Central Limit Theorem

  The Central Limit Theorem describes the characteristics of the "population of the means" which has been created from the means of an infinite number of random population samples of size (N), all of them drawn from a given "parent population". The Central Limit Theorem predicts that regardless of the distribution of the parent population:

[1] The mean of the population of means is always equal to the mean of the parent population from which the population samples were drawn.

[2] The standard deviation of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (N).

[3] [And the most amazing part!!] The distribution of means will increasingly approximate a normal distribution as the size N of samples increases.

A consequence of Central Limit Theorem is that if we average measurements of a particular quantity, the distribution of our average tends toward a normal one. In addition, if a measured variable is actually a combination of several other uncorrelated variables, all of them "contaminated" with a random error of any distribution, our measurements tend to be contaminated with a random error that is normally distributed as the number of these variables increases.

Thus, the Central Limit Theorem explains the ubiquity of the famous bell-shaped "Normal distribution" (or "Gaussian distribution") in the measurements domain.

Applet 

This applet demonstrates the gradual formation of a normally distributed population as we increase the sample size, i.e. the number N of individual random observations to be averaged.

The user can peak any of the 8 distributions representing the "parent population" (with values in the range 0 - 100) by clicking the corresponding radio-button, then he/she can select the size N of the population sample (N = 1-30) by pressing the buttons "Inc" (increase) / "Dec" (decrease). By pressing the "DRAW" button, the histogram representing the distribution of the generated population starts to be drawn. Obviously for N = 1, the histogram of the "parent population" will be obtained.

The relative population size used for the creation of the histogram can be selected by clicking one of the radio-buttons labelled "Small", "Medium", and "Large". The larger the size of this population is, the less jagged this histogram will appear, but it will take more time to be completed.

Upon the completion of the drawing of the histogram, the size of the total population will appear (at the upper-right corner of the histogram). The mean and the standard deviation of this population will also appear at the bottom of the histogram.