In recent years, understanding moduli spaces of relatively simple geometric objects has been a crucial ingredient in many advances in number theory in a subfield now called "arithmetic statistics." The perhaps most celebrated results in this direction have been the work of Bhargava-Shankar, who show that the average rank of (the finitely generated abelian group of rational points of) elliptic curves over Q is bounded. In this talk, we will discuss an application to integral points on elliptic curves. Using explicit descriptions of certain moduli spaces of genus one curves with extra structure, sometimes rationally and sometimes integrally, we show that the second moment (and the average) of the number of integral points on elliptic curves over Q is bounded. This is joint work with Levent Alpoge.