# Mean -> Frechet mean, Standard deviation ->?

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?

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By 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
Yes, I am interested in a natural generalization of standard deviation, in analogy to Frechet mean generalizing ordinary mean. A reference to the literature would be ideal. – Franck van Breugel Jul 11 '13 at 15:11
I see. Since mean (in whatever framework) is natural to define as a minimizer of a measure of spread of data/points/... it justifies the fact that the minimal such spread is called variance, as Kjetil wrote already. So the std.dev would indeed be natural to defined as a square root of the minimals sum of distances, however maybe you could say why are you interested in std.dev related to the Frechet mean? Perhaps, concentration theorem for random elements taking values in metric spaces? – Ilya Jul 11 '13 at 16:50