22.2 Sampling distribution: One mean with population standard deviation unknown

When a sample mean is used to estimate a population mean, the sample mean will vary from sample to sample: sampling variation exists, as we saw in the previous section.

When we do not know the population standard deviation \(\sigma\) (which is almost always the case), we estimate it using the sample standard deviation \(s\). Then, the standard error of the sample mean is \(\displaystyle\text{s.e.}(\bar{x}) = \frac{s}{\sqrt{n}}\). With this information, we can describe the sampling distribution of the sample mean.

Definition 22.1 (Sampling distribution of a sample mean) When the population standard deviation is unknown, the sampling distribution of the sample mean is described by:

  • an approximate a normal distribution,
  • centred around \(\mu\),
  • with a standard deviation (called the standard error of the mean) of
\[\begin{equation} \text{s.e.}(\bar{x}) = \frac{s}{\sqrt{n}}, \tag{22.1} \end{equation}\] when certain conditions are met, where \(n\) is the size of the sample, and \(s\) is the standard deviation of the individual observations in the sample (that is, the sample standard deviation).