Question

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

$\frac{x_1+x_2+...+x_q+...+x_n}{n}=\overline{x}$
$\Rightarrow x_1+x_2+...+x_q+...+x_n=n\overline{x}$

As per question, replace $x_q$ with $x_q^{\prime }$
$\therefore x_1+x_2+...+x_q^{\prime }+...+x_n=x_1+x_2+...+x_q+...+x_n+x_q^{\prime }-x_q$
$\Rightarrow x_1+x_2+...+x_q^{\prime }+...+x_n=n\overline{x}+x_q^{\prime }-x_q$

Hence, new mean is
$\frac{x_1+x_2+...+x_q^{\prime }+...+x_n}{n}=\frac{n\overline{x}-x_q+x_q^{\prime }}{n}$