Ahhh, I see what you mean, I think. Or maybe not. If I have 3 sample points, [1,1,1], then the standard deviation is 0, but where you take your Euclidean distance?

If I'm not mistaken, with 3 sample points a, b and c that have an average mu, then

  sigma = |(a, b, c) - (mu, mu, mu)|   (L2 norm in R3)

Is that the geometric interpretation you're referring to? It's neat, but the mu vector feels a bit artificial.