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Iancho PR Ranking February 2014
Posted By: Jason Lee
Date: Tuesday, 4 February 2014, at 1:38 a.m.
In Response To: Iancho PR Ranking February 2014 (Matt CohnGeier)
MCG: If you win 50% of your matches, then the standard deviation of 100 matches is 5 matches either way. So 3 standard deviations is 15 matches, or the difference between winning 65% of your matches or 35% of your matches due to random luck.
After 1,000 matches the standard deviation is only 15 matches. That means 2 standard deviations on each side are a range from 47% to 53%, and 3 standard deviations on each side are a range from 45.5% to 54.5%.
Matt has everything correct here  I just want to see if I can clarify (or make things more complicated).
Let's say you flip a fair coin 100 times and record the number of heads. Your expected number of heads is 50, but the standard deviation of the number of heads is 5 because this is a Bernoulli trial, whose standard deviation is sqrt(np(1p)). In this case, n = 100, p = 0.5, so stddev = sqrt( 100(1/2)(1/2) ) = 5.
When n is large, this Bernoulli trial can be approximated by a Normal distribution. This is (in general) due to the central limit theorem. A random variable with a normal distribution will yield approximately 95% of the values within two standard deviations of the mean. In this case, you would expect 95% of observations to lie within 40% and 60%. I'm glossing over stuff like continuity corrections, so what I'm saying isn't exactly right.
Alright, what does this mean in practice? Say you perform a trial where you flip a fair coin 100 times and record the number of heads. 95% of the time you perform this experiment, you would expect to see between 40 and 60 heads. That means 5% of the time, even though you are flipping a fair coin, you expect to see below 40 or above 60 heads.
Matt points out that the standard deviation will be lower when your winning percentage is higher, but it doesn't drop that much. The upshot is: Record data for enough people, and you are going to have a sizeable number of people whose results are rather significantly far away from their true ability, just by random chance. This is just basic probability  you can't escape it.
JLee

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