When to average in the lab for indirect measurements?

Averaging destroys information. Do it as late as is practical in your analysis.

A commenter points out that, if you are making many position/velocity measurements of the same object as it moves once, a simple linear regression is more robust as a velocity estimator than an ensemble of point-by-point analyses.

Calculate velocity ten times and then average it. If you were looking for average distance, then average the distance, if you were looking for average times, then average the times.

What you are doing doesn't even give the same number as the average velocity: