Welcome back to Hack a Stat! This new chapter of Learn a Stat is dedicated to an advanced statistic that is as interesting as dangerous: the Player Efficiency Rating, a.k.a. PER.

Introduction

No matter how hard man tries, he has never managed to resist the temptation to classify anything in the simplest way possible: he knows that it is impossible to reduce everything to a single rating scale, he knows that it is difficult to consider all aspects and concentrates in a single value, he knows that information will be lost. But he is stronger than him, in the end, he will try to find a single measure to evaluate something. The PER is a perfect example of this aspect of the human mind: trying to quantify in a single value all player’s contributions (positive and negative).
So the PER is certainly a comfortable and fascinating statistic, but at the same time dangerous: it provides us with a unique rating scale for the players, but doing so different information is lost or omitted.

Definition and starting data

PER stands for Player Efficiency Rating. The PER borns from John Hollinger‘s mind, who wanted to evolve the player’s Efficiency to a more detailed level. In fact, PER is the sum of all the positive and negative player’s contributions; however, it takes into account the minutes played and all the players’ performance in the League. Hollinger’s statistic then takes the concept of Efficiency (sum of all positive and negative contributions) and improve it, creating a unique comparison scale: this aspect is crucial, given that Efficiency gives us information about the individual player’s performance, but it does not take into account other factors, the player’s minutes, the average pace and the other players’ performances.

We need a lot of data to calculate PER. Let’s start with player’s stats:

  • Minutes played [MP];
  • Field goals made [FGM] and attempted [FGA];
  • 3-point made [3PM];
  • Free throws made [FTM] and attempted [FTA];
  • Assist [Ast];
  • Turnover [TO];
  • Offensive [OR] and defensive rebounds [DR];
  • Steals [ST];
  • Blocks[BLK];
  • Personal fouls [PF];

We need also these team stats:

  • Team field goals made [TeFGM];
  • Team assist [TeAst];
  • Team Pace [TePace];

Also, we need these League stats (average or sum of all the teams’ stats):

  • League field goals made [LgFGM] and attempted [LgFGA];
  • League free throws made [LgFTM] and attempted [LgFTA];
  • League assist [LgAst];
  • League points [LgPts];
  • League offensive rebounds [LgOR];
  • League turnovers [LgTO];
  • League Pace [LgPace];
  • League Defensive Rebound Percentage [LgDR%];

Collected all of them, we can proceed with the calculations.

Formulas and calculation

The calculation of the PER can be divided into three different parts: a first part in which the raw value of the player’s efficiency [gPER] is calculated, a second part in which this value is adjusted with the team pace, thus obtaining a correct PER [cPER] and finally a third part, where redistribution on a single scale takes place. Of the three, the heaviest in terms of calculations is the first: we need several steps to get the gPER.

We divide the calculation of the raw PER into twelve parts. Before going to analyze them, let’s define three coefficients that we will need during the calculation of the gPER. The first is called “Factor” by Hollinger:

\boldsymbol{Factor=\frac{2}{3}-\frac{0,5\cdot \frac{LgAst}{LgFGM}}{2\cdot \frac{LgFGM}{LgFTM}}}

This coefficient takes into account the performance of the whole league and provides a value that takes into account the assisted baskets and the free throws ratio. In this way, a different weight will be given to the shots made by the player. For example, in a league where many free throws are made, a basket from the field will have more weight, because it is rarer. Or if there are many assists, field shots will lose some importance, because they are judged to be slightly simpler. However, these are small variations compared to the actual value of shots taken. The second factor is the VOP, i.e. the points achieved per possession:

\boldsymbol{VOP=\frac{LgPts}{LgFGA+0,44\cdot LgFTA-LgOR+LgTO}}

This second coefficient will be used for weighing some game aspects. Finally, the third is the league Defensive Rebound Percentage.

Once these three coefficients have been defined, we can move on to the calculation of the raw PER: let’s start with the first of the twelve terms that compose it.

\boldsymbol{gPER_{1}=\left (2-Factor\cdot \frac{TeAst}{TeFGM} \right )\cdot FGM}

With this first term, the statistical weight of the field shots made is calculated: the two in brackets indicates the points generated for each shot made (the triples will be accounted later), but this value is reduced by a coefficient that takes into account shooting performance of the whole league and team. In this way, depending on the case, the field shots made by the player will be worth more or less, based on team and league performances.

