Key Concepts

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

Glossary

central limit theorem (for sums): 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]\sum X \sim N (n \mu, \sqrt{n}\sigma)[/latex]. If the size (n) of the sample is sufficiently large, then the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample sums will equal n times the population mean.