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A data consists of n observations `x_(1), x_(2), ..., x_(n). If underset(i = 1)overset(n)Sigma (x_(i) + 1)^(2) = 9n and underset(i = 1) overset(n) Sigma(x_(i) - 1)^(2) = 5n`, then the standard deviation of this data is

A

2

B

`sqrt(7)`

C

5

D

`sqrt(5)`

Text Solution

Verified by Experts

The correct Answer is:
D
We have, `sum_(i=1)^(n)(x_(i)+1)^(2)=9n " ...(i)" `
` and sum_(i=1)^(n)(x_(i)-1)^(2)=5n " ...(ii)" `
On substracting Eq. (ii) from Eq. (i) is, we get
`rArr sum_(i=1)^(n){(x_(i)+1)^(2)-(x_(i)-1)^(2)}=4n`
`rArr sum_(i=1)^(n)4x_(i) =4n rArr sum_(i=1)^(n)x_(i)=n rArr (sum_(i=1)^(n)x_(i))/(n)=1`
` therefore " mean " (bar(x))=1`
Now, standard deviation `=sqrt((sum_(i=1)^(n)(x_(i)-bar(x))^(2))/(n))=sqrt((sum_(i=1)^(n)(x_(i)-1)^(2))/(n))`
`=sqrt((5n)/(n))=sqrt(5)`
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