Joined: May '09
Location: India
Age: 31 (M)
Posts: 4873

Variance Variance is a term commonly used in poker forums but misunderstood by many beginners.

Statistically speaking variance is a measure of how much the values in a data/distribution differ from each other. It can be calculated by taking mean of squares of deviation, so called as 'mean square deviation' Now variance has units squared of mean's units which makes it inappropriate for data analysis, so another parameter standard deviation is used.

Standard deviation is square root of variance so called as 'root mean square deviation' Variance = Standard Deviation^ 2 or standard deviation = Variance^0.5 Like variance , standard deviation is a measure of dispersion in the data.

In terms of poker, variance/standard deviation is a measure of deviation from the expected results due to chance /luck.

Significance of Standard deviation : Standard deviation can be used to calculate the range within which a given % of values in the data will lie. For a normal distribution(symmetrical bell shaped curve, mean=mode=median), where M is mean, s is standard deviation image 68.2% values will be within M+/- s 95.4% values will lie within M+/-2s 99.7% values will lie within M+/-3s For normal research purposes, 95.4% is considered standard but it means 1 in 20 values will be excluded which is a lot in poker. So we'll stick to 99.7%

For other kind of distributions, range can be found using Chebyshev's inequality

Role of variance in showdowns/allins Consider AA vs 66, AA will win around 80% and 66 will win around 20%(ignoring split pots) Now many beginners hav the misconception if they play 10 hands, AA shud win 8 times.

Win rate of 80% means out of an infinite number of hands AA vs 66, AA will win 80% times. From this infinite number of hands if we randomly take a smaller sample (size n), most of the times win rate will not be 80% because of 'sampling error' due to luck/chance leading to variance.

Now if we take an infinite number of randomly selected samples of size 'n' (AA vs 66), then acc to Central limit Theorem- the distribution of win rate of AA will be a normal/near- normal distribution with mean - 80% and standard deviation 's' given by standard error of the proportion. Win rate of AA (p) = 80% Win rate of 66 (q) = 20% s= standard error of proportion = (pq/n)^0.5

e.g. if we take a non-biased sample of 100 hands, AA vs 66. (as the hands have been dealt randomly by the poker site, the sample can be considered representative.) s= (80*20/100)^0.5 = 4%

Which means, for a sample of 100 hands if the hands are dealt randomly, 99.7% times win rate of AA will be between 68%-92% and for 66 it will be 8%-32%. This still excludes around .3% of values, which means u'll get win rate outside of this range around 1 in 350 times. further more - 's' is inversely proportional to n^0.5 or variance is inversely proportional to sample size. this implies as sample size increases, sampling error decreases and so does the variance

Similarly over 100 hands of 50-50 flips, s = 5% Which implies - variance in flips is more than variance in dominated hands over same sample size. Consequently the games involving more frequent flips will have more variance.

effect of variance on winrates Variance due to software is only 1 aspect of variance, there's more variance due to playing style changes/opponents, which causes more swings in poker. (Example below is only to show effect of variance, numbers will vary for different win rate, s.d. combinations) In micros, its not unusual to have a win rate of 5 bb/100 hands with 80bb/100 hand standard deviation. Lets assume this win rate is over a sample size of 100k hands.

If we calculate true win rates of these players, we'll find that 95.4% will have true win rate between 0-10 bb/100 hands Now as the distribution is symmetrical, it implies that -> 2.3% of players with observed win rate of 5bb/100 s.d.- 80bb/100 over 100k hands are in long term losing players.

Similarly -> 2.3% of players with true win rate of 5bb/100 s.d.- 80bb/100 will come as losing players over 100k hands.

Conclusions/cliffs:- -missy variance is a bitch - Sample size/Volume the bitch

Edited by SuperNoob (Wednesday, March 28, 2012 @ 16:43 GMT)

Joined: Nov '09
Location: Canada
Age: 48 (M)
Posts: 2367

Good post SN. What is surprising in the simulations is how players of equal skill will not run as well as each other, with some being significant winners and others losers. Variance.

BTW, expect to be accused of being paid by the poker sites shortly....

I agree, but I think need comparison. How statistic bring poker player to be winner ! and why ? (reasons ) to make final conclusions, so we have basic guidance to make decision at table. How pot odds, implied odds, outs. etc ? fail cause : variance too ?

I had a day when I always lost when I had 13 outs at post turn. is 13 outs too bad to play ?

Joined: May '11
Location: United Kingdom
Age: 52 (M)
Posts: 1372

Hell I really wish I'd studied maths more and REALLY understood your post.

I think I have an instinctual feel for probability and statistics. Paradoxically, instinctual enough to let me know that instinct alone isn't enough to let me fully understand it. In fact it's likely to lead me astray.

