### Theory:

Consider the following data table and let us try to construct a frequency table with histogram, and a frequency table without a histogram using the same data table.

Problem:

A school management decides to conduct a sports meet for the academic year. The management discusses with the sports teachers of the school and comes up with an action plan.

According to the action plan, the number of students in the school will be classified considering their ages and competing against students of the same age group.

The data table is as follows:

 Age groups $$5$$ - $$8$$ $$8$$ - $$11$$ $$11$$ - $$14$$ $$14$$ - $$17$$ Number of students $$75$$ $$89$$ $$70$$ $$100$$

Construct the following using the given data table:

1. A frequency polygon using a histogram

2. A frequency polygon without using a histogram
A frequency polygon using a histogram
Step $$1$$: Find the mid-point of each of the intervals, and make a tabular column.

 Age groups Mid-point Number of students $$5$$ - $$8$$ $$6.5$$ $$75$$ $$8$$ - $$11$$ $$9.5$$ $$89$$ $$11$$ - $$14$$ $$12.5$$ $$70$$ $$14$$ - $$17$$ $$15.5$$ $$100$$

Step $$2$$: Draw a histogram of the given data Step $$3$$: Mark the mid-points on each of histogram's frequency rectangles. Step $$4$$: Join the mid-points with a straight line. Step $$5$$: If the class intervals before the first rectangle and after the last rectangles are unknown, always assume an equivalent class intervals both preceding and succeeding the rectangles.

The assumed intervals are called imagined class intervals.

Now, calculate the mid-point of the imagined class intervals and extend a line up to the imagined class intervals' mid-points. A frequency polygon without using a histogram
Step $$1$$: Find the mid-point of each of the intervals, and make a tabular column similar to creating frequency polygons with histograms.

Step $$2$$: Mark the mid-points on the graph. Step $$3$$: Join the mid-points with a straight line. Step $$4$$: If the class intervals before the first rectangle and after the last rectangles are unknown, always assume an equivalent class intervals both preceding and succeeding the rectangles.

The assumed intervals are called imagined class intervals.

Now, calculate the mid-point of the imagined class intervals and extend a line up to the imagined class intervals' mid-points. The required frequency polygon is now constructed.