One-Way ANOVA — Previous Year Questions

📊 Topic Overview

Metric	Value
Papers Appeared	22 of 27 (81.5%)
Priority	🔴 VERY HIGH — virtual guarantee in Section A
Section Distribution	Sec A: ~20
Most Repeated	Compute one-way ANOVA for given data
Probability	Extremely High — prepare as your primary Section A computation

🎯 Must-Know Concepts

1. Purpose: ANOVA tests whether three or more group means differ significantly
2. F-ratio = Between-group variance / Within-group variance
3. Steps: compute SS total, SS between, SS within; then df, MS, F
4. Assumptions: normality, homogeneity of variance, independent samples
5. Advantages and limitations of ANOVA
6. Interpretation: F obtained vs F critical from table

---

🔥 Core Computation Procedure (Section A — ~450 words)

Question type: "Compute one-way ANOVA for the following data and indicate whether the groups differ significantly."

Step-by-Step ANOVA Computation

Given: Scores from k groups (typically 3 groups with n subjects each)

Step 1: Calculate the Grand Total and Grand Mean

• Grand Total (ΣX) = Sum of all scores
• Grand Mean (X̄G) = ΣX / N (N = total subjects across all groups)

Step 2: Calculate Group Totals and Means

• For each group j: Tj = ΣXj, X̄j = Tj / nj

Step 3: Calculate Sum of Squares (SS)

SS Total (total variability): $SS_{Total} = \sum X^2 - \frac{(\sum X)^2}{N}$

SS Between (variability due to group differences): $SS_{Between} = \sum_j \frac{T_j^2}{n_j} - \frac{(\sum X)^2}{N}$

SS Within (variability within each group): $SS_{Within} = SS_{Total} - SS_{Between}$

Step 4: Degrees of Freedom (df)

• df Between = k − 1 (k = number of groups)
• df Within = N − k
• df Total = N − 1

Step 5: Mean Squares (MS)

• MS Between = SS Between / df Between
• MS Within = SS Within / df Within

Step 6: F-ratio $F = \frac{MS_{Between}}{MS_{Within}}$

Step 7: ANOVA Summary Table

Source	SS	df	MS	F
Between Groups	SS_B	k−1	MS_B	MS_B/MS_W
Within Groups	SS_W	N−k	MS_W	—
Total	SS_T	N−1	—	—

Step 8: Decision

• Find F critical from F-distribution table at df1 = k−1, df2 = N−k at α = 0.05
• If F obtained ≥ F critical: Reject H₀ → Groups differ significantly
• If F obtained < F critical: Retain H₀ → Groups do not differ significantly

Null and Alternative Hypothesis

• H₀: μ₁ = μ₂ = μ₃ (all group means are equal)
• H₁: At least one group mean is different

---

📝 Conceptual Questions

Assumptions of ANOVA (Section B — ~250 words)

ANOVA rests on several assumptions:

1. Normality: Scores within each group are normally distributed. ANOVA is relatively robust to violations of normality when sample sizes are equal and reasonably large.
2. Homogeneity of variance: The variance within each group is approximately equal (homoscedasticity). Levene's test can be used to check this assumption.
3. Independence of observations: Scores from different subjects are independent of each other.
4. Random sampling: Participants are randomly assigned to groups (in true experiments).

Advantages of ANOVA:

• Can compare three or more groups simultaneously, avoiding the inflation of Type I error that occurs with multiple t-tests.
• Partitions total variability into meaningful components (between-group and within-group).
• Extensible to complex designs (two-way ANOVA, mixed designs).

Limitations of ANOVA:

• Assumes normality and homogeneity of variance.
• Only tells us that at least one group differs; does not identify which groups differ (post-hoc tests needed).
• Sensitive to outliers.

---

🧠 Memory Aid

ANOVA Steps (GSSDF-MF-D):

• Grand total & grand mean
• SS Total
• SS Between & SS Within
• Degrees of freedom
• Formulate MS
• F-ratio
• Decision (compare with F critical)

---

📚 Related Content

• One-Way ANOVA
• Two-Way ANOVA
• Parametric Tests: t, F, Pearson

✅ Self-Assessment

1. What does the F-ratio represent in ANOVA?
2. What are the degrees of freedom between and within for 3 groups with 5 subjects each?
3. If F obtained = 4.2 and F critical = 3.89, what is your conclusion?
4. Why use ANOVA instead of multiple t-tests?
5. What assumptions must be met before applying ANOVA?

💡 Exam Tips

• Always draw the ANOVA summary table — examiners award marks for the structured table format.
• Show every calculation step: GT, group totals, SS formulas, df, MS, F.
• State H₀ and H₁ before computing.
• Look up F critical and state the conclusion explicitly.
• Computation questions: practice with at least 5 different data sets before the exam.