I am confused about the results I am getting for an apparently simple situation. I have 2 measurements (counts), call them $S_+$ and $S_-$. Based on these I build an asymmetry defined as:
$$A = \frac{S_+-S_-}{S_++S_-}$$
The parameter I need to extract experimentally, call it $x$ behaves like $\frac{dx}{x} = \frac{dA}{A}$ where $dx$ and $dA$ are the uncertainties on $x$ and $A$ (ignore systematic uncertainties for now). $x$ is fixed (given by the physics process I am studying) and let's say I want to extract $x$ with $10\%$ relative uncertainty i.e. $\frac{dx}{x} = \frac{dA}{A} = \frac{1}{10}$. I have 2 situations (I will give the actual numbers I get). In the first one I have:
$$S_+ = 0.0484 N$$
$$S_- = 0.0324 N$$
where $N$ is the number of initial events and $S_+$ and $S_-$ are the events I am actually measuring. In the second case I have:
$$S_+ = 0.0085 N$$
$$S_- = 0.0027 N$$
Using the formula above, in the first case I am getting $A_1 = 0.198$ and in the second case I am getting $A_2 = 0.519$. If I do an error propagation, I end up with the formula:
$$dA = \frac{2}{(S_++S_-)^2}\sqrt{S_+S_-^2+S_+^2S_-}$$
from which I get $dA_1 = \frac{3.448}{\sqrt{N}}$ and $dA_2 = \frac{8.083}{\sqrt{N}}$. So I get $\frac{dA_1}{A_1} = \frac{17.4}{\sqrt{N}}$ and $\frac{dA_2}{A_2} = \frac{15.6}{\sqrt{N}}$. Which means that in the first case I need about $N_1 = 30276$ events and in the second case I need $N_2 = 24336$ events. But this doesn't make sense to me. For a fixed $N$, in the first case the number of events I am actually measuring are about an order of magnitude bigger than in the second case. Given that I am only looking at the statistical uncertainty, I would expect to need ~100 times more events in the second case, to reach the same uncertainty on the parameter of interest i.e. $N_2 \sim 100N_1$. What am I doing wrong? Shouldn't I use that error propagation on $A$? What should I do such that the uncertainty on $x$ reflects that fact that in the second case I have much lower statistics? Thank you!