Question

For two data sets, each of size $5$, the variances 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. $\frac{11}{2}$
• B. $6$
• C. $\frac{13}{2}$
• D. $\frac{5}{2}$

Answer 1

Answer: (A)

$\frac{11}{2}$

Answer 2

$\sigma^{2}_{x} = 4$ $\sigma ^{2}_{y} = 5$ $\bar{x} = 2$ $\bar{y} = 4$ $\frac{\sum x_{i}}{5} = 2\quad\sum x_{i} = 10; \sum y_{i} = 20$ $\sigma ^{2}_{x} = \left(\frac{1}{2}\sum x_{i}^{2}\right)-\left(\bar{x}\right)^{2} = \frac{1}{5}\left(\sum y_{i}^{2}\right) - 16$ $\sum x_{i}^{2} = 40$ $\sum y_{i}^{2} = 105$ $\sigma ^{2}_{z} = \frac{1}{10} \left(\sum x_{i}^{2}+\sum y_{i}^{2}\right) - \left(\frac{\bar{x}+\bar{y}}{2}\right)^{2} = \frac{1}{10} \left(40+105\right) -9 = \frac{145-90}{10} = \frac{55}{10} = \frac{11}{2}$