The Hot Hand In Does success breed success?

Student: Teun Jorink Studentnumber: 10047697 Supervisor: Stephanie Chan University of Amsterdam Faculty of Economics and Business BSc Economics 27th of January 2016

Abstract

There is widespread belief among basketball fans that basketball players are more likely to hit a shot after making the previous shot, than after missing it. This study uses data from all shots from the 2014-2015 NBA season to investigate the hot hand in basketball. Analysis of the results for all attempts shows no evidence in favor of the hot hand. To take shot distance into account, three shots have also been looked at but the results for these also don’t support hot hand theory. The same can be said for pairs of free throws, and when the hot hand is regarded as a team phenomenon by analyzing field goal attempts per team. Most players shoot better after missing the previous shot than after making it.

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Statement of Originality

This document is written by student Teun Jorink who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Introduction

“He can’t miss!” and “he is absolutely feeling it!” are some of the phrases you will have heard the commentator say if you watched the basketball game between the and the Sacramento Kings on the nineteenth of January 2015. Warriors-player set an NBA record for most points in a quarter, scoring 37 points by hitting all of his 13 shots, including 9 three- pointers, and both of his 2 free throws. Klay Thompson had the “hot hand”.

The hot hand refers to the belief that the performance of a player during a particular period is significantly better than expected on the basis of the player’s overall record (Gilovich, Vallone & Tversky, 1985). The opposite of the hot hand would be the cold hand, which means that the player is shooting significantly worse than expected. Another term for having the hot or cold hand is “streak shooting”, which works in both the positive and the negative direction. Players that are considered streak shooters have the reputation to shoot either really well or really bad more often than other players, being able to hit almost every shot in one game only to go on a long cold streak the next game.

In one of the first and most influential studies on this topic, Gilovich et al. (1985) conducted a survey regarding streak shooting among 100 basketball fans from Stanford and Cornell University. All of them played basketball at least occasionally, and 65% played regularly. The results of the survey show strong support for the belief in the hot hand. They found that 91% of fans agreed that a player has “a better chance of making a shot after having just made his last two or three shots than he does after having just missed his last two or three shots”. For free throws they found the same effect, 68% believed that the probability of hitting the second is larger when the first free throw was made than when it was missed. The survey also reveals that basketball fans believe that the outcome of previous shots influences decision making. Almost all fans (96%) agreed that players tend to take more shots than they normally would after hitting a series of shots, and 84% thought that it is important to pass the ball to a player who had just made several consecutive shots. Apparently, there is a widespread belief among basketball fans that the outcome of a shot is dependent on the outcome of previous shots, such that success breeds success and failure breeds failure.

The psychological state of a player plays an important role in the common belief in the hot hand. Hooke (1989) describes the influence of the outcome of previous shots on confidence well: “In almost every competitive activity in which I’ve ever engaged (baseball, basketball, golf, tennis, even duplicate bridge), a little success generates in me a feeling of confidence which, as long as it lasts, makes me do better than usual. Even more obviously, a few failures can destroy this confidence, after which for a while I can’t do anything right”.

For this study, data from all shots from the entire 2014-2015 NBA season will be used in order to test the hot hand hypothesis that players are more likely to hit a shot after having just made one or more shots, than after having just missed one or more shots. Pairs of free throws will also be looked at to explore differences in free throw percentages after having made or missed the first free throw.

Preliminary hot hand phenomenon research

The first to study the hot hand phenomenon were Gilovich et al. (1985). Their survey showed that basketball fans believe in streak shooting, as described in the introduction. In the next part of their

3 study, they investigate whether these beliefs are supported by empirical data. They first use data for all field goal attempts from 48 home games from nine players of the 1980-1981 team. An analysis of conditional probabilities showed that for eight of the nine players, the probability of hitting a shot was larger after having missed the previous shot(s) than after having made the previous shot(s). This is the opposite of what the hot hand theory would predict. A runs test, where every sequence of consecutive made or missed shots counts as a run, also shows no evidence that hits and misses cluster together like the streak shooting hypothesis suggests. In fact, five of the nine players had more runs than expected under the assumption that the outcomes of all shots were independent of each other.

Every shot during regular play in a basketball game is subject to different conditions. Defensive pressure and the difficulty of the shot (lay-ups are easier than 3-point shots for example) are some examples of factors influencing the likelihood of a shot to be made. In order to eliminate these factors, Gilovich et al. (1985) did a controlled shooting experiment with players from both the men and women basketball teams from Cornell University. Analyzing the shots in the same way as they did with the data on the field goal attempts from the Philadelphia 76ers players again shows no evidence supporting the hot hand hypothesis. Only one of the 26 players participating in the experiment displayed results significantly supporting the hot hand.

Pairs of free throws offer another way to test dependence between successive shots without having to deal with defensive pressure and shot selection. Gilovich et al. (1985) used data from all pairs of free throws by nine players from the 1980-1981 and 1981-1982 seasons to test whether players were more likely to hit the second free throw after having made the first free throw, than after having missed the first one. Four of the nine players performed better when they made the first free throw and the other five performed worse, but none of the correlations between the first and second free throws were significant.

Wardrop (1995) expanded on this in his paper. He argued that it is highly unlikely that a basketball fan can remember 2 x 2 tables (hit or miss for the first and second shot) for all players they have seen shooting free throws. Instead, it is more reasonable to assume a basketball fan has a 2 x 2 table in mind for all players combined. The free throw data for the Boston Celtics players shows that the players hit 79% of their second free throws after they had hit the first one as well, compared to 74% after having missed the first one. Thus, if fans have this 2 x 2 table for all players together available, they do see a pattern that supports the hot hand theory, which could explain why 68% of the basketball fans interviewed by Gilovich et al. (1985) believed that the probability of hitting the second free throw is larger when the first free throw was made than when it was missed.

The 2 x 2 table for all players combined is not appropriate for analyzing the hot hand theory though, and you would need the 2 x 2 tables for individual players to do so. This is an example of Simpson’s paradox (Simpson, 1951). Wardrop (1995) looked deeper into the results for 2 players: and Rick Robey. Bird hit 86% of his free throws and Robey 57%. Contrary to the hot hand hypothesis, both players shot slightly worse after having made the first free throw than after having missed it. The 2 x 2 table for both player combined shows however that they hit 81% of their second free throws after a hit and 74% after a miss, which would support the hot hand hypothesis. This paradoxical outcome is down to the percentages in the collapsed table being weighted averages. Since Bird is a much better free throw shooter than Robey, he is more likely to hit his first free throw

4 and therefore the shooting percentage for Bird’s second free throw after hitting the first one has more relative weight than Robey’s percentage, leading to a higher weighted average. Of course, this works the other way around for the free throw percentage of the second free throw after missing the first, since Robey will miss a higher percentage of his first free throws than Bird. So if basketball fans only have the collapsed table for all players together available, they may see a pattern although actually none exists. “In other words, aggregation does not provide any evidence for the existence of the hot hand, but merely helps us to understand why the fans believe what they do” (Bar-Eli, Avugos & Raab, 2006).

Larkey, Smith & Kadane (1989) tried to challenge the conclusions from Gilovich et al. (1985). They looked at the shooting outcomes of 18 outstanding NBA players in 39 games from the 1987-1988 season. None of the players had a significant serial correlation between the conditional probabilities of hitting a shot after having made or missed the last shot(s), so this analysis shows no support of the hot hand. Larkey et al. (1989) argued however that it’s not the individual performance over the course of the entire game that should be looked at, but that the hot-hand is a phenomenon that occurs when a player takes successive shots in a short amount of time (i.e. they use sequences of 20 field goal attempts by all players on the field in their data set). In this contextual approach, one player appears to be a streak shooter: Vinnie Johnson from the , who already had the reputation to be a streak shooter. Therefore, they concluded, it is OK to believe in the hot hand.

Tversky & Gilovich (1989) responded to this by looking at the Vinnie Johnson case and found that the outcome was based on a single run of 7 consecutive made shots within a sequence of 20 shots, which they didn’t find convincing proof for the existence of the hot hand. When looking at the videotape of the game where Johnson hit those 7 shots, they found Larkey et al. (1989) made a mistake in the data and one of the hits was actually a miss. When the mistake was corrected, Johnson was not a streak shooter anymore.

Perhaps more interestingly, in the same paper Tversky & Gilovich (1989) also tested the locality hypothesis; the claim by Larkey et al. (1989) that the hot-hand is a short-lived phenomenon. Using the same data Larkey et al. (1989) used, Tversky & Gilovich (1989) looked at the serial correlation for all pairs of successive shots that are separated by at most one shot by another player on the same team. The results are not significant for any of the 18 players and don’t differ much from the global serial correlation, for which all shots of the entire game had been taken into account. Similar results were obtained when the locality analysis was restricted to shots that are separated by at most 3, 2, or 0 shots by another teammate. These results don’t support the hypothesis that the hot hand phenomenon occurs when a player takes successive shots in a short time period.

Adams (1992) also identified that Gilovich et al. (1985) hadn’t looked at the time factor in shooting success. The feeling of having the hot hand tends to diminish over time or when it is interrupted, which is one of the reasons why coaches take time-outs; to ‘break momentum’. Adams (1992) argues that if the hot hand theory is valid, and that the feeling of having the hot hand diminishes over time, the probability of hitting a shot should be larger the sooner it is taken after a successful shot. He tested this hypothesis using shots from 19 NBA games and comparing the time between a made field goal attempt and a made field goal attempt (hit-hit interval) with the time between a made field goal attempt and a missed field goal attempt (hit-miss interval). The theory of time-dependent success suggests that the hit-miss interval should be larger than the hit-hit interval, because the probability

5 of a shot being successful after a hit diminishes over time. The means of 83 players’ intervals showed however that the mean hit-hit interval was slightly larger than the mean hit-miss interval, although the difference was not statistically significant. This doesn’t support the theory of a time-dependent hot hand.

Bar-Eli, Avugos and Raab (2006) didn’t do any empirical research in their paper, but they reviewed the most important research about the hot hand that had been done in the 20 years prior to their paper. Besides basketball, they also looked at papers about the hot hand in baseball, golf, darts, pocket billiards, tennis, volleyball, bowling and even horseshoe pitching. Out of the 25 studies they looked at, 13 of them didn’t support the hot hand theory and 11 studies showed some support in favor of the hot hand theory. There was also one inconclusive study, which was the one by Wardrop (1995) discussed earlier in this paper. When they closely examined the results of the studies, Bar-Eli et al. argued that “demonstrations of hot hands per se are rare and often weak ”.

Out of the 11 papers that supported the existence of the hot hand, 8 papers focused on different sports than basketball. The most supportive results were found in sports that are of an individual nature, namely horseshoe pitching (Smith, 2003), bowling (Dorsey-Palmateer & Smith, 2004) and tennis (Klaassen & Magnus, 2001). On the other hand, no support for the hot hand was found in other studies about sports in relatively pure settings, where factors like defensive pressure and shot selection are not present. This was the case for golf (Clark, 2003a, b; Clark, 2005), darts (Gilden & Wilson, 1995), and the annual three-point contest during the NBA All-Star Weekend (Koehler & Conley, 2003). Except for Stern (1995), who analyzed players as a group instead of individually, no evidence was found for the hot hand in baseball (Albert & Bennett, 2001; Albright, 1993; Frohlich, 1994; Siwoff et al., 1988; Vergin, 2000).

The data

Play-by-play data for all 1230 regular season games and all 81 play-off games from the 2014-2015 NBA season, collected by NBAstuffer.com, will be used for this study. This contains data for 219,266 field goal attempts and 60,248 free throws from 490 NBA players. Out of these field goal attempts, 59,276 shots were three-point field goal attempts. All shots from more than 30 feet away from the basket will be taken out of the data, because these are the typical desperation heaves at the end of a quarter that have a very small chance of going in and are not very informative for analyzing the hot hand. This leads to 983 shots being taken out of the data, and that leaves us with 218,283 field goal attempts of which 58,293 were from three-point range.

I also look at free throws, but only pairs of free throws will be looked at. Two free throws is the most common amount of free throws to be awarded to players, but it is also possible to get one or three free throws. When a player is fouled on a shot attempt but still hits the shot, he is awarded one free throw as a . Other scenarios where a player gets one free throw are technical fouls (inappropriate behavior towards referees, hanging on the rim too long after dunking, delay of game etc.) or defensive three second violations (staying too long in the painted area close to the basket) by the opposing team. The team that is awarded the free throw can then choose the player who they want to take the free throw. Situations where a player is awarded only one free throw are useless for hot hand analysis, because there is no second free throw of which the outcome could depend on the outcome of the first free throw. Therefore, situations where one free throw was awarded have been taken out of the data.

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Two free throws are awarded when a player is fouled on a shot attempt from within the three point line. Also, when a team collectively has committed four fouls in a quarter, or one in the last two minutes of a quarter, every next foul (except for offensive fouls) will lead to two free throws for the other team. This is called the bonus or penalty situation. When a player is fouled on a three-point shot, he is awarded three free throws, this happened 644 times in the 2014-2015 season. These cases have also been taken out, because analyzing the free throws gets much more complicated when some players have 3 x 3 tables rather than the 2 x 2 tables mentioned before. The third free throw also isn’t very informative because most players had very few third free throws, so the sample size would be too small. There are 25,887 pairs of free throws left to analyze.

The hot hand theory suggests that one-sided tests should be used for this study in order to test for enhanced performance after success, but since some earlier studies found that this enhanced performance did not take place and some players actually performed worse, I will use two-sided tests in this study.

Results and analysis: All shots

The field goal attempts by all NBA players who took at least 350 shots during the 2014-2015 season will be looked at first. The requirement of 350 shots is a bit arbitrary, as for example 300 or 400 could also have been used without harming or improving the study much, but 350 shots is an appropriate number to include many players without having too few observations for a player. This way, all but 4 players averaged more than 5 field goal attempts per game, with the average being 9.85. All observations of shots further than 30 feet away from the basket have been taken out. Table 1 shows the conditional probabilities and serial correlations for the 20 players that took the most shots, and 6 players that are considered to be streak shooters. Results for all 259 players that met the requirement of 350 shots can be found in Appendix 1. Players are sorted by total amount of shots, in descending order.

P(hit) represents the probability for each player that a shot went in, so that it is their average shooting percentage over the course of the season if you multiply it by 100%. The other columns with probabilities display the probability that a shot is made given a certain circumstance. For example, P(hit|3 misses) shows the probability the player in question hits a shot, given that he has missed his last 3 shots. The number of shots upon which the probabilities are based are given within parentheses. The result of the first shot in a game is regarded to be independent from the last shot of the previous game. This means that the number of shots in the P(hit|1 miss) and P(hit|1 hit) columns don’t add up to the number of shots in the P(hit) column, because the first shot of each game is not included in any of these two columns. The difference between the total number of shots from the P(hit|1 miss) and P(hit|1 hit) columns and the P(hit) column is therefore the number of games the player participated in.

The serial correlations show to which extent the outcome of a shot depends on the outcome of the previous shot. Just like the conditional probabilities, the outcome of the first shot of each game is not dependent on the outcome of the last shot of the previous game. If the outcomes of consecutive shots are positively correlated, a hit is more likely to be followed by a hit and a miss is more likely to be followed by a miss. Therefore, hot hand theory suggests a positive serial correlation.

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Table 1: Probability of making a shot conditioned on the outcome of previous shots, and the serial correlations. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Serial Player Team P(hit|3 misses) P(hit|2 misses) P(hit|1 miss) P(hit) P(hit|1 hit) P(hit|2 hits) P(hit|3 hits) Correlation LeBron James CLE .505 (192) .521 (430) .503 (916) .469 (1812) .431 (807) .419 (332) .398 (133) -.072 ** HOU .398 (249) .445 (481) .449 (923) .441 (1760) .438 (739) .424 (311) .456 (125) -.010 GSW .500 (168) .510 (386) .505 (852) .483 (1759) .457 (806) .439 (353) .381 (147) -.048 Klay Thompson GSW .500 (198) .459 (401) .469 (818) .461 (1629) .461 (713) .461 (317) .446 (139) -.008 LaMarcus Aldridge POR .447 (190) .481 (393) .467 (784) .456 (1525) .457 (665) .433 (293) .463 (123) -.010 Russell Westbrook OKC .408 (233) .419 (425) .431 (794) .427 (1466) .425 (605) .412 (245) .426 ( 94) -.006 DAL .443 (174) .493 (371) .467 (752) .448 (1461) .434 (624) .471 (259) .500 (118) -.033 POR .476 (208) .433 (397) .434 (758) .434 (1442) .419 (597) .387 (240) .420 ( 88) -.015 CLE .430 (142) .505 (325) .490 (702) .464 (1417) .437 (627) .439 (262) .454 (108) -.053 LAC .500 (150) .498 (315) .500 (664) .504 (1416) .519 (671) .514 (327) .465 (157) .019 LAC .543 (127) .498 (287) .498 (638) .488 (1355) .486 (623) .441 (288) .413 (121) -.012 NOP .578 ( 90) .579 (233) .551 (557) .537 (1283) .534 (654) .546 (324) .534 (161) -.018 CHI .548 (115) .563 (279) .523 (606) .494 (1267) .473 (573) .462 (249) .453 (106) -.050 WAS .467 (150) .466 (313) .467 (645) .444 (1260) .423 (529) .393 (214) .342 ( 79) -.043 Marc Gasol MEM .519 (106) .536 (252) .527 (594) .482 (1246) .445 (560) .421 (233) .422 ( 90) -.082 ** Nikola Vucevic ORL .557 ( 97) .520 (227) .539 (538) .524 (1205) .508 (593) .491 (283) .446 (130) -.031 NOP .440 (150) .440 (302) .449 (608) .446 (1199) .451 (508) .458 (214) .489 ( 92) .002 HOU .470 (132) .498 (299) .454 (613) .423 (1179) .382 (466) .376 (170) .365 ( 63) -.072 * ATL .551 (107) .500 (230) .506 (506) .534 (1173) .543 (575) .558 (276) .544 (136) .037 DAL .496 (131) .487 (277) .462 (572) .458 (1146) .447 (492) .446 (204) .400 ( 85) -.014 SAC .429 (133) .452 (279) .444 (554) .458 (1110) .471 (488) .502 (221) .519 (108) .027 Jamal Crawford LAC .426 (148) .416 (291) .412 (549) .398 (1000) .367 (373) .386 (127) .326 ( 46) -.045 J.R. Smith CLE .425 (113) .433 (240) .411 (482) .421 ( 943) .410 (373) .373 (142) .447 ( 47) -.001 NYK .543 ( 94) .520 (204) .484 (419) .446 ( 802) .397 (343) .380 (129) .413 ( 46) -.088 * Brandon Jennings DET .333 ( 78) .374 (147) .416 (293) .404 ( 540) .403 (206) .405 ( 79) .452 ( 31) -.014 Nick Young LAL .364 ( 77) .346 (136) .396 (265) .367 ( 471) .317 (164) .265 ( 49) .250 ( 12) -.080 The number of shots upon which each probability is based is given within parentheses. Players are sorted by total amount of shots, in descending order. * p < 0.05, ** p < 0.01.

