead is for every given player if their hot and cold shooting %s deviate significantly from the expected random noise. It turns out that there were 355 player-entries in this sample to take at least 100 free throws and have a less than 100% percentage for both the first and second (because 100% success does weird things to statistical noise). If we take the standard deviation of every one of their percentages and divide it into the difference between overall and hot/cold, we obtain a Z score. Generally speaking a Z score of 2 or higher indicates statistically significant deviation, but in a sample of 355 we would expect non-zero player-entries to display those scores. What we can do instead of that is sum all the Z scores: a sample with nothing but Gaussian noise should sum to zero (or nearly so, about 4). The results are:
Hot: 168
Cold: 25(!!!)
In a stunning twist, even missing the first free throw has a positive impact on making the second one! How can this be?
1. Free throws are always taken from the same distance, but that doesn't mean that distance is inherently known by players. Especially in the flow of a game, where players are liable to take shots of many kinds in between free throws, it's very plausible that the first free throw can act as a range finder. "That one hit the front iron, I'll shoot this one a little harder."
2. Free throws like most tasks are a question of focus. "Oh shoot, I missed the first one! I'll bear down on the next one."
3. Fluke. It's a very significant effect, but as is always the case in empiricism there's no guarantee it's not just noise.
Alright, so we have two significant effects, let's look at the graphs...
http://img.photobucket.com/albums/v456/johnnyoldschool/NBAHotHandFreeThrows_zpsc330b8d7.png
BOGGLE?!? There should be 48 apparent outliers each, hot has only 11 and cold has only 6, both on the positive side. My first thought was that free throw percentage happens to be too high for three standard deviations to be possible (because it can only go up to 100%), and that is the case for some player entries, but that still leaves 226 cases where it should be possible for hot and 95 for cold (because cold shots are less frequent their standard deviation is higher). That works out to 1 apiece in +3z, so that's pretty close at 1 and 0, but our +2z is still a shit show: we should have 20 and 12.5, we do have 10 and 6. How can there be both a positive effect and dramatically reduced positive outliers?
1. Overcorrection. "That one grazed the front iron on the way in, I'll shoot this one a little harder OH GEEZ TOO HARD!"
2. Focus. "Heck yeah, I made the first one! Who cares if I make the second..."
3. Fluke.
.
Bottom line:
Taking the first of two free throws (make or miss) has a positive effect on making the second.
Making the first of two free throws has an additional positive effect.
But in all cases the second of two free throws shows self-correcting forces on the distribution, pushing all shooters towards the mean.