Standard Deviation

Standard Deviation (σ)

A player wins +100 BB over 1,000 hands and concludes their win-rate is +10 BB/100. The actual true win-rate could sit anywhere from -3 BB/100 to +23 BB/100 — a 95% confidence interval spanning 26 BB. Where does that uncertainty come from? Standard deviation. Without a solid grasp of sigma, you'll systematically overestimate your skill during heaters and underestimate it during downswings. Sigma is the math behind how far your results can drift from your true mean.

What It Actually Is

Standard deviation (sigma, σ) measures how spread out values are around an average. When results cluster close to the mean, sigma is small. When they scatter widely, sigma is large.

A simple analogy: measure the height of 10 people. The average is 175 cm. If everyone falls between 173 and 177 cm, sigma is tiny — around 1.5 cm. If heights range from 160 to 195 cm, sigma is large, around 10 cm. Poker works the same way. Your average win-rate might be +5 BB/100, but individual sessions can swing anywhere from -50 to +60 BB per 100 hands. Sigma describes that spread.

In gambling, sigma determines:

• How deep your downswings and upswings will run
• What bankroll you need to survive variance
• How many hands you need to reliably evaluate your skill level
• Your Risk of Ruin at any given stake size

Understanding sigma solves a core problem: emotional reactions to short-term results. Down 200 BB in a session? That's a catastrophe? No, it's completely normal with a sigma of 100 BB/100 over 300 hands. Up 500 BB in a week? You're a crusher? Not necessarily. That's statistically plausible for an average regular.

The Formula: How to Calculate Standard Deviation

The basic math. Given N results x₁, x₂, ..., xₙ with mean μ:

variance: V = Σ(xᵢ - μ)² / (N - 1)

sigma: σ = √V

For poker specifically: break sessions into 100-hand blocks (or use hand-by-hand data). Take each block's result in BB. Calculate the mean, that's your win-rate. Calculate deviations from the mean, square them, average them. The square root of that is your sigma.

PokerTracker 4 and Holdem Manager 3 calculate sigma automatically. Open "All Hands" then "Reports" then "Sessions" and you'll see your BB/100 win-rate plus or minus sigma. Without this basic statistic, analyzing results is flying blind.

For sports betting, sigma is calculated per bet. If you're betting $100 per wager with a +3% edge, typical sigma per bet is $98–100. That means the actual result of a 50-bet run could land anywhere from -$1,000 to +$1,300 against an EV of +$150.

Use the variance simulator to model possible outcomes based on your own numbers. It's the best way to build genuine intuition for what sigma means in the context of your actual game.

Sigma in Cash Poker

Typical sigma values for nlhe online in 2024–2026:

NL10–NL50 (micro and low stakes):

• 6-max: sigma 90–110 BB/100
• 9-max: sigma 70–90 BB/100
• Zoom (fast-fold cash): sigma 100–130 BB/100

NL100–NL500 (mid stakes):

• 6-max: sigma 100–130 BB/100
• 9-max: sigma 80–100 BB/100

NL1K+ (high stakes):

• 6-max: sigma 110–140 BB/100
• 9-max: sigma 90–120 BB/100

Why does sigma increase with stakes? At high stakes, players open wider, run more flips (50/50 spots), and get into more 3-bet/4-bet wars. All of that cranks up variance even at the same win-rate.

Why is 6-max higher than 9-max? Fewer players at the table means more hands played from the blinds, more top-pair-versus-top-pair collisions, and more all-in situations preflop. 9-max runs tighter, so variance is lower.

Zoom (PokerStars) and fast-fold poker produce sigma roughly 15–25% higher than regular cash games. The reason: the quick pace leaves no time to read opponents between hands, so you're playing against average ranges, which amplifies variance.

Sigma in MTTs (Tournaments)

In tournaments, sigma is measured as a percentage of the buy-in. Typical figures:

• Micro MTT ($1–5 buy-in): sigma 200–250% of buy-in
• Low MTT ($10–30): sigma 250–300%
• Mid MTT ($50–200): sigma 300–400%
• High MTT ($500–2000): sigma 400–500%

Those numbers are dramatically higher than cash poker. Why? Because 85–95% of tournaments end in a zero (busting before the money), only 5–15% produce a positive result, and the bulk of earnings concentrates in top-3 finishes. The distribution looks nothing like a normal curve. It's heavily right-skewed with a long tail of big scores.

