Friday, April 26, 2013

Free Throws and Hand Size: Part 2


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.


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.

adj. R^2
St. error
Hand size (in)
Hand size (in)
Hand size (in)
Height (in)
Hand size (in)
Height (in)

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.

adj. R^2
St. error
Hand size (in)
Hand size (in)


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.

1 comment:

  1. Good work. Saved me the time lol as I was going to pursue a nearly identical analysis. However, I wanted to include some female players but could only locate reliable measurements for Brittney Griner. Likewise, I intended to include wingspan (as a proxy for arm length) and an approximated plane of vision (seeing the hoop as an increasingly smaller oval as height increases) as well as college FT% for the NBA players to account for possible practice effects.