Math and Shooting Efficiency

The other day I got a question from a reader. The answer turned out to be interesting, and I think it illustrates why offensive efficiency is not as simple as many make it out to be.

Warning: This post uses math to demonstrate that Reggie is really good at basketball.

First, the question:

I had a quick question for you regarding PPS that I was hoping you might have a moment to expound upon. I was recently looking up statistics for Reggie Evans and noticed that his PPS are well above average in spite of a below average TS% and Efg%. Perhaps I'm misunderstanding how these numbers relate to/impact one another, but how is this possible? Does it just reward him for not shooting too much (given that his %'s are below avg. and his shots would be better allotted to a teammate?). 

Hey, really good question. On the surface of it, it sure looks like Reggie is a very inefficient scorer. With a career TS% of 50.1% (53% or so is average for PFs), it seems like letting Reggie shoot should be one of the team's last resorts. In fact, however, Reggie is a fantastically efficient scorer, and the key to it is that Reggie is ridonkulously good at drawing fouls. Let's look at some math to figure out why (definitions are from the basketball-reference glossary).

First, there's the matter of how we track free throw attempts. This site tracks them (like most things) on a per-48 basis. But whether you do this on a per-game or per-36/48/minute basis doesn't really matter here, because the factor this leaves out is shot attempts. One might at first glance look at Reggie's career average of 6.4 FTA/48 and think "Meh." That is, until you look at the fact that his career average field goal attempts per 48 is a paltry 7.4. In other words, he has consistently gotten to the line as much or more than most power forwards with half the shot attempts. That is spectacularly good, and makes up for the fact that he isn't a very good free throw shooter. By sheer volume of free throws (relative to his shots), he scores a lot of points per shot attempt.

Fair enough, you say. But isn't eFG% and TS% supposed to capture the added efficiency of free throws?

Effective Field Goal Percentage: (FG + 0.5 * 3P) / FGA

True Shooting Percentage: PTS / (2 * (FGA + 0.44 * FTA))

What should immediately become obvious is that eFG% accounts for the extra weight of three-point field goals, but does not account for free throws in any way, because it ignores both points and free throw attempts. In other words, a player who shoots 50/40/80 on FGs/3FGs/FTs has the exact same eFG% as a player who shoots 50/40/20, and for any given player, whether he shoots 5, 10, 20, or hundreds of free throws has no effect on his eFG%.

What about True Shooting? True Shot Attempts (that's the (FGA + 0.44 * FTA) part of the equation) clearly incorporates free throws. And looking at this formula, we can see that clearly, as FTA increases, TS% goes down (because FTA is in the denominator). And as PTS goes up, TS% goes up (because PTS is in the numerator). Let's look at how this affects two imaginary players:

Player A: 8/15 total FG (1/3 on 3FG), 7/10 FT, 26 PTS

TS% = 26 / (2 * (15 + 0.44 * 10) = 26 / (2*19.4) = 26/38.8 = 67%

Player B: 8/15 total FG (1/3 on 3FG), 12/20 FT, 31 PTS

TS% = 31 / (2 * (15 + 0.44 * 20) = 31 / (2*23.8) = 31/47.6 = 65.1%

Notice that player B has a lower TS% than Player A, despite the fact that he scored 31 points on 15 shots, while A scored 26 points on 15 shots (but by the way, holy hell, somebody get these guys under contract). Which player's production would you rather have? (Note that you don't get to choose C: "I want B to hit his free throws as well as A"). Which player is more "efficient"? Clearly, B was more efficient than A, but if you only looked at true shooting, you'd overlook that fact.  This is because the formula for True shooting penalizes a player for every free throw attempt, just as it rewards a player for every free throw made (i.e. point). This means that there is a "break-even" point of free-throw percentage where the size of FTA won't matter. As it happens, that break-even point is TS% * .88. In other words, if your current true shooting percentage is 59%, hitting 52.3% of your future free throws will "break-even", meaning that if you hit 50% your TS% will go down. Let's further illustrate with hypothical player C.

