Pop Quiz: Who are the top ten rebounders in the NBA for the 2016-17 NBA Season?

If you said…

1. Andre Drummond
2. DeAndre Jordan
3. Hassan Whiteside
4. Rudy Gobert
5. Karl-Anthony Towns
6. Dwight Howard
7. Anthony Davis
8. Russell Westbrook
9. Marcin Gortat
10. DeMarcus Cousins

… you might be right. These are the players that have the most rebounds for the 2016-17 NBA season. But that wasn’t the question. The question was who are the top ten rebounders? The question isn’t about who secures the most rebounds, but rather who gives the best chance for a team to secure a rebound. A player that secures the most rebounds may be a player designed to be in the right place at the right time with a slate of low-percentage shooting teammates.

In this article, we will break down the reason why we need to perform the advanced analytic known as Rebound Rate as opposed to straight counting; and then take a moment to actually understand the intricacies of rebounding rate. Specifically to understand where this analytic goes awry.

Counting Boards: Raw Numbers vs. Rates

For instance… who is a better rebounder: Rudy Gobert (Utah) or Alan Williams (Phoenix)? Gobert obtained 1,035 rebounds while Williams secured 292 rebounds. We’d easily see that Gobert has 700+ more rebounds. However, Gobert averaged 33.9 minutes per game over 81 games. Williams? 15.1 minutes over 47 games.

Comparing time available to play, Gobert averaged 0.3769 rebounds per minute while Williams averaged 0.4114 rebounds per minute. Now we see that Williams has a better rate. This would suggest that Williams may have a better rebounding rate than Gobert. Unfortunately this would be true if the Phoenix Suns and the Utah Jazz were identical in every which way this past season. Hint… they weren’t. In fact, Utah missed 3,482 field goals compared to 3,990 for Phoenix. Similarly, Utah’s opponents missed 3,745 field goals compared to 3,770 for Phoenix.

Not including free throws that result in rebounds, we see that Utah has at least 7,227 chances for rebounds while Phoenix has 7,760 chances. That’s at least 6.5 more chances at rebounds per game for Phoenix over Utah; 0.1354 more chances per minute. So does Alan Williams look all that much better?

Moral of the Classic Rebounding Story

The moral of this story is that straight counting is misleading. A player with more rebounds is not necessarily better; they may either have more minutes or more opportunities than their counterparts. Therefore a more sluggish rebounder may obtain more rebounds than a more reliable rebounder. A rebounder is limited by the number of opportunities they are afforded in a game. Hence some metric must be devised to help understand how well a rebounder rebounds given the opportunities in a game. One such metric is the Rebounding Rate.

Rebound Rate

The rebound rate of a player is a function of playing time and number of rebounds. It is a box score calculation. That is, possession data is not used. Keep this in mind as we go along.

The rebounding rate is given as follows:

Let’s break down this formula. The first part we are interested in is the actual percentage of rebounds obtained by a player of interest. This part is given by:

Rebounds / (Team Total Rebounds + Opposing Team Total Rebounds)

This is a percentage of all possible rebounds obtained by the player out of all possible rebounds. Think of this as the raw percentage of rebounds for a player. This is a misleading ratio as there are only a handful of players that get large amounts of minutes. Therefore, the second part we are interested is a time scaling. This part is given by:

Minutes Played / (Team Minutes Played / 5)

The value Team Minutes Played / 5 is a reduction from the total of minutes played by the team to the number of minutes of the game. We cannot say 48 minutes (or for a season; 3,936 minutes) as some games go into overtime. Hence we have the percentage of minutes played by a player.

Now we divide these two numbers! The resulting value is the percentage of all rebounds by a particular player per minute compared to the number of possible rebounds per minute possible. Hence, a rebounding rate.

At this point, the value of 100 merely moves the rate to a percentage… More on this later*. Despite this adjustment, the rebounding rate reflects the rate of a player obtaining rebounds compared to the rate of which rebounds are obtained.

So Who’s the Best Rebounders?

