Central Limit Theorem

Renjin Tu

A series of theorems in mathematical statistics called the central limit theorems provide theoretical justification for approximating the true sampling distribution of many sample statistics with the normal distribution. This entry discusses one such theorem for the sample mean. Similar theorems exist for sample median, sample standard deviation, and sample proportion. The word central in the name of the theorem means ‘fundamental.’ The central limit theorem for the sample mean states that for a large sample size, the sampling distribution of the sample mean X is approximately normal, no matter what the population distribution looks like. The approximation becomes better with increasing sample size. This surprising fact was proved in fairly general form in 1810 by Pierre-Simon Laplace. The graphs in Figures 1 through 5 show the idea of the Central Limit Theorem. Figure 1 is a histogram of a random sample data from a normally distributed Figure ...

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