Question

If the mean and standard deviation of 75 observations is 40 and 8 respectively, find the new mean and standard deviation if

(i) Each observation is multiplied by 5.

(ii) 7 is added to each observation.

Answer 1

Given: $\overline{X}=40,\sigma =8,N=75$

(i) A "change of scale" affects the value of both $\overline{X}$ and $\sigma$. A change of scale occurs when each observation is multiplied by a constant. As a result to that, $\overline{X}$ and $\sigma$ also get multiplied by that constant.

In our question, that constant is 5.

Hence,

New $\overline{X}=$ Old $\overline{X}\times 5$

$=40\times 5$

$=200$

and,

New $\sigma =$ Old $\sigma \times 5$

$=8\times 5$

$=40$

(ii) A "change of origin" affects the value of $\overline{X}$ but not that of $\sigma$. A change of origin occurs when each observation is increased by a constant. As a result to that, $\overline{X}$ also increases by the same constant, while $\sigma$ remains unchanged.

In our question, that constant is 7.

Hence,

New $\overline{X}=$ Old $\overline{X}+7$

$=40+7$

$=47$

while,

New $\sigma =$ Old $\sigma$

$=8$