Why True Shooting Percentage and Effective Field Goal Percentage Are Wrong (and Why Our Metric Is Right)

Posted: March 5, 2013 by nunuspeaks in NBA 2013

This article attempts to advance a new metric, Weighted Field Goal Percentage, which accounts for every shot (free throw, 2-point shot, and 3-point shot) taken by a player in a given game.  The metric weighs these shots (free throw, least; 3-pointer most) and generates a shooting percentage based on a player’s overall shooting performance.  For reasons that are explained below, we believe this metric is more useful than existing metrics (eFG% and TS%).

eFG% IMPLIES THE IMPOSSIBLE IS POSSIBLE:

Modern-day Pythagoras

Derrick Rose: Modern-day Pythagoras

When Derrick Rose said that he would not return to the Bulls’ starting lineup until he was “110%,” many were quick to note that, interpreted literally, this mean that Derrick Rose could never return to the Bulls.   You cannot be 110%, right?  This kind of common sense observation led me to recognize a flaw—the very same flaw (exceeding 100%)—in a metric called “effective field goal percentage.”  The rubric was designed to address the grievance that gifted three-point shooters are underappreciated if field goal percentage measures two-pointers and three-pointers equally.  Put another way, Tyson Chandler’s impressive field goal percentage would certainly be another level of awesome if he were 60% from beyond the arc.

Brace yourself, here comes math.

The basic calculation for eFG% is this:  (FG + 0.5 * 3P) / FGA.  Broken down into actual human language, what that formula means is that every made 3-point shot is divided by 2 and then added to the total number of made field goals.  The idea is to give efficient 3-point shooters recognition for their success in making long-range shots.  But if you look closely, you’ll notice that we didn’t do anything to the bottom-half of the equation (or denominator, for those not completely allergic to math).  Consider also that “FG” in the eFG% formula already includes made 3-point shots, and we have a problem.

JJ Redick makes a lot of three's.

JJ Redick makes a lot of three’s.

Let’s take an actual, real-world example.  On January 22, 2013, J.J. Redick shot the lights out against the Pistons.  Redick was nearly perfect, shooting 9-10 from the floor, including an impressive 5-6 from the 3-point line.   Redick was also 3-4 from the Free Throw line, but since eFG% doesn’t factor in Free Throws–at all–we leave that number aside.  So, when we plug in Redick’s numbers to the eFG% formula, Redick’s eFG% is (9 + .5 * 5) / 10.  You multiply the result by 100, and you get . . . . 115%.   Unless Derrick Rose is the accidental architect of a revolution in mathematics, something is off here.  What is this number (eFG%) possibly supposed to tell us?  Redick was undeniably great that night, but he absolutely was not perfect. He missed one shot.  And in any case, he certainly wasn’t better than perfect.  Right?

TRUE SHOOTING PERCENTAGE: SIMILARLY BUSTED

Another obvious drawback with the eFG% calculation is that it simply does not factor in free throws.   True shooting percentage, on the other hand, attempts to measure free throws.  The formula for TS% is:

TS% = \frac{PTS}{(2 * (FGA + 0.44 * FTA)) }

But the metric is . . . odd, to say the least.  First, TS% does not distinguish, as eFG% at least attempts to do, between a made 2-point shot and a made 3-point shot.  Rather, a player’s total points are lumped together.  Second, free throw attempts are weighted at .44.  This is the most puzzling choice in the formula.  The supposed rationale for this number is that free throws are not a full possession. But . . . are we measuring possessions?  If so, then presumably we should distinguish between 3-point shots and 2-point shots for TS%.  For example, it takes 6 possessions to get 12 points if you make only 2-point shots, but it takes only 4 possessions if you make all 3-pointers. And, anyway, is a FTA 44% of a field goal attempt, or is it more accurately understood as 33% of a 3-point attempt?

Did I mention that the “Derrick Rose Effect” is still in play with TS%?  Don’t worry if you didn’t catch it, but JJ Redick’s great night against against Detroit comes out as once again better than perfect under the TS% rubric.  Redick had 26 pts.  The denominator, working inside-out, is ((.44 x 4) + 10 ) x 2, which equals 23.52.  So . . . JJ Redick’s TS% is 26 / 23.52,  or 111%.  The use of the word “true” in “true” shooting percentage officially strains the imagination.

