- Download or select some data for the area of interest. If this data is unavailable, free world data available from DIVA-GIS, hosted by the University of California, Davis.
- Determine what projections are supported for this area. One option to do this is the EPSG Geodetic Parameter Registry.
- Type in the name of the country, or the coordinates for the area of interest in degrees and search. The search returns both Geographic and Projected coordinate systems for the area of interest.
- Study the ArcGIS Desktop reference document How to: Select a suitable map projection or coordinate system. The article is linked to a list of supported map projection. Check the properties of the projections on the list, and select one or more projections for the area of interest that preserves the properties of the data that are essential to the project. Apply that projection, and also pick others that apply to the area of interest. Conformal, Equal Area, and Equidistant projections are most commonly used, in that order.
- Create several data layers in these different coordinate systems, then compare the coordinate extents of the layers in ArcMap with the coordinates of the unknown data. Shapefiles are preferable for this process because they are quick and easy to gererate.
Be alert to differences in units. For example, an extent with units of meters has coordinates that are approximately 1/3 the size of the coordinate values in feet.
This involves some effort, but once familiar with the coordinate extents for the data in various projections for the area of interest, the numbers become more recognizable, and the correct projection for unknown data may be more readily identified.
If this procedure in unsuccessful, an alternative is to create a custom projection file. Please see the following article:
How To: Create a custom projection file with units of feet or meters in ArcMap