Any famous approach for analyzing the randomness of data?
In terms of a measure of randomness, the usually accepted measure is Shannon entropy. In this sense for example 2 random bits which are 00 1/2 of the time, 01 1/4 of the time, 10 1/8 of the time and 11 1/8 of the time would represent 1.75 bits of random (from $\frac12+2\times\frac14+3\times\frac18+3\times\frac18$).
In your example then, if we sample $N$ bytes and byte $i$ comes up $b_i$ times we compute $p_i=b_i/N$ and then sum
$$-\sum_ip_i\log_{256}p_i$$
and that is the &quot;proportion&quot; of a byte's worth of random that you are getting with each byte. By taking $N$ larger you get a tighter confidence interval for your estimate and I'll defer to better statisticians on the details.
BUT note that if you generate bytes just by listing them 0, 1, 2,..., 255, 0, 1, 2... then this will score (very close to) 1 even though most people would say this is not very random at all.
For a measure of how likely a sample is to be producing uniform random bytes, you should perform a Pearson $\chi^2$ test on the data.
For more general randomness tests also consider Marsaglia's Diehard tests.