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I have a shapefile of U.S. counties in USA Contiguous Albers Equal Area Conic. I want to do distance calculations. For example, distances from one county to another.

I do not know how to interpret the coordinates. I understand that the distance is in meters.

The ultimate goal is to use the distance as a part of a statistical procedure. Specifically, I am using it in a regression discontinuity design, in which I restrict the sample to counties that are a given distance from a particular line in the U.S. (e.g., the Central Time Zone). So, I'd like the distance to be fairly accurate because I am restricting the sample to be say 20 miles on either side of this line, and the distance should be accurate from a symmetrical perspective.

How may I recover a latitude and longitude?

I wanted to recover them so that I can apply the distance formula for coordinates by hand to get the distances in miles in a way that I can interpret. I've come across projections and UTM when searching, but I do not know what that means.

I use ArcMap 10.1.

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I would not recommend trying to recover a latitude and longitude in order to "apply the distance formula for coordinates by hand to get the distances in miles" because the length of a degree of longitude varies depending on its latitude i.e. 0 m at poles, and about 110 km (= 70 miles) at around 30 degrees N or S.

I think you will be far better to get your distances in meters direct from using Near with your Albers coordinate system and then multiply them by 39.37 to get inches, and divide that by 63,360 to get miles.

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While there are equidistant projections, they preserve distance only from some standard point or line.

To get accurate distances, over a large area such at the United States, your best option is to repoject the data to a geographic coordinate system, and then use the Haversine formula to derive the distance.

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