basic statistics introduction to statistics presentation of data.pptx

Copyright © 2020 Pearson Education Ltd.
A LWAY S L E A R N I N G Slide 1
Organizing and
Visualizing Variables
Chapter 2

Objectives
In this chapter you learn:
 How to organize and visualize categorical
variables.
 How to organize and visualize numerical
variables.
 How to summarize a mix of variables.
 How to avoid making common errors when
organizing and visualizing variables.

Organizing Data Creates Both
Tabular And Visual Summaries
 Summaries both guide further exploration and
sometimes facilitate decision making.
 Visual summaries enable rapid review of larger
amounts of data & show possible significant
patterns.
 Often, the Organize and Visualize step in
DCOVA occur concurrently.
DCOVA

Categorical Data Are Organized By
Utilizing Tables
Categorical
Data
Tallying Data
Summary
Table
DCOVA
One
Categorical
Variable
Two
Categorical
Variables
Contingency
Table

Organizing Categorical Data:
Summary Table
 A summary table tallies the frequencies or percentages of items in a set
of categories so that you can see differences between categories.
Devices Used To Watch Movies or TV Shows Percent
Television Set 49%
Tablet 9%
Smartphone 10%
Laptop / Desktop 32%
DCOVA
Devices Millennials Use to Watch Movies or Television Shows
Source: Data extracted and adapted from A. Sharma, “Big Media Needs to Embrace
Digital Shift Not Fight It,” Wall Street Journal, June 22, 2016, p. 1-2.

A Contingency Table Helps Organize
Two or More Categorical Variables
 Used to study patterns that may exist between
the responses of two or more categorical
variables.
 Cross tabulates or tallies jointly the responses
of the 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.
DCOVA

Contingency Table - Example
 A random sample of 400
invoices is drawn.
 Each invoice is categorized
as a small, medium, or large
amount.
 Each invoice is also
examined to identify if there
are any errors.
 This data are then organized
in the contingency table to
the right.
DCOVA
No
Errors Errors Total
Small
Amount
170 20 190
Medium
Amount
100 40 140
Large
Amount
65 5 70
Total 335 65 400
Contingency Table Showing
Frequency of Invoices Categorized
By Size and The Presence Of Errors

Contingency Table Based On
Percentage Of Overall Total
No
Errors Errors Total
Small
Amount
170 20 190
Medium
Amount
100 40 140
Large
Amount
65 5 70
Total 335 65 400
DCOVA
No
Errors Errors Total
Small
Amount
42.50% 5.00% 47.50%
Medium
Amount
25.00% 10.00% 35.00%
Large
Amount
16.25% 1.25% 17.50%
Total 83.75% 16.25% 100.0%
42.50% = 170 / 400
25.00% = 100 / 400
16.25% = 65 / 400
83.75% of sampled invoices
have no errors and 47.50%
of sampled invoices are for
small amounts.

Percentage of Row Totals
No
Errors Errors Total
Small
Amount
170 20 190
Medium
Amount
100 40 140
Large
Amount
65 5 70
Total 335 65 400
DCOVA
No
Errors Errors Total
Small
Amount
89.47% 10.53% 100.0%
Medium
Amount
71.43% 28.57% 100.0%
Large
Amount
92.86% 7.14% 100.0%
Total 83.75% 16.25% 100.0%
89.47% = 170 / 190
71.43% = 100 / 140
92.86% = 65 / 70
Medium invoices have a larger
chance (28.57%) of having
errors than small (10.53%) or
large (7.14%) invoices.

Percentage Of Column Totals
No
Errors Errors Total
Small
Amount
170 20 190
Medium
Amount
100 40 140
Large
Amount
65 5 70
Total 335 65 400
DCOVA
No
Errors Errors Total
Small
Amount
50.75% 30.77% 47.50%
Medium
Amount
29.85% 61.54% 35.00%
Large
Amount
19.40% 7.69% 17.50%
Total 100.0% 100.0% 100.0%
50.75% = 170 / 335
30.77% = 20 / 65
There is a 61.54% chance
that invoices with errors are
of medium size.

