Section 2 - 3A:
Class Frequency Polygons
and Relativity Frequency Polygons
for Discrete Quantitative Data
We can use class boundaries to represent a class of data
In the past section we created Class Frequency Histograms. Data from a range of numbers was
counted as being in the same class. Each class was represented by one bar. To avoid the gaps
between the bars each bar, or class, was represented by 2 class boundaries.
Class
Boundaries
x: number
of units freq (x)
.5 – 3.5
1–3
2
3.5 – 6.5
4–6
5
6.5 – 9.5
7–9
4
Freq (x)
5
4
3
2
1
.5
6.5
9.5
3.5
Number of Units
The .5 – 3.5 class , or bar , represents the numbers 1, 2, 3
The 3.5 – 6.5 class , or bar , represents the numbers 4, 5, 6
The 6.5 – 9.5 class , or bar , represents the numbers 7, 8, 9
We can also use a Class Midpoint to represent a class of data
The second class in the table above is written as 4 – 6 and stands for any child whose age is 4, 5
or 6 years old. Any child with one of those ages is counted as being in that class. The frequency for
that class is 5. We cannot say exactly what ages the 5 children have. They could all be 4 years old.
They could all be 5 or 6 years old. Many different combinations could result in a frequency of 5.
If we wanted to select one single number to represent the class 4 – 6 then class boundaries will
not work. Choosing 4 to represent 4, 5, 6 would let the lowest number in the class represent the
class. That is not fair. Choosing 6 to represent 4, 5, 6 would let the highest number in the class
represent the class. That is is not fair. The number 5 is in the middle of the class containing 4, 5, 6.
We call the the number in the middle of the class the class midpoint. We will select the class
midpoint of 5 to represent the class 4 5, 6 because it is in the middle.
The single best one number that can represent 4, 5, 6 is the number in the middle of the class.
The Class Midpoint is used as the best single point to represent the entire class.
Section 2 - 3A Lecture
Page 1 of 7
© 2015 Eitel
Finding the Class Midpoint
A class midpoint is the best single value that represents the entire class. The class midpoint is the
average of the lower and upper class limit.
The class midpoint is found by adding the lower and upper class limits together
and then dividing that total by 2.
class midpoint =
(lower class limit + upper class limit)
2
Example 1
Class midpoints
Class
midpoints
Age of
child in
years
Freq
(x)
CM =
0 +4
=2
2
2
0–4
2
CM =
5 +9
=7
2
7
5–9
8
CM =
10 +14
= 12
2
12
10 – 14
15
CM =
15 +19
= 17
2
17
15 – 19
8
CM =
20 + 24
= 22
2
22
20 – 24
2
Example 2
Class midpoints
Class
midpoints
Miles
driven
Freq
(x)
2.5
1–4
10
6.5
5–8
15
10.5
9 – 12
8
14.5
13 – 16
5
18.5
17 – 20
2
1 +4
= 2.5
2
5 +8
CM =
= 6.5
2
9 +12
CM =
= 10.5
2
13 +16
CM =
= 14.5
2
CM =
CM =
Section 2 - 3A Lecture
17 + 20
= 18.5
2
Page 2 of 7
© 2015 Eitel
Using Class Midpoints to create Class Frequency Polygons
The Class Frequency Table below was used to create the Class Frequency Histogram below.
Class
Boundaries
Freq (x)
6
x: number freq (x)
of units
5
.5 – 3.5
1–3
3
3.5 – 6.5
4–6
4
3
6.5 – 9.5
7–9
1
2
9.5 – 12.5
10 – 12
6
12.5 – 1.5
13 – 15
5
4
1
.5
3.5
6.5
9.5
12.5 15.5
Number of Units Enrolled In
Find the class midpoints for each class in the Class Frequency Table. The midpoint represents the x
value that represents the class. The frequency for the class is represented by the height based on the
scale on the y axis. Each class or rectangle can be represented by a point graphed as an ordered
pair with coordinates ( class midpoint, frequency of the class)
Class
Midpoints
x: number freq (x)
of units
Freq (x)
6
1–3
3
5
4–6
4
3
8
7–9
1
2
10 – 12
6
14
13 – 15
5
Section 2 - 3A Lecture
(14, 5)
5
2
11
(11, 6)
4
(5, 4)
(2, 3)
(8, 1)
1
2
Page 3 of 7
5
8
11
14
Number of Units Enrolled In
© 2015 Eitel
Using Class Midpoints to create Class Frequency Polygons
Graph a point (class midpoint, frequency of the class)
for each class and frequency in the table.
Connect the points with line segments.
Class
Midpoints
Freq (x)
6
x: number freq (x)
of units
(11, 6)
5
2
1–3
3
5
4–6
4
3
8
7–9
1
2
11
10 – 12
6
14
13 – 15
5
(5, 4)
(14, 5)
4
(2, 3)
1
(8, 1)
5
8
11
14
Number of Units Enrolled In
2
Add a starting point at ( first lower class limit, 0)
and a ending point at ( last upper class limit, 0).
