ANOVA Calculator (One-Way)

Perform one-way Analysis of Variance to compare means across multiple groups. Determine if at least one group differs significantly. Includes F-test, effect size, and detailed statistical interpretation.

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ANOVA Formulas: F-Statistic: F = MSB / MSW Mean Square Between: MSB = SSB / df_between df_between = k - 1 (k = number of groups) Mean Square Within: MSW = SSW / df_within df_within = N - k (N = total observations) Sum of Squares: SST = Σ(X - Grand Mean)^2 (total variation) SSB = Σ n_i(X̄_i - Grand Mean)^2 (between groups) SSW = SST - SSB (within groups) Effect Size (Eta-squared): η^2 = SSB / SST Decision Rule: If F > F_critical → Reject H₀ (significant) If p-value < alpha → Reject H₀
Example 1 (Teaching Methods): Compare test scores from 3 teaching methods Group 1 (Method A): 85, 87, 83, 86, 84 Group 2 (Method B): 92, 90, 91, 93, 89 Group 3 (Method C): 78, 80, 79, 81, 77 Calculations: Group 1: n=5, Mean=85, SD=1.58 Group 2: n=5, Mean=91, SD=1.58 Group 3: n=5, Mean=79, SD=1.58 Grand Mean = (425 + 455 + 395) / 15 = 85 SSB = 5[(85-85)^2 + (91-85)^2 + (79-85)^2] = 360 SSW = 4(1.58^2) * 3 = 30 SST = 390 F = (360/2) / (30/12) = 180/2.5 = 72 df1=2, df2=12, F_crit(0.05)=3.89 Result: F=72 > 3.89 → HIGHLY SIGNIFICANT Conclusion: Teaching methods produce significantly different results. Method B appears best. Example 2 (Drug Dosages): Group 1: 5, 6, 7, 5, 8 Group 2: 8, 9, 7, 10, 9 Group 3: 6, 7, 8, 7, 9 Mean1=6.2, Mean2=8.6, Mean3=7.4 Grand Mean=7.4 F=4.87, df1=2, df2=12 p < 0.05 → Significant difference in dosages

What is ANOVA?

ANOVA (Analysis of Variance) is a statistical test that compares means across three or more groups to determine if at least one group mean differs significantly. Instead of multiple t-tests, ANOVA analyzes variance within groups vs between groups. F-statistic measures this ratio: larger F suggests group means differ.

How does ANOVA work?

ANOVA partitions total variance into: (1) Between-group variance (differences among group means), (2) Within-group variance (variation within each group). F-ratio = Between-group variance / Within-group variance. Large F means group differences exceed random variation. If F > critical value, at least one group differs significantly.

What is the F-statistic in ANOVA?

F-statistic = MSB / MSW (Mean Square Between / Mean Square Within). MSB measures variance between group means. MSW measures average variance within groups. F ≈ 1 suggests no group differences. Large F suggests significant differences. Example: F=5.2 with df1=2, df2=27, p<0.05 indicates significant group differences.

What are degrees of freedom in ANOVA?

Two degrees of freedom: df1 (between) = k - 1 (k = number of groups), df2 (within) = N - k (N = total observations). Example: 3 groups with 10 observations each gives df1 = 3-1 = 2, df2 = 30-3 = 27. These determine the F-distribution for significance testing.

What assumptions must be met for ANOVA?

Three key assumptions: (1) Independence: observations are independent, (2) Normality: data in each group approximately normally distributed, (3) Homogeneity of variance: groups have similar variances (Levene's test checks this). Violating assumptions can invalidate results. Consider non-parametric Kruskal-Wallis test if assumptions fail.

How do you interpret ANOVA results?

If p-value < alpha (usually 0.05): Reject null hypothesis → at least one group mean differs significantly. If p >= alpha: Fail to reject → no significant differences detected. ANOVA doesn't tell which groups differ; use post-hoc tests (Tukey, Bonferroni) to identify specific group differences.

What is the difference between one-way and two-way ANOVA?

One-way ANOVA has one independent variable (factor) with multiple levels. Example: comparing test scores across 3 teaching methods. Two-way ANOVA has two independent variables and tests main effects and interaction. Example: teaching method AND class size effects on test scores. This calculator performs one-way ANOVA.

What are post-hoc tests and when are they needed?

Post-hoc tests (Tukey HSD, Bonferroni, Scheffé) identify which specific groups differ after significant ANOVA. ANOVA only tells "at least one differs" but not which ones. Post-hoc tests make pairwise comparisons while controlling Type I error. Only perform after significant ANOVA F-test.

What is Sum of Squares (SS) in ANOVA?

Sum of Squares measures variation: SST (total) = total variation in data, SSB (between) = variation due to group differences, SSW (within) = variation within groups. SST = SSB + SSW. Larger SSB relative to SSW indicates groups differ. Example: SSB=100, SSW=50 suggests strong group effect.

What are real-world applications of ANOVA?

Medical research (comparing treatments), agriculture (crop yield across fertilizers), manufacturing (quality control across machines), education (teaching methods comparison), psychology (therapy effectiveness), marketing (ad campaign performance), clinical trials (drug dosages), sports science (training programs). Any comparison of 3+ groups.