Central Limit Theorem

Consider a school that accepts students if and only if their score is 81% or higher. The school will never under any circumstances accept students with scores less than 81%.  We also assume that the maximum score is 100%.

If we plot the probability distribution function (PDF), we get something like the blue curve shown in the figure above (you may need to run the video to see the blue curve). It is evident that the probability is zero when a student score is below 81%. Just after 81%, there is a sharp increase in the probability. As the scores approach the high end the PDF decreases and becomes zero above 100%.

Evidently the distribution is not Gaussian (Bell shaped).

 

Let’s now do the following:

1- Divide the students to classes, such that each class contains the same number of students. For each class, we calculate the average score (sum of scores divided by the number of students in the class).

2- Record the average of each class.

3- Plot the histogram. For example, we counts the classes that score 83%, say we got 5. W then place (83%, 5) as a data point. We repeat the process for each score.

4- Now we let the the numbers increase indefinitely. This is a common approach in mathematics to convert a discrete system to a continuum. We consider the number of sections to be very large. Additionally, we let the number of students per each class be very large as well; then we plot the histogram.

As the numbers increase the histogram will approach the nice Gaussian (bell-shaped) curve.

But why does this happen?

The short answer is: smoothing.

The operation used to describe the probability distribution of sum of (independent) random variables is the “convolution operation. As we add variables (increase the number of students), the PDF becomes a recursive convolution. Convolution has the important property of smoothing the response and eventually becomes the Gaussian curve.