If we take Rebound Rate as a metric of the best rebounders; after all they have the “highest rates of obtaining a rebound,” then we obtain this list:

1. Andre Drummond
2. DeAndre Jordan
3. Hassan Whiteside
4. Dwight Howard
5. Tyson Chandler
6. Alan Williams
7. Rudy Gobert
8. Thomas Robinson
9. Andrew Bogut
10. Joakim Noah

Yikes. That’s five players who couldn’t even reach 50 games in a season. Well, maybe they are the best rebounders when they are healthy. But how do we really know? How do we know that Alan Williams is a better rebounder than Rudy Gobert? (I don’t buy that, by the way) The way we do that is analyze this metric.

Case Study of Rebound Rate: Silly Example

Let’s start with a childish example. Consider a single game that gets out of hand. Let’s suppose the following stats for a game Team A won 84 – 42 over Team B:

Team A: 42 – 87 FG; 0 – 0 FT; 0 – 0 3FG; 12 OREB; 43 DREB; 0 TO

Team B: 21 – 64 FG; 0 – 0 FT; 0 – 31 3FG; 0 OREB; 23 DREB; 0 TO

What an ugly game… Now suppose that Player Q on Team A comes in with one minute remaining in this game and rebounds three garbage attempts from beyond the arc. Team A responds by running up the score and makes a couple baskets in this minute. What is player Q’s rebound rate?

By the formula above, we obtain a numerator of 100 * 3 Rebounds * 48 minutes possible = 14,400. Similarly, the denominator is 1 Minute Played * ( 55 Team A Boards + 23 Team B Boards) = 78. This results in 184.615%. How do we interpret this?

We witnessed player Q pick up three boards on three total rebounds chances over a minute of the game. In total, Player Q picked up a mere 3.8462% of all possible rebounds. Going back NBA D/G-League’s to Steve Weinman’s explanation:

It measures the percentage of available rebounds a player obtained while on the court, dividing a player’s rebounds by the total number grabbed by both teams while he was in the game. Also called, drumroll, please, rebound percentage, this metric controls for pace, minutes and shooting percentages and offers a fair basis of comparison for players and teams across the league. The best rebounders are those who collect boards with the greatest frequency given their opportunities to do so.

Now is this really a percentage? We just obtained a player with over 100% rebounding percentage.

Rebound Rate’s Upper and Lower Bounds

So now that we have an example where rebounding rate is not a percentage; let’s look at the range of possible scores.

Minimum Value

To obtain a minimum value, a player must never get a rebound. Since all numbers are positive with no subtractions, we cannot have a negative rebound rate. Hence the minimum is zero. This is good!

Maximum Value

Now the maximum value leads us into some severe problems. For instance, a player cannot get more rebounds than the number of rebounds in a game. Let’s maximize that and say he gets all rebounds. Then, the rebound rate becomes 100*Team Minutes Played / (5 * Minutes Played). If this player misses a single minute, his rebounding rate goes above one. This is possible as all field goal attempts are made while the player is sitting out.

*This proves that the rebounding rate is not a percentage; contradicting the commentary by Weiman in his 2011 article. So since this isn’t really a rate either… more on this soon**… then what are we missing?

Fraction Adding

In fact, let’s look at all players on the same team. Since the total number of rebounds is constant for a game and the total number of minutes played by a team is constant, we can factor theses out. We can even factor out the constants 5 and 100. This leaves us with the fraction

Rebounds / Minutes Played

for each player on the team. According to the description above from the NBA, we should be able to add up these rates to obtain 1. We already know this is not true, but this addition amounts to the following for, say, ten players:

R1 / M1 + R2 / M2 + … + R10 / M10

**This analytic assumes that this sum is the same as

(R1 + R2 + … R10) / (M1 + M2 + … + M10)

Which… is not the case. Due to this flaw, players that are successful rebounders who see limited minutes skyrocket up the charts. And if we take a look above; we see exactly that. However, this is not the only major flaw…

Rebounding Rate Assumes Uniformity.

We are missing the rate of rebounds. In the Rebound Rate metric, it is assumed that all rebounds are uniformly distributed across the game. This is relayed through the fact that part one of the equation (proportion of rebounds) is merely multiplied against the percentage of playing time. This is the primary reason that can legitimately obtain rebounding rates/percentages over 100%.

Instead, we should find the rate of which a player rebounds against his chances, relative to the chances during his part of the game. In fact, minutes should have nothing to do with rebounding rates as it is the obvious inflation device in this metric.

