  Let $o(G)$ be the average order of a finite group $G$. In this paper, we
prove that if $o(G)<\frac{31}{12}$\,, then $G$ is supersolvable. Moreover, we
have $o(G)=\frac{31}{12}$ if and only if $G\cong A_4$. We also classify finite
groups $G$ satisfying $o(G)<\frac{31}{12}$\,.
