Calculate F-statistic, p-value, and effect sizes for one-way Analysis of Variance (ANOVA). Includes post-hoc Tukey HSD test for pairwise comparisons with step-by-step calculations.
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Perform one-way ANOVA to test if there are significant differences between the means of three or more independent groups. Calculate F-statistic, p-value, effect sizes (η², ω², Cohen's f), and conduct post-hoc Tukey HSD tests with complete step-by-step calculations.
ANOVA (Analysis of Variance) is a statistical method used to compare means across three or more groups simultaneously. Unlike multiple t-tests, ANOVA controls the Type I error rate by testing whether any group means differ significantly. The method partitions total variance into between-group variance (differences among group means) and within-group variance (individual differences within groups). If between-group variance significantly exceeds within-group variance, we conclude that at least one group mean differs.
F-Statistic Formula
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Use ANOVA whenever you're comparing 3 or more groups. Running multiple t-tests inflates the Type I error rate (false positives). With 3 groups at α=0.05, three t-tests give ~14% overall error rate, while ANOVA maintains 5%. ANOVA is the standard approach for multi-group comparisons.
A significant F-test (p < α) tells you that at least one group mean differs significantly from the others, but not which specific groups differ. That's why post-hoc tests like Tukey HSD are needed to identify which pairs of groups have significant differences.
One-way ANOVA assumes: (1) Independence - observations are independent within and across groups, (2) Normality - data in each group is approximately normally distributed, (3) Homogeneity of variance - all groups have similar variances. ANOVA is robust to moderate violations of normality with larger samples.
Tukey's Honestly Significant Difference (HSD) is a post-hoc test used after a significant ANOVA to determine which specific group pairs differ. It controls the family-wise error rate while making all pairwise comparisons, making it ideal for balanced designs.
Eta-squared (η²) measures effect size - the proportion of total variance explained by group membership. Interpretation guidelines: η² < 0.01 = negligible, 0.01-0.06 = small, 0.06-0.14 = medium, > 0.14 = large effect. For example, η² = 0.20 means 20% of variance in the outcome is due to group differences.