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Perhaps the most noticeable result from Table 1 is that 22 out of the 26 serial correlations are negative, although just 4 of them are statistically significant at the 5% level. Negative serial correlations are the consequence of these 22 players having a higher probability to hit a shot after having missed their previous shot, than to hit a shot after having hit their previous shot. The 6 players at the bottom of the table that are considered to be streak shooters (Rudy Gay, Jamal Crawford, J.R. Smith, Carmelo Anthony, Brandon Jennings and Nick Young) don’t show a different pattern when it comes down to serial correlations or the probability of hitting a shot after having just missed or hit a shot. Gay, Smith, and Jennings did however show their best shooting percentage after having hit their last 3 shots, although that might not be that informative because of the lack of shots that meet this condition. For example, Brandon Jennings only took 31 shots that meet this condition, which means that had he hit one more of these shots, it would have improved his shooting percentage by more than 3%.

Table 2: Z-values for tests on the equality of proportions for all shots. (1) (2) (3) (4) Z-value Z-value Z-value P(hit|3 hits) – P(hit|2 hits) – P(hit|1 hit) – Player P(hit|3 misses) P(hit|2 misses) P(hit|1 miss) LeBron James -1.90 -2.80 ** -2.99 ** James Harden 1.08 -0.57 -0.41 Stephen Curry -2.12 * -1.94 -1.96 Klay Thompson -0.98 0.05 -0.31 LaMarcus Aldridge 0.28 -1.23 -0.37 Russell Westbrook 0.30 -0.17 -0.22 Monta Ellis 0.97 -0.55 -1.20 Damian Lillard -0.88 -1.14 -0.56 Kyrie Irving 0.38 -1.58 -1.93 Blake Griffin -0.61 0.39 0.68 Chris Paul -2.05 * -1.38 -0.43 Anthony Davis -0.67 -0.78 -0.61 Pau Gasol -1.41 -2.32 * -1.72 John Wall -1.82 -1.68 -1.48 Marc Gasol -1.35 -2.54 * -2.80 ** Nikola Vucevic -1.65 -0.64 -1.06 Tyreke Evans 0.74 0.39 0.06 Josh Smith -1.38 -2.55 * -2.36 * Al Horford -0.11 1.30 1.21 Dirk Nowitzki -1.39 -0.90 -0.47 Rudy Gay 1.39 1.13 0.88 Jamal Crawford -1.20 -0.57 -1.35 J.R. Smith 0.26 -1.15 -0.02 Carmelo Anthony -1.44 -2.49 * -2.43 * Brandon Jennings 1.16 0.46 -0.30 Nick Young -0.77 -1.03 -1.65 * p < 0.05, ** p < 0.01

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Comparing the proportions from the P(hit|3 hits) column with the P(hit|3 misses) column, the P(hit|2 hits) column with the P(hit|2 misses) column, and the P(hit|1 hit) column with the P(hit|1 miss) column leads to more evidence against the hot hand hypothesis. This is done by the test of proportions, which tests the equality of proportions. Doing this for all 26 players in Table 1 gives 78 z- values, and these are displayed in Table 2. A positive z-value means that a player shot better after hitting for example his last 3 shots, than after missing his last 3 shots. All of the 20 positive z-values that correspond to positive differences in probabilities in these comparisons are statistically insignificant though, and 11 of the negative t-values are statistically significant.

Analysis of the shots of all 259 players with at least 350 shots yields the same results. Out of the 20 significant serial correlations, 17 are negative and the weighted average (where the weight is the number of shots taken by a player) serial correlation is -.018, which is significant at the 1% level. 174 players shot better when they had missed the last shot than when they had made it. Comparing columns in the same way as mentioned in the previous paragraph for all players as a group, only the comparison between P(hit|1 hit) and P(hit|1 miss) leads to a significant negative difference (z = - 3.97). The other 2 comparisons are also negative, but not significant (z = -1.95 for comparing P(hit|2 hits) and P(hit|2 misses), and z = -1.81 for comparing P(hit|1 hit) and P(hit|1 miss)). This could be down to players with a low shooting percentage being more prevalent in the columns of P(hit|3 misses) and P(hit|2 misses), and players with a high shooting percentage in the columns of P(hit|2 hits) and P(hit|3 hits).

Runs test: All shots

The second test that is performed to empirically analyze the hot hand phenomenon is the runs test. Sequences of consecutive hits and misses are considered to be one run. For example, a series of shots like HHH M HH MMMM consists of 4 runs. Hot hand theory suggests that hits and misses cluster together, hence the observed number of runs should be smaller than the number of runs that is expected under the assumption that the outcome of a shot is independent of previous shots. This is closely related to serial correlation, and therefore fewer runs than expected indicates positive serial correlation and vice-versa. The formula for the expected number of runs is

.

Table 3 displays the results for this runs test. As expected, the same players that had a significant negative serial correlation also had significantly more runs than expected. Out of these 26 players, 20 players had more runs than expected, contrary to the hot hand hypothesis.

Stationarity: All shots

Gilovich et al. (1985) note in their paper that it could be argued that the runs test and the serial correlations test are not strong enough to detect occasional hot runs, embedded in longer runs of normal shooting performances. They tried to create a more sensitive test of stationarity, or a constant hit rate. The shooting record of each player is divided into sets of four shots. Each set is then classified as a set of low, moderate or high performance. Sets with zero or one hit are classified as low performance, sets with two hits as moderate performance, and sets with three or four hits as high performance.

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Table 3: Runs test for all field goal attempts. (1) (2) (3) (4) (5) (6) Number Expected Player Hits Misses of runs number of runs Z-value LeBron James 850 962 971 903.54 3.18 ** James Harden 776 984 879 868.71 0.50 Stephen Curry 849 910 920 879.44 1.94 Klay Thompson 751 878 820 810.55 0.47 LaMarcus Aldridge 696 829 776 757.70 0.94 Russell Westbrook 626 840 721 718.38 0.14 Monta Ellis 654 807 742 723.49 0.98 Damian Lillard 626 816 712 709.48 0.13 Kyrie Irving 658 759 741 705.90 1.88 Blake Griffin 714 702 702 708.95 -0.37 Chris Paul 661 694 686 678.10 0.43 Anthony Davis 689 594 643 638.98 0.23 Pau Gasol 626 641 661 634.41 1.49 John Wall 560 700 644 623.22 1.19 Marc Gasol 601 645 672 623.22 2.77 ** Nikola Vucevic 631 574 614 602.15 0.68 Tyreke Evans 535 664 591 593.56 -0.15 Josh Smith 499 680 612 576.61 2.11 * Al Horford 626 547 558 584.84 -1.58 Dirk Nowitzki 525 621 581 569.98 0.66 Rudy Gay 508 602 537 552.02 -0.91 Jamal Crawford 398 602 497 480.19 1.11 J.R. Smith 397 546 460 460.73 -0.05 Carmelo Anthony 358 444 425 397.39 1.97 * Brandon Jennings 218 322 257 260.99 -0.36 Nick Young 173 298 239 219.91 1.89 * p < 0.05, ** p < 0.01.

The proportions of low, moderate and high performance sets expected by chance based on their overall shooting record can be calculated by using the formula for binomial probabilities. This formula is . The number of trials is represented by n, which is

4 in this case, and x is the number of successes, so in this case the number of shots made out of the set of four shots. The p stands for the shooting percentage of the player in question. b(x; n, p) is then the binomial probability that the player hits exactly x shots in the set of 4 shots, given that his probability of hitting a shot is p. The expected proportion of low sets is the sum of the binomial probabilities for x = 0 and x = 1, and the expected proportion of high sets is the sum of the binomial probabilities for x = 3 and x = 4. So a player whose probability of hitting a shot is 0.4, has expected proportions of low, moderate and high performance sets of 0.475, 0.346 and 0.179, respectively.

Multiplying these proportions with the total number of sets for each player leads to the expected number of low, moderate and high sets. This was done for all 259 players with at least 350 field goal attempts, and repeated four times for each player, starting the dividing into sets of four shots at the

11 first, second, third and fourth shot. In order to find support for the hypothesis that the shooting performances are non-stationary, players should have more sets of high performance than expected by chance.

The χ² test for goodness of fit leads to 1036 χ²-values, of which only 21 are significant. The formula for the χ²-value is

. Out of these 21 cases where the number of low, moderate and high sets deviated significantly from what was expected, only 2 are down to a higher amount of high runs than expected. The other 19 are down to more moderate sets than expected, and less low and high runs than expected. The player that shows the most signs of streakiness is Jusuf Nurkic, who had on average 5 more high sets than expected. Only one of his χ²-values was significant though. Looking at his serial correlation in Appendix 1, he also showed a serial correlation of .109, which was almost significant at the 5% level with a t-value of 1.95, suggesting that he is a streak shooter. The general picture however shows no support for the hot hand. Because of the size of the tables, the results for the test of stationarity for all 26 players included in Table 1 can be found in Appendix 2.

Results and analysis: Three point shots

The three point shot is becoming increasingly important in the NBA, as the total amount of three pointers taken has been increasing every year. None of the hot hand research so far seems to have focused on whether the results change if you look at long distance shots though. Players that are considered to be streak shooters are typically players that take many three pointers, and are practically never players that mainly take shots close to the basket. The commentator of a game is not likely to say that a player is ‘on fire’ after hitting 3 dunks or lay-ups, but he probably will make a hot hand related statement when a player hits 3 consecutive three point shots. One could also argue that it is more logical for a player to consider himself to have the hot hand after successfully shooting a few three pointers than after hitting a few relatively easy shots, because he will expect to make most of the easy shots anyway. It is therefore interesting to analyze whether three point field goal attempts are more supportive of the hot hand than analysis of all shots has proven to be.

The results for the three point shots, displayed in Table 4, have been obtained in the same way as the results for all shots. Again, all shots further away from the basket than 30 feet have been taken out, even though these are all three point shots. Table 4 includes the 22 players that took the most three pointers in the 2014-2015 NBA season, and the 4 streak shooters mentioned before, who can be found at the bottom of the table. The other 2 streak shooters, J.R. Smith and Jamal Crawford, took enough three pointers to be part of the 22 most frequent three point shooters. In Appendix 3, results for all 164 players that took at least 150 three point shots can be found.

Looking at the serial correlations in Table 4, there is only one player with a significant correlation; Damian Lillard. This serial correlation is positive, so Lillard can be regarded as a streak shooter from three point range. There are 9 players in the table with a positive serial correlation, and 17 with a negative serial correlation so this still doesn’t support hot hand theory. If the serial correlations for these 26 player are compared with their serial correlations of when all shots were taken into account, the difference is positive for 17 players. Even though this is a comparison of mainly insignificant serial correlations, it does suggest that the hot hand hypothesis fares slightly better

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Table 4: Probability of making a three point shot conditioned on the outcome of previous three point shots, and the serial correlations. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Serial Player Team P(hit|3 misses) P(hit|2 misses) P(hit|1 miss) P(hit) P(hit|1 hit) P(hit|2 hits) P(hit|3 hits) Correlation Stephen Curry GSW .379 (87) .414 (191) .447 (414) .443 (857) .436 (342) .435 (131) .440 (50) -.011 Klay Thompson GSW .492 (65) .428 (152) .416 (329) .431 (687) .462 (260) .429 (105) .324 (37) .045 James Harden HOU .337 (89) .322 (171) .376 (343) .378 (658) .378 (217) .348 ( 66) .474 (19) .002 HOU .407 (81) .359 (170) .364 (343) .354 (650) .325 (209) .404 ( 57) .333 (18) -.040 Damian Lillard POR .305 (95) .299 (184) .297 (323) .336 (589) .391 (179) .419 ( 62) .375 (24) .096 * ATL .447 (47) .482 (112) .484 (258) .466 (558) .458 (212) .397 ( 73) .368 (19) -.027 J.R. Smith NYK .371 (62) .375 (128) .383 (269) .387 (550) .381 (194) .406 ( 64) .348 (23) -.002 JJ Redick LAC .468 (47) .427 (110) .438 (251) .430 (546) .397 (204) .309 ( 68) .143 (14) -.042 SAS .365 (52) .376 (117) .377 (244) .408 (495) .439 (164) .423 ( 52) .667 (12) .062 TOR .288 (52) .331 (121) .326 (242) .338 (462) .314 (137) .256 ( 39) .375 ( 8) -.013 Wesley Matthews POR .460 (50) .434 (113) .398 (231) .389 (445) .396 (154) .442 ( 52) .389 (18) -.002 Robert Covington PHI .271 (48) .343 (108) .383 (227) .377 (443) .367 (147) .442 ( 43) .385 (13) -.016 LeBron James CLE .391 (46) .366 (112) .346 (231) .329 (438) .269 (119) .185 ( 27) .250 ( 4) -.079 CJ Miles IND .286 (63) .314 (121) .338 (231) .352 (437) .387 (137) .468 ( 47) .368 (19) .050 Kentavious Caldwell-Pope DET .364 (44) .337 (104) .363 (223) .349 (436) .336 (131) .316 ( 38) .250 (12) -.028 Kyrie Irving CLE .400 (30) .329 ( 79) .362 (188) .421 (435) .438 (160) .452 ( 62) .500 (26) .077 LAC .390 (41) .388 (103) .349 (218) .349 (435) .313 (128) .375 ( 32) .400 (10) -.037 Draymond Green GSW .393 (28) .276 ( 76) .338 (195) .324 (417) .301 (123) .172 ( 29) .000 ( 4) -.039 CLE .290 (31) .341 ( 88) .350 (203) .370 (413) .414 (133) .367 ( 49) .500 (14) .064 Jamal Crawford LAC .400 (45) .396 (106) .350 (220) .329 (413) .287 (115) .310 ( 29) .125 ( 8) -.064 Chris Paul LAC .409 (22) .422 ( 64) .440 (182) .400 (410) .366 (134) .286 ( 42) .182 (11) -.074 TOR .298 (47) .317 (104) .332 (214) .333 (408) .308 (120) .273 ( 33) .143 ( 7) -.024 Nick Young LAL .318 (22) .385 ( 52) .391 (110) .373 (220) .368 ( 68) .391 ( 23) .444 ( 9) -.023 Rudy Gay SAC .444 ( 9) .371 ( 35) .374 ( 91) .369 (214) .414 ( 58) .286 ( 21) .000 ( 3) .040 Brandon Jennings DET .263 (19) .341 ( 44) .380 (100) .365 (208) .377 ( 69) .440 ( 25) .300 (10) -.003 Carmelo Anthony NYK .188 (16) .282 ( 39) .321 ( 84) .349 (175) .353 ( 51) .429 ( 14) .333 ( 3) .032 The number of shots upon which each probability is based is given within parentheses. Players are sorted by total amount of shots, in descending order. * p < 0.05, ** p < 0.01.

13 when only 3 point shots are taken into account. Besides Lillard, the most striking example of this is Carmelo Anthony, who went from having a significant negative serial correlation, to having an insignificant positive serial correlation.

The test on the equality of proportions has also been done for three point shots, and the results for this are displayed in Table 5. Comparing the proportions of the P(hit|3 hits) column with the P(hit|3 misses) column may not be very informative for the three point shots, simply because there are too few observations for most players. There is however one significant result, JJ Redick shoots significantly better after missing his last 3 three point shots than after making his last 3 three point shots. Comparisons between the P(hit|2 hits) column with the P(hit|2 misses) column yield no significant results. Not surprisingly, when the P(hit|1 hit) column and the P(hit|1 miss) column are compared with each other, the only significant result is for Damian Lillard, who shoots significantly better after a hit than after a miss.

