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How to calculate basketball plus minus

Basketball Statistics Definitions

Here are some explanations of various abbreviations and statistics that we track in our iPad/iPhone stats app...

2Pt the number of 2 point shots made
2PtA the number of 2 point shot attempts
2Pt% the percentage of 2 point shots made, 2Pt% = (2Pt / 2PtA) x 100
3Pt the number of 3 point shots made
3PtA the number of 3 point shot attempts
3Pt% the percentage of 3 point shots made, 3Pt% = (3Pt / 3PtA) x 100
FG the total number of field goals made, FG = 2Pt + 3Pt
FGA the total number of field goals attempted, FGA = 2PtA + 3PtA
FG% the percentage of 3 point shot made, FG% = (FG / FGA) x 100
EFG% the Effective Field Goal percentage gives more credit to made 3 point field goals since they yield 1.5 times the scoring of a 2 point field goal, EFG% = (2Pt + 1. 5 x 3Pt) / FGA
FT the number of free throws made
FTA the number of free throw attempts
FT% the percentage of free throws made, FT% = (FT / FTA) x 100
Pts the total number of points scored by field goals and free throws, Pts = 2 x 2Pt + 3 x 3Pt + FT
Layup the number of layups made. This is determined by tapping one of the layup buttons on the Track Stats screen after a 2 point basket has been made.
LayupA the number of layups attempted. This is determined by tapping one of the layup buttons on the Track Stats screen after a 2 point basket has been missed.
Layup% the percentage of layups made, Layup% = (Layup / LayupA) x 100
Paint Pt the number of points scored in the free throw lane (the paint)
PaintA the number of shot attempts taken from within the free throw lane (the paint)
TO Pts the number of points scored immediately after a turnover. This is determined by the user first selecting a turnover. Then if the next statistical event is a made basket of any type by the opposing team turnover points are recorded.
OReb the number of offensive rebounds
DReb - the number of defensive rebounds
Reb the total number of rebounds, Total Reb = Off Reb + Def Reb
Ast the number of assists
TO the number of turnovers
Force TO the number of turnovers that are forced by the defensive player or team
Dflc the number of deflections by a defensive player or team
Dflc Vic the number of times that an offensive player or team has a pass deflected. You must turn on Ask for Deflection Victim in Team Game Rules under Setup to track this statistic.
Stl the number of steals by a defensive player or team
Blk the number of blocks by a defensive player or team
Blk Vic the number of times that an offensive player or team has a shot blocked. You must turn on Ask for Block Victim in Team Game Rules under Setup to track this statistic.
Chrg the number of time a defensive player or team takes a charge
Recov the total number of steals, blocks and charges taken by a player or team, Recov = Stl + Blk + Chrg
Foul the number of personal fouls committed
Fouled the number of times a player is fouled. You must turn on Ask for Foul Victim in Team Game Rules under Setup to track this statistic.
Force Rush the number of times a defensive player forces an offensive player to rush his shot
Force Jmp the number of times a defensive player forces a jump ball. This is recorded whenever the Jump Ball button is tapped on the Track Stats screen.
Jmp Victim the number of times an offensive player is forced into a jump ball. You must turn on Ask for Jump Ball Victim in Team Game Rules under Setup to track this statistic.
Minutes the total number of minutes a player is in the game. To track this accurately you must make substitutions at the correct clock time.
VPS the Danny Miles Value Point System is a player ranking system developed by Coach Danny Miles of Oregon Tech, VPS = (Pts + Reb + 2 x Ast + 2 x Recov) / (2 x Field Goals Missed + Free Throws Missed + 2 x Fouls + 2 x TO)
+/- the plus/minus statistic is a measure of the point differential when players are in and out of a game. It is calculated by taking the difference in the score when the player enters the game and subtracting it from the score when the player exits the game. These differences are added up over the entire game to give the score. Thus, it is a measure of the impact a player has on the score of the game when he is in the game without measuring any specific statistic.
Effic a measure of a player's efficiency, Effic = Pts + Rebs + Ast + Stl + Blk (TO + FG Misses + FT Misses)
Bench Pt the number of points scored by the non-starters
Def Stop the number of possessions a team prevents the offense from scoring any points, Def Stops = DReb + Opponents TO
Pt/Possess the average number of points scored per possession
OReb% the percentage of available rebounds the offensive team gets, OReb% = OReb / (OReb + Opponents DReb)
DReb% the percentage of available rebounds the defensive team gets, DReb% = DReb / (DReb + Opponents OReb)

