18%<\/strong> of the time.<\/li>\n<\/ul>\n\n\n\nIf you play this situation 100 times with $100 in the pot each time: <\/p>\n\n\n\n
$$EV = (0.82 \\times 100) – (0.18 \\times 100) = 82 – 18 = +64$$<\/p>\n\n\n\n
So on average, you should win $64 per hand<\/strong>.<\/p>\n<\/div>\n\n\n\n\n
But variance means you won\u2019t always see that average. Over 10 trials, you might lose 3 or 4 times in a row despite being the heavy favorite. This doesn\u2019t mean your play was wrong\u2014it\u2019s just variance.<\/p>\n\n\n\n
Example: Variance in a Tournament<\/h2>\n\n\n\n In tournaments, variance is even higher because payouts are top-heavy.<\/p>\n\n\n\n
Suppose you enter a $100 tournament with 100 players:<\/p>\n\n\n\n
\n1st place = $3,000<\/li>\n\n\n\n 2nd place = $1,500<\/li>\n\n\n\n 3rd place = $750<\/li>\n\n\n\n Remaining 10 spots = $200 each<\/li>\n<\/ul>\n\n\n\nIf your skill gives you a 10% chance to finish top 3<\/strong> and a 20% chance to cash<\/strong>, your EV might be: <\/p>\n\n\n\n$$EV = (0.05 \\times 3000) + (0.03 \\times 1500) + (0.02 \\times 750) + (0.20 \\times 200) – 100$$<\/p>\n\n\n\n
$$EV = 150 + 45 + 15 + 40 – 100 = +150$$<\/p>\n\n\n\n
So each entry is worth +$150 in the long run.<\/p>\n\n\n\n
But variance means you could easily go 20 tournaments in a row without cashing<\/strong> before hitting a big win. This is why bankroll management is stricter for tournaments than cash games.<\/p>\n\n\n\n\n
\n
Coping With Variance<\/h2>\n\n\n\n\nFocus on Decisions, Not Outcomes<\/strong>Use Proper Bankroll Management<\/strong>\n20\u201330 buy-ins for cash games<\/li>\n\n\n\n 50\u2013100 buy-ins for tournaments<\/li>\n<\/ul>\n\n\n\n\nKeep Records and Review Play<\/strong>Stay Calm During Swings<\/strong>\n
\n
\n
Why Variance Is Actually Good<\/h2>\n\n\n\n Variance keeps poker attractive. If the best player always won every hand, weaker players would quit, and the game would dry up. Luck in the short term gives everyone a chance\u2014while math ensures that, in the long term, skillful players rise to the top.<\/p>\n<\/div>\n<\/div>\n\n\n
\n
Understanding Variance: Win Rate in BB\/100<\/h2>\n\n\n\n In poker, your win rate<\/strong> is often expressed as bb\/100<\/strong>, meaning the average number of big blinds you win per 100 hands.<\/p>\n\n\n\n\nExample: You\u2019re a solid cash game player with a win rate of +5 bb\/100<\/strong>.<\/li>\n\n\n\nThat means, on average, for every 100 hands you play, you win 5 big blinds.<\/li>\n<\/ul>\n\n\n\nBut here\u2019s the catch: variance means that your actual results will swing wildly above and below that number in the short term.<\/p>\n\n\n\n
2. Standard Deviation in Poker<\/h1>\n\n\n\n\nStandard deviation (\u03c3)<\/strong>, tells you how wide the swings are around your average results.<\/p>\n<\/blockquote>\n\n\n\nFor example:<\/p>\n\n\n\n
