Relative Frequency Examples - Word Problems
Relative Frequency is defined as the frequency or how often an event occurs to the total number of outcomes in an experiment. It is actually a ratio or proportion of the frequency count for a subgroup of the population to the frequency count of the total population. Relative frequency can be calculated using the below mentioned formula

$Relative\ frequency$ = $\frac{class\ frequency}{sum\ of\ all\ frequencies}$

Relative frequency of a number of events can be calculated by forming a relative frequency table. The first column of the table shall contain the category names and the second column shall contain the counts. The total number of counts is counted and placed at the end of the count column. One more column is added to the right and then the relative frequency is calculated by dividing each of the count by the total counts added up. This gives us the relative frequency of each of the category. For example India cricket team has won $10$ out of $12$ matches in World Cup, so the relative frequency of winning would be:

$Relative\ frequency$ = $\frac{Number\ of\ matches\ won}{Total\ number\ of\ matches\ played}$

Relative frequency = $\frac{10}{12}$ = $0.833333$ = $83.33 \%$

Word Problems

Example 1: 

Construct the relative frequency table for the following data: $1, 4, 9, 6, 3, 3, 5, 7, 9, 2, 8, 2, 5, 8, 5, 8, 1, 10, 4, 2$

Solution: 

The relative frequency table contains three columns. First column contains the digits given represented by $x$, the second column contains the frequency $f$ which is the number of times each of the digit is given in the data set and the third column contains the relative frequency which we need to calculate using the formula:

Relative frequency = $\frac{f}{n}$

Where, $f$ is the frequency and n is the total frequency given

x f Relative Frequency
1 2 $\frac{2}{20}$ = 0.1
2 3 $\frac{3}{20}$ = 0.15
3 2 $\frac{2}{20}$ = 0.1
4 2 $\frac{2}{20}$ = 0.1
5 3 $\frac{3}{20}$ = 0.15
6 1 $\frac{1}{20}$ = 0.05
7 1 $\frac{1}{20}$ = 0.05
8 3 $\frac{3}{20}$ = 0.15
9 2 $\frac{2}{20}$ = 0.1
10 1 $\frac{1}{20}$ = 0.05
Total : 20

Thus, the above relative frequency table gives us the relative frequency or the proportion or percentage that each of the digits present in the data set.
Example 2:

The project grades obtained by the students in a class is given below:

Grades Male Female Total
A 10 20 30
B 15 12 27
C 8 24 32
D 18 11 29
F 9 16 25
Total 60 83

a) What percentage of the students earned grade $B$?

b) What percentage of the students earned the grade $A$ or grade $B$?

c) What percentage of the students was male?

d) What percentage of the students was female?

e) What percentage of the male students secured grade $A$?

f) What percentage of the female students got the grade $C$?

Solution: 

a) The total male and female students are $143$ in number out of which the number of students who secured grade $B$ is $27$. Therefore, the relative frequency gives us the percentage of the students who earned grade $B$. It is 

$ \%$ of students securing grade $B$ = Number of students who got grade $B$

Total number of students in the class

= $27 \times$ $\frac{100}{143}$

= $18.88 \%$

b) The total male and female students are $143$ in number out of which the number of students who secured grade $B$ is $27$ and the number of students securing grade $A$ is $30$. Therefore, the relative frequency gives us the percentage of the students who earned grade $B$ or grade $A$. It is 

$ \%$ of students securing grade $B$ = Number of students who got grade $B$

Total number of students in the class

= $27 \times$ $\frac{100}{143}$

= $18.88 \%$

$ \%$ of students securing grade $A$ = Number of students who got grade $A$

Total number of students in the class

= $30 \times$ $\frac{100}{143}$

= $20.98 \%$

Thus, the percentage of the students earned the grade $A$ or grade $B$ = $18.88 + 20.98$ = $39.86 \%$

c) The total male and female students are $143$ in number out of which the number of students who are male is $60$. Therefore, the relative frequency gives us the percentage of the students who are male. It is

$ \%$ of male students = $\frac{Number\ of\ male\ students}{Total\ number\ of\ students}$

= $60 \times$ $\frac{100}{143}$

= $41.96 \%$

d) The total male and female students are $143$ in number out of which the number of students who are female is $83$. Therefore, the relative frequency gives us the percentage of the students who are female. It is

$ \%$ of female students = $\frac{Number\ of\ female\ students}{Total\ number\ of\ students}$

