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.

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.

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.

Free Throws and Hand Size: Part 2

Introduction

Last year, I looked data between hand size and free-throw percentage to settle the age-old debate about whether or not hand size has a negative association with free throw shooting. Truthfully, I just wanted to stake my claim to this study area because only recently (starting in 2011) has hand size been objectively measured for an acceptable size of the NBA population thanks to the predraft camp, and I did this barely after the 2012 season even started. Now with another season done, there's a whole crop of new rookies, as well as more free throw attempts from guys drafted in the two seasons before that.

When Rondo steps up to the foul line, the commentator will often mention how his large hands make it difficult to shoot free throws and jump shots in general. When Shaq was playing, he was the go-to guy for this excuse. If your hands are too big, the reasoning goes, you are unable to properly grip the ball, and end up awkwardly shooting the projectile as if it were a tennis ball, not a basketball. And so Rondo, born with rare physical gifts, was doomed from the start, and thus was a bad free throw shooter from birth. It's a good story, but it's anecdotal evidence and there are no numbers to back up the claim.

Study methodology

I think there are two components to this myth. One is that large hands make it more difficult to shoot free throws (or jump shots.) The other is that large hands make is impossible to be a good free throw shooter, or at least places a ceiling on your ability to shoot. Both are important to consider. For free throws being more difficult, trend or regression analysis is needed: is hand size a significant predictor of free-throw percentage? For placing a ceiling on free throw ability, even simple graphing can get the job done because you can identify how many players shoot well with large hands.

Although this myth involves overall shooting, only free throws are considered because they're objective, isolated events, which is rare in basketball. The excellent resource DraftExpress was used to compile hand length, as well as other information. Basketball-reference was used to collect free throw totals because of its snazzy season finder, which subsets by many factors like player year. Positions were tabulated for each player -- 1 for point guards, 5 for centers, 1.5 for players who played both point and shooting guard, etc. -- where they were determined by a combination of common sense, position listed on b-ref, and looking through lineup combinations for more obscure players on 82games. For height, I used height without shoes because shoes add uneven amount of inches to players, while for age I used day of birth age, meaning it's in decimal form based upon days, not simple calender year.

I will note here that even measured from the hallowed predraft camp are not flawless, immutable -- height changes based on time of day, where a person is the morning is taller than at night; measurement procedures are not perfect and I've seen guys tested again the following year and they somehow have larger hands even at age 21; and even things like age are not as absolute as one would think, since Shabazz recently aged an entire year when someone found his real birth certificate ("Dad, how many lies have I been living?") As an additional note, hand width is sporadically measured, and I wish it were measured more often but I can only do this analysis with hand length.

Results

There were 137 players with hand measurements who took a free throw for a total of 20,167 free throws -- and this includes 75 players with at least 50 free throws taken. At this point, the best and simplest thing to do is graph hand length and free-throw percentage. (While doing a research study, this is what's known as data exploration and should always be done to look for patterns.) The results are below for all players whose hand lengths were measured at the predraft camp from 2011 to the present with a minimum of 50 free throw attempts in regular season games. The immediate reaction here is that it's a huge mess of data points with only a weak trend. I coded position with a color, as best I could, so you can parse the data within positions. There are a few young guys near 80%, above average, but they span nearly the whole range of hand sizes, and there are players with smaller than average hands who are poor shooters. Also, you're probably wondering about the outliers: Greg Smith (Houston Rockets) has giant hands, though for a power forward he's not terrible, and, yeah, Drummond is the guy south of 40%, but his hands are pretty average for a center.

Since there's an obvious correlation of height and hand size, and a correlation of height and position, one can't simply look at hand size and free-throw percentage. Breaking down the results by position, however, the results are just as noisy. I've produced the same graphs but focusing on the five positions. It's not a big sample size, but as can be seen in the graph below for point guards there's no trend. Lillard, oddly enough, has the biggest hands out of the group, but he's the second best free-throw shooting, as well as good marksman from three-point range -- and his hands are 9.75 inches wide, so it's not the case of "skinny" hands muddying the results.

