  Fr\'echet mean and variance provide a way of obtaining mean and variance for
general metric space valued random variables and can be used for statistical
analysis of data objects that lie in abstract spaces devoid of algebraic
structure and operations. Examples of such spaces include covariance matrices,
graph Laplacians of networks and univariate probability distribution functions.
We derive a central limit theorem for Fr\'echet variance under mild regularity
conditions, utilizing empirical process theory, and also provide a consistent
estimator of the asymptotic variance. These results lead to a test to compare k
populations based on Fr\'echet variance for general metric space valued data
objects, with emphasis on comparing means and variances. We examine the finite
sample performance of this inference procedure through simulation studies for
several special cases that include probability distributions and graph
Laplacians, which leads to tests to compare populations of networks. The
proposed methodology has good finite sample performance in simulations for
different kinds of random objects. We illustrate the proposed methods with data
on mortality profiles of various countries and resting state Functional
Magnetic Resonance Imaging data.