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Box Plus/Minus (BPM) : nbadiscussion

Hey everyone,

I've really enjoyed doing this series so far that I'm calling Basketball, Stat! And hopefully you've enjoyed it as well. In case you missed the first two posts, I'm doing a series that deep dives into different advanced stats that are used as all-in-one measures of player performance. Previously I looked at PER and RAPM, and this week we'll be looking at Box Plus/Minus, or BPM.

In addition to these, I'm also doing a podcast series by the same name that summarizes and discusses what I'm breaking down here. I won't go through all the nuts and bolts in the pod, but check it out if you want a slightly higher level version.

But with that, let's get into the breakdown of BPM!

The Basics

Despite its name, Box Plus/Minus is not really in the same family of Plus/Minus stats that we looked at with simple Plus/Minus, APM, and RAPM. These Plus/Minus stats generally have simple Plus/Minus at the heart of them, whereas BPM is a stat derived from a player's box score stats, such as points, rebounds, and assists, combined with team performance. Perhaps the best way to explain the difference is this: APM and RAPM use the team's performance depending on players on the court to find a player's value, but BPM uses RAPM to find the value of box score stats, and then translates that into player value.

BPM was created by Dan Myers, and one of the interesting choices he made in the creation of the stat was to intentionally limit it to stats that were tracked as of the 1973-74 season. His article about the stat on Basketball Reference suggests that he could have improved the accuracy of the stat had he not made this choice, but he prioritized something that could be historically useful over improved accuracy.

The Foundation

BPM is closely akin to a stat called Statistical Plus/Minus (SPM), which we should probably discuss first. If you recall from our breakdown of RAPM, Dan Rosenbaum didn't stop once he found Pure APM. He recognized that the stat was noisy and had too much statistical error, so he continued to tinker with it in his definitive article on the stat. (Here's a one-sentence refresh on APM: A play-by-play linear regression is run on the margin of stints of play to solve for player coefficients that equate to each player's Pure APM.)

So from here, Dan runs another regression, but this time, rather than finding the values for each player, he's looking for the value of the standard box score stats. He sets Pure APM equal to an equation that consists of box score stats multiplied by coefficients. He then solves for these coefficients that set the value for each box score stat, such as points per 40 minutes, DRB per 40 minutes, ORB per 40 minutes, 3PA per 40 minutes, etc. He uses 14 of these stats in total, and solves for the coefficient for each of them. So rather than relying on APM to dictate player value, he uses APM to assign values to box score stats. He then applies those values to each player's box score stats to calculate SPM, where he found a much smaller standard error compared to APM.

So how does SPM relate to BPM? BPM is a variation of SPM that uses advanced box score stats wherever possible. For example, where SPM used DRB per 40 minutes, BPM uses DRBD% instead. In addition, BPM’s statistical regression is built using RAPM values rather than APM, and RAPM has proven to be a much more reliable measurement than APM (If you're unfamiliar with RAPM vs APM, please refer to my previous article). Essentially, BPM is an improvement on almost every aspect of SPM.

The Formula

So let's talk about exactly what all goes into BPM! There is a simple formula that we can just plug numbers into and generate a BPM number, but if you've been following along, the purpose of this is to understand the "why" of the formula. So we’ll get there, but let's talk about what BPM is made of first.

As with SPM, the formula for BPM is going to consist of coefficients that are multiplied by players' box score stats, and that will give us BPM. But what are these coefficients and where do they come from? We start with RAPM. Dan Myers used a 14-year RAPM sample from Jeremias Engelmann and used 961 players from that sample. These 961 players were the highest "weighted" players in the sample, meaning they played a substantial amount of possessions as to not negatively impact the model.

