  Let $p$ be a prime, let $V/\mathbb{F}_p$ be an absolutely irreducible affine
variety inside the affine $r$-space. In this paper, we consider the problem of
how often a box $B$ will contain the expected number of points. In particular,
we give a lower bound on the volume of $B$ that guarantees almost all
translations of $B$ in the $r$-space contain the expected number of points.
This shows that the Weil estimate holds in smaller regions in an "almost all"
sense.
