Background for what RAPM is: +/- was a revolution for the NBA because it allowed a completely new method at evaluating players. You look at how a team scores and defends with you on the court and without you. When you set players as variables, you can use regression to calculate player impact. It's a full scope view of what matters in a game: outscoring your opponent. However, it's noisy for a number of reasons. One is that some player combinations are rare (this is known as collinearity.) Another is that the models don't deal well with players with low minutes, as they don't have enough of a sample for an accurate estimate and will often produce a ludicrous result just to "fit" the data better.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
The NBA lockout that delayed the 1999 season put a serious dent in the league and wasn't aided by the retirement of Michael Jordan and the dismantling of the Bulls. It was a transition year, with the old guard dropping off with the exception of iron-man Karl Malone, while the next generation was still developing and coming into their own. With a shortened season opening in February, games jammed together with too many back-to-back's, a team known for professionalism, led by a second year Duncan and a 33 year-old David Robinson, won the championship. This was Shaq pre-Phil Jackson but post-Jordan; it's a truly lost season.
Nevertheless, the games were played, and we can't ignore what happened. Surprisingly, by NPI RAPM, the best player in 1999 was ... David Robinson, who had a monster defensive impact. Just a season before Duncan and Robinson were neck-and-neck on a per possession basis, but the Admiral leaps ahead here. Though with a 24.9 PER and a league-leading 0.261 Win Shares per 48 minutes, and stats that did not drop in the postseason, perhaps it shouldn't be surprising. However, he only played 31.7 minutes in the regular season, lending credence to an argument for another candidate like Alonzo Mourning. Jaren Jackson is probably the oddest name near the top of the list. He was a Spurs teammate but only scored 6.4 points per game. And he's why we turn to prior-informed versions....
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
With two seasons as seed data behind it, RPI RAPM (an adjusted ridge-regression model given prior information) is a more powerful tool. After a dominating 1998, Shaq loses the top spot due to poorer defense. This was probably Alonzo's greatest season with award-caliber defense coupled with potent scoring (20 points in a slow-paced league and a 56 TS%.) He was second in MVP voting -- justified here. As for some surprising results, Blaylock continues to be a plus/minus star, and Rasheed Wallace rockets to the top on the deep but strong Blazers team that nearly defeated the Lakers in 2000. Charles Barkley also continues his one-sided results: his offensive RAPM rises to 6.16 but his defense slips to -2.64.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
Fourth in MVP voting was Iverson. He's 75th here with good offensive value despite his poor shooting efficiency, but that's submarined by his porous defense, according to RAPM. Duncan was third, and he's backed-up by a very strong +5.2 mark. Rookie of the year Vince Carter, however, climbs from the rookie prior of -2 (given to all rookies) to +3.21 overall. He heads a pretty strong rookie class including Pierce, Brad Miller, and Dirk Nowitzki. Jason Williams received some rookie love, but by this method he was a significantly negative player. For validation of the model, 16 of the top 24 players were on all-NBA teams -- or perhaps that's validation of the all-NBA teams. The lowest rated all-NBA player was the young Kobe Bryant at -1.2 and it wasn't close: he was ranked 267th and the next lowest all-NBA player was McDyess at 117.
The aforementioned Barkley was the best offensive player, according to RPI RAPM, and that's not just because of his past heroics: his non-prior informed RAPM was second. Shaq's stats were muted by fewer minutes and the slogging pace of the lockout season, but, with the first of his strong of 30+ PER seasons, he's second in RPI RAPM on offense, barely trailing Barkley. Malone rounds out the top three, and the next highest players are more intriguing: Reggie's three-point bombing is probably underrated by most box score metrics at fourth here, Grant Hill's fifth, and Jeff Hornacek, again, has a strong showing at sixth followed by Blaylock. Both Hornacek and Blaylock are players that appear to be underrated by a method that wasn't available in the 90's. Mutombo, David Robinson, and Mourning were top three by defense. Mourning won the Defensive Player of the Year award, which wasn't a bad choice, necessarily, but Mutombo has a huge lead in RAPM. Rasheed was fourth, and his value was demonstrated later when he was traded to the Pistons to complete one of the greatest defensive teams ever. Jaren Jackson appears to be an anomalous result, ranking fifth, but he was San Antonio's first "three-and-D" player. He concentrated on defense, and his value does not show up in box scores. His NPI RAPM wasn't entirely misleading. Trailing him were Shawn Bradley, Olajuwon, and Garnett. Gigantic size might actually be underrated because Yao Ming's defense looked better under the RAPM microscope and Bradley joins Gheorghe Muresan as another 7' 7" player with strong defensive results.
