Question

Let $\overline{x}$ be the mean of $x_1,x_2,...,x_n$ and $\overline{y}$ be 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 $\overline{z}=?$
(a) $(\overline{x}+\overline{y})$
(b) $\frac{1}{2}(\overline{x}+\overline{y})$
(c) $\frac{1}{n}(\overline{x}+\overline{y})$
(d) $\frac{1}{2n}(\overline{x}+\overline{y})$

Answer 1

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

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

$$\begin{gathered} \mathrm{Given}: \\ \overline{x}=\frac{x_1+x_2+.....x_n}{n} \\ \Rightarrow x_1+x_2+......+x_n=n\overline{x} \\ \mathrm{and} \\ \overline{y}=\frac{y_1+y_2+......+y_n}{n} \\ \Rightarrow y_1+y_2+......+y_n=n\overline{y} \\ \therefore \overline{z}=\frac{n\overline{x}+n\overline{y}}{2n} \\ =\frac{1}{2}\left(\overline{x}+\overline{y}\right) \end{gathered}$$