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 $a-bn\mathbb{E}H$ and multiplies its variance by $n^2$. A mean-zero control variate changes variance but cannot correct this missing normalization factor.