Converting SD to win rate confidence interval

[Paul2432]

I know enough stats to be dangerous, so hopefully someone can tell me if I am doing this right.

Suppose a player has a win rate of 3 BB/100 and an SD of 15 BB/100 over 40,000 hands.

The SD/100 hands can be converted to an SD/40000 can by multiplying by the square root of (40000/100). Hence:

SD/40000 = 15 x (40000/100)^0.5 = 300 BB

Now we can apply this to our win rate by dividing by (40000/100). This gives us a SD for our win rate of 0.75.

So our win rate within two SD is 1.5 to 4.5 BB/100.

This is all well and good. The problem I have is that if the number of hands is smaller, say 15000, then the two SD range on the win rate is larger to the point that the upper bound is an impossibly high figure (say 6 BB/100). How do I account for this when figuring my two SD interval for my win rate when I don't have a huge number of hands?

Or is my methodology all screwed up?

Thanks,
Paul

[pzhon]

[ QUOTE ]

This is all well and good. The problem I have is that if the number of hands is smaller, say 15000, then the two SD range on the win rate is larger to the point that the upper bound is an impossibly high figure (say 6 BB/100). How do I account for this when figuring my two SD interval for my win rate when I don't have a huge number of hands?


[/ QUOTE ]
Your methodology is fine. If you don't have many hands, your confidence interval should be very wide. You only have strong evidence against a very high win rate or a high loss rate.

You mention that you feel 6 BB/100 is impossible. You are saying that you have in mind some prior distribution of the possible win rates. You can update this prior distribution using the data you have. If you trust the estimate of the standard deviation and want to focus on the win rate, you can multiply your prior distribution by the probability density function for a normal distribution with (in the first example) mean 3 and standard deviation 0.75, then renormalize to probability 1. This is your updated distribution. If you initially felt that 6 BB/hour was very unlikely, it should still seem unlikely according to the updated distribution.

If you have few hands, the updated distribution will be very close to the original. You have not greatly reinforced your estimate of the probability of one win rate versus another. If you have many hands, this may reinforce the possibility that you win 3 BB/100 over the possibility that you win 2 BB/100 by a factor of 10.