David Sparks is the Arbitrarian. His statistics column appears every Thursday here at Hardwood Paroxysm. This week he turns his attention towards the metric of minutes and rotations. Enjoy.

What does it mean to describe a basketball team's rotation? Most commonly, teams are said to have somewhere between an eight and ten-man rotation, implying that N number of players (8, 9, 10, etc.) see significant playing time in a typical game. But what, more specifically, does this mean? What is "significant playing time"? What is a typical game? How can we know, from observation, how many players compose a team's rotation? Do all teams use the same number of players in their rotations? Does the size of a team's rotation vary much over the course of the season?

One difficulty with identifying team rotations is that it isn't as simple as counting then number of players who appear in any given game. There is a subtle difference between identifying the number of players who might be used in a game versus the number of players in a rotation, and that difference mainly has to do with playing time.

As such, it is common to see a threshold of playing minutes employed to identify where the rotation ends. Perhaps the rotation is all players who see more than 10 minutes of playing time in a game... but perhaps the number should be eight minutes. Regardless of the cutoff employed, this method will give an authoritative-sounding answer, but using a minutes cutoff only means that rotation size is a function of the threshold chosen, which is a telltale sign of arbitrariness.

For example, imagine two teams: a team in which five starters play 40 minutes each, and then four additional players play 10 minutes each. Using a 10-minute rotation threshold, we would say that this team has a 9-player rotation. The second team has five starters play 41 minutes each, and four more players play 8.75 minutes each. Again employing the 10-minute cutoff, we note that this team has a 5-player rotation. Certainly there is a difference between the rotations of these two teams-- the second team's minutes are slightly more highly concentrated among the starters--but is the difference equal to a 4-man difference in rotation size? I would submit that it is not. This is a somewhat contrived and extreme example, but I hope it highlights the way in which such arbitrary definitions can be misleading.

Estimating the concentration of playing time

Fortunately, there exists a formalized measure of concentration which we can apply to basketball box score data. The Herfindahl Index is typically applied to the size of firms within an industry, but we can apply it to playing time of players within a game.

Essentially, we eschew choosing an arbitrary minutes threshold in favor of measuring playing time concentration. This avoids the robustness problems of a threshold and gives a continuous (non-exclusively integer) measure of each team's rotation in a given game, arguably increasing both accuracy and precision.

Below I apply the Herfindahl measure of concentration to three very different games:

In the first game, Indiana played only six players in total. Of those six players, five played more than 40 minutes, and the sixth played 15. The Herfindahl Index of concentration for this game is a very high 0.18.

In the second game, 12 players saw 12 or more minutes, with the most playing time being 23 minutes, which is less than half of a game. As you might expect, the Herfindahl Index is much lower here (the index increases with concentration), just 0.09.

The third game sees Minnesota employ 14 players in their quest to defeat Houston. Based on this alone, you might expect even less concentration than in the Sacramento game. However, the distribution of minutes here is much less uniform than above. Seven players saw over 20 minutes of action, the other seven saw less then ten minutes, and four of these had almost negligible floor time. As a result the Herfindahl Index is somewhere between the two games above, at 0.12.

Translating concentrations into rotations

Numbers like 0.18, 0.09, and 0.12 are useful in that they make comparisons easy and consistent, but they bear little resemblance to anything like the rotation numbers we might hope to see, which we're expecting to be in the high single digits, or low double digits.

The conversion from index to rotation, fortunately, is a simple one. Merely take the inverse of the Herfindahl Index, as calculated above, to find the Rotation Index (RX). In our examples above, Herfindahl Indices of 0.18, 0.09, and 0.12 translate to rotations of size 5.55, 11.72, and 8.58, respectively. I think that you will agree that these numbers look very much like what you would subjectively conclude after viewing the above minute distributions.

Indiana's six players, with five dominating, ought to come out somewhere between five and six. Sacramento's nearly even distribution among 12 players should be very close to 12 (although not exactly, because playing times were not identical across the board). Seven Minnesota players saw substantial playing time, and the remainder, collectively (with 32 minutes between them), played enough to count for about 1.5 players if considered as a unit.

