Q78 / 180 · KCET 2022 · Mathematics
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$\sqrt{6}$
- C$2 \sqrt{\frac{10}{3}}$
- Correct answerD$2 \sqrt{6}$
Solution
Given, numbers are $-1,0,1, k$.
Standard deviation, $\sigma=\sqrt{5}$
$$
\sigma^2=\frac{\Sigma x_i^2}{n}-\left(\frac{\Sigma x_i}{n}\right)^2
$$
$$
\begin{aligned}
& \Rightarrow \quad 5=\frac{1+0+1+k^2}{4}-\left(\frac{-1+0+1+k}{4}\right)^2 \\
& \Rightarrow \quad 5=\frac{2+k^2}{4}-\frac{k^2}{16} \Rightarrow 80=8+4 k^2-k^2 \\
& \Rightarrow \quad 72=3 k^2 \Rightarrow k^2=24 \Rightarrow k=2 \sqrt{6} .
\end{aligned}
$$
Hence, the correct answer is option: 'D'.