Stats, Cont....

Sorry for the relative lull in posting. I just got back from New York and am still getting back into the swing of things. I wanted to give kind of idea of how long it takes stats to converge. In general, when you have a Bernoulli variable x with mean p (i.e., a variable that takes a value of 0 or 1, as in 1 if PFR, 0 if no PFR), its variance (the expectation of its square of its deviation from its mean) is equal to (if you don't understand a step of what's below, or you think I did something incorrect, feel free to comment):

E[(x-p)^2]

= E[x^2 + p^2 - 2xp]

= E[x^2] + p^2 - 2 E[xp]

=  p*(1)^2 + (1-p)*0^2 + p^2 - 2 p^2

=  p - p^2 = p (1-p)

Now consider the average of a series of Bernoulli trials, which is what the PFR/VP$P stats are. The variance of this would be, where S is the total number of successes in n trials (go here for more details):

E[(S/n-p)^2]

= p (1-p)/n

So the standard deviation would be:

(p (1-p)/n)^(1/2)

Taking advantage of the Central Limit Theorem, we can assume that the distribution of the PFR stat will be approximately normal around its mean, with a standard deviation equal to the standard deviation given above. A good rule of thumb is going two standard deviations away from the mean in either direction gives you a 95% confidence interval. So say you have 100 hands on a guy and you have a 20% PFR. Then p = .2 and n = 100, so the standard deviation is:

(.2*.8/100)^(1/2) = .4/10 = .04

That is, you can be 95% sure that his true PFR is between .16 and .24, which is pretty good for only 100 hands. 

-BRUECHIPS