Table 5: Z-values for tests on the equality of proportions for three point shots. (1) (2) (3) (4) Z-value Z-value Z-value P(hit|3 hits) – P(hit|2 hits) – P(hit|1 hit) – Player P(hit|3 misses) P(hit|2 misses) P(hit|1 miss) Stephen Curry 0.70 0.38 -0.31 Klay Thompson -1.65 0.01 1.10 James Harden 1.13 0.39 0.04 Trevor Ariza -0.58 0.60 -0.93 Damian Lillard 0.65 1.75 2.14 * Kyle Korver -0.58 -1.13 -0.58 J.R. Smith -0.20 0.42 -0.03 JJ Redick -2.19 * -1.58 -0.89 Danny Green 1.90 0.58 1.25 Lou Williams 0.50 -0.87 -0.25 Wesley Matthews -0.52 0.10 -0.04 Robert Covington 0.80 1.14 -0.31 LeBron James -0.56 -1.79 -1.47 CJ Miles 0.69 1.87 0.95 Kentavious Caldwell-Pope -0.74 -0.23 -0.52 Kyrie Irving 0.75 1.49 1.44 Matt Barnes 0.06 -0.14 -0.69 Draymond Green -1.55 -1.10 -0.70 Kevin Love 1.36 0.31 1.18 Jamal Crawford -1.49 -0.85 -1.17 Chris Paul -1.31 -1.42 -1.32 Kyle Lowry -0.85 -0.48 -0.44 Nick Young 0.67 0.05 -0.31 Rudy Gay -1.41 -0.66 0.49 Brandon Jennings 0.21 0.82 -0.04 Carmelo Anthony 0.73 1.01 0.38 * p < 0.05, ** p < 0.01

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Out of the 164 players who shot at least 150 three pointers, only 8 players have a significant serial correlation. Only Damian Lillard’s correlation is positive, the other 7 are all negative. The weighted mean serial correlation is -.025, which is significant at the 1% level. When the columns in Table 5 are compared in the same way as mentioned above for all 164 players as a group, it shows that the players shoot significantly better after missing 3 consecutive shots than after making 3 consecutive shots (z = -2.13), and better after a miss than after a hit (z = -3.53). The comparison between shooting percentages after 2 hits and 2 misses is also negative, but not significant (z = -1.93). Again though, players with lower shooting percentages are likely to be relatively dominant in columns of P(hit|3 misses) and P(hit|2 misses), and players with a high shooting percentage in the columns of P(hit|2 hits) and P(hit|3 hits).

Table 6: Runs test for three point shots. (1) (2) (3) (4) (5) (6) Number Expected Player Hits Misses of runs number of runs Z-value Stephen Curry 380 477 431 424.01 0.48 Klay Thompson 296 391 324 337.93 -1.08 James Harden 249 409 321 310.55 0.87 Trevor Ariza 230 420 306 298.23 0.67 Damian Lillard 198 391 240 263.88 -2.21 * Kyle Korver 260 298 290 278.71 0.96 J.R. Smith 213 337 267 262.02 0.45 JJ Redick 235 311 285 268.71 1.42 Danny Green 202 293 229 240.14 -1.04 Lou Williams 156 306 210 207.65 0.24 Wesley Matthews 173 272 209 212.49 -0.35 Robert Covington 167 276 217 209.09 0.80 LeBron James 144 294 211 194.32 1.81 CJ Miles 154 283 193 200.46 -0.78 Kentavious Caldwell-Pope 152 284 208 199.02 0.95 Kyrie Irving 183 252 199 213.03 -1.38 Matt Barnes 152 283 203 198.77 0.45 Draymond Green 135 282 192 183.59 0.94 Kevin Love 153 260 184 193.64 -1.02 Jamal Crawford 136 277 195 183.43 1.29 Chris Paul 164 246 211 197.80 1.36 Kyle Lowry 136 272 186 182.33 0.41 Nick Young 82 138 108 103.87 0.60 Rudy Gay 79 135 96 100.67 -0.69 Brandon Jennings 76 132 93 97.46 -0.67 Carmelo Anthony 61 114 75 80.47 -0.91 * p < 0.05, ** p < 0.01.

Runs test: Three point shots

Just like for the data set with all shots within 30 feet, a runs test has been performed for all three point shots within 30 feet from the basket from the same players noted in Table 4. The results for this runs test are displayed in Table 6. Based on the serial correlations, it was to be expected that

15 only Damian Lillard had significantly less runs than expected under the assumption that shots are independent of each other. Only 10 of these 26 players had less runs than expected and 16 teams had more runs than expected, which doesn’t support the hot hand hypothesis.

At the moment of writing, J.R. Smith is probably the most notorious streak shooter in the NBA. He has the reputation to be unstoppable at times, hitting seemingly impossible shots, and he also just can’t seem to hit a shot in some games. Looking at the results from Tables 1 to 6, he doesn’t appear to be a streak shooter at all though. All of his serial correlations are close to zero, suggesting that the outcome of his shots are independent of one another. The champions of the 2014-2015 NBA season, Golden State Warriors, had the ‘Splash Brothers’ at their disposal. This duo consists of Stephen Curry and Klay Thompson, who was mentioned in the introduction because he set a new NBA record for most points in a quarter. They are two of the best three point shooters in the NBA, and while they are not necessarily considered streak shooters, they are commonly said to have the hot hand during a game by the commentators. Their results don’t show any support for the hot hand either. The same holds for the other streak shooters with a similar reputation to J.R. Smith.

Stationarity: Three point shots

A test for stationarity has also been performed for three point shots, but this time the shooting records of each player had been divided into sets of three shots. The shooting percentage for three point shots is lower than for all shots, and to reflect this sets of shots with zero hits are classified as low performance, sets with one hit as moderate performance, and sets with two or three hits as high performance. This has been done for all players in Table 3 and 4, but the only significant χ²-value is down to Jamal Crawford having more moderate sets than expected, and less low and high sets than expected. The results for this test for all 26 players that are included in Table 4 can be found in Appendix 4.

Results and analysis: Teams

Momentum swings are often observed during basketball games. In almost every game you will see a team score a few times in a row with the opponent not being able to score, and every now and then a team even wins a game after being in a seemingly hopeless situation earlier in the game. In the discussion part of their paper, Gilovich et al. (1985) note that the intuition that a player has the hot hand may not only be based on his shooting performance, but also on factors such as his defense, hustling and passing. Good performances in these parts of the game may be overgeneralized to shooting and therefore enhance the perception that a player is “hot”. Even though these other factors don’t necessarily improve the shooting performance of the player in question, it perhaps does improve the performance of his teammates.

Since basketball is a team-based game, success of a player might not only boost his own confidence level, but also that of the other players on a team. Hitting the shot is obviously the most crucial part of completing an offensive possession successfully, but the other players on the team contributed to this in several ways, such as setting screens, creating space and passing the ball. Therefore, one could look at a shot as it being a team effort and that along the lines of hot hand theory, success breeds success and failure breeds failure across the entire team. This leads to the question: “do teams get hot?”

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Table 7 shows the results for the serial correlations and the runs test for all 30 NBA teams. 21 teams show a negative serial correlation between shots, and 9 a positive serial correlation. All 5 correlations that are significant are negative. The runs test results in the same pattern, 21 teams had more runs and 9 teams had fewer runs than expected. The same 5 teams that had significant negative serial correlations also had significantly more runs than expected under the hypothesis that successive shots are independent of each other. According to these results, hits and misses of a team don’t cluster together and thus don’t support the hot hand hypothesis. If anything, the results rather show that it’s the other way around.

Table 7: Serial correlations and runs test for all NBA teams. (1) (2) (3) (4) (5) (6) (7) Expected Serial Number number of Team Correlation Hits Misses of runs runs Z-value .009 3711 4340 3968 4001.93 -0.76 -.010 3331 4183 3743 3709.70 0.78 Boston Celtics -.007 3286 3974 3633 3598.40 0.82 .002 2912 3980 3354 3364.25 -0.25 .000 3419 4342 3827 3826.61 0.01 -.011 3780 4577 4182 4141.50 0.89 .001 3462 4014 3716 3718.62 -0.06 -.011 3094 4025 3538 3499.62 0.93 Detroit Pistons -.014 3039 3961 3495 3440.28 1.33 Golden State Warriors -.005 4207 4648 4441 4417.52 0.50 .004 3665 4547 4045 4059.63 -0.33 -.019 2996 3796 3408 3349.89 1.43 -.015 3778 4197 4039 3977.49 1.38 -.028 * 3052 3946 3528 3442.90 2.07 * -.034 ** 3486 4164 3924 3795.96 2.95 ** -.008 2883 3420 3156 3129.62 0.67 .007 3288 3942 3561 3586.42 -0.60 -.031 * 2985 3812 3456 3349.19 2.63 ** .014 3256 3842 3483 3525.81 -1.02 -.014 2879 3811 3326 3281.08 1.12 Oklahoma City Thunder .001 3181 3916 3503 3511.44 -0.20 -.010 3075 3692 3386 3356.37 0.73 Philadelphia 76ers .010 2763 3979 3229 3262.34 -0.84 -.002 3175 3828 3474 3472.06 0.05 Portland Trailblazers -.008 3339 4087 3708 3676.33 0.74 Sacramento Kings .001 3007 3571 3259 3265.82 -0.17 -.025 * 3474 3951 3795 3698.18 2.26 * -.032 ** 3254 3901 3662 3549.25 2.69 ** -.013 2899 3561 3238 3197.08 1.03 -.005 3509 4091 3800 3778.72 0.49 * p < 0.05, ** p < 0.01.

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Results and analysis: Free throws

Since free throws are shot within a ‘cleaner’ context than shots during regular play, they offer a way to test the hot hand hypothesis without having to deal with factors such as defensive pressure and shot selection. In the survey conducted by Gilovich et al. (1985), the 100 basketball fans were asked to estimate the probability that a 70% free throw shooter would hit his second free throw after missing or hitting the first one. The average estimate was 66% after missing, and 74% after hitting the first free throw.

Hot hand theory suggests that a player should hit a higher proportion of his second free throws after hitting the first, than after missing the first. As mentioned in the description of the data, all situations where either one or three free throws were awarded have been taken out. For all 291 players with at least 25 pairs of free throws, the weighted average second free throw percentage (where the weight is the number of free throws) is 3.83% higher than the weighted average first free throw percentage (76.89% – 73.06% = 3.83%), which may be down to the player getting to practice the exact same shot first. Looking at the unweighted averages, for which the number of free throws taken by each player is not taken into account, the difference between the second and the first free throw is 76.77% – 72.28% = 4.49%. Even though there is no constant hit rate between the first and second free throw, it is still interesting to look at the outcomes of second free throws, conditioned on the outcome of the first free throws.

Table 8 presents the results for all 33 players with at least 150 pairs of free throws. The third and fourth column display the probability the second free throw was hit after missing the first, and the probability the second free throw was hit after hitting the first. The fifth column again shows the correlation between the first and the second free throw. Only 2 serial correlations are significant, namely the negative correlation of and the positive correlation of . No support for the hot hand is apparent in the correlations, as 18 of them are positive and 15 are negative.

Perhaps for most NBA players taking free throws has become a routine that is not affected much by the intuition of having the hot hand. Especially for good free throw shooters, hitting the first free throw probably doesn’t boost their confidence much. There are a few players however, that are just so bad at shooting free throws that the opposing team decides to intentionally foul them, because the expected points scored from the awarded two free throws is lower than the expected points scored from regular play. This so-called ‘Hack-a-Shaq’ strategy, named after low percentage free throw shooter Shaquille O’Neal, is sometimes deployed when a team is in the penalty situation. A team is in the penalty situation when the team collectively has committed four fouls in a quarter, or one foul in the last two minutes of a quarter. Every next foul (except for offensive fouls) will then lead to two free throws for the other team, which can be rewarding for the fouling team if they are taken by a bad free throw shooter. For example, if the opposing team has a player that hits just 40% of his free throws, the expected points he will score from two free throws is 0.8. The average points per possession for NBA teams during the 2014-2015 NBA season was 1.03, so the expected points scored from these two free throws is lower and it may therefore be beneficial to foul the bad free throw shooter and give him two free throws. It may be that these bad free throw shooters react more sensitive to hitting or missing the first free throw.

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The players in this sample that during the 2014-2015 season have frequently been victims of the Hack-a-Shaq strategy are DeAndre Jordan, Dwight Howard, Andre Drummond and Josh Smith. As noted before, Dwight Howard shot his second free throw significantly better after hitting the first free throw, than after missing the first. The same can be said about the other players, but for them the results are not significant. Andre Drummond does however shoot more than 10% better after hitting than after missing the first free throw.

Table 8: Probability of hitting a free throw, and the probability of making the second free throw conditioned on the outcome of the first free throw, and the serial correlations. (1) (2) (3) (4) (5) (6) Pairs of Serial Player free throws P(hit) P(hit|first miss) P(hit|first hit) Correlation James Harden 412 .876 .833 (54) .891 (358) .061 LeBron James 298 .706 .773 (97) .721 (201) -.055 DeAndre Jordan 291 .405 .455 (189) .471 (102) .015 Russell Westbrook 277 .836 .804 (51) .867 (226) .070 DeMarcus Cousins 239 .803 .750 (52) .845 (187) .103 239 .837 .800 (40) .849 (199) .050 Marc Gasol 236 .797 .696 (46) .811 (190) .111 Blake Griffin 233 .736 .754 (61) .727 (172) -.027 Anthony Davis 212 .802 .826 (46) .819 (166) -.007 Andrew Wiggins 211 .756 .875 (56) .742 (155) -.141 * Gordon Hayward 204 .831 .824 (34) .829 (170) .006 Dwight Howard 196 .480 .388 (98) .531 (98) .143 * DeMar DeRozan 192 .828 .857 (35) .834 (157) -.024 184 .796 .667 (42) .775 (142) .105 Pau Gasol 184 .761 .833 (42) .817 (142) -.018 179 .813 .853 (34) .807 (145) -.047 LaMarcus Aldridge 175 .863 .818 (22) .856 (153) .035 174 .856 .906 (32) .894 (142) -.015 John Wall 173 .795 .806 (36) .796 (137) -.010 Damian Lillard 170 .856 .897 (29) .879 (141) -.020 Rudy Gay 168 .869 .889 (27) .887 (141) -.003 Kyrie Irving 168 .863 .783 (23) .890 (145) .111 Stephen Curry 168 .890 .909 (22) .911 (146) .002 Andre Drummond 165 .373 .296 (98) .403 (67) .111 Chris Paul 165 .912 .947 (19) .938 (146) -.012 Giannis Antetokounmpo 165 .736 .647 (34) .687 (131) .035 Dwyane Wade 161 .761 .780 (41) .775 (120) -.006 159 .748 .683 (41) .780 (118) .098 155 .871 .714 (49) .783 (106) .075 Isaiah Thomas 155 .723 .864 (22) .887 (133) .026 Josh Smith 153 .484 .481 (81) .514 (72) .032 Tristan Thompson 152 .625 .625 (56) .615 (96) -.010 Greg Monroe 151 .752 .853 (34) .692 (117) -.151 The number of shots upon which each probability is based is given within parentheses. Players are sorted by total amount of pair of free throws, in descending order. * p < 0.05, ** p < 0.01.

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Wardrop (1995) argued that the belief among the surveyed basketball fans that players shoot their second free throw better after hitting the first free throw than after missing it, stems from the inability to remind 2 x 2 tables (hits and misses for the first and second free throw) for each player. Instead, people have a 2 x 2 table in mind for all players combined. A collapsed 2 x 2 table however is not appropriate to analyze whether players shoot better after making the first free throw than after missing it. This is because the shooting results of bad free throw shooters are relatively dominant in the weighted average second free throw percentage conditioned on missing the first free throw, and the shooting results of good free throw shooters are relatively dominant in the weighted average second free throw percentage conditioned on hitting the first free throw.

When looking at the collapsed table for the 33 players from Table 6, the weighted average second free throw percentage (where the number of free throws taken by a player is again the weight) turns out to be 79.8% – 68.2% = 11.6% higher after hitting the first free throw than after missing it. This may be slightly misleading due to the presence of the Hack-a-Shaq targets in this sample, whose free throw percentages will be relatively dominant in the weighted average second free throw percentage conditioned on missing the first free throw. Apart from the 4 players in this sample, there are no other NBA players that are target of the Hack-a-Shaq strategy on a regular basis. If all 291 players with at least 25 pairs of free throws are taken into account, the difference diminishes to 7.4%. While there was no pattern in the data, Wardrop (1995) argued that there was a pattern in the fan’s data, however this was down to aggregation and not the hot hand phenomenon.

Discussion

The main flaw of this type of research is that the context of a shot is unknown, except for the shot distance there is no information about the difficulty of the shot and the defensive pressure. Gilovich et al. (1985) note that each player has an ensemble of shots that vary in difficulty, from which each shot is randomly selected. The assumption that each shot is randomly selected from this ensemble is inevitable with this kind of data, but could also lead to false conclusions if this assumption is violated.

If a player is perceived to have the hot hand, the opposing team may adjust their defense accordingly, increasing the difficulty of the ‘hot’ player’s shots. This effect is possibly smaller in the NBA than at the amateur level or any other lower competition, since NBA players are probably better aware of the opponents qualities because they play them several times a year and because they are analyzed before a game. Still, if a player hits a few consecutive shots, NBA teams may try to defend him more intensively so it forces him to pass the ball or into taking difficult shots.

