One-Way Analysis of Variance
Compare several group means at once by splitting the total variation into a between-groups and a within-groups part. The F ratio asks whether the split is more lopsided than chance allows.
What a learner can do afterwards
- Partition the total sum of squares into its two parts
- Interpret an F ratio and its two degrees of freedom
- Explain why running many pairwise t-tests inflates the error rate
1 · Read
One way analysis of variance compares several group means at once. The total variation splits into two piles: between group variation, driven by differences among the means, and within group variation, which is pure noise inside the groups. Add the two piles and you recover the total sum of squares.
The F ratio pits the two piles against each other after scaling each by its degrees of freedom. A large F means the group means spread wider than noise allows, and a small F means the differences could be luck. The F distribution carries two degrees of freedom, lives on nonnegative values with a right skew, and rejects only in the right tail for large values.
Suppose the between group sum of squares is 60 and the within group sum is 40. The total is 60 plus 40, which is 100. With four groups the first degrees of freedom is 3, and the second counts the leftover noise observations.
Running a separate t test for every pair multiplies the error rate, since each test risks a false rejection and the chance that at least one fires grows fast. Analysis of variance guards the family with one joint test. Only after it rejects do careful follow ups ask which pairs differ, with protection for the extra looks.
Split the variation, weigh it with F, and keep the error rate under one roof.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.