How to test for deviation from a uniform distribution?

Question

I have a random number generator as part of a system that - putatively - randomly selects one of 10,000 drive locations to store an incoming file fragment. Multiple file fragments can fill a drive location. For simplicity, let's say it's a short time window and thus there is no ceiling on number of fragments stored per location. Say that this system receives 1,000,000 file fragments in a given time period of operation.

I would expect a uniform distribution across the 10,000 locations if the system is working correctly (with some spread due to chance), but I am not sure how I would go about testing this. Would this simply be a $\chi^2$ test with 10,000 bins and an expected value of 100 fragments per bin? Then if I fail to reject the null hypothesis I can assume a sufficiently uniform distribution of fragments?

If the above is correct, how would I go about finding the point at which I can no longer assume uniformity without re-calculating $\chi^2$ using each bin? I can get alerts when a drive is above a threshold number of fragments relative to the number of fragments parsed. Say that if $5\%$ of my drive locations have more than $\mu+x$ fragments I can assume there is a deviation from random assignment; what is $x$?

Answer 1

What you propose looks OK. Since there is no estimated parameters, the number of degrees of freedom should be $10000 - 1$. Then you can find p-values from the $\chi^2_{9999}$ distribution.

But your last paragraph indicated the setting is different:

I can get alerts when a drive is above a threshold number of fragments relative to the number of fragments parsed.

So your situation is online, with information arriving continuously. That is often treated by control-charts, but I don't know how you would compute control limits for such a chart. Alternatively, see it as a problem of sequential testing. Here is a paper discussing a sequential version of the chi-square test.