One-Way ANOVA
So far, we have compared means using t tests. However, t tests are designed to compare only two group means.
When we need to compare three or more groups, using multiple t tests creates serious problems.
One-Way ANOVA solves this problem in a single, controlled statistical test.
Why Not Use Multiple t Tests?
Suppose we want to compare average exam scores across three teaching methods.
If we use multiple t tests:
- Method A vs Method B
- Method A vs Method C
- Method B vs Method C
Each test increases the chance of a false positive.
ANOVA controls this risk by testing all groups at once.
What Does One-Way Mean?
One-way refers to one categorical factor with multiple levels.
Examples of factors:
- Teaching method (A, B, C)
- Diet type (Low-carb, Vegan, Keto)
- Marketing strategy (Email, Ads, Social Media)
Each level represents a different group.
What Does ANOVA Test?
One-way ANOVA tests whether:
At least one group mean is different from the others
It does not tell us which groups differ — only that a difference exists.
Setting Up the Hypotheses
| Hypothesis | Statement |
|---|---|
| H₀ | All group means are equal |
| H₁ | At least one group mean differs |
Key Idea Behind ANOVA
ANOVA compares:
- Variation between groups
- Variation within groups
If between-group variation is large relative to within-group variation, the group means are unlikely to be equal.
The F Statistic (Conceptual)
The ANOVA test statistic is called the F statistic.
Conceptually:
F = (Variation between groups) ÷ (Variation within groups)
A large F value suggests real differences among group means.
Assumptions of One-Way ANOVA
- Observations are independent
- Each group is approximately normally distributed
- Variances are approximately equal across groups
ANOVA is fairly robust, but strong violations can affect results.
Real-World Example
A company tests three advertising strategies and measures average sales generated.
One-way ANOVA helps determine whether sales differ across strategies.
If ANOVA is significant, further analysis is needed to find which strategies differ.
ANOVA Decision Rule
Using the p-value approach:
- If p-value ≤ α → Reject H₀
- If p-value > α → Fail to reject H₀
What ANOVA Does NOT Tell You
- Which specific groups differ
- How large the differences are
- Cause-and-effect relationships
Post-hoc tests are required for detailed comparisons.
Common Mistakes to Avoid
- Using ANOVA for two groups only
- Ignoring assumption checks
- Interpreting significance as importance
- Forgetting follow-up analysis
Quick Check
What does one-way ANOVA test?
Practice Quiz
Question 1:
Why is ANOVA preferred over multiple t tests?
Question 2:
What does a large F statistic indicate?
Question 3:
Does ANOVA tell which groups differ?
Mini Practice
A researcher compares average stress levels across four job roles.
- What test should be used?
- What does rejecting H₀ imply?
What’s Next
In the next lesson, we will study Nonparametric Tests, which are used when ANOVA assumptions are violated.