The Central Limit Theorem

Background Modules for ChE473K
Process Design and Operations
at the
University of Texas at Austin

Gerald G. McGlamery, Jr., Ph.D., P.E.

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The central limit theorem is one of the most important concepts in statistics and is a strong driver of the results one sees in stochastic modeling. It states

If x is distributed with mean, µ, and standard deviation, s, then the mean obtained from a random sample of size n will have a distribution that approaches

A corollary of the central limit theorem can be stated as

If x is distributed with mean, µ, and standard deviation, s, then the Sx_i obtained from a random sample of size n will have a distribution that approaches

Note that the central limit theorem contains no assumption about the distribution that is sampled. It can be normal, log-normal, or something else.

The Central Limit Theorem is illustrated below:

Updated: Sunday, March 7, 2010

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