Assume that you have X_1\cdot X_n independant random variables.

If they are normally distributed we can use the Chi^2 statistical test to derive the "distance to the uniform distribution". In that case the non-central Chi^2 statistics gives you a good estimate of the mean and the variance of the scoring function \sum X_i^2

In my case, the random variables X_i follows for instance a binomial distribution with mean 1 and variance (1-2^(-16)). Using the Chi^2 test I get a good estimate of the mean of the scoring function \sum X_i^2 but unfortunately the variance is completely wrong.

Can somebody help me?

Thanks