I know there are many different definitions of “center of population”. But the Census Bureau uses two different definitions in finding the center of population of the whole U.S.: the “mean center of population” (which was in Missouri in 1990) and the “median center of population” (which was in Indiana in 1990). See this PDF file from the Census Bureau for definitions and locations of the national centers from 1790 to 1990.
I have been able to find a list from the USGS of [the geographic centers of the states](http://mac.usgs.gov/mac/isb/pubs/booklets/elvadist/elvadist.html#Geographic Centers) (based on one definition of “geographic center”). But I haven’t been able to find similar data for the centers of population for the states. Surely somebody must calculate this stuff. Mustn’t they?