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Q:  

Is there any source that reports percent of rural population in the USA by race (1965-2010)?

A:  

Rural is a hard concept. The best indicator I know of it is something out of the Economic Research Service. They have a 9-code rural/urban indicator for every county in the US - and good for your purposes, they have codes for counties for 1973, 1983, 1993, and 2003.

http://www.ers.usda.gov/data-products/rural-urban-continuum-codes.aspx

So, once you decide where you cutpoint on rural is (e.g., codes 6+ or whatever), you can classify all the counties in the US as urban or rural by decade.

e.g.,

County State; 1970s; 1980s; 1990s; 2000s;
Autauga AL;    Rural;   Rural;  Rural; not Rural

[made up example]

You can then merge this with the race distribution of counties by year from the Population Estimates website:

http://www.census.gov/popest/data/historical/pre-1980/county.html

http://www.census.gov/popest/data/historical/1980s/county.html

http://www.census.gov/popest/data/historical/1990s/county.html

http://www.census.gov/popest/data/intercensal/county/county2010.html

[note that intercensal means the data have been smoothed based on the endpoints of the two censuses, e.g., 1980 & 1990.

Post-censal estimates is what the Census Bureau does between censuses, and then 'corrects' it with a set of intercensal measures.]

Let's say, you are just interested in the size of the not non-white Hispanic population for simplicity:

County State; 1970s; 1980s; 1990s; 2000s;
Total 1965; Minority 1965; Total 1966; Minority 1966;etc.

Autauga AL;    Rural;    Rural;    Rural;   not Rural;
35,000; 7,000; 35,200; 6,940;

[made up example]

You can sum across all counties for each year and get a %rural for minorities (or blacks, etc.).

There's quite a bit of programming with what you want, but it isn't real difficult and would be a nice resource for others.

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