Can the statistical width of the avalanche effect be calculated?
Q. Is there a calculable mathematical relationship between the standard deviation and the block size, or must it always be determined empirically?


This is a well studied problem in probability.

If we consider each output bit as an independent random variable (and if they're not independent, your hash function is broken), each such random variable has a standard deviation of $0.5$ (if $0$ and $1$ have equal probabilities; again, if this is not true, your hash function is broken).  The sum of $N$ such independent samples will have a standard deviation of $0.5 \sqrt{N}$.  In your case, $N=512$; this gives us a standard deviation of $11.3137...$, which is just what you see.