Standard error of a sample mean

The sample mean is a random variable. This equation says that the size of the variability of the sample mean from a particular sample can be estimated by using the sample standard deviation. This estimated standard deviation is called the standard error of the sample mean.

Suppose we take a sample of 5 female UQ students and measure their heights, recording 172, 157, 174, 174, and 165 cm. This gives a sample mean of 168.4 cm. We do not know the population standard deviation but we can estimate it with the sample standard deviation, 7.37 cm. Then the standard error of the sample mean is 3.30 cm.

The sampling distribution of this sample mean is approximately normal. However, here we have an additional source of variability, since the sample standard deviation is also a random variable. Thus anything we do with these sample statistics will be more uncertain than if we knew the population standard deviation. To accomodate this we have to use the t distribution instead. The amount of extra variability from the t distribution will depend on how much we know about the population standard deviation, the degrees of freedom.

From t(4) we know that 95% of the time our sample mean will be within 9.16 cm of the population mean. Equivlently, we can be 95% confident that the mean height of female UQ students is within 9.16 cm of 168.4 cm, that is between 159.2 and 177.6 cm. It is in this way that the standard error of the sample mean is a measure of its precision as an estimator of the population mean.

Using the standard error in this way, we can give confidence intervals for the population mean at arbitrary confidence levels or we can test hypothesised values for the population mean.