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Where black holes meet geometry: Kobe Bryant and the next evolution of passing statistics

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A week ago, my esteemed SB Nation colleague Tom Ziller created a Black Hole Index. If you haven't already seen it, you should check it out, if for no other reason than to appreciate an aesthetically awesome chart.  The chart plots the NBA's top guards on two axes, one representing their usage rate, and the other representing their assists per shot attempt (including free throws). Not surprisingly, the chart shows Kobe Bryant as one of the biggest black holes in the league.  Considering Kobe's high usage rate, and relatively low assist numbers, this is in no way a surprise.  Nor, as Tom points out, is it particularly an insult.  For the rare player of Kobe's talent and ability, being a "black hole" is only an indictment in the court of public opinion. It passes no judgment on a player's value, no direct equivalency to his ability to help his team win games.

And yet, despite the author's plea for it to be seen simply as an observation, not a judgment, we Laker fans can't help but rise to the defense of our superstar.  The comments are filled with the familiar refrains that are seen whenever Kobe's reputation is slandered with words like "selfish" and "ball hog".  Stuf like "Assists don't accurately measure Kobe's passing ability because the Triangle prevents him from getting as many assists as other players." and  "It's unfair to measure based on assists because Kobe's teammates may not be as adept at converting a pass into a bucket".  And there's always "MJ was an even bigger black hole than Kobe."

To be clear, there's a very decent possibility that our defenses on the matter are absolutely correct.  No one can know whether Kobe's teammates convert assist opportunities as well as others, and we can be certain that Kobe's career assist totals would be a bit higher if Kwame Brown's hands weren't made out of Teflon.  It's hard to ignore the fact that Phil Jackson's Triangle based offenses rarely have anybody even remotely close to a league leading assist total, and yet the offense itself creates a high number of assists. These statements are potentially valid counter-points to the analysis presented.  They are also impossible to prove with the limited form of statistics made available to the common man.  Well, when life gives you lemons, you make lemonade, and when life gives you shitty statistics, you create better ones.  Using last night's game between the Los Angeles Lakers and the Boston Celtics, that's exactly what I tried to do.

In order to get to the heart of any possible effect the Triangle offense might have on passing statistics, we need a much more detailed snapshot of passing and player involvement than can be deduced from the box score.  So, I decided to track Kobe Bryant all game long by looking at the following factors: How many possessions did the Lakers have while Kobe was on the court?  How many times did Kobe touch the ball during the course of those possessions?  How many passes did Kobe complete as a result of his touches?  How many teammates' shot attempts were created as a direct result of those passes?  And finally, how many assists did Kobe generate on the night? 

Creating a data set of such detail is pointless without a control.  I needed to test the numbers Kobe generated against the same sort of numbers for a different player.  And what better comparison to make than the guy who is furthest away from Kobe on the Black Hole Index, Rajon Rondo, who conveniently happened to play for the other team in last night's contest.  Everything I tracked for Kobe, I tracked for Rondo as well.

I honestly had no idea what to expect, and on the surface, I was setting Kobe up to fail.  Using a game against the Celtics, who's defense is the best in the league at shutting down the Lakers free flowing offensive attack, as the test case for Kobe's passing proficiency?  Do I need to remind you of Kobe's stat line from the last game between these two teams? Using the league's most prominent anti-Kobe as a comparative test case?   These are not what you would call favorable circumstances.

After the first half, my concerns were the exact opposite.  It seemed clear I had chosen a bad game to sample, because Kobe was playing the "distributor" role to the extreme.  How foolish of me not to realize that Kobe's last performance, the 41 point, 0 assist masterpiece/debacle, would cause Kobe to come out last night and try to involve his teammates to the max.  Luckily, the second half restored my faith in the experiment, as Kobe was much, much more aggressive in looking for his own shot, enough so to counteract the freakishly passive first half performance. 

So what did I find out?  We'll start with the first half.

1st half Kobe Rondo Ratios 1st half Kobe Rondo Difference
Team Possessions 36 36 Touches/Possession 0.94 1.25 32%
Touches 34 45 Passes/Touch 0.82 0.80 -3%
Completed Passes 28 36 Shot attemts/pass 0.43 0.56 30%
Passes leading to shots 12 20 Assists/shot attempt 0.25 0.40 60%
Assists 3 8 Points per possession on 1.11 1.11
Passes leading to FT's 1 2 Points per possession off 0.71 1.63
Points while on floor 40 40 Usage 0.11 0.17
Total Possessions 43 44 Assists/pass 0.11 0.22
Points off floor 5 13


