Lesson 4 of 5 6–9% of the exam

You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

By Daniil Tarakanov Updated Oct 16, 20262026 curriculum

Chi-square and F-tests of variance

To test one variance against a value, use the chi-square statistic (n − 1)s² ÷ σ₀² with n − 1 degrees of freedom. To compare two variances, use F = s₁² ÷ s₂², putting the larger variance on top, with n₁ − 1 and n₂ − 1 degrees of freedom. Both distributions are non-negative and skewed right.

6 min cram, 30 min full read. Based on the 2026 CFA® Program Level I curriculum, Quantitative Methods, Learning Module 8. This is the 2026 version of the lesson. Sitting in February 2027 or later? Read the updated 2027 version.

Tailor this to you: take the self-assessment

Before this: t-tests of a mean, a difference in means and paired differences

On the exam

Construct hypothesis tests and determine their statistical significance, the associated Type I and Type II errors, and power of the test given a significance level

Construct tests of a single variance and of the equality of two variances, and determine their significance. Expect to compute a statistic from standard deviations and compare it with a critical value given in the stem.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

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.

Which variance test?

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; df n − 1
Two variances against each other Memorize
- F-test Memorize F = larger s² ÷ smaller s²; df n₁ − 1 and n₂ − 1

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. 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.

Chi-square test of a single variance Memorize

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

Rearrange itChi-square statistic for one variance: solve for any one

The question asks for

A sample of 20 monthly returns has a standard deviation of 3.50%. For H₀: σ² ≤ 7.84 (σ₀ = 2.80%), the chi-square statistic is closest to:

Sample size, nSample standard deviation, s%Hypothesized standard deviation, σ₀%

χ^{2} = \frac{( n - 1 ) s ^{2}}{σ _{0}^{2}}

(20 − 1) × 12.25 ÷ 7.84 = 29.6875

Chi-square statistic, χ²29.6875

On your calculator

(20−1)×3.5x²÷2.8x²=

x² square

20ENTER1−3.5ENTER××2.8ENTER×÷

ENTER enter (on the HP, puts the number on the stack)

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 test of the difference in variances divides one sample variance by the other, with the larger on top.

F-test of the equality of two variances Memorize

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

Rearrange itF-statistic for two variances: solve for any one

The question asks for

Two funds have return standard deviations of 6.00% and 4.50%. The F-statistic for equality of variances, with the larger variance on top, is closest to:

Larger standard deviation, s₁%Smaller standard deviation, s₂%

F = \frac{s _{1}^{2}}{s _{2}^{2}} (s_{1}^{2} \geq s_{2}^{2})

36 ÷ 20.25 = 1.7778

F-statistic1.7778

On your calculator

6x²÷4.5x²=

x² square

6ENTER×4.5ENTER×÷

ENTER enter (on the HP, puts the number on the stack)

InteractiveIs the fund riskier than promised?

Try this 0 of 2

Keep s = 3.5% and σ₀ = 2.8% (upper tail, 5%). Add months until you reject H₀.
Switch to the two-tailed alternative and read the critical values.

Reject H₀ (α 5%)

DecisionFail to reject H₀

χ² statistic29.69

Critical value30.144

p-value0.0559

Sample

Months n (df = n − 1)20 (df 19)

Sample standard deviation s3.5%

Promised standard deviation σ₀2.8%

The test

Alternative

Significance level α

Show working

H₀: σ² ≤ 7.84 against Hₐ: σ² > 7.84 (σ₀ = 2.8%, so σ₀² = 7.84).

χ² = (n − 1)s² ÷ σ₀² = 19 × 12.25 ÷ 7.84 = 29.69, df = n − 1 = 19.

Upper 5% critical value, df 19: 30.144.

29.69 is not in the rejection region → fail to reject H₀ (p = 0.0559).

Square both standard deviations: the test is about variances.

Chi-square statistic

19×3.5x²÷2.8x²=29.6875

x² square

19ENTER3.5ENTER××2.8ENTER×÷29.6875

ENTER enter (on the HP, puts the number on the stack)

Display: 29.6875

Variance tests use squared numbers, so the distributions start at zero and lean right.

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 1.744 to 1.943.

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

Go deeper Two-tailed chi-square tests

For H₀: σ² = σ₀² against Hₐ: 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 Hₐ 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 nonparametric 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.

Memorize this

MemorizeVariance tests

	Chi-square	F
Tests	one variance	two variances
Statistic	$\dfrac{(n-1)s^2}{\sigma_0^2}$	$\dfrac{s_1^2}{s_2^2}$ (larger on top)
df	$n-1$	$n_1-1$ and $n_2-1$
Shape	non-negative, skewed right	non-negative, skewed right
Two-tailed critical values	lower and upper	upper only, at $\alpha/2$

Memory hook Variances are squares, so both distributions start at zero and have a long right tail. Putting the larger variance on top means the F-statistic is never below 1, so you only need the upper critical value.

Know it cold Square the standard deviations first. Use $n-1$ , not $n$ . Both tests assume normal populations.

