A data consists of n observations: x 1 , x 2 , … , x n . If ∑ i = 1 n ( x i + 1 ) 2 = 11 n and ∑ i = 1 n x i - 1 2 = 7 n , then the variance of this data is

Question

A data consists of n observations: x 1 , x 2 , … , x n . If ∑ i = 1 n ( x i + 1 ) 2 = 11 n and ∑ i = 1 n x i - 1 2 = 7 n , then the variance of this data is, 5, 8, 6, 7

MARKS App · Accepted Answer

We have, ∑ i = 1 n x i + 1 2 = 11 n … … . i and ∑ i = 1 n x i - 1 2 = 7 n … i i Adding i and i i , we get 2 ∑ i = 1 n x i 2 + 1 = 18 n ⇒ ∑ i = 1 n x i 2 + 1 = 9 n ⇒ ∑ i = 1 n x i 2 + n = 9 n ⇒ ∑ i = 1 n x i 2 = 8 n ⇒ ∑ i = 1 n x i 2 n = 8 Subtracting i and i i , we get 4 ∑ i = 1 n x i = 4 n ⇒ ∑ i = 1 n x i = n ⇒ ∑ i = 1 n x i n = 1 Now, variance = 1 n ∑ i = 1 n x i 2 - ∑ i = 1 n x i n 2 = 8 - 1 = 7

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