Comments on calculating the errors for acceptances - W. Brooks

If only a given bin is considered (as is the case for bin-by-bin acceptance corrections), then the distribution of accepted events in the bin follows the binomial distribution; that is, the outcome of any trial is either a success or failure, and there is a precise probability for success, p.

If the acceptance is only measured once, the error estimate for the number of accepted events is given by the square root of the variance of the binomial distribution.
The formula for the variance is V = Np(1-p), where N is the number of trials; an unbiased estimate of the variance for a finite number of trials is

V = (N/N-1)N(r/N)(1-r/N)

where r is the number of accepted events.

The estimate for the acceptance is given by r/N, therefore, the error on the acceptance is given by

absolute acceptance error = sqrt( (r/N)(1-r/N) / (N-1) )

(This is the same function derived by John Price from the Bevington error formula for quotients, assuming r and N-r are independent variables whose errors are given by their square roots.)

This function is maximal for given N when r/N = 0.5. Therefore, an upper bound for the error is given by:

upper bound for absolute acceptance error = 0.5/sqrt(N-1)

Examples:
for an absolute acceptance uncertainty of 0.03: 0.03 = 0.5/sqrt(N-1), and N <= 278.
for an absolute acceptance uncertainty of 0.01: 0.01 = 0.5/sqrt(N-1), and N <= 2501.

For some purposes, the important quantity is the relative acceptance error. This has no upper bound, and is given by the error in the acceptance divided by the acceptance:

relative acceptance error = sqrt( (1-r/N) / ( (r/N)(N-1) ) )

Examples:
for a 10% relative acceptance uncertainty:
N = 7 at 95% acceptance
N = 101 at 50% acceptance
N = 1,901 at 5% acceptance

for a 3% relative acceptance uncertainty:
N = 59 at 95% acceptance
N = 1,112 at 50% acceptance
N = 21,112 at 5% acceptance

for a 1% relative acceptance uncertainty:
N = 527 at 95% acceptance
N = 10,001 at 50% acceptance
N = 190,001 at 5% acceptance