Goosing your Height

A fun post on reddit asked the question: “I hardly ever see a NBA player that is 6’2, is there a reason why?”

This asks two questions — first, is it actually true that there aren’t very many players that are 6’2″? Or is this just a recall bias? Second, if it’s true, why?

To verify the first question, I pulled all of the player heights from the current NBA rosters, and plotted the distribution of player heights in inches.


What is interesting about this distribution is how weirdly unsmooth it is. There are three jumps that I have marked — between 5’11” and 6’0″, 6’2″ and 6’5″ and 6’6″. Height tends to be relatively bell-curved in its distribution (once you control for gender) and so these types of jumps are very bizarre. First, they suggest that the poster is absolutely correct in thinking it’s bizarre how few 6’2″ individuals he (or she) sees — there are far less than there should be. Second, many basketball players are skewing their heights upwards to 6 foot, 6’3, and 6’6. 

As for reasons why, I’m less sure. There is the easy answer that players tend to just inflate their heights — there’s no set measure on how do the measurement (shoes on or off?) and I don’t think the league particularly cares much. What is funny is the fact that someone cares enough to fake it. These sorts of habits probably begin in high school (I certainly remember saying I was six foot when I wasn’t quite there) and they just die hard.


At the suggestion of a commentator on reddit, I pulled the pre-draft stats on height from DraftExpress. In the interest of speed, I just pulled the heights of everyone measured pre-draft, and plotted the distribution of heights.heightsDraftHist


In this case, we don’t see the same kinds of jumps — there is perhaps a little bit of funny stuff going on between 6 foot and 6’3, but that is likely more because of my bin choice (although I could be wrong). So this confirms our intuition that this is simply a reporting fact, not any underlying real selection factor. 


Paying to Win?

When Russian billionaire Mikhail Prokhorov bought the Nets in 2009, the only fact known by most American basketball fans was that he was extremely rich. Four years into his tenure as owner, his willingness to spend money has become obvious: this year alone, the Nets’ salary is $102 million, putting them at the top of the pile of NBA team salaries. This extraordinarily high salary reflects, on top of large salaries for Joe Johnson and Deron Williams,  the team’s trade for Paul Pierce and Kevin Garnett this past off-season. He himself, following the trade said that It’s not (my) way to wait 10 years for championship.”

Can an owner buy wins? Bulls fans often complain that owner Jerry Reinsdorf is particularly tight-fisted, and that given the income stream the Bulls bring in, he should open his wallet to ensure a championship team. And there really is quite a bit of variation in how much teams pay, as you can see below. The Nets top out at over 100 million, and the bargain basement Phoenix Suns set the bottom at roughly 53 million this year.  (Note: all data comes from



Does a bigger salary translate into more success? The playoffs aren’t over, so I’ll focus on regular season performance (with the strong caveat that there may be a difference between a team that excels in the regular season and a playoff-oriented team). Below is a plot of every team’s total salary this season and winning percentage. Almost all of the teams line up near the middle on a positively sloped line, except for four teams that I’ve highlighted: the Nets, Lakers, Knicks and Celtics.  It’s possible to construct a lot of stories about these four teams — the Lakers are a cautionary tale of injuries (big injuries to Steve Nash and Kobe Bryant), while the Celtics took on several big contracts from the Nets in the trade this off-season (although last season they also had a large salary hit as well).


If we exclude these four teams, we get a strong relationship. While not perfect, this simple linear fit explains about 50% of the variation in win rates. According to this model, for every million dollars spent by a team, they should expect to increase their win rate by roughly a percentage point. A big caveat, of course, is that this is only one year of data. It may be that in other years that “big salary” teams like the Knicks and Nets do much better in the regular season and justify their salaries. The evidence seems to suggest that while higher salaries can translate to better performance on average it is no guarantee for success.


Part of this may be due to restricted free agency and other parts of players contracts that make it possible for teams to hold onto the very best players. Hence, those players that teams like the Nets sign are free agents who are good, but not so good that they get held onto by their team. This would imply that drafting your superstar is extremely important (case in point: Tim Duncan). Alternatively, it may be the winner’s curse – teams bidding with incomplete information tend to overpay, and these big salary teams may be suffering from this (realistically, all players should cause the winner’s curse, but it may be that “superstars” cause more incomplete information).

