Average distance between a star and planet in an eccentric orbit

The average distance between a star and a planet in the classic two-body problem of celestial mechanics is different from the orbital semi-major axis of the planet if the average is performed over time or angle. The time- and angle-averaged distances are functions of both the semi-major axis and the orbital eccentricity and can differ from the orbital semi-major axis by as much as 50%-100%. This difference contradicts the usual statement found in most introductory and advanced textbooks that the average distance and the orbital semi-major axis are equal.