Variance is the mathematical measurement of how your actual poker results deviate from your expected value (EV) over time, it’s why you can play perfectly and still lose for weeks, or play poorly and win big in the short term.
Variance is what makes poker simultaneously thrilling and frustrating. It’s the reason a recreational player can beat Phil Ivey in a single session, but would lose consistently over thousands of hands. In mathematical terms, variance measures the dispersion of results around your expected value. High variance means wild swings, you might win or lose 20 buy-ins in a month while playing solid poker. Low variance means results closer to expectation, though in poker, true low variance doesn’t exist.
The formula is Variance = Σ[(xi, μ)² × P(xi)], where xi represents each possible outcome, μ is the expected value, and P(xi) is the probability of each outcome. For practical poker purposes, what matters more is understanding that variance decreases as sample size increases. Over 100 hands, anything can happen. Over 100,000 hands, your results will converge toward your true win rate. This convergence is why professionals focus on volume and proper bankroll management rather than short-term results.
How to Calculate Variance
While the pure mathematical formula involves summing the squared deviations from expected value, poker players care more about practical applications.
Example 1: Coin flip variance
You’re all-in preflop with A♠K♦ against Q♣Q♥ for a $1,000 pot. Your equity is approximately 45%. Your EV = 0.45 × $1,000 = $450. But you either win $1,000 or $0, never $450. The variance calculation:
- Outcome 1: Win $1,000 (45% of the time). Deviation from EV = $1,000 , $450 = $550
- Outcome 2: Win $0 (55% of the time). Deviation from EV = $0 , $450 = -$450
- Variance = (0.45 × $550²) + (0.55 × (-$450)²) = $247,500
The standard deviation (square root of variance) is $497. This means your actual result will typically be about $497 away from your $450 expectation.
Example 2: Multi-street variance
You have A♥K♥ on K♠7♣3♦ against an opponent’s range. Your equity is 75% against their calling range. In a $400 pot:
- EV = 0.75 × $400 = $300
- But you’ll win $400 (75%) or $0 (25%)
- Standard deviation ≈ $173
Over 1,000 similar spots, you expect to win $300,000 total, but your actual results could easily range from $250,000 to $350,000 due to variance.
Practical Applications
Decision Making
Variance doesn’t change the correct decision. If a play is +EV, variance only affects the short-term noise around that expectation. With A♥Q♥ facing a river bet where you need 30% equity to call and estimate you have 35%, you call despite knowing variance means you’ll lose this exact spot 65% of the time.
EV Calculation
A common error is confusing variance with EV. Your EV for calling $100 into a $300 pot with 35% equity:
- EV = (0.35 × $300) , (0.65 × $100) = $105 , $65 = +$40
This $40 profit is your long-term expectation. Variance determines how wildly your actual results swing around this number in the short term.
Common Shortcuts
The “20 buy-in downswing rule”: Even winning players can experience 20+ buy-in downswings due to variance. If you’re a 5bb/100 winner, there’s approximately a 5% chance of a 20 buy-in downswing over any 100,000 hand sample.
The “square root rule”: Standard deviation grows with the square root of hands played. If your standard deviation is 100bb per 100 hands, it’s 1,000bb per 10,000 hands, not 10,000bb.
Interaction with Other Concepts
Variance interacts with win rate to determine bankroll requirements. Higher variance games (like PLO or short-handed play) need larger bankrolls for the same risk of ruin. A 5bb/100 winner in full-ring NLHE might need 30 buy-ins, while the same winner in 6-max PLO might need 60+ buy-ins.
Pro Tip: Track your all-in EV vs actual results in tracking software. If you’re running significantly below EV (more than 10 buy-ins over 50k+ hands), you’re experiencing negative variance. This data helps separate bad play from bad luck.
When Does Variance Matter?
Variance impacts every poker decision indirectly through bankroll management and psychological factors:
1. Bankroll decisions: Higher variance games require larger bankrolls. PLO has roughly 2x the variance of NLHE. Tournaments have 5-10x the variance of cash games.
2. Game selection: If you have a small bankroll or low risk tolerance, choose lower variance formats. Full-ring cash games have less variance than 6-max. Limit has less variance than no-limit.
3. Tilt management: Understanding variance helps maintain emotional equilibrium. When you lose with AA three times in a row, knowing this happens to everyone prevents tilt.
4. Stake decisions: Professional players often play lower stakes than their skill suggests because they prioritize lower variance and steady income over maximum EV.
Common Mistakes with Variance
Mistaking variance for skill. Running hot doesn’t make you good, running cold doesn’t make you bad. Over small samples, variance dominates results. A 50,000 hand sample is barely enough to establish if you’re a winning player.
Fighting variance with bad play. After losing five flips in a row, some players start playing looser “to get even.” Variance is random, previous results don’t affect future outcomes. The dice have no memory.
Underestimating long-term variance. Even over 100,000 hands, a 5bb/100 winner can show results between -5bb/100 and +15bb/100 due to variance alone. True win rates emerge over millions of hands.
Don’t Confuse With…
Variance vs Standard Deviation: Variance is the average of squared deviations. Standard deviation is the square root of variance. In poker, we usually discuss standard deviation because it’s in the same units as our results (dollars or big blinds).
Variance vs EV: EV is your long-term expectation. Variance describes how results fluctuate around that expectation. You can have positive EV with high variance (tournaments) or positive EV with lower variance (cash games).
Hear It at the Table
Key Takeaway
Variance is the mathematical reason why poker remains profitable for skilled players, it keeps weaker players in the game by giving them winning sessions despite poor play. Understanding variance helps you make peace with short-term results and focus on making correct decisions. Your job isn’t to avoid variance but to ensure you’re on the right side of EV when the swings come.
Pro Tip: Use a variance calculator to understand your risk of ruin. Input your win rate, standard deviation, and bankroll to see the probability of going broke. Most winning players with 40+ buy-ins have less than 1% risk of ruin from variance alone.
