1. Organizing and Visualizing Variables
MATH2114 Week 7, Thursday
Organizing and Visualizing Variables
Slides Adopted & modified from “Business Statistics: A First Course” 7th Ed, by Levine, Szabat and Stephan, 2016 Pearson Education Inc.

2. Some feedback from Tuesday’s class
Do I have to buy the textbook?
We’re teaching from it, and the exam is open book
Not strictly required for any assessment
What does the assessment for this class look like?
Weblearn tests
5 Tutorials (First week is not assessed, leaving 4 assessed)
Exam
Little class participation :(

3. Objective/Goals Methods to organize variables.
Methods to visualize variables.
Methods to organize or visualize more than one variable at the same time.
Principles of proper visualizations
How not to mess it up too badly

4. Unambiguous, can be difficult to communicate and understand
Tables
Unambiguous, can be difficult to communicate and understand

5. Categorical Data DCOVA Categorical Data Summary Table
One Categorical Variable
Two Categorical Variables
Contingency Table
Tallying/Counting Data

6. Summary Table (1 variable)
DCOVA
Summary Table (1 variable)
Tallies frequencies or percentages of items in different categories
Main Reason Young Adults Shop Online
Reason For Shopping Online?
Percent
Better Prices
37%
Avoiding holiday crowds or hassles
29%
Convenience
18%
Better selection
13%
Ships directly 3%
Source: Data extracted and adapted from “Main Reason Young Adults Shop Online?”
USA Today, December 5, 2012, p. 1A.

7. Contingency Tables (2 or more variables)
DCOVA
Contingency Tables (2 or more variables)
Cross tabulates or tallies jointly the responses of the categorical variables
Used to study patterns that may exist between the responses of two or more categorical variables
For two variables the tallies for one variable are located in the rows and the tallies for the second variable are located in the columns

8. Raw frequencies DCOVA Filming Promotion Public Event Wedding Total
Record of all permits issued by City of Melbourne for public events, film/photo shoots, weddings and promotions/sampling.
Filming
Promotion
Public Event
Wedding
Total
Summer
120 85
255
286
746
Autumn
134
119
266
242
761
Winter
133 55
106 15
309
Spring
112
110
272
207
701
499
369
899
750
2517
Data sourced from City of Melbourne Open Data:
Accessed 6/9/2017, last updated August 22, 2017

9. Percentages DCOVA Row % Total % Column %
Filming
Promotion
Public Event
Wedding
Total
Summer
16.10%
11.40%
34.20%
38.30%
100%
Autumn
17.60%
15.60%
35%
31.80%
Winter
43%
17.80%
34.30%
4.90%
Spring
16%
15.70%
38.80%
29.50%
Permits issued in winter tended to be for filming
Total %
Filming
Promotion
Public Event
Wedding
Total
Summer
4.80%
3.40%
10.10%
11.40%
30%
Autumn
5.30%
4.70%
10.60%
9.60%
Winter
2.20%
4.20%
0.60%
12%
Spring
4.40%
10.80%
8.20%
28%
19.80%
14.70%
35.70%
29.80%
Column %
Filming
Promotion
Public Event
Wedding
Total
Summer
24%
23%
28.40%
38.10%
Autumn
26.90%
32.20%
29.60%
32.30%
Winter
26.70%
14.90%
11.80% 2%
Spring
22.40%
29.80%
30.30%
27.60%
100.00%
99.90%
100.10%
11.4% of permits issued were for summer weddings
Very few weddings requiring permits occurred in winter

10. Numerical Data DCOVA Numerical Data Ordered Array Cumulative
Distributions
Frequency

11. Average time (minutes) to get to university
DCOVA
Ordered Array
An ordered array is a sequence of data, in rank order, from the smallest value to the largest value.
Shows range (minimum value to maximum value)
May help identify outliers (unusual observations)
Are you a:
Average time (minutes) to get to university
Evening Person 10 15 20 25 30 35 40 45 50 60 70 80 90
Morning Person 1 65
Neither

12. Frequency Distribution
DCOVA
Frequency Distribution
The frequency distribution is a summary table in which the data are arranged into numerically ordered classes.
You must give attention to selecting the appropriate number of class groupings for the table, determining a suitable width of a class grouping, and establishing the boundaries of each class grouping to avoid overlapping.
The number of classes depends on the number of values in the data. With a larger number of values, typically there are more classes. In general, a frequency distribution should have at least 5 but no more than 15 classes.
To determine the width of a class interval, you divide the range (Highest value–Lowest value) of the data by the number of class groupings desired.

