CFA Level I Exam · Statistical Distributions for Financial Asset Prices and Returns

Student's t, Chi-Square and F Distributions for CFA Level I

Updated 7 October 2026 · Fact-checked

The t, chi-square and F distributions are sampling distributions used for inference. Use t for tests on a mean when the population variance is unknown, chi-square for a single variance, and F for comparing two variances. Each depends on degrees of freedom. Pick the distribution first, then compare the test statistic with the critical value.

Understand Student's t, Chi-Square and F Distributions

When you test a claim about a population, you work from a sample. The sample statistic has its own distribution. The normal distribution is a good start, but it needs the population standard deviation. In practice you rarely know it, so other distributions are used.

The Student's t distribution is used for inference about a mean when the population variance is unknown and you use the sample standard deviation instead. It is symmetric and bell-shaped with a mean of 0, like the standard normal, but it has fatter tails. Fatter tails mean larger critical values, so the test is more cautious. As degrees of freedom rise, the t distribution approaches the standard normal. For a sample mean, degrees of freedom = n − 1.

The chi-square distribution (χ²) is used for tests about a single population variance, assuming the population is normally distributed. It is the distribution of a sum of squared standard normal variables. It cannot be negative, and it is skewed to the right. As degrees of freedom increase, it becomes more symmetric. Degrees of freedom = n − 1 for a variance test.

The F distribution is used to compare two variances, and it appears in ANOVA for regression. It is the ratio of two independent chi-square variables, each divided by its own degrees of freedom. It is also right-skewed and cannot be negative. It has two degrees-of-freedom values: numerator and denominator. Order matters, because swapping them changes the distribution.

Degrees of freedom count the independent pieces of information left after you estimate parameters from the sample. Estimating the mean from the sample uses up one piece, which is why n − 1 appears so often.

Key formulas to remember

t-statistic for a mean: t = (x̄ − μ₀) ÷ (s ÷ √n); Degrees of freedom = n − 1. Use when the population variance is unknown.
Chi-square statistic for a variance: χ² = (n − 1) s² ÷ σ₀²; Degrees of freedom = n − 1. Assumes a normally distributed population. Right-skewed, never negative.
F-statistic for two variances: F = s₁² ÷ s₂²; Degrees of freedom: n₁ − 1 (numerator) and n₂ − 1 (denominator). For a two-sided test of equal variances, put the larger sample variance in the numerator and use the upper-tail critical value for α/2. For a one-sided test, the alternative hypothesis sets which variance is the numerator.
Degrees of freedom for a sample mean: df = n − 1; One parameter (the mean) is estimated from the sample.
Shape summary: t: symmetric, fat tails; χ² and F: right-skewed, ≥ 0; t approaches the standard normal as df rises.

How to solve Student's t, Chi-Square and F Distributions questions

Use this method for any question on these distributions. Most marks are lost by choosing the wrong distribution or the wrong degrees of freedom.

1Identify what is being tested: a mean, one variance, or the ratio of two variances.
2Choose the distribution: t for a mean with unknown population variance, chi-square for one variance, F for two variances.
3Compute degrees of freedom. Use n − 1 for t and chi-square. For F use n₁ − 1 and n₂ − 1 in the right order.
4Calculate the test statistic from the formula, using sample values.
5Compare with the critical value, noting one-tailed or two-tailed and the significance level.
6Reject the null hypothesis if the statistic falls beyond the critical value; otherwise fail to reject.
7For conceptual items, check the shape facts: symmetry, skew, tails, non-negativity.

Quickest way: Match the question to the distribution

When to use it: Use for conceptual items and for any item where you only need to name the distribution or its features.

Mean test, unknown variance: t. Variance test: chi-square. Ratio of variances: F.
Can the value be negative? Only t can. Chi-square and F cannot.
Symmetric means t. Right-skewed means chi-square or F.
Two df values means F. One df value means t or chi-square.
If a question says very large sample, t is close to the normal, so the critical values are nearly equal.
Use elimination: with three options, rule out any that break these facts.

Common mistakes in Student's t, Chi-Square and F Distributions

Using df = n instead of n − 1
Students forget the sample mean is estimated from the data.
Fix: For a mean or variance test, always subtract one from the sample size.
Saying the t distribution has thinner tails than the normal
Mixing up which curve is more cautious.
Fix: The t has fatter tails and a lower peak. Its critical values are larger than the normal's, especially at low df.
Treating chi-square or F as symmetric
Students assume every test distribution is bell-shaped.
Fix: Both are right-skewed and bounded below by zero. Only t is symmetric.
Swapping the F numerator and denominator degrees of freedom
Students read the sample sizes in the wrong order.
Fix: The df of the variance on top is the numerator df. Write it down before using the table.
Using a z-test when the population variance is unknown
The sample is moderately large, so students assume the normal is fine.
Fix: With unknown population variance, t is the correct choice. Use z when the population standard deviation is given. With a large sample and unknown variance, z may be used as an approximation, though t is still preferred.
Forgetting the normality assumption for the chi-square variance test
Students recall the formula but not the condition.
Fix: State that the population should be normally distributed; this test is sensitive to departures from normality.

Worked examples

Example 1

An analyst tests whether a fund's mean monthly return differs from 0.50%. A sample of 25 months gives a mean of 0.90% and a standard deviation of 2.00%. The population variance is unknown. What is the t-statistic? Options: A) 0.40, B) 1.00, C) 2.50.

Show the solution

The population variance is unknown, so use the t distribution.
Degrees of freedom = 25 − 1 = 24. You need this to find the critical value, but the question asks only for the statistic.
Standard error = 2.00 ÷ √25 = 2.00 ÷ 5 = 0.40.
t = (0.90 − 0.50) ÷ 0.40 = 0.40 ÷ 0.40 = 1.00.

Answer: B) t = 1.00.

Example 2

A sample of 20 observations has a variance of 9. Test the null that the population variance is 6. What is the chi-square statistic? Options: A) 14.2, B) 28.5, C) 38.0.

Show the solution

Use the chi-square test for a single variance.
Degrees of freedom = 20 − 1 = 19. You need this to find the critical value, but the question asks only for the statistic.
χ² = (n − 1) s² ÷ σ₀² = 19 × 9 ÷ 6.
19 × 9 = 171, and 171 ÷ 6 = 28.5.

Answer: B) χ² = 28.5.

Exam tips

Most items are conceptual: which distribution fits, its shape, or its degrees of freedom. Memorise the shape facts.
Check whether the population variance is given. If it is not, think t.
For F, read which sample variance is on top and match the df order.
When a calculation is needed, the options are in ascending order, so a quick estimate can eliminate two choices.
At about 90 seconds per question, do not hunt for table values unless the critical value is supplied.

Practice questions from Statistical Distributions for Financial Asset Prices and Returns

Student's t, Chi-Square and F Distributions in other exams

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

Student's t, Chi-Square and F Distributions: frequently asked questions

How is the t distribution different from the normal distribution?

Both are symmetric and centred at zero. The t has fatter tails and a lower peak, so its critical values are larger. As degrees of freedom increase, the t distribution converges to the standard normal.

What are degrees of freedom in simple terms?

They are the number of independent pieces of information left after estimating parameters from the sample. For a sample mean test, you estimate the mean from the data, so df = n − 1.

When do I use the chi-square and F distributions?

Use chi-square to test whether one population variance equals a stated value. Use F to compare two variances, and it also appears in regression ANOVA. Both are right-skewed and never negative.

Do I need to memorise t, chi-square or F tables?

No. Questions normally supply the critical value or ask for the statistic or the concept. Know how to compute the statistic and its degrees of freedom.