Let's say if I have the following stats:

Hands played: 10000 hands
Win rate: 5BB/100hands
Standard deviation: 40BB/100hands

How to calculation the range of my possible win rate? For 99% confidence and 95% confidence?

EDIT: Fixed factor of 10 problem. Divide 40 by sqrt(100) to get standard error, not sqrt(10000), since SD is reported for 100 hands, and 10000 hands means we played 100 units of 100 hands.

The standard deviation of your win RATE (or standard error) is 40/sqrt(100) = 4 BB/100 hands. Your 95% confidence interval is about +/- 1.96 standard deviations, and your 99% confidence interval is about +/- 2.58 standard deviations.

95% interval = 5 +/- 1.96*4 =~ 5 +/- 7.8 BB/100.

99% interval = 5 +/- 2.58*4 =~ 5 +/- 10.3 BB/100.

What does $30- HR over 4 tables NL200, convert to in PTbb or BB stats?

Bruce, I believe you meant to say that your standard error is

40/sqrt(100) since your standard deviation per 100 hands is 40BB and that a total of 100*100 = 10,000 hands have been played.

at 95% we have 5 +-1.96*4 BB/100 which is 5 +- 7.84 BB/100

at 99% we have 5+-2.58*4 BB/100 which is 5 +- 10.32 BB/100

Or alternatively,

Your expectation in big blinds and your standard deviation over 10,000 hands is 500BB and 40*sqrt100 respectively.

at 95% we have 500 +- 1.96*400 BB/10,000

at 99% we have 500 +- 2.58*400 BB/10,000

Here we look at one of the most important math problems in poker.

Somebody has won 500 BB after 10.000 hands = 5BB/100hands.
He has noticed a SD of 4BB/h which is 4BB * sqrt(100) = 40 BB/100h.

Now i make the following assumptions:

We all know that the true win rate in poker for the average poker player is something like -2BB/100h (because of rake, tip etc). Based on this assumption we get the following confidence intervall:

at 95%, the confidence intervall is -200 +/- 1,96 * 4 * sqrt(10.000) =~ -2 +/- 7,84 BB/100
at 99%, the confidence intervall is -200 +/- 2,58 * 4 * sqrt(10.000) =~ -2 +/- 10,32 BB/100

How can we assume that this player has after 10.000 hands already another true win rate than the average poker player has? There is some error in our thinking.

How can we do this, if we know, that somebody needs at least a sample size of :

N = 100 * (z * SD/WR) ^ 2 = 100 * (2,58 * 40/1) ^ 2 = 1,065,024 hands

to find out his true win rate in terms of 1 BB/100h exactly?

or needs at least a sample size of

N = 100 * (2,58 * 40/5) ^ 2 = 42,600 hands to prove his win rate of 5BB/100hands.

btw.
This player with 5BB/100h and 40SD/100h after 10.000 hands has still a very high chance to win nothing after
N = 100 * (3 * 40/2) ^ 2 = 360.000 hands

that means after 360,000 hands he can easily have a loss within a confidence probability of 99,73%.

Prove:
0 = -2/100 * 360.000 + 3 * 4 * sqrt(360,000)

ouch

Only to get a better feeling, what we talk about here:
I can transform the formula N = 100 * (z * SD/WR) ^ 2 also as follows

sqrt(N) = 10 * z * SD/WR
WR = 10 * z * SD/sqrt(N)

and show how 99,995% of the Normal Distribution arround any WR = X whatever looks like:

In case of SD=40/100h and 10,000 hands the Normal Distribution looks as follows:
+/- 10 * 4,5 * 40/sqrt(10.000) = +/- 45 * 40/100 = +/- 18BB/100h.

We dont assume his winrate being anything, we just calculate the probabilities of it being in a certain range, for example the probablity of it being between 5 +- 10.32, which is 99%.
By using "-2" in your formula you assume every player has the same true winrate, which is not true.

We speak here about trueness or "true win rate range". How can you assume a true win rate of 5BB/100h after 10.000h if all poker players are burdened with rake and tips etc and the true win rate of the average poker player is approx -2BB/100h. With your argument you can only convince an idiot but not somebody who has a knowledge for trueness in mathematics.

Let's wait what the experts here in this forum will answer.

1) 2) Were not talking about the average poker player, we talk about a poker player who experienced

Hands played: 10000 hands
Win rate: 5BB/100hands
Standard deviation: 40BB/100hands

I hope that somebody like BruceZ or jay_shark will give you an answer. I stop further discussions with you.

Only to give you a preliminary answer. We need a tool like a statistical significance test in poker.

http://en.wikipedia.org/wiki/Statist...thesis_testing

Thats what BruceZ and jay_shark did.

They calculated the ranges for the significance levels 1% and 5%. The null-hypothesis is that ops true winrate is not inside these ranges. Thats not exactly an statistical hypothesis test, but an confidence test, which op was looking for.