St.Dev. for 2000 Hands

[WhipMeBeatMe]

If you know the st. dev. per 100 hands, how can you calculate the st. dev. for 2,000 hands? Also, how do you calculate the 90% confidence interval for your win rate per 2,000 hands if you know your average win and the st.dev. per 2,000 hands?

[jtr]

To get the standard deviation over 2000 hands (for example) figure this as a number of 100-hand sessions. That's 20 sessions in this case. Call that number n.

Then your standard deviation over the longer number of hands will be the original sd / sqrt(n). In this case you would divide through by 4.47. If your original standard deviation per 100 hands was (let's say) 16 BB, then your new sd / 2000 hands (technically this is called the standard error) is about 3.6.

The quick way to get a confidence interval for your winrate is to remember that the actual winrate plus or minus 1.96 times the standard error gives the 95% confidence interval. So if you won at 3 BB/100 over your 2000 hands, that's 3 +/- (1.96 * 3.6). You're 95% confident that your true winrate is between -4 and 10 BB/100. Helpful huh? (Hint: you need more than 2000 hands.)

[Paul2432]

Just a small correction.

You are correct that SD / sqrt(n) = Standard Error

However, SD (y-trials) = SD (x-trials) x sqrt (y/x)

So in the example of the poster, after 2000 hands with and SD of 16 BB/100:

SD (2000 hands) = SD (100 hands) x sqrt(2000/100)

Plugging in SD after 2000 hands = 71.55 BB

So after 2000 hands you have a 95% interval of +/- 140 BB.

Note that if you divide 140 BB by 20 (the number of 100 hands) you get 7 BB. Which is the exact same result as the SE predicts.

Paul

[GrekeHaus]

I was interested in these numbers too. Knowing the formula, I did some calculations and came up with the following for all who are curious :

Number of hands required to know win rate +/- 4.0 BB/100 with 95% accuracy: 6,200
Number of hands required to know win rate +/- 2.0 BB/100 with 95% accuracy: 24,600
Number of hands required to know win rate +/- 1.0 BB/100 with 95% accuracy: 98,400
Number of hands required to know win rate +/- 0.5 BB/100 with 95% accuracy: 393,400
Number of hands required to know win rate +/- 0.2 BB/100 with 95% accuracy: 2,458,700
Number of hands required to know win rate +/- 0.1 BB/100 with 95% accuracy: 9,834,500

So, if you're a player who makes 2 BB/100, you will be up 19/20 times if you play 24,600 hands.

[Edit: I guess that would be 39/40 times]