The most important formula in statistics?

Andrew Gelman writes:

The most important formula in statistics: sqrt(12^2+5^2)=13.

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.

It's certainly a very important formula.

If we leave off the sqrt from the 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.