Question

Mean of $n$ observations $x_{1}, x_{2}, ........ , x_{n}$ is $\bar{x}.$ If an observation $x_q$ is replaced by $x'_q$ then the new mean is

• A. $\bar {x}-x_{q}+x'_{q}$
• B. $\frac{_{\left(n-1\right)\tilde{x}+x'_{q}}}{n}$
• C. $\frac{_{\left(n-1\right)\tilde{x}-x'_{q}}}{n}$
• D. $\frac{_{n\tilde{x}-x_{q}-x'_{q}}}{n}$

Answer 1

Answer: (D)

$\frac{_{n\tilde{x}-x_{q}-x'_{q}}}{n}$

Answer 2

We have,
$\bar{x}=\frac{x_{1}+x_{2}+\ldots+x_{q}+\ldots+x_{n}}{n}$
$\Rightarrow \Sigma x=n \bar{x} \ldots$ (i)
If $x_{q}$ is replaced by $x_{q}^{\prime}$, then new total will be
$\Sigma x^{\prime}=\Sigma x-x_{q}-x_{q}^{\prime}$
$\therefore$ New mean will be
$\bar{x}^{\prime} =\frac{\Sigma x^{\prime}}{n}$
$\bar{x}^{\prime} =\frac{\Sigma x-x_{q}+x_{q}^{\prime}}{n}$
$\bar{x}^{\prime}=\frac{n \bar{x}-x_{q}+x_{q}^{\prime}}{n}[$ from E (i)]