ANOVA F-Test Calculator
Compute the F-statistic for a one-way ANOVA test by entering sums of squares and degrees of freedom. Use it when comparing means across three or more groups to test whether at least one differs significantly.
About this calculator
Analysis of Variance (ANOVA) tests whether the means of three or more groups differ more than expected by chance. It partitions total variability into two sources: variation between groups (explained by group membership) and variation within groups (random error). The mean square between groups is MSB = SSB / dfB, and the mean square within groups is MSW = SSW / dfW. The F-statistic is their ratio: F = MSB / MSW = (SSB / dfB) / (SSW / dfW). A large F means between-group variation greatly exceeds within-group noise, suggesting the group means are not all equal. Degrees of freedom between equals k − 1 (where k is the number of groups) and degrees of freedom within equals N − k (where N is total observations). The computed F is compared to a critical value from the F-distribution at significance level α; if F exceeds the critical value, the null hypothesis of equal means is rejected.
How to use
Suppose three groups yield SSB = 84, SSW = 108, dfB = 2, dfW = 27. Step 1: MSB = 84 / 2 = 42. Step 2: MSW = 108 / 27 = 4. Step 3: F = 42 / 4 = 10.5. Enter these values into the calculator. At α = 0.05 with dfB = 2 and dfW = 27, the critical F-value is approximately 3.35. Since 10.5 > 3.35, we reject the null hypothesis and conclude that at least one group mean differs significantly from the others.
Frequently asked questions
What does the F-statistic in ANOVA actually measure?
The F-statistic is the ratio of between-group variance to within-group variance. A value of 1.0 means the groups vary no more than random samples from the same population would. Values substantially above 1 indicate that group membership explains a meaningful portion of total variance. The larger the F, the more confident you can be that at least one group mean differs. The exact threshold depends on the degrees of freedom and the chosen significance level α.
What are the assumptions required for a valid one-way ANOVA test?
One-way ANOVA assumes: (1) independence — observations are not related across or within groups; (2) normality — the data within each group are approximately normally distributed, though ANOVA is robust to mild violations with large samples; (3) homogeneity of variance (homoscedasticity) — all groups have similar variances, verifiable with Levene's test. Violating independence is the most serious problem and cannot be corrected post hoc. If normality or equal variance assumptions fail, consider the Kruskal-Wallis non-parametric alternative.
How do I interpret ANOVA results when the F-test is significant?
A significant F-test only tells you that not all group means are equal — it does not specify which pairs differ. To identify which groups differ, you must run post-hoc tests such as Tukey's HSD, Bonferroni correction, or Scheffé's method. These control the family-wise error rate inflated by multiple comparisons. Effect size for ANOVA is typically reported as η² (eta-squared) = SSB / SS_total, where values of 0.01, 0.06, and 0.14 are considered small, medium, and large respectively.