13. Frequency Distribution Example
DCOVA
Frequency Distribution Example
Example: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature
24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27

14. Frequency Distribution Example
DCOVA
Frequency Distribution Example
Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Find range: = 46
Select number of classes: 5 (usually between 5 and 15)
Compute class interval (width): 10 (46/5 then round up)
Determine class boundaries (limits):
Class 1: 10 but less than 20
Class 2: 20 but less than 30
Class 3: 30 but less than 40
Class 4: 40 but less than 50
Class 5: 50 but less than 60
Compute class midpoints: 15, 25, 35, 45, 55
Count observations & assign to classes

15. Frequency Distribution Example
DCOVA
Frequency Distribution Example
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Class Midpoints Frequency
10 but less than
20 but less than
30 but less than
40 but less than
50 but less than
Total

16. Relative Frequency and Percentage Distribution
DCOVA
Relative Frequency and Percentage Distribution
Class Frequency
10 but less than %
20 but less than %
30 but less than %
40 but less than %
50 but less than %
Total %
Relative
Frequency
Percentage
Relative Frequency = Frequency / Total, e.g = 2 / 20

17. Cumulative Frequency Distribution
DCOVA
Cumulative Frequency Distribution
Cumulative Frequency
Cumulative Percentage
Class
Frequency
Percentage
10 but less than % %
20 but less than % %
30 but less than % %
40 but less than % %
50 but less than % %
Total %
Cumulative Percentage = Cumulative Frequency / Total * e.g. 45% = 9/20 * 100

18. Why is this a good idea? DCOVA
It condenses the raw data into a more useful form;
It allows for a quick visual interpretation of the data;
It enables the determination of the major characteristics of the data set including where the data are concentrated and/or clustered.

19. DCOVA
Some Tips
Different class boundaries may provide different pictures for the same data (especially for smaller data sets)
Shifts in data concentration may show up when different class boundaries are chosen
As the size of the data set increases, the impact of alterations in the selection of class boundaries is greatly reduced
When comparing two or more groups with different sample sizes, you must use either a relative frequency or a percentage distribution

20. Easy to communicate and understand, ambiguous and simplified
DCOVA
Graphs
Easy to communicate and understand, ambiguous and simplified

21. Summary Table For One Variable Contingency Table For Two Variables
DCOVA
Graphical Displays
Categorical Data
Visualizing Data
Bar
Chart
Summary Table For One Variable
Contingency Table For Two Variables
Side By Side Bar Chart
Pie Chart
Pareto

22. Categorical Data DCOVA Pie Chart Bar Chart (mind the gap) Category
Count %
Anything/IDK 5
16%
Economy 4
13%
Engineering
Other
Pop Culture
Science 2 6%

23. Ordered Summary Table for Causes of Incomplete ATM Transactions
DCOVA
Pareto Chart
Used to portray categorical data
A vertical bar chart, where categories are shown in descending order of frequency
A cumulative polygon is shown in the same graph
Used to separate the “vital few” from the “trivial many”
Ordered Summary Table for Causes
of Incomplete ATM Transactions

24. DCOVA
Pareto Chart
The “Vital
Few”

25. Side-by-Side Bar Charts
DCOVA
Side-by-Side Bar Charts No
Errors
Total
Small
Amount
50.75%
30.77%
47.50%
Medium
29.85%
61.54%
35.00%
Large
19.40%
7.69%
17.50%
100.0%
Invoices with errors are much more likely to be of medium size (61.54% vs 30.77% and 7.69%)

26. Visualising Univariate Numerical Data
DCOVA
Visualising Univariate Numerical Data
Numerical Data
Ordered Array
Stem-and-Leaf
Display
Histogram
Polygon
Ogive
Frequency Distributions and
Cumulative Distributions

27. Average time (minutes) to get to university
DCOVA
Stem and Leaf Display
A stem-and-leaf display organizes data into groups (called stems) so that the values within each group (the leaves) branch out to the right on each row.
A simple way to see how the data are distributed and where concentrations of data exist
Good for small amounts of data
Morning
0|1
1|55
2|00
3|000
4|05555 5|
6|05 7| 8| 9|
Evening 0|
1|0005
2|055
3|005
4|0555
5|000000 6|
7|0
8|0
9|0
Are you a:
Average time (minutes) to get to university
Evening Person 10 15 20 25 30 35 40 45 50 60 70 80 90
Morning Person 1 65

28. DCOVA
Histogram
A vertical bar chart of the data in a frequency distribution is called a histogram.
In a histogram there are no gaps between adjacent bars.
The class boundaries (or class midpoints) are shown on the horizontal axis.
The vertical axis is either frequency, relative frequency, or percentage.
The height of the bars represent the frequency, relative frequency, or percentage.
Good for large amounts of data
Class Frequency
10 but less than
20 but less than
30 but less than
40 but less than
50 but less than
Total
Relative Frequency
Percentage

29. DCOVA
Polygon
Use midpoint of each class represent the data in that class
Connect the sequence of midpoints at their respective class percentages.
The cumulative percentage polygon, or ogive, displays the variable of interest along the X axis, and the cumulative percentages along the Y axis.
Useful when there are two or more groups to compare

30. Visualising Multivariate Numeric Data
DCOVA
Visualising Multivariate Numeric Data
Two Numerical Variables
Scatter Plot
Time-Series Plot

31. DCOVA
Scatter Plots
Scatter plots are used for numerical data consisting of paired observations taken from two numerical variables
One variable is measured on the vertical axis and the other variable is measured on the horizontal axis
Scatter plots are used to examine possible relationships between two numerical variables
Volume per day
Cost per day 23
125 26
140 29
146 33
160 38
167 42
170 50
188 55
195 60
200