With three seasons of data, outliers can be weeded out and patterns emerge. We can reevaluate past legends like David Robinson and Stockton who rate well, but sub-stars should receive more discussion. Mookie Blaylock, by NPI RAPM, was 8th in 1997, 11th in 1998, and 36th in 1999. By prior-informed regression, he was fifth in 1999. Blaylock played in the shadows of point guards like Stockton and Kidd, but he may have been better than we thought. Defense is notoriously tricky to judge by basic stats, and this is where RAPM can be most illuminating. Bo Outlaw stands out as a defensive force who wasn't heralded. Divac fares well as a two-way center, often better than his famous teammate Chris Webber. And, again, we shouldn't completely toss aside Jaren Jackson's RAPM results. He was a forgotten role player and not highly regarded, but like Shane Batter we should look beyond the box score.
Click here for the link to the spreadsheet.
Showing posts with label shaq. Show all posts
Showing posts with label shaq. Show all posts
Monday, March 3, 2014
Monday, February 3, 2014
Evaluating Hack-a-Shaq
Years ago, a giant center used to destroy teams on his own, putting up monster statistics with high accuracy from the field. He was an unmovable object with superhuman strength, and he seemed like the ultimate offensive weapon. He did have one weakness, however, his kryptonite -- free-throw shooting. With several seasons below 50% from the line, teams would intentionally foul him because it was much better than the alternative. People argued that he was so poor from the line, in fact, that he was detrimental to the team whenever the strategy was employed.
At the time, the strategy wasn't called Hack-a-Shaq because it was used against Wilt Chamberlain. The league changed some rules regarding fouls called in the last two minutes and the strategy laid dormant for decades until eccentric basketball mind Don Nelson used it against Dennis Rodman and then Shaquille O'Neal. Unfortunately, it's still an active weapon, most notably against Dwight Howard. There are ongoing debates about the merits of keeping this a part of basketball -- it's ugly and slows down the game versus if a player doesn't want this used against him, he should work on his shooting -- but what's missed most often is how effective it is as a strategy. People discuss how sending a 55% foul-shooter to the line results in a "worse" offense, but don't compare it to how efficient the team normally is and ignore the possibility of offensive rebounds.
Breaking down Hack-a-Shaq, there are four components for evaluating its potency:
1) The foul-shooter's percentage from the line
2) The chance of an offensive rebound off a miss
3) The offensive rating after the offensive rebound
4) The offensive rating of the team if you don't intentionally foul
Combining 1) through 3), you can compare the offensive rating of the intentional foul to 4). If the resultant offensive rating for a Hack-a-Shaq isn't lower than the team's offensive rating without fouling them, then it's probably not a worthwhile strategy, unless you have other motivations like stopping the clock.
The formula for a Hack-a-Shaq offensive rating is pretty simple:
100*(2*FT%/100+(100-FT%)/100*FTORB*FTORTG/100)
where
FT%: a player's free throws percentage (in the form of 60% and not 0.60)
FTORB: the team's chance of grabbing an offensive rebound after a free throw miss (in the form of 0.20, not 20%)
FTORTG: the team's offensive rating after a FTORB (per 100 possessions, i.e. 110 points)
The 100 at the beginning is to translate the number into points per 100 possessions. The first part of the equation inside the parenthesis are the expected points you get sending the player to the line, and the second part is the expected points from an offensive rebound given the chance of an available rebound (a miss) and the chance of the offensive team grabbing it. So with the formula explained, here's a simplified version (with the extra 100's):
Hack-a-Shaq ORTG = 2*FT%+(100-FT%)/100*FTORB*FTORTG
The FT% is the easier variable to find. You can simply use the player's season average, the average from his last three seasons, or his career average.
The variable FTORB, however, is trickier to find. It's not an official stat tracked by any site (least not that I could find) and it can't be deducted from conventional stats. There have been a handful of studies of offensive rebounding based on the shot type, like this early one from 82games.com or a recent one from hoopdon. The former study found a 13.9% OREB after free throw misses, while the latter using NBAWOWY's site found an average of 12%. The past couple seasons have seen an average of all offensive rebounding around roughly 26%, so there's clearly a substantial drop when rebounding after a missed free throw. (82game's higher figure, by the way, stems from the slightly higher offensive rebounding numbers in the mid-00's and because it included team rebounds.)
