Distribution of the sample mean

All summaries of sample data, graphical and numerical, vary from sample to sample. The most widely used summary statistic is a sample's mean, so this section describes the variability of sample means.

A single value that is sampled from a population has a distribution that is described by the population distribution. When a random sample of n values is sampled, the sample mean is also random, but has a distribution that is less variable than the population distribution. (Sample means 'average out' the extremes in a sample, so sample means tend to be closer to the centre of the population distribution.)

Simulation

The following diagram selects random samples of values from a normal population with mean 12 and standard deviation 2. This population distribution would be an appropriate model for various types of data:

The normal population is shown at the top with a random sample of 16 values underneath.

Click Take sample a few times to display different random samples and their means. Observe that the sample means vary from sample to sample — they have a distribution.

Now click the checkbox Accumulate and take 20 or 30 further samples. The bottom display shows the means from all samples in a stacked dot plot.

The second of these points is particularly important.

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.