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

Options

A. `(13)/(2)`
B. `(5)/(2)`
C. `(11)/(2)`
D. `6`

Answer

C

Explanation

`x_(x)^(2)=4`
`sigma_(y)^(2)=5`
`implies overline(x)=2`
`overline(y)=4`
`(sum x_(i))/(5)=2 implies sum x_(i)=10, sum y_(1)=20`
`sigma_(x)^(2)=((1)/(2) sum x_(i)^(2))-(overline(x))^(2)=(1)/(5)(sum y_(1)^(2))-16`
`sum x_(1)^(2)=40`
`sum y_(1)^(2)=105`
`sigma_(z)^(2)=(1)/(10)(sum x_(1)^(2)+sum y_(1)^(2))-((overline(x)+y)/(2))^(2)`
`=(1)/(10)(40+105)-9=(145-90)/(10)=(55)/(10)=(11)/(2)`