But the analysis doesn't end here. When, say, Dwight Howard is at the line, we cannot assume a 12% average because of two complicating factors: one being that Howard, the team's best offensive rebounder, is now far from the rim, and secondly when he's at the line players expect a missed free throw and fight harder for an offensive rebound. Reading play-by-play data, it's possible to calculate offensive rebounding values for when the infamous free throw chuckers miss. What's surprising is that despite having Houston's team offensive rebounder at the line and playing a significant portion with smallball lineups, the Rockets have rebounded at an above average rate when he misses a free throw.
With some approximations of rebounding percentages, the next step is the expected points after the rebound off the free throw miss. This issue has been studied before, and one can expect a significantly higher effective field-goal percentage off a miss than most other actions (like a made field goal on the other end.) For instance, the Rockets have an eFG% of 56.4 after a free throw offensive rebound compared to their season average of 51.9%, while the Clippers have averages of 56.8% and 52.8%, respectively. Until I get more data specifically for offensive rating after misses, I'm going to assume a conservative increase in efficiency after these misses -- 5 points per 100 possessions, compared to nearly 10 points that I've found from NBAWOWY, depending on the team/personnel.
With these numbers, one can now calculate Hack-a-Shaq offensive ratings. For example, using the averages for Howard and his team this season:
Hack-a-Shaq ORTG = 2*FT%+(100-FT%)/100*FTORB*FTORTG
Hack-a-Shaq ORTG = 2*53.3+(100-53.3)/100*0.171*(109.3+5)
Hack-a-Shaq ORTG = 115.7
Compared to Houston's rating of 109.3 for this season, it is not wise to put him on the line intentionally because you are effectively giving them a very high offensive efficiency (like the Nash-Suns at their very best) and one that's much higher than what you'd expect without fouling him. I'm sure people also want a Shaq-specific example, so during the 2001 regular season in LA he shot 51.3% (his worst season during the title stretch) with an offensive rating of 108.4. Using estimates of a 15% FTORB and a conservative FTORTG 5 points above their regular season average, this translates to a Hack-a-Shaq ORTG of 110.9 points per 100 possesions -- so yes, even during one of his poorest shooting seasons in LA, it was still an unwise strategy.
Or you can solve for what FT% Howard would need for this to be a break-even strategy (meaning, a percentage lower than this means it's an effective strategy.)
Hack-a-Shaq ORTG = 2*FT%+(100-FT%)/100*FTORB*FTORTG
109.3 = 2*FT%+100/100*FTORB*FTORTG-FT%/100*FTORB*FTORTG
109.3 - 100/100*FTORB*FTORTG = 2*FT%-FT%/100*FTORB*FTORTG
109.3 - FTORB*FTORTG = FT%*(2-1/100*FTORB*FTORTG)
FT% = (109.3 - FTORB*FTORTG) / (2-1/100*FTORB*FTORTG)
FT% = 49.7
(Even with an OREB% of 10, the break-even point is still only around 51.9%.)
Fifty-percent is a reasonably good break-even line for big men on good teams in most scenarios. Note that there's a high degree of elasticity with regards to free-throw percentage (i.e. FT% greatly changes the Hack-a-Shaq efficiency.) Free-throw OREB% has a lower elasticity, while ORTG after a missed free-throw is greatly inelastic. To visualize this, I prepared a series of charts showing how Hack-a-Shaq efficiency changes with each respective variable. Since there are three independent variables and one dependent variable (the result), it's a bit like four-dimensional graphing, so there's a chart for five different OREB rates, where the x-axis is FT% and the y-axis for the Hack-a-Shaq efficiency, and then one line each for a team's ORTG after a missed free throw.
What's surprising is that Dwight Howard gets all the attention for that Hack-a-Shaq strategy when he's comfortably past the break-even line, but both Drummond and DeAndre are two of the worst foul shooters ever and are far below the efficiency threshold. Even if their team rebounds their misses at a high rate, the result is an offense that would perform worse than the 2012 Bobcats, who won only 7 games. The Clippers' offense is a high-powered one featuring the best point guard in the guard, one of the best scoring big men, and several outside shooters, but you can stall their offense by sending DeAndre Jordan to the line. He's only had one season above 50% and needs to improve so he's not a liability whenever the team is in the penalty.
