  \textit{Non-statistical dynamics} are those for which a set of points with
positive measure (w.r.t. a reference probability measure which is in most
examples the Lebesgue on a manifold) do not have convergent sequence of
empirical measures. In this paper we show that behind the existence of
non-statistical dynamics there is some other dynamical property:
\textit{statistical instability}. To this aim, we present a general
formalization of the notion of statistical (in)stability which is interesting
in its own sake: As an application of the main theorem regarding statistical
stability we give another proof of a theorem of Avila and Bochi (\cite{AB09},
Theorem B). We also introduce a new class of non-statistical maps in the space
of Anasov-Katok diffeomorphisms of the annulus.
