Assume I have a gaussian distribution $\mathcal{N}(\mu, C)$ with mean $\mu$ and covariance $C$. I'm drawing $n$ random numbers from this distribution. Let $m$ be the mean of these numbers. Is there some formula that gives the probability that the distance $d = ||\mu - m||$ is at least $x$, i.e. $P(d \ge x)$?
The background here is that in a recent simulation, the results seemed to cluster around a very slightly different point than expected, and I'd like to calculate the probability of this happening by chance.