## 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).