The strategy is ugly and an effrontery to the beauty of the game, but it's also less effective in most situations than most people realize. With the relevant data, it's also fairly simple to calculate. (This will probably spawn another study looking at how a free throw miss offensive rebound percentage changes based on who's at the line.) For most of Shaq's career it was an inappropriate strategy because it gave his team a higher offensive rating than they had otherwise, and this is true of Howard too as his two worst seasons had free-throw percentages of 49, which is straddling the break-even line. When both guys were near 60%, it was an especially stupid strategy.
But you can still have fun with it.
At the time, the strategy wasn't called Hack-a-Shaq because it was used against Wilt Chamberlain. The league changed some rules regarding fouls called in the last two minutes and the strategy laid dormant for decades until eccentric basketball mind Don Nelson used it against Dennis Rodman and then Shaquille O'Neal. Unfortunately, it's still an active weapon, most notably against Dwight Howard. There are ongoing debates about the merits of keeping this a part of basketball -- it's ugly and slows down the game versus if a player doesn't want this used against him, he should work on his shooting -- but what's missed most often is how effective it is as a strategy. People discuss how sending a 55% foul-shooter to the line results in a "worse" offense, but don't compare it to how efficient the team normally is and ignore the possibility of offensive rebounds.
Breaking down Hack-a-Shaq, there are four components for evaluating its potency:
1) The foul-shooter's percentage from the line
2) The chance of an offensive rebound off a miss
3) The offensive rating after the offensive rebound
4) The offensive rating of the team if you don't intentionally foul
Combining 1) through 3), you can compare the offensive rating of the intentional foul to 4). If the resultant offensive rating for a Hack-a-Shaq isn't lower than the team's offensive rating without fouling them, then it's probably not a worthwhile strategy, unless you have other motivations like stopping the clock.
The formula for a Hack-a-Shaq offensive rating is pretty simple:
100*(2*FT%/100+(100-FT%)/100*FTORB*FTORTG/100)
where
FT%: a player's free throws percentage (in the form of 60% and not 0.60)
FTORB: the team's chance of grabbing an offensive rebound after a free throw miss (in the form of 0.20, not 20%)
FTORTG: the team's offensive rating after a FTORB (per 100 possessions, i.e. 110 points)
The 100 at the beginning is to translate the number into points per 100 possessions. The first part of the equation inside the parenthesis are the expected points you get sending the player to the line, and the second part is the expected points from an offensive rebound given the chance of an available rebound (a miss) and the chance of the offensive team grabbing it. So with the formula explained, here's a simplified version (with the extra 100's):
Hack-a-Shaq ORTG = 2*FT%+(100-FT%)/100*FTORB*FTORTG
The FT% is the easier variable to find. You can simply use the player's season average, the average from his last three seasons, or his career average.
The variable FTORB, however, is trickier to find. It's not an official stat tracked by any site (least not that I could find) and it can't be deducted from conventional stats. There have been a handful of studies of offensive rebounding based on the shot type, like this early one from 82games.com or a recent one from hoopdon. The former study found a 13.9% OREB after free throw misses, while the latter using NBAWOWY's site found an average of 12%. The past couple seasons have seen an average of all offensive rebounding around roughly 26%, so there's clearly a substantial drop when rebounding after a missed free throw. (82game's higher figure, by the way, stems from the slightly higher offensive rebounding numbers in the mid-00's and because it included team rebounds.)
But the analysis doesn't end here. When, say, Dwight Howard is at the line, we cannot assume a 12% average because of two complicating factors: one being that Howard, the team's best offensive rebounder, is now far from the rim, and secondly when he's at the line players expect a missed free throw and fight harder for an offensive rebound. Reading play-by-play data, it's possible to calculate offensive rebounding values for when the infamous free throw chuckers miss. What's surprising is that despite having Houston's team offensive rebounder at the line and playing a significant portion with smallball lineups, the Rockets have rebounded at an above average rate when he misses a free throw.
Player.............
|
OREB% FT miss
|
FT%
|
FT misses
|
Dwight Howard
|
0.171
|
53.3
|
129
|
Andre Drummond
|
0.091
|
40.7
|
55
|
DeAndre Jordan
|
0.172
|
41.3
|
64
|
With some approximations of rebounding percentages, the next step is the expected points after the rebound off the free throw miss. This issue has been studied before, and one can expect a significantly higher effective field-goal percentage off a miss than most other actions (like a made field goal on the other end.) For instance, the Rockets have an eFG% of 56.4 after a free throw offensive rebound compared to their season average of 51.9%, while the Clippers have averages of 56.8% and 52.8%, respectively. Until I get more data specifically for offensive rating after misses, I'm going to assume a conservative increase in efficiency after these misses -- 5 points per 100 possessions, compared to nearly 10 points that I've found from NBAWOWY, depending on the team/personnel.