\nA typical no-limit hold\u2019em cash player might have a standard deviation of 80 BB\/100 hands<\/strong>.<\/li>\n\n\n\nIf your win rate is 5 BB\/100<\/strong>, the standard deviation shows how \u201cnoisy\u201d your results will look session to session.<\/li>\n<\/ul>\n\n\n\nEven with a positive win rate, large standard deviation means you\u2019ll experience long losing streaks before your skill shows in the results.<\/p>\n\n\n\n
What This Means<\/h3>\n\n\n\n Your true expectation<\/strong> per 100 hands: But the swings around this expectation are much larger: <\/p>\n\n\n\n
$$\\text{Range per 100 hands} \\approx EV \\pm \\sigma$$<\/p>\n\n\n\n
So after 100 hands, your result might realistically be anywhere from: <\/p>\n\n\n\n
$$5 – 80 = -75 \\, \\text{bb\/100} \\quad \\text{to} \\quad 5 + 80 = +85 \\, \\text{bb\/100}$$<\/p>\n\n\n\n
That\u2019s a massive range compared to your actual edge.<\/p>\n\n\n\n
Sample Size and the Law of Large Numbers<\/h3>\n\n\n\n As you play more hands, variance smooths out<\/strong> because results converge toward the true EV.<\/p>\n\n\n\n\n
\n
\nThe formula for standard deviation after \\(N\\) hands is: <\/p>\n\n\n\n
$$\\sigma_{N} = \\frac{\\sigma}{\\sqrt{N\/100}}$$<\/p>\n<\/blockquote>\n\n\n\n
\nAfter 1,000 hands<\/strong>:<\/li>\n<\/ul>\n\n\n\n$$\\sigma_{1000} = \\frac{80}{\\sqrt{10}} \\approx 25.3 \\, \\text{bb\/100}$$<\/p>\n\n\n\n
\nAfter 10,000 hands<\/strong>:<\/li>\n<\/ul>\n\n\n\n$$\\sigma_{10000} = \\frac{80}{\\sqrt{100}} = 8 \\, \\text{bb\/100}$$<\/p>\n\n\n\n
\nAfter 100,000 hands<\/strong>:<\/li>\n<\/ul>\n\n\n\n$$\\sigma_{100000} = \\frac{80}{\\sqrt{1000}} \\approx 2.53 \\, \\text{bb\/100}$$<\/p>\n<\/div>\n\n\n\n
\n
Interpreting This<\/h3>\n\n\n\n\nAt 1,000 hands<\/strong>, you could easily be losing despite being a winning player.<\/li>\n\n\n\nAt 10,000 hands<\/strong>, your results will begin to resemble your true win rate more closely, but swings are still big.<\/li>\n\n\n\nAt 100,000 hands<\/strong>, your actual results will be very close to your expected +5 bb\/100<\/strong>, and luck plays a much smaller role.<\/li>\n<\/ul>\n\n\n\nThis is why pros always talk about the importance of volume<\/strong>\u2014only by playing a large number of hands does skill reliably beat variance.<\/p>\n<\/div>\n<\/div>\n\n\n\n3. Putting Everything So Far Together<\/h1>\n\n\n\nCalculating Your Win Rate (bb\/100)<\/h2>\n\n\n\n Your win rate tells you how many big blinds you win (or lose) per 100 hands on average.<\/p>\n\n\n\n
\n
\n
Formula:<\/strong> <\/p>\n\n\n\n$$\\text{Win rate} = \\frac{\\text{Total BB won or lost}}{\\text{Total Hands}} \\times 100$$<\/p>\n\n\n\n
Steps:<\/strong><\/p>\n\n\n\n\nRecord your total winnings\/losses in big blinds (not dollars).<\/li>\n\n\n\n Record the total number of hands you played.<\/li>\n\n\n\n Plug into the formula.<\/li>\n<\/ol>\n<\/div>\n\n\n\n\n
