Five students of a class have an average height 150 cm and variance 18 cm. A new student, whose height is 156 cm, joined them. The variance (in `cm^(2)`) of the height of these six student is

Options

A. 16
B. 22
C. 20
D. 18

Answer

C

Explanation

Let `x_(1), x_(2), x_(3),x_(4),x_(5)` be the heights of five students. Then,
We have
` " Mean,"bar(x)=(sum_(i=1)^(5)x_(i))/(5)=150 rArr sum_(i=1)^(5)x_(i)=750 " ...(i) " `
` " and variance " =(sum_(i=1)^(5)x_(i)^(2))/(n)-(bar(x))^(2)`
`rArr (sum_(i=1)^(5)x_(i)^(2))/(5)-(150)^(2)=18`
`rArr sum_(i=1)^(5)x_(i)^(2)=112590 " ... (ii)" `
Now, new mean `=(sum_(i=1)^(6)x_(i))/(6)`
`=(sum_(i=1)^(5)x_(i)+156)/(6)=(750+156)/(6)` [using Eq. (i)]
`rArr bar(x)_("new") =151`
and new variance
`= (sum_(i=1)^(6)x_(i)^(2))/(6)-(bar(x)_("new"))^(2) =(sum_(i=1)^(5)x_(i)^(2)+(156)^(2))/(6)-(151)^(2)`
`=(112590+(156)^(2))/(6)-(151)^(2) " " ` [using Eq. (ii)]
`=22821 - 22801=20`