Expanding ‘Points Per Miss’ (PPM)

Yesterday, Jerod wrote a fantastic article on Dirk Nowitzki’s historically efficient night. In his article, he took an idea that was initially launched by Bill Simmons about a stat that measures a player’s efficiency on the offensive side of the ball.

He called the stat Points Per Miss, or PPM.

The stat really speaks for itself. How many points does a player score for every one of his missed shots? Really, it’s determining if a player brings more to the table than he takes from it. Every missed shot is a blown possession for his team while every make is a converted one.

We discovered that Dirk had an outstanding single-game PPM of 16 in Game 1. For every one of his misses, the big fellow scored SIXTEEN points. Simply outstanding.

However, as we thought about the stat more, and took the insightful and analytical contributions of commenters into consideration, we realized a few things.

dirk-nowitzki-dfw-digs-dirk-videoAs one reader pointed out,

“Had Dirk missed one more shot, his PPM would be 12. Had he missed 3 more shots, still a respectable 12-18, his PPM would be 8. Since this stat uses division, the numbers cannot be compared linearly. For example, comparing PPMs of 16 and 12 is not the same comparison as PPMs of 6 and 2, even though the difference between each is 4.”

Another thing we realized was that all shots are not created equal. Should a missed free throw count just as much as a missed field goal? Should missed three pointers count as much or more as a missed two-pointer?

Eventually we came to some conclusions about how PPM could be greatly improved.

First of all, missed threes and twos should count the same. Both misses (unless your team gets an offensive rebound) end up costing your team a possession. That’s one more possession where your team could have scored, but didn’t because you failed to convert.

The other question was about free throws. If you miss the first attempt of a two-shot foul, you haven’t necessarily lost your team a possession. In fact, you still get the opportunity to get your team a point. Really, free throws could be thought of as half of a field goal…since you normally get two of them and each shot is worth half of a field goal.

However, sometimes you only get one free throw because you earned an and-one. Other times, you are fouled on a three point attempt and get three freebies. We decided to use John Hollinger’s formula from his advanced statistic True Shooting Percentage to work out the details. He figured out a free throw attempt is worth 0.44 possessions.

That leads us to our final formula for figuring out Points Per Miss, which is slightly different from yesterday but more precise and useful:

PPM = Points / (Field Goals Missed + (0.44 x Free Throws Missed))

Because Dirk didn’t miss a single free throw, his PPM would still have been an outstanding 16. If he had missed one free throw, it would have dropped to 13.95. If he had missed one more field goal, it would have dropped to 12.
Obviously, with such an outstanding game, the results are likely to change A LOT with just one more make or miss.

However, the benefits of PPM are much more relevant when looking at someone’s season or career as a whole.

We will get to that in a second, but first I’ll address what I’m sure some of you are thinking: Wow…these guys have done nothing significant…this is just a ripoff of Hollinger’s True Shooting Percentage.

Let’s compare the two and discuss a few main points.

  • PPM = Points / (Field Goals Missed + (0.44 x Free Throws Missed))
  • True Shooting Percentage = Points / (2 x (FGA + (0.44 x FTA)))

The stats are very similar to say the least.

HOWEVER…

I believe they are different enough to both be useful, even together.

Percentage vs. Points

For one thing, Hollinger’s TSP gives you a percentage that the player shot from the floor.  For instance, Dirk had a 61.2% True Shooting Percentage this year and Paul Pierce had a 62% True Shooting Percentage.  With that stat, you get an idea of how well the guy shot from everywhere.

Our stat gives you an idea of how many points that guy scored for every miss.  Dirk scored 2.85 points per miss and Pierce scored 2.79.

That difference is small, but it does give you a different sort of figure to compare the two players.

A More Complete Picture

The BIGGER difference I think is that it tells a more complete picture.

Let’s compare those two guys again.  If you just looked at TSP, you would assume that Pierce is the more efficient scorer than Dirk, but if you really look at the stats that doesn’t make any sense.

  • Pierce averaged 18.9 points, shot 86% from the line, 37% from three, and 50% on all Field Goals.
  • Dirk averaged 23 points, shot 89% from the line, 39% from three, and 52% on all Field Goals.

So you have a guy who averaged MORE points and shot a better percentage from ALL THREE areas, but somehow he has a WORSE True Shooting Percentage than the other guy. That’s because True Shooting Percentage rewards players that score a higher percentage of their points from the free throw line. Dirk only scored 23% of his points from the foul line whereas Pierce scored 25% of his points from there this season.

Here is a more clear example.

Let’s pretend that two players both scored 30 points on a given night.

