ANOVA Calculator
Perform one-way Analysis of Variance to compare means across multiple groups
Data Input
Theory & Formula
Analysis of Variance (ANOVA)
ANOVA is a statistical method used to test whether there are significant differences between the means of three or more independent groups. It partitions total variance into between-group and within-group variance.
Hypotheses
Null Hypothesis: \(H_0: \mu_1 = \mu_2 = ... = \mu_k\)
Alternative Hypothesis: \(H_1: \exists i,j: \mu_i \ne \mu_j\)
ANOVA Components
Sum of Squares Between Groups: \(\text{SSB} = \sum_{i=1}^{k} n_i(\bar{x}_i - \bar{x})^2\)
Sum of Squares Within Groups: \(\text{SSW} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (x_{ij} - \bar{x}_i)^2\)
F-Statistic: \(F = \frac{\text{MSB}}{\text{MSW}} = \frac{\text{SSB}/(k-1)}{\text{SSW}/(N-k)}\)
Interpretation
- Large F-statistic: Between-group variance is much larger than within-group variance, suggesting groups differ
- Small F-statistic: Between-group variance is similar to within-group variance, suggesting groups do not differ
- Compare the p-value with α (equivalently, the F-statistic with the critical F-value); reject H₀ when p < α.