Actuarial Statistics · Random sampling and sampling distributions

Chi-Square, t and F Distributions from Normal Samples

Updated 11 October 2026 · Fact-checked

These three distributions come from normal samples. Chi-square describes sums of squared standard normals, so (n-1)S²/σ² ~ χ²(n-1). The t distribution arises when you replace σ by S in the standardised mean. The F distribution is the ratio of two independent chi-squares, each divided by its degrees of freedom.

Understand Chi-square, t and F Distributions

Start with the standard normal Z. If you square it, you get a chi-square variable with 1 degree of freedom. If you add k independent squared standard normals, you get χ²(k). It is a Gamma distribution with shape k/2 and rate 1/2. Its mean is k and its variance is 2k.

Now take a random sample X₁,…,Xₙ from N(μ, σ²). The sample mean X̄ and the sample variance S² (divisor n-1) have two key properties. First, X̄ and S² are independent. This holds for normal samples only. Second, (n-1)S²/σ² ~ χ²(n-1). You lose one degree of freedom because the deviations from X̄ must sum to zero.

Why is it n-1? The sum Σ(Xᵢ-μ)²/σ² is χ²(n). It splits as Σ(Xᵢ-X̄)²/σ² plus n(X̄-μ)²/σ². The second part is the square of a standard normal, so it is χ²(1). The two parts are independent, so you can compare MGFs: (1-2t)^(-n/2) = M(t)·(1-2t)^(-1/2). This gives M(t) = (1-2t)^(-(n-1)/2), which is the MGF of χ²(n-1). That is the proof examiners expect you to follow.

The t distribution handles the case where σ is unknown. Z = (X̄-μ)/(σ/√n) is N(0,1). Replace σ by S and you get T = (X̄-μ)/(S/√n). Equivalently, T = Z ÷ √(W/ν), where Z ~ N(0,1) and W ~ χ²(ν) are independent. Here ν = n-1. The t has the same bell shape as the normal, but heavier tails. It tends to the normal as ν grows. The mean is 0 for ν > 1 and the variance is ν/(ν-2) for ν > 2.

The F distribution compares variances. If U ~ χ²(m) and V ~ χ²(n) are independent, then F = (U/m)/(V/n) ~ F(m, n). For two independent normal samples, (S₁²/σ₁²)/(S₂²/σ₂²) ~ F(n₁-1, n₂-1). If σ₁² = σ₂², it is just S₁²/S₂². Also, 1/F(m, n) ~ F(n, m), and T² ~ F(1, ν) when T ~ t(ν).

Key rules to remember

Chi-square definition: χ²(k) = Z₁² + Z₂² + … + Z_k², Zᵢ independent N(0,1); Mean k, variance 2k. It is Gamma(k/2, 1/2).
Sample variance: S² = Σ(Xᵢ - X̄)² ÷ (n - 1); Divisor n-1. Check which divisor a question uses.
Distribution of scaled sample variance: (n - 1)S²/σ² ~ χ²(n - 1); Needs a random sample from a normal population. Then E[S²] = σ² and Var(S²) = 2σ⁴/(n-1).
Independence: X̄ and S² are independent; True for normal samples. Not true in general.
Chi-square MGF: M(t) = (1 - 2t)^(-k/2), for t < 1/2; Use it to prove sums of independent chi-squares are chi-square with added degrees of freedom.
t distribution definition: T = Z ÷ √(W/ν) ~ t(ν), Z ~ N(0,1), W ~ χ²(ν) independent; Mean 0 for ν > 1. Variance ν/(ν-2) for ν > 2.
t statistic for the mean: (X̄ - μ) ÷ (S/√n) ~ t(n - 1); Use when σ is unknown and the population is normal.
F distribution definition: F = (U/m) ÷ (V/n) ~ F(m, n), U ~ χ²(m), V ~ χ²(n) independent; Order of degrees of freedom matters: numerator first.
Ratio of sample variances: (S₁²/σ₁²) ÷ (S₂²/σ₂²) ~ F(n₁ - 1, n₂ - 1); Two independent normal samples.
Useful links: 1/F(m, n) ~ F(n, m); T² ~ F(1, ν) if T ~ t(ν); F_α(m, n) = 1 ÷ F_(1-α)(n, m); Here F_α denotes the upper α point. Use the reciprocal rule to read lower points from tables.

How to solve Chi-square, t and F Distributions questions

Use this method for any question on these distributions. It works for probability, test and interval questions alike.

1Check the assumptions. Is the sample random and from a normal population? If not, say what approximation you use.
2Identify what is known. Is σ known or unknown? One sample or two? Mean or variance?
3Pick the pivot: Z if σ is known, t(n-1) if σ is unknown for the mean, χ²(n-1) for one variance, F(n₁-1, n₂-1) for a ratio of variances.
4Compute the statistic from the data. Write the degrees of freedom clearly.
5Rewrite the event in terms of the pivot. For a variance event P(S² > c), this means P(χ²(n-1) > (n-1)c/σ²).
6Read the table or compute in R. For lower tail points of F, use the reciprocal rule.
7State the conclusion in words and give the final number with the right inequality direction.

