Not necessarily, using k samples you can estimate the mean of i.i.d. samples with a variance of 1/k the sample variance. Note that this is independent of the sa

Not necessarily, using k samples you can estimate the mean of i.i.d. samples with a variance of 1/k the sample variance. Note that this is independent of the sample distribution, in particular it's also true if you're drawing from some fixed collection, regardless of its size. I find it harder to say something about the median, but I imagine some similar theorems hold. Unless you have some definition of quality for which you can show that you do need more samples with a bigger collection?