The total of the actual payoffs is a sum of independent random variables The standard deviation of this sum is the square root of the variance of the sum. The variance of a sum of independent random variables is the sum of the variances of each so
There are two ways to calculate the The most accurate is to use the known pot sizes and the win probabilities to calculate the actual variance. Since is the mean of the payoff the actual variance is
Then the standard deviation of is
The 95% confidence interval around the expected total payoff is
There is no multiplication by of the sample size. Where did that go? Well, there is a second way to get the variance and that is to estimate it ignoring the . We assume the random variables have the same variance and estimate that as
where is the mean actual payoff.
Then the variance is estimated as
is an estimate of the actual variance which is not exact because it treats the variation of as unknown.
The estimated standard deviation of is
is the estimated common standard deviation of the
Then the 95% confidence interval is
which is the familiar one.