Question

For two data sets each of size 5, the variance are given to be 4 and 5 and the corresponding means are given to be 2 and 4 respectively. The variance of the combined data set is

• A. 13/2
• B. 5/2
• C. 11/2
• D. 15/2

Hint:

When combining variance, the contribution of each set is not just its variance, but the sum of its variance and the square of the difference between its mean and the combined mean ($d^2$).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given two sets of data, each of size $n_1 = n_2 = 5$. We have their individual variances ($\sigma_1^2 = 4, \sigma_2^2 = 5$) and means ($\bar{x}_1 = 2, \bar{x}_2 = 4$). We must find the variance of the combined data set.

Step 2: Key Formula or Approach:
1. Combined Mean: $\bar{x}_c = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$.
2. Combined Variance: $\sigma_c^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}$, where $d_1 = \bar{x}_1 - \bar{x}_c$ and $d_2 = \bar{x}_2 - \bar{x}_c$.

Step 3: Detailed Explanation:
First, calculate the combined mean:
$\bar{x}_c = \frac{5(2) + 5(4)}{5 + 5} = \frac{10 + 20}{10} = \frac{30}{10} = 3$.
Next, calculate the deviations from the combined mean:
$d_1 = \bar{x}_1 - \bar{x}_c = 2 - 3 = -1$.
$d_2 = \bar{x}_2 - \bar{x}_c = 4 - 3 = 1$.
Now, calculate the combined variance:
$\sigma_c^2 = \frac{5(4 + (-1)^2) + 5(5 + (1)^2)}{5 + 5}$
$\sigma_c^2 = \frac{5(4 + 1) + 5(5 + 1)}{10} = \frac{5(5) + 5(6)}{10} = \frac{25 + 30}{10} = \frac{55}{10} = \frac{11}{2}$.

Step 4: Final Answer:
The combined variance is 11/2, which corresponds to option (C).