APM=Skill – A Monte-Carlo Simulation (Definitive Edition)
Using data from SC2replaystats (n=2,892,019):
https://sc2replaystats.com/stats/analytics?group_by%5B%5D=player_division&matchup%5B%5D=PvT&matchup%5B%5D=PvZ&matchup%5B%5D=TvZ&format%5B%5D=1v1&game_type%5B%5D=AutoMM&metric_options%5B%5D=games&metric_options%5B%5D=avg_player_apm&metric_options%5B%5D=avg_player_mmr&date=January%201,%202020%20-%20December%2031,%202020
APM correlates with MMR with an R^2 value of 0.997 using a 2 term polynomial equation:
APM = 0.00000119 * MMR^2 + 0.05411901 * MMR^1 + -53.28971110
We can see that clearly in this chart:
https://i.imgur.com/u9CfWkd.png
Given this equation we can input an MMR value and get the average APM of a player (assuming no scattering). First we need an MMR distribution that matches the ladder. Experimentally this was determined as:
MMR ~ N(4500, 170)
This produces players >5000 mmr at a rate of ~0.2% which matches the parameters of GM league of the NA ladder. We can verify the sample size for the simulation (n=7,000) is large enough to achieve normalization using a histogram chart:
https://i.imgur.com/eSb5rrh.png
Using data from a Blizzard analysis available here:
https://conf.splunk.com/files/2019/slides/FN1698.pdf
We know the correlation between APM and win-rate is ~0.64. Using this figure, we can estimate the variance in APM on a per-player basis such that corr(APM, MMR)=0.64. We do this by taking an MMR value, plugging it into the polynomial equation above that calculates average APM assuming no scattering, then adding scattering, which is assumed to be normally distributed. Experimentally this was determined to be:
Scattering ~ N(0, 13)
Which has achieved the desired correlation of corr(Fapm(MMR)+scattering, MMR)=0.649, And can be seen on this chart:
https://i.imgur.com/YXbOVZb.png
The spreadsheet output of the simulation is available for download (CSV format):
https://controlc.com/1775c004
Thus it is proven: APM correlates with MMR at such an insanely high rate that other factors only scatter the result at most +/- 13 APM in 68% of cases, +/- 21 in 95% of cases, +/- 36 in 99.7% of cases. So if you think you are some kind of “Strategy” or “Game Knowledge” GOD, then congrats, you netted yourself what a whopping +13 APM could have netted you instead.