  Fr\'echet means on non-Euclidean spaces may exhibit nonstandard asymptotic
rates rendering quantile-based asymptotic inference inapplicable. We show here
that this affects, among others, all circular distributions whose support
exceeds a half circle. We exhaustively describe this phenomenon and introduce a
new concept which we call finite samples smeariness (FSS). In the presence of
FSS, it turns out that quantile-based tests for equality of Fr\'echet means
systematically feature effective levels higher than their nominal level which
perseveres asymptotically in case of Type I FSS. In contrast, suitable
bootstrap-based tests correct for FSS and asymptotically attain the correct
level. For illustration of the relevance of FSS in real data, we apply our
method to directional wind data from two European cities. It turns out that
quantile based tests, not correcting for FSS, find a multitude of significant
wind changes. This multitude condenses to a few years featuring significant
wind changes, when our bootstrap tests are applied, correcting for FSS.
