On Asymptotics of Means of Non-Euclidean Data Stephan Huckemann (Goettingen) In many applications, data occur on non-Euclidean spaces. Simple examples are wind directions on the circle and geological crack directions on the sphere. More advanced ones are shapes of geometrical objects and phylogenetic trees which lead to so called stratified spaces. Since all of these spaces are in particular metric spaces, means and expected elements with respect to a squared distance can be defined. How to choose from a multitude of canonical distances, however, is often not clear. While linking the central limit theorem for large sample statistics to specific distances we find desirable properties, such as "manifold stability" keeping expected elements away from singularities on non-manifold G-spaces, less desirable properties, such that the distributional behavior at the cut locus may govern the rate of convergence, and undesirable properties such as stickiness" forcing sample means to hit singularities in finite time on non-manifold CAT(0) spaces.