Conceived and designed the experiments: GY SE. Performed the experiments: GY. Analyzed the data: GY. Contributed reagents/materials/analysis tools: GY SE. Wrote the paper: GY SE.
The authors have declared that no competing interests exist.
The long lasting debate initiated by Gilovich, Vallone and Tversky in
Current information era, brings with it exciting opportunities for exploring old and new research fields using extensive data sets that are easily accessible nowadays. It allows scientists to explore many topics on a larger scale and in a more precise quantitative way. Sports is a great example of how one can take advantage of large data sets that are available in a digital format and address interesting questions in a variety of contexts (eg.
Since the illuminating work of Wardrop
As mentioned above, Wardrop
The NBA season is divided into two: regular season and playoffs. In the regular season, each of the
There are three types of free throws in a basketball game  all come after a foul was committed. The resulting penalty is a series of either
The first step in analyzing the data was to clean it from all types of errors and inconsistencies:
The data of
In some cases two players from the same team share the same last name. In most of these cases the player ID or the initial of the first name helps in telling them apart but in several individual cases the data was still ambiguous: in all of these cases we simply ignored this data for the current analysis (sums into
All together, less than
As mentioned above there are three types of of free throws sequences: a single attempt, a sequence of two consecutive attempts and a sequence of three consecutive attempts. For all the two attempts sequences of every player,
The results were then tested for statistical significance for two measures:
Nonstationarity (NS): the change in success rate as the consecutive attempt number increases.
Conditional probability (CP): the change in success rate of the second attempt for a given results of the previous attempt (for a sequence of three free throw attempts the same was done with the third attempt as well).
Both of these measures can be studied with the aid of the hypergeometric distribution. In order to test the NS one can think of hits as “white balls” and misses as “black balls” and put them all in one urn after labeling them as first or second attempt. Since the null assumption is that there is no systematic deviation in the probability of success between the first and the second attempts, one can sample, without replacement, one half of the total number of throws (first and second attempts combined) and check how many hits (white balls) are in the sample. The null assumption implies that the number of hits in the first or second attempt should be consistent with a random sample from this hypergeometric probability distribution function.
In the case of testing the change in CP one can think of putting all the second attempts as balls in the urn (hits are the “white balls” and misses are the “black balls”). This time the number of balls that are drawn, without replacement, is the number of hits in the first throw. Once again, the null assumption, which states that the result of the second attempt is independent of the result of the first attempt, implies that the number of times one gets hits in both throws will agree with a random sample from this hypergeometric distribution function.
We describe the hypergeometric distribution function with the following parameters:
Thus, the formulation of the NS hypergeometric distribution function is,
After calculating these measures, in principal, one can calculate for each individual player, the
The first, computationally faster, approach involves estimating a “
When calculating for the aggregated data the total number of free throw attempts is large enough and the distribution of
A more accurate, though computationally intensive, approach, is permutation approach: first, we reshuffle the second throws of each individual. After reshuffling, we calculate the
We start by verifying two observations already pointed out in
A simple statistics artifact of aggregation, sometimes referred to as “Simpson's Paradox”
Kevin Martin  
2ndMiss  2ndHit  
1stMiss  1  35 
1stHit  25  159 
Dwight Howard  
2ndMiss  2ndHit  
1stMiss  59  102 
1stHit  89  127 
Both combined  
2ndMiss  2ndHit  
1stMiss  60  137 
1stHit  114  286 
Player 


Kevin Martin  0.96  −0.90 
Dwight Howard  1.37  −1.83 
Both combined  1.44  0.49 
Yet another demonstration for this kind of bias due to aggregation is presented in the
The results for the aggregated level data are presented at the top part of
Panels
Aggregated data 






74.32  73.54  74.71  76.49  75.58 
Number of throws  7807  8418  7300  7265  7651 

71.91  73.12  73.54  74.70  73.74 

76.95  77.57  77.70  79.29  78.10 
Number of throws  27765  27344  26416  25842  25550 

79.07  78.84  77.74  76.51  78.65 

78.29  79.18  80.65  84.23  83.51 

84.50  87.03  80.65  85.57  84.59 
Number of throws  258  293  310  298  370 



69.47  69.20  70.52  73.19  70.55 
Number of individuals  398  410  397  389  397 

68.21  69.53  69.80  71.63  70.13 

74.83  76.04  74.29  75.43  75.21 
Number of individuals  439  443  438  427  429 

76.82  80.67  78.06  74.26  76.20 

76.73  76.66  78.16  82.06  85.20 

84.79  85.21  78.23  83.06  83.16 
Number of individuals  95  112  121  120  132 
Aggregated data 






