Question

If r be the range and $S={\left\{\frac{1}{n-1}\sum _i{(x_i-\overline{x})}^2\right\}}^{1/2}$ be the s.d.of a set of observations $x_1,x_2,\cdots x_n$ then $S\le r{\left(\frac{n}{n-1}\right)}^{1/2}$. If this is true enter 1, else enter 0

Answer 1

Let $x_r=max(x_1,x_{2,\cdots ,x_n})$ $x_k=min(x_1,x_{2,\cdots ,x_n})$
$\therefore r=x_r-x_k$ $\Rightarrow {(x_i-\overline{x})}^2\le {(x_i-x_k)}^2$ $\because \overline{x}\ge x_k$
$\therefore S^2=\frac{1}{n-1}\sum _{i=1}^n{(x_i-\overline{x})}^2\le \frac{1}{n-1}\sum _{r=1}^n{(x_i-x_k)}^2$
$\le \frac{1}{n-1}\sum _{i=1}^n{(x_r-x_k)}^2$
$\Rightarrow S^2\le \frac{n{r}^2}{n-1}orS\le r{\left[\frac{n}{n-1}\right]}^{1/2}$