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In this article, we prove that the average rank of elliptic curves over$\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than$.885$). As a consequence of our methods, we also prove that at least fourfifths of all elliptic curves over $\mathbb{Q}$ have rank either 0 or 1;furthermore, at least one fifth of all elliptic curves in fact have rank 0. Theprimary ingredient in the proofs of these theorems is a determination of theaverage size of the $5$-Selmer group of elliptic curves over $\mathbb{Q}$; weprove that this average size is $6$. Another key ingredient is a new lowerbound on the equidistribution of root numbers of elliptic curves; we prove thatthere is a family of elliptic curves over $\mathbb{Q}$ having density at least$55\%$ for which the root number is equidistributed.

The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1