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][1] 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.

  [1]: https://math.stackexchange.com/questions/2518928/central-limit-like-theorem-for-the-distribution-of-all-possible-partitions