In poker, even the best players experience swings—sometimes winning big, sometimes losing despite making the right decisions.
These ups and downs aren’t just luck. They’re part of the game’s statistical reality.

Concepts like variance, standard deviation, and confidence intervals help players quantify these swings, separate skill from luck, and make smarter decisions over the long run.

Think of them as:

Variance: “Variance is the rollercoaster of poker—sometimes you soar, sometimes you crash, even when you play perfectly.”

Standard Deviation: “Standard deviation measures just how wild that ride can get, showing the typical swing from your expected results.”

Confidence Intervals: “Confidence intervals give you the real picture of your skill, showing the range where your true performance likely falls.”

In this article, we’ll break down these essential tools and show how understanding them can improve your poker strategy.

1. What Is Variance?

Variance is a statistical concept that describes how much actual results deviate from expected results due to randomness.

Poker:

Your expected value (EV) tells you what should happen in the long run.
Variance explains why short-term results often don’t match EV.

Formula for variance:

$$\text{Var}(X) = E[(X – \mu)^2]$$

Where:

$X$ = your actual results (winnings or losses)
$\mu$ = expected average winnings
$E$ = expectation (average over many trials)

In plain English: variance measures how “swingy” your results are compared to what you should expect.

Short-Term Variance vs. Long-Term Results

The key is sample size.

In 10 hands, anything can happen—you could lose every time with AA against KK.
In 10,000 hands, your results will begin to converge closer to the true expected value.

This is called the Law of Large Numbers, and it’s why professional players focus on long-term profit rather than short-term swings.

Example: Winning With Pocket Aces

Suppose you’re all-in pre-flop with AA vs. KK. The math says:

AA wins 82% of the time.
KK wins 18% of the time.

If you play this situation 100 times with $100 in the pot each time:

$$EV = (0.82 \times 100) – (0.18 \times 100) = 82 – 18 = +64$$

So on average, you should win $64 per hand.

But variance means you won’t always see that average. Over 10 trials, you might lose 3 or 4 times in a row despite being the heavy favorite. This doesn’t mean your play was wrong—it’s just variance.

Example: Variance in a Tournament

In tournaments, variance is even higher because payouts are top-heavy.

Suppose you enter a $100 tournament with 100 players:

1st place = $3,000
2nd place = $1,500
3rd place = $750
Remaining 10 spots = $200 each

If your skill gives you a 10% chance to finish top 3 and a 20% chance to cash, your EV might be:

$$EV = (0.05 \times 3000) + (0.03 \times 1500) + (0.02 \times 750) + (0.20 \times 200) – 100$$

$$EV = 150 + 45 + 15 + 40 – 100 = +150$$

So each entry is worth +$150 in the long run.

But variance means you could easily go 20 tournaments in a row without cashing before hitting a big win. This is why bankroll management is stricter for tournaments than cash games.

Coping With Variance

Focus on Decisions, Not Outcomes
If the math says your play was correct, a short-term loss doesn’t mean you were wrong.
Use Proper Bankroll Management
Because variance can wipe you out, pros recommend:

20–30 buy-ins for cash games
50–100 buy-ins for tournaments

Keep Records and Review Play
Tracking your hand histories and win rates will remind you that downswings are normal and temporary.
Stay Calm During Swings
Tilt (emotional decision-making) turns variance from a temporary swing into a permanent leak in your strategy.

Why Variance Is Actually Good

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—while math ensures that, in the long term, skillful players rise to the top.

Understanding Variance: Win Rate in BB/100

In poker, your win rate is often expressed as bb/100, meaning the average number of big blinds you win per 100 hands.

Example: You’re a solid cash game player with a win rate of +5 bb/100.
That means, on average, for every 100 hands you play, you win 5 big blinds.

But here’s the catch: variance means that your actual results will swing wildly above and below that number in the short term.

2. Standard Deviation in Poker

Standard deviation (σ), tells you how wide the swings are around your average results.

For example:

A typical no-limit hold’em cash player might have a standard deviation of 80 BB/100 hands.
If your win rate is 5 BB/100, the standard deviation shows how “noisy” your results will look session to session.

Even with a positive win rate, large standard deviation means you’ll experience long losing streaks before your skill shows in the results.

