What is the expected value of the sample mean x̅?

The expected value of the sample mean, denoted as E(x̅), is equal to the population mean, which is represented by the symbol μ. This concept is fundamental in statistics, particularly in inferential statistics where we make estimates about a population based on sample data.

When we select a random sample from a population and calculate the sample mean, the expected value of that sample mean reflects our best estimate of the population mean. This property is a result of the Central Limit Theorem, which states that as the sample size increases, the distribution of the sample mean will tend to be normally distributed around the population mean (μ). Thus, regardless of the population distribution, the expected value of the sample mean will equal the true population mean.

This understanding is critical as it allows statisticians and researchers to make inferences about the population mean based on the sample mean calculated from a smaller subset of data. It reinforces the idea that the sample mean is an unbiased estimator of the population mean.