Adjusted Plus/Minus (RAPM) is undoubtedly the most influential one-number metric in basketball analytics today, boosted by its unbiased nature (relatively speaking) which is impartial to playstyle and craftily captures both offense and defense on a fair playing field. Despite it strengths, however, RAPM has lots of intervening weaknesses which affect the metric’s ability to be interpreted. These include:

• The inability to assign credit based on performance.
• The “multicollinearity” problem from consistent lineup shuffling.
• The sampling issues caused by the penalization term.

For those unfamiliar, RAPM uses a “ridge” regression primarily because it is designed to introduce a slight bias in the model to offset the variability that would otherwise decrease the statistical significance of the coefficient estimates. To do this, most modern models use a penalization term (lambda) which tends to draw the coefficients closer to zero (or a prior, in the case of some hybrids) which is obtained via cross-validation, a process in which a data set is continually split into training and testing groups to find an ideal value.

Seems good enough, right? It’s certainly more “objective” than plotting the coefficients by lambda and picking a visually-appealing spot somewhere on the graph. But alas, there is one particular issue to address which hinders RAPM’s ability to be easily interpreted from season-to-season: a lambda value by cross-validation is chosen because it appeases the fit of the model. (And we know the caveats of what happens when solely prioritizing model fit.) So what happens when the model isn’t necessarily right?

The results can either “expand” or “compress” to an undesirable degree! Specifically, we’re dealing with the imperfections in how the metric deviates from its mean, meaning a lambda too high can excessively compress the coefficients and a lambda too low can excessively expand the results! So how do we solve this problem? At least today, by scaling the results based on external criteria. (This is not an original idea; Ben Taylor of Back Picks devised a variant of RAPM in which the coefficients were normalized to the same mean and standard deviation and Twitter user @SamphaStan applied the principle to multiple versions of RAPM.)

Similarly, I believed this to be a route worth exploring when scaling RAPM. But rather than standardizing the deviation of the metric from season-to-season, I considered something else: accounting for the natural fluctuations in offensive and defensive impact by proxy of results at the team level. (Remember, the NBA in 1997 is different than the NBA in 2022, and the evolutions of coaching and playstyle will affect how much individual players can impact the game.) But what exactly constitutes the “league” deviation in offense and defense?

By treating the team’s results (offensive and defensive efficiency) as the whole of the sum of its players, we can estimate the “actual” deviation by assigning equal credit to the “average” five players on the court. Doing this created a (seemingly) more “natural” curve that was unrelated to appeasing RAPM’s model fit, which is exactly what we wanted!

Notice how both offensive and defensive variation tapers off in the mid-2010s? This was an observation that intuitively felt inaccurate, as both qual and quant research tends to agree offensive superstars are creating more separation from their competitors than ever in the modern game. Interestingly enough, the scaled version of RAPM seems to “correct” this, as the “actual” highest deviation in offense for 2019 was the highest since 2008.

To summarize of the calculation process:

• Calculate the (weighted) means for both offense and defense; set new means to zero
• Calculate the (weighted) standard deviations for both offense and defense for RAPM and teams; adjust for team so it reflects the individual players
• Re-calculate RAPM so that the (weighted) standard deviation equals the estimated “actual” (weighted) standard deviation; do for both offense and defense

Listed below are the top-250 seasons on record from 1997 to 2019 among players with more than 3,000 possessions during the regular season (more or less 1,500 minutes played).

