# How to test the null hypothesis that the difference between groups is entirely due to binomial variance?

Given a large population (size $n$) separated into some groups (for simplicity of equal size $k$), each population member is assigned a $1$ (true) or a $0$ (false). The population mean is $p$.

Null hypothesis: the distribution is binomial, hence the group means follow a distribution with mean $p$ and variance $kp(1-p)$.

Alternative hypothesis: the difference between the group means is not consistent with a binomial distribution but due to some other reason.

How do I compute a p-value for the null hypothesis?

In my actual data the variance between the group means is much bigger than what one would expect under a binomial distribution. I want to use this p-value as a justification that the difference between groups is not just due to chance.