Solved Examples: Calculating SST, SSB, and SSW in One-Way ANOVA

📚 Quick Study Guide

• 🔢 SST (Total Sum of Squares): Measures the total variability in the data. Formula: $SST = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (y_{ij} - \overline{y})^2$, where $y_{ij}$ is the $j$-th observation in the $i$-th group, $\overline{y}$ is the overall mean, $k$ is the number of groups, and $n_i$ is the sample size of the $i$-th group.
• 📊 SSB (Sum of Squares Between): Measures the variability between the group means. Formula: $SSB = \sum_{i=1}^{k} n_i (\overline{y_i} - \overline{y})^2$, where $\overline{y_i}$ is the mean of the $i$-th group.
• 📉 SSW (Sum of Squares Within): Measures the variability within each group. Formula: $SSW = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (y_{ij} - \overline{y_i})^2$. It can also be calculated as $SSW = SST - SSB$.
• 💡 Key Relationship: In One-Way ANOVA, $SST = SSB + SSW$. This relationship is fundamental to understanding the partitioning of variance.
• 📝 Degrees of Freedom: For SST, $df_{SST} = N - 1$, where $N$ is the total number of observations. For SSB, $df_{SSB} = k - 1$. For SSW, $df_{SSW} = N - k$.

🧪 Practice Quiz

1. What does SST represent in One-Way ANOVA?
   1. A. The variability between group means.
   2. B. The variability within each group.
   3. C. The total variability in the data.
   4. D. The unexplained variability.
2. Which formula is used to calculate SSB?
   1. A. $\sum_{i=1}^{k} \sum_{j=1}^{n_i} (y_{ij} - \overline{y_i})^2$
   2. B. $\sum_{i=1}^{k} n_i (\overline{y_i} - \overline{y})^2$
   3. C. $\sum_{i=1}^{k} \sum_{j=1}^{n_i} (y_{ij} - \overline{y})^2$
   4. D. $\sum (X - \mu)^2 / N$
3. If SST = 100 and SSB = 60, what is the value of SSW?
   1. A. 160
   2. B. 40
   3. C. 60
   4. D. 100
4. What is the relationship between SST, SSB, and SSW?
   1. A. $SST = SSB - SSW$
   2. B. $SSB = SST + SSW$
   3. C. $SST = SSB + SSW$
   4. D. $SSW = SST + SSB$
5. What does SSW measure?
   1. A. The total variability.
   2. B. The variability between groups.
   3. C. The variability within groups.
   4. D. The explained variability.
6. In One-Way ANOVA, which of the following is true regarding degrees of freedom?
   1. A. $df_{SST} = k - 1$
   2. B. $df_{SSB} = N - k$
   3. C. $df_{SSW} = N - 1$
   4. D. $df_{SST} = N - 1$
7. If the group means are very similar, which of the following is likely to be small?
   1. A. SST
   2. B. SSB
   3. C. SSW
   4. D. All of the above