Testing for randomness of Hermitian Matrix

I'm working on an experiment which requires me to find the eigenvalues of a large random complex number Hermitian matrix. I had originally thought I could just eye-ball the degree to which my matix is truely random, by comparing a histogram with a Normal Gaussian distribution function. Is there a preferred way to do this which yields a number representing the degree of randomness?

Can I look at just the absolute values? Do I need to look at the reals and imaginaries separately.

The "experiment" is to see if I can reproduce all of the expressions and figures shown by John Derbyshire in his book, "Prime Obsession". That book is all about the Reimann Hypothesis.

I'm a "Newbie" to number theory as well as to Mathematica.

As a Newbie, to these forums, I'm concerned also whether or not I'm in the right forum. Forgive me if I'm not.

Thanks, Mauri