Distribution
of Sampling Means
A Sampling Distribution of Sample Means is a distribution using the means computed from all possible random samples of a specific size taken from a population.
If the samples are randomly selected with replacement, the sample means, for the most part, will be somewhat different from the population mean µ. These differences are caused by sampling error.
Sampling Error is the difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population.
Properties:
The standard deviation of the sample means is called the Standard Error of the Mean.
We denote the mean of the sample means as \(\mu_{\overline{x}}\) and the standard deviation of the sample means as \(\sigma_{\overline{x}}\). Therefore we have the following relationships:
\[\mu_{\overline{x}}=\mu\]
\[\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}\]
where \(n\) is the sample size.