Simple error analysis for probabilities

[cahill8]

I'm dealing with a histogram and want to use poisson errors for each bin. For example, having 7 items in a bin gives that bin an error of sqrt(7). I'm comparing four different data sets, each with different sizes. I'm scaling everything in terms of probabilities so the four data sets can be compared.

My smallest data set contains only 8 elements, of which 7 are in one bin, with an error of sqrt(7). Now when this is scaled to a probability, y=7/8 and yerr=sqrt(7)/8. However, this gives P=0.875+-0.33, which does not make sense since the probability cannot exceed 0. Is there something simple I'm missing?

 

[pmsrw3]

There are a couple things going on here. First, the Poisson approximation is only valid when the bin contains only a small fraction of the total sample. So if you have 100 elements and 7 are in one bin, the Poisson approximation is good. But if you have 8 elements and 7 are in one bin, the Poisson approximation is very very bad. You'd do better to use a binomial, where the standard error will be $Npq = \sqrt{\frac{(7/8)(1-7/8)}{8}}=0.11$. (Actually, it should be 0.125 but I'll not get into that.)

Second, there is no reason why a one standard error range cannot include impossible values. For very asymmetric distributions, it will commonly be the case.