Question

If each observation of a raw data whose standard deviation is σ is multiplied by a, then write the S.D. of the new set of observations.

Answer 1

We know, 

Standard deviation  \(\sigma = \sqrt{\frac{\Sigma(x_i-\bar X)^2}{n}}\) 

And, mean \(\bar X\) = \(\frac{1}{n}\Sigma x_i \) 

Now, multiply by a in x_i

New standard deviation,   \(\sigma_{new} = \sqrt{\frac{\Sigma(ax_i-\bar X)^2}{n}}\)  

\(\sigma_{new} = \sqrt{\frac{\Sigma(a^2x_i-\bar X)^2}{n}}\)  

\(\sigma_{new} = |a| \sqrt{\frac{\Sigma(x_i-\bar X)^2}{n}}\) 

\(\sigma_{new} = |a|\sigma_{old}\)

Hence, Proved