Lecture 3

1. 1

2. • Need to gain information from data.
• Data must be organised and reduced.
• Descriptive statistics
– Organising data into tables, charts and graphs.
– Numerical calculations.
• Single variable data
• Raw data
– Collected data before it is grouped or ranked.
2

3. Organising and graphing qualitative data in a
frequency distribution table.
Example:
The data below shows the gender of 50 employees and the
department in which they work at ABC Ltd.
HR – Human resources
Emp. no. Gender Dept. Emp. no. …..
Gender Mark. – Marketing
Dept
1 M HR 6 M Fin. – Finance
Fin. …..
M – Male
2 F Mark. 7 M Mark. …..
F – Female
3 M Fin. 8 M Fin. …..
4 F HR 9 F HR …..
5 F Fin. 10 F Fin. ….. 3

4. HR Marketing Finance
M │ │ │││
F ││ │ ││
Emp. no. Gender Dept. Emp. no. Gender Dept …..
1 M HR 6 M Fin. …..
2 F Mark. 7 M Mark. …..
3 M Fin. 8 M Fin. …..
4 F HR 9 F HR …..
5 F Fin. 10 F Fin. ….. 4

5. Organising and graphing qualitative data in a frequency
distribution table.
HR Marketing Finance
M 4 10 5
F 10 16 5
5

6. Organising and graphing qualitative data in a frequency
distribution table.
HR Marketing Finance Total
M 4 10 5 19
F 10 16 5 31
Total 14 26 10 50
6

7. Pie charts
HR Mark Fin Total
Total 14 26 10 50
14/50×360 26/50×360 10/50×360
Degrees 360
= 101 = 187 = 72
14/50×100 26/50×100 10/50×100
Percentage 100
= 28 = 52 = 20
Employees at ABC
20%
28% Human resources
Marketing
Finance
52%
7

8. Pie charts
Male Female Total
Total 19 31 50
19/50×360 31/50×360
Degrees 360
= 137 = 223
19/50×100 31/50×100
Percentage 100
= 38 = 62
Employees at ABC
38% Male
Female
62%
8

9. Bar graphs HR Marketing Finance Total
M 4 10 5 19
F 10 16 5 31
Total 14 26 10 50
Employees at ABC Employees at ABC
30 26
35
Number of workers
31
25
Number of workers
30
20 25
14 19
15 10 20
10 15
5 10
0 5
Human Marketing Finance 0
resources Male Female
9

10. Multiple bar graphs
HR Marketing Finance Total
M 4 10 5 19
F 10 16 5 31
Total 14 26 10 50
Employees at ABC Employees at ABC
Human
20
Number of workers
20
resources
Number of workers
15 Male
15
Marketing
10
10 Female
5
Finance 0
5
Human Marketing Finance
0 resources
Male Female
10

11. Stacked bar graphs
HR Marketing Finance Total
M 4 10 5 19
F 10 16 5 31
Total 14 26 10 50
Employees at ABC Employees at ABC
Number of workers
35 Finance 30
25
Number of workers
30 Female
20
25 15
Marketing
20 10 Male
15 5
10 0
Human
5 resources Human Marketing Finance
0 resources 11
Male Female

12. Definitions
Frequency Distribution
– for qualitative data displays the possible categories
along with the number of times (or frequency) each
category appears in the data set.
- for quantitative data is a summary of numerical data
prepared by dividing raw data into several non-
overlapping class intervals and then counting how
many observations (frequency) of the variable fall into
each class
Relative Frequency – for a particular category is the
portion or % of the observations within a category
12

13. Organising and graphing quantitative data in a frequency
distribution table.
• Frequency table consists of a number of classes and each
observation is counted and recorded as the frequency of
the class.
• If n observations need to be classified into a frequency
table, determine:
– Number of classes:
c  1  3,3log n
xmax  xmin
– Class width 
c
13

14. Organising and graphing quantitative data in a frequency
distribution table.
Example:
The following data represents the number of telephone calls
received for two days at a municipal call centre. The data was
measured per hour.
8 11 12 20 18 10 14 18 16 9
5 7 11 12 15 14 16 9 17 11
6 18 9 15 13 12 11 6 10 8
11 13 22 11 11 14 11 10 9
19 14 17 9 3 3 16 8 2 14

15. Frequency distribution
Number of classes  1  3,3log n
 1  3,3log 48  6,5  7
xmax  xmin 22  2
Class width    2,86  3
k 7
8 11 12 20 18 10 14 18 16 9
5 7 11 12 15 14 16 9 17 11
6 18 9 15 13 12 11 6 10 8
11 13 22 11 11 14 11 10 9
19 14 17 9 3 3 16 8 2 15

16. Frequency distribution
– first class [ xmin; ; min) class width)
2 5)32x
– second class [ 5 ;; 8  3 ) width)
5
5 5 ) class
“[“ value is included in class
8 11 12 20 18 10 14 18 16 9
5 7 11 12 15 14 16 9 17 11
6 18 9 15 13 12 11 6 10 8
“)“ value is excluded from class
11 13 22 11 11 14 11 10 9
19 14 17 9 3 3 16 8 2
16

17. Frequency distribution
Classes Count
[2;5) │││ 3
8 11 12 20 …. [5;8) |││││
| 4
5 7 11 12 …. [8;11) |│││││││││││ 11
6 18 9 15 …. [11;14) |│││││││││││││
| 13
11 13 22 11 ….
[14;17) │││││││││ 9
19 14 17 9 ….
[17;20) |││││││ 6
[20;23) ││ 2
17