Typical team rotations

The above are some extreme examples of rotation size, measured at the game level. However, "a team's rotation," while best assessed at the game level, is better represented summarily over the course of the season than with a single observation. Thus, calculating Rotation Indices for each team, each game, we can come up with an average rotation size over the course of a season, as a better indicator of typical rotation size for a given team.

Below I list, for each of 614 team-seasons in my dataset, a listing of typical rotation sizes. I include the average number of players per game who saw any playing time for comparison, as well as standard deviation of rotation size, to give an idea of how variable was the team's rotation. I have sorted the entries by rotation size, smallest to largest, and highlighted teams from the 2007-08 season, to make them easier to locate.

The first thing to note is that the 1986-87 Celtics top the list, followed by that season's Philadelphia team. Perhaps this is not surprising: the three top Celtics by minutes played that season were Hall-of-Famers, and that team featured Bill Walton, another HoF-er (though he did not play many minutes). Philadelphia featured two Hall-of-Fame players in Barkley and Erving. Small wonder that these teams gave many minutes to players with so much talent. The 2005 Phoenix team was the most successful of the D'Antoni era, and notice also that all five of the most recent Phoenix squads appear in the smallest 42 of this list (four in the top 21--age and O'Neal must have taken their toll on D'Antoni's famously short rotation in 2008).

At the other end of the spectrum, we see five recent San Antonio teams in the top 64 biggest rotations--perhaps this is a small part of the reason that PHO v SAS is always so compelling: along with their very different playing styles and pace, San Antonio uses one of the league's largest rotations, while Phoenix goes with a very small one. There are also nine Utah teams among the 42 biggest rotations, which would seem to indicate a deliberate pattern.

Conclusion

It would appear possible to develop a non-arbitrary and unbiased estimate of team rotation sizes using Herfindahl Indices of concentration of playing time. The season-average results appear to correspond well to common subjective assessments of rotation sizes, ranging from as low as just over seven to as high as just under ten. The league-typical rotation size, by this measure, also aligns with our more qualitative expectations: mean Rotation Index from 1986-2008 is 8.174, and the median is 8.038.

Based solely on these results, it is difficult to discern whether smaller or larger rotations correlate with success. Many good teams appear to have small rotations, but many other good teams appear to have large rotations. In the questionnaire below, I ask your predictions as to the relationship between rotation size and winning, and your reasoning. I hope you will respond.

How well do you feel this metric accurately captures rotation size? Do the figures assigned to each team mesh with your own impressions? Please take a moment to answer the survey, and feel free to leave any questions or observations you might have as comments on this post.

One difficulty with identifying team rotations is that it isn't as simple as counting then number of players who appear in any given game. There is a subtle difference between identifying the number of players who might be used in a game versus the number of players in a rotation, and that difference mainly has to do with playing time.

As such, it is common to see a threshold of playing minutes employed to identify where the rotation ends. Perhaps the rotation is all players who see more than 10 minutes of playing time in a game... but perhaps the number should be eight minutes. Regardless of the cutoff employed, this method will give an authoritative-sounding answer, but using a minutes cutoff only means that rotation size is a function of the threshold chosen, which is a telltale sign of arbitrariness.

For example, imagine two teams: a team in which five starters play 40 minutes each, and then four additional players play 10 minutes each. Using a 10-minute rotation threshold, we would say that this team has a 9-player rotation. The second team has five starters play 41 minutes each, and four more players play 8.75 minutes each. Again employing the 10-minute cutoff, we note that this team has a 5-player rotation. Certainly there is a difference between the rotations of these two teams-- the second team's minutes are slightly more highly concentrated among the starters--but is the difference equal to a 4-man difference in rotation size? I would submit that it is not. This is a somewhat contrived and extreme example, but I hope it highlights the way in which such arbitrary definitions can be misleading.

Estimating the concentration of playing time

Fortunately, there exists a formalized measure of concentration which we can apply to basketball box score data. The Herfindahl Index is typically applied to the size of firms within an industry, but we can apply it to playing time of players within a game.

Essentially, we eschew choosing an arbitrary minutes threshold in favor of measuring playing time concentration. This avoids the robustness problems of a threshold and gives a continuous (non-exclusively integer) measure of each team's rotation in a given game, arguably increasing both accuracy and precision.

