5.5 The Central Limit Theorem


10704 twclt ()
illustrates the central limit theorem for a user-specified underlying probability distribution

This quantlet illustrates the concept of the Central Limit Theorem by showing a histogram of the averages of a large number of samples simulated from a user-designated underlying probability distribution. That is, the user specifies a probability distribution function, then a number of samples are simulated from this distribution. For each sample, the average value is computed. Finally, a histogram of these average values is displayed. The user can increase or decrease the number of samples in the simulation to observe that as the number of samples increases, the histogram approaches the density function of a standard normal distribution.

To activate this quantlet, the user should type in the following:

  twclt()
After this, the following window should be shown:

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Here, the user can specify a probability distribution for a discrete random variable of four values (i.e. $ X = $ 1, 2, 3, or 4). He/she should enter in four values (making sure that they add up to 1), or use the default values of 0.25 for each one, then click on the OK button. The following display should follow (here for the default values):


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The lower portion of the Display window is the underlying probability distribution -- a visual representation of what the user previously typed in for the last window. The upper part of this window is a histogram of averages of 30 simulated samples from the given distribution. That is, XploRe makes a simulation of thirty samples from the given probability distribution, computes the average value for each of them, and plots them in the histogram. Since this is a simulation of a random variable, the resulting histogram from two simulations of the same number of samples from the same underlying distribution will most likely not be the same.

The Change number of samples window above asks for increasing or decreasing the number of samples in the simulation (or quitting). As the user increases the number of samples, the resulting empirical distribution function should look more and more similar to a normal distribution. If the data were subtracted by the estimated mean and divided by the estimated standard deviation, the empirical distribution function would look more and more like a standard normal distribution.

For the beginning statistics student, this graphically shows that the distribution of the mean of a sample approaches the normal distribution as the number of samples increases.