18. Frequency distribution
Classes Frequency (f)
[2;5) 3
[5;8) 4
[8;11) 11
[11;14) 13
[14;17) 9
[17;20) 6
[20;23) 2
Total 48
18

19. Frequency distribution
Classes f % frequency
[2;5) 3 3/48×100 = 6,3
[5;8) 4 4/48×100 = 8,3
[8;11) 11 11/48×100 = 22,9
[11;14) 13 27,1
[14;17) 9 18,8
[17;20) 6 12,5
[20;23) 2 4,2
Total 48 100
19

20. Frequency distribution
Classes f %f Cumulative frequency (F)
[2;5) 3 6,3 3
[5;8) 4 8,3 3+4=7
[8;11) 11 22,9 7 + 11 = 18
[11;14) 13 27,1 18 + 13 = 31
[14;17) 9 18,8 31 + 9 = 40
[17;20) 6 12,5 40 + 6 = 46
[20;23) 2 4,2 46 + 2 = 48
Total 48 100
20

21. Frequency distribution
Classes f %f F %F
[2;5) 3 6,3 3 3/48×100 = 6,3
[5;8) 4 8,3 7 7/48×100 = 14,6
[8;11) 11 22,9 18 18/48×100 = 37,5
[11;14) 13 27,1 31 64,6
[14;17) 9 18,8 40 83,3
[17;20) 6 12,5 46 95,8
[20;23) 2 4,2 48 100
Total 48 100
21

22. Frequency distribution
Classes f F Class mid-points (x)
[2;5) 3 3 (2 + 5)/2 = 3,5
[5;8) 4 7 (5 + 8)/2 = 6,5
[8;11) 11 18 (8 + 11)/2 = 9,5
[11;14) 13 31 (11 + 14)/2 = 12,5
[14;17) 9 40 15,5
[17;20) 6 46 18,5
[20;23) 2 48 21,5
Total 48
22

23. Frequency distribution
Classes f %f F %F (x)
[2;5) 3 6,3 3 6,3 3,5
[5;8) 4 8,3 7 14,6 6,5
[8;11) 11 22,9 18 37,5 9,5
[11;14) 13 27,1 31 64,6 12,5
[14;17) 9 18,8 40 83,3 15,5
[17;20) 6 12,5 46 95,8 18,5
[20;23) 2 4,2 48 100 21,5
Total 48 100
23

24. Histograms
Classes f %f
[2;5) 3 6,3
[5;8) 4 8,3
[8;11) 11 22,9 y-axis
[11;14) 13 27,1
[14;17) 9 18,8
[17;20) 6 12,5
[20;23) 2 4,2 x-axis
24

25. Histograms
Number of telephone calls per hour
at a municipal call centre
14
Number of hours
12
10
8
6
4
2
0
2 5 8 11 14 17 20 23
Number of calls
25

26. Definitions
Frequency Polygon
A line graph of a frequency distribution and offers
a useful alternative to a histogram. Frequency
polygon is useful in conveying the shape of the
distribution
Ogive
A graphic representation of the cumulative
frequency distribution. Used for approximating the
number of values less than or equal to a specified
value
26

27. Frequency polygons
Class mid-points (x) f %f
3,5 3 6,3
6,5 4 8,3
9,5 11 22,9 y-axis
12,5 13 27,1
15,5 9 18,8
18,5 6 12,5
21,5 2 4,2 x-axis
27

28. Frequency polygons
Number of telephone calls per hour
at a municipal call centre (x)
14 3,5
Number of hours
12 6,5
10
8
9,5
6 12,5
4
2
15,5
0 18,5
0.5 3.5 6.5 9.5 12.5 15.5 18.5 21.5 24.5
21,5
Arbitrary mid-points to Number of calls 28
close the polygon.

29. Ogives
Classes F %F
[2;5) 3 6,3
[5;8) 7 14,6
[8;11) 18 37,5 y-axis
[11;14) 31 64,6
[14;17) 40 83,3
[17;20) 46 95,8
[20;23) 48 100 x-axis
29

30. Ogives
Ogive of number of call received
at a call centre per hour
100
number of hours
90
% Cumulative
80
70
60
50
40
30
20
10
0
2 5 8 11 14 17 20 23
Number of calls
None of the hours had
less than 2 calls. 30

31. Ogives Ogive of number of call received
20% of the
hours had at a call centre per hour
more than
17 calls 100
number of hours
90
% Cumulative
80
per hour. 70
80% of the 60
hours had 50
less than 40
30
17 calls 20
10
0
per hour.
2 5 8 11 14 17 20 23
50% of Number ofhad less
the hours calls
than 12 calls per hour.
31

32. • Activity 1 Module Manual p 67
• Activity 2 Module Manual p 68
• Activity 3 Module Manual p 69
• Revision Exercise 1 Module Manual p 70
• Revision Exercise 2 Module Manual p 70
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33. • Revision Exercise 3 Module Manual p 71
• Revision Exercise 4 Module Manual p 72
• Concept Questions 1 -11 p 52 Elementary
Statistics
• Self Review Test p53 Elementary Statistics
• Supplementary Exercises p 54 -59
Elementary Statistics
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