  The hulls of linear and cyclic codes over finite fields have been of interest
and extensively studied due to their wide applications. In this paper, the
hulls of cyclic codes of length $n$ over the ring $\mathbb{Z}_4$ have been
focused on. Their characterization has been established in terms of the
generators viewed as ideals in the quotient ring $\mathbb{Z}_4[x]/\langle
x^n-1\rangle$. An algorithm for computing the types of the hulls of cyclic
codes of arbitrary odd length over $\mathbb{Z}_4$ has been given. The average
$2$-dimension $E(n)$ of the hulls of cyclic codes of odd length $n$ over
$\mathbb{Z}_4$ has been established. A general formula for $E(n)$ has been
provided together with its upper and lower bounds. It turns out that $E(n)$
grows the same rate as $n$.
