The t-test compares two means. But what if we have three or more groups?
We could run multiple t-tests, but that inflates the chance of error.
The solution is the Analysis of Variance (ANOVA).
ANOVA partitions the variability into two parts: between groups and within groups.
Partitioning the Variance
Total variability = variability between groups + variability within groups.
- Between groups: differences due to the factor (treatment).
- Within groups: differences due to chance or individual variation.
Symbolic formula:
$$F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$$
Formula in words:
$$F = \frac{\text{mean square between groups}}{\text{mean square within groups}}$$
Where:
- $$MS_{\text{between}} = \tfrac{SS_{\text{between}}}{df_{\text{between}}}$$
- $$MS_{\text{within}} = \tfrac{SS_{\text{within}}}{df_{\text{within}}}$$
Degrees of Freedom
- $$df_{\text{between}} = k - 1$$
- $$df_{\text{within}} = N - k$$
- $$df_{\text{total}} = N - 1$$
Where $$k$$ = number of groups, $$N$$ = total number of observations.
Example (One-way ANOVA)
Three groups of students use different study techniques:
- Group A: mean = 70
- Group B: mean = 75
- Group C: mean = 85
Suppose calculations give:
- $$SS_{\text{between}} = 300, , df_{\text{between}} = 2 \Rightarrow MS_{\text{between}} = 150$$
- $$SS_{\text{within}} = 200, , df_{\text{within}} = 12 \Rightarrow MS_{\text{within}} = 16.7$$
Then:
$$F = \frac{150}{16.7} = 9.0$$
This F value is compared to the F table at df = (2, 12).
Definition
- ANOVA: compares means across three or more groups.
- F ratio: signal-to-noise ratio (treatment effect vs. error).
Visual Placeholders
Figure L6.1 — Partitioning Variance. Total variability divided into Between vs. Within.
Figure L6.2 — One-way ANOVA Layout. Bar graph with three groups (A, B, C).
Figure L6.3 — ANOVA Summary Table. Source | SS | df | MS | F | p.
Why This Matters
ANOVA generalizes the t-test to multiple groups.
It is one of the most widely used tools in psychology, education, and medicine.
Understanding the F ratio is key: a large F means treatment differences are greater than chance variation.
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