## 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!

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