Question

The arithmetic mean of a set of 20 values is 55. What will be the new mean, if each of the 20 value is: [4 MARKS]
i) increased by 5?
ii) decreased by 5?
iii) multiplied by 5?
iv) divided by 5?

Answer 1

Each Part: 1 Mark

Let the 20 values be $x_1,x_2,\dots \dots ,x_{20}$
Sum of these 20 values $=x_1+x_2+\dots \dots +x_{20}$
Given mean of the 20 values = 55
$\Rightarrow \frac{x_1+x_2+\dots \dots +x_{20}}{20}=55$
$\Rightarrow x_1+x_2+\dots \dots +x_{20}=55\times 20=1100$
i) Each of the 20 values is increased by 5
$\therefore$ New values will be $(x_1+5),(x_2+5),\dots \dots (x_{20}+5)$
Their sum
$=(x_1+5)+(x_2+5)+\dots \dots +(x_{20}+5)$
$=(x_1+x_2+\dots \dots +x_{20})+20\times 5$
$=1100+100=1200$
Total number of values = 20
$\therefore$ The mean of new values $=\frac{1200}{20}=60$

ii) Each of the 20 values is decreased by 5
$\therefore$ New values will be $(x_1-5),(x_2-5),\dots \dots (x_{20}-5)$
Their sum = 1100 - 100 = 1000
Total number of values = 20
$\therefore$ The mean of new values $=\frac{1000}{20}=50$

iii) Each of the 20 values is multiplied by 5
$\therefore$ New values will be $5x_1,5x_2,\dots \dots 5x_{20}$
Their sum $=5\times 1100=5500$
Total number of values = 20
$\therefore$ The mean of new values $=\frac{5500}{20}$
=275

iv) Each of the 20 values is divided by 5
$\therefore$ New values will be $\frac{x_1}{5},\frac{x_2}{5},\dots \dots ,\frac{x_{20}}{5}$
Their sum $=\frac{1}{5}\times 1100=220$
Total number of values = 20
$\therefore$ The mean of new values $=\frac{220}{20}=11$