Central Limit Theorem

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 population with a mean 50 and a standard deviation 2; Figure 2 is a histogram for a sampling mean X with a sample size n = 2 from a normal distribution with a mean 50 and a standard deviation 2. Even the sample size is small; in this case, X is still normal with a mean 50 and a standard deviation 2 ffiffiffi2 p = 1.4142. In fact, when the population is normal, the sampling distribution of X is exactly normal for any sample size.

Figure 3 is a histogram of a random sample data from a chi-square distribution with 1 df. This is a right-skewed distribution; Figure 4 is a histogram for a sampling mean X for a sample size 2 from a chi-square distribution with 1 df. When sample size is small, the sampling distribution of X is still rightskewed; Figure 5 is a histogram for a sampling mean [Page 166]X for a sample size 30 from the same chi-square distribution with 1 df. In this case, the sampling distribution of X is approximately normal.

Figure 1 A Histogram of a Random Sample Data From a Normally Distributed Population With a Mean 50 and a Standard Deviation of 2

Figure 2 A Histogram for a Sampling Mean X With a Sample Size n = 2 From the Same Normal Distribution With a Mean 50 and a Standard Deviation 2

In symbol, let X be a random variable with mean m and standard deviation s, and X be the sample mean; when the sample size n is large, the standardized variable Z is approximately the standard normal variable:

This result has enabled statisticians to develop some large-sample procedures for making inferences about a population mean m even when the shape of the population distribution is unknown. Application of the [Page 167]central limit theorem requires a rule of thumb for deciding whether n is indeed sufficiently large. When the population distribution is quite skewed, only the ones for n = 30, 40, or more, the sampling distribution of X may have reasonably normal shapes. If the population distribution is somewhat skewed, then n = 10 or 15, and the sampling distribution of X may have reasonably normal shapes. The rule that many statisticians recommend is n ≥ 30.

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