& Graphical Tabular Descriptive Techniques
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Chapter 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 Interval data are numerical observations Nominal data are categorical observations Ordinal data are ordered categorical observations
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Data - the actual values of variables
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Types of data - examples
Interval data Age - income
55 42 75000 68000
Nominal Person Marital status
1 2 3 married single single
. .
. . . . Weight Computer . .
gain
+10 +5
Brand
. .
1 2 3 . .
IBM Dell IBM . .
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Types of data - examples
Interval data Age - income
55 42 75000 68000
Nominal data With nominal data, all we can do is, calculate the proportion of data that falls into each category.
Dell 11 22%
. .
. .Weight
gain
+10 IBM +5 25 . 50% .
Compaq Other 8 6 16% 12%
Total
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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 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
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Example 2.1: Providing information
Collect data
Bills 42.19 38.45 29.23 89.35 118.04 110.46 0.00 72.88 83.05 . .
(There are 200 data points
Prepare a frequency distribution
How many classes to use?
Less then 50 50 - 200 200 - 500 500 - 1,000 1,000 5,000 5,000- 50,000 More than 50,000 Number of observations Number of classes 5-7 7-9 9-10 10-11 11-13 13-17 17-20
Class width = [Range] / [# of classes] [119.63 - 0] / [8] = 14.95 15
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Largest Smallest Largest Smallest Largest Smallest Smallest Largest observation observation observation observationobservation observati observation observation Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. on
Example 2.1: Providing information
Draw a Histogram
80 Frequency 60 40 20 0 15 30 45 60 75 90 105 120 Bills
Bin
Frequency 15 71 30 37 45 13 60 9 75 10 90 18 105 28 120 14
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Example 2.1: Providing information
What information can we extract from this histogram
Relatively, About half of allA few bills are in large number the bills are small middle range the of large bills 80 71+37=108 13+9+10=32 18+28+14=60
Frequency
60 40 20 105 120 60 75 15 30 45 90
0
Bills
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Relative frequency
It is often preferable to show the relative frequency (proportion) of observations falling into each class, rather than the frequency itself. Class relative frequency = Total number of observations Relative frequencies should be used when
Class frequency
the population relative frequencies are studied comparing two or more histograms the number of observations of the samples studied are different
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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.
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Shapes of histograms
There are four typical shape characteristics
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Shapes of histograms
Negatively skewed Positively skewed
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Modal classes
A modal class is the one with the largest number of observations.
A unimodal histogram
The modal class
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Modal classes
A bimodal histogram
A modal class
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A modal class
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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
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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)
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Example 2.2 - Histograms
18 16 14 12 10 8 6 4 2 0 The center for A 18 16 14 12 10 8 6 4 2 0 The center for B
15 0 15 30 45 60 75
Return on investment A
15 0 15 30 45 60 75
Return on investment B
Interpretation: The center of the returns of Investment A is slightly lower than that for Investment B
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Example 2.2 - Histograms
18 16 14 12 10 8 6 4 2 0
Sample size =50
17 34 46
15 0 15 30 45 60 75
18 16 14 12 10 8 6 4 2 0
Sample size =50
16 26 43
15 0 15 30 45 60 75
Return on investment A
Return on investment B
Interpretation: The spread of returns for Investment A is less than that for investment B
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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
Return on investment A
15 0 15 30 45 60 75
15 0 15 30 45 60 75
Return on investment B
Interpretation: Both histograms are slightly positively skewed. There is a possibility of large returns.
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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.
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Interpreting histograms
Example 2.3: Comparing students' performance
Students' performance two in 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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Interpreting histograms
Histogram
Frequency 40 20 0 50 60 70 80 90 100 Marks(Manual)
The mathematical emphasis creates two groups, and a larger spread.
Frequency 40 20 0 50 60
Histogram
70
80
90
100
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Marks(Computer)
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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).
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Stem and Leaf Display
Split each observation into two parts. There are several ways of doing that: 42.19
Observation:
42.19
Stem Leaf 42 19
Stem Leaf 4 2
A stem and leaf display for Example 2.1 will use this method next.
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Stem and Leaf Display
A stem and leaf display for Example 2.1
Stem
0 1 2 3 4 5 6 7 8 9 10 11
Leaf
0000000000111112222223333345555556666666778888999999 000001111233333334455555667889999 0000111112344666778999 001335589 124445589 The length of each line 33566 represents the frequency 3458 022224556789 of the class defined by 334457889999 the stem. 00112222233344555999 001344446699 124557889
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Ogives
Ogives are cumulative relative frequency distributions. Example 2.1 - continued
Cumulative relative frequency for telephone bills
Class 0-15 15-30 30-45 45-60 60-75 75-90 90-105 105-200 Cum ulative Frequency frequency 71 71 37 108 13 121 9 130 10 140 18 158 28 186 14 200
Cumulative relative frequency
} }
Cum .Relative frquency 71/200= .355 108/200= .540 121/200= .605 130/200= .650 140/200= .700 158/200= .790 186/200= .930 200/200= 1.000
.700 .605.650 .540 .355
Bills
.790
.930 1.000
15
30
45
60 75
90
105 120
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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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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.
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The Pie Chart
Other 11.1% General management 14.2% Finance 20.6%
Accounting 28.9%
(28.9 /100)(3600) = 1040
Marketing 25.3%
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The Bar Chart
Rectangles represent each category. The height of the rectangle represents the frequency. The base of the rectangle is arbitrary
Bar Chart
80 Frequency 70 60 50 40 30 20 10 0 1 2 3 Area 4 5 More
73 52 36
64 28
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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 0 `89 `90 `91 `92 `93 `94
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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 Size Price there size and price recorded 1 315 Use graphical technique to describe the 229 2 335 relationship between size and price. 3
4 261 ................. .................
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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
400 300 200 100 0 0 10
Y
e ous h he rice r t ate the p gre ter The grea the
e siz
,
20
30
40
X
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Typical Patterns of Scatter Diagrams
Positive linear relationship No relationship
Negative linear relationsh
Negative nonlinear relationship
Nonlinear (concave) relationship This is a weak linear relationship. A non linear relationship seems to fit the data better.
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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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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)
G&M Post Star Sun Blue Collar White collar Professional 27 29 33 18 43 51 38 15 24 37 21 18
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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.
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Bar charts for a contingency table
Blue 40 30 20 10 0 1 2 3 4
Bluecollar workers prefer the "Star" and the "Sun".
White 50 40 30 20 10 0 1 2 3 4
Prof 60 50 40 30 20 10 0 1 2 3 4
Whitecollar workers and professionals mostly read the "Post" and the "Globe and Mail"
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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
Line Chart 1,200,000 1,000,000 800,000 600,000 400,000 200,000 0 87 88 89 90 91 92 93 94 95 96 97 98 99
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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