attribute joins - Joining averaged raster values to each feature using ArcGIS for Desktop?

I have precipitation data that I would like to average over each municipal unit in a boundary layer. First, how do I average the rainfall spatially by each administrative unit?

Second, the rainfall values are not unique to each administrative unit, and so I am not sure how to join them to the attributes table of the admin boundary layer. Although they will be unique once they are averaged, in the sense that there is one averaged value per administrative unit. If this is true, which variable do I merge on? Do I need to create something new?

I am using the end result (an average rainfall value for each municipal unit) in STATA, hence the need to export the attribute table. The program I'm using is ArcMap.