  We exhibit a range of $\ell ^{p}(\mathbb{Z}^d)$-improving properties for the
discrete spherical maximal average in every dimension $d\geq 5$. The strategy
used to show these improving properties is then adapted to establish sparse
bounds, which extend the discrete maximal theorem of Magyar, Stein, and Wainger
to weighted spaces. In particular, the sparse bounds imply that the discrete
spherical maximal average is a bounded map from $\ell^2(w)$ into $\ell^2(w)$
provided $w^{\frac{d}{d-4}+\delta}$ belongs to the Muckenhoupt class $A_2$ for
some $\delta>0.$
