Question

The mean of n items is $\overline{x}$. If the first item is increased by 1, second by 2 and so on, then the new mean is ____.

• A. $\overline{X}+n$
• B. $\overline{X}+\frac{n}{2}$
• C. $\overline{X}+\frac{n+1}{2}$
• D. None of the above

Answer 1

The correct option is C $\overline{X}+\frac{n+1}{2}$

$Let{x}_1,x_2,....x_n$be n items.
Then $\overline{x}=\frac{1}{n}\sum x_i$
Let $y_1=x_1+1,y_2=x_2+2,y_3=x_3+3,y_n=x_n+n$
Then the mean of the new series is$\frac{1}{n}\sum y_i=\frac{1}{n}\sum (x_i+i)=\frac{1}{n}\sum x_i+\frac{1}{n}(1+2+3+....+n)=\overline{X}+\frac{1}{n}\frac{n(n+1)}{2}=\overline{X}+\frac{n+1}{2}$