Individual NBA player points allowed per possession fits an approximate normal distribution

adriengriffin
(AP Photo/Mark J. Terrill)

When you make a bar graph of a numeric variable against the frequency with which certain values show up, the normal distribution (i.e., bell curve) shows up in a lot of places. If you graphed variables like the height of adult males in the U.S., or home sales prices in a given zip code for a certain year, or the weights of jars of peanut butter that come off a given assembly line, those graphs would fit a normal or approximately normal distribution. Variables like SAT score and IQ score are scoring systems designed specifically to accommodate a bell curve shape.

The reason is more or less simple: there are a lot of values for those variables that sit near the average, and the further you get away from the average — on both sides — the fewer instances you’ll see.

NBA defense fits a similar pattern. If you look at the points allowed per possession in 2013-14 (with data gathered from Synergy on March 22), the data for the 455 players fits a nearly ideal normal distribution shape. (Note: some players didn’t defend a large enough number of possessions to qualify for a ranking in the Synergy database).

Screen shot 2014-03-25 at 11.33.36 AM

There are a few hints of some right-skewing (which in this case means that the “bad” defenders to the right of the mode were slightly more prevalent than the “good” ones to its left), but other than that, the distribution is nearly perfectly normal.

It’s not a surprise that the 13-58 Milwaukee Bucks have had more bad defenders than they have good. There are plenty of caveats to be wary of here — some players are asked to do more than others, some players have small sample sizes as a Buck, etc. — but there are a few trends worthy of note, so let’s take a look at each of the Bucks relative to the overall NBA trend. The distribution below is the same distribution, but differentiated a bit more on the x-axis so as to be able to demarcate the differing levels of defense played by individual Bucks.

Screen shot 2014-03-25 at 11.51.44 AM

And here too is the color-coded key for locating your favorite Buck:

Screen shot 2014-03-25 at 11.48.12 AM

Are Nate Wolters and O.J. Mayo the two best defenders on the team in 2013-14? Probably not. But they have made the best of the situations into which they’ve been placed. Larry Drew gives them the easier assignments out on the perimeter, and he doesn’t ask them to double and/or trap as often as he does players like Brandon Knight and Giannis Antetokounmpo. Plus Wolters and Mayo responded better than players like Gary Neal and Luke Ridnour who were put in similar situations. (In my eyes, at least.)

On the end of the spectrum, it’s not difficult to see how players like Zaza Pachulia and Larry Sanders have done a better job slowing their opponents inside than Ersan Ilyasova, Miroslav Raduljica, and Jeff Adrien. Adrien, in particular, seems like a quick and bouncy help defender, but taller opposing power forwards have far too easy a time simply shooting right over his head.

In general, many of the NBA’s best and worst defenders according to this rubric are fluky because of the sample size. Players qualified for the ranking system with as few as 30 overall possessions defended. As a result, players like Solomon Hill and Danny Granger (Indiana version) were among the top defenders. And since Synergy doesn’t link to a 1-455 list of the ranked players, here below is a list from best to worst.