With these numbers, one can now calculate Hack-a-Shaq offensive ratings. For example, using the averages for Howard and his team this season:
Hack-a-Shaq ORTG = 2*FT%+(100-FT%)/100*FTORB*FTORTG
Hack-a-Shaq ORTG = 2*53.3+(100-53.3)/100*0.171*(109.3+5)
Hack-a-Shaq ORTG = 115.7
Compared to Houston's rating of 109.3 for this season, it is not wise to put him on the line intentionally because you are effectively giving them a very high offensive efficiency (like the Nash-Suns at their very best) and one that's much higher than what you'd expect without fouling him. I'm sure people also want a Shaq-specific example, so during the 2001 regular season in LA he shot 51.3% (his worst season during the title stretch) with an offensive rating of 108.4. Using estimates of a 15% FTORB and a conservative FTORTG 5 points above their regular season average, this translates to a Hack-a-Shaq ORTG of 110.9 points per 100 possesions -- so yes, even during one of his poorest shooting seasons in LA, it was still an unwise strategy.
Or you can solve for what FT% Howard would need for this to be a break-even strategy (meaning, a percentage lower than this means it's an effective strategy.)
Hack-a-Shaq ORTG = 2*FT%+(100-FT%)/100*FTORB*FTORTG
109.3 = 2*FT%+100/100*FTORB*FTORTG-FT%/100*FTORB*FTORTG
109.3 - 100/100*FTORB*FTORTG = 2*FT%-FT%/100*FTORB*FTORTG
109.3 - FTORB*FTORTG = FT%*(2-1/100*FTORB*FTORTG)
FT% = (109.3 - FTORB*FTORTG) / (2-1/100*FTORB*FTORTG)
FT% = 49.7
(Even with an OREB% of 10, the break-even point is still only around 51.9%.)
Fifty-percent is a reasonably good break-even line for big men on good teams in most scenarios. Note that there's a high degree of elasticity with regards to free-throw percentage (i.e. FT% greatly changes the Hack-a-Shaq efficiency.) Free-throw OREB% has a lower elasticity, while ORTG after a missed free-throw is greatly inelastic. To visualize this, I prepared a series of charts showing how Hack-a-Shaq efficiency changes with each respective variable. Since there are three independent variables and one dependent variable (the result), it's a bit like four-dimensional graphing, so there's a chart for five different OREB rates, where the x-axis is FT% and the y-axis for the Hack-a-Shaq efficiency, and then one line each for a team's ORTG after a missed free throw.
The strategy is ugly and an effrontery to the beauty of the game, but it's also less effective in most situations than most people realize. With the relevant data, it's also fairly simple to calculate. (This will probably spawn another study looking at how a free throw miss offensive rebound percentage changes based on who's at the line.) For most of Shaq's career it was an inappropriate strategy because it gave his team a higher offensive rating than they had otherwise, and this is true of Howard too as his two worst seasons had free-throw percentages of 49, which is straddling the break-even line. When both guys were near 60%, it was an especially stupid strategy.
But you can still have fun with it.
Monday, January 23, 2012
Free Throws and Hand Size: Are Big Hands Detrimental to Shooting?
Introduction
One of the most common discussions on free throws is how bigger hands make it harder to shoot. Shaq has such big hands, people say, that it's impossible for him to be a good free-throw shooter no matter what his form is or how often he practices. Another player, Rondo, has very large hands, especially for point guard, and people excuse his poor shooting from the foul line and his outside jumpers. If this is true, then we should see a lower percentages as a player's hand size increase.
I think one credible explanation is how the ball is held between someone with (relative to the NBA) small hands and someone with large hands. When taking a free throw most people are able to easily grip the ball with both hands: one to provide the power and one for stability. When your dimensions are like Shaq's, it's harder to balance the ball as you release. Try shooting with a tennis ball for an extreme example. You're entirely using one hand to launch it, and it's generally harder to retain precision in the typical basketball shooting motion. (Of course, the tennis ball's smaller size will help you make the shot.) It's a reasonable explanation, and the hand size-shooting myth is prevalent among NBA fans because it feels right. One can't assume this is true, however, without objectively looking into the matter.