32. DCOVA
Time Series Plot
Used to study patterns in a numeric variable over time
Numeric variable is measured on the vertical axis, time on the horizontal
Year
Number of Franchises
1996 43
1997 54
1998 60
1999 73
2000 82
2001 95
2002
107
2003 99
2004

33. DCOVA
Organising many categorical variables: Multidimensional Contingency Table
A multidimensional contingency table is constructed by tallying the responses of three or more categorical variables.
In Excel creating a Pivot Table to yield an interactive display of this type.
Dedicated statistical platforms have specialised methods for analysing/visualising this sort of data in a more sophisticated ways

34. Pivot Tables DCOVA A pivot table:
Summarizes variables as a multidimensional summary table
Allows interactive changing of the level of summarization and formatting of the variables
Allows you to interactively “slice” your data to summarize subsets of data that meet specified criteria
Can be used to discover possible patterns and relationships in multidimensional data that simpler tables and charts would fail to make apparent

35. Pivot Table Examples DCOVA
Three Dimensional Table Showing The Mean 10 Year Return % Broken Out By Type Of Fund, Market Cap, &Risk Level
Two Dimensional Table Showing The Mean 10 Year Return % Broken Out By Type Of Fund & Risk Level

36. How to work your way through the data? Data Discovery Methods
DCOVA
How to work your way through the data? Data Discovery Methods
Drill-down is perhaps the simplest form of data discovery
Results of drilling down to
the details about small
market cap value funds with
low risk

37. A quick guide in what not to do
DCOVA
A quick guide in what not to do
People can only comprehend so much information at once
Presentation plays a huge role in how useful visualisation is
It’s very easy to make data summaries that are misleading
Obscure the data
Give the wrong impression
Far more subtle
than this
Photo taken by /u/benjaminteeeee, posted on reddit.com/r/australia

38. Information Overload leads to obscured data
DCOVA
Information Overload leads to obscured data

39. DCOVA
False Impressions are easy to make “There are three kinds of lies: Lies, damned lies, and statistics”
Selective summarization
Presenting only part of the data collected
Improperly constructed charts
Potential pie chart issues
Improperly scaled axes
A Y axis that does not begin at the origin or is a broken axis missing intermediate values
Chart junk

40. Selective Summarisation Life is looking great
DCOVA
Selective Summarisation Life is looking great
Company
Change from Prior Year A
+7.2% B
+24.4% C
+24.9% D
+24.8% E
+12.5% F
+35.1% G
+29.7%

41. Selective Summarisation Life is looking great… Until you look further
DCOVA
Selective Summarisation Life is looking great… Until you look further
Company
Change from Prior Year
Year 1
Year 2
Year 3 A
+7.2%
-22.6%
-33.2% B
+24.4%
-4.5%
-41.9% C
+24.9%
-18.5%
-31.5% D
+24.8%
-29.4%
-48.1% E
+12.5%
-1.9%
-25.3% F
+35.1%
-1.6%
-37.8% G
+29.7%
+7.4%
-13.6%

42. How obvious is it that both of these summarise the same data?
DCOVA
How obvious is it that both of these summarise the same data?
Why is it hard to tell? What would you do to improve?

43. Graphical Errors: No relative Basis
DCOVA
Graphical Errors: No relative Basis
%: Students participation rate in each year level
Freq.: Number of Students who participated in the survey

44. Graphical Errors: Compressing the Vertical Axis
DCOVA
Graphical Errors: Compressing the Vertical Axis

45. Graphical Errors No Zero Point on the Vertical Axis
DCOVA
Graphical Errors No Zero Point on the Vertical Axis

46. DCOVA
Chart Junk

47. DCOVA
So much chart junk

48. Please make the chart junk stop
DCOVA
Please make the chart junk stop

49. <Cries in chart junk>
DCOVA
<Cries in chart junk>

50. It’s very easy to accidentally create distortions in Excel
DCOVA
It’s very easy to accidentally create distortions in Excel
Excel often will create a graph where the vertical axis does not start at 0
Excel offers the opportunity to turn simple charts into 3-D charts and in the process can create distorted images
Unusual charts offered as choices by Excel will most often create distorted images

51. Best Practices DCOVA Use the simplest possible visualization
Include a title
Label all axes
Include a scale for each axis if the chart contains axes
Begin the scale for a vertical axis at zero
Use a constant scale
Avoid 3D effects
Avoid chartjunk

52. Summary Methods to organize variables. Methods to visualize variables.
Methods to organize or visualize more than one variable at the same time.
Principles of proper visualizations.

53. Self-Learning Activities
Readings:
Chapter 2 study case titled: Clear Mountain State Student Surveys on p101 of the textbook and think about how to attempt the case questions by applying the concepts learnt in this chapter
USING STATISTICS The Choice Is Yours on p93 –p94
Chapter 2 Excel Guide, EG2.1 – EG2.6, on p102 – p111
Note: only read the contents relating to EXCEL, both In-Depth Excel and AnalysisToolPak; and discard the portions using PHStart.