As I said earlier, in the first half, Bryant played the role of distributor to an extreme.  He actually passed the ball a higher percentage of the time than Rondo did over the first two quarters.  However, the first thing that should stand out to you is the touches/possession ratio.  Rondo averaged 1.25 touches/possession, and Kobe was a shade under 1 touch per.  That is some small confirmation of one of the tenets of the whole "The Triangle causes less assists for an individual" argument, because, in a half in which Kobe was getting the ball to other players at a higher rate than Rondo was, he still saw significantly less of the ball than Rondo did.  Further, Rondo's passes created shot attempts at a higher frequency than Kobe's did, and Rondo's teammates did a better job of converting those shots into field goals as well.  All in all, Bryant was responsible for using an insanely low 11% of his teams possessions, as compared to 17% for Rondo, and Rondo's passes were twice as likely to generate an assist as Kobe's were.  Now for the second half:

2nd half Ratios 2nd half Kobe Rondo
Team Possessions 30 40 Touches/Possession 1.13 1.18 4%
Touches 34 47 Passes/Touch 0.38 0.77 100%
Completed Passes 13 36 Shot attemts/pass 0.23 0.31 32%
Passes leading to shots 3 11 Assists/shot attempt 0.33 0.18 -45%
Assists 1 2 Points per possession on 1.13 0.83
Passes leading to FT's 0 0 Points per possession off 1.30
Points while on floor 34 33 Usage 0.60 0.23
Total Possessions 40 40 Assists/pass 0.08 0.06
Points off floor 13 0


Whoa, talk about a tale of two Kobes.  Whereas in the first half, Kobe was even more of a distributor than Rondo was, in the second half we see a relationship between the two players that comes much closer to matching what we would expect based upon their reputation.  The number that should jump off the page at you is Kobe Bryant's second half usage, which is 60%.  That is every bit as insane and extreme as the first half's 11% was, perhaps even moreso.  In 3 out of 5 second half possessions (while Kobe was on the court), Kobe either shot the ball, turned the ball over, or went to the free throw line.  That is the very essence of black holey.  If this half were to be represented on the chart, it would need it's own special page showing the entirely different galaxy that is on the other side of the black hole.

Checking out the rest of the numbers, we see that Kobe's touches per possession increased significantly, perhaps a sign that, even when he gave the ball up in the second half, he kept finding ways to have the ball come back to him.  Also of note is that, while Rondo's overall numbers like usage, touches/possession, and passes/touch all remained fairly consistent, both his shot attempts generated/pass, and assists generated per attempt plummeted, a clear indication of the Lakers defensive improvement in the second half.  Also, in case you were wondering, Rondo played the entire second half, so his team has no "off-court" performance for the half.

Now we put it all together:

Total Ratios Total Kobe Rondo
Team Possessions 66 76 Touches/Possession 1.03 1.21 17%
Touches 68 92 Passes/Touch 0.60 0.78 30%
Completed Passes 41 72 Shot attemts/pass 0.37 0.43 18%
Passes leading to shots 15 31 Assists/shot attempt 0.27 0.32 21%
Assists 4 10 Points per possession on 1.12 0.96
Passes leading to FT's 1 2 Points per possession off 1.06 1.63
Points while on floor 74 73 Usage 0.33 0.20
Total Possessions 83 84 Assists/pass 0.10 0.14
Points off floor 18 13


I was worried about Kobe's extreme halves causing an invalid data set (and to be honest, it is still a concern), but, based on the fact that he ended up pretty close to his season averages for usage, turnovers, and assists, I feel pretty solid about the overall snapshot provided by the data.  Same thing for Rondo, and his numbers provide even more confidence because of how little the foundation statistics changed from one half to the next.  Now that we have all these crazy numbers, time for some context.

What we can see over the course of the game is confirmation of everything we would expect that is both pro and con to the concept of Kobe Bryant as a Black Hole.  On the one hand, he passes the ball much less than Rajon Rondo does when given the opportunity to have the ball.  A difference of 18% overall might not seem like much, but this difference mirrors the difference between Kobe's and Rondo's usage rate on the season, a difference that ranks Kobe as the league's highest usage player, and Rondo as one of the lowest.  On the other hand, we can see evidence of all of the ways the Triangle would prevent the assist from accurately recording how much Kobe Bryant passes the ball.  The offense prevents Kobe from seeing the ball as much as Rondo does, and it prevents his passes from leading to shot attempts.  Further, for what it's worth, and in my mind, it isn't worth much, Kobe's teammates simply did not do as good a job of converting Kobe's assist opportunities into actual assists.  But these are all vague concepts, so let's talk real numbers.  How would Kobe's overall assist totals be different if he had the same "advantages" as Rondo has in generating assists (and no, I'm not talking about the friendly Boston scorekeepers)?

Consider the following two scenarios, which I have broken down at each individual step.

