Solved Problems: Total, Between, and Within-Group Variance Calculations

📚 Understanding Total, Between, and Within-Group Variance

Variance, in general, measures how spread out a set of data is. When analyzing data grouped into different categories, we can break down the overall variance into components that explain the variation between the groups and the variation within each group. Understanding these components helps us determine if there are significant differences between the group means. Here's a breakdown:

• 📊 Total Variance: Represents the total variability in the entire dataset, irrespective of group membership. It considers every data point as part of a single, large sample.
• 🌱 Between-Group Variance: Measures the variability between the means of different groups. It assesses how much the group means differ from the overall mean. A large between-group variance suggests that the group means are quite different from each other.
• 🌳 Within-Group Variance: Measures the variability within each individual group. It represents the average of the variances within each group. A small within-group variance suggests that the data points within each group are clustered closely around their respective group means.

📜 History and Background

The concept of partitioning variance originated from the work of Ronald Fisher in the early 20th century, particularly in the context of analysis of variance (ANOVA). Fisher developed ANOVA to analyze agricultural experiments, allowing researchers to determine whether different treatments (e.g., fertilizers) had a significant effect on crop yields. ANOVA and the partitioning of variance became fundamental tools in statistical inference and experimental design.

✨ Key Principles

• ➕ Additivity: The total sum of squares (a measure of total variance) can be partitioned into the sum of squares between groups and the sum of squares within groups. Mathematically: $SS_{total} = SS_{between} + SS_{within}$
• ⚖️ Degrees of Freedom: The degrees of freedom are also additive. If you have $N$ total observations and $k$ groups: $df_{total} = N - 1$, $df_{between} = k - 1$, and $df_{within} = N - k$.
• 🧮 Mean Squares: Variance is estimated using mean squares, which are calculated by dividing the sum of squares by the degrees of freedom. $MS_{between} = \frac{SS_{between}}{df_{between}}$ and $MS_{within} = \frac{SS_{within}}{df_{within}}$.

➗ Calculating the Variances

Let's define the formulas we will be using:

• ➕ Total Sum of Squares ($SS_{total}$): This represents the overall variability in the data. $SS_{total} = \sum_{i=1}^{N} (x_i - \bar{x})^2$, where $x_i$ is each individual data point, and $\bar{x}$ is the overall mean.
• 👯 Between-Groups Sum of Squares ($SS_{between}$): This measures the variability between the group means and the overall mean. $SS_{between} = \sum_{j=1}^{k} n_j(\bar{x_j} - \bar{x})^2$, where $n_j$ is the number of observations in group $j$, $\bar{x_j}$ is the mean of group $j$, and $\bar{x}$ is the overall mean.
• 🧑‍🤝‍🧑 Within-Groups Sum of Squares ($SS_{within}$): This measures the variability within each group. $SS_{within} = \sum_{j=1}^{k} \sum_{i=1}^{n_j} (x_{ij} - \bar{x_j})^2$, where $x_{ij}$ is the $i$-th observation in group $j$, and $\bar{x_j}$ is the mean of group $j$.

✍️ Solved Examples

Example 1: Exam Scores

Suppose we have exam scores from three different classes:

• Class A: 80, 85, 90
• Class B: 70, 75, 80
• Class C: 90, 95, 100

Calculate the total, between-group, and within-group variance.

1. Calculate Means:
• Class A Mean: $\bar{x_A} = (80 + 85 + 90) / 3 = 85$
• Class B Mean: $\bar{x_B} = (70 + 75 + 80) / 3 = 75$
• Class C Mean: $\bar{x_C} = (90 + 95 + 100) / 3 = 95$
• Overall Mean: $\bar{x} = (80 + 85 + 90 + 70 + 75 + 80 + 90 + 95 + 100) / 9 = 85$
2. Calculate Sum of Squares:
• $SS_{total} = (80-85)^2 + (85-85)^2 + (90-85)^2 + (70-85)^2 + (75-85)^2 + (80-85)^2 + (90-85)^2 + (95-85)^2 + (100-85)^2 = 750$
• $SS_{between} = 3(85-85)^2 + 3(75-85)^2 + 3(95-85)^2 = 3(0) + 3(100) + 3(100) = 600$
• $SS_{within} = [(80-85)^2 + (85-85)^2 + (90-85)^2] + [(70-75)^2 + (75-75)^2 + (80-75)^2] + [(90-95)^2 + (95-95)^2 + (100-95)^2] = 50 + 50 + 50 = 150$
3. Calculate Variance:
• Total Variance = $SS_{total} / (N - 1) = 750 / (9 - 1) = 93.75$
• Between-Group Variance = $SS_{between} / (k - 1) = 600 / (3 - 1) = 300$
• Within-Group Variance = $SS_{within} / (N - k) = 150 / (9 - 3) = 25$

Example 2: Plant Growth

Three different fertilizers are tested on plant growth (in cm):

• Fertilizer X: 10, 12, 14
• Fertilizer Y: 8, 10, 12
• Fertilizer Z: 12, 14, 16

Calculate the total, between-group, and within-group variance.

1. Calculate Means:
• Fertilizer X Mean: $\bar{x_X} = (10 + 12 + 14) / 3 = 12$
• Fertilizer Y Mean: $\bar{x_Y} = (8 + 10 + 12) / 3 = 10$
• Fertilizer Z Mean: $\bar{x_Z} = (12 + 14 + 16) / 3 = 14$
• Overall Mean: $\bar{x} = (10 + 12 + 14 + 8 + 10 + 12 + 12 + 14 + 16) / 9 = 12$
2. Calculate Sum of Squares:
• $SS_{total} = (10-12)^2 + (12-12)^2 + (14-12)^2 + (8-12)^2 + (10-12)^2 + (12-12)^2 + (12-12)^2 + (14-12)^2 + (16-12)^2 = 4 + 0 + 4 + 16 + 4 + 0 + 0 + 4 + 16 = 48$
• $SS_{between} = 3(12-12)^2 + 3(10-12)^2 + 3(14-12)^2 = 3(0) + 3(4) + 3(4) = 24$
• $SS_{within} = [(10-12)^2 + (12-12)^2 + (14-12)^2] + [(8-10)^2 + (10-10)^2 + (12-10)^2] + [(12-14)^2 + (14-14)^2 + (16-14)^2] = 8 + 8 + 8 = 24$
3. Calculate Variance:
• Total Variance = $SS_{total} / (N - 1) = 48 / (9 - 1) = 6$
• Between-Group Variance = $SS_{between} / (k - 1) = 24 / (3 - 1) = 12$
• Within-Group Variance = $SS_{within} / (N - k) = 24 / (9 - 3) = 4$

📝 Practice Quiz

Test your understanding with these practice problems:

1. Problem 1: Given the following data for three groups, calculate the total, between-group, and within-group variance:

   • Group A: 5, 7, 9
   • Group B: 6, 8, 10
   • Group C: 7, 9, 11

2. Problem 2: Calculate the variances with the following values.

   • Group 1: 22, 24, 26
   • Group 2: 20, 22, 24

3. Problem 3: A researcher is studying the effectiveness of different teaching methods. The final exam scores for three groups are below. Determine the total, between-group and within-group variance.

   • Method 1: 65, 70, 75
   • Method 2: 70, 75, 80
   • Method 3: 75, 80, 85

🔑 Conclusion

Understanding total, between-group, and within-group variance is crucial for analyzing data across different groups. By partitioning the variance, we can gain insights into the sources of variability and make informed decisions based on the data. These concepts form the foundation for more advanced statistical techniques like ANOVA and are widely applicable in various fields of research and data analysis.