  • Player A was 10-20 from the floor and also finished 10-12 from the charity stripe.  He missed 12 shots, and ended up with a PPM of 2.757.  However, his True Shooting Percentage was 59%.
  • Player B was 14-24 from the floor, but only 2-4 from the foul line.  He also missed 12 shots, and his PPM remains the same – 2.757.  However, HIS True Shooting Percentage dips slightly to 58%.

Just by looking at TSP, the common person would believe that Player A was more efficient.  However, that’s not really the case because while Player B did not shoot as well from the foul line, he actually shot BETTER from the field, which actually saved his team more possessions. Both players were equally effective. PPM shows that, TSP doesn’t.

All of that to say this:

I believe that our stat provides a more accurate picture of true efficiency on the offensive end of the floor than True Shooting Percentage.

It shouldn’t really matter where a player is getting his points. Some players score by getting to the free throw line. Other players score remarkably well without getting to the line. We need a stat that measures overall effectiveness correctly, and I believe that PPM helps us to accomplish that goal a little better.

At the end of the day, Dirk and Pierce are both remarkably efficient, and both deserve lots of credit for that. Like Jerod said yesterday, I had no idea Pierce was so efficient, and he now gets a lot more respect from me. But PPM provides a little more accurate picture of true efficiency.

When I look for efficiency, I am really looking for how many points per possession/shot/miss someone averaged. Percentages may lie; PPM doesn’t.

Below is a quick look at the top fifty scorers from the NBA this season, plus Steve Nash…because I’m always interested to see where he finishes in these types of stats.

What you will notice is that the PPM tends to lean a little more towards a) big men who get their shots around the basket and b) lights out shooters. This doesn’t really tell us anything new though, does it? Big men and great shooters are more efficient scorers than slashers and athletes.

A few interesting players of note: Derrick Rose, Kobe Bryant, and Russell Westbrook all fall out of the top ten and into the mid-30s. Meanwhile, Pau Gasol takes an almost meteoric rise up to #4.

Another player who is not on the list is Chris Paul. He ended up with a 2.444 PPM, putting him higher than all point guards not named Steve Nash and Tony Parker.  However, his playoff PPM of 3.35 was unbelievable.  Anyone who watched him against the Lakers would have said he was remarkably efficient…but good grief!

You can go up and down the list and make your own conclusions. It is fun and, in many cases, quite surprising.

 

FGM FGA FTM FTA Total Points Missed FGA Missed FTA PPM
Dwight Howard 619 1044 546 916 1784 425 370 3.03504593399115
Dirk Nowitzki 610 1179 395 443 1681 569 48 2.84857317155833
Paul Pierce 507 1021 386 449 1511 514 63 2.78926382633095
Pau Gasol 593 1120 354 430 1541 527 76 2.74962529441153
Lebron James 758 1485 503 663 2111 727 160 2.64735390017557
Paul Millsap 525 988 256 338 1315 463 82 2.6348481205418
Steve Nash 399 811 227 249 1106 412 22 2.62284196547145
Ray Allen 480 978 193 219 1321 498 26 2.59304334170854
Chauncey Billups 339 794 384 419 1208 455 35 2.56802721088435
Kevin Love 482 1026 424 499 1476 544 75 2.5580589254766
Dwyane Wade 692 1384 494 652 1941 692 158 2.54884966908289
Chris Bosh 524 1056 384 471 1438 532 87 2.5215683523883
Kevin Martin 553 1267 594 669 1876 714 75 2.51137884872825
Tony Parker 555 1069 233 303 1368 514 70 2.51101321585903
Kevin Durant 711 1538 594 675 2161 827 81 2.50510062134842
Amar’e Stoudemire 744 1482 473 597 1971 738 124 2.4868779650752
Stephen Curry 505 1053 212 227 1373 548 15 2.47565813198702
David West 528 1040 264 327 1322 512 63 2.44941821685318
Zach Randolph 598 1188 300 396 1504 590 96 2.37884347716057
David Lee 496 978 210 267 1203 482 57 2.372406720833
LaMarcus Aldridge 707 1415 351 444 1769 708 93 2.36206804465096
Brook Lopez 644 1309 385 489 1673 665 104 2.35381844785863
Blake Griffin 696 1376 446 695 1845 680 249 2.33674451593292
Manu Ginobili 441 1018 357 410 1393 577 53 2.32042910447761
Wesley Matthews 440 979 266 315 1300 539 49 2.31910946196661
Carlos Boozer 431 845 171 244 1033 414 73 2.31552048776114
Carmelo Anthony 684 1503 507 605 1970 819 98 2.28506472416833
Luis Scola 569 1129 214 290 1352 560 76 2.2782421137773
Deron Williams 428 975 348 412 1309 547 64 2.27588844843174
Eric Gordon 427 949 287 348 1247 522 61 2.27206471831499
Rudy Gay 409 869 194 241 1069 460 47 2.22393276192061
Kobe Bryant 740 1639 483 583 2078 899 100 2.20360551431601
Derrick Rose 711 1597 476 555 2026 886 79 2.20035622746427
Al Jefferson 654 1319 220 289 1528 665 69 2.1974229176254
Josh Smith 497 1041 229 316 1274 544 87 2.18795081404135
Russell Westbrook 614 1390 531 631 1793 776 100 2.18658536585366
DeMar DeRozan 539 1154 327 402 1410 615 75 2.17592592592593
Luol Deng 531 1155 253 336 1430 624 83 2.16496093986556
Danny Granger 535 1260 395 466 1622 725 71 2.14482174970909
Andrea Bargnani 525 1173 287 350 1414 648 63 2.09258272657314
Monta Ellis 726 1611 340 431 1929 885 91 2.08531522961169
Nick Young 412 935 186 228 1115 523 42 2.05917116052301
Dorell Wright 485 1146 180 228 1344 661 48 1.97032780155984
Joe Johnson 514 1161 195 243 1312 647 48 1.96371909237861
Michael Beasley 561 1246 219 291 1401 685 72 1.95484735167718
Andray Blatche 426 957 220 283 1076 531 63 1.9258304696449
Stephen Jackson 435 1059 249 305 1240 624 56 1.91169215589541
Antawn Jamison 374 876 171 234 1010 502 63 1.90666767348788
John Wall 398 972 301 393 1131 574 92 1.84058065356073
Tyreke Evans 382 933 205 266 1012 551 61 1.75134985463104
Brandon Jennings 361 926 199 246 1019 565 47 1.73985794290398

