Add the variances of 2 sets of independent random variables?

[Arctic Inferno]

My question isn’t what you’d expect from a simple web search.
Let’s say that you throw a fair die (singular form of dice) 4 times and get {3, 4, 2, 3}.
The mean is 3 and the variance is 1/2.
You do this again and get {4, 3, 2, 3}. The mean and the variance are identical by pure chance.
What is the mean & variance of the 8 throws?
I don’t mean the combination of the two sets, 4x4=16 different outcomes.
What is the mean & variance of this?
{3, 4, 2, 3, 4, 3, 2, 3}
If you calculate the mean and the variance, you will get 3 and 1/2 respectively.
Instead of calculating the mean & variance of the long set of 8 throws,
can you calculate the means & variances of the 2 sets of 4 throws and average them out?
What about standard deviation?

Let’s say that you’re running a computer simulation of 10 million something.
Run a block of simulation of 5000 and calculate the mean, variance, & standard deviation,
and then discard the data!!! You run another simulation block of 5000, and so on.
At the end of the simulation, you will have 2000 means, variances, & standard deviations.
Can you simply average them out to get the mean, variance, & standard deviation of the whole thing?

[ne555]

http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
http://en.wikipedia.org/wiki/Computational_formula_for_the_variance

The result for the mean will be equivalent to a weighted average.

[Arctic Inferno]

The link that you gave me was a good start.
http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
That website pointed me to a paper written by Tony Chan.
http://www.cs.yale.edu/publications/techreports/tr222.pdf
The updating formula and the pairwise summation algorithm for the variances on page 2 was exactly what I was looking for.
A large sample size can be partitioned into small pieces, and the variances of each parts can be combined later.