1. Let the two data sets be $X$ and $Y$, each of size $5$.

   • For set $X$: $n_1=5,\quad \bar{x}=2,\quad \sigma_1^2=4$
   • For set $Y$: $n_2=5,\quad \bar{y}=4,\quad \sigma_2^2=5$

2. First find the mean of the combined data set.

   Since both sets have equal size, $\bar{z}=\frac{n_1\bar{x}+n_2\bar{y}}{n_1+n_2}$ $\bar{z}=\frac{5\cdot 2+5\cdot 4}{10}=\frac{10+20}{10}=3$

3. Use the formula for combined variance: $\sigma^2=\frac{n_1\left(\sigma_1^2+(\bar{x}-\bar{z})^2\right)+n_2\left(\sigma_2^2+(\bar{y}-\bar{z})^2\right)}{n_1+n_2}$

4. Substitute the values: $\sigma^2=\frac{5\left(4+(2-3)^2\right)+5\left(5+(4-3)^2\right)}{10}$

   Now simplify: $\sigma^2=\frac{5(4+1)+5(5+1)}{10}$ $\sigma^2=\frac{5\cdot 5+5\cdot 6}{10}$ $\sigma^2=\frac{25+30}{10}$ $\sigma^2=\frac{55}{10}=\frac{11}{2}$

5. Therefore, the variance of the combined data set is $\boxed{\frac{11}{2}}$

6. Checking options:

   • A: $\frac{5}{2}$
   • B: $\frac{11}{2}$ ✅
   • C: $6$
   • D: $\frac{13}{2}$

So the correct option is B.