Question

If $\bar{x}$ is the mean of $x_1, x_2, ........, x_n$ then for $a \neq 0$, the mean of $ax_1,ax_2 , .....,ax_n , \frac{x_1}{a} , \frac{x_2}{a} , ...... , \frac{x_n}{a}$is

• A. $\left( a + \frac{1}{a} \right) \bar{x}$
• B. $\left( a + \frac{1}{a} \right) \frac{\bar{x}}{2}$
• C. $\left( a + \frac{1}{a} \right) \frac{\bar{x}}{n}$
• D. $\frac{\left( a + \frac{1}{a} \right) \bar{x}}{2n}$

Answer 1

Answer: (B)

$\left( a + \frac{1}{a} \right) \frac{\bar{x}}{2}$

Answer 2

Given $\frac{x_1 + ....+x_n}{n} = \bar{x} \:\:\: ....(1)$ $\Rightarrow \frac{ax_{1}+....+ax_{n}}{an}= \bar{x} \:\: \Rightarrow \frac{ax_{1}+....+ax_{n}}{an}= a \bar{x} ......(2)$ Also, from (1), we have $\frac{\frac{1}{a}x_{1}+....+\frac{1}{a}x_{n}}{\frac{1}{a}n}$ $\Rightarrow \frac{\frac{x_{1}}{a}+....+\frac{x_{n}}{a}}{n}=\frac{\bar{x}}{a} \:\:\:\: ...(3)$ So, the mean of $ax_1, ........, ax_n$, $\frac{x_1}{a} ........, \frac{x_n}{a}$ is $\frac{a \bar{x}-\frac{\bar{x}}{a}}{2} =\frac{\bar{x}}{2}\left(a+\frac{1}{a}\right)$