### 7.1 The Central Limit Theorem for Sample Means (Averages)

In a population whose distribution may be known or unknown, if the size (*n*) of the sample is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size (*n*).

### 7.2 The Central Limit Theorem for Sums (Optional)

The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution, even if the original population is not normally distributed. Additionally, if the original population has a mean of *μ _{X}* and a standard deviation of

*σ*, the mean of the sums is

_{x}*nμ*and the standard deviation is $(\sqrt{n})$(

_{x}*σ*), where

_{x}*n*is the sample size.

### 7.3 Using the Central Limit Theorem

The central limit theorem can be used to illustrate the law of large numbers. The law of large numbers states that the larger the sample size you take from a population, the closer the sample mean, $\overline{x}$, gets to *μ*.