Question

If $\overline{x}$ is the mean of $x_1,x_2,...,x_n$ and $\overline{y}$ is the mean of $y_1,y_2,...,y_n$ . If $\overline{z}$ is the mean of $x_1,x_2,...,x_n$ , $y_1,y_2,...,y_n$ then $\overline{z}$ is equal to :

• A. $\overline{x}+\overline{y}$
• B. $\frac{\overline{x}+\overline{y}}{n}$
• C. $\frac{\overline{x}+\overline{y}}{2}$
• D. $\frac{\overline{x}+\overline{y}}{2n}$

Answer 1

The correct option is C $\frac{\overline{x}+\overline{y}}{2}$

Given,
$\overline{x}=\frac{x_1+x_2+...+x_n}{n}$
$\Rightarrow \overline{x}\times n=x_1+x_2+...+x_n$ $....\left(i\right)$

Similarly,
$\overline{y}=\frac{y_1+y_2+...+y_n}{n}$
$\Rightarrow \overline{y}\times n=y_1+y_2+...+y_n$ $....\left(ii\right)$

$\overline{z}=\frac{x_1+x_2+...+x_n+y_1+y_2+...+y_n}{n+n}$
$\Rightarrow \overline{z}=\frac{(\overline{x}\times n)+(\overline{y}\times n)}{2n}$ $\left(from\right(i\left)and\right(ii\left)\right)$
$\Rightarrow \overline{z}=\frac{\overline{x}+\overline{y}}{2}$