We have to start at the beginning and along the way we'll see why there are two ways to calculate the SD.

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

where

is the estimated common standard deviation of the

Then the 95% confidence interval is

which is the familiar one.