effect of sample size on precision I have two samples of the same population and wish to reduce the sample sizes, but maintain the same ratio of precision produced in the original samples. I suspect I can take the square root of the original sample size, but this produces very small samples. How, in short, can I scale down the original samples, but maintain the same ratio of precision? Any advice (and a formula!) would be much appreciated. 
Example: 
Original sample sizes for the same population (population N is c. 20,000)
N for sample A = 2537; N for sample B = 520
How do I reduce sample sizes while maintaining the same relative precision?
Gary Marks
Burton Craige Dist. professor, UNC-Chapel Hill
 A: I'm going to assume by precision you're talking about estimator standard error, where the estimator is estimating the mean. I'm also going to assume that the sample is a simple random sample without replacement (that is all units are equally likely to be selected). The formula for standard error under these assumptions:
$$
\sqrt{\frac{1}{n}-\frac{1}{N}} S_Y
$$
Where $S_Y$ is the population adjusted standard deviation for the variable being collected.
So we want to maintain the same relative precision, therefore:
$$
\frac{\sqrt{\frac{1}{n_{1,Y}}-\frac{1}{N}} S_Y}{\sqrt{\frac{1}{n_{1,X}}-\frac{1}{N}} S_X} = \frac{\sqrt{\frac{1}{n_{2,Y}}-\frac{1}{N}} S_Y}{\sqrt{\frac{1}{n_{2,X}}-\frac{1}{N}} S_X}
$$
In the above I assume that for one sample you're collecting $Y$ variables and in the other you're selecting $X$ variables (but this doesn't matter since they cancel out). I also let a subscript $1$ represent the original samples, while $2$ represents the different sample sizes.
Assuming in the new sample we select is $k$ times the old sample for the sample estimating $Y$ (i.e. $n_{2,Y} = k n_{1,Y}$), what should we select for $X$ to maintain relative precision?
$$
n_{2,X} = \frac{kn_{1,X}(N-n_{1,Y})}{(N-n_{1,X})+k(n_{1,X}-n_{1,Y})}
$$
So if you have an infinite population, or your original sample sizes were the same, then you simply modify your sample size by the same factor for both samples. If (like in your case) you start with different sample sizes and have a finite population, then this formula shows how changing the size of one sample changes the size of the other.
