ANOVA Calculator

Understanding ANOVA

Analysis of Variance (ANOVA) is a statistical method used to compare means of three or more groups simultaneously.

One-Way ANOVA

Tests whether there are significant differences between group means for a single independent variable.

Hypotheses:

• H₀ (Null): μ₁ = μ₂ = μ₃ = ... (all group means are equal)
• H₁ (Alternative): At least one group mean differs

ANOVA Table Components

Where: k = number of groups, N = total sample size

Key Formulas

Interpreting Results

• If p-value ≤ α: Reject H₀ (at least one group mean differs significantly)
• If p-value > α: Fail to reject H₀ (no significant differences between group means)
• Larger F-statistic: Indicates greater variation between groups relative to within groups

Assumptions

• Independence: Observations are independent within and between groups
• Normality: Data in each group are approximately normally distributed
• Homogeneity of Variance: All groups have equal variances (Levene's test can check this)

Post-Hoc Tests

If ANOVA is significant, conduct post-hoc tests to determine which specific groups differ:

• Tukey HSD: Controls family-wise error rate, compares all pairs
• Bonferroni: Conservative, adjusts α for multiple comparisons
• Scheffé: Most conservative, allows any contrast

Effect Size: Eta-Squared (η²)

• Small effect: η² ≈ 0.01
• Medium effect: η² ≈ 0.06
• Large effect: η² ≈ 0.14

Real-World Applications

• Medicine: Compare effectiveness of multiple treatments
• Psychology: Test differences across experimental conditions
• Agriculture: Compare crop yields under different fertilizers
• Education: Evaluate teaching methods across multiple classrooms
• Manufacturing: Compare product quality from different production lines