Question

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

• A. $10$
• B. $30$
• C. $40$
• D. $50$

Answer 1

The correct option is B $30$

Given that $\sum x_i=0$
$\Rightarrow \overline{x}=\frac{\sum x_i}{100}=0$
Now, $\frac{\sum |x_i-\overline{x}|}{100}=5$
$\Rightarrow \sum |x_i|=500$
Squaring both the sides
${\left(\sum |x_i|\right)}^2=(500{)}^2$
$\Rightarrow \sum x_i^2+2\sum _{1\le i\le j\le 100}|x_i{x}_j|=(500{)}^2$
$[\because {\left(\sum |x_i|\right)}^2=\sum x_i^2+2\sum |x_i{x}_j|]$
$\Rightarrow \frac{\sum x_i^2}{100}=\frac{(500{)}^2-2\sum |x_i{x}_j|}{100}$
$\Rightarrow \frac{\sum x_i^2}{100}=\frac{(500{)}^2-2\times 80000}{100}$
$\Rightarrow \frac{\sum x_i^2}{100}=2500-1600=900$
Standard deviation $=\sqrt{\frac{\sum (x_i-\overline{x}{)}^2}{100}}$
$=\sqrt{900}=30$