Central Limit Theorem

The sum of $n \to \infty$ independent identically distributed (i.i.d.) random variables (measurements) is normally distributed.

Independent Identically Distributed random variables

In maths, this term is so common that they usually just say i.i.d. to mean this.

Binomial Distribution

As we know, the mean of a binomial dist is $n p$ . The variance, as proved in Week 5 Homework question 34, is $n p (1 - p)$ .

Also as we learned, Binomial dist turns into the Poisson Distributions with high $n$ and low $p$ . Something that is also interesting is that both of these distributions look like a bell-shaped distributions when you have a very high $n$ - they are in fact normally distributed (but not continuous), this is because of the central limit theorem.

Sample Average

The real reason the normal distribution is so important is because of the Sample Average. If we have infinitely many samples, the sample average between these samples will be normally distributed, even if the samples aren't. This means that we end up with normal distributions showing up everywhere. In math terms:

\bar{ξ} = \frac{ξ_{1} + ξ_{2} + . . . + ξ_{n}}{n}

If $n \to \infty$ :

\bar{ξ} = \frac{ξ_{1} + ξ_{2} + . . . + ξ_{n}}{n} \to N (m, \frac{V (ξ)}{n})

Where $V$ is the true variance (which we would get from an infinitely large sample).

In practice, $n$ around 20 or 30 is sufficiently close to infinity that this becomes mostly true.

The body height of one person has the same expectation as the sample average, but the variance will not be divided by $n$ .