Theory:

Class interval can be divided into two categories:
 
Figure 1.svg
Continuous series:
When there is no break between two classes given in numerical order, it is called a continuous series.
Example:
Class \(1\): \(0\) - \(10\)
 
Class \(2\): \(10\) - \(20\)
 
Class \(3\): \(20\) - \(30\)
 
In this example, there is no gap between classes \(1\) and \(2\); similarly, there is no gap between classes \(2\) and \(3\).
 
Figure 2.svg
Discontinuous series:
When there is a break or gap between two classes given in numerical order, it is called a discontinuous series.
Example:
Class \(1\): \(0\) - \(10\)
 
Class \(2\): \(11\) - \(20\)
 
Class \(3\): \(21\) - \(30\)
 
In this example, there is a gap of \(1\) unit between classes \(1\) and \(2\); similarly, there is a gap of \(1\) unit between classes \(2\) and \(3\).
 
Figure 3.svg
Can a discontinuous series be converted into a continuous series?
 
Yes, a discontinuous series can be converted into a continuous series in a few steps.
 
Let us look at it in detail using the following example.
 
Class \(1\): \(5\) - \(20\)
 
Class \(2\): \(25\) - \(40\)
 
Class \(3\): \(45\) - \(60\)
 
Step \(1\): Consider the gap between the classes \(1\) and \(2\).
 
Class \(1\) ends with \(20\), and class \(2\) begins with \(25\).
 
Therefore, the gap between classes \(1\) and \(2\):
 
\(25 - 20 = 5\)
 
Gap between class \(1\) and class \(2 = 5\)
 
Step \(2\): Convert class \(1\) into a continuous series.
 
Class \(1\): \(5\) - \(20\)
 
Gap between class \(1\) and class \(2 = 5\)
 
\(\text{Lower boundary} = \text{Lower limit} - \text{Half of the gap}\)
 
\(= 5 - \frac{1}{2}(5)\)
 
\(= 5 - 2.5\)
 
\(= 2.5\)
 
Lower limit of class \(1 = 2.5\)
 
\(\text{Upper boundary} = \text{Upper limit} + \text{Half of the gap}\)
 
\(= 20 + \frac{1}{2}(20)\)
 
\(= 20 + 2.5\)
 
\(= 22.5\)
 
Upper limit of class \(1 = 22.5\)
 
Now, the updated limit of class \(1\) is \(2.5\) - \(22.5\).
 
Step \(3\): Apply steps \(1\) and \(2\) to all the other classes available in a series and make the required conversion.
 
Now, the continuous series converted from the discontinuous series will look like this.
 
Class \(1\): \(2.5\) - \(22.5\)
 
Class \(2\): \(22.5\) - \(42.5\)
 
Class \(3\): \(42.5\) - \(62.5\)
 
Finally, let us summarize the necessary classification of data.
 
Figure 28.svg
 
Important!
1. If a data set consists of a discontinuous data set, always convert the discontinuous series to a continuous series.
 
2. If the upper limit and the lower limit of the class interval belongs only to a single class, it is called an inclusive series. For example, \(11\) - \(20\), \(21\) - \(30\), \(31\) - \(40\) and so on. An inclusive series is also called a discontinuous series.
 
3. If the class interval's upper limit extends as the lower limit of the next class interval, then it is called an exclusive series. For example, \(10\) - \(20\), \(20\) - \(30\), \(30\) - \(40\) and so on. An exclusive series is also called a continuous series.