Chapter 3: Displaying and Describing Categorical Data

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Chapter 3 Displaying and DescribingCategorical
Data

• The three rules of data analysis wont be
  difficult to remember
  
• Make a picturethings may be revealed that are
  not obvious in the raw data. These will be things
  to think about.
  
• Make a pictureimportant features of and patterns
  in the data will show up. You may also see things
  that you did not expect.
  
• Make a picturethe best way to tell others about
  your data is with a well-chosen picture.

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Frequency Tables Making Piles

• We can pile the data by counting the number of
  data values in each category of interest.
  
• We can organize these counts into a frequency
  table, which records the totals and the category
  names.
  
• People on Titanic by Ticket Class

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Frequency Tables Making Piles (cont.)

• A relative frequency table is similar, but gives
  the percentages (instead of counts) for each
  category.

Relative Frequency of People on Titanic by Ticket
Class
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Displaying DataWhats Wrong With This Picture?

• You might think that
• a good way to show
• the Titanic data is
• with this display
• There are 2 things wrong

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The Area Principle

• The ship display makes it look like most of the
  people on the Titanic were crew members, with a
  few passengers along for the ride.
  
• When we look at each ship, we see the area taken
  up by the ship, instead of the length of the
  ship.
  
• The ship display violates the area principle
• The area occupied by a part of the graph should
  correspond to the magnitude of the value it
  represents.

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More on Displaying Data (not in Text)

• Your table or graphical display should always
  have a title or caption, so that casual readers
  can understand what is being presented at a
  glance
• An informative display often leads people to read
  your paper!
  
• If you are writing a paper, it is imperative that
  you attribute the source of your data.
  
• Keep the display easy-to-read
• Use simple fonts, colors/patterns
• Make sure that your reader distinguish between
  different categories
  
• Common problem- graphs that look good on your
  color monitor may be hard to read when printed
  with a B/W printer
  

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Bar Charts

• A bar chart displays the distribution of a
  categorical variable, showing the counts for each
  category next to each other for easy comparison.
  
• A bar chart stays true
  
  to the area principle.
  
• Thus, it is a better
• display for this data
• Dont forget a title (or at
• least a caption!)

People on Titanic by Ticket Class
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Bar Charts (cont.)

• A relative frequency bar chart displays the
  relative proportion of counts for each category.
  
• A relative frequency bar chart also stays true to
  the area principle.
  
• Replacing counts
  
  with percentages
  
  in the ship data
• Dont forget a title/caption

Percentage of Titanic Passengers in each Ticket
Class
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Pie Charts

• When you are interested in parts of the whole, a
  pie chart might be your display of choice.
  
• Pie charts show the whole
  group of cases as
  a circle.
  
• They slice the circle into
  pieces whose size
  is
  proportional to the
  fraction
  of the whole
  in each
  category.
• Dont forget a title /caption

Number of Titanic Passengers by Class
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Contingency Tables

• A contingency table allows us to look at two
  categorical variables together.
  
• It shows how individuals are distributed along
  each variable, contingent on the value of the
  other variable.
  
• Example of a contingency table of ticket class
  and survival.
  

Survival and Class of Titanic Passengers
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Contingency Tables (cont)

• Each cell of the contingency table gives the
  count for a combination of values of the two
  values.
  
• For example, the second cell in the crew column
  tells us that 673 crew members died when the
  Titanic sunk.

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Contingency Tables

• The margins of the table, both on the right and
  on the bottom, give totals and the frequency
  distributions for each of the variables.
  
• Each frequency distribution is called a marginal
  distribution of its respective variable.
  
• The marginal distribution of Survival is
• (Can also phrase as What aboard the Titanic
  Survived?)
  

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Conditional Distributions

• A conditional distribution shows the distribution
  of one variable for just the individuals who
  satisfy some condition on another variable.
  
• The following is the conditional distribution of
  ticket Class, conditional on having survived

For example 28.6 of those who survived were
from First Class How were these ages calculated?

What might be another way to phrase this?
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Conditional Distributions (cont.)

• The following is the conditional distribution of
  ticket Class. It is conditional on having
  perished
  

How were these ages calculated?
What might be another way to phrase this?
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Conditional Distributions (cont.)

• The conditional distributions tell us that there
  was a difference in class for those who survived
  and those who perished.
  
• Rather than a
• table of numbers,
• this is better
  shown with
  
  pie charts of
  
  the two
  distributions

Titanic Survivors and Non-survivors, by Class
What would these pie charts look like if class
had no influence on survival?

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Conditional Distributions (cont.)

• We see that the distribution of Class for the
  survivors is different from that of the
  non-survivors.
  
• This leads us to believe that Class and Survival
  are associated, and are not independent.
  
• The variables would be considered independent
  when the distribution of one variable in a
  contingency table is the same for all categories
  of the other variable.

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Segmented Bar Charts
Titanic Survivors and Non-survivors, by Class

• A segmented bar chart displays the same
  information as a pie chart, but in the form of
  bars instead of circles.
  
