# Confidence intervals for sample mean when estimated standard deviation is 0

I ran a Monte Carlo simulation to determine a confidence interval for the population mean based on N trials. The underlying distribution of results is not normal (the values are discrete -- 0, .5 or 1).

I used the sample standard deviation as an estimate for the population standard deviation but have realized this may be wrong as, for small N, the sample standard deviation can actually be zero.

Is there a way to come up with percentage confidence intervals around the sample mean; does it require first computing confidence intervals for the standard deviation -- and if so, how do I do that when the sampled sd itself is 0?

• You should be able to get somewhere if you have a model for the probability distribution (even if it's just a trinomial -- but a more informative model will help more). – Glen_b Feb 21 '15 at 1:22
• The problem may be simply due to the fact that you did not run enough Monte Carlo simulations. If the support of the distribution is made of the three values 0, .5, and 1, I actually fail to see how you would need simulations to compute the sample mean, equal to $\mathbb{P}(X=1)+.5\mathbb{P}(X=.5)$? – Xi'an Feb 21 '15 at 8:41