Explanation
Step - 1: Finding the mean
The n natural numbers are 1,2,3,...,n
The Sum of n natural number is n(n+1)2
Mean=n(n+1)2n=n+12
Step - 2: Finding the Variance
Variance=∑(xi)2n−(Mean)2
∑(xi)2=12+22+...+n2n
Since 12+22+...+n2=n(n+1)(2n+1)6
∑(xi)2n=n(n+1)(2n+1)6n
Variance=n(n+1)(2n+1)6n−(n+12)2
=(n+1)2×(2n+13−n+12)
=(n+1)2(4n+2−3n−36)
=n+12×n−16
=n2−112
Hence, the correct answer is Option B.
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