I'll check my average speed at the end of a ride just to see what it is but I don't obsess over it becasue as others have mentioned, a lot of different things can affect ones average speed (a ride with a lot of climbing, higher winds, lots of stop signs / lights, etc.). I do a lot of riding just for fun and towards the end of the season this year and (my year was pretty messed up, too with a 6 week long break and then not riding too many times per week thereafter) my average was getting up to 15.5mph. As I ride more next year I of course hope to improve it but I'm not going to obsess over it.
"Limits are a state of mind: break them before they break you."
--Michael Cotty