If each observation of a raw data whose variance is $\mathrm{\sigma }$ is multiplied by h, then the variance of the new set is

Let ${\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{n}}$be the raw data. Then, 
${\mathrm{\sigma }}^2=\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}} {\left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{X}}\right)}^2$
If each value is multiplied by h, then the values become ${\mathrm{hx}}_1{\mathrm{hx}}_2,\dots ,{\mathrm{hx}}_{\mathrm{n}}$.Then AM of the values is
$\frac{{\mathrm{hx}}_1+{\mathrm{hx}}_2+\cdots +{\mathrm{hx}}_{\mathrm{n}}}{\mathrm{n}}=\mathrm{h}\overline{\mathrm{X}}$
Therefore, the variance of the new set of values is
$\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}} \left({\mathrm{hx}}_1-\mathrm{h}{\overline{\mathrm{X}}}_2\right)={\mathrm{h}}^2\left(\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}} {\left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{X}}\right)}^2\right)={\mathrm{h}}^2{\mathrm{\sigma }}^2$