The mean and standard deviation of six observations are $8$ and $4,$ respectively. If each observation is multiplied by $3,$ find the new mean and new standard deviation of the resulting observations.

Vedclass pdf generator app on play store
Vedclass iOS app on app store

Let the observations be $x_{1}, x_{2}, x_{3}, x _{4}, x_{5} ,$ and $x_{6}$

It is given that mean is $8$ and standard deviation is $4$

Mean, $\bar{x}=\frac{x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}}{6}=8$       .......$(1)$

If each observation is multiplied by $3$ and the resulting observations are $y_{i},$ then

$y_{1}=3 x_{1}$ i.e., $x_{1}=\frac{1}{3} y_{1},$ for $i=1$ to $6$

New Mean, $\bar{y}=\frac{y_{1}+y_{2}+y_{3}+y_{4}+y_{5}+y_{6}}{6}$

$=\frac{3\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)}{6}$

$=3 \times 8$        .......[ Using  $(1)$ ]

$=28$

Standard deviation, $\sigma  = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^6 {{{\left( {{x_1} - \bar x} \right)}^2}} } $

$\therefore {\left( 4 \right)^2} = \frac{1}{6}\sum\limits_{i = 1}^6 {{{\left( {{x_i} - \bar x} \right)}^2}} $

$\sum\limits_{i = 1}^6 {{{\left( {{x_i} - \bar x} \right)}^2}}  = 96$            ........$(2)$

From $(1)$ and $(2),$ it can be observed that,

$\bar{y}=3 \bar{x}$

$\bar{x}=\frac{1}{3} \bar{y}$

Substituting the values of $x_{1}$ and $\bar{x}$ in $(2),$ we obtain

$\sum\limits_{i = 1}^6 {{{\left( {\frac{1}{3}{y_1} - \frac{1}{3}\bar y} \right)}^2} = 96} $

$ \Rightarrow \sum\limits_{i = 1}^6 {{{\left( {{y_1} - \bar y} \right)}^2} = 864} $

Therefore, variance of new observations $=\left(\frac{1}{6} \times 864\right)=144$

Hence, the standard deviation of new observations is $\sqrt{144}=12$

Similar Questions

The mean and standard deviation of $20$ observations are found to be $10$ and $2$ respectively. On rechecking, it was found that an observation $8$ was incorrect. Calculate the correct mean and standard deviation in each of the following cases:

If wrong item is omitted.

The variance of the data $2, 4, 6, 8, 10$ is

The mean life of a sample of 60 bulbs was 650 hours and the standard deviation was 8 hours. A second sample of 80 bulbs has a mean life of 660 hours and standard deviation 7 hours. Find the overall standard deviation.

Mean of $5$ observations is $7.$ If four of these observations are $6, 7, 8, 10$ and one is missing then the variance of all the five observations is

  • [JEE MAIN 2013]

The mean and variance of $5$ observations are $5$ and $8$ respectively. If $3$ observations are $1,3,5$, then the sum of cubes of the remaining two observations is

  • [JEE MAIN 2023]