Quote:
Originally Posted by Jared Russell
Another trivia question for you stat-gurus out there to chew on...
Given random assignment of teams to 8 divisions, what is the probability that any division's [mean OPR, 90th %ile OPR, 75th %ile OPR, 50th %ile OPR, etc.] is as strong as Newton's?
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As I suggested above, I looked into this very question.
TL;DR: I estimate that as a competitor you would expect to be in a division as stacked as Newton (or more stacked) once every 250-500 years. Further as a spectator, you would would expect to attend a Worlds with a division as stacked as Newton (or more stacked) once every 40-70 years.
A lot depends on the method you use to define what we mean when we say Newton is stacked. I decided that I would plot the MAX OPR for the Nth Percentile team for a division and compare that to the same Nth Percentile for the CMP Population as a whole.
This chart helps you see what I mean.
This chart is better because I do the subtraction (DIV OPR %tile - CMP OPR %tile).
So NOW I can propose my metric. Let's integrate the area under the curve for the above chart in some range and then normalize for the width of the range the area.
But what range do we want to use?
Here are two charts for proposed ranges.
The first is my "I think this would be a fair metric" range (55%tile to 95%tile). My thinking is that it eliminates the very top of the range because, Hey, whenever a handful of powerhouse teams end up in a division, the top is off the charts, that's just normal.
My second is "Let's tailor the metric as much as we can to be favorable to Newton" That is, pick the range so that it measures where Newton opens up a big gap compared to the CMP as a whole. Coincidentally that also is the top 24 Teams in a 75 Division so, if those teams make up the Playoff teams they'd have a pretty amazing set of 8 Alliances.
Now I can do a simulated Division Assignment. I did it (almost) like FIRST is rumored to do it. I divided the rookies teams and the non-rookies randomly but separately. For Rookies, the rule should be that teams with numbers 5726 and higher but that gave me a number of rookies that wasn't evenly divisible by 8. For reasons that are hard to explain, it was easier for my simulation if I had the number of rookies be a divisible by 8 so I added 3 teams to my "rookie group" (5686, 5690, & 5712) I don't think this changed my results significantly.
After running 2,500 simulated CMP Division Assignments (that is 20,000 Divisions), I get this when you look at the distribution of Division Stacked Metric (only showing chart for Method 1, but Method 2 was similar).
Looking at the data, it is easy to see that 2,500 CMP's are plenty when it comes to characterizing the variation involved.
Long story short.
Based on Method #1:
I estimate that as a competitor you would expect to be in a division as stacked as Newton (or more stacked) once every 250 years. Further as a spectator, you would would expect to attend a Worlds with a division as stacked as Newton (or more stacked) once every 40 years.
Based on Method #2:
I estimate that as a competitor you would expect to be in a division as stacked as Newton (or more stacked) once every 500 years. Further as a spectator, you would would expect to attend a Worlds with a division as stacked as Newton (or more stacked) once every 67 years.
So... there you have it. Any way you cut it, Newton is one wacky division this year.
Also, FIRST should probably think about how they make the divisions a bit more to avoid this kind of think in the future.
Comments welcome.
Dr. Joe J.