2.2 Summarizing Quantitative Data:
1. Determine the classes:
For quantitative data, we need to define the classes first. There are 3 steps to define
the classes for a frequency distribution:
Step 1: Determine the number of nonoverlapping classes, usually 5 to 20 classes.
Step 2: Determine the width of each class,
class width 
largest data value  smallest data value
number of classes
Note: the number of classes and the approximate class are
determined by trial and error!!
Step 3: Determine the class limits: the smallest possible data value should be larger
than or equal to the lower class limit while the largest possible data value should be
smaller than or equal to the upper class limit.
Example:
Suppose we have the following data (in days):
12
14
19
18
15
15
18
17
20
27
22
23
22
21
33
28
14
18
16
13
We applied the above procedure to this data.
Step 1:
We choose 5 to be the number of classes.
Step 2:
class width 
largest data value  smallest data value 33  12

 4.2 .
number of classes
5
Therefore, we use 5 as the class width.
Step 3:
The 5 classes we choose are
10-14
15-19
20-24
25-29
30-34
Note: the lower class limit in the first class (10) is smaller than the
smallest data value 12. Also, the upper class limit in the last class (34)
1
is larger than the largest data value 33.
2. Summarizing quantitative data:
Tabular summary:
In addition to frequency, relative frequency and percent frequency, another tabular
summary of quantitative data is the cumulative frequency distribution.
Cumulative frequency distribution: the number of data items with values less than
or equal to the upper class limit of each class.
Graphical display:
In addition to histogram, another graphical display of quantitative data is ogive.
Ogive: the number of data items with values less than or equal to the upper class
limit of each class.
Example (continue):
Classes
Frequency
Relative Frequency
Percent Frequency
10-14
4
0.2
20
15-19
8
0.4
40
20-24
5
0.25
25
25-29
2
0.1
10
30-34
1
0.05
5
Total
20
Classes
Cumulative
Frequency
Cumulative Relative
Frequency
Cumulative Percent
Frequency
 14
4
0.2
20
 19
4+8=12
0.2+0.4=0.6
20+40=60
 24
4+8+5=17
0.2+0.4+0.25=0.85
20+40+25=85
 29
4+8+5+2=19
 34
4+8+5+2+1=20
1
0.2+0.4+0.25+0.1=0.95
100
20+40+25+10=95
0.2+0.4+0.25+0.1+0.05=1 20+40+25+10+5=100
The histogram is
2
8
6
4
2
0
10
15
20
25
30
35
data
The ogive plot is
Ogive plot
cumulative frequency
20
15
10
5
0
0
5
10
15
20
25
30
35
data
Example:
Suppose we have the following data:
30
79
59
65
40
64
52
53
57
39
61
47
50
60
48
50
58
67
Suppose the number of nonoverlapping classes is determined to be 5. Please construct
the frequency distribution table (including frequency, percent frequency, cumulative
frequency, and cumulative percent frequency) for the data.
[solution:]
Approximat e class width 
3
79  30
 9.8
5
 The class width is 10.
Thus,
Class
Frequency
Percent
Frequency
Cumulative
Frequency
Cumulative
Percent
Frequency
30-39
40-49
50-59
60-69
2
3
7
5
(2/18)100=11.11
(3/18)100=16.67
(7/18)100=38.89
(5/18)100=27.78
2
5
12
17
11.11
27.78
66.67
94.44.
70-79
1
(1/18)100=5.56
18
100
Online Exercise:
Exercise 2.2.1
Exercise 2.2.2
4