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Thread: Plus-Minus explanation please

  1. #1
    I Want Pacers Autographs pristinecollector's Avatar
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    Question Plus-Minus explanation please

    I searched for the answer but couldn't find one that didn't remind me of some science class in high school. Would someone please explain to me what the Plus-Minus is all about? I see it on the Pacers website under their "team" tab, and I see it a lot in the after game analysis, but I have no idea what it means.

    Thank you!

  2. #2
    FREE LANCE MillerTime's Avatar
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    Default Re: Plus-Minus explanation please

    Heres a good reading for you:
    2005-2006 Adjusted Plus-Minus Ratings
    By David Lewin
    For a good explanation of the adjusted plus-minus ratings that follow, click here: (Winston/Sagarin)
    Or here: (Dan Rosenbaum)
    If you would like my mediocre explanation, keep reading.
    Note: The three guys responsible for the numbers in the two articles above all work for NBA teams now, so their numbers are no longer publicly available. Below I apply their method to the 2005-2006 NBA season.

    Most of the time, when people talk about basketball statistics the conversation comes back to their limits. Basketball statistics, like points, rebounds, and assists tell you something about what happened, but many people think they miss the real reasons why teams win and lose. Box score statistics are said to miss the intangibles, the guys who are winners but might not fill up the stat sheet.

    One simple, but until recently little used, way of evaluating players is seeing whether a team does better or worse with a player on the court. This captures a playerís total effect on the team, not just those things that show up in a box score. This approach, pioneered here at, is known as plus-minus rating, or on court-off court rating. On court-off court numbers are useful, but they depend heavily on who else is on the floor. When you adjust these numbers to take into account the other players on the court it is known as an adjusted plus-minus statistic.

    Iíll use an example to illustrate what an adjusted plus-minus statistic is exactly. Lets say youíre at the gym and youíre playing some two on two with your friends. The first game its you and your buddy Ted against two other guys. You win this game by 4. The next game, itís you and your other buddy Jim against the same two guys. You lose this game by 2. Finally, there is third game, but this time you sit out and Jim and Ted play. They win by 1. From these games you can tell that you are three points per game better than Jim, and that Ted is three points per game better than you, six better than Jim. These are basically adjusted plus-minus statistics.

    Now, you wouldnít generalize from just these three games. Maybe Jimís jumpshot just wasnít falling that day. Maybe you and Jim are similar players who donít play well together but work well with a guy like Ted. Itís difficult to say for sure from just three observations involving three players. However, for an NBA season we donít have just three pick-up games, we have 40,000 observations involving 700 players.

    Every period between substitutions is one small game. Who wins and by how much is the result of the players who were on the court. By comparing these different periods using a mathematical technique called regression we can get a good idea of how the team does with a given player in the game. If a player causes his team to score more than the other team when he is on the court we call him a good player. If he lets the other team score more than his team then he is a bad player. Pretty simple.

    Adjusted plus-minus statistics take into account everything a player does on the court that affects the game. This includes things that many other statistics measure, like scoring a basket, but it also includes things like setting a good screen, or playing good man-to-man defense which are difficult to track. Another set of effects that are often missed but are accounted for here are inter-player effects. For instance, if having a three-point shooter on the floor opens up court for other players and causes them to play better when one is in the game, then the three-point shooter will get credit for this even if he doesnít take any shots.

    Adjusted plus-minus statistics are descriptive statistics, just like points per game, field goal percentage, rebounds per game, etc. Points per game describes how many points, on average a player scores in a game. Field goal percentage describes the ratio of a playerís shots made to his shots attempted. Adjusted plus-minus describes the average performance of the team with a given player in the game, taking into account the other nine players on the court.

    Like points per game, or field goal percentage, adjusted plus-minus doesnít mean anything without context. For instance looking at the Los Angeles Lakersí statistics from 2005-2006 without context it would be reasonable to conclude that Slava Medvedenko is a better shooter than Kobe Bryant because he had a better shooting percentage (.500 to .450).

    Should the Lakersí focus their offense around Slava instead of Kobe? Probably not. Does this mean that field goal percentage is useless? No.

    There are a number of reasons why Kobeís shooting percentage was lower than Slavaís. For starters, Slava attempted only two shots over the course of the entire season while Kobe attempted 2173. This means that Slavaís percentage is very likely to be due to luck, while Kobeís is probably a good description of his actual ability. Another issue is that Kobe had a very different role than Slava: he took more shots, he took shots from different places, and he played a different position. Once you take this context into account, field goal percentage can provide very valuable information about shooting ability, but without it, it can look ridiculous.

