ANOVA (Analysis of Variance) Calculator
Use this ANOVA calculator to test whether the means of three independent groups are significantly different. Enter each group's observations as comma-separated numbers (for example: 10, 12, 14).
This tool performs a one-way ANOVA, reporting between-group and within-group sums of squares, mean squares, the F statistic, and the associated p-value. Results update automatically as you type.
How it works
One-way ANOVA partitions the total variability in the data into variability between groups (SSB) and variability within groups (SSW). The test statistic is:
F = MSB / MSW = (SSB / (k - 1)) / (SSW / (N - k))Where k is the number of groups and N is the total number of observations. A large F suggests that group means differ more than expected by chance. The p-value is computed from the F distribution with (k - 1, N - k) degrees of freedom.
Example
Suppose you have three groups of observations:
- Group 1: 10, 12, 14
- Group 2: 15, 17, 19
- Group 3: 20, 22, 24
Enter these into the calculator. It will compute SSB, SSW, MSB, MSW, the F statistic, and the p-value automatically. If p < 0.05 (typical alpha), you would conclude the group means are significantly different.
Assumptions
- Observations are independent within and across groups.
- Each group is normally distributed (robust for moderate deviations when sample sizes are similar).
- Homogeneity of variances (groups have similar variances).
If assumptions are violated, consider transformations, non-parametric alternatives (e.g., Kruskal-Wallis), or Welch's ANOVA for unequal variances.
Interpreting results
Key outputs:
- SSB (Between): Variability due to differences between group means.
- SSW (Within): Variability within each group.
- MSB and MSW: Mean squares (SS divided by degrees of freedom).
- F statistic: Ratio MSB/MSW. Higher values suggest stronger evidence against the null hypothesis (that all group means are equal).
- p-value: Probability of observing an F at least as extreme as the computed one under the null. Small p (e.g., < 0.05) indicates statistical significance.
Frequently Asked Questions
Can I input different numbers of observations per group?
Yes. One-way ANOVA supports unequal sample sizes. The calculator automatically accounts for the different group sizes when computing degrees of freedom and mean squares.
What if a group has non-numeric entries or blanks?
Non-numeric tokens and empty entries are ignored. Make sure each observation is a valid number. If a group becomes empty after filtering invalid entries, it will be excluded from the analysis.
How do I decide significance?
Compare the p-value to your alpha (commonly 0.05). If p < alpha, you reject the null hypothesis that all group means are equal. Remember to check assumptions and consider post-hoc tests to identify which groups differ.
Post-hoc testing
ANOVA tells you if at least one group mean differs, but not which pairs differ. If ANOVA is significant, use post-hoc comparisons (e.g., Tukey's HSD) to identify specific group differences while controlling family-wise error.
References
- Field, A., Miles, J., & Field, Z. (2012). Discovering Statistics Using R.
- Any standard statistics textbook covering one-way ANOVA and F-distribution.