1

CUMULATIVE FREQUENCY AND

OGIVES

2

AS 10.4.1 (a)Collect, organise and interpret univariatenumerical data in order to determinemeasures of dispersion, includingquartiles, percentiles and the

interquartilerangeAS 11.4.1 (a)Calculate and represent measures of central tendency and dispersion in univariate numerical data by drawingOgives

3

Ogives

• The word ogive is used to describe various smooth curved surfaces.

• S-shaped.

• Cumulative frequency curve.

4

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.

5

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

6

Activity 11. b) 34 learners

c) 82 – 34 = 48 learnersd) 77 learners

2. a)

b) i) 23 learners ii) 3 learners iii) 25 learners

Number of pets 0 1 2 3 4 5

frequency 8 6 6 3 2 1

Cumulative frequency

8 14 20 23 25 26

7

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)

8

Activity 2 – question 1

a) 28b) 59c) 21 + 10 =

31 or 90 – 59 = 31

Height, h, in cm

FreqCum. 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

90

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Activity 2 – question 2

a) The way the interval is given 0<x ≤ 10 versus 1 – 10

b) c) 15 learnersd) 16 + 11 = 27, or

140 – 113 = 27e) Couldn’t

%No of

learnersCumul. 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

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We represent data given on a frequency table by drawing– A broken line graph– A pie chart– A bar graph– A histogram– A frequency polygon

We represent data given on a cumulative frequency table by drawing a cumulative frequency graph or ogive

11

Drawing a Cumulative Frequency Curve or Ogive

• Running total of frequencies

• S – shape• Starts where

frequency is 0.

Cumulative Frequency Curve of Maths Marks in Grade 12

05

101520253035404550556065707580859095

100105110115120125130135140145

0 10 20 30 40 50 60 70 80 90 100

Marks

Fre

quen

cy

12

Activity 3

Grade 9 Maths Test

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10

Mark

Cu

mu

lati

ve

Fre

qu

en

cy

Mark

Freq. CumFreq

Co-ords

0 0 0 (0;0)

1 1 1 (1;1)

2 3 4 (2;4)

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

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Activity 4 – question 1Heights of learners in

grade 1

05

101520253035404550556065707580859095

90 100 110 120

height (cm)

cum

ula

tive f

requen

cy a) 59 learners

b) Yesc) 90 – 59 = 31

learners

14

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)

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Activity 4 – question 2 continued

Estimation of 1 minute

0

1

2

3

4

5

6

7

8

9

37.5

42.5

47.7

52.5

57.5

62.5

67.5

72.5

77.5

82.5

time in seconds

freq

uen

cy

Estimation of 1 minute

0

5

10

15

20

25

30

35

40

45

40 45 50 55 60 65 70 75 80

time in seconds

cum

ult

ive

freq

uen

cy

16

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

Q1

score of the 21st

learner

20 terms

MAverag

e of the 41st and 42nd

scores

20 terms

Q3

Score of the 62nd

learner

20 terms

17

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

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The Median and Quartiles from an Ogive

Grade 10 maths marks

02468

10121416182022242628303234363840424446485052545658606264666870727476788082848688

1 2 3 4 5 6 7 8 9 10

marks

cu

mu

lati

ve

fre

qu

en

cy

Q3 is the 62nd value Estimate of

upper quartile is read here. Q3≈8

Median is the 41½th value

Estimate of median is read

here. M ≈ 7Q1 is the 21st value

Estimate of lower quartile is read here.

Q1≈ 5

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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

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• 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 = 13,12On vertical axis find 13 then read across

to curve and then down to horizontal axis

16th percentile 4This means 16% of the class scored 4 marks or less.

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Activity 51. (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

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Activity 5 – question 3

Marks FrequencyCumulativ

e 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)

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Activity 5 – question 3 continued

Maths marks of 100 learners

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80

marks

cum

ula

tive

fre

quen

cy

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

Upper quartile is the 75½ th term. It lies in the interval 51 – 60.

Q3 ≈ 55,5