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Graphical Descriptive Techniques

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
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

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Person Marital status
1 married
2 single
3 single . .
. .
1 married
2 single
3 single C. ompu.ter Brand
1 IBM
2 Dell
3 IBM
. .
. .
Computer
Types of data – examples
Interval data
Age – income
55 75000
42 68000
. .
. .
55 75000
42 68000
Nominal data
With nominal data,
all we can do is,
calculate the proportion
of data that falls into
each category.
IBM Dell Compaq Other Total
25 11 8 6 50
50% 22% 16% 12%
25 11 8 6 50% 22% 16% 12%
Weight gain
+10
+5..
+10
+5..
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
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
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
Largest
observation
Collect data
(There are 200 data points
Prepare a frequency distribution
How many classes to use?
Number of observations Number of classes
Less then 50 5-7
50 – 200 7-9
200 – 500 9-10
500 – 1,000 10-11
1,000 – 5,000 11-13
5,000- 50,000 13-17
More than 50,000 17-20
Class width = [Range] / [# of classes]
Frequency
What information can we extract from this histogram
About half of all
the bills are small
71+37=108 13+9+10=32
A few bills are in
the middle range
Relatively,
large number
of large bills
18+28+14=60
Example 2.1: Providing information
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 Class r reelalatitvivee f rfreeqquueennccyy ==
Class frequency
Total number of observations
Class frequency
Total number of observations
Relative frequency
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.
Class width
There are four typical shape characteristics
Shapes of histograms
Positively skewed
Negatively skewed
Shapes of histograms
A modal class is the one with the largest
number of observations.
A unimodal histogram
The modal class
Modal classes
Modal classes
A bimodal histogram
A modal class A modal class
• Many statistical techniques require that the
population be bell shaped.
• Drawing the histogram helps verify the shape of
the population in question
Bell shaped 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)
Interpreting histograms

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
The center

Interpretation: The spread of returns for Investment A
is less than that for investment B
Return on investment A Return on investment B
17 16
Sample size =50 Sample size =50
34 26
46 43

Return on investment A Return on investment B
Interpretation: Both histograms are slightly positively
skewed. There is a possibility of large returns.
Example 2.2 – Histograms
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.
Providing information
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.
Interpreting histograms
Interpreting histograms
The mathematical emphasis
creates two groups, and a
larger spread.

Marks(Computer)
Frequency
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).
Stem and Leaf Display
Split each observation into two parts.
There are several ways of doing that:
42.19 42.19
Stem Leaf
42 19
Stem Leaf
4 2
A stem and leaf display for
Example 2.1 will use this
method next.
Stem and Leaf Display
Observation:
A stem and leaf display for Example 2.1

The length of each line
represents the frequency
of the class defined by
the stem.
Stem and Leaf Display

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.
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.
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.
The Pie Chart
Marketing
25.3%
Finance
20.6%
General
management
14.2%
Other
11.1% Accounting
28.9%
(28.9 /100)(3600) = 1040
The Pie Chart
Rectangles represent each category.
The height of the rectangle represents the frequency.

Total number of new products introduced in
North America in the years 1989,…,1994
Total number of new products introduced in
North America in the years 1989,…,1994
The Bar Chart
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
23 315
24 229
26 335
27 261
……………..
……………..
Solution
 The size (independent variable, X) affects
the price (dependent variable, Y)
 We use Excel to create a scatter diagram
2.5 Describing the Relationship
Between Two Variables
Y
X
The greater the house size,
the greater the price
0
100
200
300
400
0 10 20 30 40
Typical Patterns of Scatter Diagrams
Positive linear relationship No relationship Negative linear relationship
Negative nonlinear relationship
This is a weak linear relationship.
A non linear relationship seems to
fit the data better.
Nonlinear (concave) relationship
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.
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)
Contingency table
Blue Collar White collar Professional
G&M 27 29 33
Post 18 43 51
Star 38 15 24
Sun 37 21 18
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.
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”
Bar charts for a contingency table
Blue
0
10
20
30
40
1 2 3 4
White
0
10
20
30
40
50
1 2 3 4
Prof
0
10
20
30
40
50
60
1 2 3 4
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).
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
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.

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