In a future post, we’ll try to see how stable this relationship is, and how team salary relates to post-season performance.

Defense Doesn’t Pay in the NBA

Earlier this month ESPN unveiled a new advanced basketball metric called “real plus-minus” (RPM). The statistic was developed by Jeremias Engelmann and is trumpeted as an improvement over its predecessors, adjusted plus-minus and real adjusted plus-minus. One of the unique features of RPM is that it splits a player’s impact into offensive RPM (ORPM) and defensive RPM (DRPM). While ESPN has yet to unveil the exact methodology for RPM, early reactions to the stat can be read here, here and here. According to ESPN:

“RPM estimates how many points each player adds or subtracts, on average, to his team’s net scoring margin for each 100 possessions played. The RPM model also yields separate ratings for the player’s impact on both ends of the court: offensive RPM (ORPM) and defensive RPM (DRPM).”

Naturally, we were interested in visualizing the relationship between RPM and NBA salaries. With the exception of a few notable outliers (Stoudemire and Boozer), we found a strong relationship between RPM and annual salary. Once you take into account players who signed large contracts before suffering serious injuries (Stoudemire) and players on the end of lengthy contracts whose skills may have deteriorated (Pau Gasol), on average NBA teams appear to reward higher RPMs with higher salaries. However, as you’ll see in this first graphic there are a lot of players who fall well-below or well-above the line of best fit.


To understand what’s happening here we have to unpack RPM into its offensive and defensive components. The bottom line may not come as a surprise. From a salary perspective, offense trumps defense, by a lot. Merging ESPN’s RPM data with salary information from Basketball-Reference, we estimate that an additional point in ORPM is worth on average $1.4M extra a year in salary while an additional point in DRPM only earns an $700K on average. This makes defense a bargain when you consider that a point scored should be equivalent to a point prevented.

The results are even more striking when we control for position. In essence, we’re now asking for the average relationship between salary and RPM within position. Using the same specification but now with controls for position, we find that a point in ORPM is still valuable at $1.5M (and highly statistically significant) while a point in DRPM falls to $300K (and is no longer statistically significant). To reiterate this, if we compare two players at the same position we find that difference in their salaries is heavily driven by their production on the offensive end and has no connection to their defensive impact. For those who have argued that defense is under-rewarded in the NBA, this may not be a smoking gun, but it is notable.


We can also rank players in the NBA according to value by constructing their predicted salary based on ORPM, DRPM, position and tenure and comparing this to their actual salaries. To be conservative, we dropped anyone who played fewer than 50 games this season. Below is a list of the top and bottom five players by value in three categories: guards, forwards and centers.

Top 5 Players by Position

Guards Fowards Centers
Name Value Name Value Name Value
Vince Carter $6,422,378 Nick Collison $11,015,718 Audray Blatche $8,663,985
Ray Allen $5,464,209 Chris Andersen $9,654,286 Ryan Hollins $6,032,525
Goran Dragic $5,185,118 Mike Miller $8,263,340 Nazr Mohammed $5,131,789
Derek Fisher $4,658,847 Matt Barnes $7,520,160 Alexis Ajinca $4,917,366
Kyle Lowry $4,465,802 Matt Bonner $7,254,578 BeJuan Blair $4,725,142

Bottom 5 Players by Position

Guards Fowards Centers
Name Value Name Value Name Value
Joe Johnson -$13,011,606 Amare Stoudemire -$18,847,008 Dwight Howard -$10,757,813
Dwyane Wade -$12,158,871 Carlos Boozer -$12,417,897 Pau Gasol -$9,139,959
Deron Williams -$8,344,061 Rudy Gay -$11,260,331 Chris Bosh -$9,090,025
Eric Gordon -$7,023,669 Carmelo Anthony -$10,155,075 DeMarcus Cousins -$7,399,312
John Wall -$6,838,249 Blake Griffin -$8,622,789 Roy Hibbert -$5,980,450

We’ll post links to the full lists by position shortly.