Average first, spread second
The average outcome of a random event tells you where results are centred. It does not tell you how far they wander from that centre. Variance and standard deviation are the two usual ways to measure that wandering.
Dice are a good place to learn them, because every number can be checked by hand.
One die, step by step
A fair six-sided die has an average of 3.5. To measure spread, look at how far each face is from that average:
| Face | Distance from 3.5 | Distance squared |
|---|---|---|
| 1 | -2.5 | 6.25 |
| 2 | -1.5 | 2.25 |
| 3 | -0.5 | 0.25 |
| 4 | +0.5 | 0.25 |
| 5 | +1.5 | 2.25 |
| 6 | +2.5 | 6.25 |
The squared distances add up to 17.5. Since each face is equally likely, the variance is the average of these squared distances: 17.5 divided by 6, which is about 2.92.
The standard deviation is the square root of the variance, about 1.71. It is in the same units as the roll itself, which makes it easier to read: a typical roll lands roughly 1.7 away from 3.5. (Rolls are never exactly 1.7 away; it is a measure of typical distance, and it weights large distances more heavily because of the squaring.)
Why square the distances? If you simply averaged the signed distances, the negatives and positives would cancel to zero. Squaring removes the sign and also gives extra weight to results far from the middle.
Two dice
Roll two dice and add them. The average is 7. The variance of independent events adds, so two dice have variance of 2 × 2.92, about 5.83, and a standard deviation of about 2.42. This fits what you see in practice, where totals of 5 to 9 are common and 2 or 12 are rare. The frequency of each total is the topic of two-dice sums, but the spread tells you how wide the common band is.
Ten dice, and what grows with what
Ten dice have an average total of 35 and a variance of about 29.2, which gives a standard deviation of about 5.4.
Compare how things scale as the number of dice n grows:
- The average of the total grows in proportion to n.
- The variance of the total grows in proportion to n.
- The standard deviation of the total grows with the square root of n.
So the total’s standard deviation increases more slowly than its average, which means relative spread shrinks. Ten dice have an average of 35 and a standard deviation of 5.4, a ratio of about 0.15. One die has 3.5 and 1.71, a ratio close to 0.49. This is the reason the average of many rolls is much more stable than a single roll, a point central to what the law of large numbers does and does not promise.
The standard deviation of the average roll, as opposed to the total, falls with the square root of the number of rolls: about 1.71 for one die, and 1.71 divided by the square root of 10, roughly 0.54, for the average of ten.
Same average, different feel
Spread matters because many decisions are about swings and not only averages. Take two made-up games where you win or lose a fixed amount on a fair coin flip, with an average result of zero in both. In the first the stake is 1 unit. In the second it is 10 units.
- Game one: outcomes are +1 and -1, the variance is 1 and the standard deviation is 1.
- Game two: outcomes are +10 and -10, the variance is 100 and the standard deviation is 10.
Both have the same expected value of zero. Yet the second is ten times as swingy per play. Over 100 plays of game one, the standard deviation of the total is 10 units; over 100 plays of game two it is 100 units. A player with a fixed budget faces a much greater chance of running out in the second game, which connects to risk of ruin and to why bankroll concepts are about more than the average.
This is the difference described by volatility and return: the average describes expected outcome, and the spread describes how uncertain that outcome is.
Estimating spread from data
If you do not know the underlying chances and have only recorded results, you can compute the variance from the data. The usual recipe is to take the average of the results, find each result’s squared distance from it, add them up and divide by the number of results minus one rather than the number. The “minus one” corrects for the fact that the average itself was estimated from the same data. For large data sets the difference is negligible.
With few results the estimate of spread is itself unreliable. A short record of a few rolls can look much steadier or wilder than the true spread, which is another effect of limited sample size.
Reading spread in games
- Bigger spread, longer unlucky stretches. Even a game with a favourable setup, such as a casual dice contest between equal players, produces runs of losses when the spread is large. In card games this shows up as poker variance.
- Typical range. With a bell-shaped total, about two-thirds of results lie within one standard deviation of the average and about 95 percent within two. This is the use of the normal distribution for describing what an ordinary range is.
- Do not confuse spread with edge. A high-variance game is not more or less profitable on average. A game with a negative expectation loses on average whatever its variance, and high variance only changes how bumpy the path is. Spread gives no way to beat the odds.
Outcomes in games of chance stay uncertain. If gambling is becoming a problem, a local support service can help.