Playmaking, Kobe, & Point Guards: The Harmonic Mean of Points & Assists

Andre Khatchaturian —  August 9, 2013

There are players who score and only score, and there are players who score and make players around them better. There is value in a player who not only scores, but also makes others around him better by distributing and facilitating. How do we separate a great all-around player who has an impact on his whole team from a guy who’s just known for his scoring ability? Introducing the harmonic mean between points and assists.


The harmonic mean between two variables is simply (2*A*B) / (A + B), where A and B are any two variables. In this case, A and B are points and assists. The harmonic mean was first introduced in sports by Bill James, when he developed the Power-Speed number in baseball. In his formula, home runs and stolen bases were the two variables. To do well, you need a lot of both.  This makes sense because the numerator in the formula grows at a much faster rate than the denominator.

The same applies in the points-assists harmonic mean (PAHM). In order to have a high PAHM, a player must have a lot of points and a lot of assists. In other words, one would expect point guards to infiltrate the top of this list and for Carmelo Anthony to not fare too well.

After running the results the first time, I noticed that players who had played more games naturally had a higher PAHM because they had more opportunities to get points and assists. In order to avoid this problem, I divided their overall PAHM by games played to get their Per Game PAHM.

2012-13 Per Game PAHM Stats:

Screen Shot 2013 08 08 at 1.17.15 PM

A few observations when we look at this table:

  • As expected, the list is filled with point guards. In fact, 24 of the top 28 are point guards. This is again no surprise because point guards generally rack up assists and aren’t afraid of taking a shot. Also, the league is filled with stellar point guards today and this table reiterates that notion.
  • Chris Paul, like his fictional twin brother Cliff, is in fact great at assisting others as portrayed in the State Farm commercials.
  • Rajon Rondo is a bit of a surprise at No. 2. He didn’t play a full season but he made his way to the top of the list because of his high assist total. Just imagine how great he would be if he had a better scoring touch.
  • Kobe Bryant is 11th on this list (first among all shooting guards). So next time someone on Twitter or Facebook posts a “Kobe doesn’t pass” meme, you can swiftly respond with this stat. (More on Kobe in a bit.)
  • LeBron James is a freak, but we already knew that. He’s not only third overall in Per Game PAHM, but he’s a power forward! The next power forward on the list is Josh Smith and he’s ranked No. 40.
  • No surprise but Carmelo Anthony is not on this list. He was actually ranked No. 84 in the stat – no surprise from a guy who was criticized all year for not passing the ball.

PAHM and the Lakers

The Lakers ranked in the middle of the pack as a team in PAHM at No. 15. Four Lakers were in the top 100 in the stat – Kobe, Nash, Pau Gasol, and Steve Blake. Gasol is an interesting name in the list. He was ranked No. 45 overall and third among all power forwards. This fits his description as a finesse big who can distribute. Here are the rest of the Laker rankings:

Screen Shot 2013 08 08 at 10.13.42 PM

The main takeaway from this table other than the fact that Kobe is dominating in a stat most people wouldn’t think he could dominate is the limited help the team received from Chris Duhon and Darius Morris. Once again, the 2012-13 story cannot be told without first talking about injuries. Nash and Blake missed almost half the year and were not backed up well by their third and fourth stringers (nor were they expected to be.) Nash was also 5th in the league in 2011-12 in PAHM, but he fell to 21st last year.

But let’s talk about Kobe. He recorded the highest PAHM of his career this season at 9.86 and he has a career PAHM of 8.01, which is excellent for a shooting guard. One may attribute his high PAHM to his 31,617 career points, but the fact that he’s averaged near five assists per game throughout his career definitely has a lot to do with his high PAHM. If you cut Kobe’s assist by 60 percent, it lowers his PAHM to the low 5’s.

This is intriguing because as mentioned before, most people don’t look at Kobe as a distributor, when in fact he’s been fairly good at getting his teammates involved. And when Kobe gets his teammates involved, the Lakers win games.

We’ve seen the numbers regarding Kobe, assists, and Laker wins. This year when Kobe recorded three or fewer assists, the Lakers were 2-17, but were 23-10 when he recorded at least seven assists. A regression run between Kobe’s game-by-game PAHM with Laker wins reiterates this notion. Kobe’s game-by-game PAHM was statistically significant in the regression meaning it was a factor in Laker wins and losses.