= $83 \times$ $\frac{100}{143}$

= $58.04 \%$

e) The total male and female students are $143$ in number out of which the number of male students who secured grade $A$ is $10$. Therefore, the relative frequency gives us the percentage of the male students who earned grade $A$. It is 

$ \%$ of students securing grade $A$ = Number of students who got grade $A$
Total number of students in the class

= $10 \times$ $\frac{100}{143}$

= $6.99 \%$

f) The total male and female students are $143$ in number out of which the number of female students who secured grade $B$ is $12$. Therefore, the relative frequency gives us the percentage of the female students who earned grade $B$. It is 

$ \%$ of female students securing grade $B$ = $\frac{Number\ of\ female\ students\ who\ got\ grade\ B}{Total\ number\ of\ students\ in\ the\ class}$

= $12 \times$ $\frac{100}{143}$

= $8.39 \%$
Example 3: 

A survey was carried out on a population of $100$ people to find out the highest percentage of type of vehicle used. The data collected is as follows: $45$ used car, $32$ used public transports, $11$ rode bicycle, $12$ walked

Solution: 

Relative frequency of people using car as the mode of transport = $\frac{Number\ of\ people\ using\ car}{total\ number\ of\ people}$

= $\frac{45}{100}$ = $0.45$

Relative frequency of people using public transports as the mode of transport = $\frac{Number\ of\ people\ using\ public\ transports}{total\ number\ of\ people}$

= $\frac{32}{100}$ = $0.32$

Relative frequency of people using bicycle as the mode of transport = $\frac{Number\ of\ people\ using\ bicycle}{total\ number\ of\ people}$

= $\frac{11}{100}$ = $0.11$

Relative frequency of people who chose walking as the mode of transport = $\frac{Number\ of\ people\ who\ chose\ walking}{total\ number\ of\ people}$

= $\frac{12}{100}$ = $0.12$

Thus the highest percentage of the type of vehicle used is car and the percentage is $45 \%$
Example 4:

An examination was held to check whether Company $XYZ$ was producing good or defective fans. So, $200$ fans were selected randomly and tested. After testing it was observed that $40$ out of $200$ fans were defective. Find out the relative frequency

Solution: 

Number of defective fans = $40$

Total number of fans tested = $200$

Therefore, relative frequency of the defective fans = $\frac{Number\ of\ defective\ fans}{Total\ number\ of\ fans}$

= $\frac{40}{200}$ = $20 \%$ 

Relative frequency of good fans = $\frac{Number\ of\ good\ fans}{Total\ number\ of\ fans}$

= $\frac{(200 – 40)}{200}$ = $\frac{160}{200}$ = $80 \%$
Example 5: 

The following table shows us the number of males and females who buys two types of cars, either the Sports Utility Vehicle (SUV) or the Sports Car in a survey

Gender SUV Sports Car Total
Male 21 39 60
Female 135 45 180
Total 156 84 240

a) Find out the relative frequency of the number of male who buys Sports car

b) Find out the relative frequency of the number of female who buys Sports utility Vehicle

c) Find out the relative frequency of the number of females who buys both the types of car

d) Find out the relative frequency of both male and female buying Sports Utility Vehicle (SUV)

Solution: 

a) The relative frequency of the number of male who buys Sports car

= $\frac{Number\ of\ males\ who\ buys\ Sports\ Car}{Total\ Number\ of\ sample\ using\ both\ type\ of\ vehicle}$

= $\frac{39}{240}$ = $0.1625$

b) The relative frequency of the number of female who buys Sports utility Vehicle

= $\frac{Number\ of\ females\ who\ buy\ Sports\ Utility\ Vehicle}{Total\ Number\ of\ sample\ using\ both\ type\ of\ vehicle}$

= $\frac{135}{240}$ = $0.5625$

c) The relative frequency of the number of females who buys both the types of car

= $\frac{Number\ of\ females\ who\ buys\ both\ the\ type\ of\ cars}{Total\ Number\ of\ sample\ using\ both\ type\ of\ vehicle}$

= $\frac{180}{240}$ = $0.75$

d) The relative frequency of both male and female buying Sports Utility Vehicle (SUV)

= $\frac{\text{Number of both male and female buying Sports Utility Vehicle (SUV)}}{\text{Total Number of sample using both type of vehicle}}$

= $\frac{156}{240}$ = $0.65$