Lauretta J. Fox
Another method of graphical representation is the
cumulative
frequency
polygon
. The cumulative frequency polygon is a line graph which is used to picture cumulative frequencies of a set of numbers. The abscissa of each point is the upper limit of an interval in a frequency distribution. The ordinate of each point is the corresponding cumulative frequency. The graph starts at a frequency of zero for a group below the lowest interval in the distribution.
Exam ple
:
Construct a cumulative frequency polygon to represent the following scores obtained by 40 students on a mathematics test.
86
|
82
|
56
|
73
|
87
|
89
|
72
|
86
|
88
|
76
|
72
|
69
|
84
|
85
|
62
|
97
|
70
|
78
|
84
|
93
|
70
|
60
|
91
|
76
|
83
|
94
|
65
|
72
|
92
|
81
|
98
|
78
|
88
|
76
|
96
|
89
|
90
|
83
|
74
|
80
|
Solution
:
Make a frequency distribution for the scores, then draw the graph.
|
|
Cumulative
|
% of Cumulative
|
Scores
|
Frequency
|
Frequency
|
Frequency
|
96-100
|
3
|
40
|
100.0
|
9195
|
4
|
37
|
92.5
|
8690
|
8
|
33
|
82.5
|
8185
|
7
|
25
|
62.5
|
7680
|
6
|
18
|
45.0
|
7175
|
5
|
12
|
30.0
|
6670
|
3
|
7
|
17.5
|
6165
|
2
|
4
|
10.0
|
5660
|
2
|
2
|
5.0
|
(figure available in print form)
For some purposes the cumulative frequency polygon is very valuable. On the right side of the polygon is a scale of percent that parallels the scale of cumulative frequency. On the percent scale you read 25 corresponding to an abscissa of 72. This means that 25% of the scores were 72 or lower. The figure 72 is called the 25th
percentile
. The nth percentile is that score below which n percent of the scores in the distribution will fall.
To find the score that corresponds to a percentile on the graph, draw a horizontal line through the desired percent to intersect the cumulative frequency polygon. From the point of intersection draw a vertical line to the x-axis. The score at the point of intersection of the vertical line and the x-axis corresponds to the required percentile.
The fiftieth percentile is the
median
or middle score in a set of measurements. The 25th percentile is called the
lower
quartile
, and the 75th percentile is the
upper
quartile
.
Exercises
:
-
1.) During one week a dealer sold the following number of cars: Monday 12, Tuesday 15, Wednesday 5, Thursday 6, Friday 10, Saturday 12.
a.) Construct a histogram to represent the given data.
b.) Make a frequency polygon to represent the given data.
c.) Draw a cumulative frequency polygon to represent the given data.
-
2.) The heights in inches of 50 high school students are:
60
|
68
|
74
|
79
|
62
|
75
|
60
|
65
|
61
|
64
|
71
|
72
|
63
|
66
|
71
|
60
|
60
|
73
|
63
|
65
|
73
|
68
|
76
|
75
|
62
|
76
|
72
|
70
|
69
|
62
|
78
|
71
|
68
|
62
|
74
|
69
|
67
|
70
|
61
|
63
|
72
|
67
|
71
|
68
|
62
|
60
|
70
|
69
|
65
|
64
|
a.) Group the data into a frequency table.
b.) Construct a histogram to represent the data.
c.) Construct a cumulative frequency polygon.
d.) Find the median height. Find the upper and lower quartiles.
e.) Determine the 8Oth percentile.
-
3.) Forty students have the following IQ scores:
120
|
100
|
115
|
126
|
82
|
108
|
114
|
95
|
150
|
92
|
140
|
88
|
98
|
116
|
134
|
138
|
98
|
87
|
110
|
92
|
106
|
96
|
126
|
102
|
80
|
82
|
100
|
128
|
110
|
100
|
118
|
84
|
88
|
98
|
94
|
85
|
124
|
90
|
80
|
112
|
a.) Group the data into a frequency table.
b.) Construct a cumulative frequency polygon.
c.) Determine the median IQ score and the 7Oth percentile.