The Central Limit Theorem

You do not need to do anything in Excel for this video, it is essentially a lecture demonstration. I’m including the before and after files just in case you are curious to play around.

Download: CLT-Demo_init.xlsx

Download: CLT-Demo_final.xlsx

Background

The Central Limit Theorem (CLT) is an important statistical theory since it is a big reason why the Gaussian distribution is considered the “normal” distribution. Mathematically it can be stated as

In plain English: as the sample size increases, the sampling distribution of the mean becomes increasingly well described by a normal distribution with mean $\mu_x$ and standard deviation $\sigma/\sqrt{n}$. The CLT is basically describing the nature of the sampling distribution of the mean:

• the sample mean $\overline{x}$ is an unbiased estimate of the population mean $\mu_x$. In other words, the mean of the sampling distribution is also the mean $\mu_x$ of the underlying population.

• the standard error of the mean of $n$ measurements, $s(\overline{x}_n)$, can be estimated from a single sample, as

  $$s(\overline{x}_n) = \frac{s_x}{\sqrt{n}} \tag{estimate of $\sigma(\overline{x}_n)$}$$

• as the sample size increases ($n$ gets larger) the sampling distribution of the mean of the sample will become increasingly normal even if the original population is not normally-distributed

It is worth pointing out that the CLT applies also to other least-squares estimators such as regression parameter estimates. It also seems to work well in many cases to describe the behavior of the sampling distribution of the trimmed mean, even though that is not technically a least-squares estimate.

This screencast is a demo of the CLT in action, highlighting its edge cases and addressing how large a sample is necessary in order to assume that the sampling distribution of the mean is normally distributed. The screencast also highlights situations where the CLT does NOT work (so-called pathological distributions).

Tasks

No tasks for this video, just take notes while watching. This is a demo of the Central Limit Theorem and how it is affected by distribution shape. Below is an outline with time stamps (you can also see this outline, with links, by watching the video on YT).

00:00 Background: The Central Limit Theorem
03:44 CLT with Short Tailed Distributions
14:31 Effect of long tails on CLT convergence
22:18 Pathological distributions (CLT failure)