  Let $\mathcal{G}$ be the class of all finite groups and consider the function
$\psi'':\mathcal{G}\longrightarrow(0,1]$, given by
$\psi''(G)=\frac{\psi(G)}{|G|^2}$, where $\psi(G)$ is the sum of element orders
of a finite group $G$. In this paper, we show that the image of $\psi''$ is a
dense set in $[0, 1]$. Also, we study the injectivity and the surjectivity of
$\psi''$.
