Xcode is hidden in OS X 10.8.2
Since Xcode is working without a problem, it is probably installed correctly. You can unhide the application the same way you unhide the ~/Library folder on Lion and later. Open the terminal and enter the following command:

chflags nohidden /Applications/Xcode.app


Xcode should be visible in the Finder.

If you have any problems with Xcode not building or running properly I would recommend deleting Xcode and reinstalling from the App store.