In fact, if we take a look at the distribution of rebounds over a game, we find that there is a band about the uniform distribution. If rebounds occurred uniformly, we would expect that rebounds a collected at a one-percent every 28.8 second rate. This means if there are 100 rebounds in a game, we expect there to be a rebound every 28.8 seconds. Similarly, if there are 80 rebounds in a game, this equates to one rebound every 36 seconds; or 0.8 rebounds every 28.8 seconds. This turns into a slope of 0.034722 percent per minute.

Let’s illustrate with the Sacramento Kings for the 2016-17 NBA Season:

Performing a Kolmogorov-Smirnov goodness of fit test, we find that each game is effectively not uniformly distributed; however the mean is relatively close to uniformity (it fails too, but at a much tighter significance level). Regardless, this illustration shows that the player rebounding rate is flawed severely in distribution as a/b + c/d is indeed not (a+c)/(b+d); which is required to work for comparison of players across games.

Use Sampling Theory… Prepare to be Underwhelmed

Instead, we should apply some sampling theory to obtain a better idea of rebounding effectiveness of a player. Consider the total number of rebounds possible as a population. Each sampled rebound is then given a label of a player. If we consider the time a player is on the court, we effectively yield a stratification for sampling. That is, we limit the number of possible samples available over that sampling window.

To clarify, a stratified sample is an allotment of samples given to a population that may be under-represented. As an illustration, consider a school dominated by a male population: 990 males, 10 females. If 20 free lunch tickets are handed out to the students, we can give away tickets at random and almost guarantee no females obtain a ticket. Or, we can ensure that at least 1 or 2 tickets are guaranteed to go to females. This enforcement is called stratification. In our situation for rebounding, the ticket population is the number of rebounds possible for a player to obtain when they are in the game. The stratification of tickets corresponds to the number of available rebounds to that player within the game. If we compute the classical stratification, we then compute:

rebound percentage = (number of rebounds when player on court / number of rebounds total) x (number of rebounds by player / number of rebounds when player on court)

Hence the actual ability of the player is number of rebounds for that player given the number of rebounds available when that player is on the court. This indicates that the rebound percentage is a post-stratification using the uniform distribution on rebounds over the course of a game; across all 1230 games. Hence, we should really look at the proportion of rebounds that a player actually obtains.

In this case, we obtain the following top 10:

1. Danuel House 50.0000
2. Andre Drummond 25.2262
3. DeAndre Jordan 24.3769
4. Hassan Whiteside 23.9551
5. Boban Marjanovic 23.8532
6. Larry Sanders 23.5294
7. Dwight Howard 23.4064
8. Tyson Chandler 23.0145
9. Thomas Robinson 22.5252
10. Rudy Gobert 21.6618

If you notice, there are a couple players that creep in with small samples. Danuel House (WAS) grabbed 1 rebound in his only game (one minute played) when there were only two chances. Similarly, Larry Sanders (CLE) managed to play in 5 games for a total of 13 minutes played. Over that time he secured 4 of 17 rebounds.

Eliminating small samples, we find that Alan Williams (21.4286) and Dewayne Dedmon (21.1604) round out a legitimate top ten. The inclusion of Boban Marjanovic (DET) is an edge case as he was present for 545 possible rebounds, securing 130 over a 293 minute clip. The time here does not really matter for rebounding rate; as it merely indicates that Marjanovic is allowed to sample a little faster than Larry Sanders. This yields no impact on true rebounding rate as far as potential rebounds are concerned.

Not All Bad

Given the above arguments showing that rebounds are indeed not uniform and that pace does not dictate rebound percentage; the rebound rate is not a bad measure for teams only. This is because rates are uniform over the entire game. That is, there will always be zero rebounds at the beginning of the game and all rebounds at the end of the game. They are also additive as the denominator for a team is constant within a team. This means if a player gets 3 of total team rebounds and another player gets 4 of total team rebounds, then we obtain 7 of total team rebounds. The math works here as opposed to 3 of total 12 and 4 of total 17 being supposedly 7 of say 15 (assuming the players overlap for two possible rebounds). This math doesn’t work out, despite the NBA D-League reports suggesting otherwise.

However, needed a better metric for players; we should consider the pre-stratification estimator, which is unbiased for the player and shows the true merit of a player on the court; not the “rate of which they played” that can inflate their results.