**Introduction**

Statistics is the branch of mathematics that revolves around the collection of data. After the collection of data, we have to find ways to condense them in tabular form in order to study their salient features. Such an arrangement is called the presentation of data. One such form of presentation of data is through the frequency table. A frequency table or frequency distribution is a method to present raw data in the form from which one can easily understand the information contained in the raw data. The frequency table can be constructed to provide information in ways – frequency, cumulative frequency and relative frequency. Let us learn what we mean by relative frequency.

**Definition**

We know that a frequency is the number of times a value of the data occurs. Relative frequency is the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes.

**Formula**

To find the relative frequencies, divide each frequency by the total number of outcomes in the given data. This means that –

**Relative Frequency = **$\frac{Frequency\: of\: the\: Class}{Total\: Frequency}$

Let us understand it through an example.

**Example**

Given below are the ages of 25 students of Grade 8 in a school. Find out the relative frequency of each class.

15, 16, 16, 14, 17, 17, 16, 15, 15, 16, 16, 17, 15, 16, 16, 14, 16, 15, 14, 15, 16, 16, 15, 14, 15

**Solution**

We have been given the following data of the ages of 25 students of Grade 8 in a school

15, 16, 16, 14, 17, 17, 16, 15, 15, 16, 16, 17, 15, 16, 16, 14, 16, 15, 14, 15, 16, 16, 15, 14, 15

We need to find the relative frequency of each class.

Let us first arrange the data in the tabular form. We can see that there are 4 students of age 14, while there are 8 students aged 15. Similarly, we can see that there are 10 students of the age 16 while there are 3 students of the age 17. Hence, 4 is the frequency for the age 14, 8 is the frequency for the age 15, 10 is the frequency for the age 16 and 17 is the frequency for the age 17. Presenting this data in tabular form, we will have,

Age | Frequency |

14 | 4 |

15 | 8 |

16 | 10 |

17 | 3 |

Now, we know that –

** Relative Frequency = **$\frac{Frequency\: of\: the\: Class}{Total\: Frequency}$

Therefore,

Relative frequency of age 14 = $\frac{4}{25}$ = 0.16

Relative frequency of age 15 = $\frac{8}{25}$ = 0.32

Relative frequency of age 16 = $\frac{10}{25}$ = 0.4

Relative frequency of age 17 = $\frac{3}{25}$ = 0.12

Putting the above information in the tabular form we have

Age | Frequency | Relative Frequency |

14 | 4 | $\frac{4}{25}$ = 0.16 |

15 | 8 | $\frac{8}{25}$ = 0.32 |

16 | 10 | $\frac{10}{25}$ = 0.4 |

17 | 3 | $\frac{3}{25}$ = 0.12 |

Now, that we have understood what we mean by relative frequency, let us move to define the steps that help us calculate the relative frequency of a class.

**How to Calculate Relative Frequency?**

The following steps are used to find out the relative frequency of a class in a given data –

- Prepare a frequency table of the given data. If the data is small, tally marks can also be used to find out the frequency of each class.
- Now, we will add up the total frequency to obtain the sum total of all the frequencies, say N.
- Next, we will divide the frequency of each class by the sum of frequencies, N that we had obtained in the previous step.
- The individual frequency thus obtained for each class is its respective relative frequency.

Let us understand the above step using an example.

**Example**

The following data shows the marks obtained by students in their English exam. Calculate the relative frequency for variate –

15, 18, 16, 20, 25, 24, 25, 20, 16, 15, 18, 18, 16, 24, 15, 20, 28, 30, 27, 16, 24, 25, 20, 18, 28, 27, 25, 24, 24, 18, 18, 25, 20, 16, 15, 20, 27, 28, 29, 16

**Solution**

We have been given the scores obtained by 40 students in an English Exam. The scores are – 15, 18, 16, 20, 25, 24, 25, 20, 16, 15, 18, 18, 16, 24, 15, 20, 28, 30, 27, 16, 24, 25, 20, 18, 28, 27, 25, 24, 24, 18, 18, 25, 20, 16, 15, 20, 27, 28, 29, 16

Let us first arrange these scores in the form of a frequency distribution table. We will have,

Variate | Frequency |

15 | 4 |

16 | 6 |

18 | 6 |

20 | 6 |

24 | 5 |

25 | 5 |

27 | 3 |

28 | 3 |

29 | 1 |

30 | 1 |

Total | 40 |

Here we have been given that the total number of students who appeared in the exam was 40. Hence, the total frequency, N = 40.

