Question

The mean of $x_1$ and $x_2$ is $M_1$ and that of $x_1,x_2,x_3,x_4$ is $M_2$ then the mean of $a{x}_1,a{x}_2,\frac{x_3}{a},\frac{x_4}{a}$ is

• A. $\frac{M_1+M_2}{2}$
• B. $\frac{a{M}_1+\frac{M_2}{a}}{2}$
• C. $\frac{1}{2a}\left[\right(a^2-1)M_1+2M_2]$
• D. $\frac{1}{2a}\left[\right(a^2-1)M_1+M_2]$

Answer 1

Answer: (C)

$\frac{1}{2a}\left[\right(a^2-1)M_1+2M_2]$

$x_1+x_2=2M_1;x_1+x_2+x_3+x_4=4M_2$

$\Rightarrow x_3+x_4=4M_2-2M_1$

$\therefore \frac{1}{4}[a{x}_1+a{x}_2+\frac{1}{a}{x}_3+\frac{1}{a}{x}_4]$

$=\frac{1}{4}[2a{M}_1+\frac{1}{a}(4M_2-2M_1\left)\right]$

$=\frac{1}{2a}\left[\right(a^2-1)M_1+2M_2]$.