I included the same graph for the other four positions below for perusing. It's the same story as the point guard graph: there is no trend. I believe looking at free throw shooting within positions is the best way to approach this topic. In grouping by position, players have roles that are more similar so they're expected to have a certain shooting skill level. Since there are more people who are point guard size, there's more potential to find great shooters. However, it's not a perfect system, as there's probably a self-selection bias for position where a player who can't shoot gravitates toward the frontcourt. Players with big hands for their height who thus can't shoot, the reasoning could go, will move a position to compensate for their skill deficiency, and sorting by position will mask these effects. But there's more we can do once we look into the numbers.




A series of regression tests have been run on the same data graphed above (players with a minimum of 50 FTA's.) Hand size by itself is a significant predictor of free-throw percentage, but as we know that's because it's correlated with position quite well. In fact, position by itself is a much better predictor. The R^2 value shows how much of the variation is explained by the given variables, where the adjusted part gives a penalty for having more variables in the model. So in comparing the first two models, position explains 2.3 times more of the variability than hand size does. If you contend that position is the dominating factor but within positions hand size does matter, despite what you saw from the graphs, there is no evidence for this given the results in the third model. Hand size has a p-value of 0.473, meaning there's a 47.3% chance hand size has no effect on free-throw percentage when position is another variable. Another way to prove this is with an F-test between models 2 and 3. Adding the variable of hand size did not significantly improve the results (again) according to the F-test, which is a useful tool when dealing with multiple models. Height was thrown into the mix to see if it could help the variables get along, but that was also not a success. Position is the overwhelming factor, not hand size.

Model	adj. R^2	Intercept	Variables	Coefficient	St. error	P-value
1	0.07987	119	Hand size (in)	-5.42	1.99	8.05E-3
2	0.18299	80.7	Position	-3.27	0.779	7.65E-5
3	0.17759	94.0	Hand size (in)	-1.62	2.24	0.473
			Position	-2.90	0.932	2.68E-3
4	0.14540	172	Hand size (in)	-2.35	2.26	0.301
			Height (in)	-1.03	0.401	0.0123
5	0.16668	80.1	Hand size (in)	-1.67	2.27	0.465
			Position	-3.32	1.97	0.0964
			Height (in)	0.200	0.830	0.811

Perhaps there's a problem with the data because I'm looking at free-throw percentage of guys with only 50 free throws, as their percentages might be a bit wonky and the noise could muddle the results. There are a couple approaches to this problem. One is to construct a different regression model where the players are weighted by how many free throw attempts they have, and the other is to pool all free throws into hand size categories. The latter approach will be used first. There are nine hand sizes with at least 300 attempts, where most are well over 2000, covering the range of 7.75 inches to 9.75. In determining whether or not position is more important, a weighted position average was calculated for each category (for example, if there are 100 attempts from a point guard and 50 from a shooting guard, then the average position is 1.33.) To save ink space and another table, the regression results are virtually the same -- hand size by itself is a fine estimator, but position is better and the results do not significantly improve when position and hand size are done together.

Moving onto the weighted regression models, which now include 137 players because with the weighting I don't have to worry about players with low totals skewing the results, the output is interesting. Again, position is a better predictor than hand size, but the p-value in model 8 suggests even when you adjust for position hand size is not a ludicrous variable to consider. A p-value of 0.165 is not significant by any conventional means, but with the limited data set it's intriguing. If you're wondering whether or not model 8 is superior for the inclusion of hand size, there are a few tests for this situation. The most popular is the F-test where you're comparing the residual (difference between predicted and actual value) squared sums between a unrestricted model (the model with more variables, and in this case model 8) and the restricted model (number 7), with an adjustment for the number of observations and how many more variables the unrestricted model has. The F-test, however, states that the reduction in squared errors (i.e. model 8 has a fits the data better) is not statistically significant, and in fact the p-value is 0.165. Model 7 without hand size is preferred. As a last note, I repeated the analysis with height and hand size, since position is slightly subjective and there might be a self-selecting bias as well. The results were virtually the same where the p-value was 0.120.

