In this case there is quite a discrepancy between the two answers. Both points are in Oklahoma, but they are 116 miles apart.
If the Census Bureau insists that the population center be the center of gravity,
then it could be done more accurately by treating the area of the United States as a
region on a spherical shell with a unit of mass for each person. On the other hand
the Bureau could return to its definition of 1920 in which the population center is the
point of minimum aggregate travel but calculate it correctly. While it may have been a
daunting task back then, it is now a calculation that can be done easily with the data provided by the Census Bureau.