Question

If the standard deviation of the numbers $−1,0,1,k$ is $\sqrt{5}$ where $k>0$, then $k$ is equal to

• A. $4\sqrt{\frac{5}{3}}$
• B. $2\sqrt{\frac{10}{3}}$
• C. $\sqrt{6}$
• D. $2\sqrt{6}$

Answer 1

Answer: (D)

${σ}^2=5,\overline{x}=\frac{k}{4}$
$\frac{1}{4}\left(1+0+1+k^2\right)−\frac{k^2}{16}=5$
$\frac{k^2+2}{4}\frac{−k^2}{16}=5$
$\frac{4k^2+8−k^2}{16}=5$
$⇒3k^2+8=80$
$3k^2=72$
$k^2=24$
$k=Â\pm \sqrt{24}=2\sqrt{6}$