Let $X$ and $Y$ be independent random variables. I draw $N_x$ samples from $X$ and $N_y$ samples from $Y$. I am interested in finding the expected values for $X$, $Y$ and $Z = X-Y$. In order to find the last one, I will do the following. I will consider every possible pair of the samples from $X$ and $Y$, there will be $N_z = N_x N_y$ of those. For each of those pairs, I will find the difference between $X$ and $Y$, and then use that data to estimate the expected value of $Z$.
The variance of the mean is the variance of the sample, divided by the number of observations
$$\bar{\sigma}^2 = \frac{\sigma^2}{N}$$
Does this mean that asymptotically, with increasing $N$, my estimates for the expected value of $Z$ are significantly more precise than those of $X$ and $Y$? Or am I cheating by creating extra samples out of thin air. If I indeed am, what is the correct number of samples to be used for estimating the variance of the mean of $Z$ obtained via such procedure?