Which variance test?

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; df n − 1
Two variances against each other Memorize
- F-test Memorize F = larger s² ÷ smaller s²; df n₁ − 1 and n₂ − 1

Chi-square test of a single variance

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

Quick example

16 monthly returns with s = 3.0%, tested against σ₀ = 2.0%. χ² = 15 × 9 ÷ 4 = 33.75, with 15 degrees of freedom.

On your calculator

15×3x²÷2x²= 33.75

x² square

15ENTER3ENTER××2ENTER×÷ 33.75

ENTER enter (on the HP, puts the number on the stack)

F-test of the equality of two variances

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

Quick example

Standard deviations of 8% and 5%. F = 64 ÷ 25 = 2.56, larger variance on top.

On your calculator

8x²÷5x²= 2.56

x² square

8ENTER×5ENTER×÷ 2.56

ENTER enter (on the HP, puts the number on the stack)

How it is asked

Compute chi-square

25 returns, s = 3.2%. Test H₀: σ = 2.5%. Statistic?

24 × 3.2² ÷ 2.5² = 39.32. Traps: 40.96 (n instead of n − 1), 30.72 (standard deviations not variances).
Compute F

Standard deviations 7.2% and 5.4%. F?

7.2² ÷ 5.4² = 1.78. Traps: 1.33 (standard deviations), 0.56 (smaller on top).
One-tailed or two-tailed critical value

F = 1.78, df (30, 40). Hₐ: σ₁² greater than σ₂² (5%) or σ₁² not equal to σ₂² (5%)?

One-tailed critical 1.744: reject. Two-tailed (upper 2.5%) 1.943: fail to reject.
Pick the distribution

Which distribution tests whether a single variance equals a hypothesized value?

Chi-square. Traps: t (that is for means), F (that is for two variances).

Exam traps

Using n instead of n − 1 in the chi-square statistic or the degrees of freedom.
Plugging in standard deviations instead of variances. Square them first.
Putting the smaller variance on top in the F-test. The larger goes on top, and the F-statistic is at least 1.
Using the one-tailed F critical value for a test of 'equal variances'. A two-tailed test at 5% uses the upper 2.5% value.

Checkpoint

4 exam-style questions. About 90 seconds each.

1A sample of 25 returns has a standard deviation of 3.2%. To test H₀: σ = 2.5%, the chi-square statistic is closest to:
2Two samples have standard deviations of 7.2% and 5.4%. The F-statistic for equality of variances is closest to:
3A test of whether a single population variance equals a hypothesized value uses the:
4A value fund (31 months, s = 6%) and a growth fund (41 months, s = 4.5%) give F = 1.78 with 30 and 40 degrees of freedom. The upper 5% critical value is 1.744 and the upper 2.5% critical value is 1.943. For the one-sided alternative that the value fund is more volatile, tested at the 5% level, the analyst should conclude that:

Key takeaways

One variance: χ² = (n − 1)s² ÷ σ₀², df n − 1.
Two variances: F = larger s² ÷ smaller s², df (n₁ − 1, n₂ − 1).
Both tests use variances, not standard deviations, and assume normal populations.
The wording of Hₐ decides whether the critical value is one-tailed or two-tailed.

Chi-square test of a single variance

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

Quick example

16 monthly returns with s = 3.0%, tested against σ₀ = 2.0%. χ² = 15 × 9 ÷ 4 = 33.75, with 15 degrees of freedom.

On your calculator

15×3x²÷2x²= 33.75

x² square

15ENTER3ENTER××2ENTER×÷ 33.75

ENTER enter (on the HP, puts the number on the stack)

F-test of the equality of two variances

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

Quick example

Standard deviations of 8% and 5%. F = 64 ÷ 25 = 2.56, larger variance on top.

On your calculator

8x²÷5x²= 2.56

x² square

8ENTER×5ENTER×÷ 2.56

ENTER enter (on the HP, puts the number on the stack)

Next lesson Parametric vs nonparametric tests

Questions about this lesson?

Where this comes back

Parametric vs nonparametric tests Hypothesis Testing
Simple Linear Regression Quantitative Methods

Questions candidates ask

When do you use a chi-square test in CFA Level 1?

Use it to test a hypothesis about a single population variance, such as whether a fund's volatility is higher than promised. The statistic is (n − 1)s² ÷ σ₀², with n − 1 degrees of freedom. The next module uses the chi-square distribution again for tests of independence.

How do you calculate an F-statistic for two variances?

Square both sample standard deviations to get variances, then divide the larger by the smaller: F = s₁² ÷ s₂². The degrees of freedom are n₁ − 1 for the numerator and n₂ − 1 for the denominator. For standard deviations of 8% and 5%, F = 64 ÷ 25 = 2.56.

Why is the larger variance in the numerator of the F-test?

It is a convention that keeps F at 1 or above, so only the upper tail of the distribution is needed. For a two-tailed test of equal variances at 5%, compare F with the upper 2.5% critical value. Swapping the order would require a lower critical value.

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