For these 961 RAPM numbers spread over a 14-year period, Dan then tested many different iterations of different combinations of advanced statistics to find the combination that provided the least error in his regression. He then set all of these advanced stats and their coefficients equal to the various RAPM values and ran the regression, looking at the error compared to the 14-year RAPM sample. In the end, he settled on a combination of statistics that satisfied a few things: 1) low error, 2) a reasonable amount of variables (not a crazy, 50-term equation), and 3) few logical nonlinear terms. So with that understanding, let's look at the equation that Myers gives us:

Raw BPM = a*ReMPG + b*ORB% + c*DRB% + d*STL% + e*BLK% + f*AST% - g*USG%*TO% + h*USG%*(1-TOV%)*[2*(TS% - TmTS%) + i*AST% + j*(3PAr - Lg3PAr) - k] + l*sqrt(AST%*TRB%)

So now that we see all stats he settled on, let's go ahead and define them (and note that a-l are all coefficients that he gives value to, and we'll look at those values later):

ReMPG = Regressed Minutes per Game (a slightly altered MPG to avoid instances where a player plays very few games but with high minutes per game)
ORB% = Offensive rebounds divided by offensive rebounding opportunities
DRB% = Defensive rebounds divided by defensive rebounding opportunities
STL% = Steals divided by number of opponent possessions while player is on the floor
BLK% = Blocks divided by number of 2-point FGA while player is on the floor
AST% = Assists divided by teammate FGs (so excluding FGs made by the player)
USG% = Number of possessions "used" by a player divided by number of possessions while he's on the court (“used” meaning ends in a shot attempt, free throws, or turnover by the player)
TOV% = Turnovers divided by possessions "used" by the player
TS% = Points / ( 2 * TSA) where TSA = True Shooting Attempts (FG + 0. 44 * FTA)
TmTS% = Team's true shooting (same as immediately previous formula, but for entire team instead of the player)
3PAr = 3PA / FGA
Lg3PAr = 3PA / FGA (for entire league)
TRB% = Total rebounds divided by total rebounding opportunities

Alright, now let's take a look at what each of these terms is doing. And let me say up front that Myers explains this best in his Basketball Reference article that I mentioned earlier, but for my own sake at least, I'm going to walk through it myself as well.

ReMPG, ORB%, DRB%, STL%, BLK%, and AST% are all very straightforward, linear terms. Those terms are individually multiplied by their corresponding coefficients to add in to the player’s overall value. And remember that we're using "advanced" stats here, so everything is possession or rate based. But a few notes on some specific stats here:

The fact that MPG was factored in is an interesting tidbit on its own. Myers found that including it improved the accuracy of BPM, and he accounts for it by noting two things: 1) Coaches often know more and give better players more playing time, even if it's for reasons that stats or the general public can't see and 2) it serves as an indicator for strength of opponent, meaning that higher minute players are likely facing stronger competition.
STL% and BLK% are simple, linear terms that only appear here.
ORB% and DRB% are also simple, linear terms, but note that TRB% appears later in the equation, so these will ultimately impact each other to completely account for rebounding.
AST% makes several appearances throughout the formula, so this AST% term is merely an adjustment to more accurately measure the value of AST%.

USG%*TOV% is our first multi-variable term, but it equals out to turnovers committed by the player divided by possessions that he's on the court, which is still a simple, linear term.

