I believe averaging over a short period (say less than an hour, maybe even less than 24 hours) is of little use. This is because the thing that provides the biggest variation in measured position is the satellites being used and their actual position. And this varies the most over hours and days. So doing an average over say 10 minutes will provide the illusion of a very accurate position. But it is really a number of measurements close together in a larger square with less accuracy. If you came back the next day you would find another very accurate position, but in another part of the square.
+----------------+ | ... | | .. | | .. | | . . | | .. | | .. | +----------------+ Only over many days would you see measurements all over the square.
The best determination of accuracy is the number being given by the GPS device (even though the basis of this number is propriety). And this is usually small if the visible satellites are spread. If this value is small then you most probably have the best value you will get.
Remember also that when someone else comes looking for your cache, and say your GPS device is giving an accuracy of 6 metres, then the visitor could be anywhere 6 metres from the cache. So place the cache so they would have a good chance of finding it when within 6 metres of it.