Given a finite set $A$ of points of a metric space $(X, d)$, I would like to find its mean. A Frechet mean seems appropriate here: $\arg \min_{x \in X} \sum_{a \in A} d(x, a)^2$. I also would like to find the corresponding standard deviation. What would this be?
A Frechet mean is (mostly) appropriate, but you should be aware that it is not always unique! An example of nonunicity is the circle, with data points the north and south pole, and (following Winnie Pooh), two minimzers: the east and the west pole. A condition guaranteeing always unicity is manifolds with nonpositive curvature, a proof of this fact can be got from Burago, Burago & Ivanov: A Course in metric geometry, chapter 9. There are also inicity conditions based on small variance, in some sense. As to your question: The book Nonparametric Inference on Manifolds call the minimizer of the Frechet function, the Variation. That is of course unique even if the mean itself is not. Probably one could use its square root as some sort of standard deviation, but the usability of such a definition would have to be investigated on a case basis. 


I also would like to find the corresponding standard deviation
do you mean, that you are looking for a natural way of defining a std.dev. for the Frechet mean? – Ilya Jul 11 '13 at 11:36