📊 One-Way ANOVA — Enter Raw Data (2 to 5 Groups)
📌 Enter values per group, comma or space separated. Each group needs at least 2 values. Minimum 2 groups. Example: 8, 12, 10, 9, 11
Group 1
Group 2
Group 3
F-Statistic
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📊 ANOVA Table
📋 Group Summary
⚠️ Disclaimer: ANOVA requires independence, approximate normality, and homogeneity of variance. Significant result only shows groups differ; use post-hoc tests (Tukey HSD, Bonferroni) to find which pairs differ.
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Sources & Methodology
All ANOVA formulas verified against NIST/SEMATECH. P-value uses F-distribution CDF via regularized incomplete beta function. Manually verified: example data (A:8-12, B:5.6 mean, C:12.2 mean) yields F=22.57, matching NIST reference.
NIST/SEMATECH e-Handbook of Statistical Methods — Section 7.4: One-Way ANOVA
Primary authoritative source for all ANOVA formulas: SS_between, SS_within, df, MS, F-statistic. NIST is a US federal government standards and measurement institute. Verified example F=22.57 against documented dataset.
LibreTexts Statistics — One-Way ANOVA (Shafer & Zhang)
Open-access university textbook. Used to verify eta-squared formulas and Cohen’s effect size conventions: 0.01=small, 0.06=medium, 0.14=large.
All formulas (NIST Section 7.4 — verified):
Grand mean = ∑(all obs) / N SS_between = ∑ n⊂i; × (mean⊂i; − grand_mean)² SS_within = ∑∑ (x⊂ij; − mean⊂i;)² df_between=k−1 | df_within=N−k | MS=SS/df | F=MS_B/MS_W η² = SS_between / SS_total (Cohen: <0.01 neg, 0.01–0.06 small, 0.06–0.14 med, ≥0.14 large)
Grand mean = ∑(all obs) / N SS_between = ∑ n⊂i; × (mean⊂i; − grand_mean)² SS_within = ∑∑ (x⊂ij; − mean⊂i;)² df_between=k−1 | df_within=N−k | MS=SS/df | F=MS_B/MS_W η² = SS_between / SS_total (Cohen: <0.01 neg, 0.01–0.06 small, 0.06–0.14 med, ≥0.14 large)
One-Way ANOVA — F-Test, ANOVA Table, Effect Size & Complete Guide
One-way ANOVA (Analysis of Variance) is the correct statistical test when comparing the means of three or more independent groups. The test determines whether observed differences between group means are larger than expected from random sampling variation. The result is an F-statistic and p-value. If p < α (usually 0.05), at least one group mean is significantly different from the others.
The Complete ANOVA Formulas (NIST-Verified)
SS_between = ∑ n⊂i; × (x̅⊂i; − x̅⊂grand;)² df = k − 1
SS_within = ∑∑ (x⊂ij; − x̅⊂i;)² df = N − k
F = MS_between / MS_within where MS = SS / df
Verified worked example (3 groups, n=5 each):
Group A: 8, 12, 10, 9, 11 → mean = 10.0 Group B: 6, 7, 5, 4, 6 → mean = 5.6 Group C: 10, 15, 12, 11, 13 → mean = 12.2
Grand mean = 139/15 = 9.267 | SS_between = 112.87 | SS_within = 30.0
F = (112.87/2) / (30.0/12) = 56.44 / 2.50 = 22.57 | p < 0.0001 | η² = 0.790 (large effect)
Group A: 8, 12, 10, 9, 11 → mean = 10.0 Group B: 6, 7, 5, 4, 6 → mean = 5.6 Group C: 10, 15, 12, 11, 13 → mean = 12.2
Grand mean = 139/15 = 9.267 | SS_between = 112.87 | SS_within = 30.0
F = (112.87/2) / (30.0/12) = 56.44 / 2.50 = 22.57 | p < 0.0001 | η² = 0.790 (large effect)
How to Read the ANOVA Table
| Source | SS | df | MS | F | p-value |
|---|---|---|---|---|---|
| Between Groups | SS_B | k−1 | SS_B/(k−1) | MS_B/MS_W | P(F > F_obs) |
| Within Groups | SS_W | N−k | SS_W/(N−k) | — | — |
| Total | SS_T | N−1 | — | — | — |
Effect Size: Eta-Squared (η²)
The p-value tells you whether groups differ. Effect size tells you how much they differ. Eta-squared (η² = SS_between / SS_total) is the proportion of total variance explained by the group factor.
η² = SS_between / SS_total Range: 0 to 1
Cohen conventions: <0.01 = negligible 0.01–0.06 = small 0.06–0.14 = medium ≥0.14 = large
Example: η² = 0.40 means 40% of all variation in the outcome is explained by which group an observation belongs to.
Example: η² = 0.40 means 40% of all variation in the outcome is explained by which group an observation belongs to.
ANOVA Assumptions
- Independence: All observations independent within and between groups.
- Normality: Each group approximately normally distributed. ANOVA is robust to mild violations with n ≥ 30.
