Stephen Curry is a great shooter, but he ain't *that* great

Don't worry, I've got mad love for ya, Steph.

I know you're already foaming at the mouth after reading the title, but that's the only way I thought I could you lure you in for what follows. Here's a more appropriate title, but one you might have never thought twice about clicking on:

A method for predicting FG% in basketball based on small number of shots

So here's a list of the top 20 players sorted by FG% on long twos (I'm defining a "long two" as 18-23 ft). The data was acquired using the PlayIndex+ tool over at basketball-reference. Here is the exact query, if you want to see the data for yourself.

Top 20 Players by Maximum-Likelihood Estimate (MLE)

MLE is simply a statistics shorthand for "taking the straight mean" (actually, it means a lot more than that, but for our purposes, I think that's good enough).

The problem we face here is that the sample sizes vary so much between players and some are really small. Is Yi Jianlian really a 50% shooter from 18-23 ft? (If he is, somebody should probably sign him already.) What about Kris Humphries? Maybe not. Conversely, Dirk shot 48.6% on 175 FGA from this range, which is a larger sample size and about where we might expect him to be. Overall, the pool of 248 players with >25 FGA from 18-23 ft had a mean FG% of roughly 38%.

What can help us "shrink" these estimates closer to the mean is taking into account the variation between players in both FG% and FGA (i.e. sample size). One way to do this is to use a multi-level model (the other is a fully Bayesian approach, which Gelman says is roughly equivalent when there are a large number of "groups" such as this). If you're interested in this type of model, I highly recommend Gelman's "ARM" book.

In R, creating the model basically takes one step:

twos.mlm=lmer(cbind(FGM,FGA-FGM)~(1|PLAYER),family=binomial(),data=long_twos)

From that, I get a list of coefficients (called random effects) which can then be converted to our new (hopefully) more predictive FG%'s.

Before I show the new list of players and their estimates, take a look at how the spread of the histogram of FG%'s shrinks when going from the MLE to the multi-level model:

Now here's the top 50 according to their multi-level estimates:

Top 50 18-23 ft FG%

The column MULTI is the multi-level estimate.

Now it's starting to make more sense. Here's the bottom 50:

Bottom 50

Conclusions

Now, do we have any evidence that Stephen Curry is closer to a 46.9% shooter from 18-23 ft rather than truly being a 68.3% shooter according to the 63 shots he took last year? Sure we do! Just go back to 2010-11 when he shot 49.1% on 214 FGA (still great!). Or his rookie season when he shot 47% on 232 FGA (also great!). Now that 46.9% makes a lot more sense, right? (By the way, the title of this post should make some more sense right about now, too.) Stephen Curry is a great shooter, he just ain't *that* great.

Just as Stephen Curry probably isn't a near-70% shooter on long 2's, Glen Davis is probably better than a 25% player on those shots. Indeed, in 2011, Davis shot 35.8% on 226 FGA. And he was a 38.9% in 2010, but on only 36 FGA. You know, it's important to point out that just because a player takes a small number of attempts doesn't necessarily mean he'll be at the top or bottom of lists like this. Sometimes the player will "randomly" fall in the middle, too.

So, hopefully, this made some sense to you. Next time you see an analyst talking about how a player lead the league with an astronomically high FG% on 12 shots of a certain type (say in the 4th quarter of games on the road on Sundays), think about this post. Heck, maybe e-mail the guy a link to it.