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ANOVA Calculator

The ANOVA calculator tests whether the means of two or more groups differ. Enter one line of measurements per group; you get the F value, p-value, ANOVA table and the effect size η². Example: three groups with means 5, 9 and 10 give F = 9.26 and p = 0.0024.

Source: 7.4.3, NIST/SEMATECH e-Handbook – One-factor ANOVA. Updated: .

One line per group, separate numbers with spaces (decimal point or comma). At least 2 groups with 2 values each.
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Result

Test result
F(2; 15) = 9.265; p = 0.002399
Decision
p = 0.002399 is below α = 5%: the group means differ significantly (H₀ "all equal" is rejected).
Effect size
η² = 0.553 (Cohen’s f = 1.111): large effect
ANOVA table
SourceSSdfMSFp
Between groups842429.2650.002399
Within groups68154.533
Total15217
Figures per group
GroupnMeanStandard deviation
Group 1651.79
Group 2692.19
Group 36102.37
Steps
  • SS_zw = Σ nᵢ · (x̄ᵢ − x̄)² = 84; SS_in = Σ (x − x̄ᵢ)² = 68
  • MS_zw = SS_zw / (k − 1) = 84 / 2 = 42; MS_in = SS_in / (N − k) = 68 / 15 = 4.533
  • F = MS_zw / MS_in = 9.265
  • η² = SS_zw / SS_gesamt = 0.553

How it is calculated

One-way analysis of variance (ANOVA) tests whether the means of three or more groups differ, without running several t-tests. It compares the variation between the groups with the variation within the groups. If the difference between groups is clearly larger than the "noise" within them, F becomes large and p small.

How ANOVA works

  1. Sums of squares: SS between = Σ nᵢ · (x̄ᵢ − x̄)², SS within = Σ (x − x̄ᵢ)².
  2. Mean squares: MS = SS / df with df₁ = k − 1 and df₂ = N − k.
  3. F value: F = MS between / MS within.
  4. p-value: the probability of an F this large if all groups have the same mean (F distribution).

Example

Three groups of six values each have means 5, 9 and 10. The calculation gives F(2, 15) = 9.26 and p = 0.0024. Since p is below 0.05 the means differ significantly. η² = 0.55: about 55% of the total variation is due to group membership.

Assumptions

What after a significant result?

ANOVA only says that some means differ, not which. For that, post-hoc tests such as Tukey HSD or pairwise t-tests with a correction follow. The effect size η² shows how important the difference is (about 0.01 small, 0.06 medium, 0.14 large).

Frequently asked questions

When do you use an ANOVA?

When you want to compare the means of three or more independent groups, such as three teaching methods or four fertilisers. For two groups a t-test is enough; ANOVA then gives the same result.

How do you read the ANOVA p-value?

It tells how likely such a large difference between groups would be if all means were really equal. If p is below the level α (usually 0.05), the difference is called significant.

What does η² (eta squared) mean?

It is the share of total variation explained by group membership (SS between / SS total). As a rule of thumb: 0.01 small, 0.06 medium, 0.14 large.

Which group is actually different?

Only a post-hoc test such as Tukey HSD or Bonferroni-corrected t-tests shows that. ANOVA itself only shows that at least one difference exists.

What if the variances are unequal?

Then Welch’s ANOVA is more robust. The calculator points this out when the largest standard deviation is more than twice the smallest.

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