[DcifrThs]

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If the distribution is not normal but not much different from the normal distribution, then in practical terms I don't think it would make much difference. Even if it were not strictly accurate, it would be good enough and probably not worth doing all the extra work to get an accurate confidence interval.

I am still not convinced that win rates are not normally distributed around the mean.

If you want to base you confidence interval on the actual distribution, you might look into bootstrapping.

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one thing thats important to consider is the nature of the process that drives the win rate of a given player or a pool of players.

in discrete time, its easier to deal with but when we move to continuous time, the driving force could be a set of stochastic processes which COULD nullify any inferences made from using the current normal distribution as a base for analysis.

basically, if random processes drive parts of winrate (one process could be how a given person does x,y, or z and have it based on randomness or even have real life like jumps-like the poker graphs show- by making those processes brownian motions that accumulate quadratic variation at rate 1 per unit time) then we wont see the distribution as normal or even a good approximation unless ALL processes meet a few criteria:

-they all have to individually be random and not deterministic (though they can change over time, so long as its random)

-their drift/diffusion (mean/variance) must be adapted to the SAME information that drives the whole system

-they are jointly normally distributed

NOTE: these are some seroiusly strict conditions ... especially the last one.

if these are met then the distribution of the results of the process may approximate a normal distrubution with some confidence.

either way, studies have shown that almost all biological/psychological phenomenon are normally distributed or very easily and readily approximated by a normal distribution. since the win rate is driven by largely biological phenomenon, it would seem as if on a large enough scale, the results of the win rate observations would converge to a normal distribution as well.

the whole thing is interesting and i like thinking about it but im not good enough at all types of higher level math to write out a proof of this...

well, its bedtime.

Barron