What This Means

Your true expectation per 100 hands:
$$EV = +5 \, \text{bb/100}$$

But the swings around this expectation are much larger:

$$\text{Range per 100 hands} \approx EV \pm \sigma$$

So after 100 hands, your result might realistically be anywhere from:

$$5 – 80 = -75 \, \text{bb/100} \quad \text{to} \quad 5 + 80 = +85 \, \text{bb/100}$$

That’s a massive range compared to your actual edge.

Sample Size and the Law of Large Numbers

As you play more hands, variance smooths out because results converge toward the true EV.

The formula for standard deviation after $N$ hands is:

$$\sigma_{N} = \frac{\sigma}{\sqrt{N/100}}$$

After 1,000 hands:

$$\sigma_{1000} = \frac{80}{\sqrt{10}} \approx 25.3 \, \text{bb/100}$$

After 10,000 hands:

$$\sigma_{10000} = \frac{80}{\sqrt{100}} = 8 \, \text{bb/100}$$

After 100,000 hands:

$$\sigma_{100000} = \frac{80}{\sqrt{1000}} \approx 2.53 \, \text{bb/100}$$

Interpreting This

At 1,000 hands, you could easily be losing despite being a winning player.
At 10,000 hands, your results will begin to resemble your true win rate more closely, but swings are still big.
At 100,000 hands, your actual results will be very close to your expected +5 bb/100, and luck plays a much smaller role.

This is why pros always talk about the importance of volume—only by playing a large number of hands does skill reliably beat variance.

3. Putting Everything So Far Together

Calculating Your Win Rate (bb/100)

Your win rate tells you how many big blinds you win (or lose) per 100 hands on average.

Formula:

$$\text{Win rate} = \frac{\text{Total BB won or lost}}{\text{Total Hands}} \times 100$$

Steps:

Record your total winnings/losses in big blinds (not dollars).
Record the total number of hands you played.
Plug into the formula.

Example:

Hands played = 20,000
Total winnings = +4,000 big blinds

$$\text{Win rate} = \frac{4000}{20000} \times 100 = +20 \, \text{bb/100}$$

This means you win an average of 20 big blinds per 100 hands.

Calculating Your Variance

Variance tells you how much your results fluctuate around your average.

Step A: Gather results per 100 hands

Break your data into chunks of 100 hands each.
For each chunk, calculate how many BB you won or lost.

Step B: Compute the mean (your win rate)

$$\mu = \frac{\sum X_i}{n}$$

where $X_i$ = result in BB/100 for chunk $i$, and $n$ = number of chunks.

Step C: Compute variance

$$\text{Var}(X) = \frac{\sum (X_i – \mu)^2}{n}$$

Step D: Compute standard deviation (σ)

$$\sigma = \sqrt{\text{Var}(X)}$$

This σ is expressed in bb/100, just like your win rate.

Example:

Suppose you tracked 5 blocks of 100 hands each, with these results (in bb/100):
$$X = \{+40, -20, +10, +70, -10\}$$

1. Mean (μ):

$$\mu = \frac{40 + (-20) + 10 + 70 + (-10)}{5}$$

$$= \frac{90}{5} = +18 \, \text{bb/100}$$

2. Deviation:

For each session winrate $xi$, find the deviation from the mean:

$$Deviation=xi−xˉ$$

Session	BB Won $xi$	Deviation $xi−xˉ$
1	40	40−18=22
2	-20	-20−18=-38
3	10	10−18=−8
4	70	70−18=52
5	-10	-10−18=−28

Deviation

3. Square Each Deviation

Square the values from Step 3 to eliminate negatives and amplify larger deviations.(xi−xˉ)2(xi−xˉ)2

Session	Deviation	Squared Deviation $(xi−xˉ)^2$
1	22	$22^2=484$
2	-38	$(-38)^2=1444$
3	-8	$(−8)^2=64$
4	52	$52=2704$
5	-28	$(−28)^2=784$

4. Variance:

Variance is the average of the squared deviations.

$$Var(X)=\frac{484 + 1444 + 64 + 2704 + 784}{5}$$

$$ = \frac{547…}{5} = 1096$$

5. Standard Deviation (σ):

Standard deviation is the square root of the variance:

$$\sigma = \sqrt{1096} \approx 33.1 \, \text{bb/100}$$

Interpreting Results

Win rate = +18 bb/100 → you’re a winning player.
Standard deviation = 33 bb/100 → your results swing up and down by about ±33 BB for every 100 hands.

