Distribution of the Sample Mean

The distribution of the sample mean is the probability distribution of all possible sample means that could be obtained from a population. It is a theoretical distribution that is used to make inferences about the population mean based on a sample mean.

The distribution of the sample mean is a normal distribution if the population from which the sample is drawn is normally distributed. In this case, the mean of the sample means is equal to the population mean, and the standard deviation of the sample means (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size.

If the population is not normally distributed, the distribution of the sample mean can still be approximated by a normal distribution if the sample size is large enough (usually at least 30). This is known as the central limit theorem.

The distribution of the sample mean is important in statistical inference, as it is used to calculate confidence intervals and perform hypothesis tests. By knowing the distribution of the sample mean, we can make statements about the population mean with a certain level of confidence.