If the velocity is measured as a function and is a parabola,  then if two points of average are taken from opposite sides of the parabola, the average will be 0.
Average velocity will be zero, if the displacement is zero. As we all know that, velocity is the rate of change of displacement.

Consider an object starting from a point A, which moves to a point B. The, it returns to the point A again. Here, the displacement of the body is zero, as the starting point is the ending point. But, the body has taken some time to finish this task, right!!! Therefore, let the time be x. 
Though, 0 divided by anything will be zero, right!!! Therefore, the velocity of the body= 0/x =0.

Therefore, in this case, the average velocity of the body in overall will be zero.

Thank you.
Hope this will help you.

average velocity=total distance / total time taken
average speed=distance/time taken