Regularized 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.)

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; standardize across seasons.
• Re-calculate RAPM so that the (weighted) standard deviations are standardized across seasons; do for both offense and defense.

Listed below are the top-250 seasons on record from 1997 to 2019 among players with more than 2,500 possessions during the regular season (more or less 1,250 minutes played).

Player	Year	Team	G	Poss	ORAPM	DRAPM	RAPM
Kevin Garnett	2004	MIN	82	6,150	5.40	4.24	9.64
Kevin Garnett	2009	BOS	57	3,416	2.85	6.52	9.37
Draymond Green	2016	GSW	81	5,974	4.10	5.15	9.24
LeBron James	2011	MIA	79	5,916	6.09	3.07	9.16
LeBron James	2016	CLE	76	5,344	5.74	3.25	8.99
Giannis Antetokounmpo	2019	MIL	72	5,174	4.18	4.80	8.98
LeBron James	2010	CLE	76	5,728	6.81	2.04	8.86
LeBron James	2012	MIA	62	4,533	5.94	2.77	8.71
Tim Duncan	2003	SAS	81	6,058	3.74	4.81	8.55
Stephen Curry	2017	GSW	79	5,690	6.77	1.64	8.40
Kevin Garnett	2008	BOS	71	4,468	2.63	5.74	8.37
LeBron James	2009	CLE	81	5,769	5.84	2.50	8.34
LeBron James	2015	CLE	69	4,889	6.43	1.90	8.33
Tim Duncan	2005	SAS	66	4,134	2.86	5.10	7.96
Kevin Garnett	2005	MIN	82	5,946	4.38	3.34	7.73
Shaquille O’Neal	2001	LAL	74	5,655	6.82	0.73	7.55
Kevin Garnett	2003	MIN	82	6,480	4.19	3.30	7.49
Kevin Durant	2019	GSW	78	5,813	6.21	1.20	7.41
Shaquille O’Neal	2000	LAL	79	6,211	5.46	1.86	7.32
Tim Duncan	2004	SAS	69	4,741	2.62	4.67	7.30
Shaquille O’Neal	2004	LAL	67	4,830	4.67	2.58	7.24
Shaquille O’Neal	1998	LAL	60	4,273	4.96	2.25	7.20
Tim Duncan	2001	SAS	82	6,039	2.59	4.57	7.15
Manu Ginobili	2007	SAS	75	4,006	4.64	2.50	7.14
Dirk Nowitzki	2004	DAL	77	5,792	5.71	1.36	7.07
Shaquille O’Neal	1999	LAL	49	3,286	5.54	1.43	6.97
Alonzo Mourning	1999	MIA	46	3,152	2.79	4.10	6.90
Jrue Holiday	2019	NOP	67	5,212	4.98	1.91	6.89
Dirk Nowitzki	2003	DAL	80	6,118	4.59	2.27	6.86
Dwyane Wade	2010	MIA	77	5,390	6.53	0.29	6.83
LeBron James	2017	CLE	74	5,655	6.27	0.48	6.75
Dirk Nowitzki	2012	DAL	62	4,057	4.36	2.35	6.71
LaMarcus Aldridge	2018	SAS	75	4,955	2.99	3.71	6.70
Tim Duncan	2007	SAS	80	5,125	2.57	4.10	6.67
Stephen Curry	2019	GSW	69	5,058	4.58	2.09	6.66
Tim Duncan	2002	SAS	82	6,324	2.20	4.41	6.61
Dirk Nowitzki	2011	DAL	73	4,909	4.09	2.51	6.60
Stephen Curry	2015	GSW	80	5,534	5.56	1.02	6.58
Chris Paul	2015	LAC	82	5,729	4.66	1.91	6.57
Paul George	2019	OKC	77	6,172	4.28	2.28	6.56
