PYB350 Chap.2 One-Way ANOVA and Partitioning Variance
One-Way ANOVA and Partitioning Variance
One-way analysis of variance compares any number of group means by splitting the total variability in scores into a between-groups part, which carries the effect of the independent variable, and a within-groups part, which carries sampling error.
This chapter builds that partition from individual scores using the general linear model, counts the degrees of freedom for each part, assembles the source table, and computes the F ratio and eta-squared.
It finishes with the reporting line the tutorial uses, so the numbers you calculate turn into a sentence a marker can check.
The worked examples build source tables from raw scores, so a missing cell in a multiple-choice item becomes a one-line calculation, and the practice cards drill the degrees of freedom and the F notation you will report.
What this chapter covers
- 01
Why ANOVA is needed for more than two groups
- 02
Between-groups and within-groups variance
- 03
The general linear model: grand mean plus effect plus error
- 04
Degrees of freedom for total, between and within
- 05
The source table and the F ratio
- 06
Eta-squared and the APA reporting line
- 07
Filling a missing cell in a source table from the additive identities
Worked example · free
The recall study source table built from fifteen scores
- 2Between-groups sum of squares: 5 times [(13 minus 15) squared + (17 minus 15) squared + (15 minus 15) squared] = 5 times 8 = 40.
- 1Degrees of freedom: between = 3 minus 1 = 2; within = 15 minus 3 = 12; total = 14.
- 2Mean squares: 40 / 2 = 20 and 30 / 12 = 2.5, so F(2, 12) = 8.00, larger than 3.89.
- 1Effect size: eta-squared = 40 / 70 = .57, so technique explains about 57% of the variation in recall.
Key terms
- Sum of squares
- The total of squared deviations of scores or means from a reference mean, the building block of every ANOVA term.
- Mean square
- A sum of squares divided by its degrees of freedom, giving a variance estimate that can enter an F ratio.
- F ratio
- The between-groups mean square divided by the within-groups mean square, near one when the independent variable has no effect.
- Omnibus test
- The overall F test showing that the group means are not all equal, without saying which groups differ.
- General linear model
- The model that writes each score as the grand mean plus the effect of its group plus individual error.
One-Way ANOVA and Partitioning Variance FAQ
What does a significant F actually tell you?
It tells you that the group means are not all equal in the population, nothing more. It does not identify which groups differ or in which direction, so a significant one-way ANOVA is always followed by planned contrasts or post hoc comparisons. The F test is the gate; the follow-up tests answer the research question.
Why is the within-groups term called error?
Within a group every participant received the same treatment, so the differences among their scores cannot come from the independent variable. They reflect individual differences and measurement noise, which is the sampling error the between-groups variance is compared against. In the general linear model it is the leftover part of each score after the group effect.
Should I report eta-squared or omega-squared?
The Week 1 tutorial reports eta-squared, the effect's sum of squares divided by the total. The later factorial materials add omega-squared because eta-squared tends to over-estimate the population effect. Reporting eta-squared is standard for the one-way revision tasks; when JASP gives both, reporting omega-squared alongside it shows you know the difference.
How do I find a missing value in an ANOVA source table?
Use the two identities that always hold: the sums of squares add up, between plus within equals total, and so do the degrees of freedom. Each mean square is its sum of squares divided by its degrees of freedom, and F is the between mean square over the within mean square. Any one missing cell can be recovered from those relationships.
Exam move
Rebuild a source table from raw scores at least three times with different data, then practise filling a single missing cell, which is how multiple-choice items usually test the partition. Keep the general linear model sentence for one score in your notes, grand mean plus group effect plus error, since it explains every later design in the unit.
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