"for the sum or difference of independent random variables, variances add"
Here is a Python simulation with numbers drawn from a normal distribution with mean = 0 and std = 2. As expected, the actual variance observed for a sample is not exactly equal to 4, though it approaches this as N increases. What was a surprise to me is that the statement "variances add" is only approximately true for actual data. (That is, unless I've screwed up somehow). Looking back, I saw this before but ignored it.
Here are trials for various N, showing the mean, variance and standard deviations of the samples, the sums of the actual variances, and the variances of the sum and difference.
[UPDATE: I handled the printout of N's value poorly. For example, in the interpreter:
for N = 100, when what I actually did in code was