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%`

Categories: Reggie Evans, Math, shooting

Love the Rounders reference!

233 days ago

Love your shit Patrick - very interesting.

232 days ago

It seems to me that one of the main premises here is flawed. Efficiency is not related ultimately to points per shot attempt. It is related to points per possession used. Players A and B may have both taken 15 shots, but they didn't both use 15 possessions. Probably player A used 19(or 20) and Player B used 24. That is why player A really was more efficient than B. It doesn't matter from the point of winning a ball game how many possessions were used on shots and how many were used shooting free throws: you seem to be assuming that only possessions used on shots count, for some reason.

Reggie is NOT a very efficient offensive player. In fact, his efficiency is even lower than his true shooting indicates, because true shooting doesn't include turnovers, and Reggie has a very high turnover rate.

232 days ago

Have to agree with Eugene Earnshaw. While you talk a bit about the value of possessions at the bottom of the post, you erroneously ignore it when discussing hypothetical players A, B and C.

PPS is not a very good metric to use, since shots possessions, and possessions are the important thing to look at. Adjust your metric to points per possession used, and then we'd have a metric worth discussing.

232 days ago

Agree with Eugene and Al, PPP is more important than PPS.

Now I think I'm going to watch Rounders.

232 days ago

Patrick, excellent point. The reason TS% and PPP matter is that a fouled shot attempt is not counted as a FGA, so basically counting the FTAs accurately is taking the place of what would normally be the shot. Getting to line doesn't "save" your team from taking a shot. It just makes it much much easier (2 attempts with no opponents)

232 days ago

I personally think PPS is flawed. I agree that shots are weird because of how FGAs are counted when a player is fouled. But I don't think this is consistent; for instance a player who scores and gets an AND-1 is credited with an FGA.

In general I think that whether or not shots are "recorded" is something that evens out over time. You are going to have a very, very hard time convincing me that the average power forward uses fewer possessions to get 6 free throw attempts than Reggie does. If shots aren't recorded when a player is fouled, than clearly that happens to the average PF too, not just to Reggie.

PPS has plenty of flaws but some of you are comparing apples to oranges here.

232 days ago

And Eugene/Al,

I'm also wondering why you don't think the entire second half of the article covers this very point. The math clearly states that anyone time a player who shoots > 46% is fouled, the offense comes out ahead.

Clearly if one player is getting fouled a ton, at some point he becomes as efficient as a player who is fouled less by virtue of volume.

If you look at free throws in a vacuum, think of post-up players who draw lots of fouls during the course of a game vs. point guards that get fouled at the end of a tight game to stop the clock. Anyone who claims that the latter are more efficient is definitely not seeing the bigger picture.

232 days ago

I don't see the difference between getting to the line and hitting two FTAs at 46%, and taking a 2pt shot at a 46% clip, which is far below the league average for TS% (usually around .54) and the efficiency requirement that WP asks of a player to be helping his team when he takes a 2pt shot. Both use one possession, which has an opportunity cost of another player taking that possession instead. As for And-1s, TS% using .44 in its equation instead of .50 is meant to account for this as best as possible

232 days ago

Julien's question suggests that a piece of the analysis seems to be missing. (Patrick, correct me if I'm wrong!) The reasons teams (and players) need to make field goals at a higher clip to hit the break-even point is that **they turn the ball over.** Which brings up an important distinction: Just because it's inefficient for an opposing team to hack-a-Shaq a 47% free-throw shooter, does NOT mean it's efficient to actually BE a 47% free-throw shooter. Because the hack-a-Shaq technique takes the turnovers out of the equation! Whereas in the normal course of play, a 47% FT shooter will make a normal number of turnovers, and also miss too many free throws, and so will wind up on the wrong side of the efficiency break-even point.

232 days ago

Patrick: Nobody is disagreeing with you that it is good to draw fouls as long as you shoot better than 46% (or something like it). And yes, someone who gets fouled a ton at some point becomes more efficient than someone who doesn't draw fouls as much but shoots better. And yes, a strength of Evans' game is that he draws fouls.

But TS% measures all of that. When you observe that player B has lower TS%, but nevertheless argue he is more efficient, it seems to me as if you are simply making an erroneous claim. Certainly I do not understand what your logic is.

When you factor in the rate at which Reggie draws free throws, which is good, the percent he shoots them at, which is bad, and his fg%, which is bad, he just is not a very efficient offensive player. So when you say he's 'fantastically efficient', it is wrong. .506 TS% is quite inefficient, especially for a power forward. And the fact that he shoots many free throws compared to the number of shots he takes does not change that; it just means he is less inefficient than he would be if he were worse at drawing fouls.

231 days ago

This is just flat out incorrect...Player A is more efficient than player B and Reggie Evans is not an efficient scorer at all.

209 days ago