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NCERT Class XI Mathematics - Statistics - Solutions

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Question : 14
Total: 34
First n natural numbers
Solution:  
Here xi = 1, 2, 3, 4, ..., n
∴ Σxi = 1 + 2 + 3 + 4 + ... + n =
n(n+1)
2
∴ Mean (
−
x
) =
n(n+1)
2n
 =
(n+1)
2
Σxi2 = (1)2+(2)2+(3)2+(4)2 + ... + n2 =
n(n+1)(2n+1)
6
∴ Variance (σ2) =
n×
n(n+1)(2n+1)
6
−[
n(n+1)
2
]2
n2
=
(n+1)(2n+1)
6
 -
(n+1)2
4
=
(n+1)
2
(
2n+1
3
−
(n+1)
2
) =
(n+1)
2
 (
4n+2−3n+3
6
) =
(n+1)(n−1)
12
 =
n2−1
12
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