Sect. 2-1 Frequency
Distributions and Their
graphs
Objective SWBAT construct a frequency
distribution including limits ,boundaries ,
midpoints, relative frequencies, and cumulative
frequencies
Also how to construct frequency histograms,
frequency polygons, , relative frequency
histograms, and ogives.
Frequency Distribution
Definition
A frequency distribution is a table that
shows classes or intervals of data entries
with a count of the number of entries in
each class. The frequency f of a class is
the number of data entries in the class.
Example of a frequency
Distribution
Class
1- 5
6-10
11-15
16-20
21- 25
26-30
Frequency
5
6
8
8
5
4
Guidelines
Constructing a Frequency Distribution from a Data Set
1. Decide on the number of classes to include in the frequency
Distribution. The number of classes should be between five
and twenty; otherwise it may be difficult to detect any
patterns.
2. Find the class width as follows . Determine the range of the
data . Divide the range by the number of classes, and round up to
the next convenient number.
3. Find the class limits. You can use the minimum data entry as
the lower limit of the first class. To find the remaining lower
limits, add the class width to the lower limit of the preceding
class. Then find the upper limit of the first class. Remember that
classes can not overlap. Find the remaining upper class limits.
4. Make a tally mark for each data entry in the row of the
appropriate class.
5. Count the tally marks to find the total frequency f for each
class.
Example
• Constructing a frequency
distribution chart from a Data Set
• The following sample data set lists
the number of minutes 50 Internet
subscribers spent on the Internet
during their most recent session.
Construct a frequency chart that has
7 classes.
• 50 40 41 17 11 7 22 44 28 21 19 23
37 51 54 42 88
• 41 78 56 72 56 17 7 69 30 80 56 29
33 46 31 39 20
• 18 29 34 59 73 77 36 39 30 62 54 67
39 31 53 44
Lower Upper
Limit Limit
7
19
31
43
55
67
79
18
30
42
54
66
78
90
Solution
• The number4 of classes (7) is stated in the problem.
The minimum data entry is 7 and the
maximum is 88 so the range is 81.
88  7

7

81
7
 11.57
Class
Max entry  min
# of classes
7-18
19-30
31-42
range
43-54
55-66
No. of classes
67-78
79-90
12
or 12 Round up to
Tally
Frequency
6
10
13
8
5
6
2
∑ f = 50
Try it Yourself
Construct a frequency distribution using the ages of the
residents of Akhiok given in the opening on page 30.
Use six classes.
a. State the number of classes
b. Find the minimum and maximum values and the class
width
c. Find the class limits.
d. Tally the data entries
e. Write the fr4equency for each class.
Definition
The midpoint of a class is the sum of the lower and upper
limit of the class divided by 2 . The midpoint is sometimes
called the class mark.
Lower class limit + Upper class limit
Midoint =
2
The relative frequency of a class is the portion or percent of
the data that falls in that class. To find the relative
frequency of a class , divide the frequency f by the sample
size n.
Class frequency
Relative frequency =
Sample size
The cumulative frequency is the sum of the frequency for
that class and all previous classes. The cumulative
frequency for the last class is equal to the sample size n.
Example
Midpoints, Relative and Cumulative frequencies
Using the frequency distribution constructed in example 1, find
the midpoint , relative frequency, and cumulative frequency for
each class. Identify any patterns.
SOLUTION The midpoint , relative, and cumulative frequency
for the first three classes are calculated as follows.
Relative
Cumulative
Class f
Midpoint
frequency
frequency
7 – 18 6
7-18 ∕ 2 = 12.5
6 / 50 = 0.12
6
19-30
10
31 – 42 13
19+ 30 / 2 = 24.5
31+ 42 / 2 = 36.5
10/50 = 0.2
13/50 = 0.26
6+10 = 16
16+13 = 29
Frequency Distribution for
Internet usage (in minutes)
Class
7- 18
19-30
31-42
43- 54
55- 66
67-78
79- 90
Frequency
f
6
10
13
8
5
6
2
∑ f = 50
Midpoint
Relative
Frequency
12.5
24.5
36.5
48.5
60.5
72.5
84.5
0.12
0.2
0.26
0.16
0.1
0.12
0.04
Cumulative
Frequency
6
16
29
37
42
48
50
∑ f = 1
n
There are several patterns in the data set. For instance, the most common time
span that users spent online was 31 to 42 minutes.
