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: Note the data.
Two sets, each of size $5$. Variances are $4$ and $5$; means are $2$ and $4$. We need the variance of all $10$ values together.
Step 2: Find the combined mean.
$\bar{x}_c=\frac{5(2)+5(4)}{5+5}=\frac{10+20}{10}=3$.
Step 3: Measure how far each mean sits from the combined mean.
First set: $d_1=2-3=-1$. Second set: $d_2=4-3=1$.
Step 4: Recall the combined variance formula.
$\sigma_c^2=\frac{n_1(\sigma_1^2+d_1^2)+n_2(\sigma_2^2+d_2^2)}{n_1+n_2}$. This adds each set's spread plus how far its mean drifted.
Step 5: Substitute.
$\sigma_c^2=\frac{5(4+1)+5(5+1)}{10}=\frac{5(5)+5(6)}{10}=\frac{25+30}{10}=\frac{55}{10}$.
Step 6: Simplify.
$\frac{55}{10}=\frac{11}{2}$, which is option (C).
\[ \boxed{\,\tfrac{11}{2}\,} \]