The ANOVA (Analysis of Variance) Calculator tests whether the means of three or more groups differ significantly. It performs one-way ANOVA with a full sum of squares table, two-group comparison, and post-hoc Tukey HSD tests to identify which specific pairs differ.
Enter your group data (one group per line, values separated by commas) and set the significance level. The calculator produces the F-statistic, p-value, degrees of freedom, effect size (eta squared), and a complete ANOVA summary table with step-by-step working.
Your calculations will appear here
Analysis of Variance (ANOVA) tests whether the means of three or more groups are significantly different. It compares the variance between group means to the variance within groups. If the between-group variance is much larger than the within-group variance, the group means likely differ.
The F-statistic is the ratio of Mean Square Between (MSB) to Mean Square Within (MSW). MSB measures how much the group means vary around the grand mean. MSW measures the average variability within each group. A large F-statistic suggests the group means are not all equal.
The p-value gives the probability of observing an F-statistic as extreme as the calculated value, assuming all group means are equal (the null hypothesis). If p is less than the significance level (typically 0.05), we reject the null hypothesis and conclude that at least one group mean differs.
Eta squared (eta squared = SSB / SST) is a measure of effect size. It represents the proportion of total variance explained by group membership. Values of 0.01, 0.06, and 0.14 are conventionally interpreted as small, medium, and large effects respectively.
When ANOVA is significant, post-hoc tests such as Tukey HSD identify which specific pairs of groups differ. The Tukey HSD test controls the family-wise error rate, keeping the overall Type I error at the chosen alpha level across all pairwise comparisons.
Problem: Group 1: 23, 25, 28, 22, 27. Group 2: 30, 32, 29, 35, 31. Group 3: 18, 20, 22, 19, 21. Are the means significantly different at alpha = 0.05?
Solution: Grand mean = 25.4. SSB = 310, SSW = 62, SST = 372. MSB = 155, MSW = 5.167. F = 30.0. df = (2, 12). p < 0.001.
Answer: F = 30.0, p < 0.001. Reject H0, at least one group mean differs significantly. Eta squared = 0.833.
Problem: Group A: 5, 6, 7, 8. Group B: 10, 11, 12, 13. Is there a significant difference?
Solution: Mean A = 6.5, Mean B = 11.5. F = 25.0. df = (1, 6). p = 0.0024.
Answer: F = 25.0, p = 0.0024. The means are significantly different.
Problem: After finding a significant ANOVA with three groups, which pairs are different?
Solution: Calculate the HSD threshold using q critical, MSE, and group sizes. Compare each pair of means against the threshold.
Answer: Pairs where the absolute mean difference exceeds the HSD threshold are significantly different.
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