Section 3.1 Frequency distributions
A frequency distribution is a way to organize information in a table.
Example: 64 families were asked the question: “How many children do you have?” The data
collected is shown in the table below. Construct a frequency distribution of the following data.
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
6
6
7
8
8
9
Solution:
A frequency distribution is just a two column table.
The left column is called the class. If we are not told otherwise the class will contain a single
number. In our problem the class column will contain each of the numbers from 0 – 9.
The right column is called the frequency column. It contains the number of data points in each
class. I simply count up the number of 0’s to figure out the number that goes in the frequency
column next to the 0 in the class column. There are 8 zero’s in the data. I will put the number 8
next to the 0.
Frequency distribution
# of Children
# of families
0
8
1
12
2
17
3
11
4
6
5
4
6
2
7
1
8
2
9
1
Total
64
(The total row is optional, but I usually include a total row.)
Often we group data into classes to provide information about a distribution that might be
difficult to observe if ungrouped.
Example: The following data represents the family income (in thousands of dollars, rounded to
the nearest hundred) of 15 randomly selected families. Construct a frequency distribution with
a first class of 16 – 22
31
50
20
16
37
25
30
29
24
29
38
41
25
33
35
First I will fill in the first column of the frequency distribution. I want each class to have the
same width. I subtracted 22 – 16 and got 6. I start the second class one higher than the first at
23, and add 6 to get the endpoint of 29. Frequency distribution
Income (in thousands of
dollars)
16-22
23-29
30-36
37-43
44-50
Total
Number of
families
I need to organize the information in order from smallest to largest. So that I can construct the
right column.
16 – 22 16 20
23 – 29 29 25 29 25 24
30 – 36 31 30 33 35
37 – 43 37 38 41
44 – 50 50
Frequency Distribution
Income (in thousands of
dollars)
16-22
23-29
30-36
37-43
44-50
Total
Number of
families
2
5
4
3
1
15
Answer these questions with regards to the frequency distribution just constructed.
1) What are the total number observations?
(This is just the total number of families surveyed)
Answer: Total number of observations is 15
2) What is the width of each class?
(This is calculated by subtracting the first amount in consecutive classes. This seems like
an odd way to calculate width, but it is what is usually done.)
Width = 23 – 16 = 7
Answer: Class Width 7
3) What is the midpoint of the third class?
(This is calculated by adding the upper and lower limits in the class and dividing by 2)
30+36
2
Answer: Midpoint of third class is 33
4) What is the modal class? (Or what are the modal classes?)
(The modal class is the class (or classes) with the most observations)
Answer: The modal class is the 2nd class, 23 - 29
5) What would be the class limits of the next class if one were added?
Answer: The next class would be 51-57
Example: Use the frequency distribution to determine
a) The total number of observations
b) The width of each class
c) The midpoint of the second class
d) The modal class (or classes)
e) The class limits of the next class if an additional class was added)
Frequency Distribution
Class
40-49
50-59
60-69
70-79
80-89
90-99
Total
Frequency
7
5
3
2
7
1
25
a) The total number of observations
Answer: 25 (This is just the total from the last row of the frequency distribution.)
b) The width of each class
Many computations will give this answer. Just subtract the first number in any two consecutive
classes. For instance: 50 – 40 or 70 – 60 or 90 – 80
Answer: class width 10
c) The midpoint of the second class
50+59
2
Answer: midpoint of second class 54.5
d) The modal class (or classes)
Two classes are tied for the highest frequency. We call this a bi-modal distribution. (There are
tri-modal distributions too. If there are more than 3 classes tied for the highest frequency we
will say there is no modal class.)
Answer: Modal classes: 40 – 49 and 80 – 89
e) The class limits of the next class if an additional class was added) I think this is obvious.
Answer: Next class would be 100 - 109
Homework:
1) Use the following data, which show the result of 50 sixth-grade I.Q. scores
80
81
87
88
89
89
89
90
91
92
92
93
94
94
94
95
95
97
97
97
97
98
99
100
100
100
100
100
100
101
102
103
103
103
104
106
108
108
108
109
110
113
114
114
119
120
120
122
128
135
a) Construct a frequency distribution with first class 80 – 88.
b) Compute the total number of observations.
c) Compute the width of each class.
d) Find the midpoint of the third class.
e) Find the modal class (or classes).
f) Find the class limits of the next class if an additional class was added).