 

Three more things of note.

First of all, no stat is perfect. The danger with all new stats is that we think we have figured out something so good that we no longer need to watch the game to decide how good a player is or is not. No one should look at this stat and decide that Derrick Rose, Kobe Bryant, and Russell Westbrook are bad players.  However, it’s not surprising to anyone to see that they really aren’t super efficient on offense.

That’s the thing I love the most about this stat. In my opinion, it really reflects what you see. It’s more of a tangible verification than a new breakthrough. Anyone who has watched the Lakers all year would tell you that Pau is a more efficient scorer than Kobe, but even I didn’t think the difference would be that diverse.

Secondly, I think this stat is still a work in progress as far as what we can learn from it.

I have not had time to find out the league average for PPM yet, but it would be nice to figure that value out so we can really compare players. For instance, if you didn’t know anything about baseball, you would think that someone being successful at the plate only 30% of the time was awful. It isn’t until you realize what everyone is hitting that you understand that hitting .300 is fantastic.

With that said, PPM is still easy to understand because Points Per Miss is about as simplistic as it gets. Everyone who has ever watched basketball can understand that scoring 2.5 points per miss would be outstanding.

The other place this stat could really go is in terms of teamwork. I would be very interested to see how much a player’s presence on the floor affects his teammates’ PPM. Obviously LeBron James made his Cavs teammates more efficient players, but how much so? I believe this could be the next step of PPM.

Finally, I thought this comparison was telling: Reggie Miller’s career PPM was 2.632. Ray Allen’s is a paultry 2.304.

The ball is in your court, SportsGuy.

**********

Follow me on Twitter @The_Dr_Twitch



About Jon Washburn

Jon Washburn grew up in Indianapolis, IN and as such, is a diehard Pacers, Colts, and Cubs fans. When it comes to college, he cheers for Notre Dame football fan and Purdue basketball. Yes, this sounds shady, but since he grew up without cable, he learned to love Notre Dame - the only team on TV. Glenn "The Big Dog" Robinson was at Purdue when Jon was in his formative years, so he latched onto them as well. Did that make him a fair-weather fan at the time? Sure. Give him a break...he was 8...and he has stayed with those teams ever since. Currently, he lives in Charleston, SC with his wife who grew up in Cleveland. Although he is no longer physically in the Midwest, his heart will always be there. Jon goes by the name "Twitch" because he has Tourette's Syndrome. Hit him up on his twitter @jwtwitch.

Comments

  1. Why not look at points per attempt, PTS/(FGA+FTA), for scoring efficiency?

    2010-11 Reg Season and Playoffs thru 5/18: https://spreadsheets.google.com/spreadsheet/ccc?k

  2. I think this could further be extended (or perhaps a different statistic) by adding in assists somehow. A player's aid to his teams score is 'points scored'+"points scored after an assist by this player". Now that I reread that, this would be a modification or extension of PPG… let's call it PPP or Points Per Player. Now, how would you figure that number into the others?

  3. Why doesn't the metric take into account whether the FGA was a 2 or 3? While taking a 3 gives you more points, it also is (generally) riskier, so shouldn't you be punished more for missing a bunch of threes, rather an a bunch of twos? It's certainly debatable.

