Question

If the mean of a set of observations $x_1,x_2,....,....,x_{10}$ is $20$. Find the mean of $x_1+4,x_2+8,x_3+12,........,x_{10}+40$.

Answer 1

$mean$ of $x_1,x_2,x_3\dots x_{10}$ is $20.$

$\overline{x}=\frac{{\sum }_{n=1}^{10}x}{10}=20.$

${\sum }_{n=1}^{10}x=200.$

Now, $x_1+4,x_2+8,x_3+12\dots x_{10}+40.$

$[x_1,x_2,x_3\dots x_{10}]+[4+8+12\dots 40]$

We need to find the new $mean.$

$\therefore$ Sum of constant terms are in $AP$ where $a=4,d=4,n=10.$

$S_n=\frac{n}{2}[2a+(n-1\left)d\right],$

$S_n=\frac{10}{2}[8+9(4\left)\right],$

$S_n=220.$

$\therefore$ New mean $=\frac{{\sum }_{n=1}^{10}x+220}{10}=\frac{200+220}{10}=\frac{420}{10}=42$