  The Fr\'echet mean (or barycenter) generalizes the expectation of a random
variable to metric spaces by minimizing the expected squared distance to the
random variable. Similarly, the median can be generalized by its property of
minimizing the expected absolute distance. We consider the class of transformed
Fr\'echet means with nondecreasing, convex transformations that have a concave
derivative. This class includes the Fr\'echet median, the Fr\'echet mean, the
Huber loss-induced Fr\'echet mean, and other statistics related to robust
statistics in metric spaces. We study variance inequalities for these
transformed Fr\'echet means. These inequalities describe how the expected
transformed distance grows when moving away from a minimizer, i.e., from a
transformed Fr\'echet mean. Variance inequalities are useful in the theory of
estimation and numerical approximation of transformed Fr\'echet means. Our
focus is on variance inequalities in Hadamard spaces - metric spaces with
globally nonpositive curvature. Notably, some results are new also for
Euclidean spaces. Additionally, we are able to characterize uniqueness of
transformed Fr\'echet means, in particular of the Fr\'echet median.
