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I am trying to reverse geocode crime data from lat and long coordinates to zip code. I have a robust dataset with lat/long coordinates for each data point, as well as the zip code shape files.

How would I export a file that associates a zip code to each data point based on the initial lat/long coordinate?

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    US zip codes are not area features -- they are point locations and collections of linear features (sometimes just the left or right side). They also can move over time. You should be careful with analysis that maps these features to polygons, so that the arbitrary nature of the labels of these polygons doesn't send you down the wrong rabbit hole. – Vince Feb 2 '16 at 1:03
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This is known as a spatial join. You can do this in any GIS software - QGIS is free so would be a good place to start.

See Importing spreadsheets to display your lat/long coordinates on the map, then Performing Spatial Joins to associate the points with the zip codes. Then export the result to a new table.

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