  In dynamical systems, understanding statistical properties shared by most
orbits and how these properties depend on the system are basic and important
questions. Statistical properties may persist as one perturbs the system
(\emph{statistical stability} is said to hold), or may vary wildly. The latter
case is our subject of interest, and we ask at what timescale does statistical
stability break down. This is the time needed to observe, with a certain
probability, a substantial difference in the statistical properties as
described by (large but finite time) Birkhoff averages.
  The quadratic (or logistic) family is a natural and fundamental example where
statistical stability does not hold. We study this family. When the base
parameter is of Misiurewicz type, we show, sharply, that if the parameter
changes by $t$, it is necessary and sufficient to observe the system for a time
at least of the order of $|t|^{-1}$ to see the lack of statistical stability.
