Q. The mean of $n$ observations is $\overline{x}$. If one observation $x_{n + 1}$ is added, then the mean remains same. The value of $x_{n + 1}$ is
A) 0
B) 1
C) $n$
D) $\overline{x}$
Explanation
(d)
Let $x_{1},\ x_{2},\ x_{3},...,x_{n}$ be $n$ observation
$\therefore\ \ \ \overline{x} = \frac{\sum_{}^{}x_{i}}{n}$ ...(i)
New mean$= \frac{\sum_{}^{}x_{i} + x_{n + 1}}{n + 1}$
According to the question $\overline{x} = \frac{\sum_{}^{}x_{i} + x_{n + 1}}{n + 1}$
$\Rightarrow \ \ (n + 1)\overline{x} = n\overline{x} + x_{n + 1}$
$\Rightarrow \ \ \ x_{n + 1} = \overline{x}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.