Example:<\/strong><\/p>\n\n\n\n\nHands played = 20,000<\/li>\n\n\n\n Total winnings = +4,000 big blinds<\/li>\n<\/ul>\n\n\n\n$$\\text{Win rate} = \\frac{4000}{20000} \\times 100 = +20 \\, \\text{bb\/100}$$<\/p>\n\n\n\n
This means you win an average of 20 big blinds per 100 hands<\/strong>.<\/p>\n<\/div>\n<\/div>\n\n\n\nCalculating Your Variance<\/h2>\n\n\n\n Variance tells you how much your results fluctuate around your average.<\/p>\n\n\n\n
\n
\n
Step A: Gather results per 100 hands<\/strong><\/p>\n\n\n\n\nBreak your data into chunks of 100 hands each.<\/li>\n\n\n\n For each chunk, calculate how many BB you won or lost.<\/li>\n<\/ul>\n\n\n\nStep B: Compute the mean (your win rate)<\/strong> <\/p>\n\n\n\n\n$$\\mu = \\frac{\\sum X_i}{n}$$<\/p>\n<\/blockquote>\n\n\n\n
where \\(X_i\\)\u200b = result in BB\/100 for chunk \\(i\\), and \\(n\\) = number of chunks.<\/p>\n<\/div>\n\n\n\n
\n
Step C: Compute variance<\/strong> <\/p>\n\n\n\n\n$$\\text{Var}(X) = \\frac{\\sum (X_i – \\mu)^2}{n}$$<\/p>\n<\/blockquote>\n\n\n\n
Step D: Compute standard deviation (\u03c3)<\/strong> <\/p>\n\n\n\n\n$$\\sigma = \\sqrt{\\text{Var}(X)}$$<\/p>\n<\/blockquote>\n\n\n\n
This \u03c3 is expressed in bb\/100<\/strong>, just like your win rate.<\/p>\n<\/div>\n<\/div>\n\n\n\nExample:<\/h3>\n\n\n\n\n
\n
Suppose you tracked 5 blocks of 100 hands each, with these results (in bb\/100):
1. Mean (\u03bc):<\/strong><\/p>\n\n\n\n$$\\mu = \\frac{40 + (-20) + 10 + 70 + (-10)}{5}$$<\/p>\n\n\n\n
$$= \\frac{90}{5} = +18 \\, \\text{bb\/100}$$<\/p>\n\n\n\n
2. Deviation:<\/strong><\/p>\n\n\n\nFor each session winrate \\(xi\\), find the deviation from the mean:<\/p>\n\n\n\n
$$Deviation=xi\u2212x\u02c9$$<\/p>\n\n\n\nSession<\/th> BB Won \\(x<\/em>i<\/em>\\)<\/th>Deviation \\(xi<\/em>\u2212x<\/em>\u02c9\\)<\/th><\/tr><\/thead>1<\/td> 40<\/td> 40\u221218=22<\/td><\/tr> 2<\/td> -20<\/td> -20\u221218=-38<\/td><\/tr> 3<\/td> 10<\/td> 10\u221218=\u22128<\/td><\/tr> 4<\/td> 70<\/td> 70\u221218=52<\/td><\/tr> 5<\/td> -10<\/td> -10\u221218=\u221228<\/td><\/tr><\/tbody><\/table>Deviation<\/figcaption><\/figure>\n<\/div>\n\n\n\n\n
3. Square Each Deviation<\/strong><\/p>\n\n\n\nSquare the values from Step 3 to eliminate negatives and amplify larger deviations.(xi\u2212x\u02c9)2(x<\/em>i<\/em>\u2212x<\/em>\u02c9)2<\/p>\n\n\n\nSession<\/th> Deviation<\/th> Squared Deviation \\((xi\u2212x\u02c9)^2\\)<\/th><\/tr><\/thead> 1<\/td> 22<\/td> \\(22^2=484\\)<\/td><\/tr> 2<\/td> -38<\/td> \\((-38)^2=1444\\)<\/td><\/tr> 3<\/td> -8<\/td> \\((\u22128)^2=64\\)<\/td><\/tr> 4<\/td> 52<\/td> \\(52=2704\\)<\/td><\/tr> 5<\/td> -28<\/td> \\((\u221228)^2=784\\)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n4. Variance:<\/strong><\/p>\n\n\n\nVariance is the average of the squared deviations.<\/p>\n\n\n\n
$$Var(X)=\\frac{484 + 1444 + 64 + 2704 + 784}{5}$$<\/p>\n\n\n\n
$$ = \\frac{547…}{5} = 1096$$<\/p>\n\n\n\n
