Summing before taking a mean changes Monte Carlo normalization
= Summing before taking a mean changes Monte Carlo normalization
For $n$ observations, taking the mean of their scalar sum returns that sum, not their sample mean. If $a-b\overline H$ is unbiased for a target, replacing the mean by a sum gives expected value $a-bn\mathbb EH$ and multiplies its variance by $n^2$. A mean-zero <control variate> changes variance but cannot correct this missing normalization factor.