This page shows the mean and median population centers of the United States over time. The mean population center is the point at which the country would balance if each person was equal weight. The median population center divides the country into halves, with equal population on either side and above and below the point.
The data comes from this PDF document at the US Census Bureau.
Note: There are two data points for 1950, one including Alaska and Hawaii and one without. Points after 1950 include those states.
The median doesn't get as far west because the distance of the west coast that would increase the weighting of its population in an average is excluded