    The same logic needs to be applied to adjusted plus-minus statistics. Because adjusted plus-minus statistics depend on comparing how the changing of one player affects the performance of the team while holding everything else constant, sample size is important. Not just how many minutes a player played, but how broad a diversity of teammates did he play with. For teams with rigid rotations, like the Detroit Pistons, this can be problematic (as you will see).

    Despite these caveats, in general, adjusted plus-minus statistics give a good picture of which players help their team win, and thatís pretty much the whole point of the game, isnít it?

    2005-2006 Adjusted Plus-Minus Ratings
    Adjusted plus-minus is given in points per 100 possessions relative to an average player. The first table is the 25 best players in 2005-2006, the second table is the 10 worst. Must play at least 500 minutes to qualify.

    Top 25 Players
    Player Team Min. Rating
    Rasheed Wallace Pistons 70% 20.20
    LeBron James Cavaliers 85% 19.59
    Ben Wallace Pistons 73% 17.92
    Kobe Bryant Lakers 82% 14.34
    Andrei Kirilenko Jazz 66% 13.83
    Yao Ming Rockets 49% 13.63
    Ron Artest Kings 56% 11.83
    Alonzo Mourning Heat 33% 11.78
    Andre Iguodala 76ers 78% 11.60
    Brad Miller Kings 73% 11.07
    Jason Richardson Warriors 72% 10.53
    Dirk Nowitzki Mavericks 78% 10.52
    Chris Paul Hornets 71% 10.22
    Manu Ginobili Spurs 46% 10.01
    Ray Allen Supersonics 76% 9.84
    Steve Nash Suns 70% 9.63
    Dwyane Wade Heat 73% 9.57
    Kirk Snyder Hornets 33% 9.50
    Brent Barry Spurs 32% 9.43
    Jake Tsakalidis Grizzlies 18% 9.28
    Antonio McDyess Pistons 44% 9.24
    Antawn Jamison Wizards 83% 9.24
    Charles Hayes Rockets 13% 9.18
    Shaquille O'Neal Heat 45% 9.12
    Boris Diaw Suns 72% 8.86

    Bottom 15
    Player Team Min. Rating
    Desmond Mason Hornets 53% -9.62
    Chris Mihm Lakers 39% -9.66
    Alan Henderson Cavaliers 13% -10.16
    Pape Sow Raptors 15% -10.84
    Maurice Evans Pistons 29% -10.85
    Antonio Davis Knicks 19% -11.11
    Jason Kapono Heat 17% -11.52
    Jiri Welsch Bucks 22% -11.65
    Juwan Howard Rockets 64% -11.77
    Dahntay Jones Grizzlies 24% -11.85
    Primoz Brezec Bobcats 54% -11.91
    Drew Gooden Cavaliers 55% -12.52
    Michael Olowokandi Timberwolves 23% -13.39
    Carlos Delfino Pistons 18% -15.29
    Richard Hamilton Pistons 71% -17.90

    So, who was the best player in the NBA last year? Well, among those with greater than 500 minutes in 05-06, Rasheed Wallace had the highest per possession influence. However, taking a closer look at the data, you can see that there is something weird going on with the Detroit Pistons, which I will discuss later. For now lets leave the Pistons out, which means LeBron James was the best player in the NBA last year by a wide margin. Rounding out the top five, Pistons excluded, are Kobe Bryant, Andrei Kirilenko, Yao Ming, and Ron Artest.

    The numbers seem to generally align with conventional wisdom, although there are some surprises. The biggest surprise to me, actually, was how much better than everyone else LeBron really is. At nearly 20 points per 100 possessions better than an average player, LeBron is head and shoulders above the rest of the (non-Piston) league. He should have run away with the MVP. Steve Nash was a good player last year, no doubt, but the difference between LeBron and Steve Nash was bigger than the difference between Steve Nash and Sasha Vujacic.

    - Kirilenko and Ming have clearly established themselves as elite players in the league. Unfortunately Kirilenko and Yaoís sidekick, Tracy McGrady, have the same problem, they canít stay healthy.
    - Artest didnít let all the drama surrounding his trade demand effect his play. Adjusted plus-minus captures the one-on-one defense of players like Artest, Quinton Ross, and Shane Battier, an area that most statistical measures usually. When you combine Artestís offensive prowess with his elite defensive ability you get a top ten player in the league, albeit a crazy one.
    - Chris Paul produced a remarkable rookie season. You probably already knew this, but some might have wondered if his impact was really as strong as his box score stats indicated given that the team played better with him off the court according to raw plus-minus. However, this is a case where adjusting for who he had to play with and against makes all the difference. With the adjustments Paul was the 13th best player in the NBA last year (>500 min), slightly ahead of Steve Nash as the best point guard. Hereís a list of notable rookies:

