Introduction
Analysis of Variance (ANOVA) is a statistical technique used to compare means of three or more groups by analyzing the variance within and between groups. It tests whether the differences among group means are statistically significant. ANOVA is widely used in experimental designs and quality control.
One-Way ANOVA
One-way ANOVA tests the null hypothesis that all group means are equal against the alternative that at least one differs. It partitions total variance into between-group and within-group components. The test statistic is F = (Between-group variance) / (Within-group variance). If F is large, we reject the null hypothesis.
Key Points
- Assumptions: independence, normality, homogeneity of variances.
- Degrees of freedom: between groups = k-1, within groups = N-k, total = N-1.
- F-distribution is used for critical values and p-values.
Solved Example 1
Question: Three teaching methods are compared. Group sizes: 5,5,5. Sum of squares between (SSB) = 60, sum of squares within (SSW) = 120. Test at 5% significance.
Solution: F = 3.0, fail to reject null.
Explanation: df between = 3-1=2, df within = 15-3=12. MSB = 60/2=30, MSW = 120/12=10. F = 30/10=3.0. Critical F(2,12) at 5% ≈ 3.89. Since 3.0 < 3.89, we fail to reject the null hypothesis.
ANOVA Table and Interpretation
ANOVA results are summarized in a table with sources of variation, degrees of freedom, sum of squares, mean squares, and F-statistic. The F-statistic is compared to the critical value from F-distribution tables or using p-value. A significant F indicates that not all group means are equal, but post-hoc tests are needed to identify which groups differ.
Key Points
- Total SS = Between SS + Within SS.
- Mean square = Sum of squares / degrees of freedom.
- F = MSB / MSW.
- If p-value < alpha, reject null hypothesis.
Solved Example 1
Question: Given: k=4 groups, N=20, SSB=150, SSW=300. Construct ANOVA table and test at 5%.
Solution: F = 3.0, fail to reject null.
Explanation: df between = 3, df within = 16. MSB = 150/3=50, MSW = 300/16=18.75. F = 50/18.75=2.667. Critical F(3,16) at 5% ≈ 3.24. Since 2.667 < 3.24, fail to reject null.
Practice Questions
Q1. In a one-way ANOVA with 4 groups and 25 total observations, what are the degrees of freedom for between groups and within groups?
Answer: Between groups: 3, Within groups: 21
Explanation: df between = k-1 = 4-1=3. df within = N-k = 25-4=21.
Q2. If SSB = 80, SSW = 120, df between = 2, df within = 12, what is the F-statistic?
Answer: 4.0
Explanation: MSB = 80/2=40, MSW = 120/12=10, F = 40/10=4.0.
Q3. What does a significant F-test in ANOVA indicate?
Answer: At least one group mean is different from the others.
Explanation: A significant F-test rejects the null hypothesis that all group means are equal, but it does not specify which groups differ.
Common Mistakes
- Assuming ANOVA identifies which specific groups differ; it only indicates that at least one differs. Use post-hoc tests for pairwise comparisons.
- Ignoring assumptions of normality and homogeneity of variances; violations can lead to incorrect conclusions.
- Confusing degrees of freedom: between groups is k-1, within groups is N-k, not N-1 for both.
Quick Revision
- ANOVA compares means of three or more groups by analyzing variance.
- F-statistic = MSB / MSW, where MSB = SSB/(k-1) and MSW = SSW/(N-k).
- Reject null if F > critical value or p-value < alpha.
- Assumptions: independence, normality, equal variances.
- Post-hoc tests (e.g., Tukey) are needed to identify differing groups.