# Can I use sub-sample means to verify that the sample mean is approximately normal?

I have a large (N=10^6) Monte Carlo ensemble of runs of my engineering code. I want to compute a confidence interval for the means of two output variables, and falsify the hypothesis that they have the same population mean. The two variables are not normally distributed (they do have finite variances, though). Given the very large number of runs, I believe the sample means are normally distributed. To check this assumption, I'm thinking to divide the Monte Carlo ensemble in m=10^3 smaller ensembles, and compute the corresponding 10^3 samples means. I could then make a simple QQ plot of the distribution of these sample means, and see if it's approximately normal. If this is true, then a fortiori I expect the normality approximation to hold for the sample mean of the full Monte Carlo ensemble. Is this correct? After all, my samples are i.i.d., thus I can divide the full Monte Carlo ensemble in as many disjoint sets as I want, average the samples in each set, and still get i.i.d variables.

• Yes, that could work. – kjetil b halvorsen Sep 29 '15 at 10:19

• Hi, Erik, thanks for the answers! 1. Good point! $Y_1$ and $Y_2$ are indeed paired. Do you think $Y_1-Y_2$ may be normal, even if each one is not? I will check. 2. :) I agree, but you know...there are always nitpickers (a guy at a conference was asking for 10^8 samples!!!!) 3. I will definitely use boostrap from now on - I know about it, but I always forget to use it! – DeltaIV Sep 29 '15 at 14:35