 # Continuous Frequency Distribution

A frequency distribution is a comprehensive way to organize raw data of a quantitative variable. It shows how different values of a variable are distributed and their corresponding frequencies. However, we can prepare two frequency distribution tables, namely discrete frequency distribution and continuous frequency distribution table. In this article, you will learn what is continuous frequency distribution, how to make a frequency table for a continuous variable with detailed steps and examples.

## Continuous Frequency Distribution Definition

A continuous frequency distribution is a series in which the data are classified into different class intervals without gaps and their respective frequencies are assigned as per the class intervals and class width.

### How to Calculate Frequency Distribution

Let’s understand how to prepare a continuous frequency distribution here.

To prepare a frequency distribution, one should address the following five questions.

1. Should we have equal or unequal sized class intervals?

There are two situations where we can define unequal class interval sizes. They are:

(i) When we have data on income and other related variables where the range is very high.

(ii) If many values are concentrated in a small part of the range, using class intervals with equal sizes will lead to a loss of information on various values.

In all other cases, we can define class intervals of equivalent sizes in frequency distributions.

2. How many classes should we have?

Depending on the total number of observations, the number of classes could be between 6 and 15. Therefore, if we are using equal-sized class intervals, we can calculate the number of classes by dividing the range by the size of the class intervals.

3. What should be the size of each class?

We can determine the number of classes once we decide the class interval based on the range of the variable. Thus, we can notice that these two determinations are interlinked. Therefore, we cannot decide on one without deciding on the other.

4. How should we determine the class limits?

Class limits should be definite and explicitly stated. For example, we have two types of class intervals, such as:

(i) Exclusive class intervals: In this type of class interval, an observation equal to either the upper or the lower class limit is excluded from the frequency of the class.

(i) Inclusive class intervals: In this type of class interval, values equal to the lower and upper limits of a class are included in the frequency of the same class.

However, for discrete variables, we can use both exclusive and inclusive class intervals, whereas, for continuous variables, exclusive class intervals are used.

5. How should we get the frequency for each class?

We can find the frequency for each class by counting the number of values in a particular class.

The above points are necessarily to be followed in creating a continuous frequency distribution table.

A summary of the above defined process is given below:

Step 1: Determine the range of the data set.

Step 2: Divide the range by the number of the classes that we want our data in and then round up.

Step 3: Create class intervals using class width.

Step 4: Obtain the frequency for each class.

Let’s apply the above process to get the frequency distribution table for the given data.

## Continuous Frequency Distribution Table

Consider the following data.

17, 30, 37, 34, 39, 32, 30, 35, 12, 14, 12, 14, 14, 0, 25, 25, 25, 28, 47, 42, 49, 49, 45, 49, 46, 41, 60, 64, 62, 40, 43, 48, 48, 49, 49, 40, 41, 59, 51, 53, 82, 80, 85, 90, 98, 90,56, 55, 57, 55, 10, 14, 51, 50, 56, 70, 75, 64, 60, 66, 69, 62, 61, 70, 76, 70, 59, 56, 59, 57, 59, 55, 20, 22, 56, 51, 55, 56, 55, 50, 54, 66, 69, 64, 66, 60, 65, 62, 45, 47, 44, 40, 44, 65, 66, 65, 71, 82, 82, 90

The stepwise procedure to prepare a table of continuous frequency distribution is given below:

Step 1: Determine the range of the data set.

Maximum value = 98

Minimum value = 0

Range = Maximum value – Minimum value = 98 – 0 = 98

Step 2: Divide the range by the number of the classes that we want our data in and then round up.

Let the number of classes be 10.

Class width = 98/10 = 9.8

Thus, we can consider 10 as the class size.

Step 3: Create class intervals using class width.

For the above data, exclusive class intervals can be created and avoid taking the value which is equal to the upper limit of the class while writing the frequencies.

0 – 10, 10 – 20, 20 – 30, 30 – 40, 40 – 50, 50 – 60, 60 – 70, 70 – 80, 80 – 90, 90 – 100.

Step 4: Obtain the frequency for each class.

 Class interval Frequency Corresponding observations 0 – 10 1 0 10 – 20 8 10, 12, 12, 14, 14, 14, 14, 17 20 – 30 6 20, 22, 25, 25, 25, 28 30 – 40 7 30, 30, 32, 34, 35, 37, 39 40 – 50 21 40, 40, 40, 41, 41, 42, 43, 44, 44, 45, 45, 46, 47, 47, 48, 48, 49, 49, 49, 49, 49 50 – 60 23 50, 50, 51, 51, 51, 53, 54, 55, 55, 55, 55, 55, 56, 56, 56, 56, 56, 57, 57, 59, 59, 59, 59 60 – 70 19 60, 60, 60, 61, 62, 62, 62, 64, 64, 64, 65, 65, 65, 66, 66, 66, 66, 69, 69 70 – 80 6 70, 70, 70, 71, 75, 76 80 – 90 5 80, 82, 82, 82, 85 90 – 100 4 90, 90, 90, 98 Total 100

Hence, the above table shows the continuous frequency distribution for the given data values.

### Continuous Frequency Distribution Examples

Example 1:

Prepare a frequency distribution by inclusive method taking a class interval of 7 from the following data.

28, 17, 15, 22, 29, 21, 23, 27, 18, 12, 7, 2, 9, 4, 1, 8, 3, 10, 5, 20, 16, 12, 8, 4, 33, 27, 21, 15, 3, 36, 27, 18, 9, 2, 4, 6, 32, 31, 29, 18, 14, 13, 15, 11, 9, 7, 1, 5, 37, 32, 28, 26, 24, 20, 19, 25, 19, 20, 6, 9

Solution:

For the given data:

Range = Maximum value – Minimum value = 37 – 1 = 36

Number of classes = 36/7 = 5.1 {since the class size is 7 as per the given}

Thus, we can define 5 classes.

The inclusive class intervals can be written as:

0 – 7, 8 – 15, 16 – 23, 24 – 31, 32 – 39

Let’s write the frequencies of these classes.

 Class intervals Frequencies Corresponding data values 0 – 7 15 1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7 8 – 15 15 8, 8, 9, 9, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 15 16 – 23 14 16, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 23 24 – 31 11 24, 25, 26, 27, 27, 27, 28, 28, 29, 29, 31 32 – 39 5 32, 32, 33, 36, 37 Total 60

Example 2: Convert the class intervals of the following frequency distribution into continuous classes.

 Class interval 26 – 30 31 – 35 36 – 40 41 – 45 46 – 50 51 – 55 56 – 60 61 – 65 Frequency 9 5 16 12 12 26 14 6

Solution:

We can convert the given frequency distribution into continuous frequency distribution by subtracting 0.5 from the lower limit of the class intervals and adding 0.5 to the upper limit of class intervals in each interval.

 Class interval 25.5 – 30.5 30.5 – 35.5 35.5 – 40.5 40.5 – 45.5 45.5 – 50.5 50.5 – 55.5 55.5 – 60.5 60.5 – 65.5 Frequency 9 5 16 12 12 26 14 6