If I were to measure some quantitavie metric of a sample population and record its mean, and then I were to split by random selection all members of the population into two groups of equal size and record the means of each group on the same metric. Is their a statistical term/measure of the difference of those means from each other and/or the mean for the whole population.
This comes as I'm trying to analyze an experiment in which both the control and sample group where told information regarding the experiment during a pretest where I fell like the information could have been left out due to the large size of the population.
My rational behind that is the difference in the split means would decrease as the overall population increases and one could assume that both sample groups are evenly distributed.
If this value (as a random variable) has expectation $\mu$ and variance $\sigma^2$,
then the mean of an iid sample of $n$ of them would have expectation $\mu$ and variance $\frac1n\sigma^2$
and the mean of an iid sample of $\frac n2$ of them would have expectation $\mu$ and variance $\frac2n\sigma^2$
and the signed difference in means of two iid samples each of $\frac n2$ of them would have expectation $0$ and variance $\frac4n\sigma^2$. That tends towards $0$ as $n$ increases.