Player C: 11/19 FG (no 3s), 18/40 FT, 40 PTS

TS% = 40 / (2 * (19 + 0.44 * 40) = 40 / (2*36.6) = 40/73.2 = 54.6%

According to TS%, C looks like a much less efficient player than B. In fact, if he were playing center you'd say he was only about average (the average center had a true shooting of 53.7% in 2011-12). But if you could ever find a player who could do this night in and night out...well, you'd have basically found Wilt Chamberlain*.
And this, gentle readers, is why I put that simple metric of points-per-shot in the data. It's not a fancy formula. It isn't an official metric. It's literally just points divided by field goal attempts. And this metric quite clearly shows that although Reggie doesn't shoot very often, no coach or teammate should ever be mad at him when he makes the decision to do so!
Of course, things aren't quite that simple. I'll pick on Wilt (or Shaq...or Dwight) to illustrate this next point, which is that many free throws don't come as a result of shot attempts. Many of those free throws came when the opposing team was trying to foul, either to stop the clock far away from the basket, or to prevent Wilt/Dwight/Shaq/Reggie from getting an easy dunk. In other words, many of the free throws came instead of a shot for their respective team's possession. How bad does a player need to shoot for this to be an effective tactic for the defense?
John Hollinger already touched on this back when Mark Jackson embarassed himself by having the Warriors foul Dwight Howard about 8 million times:
Take Thursday night, for instance. Dwight Howard is a career 59.5 percent foul shooter and has done slightly better than that each of the past three seasons. But let's take 59.5 percent as his chances of converting any given free throw. Sending him to the line for two shots produces an expected return of 1.19 points from the foul shots, a scoring rate better than that of any offensive team in the history of basketball. Just sending him to the line time after time is one of the worst percentage moves a team could possibly make.

He goes on to point out that this is not just true for Dwight; that in fact, a player needs to be exceptionally bad for this to be a good strategy. To get a feel for why, we can look at some more math. According to David Berri, author of Wages of Wins, who derived some numbers from Dean Oliver's work on offensive efficiency:

The value of a point is 0.03260287. The value of a free throw attempt is -0.0151305; or, more precisely, 0.45032 * the value of a possession employed (or -0.0336).

Given these values, if a player took 100 free throw attempts, he would have to make 46.4085% to break-even. This is found by multiplying <code>100 * -0.01513505</code>.  Then divide this by the value of a point, 0.03260287.

So any player who make more than 46.41% of his free throws should to go to the line as much as possible.

The reasoning for this is fairly simple: most NBA teams hover at slightly over one point per possession. So, at first blush, you'd think that a player would need to hit more than 50% of his free throws to make a possession 'break-even'. This however, ignores offensive rebounds. Free Throws don't always end a possession, just as shooting doesn't always end a possession. This is why shooting at about 45% from 2-point range is "break-even" and shooting at a 50% clip from 2-point range will net you much more than one point per possession.

The real reason I decided to write this post is that if you asked 30 NBA coaches and a few hundred (thousand?) NBA journalist, I'd gather that well over 90% of them would say that Reggie Evan's free throw shooting is one of his liabilities and that he's not a very efficient scorer. And of course, Reggie would be a much better player if he could hit his free throws with more regularity (the same could be said of Shaq, or Wilt). But Oh, The Irony! Even at a woeful 52%, free throws are actually a huge strength of Reggie's game, and one of the reasons he is a very efficient scorer -- not because he's good at shooting them, but because he gets so damn many of them. And yes, of course, it makes sense to keep him off the floor in the last couple of minutes of a very close game, when you know the opponent is fouling anyway and you can force them to foul much better shooters.

But during the rest of the game, if the other team wants to play hack-a-reggie, you shouldn't complain and yank Reggie. You should say, "Hey, thanks, chump!" and let him clank in 52% of his free throws for a very efficient offense. In fact, one might even get a little contrarian and tell Reggie not to get too much better at shooting free throws. After all, if he creeps up to 60%, opponents might start getting bright ideas and decide that it isn't such a brilliant plan to hack-a-reggie. And when your opponent is doing something stupid like eating the cream of his Oreos whenever they have the nuts, you probably don't want to tell them about the secret.

*just for shits and giggles, here's Wilt's True shooting on the night where he scored 100 points (where he uncharacteristically went 28 for 32 from the line!):

TS% = 100 / (2 * (63 + 0.44 * 32)) = 100 / 2*77.08 = 64.8%