5. Standard Deviation (\u03c3):<\/strong><\/p>\n\n\n\nStandard deviation is the square root of the variance:<\/p>\n\n\n\n
$$\\sigma = \\sqrt{1096} \\approx 33.1 \\, \\text{bb\/100}$$<\/p>\n<\/div>\n<\/div>\n\n\n\n
Interpreting Results<\/h3>\n\n\n\n\nWin rate = +18 bb\/100<\/strong> \u2192 you\u2019re a winning player.<\/li>\n\n\n\nStandard deviation = 33 bb\/100<\/strong> \u2192 your results swing up and down by about \u00b133 BB for every 100 hands.<\/li>\n<\/ul>\n\n\n\nThis shows why you can have big downswings despite being profitable long-term.<\/p>\n\n\n\n
👉 In practice:<\/strong><\/p>\n\n\n\n\nWin rate<\/strong> tells you how much you should expect to earn over time.<\/li>\n\n\n\nVariance<\/strong> tells you how bumpy the road will be to get there.<\/li>\n<\/ul>\n\n\n\n3. Confidence Intervals in Poker<\/h1>\n\n\n\n\n
\n
When you calculate your win rate (bb\/100) from a sample of hands, it\u2019s just an estimate<\/strong>. Variance means your measured win rate may be higher or lower than your true<\/em> win rate.<\/p>\n\n\n\nA confidence interval (CI)<\/strong> gives you a statistical range that likely contains your true win rate.<\/p>\n\n\n\nConfidence intervals use standard deviation to express a range within which your true win rate probably lies with a certain level of certainty.<\/strong><\/p>\n<\/div>\n\n\n\n\n
\n
\n
The 68-95-99.7 rule<\/strong> (empirical rule) states:<\/p>\n\n\n\n\nAbout 68% of samples will fall within 1 standard deviation<\/strong> of the mean.<\/li>\n\n\n\nAbout 95% within 2 standard deviations<\/strong>.<\/li>\n\n\n\nAbout 99.7% within 3 standard deviations<\/strong>.<\/li>\n<\/ul>\n<\/div>\n\n\n\n\n
If your average win rate is 2.5 BB\/100 and standard deviation is 30 BB\/100, then:<\/p>\n\n\n\n
\nWith 68% confidence, your win rate falls between -27.5 and 32.5 BB\/100 in any 100-hand sample.<\/li>\n\n\n\n With 95% confidence, your win rate falls between -57.5 and 62.5 BB\/100.<\/li>\n\n\n\n This demonstrates extreme variance and why short-term results can swing wildly.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n\n\n\nFormula for Confidence Intervals<\/h2>\n\n\n\n\n
\n
\nFor large enough samples, we can use the normal distribution<\/strong> to approximate results: \n
Where:<\/p>\n\n\n\n
\n\\(\\mu\\) = observed win rate (bb\/100)<\/li>\n\n\n\n \\(\\sigma\\) = standard deviation (bb\/100)<\/li>\n\n\n\n \\(N\\) = total hands played<\/li>\n\n\n\n \\(Z\\) = z-score for the chosen confidence level\n
![\"\"](\"https:\/\/hhdealer.com\/blog\/wp-content\/uploads\/2025\/09\/variance.png\")
<\/figure><\/div><\/div>\n<\/div>\n\n\n\n![\"\"](\"https:\/\/hhdealer.com\/blog\/wp-content\/uploads\/2025\/09\/donkeybet.png\")
<\/figure><\/div><\/div>\n<\/div>\n\n\n\n![\"\"](\"https:\/\/hhdealer.com\/blog\/wp-content\/uploads\/2025\/09\/confidenceinpoker.png\")
<\/figure><\/div><\/div>\n<\/div>\n\n\n\n