Question:
We have two data sets each of size 5. The variances are 4 and 5 and the corresponding means are 2 and 4 respectively. Then the variance of the combined data set is:
We have two data sets each of size 5. The variances are 4 and 5 and the corresponding means are 2 and 4 respectively. Then the variance of the combined data set is:
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Always compute combined mean first before variance.
Updated On: Apr 30, 2026
- \(\frac{1}{2} \)
- \(\frac{5}{2} \)
- \(6 \)
- \(\frac{11}{2} \)
- \(\frac{13}{2} \)
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The Correct Option is D
Solution and Explanation
Concept:
Combined variance:
\[
\sigma^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1+n_2}
\]
where $d_i = \mu_i - \mu$
Step 1: Find combined mean.
\[ \mu = \frac{5(2) + 5(4)}{10} = 3 \]
Step 2: Find deviations.
\[ d_1 = 2 - 3 = -1,\quad d_2 = 4 - 3 = 1 \]
Step 3: Substitute values.
\[ \sigma^2 = \frac{5(4+1) + 5(5+1)}{10} = \frac{5\cdot5 + 5\cdot6}{10} = \frac{25 + 30}{10} = \frac{55}{10} = \frac{11}{2} \]
Step 1: Find combined mean.
\[ \mu = \frac{5(2) + 5(4)}{10} = 3 \]
Step 2: Find deviations.
\[ d_1 = 2 - 3 = -1,\quad d_2 = 4 - 3 = 1 \]
Step 3: Substitute values.
\[ \sigma^2 = \frac{5(4+1) + 5(5+1)}{10} = \frac{5\cdot5 + 5\cdot6}{10} = \frac{25 + 30}{10} = \frac{55}{10} = \frac{11}{2} \]
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