The sample mean is less variable than individual values in the population
The sample mean has a distribution with the following properties.
- It has a distribution that is centred on the population mean.
- Its variability decreases as the sample size increases.
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:
- The means of samples of size 1 have the same distribution as that of the population (a normal distribution with mean 12 and standard deviation 2 in the example above).
- The sample mean has a distribution that is centred on 12 (the population mean), whatever the sample size.
- The variability of the mean decreases as the sample size increases.
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.