If the standard deviation of the numbers -1, 0, 1, k is `sqrt(5) " where " k gt 0` is equal to

Options

A. `2sqrt((10)/(3))`
B. `2 sqrt(6)`
C. `4sqrt((5)/(3))`
D. `sqrt(6)`

Answer

B

Explanation

Given observations are -1, 0, 1 and k.
Also, standard deviation of these four observations`=sqrt(5)`
`therefore sqrt(((-1)^(2)+(0)^(2)+(1)^(2)+k^(2))/(4)-((-1+0+1+k)/(4))^(2))=sqrt(5)`
`[ because "if "x-(1),x_(2) ... x_(n) ` are n observation, then standard deviation `=sqrt((1)/(n)Sigma_(i=1)^(n) x_(i)^(2)-((1)/(n)Sigma_(i=1)^(n)x_(i))^(2))]`
`rArr (2+k^(2))/(4)-(k^(2))/(16)=5 " " `[squaring both sides]
`rArr (8+4k^(2)-k^(2))/(16)=5 rArr (8+3k^(2))/(16) =5`
`rArr 8+3k^(2)=80 rArr 3k^(2)=72`
`rArr k^(2)=24 rArr k=2sqrt(6) or -2sqrt(6)`
`rArr k=2sqrt(6) " "[ because k gt 0]`