(a) $\mathop {r = \max |{x_i} – {x_j}|}\limits_{\,\,\,\,\,\,\,\,\,\,\,\,\,i\, \ne j} $

${S^2} = \frac{1}{{n – 1}}\sum\limits_{i = 1}^n {{{({x_i} – \bar x)}^2}} $

${({x_i} – \bar x)^2} = {\left( {{x_i} – \frac{{{x_1} + {x_2} + ….. + {x_n}}}{n}} \right)^2}$

$ = \frac{1}{{{n^2}}}[({x_i} – {x_1}) + ({x_i} – {x_2}) + …. + ({x_i} – {x_i} – 1)$

$ + ({x_i} – {x_i} + 1) + …….$$ + ({x_i} – {x_n})] \le \frac{1}{{{n^2}}}{[(n – 1)r]^2}$,

==> ${({x_i} – \bar x)^2} \le {r^2} \Rightarrow \sum\limits_{i = 1}^n {{{({x_i} – \bar x)}^2} \le n{r^2}} $

==> $\frac{1}{{n – 1}}\sum\limits_{i = 1}^n {{{({x_i} – \bar x)}^2} \le \frac{{n{r^2}}}{{(n – 1)}}} $==> ${S^2} \le \frac{{n{r^2}}}{{(n – 1)}}$

==> $S \le r\sqrt {\frac{n}{{n – 1}}} $.