This problem requires us to calculate the variance of a combined data set using the given statistics of two individual data sets.

1. Key Concepts and Formulas

To solve this problem, we'll utilize the fundamental definitions and computational formulas for mean and variance:

• Mean ($\overline{X}$): The average of a data set. For a set of $n$ observations $X_i$, the mean is $\overline{X} = \frac{\sum X_i}{n}$.
• Variance ($\sigma^2$): A measure of the spread of data points around the mean. The most convenient computational formula for variance is $\sigma^2 = \frac{\sum X_i^2}{n} - (\overline{X})^2$. This formula can be rearranged to find the sum of squares: $\sum X_i^2 = n(\sigma^2 + \overline{X}^2)$.
• Combined Mean of two data sets: For two data sets, $X$ (size $n_1$, mean $\overline{x}_1$) and $Y$ (size $n_2$, mean $\overline{x}_2$), the mean of the combined data set ($\overline{X}_{combined}$) is $\overline{X}_{combined} = \frac{n_1 \overline{x}_1 + n_2 \overline{x}_2}{n_1 + n_2}$.
• Combined Variance of two data sets: The variance of the combined data set can be found using the general variance formula: $\sigma^2_{combined} = \frac{\sum (\text{all data points})^2}{\text{total number of data points}} - (\overline{X}_{combined})^2$. This requires calculating the sum of squares for all individual data points.

2. Step-by-Step Solution

Step 1: Understand the Given Information

We are given two data sets. Let's label them Data Set 1 and Data Set 2.

• For Data Set 1:

  • Number of observations ($n_1$) = 5
  • Variance (${\sigma_1}^2$) = 4
  • Mean ($\overline{x}_1$) = 2

• For Data Set 2:

  • Number of observations ($n_2$) = 5
  • Variance (${\sigma_2}^2$) = 5
  • Mean ($\overline{x}_2$) = 4

Our objective is to calculate the variance of the combined data set.

Step 2: Calculate the Sum of Observations for Each Data Set

The mean formula $\overline{X} = \frac{\sum X_i}{n}$ allows us to find the sum of observations ($\sum X_i$) for each set. This is crucial for calculating the combined mean.

• For Data Set 1: $\overline{x}_1 = \frac{\sum x_i}{n_1}$ Substituting the given values: $2 = \frac{\sum x_i}{5}$ Multiplying both sides by 5: $\sum x_i = 2 \times 5 = 10$

• For Data Set 2: $\overline{x}_2 = \frac{\sum y_i}{n_2}$ Substituting the given values: $4 = \frac{\sum y_i}{5}$ Multiplying both sides by 5: $\sum y_i = 4 \times 5 = 20$

Step 3: Calculate the Sum of Squares for Each Data Set

We use the computational formula for variance, $\sigma^2 = \frac{\sum X_i^2}{n} - (\overline{X})^2$, rearranged to find $\sum X_i^2 = n(\sigma^2 + \overline{X}^2)$. These sums of squares are essential for calculating the combined variance.

• For Data Set 1: ${\sigma_1}^2 = \frac{\sum x_i^2}{n_1} - (\overline{x}_1)^2$ Substituting the given values: $4 = \frac{\sum x_i^2}{5} - (2)^2$ $4 = \frac{\sum x_i^2}{5} - 4$ Adding 4 to both sides: $8 = \frac{\sum x_i^2}{5}$ Multiplying both sides by 5: $\sum x_i^2 = 8 \times 5 = 40$

• For Data Set 2: ${\sigma_2}^2 = \frac{\sum y_i^2}{n_2} - (\overline{x}_2)^2$ Substituting the given values: $5 = \frac{\sum y_i^2}{5} - (4)^2$ $5 = \frac{\sum y_i^2}{5} - 16$ Adding 16 to both sides: $21 = \frac{\sum y_i^2}{5}$ Multiplying both sides by 5: $\sum y_i^2 = 21 \times 5 = 105$

Step 4: Calculate the Mean of the Combined Data Set

First, find the total number of observations and the total sum of observations for the combined data set.

• Total number of observations ($N$) = $n_1 + n_2 = 5 + 5 = 10$.
• Total sum of observations ($\sum X_{combined}$) = $\sum x_i + \sum y_i = 10 + 20 = 30$.

Now, calculate the combined mean ($\overline{X}_{combined}$): $\overline{X}_{combined} = \frac{\sum X_{combined}}{N} = \frac{30}{10} = 3$ This combined mean will be used in the final variance calculation.

Step 5: Calculate the Variance of the Combined Data Set

First, find the sum of squares of all observations in the combined data set.

• Total sum of squares ($\sum X_{combined}^2$) = $\sum x_i^2 + \sum y_i^2 = 40 + 105 = 145$.

Now, apply the variance formula for the combined data set: $\sigma^2_{combined} = \frac{\sum X_{combined}^2}{N} - (\overline{X}_{combined})^2$ Substituting the calculated values: $\sigma^2_{combined} = \frac{145}{10} - (3)^2$ $\sigma^2_{combined} = 14.5 - 9$ $\sigma^2_{combined} = 5.5$ As a fraction, this is: $\sigma^2_{combined} = \frac{11}{2}$

3. Common Mistakes & Tips

• Incorrect Combined Mean: A common error is to simply average the individual means, e.g., $(\overline{x}_1 + \overline{x}_2)/2$. This is only correct if $n_1 = n_2$. In general, use the weighted average formula: $\overline{X}_{combined} = \frac{n_1 \overline{x}_1 + n_2 \overline{x}_2}{n_1 + n_2}$. In this specific problem, since $n_1=n_2=5$, $(\overline{x}_1 + \overline{x}_2)/2 = (2+4)/2 = 3$, which happens to be correct. However, always use the general formula for robustness.
• Confusing Variance Formulas: Always remember the computational formula $\sigma^2 = \frac{\sum X_i^2}{n} - (\overline{X})^2$ as it is usually more efficient than $\sigma^2 = \frac{\sum (X_i - \overline{X})^2}{n}$ for problems involving sums of squares.
• Alternative Combined Variance Formula: There's a direct formula for combined variance: $\sigma^2_{combined} = \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2}{n_1+n_2} + \frac{n_1 n_2}{(n_1+n_2)^2}(\overline{x}_1 - \overline{x}_2)^2$ Let's verify our answer using this formula: $\sigma^2_{combined} = \frac{5 \times 4 + 5 \times 5}{5+5} + \frac{5 \times 5}{(5+5)^2}(2 - 4)^2$ $\sigma^2_{combined} = \frac{20 + 25}{10} + \frac{25}{100}(-2)^2$ $\sigma^2_{combined} = \frac{45}{10} + \frac{25}{100}(4)$ $\sigma^2_{combined} = 4.5 + \frac{100}{100}$ $\sigma^2_{combined} = 4.5 + 1 = 5.5$ Both methods yield the same result, confirming our calculation. This formula can be a shortcut if remembered, but the fundamental approach used in the main solution is generally more robust as it relies directly on the definitions.

4. Summary

To find the variance of the combined data set, we first calculated the sum of observations and the sum of squares for each individual data set using their respective means and variances. Then, we combined these sums to find the total sum of observations and total sum of squares for the entire data set. Finally, we used the combined mean and total sum of squares in the standard variance formula to determine the variance of the combined data set. The result of these calculations is $5.5$ or $\frac{11}{2}$.

5. Final Answer

The variance of the combined data set is $\frac{11}{2}$. This corresponds to option (B).

The final answer is $\boxed{\text{11/2}}$