Shaquille O’Neal	2002	LAL	67	4,728	4.07	2.49	6.55
LeBron James	2014	MIA	77	5,634	6.22	0.31	6.53
Alonzo Mourning	1998	MIA	58	3,542	2.73	3.80	6.53
Kevin Garnett	2012	BOS	60	3,583	1.72	4.76	6.48
Dirk Nowitzki	2013	DAL	53	3,332	4.40	2.04	6.44
Christian Laettner	1997	ATL	82	5,703	3.48	2.94	6.42
Stephen Curry	2018	GSW	51	3,575	6.50	-0.13	6.37
Steve Nash	2007	PHX	76	5,452	6.49	-0.12	6.37
Chris Paul	2014	LAC	62	4,453	4.23	2.14	6.37
Steve Nash	2008	PHX	81	5,710	6.37	-0.05	6.33
Manu Ginobili	2005	SAS	74	4,191	3.60	2.68	6.28
Michael Jordan	1997	CHI	82	5,928	4.39	1.88	6.27
Terry Mills	1997	DET	79	3,592	4.49	1.77	6.26
Rasheed Wallace	2004	DET	68	4,428	2.13	4.12	6.25
John Stockton	2001	UTA	82	4,621	3.65	2.59	6.24
Nikola Jokic	2016	DEN	80	3,488	0.70	5.54	6.23
LeBron James	2013	MIA	76	5,549	5.67	0.55	6.22
Shaquille O’Neal	2003	LAL	67	4,991	5.14	1.07	6.22
Metta World Peace	2006	SAC	56	4,255	1.85	4.37	6.22
Mookie Blaylock	1998	ATL	70	5,003	4.17	2.01	6.18
Chris Paul	2017	LAC	61	3,889	3.61	2.55	6.16
Steve Nash	2009	PHX	74	5,109	6.62	-0.47	6.15
Nikola Jokic	2018	DEN	75	4,983	4.12	2.01	6.14
Tim Hardaway	1997	MIA	81	5,917	3.72	2.42	6.14
Andre Iguodala	2014	GSW	63	4,145	1.48	4.64	6.12
Manu Ginobili	2006	SAS	65	3,510	4.34	1.76	6.10
Kyle Lowry	2016	TOR	77	5,571	3.66	2.43	6.09
Rudy Gobert	2017	UTA	81	5,246	2.30	3.78	6.08
David Robinson	1999	SAS	49	2,905	1.71	4.36	6.07
Kevin Garnett	2000	MIN	81	6,293	3.06	2.99	6.04
Russell Westbrook	2016	OKC	80	5,661	5.22	0.80	6.02
Rasheed Wallace	1999	POR	49	2,725	2.02	3.99	6.01
Steve Nash	2010	PHX	81	5,426	6.43	-0.43	6.00
Chris Paul	2013	LAC	70	4,489	5.38	0.62	6.00
Manu Ginobili	2008	SAS	74	4,470	3.73	2.27	6.00
Jason Kidd	2005	NJN	66	4,710	3.97	2.02	5.99
John Stockton	2002	UTA	82	4,957	3.15	2.85	5.99
Mookie Blaylock	1999	ATL	48	3,224	3.90	2.08	5.98
Shaquille O’Neal	2005	MIA	73	4,815	4.01	1.95	5.97
Mike Conley	2013	MEM	80	5,168	3.88	2.09	5.97
Kevin Garnett	2010	BOS	69	4,041	0.33	5.60	5.93
Bo Outlaw	1997	LAC	82	4,224	1.45	4.45	5.90
Scottie Pippen	1997	CHI	82	5,924	4.58	1.31	5.89
Michael Jordan	1998	CHI	82	6,010	4.34	1.54	5.89
Robert Covington	2018	PHI	80	5,347	1.64	4.25	5.89
Dirk Nowitzki	2002	DAL	76	5,731	4.68	1.17	5.85
Davis Bertans	2019	SAS	76	3,454	3.61	2.23	5.84
Kobe Bryant	2008	LAL	82	6,488	5.52	0.28	5.80
Kyle Lowry	2017	TOR	60	4,469	2.87	2.93	5.80
Kevin Garnett	2013	BOS	68	3,899	-0.48	6.28	5.80
Steve Nash	2011	PHX	75	5,030	5.39	0.40	5.79
Chris Paul	2016	LAC	74	4,935	4.36	1.44	5.79
