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Assertion (A): The variance of the first n odd natural numbers is $\frac{n^{2} - 1}{3} .$
Reason (R) : The sum of the first n odd natural numbers is n² and the sum of the squares of the first n odd natural numbers is $\frac{n (4 n^{2} - 1)}{3} .$

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(A) and (R) are true. (R) is correct explanation of (A)

(A) and (R) are true, but (R) is not a correct explanation of (A)

(A) is true, but (R) is false

(A) is false, but (R) is true

answer is A.

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Detailed Solution

First n natural numbers 1.2.3........n

$Here \sum x_{i} = \frac{n (n + 1)}{2}, \sum x_{i}^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

$variance is σ^{2} = \frac{\sum x_{i}^{2}}{n} - {(\frac{\sum x_{i}}{n})}^{2} = \frac{n^{2} - 1}{12}$

$Clearly Assertion is correct, obviously Reason is correct.$

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