Another factor that could violate the assumption that players randomly select a shot is that players may take more difficult shots if they feel they have the hot hand. When a player feels he’s hot, he sometimes takes a ‘heat check’ shot. This tends to be a difficult, long range shot to test whether he is really ‘on fire’. So if players take more difficult shots after a run of hits, it will negatively affect their serial correlations and shooting percentages after one or more hits. Via the same line of thought, players that have missed several shots in a row may opt for easier shots that they are more likely to hit.

Gilovich et al (1985) also interviewed the 9 players they analyzed of the 1980-1981 Philadelphia 76ers about their beliefs regarding the hot hand. Almost all of them said they would take more shots than they normally would after a few consecutive hits, and all of the players agreed that the ball

20 should be passed to the player who appears to have the hot hand. This suggests that players indeed do not randomly select a shot from their ensemble of shots.

In a recent study, Bocskocsky, Ezekowitz and Stein (2014) tried to overcome this problem. They had access to data from the SportsVU optical tracking system, which uses 6 camera’s to take pictures of the ball, referees and players every 1/25th of a second. They use this to create a model that predicts the difficulty of a certain shot by a certain player, based on several factors that influence this. These are divided into 4 categories; game condition controls, shot controls, defensive controls and player fixed effects. Game condition controls refer to factors like time remaining and the score differential between the teams. These factors may affect the effort put in by players, pressure and fatigue. Shot controls measure the difficulty of a shot based on shot distance and the type of shot (such as a lay- up, hook shot, fade-away etc.). Defensive controls are used to measure the defensive intensity, and player fixed effects controls for differences between players. If ‘Splash Brother’ Stephen Curry and bad free throw shooter Andre Drummond take an identical three point shot, Curry is much more likely to hit it. This is captured by the player fixed effects.

Bocskocsy et al. (2014) first test whether players still believe in the hot hand. This seems to be the case, since they computed that players take shots further away from the basket, are defended more closely by the opponent, and are more likely to take the next shot if they become hot. They then go on to show that, controlling for the factors mentioned above, a player’s shooting percentage increases by 0.54 percentage points if he makes one more of his past four shots. Given the average shooting percentage in the NBA of about 45%, this comes down to an improvement of about 1.2%. If a player makes two more of his past four shots, he improves by 2.4%. These results are significant, but Bocskocsy et al. (2014) also note that the OLS regression they used as the functional form of choice might not be optimal and that it is possible that the estimates of the standard errors are imprecise, which could cause the small improvements to be insignificant. They argue however that their results at the very least cast doubt on the consensus that the hot hand is a fallacy.

Gilovich et al.’s (1985) solution to the problem that the shot selection of players is not random was to do a controlled shooting experiment with the men and women basketball teams from Cornell University. The results of the experiment didn’t significantly support the hot hand, although the hot hand hypothesis fared slightly better than under the results for the Philadelphia 76ers players. So both the studies of Bocskocsy et al. (2014) and Gilovich et al. (1985) tried to eliminate the effects of shot selection and defensive pressure, yet the results contradict each other. Perhaps the intuition of having the hot hand is stronger in a competitive situation like a basketball game than in a situation that looks more like an individual practice.

Conclusion

Analyzing shooting results from the 2014-2015 NBA season in a similar way as Gilovich et al.’s (1985) study, hardly any evidence in favor of the hot hand was found. The most notable hot hand cases were Dwight Howard’s free throws and Damian Lillard’s three pointers, but the majority of significant results was against hot hand theory. Considering all NBA players that took at least 350 shots have been looked at, it is unlikely that there are streak shooters in the NBA that have not been included in this study.

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The results of this study suggest that the hot hand is a fallacy, although recent research with access to more sophisticated data show there is actually a small improvement in shooting percentage after hitting one more of the past four shots. This improvement is so small though (about 1.08 percentage point after hitting two more of the past four shots), that it is unlikely that the public is able to spot this to the extent that it makes them believe so strongly in the hot hand as the results from Gilovich et al.’s (1985) survey suggest. Further research using new and more sophisticated ways to collect data may prove that the hot hand is not merely a cognitive illusion, but basketball fans surely overrate it. If players commonly pass the ball to the player who has the hot hand and he indeed slightly improves his shooting performance as Bocskocsky et al. (2014) found, this may still be costly to do so. The improvement is offset by shot selection and defensive pressure influences, which follows from the results in this study that show that most players shoot worse after hitting one or more shots than after missing one or more shots. Other players, who appear to be less ‘hot’, would probably have had a better chance of making the shot.

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23

Appendix

Appendix 1: Probability of making a shot conditioned on the outcome of previous shots and serial correlations for all 259 NBA players with at least 350 shots. (1) (2) (3) (4) (5) (6) (7) (8) (9) Serial Player P(hit|3 misses) P(hit|2 misses) P(hit|1 miss) P(hit) P(hit|1 hit) P(hit|2 hits) P(hit|3 hits) Correlation LeBron James .505 (192) .521 (430) .503 (916) .469 (1812) .431 (807) .419 (332) .398 (133) -.072 ** James Harden .398 (249) .445 (481) .449 (923) .441 (1760) .438 (739) .424 (311) .456 (125) -.010 Stephen Curry .500 (168) .510 (386) .505 (852) .483 (1759) .457 (806) .439 (353) .381 (147) -.048 Klay Thompson .500 (198) .459 (401) .469 (818) .461 (1629) .461 (713) .461 (317) .446 (139) -.008 LaMarcus Aldridge .447 (190) .481 (393) .467 (784) .456 (1525) .457 (665) .433 (293) .463 (123) -.010 Russell Westbrook .408 (233) .419 (425) .431 (794) .427 (1466) .425 (605) .412 (245) .426 ( 94) -.006 Monta Ellis .443 (174) .493 (371) .467 (752) .448 (1461) .434 (624) .471 (259) .500 (118) -.033 Damian Lillard .476 (208) .433 (397) .434 (758) .434 (1442) .419 (597) .387 (240) .420 ( 88) -.015 Kyrie Irving .430 (142) .505 (325) .490 (702) .464 (1417) .437 (627) .439 (262) .454 (108) -.053 Blake Griffin .500 (150) .498 (315) .500 (664) .504 (1416) .519 (671) .514 (327) .465 (157) .019 Chris Paul .543 (127) .498 (287) .498 (638) .488 (1355) .486 (623) .441 (288) .413 (121) -.012 Anthony Davis .578 ( 90) .579 (233) .551 (557) .537 (1283) .534 (654) .546 (324) .534 (161) -.018 Pau Gasol .548 (115) .563 (279) .523 (606) .494 (1267) .473 (573) .462 (249) .453 (106) -.050 John Wall .467 (150) .466 (313) .467 (645) .444 (1260) .423 (529) .393 (214) .342 ( 79) -.043 Marc Gasol .519 (106) .536 (252) .527 (594) .482 (1246) .445 (560) .421 (233) .422 ( 90) -.082 ** Nikola Vucevic .557 ( 97) .520 (227) .539 (538) .524 (1205) .508 (593) .491 (283) .446 (130) -.031 Tyreke Evans .440 (150) .440 (302) .449 (608) .446 (1199) .451 (508) .458 (214) .489 ( 92) .002 Josh Smith .470 (132) .498 (299) .454 (613) .423 (1179) .382 (466) .376 (170) .365 ( 63) -.072 * Al Horford .551 (107) .500 (230) .506 (506) .534 (1173) .543 (575) .558 (276) .544 (136) .037 Dirk Nowitzki .496 (131) .487 (277) .462 (572) .458 (1146) .447 (492) .446 (204) .400 ( 85) -.014 Paul Millsap .518 (112) .502 (255) .486 (555) .464 (1138) .437 (494) .487 (199) .444 ( 90) -.049 Andrew Wiggins .543 (129) .503 (288) .465 (589) .437 (1137) .406 (466) .446 (175) .427 ( 75) -.060 Jeff Teague .528 (125) .507 (282) .453 (569) .450 (1124) .442 (466) .443 (185) .375 ( 72) -.011 Jimmy Butler .489 (133) .493 (282) .471 (567) .461 (1118) .449 (474) .408 (196) .431 ( 72) -.022 .435 (154) .433 (289) .446 (576) .430 (1115) .404 (453) .389 (175) .364 ( 66) -.042

24

Rudy Gay .429 (133) .452 (279) .444 (554) .458 (1110) .471 (488) .502 (221) .519 (108) .027 JJ Redick .484 (124) .437 (254) .463 (525) .470 (1099) .465 (482) .483 (209) .430 ( 93) .002 Markieff Morris .366 (131) .451 (268) .445 (539) .467 (1096) .482 (475) .477 (214) .427 ( 96) .037 Kyle Lowry .424 (165) .410 (315) .417 (592) .409 (1094) .386 (428) .400 (155) .368 ( 57) -.032 Zach Randolph .432 (125) .412 (245) .478 (523) .478 (1093) .482 (488) .491 (218) .505 ( 97) .004 Gordon Hayward .397 (151) .423 (284) .445 (555) .446 (1085) .432 (454) .435 (184) .453 ( 75) -.013 Trevor Ariza .423 (142) .424 (283) .432 (574) .406 (1085) .371 (412) .424 (144) .436 ( 55) -.061 Dwyane Wade .441 (127) .451 (253) .480 (533) .470 (1083) .475 (488) .461 (219) .495 ( 97) -.005 Victor Oladipo .458 (142) .460 (285) .448 (563) .437 (1083) .415 (448) .383 (175) .355 ( 62) -.033 .529 ( 87) .520 (202) .534 (483) .511 (1074) .495 (513) .500 (232) .477 (107) -.039 Jeff Green .508 (124) .495 (283) .436 (560) .423 (1068) .406 (419) .414 (152) .426 ( 54) -.030 DeMarcus Cousins .448 (143) .440 (268) .470 (541) .468 (1065) .480 (465) .419 (210) .393 ( 84) .010 DeMar DeRozan .429 (154) .439 (305) .434 (578) .414 (1064) .393 (422) .400 (160) .426 ( 61) -.041 .337 (190) .367 (324) .404 (586) .406 (1059) .427 (410) .408 (169) .369 ( 65) .022 Avery Bradley .503 (143) .440 (291) .406 (549) .429 (1057) .440 (427) .385 (179) .328 ( 67) .034 Bradley Beal .500 (114) .519 (266) .471 (558) .423 (1048) .369 (417) .381 (147) .370 ( 54) -.102 ** Eric Bledsoe .450 (131) .474 (270) .457 (530) .450 (1041) .458 (430) .430 (179) .391 ( 69) .002 .475 (101) .506 (235) .482 (502) .466 (1031) .447 (447) .465 (185) .463 ( 80) -.035 Al Jefferson .581 ( 86) .548 (210) .530 (487) .481 (1009) .435 (457) .415 (188) .395 ( 76) -.094 ** Draymond Green .534 (103) .468 (235) .464 (500) .438 (1002) .425 (402) .440 (159) .379 ( 66) -.039 Goran Dragic .494 ( 79) .524 (189) .543 (457) .501 (1001) .466 (466) .428 (201) .392 ( 79) -.077 * Jamal Crawford .426 (148) .416 (291) .412 (549) .398 (1000) .367 (373) .386 (127) .326 ( 46) -.045 .376 (186) .360 (331) .354 (571) .371 (994) .383 (347) .367 (128) .383 ( 47) .030 Michael Carter-Williams .409 (164) .395 (299) .409 (552) .400 (993) .390 (369) .374 (131) .227 ( 44) -.019 Kevin Love .431 (109) .454 (238) .467 (512) .433 (993) .376 (402) .426 (141) .400 ( 55) -.092 ** Reggie Jackson .385 (148) .412 (277) .414 (514) .436 (991) .468 (400) .418 (177) .449 ( 69) .053 Lou Williams .382 (144) .382 (272) .413 (525) .402 (973) .390 (364) .356 (135) .304 ( 46) -.023 Mike Conley .492 (122) .447 (237) .468 (485) .447 (971) .422 (408) .403 (159) .400 ( 60) -.047 Wilson Chandler .397 (126) .437 (247) .458 (507) .428 (971) .389 (386) .365 (137) .319 ( 47) -.069 * Kemba Walker .399 (158) .384 (292) .410 (544) .392 (964) .355 (358) .364 (121) .442 ( 43) -.055 Kentavious Caldwell-Pope .450 (140) .420 (276) .419 (523) .404 (964) .362 (359) .393 (122) .432 ( 44) -.057

25

Andre Drummond .490 ( 98) .505 (206) .477 (428) .515 (960) .549 (450) .516 (221) .484 ( 95) .072 * .456 ( 90) .507 (223) .440 (464) .459 (960) .474 (411) .473 (182) .494 ( 79) .035 Corey Brewer .563 (103) .467 (229) .453 (483) .433 (958) .402 (378) .433 (141) .518 ( 56) -.051 .430 (114) .463 (229) .478 (473) .467 (946) .457 (405) .428 (173) .437 ( 71) -.021 Dion Waiters .452 (135) .403 (263) .415 (511) .397 (945) .364 (354) .402 (122) .319 ( 47) -.051 J.R. Smith .425 (113) .433 (240) .411 (482) .421 (943) .410 (373) .373 (142) .447 ( 47) -.001 Kawhi Leonard .425 (106) .461 (219) .463 (451) .479 (931) .496 (409) .484 (188) .410 ( 83) .033 Enes Kanter .638 ( 58) .600 (165) .548 (409) .519 (928) .511 (444) .543 (210) .519 (106) -.036 Tony Parker .484 ( 91) .470 (200) .468 (434) .474 (922) .484 (413) .489 (188) .535 ( 86) .017 .600 ( 65) .579 (164) .567 (406) .525 (918) .505 (438) .477 (195) .385 ( 78) -.062 Ty Lawson .462 ( 91) .495 (214) .475 (463) .439 (917) .409 (379) .369 (149) .302 ( 53) -.066 Tim Duncan .556 ( 72) .528 (176) .513 (398) .520 (907) .527 (425) .495 (202) .511 ( 90) .015 Arron Afflalo .438 (112) .402 (219) .429 (452) .422 (894) .407 (361) .391 (138) .308 ( 52) -.022 Brandon Knight .479 (119) .452 (241) .444 (477) .422 (888) .399 (348) .400 (130) .367 ( 49) .045 Aaron Brooks .405 (111) .396 (222) .414 (444) .420 (870) .404 (332) .383 (120) .425 ( 40) -.011 Marcin Gortat .657 ( 35) .602 (113) .595 (328) .573 (861) .562 (441) .588 (226) .592 (120) -.032 Jarrett Jack .482 ( 83) .466 (193) .446 (417) .452 (856) .445 (353) .493 (140) .569 ( 58) -.001 .425 ( 87) .430 (186) .457 (396) .473 (855) .486 (356) .487 (150) .500 ( 64) .029 Greg Monroe .582 ( 67) .552 (172) .511 (397) .496 (853) .491 (387) .492 (177) .442 ( 77) -.020 Mo Williams .336 (134) .377 (244) .417 (460) .401 (850) .373 (322) .307 (114) .324 ( 34) -.045 Isaiah Thomas .385 (122) .405 (232) .424 (448) .416 (846) .422 (327) .417 (132) .423 ( 52) -.002 Gerald Henderson .511 ( 92) .455 (209) .428 (425) .437 (846) .452 (341) .462 (143) .435 ( 62) .023 Chandler Parsons .476 (105) .457 (208) .461 (414) .460 (844) .449 (363) .438 (153) .466 ( 58) -.012 Giannis Antetokounmpo .457 ( 70) .485 (165) .504 (385) .485 (844) .470 (372) .471 (153) .433 ( 60) -.034 Deron Williams .364 (143) .345 (252) .374 (455) .390 (836) .417 (307) .400 (125) .417 ( 48) .044 DeMarre Carroll .479 ( 71) .467 (165) .496 (379) .487 (830) .479 (365) .442 (156) .419 ( 62) -.017 Ben McLemore .456 ( 90) .439 (198) .428 (404) .439 (822) .476 (336) .497 (147) .426 ( 68) .048 CJ Miles .359 (117) .406 (234) .400 (443) .402 (818) .408 (306) .397 (116) .395 ( 43) .009 Dennis Schroder .478 ( 92) .455 (200) .428 (407) .421 (807) .386 (308) .402 (107) .417 ( 36) -.041 Carmelo Anthony .543 ( 94) .520 (204) .484 (419) .446 (802) .397 (343) .380 (129) .413 ( 46) -.088 * Danny Green .431 (102) .400 (205) .429 (406) .429 (797) .424 (304) .420 (112) .436 ( 39) -.004

