Central Limit Theorem

The Central Limit Theorem states that sample means will be normally distributed, regardless of the nature of the underlying population.  It forms the basis for statistics such as the standard error of the mean.

The link between theory and practice can be demonstrated by a simple experiment.  An arbitrary distribution was contrived to look as unlike the nice, tidy bell shaped curve of the normal distribution.  Some poor hapless computer, given the task of running a programme to draw samples from this mess of numbers.

The mean and standard deviation of this population were 5.5 and 3.5 respectively.

The program drew 1,000 samples of 10, the mean value of these samples was 5.51 with a standard deviation of 1.06, the distribution is shown below:

The formula the standard error of the mean is:

The standard error of the mean is the standard deviation of the sample means, for a sample of 10 drawn from a population with a standard deviation of 3.5, the standard error would be:

StErr = 3.50/√10 = 3.50/3.16 = 1.11

The result of the experiment is close to that predicted by the theory.

In the context of Monte-Carlo techniques, the Central Limit Theorem is the basis for estimating the quality of output of a given process.

Page updated: 18-Feb-2008

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