For a sample size of I simulated the roll of a dice with replications.
My sample mean and variance of the mean were
. Is this because my sample size was too small, and thats why they differ by a lot?
Suppose you have a fair die, and you roll it 10 times and compute the mean of the rolls.
On a single roll the mean is 3.5 and the sd is ~1.708. So for the mean of 10 rolls the mean is 3.5 and the se of that mean is ~1.708/sqrt(10) ~=0.540
Now you conduct 200 replications, and the grand mean will be 3.5 and the se of such a grand mean will be ~=0.540/sqrt(200) ~=0.0382.
Therefore if the die is fair the mean of your sample of 200 means should be in the interval [3.42, 3.58] ~95% or the time (200 being large enough for us to use a normal approximation). Your grand mean is not in this interval so we may conclude that it is likley that the die is not fair.
You can also compare the experimental variance 1.28 with the fair die prediction of (0.540)^2~=0.292 using the F test.