A data consists of $ n $ observations: $ x_{1}, x_{2}, \ldots, x_{n} . $ If $ \sum_{i=1}^{n}\left(x_{i}+1\right)^{2}=11 n $ and $ \sum_{i=1}^{n}\left(x_{i}-1\right)^{2}=7 n $, then the variance of this data is

Question

A data consists of $ n $ observations: $ x_{1}, x_{2}, \ldots, x_{n} . $ If $ \sum_{i=1}^{n}\left(x_{i}+1\right)^{2}=11 n $ and $ \sum_{i=1}^{n}\left(x_{i}-1\right)^{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

Search any question & find its solution

Looking for more such questions to practice?