How To Find Relative Frequency

The word 'relative' is used to indicate that an event is being considered in relation or in proportion to something else.

Frequency is a way to measure how often a particular event occurs. Relative frequency, on the other hand, is a way to measure how often a particular event occurs against total occurrences.

Example:

If 8 out of 20 students travel to school by car, then the frequency for traveling to school by car would be \(8\)

The relative frequency of the same would be:

\[\dfrac 8{20} = 40\%\]

In this mini-lesson, you will also be acquainted with cumulative frequency, while learning about the relative frequency of data.  

Lesson Plan 

Relative frequency can be defined as the number of times an event occurs divided by the total number of events occurring in a given scenario.

To calculate the relative frequency two things must be known:

• Number of total events/trials
• Frequency count for a category/subgroup

\(\text {Relative Frequency}\)=\(\dfrac { Subgroup \: frequency} {Total \:frequency}\)

Example:

Ellie surveys a group of students in her school to learn about their favorite sport. The data collected has been presented below. What will be the relative frequency for volleyball?

The total number of students in the data set is found by adding the heights of all the bars
\[= 10 + 20 + 25 + 5 + 10 = 70\]

The number of students who like volleyball \(= 20\)

So, the relative frequency for the Volleyball will be \(\dfrac {20}{70} = 28.57\) (to two decimals)

The above frequency can also be expressed simply as a fraction \(\dfrac 27\)   

How to Find the Cumulative Frequency?

In statistics, the frequency of the first-class interval is added to the frequency of the second class, and this sum is added to the third class and so on. Frequencies that are obtained this way are known as cumulative frequency (c.f.).  

Example:

Robert is the sales manager of a toy company. On checking his quarterly sales record, he can observe that by the month of April, a total of 83 toy cars were sold.

Month	Number of toy cars sold (Frequency)	Total number of toy cars sold (Cumulative Frequency)
January	20	20
February	30	20 + 30 = 50
March	15	50 + 15 = 65
April	18	65 + 18 = 83

Note how the last cumulative total will always be equal to the total for all observations since all frequencies will already have been added to the previous total. Here, 83 = 20 + 30 + 15 +18

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To find the relative frequency, divide the frequency(f) by the total number of data values (n).

The formula for the relative frequency is given as: \(\dfrac {f}{n}\)

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There is a major difference between relative frequency and probability.

Probability is the measure of an expected event or an event that might occur. It is useful in the cases when each outcome is equally likely. Relative frequency on the contrary measures an actual event that has already occurred. 

One can say the probability is theoretical while relative frequency is more practical.

Let's take an example of cards, to get this concept even better.

As we all know there are 52 cards, out of which 26 cards are black.  

If you had one single chance to randomly pick up a card out of 52 cards, the probability of the black card getting picked will be

\begin{align*} \text {Probability} &= \dfrac{ Number\: of\: black \:cards } {Total\: number\: of \:cards}\\&=\dfrac {26}{52}\\ &=\dfrac12 \\ &=0.5\%\end{align*}

However, to calculate relative frequency, we will need two data sets:

1. Number of times the black cards actually got picked 

2. Total trials done for picking a card

So, if out of 10 trials, black cards got picked only 4 times the relative frequency for the black card is

\begin{align*} \text {Relative frequency} &= \dfrac{ Number\: of\: \:times \:black \:cards \:got\: picked} {Total\: number\: of \:trials\:done}\\&=\dfrac {4}{10}\\  &=40\%\end{align*}

Let us learn how to use the relative frequency formula to solve the below-given problems.

1. There are 16 students in a class, 6 boys and 10 girls. Find out the frequency as well as the relative frequency of boys?

Solution:

Frequency of boys \(\text {(f)}\) = total number of boys  \(=6\)

For relative frequency, we will use the general formula of the relative frequency \(=\dfrac {f}{n}\)                                                  

\[\begin{align*} \text {Relative frequency} &= \dfrac{ Boys \:count} {Total\: student \:count}\\&=\dfrac {6 }{16}\\  &=37.5\%\end{align*}\]

We can express relative frequency both as a fraction \(\dfrac {6 }{16}\) and a percentage \(37.5\%\)

Example 3

 

 

Anna has a packet containing \(20\) candies. Her favorites are the yellow ones and the red ones. The table below shows the frequency of each different candy selected as she picked all \(20\) sweets one by one and finished them all.

Candy color	Yellow	Red	Green	Brown
Frequency	\(6\)	\(6\)	\(3\)	\(5\)

A) What is the relative frequency of the picked candy to be one of her favorites?

B) What is the relative frequency for the brown candy

Solution

Relative frequency = number of times an event has occurred / number of trials

A) Relative frequency of the picked candy to be one of her favorites

\(=\dfrac { Frequency\hspace{2mm} of\hspace{2mm} yellow + Frequency\hspace{2mm} of\hspace{2mm} red\hspace{2mm} candy} {20} = \dfrac {12}{20} = 60\%\)

B) Relative frequency of the brown candy

\(=\dfrac { Frequency\hspace{2mm}of \hspace{2mm} brown \hspace{2mm}candy} {20}\) = \(\dfrac {5}{20}\) = \(25\%\)

\(\therefore\) The relative frequency for Anna's favorite candy is \(60\%\) and for the brown candy it is \(25\%\)

Example 4

 

 

A tire company runs a durability test on its batch of \(600\) tires. The test result concludes that \(560\) out of a batch of \(600\) tires will withstand \(10,000\) miles of normal wear. Find the relative frequency of tires that will not last \(10,000\) miles.

Solution

Relative frequency = number of times an event has occurred / number of trials

\(40\) tires failed the durability test.

The relative frequency of the \(40\) tires

\[=\dfrac{40}{600}\\=6.66\%\]

\(\therefore\) The relative frequency of tires that will not last \(10,000\) miles is \(6.66\%\)

Example 5

 

 

A car dealer wants to calculate the total sales for the past month and wants to know the monthly sales in percentage after weeks 1, 2, 3, and 4. Create a relative cumulative frequency table and present the information that the dealer needs.

Week	No. of Cars Sold
1	10
2	17
3	14
4	11

Solution

First total up the sales for the entire month:

10 + 17 + 14 + 11 = 52 cars

Then find the relative frequencies for each week by dividing the number of cars sold that week by the total:

• The relative frequency for the first week is: 10/52 = 0.19
• The relative frequency for the second week is: 17/52 = 0.33
• The relative frequency for the third week is: 14/52 = 0.27
• The relative frequency for the fourth week is: 11/52 = 0.21

To find the relative cumulative frequencies, start with the frequency for week 1, and for each successive week, total all of the previous frequencies

Week	Cars sold	Relative Frequency	Relative Cumulative Frequency
1	10	0.19	0.19
2	17	0.33	0.19 + 0.33 = 0.52
3	14	0.27	0.52 + 0.27 = 0.79
4	11	0.21	0.79 + 0.21 = 1

*Note that the first relative cumulative frequency is always the same as the first relative frequency, and the last relative cumulative frequency is always equal to 1.

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