  Let $(M,g)$ be a Riemannian manifold. If $\mu$ is a probability measure on
$M$ given by a continuous density function, one would expect the Fr\'{e}chet
means of data-samples $Q=(q_1,q_2,\dots, q_N)\in M^N$, with respect to $\mu$,
to behave ``generically''; e.g. the probability that the Fr\'{e}chet mean set
$\mbox{FM}(Q)$ has any elements that lie in a given, positive-codimension
submanifold, should be zero for any $N\geq 1$. Even this simplest instance of
genericity does not seem to have been proven in the literature, except in
special cases. The main result of this paper is a general, and stronger,
genericity property: given i.i.d. absolutely continuous $M$-valued random
variables $X_1,\dots, X_N$, and a subset $A\subset M$ of volume-measure zero,
$\mbox{Pr}\left\{\mbox{FM}(\{X_1,\dots,X_N\})\subset M\backslash A\right\}=1.$
We also establish a companion theorem for equivariant Fr\'{e}chet means,
defined when $(M,g)$ arises as the quotient of a Riemannian manifold
$(\widetilde{M},\tilde{g})$ by a free, isometric action of a finite group. The
equivariant Fr\'{e}chet means lie in $\widetilde{M}$, but, as we show, project
down to the ordinary Fr\'{e}chet sample means, and enjoy a similar genericity
property. Both these theorems are proven as consequences of a purely geometric
(and quite general) result that constitutes the core mathematics in this paper:
If $A\subset M$ has volume zero in $M$ , then the set $\{Q\in M^N :
\mbox{FM}(Q) \cap A\neq\emptyset\}$ has volume zero in $M^N$. We conclude the
paper with an application to partial scaling-rotation means, a type of mean for
symmetric positive-definite matrices.
