What is NBA Real Plus Minus?

In the NBA real plus minus (rpm) is a metric devised by ESPN with the goal of determining the relative value of NBA players. Countless, countless hours are spent by analysts, TV personalities, and team employees arguing and trying to determine the value of individual players league wide. For teams, measuring the quality of an individual player is the holy grail of basketball analytics; accurately measuring player quality will allow a team to expertly maneuver trades and acquisitions in order to create the best team possible.

But for us as fans, half the fun of basketball conversations is arguing over which players are better than one another. Mostly, this is done by the eye test: I probably think player A is better than player B because I’ve watched each play enough that my gut tells me it is true. The real plus minus NBA metric is a statistically driven approach designed to assign a single number to players that captures their overall quality. Comparing two players then simply boils down to who has the better real plus minus.

Unfortunately, though, real plus minus is complicated because it was designed by statisticians. In this article, I will try to pair an understandable explanation of how the statistic is computed with a graphic showing how real plus minus builds upon other simpler statistics. Hopefully this approach will demystify RPM.

Lastly, RPM is not without its flaws – just as any metric for any real world quality. So to conclude this article, I’ll include a discussion of RPM’s flaws and criticisms and try to either explain how they may be addressed or why they are difficult to fix.

2023 Top 10 NBA Real Plus Minus

First, to look at how good a stat is, it can be helpful to look at how it does identifying the best NBA players. In 2023, the top 10 players in RPM in the NBA are

1. Joel Embiid, 10.5
2. Jayson Tatum, 9.01
3. LeBron James, 7.63
4. Nikola Jokic, 7.39
5. Luka Doncic, 7.25
6. Damian Lillard, 7.08
7. Kyrie Irving, 7.08
8. Anthony Davis, 6.95
9. Evan Mobley, 6.76
10. Franz Wagner, 6.71

Through the top 5, this list is pretty indisputable. Joel Embiid and Nikola Jokic are the two in consideration for MVP. Lebron is an all time great still playing well. Tatum and Doncic are probably the two best young stars in the league. After this, the list gets a little more interesting but still seems to identify some really good players.

The Goal of Real Plus Minus

One reason sports are so compelling is because they pair simplicity with complexity. Most games are pretty simple. Get the basketball through the hoop. Move the football as far forward as possible. Hit the tiny dimpled ball into an exceedingly small hole with a crooked metal stick (don’t worry we put traps and tricks along the way. And you have to wear pants).

Though the rules in sports are often extremely simple, the complexity in sports is enough that the business of talking about sports is worth hundreds of billions of dollars. Even better, sometimes the simplest questions generate the most discussion. In particular, year after year in every sport, the most common debate is whether one player is better or worse than another.

Tackling this question necessarily involves some combination of box score statistics, team success, and the eye test. Let us narrow our focus to the NBA. The goal of the game is simple: score more points than your opponent. However, is a player who averages 20 points and 10 rebounds better or worse than a guy who averages 30 points and 5 rebounds? Even worse, a player can affect a play significantly without recording any box score data. For example, sometimes a great outlet pass is the key to a successful fast break. Sometimes a well-set pick leaves a teammate wide open for easy points.

Because of the complexity of play but the simplicity of the goal (score more points than your opponent), many analysts have searched for a holy grail metric which measures how much an individual player contributes to scoring more points than your opponent. This is exactly the goal of RPM.

Real plus minus is split into two components: offensive RPM and defensive RPM. Offensive real plus minus is meant to measure how many more points your team will score per 100 possessions with you on the court versus a league-average player. Defensive RPM is how many fewer points your team allows on defense per 100 possessions with you on the court versus a league-average player. Real plus minus is then offensive and defensive plus minus added to one another.

RPM is meant to directly measure how much individual players help teams win. If it achieved this goal perfectly, there would be no more debate about which players were the best – we would know. But, we can’t directly measure how much individual players help teams win so we have to estimate that number. In the remainder of this article, we’ll go through the RPM calculation to show you how the statisticians behind the metric designed it to be as close as possible to the truth.

How is Real Plus Minus Calculated / Real Plus Minus Formula

Real plus minus is like an ogre; it has layers. RPM starts from a simple box-score statistic, plus minus, and builds upon this idea and adjusts this idea until it accurately captures a player’s impact on the game. So, we’ll start at the innermost layer and work towards the outside all the while explaining why the addition of each layer is necessary.

The innermost layer, the core, the kernel of real plus minus is your basic +/- stat shown in box scores. Plus minus for an individual player is nothing more than how many points their team scored while they were on the court minus how many points the opponent scored while they were on the court. For example, a plus minus of +8 means that during my minutes, my team outscored my opponent by eight points. The general idea is that if a player has a large positive plus minus, then the minutes and effort that player contributed had a net positive effect on that team’s shot at winning the game.

