What is one-way ANOVA?
One-way ANOVA extends the 2-sample t-test to three or more groups that differ in one factor. It is the statistical follow-up to stratification: the plot shows a difference between groups, and ANOVA tells you whether that difference is larger than the noise within the groups.
Reading the results
- F and p-value: p below α means at least one mean differs.
- Welch's ANOVA: the same question without assuming equal variances.
- Levene / Brown-Forsythe test: whether the group variances differ.
- Tukey intervals: which pairs differ, and by how much, with a controlled overall error rate.
- Grouping letters: groups sharing a letter are not significantly different.
ANOVA example
Coating thickness was measured on 8 parts from each of four spray nozzles. ANOVA gives F = 19.2 and p below 0.001; the nozzle explains 67% of the variation, and the variances are similar (Levene p = 0.21). Tukey's test shows three levels: nozzle C is thickest (52.4 µm, group A), nozzle B is in between (51.1 µm, group B), and nozzles A and D are thinnest and not different from each other (49.8 and 49.6 µm, group C). The next step: find out what makes nozzle C deposit more, and whether nozzle B is drifting the same way.
Beyond one-way ANOVA
When two or more factors may matter at the same time, plan a designed experiment so you can also see their interactions. When the groups are measurement operators and parts, use a Gage R&R study, which is a specialized two-way ANOVA.
How to use this ANOVA tool
- Name the groups. Add one group for each level of the factor: machines, suppliers, shifts or methods.
- Paste the data. Paste one column per group from Excel. The columns may have different lengths.
- Read the ANOVA. Check the F test p-value and R², and Welch's test if the variances differ.
- Find which groups differ. Use the Tukey intervals and grouping letters to see which pairs are different.
- Export. Export the plots, ANOVA table and comparisons to PDF or Excel.
Frequently asked questions
What is ANOVA?
Analysis of variance (ANOVA) tests whether the means of three or more groups are equal. It compares the variation between the group means with the variation within the groups. A large F ratio, with a small p-value, means at least one group mean differs.
Why not run several t-tests?
Each t-test has a 5% risk of a false alarm. With four groups there are six pairs, so the chance of at least one false alarm rises to about 26%. ANOVA tests all groups at once, and Tukey comparisons control the overall error rate when you then compare pairs.
What does the Tukey test tell me?
After a significant ANOVA, Tukey's honestly significant difference (HSD) test compares every pair of groups while keeping the overall error rate at α. An interval that does not include zero is a significant difference. The grouping letters summarize the result: groups that share a letter are not significantly different.
What are the assumptions of ANOVA?
Independent observations, roughly normal data within each group, and similar variances across groups. ANOVA is robust to moderate non-normality with balanced groups. The tool runs the Brown-Forsythe (Levene) test for equal variances and also reports Welch's ANOVA, which does not assume equal variances.
When should I use Welch's ANOVA?
Use Welch's ANOVA when the group variances differ (Levene p below 0.05), especially with unequal group sizes. When variances are similar, the classic F test and Welch's test give similar answers.
What is R² in ANOVA?
R² is the share of the total variation explained by the differences between groups. It tells you how much the factor matters in practice, while the p-value only tells you whether the difference is real.