The second is very easy:

\boldsymbol{gPER_{2}=3PM}

The triples scored by the player are taken into account so that the difficulty of this shot is considered.

The third is related to assist:

\boldsymbol{gPER_{3}=\frac{2}{3}\cdot Ast}

The assists are multiplied by a 2/3 coefficient, to give a smaller value to it compared to a made shot.

Then the fourth considers free throws:

\boldsymbol{gPER_{4}=FT\cdot 0,5\cdot \left [ 1+\left ( 1-\frac{TeAst}{TeFGM}+\frac{2}{3}\cdot \frac{TeAst}{TeFGM} \right ) \right ]}

We can make the same consideration of the field shots.

The next three terms will be accounted for as negative; let’s start with the fifth:

\boldsymbol{gPER_{5}=VOP\cdot TO}

A high VOP can indicate two things: games with high scores or games at low paces. In both cases, the turnovers play a more central role, since they would decrease the VOP. That’s why the player’s turnovers are multiplied by this coefficient, to give greater weight to this contribution when it is more incisive.

For the sixth term, we are again considering the shots taken by the player but, in this case, we are interested in the missed ones.

\boldsymbol{gPER_{6}=VOP\cdot LgDR\%\cdot (FGA-FGM)}

The number of missed shots is multiplied by the VOP: the missed shots have a greater weight in a league with few missed shots. It is also multiplied by the LgDR% because you want to correlate the number of missed shots with the defensive rebounds since a missed shot followed by an offensive rebound is less harmful than one that concludes team possession.

The seventh term takes up the missed free throws:

\boldsymbol{gPER_{7}=VOP\cdot 0,44\cdot \left [0,44+(0,56\cdot LgDR\%) \right ]\cdot (FTA-FTM)}

We then return to adding positive terms with the eighth:

\boldsymbol{gPER_{8}=VOP\cdot (1-LgDR\%)\cdot DR}

Defensive rebounds have more weight when few are caught in the league. Here then the value is multiplied by (1-LgDR%), or the percentage of offensive league rebounds, and also for the VOP, which is always taken into account for the points for possession discussion.

Same speech for the ninth term, linked instead to the offensive rebounds:

\boldsymbol{gPER_{9}=VOP\cdot LgDR\%\cdot OR}

Steals have equal and opposite value to turnovers: the tenth term has the same formula as the fifth, but in this case, it is obviously considered as positive.

\boldsymbol{gPER_{10}=VOP\cdot ST}

The eleventh term is also positive and is related to the blocks.

\boldsymbol{gPER_{11}=VOP\cdot LgDR\%\cdot BLK}

The presence of LgDR% is linked to the correlation between defensive rebounds and blocks described in the post of the Individual Defensive Rating.

Finally, the last term, again negative, accounts for the personal fouls, taking into account the free throws ratio and the fouls committed in the whole league.

\boldsymbol{gPER_{12}=PF\cdot \left [ \frac{LgFTM}{LgPF}-0,44\cdot \frac{LgFTA}{LgPF}\cdot VOP \right ]}

Once we have calculated all the terms, we can sum them together and then divide them by the minutes played, thus obtaining the notorious raw PER.

\boldsymbol{gPER=\frac{1}{MP}\cdot (gPER_{1}+gPER_{2}+gPER_{3}+gPER_{4}-gPER_{5}-gPER_{6}-gPER_{7}+gPER_{8}+gPER_{9}+gPER_{1}+gPER_{11}-gPER_{12})}

We can define the raw PER as an efficiency per minute, in which, however, different weights are given to the different contributions. Division by minutes played is the operation that allows taking into account the player’s presence on the court.

We can then move on to the second part of the calculation of the PER: the Pace correction.

\boldsymbol{cPER=gPER\cdot \frac{LgPace}{TePace}}

To make this correction, the gPER is multiplied by the ratio of the league and team Pace; doing this, the raw PER is compared with the average league pace, thus purifying it from the pace differences of the various teams.

We have reached the last part: the creation of the univocal evaluation scale.
To perform the last calculation we need to first calculate the weighted average of all players’ cPERs in the league, weighted on the minutes played. We use a weighted average as we want to give greater importance to the performance with many minutes played: it is not uncommon for those who play 5/6 minutes to obtain very high PER, but using the weighted average their influence on the summation will be marginal.

And after this value is obtained, the univocal scale can be created with the following formula (the weighted average just explained is at the denominator).