WEIGHTED FIELD GOAL PERCENTAGE:  A VICTORY FOR COMMON SENSE

Here is what I propose: a rubric that takes into account every shot a player makes but weighs those shots according to their value.  We start with the notion that a made 3-pointer gives you 3 out of a possible 3 shots.  Since a 2-point shot gives you 2 out of 3, we weight it accordingly.  Ditto for free throws, only they yield 1 out of 3.  What does this mean?  Think of it this way:  if you added every shot a guy took on a given night and made no distinction between free throws, 2-pointers, and 3-pointers, you’d have an accurate shooting percentage.  But . . . what would this number really tell you?  A better number would provide some distinctions between 1’s, 2’s, and 3’s.  Our new metric does this, by weighing 1’s, 2’s, and 3’s.  Here’s the formula:

(FT x (3/9)) + ((FG – 3ptFG) x (6/9)) + (3pt FG)

——————————————————                              x 100 =   Weighted Field Goal %

(FTA x (3/9)) + ((FGA – 3pt FGA) x (6/9) + (3pt FGA)

Congratulations if you’re still with us.  It’s been brutal, but we’re almost done.  In human terms, all we’re doing there is adding up the number of shots made and dividing that number the number of shots taken, but we’re assigning those shots a weight (1/3, 2/3, or 3/3) based on whether the shot is a free throw, a 2-point field goal, or a 3 point field goal.  And notice we are weighing shot attempts as well.  This is important, as we will see, because it means our percentage can never exceed 100%.  A small victory for common sense.  So what does this number tell us?  What has this all been for?  Let’s plug in J.J. Redick’s numbers to see what we get.   On top of shooting 90% from the field and going 5-6 from the 3-point line (83%), Redick also shot 3-4 from the stripe (75%).  But these 3 shots from the line are not factored in anywhere in conventional Field Goal Percentage.   Conventional Field Goal percentage does not give extra weight to Redick’s 5 made 3-pointers, either.  So what we’re after is a number that factors in both 3-point shooting and free-throw shooting.

When we run Redick’s numbers through the WFG% equation, Redick comes out shooting 87% overall.  What does this number mean?  A couple of things.  First of all, when we say Redick shot 90% from the field on 9-10 shooting, we are speaking in terms of conventional field goal percentage.  We aren’t, in other words, factoring in free throws. Nor are we weighing three’s more heavily than other shots.  And Redick missed a free throw.  Our WFG% counts that miss.   And our WFG% values 3-point shots (both in makes and attempts) more than other shots.  Redick was a perfect 4-4 on 2-point shots.  But we don’t weigh those shots as heavily as we weigh his 5 made threes.  Redick’s one miss from the field was a 3-point shot, so that miss necessarily counts against him more than a two-point miss would have. There’s probably a lesson in here about how you can’t have it both ways (extra credit for made three’s but no extra penalty for missed three’s), but we leave that for another day.

Our metric is a work in progress, no doubt, but we think it has certain obvious advantages to the existing metrics (eFG% and TS%) that claim to measure basically the same things.   For those of you who would like to play with the metric, here’s link to an Excel spreadsheet that will do the math for you.

(part 2 of this article, which addresses feedback to the original article, can be found here)

—C. Smith
@nunuspeaks

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Comments
  1. Chris says:

    It makes no sense whatsoever to emphasize a missed 3 over a missed 2. Basketball is a possession game. When I make a 3, I got 3 points on 1 possession. When I make a 2, I got 2 points on 1 possession. If I miss a 3, I got 0 points. When I miss a 2, I got 0 points. Efg% and TS% measure efficiency on a possession basis, excluding turnovers, and is results orientated. What you are doing is correcting a design mistake, not a substantial one. I will agree nonetheless, that the TS% formula should be redone, as the .44 basis is just stupid.

  2. Jeremy says:

    What you/we really want to know is how many points a player scores relative to the number of possessions that they use up trying to score. It isn’t about what percentage of their shots go through the hoops, it’s about how efficiently they score.

