Question

The mean of $20$ numbers is $18.$ If $3$ is added to each of the first ten numbers, find the mean of the new set of $20$ numbers.

Answer 1

Given: Mean of $20$ numbers is $18.$

Let the numbers be $x_1,x_2,x_3,........x_{20}.$

We know:
Mean $=\frac{\text{Sum of observations}}{\text{Number of observations}}$
$\therefore \frac{x_1+x_2+.......+x_{20}}{20}=18$

$x_1+x_2+.......+x_{20}=360\dots \dots \dots \left(i\right)$
When $3$ is added to first $10$ numbers then,

New numbers are: $(x_1+3),(x_2+3),.....(x_{10}+3),x_{11},.....x_{20}$
New Mean $=\frac{(x_1+3)+(x_2+3)+........+(x_{10}+3)+x_{11}+........+x_{20}}{20}$

$=\frac{(x_1+x_2+......+x_{10})+(3\times 10)+x_{11}+........+x_{20}}{20}$

$=\frac{(x_1+x_2+.......+x_{20})+30}{20}$

$=\frac{360+30}{20}$ [Using $eq.\left(i\right)$]

$=\frac{390}{20}=19.5$