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I'm writing a test for a probabilistic data structure that I've implemented. Since its probabilistic, its performance is different every time, and in particular, the performance varies much more widely (as a percentage) when the number of items is small. I want the test to fail if the performance is so bad that it's unlikely to have been that bad by chance. Basically, I want a significance test.

In the test, I check the structure n times. Each check may fail with probability p. If the total number of failures is more than twice the standard deviation greater than the mean (n*p), I want to fail the test. Now, I know n and I can calculate p. How can I use n and p to get the standard deviation?

I know it can be easily estimated by simulation, but there are about a hundred combinations of n and p, so I would prefer a way to just calculate it. Unfortunately, I haven't found an answer, or if I have it was too complex for this layman to notice.

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When you say "the total number of failures is greater than twice the standard deviation..." you should probably add "...more than the expected number of failures $np$..." – Henry Jun 10 '11 at 22:23
Good question! I hope it's okay that I removed the "probability-theory" tag. – Elliott Jun 10 '11 at 22:30
Thanks, Henry. It was poorly phrased. – Adam M. Jun 10 '11 at 22:32
up vote 4 down vote accepted

This is the binomial distribution.

If you know $p$ before the test and the checks are independent then the standard deviation is $\sqrt{ n p (1-p)}$

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Excellent. This does indeed agree with my simulations. Thank you! – Adam M. Jun 10 '11 at 22:27
Similarly to calculate mean we simply multiply p and n. – user75917 May 4 '13 at 10:43

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