Population-weighted center of a state

I have a list of states, major cities in each state, their populations, and lat/long coordinates for each. Using this, I need to calculate the latitude and longitude that corresponds to the center of a state, weighted by where the population lives.

For example, if a state has two cities, A (population 100) and B (population 200), I want the coordinates of the point that lies 2/3rds of the way between A and B.

I'm using the SAS dataset that comes installed called maps.uscity. It also has some variables called "Projected Logitude/Latitude from Radians", which I think mi开发者_开发知识库ght allow me just to take a simple average of the numbers, but I'm not sure how to get them back into unprojected coordinates.

More generally, if anyone can suggest of a straightforward approach to calculate this it would be much appreciated.

The Census Bureau has actually done these calculations, and posted the results here: http://www.census.gov/geo/www/cenpop/statecenters.txt

Details on the calculation are in this pdf: http://www.census.gov/geo/www/cenpop/calculate2k.pdf

To answer the question that was asked, it sounds like you might be looking for a weighted mean. Just use PROC MEANS and take a weighted average of each coordinate:

/* data from http://www.world-gazetteer.com/ */
data AL;
input city $10 pop lat lon;
datalines;
Birmingham 242452 33.53 86.80
Huntsville 159912 34.71 86.63
Mobile     199186 30.68 88.09
Montgomery 201726 32.35 86.28
;
proc means data=AL;
weight pop;
var lat lon;
run;

Itzy's answer is correct. The US Census's lat/lng centroids are based on population. In constrast, the USGS GNIS data's lat/lng averages are based on administrative boundaries.

The files referenced by Itzy are the 2000 US Census data. The Census bureau is in the processing of rolling our the 2010 data. The following link is a segway to all of this data.

http://www.census.gov/geo/www/tiger/

I can answer a lot of geospatial questions. I am part of a public domain geospatial team at OpenGeoCode.Org

I believe you can do this using the same method used for calculating the center of gravity of an airplane:

1. Establish a reference point southwest of any part of the state. Actually it doesn't matter where the reference point is, but putting it SW will keep all numbers positive in the usual x-y send we tend to think of things.
2. Logically extend N-S and E-W lines from this point.
3. Also extend such lines from the cities.
4. For each city get the distance from its lines to the reference lines. These are the moment arms.
5. Multiply each of the distance values by the population of the city. Effectively you're getting the moment for each city.
6. Add all of the moments.
7. Add all of the populations.
8. Divide the total of the moments by the total of the populations and you have the center of gravity with respect for the reference point of the populations involved.