Quickest way: Match the question to the pivot in seconds

When to use it: Use this in MCQs and when you are short of time on written parts.

Look for the unknown: a mean with σ unknown means t; a variance means χ²; two variances mean F.
Degrees of freedom are n-1 for each sample variance. Do not use n.
For χ² probabilities, multiply S² by (n-1)/σ² first.
For moments, use the shortcuts: Var of χ²(k) is 2k; Var of S² is 2σ⁴/(n-1); t(ν) variance is ν/(ν-2).
If the table gives only upper points of F, flip the degrees of freedom and take the reciprocal.

Common mistakes in Chi-square, t and F Distributions

Using n instead of n-1 as the degrees of freedom for (n-1)S²/σ² or for the t statistic.
Students remember the sample size and forget that X̄ is estimated from the same data.
Fix: Write the pivot with its degrees of freedom before reading any table: χ²(n-1) or t(n-1).
Using the normal table when σ is unknown and n is small.
The t and normal curves look alike, so students treat them as the same.
Fix: If S replaces σ, use t(n-1). The t has heavier tails, so its critical values are larger than the normal ones.
Claiming X̄ and S² are independent for any distribution.
Students remember the result without its condition.
Fix: State that independence holds for a random sample from a normal distribution. Say so in written answers.
Reversing the numerator and denominator degrees of freedom in F(m, n).
Tables are read by row and column and the order is easy to swap.
Fix: Numerator d.f. belongs to the variance in the numerator of the ratio. Write F(n₁-1, n₂-1) explicitly.
Forgetting to scale S² by (n-1)/σ² before using chi-square tables.
Students treat S² itself as chi-square.
Fix: S² is not chi-square. Only (n-1)S²/σ² is. Convert first, then look up.
Thinking the chi-square distribution is symmetric, so using ±critical values.
Habit from the normal and t distributions.
Fix: Chi-square and F are skewed right. Lower and upper points must be read separately.

Worked examples

Example 1

A random sample of n = 10 is taken from N(μ, σ²) with σ² = 4. Find the probability that the sample variance S² exceeds 7.5 (given that for χ²(9), the upper 25% point is 11.39, the upper 10% point is 14.68 and the upper 5% point is 16.92).

Show the solution

The pivot is 9S²/σ² ~ χ²(9).
S² > 7.5 is equivalent to 9S²/4 > 9 × 7.5 ÷ 4.
Compute 9 × 7.5 ÷ 4 = 67.5 ÷ 4 = 16.875.
So we need P(χ²(9) > 16.875).
The upper 10% point is 14.68 and the upper 5% point is 16.92. Since 14.68 < 16.875 < 16.92, the probability lies between 0.05 and 0.10.
Because 16.875 is just below the 5% point 16.92, the probability is slightly above 0.05. The tables given cannot support a more precise figure. In R, pchisq(16.875, 9, lower.tail = FALSE) gives the exact value.

Answer: P(S² > 7.5) = P(χ²(9) > 16.875), which is slightly above 0.05 (between 0.05 and 0.10), because 16.875 lies just below the 5% point 16.92.

Exam tips

Write the assumption 'random sample from a normal population' in every written answer. Examiners award a mark for it.
Learn the MGF proof of (n-1)S²/σ² ~ χ²(n-1). It is a standard written question.
In MCQs, check the degrees of freedom before the arithmetic. Wrong d.f. options are common distractors.
In the Paper B computer exam, use R functions such as pchisq, qt and qf. Show the function call, its arguments and the interpretation.
For F tables that give only upper points, practise the reciprocal rule until it is automatic.

Practice questions from Random sampling and sampling distributions

Chi-square, t 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.

Chi-square, t and F Distributions: frequently asked questions

Why is (n-1)S²/σ² chi-square with n-1 degrees of freedom?

The sum Σ(Xᵢ-μ)²/σ² is χ²(n). It splits into Σ(Xᵢ-X̄)²/σ² and n(X̄-μ)²/σ², which are independent. The second part is χ²(1). Dividing MGFs leaves χ²(n-1) for the first part.

What is the difference between the t and normal distributions?

Both are symmetric about 0 and bell-shaped. The t has heavier tails because S varies from sample to sample. As the degrees of freedom increase, t(ν) approaches N(0,1).

How do I use t tables when the variance is unknown?

Compute T = (X̄-μ)/(S/√n). Use n-1 degrees of freedom. Find the row for the d.f. and the column for the tail probability. For a two-sided test at 5%, use the 2.5% upper point.

When do I use the F distribution?

Use it for the ratio of two independent sample variances from normal populations. Under equal population variances, S₁²/S₂² ~ F(n₁-1, n₂-1). It is also used in regression ANOVA.