The raw data x₁, x₂, , x_ n is an A.P. with common difference d and first term 0 . x and ^2 are mean and variance of x_ i , i =1,2, n , then ^2 is

Step-by-step solution

aligned x = & x₁+x₂+ +x_ n n & = n 2 [2 x₁+( n -1) d ] n & = ( n -1) d 2 x_ i ^2 & =x₁^2+x₂^2+ +x_ n ^2 & =0+ d ^2+(2 ~d )^2+ +[( n -1) d ]^2 & = d ^2 [1+2^2+3^2+ +( n -1)^2 ] & = d ^2 [ n ( n -1)(2 n -1) 6 ] ^2 & = 1 n x_ i ^2-( x )^2 & = d ^2( n -1)(2 n -1) 6 - [ ( n -1) d 2 ]^2 & = d ^2( n -1) 2 ( 2 n -1 3 - n -1 2 ) & = d ^2( n -1) 2 ( n +1 6 )= ( n ^2-1 ) d ^2 12 aligned