Below I apply the Herfindahl measure of concentration to three very different games:

In the first game, Indiana played only six players in total. Of those six players, five played more than 40 minutes, and the sixth played 15. The Herfindahl Index of concentration for this game is a very high 0.18.

In the second game, 12 players saw 12 or more minutes, with the most playing time being 23 minutes, which is less than half of a game. As you might expect, the Herfindahl Index is much lower here (the index increases with concentration), just 0.09.

The third game sees Minnesota employ 14 players in their quest to defeat Houston. Based on this alone, you might expect even less concentration than in the Sacramento game. However, the distribution of minutes here is much less uniform than above. Seven players saw over 20 minutes of action, the other seven saw less then ten minutes, and four of these had almost negligible floor time. As a result the Herfindahl Index is somewhere between the two games above, at 0.12.

Translating concentrations into rotations

Numbers like 0.18, 0.09, and 0.12 are useful in that they make comparisons easy and consistent, but they bear little resemblance to anything like the rotation numbers we might hope to see, which we're expecting to be in the high single digits, or low double digits.

The conversion from index to rotation, fortunately, is a simple one. Merely take the inverse of the Herfindahl Index, as calculated above, to find the Rotation Index (RX). In our examples above, Herfindahl Indices of 0.18, 0.09, and 0.12 translate to rotations of size 5.55, 11.72, and 8.58, respectively. I think that you will agree that these numbers look very much like what you would subjectively conclude after viewing the above minute distributions.

Indiana's six players, with five dominating, ought to come out somewhere between five and six. Sacramento's nearly even distribution among 12 players should be very close to 12 (although not exactly, because playing times were not identical across the board). Seven Minnesota players saw substantial playing time, and the remainder, collectively (with 32 minutes between them), played enough to count for about 1.5 players if considered as a unit.

Typical team rotations

The above are some extreme examples of rotation size, measured at the game level. However, "a team's rotation," while best assessed at the game level, is better represented summarily over the course of the season than with a single observation. Thus, calculating Rotation Indices for each team, each game, we can come up with an average rotation size over the course of a season, as a better indicator of typical rotation size for a given team.

Below I list, for each of 614 team-seasons in my dataset, a listing of typical rotation sizes. I include the average number of players per game who saw any playing time for comparison, as well as standard deviation of rotation size, to give an idea of how variable was the team's rotation. I have sorted the entries by rotation size, smallest to largest, and highlighted teams from the 2007-08 season, to make them easier to locate.

The first thing to note is that the 1986-87 Celtics top the list, followed by that season's Philadelphia team. Perhaps this is not surprising: the three top Celtics by minutes played that season were Hall-of-Famers, and that team featured Bill Walton, another HoF-er (though he did not play many minutes). Philadelphia featured two Hall-of-Fame players in Barkley and Erving. Small wonder that these teams gave many minutes to players with so much talent. The 2005 Phoenix team was the most successful of the D'Antoni era, and notice also that all five of the most recent Phoenix squads appear in the smallest 42 of this list (four in the top 21--age and O'Neal must have taken their toll on D'Antoni's famously short rotation in 2008).

At the other end of the spectrum, we see five recent San Antonio teams in the top 64 biggest rotations--perhaps this is a small part of the reason that PHO v SAS is always so compelling: along with their very different playing styles and pace, San Antonio uses one of the league's largest rotations, while Phoenix goes with a very small one. There are also nine Utah teams among the 42 biggest rotations, which would seem to indicate a deliberate pattern.

Conclusion

It would appear possible to develop a non-arbitrary and unbiased estimate of team rotation sizes using Herfindahl Indices of concentration of playing time. The season-average results appear to correspond well to common subjective assessments of rotation sizes, ranging from as low as just over seven to as high as just under ten. The league-typical rotation size, by this measure, also aligns with our more qualitative expectations: mean Rotation Index from 1986-2008 is 8.174, and the median is 8.038.

Based solely on these results, it is difficult to discern whether smaller or larger rotations correlate with success. Many good teams appear to have small rotations, but many other good teams appear to have large rotations. In the questionnaire below, I ask your predictions as to the relationship between rotation size and winning, and your reasoning. I hope you will respond.

How well do you feel this metric accurately captures rotation size? Do the figures assigned to each team mesh with your own impressions? Please take a moment to answer the survey, and feel free to leave any questions or observations you might have as comments on this post.