PPP Rank
Solomon Hill 0.50 1
Danny Granger (LAC) 0.60 2
Victor Claver 0.61 3
Stephen Jackson 0.62 4
Brook Lopez 0.65 5
Otto Porter 0.65 5
Nick Calathes 0.65 5
Willie Green 0.68 8
Glen Rice 0.68 8
Lorenzo Brown 0.68 10
Kobe Bryant 0.68 10
Will Barton 0.69 12
Kenyon Martin 0.70 13
Roger Mason 0.70 13
Kyle O’Quinn 0.70 13
Danny Granger (IND) 0.71 16
Josh Harrellson 0.72 17
Mike James 0.72 17
Chris Kaman 0.72 17
Aaron Gray 0.73 20
Andre Miller (WAS) 0.73 20
Andre Miller (DEN) 0.73 20
Chris Paul 0.73 20
Marquis Teague (CHI) 0.73 20
Jason Terry 0.73 20
Hasheem Thabeet 0.74 26
Ronny Turiaf 0.74 26
David West 0.74 26
Derrick Williams (MIN) 0.74 26
Antawn Jamison 0.75 30
Kosta Koufos 0.75 30
Luc Mbah a Moute (MIN) 0.75 30
Kendrick Perkins 0.75 30
A.J. Price 0.75 30
Alexey Shved 0.75 30
C.J. Watson 0.75 30
Omer Asik 0.75 30
Omri Casspi 0.75 30
Jae Crowder 0.75 30
Roy Hibbert 0.76 40
Ed Davis 0.76 40
DeMar DeRozan 0.76 40
Orlando Johnson 0.77 43
Steve Nash 0.77 43
Andrew Bynum 0.77 45
Norris Cole 0.77 45
Chris Douglas-Roberts 0.77 45
Mike Dunleavy 0.77 45
Danny Green 0.78 49
Mike Harris 0.78 49
Kirk Hinrich 0.78 49
Kawhi Leonard 0.78 49
Andre Roberson 0.78 49
Paul Pierce 0.79 54
Tayshaun Prince 0.79 54
Henry Sims 0.79 54
Klay Thompson 0.79 54
Marcus Thornton (SAC) 0.79 54
Nick Young 0.79 54
Ronnie Brewer 0.80 60
Mario Chalmers 0.80 60
Luol Deng (CHI) 0.80 60
Nene Hilario 0.80 60
Byron Mullens (PHI) 0.80 60
Dirk Nowitzki 0.80 60
Patrick Patterson (TOR) 0.80 60
John Salmons (TOR) 0.80 60
Jonas Valanciunas 0.80 60
Shawne Williams 0.80 60
Chris Andersen 0.81 70
Ryan Anderson 0.81 70
D.J. Augustin (TOR) 0.81 70
Kevin Durant 0.81 70
Pau Gasol 0.81 70
Jordan Hamilton (DEN) 0.81 70
Ryan Hollins 0.81 70
Kevin Martin 0.81 70
O.J. Mayo 0.81 70
Ronnie Price 0.81 70
Derrick Rose 0.81 70
Robert Sacre 0.81 70
Luis Scola 0.81 70
Anderson Varejao 0.81 70
Jan Vesely (DEN) 0.81 70
Dwight Buycks 0.82 85
Stephen Curry 0.82 85
Tim Duncan 0.82 85
Diante Garrett 0.82 85
Rudy Gobert 0.82 85
Al Harrington 0.82 85
Al Horford 0.82 85
Andre Iguodala 0.82 85
Al Jefferson 0.82 85
DeAndre Jordan 0.82 85
Miles Plumlee 0.82 85
John Salmons (SAC) 0.82 85
Ishmael Smith 0.82 85
Marreese Speights 0.82 85
Greivis Vasquez (TOR) 0.82 85
Dwyane Wade 0.82 85
Deron Williams 0.82 85
Nate Wolters 0.82 85
Metta World Peace 0.82 85
Carmelo Anthony 0.83 104
Leandro Barbosa 0.83 104
Jamal Crawford 0.83 104
Paul George 0.83 104
Spencer Hawes (PHI) 0.83 104
Dwight Howard 0.83 104
Robbie Hummel 0.83 104
Courtney Lee 0.83 104
Kevin Love 0.83 104
C.J. McCollum 0.83 104
Byron Mullens (LAC) 0.83 104
Jared Sullinger 0.83 104
Isaiah Thomas 0.83 104
Russell Westbrook 0.83 104
Draymond Green 0.84 118
