  Fr\'echet means are indispensable for nonparametric statistics on
non-Euclidean spaces. For suitable random variables, in some sense, they
"sense" topological and geometric structure. In particular, smeariness seems to
indicate the presence of positive curvature. While smeariness may be considered
more as an academical curiosity, occurring rarely, it has been recently
demonstrated that finite sample smeariness (FSS) occurs regularly on circles,
tori and spheres and affects a large class of typical probability
distributions. FSS can be well described by the modulation measuring the
quotient of rescaled expected sample mean variance and population variance.
Under FSS it is larger than one - that is its value on Euclidean spaces - and
this makes quantile based tests using tangent space approximations
inapplicable. We show here that near smeary probability distributions there are
always FSS probability distributions and as a first step towards the conjecture
that all compact spaces feature smeary distributions, we establish directional
smeariness under curvature bounds.