Model	adj. R^2	Intercept	Variables	Coefficient	St. error	P-value
6	0.1318	131	Hand size (in)	-6.77	1.45	7.71E-6
7	0.2340	81.3	Position	-3.28	0.502	1.27E-9
8	0.2394	100	Hand size (in)	-2.35	1.68	0.165
			Position	-2.77	0.618	1.58E-5

Conclusions

The next time you see Rondo brick a free throw off the front of the rim, don't believe the cause is his unusually large hands. Based on a large set of players, including ones with similar sized hands or larger, that is no excuse to be such a poor foul shooter. A series of regression tests have found no evidence of hand size influencing free throw percentage.

You don't even need to mention the regression models for why Rondo doesn't have an excuse. According to Lee Jenkins of Sports Illustrated, Rondo's hands were measured by the Celtics at 9.5 inches long and 10 inches long (that article by SI is completely bizarre: there are accounts of Rondo's connect four prowess and the doctor at his birth remarking on his humongous hands.) Kawhi Leonard, the Spurs' small forward, has hands 9.75 inches long and the second widest hands in the draft database at 11.25 inches (hand widths are incomplete, but it's 167 players.) However, he shoots 80.4% from the line and he's a decent free-point shooter -- and I didn't include the playoffs where he's been slightly better. Rondo's at a pathetic 62% for his career while players with his hand length are at 68% and they're typically power forwards. For another example, Andrew Nicholson has 10 inch long hands but shoots at 80%. He's a rookie and late-comer to the game, playing in the frontcourt where he's not even expected to shoot well; Rondo has no excuse. There are also plenty of historical examples -- Jordan's known for his baseball mitt hands and he shot 83.5%; Connie Hawkins palmed the ball like a tennis ball and still shot 78%; and plenty of giants with large hands like Sabonis, Yao, and Ilgauskas were plus 80% from the line.

But perhaps there's either a small effect hidden within the results here or something overlooked. As I discussed earlier, there are two hypotheses -- hand size negatively impacts free-throw shooting and hand size puts a limit on your ability as a free-throw shooter. The former has been challenged quite effectively with a bevy of statistics. The latter may still have some truth to it, but it's not a large effect since players with large hands are still capable of shooting above the league average. But it might break down at the extremes. There's only one player with 11.25 inch long hands, Greg Smith of the Rockets, and he shoots near 60% (I can't find an official source, but it appears that's around the range of Shaq's hands.)

Obviously, you can't draw conclusions from one player; it'll be interesting when the next draft rolls around if anyone else shows up with boulder size hands. It's also hard to tell from the data if there's a free-throw percentage ceiling across the span of hand sizes. One might posit, for example, that you can't be an elite shooter with hands greater than 9.5 inches, but there are only four players (Irving, Lillard, Klay Thompson, and Isaiah Thomas) you could reasonably argue as elite shooters for their age -- again, not enough for a conclusion.

However, even though the myth is not totally extinguished, it's severely limited -- hand size is not such an overwhelming factor that it applies independently to every player with significant results, and even if your hand size is above average you can still be an above average shooter. Sorry, Rondo, but you'll have to blame something else.

Future work

As with any investigation, the study is ongoing. The hand measurements being for the NBA predraft camp doesn't mean they have to apply to the NBA; I can tabulate the college totals of these players, increasing the sample size because many players measured did not play a single minute and most who did took few free throws. I can also use hand widths, which I have ignored in this article because it limits the number of players. Perhaps hand widths are a more important factor because it has more to do with grip, and college stats can provide enough information for this. But we can't state definitively unless we actually look at the numbers.

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.

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.