In short, PAHM has several good uses:

  • It’s a great way to evaluate point guards. The higher a point guard’s PAHM, the more versatile they are in terms of scoring and distributing.
  • It’s also interesting to look at players that play other positions pop up at the top of the list because it shows that they are not only good scorers, but solid facilitators.
  • It’s a number that separates great players from great scorers. For instance, it says a lot that LeBron has a higher career PAHM (11.08) than Allen Iverson (10.0) even though the former is a power forward and the latter is a point guard.

Andre Khatchaturian


38 responses to Playmaking, Kobe, & Point Guards: The Harmonic Mean of Points & Assists

  1. “[PAHM] a great way to evaluate point guards. The higher a point guard’s PAHM, the more versatile they are in terms of scoring and distributing.”

    Don’t Rondo and J’rue ranking as 2 of the top 4 PGs on the list sort of mitigate the last part of that statement?

    I can’t believe Monta Ellis made it as high as he did, as the third highest SG behind Kobe and Harden. Interesting article with some fascinating insight. Thanks.

  2. PAHM needs to be correlated to amount of time the ball is in the hands of the player. Sure LeBron is a forward, but he also had the ball in his hands a lot as a ‘Point’ forward. Same for Kobe, he has the ball in his hands a lot too.
    Players who have a high PAHM but spend very little time with the ball in their hands are the ideal efficient players. They are the ones who make the right pass, shot or play.

  3. Andre Khatchaturian August 9, 2013 at 3:02 pm

    True – but that was a general statement since most point guards aren’t inept at scoring like Rondo. But still 13 PPG isn’t bad and that’ll probably rise now that he’s the star

  4. Very Nice, Andre!

  5. The bad thing about this metric is that it favors bad players over very good players, as long as their points and assists are relatively close together. Dwight Howard, who scores a lot but doesn’t get that many assists, is somehow worse than Chris Duhon, who doesn’t score a lot OR get many assists.

  6. Andre Khatchaturian August 9, 2013 at 5:03 pm

    Patel – you are probably right. Wish tehre was a way to check for sure though.

  7. Been away for a while, so need to catch up. I will add to the props to Pau a few threads back. He was key to 2 banners and therefore will always be key in Laker history. Also loved the comments about D Fisher, who is sorely missed for a variety of reasons.
    Not sure if everyone saw this yesterday.
    “”When it came time to try to convince Dwight to stay, we lost the best closer in the business in Dr. Buss.”
    Is there a lottery for owners? Coaches? If so, how do we qualify?

  8. Chris,
    No, this metric favors playmakers over guys who would be better classified as “finishers”. Dwight is more the latter than the former.

  9. I’m just amazed that Chis Paul had 678 assists while only playing 70 games. Twelve more games at his average and he would have over 800 assists for the season. That is insane when compared to the rest of the list.

    Regarding points, I’d look at FG% as well. LeBron scoring his 2,036 points at a 56% clip makes a huge difference over another player scoring at a much lower clip. Paul’s sheer volume of assists and LeBron’s scoring clip probably means those two are far ahead of everyone else.

  10. Of coures I looked right over Rajon Rondo in my previous comment. His assists total through 38 puts him up there with CP3.

  11. Dont discount monta – he should be a scoring pg like holiday not a sg – too small to.guard.guys like kobe or wade – I think hes gonna be a good fit in dallas

  12. Andre Khatchaturian August 9, 2013 at 7:21 pm


    Also important to note that this isn’t a better/worse stat. This isn’t Wins ABove Replacement or PER. It’s only saying that a player distributes and scores better than others.

  13. Very interesting post, Andre.

    Besides the point, but I think of LBJ as more of a “small forward” than a power forward, although he is indeed mighty.

  14. Kobe has been playing for like 18 seasons. We all know that he is not going to be remembered for sharing the ball and making his teammates better, or being unselfish.

  15. Agree with Patel re addressing amount of time ball spends in ones hands. Skewed towards initiators in Pick in roll heavy offenses. Anyway to include usage rate as a variable?

  16. I would assume touches would be a better added variable than usage, as assists don’t count against usage no?

  17. Kobe not passing has become a stigma despite his uncanny ability to find his teammates on scoring positions. Guys who can convert (Ariza, Fisher, Gasol and Bynum) were extremely successful and even Dwight… the others (Smush, Kwame and Jumaine Jones) weren’t so much.

