  We estimate the average cardinality $\mathcal{V}(\mathcal{A})$ of the value
set of a general family $\mathcal{A}$ of monic univariate polynomials of degree
$d$ with coefficients in the finite field $\mathbb{F}_{\hskip-0.7mm q}$. We
establish conditions on the family $\mathcal{A}$ under which
$\mathcal{V}(\mathcal{A})=\mu_d\,q+\mathcal{O}(q^{1/2})$, where
$\mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!}$. The result holds without any
restriction on the characteristic of $\mathbb{F}_{\hskip-0.7mm q}$ and provides
an explicit expression for the constant underlying the $\mathcal{O}$--notation
in terms of $d$. We reduce the question to estimating the number of
$\mathbb{F}_{\hskip-0.7mm q}$--rational points with pairwise--distinct
coordinates of a certain family of complete intersections defined over
$\mathbb{F}_{\hskip-0.7mm q}$. For this purpose, we obtain an upper bound on
the dimension of the singular locus of the complete intersections under
consideration, which allows us to estimate the corresponding number of
$\mathbb{F}_{\hskip-0.7mm q}$--rational points.