26

Evan Turner .429 ( 98) .448 (201) .435 (405) .425 (790) .411 (299) .413 (109) .395 (43) -.023 Serge Ibaka .552 ( 87) .503 (183) .476 (380) .476 (786) .471 (342) .453 (148) .458 (59) -.006 Luol Deng .466 ( 58) .486 (146) .507 (365) .470 (785) .437 (348) .464 (140) .475 (59) .070 Terrence Ross .488 ( 86) .449 (187) .430 (395) .411 (779) .406 (298) .351 (114) .361 (36) -.024 .518 ( 85) .505 (190) .466 (384) .459 (777) .480 (323) .514 (140) .552 (67) .014 Gerald Green .451 ( 82) .446 (184) .448 (395) .420 (772) .382 (304) .330 (109) .353 (34) -.067 Ryan Anderson .425 (106) .399 (218) .388 (405) .402 (759) .436 (289) .423 (123) .451 (51) .049 .388 (116) .379 (214) .414 (403) .415 (755) .411 (287) .336 (107) .333 (30) -.003 Marcus Morris .400 ( 85) .425 (179) .427 (370) .438 (754) .442 (303) .480 (125) .400 (55) .015 .387 ( 75) .433 (164) .464 (366) .452 (754) .443 (305) .466 (118) .447 (47) -.022 Robert Covington .423 (104) .410 (212) .403 (407) .397 (753) .401 (277) .450 (100) .486 (37) -.002 Wesley Matthews .438 ( 80) .483 (174) .485 (379) .448 (752) .412 (313) .381 (118) .450 (40) -.073 Rodney Stuckey .513 ( 76) .462 (171) .457 (372) .441 (752) .430 (309) .359 (128) .295 (44) -.027 O.J. Mayo .456 ( 79) .446 (177) .444 (381) .419 (750) .404 (292) .360 (111) .333 (36) -.040 Greivis Vasquez .446 ( 74) .392 (171) .451 (381) .409 (750) .360 (283) .436 ( 94) .359 (39) -.091 * Chris Bosh .465 ( 86) .483 (180) .484 (378) .460 (745) .452 (323) .471 (138) .443 (61) -.032 .529 ( 70) .494 (162) .499 (373) .446 (744) .414 (309) .395 (119) .273 (44) -.084 * Carlos Boozer .452 ( 62) .482 (139) .521 (330) .499 (743) .503 (342) .456 (158) .391 (64) -.018 Courtney Lee .444 ( 63) .477 (155) .451 (344) .463 (741) .466 (309) .438 (128) .458 (48) .016 Kenneth Faried .548 ( 62) .543 (151) .489 (329) .510 (732) .549 (328) .529 (155) .485 (68) .060 Kyle Korver .500 ( 78) .429 (163) .453 (340) .470 (732) .459 (303) .435 (115) .487 (39) .006 Matt Barnes .427 ( 82) .460 (174) .452 (361) .436 (723) .419 (272) .443 ( 97) .500 (34) -.032 Timofey Mozgov .585 ( 41) .495 (105) .550 (280) .543 (714) .544 (333) .507 (148) .500 (60) -.007 Tim Hardaway Jr. .430 (107) .415 (205) .395 (385) .391 (713) .394 (259) .402 ( 92) .387 (31) -.001 Roy Hibbert .412 ( 68) .481 (160) .478 (347) .446 (710) .411 (287) .436 (110) .364 (44) -.067 .391 (133) .389 (234) .392 (411) .375 (709) .350 (263) .396 ( 91) .389 (36) -.042 Elfrid Payton .463 ( 95) .402 (179) .422 (353) .426 (707) .445 (272) .418 (110) .268 (41) .023 Brandon Bass .509 ( 53) .577 (137) .529 (314) .500 (702) .483 (302) .448 (125) .283 (46) -.045 Boris Diaw .411 ( 56) .453 (137) .472 (326) .462 (702) .465 (288) .513 (117) .490 (51) -.007 Donatas Motiejunas .540 ( 50) .528 (123) .526 (306) .505 (697) .498 (321) .489 (141) .525 (61) -.028 Wesley Johnson .495 ( 97) .426 (188) .420 (357) .415 (689) .426 (256) .352 ( 91) .286 (28) .006

27

Jonas Valanciunas .514 ( 35) .538 ( 91) .574 (258) .569 (689) .579 (347) .583 (175) .628 ( 86) .006 David West .485 ( 66) .490 (151) .485 (328) .471 (686) .462 (292) .479 (119) .367 ( 49) -.022 Nikola Mirotic .461 ( 89) .420 (181) .411 (358) .397 (685) .379 (240) .338 ( 74) .304( 23) -.032 Marreese Speights .422 (45) .483 (118) .490 (292) .485 (682) .497 (306) .457 (138) .463 ( 54) .007 Nene .458 (59) .477 (128) .523 (302) .504 (677) .507 (298) .515 (134) .483 ( 58) -.017 Zach LaVine .521 (71) .474 (156) .448 (337) .426 (671) .409 (259) .379 ( 95) .448 ( 29) -.039 Nicolas Batum .371 (89) .421 (190) .380 (353) .395 (669) .392 (240) .402 ( 82) .357 ( 28) .012 .391 (69) .432 (155) .449 (332) .428 (664) .388 (263) .400 ( 95) .378 ( 37) -.061 Andre Iguodala .491 (55) .472 (127) .478 (295) .473 (659) .504 (266) .486 (111) .432 ( 44) .026 Nerlens Noel .486 (72) .465 (155) .462 (316) .462 (653) .462 (262) .449 (107) .465 ( 43) .000 Mario Chalmers .342 (73) .380 (158) .400 (330) .409 (651) .417 (242) .409 ( 93) .459 ( 37) .018 Jeremy Lin .354 (82) .377 (159) .399 (321) .425 (650) .443 (255) .421 (107) .415 ( 41) .045 Tyler Zeller .575 (40) .566 (106) .516 (252) .550 (646) .557 (309) .612 (147) .605 ( 76) .041 .590 (39) .563 (119) .472 (286) .470 (638) .465 (271) .504 (117) .400 ( 55) -.007 Luc Mbah a Moute .476 (84) .402 (169) .411 (331) .395 (635) .367 (237) .321 ( 81) .333 ( 24) -.044 Bojan Bogdanovic .441 (59) .427 (131) .447 (291) .447 (635) .435 (262) .481 (108) .440 ( 50) -.012 D.J. Augustin .438 (64) .419 (148) .429 (319) .405 (634) .402 (234) .375 ( 88) .355 ( 31) -.028 Iman Shumpert .357 (70) .418 (158) .395 (319) .400 (630) .400 (230) .434 ( 83) .333 ( 33) .005 Solomon Hill .414 (58) .432 (139) .459 (318) .397 (629) .354 (229) .370 ( 73) .360 ( 25) -.106 * Manu Ginobili .456 (68) .420 (150) .422 (313) .424 (627) .397 (237) .354 ( 79) .375 ( 24) -.025 Kevin Martin .477 (86) .450 (169) .457 (337) .427 (626) .388 (250) .440 ( 91) .472 ( 36) -.069 Ersan Ilyasova .484 (62) .462 (132) .497 (300) .458 (618) .417 (254) .452 ( 93) .472 ( 36) -.079 Tristan Thompson .581 (31) .522 ( 92) .525 (238) .549 (617) .563 (279) .569 (123) .633 ( 49) .038 .360 (86) .365 (159) .418 (318) .412 (616) .408 (233) .395 ( 86) .419 ( 31) -.011 Anthony Morrow .582 (55) .515 (130) .488 (289) .464 (614) .426 (251) .402 ( 87) .429 ( 28) -.062 DeAndre Jordan .700 (10) .730 ( 37) .730 (159) .713 (613) .737 (358) .764 (208) .795 (122) .008 Dwight Howard .654 (26) .575 ( 80) .582 (220) .589 (604) .577 (326) .565 (168) .548 ( 84) -.005 .382 (68) .355 (138) .404 (287) .415 (600) .435 (223) .494 ( 85) .514 ( 37) .031 PJ Tucker .469 (64) .472 (142) .462 (299) .438 (596) .438 (219) .444 ( 81) .345 ( 29) -.023 .597 (62) .489 (137) .468 (295) .449 (595) .434 (244) .495 ( 95) .429 ( 42) -.033 Norris Cole .464 (56) .456 (136) .419 (291) .414 (592) .428 (222) .414 ( 87) .313 ( 32) .009

28

Jason Smith .389 (54) .394 (127) .408 (282) .434 (592) .487 (228) .449 ( 98) .351 (37) .079 Jose Juan Barea .421 (57) .425 (127) .460 (287) .424 (590) .380 (221) .389 ( 72) .435 (23) -.080 .578 (45) .519 (106) .490 (247) .499 (579) .498 (259) .509 (112) .478 (46) .008 .605 (38) .509 (116) .438 (272) .425 (577) .427 (211) .446 ( 74) .375 (24) -.011 Beno Udrih .548 (31) .413 ( 92) .471 (240) .482 (575) .476 (246) .471 (102) .535 (43) .005 Jae Crowder .479 (48) .508 (122) .464 (278) .426 (573) .388 (214) .271 ( 70) .200 (15) -.076 Evan Fournier .458 (48) .482 (114) .507 (276) .444 (570) .364 (236) .427 ( 82) .394 (33) -.143 ** Zaza Pachulia .692 (39) .574 (115) .469 (260) .451 (568) .459 (229) .505 ( 91) .474 (38) -.011 Mike Dunleavy .410 (39) .500 (112) .452 (263) .443 (566) .412 (228) .381 ( 84) .333 (27) -.040 .367 (79) .420 (157) .401 (299) .405 (561) .414 (203) .377 ( 77) .280 (25) .012 Patrick Patterson .451 (51) .431 (116) .450 (260) .455 (561) .488 (217) .477 ( 88) .500 (30) .038 Chris Kaman .440 (25) .586 ( 87) .534 (234) .515 (557) .508 (248) .509 (110) .583 (48) -.026 .386 (70) .426 (141) .443 (282) .444 (552) .454 (227) .484 ( 91) .436 (39) .011 .564 (39) .457 (105) .431 (253) .422 (550) .433 (215) .368 ( 87) .400 (30) .002 Amir Johnson .652 (23) .621 ( 66) .615 (195) .580 (548) .558 (274) .496 (129) .444 (54) -.057 .403 (67) .428 (145) .422 (296) .377 (546) .339 (189) .268 ( 56) .000 (12) -.084 Hollis Thompson .338 (68) .364 (132) .413 (276) .413 (542) .421 (195) .446 ( 74) .469 (32) .008 Brandon Jennings .333 (78) .374 (147) .416 (293) .404 (540) .403 (206) .405 ( 79) .452 (31) -.014 .397 (63) .457 (127) .464 (261) .480 (540) .470 (234) .450 (100) .359 (39) .007 Kelly Olynyk .273 (33) .393 ( 84) .500 (234) .478 (540) .469 (239) .455 (101) .400 (40) -.031 Devin Harris .420 (50) .434 (113) .438 (256) .420 (538) .416 (202) .408 ( 76) .444 (27) -.022 .354 (65) .433 (141) .413 (276) .416 (534) .419 (198) .395 ( 76) .370 (27) .006 George Hill .475 (59) .471 (119) .478 (253) .478 (533) .464 (237) .444 ( 99) .405 (42) -.014 .449 (78) .414 (152) .390 (292) .383 (532) .310 (184) .294 ( 51) .357 (14) -.082 Amar'e Stoudemire .615 (26) .605 ( 76) .580 (207) .551 (523) .548 (252) .600 (120) .607 (61) -.032 Rasual Butler .438 (48) .464 (110) .454 (249) .423 (515) .351 (194) .421 ( 57) .471 (17) -.104 * Tony Allen .429 (35) .517 ( 89) .507 (221) .494 (512) .491 (218) .483 ( 89) .406 (32) -.016 Gorgui Dieng .435 (23) .538 ( 78) .553 (215) .508 (508) .477 (220) .449 ( 89) .485 (33) -.076 Marco Belinelli .436 (55) .391 (110) .449 (243) .434 (505) .394 (193) .313 ( 64) .333 (18) -.055 Mike Scott .541 (37) .480 (100) .451 (235) .440 (504) .403 (191) .397 ( 68) .333 (24) -.048 Langston Galloway .456 (79) .385 (148) .387 (271) .404 (503) .422 (187) .447 ( 76) .469 (32) .035

29

Gary Neal .417 (72) .374 (147) .365 (274) .375 (502) .351 (174) .407 ( 59) .333 (24) -.015 Jameer Nelson .412 (34) .485 (103) .441 (247) .406 (502) .385 (192) .371 ( 70) .200 (25) -.056 Marcus Smart .400 (65) .382 (136) .364 (264) .376 (502) .355 (169) .302 ( 53) .286 (14) -.009 Henry Sims .489 (47) .457 (105) .454 (227) .474 (502) .512 (203) .500 ( 82) .469 (32) .059 Joakim Noah .424 (33) .490 ( 96) .470 (234) .441 (501) .410 (188) .460 ( 63) .476 (21) -.061 Marvin Williams .535 (43) .455 ( 99) .466 (236) .425 (494) .383 (180) .364 ( 55) .222 (18) -.083 Mason Plumlee .643 (14) .614 ( 57) .551 (167) .577 (492) .600 (240) .602 (118) .627 (59) .049 Channing Frye .579 (57) .441 (118) .398 (244) .392 (487) .399 (168) .391 ( 64) .240 (25) .001 Derrick Williams .514 (37) .473 ( 91) .463 (216) .447 (485) .425 (200) .405 ( 79) .400 (30) -.038 Michael Kidd-Gilchrist .571 (42) .495 ( 97) .511 (229) .465 (482) .409 (198) .378 ( 74) .346 (26) -.102 * Kevin Seraphin .684 (19) .561 ( 66) .535 (185) .512 (480) .507 (211) .505 ( 91) .475 (40) -.028 Otto Porter .429 (28) .570 ( 86) .458 (214) .451 (477) .418 (182) .371 ( 70) .409 (22) -.041 Shaun Livingston .476 (21) .471 ( 68) .486 (183) .507 (473) .552 (192) .494 ( 83) .433 (30) .066 Nick Young .364 (77) .346 (136) .396 (265) .367 (471) .317 (164) .265 ( 49) .250 (12) -.080 Alan Anderson .448 (29) .494 ( 77) .473 (201) .465 (471) .435 (191) .514 ( 72) .500 (32) -.038 .362 (58) .389 (131) .353 (255) .345 (470) .367 (139) .317 ( 41) .333 ( 9) .014 .444 (18) .438 ( 64) .492 (193) .472 (470) .454 (194) .461 ( 76) .484 (31) -.039 .458 (24) .554 ( 56) .616 (164) .601 (469) .586 (227) .602 (103) .711 (45) -.030 .625 (16) .581 ( 43) .637 (135) .665 (469) .678 (255) .699 (136) .685 (73) .042 .531 (49) .495 (105) .470 (215) .510 (467) .551 (225) .573 (117) .516 (62) .081 Ray McCallum .395 (43) .429 ( 98) .461 (219) .443 (465) .432 (183) .391 ( 69) .348 (23) -.030 Quincy Pondexter .355 (31) .461 ( 89) .425 (212) .420 (464) .420 (174) .431 ( 65) .417 (24) -.005 Trevor Booker .600 (20) .549 ( 71) .489 (186) .491 (460) .505 (196) .506 ( 85) .474 (38) .016 JJ Hickson .636 (33) .494 ( 81) .472 (197) .476 (458) .432 (190) .471 ( 70) .556 (27) -.041 Terrence Jones .324 (37) .400 ( 85) .475 (198) .493 (456) .519 (208) .480 ( 98) .442 (43) .045 Jared Dudley .462 (26) .465 ( 71) .474 (192) .468 (453) .446 (184) .529 ( 68) .586 (29) -.028 .517 (29) .472 ( 72) .497 (177) .538 (452) .602 (211) .579 (107) .549 (51) .105 * Brian Roberts .254 (71) .287 (122) .358 (229) .395 (448) .456 (149) .466 ( 58) .583 (24) .098 .645 (31) .500 ( 86) .462 (195) .474 (447) .476 (187) .421 ( 76) .345 (29) .014 .442 (43) .426 ( 94) .436 (218) .387 (444) .324 (148) .250 ( 40) .143 ( 7) -.112 * Lance Thomas .423 (52) .398 (108) .404 (225) .413 (441) .427 (157) .439 ( 57) .429 (21) .022

