# One-sample variance test

Use the one-sample variance test to compare the variance of a sample to a reference. Run it in Excel using the XLSTAT add-on statistical software.

## Use of the one-sample variance test

Use this tool to compare the variance of a normally-distributed sample with a given value.

## Description of the one-sample variance test

Let us consider a samle of n independent normally distributed observations. One shows that the sample variance, s² follows a scaled chi-squared distribution with n-1 degrees of freedom.

s² ~ [σ²/(n-1)] * Χ²_{n-1}

where s² is the theoretical sample variance. This allows us to compute a confidence interval around the variance.

To compare this variance to a reference value, a parametric test is proposed. It is based on the following statistic:

Χ_{0}² = (n-1) s²/σ_{0}²

which follows a chi-square distribution with n-1 degrees of freedom.

This test is said to be parametric as its use requires the assumption that the samples are distributed normally. Moreover, it also assumed that the observations are independent and identically distributed. The normality of the distribution can be tested beforehand using the normality tests.

Three types of test are possible depending on the alternative hypothesis chosen:

For the two-tailed test, the null H0 and alternative Ha hypotheses are as follows:

- H0 : s² = s
_{0}² - Ha : s² ≠ s
_{0}²

In the left one-tailed test, the following hypotheses are used:

- H0 : s² = s
_{0}² - Ha : s² < s
_{0}²

In the right one-tailed test, the following hypotheses are used:

- H0 : s² = s
_{0}² - Ha : s² > s
_{0}²

Included in