--- title: "Variance" url: https://gagamblinghelpline.org/terms/variance/ description: "What variance measures, a worked per-trial derivation for an even-money wheel bet, and how expected result and spread scale differently as trials accumulate." --- # Variance ## Topic chapters - [Probability](https://gagamblinghelpline.org/terms/#C) (10) - [Lotteries and Draws](https://gagamblinghelpline.org/terms/#C) (4) - [Odds and Prices](https://gagamblinghelpline.org/terms/#B) (8) - [House Advantage](https://gagamblinghelpline.org/terms/#A) (10) - [Variance and Streaks](https://gagamblinghelpline.org/terms/#B) (11) - [Games and Rules](https://gagamblinghelpline.org/terms/#A) (7) ## Long-form chapters - [Probability and expected value](https://gagamblinghelpline.org/topics/probability-and-expected-value/) - [How the house edge is calculated](https://gagamblinghelpline.org/topics/how-house-edge-is-calculated/) - [Odds notation and implied probability](https://gagamblinghelpline.org/topics/odds-notation-and-implied-probability/) - [Variance, spread and why streaks are ordinary](https://gagamblinghelpline.org/topics/variance-and-streaks/) - [Independence and the gambler's fallacy](https://gagamblinghelpline.org/topics/independence-and-the-gamblers-fallacy/) - [Return to player and lottery prize funds](https://gagamblinghelpline.org/topics/return-to-player-and-lottery-prize-funds/) - [How games of chance are designed](https://gagamblinghelpline.org/topics/how-games-of-chance-are-designed/) - [A history of lotteries and betting](https://gagamblinghelpline.org/topics/history-of-lotteries-and-betting/) - [The development of gambling regulation](https://gagamblinghelpline.org/topics/development-of-gambling-regulation/) ## Letters carrying entries [A](https://gagamblinghelpline.org/terms/#A) [B](https://gagamblinghelpline.org/terms/#B) [C](https://gagamblinghelpline.org/terms/#C) [D](https://gagamblinghelpline.org/terms/#D) [E](https://gagamblinghelpline.org/terms/#E) [F](https://gagamblinghelpline.org/terms/#F) [G](https://gagamblinghelpline.org/terms/#G) [H](https://gagamblinghelpline.org/terms/#H) [I](https://gagamblinghelpline.org/terms/#I) [J](https://gagamblinghelpline.org/terms/#J) [K](https://gagamblinghelpline.org/terms/#K) [L](https://gagamblinghelpline.org/terms/#L) [M](https://gagamblinghelpline.org/terms/#M) [N](https://gagamblinghelpline.org/terms/#N) [O](https://gagamblinghelpline.org/terms/#O) [P](https://gagamblinghelpline.org/terms/#P) [Q](https://gagamblinghelpline.org/terms/#Q) [R](https://gagamblinghelpline.org/terms/#R) [S](https://gagamblinghelpline.org/terms/#S) [T](https://gagamblinghelpline.org/terms/#T) [U](https://gagamblinghelpline.org/terms/#U) [V](https://gagamblinghelpline.org/terms/#V) [W](https://gagamblinghelpline.org/terms/#W) [Z](https://gagamblinghelpline.org/terms/#Z) Twenty-four of twenty-six letters hold at least one entry. X and Y hold none. ## Figures at a glance - Single number, 37 pockets, paid 35 to 1 **2.70 %** per unit staked - Single number, 38 pockets, paid 35 to 1 **5.26 %** per unit staked - Dice sequence, pass on 7 or 11 **1.41 %** per unit staked - Two-way market priced 1.91 each side **4.50 %** per unit staked Each figure is derived on the chapter that states it. ## Editorial policy Figures are derived on the page that states them, and each is printed with the base it is measured against. [Read the method](https://gagamblinghelpline.org/method/) Variance and Streaks Variance is the average squared distance of results from their mean; its square root, the standard deviation, is the typical distance in the original units. per trial Expected value locates the centre of a distribution. Variance describes how far individual results sit from that centre. Both are needed, because a bet with a small negative expectation and a wide spread can produce long stretches of results that look nothing like its expectation, and a bet with the same expectation and a narrow spread cannot. ### Working the per-trial figure Take the even-money wheel bet: one unit staked, returning +1 with probability 18/37 and −1 with probability 19/37. Because every outcome is +1 or −1, every squared outcome is exactly 1, which makes the arithmetic short. ``` mean m = (18/37)(+1) + (19/37)(-1) = -0.027027 mean of X^2 = (18/37)(1) + (19/37)(1) = 1.000000 variance = 1.000000 - (-0.027027)^2 = 1.000000 - 0.000730 = 0.999270 per trial standard dev. = sqrt(0.999270) = 0.999635 units ``` ### How the two quantities scale Across *n* independent trials the expected result multiplies by *n*, while the standard deviation multiplies only by the square root of *n*. That difference in growth rate is the whole of the long run. ``` trials expected result typical spread ratio 100 -2.70 units 10.00 units 0.27 1,000 -27.03 units 31.61 units 0.86 10,000 -270.27 units 100.00 units 2.70 100,000 -2,702.70 units 316.23 units 8.55 ``` At a hundred trials the spread is nearly four times the expected loss, so finishing above the starting point is entirely ordinary. At ten thousand trials the expected loss sits 2.70 standard deviations below zero and finishing above the starting point has become distinctly unusual. At a hundred thousand it is 8.55 standard deviations below, which is no longer a close question. Nothing about the game changed between those rows; only the count of trials did. ## Related entries [Standard deviation](https://gagamblinghelpline.org/terms/#S) The typical distance of a result from the average, growing with the square root of the count of trials. VARIANCE AND STREAKS [Long run](https://gagamblinghelpline.org/terms/#L) The count of trials at which expected loss grows large relative to the spread of results around it. VARIANCE AND STREAKS [Streak](https://gagamblinghelpline.org/terms/#S) A run of identical consecutive results, whose probability falls geometrically with length but which is ordinary across many trials. VARIANCE AND STREAKS [Volatility](https://gagamblinghelpline.org/terms/#V) How widely a game's results spread around its return, set separately from the return itself. VARIANCE AND STREAKS [Gambler's ruin](https://gagamblinghelpline.org/terms/#G) The result that a player with finite funds facing a negative expectation and unlimited trials is eventually certain to reach zero. VARIANCE AND STREAKS [Bankroll](https://gagamblinghelpline.org/terms/#B) The total amount set aside for play, against which the spread of results over many trials is measured. VARIANCE AND STREAKS