The mean varies from sample to sample and has a distribution

As with graphical summaries of sample data, summary statistics vary from sample to sample. The most important summary statistic of a numerical sample is its mean and we will concentrate on sample means in this section.

Since sample means 'average out' the extremes in a sample, they are less variable than the individual measurements in the population.

The sample-to-sample variability of sample means can be described with a distribution. (This is equivalent to considering sample means to be single values randomly selected from a 'population' of sample means.)

The top half of the following diagram shows a random sample of 20 values from a normal population (with mean 12 and standard deviation 2). The population distribution is shown in grey. The sample mean is also displayed.

Click Take sample to display a few random samples and their means. Observe that the sample means are more closely concentrated round the population mean, 12, than the individual sample values.

Now click the checkbox Accumulate and take 20 or 30 further samples. The lower display shows the means from all samples in a jittered dot plot. The distribution of the means can be described in a similar way to the distribution of the original sample. Its spread is lower than that of the underlying population.

You may click on the crosses representing the means in the lower jittered dot plot; the sample that generated that mean is displayed above. Look at the samples that gave rise to the highest and lowest means.