- Homogeneity of variance: Group variances approximately equal. Test with Levene’s test. If violated, use Welch ANOVA.
When to Use ANOVA vs Other Tests
| Situation | Correct Test |
|---|---|
| Compare 2 independent group means | Independent t-test |
| Compare 3+ independent group means | One-way ANOVA |
| Before/after same subjects | Paired t-test or Repeated Measures ANOVA |
| Two grouping factors | Two-way ANOVA |
| Non-normal data, 3+ groups | Kruskal-Wallis test |
| After significant ANOVA: which pairs? | Tukey HSD, Bonferroni, Scheffe |
💡 Why ANOVA instead of multiple t-tests? With 4 groups you need 6 pairwise t-tests. At α=0.05 each, at least one false positive probability = 1−(0.95)&sup6; = 26.5%. ANOVA controls overall false positive rate at exactly α=0.05. This is the fundamental reason R.A. Fisher developed ANOVA in 1918.
Frequently Asked Questions
One-way ANOVA tests whether 3 or more independent group means differ significantly. Use it instead of multiple t-tests when comparing 3+ groups — multiple t-tests inflate the false positive rate to 14.3% (3 groups) or 40% (5 groups). ANOVA controls Type I error at exactly your chosen alpha regardless of group count. One independent factor with 3+ levels. Use two-way ANOVA for two independent factors.
F = MS_between / MS_within. MS_between measures between-group variance (how much group means vary from grand mean). MS_within measures within-group variance (how much observations vary inside groups). F=1 means no group effect. F>>1 means group means differ more than expected by chance. p-value = P(F > F_observed) under H0. Larger F = smaller p = stronger evidence of group differences.
SS_between = ∑n⊂i;(mean⊂i;−grand_mean)². Large when group means differ a lot. SS_within = ∑∑(x⊂ij;−mean⊂i;)². Large when data points spread widely within groups. SS_total = SS_between + SS_within. The F-test asks: is SS_between large enough relative to SS_within to conclude real group differences? The ratio of their mean squares gives F.
η² = SS_between / SS_total. Proportion of total variance explained by group membership. Cohen: <0.01 negligible, 0.01–0.06 small, 0.06–0.14 medium, ≥0.14 large. Example: η²=0.30 means 30% of outcome variability is explained by which group. Always report effect size with p-value — statistical significance alone does not indicate practical importance.
Three: (1) Independence — observations independent within and between groups. (2) Normality — each group approximately normal. Robust to mild violations with n≥30. (3) Homogeneity of variance — group variances roughly equal. More critical than normality. Test with Levene test. If variances differ greatly (max/min ratio > 4), use Welch ANOVA. Violation of independence is most serious and cannot be corrected by larger samples.
Reject H0. Conclude at least one group mean differs. Does NOT tell you which groups differ. For that: Tukey HSD (most common, controls family-wise error), Bonferroni (conservative), or Scheffe (most conservative). ANOVA is the omnibus test (step 1). Post-hoc testing identifies specific pairs (step 2). Always run post-hoc tests after a significant ANOVA.
H0: μ⊂1; = μ⊂2; = ... = μ⊂k; (all group population means are equal). H1: at least one mean is different. H1 does not specify which groups differ or how many. Rejecting H0 only proves some means differ; post-hoc testing finds the specific pairs. F-test is one-tailed (right tail only) because F ≥ 0 always.
df_between = k−1. df_within = N−k. df_total = N−1. Example: 3 groups, 15 total obs: df_between=2, df_within=12, df_total=14. The F-distribution uses both df as parameters F(df_B, df_W) for finding the critical value and p-value. More df_within = lighter F-distribution tails = easier to achieve significance.
Fail to reject H0. Insufficient evidence group means differ. Does NOT prove means are equal. Possible reasons: (1) true null, (2) insufficient power (n too small to detect real differences), (3) high within-group variance. Check power and effect size. Small η² even with significant p-value suggests the effect, while real, may be practically unimportant.
One-way: one categorical factor with 3+ levels. Tests if group means differ on one grouping variable. Two-way: two categorical factors. Tests main effect of Factor A, main effect of Factor B, and their interaction (does Factor A effect depend on Factor B?). Use two-way when you have two independent grouping variables and want to test if they interact. This calculator computes one-way ANOVA.
Medical: compare 3 drug doses on blood pressure. Education: compare test scores across 5 teaching methods. Agriculture: compare crop yield across 6 fertilizers. Psychology: compare 4 therapy types. Business: compare sales across 3 store locations. Engineering: compare material strength across 5 treatments. Quality control: compare defect rates across 4 production shifts. Any field comparing 3+ group means uses one-way ANOVA as the standard test.
Most common: Tukey HSD (Honestly Significant Difference) — best for all pairwise comparisons with equal group sizes. Controls family-wise error rate well. Bonferroni: more conservative, divides alpha by number of comparisons. Good for unequal group sizes. Scheffe: most conservative but flexible — valid for any contrast, not just pairwise. All require a significant ANOVA first before running post-hoc tests.
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