FRM Exam Part I · Hypothesis Testing

Chi-Square and F-Tests for Variance Explained

Updated 11 October 2026 · Fact-checked

To test one variance against a hypothesised value, use the chi-square statistic χ² = (n − 1)s² ÷ σ₀² with n − 1 degrees of freedom. To compare two variances, use F = s₁² ÷ s₂² with (n₁ − 1, n₂ − 1) degrees of freedom. Compare the statistic with the critical value.

Understand Chi-Square and F-Tests for Variance

A mean test asks where the centre of a distribution sits. A variance test asks how spread out it is. In risk work the spread is often the point: you may want to know whether a portfolio's volatility has really changed, or whether two funds have different risk.

The chi-square test checks a single population variance against a hypothesised value σ₀². The sample variance s² is random, so it will not equal σ₀² exactly. If returns are normally distributed, the quantity (n − 1)s² ÷ σ₀² follows a chi-square distribution with n − 1 degrees of freedom when the null is true. A value far from its centre suggests the true variance differs from σ₀².

The F-test compares the variances of two independent samples. If both populations are normal, the ratio s₁² ÷ s₂² follows an F distribution with numerator degrees of freedom n₁ − 1 and denominator degrees of freedom n₂ − 1 when the two population variances are equal. A ratio near 1 supports equal variances. A ratio far above 1 (or far below it) does not.

Both distributions take only non-negative values. The chi-square is skewed to the right, so it is not symmetric like the normal or t. That means a two-tailed test uses two different critical values, one in each tail. The F distribution is also right-skewed, and its degrees of freedom order matters: swapping numerator and denominator gives a different distribution.

These tests are sensitive to the normality assumption. If returns have fat tails, the tests can reject too often. Keep that limitation in mind when a question asks about validity.

Key formulas to remember

Chi-square statistic for one variance: χ² = (n − 1) s² ÷ σ₀²; Degrees of freedom = n − 1. Assumes a normal population. σ₀² is the variance under the null.
F-statistic for two variances: F = s₁² ÷ s₂²; Degrees of freedom: numerator n₁ − 1, denominator n₂ − 1. Samples must be independent and from normal populations.
Hypotheses for a single variance: H₀: σ² = σ₀² versus H₁: σ² ≠ σ₀² (or > or <); One-tailed alternatives use one critical value; two-tailed uses a lower and an upper value.
Hypotheses for two variances: H₀: σ₁² = σ₂² versus H₁: σ₁² ≠ σ₂² (or >); Convention: put the larger sample variance on top so F ≥ 1, then use the upper critical value.
Decision rule: Reject H₀ if the statistic falls in the rejection region; Equivalent to rejecting when the p-value is below the significance level.

How to solve Chi-Square and F-Tests for Variance questions

Use the same sequence for any variance-test question. Decide first whether you have one sample or two.

1State H₀ and H₁. Work with variances, not standard deviations. Square any standard deviation you are given.
2Pick the test: one sample against a number means chi-square; two samples against each other means F.
3Compute the statistic: χ² = (n − 1)s² ÷ σ₀² or F = s₁² ÷ s₂².
4Find the degrees of freedom: n − 1 for chi-square, (n₁ − 1, n₂ − 1) for F.
5Get the critical value from the table at the stated significance level, paying attention to one-tailed or two-tailed.
6Compare. Reject H₀ if the statistic is beyond the critical value.
7State the conclusion in words, for example that there is evidence the volatility differs from the target.

Quickest way: Square, plug in, compare

When to use it: Use when the question gives sample standard deviations and a table value, or asks which conclusion follows.

Square the standard deviations immediately.
For F, divide larger variance by smaller so the ratio is at least 1, then compare with the upper critical value.
For chi-square, compute (n − 1)s² ÷ σ₀². If it is above 1 times (n − 1) the sample variance exceeds the null value, so you know the direction before looking at the table.
Eliminate options that use the wrong degrees of freedom, such as n instead of n − 1.

