Illustrating the Central Limit Theorem

Quite often, when we do experiments in a lab, we are told to take the average of several observations. Let us suppose we carried out a particular observation $n$ times and obtained the $n$ readings $(x_1,x_2,\ldots, x_n).$. The average, which we denote by $z$, is

(1)
\begin{align} z=\frac{(x_1+x_2+\cdots+x_n)}{n} \ . \end{align}

How does one model this mathematically? We assume that each measurement is

1. independent of the others and
2. identical to all other measurements.

This is captured mathematically by saying that the $x_i$ are independent and identically distributed (i.i.d.) random variables. What is the probability distribution that we should assign to the random variable? That clearly depends on the details of the experiment being carried out. The Central Limit Theorem (CLT) tells us we need not worry about the details of the probability distribution — all we need to know is its mean and variance.

More to come

### Statement of the Theorem

This page will (pictorially) illustrate the Central Limit theorem through a couple of examples.