This formula is so, so important. It tells you that when you have two sources of variation, only the larger one matters (unless the variances are very close to each other). It comes up all the time in multilevel modeling.
The most important formula in statistics: sqrt(12^2+5^2)=13
Good point.
So is all mathematics a series of footnotes to Pythagoras, as Whitehead said about philosophy and Plato?
The Pythagorean Theorem is not due to Pythagoras. Stigler's Law and all that. Also see Exercise 3.1 of Gelman and Hill.
simple and important indeed. for two independent sources of variation though?
It's certainly a very important formula.
If we leave off the sqrt from your formula, to make it (12^2+5^2)=13, we get to say "the variance of the sum is the sum of the variances", which is just a nice mnemonic thing to say.
A variant of this formula also tells you how to combine multiple estimates. If you have two estimates of a particular fact — say, two presidential polls — you combine them basis the inverse of the variances.
This is a really good teaching case.
Students DO NOT get it when issues like this are explained using only algebraic symbols.
Using actual numbers usually helps to bring the point across.
this reminds me of the issue that the pooled variance is determined mainly by the greater sample variance(given the sample sizes don't differ too much), thus a test based on the pooled sample variance might not be proper.
anyway I like this nice simple post.