Now, we know that

**Relative Frequency = **$\frac{Frequency\: of\: the\: Class}{Total\: Frequency}$

Therefore, in order to obtain the relative frequency of each variate, we will have to divide each frequency by the total frequency, i.e. we will have to divide each frequency by 40.

We will then have,

Variate | Frequency | Relative Frequency |

15 | 4 | $\frac{4}{40}$ = 0.1 |

16 | 6 | $\frac{6}{40}$ = 0.15 |

18 | 6 | $\frac{6}{40}$ = 0.15 |

20 | 6 | $\frac{6}{40}$ = 0.15 |

24 | 5 | $\frac{5}{40}$ = 0.125 |

25 | 5 | $\frac{5}{40}$ = 0.125 |

27 | 3 | $\frac{3}{40}$ = 0.075 |

28 | 3 | $\frac{3}{40}$ = 0.075 |

29 | 1 | $\frac{1}{40}$ = 0.025 |

30 | 1 | $\frac{1}{40}$ = 0.025 |

Total | 40 |

**Probability Frequency vs Relative Frequency**

Can we say that the probability and the relative frequency are the same or do they differ from each other? Probability and the relative frequency are certainly not the same. Let us understand how probability is different from the relative frequency. We know that Probability is the measure of an expected event or an event that might occur. This means that probability is useful in the cases when each outcome is equally likely. On the other hand, Relative frequency on the contrary measures an actual event that has already occurred. In other words, while relative frequency is a practical approach, the probability is a theoretical concept.

Probability | Relative Frequency |

Probability is the measure of an expected event or an event that might occur. | Relative frequency is the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes. |

It is useful in the cases when each outcome is equally likely. | It measures an actual event that has already occurred. |

It is a theoretical concept | It is a practical approach |

Let us understand this through an example.

We know that in a deck of 52 cards, 26 of the cards are white while the other 26 cards are black. Suppose, we wish to draw a white car from the deck. What would be the probability of this draw?

We know that Probability = $\frac{Favourable\: Number\: of\: Outcomes}{Total\: Outcomes}$

Now, in this case, we will have

Favourable number of outcomes = 26 ( as we have 26 white cards)

Total number of outcomes = 52 ( as we have 52 cards in total )

Therefore, the probability for drawing a white card from a deck of 52 cards will be –

Probability = $\frac{26}{52}$ = $\frac{1}{2}$ = 0.5

Now, if we wish to find the relative frequency, we will be requiring two data sets – a) Number of times the white card was actually picked b) Total trials done for picking a card. Therefore, if we assume that there were 20 trials and the white card was picked for 6 times, the relative frequency of the white card will be –

Relative Frequency = $\frac{Number\: of\: times\: the\: white\: card\: got\: picked}{Total\: number\: of\: trials}$

Hence,

Relative Frequency = $\frac{6}{20}$ = 0.3

**Cumulative Relative Frequency**

Cumulative relative frequency is the accumulation of the previous relative frequencies. How do we obtain the cumulative relative frequency of a class? The following steps are followed to find the relative frequency –

- Prepare a frequency table of the given data. If the data is small, tally marks can also be used to find out the frequency of each class.
- Now, we will add up the total frequency to obtain the sum total of all the frequencies, say N.
- Next, we will divide the frequency of each class by the sum of frequencies, N that we had obtained in the previous step.
- Add all the previous relative frequencies to the current relative frequency.
- The last value is equal to the total of all the observations.

Let us understand it through an example.

**Example**

Prepare a cumulative relative frequency table of the following ages ( in years ) of 25 students in a school.

13, 14, 13, 12, 14, 13, 14, 15, 13, 14, 13, 14, 16, 12, 14, 13, 14

Solution

We have been given the data of 25 students in a school. Let us first prepare the frequency distribution table for the given data. We will have

Age | Frequency |

12 | 4 |

13 | 10 |

14 | 8 |

15 | 1 |

16 | 1 |

17 | 1 |

Next, let us find the relative frequency of each class. We will have,

Age | Frequency | Relative Frequency |

12 | 4 | $\frac{4}{25}$ = 0.16 |

13 | 10 | $\frac{10}{25}$ = 0.4 |

14 | 8 | $\frac{8}{25}$ = 0.32 |

15 | 1 | $\frac{1}{25}$ = 0.04 |

16 | 1 | $\frac{1}{25}$ = 0.04 |

17 | 1 | $\frac{1}{25}$ = 0.04 |

Total | 25 |

Now, we will find the cumulative relative frequency of each class.