James Johnson 0.84 118
Darius Johnson-Odom 0.84 118
David Lee 0.84 118
Luc Mbah a Moute (SAC) 0.84 118
Quincy Miller 0.84 118
Nazr Mohammed 0.84 118
Joakim Noah 0.84 118
Victor Oladipo 0.84 118
Zaza Pachulia 0.84 118
John Wall 0.84 118
Lavoy Allen 0.84 118
Louis Amundson 0.84 118
Andrea Bargnani 0.84 118
Brandon Bass 0.84 118
Tyson Chandler 0.84 118
DeMarcus Cousins 0.84 118
Jordan Crawford (BOS) 0.84 118
Anthony Davis 0.84 118
LeBron James 0.84 118
Kevin Garnett 0.84 118
Matt Barnes 0.85 139
Ken Bazemore (LAL) 0.85 139
Eric Bledsoe 0.85 139
Andrew Bogut 0.85 139
Carlos Boozer 0.85 139
Avery Bradley 0.85 139
Michael Carter-Williams 0.85 139
Nick Collison 0.85 139
Jordan Crawford (GSW) 0.85 139
Gerald Green 0.85 139
James Harden 0.85 139
Xavier Henry 0.85 139
Kris Humphries 0.85 139
Damian Lillard 0.85 139
John Lucas 0.85 139
Patrick Mills 0.85 139
Markieff Morris 0.85 139
Timofey Mozgov 0.85 139
Nate Robinson 0.85 139
Kevin Seraphin 0.85 139
Lance Stephenson 0.85 139
Jeff Taylor 0.85 139
Jeff Teague 0.85 139
LaMarcus Aldridge 0.86 162
Gustavo Ayon 0.86 162
Marco Belinelli 0.86 162
Bismack Biyombo 0.86 162
Trevor Booker 0.86 162
Caron Butler (OKC) 0.86 162
Jimmy Butler 0.86 162
Matthew Dellavedova 0.86 162
Jamaal Franklin 0.86 162
George Hill 0.86 162
Perry Jones 0.86 162
Shawn Marion 0.86 162
Ray McCallum 0.86 162
Paul Millsap 0.86 162
Marcus Morris 0.86 162
Daniel Orton 0.86 162
Nikola Pekovic 0.86 162
Mirza Teletovic 0.86 162
Marcus Thornton (BKN) 0.86 162
Jeff Green 0.87 181
Blake Griffin 0.87 181
Jrue Holiday 0.87 181
Reggie Jackson 0.87 181
Ty Lawson 0.87 181
Josh McRoberts 0.87 181
Jodie Meeks 0.87 181
Anthony Morrow 0.87 181
Jermaine O’Neal 0.87 181
Greg Oden 0.87 181
Chandler Parsons 0.87 181
Phil Pressey 0.87 181
Dennis Schroeder 0.87 181
Donald Sloan 0.87 181
Greg Stiemsma 0.87 181
Hollis Thompson 0.87 181
Nikola Vucevic 0.87 181
Bradley Beal 0.87 181
Chauncey Billups 0.87 181
Steve Blake (LAL) 0.87 181
Darren Collison 0.87 181
Dante Cunningham 0.87 181
Andre Drummond 0.87 181
Arinze Onuaku 0.88 204
Trevor Ariza 0.88 204
Patrick Beverley 0.88 204
Trey Burke 0.88 204
Alec Burks 0.88 204
Mike Conley 0.88 204
Luol Deng (CLE) 0.88 204
Chuck Hayes (TOR) 0.88 204
Gerald Henderson 0.88 204
Jonas Jerebko 0.88 204
Jeremy Lin 0.88 204
Shaun Livingston 0.88 204
Shelvin Mack 0.88 204
Kendall Marshall 0.88 204
Cartier Martin 0.88 204
Jason Maxiell 0.88 204
Arnett Moultrie 0.88 204
Shabazz Muhammad 0.88 204
Nemanja Nedovic 0.88 204
Andrew Nicholson 0.88 204
Mason Plumlee 0.88 204
Quincy Pondexter 0.88 204
Thomas Robinson 0.88 204
Cody Zeller 0.88 204
Alan Anderson 0.89 228
Jerryd Bayless (BOS) 0.89 228
DeMarre Carroll 0.89 228
