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The One-Way ANOVA (Analysis of Variance) Calculator tests whether the means of three or more independent groups differ significantly. ANOVA is one of the most widely used statistical methods in scientific research, comparing variability between groups to variability within groups to produce an F-statistic.
This calculator computes the complete ANOVA table including sum of squares, degrees of freedom, mean squares, and the F-statistic for up to three groups with up to five observations each.
One-way ANOVA partitions the total variability in data into two components:
The core formulas are:
$$SS_{Between} = \sum_{i=1}^{k} n_i (\bar{X}_i - \bar{X}_{grand})^2$$
$$SS_{Within} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2$$
$$MS_{Between} = \frac{SS_{Between}}{k - 1}, \quad MS_{Within} = \frac{SS_{Within}}{N - k}$$
$$F = \frac{MS_{Between}}{MS_{Within}}$$
Where \(k\) is the number of groups, \(n_i\) is the size of group \(i\), \(N\) is the total sample size, \(\bar{X}_i\) is the mean of group \(i\), and \(\bar{X}_{grand}\) is the overall mean. A large F-statistic indicates that between-group variability exceeds within-group variability, suggesting the group means are not all equal.
The null hypothesis states that all group means are equal (\(H_0: \mu_1 = \mu_2 = \mu_3\)). If F exceeds the critical value from the F-distribution at the chosen significance level, we reject \(H_0\).
Interpreting the ANOVA results involves examining several components:
Compare your F-statistic to critical F-values from an F-distribution table using dfbetween and dfwithin. For example, with df(2, 6), the critical F at α = 0.05 is approximately 5.14.
Inputs
Results
Three groups of students taught by different methods. The large F-statistic (22.5) strongly suggests significant differences among the methods.
Inputs
Results
Plant heights measured under three fertilizer conditions. F = 16.0 indicates significant differences in growth across fertilizer types.
One-way ANOVA assumes: (1) Independence — observations are independent of each other, (2) Normality — data within each group are approximately normally distributed, and (3) Homogeneity of variances — all groups have roughly equal variance (testable via Levene's test). ANOVA is robust to moderate violations of normality with equal sample sizes.
A significant F-statistic only tells you that at least one group mean differs from the others — it does not specify which groups differ. To identify specific pairwise differences, you need post-hoc tests such as Tukey's HSD, Bonferroni correction, or Scheffé's method.
When comparing three or more groups, performing multiple t-tests inflates the Type I error rate. With k groups, you would need k(k−1)/2 pairwise comparisons. ANOVA controls the overall error rate by testing all groups simultaneously in a single omnibus test.
F = MSBetween/MSWithin. An F close to 1 suggests no difference between group means. As F increases, evidence against the null hypothesis grows. Compare F to the critical value from F-distribution tables at your chosen α level with the appropriate degrees of freedom.
Eta-squared (η²) = SSBetween / SSTotal is a measure of effect size representing the proportion of total variance explained by group membership. Values of 0.01, 0.06, and 0.14 are considered small, medium, and large effect sizes respectively.
Yes, one-way ANOVA works with unequal group sizes (unbalanced designs). However, unequal sizes reduce statistical power and make the test more sensitive to violations of the homogeneity of variance assumption. When variances are unequal with unbalanced groups, consider Welch's ANOVA instead.
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