Variance
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.