1 Graphical Descriptive Techniques Chapter 2 2 2.1 Introduction Descriptive statistics involves the arrangement, summary, and presentation of data, to

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Graphical Descriptive Techniques

Graphical Descriptive Techniques

Chapter 2Chapter 2

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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

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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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Types of data - examples

Interval data

Age - income55 7500042 68000

. .

. .

Age - income55 7500042 68000

. .

. .Weight gain+10+5..

Weight gain+10+5..

Nominal

Person Marital status1 married2 single3 single. .. .

Person Marital status1 married2 single3 single. .. .Computer Brand

1 IBM2 Dell3 IBM. .. .

Computer Brand1 IBM2 Dell3 IBM. .. .

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Types of data - examples

Interval data

Age - income55 7500042 68000

. .

. .

Age - income55 7500042 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%

IBM Dell Compaq Other Total 25 11 8 6 50 50% 22% 16% 12%

Weight gain+10+5..

Weight gain+10+5..

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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

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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

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2.3 Graphical Techniques forInterval 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

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Largest observation

Collect dataBills42.1938.4529.2389.35118.04110.460.0072.8883.05

.

.

(There are 200 data points

Prepare a frequency distributionHow many classes to use?Number of observations Number of classes

Less then 50 5-750 - 200 7-9200 - 500 9-10500 - 1,000 10-111,000 – 5,000 11-135,000- 50,000 13-17More than 50,000 17-20

Class width = [Range] / [# of classes]

[119.63 - 0] / [8] = 14.95 15Largest observationLargest observation

Smallest observationSmallest observationSmallest observationSmallest observation

Largest observation

Example 2.1: Providing information

10

0

20

40

60

80

15 30 45 60 75 90 105 120

Bills

Fre

qu

en

cy

Draw a HistogramBin Frequency

15 7130 3745 1360 975 1090 18

105 28120 14


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0

20

40

60

8015 30 45 60 75 90 10

5

120

Bills

Fre

qu

ency

What information can we extract from this histogramAbout half of all the bills are small

71+37=108 13+9+10=32

A few bills are in the middle range

Relatively,large numberof large bills

18+28+14=60


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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 relative frequency = Class frequency

Total number of observations

Class frequency

Total number of observations

Relative frequency

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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

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There are four typical shape characteristics

Shapes of histograms

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Positively skewed

Negatively skewed

Shapes of histograms

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A modal class is the one with the largest number of observations.

A unimodal histogram

The modal class

Modal classes

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Modal classes

A bimodal histogram

A modal class A modal class

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• 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

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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

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Example 2.2 - Histograms

18-16-14-12-10- 8- 6- 4- 2- 0-

18-16-14-12-10- 8- 6- 4- 2- 0-

-15 0 15 30 45 60 75 -15 0 15 30 45 60 75

Return on investment A Return on investment B

Interpretation: The center of the returns of Investment AThe center of the returns of Investment Ais slightly lower than that for Investment Bis slightly lower than that for Investment B

The center for B

The center for A

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18-16-14-12-10- 8- 6- 4- 2- 0-

18-16-14-12-10- 8- 6- 4- 2- 0-

-15 0 15 30 45 60 75 -15 0 15 30 45 60 75

Interpretation: The spread of returns for Investment AThe spread of returns for Investment Ais less than that for investment Bis 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


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18-16-14-12-10- 8- 6- 4- 2- 0-

18-16-14-12-10- 8- 6- 4- 2- 0-

-15 0 15 30 45 60 75 -15 0 15 30 45 60 75Return on investment A Return on investment B

Interpretation: Both histograms are slightly positively Both histograms are slightly positively skewed. There is a possibility of large returns.skewed. There is a possibility of large returns.


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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

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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.


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Histogram

02040

50 60 70 80 90 100

Marks(Manual)

Fre

qu

en

cy

Histogram

02040

50 60 70 80 90 100

Marks(Manual)

Fre

qu

en

cy

Histogram

02040

50 60 70 80 90 100

Marks(Computer)

Fre

qu

en

cy

Histogram

02040

50 60 70 80 90 100

Marks(Computer)

Fre

qu

en

cy


The mathematical emphasiscreates two groups, and a larger spread.

