Averages and Midpoints

Another student explained that the average (mean) of 2, 9, 3, and 6 is 5 because the midpoint between 2 and 6 is 4 and the midpoint between 9 and 3 is 6. Since the midpoint between 4 and 6 is 5, then the average (mean) of the four numbers is 5. Does this method of taking the average (mean) of two pairs of numbers and then averaging the two averages always work? Explain. Does it work with bigger sets of numbers? Why or why not?

(Source: Kamii, C., Pritchett, M., & Nelson, K. (1996). Fourth graders invent ways of computing averages. Teaching Children Mathematics, 3(2), 78-82.)