Edited by zeroster (Thursday, March 29, 2012 @ 03:28 GMT)

Joined: May '08
Location: Netherlands
Age: 46 (M)
Posts: 6197

Ohhhh btw, @supernoob, wanted to ask you before: Wot kind of education you had? Somekind of math incl programmeur right? Friend of mine works at software branch, nowadays they let write the software in india, much cheaper and many indian software writers are extremely smart with that. Sometimes he has to go there to top some things up, but mostly everything is skype.

Joined: Apr '09
Location: Portugal
Age: 37 (M)
Posts: 4772

Variance is a concept that everyone who plays poker have always present, one way or another. Even not knowing it, we tend to make our decisions having some agreement with this concept. That is why we feel confident in going all-in pre-flop with AA, exactly the same confidence of playing this hand in slow play. But, like you explained, even being the hand with bigger percentage of winnings, it doesn’t mean it will be as we see in statistics. No one is taking note of how many times have already won with AA to know when should they not play it because it will be losing time.

Joined: May '09
Location: India
Age: 31 (M)
Posts: 4873

sorry if its a bit hard to understand for some, i tried to keep things simple and further explanations would hav resulted in a very long post

Posted by Serpang: I agree, but I think need comparison. How statistic bring poker player to be winner ! and why ? (reasons ) to make final conclusions, so we have basic guidance to make decision at table. How pot odds, implied odds, outs. etc ? fail cause : variance too ?

sorry but i dont understand your question, can u be more specific? and pot odds/implied odds dont fail, they just appear to fail over small samples.

13 outs is approx 28%. if ur sample size is small, u can lose all

Posted by doomdy: Ohhhh btw, @supernoob, wanted to ask you before: Wot kind of education you had? Somekind of math incl programmeur right?

Joined: Jan '10
Location: Indonesia
Age: 54 (M)
Posts: 1504

Posted by SuperNoob: sorry if its a bit hard to understand for some, i tried to keep things simple and further explanations would hav resulted in a very long post

Posted by Serpang: I agree, but I think need comparison. How statistic bring poker player to be winner ! and why ? (reasons ) to make final conclusions, so we have basic guidance to make decision at table. How pot odds, implied odds, outs. etc ? fail cause : variance too ?

sorry but i dont understand your question, can u be more specific? and pot odds/implied odds dont fail, they just appear to fail over small samples.

13 outs is approx 28%. if ur sample size is small, u can lose all

Posted by doomdy: Ohhhh btw, @supernoob, wanted to ask you before: Wot kind of education you had? Somekind of math incl programmeur right?

i'm a physician

sorry but i dont understand your question, can u be more specific?

When will we use statistic to make decision at table since variance could be make statistic fail ? How many % fail possibility ? example : I didn't know my opponent ( we just start s n g ), and I had 13 outs when my opp bet around 1/2 our total chips post turn, others fold. 13 outs approx 28% , small chance to win? should I fold ? How many out is feasible to play ?

Actually, I did not trust calculate pot odds/ implied odds and out bring me to be winner - disappointed me too much.But I TRUST statistic bring me to make good decision in my real life ( my job, gambling - baccarat ).

Do you have statistic data about winning decision ( even winning player ) by pot odds, implied odds and outs ? Is it help us to survive or bring us to win ? Just survive but have small stake just hope luck or miracle to win MTT, at s n g still possible.

pot odds/implied odds dont fail, they just appear to fail over small samples.

Yeah, ITM 10% - 15% still possible to be winning player. but really hurt if it happen at bubble and we out cause that pot out, implied out and outs analysis. (so far use instinct is better for me )

Edited by Serpang (Thursday, March 29, 2012 @ 15:55 GMT)

Joined: May '09
Location: India
Age: 31 (M)
Posts: 4873

@ serpang it depends on pot, stack sizes and bet to determine how many outs u need for the play to be profitable. e.g. if u have 28% chances to win, u can call bets upto 1/2 pot assuming there is no betting on river. i suggest u read about pot odds, there are many gud articles available for free.

Joined: Jan '10
Location: Indonesia
Age: 54 (M)
Posts: 1504

Posted by SuperNoob: @ serpang it depends on pot, stack sizes and bet to determine how many outs u need for the play to be profitable. e.g. if u have 28% chances to win, u can call bets upto 1/2 pot assuming there is no betting on river. i suggest u read about pot odds, there are many gud articles available for free.

Thanks, I forgot ask again . Looks like pot odds most profitable to play cash game, cause produce direct profit / money. So far I didn't see good pot odds decision at my MTT ( many time I just saw opposite, lol ). Hope someday me or someone make good video or hand history to clearly discussing about it. I try to read again article about pot odd and implied odd.

Joined: Apr '09
Location: Australia
Age: 33 (M)
Posts: 6471

Posted by Serpang: So far I didn't see good pot odds decision at my MTT (

pot odds are a lot less important in an MTT - because there are a lot of other factors that affect the value of your chips. In cash game - it is pure maths - if you lose (as a favorite) you can simply reload your stack - the value of your chips never changes.