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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

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Split each observation into two parts.

There are several ways of doing that:

42.19 42.19

Stem Leaf 4219

Stem Leaf4 2

A stem and leaf display forExample 2.1 will use thismethod next.


Observation:

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A stem and leaf display for Example 2.1Stem Leaf

0 00000000001111122222233333455555566666667788889999991 0000011112333333344555556678899992 00001111123446667789993 0013355894 1244455895 335666 34587 0222245567898 3344578899999 0011222223334455599910 00134444669911 124557889

The length of each linerepresents the frequency of the class defined by the stem.


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Ogives

Cumulative relative frequency

Bills

Cumulative relative frequency

Bills

Cumulative relative frequency for telephone billsCumulative Cum.Relative

Class Frequency frequency frquency0-15 71 71 71/200=.355

15-30 37 108 108/200=.54030-45 13 121 121/200=.60545-60 9 130 130/200=.65060-75 10 140 140/200=.70075-90 18 158 158/200=.79090-105 28 186 186/200=.930

105-200 14 200 200/200=1.000

Cumulative relative frequency for telephone billsCumulative Cum.Relative

Class Frequency frequency frquency0-15 71 71 71/200=.355

15-30 37 108 108/200=.54030-45 13 121 121/200=.60545-60 9 130 130/200=.65060-75 10 140 140/200=.70075-90 18 158 158/200=.79090-105 28 186 186/200=.930

105-200 14 200 200/200=1.000

}}

15

.355

30

.540

45

.605

60

.650

75

.700

90

.790

105

.930

120

1.000

Ogives are cumulative relative frequency distributions.

Example 2.1 - continued

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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.

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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.

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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

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Marketing25.3%

Finance20.6%

General management14.2%

Other11.1% Accounting

28.9%

(28.9 /100)(3600) = 1040

The Pie Chart

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Rectangles represent each category.

The height of the rectangle represents the frequency.

The base of the rectangle is arbitrary

Bar Chart

0

10

20

30

40

50

60

70

80

1 2 3 4 5 More

Area

Fre

qu

en

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73

5236

64

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The Bar Chart

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Use bar charts also when the order in which nominal data are presented is meaningful.

0

5,000

10,000

15,000

20,000

‘89 ‘90 ‘91 ‘92 ‘93 ‘94

Total number of new products introduced in North America in the years 1989,…,1994Total number of new products introduced in North America in the years 1989,…,1994

The Bar Chart

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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 Price23 31524 22926 33527 261……………..……………..

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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

0

100

200

300

400

0 10 20 30 40

Y

X

The greater the house siz

e,

the greater the price

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Typical Patterns of Scatter DiagramsPositive linear relationship Negative linear relationshipNo relationship

Negative nonlinear relationship

This is a weak linear relationship.A non linear relationship seems to fit the data better.

Nonlinear (concave) relationship

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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.

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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 ProfessionalG&M 27 29 33Post 18 43 51Star 38 15 24Sun 37 21 18

Blue Collar White collar ProfessionalG&M 27 29 33Post 18 43 51Star 38 15 24Sun 37 21 18

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SolutionIf 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

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Blue

0

10

20

30

40

1 2 3 4

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

White

0

10

20

30

40

50

1 2 3 4

Prof

0

10

20

30

40

50

60

1 2 3 4

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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).

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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

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Line Chart

0200,000400,000600,000800,000

1,000,0001,200,000

87 88 89 90 91 92 93 94 95 96 97 98 99

For the first five years – total tax was relatively flatFrom 1993 there was a rapid increase in tax revenues.

Line charts can be used to describe nominal data time series.

Line Chart

Documents

1 Graphical Descriptive Techniques Chapter 2 2 2.1 Introduction Descriptive statistics involves the arrangement, summary, and presentation of data, to