1. Graphical & Tabular Descriptive Techniques
Chapter 2 & 3
Graphical & Tabular Descriptive Techniques

2. 2.1 Introduction Descriptive statistics methods make use of
graphical techniques
numerical descriptive measures.
The methods presented apply to both
the entire population
the population sample

3. 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
. .

4. 2.2 Types of data and information
A variable - a characteristic of population or sample that is of interest for us.
Age, income
Marital status, weight gain
The waiting time for medical services
Data - the actual values of variables
Interval data are numerical observations
Nominal data are categorical observations

5. Types of data - examples
Interval data
Nominal data
With nominal data,
all we can do is to calculate the proportion or number of data that falls into each category.
Age - income
. .
Weight gain
+10 +5 .
IBM Dell Compaq Other Total
50% % % %

6. Graphical & Tabular Techniques for Nominal Data…
The only allowable calculation on nominal data is to count the frequency of each value of the variable.
We can summarize the data in a table that presents the categories and their counts called a frequency distribution.
A relative frequency distribution lists the categories and the proportion with which each occurs.
Refer to Example 2.1

7. Example 2.1 1. Budweiser Light 2. Busch Light 3. Coors Light
4. Michelob Light
5. Miller Lite
6. Natural Light
7. Other brand
The responses were recorded using the codes.
Construct a frequency and relative frequency distribution for these data and graphically summarize the data by producing a bar chart and a pie chart.

8. Example Xm02-01*

9. Frequency and Relative Frequency Distributions
“Countif” in excel is useful to count number of each category.

10. Nominal Data (Frequency)
Bar Charts are often used to display frequencies…

11. Nominal Data (Relative Frequency)
Pie Charts show relative frequencies…

12. Nominal Data It all the same information, (based on the same data).
Just different presentation.

13. Graphical Techniques for Interval Data
There are several graphical methods that are used when the data are interval (i.e. numeric, non-categorical).
The most important of these graphical methods is the histogram.
The histogram is not only a powerful graphical technique used to summarize interval data, but it is also used to help explain probability distribution.

14. Building a Histogram… Collect the Data  (Example 2.4)
Create a frequency distribution for the data…
How?
a) Determine the number of classes to use…
Refer to Table 2.6:
With 200 observations, we should have between 7 & 10 classes…

15. Building a Histogram… Collect the Data 
Create a frequency distribution for the data…
How?
a) Determine the number of classes to use. [8]
b) Determine how large to make each class…
Look at the range of the data, that is,
Range = Largest Observation – Smallest Observation
Range = $ – $0 = $119.63
Then each class width becomes:
Range ÷ (# classes) = ÷ 8 ≈ 15

16. Building a Histogram… Collect the Data 
Create a frequency distribution for the data…
How?
a) Determine the number of classes to use. [8] 
b) Determine how large to make each class. [15] 
c) Place the data into each class…
each item can only belong to one class;
lower limit of each class ≤ observations of each class < their upper limits.
lower limit of each class < observations of each class ≤ their upper limits.

17. Building a Histogram… (Data -> Data Analysis -> Histogram)
Collect the Data 
Create a frequency distribution for the data. 
3) Draw the Histogram…

18. Building a Histogram… Collect the Data 
Create a frequency distribution for the data. 
Draw the Histogram. 

19. Interpret… (18+28+14=60)÷200 = 30% about half (71+37=108)
i.e. nearly a third of the phone bills
are $90 or more.
about half (71+37=108)
of the bills are “small”,
i.e. less than $30
There are only a few telephone
bills in the middle range.

20. Shapes of Histograms… Symmetry
A histogram is said to be symmetric if, when we draw a vertical line down the center of the histogram, the two sides are identical in shape and size:
Frequency
Frequency
Frequency
Variable
Variable
Variable

21. Shapes of Histograms… Skewness
A skewed histogram is one with a long tail extending to either the right or the left:
Frequency
Frequency
Variable
Variable
Positively Skewed
Negatively Skewed

22. Shapes of Histograms… Unimodal Bimodal Modality
A unimodal histogram is one with a single peak, while a bimodal histogram is one with two peaks:
Frequency
Frequency
Variable
Variable
A modal class is the class with
the largest number of observations

23. Shapes of Histograms… Bell Shape
A special type of symmetric unimodal histogram is a bell shaped histogram:
Many statistical techniques require that the population be bell shaped.
Drawing the histogram helps verify the shape of the population in question.
Frequency
Variable
Bell Shaped