72.45  72.38  72.63  75.33  73.81 
Number of records  7800  7350  6990  6537  6709 

78.71  79.48  79.52  80.63  79.62 
Number of records  19965  19994  19426  19305  18841 



71.94  74.20  72.28  74.59  73.59 
Number of individuals  418  424  414  405  408 

76.53  77.05  75.79  75.97  76.42 
Number of individuals  430  435  425  421  422 
The success rates of free throws attempts are summarized in







13.62  12.08  11.12  12.39  11.52 

13.48  12.25  10.31  10.9  11.13 

11.13  12.48  11.86  9.15  9.89 

3.71  4.13  4.04  1.87  2.58 

2.1e04  3.6e05  5.3e05  6.1e02  9.9e03 

5.2e04  1.2e04  1.4e04  7.6e02  1.5e02 
In the left panels (with the red bars) of
The reasons for this effect could be easily justified as an “alignment gauge” for the hand of the shooter. The time taken by the player until the second attempt also allows for rest and more concentration before the second and third shots are taken. Needless to say that this tendency cannot go on forever and it may be interesting to quantify this feature further using targeted experiments.
Panels b and d of
As for the statistical significance of this result, one is referred to
The interpretation of this result is essentially that the results are unlikely to emerge from a collection of uncorrelated sequences each with a constant probability of success and no auto correlation. But statistical significance is borderline in some cases and the question about its' origin remain: does it mean that we found proofs that“success breed success” or can something else explain the observed “hot hand” pattern? Recently, for the change in the conditional probability, similar results were obtained in
In principle, “better and worse” periods for individual players can cause the same effect as “success breeds success” on the success rate of the second free throw attempt and are hard to separate apart. To illustrate the reasons for this difficulty, two of the simplest descriptions of each possibility are compared:
“Better” and “worses” periods: switching with random (or constant) periods of time spent in each of the two states where each state is a Bernoulli independent repeated trials with probability of success
Positive/negative one step feedback: the probability of success given the last trial was a success is
Both options can be viewed as a system where each player behaves essentially like two players sharing the overall sequence with different probabilities of success. This point of view connects back to the first observation (effect of aggregation) and to
In order to distinguish between the two there is a need for an analysis of the time series of results which will help in identifying a characteristic timescale in which the probability of success changes and then to try and link it to one of the possibilities presented above (see
1.
2.
This plot shows the individual
We conclude that these two points suggest that the observed pattern interpreted as “hot hand” in the analyzed data is in large part a consequence of better and worse periods.
It is believed that the last quarter of a basketball game is very different from the other quarters. The game is often interrupted and special tactics apply to this period. An interesting question is if and how the features we have seen so far are affected by this. For that purpose, we have divided the data into two parts: 1) quarters
Strong evidence for the existence of a “hot hand” phenomenon in free shots of NBA players were found. More precisely, several statistically nontrivial features of the data were found and can be summed into one concept: heterogeneity. The heterogeneous behavior was found both in “space” (across players) and time (along one season). In particular it has been shown that
If one looks at the aggregated data he/she is likely to observe patterns that do not necessarily exist at the individual level.
The probability of success increases with the order of throw attempt in a sequence (NS).
Even if one looks at each individual sequence separately, “hot hand” patterns are still visible (CP): probability of success following a success is higher than the probability of success following a failure.
These patterns could have resulted from “better and worse” periods and not necessarily from positive/negative feedback loops.
These statistical features per se are not so surprising when studying performance of human subjects. Nevertheless, due to the intensive debate in the last
From the basketball fan/professional perspective, it could be beneficial if the NS result which implies that the probability of success increases with the shot number would be further exploited. To start with, this could be added as one more statistical feature that is calculated and presented throughout basketball matches, but more importantly, one can study further the psychological and physiological reasons behind it and maybe come up with techniques that will help the player to improve the first trial(s) in a sequence of trials.
Although the phenomenon that was studied here is taken from the world of sports, the implications are much more far reaching and should bear in one's mind when analyzing data of any kind. In spite of the fact that it is known for many years that aggregated data (or mean behavior) can deviate significantly from the microscopic dynamics underlying it (e.g.
The current example of the “hot hand” phenomenon serves as a fascinating one since it is possible to trace back the fundamental reason for the deviations between the observed macro patterns and the underlying micro processes causing them to
(EPS)
(PDF)
Data for all free throws taken during 2005/6–2009/10 in the NBA.
(DAT)
We would like to thank Steven H. Kleinstein, Jonathan Belmaker and the anonymous reviewers for very useful comments. This work was supported in part by the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center.