One-Way Analysis of Variance with Step-by-Step Derivation
Enter group data (each group separated by semicolon)
Example: 85,90,88,92,87;78,82,80,85,79;92,95,90,98,94 for 3 groups
Results
F-Statistic
p-value
df Between
df Within
Critical Value (α=0.05)
ANOVA Table
Source
SS
df
MS
F
Between Groups
Within Groups
Total
Step-by-Step Derivation
ANOVA Formula
SS_total = Σ(xᵢⱼ - x̄_grand)²
SS_between = Σnⱼ(x̄ⱼ - x̄_grand)²
SS_within = Σ(xᵢⱼ - x̄ⱼ)²
F = MS_between / MS_within, MS = SS/df
One-way ANOVA tests if means of three or more independent groups are significantly different.
⚠ANOVA requires at least 3 groups. If significant, post-hoc tests (like Tukey HSD) can identify which groups differ.
What is ANOVA?
Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. One-way ANOVA compares means from three or more independent groups to determine if at least one group mean is significantly different.
Between-Group Variance
Variance between group means - how much groups differ from each other
Within-Group Variance
Variance within groups - how much individuals differ within same group
F-Statistic
Ratio of between-group variance to within-group variance
Null Hypothesis
H₀: μ₁ = μ₂ = μ₃ = ... = μₖ (all group means are equal)
💡 Example: 3 groups of test scores: [85,90,88,92,87], [78,82,80,85,79], [92,95,90,98,94]. Calculate F-statistic to compare means.
Analysis of Variance (ANOVA) tests if there is a statistically significant difference between means of three or more independent groups using F-test.
How to interpret F-statistic?▼
F-statistic compares between-group variance to within-group variance. Higher F = more likely group means differ. Compare to critical value or use p-value.
What are null and alternative hypotheses?▼
H₀: All group means are equal. H₁: At least one group mean is different. If p < α, reject H₀ and conclude there is a significant difference.
Assumptions for ANOVA?▼
Independence of observations, normality (each group normally distributed), and homogeneity of variances (equal variances).
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