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, the mean of $20$ numbers is $18$.

Step 1: Find the value of sum of terms

$\mathrm{Mean}\left(\overline{X}\right)=\frac{\mathrm{Sum}\mathrm{of}\mathrm{terms}}{\mathrm{Number}\mathrm{of}\mathrm{terms}}$

$18=\frac{x_1+x_2+x_3+...+x_{20}}{20}$

$\Rightarrow x_1+x_2+x_3+...+x_{20}=360$

Step 2: Find the new value of sum of terms

If $3$ is added for first ten numbers, then $30$ is added to the total sum, then we get

$x_1+x_2+x_3+...+x_{20}+30$

$=360+30$

$=390$

Step 3: Find the value of sum of terms

New mean, $\left(\overline{X}\right)=\frac{390}{20}$

$=19.5$

Hence, the value of new mean is $19.5$