  In their work [9], Feng Luo and Richard Stong introduced the concept of the
average edge order, denoted as $\mu_0(K)$. They demonstrated that if
$\mu_0(K)\leq \frac{9}{2}$ for a closed $3$-manifold $K$, then $K$ must be a
sphere. Building upon this foundation, Makoto Tamura extended similar results
to $3$-manifolds with non-empty boundaries in [10, 11]. In our present study,
we extend these findings to normal $3$-pseudomanifolds. Specifically, we
establish that for a normal $3$-pseudomanifold $K$ with singularities,
$\mu_0(K)\geq\frac{30}{7}$. Moreover, equality holds if and only if $K$ is a
one-vertex suspension of $\mathbb{RP}^2$ with seven vertices. Furthermore, we
establish that when $\frac{30}{7}\leq\mu_0(K)\leq\frac{9}{2}$, the
$3$-pseudomanifold $K$ can be derived from some boundary complexes of
$4$-simplices by a sequence of possible operations, including connected sums,
bistellar $1$-moves, edge contractions, edge expansions, vertex folding, and
edge folding. In the end, we discuss some normal $3$-pseudomanifolds exhibiting
higher average edge orders.
