The document describes descriptive statistics and methods for presenting qualitative and quantitative data. It discusses frequency distributions, relative frequencies, percentages and graphs including bar charts, pie charts, and line graphs. Examples show how to construct these graphs and calculate values for datasets. Exercises provide practice creating frequency tables, determining relative frequencies and percentages, and representing data using pie charts.

1. Chapter 2:
Descriptive Statistics
Part one
ABDIRAHMAN HAYBE

2. 2.0 DESCRIPTIVE DATA
2.1.Presentation of qualitative data: tables, bar chart (simple,
component and multiple), pie chart and line graph; benefits and
interpretation.
2.2.Presentation of quantitative data: stem and leaf display, frequency
table, histogram, polygon, frequency curve, ogive and box plot; benefits
and interpretation.
2.3.Central tendency measurement: mean, mode and median;
weighted mean.
2.4.Dispersion measurement: range, quartile, percentile, interquartile
range, mean deviation, variance, standard deviation, coefficient of
variation.
2.5.Mean, variance and standard deviation for grouped data.
2.6.Measure of skewness and kurtosis: Pearson’ coefficient of
skewness.
2

3. Introduction
Raw data - Data recorded in the sequence in which
they were originally collected,
before being processed or ranked.
Array data - Raw data that are arranged in
ascending or descending order.
3

4. Quantitative raw data
Example 1
4
∙ These types of raw data shown in Examples 1 is also called
ungrouped data.

5. Organizing and Graphing
Qualitative Data
• Frequency Distributions / Table
• A frequency distribution for qualitative data lists all
categories and the number of elements that belong to
each of the categories.
• It exhibits the frequencies are distributed over various
categories
• Also called a frequency distribution table or simply a
frequency table.
– The number of students who belong to a certain
category is called the frequency of that category.
5

6. Frequency Distributions / Table
6

7. Relative Frequency and Percentage
Distribution
• A relative frequency distribution is a listing of all
categories along with their relative frequencies
(given as proportions or percentages).
• It is commonplace to give the frequency and relative
frequency distribution together.
• Calculating relative frequency and percentage of a
category
7

8. Relative Frequency of a Category
Relative Frequency of a category = Frequency of that category
Sum of all frequencies
Percentage = (Relative Frequency)* 100
8
SQQS1013 W2 L3

9. Frequency Distribution Table
W W P Is Is P Is W St Wj
Is W W Wj Is W W Is W Wj
Wj Is Wj Sv W W W Wj St W
Wj Sv W Is P Sv Wj Wj W W
St W W W W St St P Wj Sv
Example 3
A sample of UUM staff-owned vehicles produced by
Proton was identified and the make of each noted. The
resulting sample follows (W = Wira, Is = Iswara, Wj =
Waja, St = Satria, P = Perdana, Sv = Savvy):
Construct a frequency
distribution table for
these data with their
relative frequency and
percentage.
9

10. Example 3: Solution
Frequency
Relative
Frequency
Percentage (%)
Wira 19
Iswara
Perdana
Waja
Satria
Savvy
Total
10

11. Example 3: Solution
Solution:
Category
Frequency
Relative
Frequency
Percentage (%)
Wira 19
Iswara 8
Perdana
Waja
Satria
Savvy
Total
11

12. Example 3: Solution
Solution:
Category
Frequency
Relative
Frequency
Percentage (%)
Wira 19
Iswara 8
Perdana 4
Waja 10
Satria 5
Savvy 4
Total 50
12

13. Example 3: Solution
Solution:
Category
Frequency
Relative
Frequency
Percentage (%)
Wira 19
Iswara 8
Perdana 4
Waja 10
Satria 5
Savvy 4
Total 50
19/50 = 0.38
0.16
0.38*100 = 38
13

14. Example 3: Solution
Solution:
Category
Frequency
Relative
Frequency
Percentage (%)
Wira 19
Iswara 8
Perdana 4
Waja 10
Satria 5
Savvy 4
Total 50
19/50 = 0.38
0.20
0.10
0.16
0.08
0.08
0.38*100 = 38
0.16*100 = 16
0.08*100 = 8
0.20*100 = 20
0.10*100 = 10
0.08*100 = 8
100
1.00
14

15. Graphical Presentation of
Qualitative Data
• Bar Graphs
• A graph made of bars whose heights represent the frequencies of
respective categories.
• Such a graph is most helpful when you have many categories to
represent.
• Notice that a gap is inserted between each of the bars.
• It has
• => simple/ vertical bar chart
• => horizontal bar chart
• => component bar chart
• => multiple bar chart
15

16. Simple/ Vertical Bar Chart
• To construct a vertical bar chart, mark the various
categories on the horizontal axis and mark the
frequencies on the vertical axis
• Refer to Figure 2.1 and Figure 2.2,
16

17. Simple/ Vertical Bar Chart
Figure 2.1
Figure 2.2
17

18. Horizontal Bar Chart
• To construct a horizontal bar chart, mark the various
categories on the vertical axis and mark the frequencies
on the horizontal axis.
• Example 4: Refer Example 3.
18

19. Example 4: Solution
Figure 2.3
19

20. Horizontal Bar Chart
∙ Another example of horizontal bar chart: Figure 2.4
Figure 2.4: Number of students at Diversity College
who are immigrants, by last country of
permanent residence.
20

21. Component Bar Chart
• To construct a component bar chart, all categories are
in one bar and each bar is divided into components.
• The height of components should be tally with the
representative frequencies.
• Example 5:
• Suppose we want to illustrate the information below,
representing the number of people participating in the
activities offered by an outdoor pursuits centre during
June of three consecutive years.
21

