Question

Let $r$ be the range and $S^2=\frac{1}{n−1}{∑}_{i=1}^n{\left(x_i−\stackrel{ˉ}{x}\right)}^2$ be the S.D. of a set of observations $x_1,x_2,….x_n$, then

• A. $S≤r\sqrt{\frac{n}{n−1}}$
• B. $S=r\sqrt{\frac{n}{n−1}}$
• C. $Sâ‰\yen r\sqrt{\frac{n}{n−1}}$
• D. None of these

Answer 1

Answer: (A)

We have $r=\underset{iî€=j}{\max }âˆ\pounds x_i−x_jâˆ\pounds$ and
$S^2=\frac{1}{n−1}\underset{i=1}{\overset{n}{∑}}{\left(x_i−\stackrel{ˉ}{x}\right)}^2$
Now, consider ${\left(x_i−\stackrel{ˉ}{x}\right)}^2={\left(x_i−\frac{x_1+x_2+….+x_n}{n}\right)}^2$
$=\frac{1}{n^2}\left[\left(x_i−x_1\right)+\left(x_i−x_2\right)+….+\left(x_i−x_i−1\right)\right]$
$+\left(x_i−x_i+1\right)+…+\left(x_i−x_n\right)]≤\frac{1}{n^2}[(n−1)r{]}^2$
$⇒{\left(x_i−\stackrel{ˉ}{x}\right)}^2≤r^2$
$⇒\underset{i=1}{\overset{n}{∑}}{\left(x_i−\stackrel{ˉ}{x}\right)}^2≤n{r}^2$
$⇒\frac{1}{n−1}\underset{i=1}{\overset{n}{∑}}{\left(x_i−\stackrel{ˉ}{x}\right)}^2≤\frac{n{r}^2}{(n−1)}$
$⇒S≤r\sqrt{\frac{n}{n−1}}$