\boldsymbol{PER=15\cdot \frac{cPER}{\frac{\sum_{i=1}^{n}cPER_{i}\cdot min_{i}}{\sum_{i=1}^{n}min_{i}}}}

By making the ratio between the individual player’s cPER and the league one, you get his contribution parameterized with the performance of all the league: multiplying this ratio by 15, we set the reference value as 15. This means that the average of all PERs will always be 15 and this value will be the basis for comparisons.

How to read and analyze

For this example, I created a mini-league, made up of four teams, which are made up of two players. In the mini-league you play 1 vs 1 in 40-minute games. In this way, we will have a clear view of the calculation, which I consider fundamental for understanding.

Another assumption, PER can be calculated both for the single game and for the whole season: you just need to use the right data. Here are the scores of the four teams and the sum of the team statistics to obtain the league values.

The first step is to calculate the league coefficients:

Then, the raw PER can be found:

Bamforth and Fesenko have captured many defensive rebounds but their gPER8s remain around one. Opposite speech for gPER9s (linked to offensive rebounds): for Fesenko, this value is very high. This variation is due to the league Defensive Rebound Percentage: in this example, the LgDR% is very high. Therefore, the offensive rebound is rare and, consequently, also its relative gPER. If the LgDR% was lower (usually around 0.70-0.75), the offensive rebounds would lose some of their importance, while the defensive ones would gain.

The same can be said for the blocks: Fesenko has made five and obtains a high gPER11: this is because LgDR% is very high.

As for the shots terms, as you can see among the players who have taken the highest number of shots the final values of gPER1 are very similar.

After obtained them, we need to correct the values with the Pace:

The best in gPER was by Orelik, followed by Randolph and Fesenko: the first two, however, played at a much higher pace than the center. Through the pace correction, we note that Fesenko is the best because he made the most with a lower number of possessions. This aspect is absolutely not considered in Efficiency and is a big limit of this statistic.

We need to make only the last part of the procedure: calculate the weighted average of cPER to get the PER:

Fesenko is the best with a PER of 42, followed by Orelik and Randolph.

But what does it mean to have a PER of 42? Well, obviously the higher the value, the better the player’s contribution. But the real peculiarity of this statistic is that the final formula creates a fixed rating scale and the numbers always assume the same meaning. So to know what it means to have a PER of 42, just consult the following classification drawn up by Hollinger himself:

More than MVP: 43 o più
MVP: 33 – 42,9
Almost MVP: 30 – 32,9
Less than an MVP: 25 – 29,9
All-Star player: 22,5 – 24,9
Possible All-Star player: 20 – 22,4
Second offensive option: 18 – 19,9
Third offensive option: 16,5 – 17,9
Average player: 15 – 16,4
Bench player: 13 – 14,9
Always sit on the bench: 11 – 12,9
Barely a player: 9 – 10,9
Not a player at all: 0 – 8,9

By consulting the list, we understand Fesenko played really well, as an MVP.

In this example, we have used the scores of a single game, therefore the PERs assume very different values. In fact, PER is a statistic that makes sense over time, when players have accumulated many minutes and contributions. For a single game, it is not very reliable, but it is still a more reliable value than normal Efficiency. We can make a quick comparison Efficiency – PER:

Observing the Efficiencies, the classification would be the same as the raw PER: thanks to the corrections of the pace and the redistribution on a univocal scale we have a more accurate vision of the efficiency. Orelik played very well, but Fesenko, actually, was more effective than him.

Now, one question can arise: does an NBA player with a PER of 20 have the same efficiency as a Serie A player with the same PER? No.

It’s true, the rating scale is univocal and always has the same meaning, but only within the league. In fact, when we calculate the weighted average of all the cPERs we do not add the values of all the players in the world, but only those of their league. Two players with the same PER value belonging to two different leagues will have the same impact and efficiency in their respective leagues, but all in relation to the average level of the competition. Comparing the PERs of different leagues is not a wise move.

Before concluding, one last consideration: observing the calculations and the example we can see that the PER does not take into consideration different defensive contributions. Indeed, Hollinger’s statistics take into account the tangible defensive contributions (steals, defensive rebounds, and blocks), while completely ignoring all those semi-tangible or intangible terms that we discovered during the study of the individual Defensive Rating.
PER, therefore, takes into account all the player’s contributions, but we can say that it is a statistic more focused on the offensive phase than on the defensive one.

This Learn a Stat ends here. See you soon, your friendly neighborhood Cappe!

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