We can calculate the probability of one race having higher skilled players. Activeplayer.io says there are 400k active sc2 players. That means serral is a sigma ~4.5. Given the MMR distribution of the ladder (u=2900,o=882), serral is a sigma 4.6. So the figures corroborate one another. Now the question is what the probability is that we can generate 400k players at random using the ladder population parameters (aka N(u=2900,o=882)) and we randomly assign players to each race (assuming race selection is totally random), what’s the probability of having equally skilled top-tier players? Well this chart shows exactly that:
https://i.imgur.com/luhYoAx.png
X axis the difference between groups. Y axis is the probability of occurring (0.0 = 0%, 1.0 = 100%). This is a cumulative probability chart, meaning it’s the probability of >= X occurring. The blue line is the largest difference between the highest ranked player in each group. The red is the difference in the average of the top 5. X axis is measured in standard deviations, so 1 standard deviation is equivalent to 882 mmr aka a 99% win-rate.
If we look at the blue line at 0.5 on the Y axis, we can see there is a 50% chance a >=0.3 standard deviation difference in skill will occur between the top players for each race. That’s 265 mmr. So there’s a 50% chance the SC2 population could produce a >=265 skill difference between the best and worst performing top player from each race. That’s equivalent to an 82% win-rate in favor of the better player.
Translation, it’s highly likely that professional Protoss are simply worse at the game compared to their competitors.
What’s rather striking about this number of 265 is that it’s remarkably similar to the amount of MMR advantage that Protoss has in grandmaster. This mmr advantage can be calculated 3 ways. The easiest way is to simply count up from the bottom of grandmaster how many protoss you’d need to demote to equalize GM to 33/33/33. Then you take the difference from that protoss’ mmr to the bottom of GM, and that’s the advantage protoss has. It’s about 300. In other words we’ve calculated the size of the advantage of protoss using a new, 4th metric, aka the population parameters of the sc2 ladder & the likely difference in skill between outliers. Isn’t statistics fun.
So this is how SC2 balance actually works. Serral is simply ~0.3 standard deviations more talented than his Protoss competitors, and, because the balance counsel is incapable of understanding basic math, they’ve nerfed Zerg to equalize Serral’s advantage, which resulted in Protoss having an advantage across the ladder equal in size to the difference in skill between Serral and his nearest Protoss competitor.
This opens up another way to calculate the imba. Sigma ~4.5 outliers should occur too frequently because toss is given 300 mmr for free. How would the mmr of top ranking protoss have to be adjusted to have the proper distribution of outliers? I could calculate it just to flex, but it’s just over kill at this point.