Why the average is not the whole story
An average tells you where results tend to land over many repetitions. It says nothing about how far any single result, or any short run of results, might stray from that average. Two games can share the same average and still feel completely different to play. Variance and standard deviation are the tools that put a number on that difference.
This article builds the idea from the ground up using coins and dice, where every figure can be checked by hand. Nothing here is a method for winning. The aim is to understand why results bounce around, and how much.
The average first: expected value
The starting point is the expected value, the long-run average result of one trial. For a fair six-sided die the faces 1 to 6 are equally likely, so the expected value is (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. You will never roll a 3.5, but over a great many rolls the average drifts towards it.
For a coin, score heads as 1 and tails as 0. The expected value is 0.5.
Variance: the average squared distance
Variance measures how far outcomes sit from the average, on average. The recipe has three steps:
- Find the distance of each outcome from the mean.
- Square each distance, so that misses above and below do not cancel out and big misses count extra.
- Take the probability-weighted average of those squares.
For the fair die the mean is 3.5. The distances are -2.5, -1.5, -0.5, 0.5, 1.5 and 2.5. Squared, they are 6.25, 2.25, 0.25, 0.25, 2.25 and 6.25, which add up to 17.5. Dividing by 6 gives a variance of about 2.92 (exactly 35/12).
For the coin scored 0 or 1, each outcome is 0.5 away from the mean, so each squared distance is 0.25 and the variance is 0.25.
Standard deviation: back to the original units
Because variance is built from squares, its units are squared too, which is awkward to picture. The standard deviation is simply the square root of the variance, which returns the figure to the original units.
- Fair die: the square root of 2.92 is about 1.71.
- Coin scored 0 or 1: the square root of 0.25 is 0.5.
A rough reading of the die figure is that a typical roll lands within about 1.7 of 3.5. That is a loose description rather than a rule, but it conveys the scale.
Adding up many trials
The useful property is how spread behaves when you add independent trials together. Variances add. Standard deviations do not; they grow with the square root of the number of trials.
Take 100 fair coin flips and count the heads:
- The expected number of heads is 100 x 0.5 = 50.
- The variance is 100 x 0.25 = 25.
- The standard deviation is the square root of 25, which is 5.
So a typical result is within a handful of heads of 50. Now take 400 flips. The expected number is 200, the variance is 100 and the standard deviation is 10. Four times as many flips doubled the absolute spread, because the square root of 4 is 2.
Here is the part that surprises people. The absolute spread grows, but the relative spread shrinks. A standard deviation of 5 on 100 flips is 5% of the number of flips. A standard deviation of 10 on 400 flips is 2.5%. This is the arithmetic behind the law of large numbers: averages settle down even though the raw count of heads can wander further from its target.
The same pattern holds for dice. For the total of two dice the variance is 2 x 35/12 = 35/6, about 5.83, and the standard deviation is about 2.42. The familiar pattern of two dice totals clustering around seven is a picture of exactly this spread.
A casino-style illustration
Consider a bet that pays one unit on a win and costs one unit on a loss, on a single-zero roulette wheel with 37 pockets, such as a bet on red. There are 18 winning pockets and 19 others, so the chance of winning is 18/37.
- Expected result per bet: (18/37) x 1 + (19/37) x (-1) = -1/37, about -0.027 units.
- Every outcome is plus or minus one, so the average of the squares is 1. Subtracting the squared mean (about 0.0007) leaves a variance just under 1, and a standard deviation of about 1.
After 100 such bets, treated as independent, the expected result is about -2.7 units, but the standard deviation of the total is about 10 units. The swing is several times larger than the expected loss. That is why short sessions can end ahead or behind by amounts that look nothing like the house edge, and why the edge only shows up reliably over very long runs. It does not mean the game can be beaten; the expected result is still negative, and outcomes are uncertain in both directions.
What standard deviation does and does not tell you
Used carefully, it is a good summary of scale. It has limits.
- It does not cap results. Results beyond two or three standard deviations are rarer but entirely possible.
- It describes spread, not direction. It cannot say whether the next result will be high or low.
- For skewed games, with many small losses and an occasional large win, a single figure hides the shape. A game with rare big payouts can have the same average and a much larger standard deviation than one with steady small ones.
- It assumes you know the true probabilities. In real settings you often only have an estimate from a limited sample.
For many sums of independent trials the shape of the results is close to a bell curve, which is why a standard deviation is so often paired with the normal distribution. About two thirds of results typically fall within one standard deviation of the mean and about 95% within two. These are rules of thumb for bell-shaped results, not guarantees for any single run.
Putting it in context
Variance explains why a few plays prove very little. A fair coin can easily give 7 heads in 10 flips without being biased, and a streak of losses on a game with a house edge does not mean the game changed. Past results do not shift the next result, which is the point behind the gambler’s fallacy.
Gambling carries a house edge and outcomes are uncertain, with real financial risk. Understanding variance is not a way to reduce that risk to zero. If gambling is causing you or someone you know difficulty, a local support service can help.