My physics teacher told me once about Arthur Eddington's famous observation that the ratio of scaling factors of the electromagnetic and gravitational forces was the same order as the uncertainty of N where N is the number of particles in the universe. From then on, I wondered what "uncertainty" could possibly mean. He couldn't really explain it to me, or I didn't understand it (or both).

I can see that for "random" or acausal (stochastic) phenomena, the "absolute" uncertainty in a measurement is equal to the square root of that quantity of measurements. What is that about? I think I understand square roots and squaring just fine, and am looking for a little help in explanation.

In example of radioactive decay, you can calculate the probability for each number of decays within one second, it follows a poisson distribution. And you can evaluate the standard deviation - it is sqrt(N).

Many phenomena which involve counting have an uncertainty of sqrt(N).