Question

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

Answer 1

Standard deviation , $\sigma =\sqrt{\frac{\sum (x_i-\overline{x})}{n}}$

Here, $\overline{x}$ represents the arithmetic mean

Multiplying each $x_i$ by a:

${\overline{x}}_{new}=\frac{1}{n}\sum a{x}_i=a\times \frac{1}{n}{\sum }_i{x}_i=a{\overline{x}}_{old}$

New standard deviation , $\sigma =\sqrt{\frac{\sum (a_i-\overline{a}{)}^2}{n}}$

$=\sqrt{\frac{{\sum }_i{a}^2.(x_i-\overline{x}{)}^2}{n}}=\left|a\right|\sqrt{\frac{{\sum }_t(x_i-\overline{x}{)}^2}{n}}=\left|a\right|.\sigma$