Draymond Green	2015	GSW	79	5,220	1.68	4.10	5.78
Joel Embiid	2019	PHI	64	4,672	2.84	2.93	5.76
Mookie Blaylock	1997	ATL	78	5,603	4.56	1.19	5.75
Andrei Kirilenko	2006	UTA	69	4,826	2.96	2.76	5.72
LeBron James	2008	CLE	75	5,831	4.45	1.22	5.67
Tim Duncan	2006	SAS	80	5,188	1.37	4.31	5.67
Victor Oladipo	2018	IND	75	5,225	2.22	3.42	5.64
Dwight Howard	2010	ORL	82	5,469	2.77	2.80	5.57
Dwyane Wade	2007	MIA	51	3,767	4.66	0.90	5.55
Dwight Howard	2011	ORL	78	5,630	1.37	4.16	5.53
Dirk Nowitzki	2005	DAL	78	5,962	3.47	2.05	5.52
Lamar Odom	2009	LAL	78	4,640	1.96	3.53	5.50
Alonzo Mourning	1997	MIA	66	4,332	1.56	3.95	5.50
Andrei Kirilenko	2004	UTA	78	5,363	3.63	1.86	5.49
Kevin Garnett	1999	MIN	47	3,467	2.25	3.24	5.49
Chris Paul	2011	NOH	80	5,438	4.17	1.30	5.46
Mike Conley	2019	MEM	70	4,782	3.95	1.50	5.46
Kemba Walker	2018	CHA	80	5,669	3.24	2.22	5.46
Gary Payton	2000	SEA	82	6,769	4.79	0.65	5.44
Nick Collison	2013	OKC	81	3,150	2.53	2.90	5.43
Dwyane Wade	2012	MIA	49	3,184	4.90	0.52	5.42
Steve Nash	2006	PHX	79	5,679	5.95	-0.54	5.41
Tim Hardaway	1998	MIA	81	5,705	4.41	0.99	5.40
Jeff Hornacek	1997	UTA	82	5,007	4.12	1.27	5.39
Danny Green	2019	TOR	80	4,667	3.82	1.57	5.39
Chris Paul	2018	HOU	58	3,792	3.69	1.70	5.39
Kyle Korver	2018	CLE	73	3,313	3.58	1.81	5.39
Amir Johnson	2013	TOR	81	4,439	3.15	2.25	5.39
Kevin Garnett	2011	BOS	71	4,276	-0.05	5.44	5.39
Kawhi Leonard	2016	SAS	72	4,702	1.31	4.07	5.38
Dikembe Mutombo	2000	ATL	82	5,750	0.06	5.32	5.38
David Robinson	2000	SAS	80	4,867	1.72	3.65	5.37
Grant Hill	1999	DET	50	3,444	4.19	1.16	5.36
Dwyane Wade	2011	MIA	76	5,471	5.07	0.27	5.35
Kevin Garnett	2006	MIN	76	5,537	2.82	2.52	5.34
Chris Paul	2012	LAC	60	4,116	4.10	1.23	5.33
Eddie Jones	2000	CHH	72	5,581	2.73	2.58	5.32
Kawhi Leonard	2015	SAS	64	4,029	1.68	3.62	5.30
Dirk Nowitzki	2007	DAL	78	5,343	4.43	0.86	5.29
LeBron James	2019	LAL	55	4,218	3.14	2.15	5.29
Damian Lillard	2019	POR	80	5,984	4.62	0.65	5.27
Kevin Garnett	1998	MIN	82	6,313	2.27	2.97	5.25
Draymond Green	2017	GSW	76	5,262	1.18	4.05	5.23
Dwight Howard	2012	ORL	54	3,891	1.01	4.22	5.23
John Stockton	2000	UTA	82	4,696	2.73	2.46	5.20
Karl Malone	1998	UTA	81	5,788	5.07	0.13	5.19
Rashard Lewis	2009	ORL	79	5,609	2.93	2.24	5.18
Kawhi Leonard	2017	SAS	74	4,891	4.14	1.00	5.15
Jimmy Butler	2018	MIN	59	4,325	3.16	1.96	5.13
Eric Bledsoe	2019	MIL	78	4,948	1.92	3.22	5.13
Shawn Bradley	2001	DAL	82	3,860	-0.59	5.71	5.12