Tables Used For Organizing
Numerical Data
Numerical Data
Ordered Array
DCOVA
Cumulative
Distributions
Frequency
Distributions

Organizing Numerical Data:
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).
Age of
Surveyed
College
Students
Day Students
16 17 17 18 18 18
19 19 20 20 21 22
22 25 27 32 38 42
Night Students
18 18 19 19 20 21
23 28 32 33 41 45
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.
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
DCOVA

 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: 58 - 12 = 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.




 Compute class midpoints: 15, 25, 35, 45, 55.
 Count observations & assign to classes.
DCOVA

Organizing Numerical Data: Frequency
Distribution Example
Class Midpoints Frequency
10 but less than 20 15 3
Total 20
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
DCOVA

Class Frequency
10 but less than 20 3 .15 15%
20 but less than 30 6 .30 30%
30 but less than 40 5 .25 25%
40 but less than 50 4 .20 20%
50 but less than 60 2 .10 10%
Total 20 1.00 100%
Relative
Frequency Percentage
Organizing Numerical Data: Relative &
Percent Frequency Distribution Example
DCOVA
Relative Frequency = Frequency / Total, e.g. 0.10 = 2 / 20

10 but less than 20 3 15% 3 15%
20 but less than 30 6 30% 9 45%
30 but less than 40 5 25% 14 70%
40 but less than 50 4 20% 18 90%
50 but less than 60 2 10% 20 100%
Total 20 100% 20 100%
Organizing Numerical Data: Cumulative
Class Percentage
Cumulative
Percentage
Cumulative Percentage = Cumulative Frequency / Total * 100 e.g. 45% = 100*9/20
Frequency
Cumulative
Frequency
DCOVA

Why Use a Frequency Distribution?
 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 /
clustered.
DCOVA

Frequency Distributions:
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.
DCOVA

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

Visualizing Categorical Data:
The Bar Chart
 The bar chart visualizes a categorical variable as a series of bars. The
length of each bar represents either the frequency or percentage of
values for each category. Each bar is separated by a space called a gap.
DCOVA
Devices
Used to
Watch
Percent
Television Set 49%
Tablet 9%
Smartphone 10%
Laptop /
Desktop
32%

The Pie Chart
 The pie chart is a circle broken up into slices that represent categories.
The size of each slice of the pie varies according to the percentage in
each category.
DCOVA
Devices
Used to
Watch
Percent
Television Set 49%
Tablet 9%
Smartphone 10%
Laptop /
Desktop
32%

The Doughnut Chart DCOVA
 The doughnut chart is the outer part of a circle broken up into pieces
that represent categories. The size of each piece of the doughnut varies
according to the percentage in each category.
Devices
Used to
Watch
Percent
Television Set 49%
Tablet 9%
Smartphone 10%
Laptop /
Desktop
32%

The Pareto Chart
 Used to portray categorical data (nominal
scale).
 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.”
DCOVA

The Pareto Chart (con’t) DCOVA
Cumulative
Cause Frequency Percent Percent
Warped card jammed 365 50.41% 50.41%
Card unreadable 234 32.32% 82.73%
ATM malfunctions 32 4.42% 87.15%
ATM out of cash 28 3.87% 91.02%
Invalid amount requested 23 3.18% 94.20%
Wrong keystroke 23 3.18% 97.38%
Lack of funds in account 19 2.62% 100.00%
Total 724 100.00%
Source: Data extracted from A. Bhalla, “Don’t Misuse the Pareto Principle,” Six Sigma Forum
Magazine, May 2009, pp. 15–18.
Ordered Summary Table For Causes
Of Incomplete ATM Transactions

The Pareto Chart (con’t) DCOVA
The “Vital
Few”

