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

Standard deviation,

\[\sigma = \sqrt{\frac{\sum_i \left( x_i - x \right)^2}{n}}\]

Here, 

\[\bar{ x} \] represents the arithmetic mean.
Multiplying each  \[x_i\] bay \[a\] : \[x_{new} = \frac{1}{n}\sum_i a . x_i \]
\[ = a \times \frac{1}{n} \sum_i x_i \]
\[ = a . x_{old}\]

\[\text{ New standard deviation,}  \sigma_{new} = \sqrt{\frac{\sum_i \left( a . x_i - a . x \right)^2}{n}}\]
\[ = \sqrt{\frac{\sum_i a^2 . \left( x_i - x \right)^2}{n}}\]
\[ = \left| a \right|\sqrt{\frac{\sum_i \left( x_i - x \right)^2}{n}}\]
\[ = \left| a \right| . \sigma\]