CUMULATIVE
FREQUENCY AND
OGIVES
1
AS 10.4.1 (a)
Collect, organise and interpret univariate
numerical data in order to determine
measures of dispersion, including
quartiles, percentiles and the interquartile
range
AS 11.4.1 (a)
Calculate and represent measures of
central tendency and dispersion in
univariate numerical data by drawing
Ogives
2
Ogives
• The word ogive is used to describe
various smooth curved surfaces.
• S-shaped.
• Cumulative frequency curve.
3
Cumulative Frequency Table
• In a frequency table you keep count of
the number of times a data item occurs
by keeping a tally. The number of times
the item occurs is called the frequency of
that item.
• In a frequency table you can also find a
‘running total’ of frequencies. This is
called the cumulative frequency. It is
useful to know the running total of the
frequencies as this tells you the total
number of data items at different stages
in the data set.
4
Cumulative Frequency Table showing the
marks obtained by students in a test
Mark
Frequency
Cumulative
frequency
This tells you that
1
1
1
1 students scored 1
mark
2
3
3+1=4
4 students scored
marks of 2 or less
3
4
4+4=8
8 students scored
marks of 3 or less
4
6
6+8=14
5
9
9+14=23
6
11
11+23=34
7
15
15+34=49
8
18
18+49=67
9
10
10+67=77
10
5
5+77=82
Total
82
Check that the final
total in the cumulative
column is the same as
the total number of
students
5
Activity 1
1. b) 34 learners
c) 82 – 34 = 48 learners
d) 77 learners
2. a)
Number of pets
frequency
Cumulative frequency
b) i) 23 learners
ii) 3 learners
iii) 25 learners
0
8
8
1
6
14
2
6
20
3
3
23
4
2
25
5
1
26
6
A cumulative frequency table
can be drawn up from:
• Ungrouped data (see page 3)
• Grouped discrete data (see page 4)
• Grouped continuous data (see page
5)
7
Activity 2 – question 1
Height, h, in
cm
Freq
Cum.
Freq.
90<h ≤95
5
5
95<h ≤100
9
14
100<h ≤105
17
31
105<h ≤110
28
59
110<h ≤115
21
80
115<h ≤120
10
90
a) 28
b) 59
c) 21 + 10 = 31
or
90 – 59 = 31
90
8
Activity 2 – question 2
%
No of
learners
Cumul.
Freq.
0<h ≤10
0
0
10<h ≤20
2
2
20<h ≤30
6
8
30<h ≤40
7
15
40<h ≤50
14
29
50<h ≤60
20
49
60<h ≤70
35
84
70<h ≤80
29
113
80<h ≤90
16
129
90<h ≤100
11
140
Total =
140
a) The way the
interval is given
0<x ≤ 10 versus
1 – 10
b)
c) 15 learners
d) 16 + 11 = 27, or
140 – 113 = 27
e) Couldn’t
9
We represent data given on a
frequency table by drawing
–
–
–
–
–
A
A
A
A
A
broken line graph
pie chart
bar graph
histogram
frequency polygon
We represent data given on a
cumulative frequency table by
drawing a cumulative frequency
graph or ogive
10
Drawing a Cumulative Frequency Curve or Ogive
Frequency
Cumulative Frequency Curve of Maths Marks in
Grade 12
• Running total
of frequencies
• S – shape
• Starts where
frequency is 0.
145
140
135
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
0
10
20
30
40
50
60
70
80
90
100
Marks
11
Activity 3
Mark
Freq.
