Why Understanding Variance is Important

In poker, even if you have a positive expectation (+EV), you may still lose money in the short term due to luck fluctuations. Variance describes the magnitude of these fluctuations. Without understanding variance, you might doubt your skills during a downswing or overestimate your ability during an upswing. By calculating standard deviation and required sample size, you can more accurately assess your true winrate and avoid emotional decisions.

Basic Concepts

• Winrate: Usually expressed in big blinds per 100 hands (bb/100). For example, 5 bb/100 means an average profit of 5 big blinds per 100 hands.
• Standard Deviation: Measures the volatility of winrate per 100 hands. In typical Texas Hold'em cash games, the standard deviation is around 80–120 bb/100. A higher standard deviation indicates more severe short-term fluctuations.
• Sample Size: The number of hands played. The larger the sample size, the more accurate the estimate of your true winrate.

Step-by-Step Procedure

Step 1: Collect Data

You need to record the number of hands and profit (in bb) for each session. Poker tracking software (such as PokerTracker or Hold'em Manager) can automatically generate this data.

Step 2: Calculate Winrate per 100 Hands

Winrate = (Total Profit / Total Hands) × 100.
For example: Profit 500 bb, hands 10,000 → Winrate = (500 / 10,000) × 100 = 5 bb/100.

Step 3: Calculate Standard Deviation

Standard deviation is slightly more complex. You need a sequence of winrates per 100 hands. If your data is per session, you must first normalize it. The formula is:
Standard deviation = sqrt( (Σ(xi – μ)²) / (n – 1) ), where xi is the winrate per 100 hands for each session, μ is the average winrate, and n is the number of sessions.
In practice, most tracking software gives the standard deviation directly. For manual calculation, you can use Excel's STDEV.S function.

Step 4: Calculate Standard Error

Standard Error = Standard Deviation / sqrt(N), where N is the number of sessions (or number of hands divided by 100). It measures the uncertainty of your winrate estimate.

Step 5: Determine Required Sample Size

To obtain a winrate estimate with a certain confidence level, the required sample size can be calculated using the formula:
Required Hands = ( (z × σ) / Error )² × 100
where z is the z-score (1.96 for 95% confidence level), σ is the standard deviation, and Error is the acceptable absolute error (e.g., ±1 bb/100).
For example: Assuming standard deviation σ = 100 bb/100, desired error ±1 bb/100, and 95% confidence level, then Required Hands = (1.96×100/1)² × 100 = (196)² × 100 ≈ 3,841,600 hands. This seems huge, but if you accept a looser error (e.g., ±5 bb/100), you only need about 38,400 hands.

Step 6: Practical Application

• Calculate the required sample size based on your actual standard deviation.
• Periodically check whether your winrate falls within the confidence interval. For example, with a current sample of 50,000 hands, winrate 5 bb/100, standard deviation 100, then Standard Error = 100/sqrt(500) ≈ 4.47, 95% confidence interval = ±1.96×4.47 ≈ ±8.76 bb/100, i.e., [-3.76, 13.76]. This interval is wide, meaning the winrate estimate is still unreliable.

Common Mistakes

• Ignoring variance: Judging yourself solely based on short-term results. For example, a positive result over 20,000 hands may just be luck.
• Using the wrong standard deviation: Different game types (e.g., cash vs. tournaments) have very different standard deviations. Tournament variance is usually much higher than cash games.
• Drawing conclusions from too small a sample: With fewer than 100,000 hands, the confidence interval for winrate is often large, so interpret cautiously.
• Not accounting for rake: Winrate should be adjusted for rake, otherwise you overestimate actual profit.

Advanced Tips

• Use confidence interval charts: Plot a band showing how the confidence interval changes as sample size increases; you can see the uncertainty shrink.
• Distinguish short-term from long-term: In the short term (e.g., 10,000 hands) results are almost entirely variance-driven; only in the long term (e.g., 500,000 hands) does skill become apparent.
• Track multiple accounts: If you play different stakes or game types, calculate standard deviations separately.
• Combine with bankroll management: Set buy-in amounts based on your standard deviation and risk tolerance. For example, if standard deviation is high, you need more buy-ins as a safety buffer.

Summary

Variance is central to understanding the true nature of poker profits. By calculating your winrate standard deviation and required sample size, you can scientifically evaluate your performance amidst luck and avoid being misled by short-term swings. Remember: a sufficiently large sample (typically tens to hundreds of thousands of hands) is needed to reliably estimate your true winrate. Use tools to record and analyze data, separate emotions from decisions, and that is the key to improvement.