Vince Carter 0.89 228
Samuel Dalembert 0.89 228
Boris Diaw 0.89 228
Goran Dragic 0.89 228
Reggie Evans (BKN) 0.89 228
Randy Foye 0.89 228
Channing Frye 0.89 228
Marc Gasol 0.89 228
Taj Gibson 0.89 228
Derrick Favors 0.89 228
Archie Goodwin 0.89 228
Ben Gordon 0.89 228
Marcin Gortat 0.89 228
John Jenkins 0.89 228
Terrence Jones 0.89 228
E’Twaun Moore 0.89 228
Patrick Patterson (SAC) 0.89 228
Austin Rivers 0.89 228
Terrence Ross 0.89 228
Larry Sanders 0.89 228
Tristan Thompson 0.89 228
Louis Williams 0.89 228
D.J. Augustin (CHI) 0.90 253
Giannis Antetokounmpo 0.90 253
Shane Battier 0.90 253
DeJuan Blair 0.90 253
Chris Bosh 0.90 253
Will Bynum 0.90 253
Tyreke Evans 0.90 253
Raymond Felton 0.90 253
Rudy Gay (SAC) 0.90 253
Tyler Hansbrough 0.90 253
Manny Harris 0.90 253
Gordon Hayward 0.90 253
Cory Joseph 0.90 253
Robin Lopez 0.90 253
Kyle Lowry 0.90 253
C.J. Miles 0.90 253
Greg Monroe 0.90 253
J.J. Redick 0.90 253
Ramon Sessions (CHA) 0.90 253
Iman Shumpert 0.90 253
Josh Smith 0.90 253
Garrett Temple 0.90 253
Anthony Tolliver 0.90 253
Kemba Walker 0.90 253
Thaddeus Young 0.90 253
Tyler Zeller 0.90 253
Devin Harris 0.91 279
John Henson 0.91 279
J.J. Hickson 0.91 279
Kyrie Irving 0.91 279
Jarrett Jack 0.91 279
Amir Johnson 0.91 279
Ryan Kelly 0.91 279
Kyle Korver 0.91 279
Rashard Lewis 0.91 279
Khris Middleton 0.91 279
Rajon Rondo 0.91 279
Ricky Rubio 0.91 279
P.J. Tucker 0.91 279
Greivis Vasquez (SAC) 0.91 279
Alexis Ajinca 0.91 279
Ray Allen 0.91 279
Luke Babbitt 0.91 279
Reggie Bullock 0.91 279
Wilson Chandler 0.91 279
Earl Clark 0.91 279
Jared Dudley 0.91 279
Manu Ginobili 0.91 279
Serge Ibaka 0.92 301
Brandon Jennings 0.92 301
Jeremy Lamb 0.92 301
Donatas Motiejunas 0.92 301
Tony Parker 0.92 301
Zach Randolph 0.92 301
Jason Smith 0.92 301
Ekpe Udoh 0.92 301
Pero Antic 0.92 301
Jose Barea 0.92 301
Harrison Barnes 0.92 301
Nicolas Batum 0.92 301
Corey Brewer 0.92 301
Jose Calderon 0.92 301
Glen Davis 0.92 301
Monta Ellis 0.92 301
Tony Allen 0.93 317
Eric Gordon 0.93 317
Maurice Harkless 0.93 317
Jordan Hill 0.93 317
Shane Larkin 0.93 317
Ian Mahinmi 0.93 317
Toure’ Murry 0.93 317
Kelly Olynyk 0.93 317
Pablo Prigioni 0.93 317
Kyle Singler 0.93 317
Gerald Wallace 0.93 317
Quincy Acy 0.93 317
Darrell Arthur 0.93 317
Steve Blake (GSW) 0.93 317
Chase Budinger 0.93 317
Jason Collins 0.93 317
Jordan Farmar 0.93 317
Gary Neal 0.94 334
Jameer Nelson 0.94 334
Steve Novak 0.94 334
Mike Scott 0.94 334
Tony Snell 0.94 334
Tiago Splitter 0.94 334
Dion Waiters 0.94 334
Jeff Withey 0.94 334
Brandan Wright 0.94 334
Steven Adams 0.94 334
Nando De Colo 0.94 334
Derek Fisher 0.94 334
Jimmer Fredette 0.94 334
Enes Kanter 0.95 347