Side By Side Bar Charts
 The side by side bar chart represents the data from a contingency
table.
DCOVA
No Errors
Errors
0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0%
Invoice Size Split Out By Errors
& No Errors
Large Medium Small
Invoices with errors are much more likely to be of
medium size (61.5% vs 30.8% & 7.7%).
No
Errors Errors Total
Small
Amount
50.75% 30.77% 47.50%
Medium
Amount
29.85% 61.54% 35.00%
Large
Amount
19.40% 7.69% 17.50%
Total 100.0% 100.0% 100.0%

Doughnut Charts
 A Doughnut Chart can be used to represent the data from a contingency table.
DCOVA
Invoices with errors are much more likely to be of
medium size (61.5% vs 30.8% & 7.7%).
No
Errors Errors Total
Small
Amount
50.75% 30.77% 47.50%
Medium
Amount
29.85% 61.54% 35.00%
Large
Amount
19.40% 7.69% 17.50%
Total 100.0% 100.0% 100.0%
30.8%
61.5%
7.7%
30.8%
29.9%
19.4%
Invoice Size & Errors
Inner Ring With Errors, Outer Ring No
Errors
Small Medium Large

Visualizing Numerical Data
By Using Graphical Displays
Numerical Data
Ordered Array
Stem-and-Leaf
Display Histogram Polygon Ogive
Frequency Distributions
and
Cumulative Distributions
DCOVA

Stem-and-Leaf Display
 A simple way to see how the data are distributed
and where concentrations of data exist.
METHOD: Separate the sorted data series
into leading digits (the stems) and
the trailing digits (the leaves).
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.
Stem Leaf
1 67788899
2 0012257
3 28
4 2
Age of College Students
Day Students Night Students
Stem Leaf
1 8899
2 0138
3 23
4 15
Age of
Surveye
d College
Students
Day Students
16 17 17 18 18 18
19 19 20 20 21 22
22 25 27 32 38 42
Night Students
18 18 19 19 20 21
23 28 32 33 41 45
DCOVA

Visualizing Numerical Data:
The 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.
DCOVA

The Histogram
Class Frequency
10 but less than 20 3 .15 15
20 but less than 30 6 .30 30
30 but less than 40 5 .25 25
40 but less than 50 4 .20 20
50 but less than 60 2 .10 10
Total 20 1.00 100
Relative
Frequency Percentage
0
2
4
6
8
5 15 25 35 45 55 More
Frequency
Histogram: Age Of Students
(In a percentage
histogram the vertical
axis would be defined to
show the percentage of
observations per class).
DCOVA
Histogram: Temperature

The Percentage Polygon
 A percentage polygon is formed by having the midpoint of
each class represent the data in that class and then connecting
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.
DCOVA

The Frequency Polygon DCOVA
Useful When Comparing Two or More Groups

The Percentage Polygon
DCOVA

The Cumulative Percentage Polygon (Ogive)
DCOVA
Useful When Comparing Two or More Groups

DCOVA
The Cumulative Percentage Polygon (Ogive)

Visualizing Two Numerical Variables
By Using Graphical Displays
Two Numerical
Variables
Scatter
Plot
Time-
Series
Plot
DCOVA

Visualizing Two Numerical
Variables: The Scatter Plot
 Scatter plots are used for numerical data consisting of paired
observations taken from two numerical variables.
 One variable’s values are displayed on the horizontal or X
axis and the other variable’s values are displayed on the
vertical or Y axis.
 Scatter plots are used to examine possible relationships
between two numerical variables.
DCOVA

Scatter Plot Example
Volume
per day
Cost per
day
23 125
26 140
29 146
33 160
38 167
42 170
50 188
55 195
60 200
Cost per Dayvs. Production Volume
0
50
100
150
200
250
20 30 40 50 60 70
Volume per Day
C
ost
per
D
ay
DCOVA

 A Time-Series Plot is used to study
patterns in the values of a numeric
variable over time.
 The Time-Series Plot:
 Numeric variable’s values are on the
vertical axis and the time period is on
the horizontal axis.
Visualizing Two Numerical
Variables: The Time Series Plot
DCOVA

