Originally Posted by
uh_no
While somewhat enlightening, the chosen number of minutes (15 and 10) are somewhat arbitrary and make the analysis somewhat contrived.
The question we want to ask is, what determines whether someone is in the rotation or not? Sometimes that is very fuzzy. It would seem that if you ordered players on a team by playing time, there should be a point when you say "this guy was in the rotation and this guy was not" You used 15 and 10 minutes, but how do we know that the team didn't have a handful of people at 9 minutes? In essence, we don't.
I think a better way to do this would be to ask the question "what does the distribution of minutes look like across the team"
To do this, we'd start by ordering players by minutes played and graphing the minutes played vs player, which should yield some monotonically decreasing graph. While we could look at this graph and figure out how minutes are used, we'd like 1 or 2 individual metrics to describe usage.
The next step is to calculate the expected value, in other words, from a high level, how for a given minute, what is the average index of the player who used that minute. You do this by sorting the values for minutes played for each player on the team (highest to lowest) and then for each minute value, mutliply the value by its place in the list (1 for the highest minute value, 2 for the next highest....N for the Nth highest) finally dividing the sum of all those by the number of players:
(1*minutes(1)+2*minutes(2)+....N*minutes(N))/200
The higher this number, the more deep the minutes are distributed. This is a good starting point. For teams such as our 2010 title team, the number will be somewhere around 3.5. A team with 10 guys playing 20 minutes scores 5.5. A team with 5 guys playing 40 minutes each scores a 3.
This can be broken, though, as we can (however unlikely) have a team where one or two players get a huge number of minutes while the next 8 get medium minutes which looks identical to a team where the top 7 players play all the minutes equally per the metric i just derived...so how do we come up with a metric which differentiates these two cases? We can compute the kurtosis of minutes away from the top minute getter. This is effectively just the same as the above number, except we take the 4th power of the index
4throot((1*minutes(1)+2^4*minutes(2)+...N^4*minute s(N))/200)
Similar to the above number, the higher the number, the more deeply they are distributed. But in this case, we effectively nullify the amount that the starters play and get a good idea of how deeply the bench is used.
Between these two numbers we will get an idea of both overall how are the minutes distributed, as well as how deeply the bench is used.
If I get a chance I will run a couple teams through this metric to get some baseline numbers and see if we can actually gain something out of this.
This has the added benefit of giving a much more fine grained output than simply number of players who played x minutes.