Let $K \in \R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K \cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean width difference $W(łK)-W((łK)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $ł\to \infty$.