Question

Mean of $n$ observations $x_1,x_2,....,x_n$ is $\overline{x}$. If an observation $x_q$ is replaced by $x_q^{\prime }$ then the new mean is

• A. $\overline{x}-x_q+x_q^{\prime }$
• B. $\frac{(n-1)\overline{x}+x_q^{\prime }}{n}$
• C. $\frac{(n-1)\overline{x}-x_q^{\prime }}{n}$
• D. $\frac{n\overline{x}-x_q+x_q^{\prime }}{n}$

Answer 1

The correct option is D $\frac{n\overline{x}-x_q+x_q^{\prime }}{n}$

Given : Mean of $n$ observations $x_1,x_2,....x_n$ is $\overline{x}$

$⟹\frac{\sum _{i=1}^n{x}_i}{n}=\overline{x}$ ....... $\left(i\right)$

New mean $=\frac{x_1+...x_{q-1}+x_{q^{\prime }}+x_{q+1}+....x_n}{n}$

$=\frac{x_1+...+x_{q^{\prime }}+...+x_n+x_q-x_q}{n}$

$=\frac{\sum _{i=1}^n{x}_i-x_q+x_q^{\prime }}{n}$

$=\frac{n\overline{x}-x_q+x_q^{\prime }}{n}$ ....... From $\left(i\right)$