Ogive (Cumulative frequency curve)
Ogive is a graphic presentation of
the cumulative frequency distribution of continuous series. In a continuous
series, if the upper limit (or the lower limit) of each class interval is taken
as x-coordinate and its corresponding c.f. as y-coordinate and the points are
plotted in the graph, we obtain a curve by joining the points with freehand.
Such a curve is known as ogive or cumulative frequency curve.
Since ‘less than’ c.f. and ‘more than’
c.f. are the two types of cumulative frequency distribution, there are two
types of ogive. They are less than ogive and more than ogive.
1. Less than
ogive (or less than cumulative frequency curve)
When the upper limit of each class interval is taken as x-coordinate and its corresponding frequency as y-coordinate, the ogive so obtained is known as less than ogive (or less than cumulative frequency curve). Obviously, less than ogive is an increasing curve, sloping upwards from left to right and has the shape of an elongated S.
2. More than
ogive (or more than cumulative frequency curve)
When the lower limit of each class interval is taken as
x-coordinate and its corresponding frequency as y-coordinate, the ogive so
obtained is known as more than ogive (or more than cumulative frequency curve).
More than ogive is a decreasing curve sloping downward from left to right and
has the shape of an elongated S, upside down.
Construction of less than ogive and more than ogive:
To construct a ‘less than’ or ‘more than’
ogive, we proceed as follows:
1) Make a less than cumulative frequency table.
2) Choose the suitable scale and make the upper class limit of each
class interval along x-axis and cumulative frequencies along y-axis.
3) Plot the coordinates (upper limit, less than c.f.) on the graph.
4) Join the point by freehand and obtain a less than ogive.
Worked Out Examples
Example 1: The table given below shows
the marks obtained by 80 students in science. Construct (i) less than
ogive (ii) more than ogive.
Marks |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |
No. of students |
3 |
8 |
17 |
29 |
15 |
6 |
2 |
Solution: Here,
Less than cumulative frequency table:
Marks |
No. of students |
Upper limit |
Less than c.f. |
0-10 10-20 20-30 30-40 40-50 50-60 60-70 |
3 8 17 29 15 6 2 |
10 20 30 40 50 60 70 |
3 (less than
10) 11 (less
than 20) 28 (less
than 30) 57 (less
than 40) 72 (less
than 50) 78 (less than
60) 80 (less
than 70) |
The less than ogive graph,
More
than cumulative frequency table:
Marks |
No. of students |
Upper limit |
More than c.f. |
0-10 10-20 20-30 30-40 40-50 50-60 60-70 |
3 8 17 29 15 6 2 |
0 10 20 30 40 50 60 |
80 (more
than 0) 80-3 = 77
(more than 10) 77-8 = 69
(more than 20) 69-17 = 52
(more than 30) 52-29 = 23
(more than 40) 23-15 = 8
(more than 50) 8-6 = 2
(more than 60) |
If we draw the both less than ogive and more than ogive of
a distribution on the same graph paper, they intersect at a point. The foot of
the perpendicular drawn from the point of intersection of two ogives to the
x-axis gives the value of median of the distribution.
For example,
Marks |
f |
Upper limit |
Less than c.f. |
Lower limit |
More than c.f. |
0-10 10-20 20-30 30-40 40-50 50-60 60-70 |
3 8 17 29 15 6 2 |
10 20 30 40 50 60 70 |
3 11 28 57 72 78 80 |
0 10 20 30 40 50 60 |
80 77 69 52 23 8 2 |
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