Player	Year	Team	G	Poss	ORAPM	DRAPM	RAPM
Kevin Garnett	2004	MIN	82	6150	4.59	4.38	8.97
Kevin Garnett	2009	BOS	57	3416	2.10	5.80	7.90
LeBron James	2010	CLE	76	5728	5.96	1.77	7.73
LeBron James	2011	MIA	79	5916	4.77	2.89	7.66
Kevin Garnett	2008	BOS	71	4468	2.48	4.99	7.48
LeBron James	2015	CLE	69	4889	6.04	1.42	7.46
Giannis Antetokounmpo	2019	MIL	72	5174	3.57	3.81	7.38
Tim Duncan	2003	SAS	81	6058	3.48	3.89	7.37
LeBron James	2012	MIA	62	4533	4.84	2.51	7.35
Tim Duncan	2004	SAS	69	4741	2.23	4.83	7.06
Draymond Green	2016	GSW	81	5974	2.98	4.05	7.03
LeBron James	2016	CLE	76	5344	4.29	2.53	6.82
Tim Duncan	2005	SAS	66	4134	2.43	4.38	6.81
Shaquille O’Neal	1998	LAL	60	4273	4.55	2.23	6.78
Shaquille O’Neal	2004	LAL	67	4830	3.96	2.66	6.63
Kevin Garnett	2005	MIN	82	5946	3.73	2.87	6.60
Kevin Garnett	2003	MIN	82	6480	3.90	2.67	6.57
LeBron James	2009	CLE	81	5769	4.30	2.23	6.52
Kevin Durant	2019	GSW	78	5813	5.50	0.89	6.39
Shaquille O’Neal	2001	LAL	74	5655	5.73	0.63	6.36
Dirk Nowitzki	2004	DAL	77	5792	4.85	1.41	6.26
Alonzo Mourning	1999	MIA	46	3152	1.95	4.30	6.25
Stephen Curry	2017	GSW	79	5690	5.20	1.03	6.23
Dirk Nowitzki	2003	DAL	80	6118	4.28	1.84	6.11
Alonzo Mourning	1998	MIA	58	3542	2.31	3.78	6.10
Rasheed Wallace	2004	DET	68	4428	1.81	4.26	6.07
Shaquille O’Neal	2000	LAL	79	6211	4.12	1.89	6.01
Tim Duncan	2001	SAS	82	6039	2.19	3.81	6.00
Stephen Curry	2015	GSW	80	5534	5.22	0.76	5.98
Dwyane Wade	2010	MIA	77	5390	5.71	0.26	5.97
Steve Nash	2008	PHX	81	5710	6.00	-0.04	5.96
Chris Paul	2015	LAC	82	5729	4.37	1.43	5.80
Jrue Holiday	2019	NOP	67	5212	4.33	1.47	5.80
Mookie Blaylock	1998	ATL	70	5003	3.75	2.00	5.75
Shaquille O’Neal	1999	LAL	49	3286	4.23	1.50	5.73
Kevin Garnett	2012	BOS	60	3583	1.40	4.32	5.72
Dirk Nowitzki	2012	DAL	62	4057	3.55	2.13	5.68
Shaquille O’Neal	2003	LAL	67	4991	4.79	0.87	5.66
Dirk Nowitzki	2011	DAL	73	4909	3.21	2.36	5.57
Stephen Curry	2019	GSW	69	5058	3.95	1.61	5.56
Manu Ginobili	2008	SAS	74	4470	3.52	1.98	5.50
Michael Jordan	1998	CHI	82	6010	3.93	1.53	5.46
Dirk Nowitzki	2013	DAL	53	3332	3.73	1.73	5.46
Kobe Bryant	2008	LAL	82	6488	5.20	0.25	5.45
Shaquille O’Neal	2002	LAL	67	4728	3.47	1.98	5.45
Paul George	2019	OKC	77	6172	3.67	1.77	5.44
Metta World Peace	2006	SAC	56	4255	1.33	4.10	5.43
Tim Duncan	2002	SAS	82	6324	1.87	3.51	5.39
Christian Laettner	1997	ATL	82	5703	2.47	2.91	5.38
Manu Ginobili	2005	SAS	74	4191	3.06	2.30	5.37
LeBron James	2014	MIA	77	5634	5.07	0.26	5.33
LeBron James	2013	MIA	76	5549	4.80	0.48	5.28
LeBron James	2008	CLE	75	5831	4.19	1.07	5.26
Steve Nash	2010	PHX	81	5426	5.62	-0.36	5.26
Tim Duncan	2007	SAS	80	5125	1.71	3.55	5.26
John Stockton	2001	UTA	82	4621	3.08	2.17	5.25
Manu Ginobili	2007	SAS	75	4006	3.08	2.17	5.25
Bo Outlaw	1997	LAC	82	4224	0.79	4.45	5.24
