The standard deviation of a, a+d, a+2d, .... a + 2nd is

$\begin{array}{rcl} & & \mathrm{a}, \mathrm{a}+\mathrm{d}, \mathrm{a}+2\mathrm{d}, ....... \mathrm{a}+2\mathrm{nd} \mathrm{are} \mathrm{in} \mathrm{A}.\mathrm{P}\\ \mathrm{Mean} & =& \frac{\mathrm{a}+\mathrm{a}+2\mathrm{nd}}{2}=\mathrm{a}+\mathrm{nd}\\ {\mathrm{\sigma }}^2& =& \frac{\sum {({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}})}^2}{2\mathrm{n}+1}\\ & =& \frac{{(1^2+2^2+3^2+.......+{\mathrm{n}}^2)}^2}{2\mathrm{n}+1} {\mathrm{d}}^2\\ & =& \frac{2\mathrm{n}(\mathrm{n}+1) (2\mathrm{n}+1)}{6(2\mathrm{n}+1)} {\mathrm{d}}^2\\ & =& \frac{\mathrm{n}(\mathrm{n}+1)}{3} {\mathrm{d}}^2\\ \mathrm{\sigma }& =& \sqrt{\frac{\mathrm{n}(\mathrm{n}+1)}{3}} \mathrm{d}\end{array}$