Methodology
In order to systematically test the hypothesis that larger hands lower a player's free-throw percentage, you need lots of data. Fortunately, at the pre-draft combine they started collecting hand size from virtually every player. This was only started in 2010, but that still leaves a huge pool of guys to analyze. I could have also included players who have had measurements released to the public by the team or media; you can find plenty of these "facts" online. However, those are self-selected data and I would be compiling them one at a time, which is too slow of a rate for me.
Over the past two years there were 155 players with hand measurements, but only 21 with over 50 free throw attempts in NBA games. The hand sizes ranged from 11.25 inches long to 7.25 among players. Hand width was also measured but not for every player, so I didn't include it in the study. I did, however, use a few other explanatory variables. Since hand size correlates with height and therefore position, those were both used in the data. I also had age as a variable because players typically improve from the line as they get older, although these players are all rookies or sophomores. The height used was the pre-draft's height without shoes because the inflated with shoes number or whatever a team lists has more variation -- some guys gain an inch for their shoes, and some two.
Results
The first results I'll discuss are for players with over 50 attempts and treating each player as a separate datum. Free-throw percentage was the dependent variable, and hand size, height, position, and age were tested as the independent variables in a linear regression. I tried every combination of the independent variables, and in every case hand size statistically insignificant. Even with just one independent variable the p-value for hand length -- it's the probability that hand length has zero effect on FT% -- was only 0.111. For reference, 0.05 is taken as statistically significant, and hand size correlates with position well enough that you'd think there'd be enough of a correlation for a better result. By contrast, using just height, the p-value was 0.016 and the R-squared value 0.27, meaning 27% of the variation in free-throw shooting percentage was explained by height alone.
The table below contains the 21 players with over 50 attempts arranged by hand size. There's a bit of a pattern where most of the best shooters have smaller hands, but it's not perfect. Aminu, in particular, has huge hands but he's shot well from the line. Likewise, Vasquez has tiny hands for an NBA player but he's been terrible. I also want to note that 21 samples are not large enough for a conclusive study, and in couple years there should be enough data. However, there is no overwhelming evidence to support the claim that hand size has a negative effect on free-throw shooting, and I think that's a fascinating preliminary result.
Perhaps a better method is to pool each free throw attempt into a hand size measurement. That way a rookie who only plays in a blowout will have his ten attempts be used with the rest of the group of players with his hand length. Luckily, hands were measured in discrete units instead of something more exact like 8.425" and there are only nine categories. The linear regression based on one independent variable, hand size, output a p-value of 0.326 and an R-squared of 0.137. Basically, there was no correlation, and the graph below illustrates the point. There isn't any designation of point guard versus centers, but pundits claim someone like Rondo shoots so poorly because of his large hands. One problem, however, with a smaller than ideal data set is that players with high attempts like John Wall skew the results, and that's why you're seeing such different results for each size.
Another approach I tried was grouping the results by position and applying linear regression to each position. Unfortunately, there aren't enough players yet with hand measurements to subdivide the data even when lowering the free throw requirement to 20 attempts. The results, however, were so one-sided that I doubt more data would reveal a statistical significance between hand size within positions. Point guards, for example, a group known for their sweet shooting, had a tiny R-squared value of 0.0529 and a p-value for the hand length coefficient of 0.523. R-squared, to reiterate, is saying hand length only explains 5% of the variation in free-throw percentage among point guards, and that's a abhorrent result. There were only 10 point guards to qualify, but the same was true of each position. Again, there was no evidence to conclude that hand size was a significant variable in determining free-throw percentage. To illustrate the lack of association between size and free throws, Iman Shumpert was more of a combo guard so he wasn't included, but he has larger hands than any point, yet he's at 22/24 for the year.
Conclusion
The NBA athletes we see on TV are undeniably talented and earning more money than most of us will ever accrue in our lives. When we see a player making $10 million a year miss a free throw -- a shot even little kids can make -- we're angry and baffled. Surely he's been practicing for years, so why is he shooting 60% for the season? In asking that question some seek answers about how some players aren't able to make them during the flow of the game or that these athletes' hands are too big to accurately shoot. We can find examples like Duncan, who clearly has worked on his shot but often shoots under 70%, to fit our theory and confirm our own perceptions, but there are also counter-examples. Pau Gasol has enormous hands and palms the ball with ease, but he has an accurate jump shot and cleared 80% for the season multiple times. John Stockton is a point guard with large mitts, and he's at 82.6% for his career with a good three-point shot. Michael Jordan, known for using his big hands to dunk with flair, is even better at 83.5%.
