Let A_r be a real-valued normal random variable whose mean is an integer A.
Let A_i be the rounding of A_r such that A_r = A_i + e, where e is an uniformly distributed random variable taking the value from -0.5 to 0.5.
Now taking variance,
What would be the cov(A_i,e)? Will it be less than -1/12 so that Var(A_r) is always less than Var(A_i)?