No, the standard deviation by itself will tell you next to nothing about the nature of the distribution. Something like the two-sample Kolmogorov-Smirnov test might help you.
doc kstest2
The problem remains of defining what true randomness is. If your numbers are generated by a computer, then what you get are pseudo-random numbers. For such numbers, the "randomness" breaks down once you have very many samples, or when you consider many dimensions. No all random numbers are created equal.