    Notable Rookies
    Player Team Min. Rating
    Chris Paul Hornets 71% 10.22
    Charles Hayes Rockets 13% 9.18
    Nate Robinson Knicks 39% 7.09
    Monta Ellis Warriors 22% 6.00
    Luther Head Rockets 58% 4.97
    Marvin Williams Hawks 49% 4.74
    Raymond Felton Bobcats 61% 4.35
    Andrew Bogut Bucks 59% 4.09
    Charlie Villanueva Raptors 59% 2.56
    Sarunas Jasikevicius Pacers 39% 1.45
    Ryan Gomes Celtics 35% 1.31
    Channing Frye Knicks 40% -1.40
    Jose Calderon Raptors 37% -1.71
    Danny Granger Pacers 44% -3.14
    Francisco Garcia Kings 33% -3.44
    Salim Stoudamire Hawks 31% -3.69
    Joey Graham Raptors 40% -5.24
    Deron Williams Jazz 58% -5.80
    Johan Petro Supersonics 32% -6.34
    Jarrett Jack Trail Blazers 40% -9.41

    The Allen Iverson Trade
    The biggest story in basketball lately is Allen Iversonís move to Denver. There has been a lot of speculation as to whether the Nuggets are now a title contender, or if this trade even makes them better. Iverson is clearly a great scorer, but he tends to use a huge number of possessions inefficiently, and doesnít play much defense. I was very surprised to see John Hollinger gloss over these facts in his recent review of the trade, but clearly he buys into the conventional wisdom that although Iverson is flawed his scoring ability is enough to make his teams better.

    Reality tells a bit of a different story. In 2005-2006 Iverson made his team 7.38 points per 100 possessions better with him on the court (as opposed to replacing him with an average player). This is good, 42nd in the league, but not as good as Iversonís reputation would lead you to expect. Andre Miller closely followed Iverson, ranking 52nd with a value of 6.62. In 2004-2005 (full report on that data coming next week) Iverson actually had an impact of -4.41 compared to an average player. Miller was more consistent year to year, with a value of 5.62. This suggests that if both players play as they have in the past then the Nuggets will be at best as good as they were before the trade.

    Itís important to remember that all statistics tell us what a player did, not what he will do. Knowing Iverson averaged 33 points per game last year with the Sixers last year does not mean heíll average 33 with the Nuggets this year. He might, but he might not. Same goes for adjusted plus-minus. While Iverson was only a slightly above average player over the last two years (which doesnít mean he didnít have any value, he did, he played a lot of minutes that otherwise would have gone to players worse than average), itís difficult to say for sure how a move to the Nuggets will affect his play. It is certainly possible that Iverson will change his game and cause the Nuggets to be a better team. All I can say is that past returns indicate Iverson is unlikely to be much of an upgrade over Andre Miller.

    The Detroit Pistons, and Other Possible Sources of Error

    Detroit Pistons
    Player Team Min. Rating
    Rasheed Wallace Pistons 70% 20.20
    Ben Wallace Pistons 73% 17.92
    Antonio McDyess Pistons 44% 9.24
    Lindsey Hunter Pistons 9% 6.52
    Chauncey Billups Pistons 74% 3.72
    Carlos Arroyo Pistons 15% 3.33
    Tayshaun Prince Pistons 73% -3.25
    Maurice Evans Pistons 29% -10.85
    Carlos Delfino Pistons 18% -15.29
    Richard Hamilton Pistons 71% -17.90

    One look at the values for the Detroit Pistons starting five tells you something is up. The Wallaces rate at the very top of the league, Chauncey Billups is above average, Tayshaun Prince is below average, and Richard Hamilton is among the worst in the league. This is not what you would have expected. Also surprising is Antonio McDyessí rating as the 18th best player in the league.

    In 2005-2006 the Pistons were a historically unique team. According to they played their starting five together for 1674 minutes. The New Jersey Nets were the only other team in the league to have one lineup play at least 1000 minutes together. Most teams didnít even have a 5-man unit with 500 minutes. In addition, the Pistons played another 314 minutes with the same lineup only McDyess subbed in for one of the Wallaces. The Detroit starting fiveís individual minutes played range from 2776 to 2922. In order to separate the value of players who play together you have to look at the time in which they did not share the court and in this case, that is a very small sample.

    Take Ben Wallace and Richard Hamilton for instance. They shared the court for about 85% of their minutes. During these shared minutes the Pistons were successful. However, Wallace has on of the highest ratings in the league while Hamilton has one of the lowest. This is because the system gives Ben Wallace most of the credit for what those two accomplished while they were on the court together. Why does it do this? Because during the short period when Hamilton was in the game and Ben Wallace wasnít the Pistons did worse than usual, and during the short period that Ben was in the game and Hamilton wasnít the Pistons played well. This is a reasonable inference, but when itís based on only two or three hundred minutes then it is not reliable. Luckily, the Pistons are the only team for whom this is a major issue because their substitution patterns were so dramatically different than any other team in the league.

