Section 2 – 3B :
Cumulative Frequency Line Graphs (Ogive Graphs)
for Discrete Quantitative Data
In past sections have used Class Frequency Tables to create Cumulative Frequency Tables.
We will now use a Cumulative Frequency Table to create a Cumulative Frequency Line Graph.
A Class Frequency
Table
Hours a
week of
freq (x)
home
work
A Cumulative
Frequency Table
Hours a
week of
home
work
Cumulative
freq (x)
0–4
2
x ≤4
2
5–9
6
x ≤9
8
10 – 14
9
x ≤ 14
17
15 – 19
15
x ≤ 19
32
20 – 24
16
x ≤ 24
48
An Example of a
Class Frequency Polygon
An Example of a
Cumulative Frequency Line Graph
50
40
cumulative
freq (x)
Freq (x)
40
35
30
25
20
15
10
5
30
20
10
0
0 2 7
12 17 22 24
Hours a week of homework
Section 2–3B Lecture
0
Page 1 of 4
4
9
14
19
24
Hours a week of homework
© 2015 Eitel
Cumulative Frequency Line Graph or “Ogive” graphs
In this section we will graph Cumulative Class Frequency Line Graphs using a Cumulative
Frequency Table. Another name for the Cumulative Frequency Line Graph is an “Ogive”
graph. The graph is best used when you want to express a cumulative total. The relative slopes
from point to point indicate greater or lesser increases. Steep slopes Indicate larger changes than
more gradual slopes.
Cumulative Class Frequency Line Graphs are never normal in shape. The heights of the points
start at a lower height and each successive point has a higher height as the cumulative total
frequency gets larger. The last height is the total number of all the data bits.
Steps in Creating a Cumulative Frequency Polygon
The graph starts at the point ( 0 , 0 ). Additional points are then plotted on the graph using x and
y values. The x values are the Upper Class Limits for each class in the Cumulative Frequency
Table and the y values are the Cumulative Frequencies in the Table. Line segments then
connect the point at the origin point to each point in order. The graph ends at the point
directly above the last upper class limit.
Steps in Creating a Cumulative Frequency Line Graph.
from a Cumulative Frequency Table
1. Put numbers on the vertical axis that will be used to represent the cumulative totals
2. Put numbers on the horizontal axis that will be used to represent the x values in the
first column of the Cumulative Frequency Table.
3. Plot a point at point (0 , 0)
4. Plot points using ordered pairs from the Cumulative Frequency Table
( x 1 , cumulative freq 1 ) ( x 2 , cumulative freq 2 ) ( x 3 , cumulative freq 3 ) etc.
5. Starting at the point (0,0) draw line segments connecting the points in order.
Section 2–3B Lecture
Page 2 of 4
© 2015 Eitel
Example 1
The Class Frequency Table
Hours a
week of
freq (x)
home
work
The Cumulative Frequency Table
Hours a
week of Cumulative
home
freq (x)
work
10 –13
2
x ≤ 13
2
14 – 17
6
x ≤ 17
8
18 – 21
9
x ≤ 21
17
22 – 25
15
x ≤ 25
32
26 –29
16
x ≤ 29
48
Steps in graphing a Cumulative Frequency Line Graph.
1. Start at the point (0 , 0)
2. Plot points (13 , 2) at (17 , 8) at (21 , 17) at (25 , 32) and at (29 , 48)
3. Starting at the point (0,0) draw line segments connecting the points in order.
The Cumulative Frequency Table
The Cumulative Frequency Line Graph
Hours a
week of
home
work
Cumulative
freq (x)
x ≤ 13
2
x ≤ 17
8
x ≤ 21
17
10
x ≤ 25
32
0
50
cumulative
freq (x)
40
30
20
17 values less than
or equal to 21
2 values less than or equal to 13
0
x ≤ 29
48
Section 2–3B Lecture
13
17
21
25
29
Hours a week of homework
Page 3 of 4
© 2015 Eitel
Example 2
1. Start at the point (0 , 0)
2. Plot points at (4 , 3) (9 , 10) (14 , 16)
(19 , 20) and (24 , 42)
3. Starting at the point (0, 0) draw line segments connecting the points in order.
The Cumulative Frequency Table
x:
age of
child
Cumulative
Freq (x)
x≤4
3
x ≤9
10
25
20
cumulative
freq (x)
x ≤ 14
The Cumulative Frequency line graph
17
15
10
9 values less
than or equal to 10
5
x ≤ 19
20
x ≤ 24
22
0
4
x≤4
1. Start at the point (0 , 0)
9
14
19
24
Age of Child
Example 3
2. Plot points at (1 , 5) (2 , 12) (3 , 15) (4 , 23) and (5 , 25)
3. Starting at the point (0,0) make line segments connecting the points in order.
The Cumulative Frequency Table
The Cumulative Frequency Line Graph
x:
number
of days a
week at
school
Cumulative
freq (x)
x ≤1
5
x ≤2
12
x ≤3
15
5
x ≤4
23
0
x ≤5
25
25
Section 2–3B Lecture
cumulative
freq (x)
20
15
15 values less
than or equal to 3
10
0
Page 4 of 4
1
2
3
4
5
Days a week at school
© 2015 Eitel