## Central Limit Theorem

The Central Limit Theorem says that as sample size n increases, the sum (or mean) of the n observations in the sample will be approximately normally distributed regardless of the distribution of the observations making up the sample! The approximation gets better with increasing sample size. Also, if the original observations are drawn from a skewed distribution (such as the exponential distribution), a larger sample size will be needed in order to achieve as good an approximation to the normal distribution as would be obtained if the observations had been drawn from a non-skewed distribution (e.g., the uniform distribution).

The applet below is designed to demonstrate this phenomenon. Two distributions of random variables may be simulated, uniform or exponential. Pressing the "Go" button performs a simulation and displays a histogram of the results. Setting the sample size to be 1 allows you to see the shape of the distribution from which the individual observations are being drawn. The uniform distribution results in a fairly even number of random deviates falling in each bin of the histogram. This is an example of a non-skewed distribution. The exponential distribution results in more observations falling in bins to the left (relatively smaller numbers), with the heights of the bins falling off as one moves to the right. This is an example of a skewed distribution.

Despite the drastically different shape of these two distributions, plotting sums rather than individual observations shows that the sums are distributed approximately normally, with the approximation becoming better as the number of elements in the sum increases. Just increase the sample size from 1 to 10 or 100 to see this effect. Also note that you need larger sample sizes in the case of the exponential distribution to achieve approximate normality. With the uniform distribution you actually only need samples of size 2 to get close to normality!