24. Example (Histogram1) The weights in pounds of a sample of 25 workers:
164, 148, 137, 157, 173, 156, 177, 172, 169, 165, 145, 168, 163, 162, 174, 152, 156, 168, 154, 151, 174, 146, 134, 140, and 171.
Q1. Construct Histogram (Frequency Distribution).
Q2. Construct Relative Frequency Distribution.
Q3. How many observations satisfying W<170?
Q4. How many observations satisfying W>160?
Q5. What is the relative frequency satisfying W>150?
Q6. What is a variable?
If 25 workers are population, the relative frequency distribution can be regarded as the probability distribution.
Q6. What is the probability of selecting a worker whose weight is larger than 160 pounds?
Q7. What is the probability of selecting a worker with W<=180?

25. Stem and Leaf Display (Add in -> Data plus )
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).

26. Split each observation into two parts.
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.

27. Stem and Leaf Display
A stem and leaf display for Example 2.4 Stem Leaf
The length of each line represents the frequency of the class defined by
the stem.

28. } 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

29. 2.4 Describing the Relationship Between Two Variables
We are interested in the relationship between two interval variables.
Example 2.9
A real estate agent wants to study the relationship between house price and house size
Twelve houses recently sold are sampled and their size and price recorded
Use graphical technique to describe the relationship between size and price.
Size Price
315
229
335
261
……………..

30. 2.4 Describing the Relationship Between Two Variables (interval and nominal)
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

31. 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.

32. Relationship between Two Nominal Variables…
A cross-classification table (or cross-tabulation table) is used to describe the relationship between two nominal variables: a contingency table.
A cross-classification table lists the frequency of each combination of the values of the two variables…

33. Example 2.10
A sample of newspaper readers was asked to report which newspaper they read: Globe and Mail (1) Post (2), Star (3), Sun (4), and to indicate whether they were blue-collar worker (1), white-collar worker (2), or professional (3).
The responses are stored in file Ex2.10

34. Example 2.10
By counting the number of times each of the 12 combinations occurs, we produced the Table 2.9.
Occupation
Newspaper Blue Collar White Collar Professional Total
G&M
Post
Star
Sun
Total

35. Example 2.10
If occupation and newspaper are related, then there will be differences in the newspapers read among the occupations. An easy way to see this is to covert the frequencies in each column to relative frequencies in each column. That is, compute the column totals and divide each frequency by its column total.
Occupation
Newspaper Blue Collar White Collar Professional
G&M 27/120 = /108 = /126 = .26
Post 18/120 = /108 = /126 = .40
Star 38/120 = /108 = /126 = .17
Sun 37/120 = /108 = /126 = .16

36. 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.

37. Example 2.10
Interpretation: The relative frequencies in the columns 2 & 3 are similar, but there are large differences between columns 1 and 2 and between columns 1 and 3.
This tells us that blue collar workers tend to read different newspapers from both white collar workers and professionals and that white collar and professionals are quite similar in their newspaper choice.
similar
dissimilar

38. 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”

39. Graphing the Relationship Between Two Nominal Variables…
Use the data from the cross-classification table to create bar charts…
Professionals tend to read the Post more than twice as often as the Star or Sun…

40. Describing Time Series Data
Observations measured at the same point in time are called cross-sectional data.
Observations measured at successive points in time are called time-series data.
Time-series data graphed on a line chart, which plots the value of the variable on the vertical axis against the time periods on the horizontal axis.

41. Example 2.8
We recorded the monthly average retail price of gasoline since 1978.
Xm02-08
Draw a line chart to describe these data and briefly describe the results.

42. Example 2.8 How can we describe gasoline price?

43. Example 2.9 Price of Gasoline in 1982-84 Constant Dollars
Xm02-09
Remove the effect of inflation in Example 2.8 to determine whether gasoline prices are higher than they have been in the past after removing the effect of inflation.

44. Example 2.9: Inflation adjusted gasoline price

45. Summary I…
Factors That Identify When to Use Frequency and Relative Frequency Tables, Bar and Pie Charts
Objective: Describe a single set of data.
Data type: Nominal
Factors That Identify When to Use a Histogram, Ogive, or Stem-and-
Leaf Display
Data type: Interval
Factors that Identify When to Use a Cross-classification Table
Objective: Describe the relationship between two variables.
Factors that Identify When to Use a Scatter Diagram

46. Summary II… Interval Data Nominal Single Set of Data Histogram
Frequency and Relative Frequency Tables, Bar and Pie Charts
Relationship Between
Two Variables
Scatter Diagram
Cross-classification Table, Bar Charts