In my experience, +/- 1 degree is the normal sensor noise. Averaging will certainly help, but how much improvement you can expect is hard to estimate. There are a couple of tilt sensors on the market that claim 0.01 degree accuracy, but those appear to average over thousands of readings.
I've been looking for a while now, and so far, there don't seem to be any consumer-grade sensors that do better than about 1 degree in single-reading accuracy, and that is after very careful calibration in a disturbance-free environment.
Incidentally, if the 0/360 degree transition is giving you a headache, note that there is a procedure called the "circular mean" that solves the problem. See this page: laketyersbeach.net.au/windaveraging.html