Displaying and Describing

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Chapter 3

• Displaying and Describing
• Categorical Data

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The Three Rules of Data Analysis

• 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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Case Study - Titanic

• At 1140 on the night of April 14, 1912,
  Frederick Fleets cry of Iceberg, right ahead
  signal the beginning of a nightmare that has
  become legend.
• By 215 am the Titanic thought by many to be
  unsinkable, had sunk, leaving more than 1,500
  passengers and crew members on board to meet
  their icy fate.

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Data passengers and crew aboard the Titanic
Survival Age Sex Class
Dead Adult Male Third
Dead Adult Male Crew
Dead Adult Male Second
Dead Adult Male Crew
Dead Adult Male Crew
Dead Adult Female Second
Alive Adult Female First
Dead Child Male First
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Titanic (cont.)

• Each case (row) of the data table represents a
  person on board the ship.
• The variables are

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Titanic Variables (cont.)

• Survival
• Age
• Sex
• Ticket Class

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Where from here?

• We will study categorical variables.
• Interesting when we look at how categorical
  variables work together.
• Example of questions
• - What percent of people were in first class?
  What in second class?
• - Was the percent of survivors higher in first
  class than in second class?

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What are the Ws?

• Who
• What
• When
• Where
• hoW
• Why

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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

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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.

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Answer to first question on slide 3-7

• What percent of people were in first class? What
  in second class?
• There were people in first class.
• There were people in second class.
• There were people on board.
• So, there were
• people in first class and

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Whats Wrong With This Picture?

• The length of the ship is the count in each
  class.
• 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 makes it look like most of the
  people on the Titanic were crew members, with a
  few passengers along for the ride.

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The Area Principle

• The ship display violates the area principle
• The area occupied by the graph should correspond
  to

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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 a better display for the ship data is

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Bar Charts (cont.)

• A relative frequency bar chart displays the
  relative
• A relative frequency bar chart also stays true to
  the area principle.
• Replacing counts with percentages in the ship
  data

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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.

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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 we can examine the class of ticket and
  whether a person survived the Titanic

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Contingency Tables (cont.)

• 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.
• Ex. The marginal distribution of Survival is

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Contingency Tables (cont.)

• Each cell of the table gives the count for a
  combination of values of the two values.

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Contingency Table (cont.)

• How many crew members died when the Titanic sunk?
• What percentage of crew members died when the
  Titanic sunk?
• How many first class passengers 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 conditional distribution of ticket Class,
  conditional on having survived

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Conditional Distributions (cont.)

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

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Conditional Distributions (cont.)

• The conditional distributions tell us that there
  is a difference in class for those who survived
  and those who perished.
• This can be shown with pie charts of the two
  distributions

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Conditional Distributions (cont.)

• Better yet use side bar charts

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Conditional Distributions (cont.)

• 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.
• 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, that they are not independent.

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Segmented Bar Charts

• 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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Recall Conditional distributions
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Answer to second question on slide 3-7

• Was the percent of survivors higher in first
  class than in second class?
• The Who here is restricted
• There were survivors in first class.
• There were survivors in second class.
• There were a total of survivors in
  the ship.
• So, the percent of survivors in first class is
• Thus, the percent of survivors in first class
  were

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Recall Contingency table
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Change question

• What percent of first and second class passengers
  survived?
• There were survivors in first class.
• There were first class passengers.
• So,
  in first class survived.
• There were survivors in second class.
• There were second class passengers.
• So, of
  passengers in second class survived.

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What Can Go Wrong?

• Dont violate
• 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.)

• 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 percentages. Pay
  particular attention to the wording of the
  context.
• Dont forget to look at the variables separately
  too. Examine 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 case - dont claim something
  you cant.

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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.
• If conditional distributions of one variable are
  the same for every category of the other, the
  variables are independent.

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Exercise 3.25 - Seniors

• Prior to graduation, a high school class was
  surveyed about its plans. The following table
  displays the results for white and minority
  students (included African American, Asian,
  Hispanic and Native American)

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Exercise 3.25 (cont.)

• What percent of the graduates are white?
• What percent of the graduates are planning to
  attend a 2-year college?
• What percent of the graduates are white and
  planning to attend a 2-year college?
• What percent of the white graduates are planning
  to attend a 2-year college?
• What percent of the graduates planning to attend
  a 2-year college are white?

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Answers Exercise 3.25

• First, get a table with marginal totals

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Answer Exercise 3.25 (cont.)

• What percent of the graduates are white?

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Answer Exercise 3.25 (cont.)

• What percent of the graduates are planning to
  attend a 2-year college?

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Answer Exercise 3.25 (cont.)

• What percent of the graduates are white and
  planning to attend a 2-year college?

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Answer Exercise 3.25 (cont.)

• What percent of the white graduates are planning
  to attend a 2-year college?

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Answer Exercise 3.25 (cont.)

• What percent of the graduates planning to attend
  a 2-year college are white?

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Exercise 3.27 Seniors (cont.)

• Find the conditional distributions (percentages)
  of plans for the white students.
• Find the conditional distributions (percentages)
  of plans for the minority students.
• Create a graph comparing the plans of white and
  minority students
• Do you see any important differences in the
  post-graduation plans of white and minority
  students? Write a brief summary of what these
  data show, including comparisons of conditional
  distributions

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Answer Exercise 3.27 Seniors

• Find the conditional distributions (percentages)
  of plans for the white students.

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Answer Exercise 3.27 Seniors (cont.)

• Find the conditional distributions (percentages)
  of plans for the minority students.

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Answer Exercise 3.27 Seniors (cont.)

• Create a graph comparing the plans of white and
  minority students.

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Answer Exercise 3.27 Seniors (cont.)

• Segmented bar chart

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• Alternatively side bar charts

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Answer Exercise 3.27 Seniors (cont.)

• Do you see any important differences in the
  post-graduation plans of white and minority
  students? Write a brief summary of what these
  data show, including comparisons of conditional
  distributions
• -
• - Caution should be used with the percentages for
  Minority graduates, because the total is so small
  each graduate is almost 2.
• -