Key Concepts

• The central limit theorem (for means) states that even if a population distribution is non-normal or the shape is unknown, the shape of the sampling distribution of the sample mean will be approximately normal if the sample size is large enough.
• The mean of the sampling distribution of the sample means is equal to the mean of the population.
• The standard deviation of the sampling distribution of the sample mean is the population mean divided by the square root of the sample size.  This is also called the standard error of the mean.

Glossary

central limit theorem (for means): given a random variable (RV) with known mean μ and known standard deviation, σ, if the size (n) of the sample is sufficiently large, then [latex]\overline{X} \sim N (M, \frac{\sigma}{\sqrt{n}})[/latex]. If the size (n) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The standard deviation of the distribution of the sample means, [latex]\frac{\sigma}{\sqrt{n}}[/latex], is called the standard error of the mean.

sampling distribution: given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution

standard error of the mean: the standard deviation of the distribution of the sample means, or [latex]\frac{\sigma}{\sqrt{n}}[/latex]