Our next term, which is Myers' scoring term, starts to get very interesting. Let's look at it as a whole first, then break it down: h*USG%*(1-TOV%)*[2*(TS% - TmTS%) + i*AST% + j*(3PAr - Lg3PAr) - k]

USG%*(1-TOV%) can be translated as, "Shooting USG%." It is the percentage of the team's possessions in which the player is shooting, in other words, USG% with turnovers factored out. He multiplies this by the entire following scoring term.
2*(TS% - TmTS%) accomplishes a couple things: 1) it translates TS% and TmTS% into points per shooting possession by multiplying by two, and 2) it compares the player's TS% to the team's average TS% (average without the player). So if our player has a higher TS% than his team's average without him, this is a positive term. If it's lower, it's a negative term, which makes sense, as his less efficient shots would theoretically be hurting his team, since they would otherwise get a higher efficiency shot had the player not taken the less efficient shots.
AST% is factored in again here, which is interesting considering Myers referred to this as a scoring stat. But he explains that a player's assists are worth more if he's also contributing to scoring, which is why it's factored in here.
3PAr - Lg3PAr is considering whether the player is shooting threes more often than the league average. Given that we already factored in TS% and that this is the rate of 3PA and not a scoring stat, this is accounting for the value of stretching the floor. I absolutely love that he does this! Spacing is an extremely valuable factor on offensive and this was one of the first (if not the first) advanced all-in-one stat to account for this.
And finally, k is a constant that we'll give value to shortly.

And that brings us to our last term: sqrt(AST%*TRB%). I love the way Myers explains this stat, so I'll just share what he wrote about it:

Finally, a positive interaction term between rebounding and assists is included. This can be interpreted a number of ways – athleticism interacting with basketball awareness, size interacting with basketball skills, etc. This term was highly significant, and helped the overall fit of the regression quite a bit. Using the square root of the interaction both makes sense theoretically (maintaining a denominator of opportunities or possessions) and empirically (it is more significant and helps the overall fit of the regression more).

And that gives us our Raw BPM! Afterwards, he adjusts at the team level, and we will get there in a second, but let's first look at the value of the coefficients.

Coeff.	Term	BPM Value	Variable Format
a	Regr.MPG	0.123391	48.0
b	ORB%	0.119597	100.0
c	DRB%	-0.151287	100.0
d	STL%	1.255644	100.0
e	BLK%	0. 531838	100.0
f	AST%	-0.305868	100.0
g	TO%*USG%	0.921292	.000*100.0
h	Scoring	0.711217
	USG%		100.0
	TO%		.000
	TS% & TmTS%		.000
i	AST Interaction	0.017022	100.0
j	3PAr Interaction	0.297639	.000
k	Threshold Scoring	0.213485
l	sqrt(AST%*TRB%)	0.725930	100.0*100.0

A quick note on two coefficients: AST% and DRB%. You'll notice they're both negative, and that might leave you scratching your head as to why, considering those are positive contributions. Remember that AST% is factored in multiple places, and the latter positive terms outweigh the initial negative. The same is true for DRB% when you look at TRB% factored in at the end.

So with that, let's look at the team adjustment that Myers makes:

BPM_Team_Adjustment = [Team_Rating*120% - S(Player_%Min*Player_RawBPM)]/5

There's really two big terms at work here:

The Team_Rating term is the net rating of the team, adjusted for strength of schedule.
S(Player_%Min*Player_RawBPM) is the total of a team's player's Raw BPMs, but adjusted for the percentage of minutes played. So if a player has a Raw BPM of 20, but played 10% of the team's overall minutes, his value here is going to be two. Then you add in that value for each player on the team.

Once you have the BPM_Team_Adjustment for a team, that number is then added to every player's Raw BPM on that team. So let's say we have a Team_Rating of +10, and the minute-weighted sum of the players' Raw BPM on the team is +6, that would give us an equation of BPM_Team_Adjustment = [10*1.2 - 6]/5 = 1.2. We would then add 1.2 to every player's Raw BPM that's on the team we just looked at.

Let's take a slight detour to talk about the 120% number, because it stands out here as a somewhat arbitrary number at first. When watching a game where a good team falls behind, you may often find yourself assuming that the trailing team is going to make a comeback. It seems like it happens all of the time. Well, Jeremias Engelmann did a study and found that assumption to be well-based! He found that teams that are usually ahead are generally 20% better than their final results indicate. Conversely, teams that are usually trailing are generally 20% worse than their results. So this 120% accounts for that fact.