However, there are many, many reasons one cannot use plus minus alone to evaluate which players are good and which are not. The two foremost of these are:

1. Plus minus does not measure individual player’s contributions to the team. Rather, it measures how well the team performed as a whole when that player was on the court.
2. Plus minus does not take into account the quality of the opponent.

Let’s address the first point first. In the 2015-2016 season, the leader in plus/minus per game was Draymond Green. Draymond Green’s +/- was an astounding 13.2 points/game. That means that in every game, the Warriors outscored their opponents by 13.2 points in the minutes that Draymond was on the court. However, Draymond was decidedly not the best player in the NBA that year. He never has been. He wasn’t even the best or second best player on his own team. Rather, his plus minus was so high because he played on an all time great team. The Warriors dominated everyone that year. Thus, Draymond Green’s plus minus was artificially inflated largely due to the effect of his teammates blitzing the league.

The second point above is quite a bit less significant in the NBA because everyone’s strength of schedule tends to largely even out over the course of the year. However, consider each team’s 6-10th best players. These guys come into the game to give the starters rest. They often play against the other team’s 6-10th best players. Thus, if my #8 guy has a +/- of +1, all that means is that he is one point better than the other team’s bench players. This is significantly less impressive than a starter with a +/- of +1 that has to go against the best of the best week in and week out.

The next layer in developing real plus minus accounts for these two deficiencies.

Adjusted Plus Minus

The biggest problem with using vanilla plus minus is that it doesn’t take into account the effects of the other players on the court. Adjusted Plus Minus tries to account for this by adjusting the +/- values to the quality of the other players on the court. This is the second layer in our real plus minus onion.

Here is an example. DeAndre Jordan and Kevin Durant had a lot of overlapping minutes last year. In these minutes, the Nets probably outscored their opponents by a fairly significant margin. However, most of this dominance should be attributable to Kevin Durant, not to DeAndre Jordan. If we only used raw plus minus statistics, DeAndre Jordan will look like a pretty good player. However, after we ‘factor in’ the fact that Kevin Durant is probably the cause for this success, DeAndre Jordan’s rating – what we’ll eventually call adjusted plus minus – should be significantly lower.

OK. But that was subjective. We ‘knew’ Kevin Durant was better because we know he is better. How is a computer supposed to figure that out? A simplified version is to look at DeAndre Jordan’s +/- with Kevin Durant on the floor and his +/- with Kevin Durant off the floor and compare these two values. Interestingly, though, the difference between these two values doesn’t tell us anything about DeAndre Jordan’s value to his team, but rather it helps us evaluate how good Kevin Durant is. If DJ + KD was a plus 8 on the floor but DJ + no KD was a -1 on the floor, well that actually means that Kevin Durant was 9 points better than whoever replaced him.

Then, looking at all the different lineup combinations throughout the year should give a pretty good idea of just how much each individual player contributes.

There is a problem, though. If two players play 99% of their minutes together, it can be very difficult to figure out the difference between these two players’ qualities. In the extreme, if 100% of Kevin Durant’s minutes are shared with DeAndre Jordan’s minutes, then there is actually no way to separate out the effect of KD versus DJ. While players don’t share all their minutes with each other, coaches often tend to rotate guys in and out together so that most of their minutes will be shared. The more this happens, the more data we need to figure out the truth value of each player.

The next step on our journey towards understanding real plus minus addresses this issue induced by players playing many of their minutes together. First, a statistical aside.

Regularization and Priors

The third layer, and really the final significant layer, in working towards real plus minus is “regularized adjusted plus minus” (RAPM). I’ll take a quick detour to discuss what mathematicians and statisticians mean when they talk about things being regularized or regularization. If you don’t want to understand the math that goes into this step along the way to real plus minus, you can skip ahead to the next section and not miss anything.

In the simplest, most universal sense, regularization is the process of taking a model based on data but adding in one’s own assumptions that aren’t based on observed data. Weather predictions provide a sufficiently simple example.

If it is 100 degrees out today but the average for this time of year is 75, I may be fairly certain that tomorrow it will be close to 100. But as the days go on, I expect the temperature to return closer to 75 degrees. To do this, I have regularized my observations with prior knowledge.

Tomorrow I might predict a high of 85 degrees because I have combined observed data (it is 100 degrees today) and historical trends (the average is 75) to come up with a best guess for what is going to happen. Maybe the next day I’ll predict it is 81 degrees. Regularization is the process of combining expectations and knowledge about how a system works with real, observed data to make the most educated guess possible. The mathematical formulation of my expectations and my knowledge is what we call a prior. Bayesian statistics is all about how to incorporate priors into observed data to make educated guesses about future outcomes.

