Bankroll and variance
How many hands before you know your win rate
Solved Hand5 min read
You have played 10,000 hands and you are up 5 big blinds per 100. That result alone does not establish that you win in the long run. Poker results swing so far that a long run of hands still leaves a wide range of possible true win rates. An approximate confidence interval shows how much uncertainty remains at different sample sizes.
01How far results swing
A win rate is usually given in , big blinds won per 100 hands. Over any 100 hands, your result can land far from that average. The usual measure of the spread is the , also in big blinds per 100 hands. This post uses 80 and 100, the two figures the trainer's glossary works with.
With a standard deviation of 80 and a win rate of 5, the spread of one 100-hand result is 16 times the average you are trying to measure. You need a lot of 100-hand stretches before the swings average out.
02The margin of error
The spread of an average shrinks with the square root of the number of samples. Assuming independent results, a stable win rate and a suitable standard-deviation estimate, a normal approximation gives a 95% around the measured rate:
Margin = 1.96 × standard deviation ÷ √(hands ÷ 100)
10,000 hands at a standard deviation of 80: 1.96 × 80 ÷ √100 = 15.7
Interval = 5 ± 15.7, from −10.7 to +20.7
After 10,000 hands at 5 , the approximate interval runs from losing 10.7 to winning 20.7. It is not a hard bound on the true rate: over repeated samples, intervals built this way would cover the fixed true rate about 95% of the time under the assumptions. That range includes a strong winner, a break-even player and a clear loser.
| Hands played | Standard deviation 80 | Standard deviation 100 |
|---|---|---|
| 10,000 | −10.7 to +20.7 | −14.6 to +24.6 |
| 50,000 | −2.0 to +12.0 | −3.8 to +13.8 |
| 200,000 | +1.5 to +8.5 | +0.6 to +9.4 |
At 50,000 hands the interval still reaches below zero. Of the three sample sizes shown, only at 200,000 does it sit wholly above zero, and even then it runs from barely winning to winning well.
03The square-root rule
The margin shrinks with the square root of the hands, not the hands themselves. To halve it, you play four times as many hands. To cut it to a quarter, sixteen times as many.
That is why the jump from 10,000 to 50,000 hands helps a lot, and the jump from 50,000 to 200,000 helps less than it sounds: four times the hands for half the margin.
04Hands to show you are winning
Turn the sum round and ask how many hands it takes for the interval to clear zero, if the observed win rate stays at the stated value. This is a margin-of-error calculation, not a promise of significance or a statistical-power calculation:
- Win rate 10
- 25,000 hands
- standard deviation 80
- Win rate 5
- 98,000 hands
- standard deviation 80
- Win rate 2
- 615,000 hands
- standard deviation 80
Halve the win rate and you need four times as many hands for it to show in the results. A player winning 2 big blinds per 100 needs over half a million hands for that observed rate to clear zero under these assumptions. A real sample may show a higher or lower rate, and changes in opponents or skill can invalidate the fixed-rate model. At a standard deviation of 100, the 5 big blind player needs 154,000.
05A faster measure
Results mix two things: the quality of your decisions and the cards you were dealt. The cards account for most of the swings. Measuring decisions directly avoids judging them by the eventual outcome.
That is what grading does. The trainer compares each decision with the best one and records the , the chips the choice gave up on average. A graded decision does not depend on what the river card was, so it measures a different part of play. The deal still affects the spots you face, and the score depends on the grading model's coverage and accuracy.
The variance simulator shows the same idea for tournaments, where the swings are bigger still.
Try itThe variance simulatorRun hundreds of tournament careers at the same skill and see how far apart the results land.Try itYour graded decisionsEV loss, accuracy and breakdowns from every decision the trainer has graded.Sources
- Confidence interval, Wikipedia
- Standard error, Wikipedia
Solved Hand is a study tool with no real money. Nothing here is gambling or financial advice; see the disclaimer and how the charts and grades are made. If gambling is causing you harm, here is where to get help.