Question

Let $\overline{x}$ be the mean of $x_1,x_2,\dots ,x_n$. and $\overline{y}$ be the mean of $y_1,y_2\dots ,y_n$ If $\overline{z}$ is the mean of $x_1,x_2,\dots ,x_n,y_1,y_2,\dots y_n$, then $\overline{z}=?$

(a) $(\overline{x}+\overline{y})$

(b) $\frac{1}{2}(\overline{x}+\overline{y})$

(c) $\frac{1}{n}(\overline{x}+\overline{y})$

(b) $\frac{1}{2n}(\overline{x}+\overline{y})$

Answer 1

The correct option is (b): $\frac{1}{2}(\overline{x}+\overline{y})$

Given:
$\overline{x}=\frac{(x_1+x_2+.....x_n)}{n}$

$\Rightarrow x_1+x_2+......+x_n=n\overline{x}\dots \dots \left(i\right)$

and, $\overline{y}=\frac{(y_1+y_2+......+y_n)}{n}$

$\Rightarrow y_1+y_2+......+y_n=n\overline{y}\dots \dots \left(ii\right)$

Now, $\overline{z}=\frac{(x_1+x_2+...+x_n)+(y_1+y_2+...+y_n)}{2n}$

∴ $\overline{z}=\frac{(n\overline{x}+n\overline{y})}{2n}$ [From $eq.\left(i\right)$ and $\left(ii\right)$]

$=\frac{1}{2}(\overline{x}+\overline{y})$