Try it Yourself 2
Using the frequency distribution constructed in try it Yourself1,
find the midpoint, relative frequency, and cumulative frequency
for each class. Identify any patterns.
a. Use the formulas to find each midpoint, relative frequency,
and cumulative frequency..
b. Organize your results in a frequency distribution.
c. Identify patterns that emerge from the data.
Graphs of frequency Distributions
Definition
A frequency histogram is a bar graph that represents the
frequency distribution of a data set. A histogram has the
following properties.
1. The horizontal scale is quantitative and measures the data
values.
2. The vertical scale measures the frequencies of the classes.
3. Consecutive bars must touch.
Example : Constructing a
frequency Histogram
14
12
12.5
10
24.5
8
36.5
6
48.5
60.5
4
72.5
2
84.5
0
Time
Example 3 Constructing a
frequency Histogram
Class
7- 18
19 – 30
31 - 42
43 – 54
55 - 66
67 – 78
79 - 90
Class
Boundaries
6.5 – 18.5
18.5 - 30.5
30.5 - 42.5
42.5 - 54.5
54.5 - 66.5
66.5 - 78.5
78.5 - 90.5
Frequency
f
6
10
13
8
5
6
2
Draw a histogram for the
frequency Distribution in
Example 2 describe any
patterns.
The boundaries of the
remaining classes are shown
in the table at the left.
First find the class boundaries. The distance from the upper limit of the first
class to the lower limit of the second class is 19-18 = 1. half this distance is 0.5.
So the lower and upper boundaries of the first class are as follows.
First Class boundary = 7- 0.5 = 6.5
First class upper boundary = 18 + 0.5 = 18.5
Try it yourself
Use the frequency distribution form try it yourself 1 to construct
a frequency histogram that represents the ages of the
residents of Akhiok. Describe any patterns.
a. Find the class boundaries
b. Choose the appropriate horizontal and vertical axes.
c. Use the frequency distribution to find the height of each bar.
d. Describe any patterns for the data.
Example 4 Constructing a
frequency Polygon
To construct a frequency polygon use the same horizontal and
vertical scales that were used in the histogram labeled with
class midpoints in example 3. Then plot the points that
represent the midpoint and frequency of each class and
connect the points in order from left to right. Because the
graph should begin and end on the horizontal axis, extend the
left side to one class width before the first class midpoint and
extend the right side to one class width after the last class
midpoint. You can see that the frequency of subscribers
increase up to 36.5 minutes and then decreases.
14
12
0.5
10
12.5
24.5
8
36.5
48.5
6
60.5
4
72.5
2
84.5
90.5
0
Time on (in minutes)
Try it yourself
Construct a frequency polygon that represents the ages of
the residents of Akhiok. Describe any patterns
a. Choose the appropriate horizontal and vertical scales
b. Plot points that represent the midpoint and frequency
for each class.
c. Connect the points and extend the sides as necessary.
d. Describe any patterns fore the data.
Example 5 Constructing a
relative frequency Histogram
Draw a relative frequency histogram for the frequency
distribution in example 2.
Solution : The relative frequency histogram is shown. Notice
that the shape of the histogram is the same as the frequency
histogram constructed in example 3.
0.3
0.25
6.5
18.5
0.2
30.5t
42.5
0.15
54.5
66.5
0.1
78.5
0.05
90.5
0
Time Online (in Minutes)
Try it Yourself
Construct a relative frequency Histogram that represents the
ages of the residents of Akhiok.
a. Use the same horizontal scale as used in the frequency
histogram.
b. Revise the vertical scale to reflect relative frequencies.
c. Use the relative frequencies to find the height of each bar.
Definition
A cumulative frequency graph or ogive is a line graph that
displays the cumulative frequency of each class at its upper
class boundary.. The upper boundaries are marked on the
horizontal axis and the cumulative frequencies are marked
on the vertical axis.
GUIDELINES
Constructing an OGIVE (cumulative frequency Graph)
1. Construct a frequency Distribution that includes
cumulative frequencies as one of its columns.
2. Specify the horizontal and vertical scales. The horizontal
consists of upper class boundaries and the vertical scale
measures cumulative frequencies.
3. Plot points that represents the upper class boundaries
and their corresponding fre3quencies.
4. Connect the points in order from left to right.
5. The graph should start at the lower boundary of the first
class (cumulative frequency is zero) and should end at
the upper boundary of the last class (cumulative
frequency is equal to the sample size).
