Question

Let r be the range and $S^2=\frac{1}{n-1}{\sum }_{i=1}^n(x_i-\overline{x}{)}^2$, where S is the S.D. of a set of observations $x_1,x_2.....x_n$, then

• A. $S\le r\sqrt{\frac{n}{n-1}}$
• B. $S=r\sqrt{\frac{n}{n-1}}$
• C. $S\ge r\sqrt{\frac{n}{n-1}}$
• D. None of these

Answer 1

The correct option is A $S\le r\sqrt{\frac{n}{n-1}}$

$Wehaver=max|x_i-y_j|$
$And{S}^2=\frac{1}{n-1}\sum _{i=1}^n(x_i-\overline{x}{)}^2$
$Now,(x_i-\overline{x}{)}^2={(x_i-\frac{x_1+x_2+....+x_n}{n})}^2$
$=\frac{1}{n^2}\left[\right(x_i-x_1)+(x_i-x_2)+....+(x_i-x_i-1)+(x_i-x_i+1)+...+(x_i-x_n\left)\right]\le \frac{1}{n^2}\left[\right(n-1)r{]}^2,\Rightarrow (x_i-\overline{x}{)}^2\le r^2\Rightarrow \sum _{i=1}^n(x_i-\overline{x}{)}^2\le n{r}^2$