Vlade Divac	1999	SAC	50	3,594	2.01	3.10	5.11
Rasheed Wallace	2000	POR	81	5,436	1.56	3.53	5.10
LeBron James	2007	CLE	78	6,124	3.46	1.63	5.09
Dirk Nowitzki	2006	DAL	81	5,774	3.84	1.24	5.08
Lamar Odom	2010	LAL	82	5,127	1.20	3.87	5.07
Chauncey Billups	2009	DEN	79	5,506	4.62	0.44	5.06
Jrue Holiday	2018	NOP	81	6,242	1.92	3.14	5.06
John Stockton	1998	UTA	64	3,525	3.21	1.83	5.05
Kobe Bryant	2010	LAL	73	5,570	4.21	0.82	5.04
Kevin Garnett	2007	MIN	76	5,745	2.65	2.39	5.04
Paul Millsap	2012	UTA	64	4,060	2.63	2.41	5.04
Lamar Odom	2011	LAL	82	5,077	2.02	3.01	5.02
John Stockton	2003	UTA	82	4,337	2.89	2.12	5.01
Dirk Nowitzki	2008	DAL	77	5,319	4.24	0.76	5.00
Ray Allen	2003	SEA	76	5,456	4.02	0.98	5.00
Baron Davis	2007	GSW	63	4,616	3.69	1.32	5.00
Andre Iguodala	2015	GSW	77	4,283	1.77	3.23	5.00
Stephen Curry	2016	GSW	79	5,779	5.06	-0.07	4.99
Stephen Curry	2014	GSW	78	5,826	4.44	0.54	4.98
Steve Nash	2012	PHX	62	3,898	4.57	0.41	4.97
Dirk Nowitzki	2014	DAL	80	5,292	3.70	1.27	4.97
Manu Ginobili	2014	SAS	68	3,199	4.31	0.65	4.96
Eric Gordon	2019	HOU	68	4,483	4.55	0.40	4.95
James Harden	2018	HOU	72	5,223	5.47	-0.55	4.92
Jason Kidd	2004	NJN	67	4,765	3.29	1.63	4.92
Karl Malone	1997	UTA	82	5,772	4.77	0.14	4.91
Patrick Patterson	2017	TOR	65	3,219	2.62	2.29	4.91
Tyus Jones	2018	MIN	82	2,985	3.23	1.68	4.90
Patrick Ewing	1997	NYK	78	5,586	1.10	3.80	4.90
Tim Duncan	2013	SAS	69	4,124	1.06	3.84	4.90
Grant Hill	2000	DET	74	5,697	4.81	0.08	4.89
Dirk Nowitzki	2009	DAL	81	5,972	4.01	0.87	4.88
Vlade Divac	1998	CHH	64	3,441	1.98	2.90	4.88
Ray Allen	2001	MIL	82	6,164	4.35	0.52	4.87
Grant Hill	1998	DET	81	6,193	3.16	1.71	4.86
Nick Collison	2014	OKC	81	2,726	2.35	2.51	4.86
Karl-Anthony Towns	2018	MIN	82	5,852	4.37	0.48	4.85
Tim Duncan	2008	SAS	78	4,940	1.75	3.08	4.83
Ekpe Udoh	2012	MIL	61	2,521	1.10	3.73	4.83
Vlade Divac	2000	SAC	82	5,104	1.35	3.47	4.82
Shane Battier	2007	HOU	82	5,767	0.42	4.40	4.82
Jeff Hornacek	1998	UTA	80	4,754	4.19	0.63	4.81
Dwyane Wade	2006	MIA	75	5,643	3.81	1.00	4.81
Andrei Kirilenko	2005	UTA	41	2,529	2.93	1.87	4.80
Tim Duncan	2012	SAS	58	3,192	0.59	4.21	4.80
Kobe Bryant	2009	LAL	82	5,911	4.70	0.08	4.78
Metta World Peace	2008	SAC	57	4,376	1.71	3.07	4.78
Clint Capela	2016	HOU	77	3,067	2.45	2.31	4.76
Reggie Miller	1998	IND	81	5,293	4.22	0.53	4.75
Jusuf Nurkic	2019	POR	72	4,171	2.02	2.73	4.75
Baron Davis	2005	GSW	46	3,062	3.35	1.39	4.74
Alonzo Mourning	2000	MIA	79	5,268	1.67	3.07	4.74
