  For a prime $p$, positive integers $r,n$, and a polynomial $f$ with
coefficients in $\mathbb{F}_{p^r}$, let
$W_{p,r,n}(f)=f^n\left(\mathbb{F}_{p^r}\right)\setminus
f^{n+1}\left(\mathbb{F}_{p^r}\right)$. As $n$ varies, the $W_{p,r,n}(f)$
partition the set of strictly preperiodic points of the dynamical system
induced by the action of $f$ on $\mathbb{F}_{p^r}$. In this paper we compute
statistics of strictly preperiodic points of dynamical systems induced by
unicritical polynomials over finite fields by obtaining effective upper bounds
for the proportion of $\mathbb{F}_{p^r}$ lying in a given $W_{p,r,n}(f)$.
Moreover, when we generalize our definition of $W_{p,r,n}(f)$, we obtain both
upper and lower bounds for the resulting averages.
