In a one dimensional normal distribution, it is really handy to know that 68% of the data are within one standard deviation, 95% lies within two standard deviations, etc. My question is about the higher dimensional version of this.
Can anyone give me rules of thumb similar to the ones we use so often with standard devs, i.e., in dimension N, what percentage of a normal distribution is at Mahalanobis distance 1, 2, 3.... I kind of assume someone has come up with handy rules for this, even if they are only good approximations.
The point is to know at what Mahalanobis distance I will have excluded x% of the data.
Notes
I worked out the distribution of the Mahalanobis distance for a normal distribution in N dimensional space to be
$$ c \cdot r^{N-1} \cdot \exp\Big(\frac{-r^2}{2}\Big) $$
where $c$ is a constant. It is easy to check that the most common distance for a point will be $\sqrt{N-1}$. This means $N > 1$ is quite different than the one dimensional case because the the most frequent distance is not distance zero.
(Yes, I know one can integrate this by hand using successive integration by parts until one gets an answer in terms of $erf$ ... but (1) that sounds painful for unspecified $N$, and (2) it has got to already be known)
