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I think this is quite normal. Mobile data consume battery especially in low coverage area. You can check your Power management section in Settings menu to make sure there are no apps draining your battery.
Go to Settings -> Power Management -> Battery usage, and see what apps are causing the most drainage.
I'd suggest that you do a PC Companion phone repair, this will do a clean installation of the latest firmware and will also wipe your internal memory. Don't forget to back up your data before you proceed.
Open PC companion -> Support zone -> Phone/tablet software update -> Repair my phone/tablet (blue link style). Follow the steps given, do not connect your phone until prompted to press volume down while plugging USB cable.
http://www.sonymobile.com/gb/tools/pc-companion/