Common mistakes in Chi-Square and F-Tests for Variance

Using standard deviations instead of variances in the statistic.
Questions often quote volatility, and it is easy to plug it straight in.
Fix: Square every standard deviation before computing χ² or F.
Using n degrees of freedom instead of n − 1.
Candidates forget that one degree of freedom is used up estimating the sample mean.
Fix: Always write df = n − 1 before looking at the table.
Treating the chi-square as symmetric and using ± the same value for a two-tailed test.
The normal and t distributions are symmetric, so the habit carries over.
Fix: Use separate lower and upper critical values for chi-square, because the distribution is right-skewed.
Reversing the numerator and denominator degrees of freedom in the F-test.
The tables are indexed by two numbers and the order is easy to swap.
Fix: The numerator df belongs to the variance on top of the ratio. Write it first.
Ignoring the normality assumption.
Candidates treat these tests like the t-test, which is more robust.
Fix: Remember that both tests rely on normal populations and can be unreliable with fat-tailed returns.

Worked examples

Example 1

A risk manager believes a fund's monthly return volatility is 4%. A sample of 25 monthly returns has a standard deviation of 5%. Test H₀: σ² = 0.0016 against H₁: σ² ≠ 0.0016 at the 5% level. The chi-square critical values for 24 degrees of freedom are 12.40 (lower) and 39.36 (upper).

Show the solution

Sample variance: s² = 0.05² = 0.0025. Null variance: σ₀² = 0.04² = 0.0016.
Degrees of freedom: n − 1 = 24.
Statistic: χ² = 24 × 0.0025 ÷ 0.0016 = 0.06 ÷ 0.0016 = 37.5.
Compare: 12.40 < 37.5 < 39.36, so the statistic lies inside the acceptance region.

Answer: χ² = 37.5, which is below the upper critical value of 39.36. Do not reject H₀. There is not enough evidence at the 5% level that the variance differs from 0.0016.

Example 2

Fund A has a sample standard deviation of 6% from 21 observations. Fund B has a sample standard deviation of 4% from 31 observations. Test whether Fund A has a higher variance at the 5% level. The upper 5% critical value of F with (20, 30) degrees of freedom is 1.93.

Show the solution

H₀: σ_A² = σ_B²; H₁: σ_A² > σ_B². This is one-tailed.
Variances: s_A² = 0.0036 and s_B² = 0.0016.
Statistic: F = 0.0036 ÷ 0.0016 = 2.25.
Degrees of freedom: numerator 21 − 1 = 20, denominator 31 − 1 = 30.
Compare: 2.25 > 1.93, so the statistic is in the rejection region.

Answer: F = 2.25 exceeds 1.93, so reject H₀. There is evidence at the 5% level that Fund A's variance is higher than Fund B's.

Exam tips

Check whether the question gives standard deviations. Square them before anything else.
Look at the alternative hypothesis to decide one-tailed or two-tailed. The critical values are provided or implied, so mismatching them loses the mark.
For F questions, confirm which sample is on top and which degrees of freedom go with it.
Expect conceptual questions on assumptions: normal populations, independent samples, and sensitivity to fat tails.
You rarely need to look up tables from memory. Focus on setting up the statistic and the degrees of freedom correctly.

Practice questions from Hypothesis Testing

Chi-Square and F-Tests for Variance in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Chi-Square and F-Tests for Variance: frequently asked questions

What is the difference between the chi-square and F distributions?

The chi-square distribution has one degrees-of-freedom parameter and is used to test a single variance. The F distribution is the ratio of two chi-square variables, each divided by its degrees of freedom, so it has two parameters and is used to compare two variances. Both are right-skewed and non-negative.

Do I need to remember critical values for the FRM exam?

Usually the question provides the critical value or gives enough information to decide. What you must do yourself is compute the statistic and the degrees of freedom correctly. Focus on the method rather than memorising tables.

Why must the data be normal for these tests?

The chi-square and F results come from the fact that the scaled sample variance of a normal sample follows a chi-square distribution. If the data are fat-tailed, that link breaks and the tests can give misleading rejection rates. This is why they are less robust than tests on the mean.

Can I use the F-test for a regression?

Yes. The F-statistic in regression compares variances, such as explained and unexplained variation, to test joint hypotheses. The idea is the same ratio of scaled variances, but the degrees of freedom come from the regression setup.