Thread by @IneffectiveMath

I have prepared a small bedtime story about a thought experiment for what kind of talent distribution you ought to expect among players in an "apex" sporting league. (It is a bedtime story because it is heavily simplified and relies primarily on the pictures.)

For argument's sake let's use the nhl as our "apex" league, but there's no data here, only expectations. I have been thinking about this for the sake of setting priors for my models, so I want something "pre-data".

(If you want data instead, you can also see some compiled by @hockeystatisti1
here:





)hockey-statistics.com/2021/08/07/tal…
hockey-statistics.com/2021/08/14/tal…

Imagine a large population of hockeyers, something like "everybody who has non-preposterous thoughts about playing in the apex league some day". You'd expect this population to have a normal distribution of hockeying ability.

Here I've chosen the 0 on my hockeying scale to be average, with no meaning attached to the x scales or the y-scales. We're using numbers here because we can't avoid it, really this thread is about shapes. That's the starting point here, the normal shape.

The process of going from "I know how to hockey" to "I am nhl player" is a convoluted one, but we still expect it to follow some broadly sensible pattern where the "good enough" players mostly make and not good enough players mostly don't.

Let's fix our scale of "nhl calibre" at x = 1, and then assume that your chance of making "the jump" is a logistic function centred there. That's what logistic functions are for, to turn "goodness of thing" into "chance of result".

So if you're (exactly) "nhl calibre" you have a 50% chance of making, if you're considerably better than that, you're a lock; if considerably worse, you have only a tiny chance of eking out some strange role somewhere.

So the expected talent distribution of the apex league itself should have a shape that follows the shape of the larger population multiplied by the chance of making it to the smaller "apex" population, which looks like this:

The most common ability (blue line) is a little better than "replacement level" (grey). The /average/ ability (red line) is higher still. The shape of the talent distribution is skewed to the right, towards the better players.

Or, zoomed in to show the pattern near the interesting bits more.

57% of the league is "below average"; that sliver between median and mean is 7% of the league.

The location parameters (the big pop centres at 0, the logistic centres at 1) are simple enough but then both distros have shape parameters (how steep, basically, for both) and by tinkering with them you can alter the slivers without changing the overall shape.

If you wanted something more sophisticated, you could change the chance-of-making-it function to target icetime instead. That would presumably skew even more towards the stronger players, since they (mostly) play more.