Why is the Uncertainty of Counting Phenomena Equal to the Square Root of N?

[Mk]

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).

For inherently random phenomena that involve counting individual events or occurrences, we measure only a single number N. This kind of measurement is relevant to counting the number of radioactive decays in a specific time interval from a sample of material. It is also relevant to counting the number of Lutherans in a random sample of the population. The (absolute) uncertainty of such a single measurement, N, is estimated as the square root of N. As example, if we measure 50 radioactive decays in 1 second we should present the result as 50±7 decays per second. (The quoted uncertainty indicates that a subsequent measurement performed identically could easily result in numbers dif- fering by 7 from 50.)

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.

 

[mfb]

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).