I loved this series of posts by Benjamin Morris (who now writes for FiveThirtyEight) on Dennis Rodman. The “major prongs” of his argument, as he calls it, are as follows:

Rodman was a better rebounder than you think:Rodman’s ability as a rebounder is substantially underrated. Rodman was a freak, and is unquestionably — by a wide margin — the greatest rebounder in NBA history. In this section I will use a number of statistical metrics to demonstrate this point (preview factoid: Kevin Garnett’s career rebounding percentage is lower than Dennis Rodman’s career *offensive* rebounding percentage). I will also specifically rebut two common counterarguments: 1) that Rodman “hung out around the basket”, and only got so many rebounds because he focused on it exclusively [he didn’t], and 2) that Bill Russell and Wilt Chamberlain were better rebounders [they weren’t].Rodman’s rebounding was more valuable than you think:The value of Rodman’s rebounding ability is substantially underrated. Even/especially by modern efficiency metrics that do not accurately reward the marginal value of extra rebounds. Conversely, his lack of scoring ability is vastly overrated, even/especially by modern efficiency metrics that inaccurately punish the marginal value of not scoring.Rodman was a bigger winner than you think:By examining Rodman’s +/-with respect to wins and losses — i.e., comparing his teams winning percentages with him in the lineup vs. without him in the lineup — I will show that the outcomes suggest he had elite-level value. Contrary to common misunderstanding, this actually becomes *more* impressive after adjusting for the fact that he played on very good teams to begin with.Rodman belongs in the Hall of Fame [or not]: [Note this section didn’t go off as planned. Rodman was actually selected for the HoF before I finished the series, so section 4 is devoted to slightly more speculative arguments about Rodman’s true value.] Having wrapped up the main quantitative prongs, I will proceed to audit the various arguments for and against Rodman’s induction into the Hall of Fame. I believe that both sides of the debate are rationalizable — i.e., there exist reasonable sets of preferences that would justify either outcome. Ultimately, however, I will argue that the most common articulated preferences, when combined with a proper understanding of the available empirical evidence, should compel one to support Rodman‘s induction. To be fair, I will also examine which sets of preferences could rationally compel you to the opposite conclusion.

This part of the series regarding so-called “ambicourtedness” – the ability to dominate rebounding at both ends of the court – blew my mind.

The key point here is that, normally, you can be a great offensive rebounder, or you can be a great defensive rebounder, but it’s very hard to be both. Unless you’re Dennis Rodman:

To prepare the data for this graph, I took the top 1000 rebounding seasons by total rebounding percentage (the gold-standard of rebounding statistics, as discussed insection (a)), and ranked them 1-1000 for both offensive (ORB%) and defensive (DRB%) rates. I then scored each season by the

higher(larger number) ranking of the two. E.g., if a particular season scored a 25, that would mean that it ranks in the top 25 all-time for offensive rebounding percentageandin the top 25 all-time for defensive rebounding percentage (I should note that many players who didn’t make the top 1000 seasons overall would still make the top 1000 for one of the two components, so to be specific, these are the top 1000 ORB% and DRB% seasonsof the top 1000 TRB%seasons).This score doesn’t necessarily tell us who the best rebounder was, or even who was the most balanced, but it should tell us who was the strongest in the weakest half of their game (just as you might rank the off-hand of boxers or arm wrestlers). Fortunately, however, Rodman doesn’t leave much room for doubt: his 1994-1995 season is #1 all-time

on both sides. He has 5 seasons that are dual top-15, while no other NBA player has even a single season that ranks dual top-30. The graph thus shows how far down you have to go to find any player withnnumber of seasons at or below that ranking: Rodman has 6 seasons register on the (jokingly titled) “Ambicourtedness” scale before any other player has 1, and 8 seasons before any player has 2 (for the record, Charles Barkley’s best rating is 215).