    Kobe’s shoot-first mentality is just a product of his extreme confidence in himself to score. Sure he takes bad shots but he also takes big shots and game-winning shots like no other since MJ. Thats what makes them primetime superstars and not the stars that choose to pass when it counts.

  18. I wonder how “Magic” would fair in this formula…Jerry West…Oscar Robertson…

  19. LeBron James plays PG in the Heats offense. Kobe plays PG in the Lakers offense. It’s not rocket science. The guy who has the ball and makes the plays is the PG of the offense.

  20. The fact that they do given that’s not their actual position is astounding.

  21. Andre Khatchaturian August 10, 2013 at 1:59 pm

    To Anonymous:

    I just computed those 3 guys and MJ for their careers:
    Jordan: 8.9 (Kobe is 8.0 lifetime)
    West: 10.73
    Robertson: 13.88
    Magic: 14.23

  22. The second Aaron comment was a fake Aaron.

  23. Andre Khatchaturian August 10, 2013 at 4:34 pm

    Aaron – that was me lol I was trying to address you but I accidentally wrote your name in the Name box. My bad. I just noticed.

  24. Andre,

    This is interesting stuff. Food for thought. As a suggestion, maybe you could pull together some statistics re: the relative defensive performance of players in the NBA–especially, I would think, for players on the perimeter.

    As you may have noticed, there is a great deal of discussion on this post as to the relative merits of the defensive prowess of the Lakers’ players. Perhaps some insightful statistics would give us a good idea of what kind of defensive team the Lakers actually have on their hands for this coming season.

    It’s just a thought. Thanks again for the great work.

  25. Andrew Andrekopoulos August 10, 2013 at 7:57 pm

    Very Intereasting Article. Thank you!

  26. This is a very provocative column. I am not sure I agree that each assist should count as “1”. Points scored are straight forward but assists could mean 2 or 3 points scored. As a result, assists are undervalued in terms of points scored. If you change your formula to reflect this the results are closer to what you might have thought they would be in the first place. For example, let’s say each assist resulted in 2 points scored. Then the top 10 in order are:
    LeBron James
    Russell Westbrook
    Chris Paul
    Tony Parker
    Stephen Curry
    Deron Williams
    Rajon Rondo
    Jrue Holiday
    Kobe Bryant
    John Wall

    I think this is a better reflection of the value of each player to their team and relative to each other. If Rondo was truly a top 2 or 3 point guard, Ainge would not keep trying to trade him and Rivers might still be in Boston.

  27. Baylor, your suggestion is a weighted valuation of every asset. Granting an assist does result in atleast 2 points, because an assist doesn’t get counted unless the basket was converted. What about if the assist guy makes the perfect pass on a perfect situation but the finisher fumbles or misses it?

    Metrics are nice measurement sticks but thats what they are. They should not be above and beyond the measurements of instinct, will determination and mental toughness. Thats why I’ll never be a metric/analytic guy in basketball. There’s just too much going on in the court that counts more than actual statistics.

  28. If one of two variables is much larger than the other, the PAHM works out to be a little less than twice the smallest of the two. So in the case of big time scorers the number is roughly twice that person’s assists. It’s good way to evaluate a point guard because if they score enough the number approaches twice their assist count.

  29. (Real Aaron from his fiancé’s iPhone)…Baylor fan with some genius breakdown of advanced statistics everywhere. Haha. If I were you I would publish that brilliant breakthrough. Genius. Seriously genius. That would help eluminate the creator over the ball mover. For instance LBJ creates more three point shots than a Luke Walton. Overall. Although of course man LBJ kick outs are moves to one or two more players before the three pointer is taken. But overall it helps differentiate the value of the assist.

  30. Andre Khatchaturian August 12, 2013 at 12:20 am

    Regarding multiplying assists by two:

    I did give that thought before doing the study, however, simply multiplying assists by two is blatantly inaccurate and wrong. Some assists are worth three points. Some are two. Just assuming all of them are two would be incorrect. Of course weighing the assists by if they ended up being 2s or 3s would be more accurate, but seeing that that information is not readily available on basketball-reference or, it would be difficult to compile that info. Aaron and Baylor fan…if you two are willing to sift through every single game’s play by play to find that information then I will be impressed lol.

    Another way to do it was to use Assists and FGM as your variables, but this completely ignores FTM. So why not include (FGM + FTM) as a variable? Because then you’re giving equal weight to FGM and FTM which should not be the case. Simply using FGM also ignores three pointers.

