Every ten years a new population center of the country is determined by geographers at the Federal Census Bureau, using data from the latest recently completed national census.
The Center for Land Use Interpretation (your tax dollars at work) report:
The official Center of Population for the USA is determined mathematically as the place where an imaginary, flat, weightless and rigid map of the United States would balance perfectly if all the 310,570,000 or so residents counted in the 2010 census were of identical weight. The previous official population center of the USA, based on the results of the 2000 US Census, is 2.8 miles east of Edgar Springs, Missouri. It moved 35 miles west-southwest from a point near Steelville, Missouri (its position determined by the 1990 census). The new Population Center, near Plato, is 25 miles southwest of Edgar Springs, reflecting the general migration of population from the wintery urban northeast to the sunbelt of the Southwest. Currently the federal government is working with local representatives at Plato to determine a location for a monument to mark the new center in a place that is accessible to the public.
A nice map from Wikipedia shows the population center for every census since the first census in 1790.
It raises several interesting questions for a multi-curricular investigation.. for example, note the change from 1920-1930 or 1940-1950 as compared to 1930-1940. Now I read "The Grapes of Wrath", and all those Okies heading west should have made a big jump, ...shouldn't it? But it seems prosperity brought a greater incentive to move west than poverty.
And what accounts for the really small jump from 1910 to 1920? Yes there was a war on...but there was in the forties also.
If you are interested in expanding this analysis, you might explore another site operated by USA Today which has separate offerings for population centers of various descriptions (whites, Blacks, American Indians (NOT Oklahoma if that was your guess)etc..). You can also pop open a page with statistics about the various states and the nation as a whole, but it doesn't include the 2010 census results. The data includes some interesting (read scary) data on poverty numbers by age..
So, the next time you are cruising west on 44 across Missouri, set the old Tom Tom on Plato, Missouri and swing by. Cannon's Cafe on Hwy 32 in Plato should have Coffee on, and if not Weber's is just up the road. Can't vouch for the pie yet, but when I go through they will almost certainly give me a big slice for free for this unsolicited plug. Are you listening Plato?
I also found an animated link for US centers, but only up to 2000.. At the same site I found another animation for the MEDIAN center of population of the US. Very different type of motion, and the current median is located well east of the average position... still back in Indiana (although the 2010 census may put it over the line into Illinois, or maybe down into Kentucky... I'm rooting for a location that will be, like me, near beautiful Possom Trot, Ky.
Addendum: After Arjen's post (below) I found the following on a Wikipedia page about Centers of population: