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
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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
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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)
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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
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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
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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.
21
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