More:

The problem, though, is that there are both competitive and physical limitations to how much someone can really excel at both simultaneously. Not the least of which is that offensive and defensive rebounds literally take place on opposite sides of the floor, and not everyone gets up and set for every possession. Thus, if someone wanted to cheat toward getting more rebounds on the offensive end, it would likely come, at least in some small part, at the expense of rebounds on the defensive end. Similarly, if someone’s playing style favors one, it probably (at least slightly), disfavors the other. Whether or not that particular factor is in play, at the very least you should expect a fairly strong regression to the mean: thus, if a player is excellent at one or the other, you should expect them to be not as good at the other, just as a result of the two not being perfectly correlated. To examine this empirically, I’ve put all 1000 top TRB% seasons on a scatterplot comparing offensive and defensive rebound rates:

Clearly there is a small negative correlation, as evidenced by the negative coefficient in the regression line. Note that technically, this shouldn’t be a linear relationship overall – if we graphed every pair in history from 0,0 to D,R, my graph’s trendline would be parallel to the tangent of that curve

as it approaches Dennis Rodman. But what’s even more stunning is the following:Rodman is in fact not only an outlier, he is such a ridiculously absurd alien-invader outlier that when you take him out of the equation, the equation changes drastically: The negative slope of the regression line nearly doubles in Rodman’s absence. In case you’ve forgotten, let me remind you that

Rodman only accountsfor 12 data points in this 1000 point sample:If that doesn’t make your jaw drop, I don’t know what will! For whatever reason, Rodman seems to be supernaturally impervious to the trade-off between offensive and defensive rebounding. Indeed, if we look at the same graph with only Rodman’s data points, we see that, for him, there is actually an extremely steep, upward sloping relationship between the two variables:In layman’s terms, what this means is that Rodman comes in varieties of Good, Better, and Best — which is how we would expect this type of chart to look if there were no trade-off at all. Yet clearly the chart above proves that such a tradeoff exists! Dennis Rodman almost literally defies the laws of nature (or at least the laws of probability).

Rodman was better at rebounding than Jordan ever was at scoring. I dare you to read through this section and not have the hairs at the back of your neck stand up:

For the uninitiated, the main stat I will be using for this analysis is “rebound rate,” or “rebound percentage,” which represents the percentage of available rebounds that the player grabbed while he was on the floor. Obviously, because there are 10 players on the floor for any given rebound, the league average is 10%. The defensive team typically grabs 70-75% of rebounds overall, meaning the average rates for offensive and defensive rebounds are approximately 5% and 15% respectively. This stat is a much better indicator of rebounding skill than rebounds per game, which is highly sensitive to factors like minutes played, possessions per game, and team shooting and shooting defense. Unlike many other “advanced” stats out there, it also makes perfect sense intuitively (indeed, I think the only thing stopping it from going completely mainstream is that the presently available data can technically only provide highly accurate “estimates” for this stat. When historical play-by-play data becomes more widespread, I predict this will become a much more popular metric).

Dennis Rodman has dominated this stat like few players have dominated any stat. For overall rebound % by season, not only does he hold the career record, he led the league 8 times, and holds the top 7 spots on the all-time list (red bars are Rodman):

^{Note this chart only goes back as far as the NBA/ABA merger in 1976, but going back further makes no difference for the purposes of this argument. As I will explain in my discussion of the “Wilt Chamberlain and Bill Russell Were Rebounding Gods” myth, the rebounding rates for the best rebounders tend to get worse as you go back in time, especially before Moses Malone.}

As visually impressive as that chart may seem, it is only the beginning of the story. Obviously we can see that the Rodman-era tower is the tallest in the skyline, but our frame of reference is still arbitrary: e.g., if the bottom of the chart started at 19 instead of 15, his numbers would look even more impressive. So one thing we can do to eliminate bias is put the average in the middle, and count percentage points above or below, like so:With this we get a better visual sense of the relative greatness of each season. But we’re still left with percentage points as our unit of measurement, which is also arbitrary: e.g., how much better is “6%” better? To answer this question, in addition to the average, we need to calculate the standard deviation of the sample (if you’re normally not comfortable working with standard deviations, just think of them as standardized units of measurement that can be used to compare stats of different types, such as shooting percentages against points per game). Then we re-do the graph using standard deviations above or below the mean, like so:

^{Note this graph is actually exactly the same shape as the one above, it’s just compressed to fit on a scale from –3 to +8 for easy comparison with subsequent graphs. The SD for this graph is 2.35%.}