    An example to show the most immediate problem with your formula:

    Player A is 1-3 FG including 0-1 3FG, for a WFG% of 28.57%.
    Player B is 1-3 FG including 0-2 3FG, for a WFG% of 25.00%.

    Why should Player B be penalized for the fact that he missed an extra 3, rather than an extra 2? Why is missing a 3 more costly to the team than missing a 2? If you’re factoring in offensive rebounding percentages and foul rates and quality of the 2 and everything else then maybe, but you aren’t factoring those things in.

    A second example with the same issue at its core:

    Player C is 5-10 FG, 2-4 3FG, 6-7FT, for a WFG% of 58.06%.
    Player D is 5-10 FG, 3-6 3FG, 6-7FT, for a WFG% of 57.58%.
    Player E is 5-10 FG, 4-8 3FG, 6-7FT, for a WFG% of 57.14%.

    So by WFG%, C > D > E, at least in these performances.
    Yet on the exact same number of shots, Player C scored 18, D scored 19, and E scored 20.
    Again, it’s punishing the guys who shot more threes by counting their missed threes more heavily than their missed twos – and theirs no reason for that.

    The key is that what you should be trying to measure is the number of points scored relative to the number of possessions used up trying to score them. That’s not done entirely satisfactorily in eFG (no FTs) or TS% (the .44 thing instead of measure possessions explicitly), so there’s room to improve, but WFG% as proposed is a step in the wrong direction. You’ve taken issue with the incorrect use of the term ‘percentage’ – the solution is renaming those terms (because you’re right, they aren’t percentages), not adjusting them until they are percentages at the cost of informational value.

    • nunuspeaks says:

      So what would you say TS% and eFG% are, if not percentages? What does it mean to say JJ Redick had an eFG% of 115% on a given night? Like I said, the WFG isn’t perfect, but we actually think it’s a step in the right direction, considering that it attempts to weigh all three shots (1, 2, 3). Your example (using the spreadsheet!) is helpful to show that there IS a penalty involved for taking MORE three’s. And it’s true that we didn’t exactly anticipate that behavior. Our goal was simply to avoid saying “Redick shot 115%.”

      However, it has yet to be shown that such a penalty is *wrong.* I’ll admit that the *reason* it exists (for us) is to make the numbers reflect a percentage that cannot exceed 100%. But, like you said, if you factor in rebounding, fouling, opportunity cost on possessions, etc, then you may be able to justify the penalty. We obviously can’t make that argument definitively yet, but the possibility is there.

      Your comment is very helpful, and I guess my question in return would be: how would you fix WFG? Presumably what is needed is a different weight for 3pt attempts as opposed to 3pt makes. The problem is, if you simply count a 3pt attempt as a 2pt attempt, you will get the “Derrick Rose Effect” again (see here https://escobarmag.wordpress.com/2013/03/06/weighted-field-goal-percentage-part-2/).

      We think this metric (WFG) has the potential to be quite useful and welcome any revisions.

      –Chris

      • Jeremy says:

        Well eFG% is an effective FG% – it’s right in the name. JJ Redick shooting a 115% eFG% for a night means his shooting had the same effect as if he’d shot 115% on all twos.
        It’s not a measure of what percentage of his shots he made or anything like that, it’s a measure of the effect of those shots – and the effect of taking shots when you include threes can be greater than the effect of shooting 100% on twos.

        TS% is the same thing only (rightly) factoring in free throws and using (an estimate of) possessions as the denominator.

        Again, basketball is a game of 1) maximizing the number of possessions you have relative to your opponent, 2) maximizing points scored given the possessions you have, 3) minimizing the points your opponents score given the possessions they have. If this isn’t obvious, really stop and think about it for a little bit, because it’s completely key. Those are the three aspects of the game, and that’s the approach you should take when thinking about the game.

        FG% is a measure of what percentage of your shots from the field go in, and gives us a first-pass proxy for how efficiently someone scores (where efficiency is scoring more points per possession).
        eFG% improves on it by factoring in threes to reflect the fact that not all shots are worth the same amount of points, and thus gives a better measure of efficiency.
        TS% improves on it again by factoring in that there are other ways to score than from the field (ie FTs), and that they can use up possessions without counting as a FGA, and adjusts the denominator accordingly, again giving an even better measure of efficiency.

