I attempted to check the accuracy of my Garmin Etrex Venture HC which tells me to have an accuracy of about 3 m under favourable conditions. In order to do so I putted the instrument on a fixed position while it was recording this position at a sampling rate of 1 second for about 700 seconds. This file was extracted in GPX format and transformed by means of http://www.gpsvisualizer.com/ into plain text format. This format can be read by MATLAB for processing the data.
The results were quite encouraging. The standard deviation around the mean position of latitude, longitude and height were all about 1.5 m within the recording time of about 700 s. This means that the accuracy given by the GPS navigator stands for the diameter of the error sphere with a probabilty of about 67%.
An other interesting result is the time scale at which the measured positions change significantly. This time scale I could estimate very roughly only because it is of the order of several hundred seconds. Therefore the performed test of 700 s duration was too short in order to obtain a reliable result. I am going to repeat the test with a longer record to obtain a more accurate result for the auto correlation time. However, the estimated order of magnitude of this time scale provides already now the conclusion that it requires at least a hour of position record for a reliable estimate of a mean position. Therefore it makes no sense to obtain a more accurate position by averaging them over say one minute, i. e 60 position measurements with a sampling time of 1 s.
A better method to average postion measurements of a GPS recorder is to repeat position measurements at the same location after a minimum time interval of at least 1 hour. These postion measurements are statistically independent and the error of the estimated mean position is proportional to the standard deviation divided by the square root of the number of statistically independed measurements.