We end this section with another important result that is stated here without proof.

Independence of sample mean and variance

If \(\{X_1, X_2, \dots, X_n\}\) is a random sample from a \(\NormalDistn(\mu, \sigma^2)\) distribution, the sample variance, \[ S^2 \;=\; \frac {\sum_{i=1}^n {(X_i - \overline{X})^2}} {n-1} \] is independent of the sample mean, \(\overline{X}\).

Although we cannot prove independence with the statistical theory that we have covered so far, it can be demonstrated with a simulation.

Simulation to demonstrate independence

The diagram below shows a random sample from a \(\NormalDistn(\mu=12,\; \sigma^2 = 2^2)\). The sample mean and standard deviation are plotted against each other on the scatterplot on the right.

Click Accumulate and take a hundred or more samples. (Hold down the Take sample button.) There is a fairly circular scatter of crosses in the scatterplot — the sample standard deviation is as likely to be high when the sample mean is low as when the sample mean is high. The scatterplot therefore supports the fact that the sample mean and sample standard deviation are independent.

(We have not yet introduced the correlation coefficient, but if you already understand the concept, you should note that the correlation coefficient between the sample means and standard deviations is close to zero.)

Note that independence of the mean and standard deviation also implies independence of the mean and variance.