Practical implication: an MTT bankroll needs a minimum of 100 buy-ins at micro stakes and 200+ at high stakes. The rule of thumb: 4 × sigma² / win-rate = minimum bankroll for 5% Risk of Ruin.

Plugging in an MTT with +20% ROI and 300% sigma: bankroll = 4 × 9 / 0.2 = 180 buy-ins. That's exactly where the "200 buy-ins for MTT" standard in poker literature comes from.

Running the numbers through a variance simulator makes this concrete: with +20% ROI after 1,000 tournaments, your actual ROI sits somewhere between -10% and +50% at 95% confidence. That means even after 1,000 runs you still don't know your true edge. At 3,000 tournaments, the range tightens to roughly +5% to +35%.

Sigma in Sports Betting

Sigma in sports betting depends on bet size and odds:

sigma_bet = stake × √(probability × (1 - probability))

For a typical -110 bet (52.4% implied probability): sigma = stake × 0.499, roughly half the stake.

Concretely: a $100 bet at -110 produces sigma of $50 per bet. Variance is quadratic, so V = 2500.

Portfolio sigma across multiple bets:

When bets are independent: sigma_portfolio = √(sum of individual variances).

Concretely: 10 bets at $100 with sigma $50 each gives sigma_portfolio = √(10 × 2500) = $158. Not $500 (i.e., 10 × $50), substantially less, thanks to averaging.

That's the core logic of diversification. The more independent bets you run, the smaller the portfolio variance relative to total volume. It's exactly why sharps play 200–500 bets per year instead of 20 big ones.

Correlations break this math. If 10 bets are correlated, say, all on the same team across different games, the real portfolio sigma can approach the sum of individual sigmas rather than their square root. Sharps are meticulous about avoiding correlated exposure.

Sigma in Blackjack Card Counting

In card counting, sigma is measured in betting units per hour of play.

Typical figures for Hi-Lo:

• 1 hour at a single table: sigma 2–2.5 units
• 4-hour session: sigma 4–5 units (scaling as √4 = 2×)
• 100 hours of play: sigma 20–25 units

With a +1.5% edge and sigma of 2.5 units/hour, a player earns roughly 1.5 units per hour on average, but the variance is brutal. A run of 50 losing hours straight is possible; longer than 80 is rare.

That's exactly why card counters work in rotating teams, spreading action across multiple casinos simultaneously. Reducing portfolio sigma through multi-table play works in blackjack the same way it does in poker.

Confidence Intervals via Sigma

The main practical value of sigma is calculating confidence intervals (CI).

For a normal distribution:

• 68% of results fall within ±1σ of the mean
• 95% within ±2σ (more precisely 1.96σ)
• 99.7% within ±3σ

To estimate win-rate from a sample of N hands: CI_95% = win_rate ± 1.96 × sigma / √N

Concretely: a win-rate of +5 BB/100 after 10,000 hands with sigma 100. CI = ±1.96 × 100 / √10,000 = ±1.96 BB/100. Your true win-rate sits somewhere between +3.04 and +6.96 BB/100 with 95% confidence.

After 100,000 hands: CI = ±0.62 BB/100, range +4.38 to +5.62. Much tighter.

After 1,000,000 hands: CI = ±0.2 BB/100. That's genuine convergence to your true value.

The practical takeaway: until you have 50,000–100,000 hands, you don't actually know your real win-rate. Every result is noise plus a possible real edge, and without understanding confidence intervals you'll constantly misjudge yourself in both directions.

Relationship with Risk of Ruin

Sigma feeds directly into the Risk of Ruin formula:

RoR = ((1 - winrate/σ) / (1 + winrate/σ))^(bankroll/σ)

The higher sigma is at a given win-rate, the higher your RoR. Double sigma while keeping win-rate constant and you need four times the bankroll to maintain the same Risk of Ruin level, sigma enters as a square.

