I have count data for each of a sample of individuals (it's the number of times each independent individual performed a certain behaviour during a standardised observation of that individual). How can I calculate a confidence interval for the population mean? All over the internet there are Poisson methods for calculating the CI for a single count, but I can't find a method for a sample of separate counts. R code would be appreciated!

Edit: information about my data is useful. I have ten cells of data: Gender, by a five level ordinal variable, with small n in each cell. I am not certain the data is Poisson. I have done a possible dodgy Poisson regression which shows a significant effect of Var and an interaction with Gender. I would like to plot means with CIs to see if I believe the model.

Gender  Var  Counts
M       1    0 2 1 1 13 6 11 5 1
M       2    4 6 0 2 0 6 7
M       3    0 0
M       4    0 1
M       5    3 3 2 0 4 8
F       1    3 1 4 7 5 0 13 1
F       2    4 6
F       3    0 6
F       4    1 0 0
F       5    0 0 0 0 1 2
  • $\begingroup$ How many individuals / counts do you have? For large N (not necessarily large counts), the sampling distribution of mean / lambda is normal. Are you sure your data are exactly Poisson? $\endgroup$ Commented Dec 10, 2015 at 17:40
  • 2
    $\begingroup$ Thanks. Not certain data is Poisson. See edit for data. $\endgroup$
    – Amorphia
    Commented Dec 10, 2015 at 18:29

1 Answer 1


Here'a a quick display of the data:

enter image description here

If you're prepared to assume that within each Gender/Var combination the counts are Poisson with the same mean, then this is reasonably straightforward.

The variance of a sum of independent Poissons is Poisson, so you can form a confidence interval for the expected sum within each Var/Gender category using methods you say you're already familiar with. Since the mean is the sum divided by a constant, you simply divide the endpoints of your confidence interval for the expected sum by that constant. Done!

However, there's a fairly clear suggestion of more variation within groups than the Poisson model would indicate ("overdispersion").

Some possibilities - you could look at fitting an overdispersed Poisson model, or you could look at a negative binomial model, and then in either case obtain a confidence interval for each subgroup mean from that.

For example, here's the fitted means $\pm 1$ standard error of the mean for a main effects overdispersed Poisson model with log link (done in R):

enter image description here

I'd expect similar results from a negative binomial model. (You would possibly prefer a model with an interaction term though the results would look fairly similar to these.)

These standard errors would be about 1.8 times as big as you'd get by assuming a straight Poisson model.



plot(Counts~jitter(VarGen),pch=as.character(Gender), data=a,col=Gender)


  • $\begingroup$ Many thanks for this extremely valuable answer! I had to put this project on ice and so haven't looked at this for a long time (sorry for the delay in acceptance) but this will be very useful when I get back to this. $\endgroup$
    – Amorphia
    Commented May 18, 2016 at 9:56

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