The inclusion of hand size measurements for nearly every drafted player recently will lead to a better data set, and at the end of the year I'll redo the regression to see if anything changes. There are a couple other hypotheses that need to be tested. Maybe there's a hand size limit beyond which percentage plunges, but we don't have enough information for that yet. The players in the data are all very young and the results could change once they improve or, even their hand size is so problematic, flatline at a poor number. Within a couple of years we could get a definitive answer as more guys get measured and the ones that already have rack up attempts. However, based on the data right now, there is no evidence to support the hypothesis that hand size negatively affects free-throw percentage.
One of the most common discussions on free throws is how bigger hands make it harder to shoot. Shaq has such big hands, people say, that it's impossible for him to be a good free-throw shooter no matter what his form is or how often he practices. Another player, Rondo, has very large hands, especially for point guard, and people excuse his poor shooting from the foul line and his outside jumpers. If this is true, then we should see a lower percentages as a player's hand size increase.
I think one credible explanation is how the ball is held between someone with (relative to the NBA) small hands and someone with large hands. When taking a free throw most people are able to easily grip the ball with both hands: one to provide the power and one for stability. When your dimensions are like Shaq's, it's harder to balance the ball as you release. Try shooting with a tennis ball for an extreme example. You're entirely using one hand to launch it, and it's generally harder to retain precision in the typical basketball shooting motion. (Of course, the tennis ball's smaller size will help you make the shot.) It's a reasonable explanation, and the hand size-shooting myth is prevalent among NBA fans because it feels right. One can't assume this is true, however, without objectively looking into the matter.
Methodology
In order to systematically test the hypothesis that larger hands lower a player's free-throw percentage, you need lots of data. Fortunately, at the pre-draft combine they started collecting hand size from virtually every player. This was only started in 2010, but that still leaves a huge pool of guys to analyze. I could have also included players who have had measurements released to the public by the team or media; you can find plenty of these "facts" online. However, those are self-selected data and I would be compiling them one at a time, which is too slow of a rate for me.
Over the past two years there were 155 players with hand measurements, but only 21 with over 50 free throw attempts in NBA games. The hand sizes ranged from 11.25 inches long to 7.25 among players. Hand width was also measured but not for every player, so I didn't include it in the study. I did, however, use a few other explanatory variables. Since hand size correlates with height and therefore position, those were both used in the data. I also had age as a variable because players typically improve from the line as they get older, although these players are all rookies or sophomores. The height used was the pre-draft's height without shoes because the inflated with shoes number or whatever a team lists has more variation -- some guys gain an inch for their shoes, and some two.
Results
The first results I'll discuss are for players with over 50 attempts and treating each player as a separate datum. Free-throw percentage was the dependent variable, and hand size, height, position, and age were tested as the independent variables in a linear regression. I tried every combination of the independent variables, and in every case hand size statistically insignificant. Even with just one independent variable the p-value for hand length -- it's the probability that hand length has zero effect on FT% -- was only 0.111. For reference, 0.05 is taken as statistically significant, and hand size correlates with position well enough that you'd think there'd be enough of a correlation for a better result. By contrast, using just height, the p-value was 0.016 and the R-squared value 0.27, meaning 27% of the variation in free-throw shooting percentage was explained by height alone.
The table below contains the 21 players with over 50 attempts arranged by hand size. There's a bit of a pattern where most of the best shooters have smaller hands, but it's not perfect. Aminu, in particular, has huge hands but he's shot well from the line. Likewise, Vasquez has tiny hands for an NBA player but he's been terrible. I also want to note that 21 samples are not large enough for a conclusive study, and in couple years there should be enough data. However, there is no overwhelming evidence to support the claim that hand size has a negative effect on free-throw shooting, and I think that's a fascinating preliminary result.
