There is a lot more discussion here. The post calls this the "Pythagorean Theorem of Statistics", since an equivalent formulation is:

I don't want to detail the proof, but I did fool around a bit in R to explore this:

Here is what it prints:

So, if we add or subtract a constant C, the variance is unchanged. But if we multiply by C, the variance is multiplied by C

^{2}; and if we divide by C, the variance is divided by C

^{2}.

Now consider a second set of numbers from rnorm. The first vector has a mean of 5 and sd of 2 (variance of 4), while the second has a mean of 4 and sd of 3 (variance of 9).

Our simulation confirms the rule that the variances add.

And finally, look at multiplication:

The variance depends on the mean of the distributions. Here, the variances of u and v (as well as u + 1,2..3) are always 1. For means of:

I found an expression here:

That is:

In the cases above (variance is unchanged and equal to 1) we have:

Looks correct.

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