I'd be happy for simply a reference or even search terms as I feel like this has to be known*.

Suppose we have a known probability distribution $X$ and a fixed integer $n$. I am interested in the following distribution on all the set of partitions of $X$ interest into n (non-empty, if need be) sets. If $P = \amalg X_i$ is such a partition and $m_i = E(X_i)$, then the distribution's value at $P$ is $ X_P = \frac{\sum m_i}{n}$.

Is there something like the Central Limit Theorem for this case? If I take a random partition $P$, I'd like to know something like its p-value - what do we expect for $P$ and how much variation can we expect?

This seems to be a way to define or test whether or not a division of $X$ into groups is meaningful: it is meaningful when the $X_P$ differs from a generic grouping by some fixed amount. That's why I would think this is known.

Thanks!

I first asked this question yesterday on mathstackexchange here and didn't think it was appropriate since it's not a research related question. However, the suggested question/answers to this on mathoverflow and the little attention I have received on mathstatckexchange suggests this might be a more appropriate site. In either case, I will remove one of the two based on feedback.