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One-Way ANOVA Calculator (With Steps)

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One-Way ANOVA Calculator

One-way analysis of variance (ANOVA) tests whether the means of two or more independent groups are all equal. It compares the variation between the group means with the variation inside the groups; when the between-group spread is large relative to the within-group spread, the F statistic is big and at least one mean differs.

Enter each group on its own line. The calculator reports each group’s size, mean, standard deviation and within-group sum of squares, the grand mean, SS between and SS within with their degrees of freedom, the mean squares, F, the p-value from the F distribution, the critical F and a complete ANOVA table you can paste into a report. Eta squared (η²) gives the proportion of total variation explained by group membership.

ANOVA assumes roughly normal data with similar variances across groups. A significant result says the means are not all equal, not which ones differ; follow it with pairwise comparisons such as Tukey’s HSD. With only two groups, one-way ANOVA is equivalent to the pooled two-sample t-test (F = t²).

At least 2 groups with 2+ values each.

Result and step-by-step solution

F(2, 27) = 2.3575
p = 0.1138
Fail to reject H₀ at α = 0.05. The group means are not significantly different; the variation between groups is consistent with chance.
  1. State the hypotheses
    H₀: μ₁ = μ₂ = … = μ3. H₁: at least one mean differs.
  2. Group summaries
    GroupnMeanSDSS within
    Group 11083.48.4354640.4
    Group 21089.34.8086208.1
    Group 31084.75.2926252.1
    Grand mean = 85.8 (N = 30)
  3. Sum of squares between groups
    SSB = Σ ni(x̄i − x̄)² = 192.2, dfB = k − 1 = 2
  4. Sum of squares within groups
    SSW = Σ Σ(x − x̄i)² = 1100.6, dfW = N − k = 27
  5. Mean squares and F
    MSB = 192.2 ÷ 2 = 96.1; MSW = 1100.6 ÷ 27 = 40.763; F = MSB ÷ MSW = 2.3575
  6. ANOVA table
    SourceSSdfMSFp
    Between192.2296.12.35750.1138
    Within1100.62740.763
    Total1292.829
  7. Decision
    p = 0.1138 ≥ α = 0.05. Critical F0.05(2, 27) = 3.3541.
Effect size η²0.149

Frequently asked questions

What does a significant ANOVA result mean?
At least one group mean differs from the others. It does not identify which; run post-hoc pairwise tests to find out.
What are the degrees of freedom in one-way ANOVA?
Between groups: k − 1. Within groups: N − k, where k is the number of groups and N the total number of observations.
Do the groups need equal sizes?
No. Unequal sizes are fine, although the test is more sensitive to unequal variances when sizes differ a lot.
How is η² interpreted?
It is SS between divided by SS total, the share of variation explained by the groups. Around 0.01 is small, 0.06 medium and 0.14 large.

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Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.

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