Cum
Freq
Coords
0
0
0
(0;0)
1
1
1
(1;1)
90
2
3
4
(2;4)
80
3
4
8
(3;8)
4
6
14
(4;14)
5
9
23
(5;23)
6
11
34
(6;34)
7
15
49
(7;49)
8
18
67
(8;67)
9
10
77
(9;77)
10
5
82
(10;82)
Tot=82
Cumulative Frequency
Grade 9 Maths Test
70
60
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
Mark
12
Activity 4 – question 1
cumulative frequency
Heights of learners in
grade 1
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
a) 59 learners
b) Yes
c) 90 – 59 = 31
learners
90
100
110
120
height (cm)
13
Activity 4 – question 2
Time (in
sec)
Freq
Cum
Freq
Co-ords
35<x ≤ 40
0
0
(40;0)
40<x ≤ 45
2
2
(45;2)
45<x ≤ 50
7
9
(50;9)
50<x ≤ 55
8
17
(55;17)
55<x ≤ 60
8
25
(60;25)
60<x ≤ 65
6
31
(65;31)
65<x ≤ 70
5
36
(70;36)
70<x ≤ 75
5
41
(75;41)
75<x ≤ 80
4
45
(80;45)
14
Activity 4 – question 2 continued
Estimation of 1 minute
Estimation of 1 minute
45
cumultive frequency
9
8
6
5
4
3
2
1
time in seconds
82.5
77.5
72.5
67.5
62.5
57.5
52.5
47.7
42.5
0
37.5
frequency
7
40
35
30
25
20
15
10
5
0
40 45 50 55 60 65 70 75 80
time in seconds
15
THE MEDIAN AND QUARTILES FROM
A CUMULATIVE FREQUENCY TABLE
Suppose we have the marks of 82 learners. We
can divide the marks into four groups
containing the same number of marks in the
following way:
20
terms
M
Q1
Average
score
20
of the
of the terms
41st and
st
21
42nd
learner
scores
20
terms
Q3
Score
of the
62nd
learner
20
terms
16
These values can be found in the cumulative
frequency table by counting the data items:
Mark
Frequency
Cumulative
frequency
1
1
1
2
3
4
3
4
8
4
6
14
5
9
23
6
11
34
7
15
49
8
18
67
9
10
77
10
5
82
Total
82
The 21st student
is here.
Q1= 5
The 41st and
42nd students are
here.
Median = 7
The 62nd
student is here.
Q3= 8
17
The Median and Quartiles from
an Ogive
Q3 is the
62nd value
Q1 is the
21st value
cumulative frequency
Median is the
41½th value
Grade 10 maths marks
Estimate of
upper quartile is
read here. Q3≈8
88
86
84
82
80
78
76
74
72
70
68
66
64
62
60
58
56
54
52
50
48
46
44
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
Estimate of
median is read
here. M ≈ 7
1
2
3
4
5
6
marks
7
8
9
10
Estimate of
lower quartile
is read here.
Q1≈ 5
18
Percentiles
• Deciles: They divide the data set into 10
equal parts
• Percentiles : They divide the data set into
100 equal parts
• The median is the 50th percentile. This
means 50% of the data items are below the
median
• Q1 = 25th percentile. This means 25% of
the data items are below Q1
• Q3 = 75th percentile. This means 75% of the
data items are below Q3
19
• Percentiles should only be used with
large sets of data.
Example:
The 16th percentile of the data on the
previous page is found like this:
16% of 82 = 13,12
On vertical axis find 13 then read across
to curve and then down to horizontal
axis
16th percentile  4
This means 16% of the class scored 4
marks or less.
20
Activity 5
1. (a) 10% of 82 = 8,2
10th percentile ≈ 3
90% of 82 = 73,8
90th percentile ≈ 9
(b) 80% of the marks lie between 3 and 9.
(c) 50% of the class got 7 or less out of 10
for the test.
2. (a) 50th
(b) 25th (c) 75th
21
Activity 5 – question 3
Marks
Frequency
Cumulative
frequency
Points
1 – 10
1
1
(10;1)
11 – 20
2
3
(20;3)
21 – 30
13
16
(30;16)
31 – 40
24
40
(40;40)
41 – 50
32
72
(50;72)
51 – 60
16
88
(60;88)
61 – 70
11
99
(70;99)
71 – 80
1
100
(80;100)
22
Activity 5 – question 3 continued
Maths marks of 100
learners
c)
Median is the 50½ th
term. It lies in the
interval 41 – 50. Median
≈ 45,5
f)
Lower quartile is the
25½ th term. It lies in
the interval 31 – 40.
Q1 ≈ 35,5
100
cumulative frequency
90
80
70
60
50
40
30
20
10
0
0 10 20 30 40 50 60 70 80
Upper quartile is the
75½ th term. It lies in
the interval 51 – 60.
Q3 ≈ 55,5
marks
23