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We prove that when all elliptic curves over $\mathbb{Q}$ are ordered byheight, the average size of their 4-Selmer groups is equal to 7. As aconsequence, we show that a positive proportion (in fact, at least one fifth)of all 2-Selmer elements of elliptic curves, when ordered by height, do notlift to 4-Selmer elements, and thus correspond to nontrivial 2-torsion elementsin the associated Tate--Shafarevich groups.

The average number of elements in the 4-Selmer groups of elliptic curves is 7