Variance and Standard Deviation Variance & Standard Deviation
Statistical values that show how widely results are spread around the average. In slots, they are the math behind volatility: even at the same RTP, a larger standard deviation means much more uneven results.
| English | Variance & Standard Deviation |
|---|---|
| Classification | Math & Probability |
| English | Variance / Standard Deviation |
| Category | Math / Probability |
| Variance formula | E[X²] - (E[X])² |
| Standard deviation | Square root of variance |
| Slot standard deviation range | Roughly several times to tens of times the bet or more per spin (varies by game) |
| Related concepts | Volatility, law of large numbers, risk of ruin |
Overview
Variance is a value showing how far results are spread from the average, and standard deviation is the square root of the variance. Both express “how bumpy it is” as a number.
The standard deviation is the mathematical core of what slot players call volatility. If RTP tells you “how much is returned on average,” the standard deviation tells you “how widely results swing around that average.”
How to calculate it
If X is the result of one spin (the multiplier received relative to the bet), it is calculated as follows.
| Value | Formula | Meaning |
|---|---|---|
| Mean (expected value) | E[X] = Σ (multiplier × probability) | Average return per spin, equal to the RTP |
| Variance | E[X²] – (E[X])² | Average of the squared deviations from the mean |
| Standard deviation | √variance | Typical size of deviation from the mean |
Same RTP, different standard deviation
Let’s compare two hypothetical games that both have an RTP of exactly 96%.
| Games | How wins work | RTP | Standard deviation |
|---|---|---|---|
| A (low volatility) | 2x with 48% probability | 96% | About 1.0 |
| B (high volatility) | 100x with 0.96% probability | 96% | About 9.75 |
Both games return exactly 96% in the long run, but B’s standard deviation is about 10 times A’s. A pays out small wins about once every two spins, while B pays out big wins about once every 100 spins. Real slots mix in a much wider variety of win amounts, but the principle is the same.
In the simulation
We simulated each game 20,000 times. The starting balance was 100 bets’ worth (100 units), betting 1 unit per spin.
| Spin count | A ruin probability | A profit probability | B bust probability | B profit probability |
|---|---|---|---|---|
| 200 spins | 0% | 25.9% | 52.6% | 27.2% |
| 1,000 spins | 4.3% | 9.7% | 77.8% | 19.1% |
| 5,000 spins | 95.4% | 0.2% | 92.2% | 7.4% |
Low-volatility A rarely loses much in the short run, but the longer you play, the closer you get to the 96% RTP, so you end up losing most of it. High-volatility B wipes out more than half of players early on, but those who survive keep a lasting chance of profit thanks to big wins. In other words, the standard deviation determines the shape of how you win and lose.
Standard deviation and playing time
The standard deviation of the average result over many spins shrinks to σ ÷ √N (σ is the standard deviation of one spin, N is the number of spins). That’s why a game with a larger standard deviation needs many more spins to get close to its RTP. You can read more about this principle in the law of large numbers entry.
Why providers don’t publish it
Most providers publish the RTP but not the standard deviation. Instead, they show a volatility rating such as “low, medium, high,” or a lightning-bolt icon. Some markets’ certification documents do include the standard deviation or a volatility index. As a player, a good rule of thumb is that the higher the max win and the lower the hit frequency, the larger the standard deviation tends to be.
Summary
| Metric | What it tells you |
|---|---|
| RTP (average) | How much is returned in the long run |
| Standard deviation | How widely results swing |
| Hit Rate | How often you win |
| Max win | The most you can receive in a single spin |