Chris Paul	2014	LAC	62	4453	3.45	1.77	5.22
Kevin Garnett	2010	BOS	69	4041	0.29	4.84	5.13
Jason Kidd	2005	NJN	66	4710	3.37	1.74	5.11
Kevin Garnett	2000	MIN	81	6293	2.20	2.89	5.10
Chris Paul	2013	LAC	70	4489	4.56	0.53	5.10
Shaquille O’Neal	2005	MIA	73	4815	3.41	1.68	5.09
Kevin Garnett	2011	BOS	71	4276	-0.03	5.11	5.08
Michael Jordan	1997	CHI	82	5928	3.22	1.83	5.06
Mike Conley	2013	MEM	80	5168	3.29	1.77	5.06
Andre Iguodala	2014	GSW	63	4145	1.20	3.86	5.06
Tim Hardaway	1997	MIA	81	5917	2.67	2.38	5.05
LeBron James	2017	CLE	74	5655	4.79	0.25	5.05
Mookie Blaylock	1999	ATL	48	3224	2.87	2.18	5.05
Terry Mills	1997	DET	79	3592	3.31	1.72	5.03
Tim Duncan	2006	SAS	80	5188	0.99	4.03	5.02
Andrei Kirilenko	2004	UTA	78	5363	3.08	1.92	5.00
Dwight Howard	2011	ORL	78	5630	1.08	3.91	4.99
Tim Hardaway	1998	MIA	81	5705	4.00	0.97	4.97
John Stockton	2002	UTA	82	4957	2.68	2.27	4.95
Dirk Nowitzki	2002	DAL	76	5731	3.99	0.93	4.93
Kevin Garnett	2013	BOS	68	3899	-0.40	5.31	4.90
Kevin Garnett	1999	MIN	47	3467	1.49	3.40	4.89
Dwight Howard	2010	ORL	82	5469	2.43	2.42	4.84
Kevin Garnett	1998	MIN	82	6313	1.85	2.96	4.81
Alonzo Mourning	1997	MIA	66	4332	0.88	3.93	4.81
Dikembe Mutombo	2000	ATL	82	5750	-0.18	4.97	4.78
Karl Malone	1998	UTA	81	5788	4.66	0.11	4.77
Manu Ginobili	2006	SAS	65	3510	3.12	1.65	4.77
Davis Bertans	2019	SAS	76	3454	3.04	1.73	4.77
Dirk Nowitzki	2005	DAL	78	5962	2.95	1.76	4.71
Andrei Kirilenko	2006	UTA	69	4826	2.13	2.59	4.71
Stephen Curry	2018	GSW	51	3575	4.83	-0.12	4.71
Ben Wallace	2004	DET	81	5726	-0.13	4.80	4.67
Dirk Nowitzki	2008	DAL	77	5319	3.99	0.67	4.66
Dwight Howard	2012	ORL	54	3891	0.82	3.83	4.65
Draymond Green	2015	GSW	79	5220	1.57	3.06	4.64
Scottie Pippen	1997	CHI	82	5924	3.38	1.25	4.63
David Robinson	2000	SAS	80	4867	1.14	3.48	4.62
Steve Nash	2011	PHX	75	5030	4.22	0.38	4.61
LaMarcus Aldridge	2018	SAS	75	4955	2.05	2.56	4.61
John Stockton	1998	UTA	64	3525	2.80	1.82	4.61
Nikola Jokic	2016	DEN	80	3488	0.25	4.36	4.61
Nick Collison	2013	OKC	81	3150	2.15	2.46	4.61
Joel Embiid	2019	PHI	64	4672	2.30	2.29	4.60
Lamar Odom	2009	LAL	78	4640	1.45	3.15	4.59
Amir Johnson	2013	TOR	81	4439	2.67	1.91	4.57
Vlade Divac	1999	SAC	50	3594	1.29	3.25	4.54
Ray Allen	2003	SEA	76	5456	3.74	0.79	4.53
Kyle Lowry	2016	TOR	77	5571	2.63	1.88	4.50
Mike Conley	2019	MEM	70	4782	3.36	1.14	4.50
Mookie Blaylock	1997	ATL	78	5603	3.36	1.13	4.49
Chris Paul	2011	NOH	80	5438	3.27	1.22	4.49
Eddie Jones	2000	CHH	72	5581	1.94	2.53	4.48
Jason Kidd	2004	NJN	67	4765	2.80	1.68	4.48
Dwyane Wade	2012	MIA	49	3184	3.99	0.47	4.47
Chris Paul	2012	LAC	60	4116	3.34	1.12	4.46
Russell Westbrook	2016	OKC	80	5661	3.88	0.57	4.45
Steve Nash	2009	PHX	74	5109	4.86	-0.41	4.45
Damian Lillard	2019	POR	80	5984	3.99	0.45	4.44
