Chi-square and F-tests of variance

Why test a variance?

In finance the variance is the risk. A fund may promise to keep volatility below a stated level, or you may suspect that one strategy is riskier than another. Both questions are tests about variances, and both distributions involved are built from squared numbers.

What is being tested?squared numbers, so the distributions start at zero and lean right

• One variance against a value Memorize
  • Chi-square test Memorize χ² = (n − 1)s² ÷ σ₀²; n − 1 degrees of freedom
• Two variances against each other Memorize
  • F-test Memorize F = larger s² ÷ smaller s²; n₁ − 1 and n₂ − 1 degrees of freedom

How do you test one variance?

Memorize

The chi-square distribution is non-negative and skewed right, and its shape depends on the degrees of freedom (df). Its critical values are tabulated, and the lower and upper tails are not mirror images. The test of a single variance compares the sample variance with a hypothesized value. The sample-size rules for choosing t or Z do not apply to chi-square and F tests.

$$\chi^2 = \frac{(n-1)\,s^2}{\sigma_0^2},\quad df = n - 1$$

How do you compare two variances?

Memorize

The F-distribution is also non-negative and skewed right, with two degrees-of-freedom values: one for the numerator and one for the denominator. The shape of the F-distribution depends on both degrees of freedom. The test of the difference in variances divides one sample variance by the other.

$$F = \frac{s_1^2}{s_2^2}\quad(s_1^2 \ge s_2^2),\quad df = (n_1 - 1,\ n_2 - 1)$$

Putting the larger variance on top is the usual convention: F is then at least 1, and a two-tailed test needs only the upper critical value at α ÷ 2. If the question fixes the order instead (for example, after ÷ before), a two-tailed test needs both a lower and an upper critical value, and you reject if F falls outside them.

Move the statistic into the tail on each tab. On the F tab, switch between one-tailed and two-tailed and watch the critical value move from 1.783 to 1.994.

Exam tip Standard deviations come in the stem; both tests need variances. Square first, then plug in.

Exam tip In a spreadsheet: CHISQ.INV.RT(0.05, 19) gives the upper 5% chi-square value 30.144; F.INV.RT(0.025, 25, 40) gives 1.994; F.INV(0.025, 25, 40) gives the lower value 0.472.

Go deeper Two-tailed chi-square tests

For $H_0$: σ² = σ₀² against $H_A$: σ² ≠ σ₀², there are two rejection regions, and because the chi-square curve is lopsided you need two different critical values: a lower one with α ÷ 2 in the left tail and an upper one with α ÷ 2 in the right tail. For a one-sided $H_A$ you need only one of them: the upper value for “greater than”, the lower value for “less than”.

Go deeper The normality assumption

Both tests assume the underlying populations are normal, and they are far more sensitive to departures from normality than the t-test for a mean. If returns show heavy tails, the result can mislead, and a non-parametric approach or a larger sample is safer.

Go deeper The F-statistic returns in regression

The F-statistic reappears in the analysis of variance for a regression, where it compares the variance explained by the model with the variance left unexplained. The mechanics are the same: a ratio of variances, with its own pair of degrees of freedom.