Question

If the arithmetic mean of the observations $x_1,x_2,x_3,...,x_n$ is $1$, then the arithmetic mean of $\frac{x_1}{k},\frac{x_2}{k},\frac{x_3}{k},...,\frac{x_n}{k}(k>0)$ is :

• A. greater than 1
• B. less than 1
• C. equal to 1
• D. none of these

Answer 1

The correct option is D none of these

We have,
$\frac{x_1+x_2+x_3+...+x_n}{n}=1$
$\Rightarrow x_1+x_2+x_3+...+x_n=n$ (Given)
$\therefore$ Required arithmetic mean $=\frac{\frac{x_1}{k}+\frac{x_2}{k}+\frac{x_3}{k}+...+\frac{x_n}{k}}{n}=\frac{1}{k}\left(\frac{x_1+x_2+x_3+...+x_n}{n}\right)=\frac{1}{k}$

$ifk<1\to \frac{1}{k}>1$

$ifk>1\to \frac{1}{k}<1$

$ifk=2\to k=1$

No information of x is given with respect to 1.