  Most systematic tables of data associated to ranks of elliptic curves order
the curves by conductor. Recent developments, led by work of Bhargava-Shankar
studying the average sizes of $n$-Selmer groups, have given new upper bounds on
the average algebraic rank in families of elliptic curves over $\mathbb{Q}$
ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$
ordered by height in which we compute ranks and $2$-Selmer group sizes, the
distributions of which may also be compared to these theoretical results. A
striking new phenomenon observed in these databases is that the average rank
eventually decreases as height increases.
