Question

The mean of six numbers is $23$. If one of the numbers is excluded, the mean of the remaining numbers is $20$. Find the excluded number

Answer 1

Step 1: Find the value of sum of terms

Given, the mean of six numbers items is $\overline{X}=23$.

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

$\frac{x_1+x_2+x_3+x_4+x_5+x_6}{6}=23$

$x_1+x_2+x_3+x_4+x_5+x_6=138$

Step 2: Find the new value of sum of terms. If one of the numbers is excluded.

Let excluded number$=x_1$

$=x_2+x_3+x_4+x_5+x_6$

$=138-x_1$

Step 3: Find the excluded number.

Given, the mean of the remaining numbers is $20$. Which means the mean of five numbers is $20$.

I.e.,

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

$\frac{138-x_1}{5}=20$

$\Rightarrow 138-x_1=100$

$\Rightarrow x_1=138-100$

$\therefore x_1=38$

Hence, the value of excluded number is $38$.