 # Relative Frequency

In general terms, frequency is defined as the number of times an event occurs in a random experiment.

Relative frequency is the ratio of the observed frequency and the total frequency for any experiment conducted randomly. If many relative frequencies are calculated from the same experiment, they will not be equal. However, the sum of all relative frequencies will be one.

For instance, in a game of chess, you win 7 out of 10 times. So, the relative frequency of your victory is 7/10 or 70%.

### How to Calculate Relative Frequency?

By dividing the number of times an event occurs by the total number of trials, we shall obtain the relative frequency for an experiment or survey.

With the help of the example given below, we shall try to find the formula for calculating relative frequency.

 Shoe Size Frequency 34-35 2 35-36 4 36-37 3 37-38 6

This table shows the shoe sizes of 15 people. In order to calculate the relative frequencies for each of them, we need to follow the steps given below.

1. We have to divide the given frequency by the total number of frequencies or the sum of all frequencies (15 in this case).

2. In order to express it as a percentage, the obtained fraction can be multiplied by 100.

In this case, if we have to find the relative frequency for the people whose shoe size is in the range 36-37, we shall divide the given frequency in the table (3) by the sum of all frequencies (15). This gives us 3/15 or 1/5. In the form of a percentage, this would be 20%.

Relative frequency = Frequency of an event/ Total number of trials

### Cumulative Relative Frequency Distribution

When a data set is compiled in the form of a table, the cumulative frequency of each item is the sum of the frequencies of various other data items that come before it.

The cumulative relative frequency of each item is the sum of the relative frequencies of all the items that are mentioned before it. It depends not only on the frequency of each response or item but also on the relationship of other data items with each other.

Let us look at the previous example to understand it better.

 Shoe Size Frequency Relative Frequency Cumulative Relative Frequency 34-35 2 2/15 2/15 35-36 4 4/15 6/15 (4/15+2/15) 36-37 3 3/15 9/15 (4/15+2/15+3/15) 37-38 6 6/15 15/15 (4/15+2/15+3/15+6/15)

After calculating the relative frequencies in the third column, we have added the relative frequency of each data item with the relative frequency of the preceding data items to get the cumulative relative frequency.

### Examples of Relative Frequency:

Question 1: Joisy wanted to know if her coin was biased or not. She flipped in 50 times and got heads 33 times. Calculate the relative frequencies for the occurrence of heads and tails.

Solution:  In the experiment above, the head appears 33 times, which means the tail appears 17 times.

So, going by the formula of relative frequency:

1. For heads = 33/50 or 66%

2. For tails = 17/50 or 34%

Question 2: Two dice are rolled simultaneously. Find the relative frequency of the following events.

1. Occurrence of a doublet

2. Occurrence of 10 as the sum of numbers on both the dice

Solution: On rolling two dice, we can have the following sample space:

(1,1  1,2  1,3  1,4  1,5  1,6

2,1  2,2  2,3  2,4  2,5  2,6

3,1  3,2  3,3  3,4  3,5  3,6

4,1  4,2  4,3  4,4  4,5  4,6

5,1  5,2  5,3  5,4  5,5  5,6

6,1  6,2  6,3  6,4  6,5  6,6 )

The total number of observations is 36.

1. In this, doublets occur six times (1,1  2,2  3,3  4,4  5,5  6,6). So the relative frequency will be 6/36 or 1/6.

2. 10 occurs as the sum in 3 cases (6, 4  5,5  4,6). So, the relative frequency of obtaining 10 as the sum of the numbers on both the dice will be 3/36 or 1/12.

Find Out if The Batch Will Be Approved or Not.

Question 3: In an election, 20,000 votes were polled. Out of these, 9268 people cast their votes in favor of the ruling government. 3456 people were unsure, so they exercised the option of ‘None of the above’. There were two opposition parties. The votes polled in favor of one of them was three times the number of votes polled in favor of the other. Calculate the relative frequency of the votes polled in favor of the two opposition parties.

1. What is Meant By a Range of Probabilities?

Let us consider an example to illustrate this.

A die is cast. What is the probability the number on the top face will be less than seven?

Since there are six elements in our sample space and the number of trials is also 6, so the probability becomes 6/6 or 1.

So, for a sure event, the probability will always be 1.

Now, let us find the probability that the top face of the same die will have a number greater than 7.

Since there are zero elements in our sample space and the number of trials is also 6, so the probability becomes 0/6 or 0.

So, for an impossible event, the probability will always be 0.

Therefore, all probabilities lie in the range of 0-1. Any event that has a probability of 0.5 or 50% is as likely to occur as it is not.

Hence, 0<P(E)<1

2. What are Complementary Events in Probability?

Any two events of an experiment for which the sum is 1 are called complementary events. They cover all the possible outcomes of an experiment.

For example, from a deck of cards, if a card is drawn at random, then the drawing a red card and drawing a black card will be called complementary events as (26/52+26/52)=52/52=1.

P(event occurring)= 1- P(event not occurring)