Vlade Divac	1998	CHH	64	3441	1.56	2.89	4.44
Grant Hill	1998	DET	81	6193	2.74	1.69	4.43
Danny Green	2019	TOR	80	4667	3.23	1.19	4.43
Lamar Odom	2011	LAL	82	5077	1.59	2.83	4.42
Gary Payton	2000	SEA	82	6769	3.58	0.82	4.40
Kobe Bryant	2010	LAL	73	5570	3.69	0.71	4.40
Lamar Odom	2010	LAL	82	5127	1.05	3.34	4.40
John Stockton	2003	UTA	82	4337	2.69	1.72	4.40
Kevin Garnett	2006	MIN	76	5537	2.03	2.36	4.39
Rasheed Wallace	2000	POR	81	5436	1.01	3.38	4.39
Jeff Hornacek	1998	UTA	80	4754	3.77	0.61	4.39
John Stockton	2000	UTA	82	4696	1.94	2.43	4.37
Reggie Miller	1998	IND	81	5293	3.81	0.52	4.33
Tim Duncan	2008	SAS	78	4940	1.65	2.68	4.33
Paul Millsap	2012	UTA	64	4060	2.14	2.19	4.33
Grant Hill	1999	DET	50	3444	3.11	1.21	4.33
Nikola Jokic	2018	DEN	75	4983	2.95	1.37	4.32
Tim Duncan	2012	SAS	58	3192	0.48	3.82	4.30
Metta World Peace	2008	SAC	57	4376	1.62	2.67	4.29
Shawn Bradley	2001	DAL	82	3860	-0.46	4.75	4.29
Patrick Ewing	1997	NYK	78	5586	0.50	3.78	4.28
Kawhi Leonard	2015	SAS	64	4029	1.57	2.71	4.28
Chris Paul	2016	LAC	74	4935	3.19	1.08	4.27
James Harden	2015	HOU	81	6056	5.13	-0.88	4.25
LeBron James	2019	LAL	55	4218	2.59	1.66	4.25
Chris Paul	2017	LAC	61	3889	2.61	1.64	4.25
Dwyane Wade	2011	MIA	76	5471	3.98	0.26	4.24
Tracy McGrady	2003	ORL	75	5868	3.84	0.38	4.23
Steve Nash	2007	PHX	76	5452	4.31	-0.10	4.22
Shane Battier	2006	MEM	81	5196	0.41	3.81	4.22
Jeff Hornacek	1997	UTA	82	5007	3.00	1.21	4.21
Toni Kukoc	1998	CHI	74	4249	1.42	2.78	4.20
Kevin Garnett	1997	MIN	77	5742	0.24	3.95	4.19
Robert Horry	1998	LAL	72	4337	2.32	1.86	4.19
Dwyane Wade	2008	MIA	51	3836	4.03	0.16	4.19
Eric Gordon	2019	HOU	68	4483	3.92	0.25	4.18
Vlade Divac	2000	SAC	82	5104	0.85	3.32	4.17
Kyle Korver	2015	ATL	75	4866	3.12	1.04	4.17
Rashard Lewis	2009	ORL	79	5609	2.16	2.00	4.16
Tim Duncan	2013	SAS	69	4124	0.91	3.25	4.15
Detlef Schrempf	1998	SEA	78	5199	3.64	0.50	4.14
Ray Allen	2010	BOS	80	5481	3.28	0.85	4.13
Ray Allen	2001	MIL	82	6164	3.67	0.46	4.12
Russell Westbrook	2015	OKC	67	4769	4.50	-0.40	4.11
Shane Battier	2007	HOU	82	5767	0.28	3.82	4.10
Steve Nash	2012	PHX	62	3898	3.72	0.37	4.09
Nene	2011	DEN	75	4550	0.69	3.38	4.08
Andre Iguodala	2015	GSW	77	4283	1.66	2.41	4.08
Stephen Curry	2014	GSW	78	5826	3.62	0.45	4.07
Dirk Nowitzki	2014	DAL	80	5292	3.02	1.05	4.07
Alonzo Mourning	2000	MIA	79	5268	1.10	2.96	4.06
Manu Ginobili	2014	SAS	68	3199	3.51	0.54	4.05
Paul Pierce	2008	BOS	80	5524	3.39	0.65	4.04
Chris Paul	2010	NOH	45	3303	3.87	0.17	4.04
Baron Davis	2005	GSW	46	3062	2.85	1.19	4.04
Paul Pierce	2005	BOS	82	5874	2.60	1.44	4.03
Dikembe Mutombo	1999	ATL	50	3314	-1.66	5.70	4.03
Ben Wallace	2006	DET	82	5299	-0.13	4.14	4.01
Rudy Gobert	2017	UTA	81	5246	1.53	2.47	4.00
Kobe Bryant	2001	LAL	68	5460	3.18	0.80	3.97
Eric Bledsoe	2019	MIL	78	4948	1.43	2.53	3.96
