# Central Limit Theorem Illustrated with Four Probability Distributions

This Demonstration shows an amazing result, the central limit theorem: by sampling a population with an unknown distribution, either finite or infinite, the sampling distribution of the mean value of each sample will be normal, with estimated mean close to the distribution mean , and variance close to the distribution variance divided by the square root of the sample size . This result requires that the sample size be large [1].
We consider four different probability distributions for as a function of between the interval and ; outside this interval, the probability is set to 0. The are , , and one that is discrete; is the constant that enables the integral of the probability distribution to be normalized; that is, .
Use the "trials T" slider to set the number of samples, each one of size , sampled from the corresponding .
The top plot shows the plot of the selected ; a red line indicates the position of the probability distribution . On the right, the values of and of are indicated.
The bottom plot shows the histogram obtained from the sample; a Gaussian distribution with parameters and is overlaid on the histogram. A red line indicates the position of the probability distribution . On the right, the values of and of the are indicated.

### DETAILS

In the four snapshots, ; and are evaluated:
Snapshot 1: a small and small result in a poor approximation of a Gaussian; this is also shown by the discrepancy between estimated and real values
Snapshot 2: a large and small result in a poor approximation of a Gaussian, but an accordance between estimated and real values can be found
Snapshot 3: a small and large result in a good approximation of a Gaussian, and the estimated and real values are close to each other
Snapshot 4: a large and a large result in a very good approximation of a Gaussian, and the estimated and real values are close to each other
Reference
[1] R. E. Walpole, R. H. Myers, S. L. Myers and K. Ye, Probability & Statistics for Engineers & Scientists, 9th ed., Boston: Prentice Hall, 2012.

### PERMANENT CITATION

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