Based on the following statements, choose the correct option Statement-I: The variance of the first n even natural numbers is n^2-1 4 . Statement-II: The difference between the variance of the first 20 even natural numbers and their arithmetic mean is 112.

Step-by-step solution

aligned & S =2+4+ .+2 n=2 n(n+1) 2 =n(n+1) & Mean of first n natural numbers = n(n+1) n =(n+1) & Variance = 1 n _ i=1 ^n (x_i )^2-( x )^2 & = 1 n (2^2+4^2+ .+4 n^2 )-( x )^2 & = 4 n n(n+1)(2 n+1) 6 -(n+1)^2= n^2-1 3 aligned So, statement I is false Now, mean of first 20 even natural numbers =20+1=21 Variance of first 20 even natural numbers = (20)^2-1 3 =133 Difference =133-21=112 So, statement II is true.