Player Efficiency Rating (PER): Good Stat or Bad Stat?
Much like Total QBR for quarterbacks, player efficiency rating is an attempt to put a single number on the quality of a basketball player. If you’ve been following along, we’ve already discussed how real plus minus attempts to do just that: assign a number to a basketball player’s skill. However, unlike real plus minus which measures quality by looking at how the team performs when a player is on/off the court, player efficiency rating tries to answer the same question by looking at an individual player’s box score statistics to see what they accomplished on their own.
Really, these statistics are two sides of the same coin. Good players will both help their team out-score their opponents (as measured by real plus minus) and they will accrue meaningful box score statistics that impact the outcome (as measured by PER).
In this article, we’ll show how player efficiency rating was developed and how it is calculated. Then, we’ll discuss its merits and deficiencies. First, let’s talk about the general method.
The Goal of PER
Outside of all the discussion about pace and per minute statistics in the NBA, the game is really about efficiency per possession. One team gets the ball, then the other team does. Each team gets the ball nearly an identical amount of times. So, basketball as a game is all about how efficiently you use your possessions. There is no mechanism in the game to give one team more possessions than another (assuming, of course, that one considers an offensive rebound as continuing a possession rather than an entirely new possession to begin with).
Thus, the score of the game is entirely dependent on how efficient each team is on their own possessions. The goal of every player, then, should be to maximize their team’s points per possession and minimize their opponents’ points per possession.
PER has the goal of rating players by how efficient they are on the possessions they contribute to. This is the reason for the name player efficiency rating; the efficiency in question is a per possession efficiency. Box score statistics can all be translated into quantities that measure how a player impacts either their team’s efficiency or their opponent’s efficiency. If you make a 3 pointer, that constitutes an efficient possession because you score more points than on an average possession. Record a steal? Great, you’ve ended one of your opponent’s possessions so that their efficiency decreases.
The calculation – the formula – for player efficiency rating is extremely complicated. It is in fact complicated enough that I won’t include it here (see, for example, the Wikipedia page for PER for the precise formula). However, each component of the formula attempts to capture in some way how a player impacts their team’s efficiency or their opponent’s efficiency.
Player Efficiency Rating in 3 Parts
The way I like to understand player efficiency rating is as a sum of three distinct parts, each measuring a different way a player changes his team’s offensive or defensive efficiency. These three parts are:
- Decreasing your team’s efficiency by losing possession
- This happens through turnovers, missed shots, missed free throws, and offensive fouls.
- Increasing your team’s efficiency by gaining possession
- This happens through steals, rebounds, and blocks
- Increasing your team’s efficiency by scoring and creating points
The most important thing to understand before going further into each of these three parts is that every possession has an average number of points scored, which we’ll call PPP (points per possession). This quantity is important to understand because it helps compare possession based events (steals, turnovers, rebounds, etc.) to scoring based events. For the last few years, the average points per possession have been about PPP=1.1. So, if you turn the ball over, we can now measure the impact of that event in ‘points’; a turnover loses your team about 1.1 points.
In the next three sections we’ll talk about increasing and decreasing efficiency in terms of ‘points lost/gained’ by an action. We begin by developing an intermediate stat which is ‘unadjusted’ player efficiency rating.
Decreasing Efficiency by Losing Possession.
This one is pretty straightforward, if you cause your team to lose possession, you get the credit for the lost 1.1 PPP your team has accrued. If you turn the ball over, you’re on the hook for the full 1.1 points. If you miss a shot, though, you’ve actually lost a bit less because there is a chance that your team gets an offensive rebound. So, if you’re docked 1.1 points for a turnover, you’re docked a bit less for a missed shot. The third way you can lose your team the possession is by missing free throws. Missing free throws again has a slightly different value than a turnover or a missed shot because, for example, missing the second of two free throws is not truly that detrimental whereas missing the first of a one-and-one is essentially equivalent to a missed field goal.
The player efficiency rating formula combines the value of each of these three negative events together. This sum constitutes the first part of the player efficiency rating. The first part of our flow chart which is due to these considerations is shown below.
Increasing Efficiency by Gaining Possessions
Almost identically, any time your actions cause your team to gain possession, you should be rewarded for that fact. Recording a steal is the defensive equivalent of a turnover. So, steals are worth 1.1 points because you decrease your opponents expected offensive performance by that amount. The other ways you can gain possession for your team are with a block or an offensive/defensive rebound.
Again, much like before, these other three effects have less than the full 1.1 points of value. Offensive and defensive rebounds are adjusted down from the full 1.1 points by an amount corresponding roughly to the probability that one of your teammates would have recorded that same rebound if you hadn’t. Think of it like this: usually on defensive rebounds there are four defenders around the ball that all could have gotten the DRB if they wanted but only one actually gets that rebound. You shouldn’t get as much credit as for a steal for that. Blocks are also adjusted down from 1.1 points slightly because there is no guarantee that your team retains possession from a block.
The value of each of these events is estimated using offensive and defensive rebounding percentages and a value is assigned to each. This leads to the second component of player efficiency rating: creating possessions.
Efficiency via Scoring and Assists
The third component of player efficiency rating is all about putting points on the board. Whether you’re helping your team through actual scoring or hurting it via giving up FT attempts via fouls, actions with a direct impact on the score will be captured in this efficiency metric.
If you make a field goal, you get credit for those two points minus a small amount which attempts to capture the fact that your teammate’s actions – an assist, a good screen, etc. – also contributed to those two points. If you make a three or a free throw, you get credited similarly for some but not all of the points you put on the board.