And that is how BPM is calculated! But there's one other thing we can do here if we want to go a step further. Myers' RAPM dataset from Engelmann had both ORAPM and DRAPM, so he ran another regression just for ORAPM to get another set of coefficients. So you can take the same Raw BPM formula, and use a separate set of coefficients to solve for OBPM. Here are the OBPM coefficients:

Coeff.	Term	OBPM Value	Variable Format
a	Regr.MPG	0.064448	48.0
b	ORB%	0.211125	100.0
c	DRB%	-0.107545	100.0
d	STL%	0.346513	100.0
e	BLK%	-0.052476	100.0
f	AST%	-0.041787	100.0
g	TO%*USG%	0.932965	.000*100.0
h	Scoring	0.687359
	USG%		100.0
	TO%		.000
	TS% & TmTS%		.000
i	AST Interaction	0.007952	100.0
j	3PAr Interaction	0.374706	.000
k	Threshold Scoring	-0.181891
l	sqrt(AST%*TRB%)	0. 239862	100.0*100.0

From there, you perform the same team adjustment to find OBPM. Then, DBPM is simply equal to BPM - OBPM. And now we're really finished with the formula!

Bonus Round! - Value Over Replacement Player

As we mentioned in the beginning, even though BPM uses MPG as part of the formula, it is a per 100 possessions based stat, so it doesn't account for playing time. However, Myers doesn't leave us hanging! He also presents us with another step we can take in calculating Value Over Replacement Player, or VORP.

VORP = [BPM – (-2.0)] * (% of minutes played)*(team games/82)

VORP is supposed to measure how many points per 100 possessions a player is valued at over an average player at that position. It's a good approximation for how much value a player is providing to his team during a season, and it's a number that you can compare linearly. As Myers points out, a player with a VORP of 4. 0 contributed twice as much to his team compared to a player with a VORP of 2.0.

Historical Examples

Given that BPM is still a box score based stat, I wanted to do something similar to what we looked at with PER. So let's first take a look at the top 100 BPM seasons since the 1973-74, with a minimum of 40 games played.

The top 10 looks really good. There are multiple LeBron and Jordan seasons, which is usually a good sign at the top. Westbrook, Curry, Harden, and interestingly Chris Paul round out the rest of the top 10 seasons. Overall, I think this is a great representation. All of these individual seasons were some of the most incredible regular season performances of all time. The Chris Paul season is probably the one that most people would take issue with, but in general, I think this is a good list.

When we expand to the top 100 however, we start to see some cracks. Similarly to PER, we have no sign of Stockton or Nash in the top 100. I don't believe that alone is a deal-breaker, but when you consider there are four Westbrook seasons and five Chris Paul seasons in the top 100, having Stockton and Nash missing seems to be an issue. At least we have Magic here this time, so not all is lost!

With PER, we saw defensive players severely underrated. Myers told us up front that he believe BPM doesn't accurately reflect defensive contribution, but I expect DBPM to do a much better job than PER. So let's take a look at DRPM historically.

The top 10 is...interesting. We have some very great signs though, with Ben Wallace and Marcus Camby having multiple appearances. There's no way we would expect either of them to appear in the top anything for PER, but they're clearly elite defensive players, and DBPM captures that. However, the rest of the list is a little more suspect with Ekpe Udoh and Greg Stiemsma both making appearances.

The rest of the top 100 has some notable absences, such as Kawhi, Bruce Bowen, Rodman, and Pippen, while Shawn Bradley and Myles Turner both make appearances. Myers mentioned that DRPM tended to favor big men, and that seems to clearly be the case here.

Strengths and Weaknesses

Overall, going through this exercise gave me a new respect for BPM. I was aware of its defensive deficiencies beforehand, but the method by which Myers created BPM was extremely well thought out and well tested. It truly is an enormous advancement for box-score-based stats. While it's not a true successor of PER, it's a better alternative in almost every respect.

Still, BPM suffers from the common box score stat deficiencies, namely defense. It absolutely is an improvement over prior box score based statistics as it captures, if not overstates, big man defense. This is a big step forward, but perimeter defense still seems to be largely undervalued here.