This shows why you can have big downswings despite being profitable long-term.

👉 In practice:

Win rate tells you how much you should expect to earn over time.
Variance tells you how bumpy the road will be to get there.

3. Confidence Intervals in Poker

When you calculate your win rate (bb/100) from a sample of hands, it’s just an estimate. Variance means your measured win rate may be higher or lower than your true win rate.

A confidence interval (CI) gives you a statistical range that likely contains your true win rate.

Confidence intervals use standard deviation to express a range within which your true win rate probably lies with a certain level of certainty.

The 68-95-99.7 rule (empirical rule) states:

About 68% of samples will fall within 1 standard deviation of the mean.
About 95% within 2 standard deviations.
About 99.7% within 3 standard deviations.

If your average win rate is 2.5 BB/100 and standard deviation is 30 BB/100, then:

With 68% confidence, your win rate falls between -27.5 and 32.5 BB/100 in any 100-hand sample.
With 95% confidence, your win rate falls between -57.5 and 62.5 BB/100.
This demonstrates extreme variance and why short-term results can swing wildly.

Formula for Confidence Intervals

For large enough samples, we can use the normal distribution to approximate results:
$$CI = \mu \pm Z \times \frac{\sigma}{\sqrt{N/100}}$$

Where:

$\mu$ = observed win rate (bb/100)
$\sigma$ = standard deviation (bb/100)
$N$ = total hands played
$Z$ = z-score for the chosen confidence level
- 95% confidence → $Z \approx 1.96$
- 99% confidence → $Z \approx 2.58$

Expanded Example

Suppose you’ve played 20,000 hands and tracked:

Observed win rate = +5 bb/100
Standard deviation = 80 bb/100

Step 1: Compute standard error (SE)

$$SE = \frac{\sigma}{\sqrt{N/100}} = \frac{80}{\sqrt{200}} \approx 5.66$$

So the uncertainty in your win rate estimate is about ±5.7 bb/100.

Step 2: Compute 95% CI

$$CI = 5 \pm 1.96 \times 5.66$$

$$CI = 5 \pm 11.1$$

$$CI = [-6.1, +16.1] \, \text{bb/100}$$

Step 3: Interpret

Your observed win rate is +5 bb/100.
But with 95% confidence, your true win rate lies somewhere between –6.1 and +16.1 bb/100.
This means that while you look like a winning player, statistically you could still be break-even or even a small loser until more hands are played.

Step 4: Increase Sample Size

What if you play 200,000 hands instead of 20,000?

$$SE = \frac{80}{\sqrt{2000}} \approx 1.79$$

Now:

$$CI = 5 \pm 1.96 \times 1.79$$

$$CI = 5 \pm 3.5 = [1.5, 8.5] \, \text{bb/100}$$

Now your confidence interval is much tighter. You can be statistically confident you are a winning player, with a true win rate between 1.5 and 8.5 bb/100.

Why Confidence Intervals Matter

Small samples lie. A 10,000-hand winning streak may still fall within the variance of a losing player.
Larger samples reduce uncertainty. The more hands you play, the narrower your CI becomes, giving a clearer picture of your skill.
Confidence intervals help manage expectations. They prevent overconfidence after short-term good runs and reduce despair during bad runs.

Step 1 — compute the standard error (SE)

We are given σ = 80 bb/100 (this is the SD of results measured per 100-hand block).
Number of 100-hand blocks: $k=N/100=20000/100=200$.
$$SE = \frac{\sigma}{\sqrt{k}} = \frac{80}{\sqrt{200}}$$

Compute step-by-step:

$\sqrt{200} = 14.14$.
$SE = 80 \div 14.14 = 5.66$ (bb/100).

So SE ≈ 5.6569 bb/100.

Step 2 — apply the 68–95–99.7 rule to your estimate (μ = +5 bb/100)

Use multiples of SE around the observed mean.

