Samples from a black box Say I have a black-box that contains a lot of numbers (integer numbers). The quantity is very very large and the range of the numbers is finite but unknown (the numbers can be repetitive). I want to know what would be the minimum amount of samples I can randomly pick up from the box so that the values are representative from the black box. For that I do the following: I pick up an initial set of 10 numbers, then I determine the minimum, maximum, median, and average (statistical parameters). Next I pick up a new set of 20 numbers (whole new set), and again determine the statistical parameters. I do this procedure with 30, 40, ....., 1000 samples. In each step I pick up a whole new set (having returned the previous set to the box and "mix it"). Then, I would say that the minimum amount of samples is when I see that the statistical parameters “does not change much” with the increase of the number of samples.
Now my question is, what if I modify the previous procedure in this way: I pick up an initial set of 10 numbers, then I determine the minimum, maximum, median, and average. Next I pick up an additional set of 10 numbers, so that with the previous set I can have a set of 20 numbers, and again determine the statistical parameters. Next, pick up an additional 10 numbers so that with the previous set I can have a set of 30 numbers, and again determine the statistical parameters. I do this procedure with 30, 40, ....., 1000 samples. In this case in each step I am just pick up 10 additional numbers, instead of picking up the whole set. Again here, the minimum amount of samples is when I see that the statistical parameters “does not change much” with the increase of the samples.
I have seen that doing the first and second procedure, I get different minimal amount of samples, so which procedure would be the right one for my purpose, the first or the second?
Any reference that can help me to understand better it would be appreciated.

Either set of 1000 samples is equally representative of the underlying population. It doesn't matter whether you pick them 10 at a time or all at once. In the sequential sampling, all you're seeing is that when you add 10 samples to a sample of 990, you'll wind up with something that looks very much like the 990 samples you already had, and that if you had selected 1000 entirely new samples, you wouldn't get something that looked as much alike (simply because you're not keeping 99% of the data the same). But that doesn't mean that your 1000 samples selected 10 at a time is any more representative than 1000 samples selected all at once. Either way, all you have at the end is a list of 1000 random samples - those samples cannot be distinguished in any way (from the list of samples, one cannot distinguish how they were picked), so they're indistinguishable in terms of their representativeness as well.