• Here is the segmented bar chart for ticket Class
  by Survival status

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What Can Go Wrong?

• Dont violate the area principle.
• While some people might like the pie chart on the
  left better, it is harder to compare fractions of
  the whole, which a well-done pie chart does.

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What Can Go Wrong? (cont.)

• Keep it honestmake sure your display shows what
  it says it shows.
  
• This plot of the percentage of high-school
  students who engage in specified dangerous
  behaviors has a problem. Can you see it?

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What Can Go Wrong? (cont.)

• Dont confuse similar-sounding percentagespay
  particular attention to the wording of the
  context.
  
• Dont forget to look at the variables separately
  tooexamine the marginal distributions, since it
  is important to know how many cases are in each
  category.
• Be sure to use enough individuals!
• Do not make a report like We found that 66.67 of
  the rats improved their performance with
  training. The other rat died.
  

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What Can Go Wrong? (cont.)

• Dont overstate your casedont claim something
  you cant.
  
• Dont use unfair or silly averagesthis could
  lead to Simpsons Paradox, so be careful when you
  average one variable across different levels of a
  second variable.

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Simpsons Paradox Example

• Chris has a 3.20 SFSU GPA
• Sean has a 3.34 SFSU GPA
• Who seems to do better in SFSU classes, Chris or
  Sean?
  
• Who do you think is likely to get a better grade
  in DS412 (Operations Management)
  
• What else might be useful to know?

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Simpsons Paradox, Continued

• Is comparing Chriss and Seans GPAs a fair
  assessment of their abilities to get good grades
  in particular classes?
  
• What data is shared in common?
• Who do you think is likely to get a better grade
  in DS412 (Operations Management) now?
  

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What have we learned?

• We can summarize categorical data by counting the
  number of cases in each category (expressing
  these as counts or percents).
  
• We can display the distribution in a bar chart or
  pie chart.
  
• And, we can examine two-way tables called
  contingency tables, examining marginal and/or
  conditional distributions of the variables.

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Additional Examples

• Pr9 A May 2001 Gallup Poll found that many
  Americans believe in supernatural phenomena.
  The poll was based on telephone responses from
  1012 randomly selected adults.
• Is it reasonable to conclude 66 of those polled
  believe in either Ghosts or Astrology?
  
• Can you tell what of people did not believe in
  any of these phenomena? Explain.
  
• What is an appropriate graph?

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Chart for Example 9
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Additional Examples

• Pr 22 A survey of autos in the student lot and
  staff lot at SFSU classified cars by country of
  origin
  
• What of cars surveyed where foreign?
• What of the American cars were student owned?
• What of students own American cars?
• What is the marginal distributor of origin?
• What is the conditional distribution of origin by
  owner type?
  
• Do you think that car origin is independent of
  owner type? Use a graph to explain your argument

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Example 22 continued

• First create
• Contingency Table Origin of Students' and
  Staffs' Cars
  

• Answers
• a) (45102)/359 41
• b) We have 212 American cars of which 107 were
  student driven, 50.5
  
• c) We have 195 students, of whom 107 drove
  American cars, 55
  
• (note that b and c look similar but are not the
  same question,
  
• nor do they have the same answer!)
• Marginal Dist of care origin
• American 212/359 59, Europe 13, Asian
  28
  

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Example 22 continued

• f) Now find Conditional Distributions for each
  driver type
  
• Does Car Origin Seem independent of type of
  driver?
  
• What is an even better way to show this?

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Example 22- Extension

• Pick an appropriate graph to show the difference
  between the two distributions
  

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Example Old Test Problem

• The Bookstore will run a contest where the prize
  is a pair of concert tickets and must decide
  between the upcoming Rolling Stones and the Black
  Eyed Peas concerts. They want to figure out
  which will generate more excitement, so they paid
  a student to run a survey of students and staff
  (profs, administrators, and other non-students)
  on preferences. The student produced the
  following table
• A) What percent of people surveyed preferred
  tickets to the Rolling Stones?
  
• B) What percent of people surveyed were Staff?
• C) What percent of the students preferred tickets
  to the Rolling Stones?
  
• D) What percent of those who preferred tickets to
  the Rolling Stones were students?
  
• E) What percent of respondents were students and
  preferred tickets to the Rolling Stones?
  
• F) Does it appear that preference for concert
  tickets is independent of a respondents being a
  staff or a student? Show why using calculations
  or graphics

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Example Old Test Problem

• First, it is probably useful to fill out the
  contingency table with the marginal
  distributions
  
• A) What percent of people surveyed preferred
  tickets to the Rolling Stones?
  
• B) What percent of people surveyed were Staff?
• C) What percent of the students preferred tickets
  to the Rolling Stones?
  
• D) What percent of those who preferred tickets to
  the Rolling Stones were students?
  
• E) What percent of respondents were students and
  preferred tickets to the Rolling Stones?
  
• F) Does it appear that preference for concert
  tickets is independent of a respondents being a
  staff or a student? Show why using calculations
  or graphics