Player
|
Free-throw %
|
Hand length (in)
|
Height w/out shoes
|
Larry Sanders
|
57.1
|
9.75
|
6' 9.25"
|
Al Farouq Aminu
|
79.6
|
9.5
|
6' 7.25"
|
Ekpe Udoh
|
64.7
|
9.5
|
6' 8.75"
|
DeMarcus Cousins
|
68.7
|
9.25
|
6' 9.5"
|
Ed Davis
|
59.0
|
9.25
|
6' 9"
|
Marshon Brooks
|
77.4
|
9
|
6' 4.25"
|
Trevor Booker
|
65.3
|
9
|
6' 6.25"
|
Wesley Johnson
|
68.6
|
9
|
6' 6.25"
|
Derrick Favors
|
58.6
|
8.75
|
6' 8.75"
|
Evan Turner
|
50.6
|
8.75
|
6' 5.75"
|
Greg Monroe
|
66.4
|
8.75
|
6' 9.75"
|
Xavier Henry
|
63.5
|
8.75
|
6' 5.25"
|
Eric Bledsoe
|
74.4
|
8.5
|
6' 0.25"
|
Gordon Hayward
|
71.9
|
8.5
|
6' 6.75"
|
Manny Harris
|
76.3
|
8.5
|
6' 4"
|
Paul George
|
77.9
|
8.5
|
6' 7.75"
|
John Wall
|
76.7
|
8.25
|
6' 2.75"
|
Kyrie Irving
|
80.0
|
8.25
|
6' 1.75"
|
Jordan Crawford
|
82.9
|
8
|
6' 3"
|
Kemba Walker
|
74.5
|
8
|
5' 11.5"
|
Greivis Vasquez
|
61.5
|
7.75
|
6' 4.75"
|
Perhaps a better method is to pool each free throw attempt into a hand size measurement. That way a rookie who only plays in a blowout will have his ten attempts be used with the rest of the group of players with his hand length. Luckily, hands were measured in discrete units instead of something more exact like 8.425" and there are only nine categories. The linear regression based on one independent variable, hand size, output a p-value of 0.326 and an R-squared of 0.137. Basically, there was no correlation, and the graph below illustrates the point. There isn't any designation of point guard versus centers, but pundits claim someone like Rondo shoots so poorly because of his large hands. One problem, however, with a smaller than ideal data set is that players with high attempts like John Wall skew the results, and that's why you're seeing such different results for each size.
Hand length (in)
|
Free-throw %
|
9.75
|
58.2
|
9.5
|
73.6
|
9.25
|
65.8
|
9
|
71.6
|
8.75
|
59.9
|
8.5
|
74.3
|
8.25
|
75.6
|
8
|
81.3
|
7.75
|
62.9
|
Another approach I tried was grouping the results by position and applying linear regression to each position. Unfortunately, there aren't enough players yet with hand measurements to subdivide the data even when lowering the free throw requirement to 20 attempts. The results, however, were so one-sided that I doubt more data would reveal a statistical significance between hand size within positions. Point guards, for example, a group known for their sweet shooting, had a tiny R-squared value of 0.0529 and a p-value for the hand length coefficient of 0.523. R-squared, to reiterate, is saying hand length only explains 5% of the variation in free-throw percentage among point guards, and that's a abhorrent result. There were only 10 point guards to qualify, but the same was true of each position. Again, there was no evidence to conclude that hand size was a significant variable in determining free-throw percentage. To illustrate the lack of association between size and free throws, Iman Shumpert was more of a combo guard so he wasn't included, but he has larger hands than any point, yet he's at 22/24 for the year.
Conclusion
The NBA athletes we see on TV are undeniably talented and earning more money than most of us will ever accrue in our lives. When we see a player making $10 million a year miss a free throw -- a shot even little kids can make -- we're angry and baffled. Surely he's been practicing for years, so why is he shooting 60% for the season? In asking that question some seek answers about how some players aren't able to make them during the flow of the game or that these athletes' hands are too big to accurately shoot. We can find examples like Duncan, who clearly has worked on his shot but often shoots under 70%, to fit our theory and confirm our own perceptions, but there are also counter-examples. Pau Gasol has enormous hands and palms the ball with ease, but he has an accurate jump shot and cleared 80% for the season multiple times. John Stockton is a point guard with large mitts, and he's at 82.6% for his career with a good three-point shot. Michael Jordan, known for using his big hands to dunk with flair, is even better at 83.5%.
The inclusion of hand size measurements for nearly every drafted player recently will lead to a better data set, and at the end of the year I'll redo the regression to see if anything changes. There are a couple other hypotheses that need to be tested. Maybe there's a hand size limit beyond which percentage plunges, but we don't have enough information for that yet. The players in the data are all very young and the results could change once they improve or, even their hand size is so problematic, flatline at a poor number. Within a couple of years we could get a definitive answer as more guys get measured and the ones that already have rack up attempts. However, based on the data right now, there is no evidence to support the hypothesis that hand size negatively affects free-throw percentage.
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