22. Example 5:
2004 2005 2006
Climbing 21 34 36
Caving 10 12 21
Walking 75 85 100
Sailing 36 36 40
Total 142 167 191
22

23. Example 5: Solution
Figure 2.5
23

24. Multiple Bar Chart
• To construct a multiple bar chart, each bar that is
representative of any categories are gathered in groups.
• The height of the bar represents the frequencies of
categories.
• Useful for making comparisons (two or more values).
• Example 6: Refer example 5.
24

25. Example 6: Solution
Figure 2.6
25

26. Horizontal Bar Chart
∙ Another example : Figure 2.7
Figure 2.7: Preferred snack choices of students at UUM.
26

27. Pie Chart
– A circle divided into portions that represent the relative
frequencies or percentages of a population or a
sample belonging to different categories.
– An alternative to the bar chart and useful for
summarizing a single categorical variable if there
are not too many categories.
– The chart makes it easy to compare relative sizes of
each class/category.
27

28. Pie Chart
– The whole pie represents the total sample or population. The
pie is divided into different portions that represent the different
categories.
– To construct a pie chart, we multiply 360 by the relative
frequency for each category to obtain the degree measure or
size of the angle for the corresponding categories.
– Example 7 (Table 2.6 and Figure 2.8):
28

29. Example 7: Solution
Figure 2.8
29

30. Example 8: Solution
Example 8 (Table 2.7 and Figure 2.9):
Movie
Genres
Frequency Relative
Frequency
Angle Size
Comedy
Action
Romance
Drama
Horror
Foreign
Science
Fiction
54
36
28
28
22
16
16
200 1.00 360o
0.27
0.18
0.14
0.14
0.11
0.08
0.08
360*0.27=97.2O
360*0.18=64.8O
360*0.11=39.6O
360*0.14=50.4O
360*0.08=28.8O
360*0.08=28.8O
360*0.14=50.4O
30

31. Example 8: Solution
Figure 2.9 31

32. Line Graph/Time Series Graph
• A graph represents data that occur over a specific
period time of time.
• Line graphs are more popular than all other graphs
combined because their visual characteristics reveal
data trends clearly and these graphs are easy to
create.
• When analyzing the graph, look for a trend or pattern
that occurs over the time period.
32

33. Line Graph/Time Series Graph
• Example is the line ascending (indicating an increase
over time) or descending (indicating a decrease over
time).
• Another thing to look for is the slope, or steepness, of
the line. A line that is steep over a specific time period
indicates a rapid increase or decrease over that period.
• Two data sets can be compared on the same graph
(called a compound time series graph) if two lines are
used.
• Data collected on the same element for the same
variable at different points in time or for different periods
of time are called time series data. 33

34. Line Graph/Time Series Graph
• A line graph is a visual comparison of how two
variables—shown on the x- and y-axes—are related or
vary with each other. It shows related information by
drawing a continuous line between all the points on a
grid.
• Line graphs compare two variables: one is plotted along
the x-axis (horizontal) and the other along the y-axis
(vertical).
• The y-axis in a line graph usually indicates quantity (e.g.,
RM, numbers of sales litres) or percentage, while the
horizontal x-axis often measures units of time. As a
result, the line graph is often viewed as a time series
graph
34

35. Time Series Graph
Example 9
A transit manager wishes to use the following data for a
presentation showing how Port Authority Transit
ridership has changed over the years. Draw a time series
graph for the data and summarize the findings.
Year
Ridership
(in millions)
1990
1991
1992
1993
1994
88.0
85.0
75.7
76.6
75.4
35

36. Example 9: Solution
Solution:
The graph shows a decline in ridership through 1992 and
then leveling off for the years 1993 and 1994.
36

37. Lets Exercise
Exercise 1
1.The following data show the method of payment by 16
customers in a supermarket checkout line. Here, C =
cash, CK = check, CC = credit card, D = debit and O =
other.
C CK CK C CC D O C
CK CC D CC C CK CK CC
a.Construct a frequency distribution table.
b.Calculate the relative frequencies and percentages for all
categories.
c.Draw a pie chart for the percentage distribution.
37

38. Exercise 1: Solution
1.a). Frequency distribution table, relative
frequencies, percentages and angle sizes of all
categories.
Method of
payment
Frequency, f
Relative
frequency
Percentage
(%)
Angle
Size (o)
Cash
Check
Credit Card
Debit
Other
4
5
4
2
1
Total 16
0.2500
0.3125
0.2500
0.1250
0.0625
1
25.00
31.25
25.00
12.50
100
6.25
90
112.5
90
45
22.5
360
38

39. Exercise 1: Solution
b). Pie Chart
39

40. Exercise 2
Exercise 2:
The frequency distribution table represents the sale of
certain product in ZeeZee Company.
Each of the products was given the frequency of the
sales in certain period.
Find the relative frequency and the percentage of each
product.
Then, construct a pie chart using the information.
40

41. Exercise 2: Solution
1.a). Frequency distribution table, relative
frequencies, percentages and angle
sizes of all categories.
Type of
product
Frequency
Relative
Frequency
Percentage
(%)
Angle
Size (o)
A 13
B 12
C 5
D 9
E 11
Total 50
0.24
0.26
0.10
0.18
0.22
1.00
26
24
10
18
22
100
93.6
86.4
36.0
64.8
79.2
360
41