In the paper "Applications of Quaternions to Computation with Rotations" by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by integrating quaternions over the 3-sphere).
How can I make sense of this result? If rotation angle around a given axis can be 0..360°, should not the mean be 180? or 0 if it can be −180..180°? or 90° if you take absolute values?