The mean and S.D of $1,2,3,4,5,6$  is 

$\overline{x}=\frac{1+2+3+4+5+6}{6}=\frac{21}{6}=\frac{7}{2}$

If $S.D=\sigma ,{\sigma }^2+x^{-2}=\frac{1}{6}\Sigma x_i^2$

$\frac{1}{6}\Sigma x_i^2=\frac{1}{6}.\frac{1}{6}n(n+1)(2n+1)=\frac{1}{6}.\frac{6\times 7\times 13}{6}=\frac{91}{6}$

$\begin{array}{l}\Rightarrow {\sigma }^2+\frac{49}{4}=\frac{91}{6}\\ \Rightarrow {\sigma }^2=\frac{91}{6}-\frac{49}{4}\end{array}$

$\begin{array}{l}=\frac{182-147}{12}\\ =\frac{35}{12}\\ \therefore \sigma =\sqrt{\frac{35}{12}}\end{array}$