30

Matthew Dellavedova .367 (49) .370 (108) .341 (223) .359 (440) .333 (132) .257 ( 35) .111 ( 9) -.008 KJ McDaniels .500 (48) .477 (109) .430 (228) .396 (434) .347 (147) .275 ( 40) .125 ( 8) -.083 CJ McCollum .382 (34) .489 ( 88) .437 (199) .442 (432) .479 (169) .521 ( 71) .500 (32) .042 Shane Larkin .483 (29) .468 ( 79) .460 (198) .434 (431) .367 (158) .358 ( 53) .389 (18) -.093 .537 (41) .419 ( 93) .443 (212) .400 (430) .364 (143) .262 ( 42) .100 (10) -.080 Andrew Bogut .842 (19) .536 ( 56) .573 (157) .563 (430) .569 (188) .500 ( 80) .469 (32) -.004 .650 (20) .631 ( 65) .514 (175) .489 (427) .478 (186) .438 ( 80) .567 (30) -.036 Jason Thompson .333 (24) .479 ( 71) .455 (178) .471 (427) .485 (169) .464 ( 69) .385 (26) .030 Rudy Gobert .563 (16) .533 ( 45) .617 (141) .604 (427) .602 (206) .600 ( 90) .611 (36) -.015 Tony Wroten .444 (63) .481 (129) .421 (235) .408 (426) .385 (161) .368 ( 57) .286 (21) -.036 Omer Asik .667 (24) .540 ( 63) .547 (170) .509 (422) .465 (172) .500 ( 58) .500 (22) -.082 CJ Watson .424 (33) .438 ( 80) .462 (195) .438 (413) .435 (161) .424 ( 59) .500 (20) -.027 Tony Snell .425 (40) .419 ( 86) .441 (188) .421 (413) .430 (151) .455 ( 55) .524 (21) -.011 Dante Exum .417 (36) .385 ( 91) .372 (207) .351 (413) .306 (124) .257 ( 35) .286 ( 7) -.067 James Johnson .357 (14) .551 ( 49) .567 (141) .585 (410) .582 (201) .582 ( 98) .591 (44) .015 Cory Joseph .560 (25) .516 ( 62) .500 (160) .517 (410) .511 (174) .565 ( 69) .581 (31) .012 .526 (38) .482 ( 85) .474 (192) .440 (402) .441 (152) .466 ( 58) .619 (21) -.033 Ramon Sessions .378 (37) .463 ( 95) .395 (200) .377 (400) .378 (127) .325 ( 40) .250 (12) -.017 Kent Bazemore .409 (22) .441 ( 68) .465 (172) .428 (400) .389 (144) .422 ( 45) .429 (14) -.077 Steven Adams .667 (18) .630 ( 54) .560 (150) .545 (398) .545 (178) .519 ( 77) .571 (28) -.015 Kirk Hinrich .326 (43) .453 ( 95) .420 (205) .379 (398) .355 (121) .333 ( 27) .250 ( 4) -.063 .621 (29) .536 ( 69) .528 (180) .489 (397) .467 (180) .444 ( 81) .471 (34) -.061 .362 (47) .444 (108) .403 (216) .373 (397) .371 (132) .391 ( 46) .294 (17) -.031 .520 (25) .477 ( 65) .497 (169) .442 (394) .372 (148) .318 ( 44) .364 (11) -.126 * Anthony Tolliver .412 (34) .470 ( 83) .435 (184) .410 (393) .363 (135) .459 ( 37) .231 (13) -.072 .556 (27) .397 ( 73) .433 (180) .411 (392) .425 (134) .563 ( 48) .545 (22) -.008 .531 (32) .451 ( 82) .420 (181) .433 (390) .479 (144) .492 ( 61) .560 (25) .059 Kosta Koufos .625 (16) .521 ( 48) .527 (150) .512 (389) .497 (151) .635 ( 52) .640 (25) -.030 Al-Farouq Aminu .464 (28) .423 ( 71) .402 (169) .423 (388) .437 (142) .481 ( 52) .333 (21) .035 Hassan Whiteside .500 (12) .649 ( 37) .656 (128) .628 (387) .606 (213) .624 (109) .649 (57) -.051 Charlie Villanueva .516 (31) .475 ( 80) .443 (183) .417 (386) .445 (137) .404 ( 52) .353 (17) .003

31

John Henson .636 (11) .667 ( 42) .597 (134) .570 (386) .560 (182) .506 ( 81) .500 (34) -.037 Darrell Arthur .465 (43) .391 ( 92) .394 (188) .405 (383) .387 (137) .435 ( 46) .471 (17) -.007 Jusuf Nurkic .318 (44) .353 ( 85) .389 (175) .446 (383) .497 (147) .508 ( 61) .500 (26) .109 .319 (47) .354 ( 99) .351 (191) .396 (376) .438 (128) .327 (49) .429 (14) .087 Isaiah Canaan .365 (52) .404 (104) .379 (195) .389 (375) .404 (136) .408 (49) .316 (19) .025 Cody Zeller .524 (21) .453 ( 64) .472 (163) .461 (373) .439 (148) .538 (52) .591 (22) -.033 Rodney Hood .462 (39) .438 ( 89) .412 (182) .419 (370) .420 (138) .462 (52) .450 (20) .008 Carl Landry .522 (23) .456 ( 57) .504 (141) .515 (369) .515 (163) .500 (72) .517 (29) .012 Ish Smith .439 (57) .383 (107) .351 (194) .390 (367) .447 (132) .407 (54) .381 (21) .097 Brandan Wright .800 (10) .676 ( 37) .595 (111) .644 (362) .676 (179) .706 (85) .698 (43) .083 .550 (40) .500 ( 86) .489 (178) .454 (361) .439 (155) .365 (63) .350 (20) -.050 Jose Calderon .479 (48) .398 ( 93) .407 (182) .415 (354) .454 (130) .453 (53) .286 (21) .047 Donald Sloan .438 (48) .485 ( 99) .389 (180) .412 (352) .448 (125) .429 (49) .368 (19) .059 David Lee .500 (20) .482 ( 56) .489 (135) .500 (352) .484 (157) .500 (68) .600 (30) -.005 Alex Len .409 (22) .466 ( 58) .438 (137) .509 (352) .554 (148) .537 (67) .467 (30) .116 * Joe Ingles .286 (21) .418 ( 55) .458 (153) .418 (352) .407 (123) .359 (39) .333 ( 9) -.051 Jerami Grant .381 (42) .344 ( 90) .320 (181) .354 (350) .390 (105) .394 (33) .333 ( 9) .071 Weighted means .457 .458 .459 .453 .449 .451 .447 -.018 ** The number of shots upon which each probability is based is given within parentheses. Players are sorted by total amount of shots, in descending order. * p < 0.05, ** p < 0.01.

32

Appendix 2: Stationarity for all shots. Player LeBron James James Harden Total Shots 1812 1760 P(hit) 0.47 0.44 E(low) 0.36 0.41 E(moderate) 0.37 0.36 E(high) 0.27 0.23 Division Division 1 2 3 4 1 2 3 4 Total Low 151 167 159 161 172 177 176 171 Total Mod 190 162 178 171 170 163 157 168 Total High 112 123 115 120 98 99 106 100

E(Frequency Low) 163.18 162.82 162.82 162.82 178.61 178.20 178.20 178.20 E(Frequency Mod) 168.58 168.21 168.21 168.21 160.42 160.06 160.06 160.06 E(Frequency High) 121.24 120.97 120.97 120.97 100.97 100.74 100.74 100.74

χ²-value 4.34 0.37 0.95 0.07 0.90 0.09 0.36 0.69 P-value 0.11 0.83 0.62 0.96 0.64 0.95 0.84 0.71 Obs. – Exp. High -9.24 2.03 -5.97 -0.97 -2.97 -1.74 5.26 -0.74

Player Stephen Curry Klay Thompson Total Shots 1759 1629 P(hit) 0.48 0.46 E(low) 0.34 0.37 E(moderate) 0.37 0.37 E(high) 0.29 0.26 Division Division 1 2 3 4 1 2 3 4 Total Low 140 145 143 145 151 151 145 147 Total Mod 176 172 173 175 152 159 161 161 Total High 123 122 123 119 104 97 100 98

E(Frequency Low) 148.80 148.80 148.80 148.80 151.86 151.86 151.49 151.49 E(Frequency Mod) 164.23 164.23 164.23 164.23 150.78 150.78 150.40 150.40 E(Frequency High) 125.97 125.97 125.97 125.97 104.36 104.36 104.11 104.11

χ²-value 1.43 0.59 0.76 1.19 0.02 0.97 1.19 1.24 P-value 0.49 0.74 0.68 0.55 0.99 0.61 0.55 0.54 Obs. – Exp. High -2.97 -3.97 -2.97 -6.97 -0.36 -7.36 -4.11 -6.11

33

Player LaMarcus Aldridge Russel Westbrook Total Shots 1525 1466 P(hit) 0.46 0.43 E(low) 0.38 0.43 E(moderate) 0.37 0.36 E(high) 0.25 0.21

Division Division 1 2 3 4 1 2 3 4 Total Low 142 135 133 140 153 150 158 157 Total Mod 143 161 161 151 128 150 131 127 Total High 96 85 86 89 85 66 77 81

E(Frequency Low) 145.00 145.00 144.62 144.62 157.05 157.05 157.05 156.63 E(Frequency Mod) 140.71 140.71 140.34 140.34 131.46 131.46 131.46 131.10 E(Frequency High) 95.29 95.29 95.04 95.04 77.48 77.48 77.48 77.27

χ²-value 0.10 4.73 4.83 1.34 0.93 4.63 0.01 0.31 P-value 0.95 0.09 0.09 0.51 0.63 0.10 0.99 0.86 Obs. – Exp. High 0.71 -10.29 -9.04 -6.04 7.52 -11.48 -0.48 3.73

Player Monta Ellis Damian Lillard Total Shots 1461 1442 P(hit) 0.45 0.43 E(low) 0.39 0.42 E(moderate) 0.37 0.36 E(high) 0.24 0.22

Division Division 1 2 3 4 1 2 3 4 Total Low 149 144 145 143 149 156 143 148 Total Mod 131 137 131 139 140 125 144 133 Total High 85 84 88 82 71 79 73 78

E(Frequency Low) 144.12 144.12 143.72 143.72 150.19 150.19 150.19 149.78 E(Frequency Mod) 133.89 133.89 133.52 133.52 130.35 130.35 130.35 129.99 E(Frequency High) 86.99 86.99 86.75 86.75 79.45 79.45 79.45 79.23

χ²-value 0.27 0.18 0.08 0.49 1.62 0.45 2.30 0.11 P-value 0.87 0.92 0.96 0.78 0.44 0.80 0.32 0.95 Obs. – Exp. High -1.99 -2.99 1.25 -4.75 -8.45 -0.45 -6.45 -1.23

34

Player Kyrie Irving Blake Griffin Total Shots 1417 1416 P(hit) 0.46 0.50 E(low) 0.37 0.31 E(moderate) 0.37 0.37 E(high) 0.26 0.32

Division Division 1 2 3 4 1 2 3 4 Total Low 122 127 116 124 104 103 111 113 Total Mod 152 136 153 139 145 136 125 121 Total High 80 91 84 90 105 114 117 119

E(Frequency Low) 130.19 130.19 129.82 129.82 108.38 108.08 108.08 108.08 E(Frequency Mod) 131.40 131.40 131.03 131.03 132.73 132.36 132.36 132.36 E(Frequency High) 92.41 92.41 92.14 92.14 112.88 112.57 112.57 112.57

χ²-value 5.41 0.26 5.87 0.80 1.86 0.36 0.66 1.57 P-value 0.07 0.88 0.05 0.67 0.39 0.84 0.72 0.46 Obs. – Exp. High -12.41 -1.41 -8.14 -2.14 -7.88 1.43 4.43 6.43

Player Chris Paul Anthony Davis Total Shots 1355 1283 P(hit) 0.49 0.54 E(low) 0.33 0.26 E(moderate) 0.37 0.37 E(high) 0.29 0.37

Division Division 1 2 3 4 1 2 3 4 Total Low 105 112 109 110 87 86 75 71 Total Mod 135 132 128 129 118 109 130 144 Total High 98 94 101 99 115 125 115 105

E(Frequency Low) 111.87 111.87 111.87 111.87 82.92 82.92 82.92 82.92 E(Frequency Mod) 126.60 126.60 126.60 126.60 118.69 118.69 118.69 118.69 E(Frequency High) 99.53 99.53 99.53 99.53 118.39 118.39 118.39 118.39

χ²-value 1.00 0.54 0.11 0.08 0.30 1.27 1.93 8.63 P-value 0.61 0.76 0.95 0.96 0.86 0.53 0.38 0.01 Obs. – Exp. High -1.53 -5.53 1.47 -0.53 -3.39 6.61 -3.39 -13.39

35

Player Pau Gasol John Wall Total Shots 1267 1260 P(hit) 0.49 0.44 E(low) 0.32 0.40 E(moderate) 0.37 0.37 E(high) 0.30 0.23

Division Division 1 2 3 4 1 2 3 4 Total Low 96 99 91 99 126 124 125 121 Total Mod 128 122 140 127 117 117 110 124 Total High 92 95 85 90 72 73 79 69

E(Frequency Low) 101.57 101.57 101.57 101.57 126.03 125.63 125.63 125.63 E(Frequency Mod) 118.47 118.47 118.47 118.47 115.23 114.86 114.86 114.86 E(Frequency High) 95.96 95.96 95.96 95.96 73.74 73.51 73.51 73.51

χ²-value 1.24 0.18 6.27* 1.05 0.07 0.06 0.62 1.17 P-value 0.54 0.91 0.04 0.59 0.97 0.97 0.73 0.56 Obs. – Exp. High -3.96 -0.96 -10.96 -5.96 -1.74 -0.51 5.49 -4.51

Player Marc Gasol Nikola Vucevic Total Shots 1246 1205 P(hit) 0.48 0.52 E(low) 0.34 0.28 E(moderate) 0.37 0.37 E(high) 0.29 0.35

Division Division 1 2 3 4 1 2 3 4 Total Low 101 100 96 100 86 84 75 75 Total Mod 123 128 136 126 110 109 122 124 Total High 87 83 79 84 105 108 103 101

E(Frequency Low) 105.57 105.57 105.57 105.23 83.64 83.64 83.37 83.37 E(Frequency Mod) 116.33 116.33 116.33 115.96 112.37 112.37 112.00 112.00 E(Frequency High) 89.10 89.10 89.10 88.81 104.99 104.99 104.64 104.64

χ²-value 0.63 1.88 5.34 1.39 0.12 0.19 1.76 2.25 P-value 0.73 0.39 0.07 0.50 0.94 0.91 0.42 0.32 Obs. – Exp. High -2.10 -6.10 -10.10 -4.81 0.01 3.01 -1.64 -3.64

36

Player Tyreke Evans Josh Smith Total Shots 1199 1179 P(hit) 0.45 0.42 E(low) 0.40 0.44 E(moderate) 0.37 0.36 E(high) 0.24 0.21

Division Division 1 2 3 4 1 2 3 4 Total Low 124 125 116 125 120 123 123 131 Total Mod 98 99 118 102 123 120 119 98 Total High 77 75 65 72 51 51 52 65

E(Frequency Low) 118.76 118.76 118.76 118.76 128.03 128.03 128.03 128.03 E(Frequency Mod) 109.54 109.54 109.54 109.54 105.11 105.11 105.11 105.11 E(Frequency High) 70.69 70.69 70.69 70.69 60.86 60.86 60.86 60.86

χ²-value 2.01 1.60 1.18 0.87 5.14 3.90 3.32 0.83 P-value 0.37 0.45 0.56 0.65 0.08 0.14 0.19 0.66 Obs. – Exp. High 6.31 4.31 -5.69 1.31 -9.86 -9.86 -8.86 4.14

Player Al Horford Dirk Nowitzki Total Shots 1173 1146 P(hit) 0.53 0.46 E(low) 0.26 0.38 E(moderate) 0.37 0.37 E(high) 0.36 0.25

Division Division 1 2 3 4 1 2 3 4 Total Low 84 81 79 81 116 112 105 105 Total Mod 98 109 107 102 94 95 116 121 Total High 111 103 106 109 76 79 65 59

E(Frequency Low) 77.28 77.28 77.02 77.02 108.05 108.05 108.05 107.67 E(Frequency Mod) 108.88 108.88 108.51 108.51 105.75 105.75 105.75 105.38 E(Frequency High) 106.84 106.84 106.47 106.47 72.20 72.20 72.20 71.95

χ²-value 1.83 0.32 0.07 0.66 2.09 1.88 1.80 4.71 P-value 0.40 0.85 0.96 0.72 0.35 0.39 0.41 0.09 Obs. – Exp. High 4.16 -3.84 -0.47 2.53 3.80 6.80 -7.20 -12.95

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Player Rudy Gay Jamal Crawford Total Shots 1110 1000 P(hit) 0.46 0.40 E(low) 0.38 0.48 E(moderate) 0.37 0.34 E(high) 0.25 0.18

Division Division 1 2 3 4 1 2 3 4 Total Low 116 106 106 100 110 116 117 118 Total Mod 87 101 101 109 105 96 88 89 Total High 74 70 70 67 35 37 44 42

E(Frequency Low) 104.86 104.86 104.86 104.48 119.66 119.19 119.19 119.19 E(Frequency Mod) 102.39 102.39 102.39 102.02 86.11 85.76 85.76 85.76 E(Frequency High) 69.75 69.75 69.75 69.50 44.23 44.05 44.05 44.05

χ²-value 3.76 0.03 0.03 0.76 6.85* 2.43 0.10 0.23 P-value 0.15 0.98 0.98 0.68 0.03 0.30 0.95 0.89 Obs. – Exp. High 4.25 0.25 0.25 -2.50 -9.23 -7.05 -0.05 -2.05

Player J.R. Smith Carmelo Anthony Total Shots 943 802 P(hit) 0.42 0.45 E(low) 0.44 0.40 E(moderate) 0.36 0.37 E(high) 0.20 0.24

Division Division 1 2 3 4 1 2 3 4 Total Low 98 101 104 103 76 81 86 77 Total Mod 94 90 87 79 84 72 65 78 Total High 43 44 44 53 40 47 49 44

E(Frequency Low) 103.23 103.23 103.23 103.23 79.38 79.38 79.38 78.98 E(Frequency Mod) 83.78 83.78 83.78 83.78 73.29 73.29 73.29 72.92 E(Frequency High) 47.99 47.99 47.99 47.99 47.33 47.33 47.33 47.10

χ²-value 2.03 0.84 0.46 0.80 2.85 0.06 1.55 0.61 P-value 0.36 0.66 0.79 0.67 0.24 0.97 0.46 0.74 Obs. – Exp. High -4.99 -3.99 -3.99 5.01 -7.33 -0.33 1.67 -3.10

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Player Brandon Jennings Nick Young Total Shots 540 471 P(hit) 0.40 0.37 E(low) 0.47 0.53 E(moderate) 0.35 0.32 E(high) 0.18 0.14

Division Division 1 2 3 4 1 2 3 4 Total Low 62 60 64 63 60 59 57 61 Total Mod 48 50 41 44 41 42 47 40 Total High 25 24 29 27 16 16 13 16

E(Frequency Low) 63.29 62.82 62.82 62.82 62.29 62.29 62.29 62.29 E(Frequency Mod) 46.94 46.59 46.59 46.59 37.91 37.91 37.91 37.91 E(Frequency High) 24.77 24.59 24.59 24.59 16.80 16.80 16.80 16.80

χ²-value 0.05 0.39 1.48 0.38 0.37 0.65 3.49 0.18 P-value 0.97 0.82 0.48 0.83 0.83 0.72 0.17 0.91 Obs. – Exp. High 0.23 -0.59 4.41 2.41 -0.80 -0.80 -3.80 -0.80 Players are sorted by total amount of shots, in descending order. * p < 0.05, ** p < 0.01.

