I Statistic uncertainties in cross section plots - how to calculate?

[Amy_93]

TL;DR Summary

TL;DNR: I am not sure how to calculate the statistic uncertainties for equations like N_sig=M*(N_meas-N_bkg), assuming Poisson distributions

Hi there,

I hope I chose the right forum for my question.

So, basically, I'm doing an analysis measuring the number of signal particles in a certain momentum bin i, and doing two corrections:

N_{sig, i}=M*(N_{meas, i}-N_{bkg, i})

Here, M is a matrix covering PID correction and PID efficiencies, and N_{bkg, i} is the number of background events in this bin (based on MC).
Now, there's obviously also statistical uncertainties in M and N_{bkg, i} that I want to calculate in include in the error bar:

N_{sig, i}=N_{sig, i, mean}±δN_{sig, i}

But, how?

Assuming that M and N_{bkg, i} follow a Poisson distribution, the standard derivation for N_{bkg, i} would read as

σ_{Nbkg, i}=√N_{bkg, i}

but this is only the standard deviation of events from the sample mean, right? To account for the fact that I'm only looking at a sample and not all possible collision events in the world I expected to need an expression like

σ_{Nbkg, i}/√N

and this is also what I find in textbooks, but here I'm lost.

- Is this equation even the right one to start with?
- If so, is N simply the total number of MC background events used to calculate N_{bkg, i}?Thanks for ponting me in the right direction,
Amy

 

[Vanadium 50]

I am confused. Why is M Poisson?

The usual trick is to express what you want entirely in terms of independent measured quantities, which are all then Poisson.

 

[mfb]

σ_{Nbkg, i}=√N_{bkg, i}

What happens outside this bin is irrelevant for the Poisson statistics in that bin. Note that N here is your full set of MC events in that bin. Your background estimate will likely use a scaled version of that, so you need to scale the uncertainty correspondingly.

Is M diagonal? If yes, then you have k independent problems and you can just use standard error propagation to combine the different sources. If it's not diagonal, then you need some unfolding procedure. All the standard procedures also come with a way to estimate uncertainties.

 

[hutchphd]

I am unclear. What is the dimension of the matrix M? The transformation it accomplishes sums data for bin i over various PID settings? Please be more explicit
What is "MC"

 

[Amy_93]

Ah thank you guys so much, data unfolding is exactly what I was looking for, I can take it from there :) Thanks again