The sample mean is less variable than individual values in the population

The sample mean has a distribution with the following properties.

It is possible to quantify these bullet-points more precisely. The distribution of a sample mean, , is centred on the population mean,

 =  μ

The standard deviation of the distribution of the sample mean is

 = 

where the sample size is n.

The following diagram is similar to that on the previous page. The top half of the diagram shows a jittered dot plot of a random sample from a normal population with mean 12 and standard deviation 2. The bottom half shows the sample mean.

Set the checkbox Accumulate then click Take sample a few times to see the variability of the means of samples of size 16.

Use the pop-up menu to change the sample size, then repeat the sampling to investigate the effect of sample size on the distribution of the sample mean. Verify that:

Finally, click the checkbox below to superimpose the theoretical distribution of the sample mean on the diagram above. Observe how this changes with the sample size.