There is one further, major, problem with our graph: As strange as it may sound, Dennis Rodman’s own stats are skewing the data in a way that biases the comparison against him. Specifically, with the mean and standard deviation set where they are, Rodman is being compared to himself as well as to others. E.g., notice that most of the blue bars in the graph are below the average line: this is because the average includes Rodman. For most purposes, this bias doesn’t matter much, but Rodman is so dominant that he raises the league average by over a percent, and he is such an outlier that he alone nearly doubles the standard deviation. Thus, for the remaining graphs targeting individual players, I’ve calculated the average and standard deviations for the samples from the other players only:

^{Note that a negative number in this graph is not exactly a bad thing: that person still led the league in rebounding % that year. The SD for this graph is 1.22%.}

But not all rebounding is created equal: Despite the fact that they get lumped together in both conventional rebounding averages and in player efficiency ratings, offensive rebounding is worth considerably more than defensive rebounding. From a team perspective, there is not much difference (although not necessarily *no* difference – I suspect, though I haven’t yet proved, that possessions beginning with offensive rebounds have higher expected values than those beginning with defensive rebounds), but from an individual perspective, the difference is huge. This is because of what I call “duplicability”: simply put, if you failed to get a defensive rebound, there’s a good chance that your team would have gotten it anyway. Conversely, if you failed to get an offensive rebound, the chances of your team having gotten it anyway are fairly small. This effect can be very crudely approximated by taking the league averages for offensive and defensive rebounding, multiplying by .8, and subtracting from 1. The .8 comes from there being 4 other players on your team, and the subtraction from 1 gives you the value added for each rebound: The league averages are typically around 25% and 75%, so, very crudely, you should expect your team to get around 20% of the offensive and 60% of the defensive rebounds that you don’t. Thus, each offensive rebound is adding about .8 rebounds to your team’s total, and each defensive rebound is adding about .4. There are various factors that can affect the exact values one way or the other, but on balance I think it is fair to assume that offensive rebounds are about twice as valuable overall.To that end, I calculated an adjusted rebounding % for every player since 1976 using the formula (2ORB% + DRB%)/3, and then ran it through all of the same steps as above:

Mindblowing, really. But before putting this graph in context, a quick mathematical aside: If these outcomes were normally distributed, a 6 standard deviation event like Rodman’s 1994-1995 season would theoretically happen only about once every billion seasons. But because each data point on this chart actually represents a maximum of a large sample of (mostly) normally distributed seasonal rebounding rates, they should instead be governed by the Gumbel distribution for extreme values: this leads to a much more manageable expected frequency of approximately once every 400 years (of course, that pertains to the odds of someone like Rodman coming along in the first place; now that we’ve had Rodman, the odds of another one showing up are substantially higher). In reality, there are so many variables at play from era to era, season to season, or even team to team, that a probability model probably doesn’t tell us as much as we would like (also, though standard deviations converge fairly quickly, the sample size is relatively modest).

Rather than asking how abstractly probable or improbable Rodman’s accomplishments were, it may be easier to get a sense of his rebounding skill by comparing this result to results of the same process for other statistics. To start with, note that weighting the offensive rebounding more heavily cuts both ways for Rodman: after the adjustment, he only holds the top 6 spots in NBA history, rather than the top 7. On the other hand, he led the league in this category 10 times instead of 8, which is perfect for comparing him to another NBA player who led a major statistical category 10 times — Michael Jordan:

^{Red bars are Jordan. Mean and standard deviation are calculated from 1976, excluding MJ, as with Rodman above.}As you can see, the data suggests that Rodman was a better rebounder than Jordan was a scorer. Of course, points per game isn’t a rate stat, and probably isn’t as reliable as rebounding %, but that cuts in Rodman’s favor. Points per game should be more susceptible to varying circumstances that lead to extreme values. Compare, say, to a much more stable stat, Hollinger’s player efficiency rating:

Actually, it is hard to find any significant stat where someone has dominated as thoroughly as Rodman. One of the closest I could find is John Stockton and the extremely obscure “Assist %” stat:

^{Red bars are Stockton, mean and SD are calculated from the rest.}Stockton amazingly led the league in this category 15 times, though he didn’t dominate individual seasons to the extent that Rodman did. This stat is also somewhat difficult to “detangle” (another term/concept I will use frequently on this blog), since assists always involve more than one player. Regardless, though, this graph is the main reason John Stockton is (rightfully) in the Hall of Fame today.

Read the rest of the series here.