Time Series Plot Example
0
20
40
60
80
100
120
2008 2010 2012 2014 2016 2018
Number
of
Franchises
Year
Number of Franchises, 1996-2004
Year
Number of
Franchises
2009 43
2010 54
2011 60
2012 73
2013 82
2014 95
2015 107
2016 99
2017 95
DCOVA

 A multidimensional contingency table is constructed by
tallying the responses of three or more categorical variables.
 Can be used to discover possible patterns and relationships in
multidimensional data that simpler tables and charts would fail to
make apparent.
 As a practical rule, tables should be limited to no more than
three or four variables.
 In typical use, these tables:
 Extend contingency tables to two or more row or column variables, or
 Replace the frequencies found in a contingency table with summary
information about a numeric variable.
Organizing A Mix Of Variables: The
Multidimensional Contingency Table
DCOVA

A Multidimensional Contingency Table
Tallies Responses Of Three or More
Categorical Variables
Two Dimensional Table Showing
Fund Type and Risk Level for
sample of 479 retirement funds.
DCOVA
Three Dimensional Table
Showing Fund Type, Market
Cap, and Risk Level for the
sample of the 479 retirement
funds.

Excel, Minitab, and JMP Can Be Used To
Create Multidimensional Contingency Tables
 In Excel creating a Pivot Table yields an interactive
display of this type.
 In JMP you can create a table that is also interactive.
 In Minitab you can create such a table but it is not
interactive.
 JMP and Minitab have many specialized statistical &
graphical procedures (not covered in this book) to
analyze & visualize multidimensional data.
DCOVA

Drilling-Down On A Table Reveals
The Data The Table Summarizes
 Clicking a cell in an Excel table displays the
rows of data associated with that cell.
 Clicking a cell in a JMP table highlights those
the rows of data that are the source for that cell.
 Drill-down is perhaps the simplest form of data
discovery.
DCOVA

Drill-Down Reveals The Data
Underlying A Higher-Level Summary
DCOVA
Results of drilling down to
the details about small value
funds with low risk revealing
the ten-year return ranges from
4.83% to 9.44%.

Displays To Visualize A Mix Of Many
Variables
 Displays are more useful than a multidimensional
contingency table with many row and column
variables.
 The data (not just summary statistics) can be
shown for numerical variables.
 Multiple numerical variables can be presented in
one summarization.
 Visualizations can reveal patterns that can be
hard to see in tables.
DCOVA

Colored Scatter Plots Visualize Both
Numerical Variables & Categorical Variable(s)
DCOVA
Observations:
Large Market Capitalization Funds (red dots)
1. Relatively have best returns and lowest expense ratios.
2. Some have either low returns or high expense ratios or both.
JMP Colored
Scatter Plot

Bubble Charts Extend Scatter Plots
 Use the size of points (called bubbles) to
represent the value of an additional variable.
 In Excel and Minitab the additional variable
must be numerical.
 In JMP the variable can be either numerical or
categorical.
DCOVA

DCOVA
An Excel PivotChart Visualizes Specific
Categories From A PivotTable Summary
Low Risk Small
Market Cap Funds
Have The Highest Mean
Return Among Low Risk
Funds.

Treemaps Are Graphical Displays Of
Contingency Tables
DCOVA
Excel Treemap:
Size of tiles correspond to
the frequency in a cell.
JMP Treemap:
Size of tiles correspond to the value
of the numeric variable Market Cap.

DCOVA
Sparklines Are Compact Time-Series
Visualizations Of Numerical Variables
Movie
revenues
by week
per month

Filtering and Querying Data
 Two operations associated with preparing tabular
or visual summaries are Data Filtering and
Querying.
 Data filtering selects rows of data that match
criteria; specified value(s) for specific variable(s).
 Data Querying is similar but may not select all of
the columns of the matching rows.
 Excel, JMP, and Minitab all have various filtering &
query features.
DCOVA

Example Of JMP & Minitab Filtering /
Querying
DCOVA
Selecting all rows in value retirement funds that
have ten-year return percentages that are greater
than or equal to 9.

