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

$\sigma^{2}_{x} = 4 , \sigma^{2}_{y} = 5 , x = 2, y=4$ $\frac{1}{5} \sum x_{i}^{2} - \left(2\right)^{2} = 4; \frac{1}{5} \sum y_{i}^{2} - \left(4\right)^{2} = 5$ $\sum x_{i}^{2} = 40 ; \sum y_{i}^{2} = 105$ $\Rightarrow\sum\left(x_{i}^{2} + y_{i}^{2}\right) = 145$ $\Rightarrow \sum\left(x_{i} + y_{i}\right) = 5\left(2\right) + 5\left(4\right) = 30$ Variance of combined data $= \frac{1}{10} \sum\left(x_{i}^{2} + y_{i}^{2}\right) - \left(\frac{1}{10} \sum \left(x_{i} + y_{i}\right)\right)^{2}$ $= \frac{145}{10} - 9 = \frac{11}{2}$