I AM running a simulation of one million hands, and I know how to calculate the variance; and I'm counting splits as one hand. My problem is that I'm solving for the standard deviation of one HAND, but then trying to use that to find the standard deviation of a hundred WAGERS, which could number anywhere from one to, say, 1.5 per hand. Therefore, I can't use the usual s.d.*sqrt(hands), because the number of hands will be a variable. I also can't do s.d.*sqrt(hands/avg # bets per hand), because there is a covariance between the number of hands and the average number of bets placed per hand.

To put it another way, I don't play exactly H/avg hands, but rather H/avg +- something, which gives an additional term to the standard deviation. If I finish in less than H/avg hands, that means I did better because I doubled and split more often. So there's a negative correlation, and I need to account for that covariance to get the exact standard deviation of placing N bets.