Depending on your level of accuracy I would suggest the following:
Low level Accuracy
Use QGIS for Android (free and has a gps toolbar) along with your tablet or handheld native gps
High Level Accuracy
Use QGIS for Android along with your vendor gps
With qualifications and suitably modified, this is correct: the standard deviation measures spread, which is inversely related to precision, while the vectors (not distances) to the reference points measure inaccuracy.
Discussion
With only five readings per location several problems will arise:
The standard deviations of the coordinates will vary--by a factor of three or more--due only to random variation. Thus you might obtain a wide range of standard deviations, leaving uncertainty about what the precision really is.
Unless each group of readings was taken at widely spaced times, they could be autocorrelated: the full amount of possible variation might be much larger than observed. This is because some components of the positional error "drift" only slowly in time.
One important component of error, the numbers and positions of available satellites, might not be sufficiently well represented in such a small dataset. This could be evidenced by very large variation in some of the readings (that is, a distinctly non-normal distribution of values relative to the reference points).
The first problem is overcome using an Analysis of Variance (ANOVA) to obtain a single estimate of the overall variance, which would be based on 20*(5-1) = 80 degrees of freedom, which is large enough to give a reasonable estimate.
ANOVA will not overcome the second problem, which requires customized techniques to estimate the autocorrelation (assuming each reading includes a time stamp).
The third problem requires more careful analyses, starting with a (routine) distributional analysis of the ANOVA residuals. That is a discussion that would take us too far afield here.
Additional comments
Average distances normally would not be used to estimate inaccuracy: they actually indicate imprecision. Instead, the difference between the centroid (average point) of each group of readings and its reference point would be more appropriate.
More sophisticated techniques of multivariate statistical analysis could be brought to bear to analyze the variance-covariance matrices of the readings, if there is a possibility that the x- and y-coordinates are correlated. If that possibility is ruled out by examination of the dataset and on physical grounds (which is the default assumption by many people) then you can just analyze the 20*5*2 = 200 residuals of the separate coordinates (which are their values relative to the reference locations). In particular, ANOVA software will automatically include a test (the F Test) to determine whether there is evidence of any inaccuracy at all.
Best Answer
To perform post-processing you typically need a GPS that can record raw pseudorange and carrier cycle data. Very few consumer level GPS support this functionality, and those that do require software modification. I am not sure about android, most likely it would depend on the GPS chip that the phone is using.