One-Way ANOVA Guide
One-way ANOVA compares variation among group means with variation inside the groups. It is an omnibus test: the result can indicate that the common-mean model fits poorly without identifying which specific group pairs differ.
Method
- Enter one independent numeric group per line.
- Calculate each group mean and the grand mean across all observations.
- Split total variation into between-group and within-group sums of squares.
- Divide each sum of squares by its degrees of freedom to obtain mean squares.
- Compute F = MS between ÷ MS within and use the F distribution with df₁ = k − 1 and df₂ = N − k for the upper-tail p-value.
Worked example
For the sample groups with means 5, 9 and 10, SS between = 84 and SS within = 68.
The resulting F is about 9.2647 with df (2, 15), p about 0.002399 and eta squared about 0.5526.
Notes
- ANOVA does not by itself identify which pairs of means differ.
- Independent observations, outliers, distribution shape and spread across groups matter to model suitability.
- Eta squared is a descriptive proportion of total variation associated with the grouping in the entered sample, not proof of causation.
What an F test can tell you, and what it cannot
Suppose three independent groups have observed values 4, 5, 6; 5, 6, 7; and 8, 9, 10. Their sample means are 5, 6 and 9, so the between-group differences are visibly larger than some within-group differences. One-way ANOVA formalizes that comparison by dividing a between-group mean square by a within-group mean square to obtain an F statistic.
Even a statistically small p-value for the overall F test does not identify which specific pair of groups differs. Post-hoc comparisons and multiple-testing control may be needed, and assumptions about independent observations, residual behavior and variance should be checked first. NIST's ANOVA reference explains the decomposition and F test rather than treating the calculation as automatic proof of causation.
ANOVA compares group means through between- and within-group variation
One-way ANOVA asks whether several group means can reasonably be treated as equal under its model. The F statistic compares variability between group means with variability inside the groups. A larger ratio provides more evidence against the equal-means null hypothesis.
The test is omnibus: a significant result indicates that not all means are equal, but it does not identify which specific groups differ.
Check assumptions and follow-up questions separately
Independence comes from the study design, while strong outliers, severe non-normality in very small groups or large variance differences can affect the classical ANOVA approximation. Similar sample sizes can make the method more robust to some departures, but assumptions should still be reviewed.
After a significant omnibus test, use a suitable post-hoc procedure when pairwise differences matter. Report group summaries and an effect-size measure such as eta squared so practical magnitude is not reduced to the p-value alone.
Authoritative references
Primary or authoritative references for the external standard, scoring rule or identifier convention used on this page.
Editorial review and limitations
Continue with the related tools
Use the linked tools for your own inputs and scenarios.
- T-Test Calculator — Run one-sample, Welch two-sample or paired two-tailed t tests with t statistic, degrees of freedom and p-value.
- P-Value Calculator — Convert a z, t or chi-square test statistic into the matching one- or two-tailed p-value with explicit distribution assumptions.
- Standard Deviation Calculator — Calculate population and sample variance and standard deviation from a numeric data set.