### Theory:

Class interval can be divided into two categories:

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 $$3$$.

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 $$3$$.

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.

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.