    (copied and re-uploaded from Chris's spreadsheet)
    https://spreadsheets.google.com/spreadsheet/ccc?k

    • Shooting a 2 inherently increases reward due to the ability to draw more fouls and thus shoot more free throws. Which also makes me believe shooting 50% from inside the arc is not the same as shooting 33% from outside due to the assumed increase in free throw attempts.

      Any idea as to how to quantify the benefit of shooting 2's? A simple formula would be FTA/(FGA-3PA), but that excludes the rare foul on a 3 point attempt and is probably missing something else. Also, what would the 3P% need to be when taking into account the opportunity cost of missed free throws? (I think 82games.com calculated what it needs to be at some point, but I can't remember what it ways)

    • The metric DOES take into account 2's versus 3's because your overall POINTS are affected by 2's and 3's.
      For instance, let's say 2 guys both go 7 for 8. One guy makes 7 threes, and the other guy makes all twos. Player one would have a PPM of 21 and player 2 would only have a PPM of 14.
      You don't really need to make a special formula for them because they are already affecting the PPM dramatically.

  4. Just a thought on another potential method of weighing different types of shots/misses…

    Rather than thinking about a missed shot as a loss of possession, one could view it as a loss of potential/expected points. A free throw may be worth .44 possessions, but it's a also point that most players in the league should be able to get. We can find the "expected" number of points for each type of shot by looking at league wide 3 point, 2 point, and free throw percentages. I don't have time right now to find the exact numbers for this, but using the medians as a point of reference, I found the following:

    .491 (league 2P%, I roughly took out the 3 pointers by using median FGA and 3PA numbers) * 2 (potential points for a 2 point shot) = .982 expected points
    .355 (league 3P%) * 3 = 1.062
    .765 (league FT%) * 1 = .765

  5. WPPM (weighted points per miss) would therefore be Points / ((.982)*(Missed 2 Pointers)+(.1.062)*(Missed 3 Pointers)+(.765)*(Missed Free Throws))

    Not quite as pretty, but I think taking into account both the relative difficulty of the shots and their different values is an interesting idea. One can also think of WPPM as the ratio of the points a player scores to the points he was unable to score.

    Also, his is kind of unrelated, but it just came to mind while writing this – it might even be interesting to use a completely different statistic, also assessing a player's efficiency by looking at a breakdown of his shot selection, measuring the percentage of "expected" points a player was able to score:

    EP% (expected points percentage) = Points/((.982)*(Attempted 2 Pointers)+(.1.062)*(Attempted 3 Pointers)+(.765)*(Attempted Free Throws))

  6. Using this to look at Dirk's 48 point game…

    Dirk: 48 P, 15 2PA, 24 FTA – 33.09 expected points, he scored 145.1% of that
    LeBron last night for comparison: 35 P, 25 2PA, 1 3PA, 13 FTA, 35.557 expected points, he scored 98.4% of that
    Rose last night for comparison: 23 P, 18 2PA, 9 3PA, 7 FTA, 32.589 expected points, he scored 70.6% of that

    Looking at the seasons of those same three players…

    Dirk: 1681 P, 1011 2PA, 168 3PA, 443 FTA = 111.3% of expected points
    LeBron: 2111 P, 1206 2PA, 279 3PA, 663 FTA = 106.2% of expected points
    Rose: 2026 P, 1212 2PA, 385 3PA, 555 FTA = 100.1% of expected points

  7. Obviously it's not perfect (especially since my league wide percentages aren't even correct), and maybe this has been done before, but I think it's a really interesting way of looking at efficiency, similar to what you've done here. It also has a very intuitive and understandable meaning. I'd like to see a few more players, but you mention with PPM, this confirms what we thought we've seen all season – Dirk is very efficient, Lebron is pretty efficient, and Rose struggles a bit with his efficiency.

  8. tonythestatmaster says:

    What was Michael Jordan's career PPM?

  9. You make the point in your example that player B "…shot BETTER from the field, which actually saved his team more possessions", however I don't think that is the case. Assuming both players only shot 2pt field goals and there were no "and-1's", both used 26 possessions and missed 10 FGs, leading to 10 loose ball opportunities. Both players A & B also missed 2 shots from the charity stripe. A miss on the 1st FT has no affect on the possession, but a miss on the 2nd leads to a loose ball, whereas a made shot simply hand the ball over to the opposing team. Of course the player should always try to make the FT, but if they're going to miss one, it would be more beneficial for them to miss the 2nd shot. If we assume the FT rebound goes to the opposite team, players A & B saved their teams the same number of possessions.

    I too think PPA would be a better measure of efficiency.

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