1. Graphical Descriptive Techniques
Chapter 2
Graphical Descriptive Techniques

2. 2.1 Introduction
Descriptive statistics involves the arrangement, summary, and presentation of data, to enable meaningful interpretation, and to support decision making.
Descriptive statistics methods make use of
graphical techniques
numerical descriptive measures.
The methods presented apply to both
the entire population
the population sample

3. 2.2 Types of data and information
A variable - a characteristic of population or sample that is of interest for us.
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 - examples
Interval data
Nominal data
With nominal data,
all we can do is,
calculate the proportion
of data that falls into
each category.
Age - income
. .
Weight gain
+10 +5 .
IBM Dell Compaq Other Total
50% % % %

6. Types of data – analysis
Knowing the type of data is necessary to properly select the technique to be used when analyzing data.
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

7. Cross-Sectional/Time-Series Data
Cross sectional data is collected at a certain point in time
Marketing survey (observe preferences by gender, age)
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

8. 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

9. 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

10. Example 2.1: Providing information
Draw a Histogram

11. Example 2.1: Providing information
What information can we extract from this histogram
Relatively,
large number
of large bills
About half of all
the bills are small
A few bills are in
the middle range 80
71+37=108
=32
=60 60
Frequency 40 20 15 30 45 60 75 90
105
120
Bills

12. Relative frequency
It is often preferable to show the relative frequency (proportion) of observations falling into each class, rather than the frequency itself.
Relative frequencies should be used when
the 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

13. 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. Then,
several classes are combined together to form a wider and “more populated” class.
It is possible to form an open ended class at the higher end or lower end of the histogram.

14. Shapes of histograms Symmetry
There are four typical shape characteristics

15. Shapes of histograms
Skewness
Negatively skewed
Positively skewed

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

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

18. Bell shaped histograms
Many statistical techniques require that the population be bell shaped.
Drawing the histogram helps verify the shape of the population in question

19. Interpreting histograms
Example 2.2: Selecting an investment
An investor is considering investing in one out of two investments.
The returns on these investments were recorded.
From the two histograms, how can the investor interpret the
Expected returns
The spread of the return (the risk involved with each investment)

20. Example 2.2 - Histograms Return on investment A Return on investment B
The center for B
The center
for A
18-
16-
14-
12-
10- 8- 6- 4- 2- 0-
18-
16-
14-
12-
10- 8- 6- 4- 2- 0-
Return on investment A
Return on investment B
Interpretation: The center of the returns of Investment A is slightly lower than that for Investment B

21. Example 2.2 - Histograms Return on investment A Return on investment B
Sample size =50
Sample size =50
18-
16-
14-
12-
10- 8- 6- 4- 2- 0-
18-
16-
14-
12-
10- 8- 6- 4- 2- 0- 17 16 34 26 46 43
Return on investment A
Return on investment B
Interpretation: The spread of returns for Investment A is less than that for investment B

22. Example 2.2 - Histograms Return on investment A Return on investment B
18-
16-
14-
12-
10- 8- 6- 4- 2- 0-
18-
16-
14-
12-
10- 8- 6- 4- 2- 0-
Return on investment A
Return on investment B
Interpretation: Both histograms are slightly positively skewed. There is a possibility of large returns.

23. Providing information
Example 2.2: Conclusion
It seems that investment A is better, because:
Its expected return is only slightly below that of investment B
The risk from investing in A is smaller.
The possibility of having a high rate of return exists for both investment.

24. 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 development of 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.

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

26. Stem and Leaf Display
This is a graphical technique most often used in a preliminary analysis.
Stem and leaf diagrams use the actual value of the original observations (whereas, the histogram does not).

27. 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.

28. 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.

29. } 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

30. 2.4 Graphical Techniques for Nominal data
The only allowable calculation on nominal data is to count the frequency of each value of a variable.
When the raw data can be naturally categorized in a meaningful manner, we can display frequencies by
Bar charts – emphasize frequency of occurrences of the different categories.
Pie chart – emphasize the proportion of occurrences of each category.

31. The Pie Chart
The pie chart is a circle, subdivided into a number of slices that represent the various categories.
The size of each slice is proportional to the percentage corresponding to the category it represents.

32. The Pie Chart
Example 2.4
The student placement office at a university wanted to determine the general areas of employment of last year school graduates.
Data was collected, and the count of the occurrences was recorded for each area.
These counts were converted to proportions and the results were presented as a pie chart and a bar chart.

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

34. The Bar Chart Rectangles represent each category.
The height of the rectangle represents the frequency.
The base of the rectangle is arbitrary 73 64 52 36 28

35. 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

36. 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
Twelve houses recently sold are sampled and there size and price recorded
Use graphical technique to describe the relationship between size and price.
Size Price
315
229
335
261
……………..

37. 2.5 Describing the Relationship Between Two Variables
Solution
The size (independent variable, X) affects the price (dependent variable, Y)
We use Excel to create a scatter diagram Y
The greater the house size,
the greater the price X

38. 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.

39. 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.

40. 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)

41. 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.

42. 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”

43. 2.6 Describing Time-Series Data
Data can be classified according to the time it is collected.
Cross-sectional data are all collected at the same time.
Time-series data are collected at successive points in time.
Time-series data is often depicted on a line chart (a plot of the variable over time).

44. 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

45. Line Chart For the first five years – total tax was relatively flat
From 1993 there was a rapid increase in tax revenues.
Line charts can be used to describe nominal data time series.