# Distribution of means

Please help with a problem of practical application (explained by an example):

Suppose there are 10 objects and 30 values (or scores). Suppose also there are 1000 students randomly selected from a large population. Now each student is required to give one of the 30 scores to each of the 10 objects. It is very likely the student gives different scores to the 10 different objects.

Obviously, each given object obtains 1000 scores from the 1000 students, respectively. Thus for each object, we (1) can calculate the mean of the 1000 scores obtained; (2) have a frequency distribution over the 30 scores. Note that, as the 10 objects are independent, the 10 frequency distributions will probably be different.

However, in our practical application, rather than considering the 10 frequency distributions, what we are really interested in is the relationship of the 10 objects, therefore, of the 10 means. Therefore, the question is:

   Does it make sense to talk about “the distribution of the 10 means”?


I understand the basic concept of “a distribution of sample means” and I know “Central Limit Theorem”. Here the concept and Theorem seem not to be applicable to “the distribution of the 10 means” (as they are 10 independent objects, rather than one, with 10 different distributions). So what theory would support “the distribution of the 10 means”? Or should there not be a so-called “distribution of the 10 means” at all?

Thanks and best regards.

Diana

-

## 1 Answer

what we are really interested in is the relationship of the 10 objects, therefore, of the 10 means

This means you are interested in covariance, which for a random vector of size $10$ takes the form of a $10\times 10$ covariance matrix.

Does it make sense to talk about “the distribution of the 10 means”?

Yes, if you take mean scores over 30-student samples, you can consider their joint distribution.

-