2) These are the numbers of electronic cigarettes sold at a Walmart over the last 20 days:
60, 45, 80, 35, 45, 25, 40, 60, 90, 36
50, 43, 76, 35, 94, 15, 77, 20, 84, 55
a) Construct a frequency distribution with first class 10 – 22.
b) Compute the total number of observations.
c) Compute the width of each class.
d) Find the midpoint of the third class.
e) Find the modal class (or classes).
f) Find the class limits of the next class if an additional class was added).
3) The number of goals scored by the Thunderbird Chief’s soccer team of the last 14 games are as
follows:
2, 3, 1, 2, 1, 3, 2, 3, 4, 5, 4, 2, 2, 3
Construct a frequency distribution with class width of 1, like in example 1.
4) Create a frequency distribution for the following 15 scores, like in example 1:
46,46,46,47,49,50,50,51,53,54,54,56,56,56,57
5) The number of goals scored by the Thunderbird Chief’s soccer team of the last 14 games are as
follows:
2, 3, 1, 7, 1, 3, 0, 3, 4, 5, 4, 6, 2, 3
a) Construct a frequency distribution with first class 0-2.
b) Compute the total number of observations.
c) Compute the width of each class.
d) Find the midpoint of the third class.
e) Find the modal class (or classes).
f) Find the class limits of the next class if an additional class was added).
6) Use the following data to answer the questions below.
46,46,46,47,49,50,50,51,53,54,54,56,56,56,57
a) Construct a frequency distribution with first class 0-2.
b) Compute the total number of observations.
c) Compute the width of each class.
d) Find the midpoint of the third class.
e) Find the modal class (or classes).
f) Find the class limits of the next class if an additional class was added).
7) The community of Anthem is planning to improve the local park. The responses of 18
families who were asked how many times per year they visited the park per year are shown
below. Construct a frequency distribution with class width 1.
20
24
25
20
24
26
20
25
26
21
25
26
21
25
27
23
25
28
8) A car dealership is interested in the number of cars sold daily. A sample is taken over 25
days to obtain the number of cars sold each day. Construct a frequency distribution with class
width 1.
1
2
4
5
7
1
3
4
5
8
2
3
4
6
9
2
3
4
6
10
2
4
4
6
12
9) The community of Anthem is planning to improve the local park. The responses of 18
families who were asked how many times per year they visited the park per year are shown
below.
20
24
28
20
24
28
20
25
29
21
25
30
21
25
31
a) Construct a frequency distribution with first class 20-23.
b) Compute the total number of observations.
c) Compute the width of each class.
d) Find the midpoint of the third class.
e) Find the modal class (or classes).
f) Find the class limits of the next class if an additional class was added).
23
25
32
10) A car dealership is interested in the number of cars sold daily. A sample is taken over 25
days to obtain the number of cars sold each day.
1
2
4
5
7
1
3
4
5
8
2
3
4
6
9
2
3
4
6
10
2
4
4
6
12
a) Construct a frequency distribution with first class 1-4.
b) Compute the total number of observations.
c) Compute the width of each class.
d) Find the midpoint of the third class.
e) Find the modal class (or classes).
f) Find the class limits of the next class if an additional class was added).
Answers:
1a)
IQ (Class)
80-88
89-97
98-106
107-115
116-124
125-133
134-142
total
Number of students (Frequency)
4
17
15
8
4
1
1
50
1b) 50 1c) 9 1d) 102 1e) 89-97 1f) 143-151
3)
Number of goals scored (Class)
1
2
3
4
5
total
Number of times (Frequency)
2
5
4
2
1
14
5a)
Number of goals scored (Class)
0-2
3-5
6-8
Total
Number of times (Frequency)
5
7
2
14
5b) 14 5c) 3 5d) 7 5e) 3-5 5f) 9-11
7)
Number of visits (Class)
20
21
22
23
24
25
26
27
28
Total
Number of families (Frequency)
3
2
0
1
2
5
3
1
1
18
9a)
Number of visits (Class)
20-23
24-27
28-31
32-35
Total
Number of families (Frequency)
6
6
5
1
18
9b) 18 9c) 4 9d) 29.5 9e) 20-23 and 24-27 9f) 36-39