Concrete example. A cash game player with a +3 BB/100 win-rate:

• sigma 80 BB/100: 30 buy-ins needed for 5% RoR
• sigma 100 BB/100: 47 buy-ins needed for 5% RoR
• sigma 120 BB/100: 67 buy-ins needed for 5% RoR

In other words, your format choice. 6-max vs. 9-max, Zoom vs. regular cash, meaningfully shifts your required bankroll, and the mechanism is entirely the difference in sigma.

More on the sigma-bankroll relationship in the Risk of Ruin and Kelly Criterion articles.

Sample Size: How Much Data Do You Actually Need

The sample size required for a reliable win-rate estimate depends on the win-rate/sigma ratio. The bigger the gap, the smaller the sample you need.

Cash game poker (sigma 100 BB/100):

A thin win-rate demands far more data. That's exactly why regulars at NL500 running 1–2 BB/100 can't pin down their edge even after years of play. They simply can't log enough hands fast enough.

MTT (sigma 300%):

• ROI +10%: 3,600 tournaments needed for CI ±5%
• ROI +30%: 400 tournaments needed

Sports betting (sigma on a $50 bet with $100 stake):

• ROI +1%: 10,000+ bets
• ROI +3%: 1,100 bets
• ROI +5%: 400 bets

Use the variance simulator to model your own sample and see exactly where you stand on the road to statistical confidence.

Common Mistakes

1. Overvaluing a small sample. A player wins +50 BB over 1,000 hands and concludes their win-rate is +5 BB/100. At a 95% CI, the actual win-rate sits somewhere between -15 and +25. That's noise, not proof of skill.

2. Ignoring sigma when choosing a format. A player moves from 9-max to 6-max chasing the same win-rate but doesn't account for sigma running 20% higher in the shorter format. Real Risk of Ruin grows proportionally.

3. Confusing variance with sigma. Variance is sigma squared, not sigma itself. If sigma is 100, variance is 10,000. Mix those up and your bankroll calculations go sideways fast. Always confirm which units a figure is expressed in.

4. Underestimating tail events. A normal distribution puts 99.7% of results within ±3σ. Real poker results have fat tails: extreme upswings and downswings happen more often than the theoretical model predicts.

5. Ignoring correlations. Portfolio sigma is calculated as the square root of summed variances only when bets are independent. In practice, plenty of bets are correlated, multiple lines on the same match, several tournaments on the same day, and real sigma ends up higher than the theoretical figure.

6. Using the same sample-size benchmarks across formats. 10,000 cash-game hands give you a solid win-rate estimate. 10,000 MTT results give you an even tighter one, but getting there takes an enormous amount of time. Don't compare sample sizes across formats directly.

Where It Falls Short

Standard deviation rests on statistical assumptions that don't always hold:

Normal distribution. CI formulas assume a Gaussian distribution. Real poker results have fat tails, extreme outcomes happen more often than the normal distribution predicts. Actual Risk of Ruin on a thin bankroll is frequently higher than the calculated figure, precisely because of those tails.

Stable win-rate. Sigma is calculated from historical data, but your win-rate shifts over time: your skill grows, the competition changes, you pick up new formats. Historical sigma doesn't predict future sigma.

Independence of outcomes. Every hand or bet is treated as independent. In practice, emotional decisions, tilt, and chains of choices within a single session create correlations. Real-world sigma is effectively higher than the theoretical figure.

Process stationarity. The game evolves: new strategies, new players, new tools. Sigma from five years ago may look nothing like today's.

Distributional skew. MTTs and sports betting have heavily skewed distributions with a long tail of big scores. Sigma describes spread, not the shape of the distribution. For the full picture you need the median, mode, and a skewness estimate.

Data accuracy. Sigma is only as good as your records. If the data has gaps, not every hand tracked, formats miscategorized, your measured sigma will drift from the real one.

Despite these limitations, sigma is a foundational concept for anyone playing poker or betting seriously. Without understanding variance, every decision about bankroll size, format selection, and skill assessment gets made in the dark.