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I would like to be able to have a file with U.S. Census data pre-aggregated at some level of the Military Grid Reference System (MGRS). Ideally at thier 10km precision level, but I'd take whatever I could get.

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Unfortunately, I don't think this exists as a pre-aggregated data file.

However you could use a tool like census_area to get a close approximation. With this tool, you could download block or block group level data and aggregate them your desired geographies.

(I'm one of the authors of this library)

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    The library will do it for you. You'll define your grid, and the library will find all the blocks that fall within a grid cell. – fgregg May 8 '17 at 19:13

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