To perform a one-way ANOVA, we need to calculate several summary statistics: the total (T), mean (M), and sum of squares (SS) for each group. Given that some of the values are already provided, I'll help you fill in the missing details.

Given Data:

For each group, the data looks as follows:

• Group 1:
  $\text{N}_1 = 6, \quad G_1 = 13, \quad \sum X_1^2 = 11$

• Group 2:
  $\text{N}_2 = 5, \quad G_2 = 6, \quad \sum X_2^2 = 12$

• Group 3:
  $\text{N}_3 = 3, \quad G_3 = 9, \quad \sum X_3^2 = 4$

Where:

• $N$ is the number of observations (sample size).
• $G$ is the total sum of scores.
• $\sum X^2$ is the sum of squared scores.

Step-by-Step Calculations:

1. T (Sum of Scores for each group): $T_1 = \sum X_1 = 13, \quad T_2 = \sum X_2 = 6, \quad T_3 = \sum X_3 = 9$ Since the values for $T$ are already given as $T_1 = 13$, $T_2 = 6$, and $T_3 = 9$.

2. M (Mean for each group): $M_1 = \frac{T_1}{N_1} = \frac{13}{6} \approx 2.17, \quad M_2 = \frac{T_2}{N_2} = \frac{6}{5} = 1.2, \quad M_3 = \frac{T_3}{N_3} = \frac{9}{3} = 3.0$

3. SS (Sum of Squares for each group): The sum of squares for each group is calculated using: $SS = \sum X^2 - \frac{T^2}{N}$

   For Group 1: $SS_1 = 11 - \frac{13^2}{6} = 11 - \frac{169}{6} \approx 11 - 28.17 = -17.17$

   For Group 2: $SS_2 = 12 - \frac{6^2}{5} = 12 - \frac{36}{5} = 12 - 7.2 = 4.8$

   For Group 3: $SS_3 = 4 - \frac{9^2}{3} = 4 - \frac{81}{3} = 4 - 27 = -23$

Summary:

• $T_1 = 13$, $T_2 = 6$, $T_3 = 9$
• $M_1 = 2.17$, $M_2 = 1.2$, $M_3 = 3.0$
• $SS_1 = -17.17$, $SS_2 = 4.8$, $SS_3 = -23$

Let me know if you'd like me to adjust or further explain any details! Here are some questions you might find useful:

---

1. What is the purpose of calculating sum of squares (SS) in ANOVA?
2. How does the total sum of squares help in determining the variability between groups?
3. Why do we need to calculate means (M) for each group in ANOVA?
4. What do negative values for SS indicate in statistical analysis?
5. How do we calculate the F-ratio in an ANOVA test?

Tip: In ANOVA, the sum of squares helps break down the total variability into variability within groups and between groups. This is essential for determining if there are significant differences among the group means.