Question

The variance of the numbers $2,3,11$ and $x$ is $\frac{49}{4}$. Find the value of $x$.

• A. $6,\frac{14}{3}$
• B. $6,\frac{14}{5}$
• C. $6,\frac{16}{3}$
• D. $4,\frac{13}{5}$

Answer 1

Answer: (A)

From given data, we make the following table.

$x$	$x^2$
$2$	$4$
$3$	$9$
$11$	$121$
$x$	$x^2$
$\Sigma x=16+x$	$\Sigma x^2=134+x^2$

But we know that, variance =$\frac{\Sigma x^2}{n}-{\left(\frac{\Sigma x}{n}\right)}^2$
$\Rightarrow \frac{134+x^2}{4}-{\left(\frac{16+x}{4}\right)}^2=\frac{49}{4}$ (given)
$\Rightarrow \frac{134+x^2}{4}-\frac{\left(256+x^2+32x\right)}{16}=\frac{49}{4}$
$\Rightarrow \frac{3x^2-32x+280}{16}=\frac{49}{4}$
$\Rightarrow 3x^2-32x+280=196$
$\Rightarrow 3x^2-32x+84=0$
$\Rightarrow \left(x-6\right)\left(3x-14\right)=0$
$\Rightarrow x=6,x=\frac{14}{3}$
Therefore, the values of $x$ are $6$ and $\frac{14}{3}$.