Test Your Knowledge: One-Way ANOVA Sums of Squares Calculation Questions

📚 Quick Study Guide

• 🔢 One-Way ANOVA: This statistical test compares the means of two or more groups using variance.
• ➕ Total Sum of Squares (SST): Measures the total variability in the data. Formula: $SST = \sum_{i=1}^{n} (Y_i - \overline{Y})^2$, where $Y_i$ is each individual score and $\overline{Y}$ is the grand mean.
• 🏢 Sum of Squares Between Groups (SSB): Measures the variability between the group means. Formula: $SSB = \sum_{j=1}^{k} n_j(\overline{Y}_j - \overline{Y})^2$, where $n_j$ is the sample size of group j, $\overline{Y}_j$ is the mean of group j, and $\overline{Y}$ is the grand mean.
• 📉 Sum of Squares Within Groups (SSW): Measures the variability within each group. Formula: $SSW = \sum_{j=1}^{k} \sum_{i=1}^{n_j} (Y_{ij} - \overline{Y}_j)^2$, where $Y_{ij}$ is each individual score within group j and $\overline{Y}_j$ is the mean of group j.
• 🤝 Relationship: $SST = SSB + SSW$
• 📐 Degrees of Freedom:
  • Total: $df_T = N - 1$, where N is the total number of observations.
  • Between Groups: $df_B = k - 1$, where k is the number of groups.
  • Within Groups: $df_W = N - k$
• 💡 Key Tip: Always double-check your calculations, especially when dealing with multiple groups and sample sizes.

Practice Quiz

1. Which of the following formulas represents the Total Sum of Squares (SST) in a One-Way ANOVA?

   1. $SST = \sum (X_i - \overline{X})^2$
   2. $SST = \sum (X_i - \mu)^2$
   3. $SST = \sum n_i(\overline{X}_i - \overline{X})^2$
   4. $SST = \sum (X_i - \overline{X}_i)^2$

2. What does the Sum of Squares Between Groups (SSB) measure?

   1. The total variability in the data.
   2. The variability within each group.
   3. The variability between the group means.
   4. The error in the experiment.

3. If SST = 100 and SSB = 60, what is the value of SSW?

   1. 40
   2. 160
   3. 6000
   4. -40

4. In a One-Way ANOVA, if there are 4 groups, what is the degrees of freedom between groups ($df_B$)?

   1. 3
   2. 4
   3. 5
   4. N-4

5. Which of the following is NOT a component of the total variance in One-Way ANOVA?

   1. Sum of Squares Error (SSE)
   2. Sum of Squares Within Groups (SSW)
   3. Sum of Squares Between Groups (SSB)
   4. Total Sum of Squares (SST)

6. What is the formula for degrees of freedom total ($df_T$) if N is the total number of observations?

   1. N
   2. N + 1
   3. N - 1
   4. N - k

7. Which of the following is the correct representation of the relationship between SST, SSB and SSW?

   1. SST = SSB - SSW
   2. SSB = SST + SSW
   3. SSW = SST + SSB
   4. SST = SSB + SSW