Excel Slicers Filter & Query Data
From A Pivot Table
 A slicer is a panel of clickable buttons
superimposed over a worksheet.
 Each button in a slicer represents a unique
value of a variable found in a the source data of
a PivotTable.
 By clicking buttons in the slicer panels, you
query the data.
DCOVA

DCOVA
Example Of Slicers For The
Retirement Funds Workbook
With the four slicers below, you can ask questions such as:
1. What are the attributes of the fund(s) with the lowest expense ratio?
2. What are the expense ratios associated with large market cap value
funds that have a star rating of five?

DCOVA
Answering The Questions
What are the
attributes of the
fund(s) with the
lowest expense ratio?
The updated
PivotTable (not shown
below) reveals only
one such fund.
What are the expense ratios
associated with large market cap value
funds that have a star rating of five?
The expense ratios for these funds are:
0.83, 094, 1.05, 1.09, 1.18, and 1.19.

The Challenges in Organizing
and Visualizing Variables
 When organizing and visualizing data need to
be mindful of:
 The limits of other’s ability to perceive and
comprehend.
 Presentation issues that can undercut the usefulness
of methods from this chapter.
 It is easy to create summaries that:
 Obscure the data or
 Create false impressions.
 Contain Chartjunk
DCOVA

An Example Of Obscuring Data,
Information Overload
DCOVA

False Impressions Can Be
Created In Many Ways
 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.
 Chartjunk.
DCOVA

An Example of Selective Summarization, These
Two Summarizations Tell Totally Different Stories
DCOVA
Company
Change
from
Prior
Year Company Year 1 Year 2 Year 3
A +7.2% A -22.6% -33.2% +7.2%
B +24.4% B -4.5% -41.9% +24.4%
C +24.9% C -18.5% -31.5% +24.9%
D +24.8% D -29.4% -48.1% +24.8%
E +12.5% E -1.9% -25.3% +12.5%
F +35.1% F -1.6% -37.8% +35.1%
G +29.7% G +7.4% -13.6% +29.7%

How Obvious Is It That Both Pie Charts
Summarize The Same Data? DCOVA
Why is it hard to tell? What would you do to improve?

Graphical Errors:
No Relative Basis
A’s received by
students.
A’s received by
students.
Bad Presentation
0
200
300
FR SO JR SR
Freq.
10%
30%
FR SO JR SR
FR = Freshmen, SO = Sophomore, JR = Junior, SR = Senior

100
20%
0%
%
Good Presentation
DCOVA

Graphical Errors:
Compressing the Vertical Axis
Good Presentation
Quarterly Sales Quarterly Sales
Bad Presentation
0
25
50
Q1 Q2 Q3 Q4
$
0
100
200
Q1 Q2 Q3 Q4
$

DCOVA

Graphical Errors: No Zero Point
on the Vertical Axis
Monthly Sales
36
39
42
45
J F M A M J
$
Graphing the first six months of sales
Monthly Sales
0
39
42
45
J F M A M J
$
36
Good Presentations
Bad Presentation
DCOVA

Graphical Errors: Chart Junk,
Can You Identify The Junk?
DCOVA
Bad Presentation Good Presentation

Left illustration adapted from S. Watterson, “Liquid Gold—Australians Are Changing the World of Wine. Even the French Seem Grateful.” Time,
November 22, 1999, p. 68-69

Best Practices for Constructing
Visualizations
 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 or “exploded” effects & the use of chartjunk.
 Use consistent colorings in charts meant to be compared.
 Avoid using uncommon chart types including radar,
surface, bubble, cone, and pyramid charts.
DCOVA

Chapter Summary
In this chapter we covered:
 Organizing and visualizing categorical variables.
 Organizing and visualizing numerical variables.
 Summarizing a mix of variables.
 Avoiding common errors when organizing and
visualizing variables.

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