How to estimate errors on a sample with very few data points I have a very simple question to ask, but I can't figure this on my own.
I have two samples: sample A with only three data points, and sample B with hundreds of points. For each sample I measure the median of certain quantity.
Now my question is, what is the error associated to the median?
I can consider the quartiles but, what if sample A consisted of only 1 single point? This would virtually assign no error to median value. I expect my measurement more accurate for sample B.
 A: In the absence of a clearly identified aim I'll begin with some general comments in the hope that the purpose of your analysis becomes clarified. It would be nice to know if you're after an interval, hypothesis test or simply a standard error - but if the last, to what end?]
If the distribution of the population from which the sample is drawn is known, the distribution of the median may be computed. 
The density of the $r$-th order statistic for a sample of size $n$ for a continuous random variable is
$$f_{Y_r}=\frac{n!}{(r-1)!(n-r)!}[F(x)]^{r-1}[1-F(x)]^{n-r}f(x)$$
For $n=3$,
$$f_{Y_2}= 6F(x)[1-F(x)]f(x)$$
For even $n$ it's more complex, but sometimes still doable.
If the sample size is 1, the distribution of the median is trivial - it's just the distribution of a single observation.
If the density is available, it should be possible to compute the standard deviation of the distribution of the order statistic.
(If you don't know the distribution, it's also possible to get an asymptotic standard error for the median, but it relies on knowing the height of the density at the median, which - while a much weaker requirement - would seem unlikely unless you knew the distribution.)
Additionally, you can generate nonparametric intervals for a median from the order statistics, but I don't think this gets at your present problem.
