## The Mean, Standard Deviation, and Sampling Distribution of the Sample Mean

This section gives several concrete examples of calculating the exact distributions of the sample mean. The corresponding means and standard deviations are computed for demonstration based on these distributions. Next, it discusses sampling distributions of sample means when the sample size is large. It also considers the case when the population is normal. Finally, it uses the central limit theorem for large sample approximations.

### The Mean and Standard Deviation of the Sample Mean

#### Key Takeaways

• The sample mean is a random variable; as such it is written $\bar{X}$, and $\bar{x}$ stands for individual values it takes.
• As a random variable the sample mean has a probability distribution, a mean $\mu_{\bar{X}}$, and a standard deviation $\sigma_{\bar{X}}$.
• There are formulas that relate the mean and standard deviation of the sample mean to the mean and standard deviation of the population from which the sample is drawn.