  There has been some interest on how the average character degree affects the
structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, the
average number of zeros of characters of a finite group $ G $ as the number of
zeros in the character table of $ G $ divided by the number of irreducible
characters of $ G $. We show that if $ \mathrm{anz}(G) < 1 $, then the group $
G $ is solvable and also that if $ \mathrm{anz}(G) < \frac{1}{2} $, then $ G $
is supersolvable. We characterise abelian groups by showing that $
\mathrm{anz}(G) < \frac{1}{3} $ if and only if $ G $ is abelian.
