ANOVA Calculator

A one-way analysis of variance (ANOVA) tests whether the means of three or more independent groups are all equal. The null hypothesis is H₀: μ₁ = μ₂ = … = μₖ, and the alternative is that at least one mean is different. The test compares two estimates of the same variance: the spread of the group means around the grand mean (the between-group mean square, MSB) and the spread of the observations around their own group mean (the within-group mean square, MSW). If all groups share one mean the two are similar and F = MSB / MSW is close to 1. A large F means the group means lie further apart than the noise inside the groups can explain.

Use it for one numeric outcome and one factor with several levels, for example plant weight under a control and two treatments. With two groups it gives the same p-value as the pooled two-sample t-test (F = t²), so ANOVA is the natural extension of the t-test to more than two groups. Two factors at once need the two-way ANOVA calculator. A significant F only says that some means differ; the post-hoc tests below say which. New to the idea? Read ANOVA explained.

1. Choose the input type: raw data (one list per group) or summary statistics (n, mean and sample SD per group). Add another group gives you up to 10 groups.
2. Enter each group's observations separated by commas, spaces or new lines, at least 2 per group. The groups may have different sizes.
3. Pick the method, classical (assumes equal variances) or Welch (does not), and the significance level α, usually 0.05.
4. Press Calculate ANOVA. Read the F-value and p-value, the ANOVA table, the effect sizes, the variance check and the step-by-step working. When the classical test is significant, the Tukey HSD comparisons follow.
5. Load example enters the PlantGrowth data used below, and Copy link to this calculation shares your exact analysis.

The classical F and p-value are the ones printed by scipy.stats.f_oneway and by R's aov; the Welch F, degrees of freedom and p-value are those of R's oneway.test and of statsmodels' anova_oneway with unequal variances.

• SS, df and MS. Each source of variation has a sum of squares (SS), degrees of freedom (df) and a mean square (MS = SS / df). Between groups measures how far the group means lie from the grand mean; within groups is the scatter of the observations around their own group mean, the error term.
• F and p. F = MSB / MSW. The p-value is the chance of an F at least this large if all population means were equal. Only a large F counts against H₀, so the test is right-tailed. Reject H₀ when p ≤ α, which is the same as F exceeding the critical value F(α; k − 1, N − k); read it in an F table or with the F table calculator.
• Effect size. η² is the share of the total variance that goes with group membership, ω² is a less biased estimate and Cohen's f = √(η² / (1 − η²)). Cohen (1988) calls η² = 0.01, 0.06 and 0.14 small, medium and large, and f = 0.10, 0.25 and 0.40. ω² can come out slightly negative when F is below 1; read a negative value as 0. See effect size explained.
• Variance check. The Brown-Forsythe (median Levene) test under the result compares the group variances. A small p, or a largest-to-smallest SD ratio above 2, is a reason to choose Welch ANOVA.

Three fertilizers are tried on three plots each. The yields are A: 4, 5, 6; B: 6, 7, 8; C: 8, 9, 10. Type them into the calculator to reproduce every number below.

The PlantGrowth data set that ships with R (Dobson, 1983) holds the dry weight of 30 plants: ten controls and ten plants for each of two treatments. Press Load example to enter the weights.

F(2, 27) = 4.846 and p = 0.0159, so at α = 0.05 the mean weights differ. The critical value is F(0.05; 2, 27) = 3.3541. The effect sizes are η² = 0.2641, ω² = 0.2041 and Cohen's f = 0.5991: about 26% of the variation in weight goes with the condition, a large effect by Cohen's benchmarks. The Brown-Forsythe test finds no sign of unequal variances (W = 1.1192, p = 0.3412), so the classical test is appropriate; Welch's F is 5.181 with 2 and 17.128 degrees of freedom (p = 0.0174).

Only treatment 2 against treatment 1 is significant after Tukey's correction (adjusted p = 0.012). Written up: F(2, 27) = 4.85, p = .016, η² = .26. R prints the same numbers: summary(aov(weight ~ group, data = PlantGrowth)) gives F = 4.846 and p = 0.0159, TukeyHSD gives the pair table above and oneway.test the Welch result.

The classical F test pools the variances of all groups. When the group sizes are unequal and the variances differ, its error rates drift away from α, and Welch's F test, which weights each group by n / s² and adjusts the denominator degrees of freedom, is the safer choice. Check the SD ratio and the Brown-Forsythe test under the result, or use the Levene test calculator. Example with A: 12.1, 11.8, 12.4, 12.0; B: 14.2, 15.9, 13.1, 16.4, 12.8, 15.0; C: 10.0, 13.5, 8.9, 15.2, 9.4, 14.8, 11.1 (group SDs 0.25, 1.4652 and 2.6248):

The classical test misses the difference at α = 0.05 (p = 0.0533), Welch's test finds it (p = 0.0146) and the variance check confirms that the spreads differ. Decide on the method from the design and the variance check, not from whichever p-value is smaller.

• Independent observations. Each unit belongs to one group and units do not influence each other. No test can settle this; it comes from the design. Repeated measurements on the same subjects need a different model.
• Roughly normal values within each group. The F test tolerates moderate departures, especially when the groups are similar in size and not tiny. Check with a histogram or the Shapiro-Wilk test (see normality tests explained).
• Equal variances (classical F only). Use the Brown-Forsythe result and the SD ratio, or switch to Welch ANOVA.
• No extreme outliers. They inflate the within-group variance and pull the means. Screen the data with the outlier calculator.
• A numeric outcome. For ranks or strongly skewed data use the Kruskal-Wallis test (compare the two in parametric vs nonparametric tests).

A significant F says that at least one mean differs, not which. Testing every pair with separate t-tests is no answer: k groups have k(k − 1)/2 pairs, and at α = 0.05 separate t-tests without correction give about 14.26% probability of at least one false positive for 3 groups (3 pairs) and 40.13% for 5 groups (10 pairs).

• Tukey HSD compares every pair while holding the family-wise error rate at α. It appears under the result when the classical test is significant; the Tukey HSD calculator also accepts group means with the MSE of any ANOVA.
• Bonferroni or Holm adjust the p-values of a smaller, planned set of comparisons; use the Bonferroni correction calculator.
• Games-Howell is the usual choice when the variances are unequal. This site does not offer it; SPSS lists it under Post Hoc and R has it in the rstatix package.

The reasoning behind these rules is in multiple comparisons explained.