Kevin Garnett	1997	MIN	77	5,742	0.78	3.97	4.74
Ray Allen	2010	BOS	80	5,481	3.75	0.98	4.73
Tim Duncan	2016	SAS	61	3,030	0.75	3.98	4.73
Kyle Korver	2015	ATL	75	4,866	3.33	1.40	4.72
Paul Pierce	2005	BOS	82	5,874	3.05	1.67	4.72
Damian Lillard	2017	POR	75	5,511	6.09	-1.38	4.70
Kobe Bryant	2001	LAL	68	5,460	3.76	0.93	4.70
Vlade Divac	1997	CHH	81	5,290	2.65	2.05	4.70
Joe Ingles	2019	UTA	82	5,467	1.75	2.94	4.70
Jeff Foster	2005	IND	61	2,974	0.86	3.84	4.70
Arvydas Sabonis	1999	POR	50	2,531	2.19	2.49	4.68
Nikola Jokic	2017	DEN	73	4,264	4.24	0.42	4.67
George Hill	2017	UTA	49	2,985	1.42	3.23	4.65
Steve Nash	2005	PHX	75	5,228	5.79	-1.15	4.64
Patrick Patterson	2016	TOR	79	3,957	4.30	0.34	4.64
Tracy McGrady	2002	ORL	76	5,729	3.19	1.45	4.64
Tim Duncan	1999	SAS	50	3,686	2.65	1.99	4.64
Toni Kukoc	1998	CHI	74	4,249	1.84	2.79	4.64
Greg Ostertag	1997	UTA	77	3,425	1.84	2.81	4.64
Russell Westbrook	2014	OKC	46	2,933	3.60	1.03	4.63
Joel Embiid	2018	PHI	63	4,034	2.87	1.76	4.63
Patrick Beverley	2014	HOU	56	3,500	2.82	1.81	4.63
Shane Battier	2006	MEM	81	5,196	0.56	4.07	4.63
Chris Paul	2010	NOH	45	3,303	4.43	0.19	4.62
Robert Horry	1998	LAL	72	4,337	2.74	1.88	4.62
Tim Duncan	2009	SAS	75	4,712	1.62	3.00	4.62
Dirk Nowitzki	2001	DAL	82	6,192	2.12	2.49	4.61
Brook Lopez	2019	MIL	81	5,069	1.07	3.54	4.61
Rudy Gobert	2018	UTA	56	3,700	-0.11	4.72	4.61
Tracy McGrady	2003	ORL	75	5,868	4.13	0.47	4.60
John Stockton	1999	UTA	50	2,617	2.69	1.91	4.60
Al-Farouq Aminu	2019	POR	81	4,828	2.72	1.88	4.59
Jimmy Butler	2017	CHI	76	5,654	2.02	2.58	4.59
Nick Collison	2012	OKC	63	2,546	1.84	2.75	4.59
Rasheed Wallace	2005	DET	79	4,979	1.34	3.25	4.59
Rasheed Wallace	2002	POR	79	5,623	1.51	3.08	4.58
Detlef Schrempf	1998	SEA	78	5,199	4.06	0.51	4.57
Chris Paul	2009	NOH	78	5,607	4.89	-0.34	4.55
Kevin Durant	2014	OKC	81	6,342	4.30	0.24	4.54
Vince Carter	2013	DAL	81	4,250	2.62	1.91	4.53
Toni Kukoc	1997	CHI	57	3,162	3.24	1.28	4.52
Shaquille O’Neal	2006	MIA	59	3,507	2.30	2.18	4.49
David Robinson	2003	SAS	64	3,189	-0.07	4.56	4.49
Ben Wallace	2004	DET	81	5,726	-0.16	4.64	4.49
Manu Ginobili	2011	SAS	80	4,880	3.95	0.53	4.48
Derek Fisher	2002	LAL	70	3,887	2.97	1.51	4.48
Paul Millsap	2013	UTA	78	4,560	1.99	2.49	4.48
Nene	2011	DEN	75	4,550	0.88	3.60	4.48
Dirk Nowitzki	2010	DAL	81	5,954	2.84	1.63	4.47
Dwyane Wade	2008	MIA	51	3,836	4.28	0.18	4.46
Manu Ginobili	2015	SAS	70	3,217	2.61	1.85	4.46

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