Age | Frequency | Relative Frequency | Cumulative Relative Frequency |

12 | 4 | 0.16 | 0.16 |

13 | 10 | 0.4 | 0.56 |

14 | 8 | 0.32 | 0.88 |

15 | 1 | 0.04 | 0.92 |

16 | 1 | 0.04 | 0.96 |

17 | 1 | 0.04 | 1 |

Total | 25 |

**Solved Examples**

**Example 1** A coin is tossed 20 times and lands 15 times on heads. What is the relative frequency of observing the coin land on heads?

**Solution** We have been given that a coin is tossed 20 times and lands 15 times on heads. We are required to find the relative frequency of observing the coin land on heads. Let us summarise the information given to us –

Total number of trials = 20

Number of trials on which the coin lands on a head = 15

Now, we know that relative frequency is the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes and

Relative Frequency = $\frac{Number\: of\: times\: the\: coin\: lands\: on\: a\: head}{Total\: number\: of\: trials}$ …….. ( 1 )

Substituting the given information in the equation ( 1 ) we get,

Relative Frequency = $\frac{15}{20}$ = 0.75.

**Hence, the relative frequency of observing the coin land on heads = 0.75.**

**Example 2**** **A coin is tossed 30 times and the following outcomes are observed –

Trial | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Outcome | H | T | T | T | H | T | H | H | H | T |

Trial | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Outcome | H | T | T | H | T | T | T | H | T | T |

Trial | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

Outcome | H | H | H | T | H | T | H | T | T | T |

Find the relative frequency of observing heads after each trial. Also compare it to the theoretical probability of observing heads

**Solution**** **We have been given the following data which shows the outcomes when a coin is tossed 30 times.

Trial | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Outcome | H | T | T | T | H | T | H | H | H | T |

Trial | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Outcome | H | T | T | H | T | T | T | H | T | T |

Trial | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

Outcome | H | H | H | T | H | T | H | T | T | T |

We are first required to find the relative frequency of observing heads after each trial. The first step to this direction will be to count the number of the favourable outcomes. A favourable outcome is when is the outcome is in our event set. We will thus create a table of running count, after each trial, say t, of the number of favourable outcomes, say p, we have observed. For instance, after 20 trials, i.e. t = 20, we have that the outcome as head was observed 8 times while tail as outcome was observed 12 times. This means that the favourable outcome p = 8. We will thus have,

t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

p | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 5 | 5 |

t | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

p | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 8 | 8 | 8 |

t | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

p | 9 | 10 | 11 | 11 | 12 | 12 | 13 | 13 | 13 | 13 |

The next step will be to compute the relative frequency. We know that relative frequency is the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes. This means that,

Relative Frequency ( f ) = $\frac{p}{t}$

Therefore, we will have the relative frequency of each trial as –

t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

f | 1 . 0 0 | 0 . 5 0 | 0 . 3 3 | 0 . 2 5 | 0 . 4 0 | 0 . 3 3 | 0 . 4 3 | 0 . 5 0 | 0 . 5 6 | 0 . 5 0 |

T | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

f | 0 . 5 5 | 0 . 5 0 | 0 . 4 6 | 0 . 5 0 | 0 . 4 7 | 0 . 4 4 | 0 . 4 1 | 0 . 4 4 | 0 . 4 2 | 0 . 4 0 |

t | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

F | 0 . 4 3 | 0 . 4 5 | 0 . 4 8 | 0 . 4 6 | 0 . 4 8 | 0 . 4 6 | 0 . 4 8 | 0 . 4 6 | 0 . 4 5 | 0 . 4 3 |

From the last entry in this table we can now easily read the relative frequency after 30 trials is 0.43. Also, The relative frequency is close to the theoretical probability of 0.5. this also implies that in general, the relative frequency of an event tends to get closer to the theoretical probability of the event as we perform more trials.

**Key facts and Summary**

- A frequency is the number of times a value of the data occurs.
- Relative frequency is the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes.
- Relative frequency measures an actual event that has already occurred.
- Probability is the measure of an expected event or an event that might occur.
- In general, the relative frequency of an event tends to get closer to the theoretical probability of the event as we perform more trials.
- Cumulative relative frequency is the accumulation of the previous relative frequencies.