  We adopt a statistical point of view on the conjecture of Lang which predicts
a lower bound for the canonical height of non-torsion rational points on
elliptic curves defined over $\mathbb{Q}$. More specifically, we prove that
among the family of all elliptic curves defined over $\mathbb{Q}$ and having
positive rank, there is a density one subfamily of curves which satisfy a
strong form of Lang's conjecture.