39

Appendix 3: Probability of making a three point shot conditioned on the outcome of previous three point shots and serial correlations for all 164 NBA players with at least 150 three point shots. (1) (2) (3) (4) (5) (6) (7) (8) (9) Serial Player P(hit|3 misses) P(hit|2 misses) P(hit|1 miss) P(hit) P(hit|1 hit) P(hit|2 hits) P(hit|3 hits) Correlation Stephen Curry .379 (87) .414 (191) .447 (414) .443 (857) .436 (342) .435 (131) .440 (50) -.011 Klay Thompson .492 (65) .428 (152) .416 (329) .431 (687) .462 (260) .429 (105) .324 (37) .045 James Harden .337 (89) .322 (171) .376 (343) .378 (658) .378 (217) .348 ( 66) .474 (19) .002 Trevor Ariza .407 (81) .359 (170) .364 (343) .354 (650) .325 (209) .404 ( 57) .333 (18) -.040 Damian Lillard .305 (95) .299 (184) .297 (323) .336 (589) .391 (179) .419 ( 62) .375 (24) .096 * Kyle Korver .447 (47) .482 (112) .484 (258) .466 (558) .458 (212) .397 ( 73) .368 (19) -.027 J.R. Smith .371 (62) .375 (128) .383 (269) .387 (550) .381 (194) .406 ( 64) .348 (23) -.002 JJ Redick .468 (47) .427 (110) .438 (251) .430 (546) .397 (204) .309 ( 68) .143 (14) -.042 Danny Green .365 (52) .376 (117) .377 (244) .408 (495) .439 (164) .423 ( 52) .667 (12) .062 Lou Williams .288 (52) .331 (121) .326 (242) .338 (462) .314 (137) .256 ( 39) .375 ( 8) -.013 Wesley Matthews .460 (50) .434 (113) .398 (231) .389 (445) .396 (154) .442 ( 52) .389 (18) -.002 Robert Covington .271 (48) .343 (108) .383 (227) .377 (443) .367 (147) .442 ( 43) .385 (13) -.016 LeBron James .391 (46) .366 (112) .346 (231) .329 (438) .269 (119) .185 ( 27) .250 ( 4) -.079 CJ Miles .286 (63) .314 (121) .338 (231) .352 (437) .387 (137) .468 ( 47) .368 (19) .050 Kentavious Caldwell-Pope .364 (44) .337 (104) .363 (223) .349 (436) .336 (131) .316 ( 38) .250 (12) -.028 Kyrie Irving .400 (30) .329 ( 79) .362 (188) .421 (435) .438 (160) .452 ( 62) .500 (26) .077 Matt Barnes .390 (41) .388 (103) .349 (218) .349 (435) .313 (128) .375 ( 32) .400 (10) -.037 Draymond Green .393 (28) .276 ( 76) .338 (195) .324 (417) .301 (123) .172 ( 29) .000 ( 4) -.039 Kevin Love .290 (31) .341 ( 88) .350 (203) .370 (413) .414 (133) .367 ( 49) .500 (14) .064 Jamal Crawford .400 (45) .396 (106) .350 (220) .329 (413) .287 (115) .310 ( 29) .125 ( 8) -.064 Chris Paul .409 (22) .422 ( 64) .440 (182) .400 (410) .366 (134) .286 ( 42) .182 (11) -.074 Kyle Lowry .298 (47) .317 (104) .332 (214) .333 (408) .308 (120) .273 ( 33) .143 ( 7) -.024 Terrence Ross .452 (31) .412 ( 80) .402 (189) .373 (405) .336 (131) .343 ( 35) .300 (10) -.067 Wilson Chandler .340 (53) .333 (108) .340 (212) .341 (402) .310 (113) .286 ( 28) .200 ( 5) -.030 Jason Terry .536 (28) .439 ( 82) .370 (189) .386 (402) .417 (120) .395 ( 38) .300 (10) .046 Ben McLemore .455 (33) .361 ( 83) .389 (190) .361 (382) .315 (111) .269 ( 26) .000 ( 4) -.074 Gerald Green .333 (42) .379 ( 87) .393 (191) .360 (381) .308 (117) .167 ( 30) .500 ( 4) -.086

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Trey Burke .261 (46) .255 (98) .317 (202) .325 (379) .297 (101) .080 (25) .000 ( 2) -.020 DeMarre Carroll .522 (23) .424 (66) .401 (167) .396 (376) .374 (123) .286 (35) .125 ( 8) -.028 Joe Johnson .556 (27) .429 (70) .410 (178) .356 (374) .318 (110) .320 (25) .400 ( 5) -.092 Avery Bradley .415 (41) .317 (82) .348 (181) .352 (366) .349 (106) .379 (29) .333 ( 6) .001 Ryan Anderson .270 (37) .341 (85) .366 (186) .342 (365) .342 (114) .306 (36) .000 (10) -.024 Paul Pierce .320 (25) .424 (66) .455 (165) .414 (365) .393 (117) .469 (32) .545 (11) -.061 Greivis Vasquez .364 (11) .509 (53) .444 (160) .382 (361) .348 (115) .313 (32) .429 ( 7) -.096 Isaiah Thomas .667 (27) .467 (75) .412 (177) .365 (359) .313 (112) .269 (26) .400 ( 5) -.101 Chandler Parsons .448 (29) .447 (76) .371 (167) .376 (351) .350 (117) .257 (35) .286 ( 7) -.021 Tim Hardaway Jr. .361 (36) .317 (82) .372 (180) .346 (350) .356 (104) .471 (34) .429 (14) -.017 Nicolas Batum .350 (40) .348 (92) .319 (182) .326 (347) .326 ( 89) .381 (21) .286 ( 7) .007 Channing Frye .667 (24) .517 (60) .469 (162) .393 (346) .336 (110) .212 (33) .143 ( 7) -.132 * Eric Gordon .444 (18) .500 (58) .484 (153) .446 (345) .370 (127) .444 (36) .538 (13) -.114 Mo Williams .289 (38) .289 (76) .352 (162) .352 (338) .312 (109) .419 (31) .231 (13) -.042 Nikola Mirotic .474 (38) .364 (88) .318 (176) .311 (338) .295 ( 78) .333 (18) .600 ( 5) -.023 Arron Afflalo .240 (25) .262 (65) .316 (152) .355 (335) .346 (104) .212 (33) .286 ( 7) .032 Aaron Brooks .550 (20) .434 (53) .428 (145) .390 (331) .303 ( 99) .318 (22) .000 ( 6) -.126 * Gordon Hayward .333 (36) .289 (76) .305 (154) .366 (328) .390 (100) .379 (29) .167 ( 6) .088 Derrick Rose .231 (39) .256 (82) .305 (177) .301 (326) .314 ( 86) .231 (26) .200 ( 5) .009 Anthony Morrow .393 (28) .409 (66) .411 (146) .435 (324) .509 (106) .447 (38) .385 (13) .098 Patrick Beverley .378 (37) .400 (80) .363 (168) .356 (323) .283 ( 99) .160 (25) .000 ( 3) -.082 Brandon Knight .609 (23) .514 (70) .399 (158) .389 (321) .360 (100) .231 (26) .000 ( 4) -.039 Bradley Beal .313 (16) .415 (53) .419 (136) .399 (318) .364 (110) .333 (30) .400 ( 5) -.056 Mike Dunleavy .438 (16) .438 (48) .464 (138) .421 (316) .369 (103) .333 (30) .125 ( 8) -.095 Monta Ellis .417 (24) .338 (68) .268 (149) .297 (310) .359 ( 78) .478 (23) .333 ( 9) .094 Marcus Morris .182 (22) .296 (54) .348 (135) .365 (304) .326 ( 92) .375 (24) .625 ( 8) -.023 Mike Conley .294 (17) .327 (52) .393 (135) .386 (303) .356 ( 90) .455 (22) .333 ( 9) -.037 Jeff Green .421 (19) .407 (59) .321 (137) .333 (303) .350 ( 80) .286 (21) .500 ( 4) .030 Khris Middleton .300 (10) .235 (34) .390 (118) .399 (301) .374 ( 99) .345 (29) .333 ( 9) -.017 Patrick Patterson .292 (24) .350 (60) .348 (132) .377 (297) .435 ( 85) .409 (22) .167 ( 6) .087 Josh Smith .200 (10) .333 (45) .328 (122) .333 (297) .321 ( 84) .273 (22) .400 ( 5) -.007

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Danilo Gallinari .400 (30) .391 (69) .400 (150) .361 (294) .306 (85) .238 (21) .250 ( 4) -.094 O.J. Mayo .450 (20) .265 (49) .357 (129) .361 (291) .384 (86) .423 (26) .375 ( 8) .028 Dirk Nowitzki .280 (25) .286 (56) .362 (130) .371 (291) .345 (84) .364 (22) .600 ( 5) -.017 Anthony Tolliver .412 (17) .441 (59) .386 (140) .366 (290) .278 (79) .313 (16) .000 ( 3) -.108 Bojan Bogdanovic .385 (13) .347 (49) .320 (122) .353 (289) .379 (87) .481 (27) .556 ( 9) .062 Hollis Thompson .375 (16) .400 (50) .395 (129) .402 (286) .437 (87) .500 (30) .500 (12) .041 Andre Iguodala .375 ( 8) .512 (41) .412 (119) .360 (286) .364 (77) .250 (20) .000 ( 4) -.048 Iman Shumpert .389 (18) .244 (45) .325 (123) .352 (284) .383 (81) .400 (25) .250 ( 8) .059 Russell Westbrook .406 (32) .282 (71) .276 (145) .300 (283) .268 (71) .214 (14) .000 ( 3) -.009 Marcus Smart .333 (21) .327 (55) .368 (133) .335 (281) .268 (82) .238 (21) .200 ( 5) -.103 Harrison Barnes .200 ( 5) .231 (26) .350 (100) .395 (276) .325 (77) .294 (17) .667 ( 3) -.027 Manu Ginobili .438 (16) .396 (48) .368 (125) .353 (275) .289 (76) .375 (16) .500 ( 4) -.081 Jeff Teague .385 (13) .317 (41) .390 (118) .341 (270) .265 (68) .143 (14) .000 ( 2) -.127 Devin Harris .400 (15) .326 (43) .361 (119) .351 (268) .333 (75) .444 (18) .750 ( 4) -.029 Eric Bledsoe .385 (26) .350 (60) .310 (129) .330 (267) .350 (60) .250 (12) .000 ( 0) .040 Marvin Williams .818 (11) .474 (38) .419 (117) .360 (264) .278 (72) .353 (17) .200 ( 5) -.142 * Paul Millsap .000 ( 2) .357 (28) .369 (103) .348 (264) .342 (73) .524 (21) .375 ( 8) -.027 Kemba Walker .444 (18) .404 (57) .328 (131) .323 (263) .254 (71) .125 (16) .000 ( 2) -.078 Corey Brewer .412 (17) .277 (47) .274 (117) .286 (262) .321 (53) .333 (12) .250 ( 4) .048 Dante Exum .125 ( 8) .263 (38) .301 (113) .318 (261) .314 (70) .368 (19) .167 ( 6) .014 Randy Foye .273 (22) .368 (57) .385 (130) .364 (258) .329 (79) .273 (22) .000 ( 5) -.056 Deron Williams .308 (13) .250 (36) .343 (108) .380 (258) .385 (78) .458 (24) .429 ( 7) .043 Jimmy Butler .500 (12) .537 (41) .431 (109) .391 (258) .347 (75) .375 (16) .333 ( 3) -.085 Goran Dragic .143 ( 7) .310 (29) .367 (109) .349 (258) .301 (73) .375 (16) .500 ( 4) -.068 Vince Carter .300 (20) .241 (54) .296 (125) .296 (257) .322 (59) .357 (14) .250 ( 4) .026 Wesley Johnson .533 (15) .364 (44) .377 (114) .354 (257) .388 (67) .300 (20) .000 ( 4) .011 Jameer Nelson .308 (13) .396 (48) .348 (115) .344 (256) .316 (79) .364 (22) .200 ( 5) -.033 Marco Belinelli .400 (15) .372 (43) .364 (107) .391 (256) .386 (83) .208 (24) .250 ( 4) .022 PJ Tucker .231 (13) .370 (46) .385 (117) .345 (252) .362 (58) .125 (16) .000 ( 2) -.022 Courtney Lee .235 (17) .212 (33) .387 ( 93) .413 (252) .411 (73) .278 (18) .500 ( 2) .024 Victor Oladipo .043 (23) .184 (49) .304 (115) .343 (245) .350 (60) .364 (11) .000 ( 1) .047

42

Dion Waiters .250 (16) .347 (49) .292 (113) .298 (245) .267 (60) .167 (12) .000 ( 2) -.027 Isaiah Canaan .333 (21) .436 (55) .395 (119) .374 (243) .350 (80) .435 (23) .300 (10) -.046 Quincy Pondexter .500 ( 4) .417 (24) .409 ( 93) .374 (243) .382 (76) .370 (27) .375 ( 8) -.028 Wayne Ellington .261 (23) .271 (48) .367 (109) .372 (242) .314 (70) .214 (14) .000 ( 3) -.054 Matthew Dellavedova .417 (12) .382 (34) .389 ( 95) .383 (240) .333 (69) .267 (15) .500 ( 2) -.058 Charlie Villanueva .357 (14) .436 (39) .423 (104) .381 (239) .389 (72) .227 (22) .000 ( 4) -.034 James Jones .333 (15) .359 (39) .379 (103) .356 (239) .338 (68) .200 (15) .000 ( 1) -.041 Kyle Singler .400 (10) .394 (33) .443 ( 97) .402 (239) .348 (69) .412 (17) .600 ( 5) -.096 Tobias Harris .333 (12) .439 (41) .398 (103) .369 (236) .318 (66) .375 (16) .167 ( 6) -.081 Mario Chalmers .400 (10) .366 (41) .309 (110) .305 (236) .208 (53) .125 ( 8) .000 ( 1) -.106 Reggie Jackson .118 (17) .286 (42) .309 (110) .305 (236) .275 (51) .222 ( 9) .000 ( 1) -.035 Rasual Butler .286 ( 7) .455 (33) .423 ( 97) .391 (233) .261 (69) .273 (11) .500 ( 2) -.167 * D.J. Augustin .308 (13) .421 (38) .402 (102) .352 (233) .259 (58) .300 (10) .000 ( 2) -.145 Tyreke Evans .111 ( 9) .185 (27) .323 ( 93) .310 (229) .241 (58) .167 (12) .000 ( 2) -.087 Tony Snell .182 (11) .176 (34) .347 ( 95) .369 (222) .415 (65) .500 (24) .364 (11) .069 Nick Young .318 (22) .385 (52) .391 (110) .373 (220) .368 (68) .391 (23) .444 ( 9) -.023 Evan Fournier .545 (11) .485 (33) .427 ( 89) .386 (220) .260 (73) .176 (17) .333 ( 3) -.174 * John Wall .333 ( 9) .259 (27) .291 ( 86) .306 (219) .264 (53) .091 (11) .000 ( 1) -.029 Kawhi Leonard .300 (10) .368 (38) .295 ( 95) .358 (218) .411 (56) .500 (16) .429 ( 7) .119 Luol Deng .222 ( 9) .286 (28) .349 ( 86) .359 (217) .367 (60) .313 (16) .333 ( 3) .018 Alan Anderson .333 (12) .290 (31) .359 ( 78) .382 (217) .354 (65) .438 (16) .400 ( 5) -.005 Jae Crowder .400 (10) .281 (32) .293 ( 92) .298 (215) .260 (50) .500 ( 6) .000 ( 1) -.036 Caron Butler .500 ( 8) .360 (25) .407 ( 81) .386 (215) .400 (60) .588 (17) .429 ( 7) .008 Rudy Gay .444 ( 9) .371 (35) .374 ( 91) .369 (214) .414 (58) .286 (21) .000 ( 3) .040 Mike Scott .182 (11) .286 (35) .323 ( 93) .327 (214) .286 (49) .333 ( 9) - ( 0) -.038 Ersan Ilyasova .600 (10) .382 (34) .358 ( 95) .371 (213) .373 (59) .357 (14) .667 ( 3) .015 Jodie Meeks .263 (19) .175 (40) .283 ( 92) .349 (212) .419 (62) .435 (23) .333 ( 9) .142 .000 ( 2) .500 (20) .425 ( 80) .373 (209) .340 (50) .500 (10) .000 ( 1) -.085 Brandon Jennings .263 (19) .341 (44) .380 (100) .365 (208) .377 (69) .440 (25) .300 (10) -.003 Patty Mills .400 (15) .366 (41) .352 ( 91) .375 (208) .444 (63) .333 (24) .333 ( 6) .094 Jared Dudley .000 ( 2) .429 (21) .392 ( 74) .398 (206) .404 (57) .533 (15) .750 ( 4) .012

