The mean of `n` observations is ` X ` . If the first item is increased by 1, second by 2 and so on, then the new mean is (a) ` X +n` (b) ` X +n/2` (c) ` X +(n+1)/2` (d) None of these

Options

A. `overline(X) +n`
B. `overline(X)+(n)/(2)`
C. `overline(X)+(n+1)/(2)`
D. None of these

Explanation

Let `x_(1),x_(2),..,x_(n)` be n items. Then `overline(X)=(1)/(n) sumx_(i)`. Let
`y_(1)=x_(1)+1,y_(2)=x_(2)+2,y_(3)=x_(3),..,y_(n)=x_(n)+n`.
Then the mean of the new series is
`(1)/(n) sum (x_(i)+i)=(1)/(n)sum x_(i)+(1)/(n)(1+2+3+..+n)`
`=overline(X)+(1)/(n)(n(n+1))/(2)`
`=overline(X)=(n+1)/(2)`