The last box score stat that goes into the ‘scoring points’ component of PER is the personal fouls. If you record a foul on offense, it is a turnover. On defense, you have a chance of sending your opponents to the line which may lead to points. Player efficiency rating therefore contains a factor which estimates how much impact your fouls have on the game. All these taken together leads to the third component: scoring points.
Final Adjustments
All these measures combine to give one number which is more or less an estimate of how many points you contributed to your team relative to an average player given the same opportunities. But, the goal of PER is to deliver a metric which measures a player’s overall efficiency so the current form needs to be adjusted a bit.
First, we take the unadjusted player efficiency rating and divide by minutes played so that players with more minutes don’t automatically ‘look better’. This transforms the metric into a rate statistic which is kind of like ‘points added per minute’.
Second, the per minute number is adjusted for your team’s pace. Teams that get more possessions over the course of a normal 48 minute game will lead to naturally lead to more accrued stats and, as a result, their players will look better. To counteract this, the ‘points added per minute’ is transformed into ‘points added per possession’ by multiplying by an estimate of a team’s possessions per minute. At this point, the statistic truly could be called player efficiency rating because it is a measure of how many points each player adds to their team’s value per possession.
However, one final step is taken so that PER can be compared meaningfully across years. In a given year, everyone’s PER is adjusted by multiplying by some constant so that the league-wide average PER is 15.
At the end of the road, we’re left with a final stat that looks like this:
My Criticisms
Don’t get me wrong, I think PER is basically on the right track and has the potential to be a really good metric. But as it stands, it’s just off in enough ways that I think it squarely misses the mark. So, with apologies to John Hollinger, here are a few shortcomings of PER that I’ve noticed.
1) First and foremost, there is no adjustment for the quality of teammates and opponents. Players on good teams will naturally record much better stats. Naturally, this will lead to them being rated as much more efficient. Similarly, playing against quality opponents will lead to worse stats. While the strength of opponents problem should generally even out over the course of the season, both these factors are worth considering and controlling for.
2) If you look at Wikipedia’s list of highest career player efficiency rating, you’ll notice the top two are the two GOATS, Lebron James and Michael Jordan. After them, though, is a bunch of centers. David Robinson is surely a great all-time player, but nowhere near the 5th best.
Efficiency needs to be measured relative to the difficulty of the action. Centers will take a higher percentage of easy shots and so will be measured as unduly efficient. Centers will also have a significantly larger percentage of rebounds based on positioning alone. To see what kind of value a center provides, the impact they provide by rebounding should be measured against the average center, not against league average. Changing these two factors should lead to a PER that doesn’t inflate the impact of centers nearly as much.
3) Defensive and offensive fouls have a decidedly different impact on the game. Offensive fouls should be the exact same as a turnover (which they are in the box score). However, defensive fouls are much more complex. Sometimes, like on a fast break, a defensive foul is a good play and you shouldn’t be penalized for it. Moreover, the act of drawing a foul on defense should be rewarded in the same way that a steal should. The point being, fouls should be incorporated into the formula more carefully.
4) Players who have a higher usage rate will tend to have a lower efficiency. That is, players who shoot and pass more will likely be less efficient than otherwise because they are forced to perform in sub-optimal settings. If the goal of PER is to actually give a measure on efficiency, then this is not a problem. However, if PER is being used as a substitute for player quality, then some adjustment needs to be made for players with high usage rates.
5) Perhaps I am reading the formula wrong, but I think too many points are being given for a made basket. Currently, making a 2 point shot is given credit for the 2 points minus some small fraction corresponding to the net effect of the teammates. To illustrate my point, let’s ignore that and assume the formula gave the full 2 points for a made field goal.
Missing a field goal, on the other hand, is worth negative 1.1 points, the value associated to ending a possession. Thus, making a two point basket is worth 3.1 points more than missing it. That can’t possibly be correct.
Instead of crediting a player the full two points for making a basket, you should in fact add only 0.9 points. This 0.9 points corresponds to the value added relative to an average possession. The same should be true for three pointers made. Because of this, I believe that the formula incorrectly captures the effect of scoring.
6) The way the stat is normalized to a ‘per possession’ basis is a little strange. The current way is the best possible method if you want to estimate strictly from the box score how many possessions a player was on the court for. However, if we’re going to improve this stat, we should actually go through the play-by-play, count the number of possessions a player is on the court for (the official NBA stats page actually does this), and use that number for the normalization.
7) My smallest complaint is that PER is not really interpretable. This isn’t really a complaint about the stat, just about style. Normalizing to live on a scale with the average being 15 doesn’t really make sense. If I wasn’t told, I wouldn’t know if PER=25 is merely a good player or an all time great player. More standard adjustments like normalizing so that average is 100 or 0 is probably a better idea.
Better yet, if we don’t normalize the stat at all, then there may even be some helpful interpretations like ‘average points added per possession’. As is, the number itself doesn’t really tell you anything. Only the comparison between two players’ efficiency ratings tells you something.
8) Finally, box scores don’t tell the whole story. Players can impact the game without necessarily accruing stats in the bottom line. These types of players (2020-2021 PJ Tucker?) will be dramatically underrated by PER. I don’t have a very good way to fix this, so this is not really a criticism but rather an observation.
If you liked this article discussing the ideas behind the development of advanced sports statistics, check out our other articles discussing different stats:
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