We also see some of the assist-first guards undervalued, such as Nash and Stockton. Even though Magic had multiple top 100 seasons, BPM tended to favor Westbrook-type play comparatively, and most basketball minds would argue Magic was a much better player.

Conclusion

As I mentioned above, I have a renewed respect for BPM after doing this exercise. As an overall stat, it's useful to look at and should definitely take precedent over PER. I even think it's valuable to compare players using BPM, particularly when converting to VORP in a season. But big men will be overrated on defense, and pass-first guards will be seemingly underrated. Still, BPM was an enormous advancement on box score stats!

The NBA introduces a "plus-minus" indicator - Basketball

In the NBA regular season that has just started, official match reports now indicate a new statistical indicator "plus-minus".

As in other team sports (hockey, football), it shows how much more points the player's team scored (or conceded) in those moments when he was on the court.

For example, Andrey Kirilenko's plus-minus score was +18 in the match against Golden State, the third result in the team. And the leader in the league after the first day was his partner, point guard Daron Williams, "+24", writes "Sport-Express".

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Handicap in basketball betting

Handicap in basketball expresses the final score difference and is designed to equalize the chances of teams. Obviously, in the matches of the leader and the outsider, the favorite will win. The odds for this outcome do not exceed 1.2-1.3. It is unprofitable to play on meager quotes, since it will take 3-4 successful transactions to win back losses. Therefore, it is profitable to bet on handicaps for 1.9and more than the net gain of 1.25.

Basketball handicap example

Let's take an equiprobable handicap of 18.5 points in the example match. An equiprobable handicap means that the passage of a positive and a reflective value is evaluated equally - 50 to 50.

Negative handicap (with a minus sign). F1 -18.5 – the home team must win by at least 19 points. If the score difference is 18 or less or the team lost, the bet is settled as a loss.

Plus handicap (with plus sign). F2 +18.5 – guests must not lose by 19 or more points. To pass the bet, you need to win or lose, but with a difference of no more than 18.

Add or subtract to the points scored by one of the opponents the number of points indicated in the handicap. If after the calculations the team continues to lead, then the bet will win.

Whole handicap. Ф(-5) or Ф(+5) – the principle remains unchanged, but there is a possibility of a return (calculation of the bet with a coefficient of 1.0) if the difference in the score is 5 points. If the gap is less or more than 5 points, a negative or positive handicap will play.

Zero handicap. Ф(0) - means that in case of a draw there will be a return. When the team on which they took the zero handicap wins, the bet will also pass. If the team loses, the deal will fail. This is the same bet on winning, but with the possibility of getting money back if the outcome is a draw.

Which events are suitable for betting on handicap

The schedule for a meeting of equal teams contains close handicaps - 3.5 or 4.5. Ignore such events except in two cases:

when taking a handicap instead of a clear win to increase the odds;
when identifying an overestimated outcome (Value betting).

Confrontation of opponents of the same strength does not provide for a significant difference in the score. The game is tight, and the determination of the winner is postponed until the end. 5 minutes before the final siren, basketball players break the rules, counterattack and throw three-pointers more often.

The team can lead in the course of the match by 4-6 points, but fail to win back the ending and lose.

Close handicap in basketball (up to 5-6 points) is an unprofitable choice in tense confrontations. The ending is unpredictable. Each opponent is able to break away by 6 points if the score is equal. Choose games with clear favorites and underdogs, or look for values in close handicaps.

Analysis of statistics

Analyze the totals along with the handicap market. Also study the percentage of shots made: from the field, from behind the arc, free throws. Grassroots teams (with a small total) rarely finish matches with a margin of more than 6 points. High-performing clubs, on the other hand, often win or lose resoundingly.

If athletes have low 3-pointers and half of their free throws miss, they will “fail” on a bad day. Model an upcoming event based on statistics even before you get acquainted with the odds of a bookmaker.

Summary

Basketball handicap betting only in theory looks easy. In practice, you need to carefully analyze the events and predict the scenario of the match. But this does not guarantee success, because it is impossible to take into account all factors.