- #1

kelly0303

- 579

- 33

$$A = \frac{S_+-S_-}{S_++S_-}$$

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!