    A reasonable way to correct for this would be to just assign each member of Pistons starting five equal credit for the time the unit spent on the floor together. Doing this results in the following values for the Pistons starting five: R. Wallace 10.49, B. Wallace 9.9, Billups 3.94, Prince 1.00, Hamilton -4.86. So Hamilton is still negative, even when given one fifth of the credit for Detroitís dominant starting five. These values seems more reasonable to me, but in terms of sheer descriptive accuracy the regressed values are the best.

    This problem can occur on a smaller level when two players play almost all of their minutes together. One example of this is Gilbert Arenas and Antwan Jamison of the Wizards.

    Player Team Min. Rating
    Antawn Jamison Wizards 83% 9.24
    Gilbert Arenas Wizards 85% 2.58

    Arenas was on the floor for 89% of Jamisonís minutes. The Wizards outscored their opponents by 240 points (in 2920 minutes) when they were both on the floor. When just Jamison was on the floor they outscored their opponents by 4 (363 minutes). When just Arenas was on the floor the Wizards were outscored by 35 (460 minutes).

    The inference the system makes, that Jamison was the main reason why the Wizards were good when both players were on the floor, is reasonable, but based on a small sample size. So, as I did with the Pistonsí starting five Iíll give Arenas and Jamison equal credit for what happened when they shared the court. Doing this gives Jamison a value of 6.20 and Arenas a value of 5.38. Point being even in cases where there is heavy overlap the regression distributes credit in the most accurate way, and any difference from what might be expected is due to differences between expectations and reality.

    Notes: Dan Rosenbaum chose to add a clutch/garbage time adjustment to his system that weighted different periods of the game differently. I chose not to do this because, while Danís weights were probably reasonable, they were subjective and I want these numbers to be purely objective and descriptive. I believe Sagarin-Winston model uses expected win probabilities to determine how much each player changed the teamís chance of winning. I am exploring adding this to my model.

    Also see: the full list of ratings for all players.

    Coming in future columns:
    - Adjusted plus-minus recap for 2004-2005
    - Adjusted plus-minus ratings for the first half of 2006-2007
    - Individual offensive and defensive adjusted plus-minus ratings
    - Player wins and losses based on adjusted plus-minus ratings

    Special Thanks To:
    Dan Rosenbaum
    Roland Beech
    Vittorio Addona

    David Lewin is a 19 year-old college sophomore from Wayland, Massachusetts. He currently attends Macalester College in St. Paul, Minnesota where he plays football. He has contributed research and articles to Pro Football Prospectus 2006 and in addition to David is interested in a career in sports when he graduates and is always looking for interesting summer opportunities. A list of his articles can be found at He can be reached at
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  3. #3
    100 Miles from the B count55's Avatar
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    Default Re: Plus-Minus explanation please

    Quote Originally Posted by pristinecollector View Post
    This quote is hidden because you are ignoring this member. Show Quote
    I searched for the answer but couldn't find one that didn't remind me of some science class in high school. Would someone please explain to me what the Plus-Minus is all about? I see it on the Pacers website under their "team" tab, and I see it a lot in the after game analysis, but I have no idea what it means.

    Thank you!
    More simply, it's the point differential when any individual player is on the floor. For example, at the start of the game, the score is tied. Let's say that Roy Hibbert plays for the first 6 minutes, and when he leaves the Pacers are up 14-10. Roy gets credited as having a +4 for that stretch. This is done for every other stint he plays, and compiled for a number at the end of the game. If Roy played four stretches, and the Pacers were +4, -2, -5, and +1 in those periods, then Roy would be credited with a -2 for the game, or the Pacers were outscored by 2 when Hibbert was on the floor.

    It's a stat that's always been heavily used in Hockey that's been adopted by the NBA. In hockey, it tends to be more useful because guys play as part of lines, generally being on the ice with the same teammates most of the time. It's marginally useful in the NBA, but it can be pretty misleading because of changing lineups and situations.

  4. #4
    I Want Pacers Autographs pristinecollector's Avatar
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    Default Re: Plus-Minus explanation please

    Very clear explanations you two. I understand now. Thank you very much.

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    Administrator/ The Real Jay Jay@Section12's Avatar
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    Default Re: Plus-Minus explanation please

    Quote Originally Posted by count55 View Post
    This quote is hidden because you are ignoring this member. Show Quote
    In hockey, it tends to be more useful because guys play as part of lines, generally being on the ice with the same teammates most of the time. It's marginally useful in the NBA, but it can be pretty misleading because of changing lineups and situations.
    Thus, five-man +/- is much more useful, and has a decent version of it.
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