Example 6; Constructing an Ogive
60
50
6.5
18.5
40
30.5
30
42.5
54.5
20
66.5
78.5
10
90.5
0
Category 1
Constructing an Ogive
Draw an Ogive for the frequency distribution in example 2.
Estimate how many subscribers spent less than 60 minutes
during their last session. Also use the graph to estimate when
the greatest increase in usage occurs.
Upper
Class
f
Boundaries
18.5
30.5
42.5
54.5
66.5
78.5
90.5
6
10
13
8
5
6
2
Cumulative
Frequencies
6
16
29
37
42
48
50
Using the frequency distribution you
can construct the ogive shown . The
upper class boundaries, frequencies,
and the cumulative frequencies are
listed in the table. Notice that the
graph starts at 6.5 where the
cumulative frequency is zero and
the grasph ends at 90.5 where the
cumulative frequency is 50.
From the ogive you can see that
subscribers spent less than 60
minutes online during their last
session.
Try it yourself
Construct an ogive that represents the ages of the
residents of Akhiok. Estimate the number of residents
who are less than 45 years old.
a. Specify the horizontal and vertical scales.
b. Plot the points given by the upper class boundaries and
the cumulative frequencies.
c. Construct the graph.
d. Estimate the number of residents who are less than 45
years old.
Example 7: Using technology to
construct Histograms
• Use a calculator to construct a Histogram for the frequency
distribution in Example 2.
Solution : Excel and TI – 83 have features for graphing
histograms..
Homework 1-8, 9-31odd, ex.cr.
33 Pgs.41-45
Section 2.1
Frequency Distributions
and Their Graphs
Frequency Distributions
Minutes Spent on the Phone
102 124
71 104
103 116
105
97
109
99
108
112
85
107
105
86
118
122
67
99
103
82
87
95
87 100
78 125
101
92
Make a frequency distribution table with five classes.
Key values:
Minimum value =
Maximum value =
67
125
Steps to Construct a
Frequency Distribution
1. Choose the number of classes
Should be between 5 and 15. (For this problem use 5)
2. Calculate the Class Width
Find the range = maximum value – minimum. Then divide
this by the number of classes. Finally, round up to a
convenient number. (125 - 67) / 5 = 11.6 Round up to 12.
3. Determine Class Limits
The lower class limit is the lowest data value that belongs in
a class and the upper class limit is the highest. Use the
minimum value as the lower class limit in the first class. (67)
4. Mark a tally | in appropriate class for each data value.
After all data values are tallied, count the tallies in each
class for the class frequencies.
Construct a Frequency Distribution
Minimum = 67, Maximum = 125
Number of classes = 5
Class width = 12
Class Limits
Tally
67
78
79
90
91
102
103
114
115
126
Do all lower class limits first.
3
5
8
9
5
Frequency Histogram
Boundaries
Class
66.5 - 78.5
67 - 78
3
78.5 - 90.5
79 - 90
5
90.5 - 102.5
91 - 102
8
102.5 -114.5
103 -114
9
114.5 -126.5
115 -126
5
Time on Phone
9
9
8
8
7
6
5
5
5
4
3
3
2
1
0
66.5
78.5
90.5
102.5
minutes
114.5
126.5
Frequency Polygon
Class
67 - 78
Time on Phone
3
9
9
79 - 90
91 - 102
5
8
8
8
7
6
103 -114
9
5
115 -126
5
3
5
5
4
3
2
1
0
72.5
84.5
96.5
108.5
minutes
120.5
Mark the midpoint at the top of each bar. Connect consecutive midpoints.
Extend the frequency polygon to the axis.
Other Information
Midpoint: (lower limit + upper limit) / 2
Relative frequency: class frequency/total frequency
Cumulative frequency: number of values in that class or in lower
Midpoint
Class
Relative
Frequency
(67 + 78)/2
3/30
Cumulative
Frequency
67 - 78
3
72.5
0.10
3
79 - 90
5
84.5
0.17
8
91 - 102
8
96.5
0.27
16
103 - 114
9
108.5
0.30
25
115 - 126
5
120.5
0.17
30
Relative Frequency Histogram
Relative frequency
Time on Phone
Relative frequency on vertical scale
minutes
Ogive
An ogive reports the number of values in the data set that
are less than or equal to the given value, x.
Cumulative Frequency
Minutes on Phone
30
30
25
20
16
10
8
3
0
0
66.5
78.5
90.5
102.5
minutes
114.5
126.5