I have some poisson distributed data, where I need to construct a 95% confidence interval for an average.
The sample is 50,
The average is: 15891
The standard deviation is 6733
How do I construct the confidence based on this?
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3$\begingroup$ If your data is Poisson distributed then mean = variance and it does not seems to be like this in your case... $\endgroup$– TimCommented Dec 3, 2015 at 11:21
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$\begingroup$ The data is given, not something I've calculated. It should be correct for me to make a confidence interval on the average? $\endgroup$– user100002Commented Dec 3, 2015 at 11:25
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By the Central Limit Theorem, your average is - under certain assumptions - normally distributed. These assumptions will "usually" hold, let's assume they do here. $n=50$ is usually enough for an asymptotic treatment.
You can calculate the standard error of the mean like this:
$$ SE = \frac{s}{\sqrt{n}} = \frac{6733}{\sqrt{50}} \approx 952 $$
Your confidence interval is therefore
$$ 15891\pm 1.96\times 952 \approx [14025,17757].$$
(The factor of $\pm 1.96$ comes from the 2.5% and 97.5% quantile of the standard normal distribution.)
However, this is the general case. As Tim notes, data with an average of 15891 and a standard deviation of 6733 does not appear to be Poisson distributed, where you would expect the mean and variance to be equal (up to sampling variation). If you actually have Poisson data, this thread may be helpful. Essentially, for "sufficiently large" samples ($n=50$ should qualify), you would take the route outlined here, relying on asymptotic normality. For smaller samples, there are specialized techniques linked to there.
answered Dec 3, 2015 at 12:25