James Anderson 0.95 347
Aron Baynes 0.95 347
Michael Beasley 0.95 347
Andray Blatche 0.95 347
Aaron Brooks (HOU) 0.95 347
Brandon Davies 0.95 347
Wesley Johnson 0.95 347
Thabo Sefolosha 0.95 347
J.R. Smith 0.95 347
Amar’e Stoudemire 0.95 347
Mo Williams 0.95 347
Richard Jefferson 0.96 359
Michael Kidd-Gilchrist 0.96 359
Brian Roberts 0.96 359
Jason Thompson 0.96 359
Hedo Turkoglu 0.96 359
Jeff Adrien (CHA) 0.96 359
Arron Afflalo 0.96 359
Al-Farouq Aminu 0.96 359
Dewayne Dedmon 0.96 359
Gorgui Dieng 0.96 359
Reggie Evans (SAC) 0.96 359
Evan Fournier 0.96 359
Wesley Matthews 0.97 371
Travis Outlaw 0.97 371
Ramon Sessions (MIL) 0.97 371
Rodney Stuckey 0.97 371
Jarvis Varnado 0.97 371
Matt Bonner 0.97 371
Jeremy Evans 0.97 371
Kentavious Caldwell-Pope 0.98 378
Alex Len 0.98 378
Martell Webster 0.98 378
Dorell Wright 0.98 378
Cole Aldrich 0.99 382
Elton Brand 0.99 382
Landry Fields 0.99 382
Tobias Harris 0.99 382
Chris Johnson 0.99 382
Andrei Kirilenko 0.99 382
Doron Lamb 0.99 382
Derrick Williams (SAC) 0.99 382
Marvin Williams 0.99 382
Rudy Gay (TOR) 0.99 382
MarShon Brooks 1.00 392
Rasual Butler 1.00 392
Vitor Faverani 1.00 392
Francisco Garcia 1.00 392
Chuck Hayes (SAC) 1.00 392
Bernard James 1.00 392
Brandon Knight 1.00 392
Meyers Leonard 1.00 392
Jon Leuer 1.00 392
Ben McLemore 1.00 392
Darius Miller 1.00 392
Mike Miller 1.00 392
Luke Ridnour 1.00 392
Peyton Siva 1.00 392
Aaron Brooks (DEN) 1.01 406
Kenneth Faried 1.01 406
Tim Hardaway 1.01 406
Ersan Ilyasova 1.01 406
Jeremy Tyler 1.01 406
Jan Vesely (WAS) 1.01 406
Jeff Ayres 1.02 412
Caron Butler (MIL) 1.02 412
Luigi Datome 1.02 412
Alonzo Gee 1.02 412
Sergey Karasev 1.02 416
Brandon Rush 1.02 416
Evan Turner (IND) 1.02 416
Beno Udrih 1.02 416
Tony Wroten 1.02 416
Kent Bazemore (GSW) 1.03 421
Anthony Bennett 1.03 421
Joe Johnson 1.03 421
Gal Mekel 1.04 424
Miroslav Raduljica 1.04 424
Anthony Randolph 1.04 424
Evan Turner (PHI) 1.04 424
Elliot Williams 1.04 424
Joel Freeland 1.05 429
Carl Landry 1.05 429
Tornike Shengelia 1.05 429
Julyan Stone 1.05 429
Spencer Hawes (CLE) 1.07 433
Tyshawn Taylor 1.07 433
Jerryd Bayless (MEM) 1.08 435
Jordan Hamilton (HOU) 1.08 435
Toney Douglas (GSW) 1.10 437
Darius Morris (PHI) 1.10 437
Ian Clark 1.11 439
Drew Gooden 1.11 440
Udonis Haslem 1.11 440
Charlie Villanueva 1.11 440
Wayne Ellington 1.12 443
Chris Copeland 1.13 444
Chris Singleton 1.13 444
Greg Smith 1.13 444
Jannero Pargo 1.14 447
Earl Watson 1.14 447
Toney Douglas (MIA) 1.19 449
Darius Morris (MEM) 1.19 449
Jeff Adrien (MIL) 1.21 451
Jamaal Tinsley 1.21 451
Eric Maynor 1.23 453
Dionte Christmas 1.30 454
Marquis Teague (BKN) 1.39 455

 

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