Question 1

What is the F-distribution used for?

Accepted Answer

The F-distribution is used in: (1) One-way ANOVA: F = MS_between / MS_within tests whether group means are equal. (2) Two-way ANOVA and factorial designs. (3) Regression F-test: overall test of whether any predictors explain variance. (4) Equality of variances: F = s₁²/s₂² tests if two population variances are equal. (5) Nested models: comparing restricted vs full models. Any ratio of two variance estimates from normally distributed data follows an F-distribution.

Question 2

How do I interpret the F-statistic?

Accepted Answer

F = (explained variance) / (unexplained variance). F = 1 means explained and unexplained variance are equal (consistent with H₀). F >> 1 means the explained variance greatly exceeds what you would expect by chance. The p-value P(F(d1,d2) > observed F) measures how likely such a large F would be if H₀ were true. p < 0.05 means the result is statistically significant at α = 0.05.

Question 3

How are degrees of freedom determined in ANOVA?

Accepted Answer

In one-way ANOVA with k groups and N total observations: d1 = k − 1 (numerator = between-groups df). d2 = N − k (denominator = within-groups df). Total df = N − 1 = d1 + d2. Example: 3 groups of 10 subjects each (N=30): d1 = 2, d2 = 27. In two-way ANOVA, d1 differs for each factor and interaction. In regression with p predictors and N observations: d1 = p, d2 = N − p − 1.

Question 4

What are F-distribution critical values?

Accepted Answer

The critical value F*(d1, d2, α) is the value such that P(F > F*) = α. If the observed F exceeds F*, you reject H₀ at significance level α. Common critical values: For F(2, 27) at α = 0.05: F* ≈ 3.35. For F(5, 100) at α = 0.05: F* ≈ 2.31. Higher d1 or d2 (more data) reduces the critical value, making it easier to achieve significance.

Question 5

What is the relationship between F, t, and chi-square distributions?

Accepted Answer

F(1, d2) = t(d2)² — the square of a t-distributed variable with d2 df follows F(1, d2). F(d1, ∞) = χ²(d1) / d1 — as d2 → ∞, d1×F → chi-square. F(1, ∞) = z² where z is standard normal. These relationships mean that t-tests and chi-square tests are special cases of the F-test, and the F-distribution table can be used to look up p-values for all of them.

Question 6

Is the F-test robust to non-normality?

Accepted Answer

The ANOVA F-test is moderately robust to violations of normality, especially with larger sample sizes (CLT applies) and roughly equal group sizes. It is more sensitive to violations of homogeneity of variance (equal variances assumption), particularly with unequal group sizes. For non-normal data, consider the Kruskal-Wallis non-parametric test. For unequal variances with normal data, use Welch's ANOVA.

Test	d1 (numerator)	d2 (denominator)	Formula
One-way ANOVA	k − 1	N − k	k = groups, N = total obs
Two-way ANOVA (factor A)	a − 1	N − ab	a = levels of A
Regression (overall)	p	N − p − 1	p = predictors
Equality of 2 variances	n₁ − 1	n₂ − 1	Test: F = s₁²/s₂²
Levene's test	k − 1	N − k	Same as ANOVA

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