Question

Let $x_1,x_2\dots x_{100}$ be $100$ observations such that $\sum _{i=1}^{100}{x}_i=0,\sum _{1\le i<j\le 100}\left|x_i{x}_j\right|=80000$ and mean deviation about their mean be $5.$ Then their standard deviation is

• A. 30
• B. 30.00
• C. 30.0

Answer 1

Answer: (A), (B), (C)

We have, $\overline{x}=\frac{\sum _{i=1}^{100}{x}_i}{100}=0(\because \sum _{i=1}^{100}{x}_i=0)$
Now, $\frac{\sum _{i=1}^{100}|x_i-\overline{x}|}{100}=5$
$\Rightarrow \sum _{i=1}^{100}|x_i|=500$
Squaring on both sides, we get
${\left(\sum _{i=1}^{100}|x_i|\right)}^2=(500{)}^2$
$\Rightarrow \sum _{i=1}^{100}|x_i{|}^2+2\sum _{1\le i<j\le 100}\left|x_i{x}_j\right|=250000$
$\Rightarrow \frac{\sum _{i=1}^{100}|x_i{|}^2}{100}=\frac{250000-2{\sum }_{1\le i<j\le 100}\left|x_i{x}_j\right|}{100}$
$\Rightarrow \frac{\sum _{i=1}^{100}|x_i{|}^2}{100}=\frac{250000-160000}{100}$
$\Rightarrow \frac{\sum _{i=1}^{100}|x_i{|}^2}{100}=900$
$\therefore$ Required standard deviation
$=\sqrt{\frac{\sum _{i=1}^{100}|x_i-\overline{x}{|}^2}{100}}=\sqrt{\frac{\sum _{i=1}^{100}|x_i{|}^2}{100}}=30$