Perform one-way ANOVA (Analysis of Variance) to determine if there are statistically significant differences between three or more group means.
Last updated: March 2026
Each line represents one group of measurements.
| Source | SS | df | MS |
|---|---|---|---|
| Between | 32.6667 | 2 | 16.3333 |
| Within | 9.0000 | 9 | 1.0000 |
| Total | 41.6667 | 11 | — |
Measures the proportion of total variance explained by the groups:
ANOVA (Analysis of Variance) is a statistical method used to test whether the means of three or more independent groups are significantly different from each other. It generalizes the t-test to handle multiple groups simultaneously while controlling the false positive rate.
The method works by partitioning the total variation in the data into two components: the variation between group means (which we're interested in) and the variation within groups (which is due to random fluctuation). The ratio of these two variations, called the F-statistic, tells us whether the group differences are larger than we'd expect by chance alone.
A one-way ANOVA tests a single factor (like treatment type) across multiple groups. If the p-value is less than 0.05, we conclude there's a statistically significant difference between at least two groups—though ANOVA doesn't tell us which pairs differ (that requires post-hoc tests like Tukey's HSD).
Input your data with each group on a separate line. Values within a group should be separated by commas or spaces.
The table shows the sum of squares (SS), degrees of freedom (df), and mean squares (MS) for between and within group variation.
If the p-value is less than 0.05, reject the null hypothesis. There is a significant difference between groups.
Testing fertilizer effectiveness across three types:
With F = 11.66, we have strong evidence to reject the null hypothesis that all fertilizer types produce equal yields. There is a statistically significant difference between at least two fertilizer types.
Next step: Use post-hoc tests (Tukey, Bonferroni) to determine which specific pairs differ.
Use ANOVA when comparing three or more groups. Running multiple t-tests increases false positive risk (multiple comparisons problem). ANOVA controls this with a single test.
The F-statistic is the ratio of between-group variance to within-group variance. A large F-value suggests group means differ more than expected by chance, yielding a small p-value.
ANOVA is fairly robust to moderate violations, especially with larger samples. For severe non-normality, consider Kruskal-Wallis (non-parametric alternative) or data transformation.
Significant ANOVA means at least two groups differ, but not which ones. Use post-hoc tests like Tukey HSD, Bonferroni, or Scheffé for pairwise comparisons.
Generally aim for 10-30 observations per group. Use power analysis based on expected effect size, desired power (0.80), and alpha (0.05) for specific recommendations.
ANOVA can handle unequal sample sizes, but power decreases and robustness is reduced. Keep the ratio of largest to smallest group below 2:1 if possible.
No, regular ANOVA assumes independence. Use repeated measures ANOVA or mixed models for paired/clustered data to account for correlations.
df Between = k - 1 (groups minus 1). df Within = n - k (total observations minus groups). They determine the shape of the F-distribution for statistical testing.
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