Chapter 2: Frequency Distributions and Graphs
After the data have been collected, the main tasks a statistician must accomplish
are the organization and presentation of the data. The organization must be done
in a meaningful way and the presentation should be such that an interested
reader of the study can understand the data distribution.
The data in their original form (before it has been organized) is called raw data.
All the data values obtained are divided into classes that must satisfy the
following conditions:
1. there is usually between 5 and 20 classes;
2. the classes must be mutually exclusive;
3. the classes must be exhaustive;
The frequency is the number of values in a specific class.
A frequency distribution is the organization of raw data in table form, using
classes and frequencies.
Example on page 35:
Construct a frequency distribution for the data below.
1
2
6
7
12
13
2
6
9
5
18
7
3
15
15
4
17
1
14
5
4
16
4
5
8
6
5
18
5
2
Class Limits
Tally (OPTIONAL)
Frequency
_______________________________________________________
1–3
10
4–6
14
7–9
10
10 – 12
6
13 – 15
5
16 – 18
5
______________________________________________
1
Total = 30
The types of frequency distributions that are used the most are the categorical
frequency distribution and the grouped frequency distribution.
Categorical Frequency Distributions:
The categorical frequency distribution is used for data that can be placed in
specific categories or represent values of a qualitative variable.
Example on p. 36:
Construct a frequency distribution for the data below.
A
B
B
AB
O
O
O
B
AB
B
B
B
O
A
O
A
O
O
O
AB
AB
A
O
B
A
Class
Tally
Frequency (f)
Percent
_______________________________________________________
A
5
20
B
7
28
O
9
36
AB
4
16
______________________________________________________
Sum of Frequency (n) = 25
Total percent= 100
Grouped Frequency Distributions:
When the data are numerical and their range is large, the data must be grouped
into classes that are more than one unit in length.
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In this case we have additional conditions for the classes:
4. the class width should be preferably an odd number;
5. the classes must be equal in width;
6. the classes must be continuous.
The procedure for constructing a grouped frequency distribution is described in
the next example.
Example on p. 39: The data represent the record high temperatures for each of
the 50 states. Construct a grouped frequency distribution for the data using 7
classes.
112
110
107
116
120
100
118
112
108
113
127
117
114
110
120
120
116
115
121
117
134
118
118
113
105
118
122
117
120
110
105
114
118
119
118
110
114
122
111
112
109
105
106
104
114
112
109
110
111
114
Step 1: Determine the classes

Find the highest value and the lowest value and use them to find the range.
Range = highest value in the data set – lowest value=134-100=34

Find the class width by dividing the range by the number of classes. Round
the answer up to the next whole number if there is a remainder. The class
width is the difference between the lower class limit of one class and the
lower class limit of the next class.
Class width=Range/number of classes (rounded up to the closest odd
number)=34/7 ~ 5

Use your lowest value as your starting point. Add the class width to the
starting point to get the lower limit for the next class. Keep adding until
there are 7 classes. Subtract 1 from the lower limit of the second class to
get the upper limit of the first class.
3
Step 2: Find the class boundaries
Find the class boundaries by subtracting 0.5 from each lower class limit and
adding 0.5 to each upper class limit. In future calculations we will also need to
find the midpoints Xm of each class.
Xm=(higher limit of the class+lower limit of the class)/2
Step 3: Tally the data and find the numerical frequencies from the tallies.
Step 4: Find the cumulative frequencies
The cumulative frequency for a class is the sum of the frequencies for that class
and all previous classes. To find this value, add up all the frequencies that lead
up to each class.
Now let us construct our frequency distribution:
Class Limits
Class
Boundaries
Frequency (f)
HW: p. 43-44, Ex. 7-17 (odd).
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Cumulative
Frequency
Midpoints
(Xm)