How to use the one-way anova calculator
- Give each group a name if you like, then paste its values in the box, separated by commas, spaces or new lines. Use Add a group for up to 12 groups. Two groups is the minimum.
- Set the significance level (alpha). The usual choice of 0.05 means you accept a 5% chance of a false alarm when all the groups really have the same mean.
- Read the F statistic, the p-value and the decision. The ANOVA table below shows the sums of squares, degrees of freedom and mean squares, so you can check each step against a textbook or a spreadsheet.
- Look at the group table to see which group means are high or low. Copy or print the result to keep it with your notes.
Formula
Grand mean = sum of all values / N
SS between = sum over groups of n x (group mean - grand mean)^2, df between = k - 1
SS within = sum over all values of (value - its group mean)^2, df within = N - k
F = (SS between / df between) / (SS within / df within)
p-value = area of the F distribution with (k - 1, N - k) degrees of freedom at or above F
k is the number of groups and N the total number of values. Eta squared is SS between divided by the total sum of squares and tells you what share of all the variation sits between the groups.
Three groups of five
Group A holds 4.2, 5.1, 3.9, 4.8, 5.0. Group B holds 5.6, 6.1, 5.9, 6.4, 5.2. Group C holds 4.9, 5.3, 4.6, 5.5, 5.1. The grand mean is 5.173333. SS between is 3.909333 on 2 degrees of freedom and SS within is 2.44 on 12, giving mean squares of 1.954667 and 0.203333.
F is 1.954667 / 0.203333 = 9.613115. With 2 and 12 degrees of freedom the p-value is 0.0032208, well under 0.05, and the critical F at the 0.05 level is 3.8853. So the data are not consistent with all three group means being equal. Eta squared is 0.6157, which means about 62% of the variation lies between the groups.
What the F test answers
A one-way ANOVA asks one question: could all the groups share the same true mean? It compares the spread of the group means with the spread of values inside the groups. If the group means are further apart than the within-group noise explains, F is large and the p-value is small.
A significant result does not say which groups differ. For that you need follow-up comparisons that account for making several comparisons at once. With only two groups the F test gives the same p-value as a pooled two-sample t-test, and F equals t squared; you can confirm that with the p-value and t-test calculator.
Assumptions behind the test
The test assumes the observations are independent, each group is roughly normally distributed, and the groups have similar variances. It is fairly forgiving when the groups are about the same size, and less so when sizes are very unequal and the spreads differ.
To look at the spread inside each group, use the standard deviation calculator or the descriptive statistics calculator.
How the p-value is computed
The F distribution tail area comes from the regularized incomplete beta function, evaluated with a continued fraction. The same function was compared with a separate numeric integration of the F density across many combinations of degrees of freedom, and agreed to better than 1e-10 relative error. The sums of squares were checked against numpy in a separate script.
Assumptions and limits
- Every group needs at least one value, and there must be more values in total than groups. Groups with a single value count toward the means but cannot show any spread.
- If every value inside every group is identical, the within-group variation is zero and F is not defined, so the tool reports that instead of a number.
- Values are used exactly as typed. Nothing is removed as an outlier and no data are transformed.
- This is a fixed-effects one-way design. Repeated measures, two-way designs, and tests of equal variances are not covered.
Frequently asked questions
How many groups and values can I enter?
You can enter 2 to 12 groups, with up to 5,000 values in each. A name for each group is optional and is only used to label the result tables.
What does it mean when the p-value is below my alpha?
It means that if all the group means were really equal, a spread of means as large as yours would be rare, so you reject that idea. It points to at least one group being different but does not say which.
Why do I need a follow-up test after a significant ANOVA?
The F test only tells you that the means are not all equal. Comparing pairs of groups one by one raises the chance of a false alarm, so pairwise follow-ups should use a method built to adjust for that.
What is eta squared?
Eta squared is the share of the total variation in the data that lies between the groups. A value of 0.62 means the group labels account for about 62% of the variation, and the rest is variation inside the groups.