        The natural improvement from there is to accurately measure possessions used, rather than estimating using the .44 formula – so if you want to work on that then that’s fine, but points per possession already exists.

        What does WFG% improve upon? How does it give us a better understanding of a player’s contribution to efficient scoring, ie to the points they’re scoring relative to the possessions they’re using to do it? That is, after all, what we should care about.

        It seems it’s motivated purely out of the desire to prevent percentages greater than 100%, rather than out of a desire to give us a better understanding of scoring efficiency. It succeeds in doing that, sure, but at the cost of being a worse indicator of what we actually want to know.

        An easier solution is to realize that there’s no fundamental problem with eFG% and TS% being greater than 100% (because sometimes players have scoring games that are more efficient than shooting 100% on twos), and then that’s the end of it.

  3. […] Why True Shooting Percentage and Effective Field Goal Percentage Are Wrong (and Why Our Metric Is&nb… […]

  4. beelicker says:

    Scoring-wise, the ‘base level’ FG = 2FT … 1 made FT is worth half of a made 2 so weight each mFT as counting o.5 so that making 10/10 FT = 50% scoring equivalency to making 10 2s … half the points is simply the maximum scoring equivalence you can attain making FTs vs making the same # of FGs … half the points maximum of making the same # of 2pt shots

    scoring equivalence per 2pt shot is what this is measuring … subtracting .5 per each missed FT weights your maximum scoring/shot efficiency equivalent to # of 2s on the same make/miss per shot basis as adding .5 per m3FG

    eFG = .5mFT+ mFG + 1.5m3 / FTA + FGA + 3FGA

    Fouled in the act of missing you are given 2/3 attempts to make the full scoring value of that shot

    Fouled in the act of making you also get the one pt ‘bonus’ attempt, which can also be awarded other ways to include team-accumulated triggers + make one of those to get another

    Now your points per shot efficiency stat includes FTs but why?

    I think you still want to consider FTs a separate category of shooting per se apart from FGs as far as for making raw overall/particular player skill level comparisons/decisions

    current eFG still allows combined FG shooting% to be meaningful instead of a corrupt no-information nonsense stat (w/o the adjustment for points value) which is useful for nothing at all

    Why would box scores even provide a single combined FG column/stat anyway? Separate 2 & 3 columns should be mandatory for 2 obviously incompatible values for FGs

    Also lumping FTs unweighted into one big master column with 2s & 3s would be just as ‘logically’ consistent

  5. sup says:

    first off, of course ts% is stupid. larry bird shot better from fg, ft, and 3, yet over his career SOMEHOW ends up with a slightly less ts% than bron.

    I get that it’s weighted, but it makes no fuckin sense. he shot SIGNIFICANTLY better than bron at the line, SIGNIFICANTLY better than bron from 3, his overall fg% is better, and yet somehow his “ts %” is worse? LOL. I don’t think so bro.

    even if bron shot slightly better from 2 (which is what it says on b-ref), one would assume the HUGE advantages bird had from 3 and the line would ATLEAST keep them even, if not give him a shooting advantage.

    NO ONE thinks bron is a more efficient scorer over his career than Bird. NO ONE.

    common sense dictates A LOT of things. james being more efficient over his career than bird is retarded to the 10th degree. ts % seems to reward guys for actually taking more shots and getting to the line a lot.

    it should be really simple, the points should be weighted as such:

    1 three point field goal = 1.5 two point field goals
    1 two point field goal = 1.0 two pointer
    1 free throw = 0.5 two pointers

    OR

    1 three point field goal = 1.0 three pointer
    1 two point field goal = .67 three pointer
    1 free throw = .33 three pointer

    OR

    1 three point field goal = 3.0 free throws
    1 two point field goal = 2.0 free throws
    1 free throw = 1.0 free throw

    just depends how you create the formula.

    don’t even get me started with “win shares”. win shares is the worst thing to happen to basketball stat geeks ever. not only is it inaccurate, people put faith in a BASEBALL-related stat that crossed over, and it’s quite simply the MOST complex, befuddling, otherworldly formula I’ve seen in my life.

    It makes astrophysics look simple.

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