How Do You Calculate One Sigma Confidence Intervals for Poisson Events?

[theneedtoknow]

I have been analyzing some data at work, and I have measured the occurrence rates of some event. How do I give a one sigma confidence interval to go along with it, assuming it is a Poisson event? For example, I found that something occurs 20 out of 10 000 times, something else occurs 43 out of 10 000 times, etc. How do I calculate one sigma error bars for these values? I know that for large number of events the uncertainty goes to root N, but what about for smaller numbers like, say, a rate of 5 in a 1000.

 

[DrDu]

Using a sigma confidence interval you are implicitly assuming an asymptotic normal distribution. Or do you want e.g. a 64% error interval?
In the first case, if x is the parameter of the Poisson distribution, then it's variance is x, too. The estimate of x is n, the number of events observed. Then possible 1 sigma confidence intervals are
n+-sqrt(n).
If you want better CIs, with a given significance level (rather than sigma value) p, refer to the article by Crow and Gardner:
http://www.ps.uci.edu/~markm/freq/crowgardner.ps