1. Graphical Descriptive Techniques
Chapter 2
Graphical Descriptive Techniques

2. 2.1 Introduction Descriptive statistics
summary and presentation of data, to support interpretation and decision making.
Methods
graphical techniques
numerical descriptive measures.
Apply to both
the entire population
the sample

3. 2.2 Types of data and information
A variable - a characteristic of population or sample.
Cereal choice
Capital expenditure
The waiting time for medical services
Data - the actual values of variables
Interval data are numerical observations
Nominal data are categorical observations
Ordinal data are ordered categorical observations

4. Types of data - examples
Interval data
Nominal
Age - income
. .
Person Marital status
1 married
2 single
3 single
. .
Weight gain
+10 +5 .
Computer Brand
1 IBM
2 Dell
3 IBM
. .

5. Types of data – analysis
Type of analysis allowed for each type of data
Interval data – arithmetic calculations
Nominal data – counting the number of observation in each category
Ordinal data - computations based on an ordering process

6. Cross-Sectional/Time-Series Data
Cross sectional data is collected at a certain point in time
Test score in a statistics course
Starting salaries of an MBA program graduates
Time series data is collected over successive points in time
Weekly closing price of gold
Amount of crude oil imported monthly

7. 2.3 Graphical Techniques for Interval Data
Example 2.1: Providing information concerning the monthly bills of new subscribers in the first month after signing on with a telephone company.
Collect data
Prepare a frequency distribution
Draw a histogram

8. Example 2.1: Providing information
Collect data
Prepare a frequency distribution
How many classes to use?
Number of observations Number of classes
Less then ,
1,000 – 5,
5, ,
More than 50,
Class width = [Range] / [# of classes]
[ ] / [8] =
(There are 200 data points
Largest
observation
Largest
observation
Largest
observation
Largest
observation
Smallest
observation
Smallest
observation
Smallest
observation
Smallest
observation

9. Example 2.1: Providing information
Draw a Histogram

10. Relative frequency Relative frequencies should be used when
population relative frequencies are studied
comparing two or more histograms
the number of observations of the samples studied are different
Class relative frequency =
Class frequency
Total number of observations

11. Class width
It is generally best to use equal class width, but sometimes unequal class width are called for.
Unequal class width is used when the frequency associated with some classes is too low.
several classes are combined together to form a wider class.
It is possible to form an open ended class at the higher end or lower end of the histogram.

12. Shapes of histograms There are four typical shape characteristics
Symmetric distribution

13. Shapes of histograms
Skewness
Negatively skewed
Positively skewed

14. Modal classes A unimodal histogram
A modal class is the one with the largest number of observations.
A unimodal histogram
The modal class

15. Modal classes
A bimodal histogram
A modal class
A modal class

16. Bell shaped histograms
Drawing the histogram helps verify the shape of the population in question

17. Interpreting histograms
Example 2.3: Comparing students’ performance
Students’ performance in two statistics classes were compared.
The two classes differed in their teaching emphasis
Class A – mathematical analysis and theory.
Class B – applications and computer based analysis.
The final mark for each student in each course was recorded.
Draw histograms and interpret the results.

18. Interpreting histograms
The mathematical emphasis
creates two groups, and a larger spread.

19. Stem and Leaf Display
Stem and leaf diagrams use the actual value of the original observations (whereas, the histogram does not).

20. Stem and Leaf Display Split each observation into two parts.
There are several ways of doing that:
Observation:
Stem Leaf 42 19
Stem Leaf
4 2
A stem and leaf display for Example 2.1 will use this method next.

21. Stem and Leaf Display
A stem and leaf display for Example 2.1 Stem Leaf
The length of each line represents the frequency of the class defined by
the stem.

22. } Ogives Ogives are cumulative relative frequency distributions.
Example continued
120
1.000
105
.930 90
.790 } } 75
.700 60
.650
.540
.605
.355 15 30 45

23. 2.4 Graphical Techniques for Nominal data
The raw data can be categorized and we can display frequencies by
Bar charts
Pie chart

24. The Pie Chart (28.9 /100)(3600) = 1040 Other 11.1% Accounting 28.9%
General
management
14.2%
Finance
20.6%
Marketing
25.3%

25. The Bar Chart 73 64 52 36 28

26. The Bar Chart
Use bar charts also when the order in which nominal data are presented is meaningful.
Total number of new products introduced in North America in the years 1989,…,1994
20,000
15,000
10,000
5,000
‘ ‘ ‘ ‘ ‘ ‘94

27. 2.5 Describing the Relationship Between Two Variables
We are interested in the relationship between two interval variables.
Example 2.7
A real estate agent wants to study the relationship between house price and house size
Use graphical technique to describe the relationship between size and price.
Size Price
315
229
335
261
……………..

28. 2.5 Describing the Relationship Between Two Variables
Solution
The size (independent variable, X) affects the price (dependent variable, Y)
We use scatter diagram Y
The greater the house size,
the greater the price X

29. Typical Patterns of Scatter Diagrams
Positive linear relationship
No relationship
Negative linear relationship
Negative nonlinear relationship
Nonlinear (concave) relationship
This is a weak linear relationship. A non linear relationship seems to
fit the data better.

30. Graphing the Relationship Between Two Nominal Variables
We create a contingency table.
This table lists the frequency for each combination of values of the two variables.
We can create a bar chart that represent the frequency of occurrence of each combination of values.

31. Contingency table Example 2.8
To conduct an efficient advertisement campaign, the relationship between occupation and newspapers readership is studied. The following table was created (To see the data click Xm02-08a)

32. Contingency table Solution
If there is no relationship between occupation and newspaper read, the bar charts describing the frequency of readership of newspapers should look similar across occupations.

33. Bar charts for a contingency table
Blue-collar workers prefer
the “Star” and the “Sun”.
White-collar workers and professionals mostly read the
“Post” and the “Globe and Mail”

34. 2.6 Describing Time-Series Data
Time-series data is often depicted on a line chart (a plot of the variable over time).

35. Line Chart
Example 2.9
The total amount of income tax paid by individuals in 1987 through 1999 are listed below.
Draw a graph of this data and describe the information produced

36. Line Chart For the first five years – total tax was relatively flat
From 1993 there was a rapid increase in tax revenues.