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: Find the combined mean.
Both groups have size $5$, means $2$ and $4$: \[ \bar x_c = \frac{5(2)+5(4)}{10} = \frac{30}{10} = 3. \]

Step 2: Measure how far each mean sits from $3$.
$d_1 = 2-3 = -1$ and $d_2 = 4-3 = 1$.

Step 3: Apply the combined variance formula.
\[ \sigma_c^2 = \frac{n_1(\sigma_1^2+d_1^2)+n_2(\sigma_2^2+d_2^2)}{n_1+n_2} = \frac{5(4+1)+5(5+1)}{10}. \]

Step 4: Compute.
$\tfrac{25+30}{10} = \tfrac{55}{10} = \tfrac{11}{2}$. \[ \boxed{\tfrac{11}{2}} \]