In most situations where data are modelled as a random sample from a \(\NormalDistn(\mu, \sigma^2)\) distribution, the parameter \(\mu\) is of most interest.

However very occasionally, a test about the distribution's variance, \(\sigma^2\), is needed. This could be a one-tailed test such as

H0 :   \(\sigma^2 = \sigma_0^2\)
HA :   \(\sigma^2 \gt \sigma_0^2\)

where \(\sigma_0^2\) is some constant, or a two-tailed test of the form

H0 :   \(\sigma^2 = \sigma_0^2\)
HA :   \(\sigma^2 \ne \sigma_0^2\)

Chi-squared test

The test is based on the sample variance, \(S^2\), of a random sample of \(n\) values. If H0 holds,

\[ X^2 \;=\; \frac {n-1}{\sigma_0^2} S^2 \]

has a \(\ChiSqrDistn(n - 1\;\text{df})\) distribution. The test's p-value can be found from a tail probability of this distribution.

Example

The following 20 values are a random sample from a \(\NormalDistn(\mu, \sigma^2)\) distribution.

18.68 16.28 26.02 21.57 20.54 19.45 24.55 23.03 19.34 24.69
21.31 15.22 22.81 20.53 21.01 14.98 20.52 22.39 23.37 23.23

Test whether the distribution's variance is \(\sigma^2 = 4\).

(Solved in full version)

Robustness

This tests in this section are all based on the assumption that the data are a random sample from a normal distribution. The t-test for the distribution's mean is not affected badly if the underlying distribution is non-normal, so we say that this test is robust.

However the chi-squared test test for the variance does not provide an accurate p-value if the distribution from which the data are sampled has a non-normal shape. This chi-squared test is not robust.