Theory:

In class \(IX\), we learned how to represent the data pictorial, such as bar graphs, histograms and frequency polygons. Now, we shall learn how to represent the cumulative frequency distribution graphically.
 
Let us understand the concept using an example.
Example:
Let us construct the cumulative frequency distribution for the below table.
 
Total marks\(100 - 200\)\(200 - 300\)\(300 - 400\)\(400 - 500\)
No. of students\(30\)\(15\)\(26\)\(44\)
 
Click here to learn the median of the above table and the cumulative frequency distribution of less than type and more than type.
 
First, let us draw the cumulative frequency graph for the less than type.
 
Here, \(200\), \(300\), \(400\) and \(500\) are the upper limits of the respective class intervals. To represent the data in the table graphically, we mark the upper limits of the class intervals on the horizontal axis(\(x\)-axis) and their corresponding cumulative frequencies on the vertical axis(\(y\)-axis).
 
Now, plot the points corresponding to the ordered pairs given by (upper limit, cumulative frequency distribution).
 
Therefore, the ordered pairs are \((200, 30)\), \((300,45)\), \((400,71)\), \((500,115)\).
 
Now, join the points using a free hand smooth curve. This curve is called a cumulative frequency curve or an ogive (of the less than type).
 
6.png
 
Now, let us draw the cumulative frequency graph for the more than type.
 
Here, \(100\), \(200\), \(300\), and \(400\) are the lower limits of the respective class intervals. To represent the data in the table graphically, we mark the lower limits of the class intervals on the horizontal axis(\(x\)-axis) and their corresponding cumulative frequencies on the vertical axis(\(y\)-axis).
 
Now, plot the points corresponding to the ordered pairs given by (lower limit, cumulative frequency distribution).
 
Therefore, the ordered pairs are \((100, 115)\), \((200, 85)\), \((300, 70)\), \((400, 44)\).
 
Now, join the points using a free hand smooth curve. This curve is called a cumulative frequency curve or an ogive (of the more than type).
 
3.png
 
We can determine the median of the data using the ogive graph.
 
First method:
 
Here, \(n = 115\), then \(\frac{n}{2} = \frac{115}{2} = 57.5\)
 
Now, plot the point \(57.5\) on the \(y\)-axis. From this point, draw a line parallel to the \(x\)-axis, which meets the curve at a point. From that point, draw a line perpendicular to the \(x\)-axis. The point of intersection of this point with the \(x\)-axis is the median of the data.
 
4.png
 
Second method:
 
Draw two ogive graphs. The two curves intersect at a point. From that point, draw a line perpendicular to the \(x\)-axis. This point is the median of the given data.
 
7.png