Let’s talk about baseball as a sports example. Nick Castellanos before this season had a career average somewhere around .275. This year, he came out hot and hit somewhere near .340 in the first half. If I had to predict what his batting average would be in the second half what should I guess?

There are really three options. First, you guess .275 because that is his all time batting average and he should probably regress to the mean right? Well, this probability isn’t the best guess because you aren’t taking into account his increased success so far this year; maybe he actually did get a little better as a hitter and his first half performance is evidence towards this fact.

Second, you could guess .340 because when you look at this year’s data and extrapolate forward, that is what you get. This probably isn’t the best guess because this is so wildly outside expectations for how much a guy can improve in one year. This is the unregularized, data-only approach.

To me, the correct approach is to probably guess somewhere in the high .200’s or low .300’s. This guess takes observations from this year (he is hitting better than in the past) but regularizes (tempers) it with prior knowledge (this guy probably isn’t actually a .340 hitter).

In the NBA, when we study real plus minus, we will use regularization to solve this DeAndre Jordan/Kevin Durant problem. We want to tell the model that Kevin Durant is more responsible for the success than DeAndre Jordan is in their shared minutes. This leads to regularized adjusted plus minus.

Regularized Adjusted Plus Minus

The third layer going into development of real plus minus is adding priors about how good players are.

xRAPM, or expected regularized adjusted plus minus (it is beginning to be a mouthful even for me), regularizes adjusted plus minus by guessing that players tend to be about as good as they were the year before. If we guess that Durant is better than Jordan, then the model will come up with numbers that tend to match this data.

Regularized adjusted plus minus is essentially the last step in building up real plus minus. However, there are a few small changes that ESPN made to xRAPM to improve things just a little bit more. Maybe predicting a player is exactly as good as he was last year isn’t the most accurate thing we could do. For example, old players will likely be worse than the previous year. Younger players will tend to get better. Real plus minus only adds one more layer to this onion.

Real Plus Minus

Real plus minus is essentially xRAPM with a more sophisticated prior and split into offensive and defensive components. xRAPM uses the previous year’s ranking as a best guess for the current year’s ranking. However, we can likely improve upon this by incorporating other aspects. One thing we know ESPN does is they include age to predict whether a player is likely to get better or worse than a previous year. Other aspects of ESPN’s real plus minus are not public domain knowledge, but we do know that they essentially try to improve their ‘best guess’ for how good a player is.

Real plus minus also splits a player’s rating into an offensive and a defensive efficiency. I have no idea if splitting things in this way leads to a more accurate statistic, but it definitely can lead to more insight into players. For example, in 2020-2021 Steph Curry had by far the best offensive RPM but was league-average on defense. This is a valuable data point and can provide a lot to many NBA debates.

Criticisms of Real Plus Minus

So, that is really all there is to real plus minus. Start with a traditional box score plus minus, account for who is on the court, and add in some fancy guesswork to expand the size of your dataset and call it a day. It is a pretty good idea, it is essentially the technique I used to rank NBA teams overall. But, it is of course not perfect. Real plus minus certainly has some flaws.

First and foremost, real plus minus implicitly assumes that player quality is additive. However, after watching the debacle that was the 2019-2020 Philadelphia 76ers we know this isn’t true. On their own, Al Horford, Joel Embiid, and Ben Simmons are significantly above average players. They are three all-star caliber players. But that team was way worse than the sum of these three parts. The problem was that each of these three guys had similar strengths and weaknesses so they didn’t complement each other well. So, the three of them together didn’t nearly perform as well as their skillsets would predict. This is kind of like saying analytics can’t capture team chemistry.

A second issue I would like to address is much less important for the casual fan but more important for front offices. Real plus minus does not tell you how much to pay a player; it does not tell you what a player is worth. There is a weird phenomenon in any sport with limited roster sizes: a +10 asset is not worth the same as two +5 assets. If that were the case, then in your fantasy football league, you should trade Mahomes for Big Ben + Tua. Nobody is going to do that, obviously, because you can only start one quarterback.

Similarly, in the NBA only 5 players are going to be on the court at any given time. If you need your team to match the 2015-2016 Warriors +10 points differential, then a guy who is only worth +1 is actually hurting your chances at winning the league because he ‘soaks up’ a valuable roster slot. If you can get +8 out of any individual player, then your team has a good shot at being elite. So, real plus minus is best used to determine relative rank between players instead of trying to determine contract values and roster construction.

Overall, though, real plus minus is among the most accurate metrics we have to capture NBA players’ contributions to their teams. While it has its flaws, real plus minus is about as good as you can do with just box score statistics.

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