  The Central Limit Theorem (CLT) for extrinsic and intrinsic means on
manifolds is extended to a generalization of Fr\'echet means. Examples are the
Procrustes mean for 3D Kendall shapes as well as a mean introduced by Ziezold.
This allows for one-sample tests previously not possible, and to numerically
assess the `inconsistency of the Procrustes mean' for a perturbation model and
`inconsistency' within a model recently proposed for diffusion tensor imaging.
Also it is shown that the CLT can be extended to mildly rank deficient
diffusion tensors. An application to forestry gives the temporal evolution of
Douglas fir tree stems tending strongly towards cylinders at early ages and
tending away with increased competition.