Steve Nash	2005	PHX	75	5228	4.92	-0.99	3.94
Arvydas Sabonis	1998	POR	73	4475	1.67	2.27	3.94
Dirk Nowitzki	2006	DAL	81	5774	2.76	1.16	3.93
Rasheed Wallace	2005	DET	79	4979	1.14	2.79	3.93
Kawhi Leonard	2016	SAS	72	4702	0.74	3.19	3.93
Robert Covington	2018	PHI	80	5347	0.98	2.93	3.92
Baron Davis	2004	NOH	67	5097	2.09	1.83	3.92
Grant Hill	2000	DET	74	5697	3.60	0.31	3.91
Tim Duncan	1999	SAS	50	3686	1.83	2.08	3.91
Dirk Nowitzki	2010	DAL	81	5954	2.49	1.41	3.90
Kyle Lowry	2017	TOR	60	4469	2.00	1.90	3.90
Dirk Nowitzki	2001	DAL	82	6192	1.80	2.09	3.89
Scottie Pippen	1998	CHI	44	3166	3.60	0.29	3.89
Dwyane Wade	2007	MIA	51	3767	3.10	0.78	3.88
Greg Ostertag	1997	UTA	77	3425	1.11	2.77	3.88
Tracy McGrady	2002	ORL	76	5729	2.72	1.15	3.87
Gary Payton	1998	SEA	82	5982	3.16	0.70	3.86
Tim Duncan	2009	SAS	75	4712	1.19	2.67	3.86
Vince Carter	2013	DAL	81	4250	2.22	1.62	3.85
Kevin Garnett	2007	MIN	76	5745	1.76	2.08	3.84
Manu Ginobili	2015	SAS	70	3217	2.45	1.38	3.83
Rasheed Wallace	2006	DET	80	5060	1.18	2.63	3.81
Patrick Beverley	2014	HOU	56	3500	2.30	1.51	3.81
Baron Davis	2008	GSW	82	6701	2.96	0.84	3.80
Rasheed Wallace	1998	POR	77	5505	0.81	2.99	3.80
Victor Oladipo	2018	IND	75	5225	1.44	2.36	3.80
Paul Millsap	2013	UTA	78	4560	1.69	2.11	3.80
Chauncey Billups	2009	DEN	79	5506	3.40	0.39	3.79
Vlade Divac	1997	CHH	81	5290	1.79	2.00	3.79
Steve Nash	2006	PHX	79	5679	4.27	-0.50	3.77
Kemba Walker	2018	CHA	80	5669	2.25	1.52	3.77
Bo Outlaw	2000	ORL	82	4718	0.04	3.73	3.77
Robert Horry	2000	LAL	76	3410	1.38	2.38	3.77
Chris Paul	2018	HOU	58	3792	2.61	1.15	3.76
Kyle Korver	2018	CLE	73	3313	2.52	1.23	3.75
Rasheed Wallace	2002	POR	79	5623	1.29	2.45	3.74
Josh Smith	2012	ATL	66	4441	0.64	3.10	3.74
Derek Fisher	2002	LAL	70	3887	2.54	1.20	3.74
Dirk Nowitzki	2009	DAL	81	5972	2.95	0.78	3.73
Tim Duncan	1998	SAS	82	5959	2.13	1.60	3.73
Shawn Marion	2001	PHX	79	5625	0.52	3.20	3.73
Gheorghe Muresan	1997	WAS	73	3568	0.46	3.28	3.73
LeBron James	2007	CLE	78	6124	2.30	1.42	3.72
Kevin Durant	2014	OKC	81	6342	3.50	0.20	3.71
Shaquille O’Neal	2006	MIA	59	3507	1.66	2.05	3.71
Paul Pierce	2004	BOS	80	6131	2.04	1.66	3.70
Andre Miller	2001	CLE	82	5516	2.39	1.31	3.70
Dirk Nowitzki	2007	DAL	78	5343	2.95	0.75	3.69
Dwyane Wade	2006	MIA	75	5643	2.74	0.94	3.68
Bo Outlaw	1998	ORL	82	5422	0.51	3.16	3.68
Omer Asik	2013	HOU	82	4983	0.09	3.59	3.68
Andrew Bogut	2011	MIL	65	4296	-0.87	4.54	3.67
Rasheed Wallace	2003	POR	74	5088	1.34	2.32	3.66
Jusuf Nurkic	2019	POR	72	4171	1.53	2.13	3.66
Andre Miller	2012	DEN	66	3569	2.93	0.72	3.66
Damian Lillard	2017	POR	75	5511	4.65	-1.00	3.65
Kawhi Leonard	2017	SAS	74	4891	3.05	0.60	3.65

• Data is from NBA.com; RAPM is prior-informed (regresses toward prior year’s data point rather than zero)