    While it’s seems simple to just use PTS and ASTS, it still does the job in telling the story of who is a good playmaker. At the end of the day, you still need a lot of both to have a high PAHM.

    I should have addressed this in the methodology section to clear any confusion as to why I only did points and assists. Thanks for reading the article and giving feedback, though. I appreciate it.

  31. Forgive my ignorance..but I don’t get the math. 678 then 1186 is 863,
    for Chris Paul…..are you averaging? Giving 2 points for each assist?

  32. Andre…how do you arrive at the harmonic mean? I googled it and got REALLY confused!

  33. Thanks, Andre.

    I just computed those 3 guys and MJ for their careers:
    Jordan: 8.9 (Kobe is 8.0 lifetime)
    West: 10.73
    Robertson: 13.88
    Magic: 14.23

    According to your computations Kobe is not far off from Jordan. Surprise!!!

  34. Hi Andre, these are just descriptive statistics. There is no cause and effect per se. You choose to use assists and points scored; I have never liked assists since they do not account for how many points were scored and ignore players who were fouled and shot free throws. Your method actually undervalues assists which seems to be against your initial premise. Giving 2 points is just as valid as giving 1 and much more accurate. I did run baskets scored and assists and came up with the following top 10:

    LeBron James
    Russell Westbrook
    Kobe Bryant
    James Harden
    Tony Parker
    John Wall
    Chris Paul
    Stephen Curry
    Deron Williams
    Kyrie Irving

    Frankly, I think this much better reflects the impact on scoring each of these players have. By counting each free throw as a basket it adds the dimension of players willing to take a foul to score points hence Harden jumping up to #4. My other issue with assists is that they ignore the “hockey assist”. Watch the first quarter of Magic’s summer all-star game which Darius linked. Several plays started with Magic passing to Bird and Bird passing to a cutting player. I wish there were a way to capture the person who initiates the play and is able to anticipate what the next pass will be.

  35. Forgive my ignorance..but I don’t get the math. 678 then 1186 is 863,
    for Chris Pau???l…..are you averaging?
    Giving 2 points for each assist?Andre…how do you arrive at the harmonic mean? I googled it and got REALLY confused!
    Anyone know?

  36. Andre Khatchaturian August 12, 2013 at 6:08 pm

    Dave —

    Harmonic mean of two variables is simply (2*A*B) / (A+B) where A and B are two variables (in this case points and assists)

    Baylor Fan —

    I understand what you’re saying, but it would be even more accurate if you weighted the assists properly rather than multiplying by two. As someone else previously mentioned, LeBron is great at assisting in threes, for example.

    I agree regarding the “hockey assists” though. The NBA (or someone) should start keeping track of that. Perhaps that will be a good thing to track for the Lakers at least when the season starts.

    With that said, we still need information regarding Assists on 2’s and Assists on 3’s and until that is available, I’m not comfortable in weighing assists by 2. For instance, the last list you had didn’t even have Rondo in the top 10. I think my method gives more justice for Rondo – who is a very underrated shooter. He shot 48% last year and averaged 13 points – but his primary strength is distributing. He deserves to be in the top 3 there. Once again, I’m not saying you’re wrong – but I think it’s a tom-ay-to vs tom-ah-to debate. Both ways are right but it’s a matter of preference. You can’t just say “oh this way is right, baaagh!” I can come up with arguments supporting my way and you can come up with arguments supporting yours. That said, at the end of the day I just want to know if the guy decided to get an assist on a play or decided to go for the score…you STILL need a lot of both to do well in the stat.

  37. First of all, I like the analysis and it does put light on the different abilities of the players. I look forward to more columns. As for Rondo, I will defer to Kobe and agree that Rondo needs to score more to dominate on the offensive end. There are too many games where he forces passes and turns down shots so he can get to 10 assists and a possible triple double.

  38. Andre!! I understand there are 2 variables–points and assists. Using stars and slashes means what? Are you adding together the points and assists and then dividing them?
    Are you giving 2 points for an assist? What??
    If you could explain in plain English like, “OK add points and assists and divide by ….or multiply assists by whatever and add points and divide by……..”..
    For Chris Paul….how does 678 assists and 1186 points have a “harmonic mean” of 863. I know I’m gonna feel dumb when you explain. hahhahahaha It’s drivin’ me nuts!