Let n 3 .A list of numbers x₁, x₂, x_ n has mean and standard deviation . A new list of numbers y₁, y₂, . y_ n is made as follows: y₁= x₁+x₂ 2 , y 2= x₁+x₂ 2 and y_ j =x for j=3,4, . n . The mean and the standard deviation of the list are and . Then which of the following is necessarily true?

Step-by-step solution

= x_ i n n 3 = x_ i ² n - ( x_ i n )² = y₁+y₂+y₃+ +y_ n n = x₁+x₂ 2 + x₁+x₂ 2 +x₃+x₄+ +x_ n n = x ₁+ x ₂+ . .+ x _ n n = = ( x₁+x₂ 2 )²+ ( x₁+x₂ 2 )²+x₃²+ +x_ n ² n - ( y₁ n )² array l = 1 4 (2 x₁²+2 x₂²+4 x₁ x₂ )+x₃²+ . .+x_ n ² n - ( y₁ n )² = 1 2 (x₁²+x₂² )+x₁ x₂+x₃²+ +x_ n ² n - ( x_ i n )² x₁²+x₂²+ +x₂² n - ( x_ i n )² & (2) by (1) and (2) array