Every time I teach the Central Limit Theorem, I get questions from students on why we divide the population standard deviation, sigma, by the square root of the sample size to calculate the standard deviation of the sampling distribution which we call the standard error.

Recall that the equation for the standard error is

where σ is the population standard deviation and n is the sample size.

One can find any number of precise “academic explanations” of why this is true, and I give my students links to those references. But I often get follow-up questions from students who look at those references and then ask for a simpler, clearer explanation that does not involve a lot of algebra.

So, I am going to attempt one here and in the companion video.

Let’s begin with our reliable dice as an example of a population. If we use the standard six-sided dice, and assuming the dice are fair, and the rolls are fair, then each face has an equal chance of coming up. Because there are six faces, 1,2,3,4,5,6, each face has 1/6 or 16.6666666…% chance of coming up.

Let’s also assume that each roll of the dice is a sample of size n=1 since we just have one die. We know from basic probability that we need to look at the long-term when we are looking at empirical probabilities, so let us make 10,000 rolls of our single die. So, we then have 10,000 samples of sample size n = 1.