Q29 / 50 · Manipal (MET) 2023 · Mathematics

If $\sum_{i=1}^9\left(x_i-5\right)=9$ and $\sum_{i=1}^9\left(x_i-5\right)^2=45$, then the standard deviation of the 9 items $x_1, x_2 \ldots, x_9$ is

  • A
    $9$
  • B
    $4$
  • C
    $3$
  • Correct answer
    D
    $2$

Solution

$\begin{aligned} & \text { Given, } \sum_{i=1}^{\mathcal{y}}\left(x_i-5\right)=9 \\ & \sum x_i-45=9 \Rightarrow \sum x_i=54\end{aligned}$ Also, $\sum_{i=1}^9\left(x_i-5\right)^2=45$ $\begin{aligned} \Rightarrow \quad & \sum x_i^2-10 \times 54+25 \times 9=45 \\ & \sum x_i^2=360\end{aligned}$ $\therefore$ Standard deviation $(\sigma)=\sqrt{\frac{360}{9}-\left(\frac{54}{9}\right)^2}=2$.

Hence, the correct answer is option: 'D'.

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