Center of population
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Center of population is a subject of study in the field of demographics.
The Mean center of U.S. population has been calculated for each U.S. Census since 1790. If the United States map were perfectly balanced on a point, this point would be its physical centroid. To calculate population centroid, one assigns a uniform weight to each person and zero weight to the land surface. At this centroid, the weights of all the people are balanced over the point. This measure can be somewhat practical. When you calculate this center of population, you have found an efficient location for a capital. The distance for people to travel to this central location is minimized. One must generally choose a small territory and assume the world is flat. A 3-D centroid of population across the whole globe would be near the Earth's core.
In contrast to the U.S., Australia has not seen its population centroid move drastically since the creation of the country. In 1911, the centroid was in central New South Wales; in 1996, it was only slightly farther northwest.
Likewise, the center of population in Great Britain did not move much in the 20th century. In 1901, it was in Rodsley, Derbyshire and in 1911 in Longford. In 1971 it was at Newhall and in 2000, it was in Appleby Magna, Leicestershire.
In 1993, the mean population center of Canada was found to be in the Township of Laxton, Digby and Longford, now part of the city of Kawartha Lakes, Ontario.
Capitals
The center of population for Bangladesh is close to Dhaka. Ohio's center of population is 30 miles from Columbus. The center of US population is located in Phelps County, Missouri, in the east-central part of the state, far from Washington, D.C. However, when Washington was chosen as the federal capital of the United States in 1790, the center of the U.S. population was in Kent County, Maryland, a mere 47 miles (76 km) east-northeast of the new capitol. Therefore, Congressmen from the western states must today travel inordinately far to do their business in the national capitol.
There are many other factors involved in the efficiency of travel. People from Maryland or Virginia and a host of other states can easily drive to Washington, D.C. Those from across the Rocky Mountains are already committed to flying across the country, so an extra hour of flight may not matter very much to them. Practical selection of a new site for a capital is a complex problem that depends on population density patterns and transportation networks. Plans for future development can lead to placement of a forward capital, such as Brasilia.
Median centers
Recently, the Center for the Study of Global Christianity in Massachusetts declared that the world's Christian Center of Gravity was located in Timbuktu, Mali. According to their definition of Center of Gravity, half of all Christians live west of Timbuktu and half live south of Timbuktu. Presumably they had chosen the International date line to differentiate eastern from western people.
A median center of population can be used to show where the median east vs west person lives. A person living near Louisville, Kentucky has about half of the US population living north of him and half of the US population living south of him. This measure is meaningful only along the specific coordinate axes (north-south east-west).
Sources
Bellone F and Cunningham R (1993). "All Roads Lead to... Laxton, Digby and Longford." Statistics Canada 1991 Census Short Articles Series.