Connect all the the points with line segments.
Freq (x)
6
(11, 6)
5
4
(14, 5)
(5, 4)
(2, 3)
3
2
1
(1, 0)
1
2
(8, 1)
(15, 0)
5
8
11 14 15
Number of Units Enrolled In
The polygon starts on the x axis at the point (first lower class limit, 0). The first lower class
limit is 1 for this table so the graph starts at the point (1, 0) The graph then ends on the x axis at the
point (last upper class limit, 0). The last upper class limit is 15 for this table so the graph ends at
the point (15, 0). Plot the ordered pairs that represent each class in the table ((2,3) (5,4)
(8,1) (11,6) and (14, 5) (class midpoint, frequency of the class). Connect all the points on the
graph.
The y axis Scale
The y axis is scaled to the numbers in the frequency (x) column. This graphs counts by ones.
Section 2 - 3A Lecture
Page 4 of 7
© 2015 Eitel
Example 2 of Class Frequency Polygons
Class Frequency Table
History Freq
Class
Midpoints Class
(x)
Size
4.5
0–9
1
14.5
10 – 19
4
24.5
20 – 29
6
34.5
30 – 39
2
44.5
40 – 49
5
Class Frequency Polygon
Freq (x)
6
(24.5, 6)
(44.5, 5)
5
4
3
(14.5, 4)
2
1
(34.5, 2)
(4.5, 1)
(49, 0)
(0, 0)
0
4.5
14.5
24.5
34.5
Class Size
44.5
49
The polygon starts on the x axis at the point (first lower class limit, 0). The first lower class
limit is 0 for this table so the graph starts at the point (0, 0) The graph then ends on the x axis at
the point (last upper class limit, 0). The last upper class limit is 49 for this table so the graph ends
at the point (49, 0). Plot the ordered pairs that represent each class in the table (4.5,1)
(14.5,4) (24.5,6) (34.5,2) and (44.5, 5) (class midpoint, frequency of the class). Connect all the
points on the graph.
The y axis Scale
The y axis is scaled to the numbers in the frequency (x) column. This graphs counts by ones.
This Class Frequency Polygon is not “close” to normal in shape.
Section 2 - 3A Lecture
Page 5 of 7
© 2015 Eitel
Example 3 of Class Frequency Polygon
Class Frequency Table
NFL
age at
retirement Freq (x)
in years
20 – 24
5
Class
Midpoints
22
27
25 – 29
25
32
30 – 34
40
37
35 – 39
26
42
40 – 44
4
Class Frequency Polygon
Freq (x)
40
35
30
25
20
15
10
5
(32, 40)
(27, 25)
(22, 5)
(37, 26)
(42, 4)
(44, 0)
(20, 0)
0
20 22 27 32 37 42 44
NFL age at retirement in years
The polygon starts on the x axis at the point (first lower class limit, 0). The first lower class
limit is 20 for this table so the graph starts at the point (20, 0) The graph then ends on the x axis at
the point (last upper class limit, 0). The last upper class limit is 44 for this table so the graph ends
at the point (44, 0). Plot the ordered pairs that represent each class in the table (22,5)
(27,25) (32,40) (37,26) and (42, 4) (class midpoint, frequency of the class). Connect all the points
on the graph.
The y axis Scale
The y axis is scaled to the numbers in the frequency (x) column. This graphs counts by ones.
This Class Frequency Polygon is “close” to normal in shape. The data forms a normal
distribution.
Section 2 - 3A Lecture
Page 6 of 7
© 2015 Eitel
Relative Class Frequency Polygon
The graphs of a frequency polygon and a retaliative frequency polygon are almost exactly alike.
The only difference is in the vertical scale used on the y axis. A Relative Class Frequency
Polygon has a the vertical scale on the y axis that is expressed as a Percent of the entire
data.
A Relative Class Frequency Table
Hours a
Class
Relative
week
of
Midpoints homework
Freq.
3
1–5
.10
8
6 – 10
.20
13
11 – 15
.40
18
16 – 20
.20
23
21 – 25
.10
A Relative Class Frequency Polygon
Relative
Freq (x)
.40
(13, .40)
.30
(8, .20)
.20
(18, .20)
(3, .10)
.10
(25, 0)
(1, 0)
0
1
3
8
13
18
23
Hours a week of Homework
25
The polygon starts on the x axis at the point (first lower class limit, 0). The first lower class
limit is 1 for this table so the graph starts at the point (1, 0) The graph then ends on the x axis at
the point (last upper class limit, 0). The last upper class limit is 25 for this table so the graph ends
at the point (25, 0). Plot the ordered pairs that represent each class in the table (3, .10)
(8, .20) (13,. 40) (18, .20) and (23, .10) (class midpoint, percent of the data). Connect all the
points on the graph.
The y axis scaled as a Percent of the entire data.
This Class Frequency Polygon is “close” to normal in shape. The data forms a normal
distribution.
Section 2 - 3A Lecture
Page 7 of 7
© 2015 Eitel