A data consists of n observations:
Question:

A data consists of $\mathrm{n}$ observations:

$\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \ldots, \mathrm{x}_{\mathrm{n}}$. If $\sum_{\mathrm{i}=1}^{\mathrm{n}}\left(\mathrm{x}_{\mathrm{i}}+1\right)^{2}=9 \mathrm{n} \quad$ and

$\sum_{i=1}^{n}\left(x_{i}-1\right)^{2}=5 n$, then the standard deviation of

this data is :

  1. 5

  2. $\sqrt{5}$

  3. $\sqrt{7}$

  4. 2


Correct Option: , 2

Solution:

$\sum\left(x_{i}+1\right)^{2}=9 n$ …………..(1)

$\sum\left(x_{i}-1\right)^{2}=5 n$ …………….(2)

$(1)+(2) \Rightarrow \sum\left(x_{1}^{2}+1\right)=7 n$

$\Rightarrow \frac{\sum x_{i}^{2}}{n}=6$

$(1)-(2) \Rightarrow 4 \Sigma x_{i}=4 n$

$\Rightarrow \Sigma x_{\mathrm{i}}=\mathrm{n}$

$\Rightarrow \frac{\Sigma \mathrm{x}_{\mathrm{i}}}{\mathrm{n}}=1$

$\Rightarrow$ variance $=6-1=5$

$\Rightarrow$ Standard diviation $=\sqrt{5}$

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