I've never seen this data before. It's really interesting that the population center was actually in or near D.C. in the early 1800s. I've never heard that this is a reason for it's location. Merely coincidence?
Population center is exactly why DC was selected... it's also why most state capitals were selected.
A southern capital was part of the deal on the assumption of state debts by the federal government. Washington doesn't seem that southern these days, but the District of Columbia was about 1/3 in Virginia, and Maryland was culturally speaking something of an annex of Virginia. That Mt. Vernon was nearby played a part in putting the capital there and not around (say) Port Deposit or Fredericksburg.