Roulette variance: why equal edges feel different

Variance measures how far outcomes spread around their mean. Standard European red and straight-up wagers have the same expected loss per chip, yet feel different: red produces frequent small wins; a number wager produces infrequent larger wins and many losses.

Compare one-chip results

Both means are about −0.027 chip per spin. The size and frequency of departures from that mean differ greatly.

Calculate the spread

For a net result X, variance is the average of X squared minus the square of its mean. For red, X squared is always one. Its variance is 1 − (1/37)^2, about 0.9993 squared chips. Standard deviation, the square root of variance, is about one chip.

For a straight-up wager, the average squared result is (35^2 + 36)/37. Subtract (1/37)^2 to get variance about 34.0804, with standard deviation about 5.84 chips. The rare 35-chip win contributes heavily to that spread.

For 100 independent fixed one-chip wagers, variances add. Expected profit is −100/37 for either selection, while standard deviation is about 10 chips on red and 58.38 chips on a number. These are distribution measures, not guaranteed bands around an ending balance. Changing stakes or stopping early requires another model.

Read more than the mean

The mean averages all results. The median splits ordered sessions in half. A percentile identifies a location in that ordered set. A rare large gain can pull the mean above the median, so an average alone can hide how most runs ended.

For each simulated run, drawdown measures the decline from its previous peak. Two sessions can finish with the same profit yet reach very different interim losses. Label whether a chart shows ending balance, net profit or drawdown before comparing lines.

Use simulation methodology to check sample size and stop conditions. The strategy simulator can illustrate stake-rule differences, while house edge explains why a smoother-looking progression does not establish a better expected return.