Best way to determine if a list of bytes are random?
In TCS, another approach to this problem has been via property testing of distributions, where one is to distinguish whether a distribution is (truly) uniform distribution or is "not even close" to being uniform (in a formal way). Here one gets precise bounds on the number of samples needed to decide on the question.

See, for example Section 6 of the following tutorial:
http://people.csail.mit.edu/ronitt/papers/icm.ps

In particular, one can decide whether a distribution on $[n]$ is truly uniform or is $\epsilon$-far (in total variation distance) from uniform with $O(\sqrt{n}/\epsilon^4)$ queries/samples from the said distribution. (This is also tight in the sense that $\Omega(\sqrt{n})$ samples are needed.)