±1 SE (≈68%)

$$\text{range} = \mu \pm 1\cdot SE = 5 \pm 5.6569$$

$$\text{68% CI (approx)} = [\,5 – 5.6569,\ 5 + 5.6569\,] = [\,-0.6569,\ 10.6569\,]\ \text{bb/100}$$

±2 SE (≈95%, empirical approximation)

$$\text{range} = 5 \pm 2\cdot 5.6569 = 5 \pm 11.3137$$

$$\text{95% (approx)} = [\,-6.3137,\ 16.3137\,]\ \text{bb/100}$$

±3 SE (≈99.7%)

$$\text{range} = 5 \pm 3\cdot 5.6569 = 5 \pm 16.9706$$

$$\text{99.7% (approx)} = [\,-11.9706,\ 21.9706\,]\ \text{bb/100}$$

Note: for a formal 95% confidence interval we usually use $Z=1.96$ rather than exactly 2. Using $1.96\times SE$ gives:
$$1.96\times SE = 1.96\times 5.6568542495 = 11.0874$$

$$\text{95% CI (exact)} = 5 \pm 11.0874 = [\,-6.0874,\ 16.0874\,]\ \text{bb/100}$$

Plain-English interpretation

About 68% of repeated 20,000-hand samples (with the same underlying process) would give a measured win rate within ±5.66 bb/100 of the true mean.
About 95% would lie within ≈±11.09 bb/100 (using 1.96×SE).
So with your observed +5 bb/100 after 20k hands, the 95% CI includes zero (≈ −6.09 to +16.09), meaning you cannot yet be 95% confident that your true win rate is positive — the data are still consistent with a small loss, break-even, or a decent win.

How many hands do you need to be confident?

If you want a specific margin of error $m$ (in bb/100) at confidence $Z$ (e.g. 95% → $Z=1.96$), you can solve for $N$:
$$\text{SE} = \frac{\sigma}{\sqrt{N/100}} \quad\Rightarrow\quad m = Z\cdot SE = Z\cdot\frac{\sigma}{\sqrt{N/100}}$$

Rearrange:

$$N = 100\left(\frac{Z\sigma}{m}\right)^2$$

Examples with σ = 80 bb/100, Z = 1.96 (95%):

To get m = 5 bb/100:

$Z\sigma = 1.96\times80 = 156.8$
$156.8/5 = 31.36.156$
latex^2 = 983.4496[/latex]
$N = 100\times 983.4496 = 98{,}344.96$ → ≈ 98,345 hands.

To get m = 2 bb/100:

$156.8/2 = 78.4$
$78.4^2 = 6{,}146.56$
$N = 100\times 6{,}146.56 = 614{,}656$ → ≈ 614,656 hands.

To get m = 1 bb/100:

$156.8/1 = 156.8$
$156.8^2 = 24{,}586.24$
$N = 100\times 24{,}586.24 = 2{,}458{,}624$ → ≈ 2.46 million hands.

So: to be 95% confident your true win rate is within ±5 bb/100 you need ~98k hands; to be within ±2 bb/100 you need ~615k hands. This is why large-volume players emphasize huge sample sizes.

Important caveats

The empirical rule assumes approximate normality. By the Central Limit Theorem, the sample-mean distribution becomes roughly normal for large $N$, so the approach is reasonable for big samples.
You must have a reasonable estimate of σ (itself noisy for small samples).
Hands must be independent and your playing/stakes style must be stationary — if your game changes (stakes, opponents, style), these numbers don’t apply directly.
Confidence intervals are a frequentist coverage statement: “If we repeat the whole sampling procedure many times, X% of those intervals will contain the true parameter.” Casual language like “there’s a 95% chance the true value lies in this interval” is commonly used but has a technical nuance.

Practical takeaway

Variance: Variance measures how much your results can swing over a period of time. In poker, it explains why you might lose a few big pots even when you’re playing well. High variance means bigger swings, low variance means more consistent results.
Standard Deviation: Standard deviation is a way to quantify those swings. It tells you, on average, how far your results deviate from your expected outcome. The bigger the standard deviation, the more unpredictable your short-term results.
Confidence Intervals: Confidence intervals give you a range where your “true” performance is likely to fall. For example, if your average win rate is 5 big blinds per 100 hands, a 95% confidence interval might show that the real win rate is probably between 3 and 7 big blinds per 100 hands. It’s a way to understand your skill over the noise of short-term luck.
Use ±1 SE (≈68%) for quick intuition, 95% Confidence Interval (≈1.96 SE) for standard confidence statements.
With typical poker σ (~70–100 bb/100), you need tens to hundreds of thousands of hands to narrow the Confidence Interval enough to reliably say you’re a winner.

Understanding Variance, Standard Deviation, and Confidence Intervals in Poker