The sampling distribution of the sample mean is a probability model that illustrates the different values of the sample mean that could be observed from different samples from the population.
This distribution represents the distribution of all possible sample means that could be obtained by repeatedly drawing samples from the population. It is centered around the population mean and has a standard deviation that is equal to the standard deviation of the population divided by the square root of the sample size.
The sampling distribution of the sample mean follows a normal distribution if the population from which the samples are drawn follows a normal distribution. This is known as the central limit theorem. However, even if the population is not normally distributed, the sampling distribution of the sample mean can still be approximately normal if the sample size is large enough (typically greater than 30) due to the central limit theorem.
The sampling distribution of the sample mean is an important concept in statistics as it allows us to make inferences about the population based on the characteristics of the sample. For example, it enables us to estimate population parameters, such as the population mean, based on the sample mean. It also allows us to conduct hypothesis tests and calculate confidence intervals.