  In this paper, we show that the average size of the automorphism group over
$\mathbb{F}_q$ of a smooth degree $d$ hypersurface in
$\mathbb{P}^{n}_{\mathbb{F}_q}$ is equal to $1$ as $d\rightarrow \infty$. We
also discuss some consequence of this result for the moduli space of smooth
degree $d$ hypersurfaces in $\mathbb{P}^n$.