  • In Scenario 1, I take the amount of possessions Kobe spent on the court, and normalize those possessions to Rondo by using Rondo's Touches/Possession, Shot attempts/Pass, and Assists/Shot Attempt ratios, while keeping Kobe's pass/touch ratio.  In simpler terms, we take Kobe's possessions, and put the ball in his hands as much as Rondo has, and assume that Kobe's passes will generate shots at the same rate that Rondo's passes do, and that Kobe's teammates will make the shot at the same rate that Rondo's teammates do.  In the simplest terms, we are treating the two players as if they are the same, with the only differences being how often they decide to pass, and how often they are on the court.  Under this scenario, Kobe would have generated an estimated 7 assists (rounding up) instead of the 4 assists he did generate.  Without rounding, the percentage increase is roughly 67%.
  • In Scenario 2, I take the normalization process even further by giving Kobe the same number of possessions as Rondo has.  In this scenario, Kobe and Rondo have everything the same, and the only difference between the two players is how often they decide to pass with the ball in their hands.  Under this scenario, the estimated assists increases by one to 8, and the percentage increase is 93%
Scenario 1 Scenario 2
Team Possessions 66 76
Touches 80 92
Completed Passes 48 55
Passes leading to shots 21 24
Assists 7 8
% Difference 67% 93%


Taking the numbers at face value, this presents a pretty compelling argument that Kobe's place in the Triangle plays a large role in making his black hole factor appear significantly greater than it is.  If, for example we used the smaller of the two % increases and applied it back to Mr. Ziller's Black Hole chart, Kobe would be taking residence in the area somewhere north of Derrick Rose and Russell Westbrook, much further away from the black hole than the other high usage shooting guards, and much further along the assist axis than almost any shooting guard in the league.  Yes, his usage would still be nearly off the charts (in fact, it would climb to be literally off the chart), but his place on the black hole scale would be improved.

Can we take these numbers at face value. ABSOLUTELY NOT!!  I can not emphasize enough that this entire piece is nothing more than a thought exercise, and should not be used as definitive proof of anything other than "C.A. does some crazy shit with numbers sometimes".  First off, using the sample size of one game is ridiculous.  You might as well make long term investment decisions based on what the Dow did yesterday.  Unfortunately, collecting the information required to do this type of analysis is rather labor intensive, and it makes watching a basketball game about as enjoyable as doing your taxes.  Doing it for this one game was exhausting.  Doing it for a large enough group of games as to validate the sample won't be happening until somebody from the Lakers organization offers me a full time job as an analyst.

Besides the sample size issue, the assumptions used to create our two scenarios are of the elephantine persuasion (i.e. huge).  In order to assume that, if Kobe were running a similar offense to the one that Rondo captains, he would generate shot attempts at the same rate and those shot attempts would be converted at the same rate, one must assume that Kobe is just as good a passer as Rondo is.  Such an assumption is simply impossible to prove.  Kobe is a good passer, but we have no way of knowing, short of a straight up trade that I don't think anybody in the world is interested in, that Kobe would perform as well as Rondo does in the same situation that Rondo has.  We know that the Triangle dictates that Kobe's pass to a teammate often isn't the final piece of the puzzle, and that the Celtics offense hinges much more on Rondo making that final pass.  But we have no idea whether Kobe would be able to make the final pass as well as Rondo does, if he would be able to find Ray Allen with a pass right to the sweet spot, which allows Allen to get his jump shot off that much faster and have a better chance of going in.  And I'm not even getting into any possible athletic advantages Rondo might have over Kobe as it pertains to being a point guard (speed, quickness, and huge hands are just a few factors that would probably fall in Rondo's favor there).

And finally we come to the issue of what happens to all the extra non-shots.  If Kobe has the ball in his hands more, and only some of that extra ball-handling is going towards extra passes, the rest must be going to extra possessions used.  And because there aren't any additional possessions involved, that means his usage would continue to increase past its already high level.  Applying the ratio of Kobe's used possessions to his touches against the higher touches we give Kobe in scenario 1, the extrapolated usage would be about 39%.  The usage does not increase at the same rate as the assists, because not all of the "advantages" we've taken from Rondo and given to Kobe have anything to do with usage, but really, any increase in usage would be cause for alarm.  Going back to the chart, Kobe would be further towards the sun on the x axis, but would be drifting away from the sun on the y axis.  In other words, he'd be drifting off into the abyss that is neither black hole, nor sun, probably towards the gravitational pull of an unseen planet that I can only imagine is named Mamba World.

On the one hand, this data takes the first steps in proving in a real and tangible way that Kobe Bryant's assist totals are skewed because of the offensive system he plays in, because less of his passes lead to shot attempts for his teammates, and it's not because Kobe isn't as good at passing.  On the other hand, we can see that even as Kobe is reaching his (extremely high) average in terms of "using" a lot of his team's possessions, he still sees less of the ball overall than a normal ball-dominant guard, something which lends credence to the idea that his role in the offense limits his usage just as much as it limits his assists.  The next questions to ask would be whether an increase in Kobe's touches would necessitate an increase in his ratio of passes/touches (because it might literally be impossible for Kobe to reach Rondo's touches per possession ratio without an increase in passes)? And is Kobe's high usage rate relative to the low amount of touches he receives within the offense limited by playing the triangle, or is it cause by playing in the triangle, because his role within the offense is to be a finisher?  Those questions are impossible to answer with anything more than opinions and postulates, which basically puts us right back to square one in the argument of Kobe Bryant, Black Hole.

But hey, at least we had some fun, right?

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