Question

the mean of 20 numbers is 18 . if 3 is added to each number, what is the new mean ?

Answer 1

Given :-

Mean of 20 numbers = 18

Required to find :-

• New mean ?

Formulae used :-

$\boxed{\rm{ Mean = \dfrac{Sum\;of\;observations }{No. \; of \; observations } }}$

$\boxed{\rm{ Sum \; of \; the \; observations = Mean \times No. \; of \; observations }}$

Solutions :-

Given that :-

Mean of 20 numbers = 18

If 3 is added to every number !

we need to find the new mean .

So,

Let consider ;

Mean of 20 numbers = 18

No. of observations = 20

So,

Using the formula ;

$\boxed{\rm{ Sum \; of \; the \; observations = Mean \times No. \; of \; observations }}$

Hence,

Sum of the observations = 18 x 20

Sum of the observations = 360

So,

The actual sum of the observations is 360

It is given that :-

If 3 is added to each observation of the data

So,

This implies

20 x 3 = 60

Add 60 to the actual sum of the observations

Because ;

3 is added to each and every observation

This implies ,

New sum of the observations = 360 + 60

New sum of the observations = 420

Here,

No. of observations are 20

( since , no new observation is added nor removed )

So,

Using the formula ;

$\boxed{\rm{ Mean = \dfrac{Sum\;of\;observations }{No. \; of \; observations } }}$

Hence

$\rightarrowtail{\tt{ Mean = \dfrac{ 420 }{20}}}$

$\rightarrowtail{\tt{Mean = \dfrac{42}{2}}}$

$\rightarrowtail{\tt{Mean = 21 }}$

$\Large{\leadsto{\overline{\underline{\sf{\therefore{New\;Mean = 21 }}}}}}{\bigstar}$

Answer 2

$\huge\mathfrak{Answer:}$

Mean:

• It is the average of numbers.
• Mean is given by : Sum of observations/Number of observations.

Given:

• We have been given that the mean of 20 numbers is 18.

To Find:

• We need to find the mean when 3 is added to each number.

Solution:

As it is given that the mean of 20 numbers is 18.

$\implies\sf{\dfrac{Σfixi}{Σfi} = mean}$

$\implies\sf{ \dfrac{Σfi}{20} = 18}$

$\implies\sf{ Σfixi = 360}$

Now, when 3 is added to each number, then the total numbers added = 3 × 20 = 60.

∴360 + 60 = 420

Now, the mean for the numbers after 3 is added to each of them =

$\mapsto\sf{ \dfrac{Σfixi}{Σfi} }$

$\implies\sf{ \dfrac{420}{20} }$

$\implies\sf{21}$

Hence, the required mean is 21.