Question

If each observation is multiplied by ${\scriptstyle{}^{1}/{}_{3}}$ then the mean of the new data will be:-
A.${}^{1}/{}_{3}$ times
B.3 times
C. \[{}^{1}/{}_{\sqrt{3}}\] times
D.${}^{2}/{}_{\sqrt{3}}$ times

Answer 1

Hint: Mean of a data is given by the summation of all the observation divided by the total no. of observation. Use this data to find the result.

Complete step-by-step answer:
 Let there are $n$ observation given by ${{x}_{1}},{{x}_{2}},{{x}_{3}}.......{{x}_{n}}$
Now as described in the hint the mean of the data can be give as
     $\overline{x}=\dfrac{{{x}_{1}},{{x}_{2}},{{x}_{3}}+.......{{x}_{n}}}{n}$
Now since all the observation is multiplied by $\dfrac{1}{3}$ the new observation are
     $\dfrac{{{x}_{1}}}{3};\dfrac{{{x}_{2}}}{3};\dfrac{{{x}_{3}}}{3}.......\dfrac{{{x}_{n}}}{3}$
Therefore the new mean of the data can be given as
\[{{\overline{x}}_{new}}=\dfrac{\left( \dfrac{{{x}_{1}}}{3}+\dfrac{{{x}_{2}}}{3}+\dfrac{{{x}_{3}}}{3}+\cdot \cdot \cdot \cdot \cdot \cdot +\dfrac{{{x}_{n}}}{3} \right)}{n}\]
${{\overline{x}}_{new}}=\dfrac{1}{3}\times \dfrac{\left( {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+.........+{{x}_{n}} \right)}{n}$
We know that the expression multiplied by ${3}/{3}\;$ is the old mean
     \[\therefore \ {{\overline{x}}_{new}}=\dfrac{1}{3}\times \overline{x}\]
$\therefore $ The mean of new data is $\dfrac{1}{3}$ times the mean of old data.

Option A is correct

Note: If each observation is changed by a factor that the mean will also be changed by the same factor. For example if all the observation is increased by 5 the mean is also increased by 5.
Try to prove it using the above solution.