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Serge Ibaka .833 ( 6) .621 (29) .420 (81) .376 (205) .306 (62) .214 (14) .000 (2) -.116 Pero Antic .125 ( 8) .214 (28) .259 (81) .309 (204) .220 (50) .333 ( 9) .000 (2) -.044 Solomon Hill .500 ( 8) .348 (23) .387 (80) .330 (203) .229 (48) .111 ( 9) .000 (1) -.163 Luc Mbah a Moute .222 ( 9) .333 (30) .345 (84) .308 (201) .275 (51) .167 (12) .000 (2) -.074 Pablo Prigioni .444 ( 9) .321 (28) .393 (84) .338 (201) .317 (41) .250 ( 8) .500 (2) -.074 Joe Ingles .200 ( 5) .440 (25) .380 (79) .360 (200) .400 (50) .308 (13) .000 (2) .020 Ty Lawson .333 ( 9) .400 (30) .341 (82) .352 (196) .367 (49) .455 (11) .333 (3) .026 Mirza Teletovic .235 (17) .250 (44) .289 (97) .318 (195) .333 (57) .375 (16) .400 (5) .047 Kevin Martin .278 (18) .395 (38) .446 (92) .391 (192) .371 (62) .333 (18) .200 (5) -.074 Jared Sullinger .556 ( 9) .400 (35) .352 (88) .286 (192) .156 (45) .000 ( 5) - (0) -.206 * George Hill .375 (16) .450 (40) .396 (91) .361 (191) .316 (57) .308 (13) .333 (3) -.081 Brian Roberts .286 (14) .286 (28) .297 (74) .330 (188) .356 (45) .250 (12) .333 (3) .061 Norris Cole .500 ( 6) .333 (18) .348 (66) .309 (188) .240 (50) .200 (10) .000 (2) -.117 Boris Diaw .167 ( 6) .273 (22) .351 (74) .306 (186) .189 (37) .000 ( 3) - (0) -.167 Kobe Bryant .278 (18) .341 (44) .347 (95) .300 (180) .200 (50) .200 (10) .000 (2) -.153 Kelly Olynyk .667 ( 3) .500 (18) .373 (67) .356 (180) .275 (51) .000 ( 9) - (0) -.104 Spencer Hawes .833 ( 6) .458 (24) .368 (76) .322 (180) .225 (40) .000 ( 5) - (0) -.146 Jose Juan Barea .000 ( 1) .471 (17) .362 (69) .322 (180) .410 (39) .500 (12) .600 (5) .047 Dennis Schroder .667 ( 3) .278 (18) .328 (64) .335 (179) .289 (38) .143 ( 7) - (0) -.040 Shawne Williams .143 ( 7) .348 (23) .319 (69) .362 (177) .453 (53) .381 (21) .143 (7) .137 Kirk Hinrich .364 (11) .345 (29) .351 (74) .362 (177) .405 (37) .571 ( 7) .333 (3) .053 Carmelo Anthony .188 (16) .282 (39) .321 (84) .349 (175) .353 (51) .429 (14) .333 (6) .032 Markieff Morris .500 ( 4) .444 (18) .297 (64) .326 (175) .222 (36) .143 ( 7) .000 (1) -.081 Jeremy Lin .000 ( 4) .278 (18) .397 (58) .372 (172) .213 (47) .000 ( 6) - (0) -.197 * CJ Watson .250 ( 4) .429 (21) .437 (71) .409 (171) .444 (45) .357 (14) .200 (5) .008 Damjan Rudez .500 ( 4) .368 (19) .422 (64) .402 (169) .444 (45) .500 (12) .500 (2) .022 Langston Galloway .545 (11) .519 (27) .429 (77) .363 (168) .313 (48) .462 (13) .750 (4) -.116 Chris Bosh .571 ( 7) .423 (26) .427 (75) .375 (168) .327 (49) .300 (10) .000 (1) -.101 Chris Copeland .250 (16) .343 (35) .321 (81) .313 (166) .233 (43) .375 ( 8) .000 (2) -.093 Rodney Hood .556 ( 9) .370 (27) .309 (68) .373 (166) .458 (48) .467 (15) .500 (4) .153 Austin Rivers .500 ( 2) .353 (17) .350 (60) .315 (165) .290 (31) .143 ( 7) .000 (1) -.060

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Gary Neal .333 (12) .273 (33) .286 (77) .307 (163) .361 (36) .250 ( 8) .000 ( 1) .076 Richard Jefferson .667 ( 3) .647 (17) .536 (56) .423 (163) .357 (42) .286 ( 7) - ( 0) -.177 CJ McCollum .333 ( 3) .385 (13) .474 (57) .407 (162) .388 (49) .333 (15) .200 ( 5) -.086 Zach LaVine .333 ( 6) .222 (18) .259 (58) .354 (161) .317 (41) .400 (10) .500 ( 2) .064 PJ Hairston .455 (11) .324 (34) .329 (79) .306 (160) .278 (36) .125 ( 8) .000 ( 1) -.051 Kevin Durant .375 (16) .412 (34) .400 (75) .403 (159) .456 (57) .458 (24) .200 (10) .056 Kent Bazemore .333 ( 3) .286 (14) .386 (57) .342 (158) .406 (32) .250 ( 8) - ( 0) .020 Derrick Williams .500 ( 2) .308 (13) .245 (53) .314 (156) .250 (40) .000 ( 9) - ( 0) .005 Jerami Grant .125 ( 8) .304 (23) .344 (64) .318 (154) .294 (34) .143 ( 7) - ( 0) -.050 Michael Carter-Williams .375 ( 8) .296 (27) .250 (68) .234 (154) .192 (26) .000 ( 4) - ( 0) -.061 Darren Collison .286 ( 7) .333 (24) .359 (64) .391 (151) .422 (45) .400 (10) .250 ( 4) .064 Hedo Turkoglu .750 ( 4) .667 (18) .500 (58) .413 (150) .317 (41) .300 (10) .000 ( 2) -.182 KJ McDaniels .444 ( 9) .296 (27) .294 (68) .287 (150) .226 (31) .000 ( 4) - ( 0) -.071 Weighted means .367 .363 .369 .363 .350 .345 .330 -.025 ** The number of shots upon which each probability is based is given within parentheses. Players are sorted by total amount of shots, in descending order. * p < 0.05, ** p < 0.01.

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Appendix 4: Stationarity for all three point shots. Player Stephen Curry Klay Thompson Total Shots 857 687 P(hit) 0.44 0.43 E(low) 0.17 0.18 E(moderate) 0.41 0.42 E(high) 0.42 0.40 Division Division 1 2 3 1 2 3 Total Low 50 52 44 46 46 41 Total Mod 109 107 130 89 93 94 Total High 126 126 111 94 89 93

E(Frequency Low) 49.14 49.14 49.14 42.22 42.03 42.03 E(Frequency Mod) 117.45 117.45 117.45 95.88 95.46 95.46 E(Frequency High) 118.41 118.41 118.41 90.90 90.50 90.50

χ²-value 1.11 1.58 2.34 0.94 0.46 0.12 P-value 0.57 0.45 0.31 0.63 0.79 0.94 Obs. – Exp. High 7.59 7.59 -7.41 3.10 -1.50 2.50

Player James Harden Trevor Ariza Total Shots 658 650 P(hit) 0.38 0.35 E(low) 0.24 0.27 E(moderate) 0.44 0.44 E(high) 0.32 0.29 Division Division 1 2 3 1 2 3 Total Low 53 54 52 56 55 59 Total Mod 96 89 94 101 106 93 Total High 70 76 72 59 55 64

E(Frequency Low) 52.59 52.59 52.35 58.27 58.27 58.27 E(Frequency Mod) 96.06 96.06 95.62 95.73 95.73 95.73 E(Frequency High) 70.35 70.35 70.03 61.99 61.99 61.99

χ²-value 0.00 1.01 0.09 0.52 2.07 0.15 P-value 1.00 0.60 0.96 0.77 0.35 0.93 Obs. – Exp. High -0.35 5.65 1.97 -2.99 -6.99 2.01

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Player Damian Lillard Kyle Korver Total Shots 589 558 P(hit) 0.34 0.47 E(low) 0.29 0.15 E(moderate) 0.44 0.40 E(high) 0.26 0.45

Division Division 1 2 3 1 2 3 Total Low 59 64 67 26 25 27 Total Mod 89 74 72 72 80 76 Total High 48 58 56 88 80 82

E(Frequency Low) 57.34 57.34 57.05 28.33 28.18 28.18 E(Frequency Mod) 87.11 87.11 86.66 74.15 73.76 73.76 E(Frequency High) 51.56 51.56 51.29 83.51 83.07 83.07

χ²-value 0.33 3.55 4.65 0.50 1.00 0.13 P-value 0.85 0.17 0.10 0.78 0.61 0.94 Obs. – Exp. High -3.56 6.44 4.71 4.49 -3.07 -1.07

Player J.R. Smith JJ Redick Total Shots 550 546 P(hit) 0.39 0.43 E(low) 0.23 0.18 E(moderate) 0.44 0.42 E(high) 0.33 0.40

Division Division 1 2 3 1 2 3 Total Low 41 47 39 27 34 29 Total Mod 81 68 84 82 72 80 Total High 61 68 59 73 75 72

E(Frequency Low) 42.10 42.10 41.87 33.63 33.45 33.45 E(Frequency Mod) 79.82 79.82 79.39 76.24 75.82 75.82 E(Frequency High) 61.08 61.08 60.75 72.12 71.73 71.73

χ²-value 0.05 3.11 0.51 1.75 0.35 0.82 P-value 0.98 0.21 0.77 0.42 0.84 0.66 Obs. – Exp. High -0.08 6.92 -1.75 0.88 3.27 0.27

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Player Danny Green Lou Williams Total Shots 495 462 P(hit) 0.41 0.34 E(low) 0.21 0.29 E(moderate) 0.43 0.44 E(high) 0.36 0.27

Division Division 1 2 3 1 2 3 Total Low 40 34 36 40 46 46 Total Mod 63 69 66 77 64 63 Total High 62 61 62 37 43 44

E(Frequency Low) 34.22 34.01 34.01 44.75 44.46 44.46 E(Frequency Mod) 70.77 70.35 70.35 68.44 67.99 67.99 E(Frequency High) 60.01 59.64 59.64 40.82 40.55 40.55

χ²-value 1.90 0.06 0.48 1.93 0.44 0.71 P-value 0.39 0.97 0.79 0.38 0.80 0.70 Obs. – Exp. High 1.99 1.36 2.36 -3.82 2.45 3.45

Player Wesley Matthews Robert Covington Total Shots 445 443 P(hit) 0.39 0.38 E(low) 0.23 0.24 E(moderate) 0.44 0.44 E(high) 0.34 0.32

Division Division 1 2 3 1 2 3 Total Low 37 29 33 35 31 35 Total Mod 60 72 65 67 69 66 Total High 51 47 49 45 47 46

E(Frequency Low) 33.80 33.80 33.57 35.55 35.55 35.55 E(Frequency Mod) 64.49 64.49 64.05 64.53 64.53 64.53 E(Frequency High) 49.71 49.71 49.38 46.92 46.92 46.92

χ²-value 0.65 1.70 0.03 0.18 0.89 0.06 P-value 0.72 0.43 0.99 0.91 0.64 0.97 Obs. – Exp. High 1.29 -2.71 -0.38 -1.92 0.08 -0.92

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Player LeBron James CJ Miles Total Shots 438 437 P(hit) 0.33 0.35 E(low) 0.30 0.27 E(moderate) 0.44 0.44 E(high) 0.25 0.29

Division Division 1 2 3 1 2 3 Total Low 40 40 36 42 38 44 Total Mod 71 69 77 61 68 56 Total High 35 36 32 42 39 45

E(Frequency Low) 44.15 43.85 43.85 39.38 39.38 39.38 E(Frequency Mod) 64.88 64.44 64.44 64.29 64.29 64.29 E(Frequency High) 36.97 36.71 36.71 41.33 41.33 41.33

χ²-value 1.07 0.68 4.46 0.35 0.39 1.94 P-value 0.58 0.71 0.11 0.84 0.82 0.38 Obs. – Exp. High -1.97 -0.71 -4.71 0.67 -2.33 3.67

Player Kentavious Caldwell-Pope Kyrie Irving Total Shots 436 435 P(hit) 0.35 0.42 E(low) 0.28 0.19 E(moderate) 0.44 0.42 E(high) 0.28 0.38

Division Division 1 2 3 1 2 3 Total Low 37 37 44 31 30 34 Total Mod 70 69 56 58 60 51 Total High 38 39 44 56 54 59

E(Frequency Low) 40.07 40.07 39.80 28.19 28.00 28.00 E(Frequency Mod) 64.34 64.34 63.90 61.41 60.99 60.99 E(Frequency High) 40.58 40.58 40.30 55.39 55.01 55.01

χ²-value 0.90 0.63 1.76 0.48 0.18 3.21 P-value 0.64 0.73 0.41 0.79 0.91 0.20 Obs. – Exp. High -2.58 -1.58 3.70 0.61 -1.01 3.99

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Player Matt Barnes Draymond Green Total Shots 435 417 P(hit) 0.35 0.32 E(low) 0.28 0.31 E(moderate) 0.44 0.44 E(high) 0.28 0.25

Division Division 1 2 3 1 2 3 Total Low 40 38 33 43 44 41 Total Mod 66 67 74 60 57 62 Total High 39 39 37 36 37 35

E(Frequency Low) 39.93 39.65 39.65 42.99 42.68 42.68 E(Frequency Mod) 64.33 63.89 63.89 61.74 61.29 61.29 E(Frequency High) 40.74 40.46 40.46 34.27 34.03 34.03

χ²-value 0.12 0.27 3.01 0.14 0.60 0.10 P-value 0.94 0.87 0.22 0.93 0.74 0.95 Obs. – Exp. High -1.74 -1.46 -3.46 1.73 2.97 0.97

Player Kevin Love Jamal Crawford Total Shots 413 413 P(hit) 0.37 0.33 E(low) 0.25 0.30 E(moderate) 0.44 0.44 E(high) 0.31 0.25

Division Division 1 2 3 1 2 3 Total Low 36 34 32 41 35 34 Total Mod 57 59 67 60 73 76 Total High 44 44 38 36 29 27

E(Frequency Low) 34.18 34.18 34.18 41.33 41.33 41.33 E(Frequency Mod) 60.34 60.34 60.34 60.88 60.88 60.88 E(Frequency High) 42.48 42.48 42.48 34.78 34.78 34.78

χ²-value 0.34 0.09 1.35 0.06 4.34 6.80 P-value 0.85 0.96 0.51 0.97 0.11 0.03 Obs. – Exp. High 1.52 1.52 -4.48 1.22 -5.78 -7.78

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Player Chris Paul Kyle Lowry Total Shots 410 408 P(hit) 0.40 0.33 E(low) 0.22 0.30 E(moderate) 0.43 0.44 E(high) 0.35 0.26

Division Division 1 2 3 1 2 3 Total Low 28 25 27 37 41 39 Total Mod 58 66 61 65 57 62 Total High 50 45 48 34 37 34

E(Frequency Low) 29.38 29.38 29.38 40.30 40.00 40.00 E(Frequency Mod) 58.75 58.75 58.75 60.44 60.00 60.00 E(Frequency High) 47.87 47.87 47.87 35.26 35.00 35.00

χ²-value 0.17 1.72 0.28 0.66 0.29 0.12 P-value 0.92 0.42 0.87 0.72 0.87 0.94 Obs. – Exp. High 2.13 -2.87 0.13 -1.26 2.00 -1.00

Player Nick Young Rudy Gay Total Shots 220 214 P(hit) 0.37 0.37 E(low) 0.25 0.25 E(moderate) 0.44 0.44 E(high) 0.31 0.31

Division Division 1 2 3 1 2 3 Total Low 20 17 18 20 20 18 Total Mod 30 33 30 28 26 28 Total High 23 23 24 23 25 24

E(Frequency Low) 18.02 18.02 17.77 17.82 17.82 17.57 E(Frequency Mod) 32.12 32.12 31.68 31.29 31.29 30.85 E(Frequency High) 22.86 22.86 22.55 21.88 21.88 21.58

χ²-value 0.36 0.08 0.18 0.67 1.60 0.55 P-value 0.84 0.96 0.91 0.72 0.45 0.76 Obs. – Exp. High 0.14 0.14 1.45 1.12 3.12 2.42

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Player Brandon Jennings Carmelo Anthony Total Shots 208 175 P(hit) 0.37 0.35 E(low) 0.26 0.28 E(moderate) 0.44 0.44 E(high) 0.30 0.28

Division Division 1 2 3 1 2 3 Total Low 20 22 16 20 16 17 Total Mod 25 23 32 17 25 25 Total High 24 24 20 21 17 15

E(Frequency Low) 17.64 17.64 17.38 16.03 16.03 15.76 E(Frequency Mod) 30.46 30.46 30.02 25.74 25.74 25.29 E(Frequency High) 20.90 20.90 20.60 16.23 16.23 15.95

χ²-value 1.75 3.37 0.26 5.35 0.06 0.16 P-value 0.42 0.19 0.88 0.07 0.97 0.92 Obs. – Exp. High 3.10 3.10 -0.60 4.77 0.77 -0.95 Players are sorted by total amount of shots, in descending order. * p < 0.05, ** p < 0.01.

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