F-Table

What Is an F-Table?

An F-table displays critical values for the F-distribution, which is the ratio of two chi-square distributions divided by their respective degrees of freedom. The F-distribution is primarily used in Analysis of Variance (ANOVA) and tests comparing two variances.

$F = \frac{\chi^2_1/df_1}{\chi^2_2/df_2} = \frac{s_1^2/\sigma_1^2}{s_2^2/\sigma_2^2}$

Understanding the Two Degrees of Freedom

The F-distribution has two degrees of freedom parameters:

• df₁ (numerator): Associated with the variance in the numerator
• df₂ (denominator): Associated with the variance in the denominator

In ANOVA context:

• df₁ = number of groups - 1 (between-group variation)
• df₂ = total N - number of groups (within-group variation)

How to Read the F-Table

1. Select the appropriate α level (0.05, 0.01, etc.)
2. Find your df₁ (numerator) in the column headers
3. Find your df₂ (denominator) in the row labels
4. The intersection gives the critical F-value

Example: For a one-way ANOVA with 3 groups and 30 total observations at α = 0.05:

• df₁ = 3 - 1 = 2
• df₂ = 30 - 3 = 27
• Critical F(2, 27) ≈ 3.35

When to Use the F-Distribution

Analysis of Variance (ANOVA)

Testing whether group means differ significantly:

$F = \frac{MS_{between}}{MS_{within}} = \frac{\text{Between-group variance}}{\text{Within-group variance}}$

Variance Ratio Test

Comparing two population variances:

$F = \frac{s_1^2}{s_2^2}$

where s₁² > s₂² (larger variance in numerator).

Regression Analysis

Testing overall model significance:

$F = \frac{MS_{regression}}{MS_{residual}}$

Key Properties

• Always positive: F values are ≥ 0
• Right-skewed: Especially for small degrees of freedom
• Not symmetric: F(df₁, df₂) ≠ F(df₂, df₁)
• Reciprocal relationship: F₁₋α(df₁, df₂) = 1/Fα(df₂, df₁)

Common ANOVA Critical Values (α = 0.05)

Decision Rules

For a right-tailed test (typical for ANOVA):

• If F_calculated > F_critical: Reject H₀
• If F_calculated ≤ F_critical: Fail to reject H₀

For a two-tailed variance ratio test:

• Use both F_α/2 and F₁₋α/2 critical values
• Reject if F falls in either tail

Example: One-Way ANOVA

Testing whether teaching methods affect exam scores:

• 4 methods, 10 students each (N = 40)
• df₁ = 4 - 1 = 3
• df₂ = 40 - 4 = 36
• At α = 0.05, F_critical ≈ 2.87
• If F_calculated = 4.52, reject H₀ (methods differ)

Related Tools

• One-Way ANOVA Calculator — Compare multiple groups
• Two-Way ANOVA Calculator — Two factors
• F-Distribution Calculator — Find probabilities
• Levene's Test Calculator — Test variance equality

Summary

The F-table is fundamental for ANOVA and variance comparison tests. Unlike distributions with a single degrees of freedom parameter, the F-distribution requires both numerator and denominator degrees of freedom. Critical values depend on both parameters, so tables are typically organized by significance level with separate pages or tabs for different α values. Understanding how to navigate the F-table—matching both df parameters to your analysis—is essential for proper hypothesis testing in experimental and observational studies.