. Chapter 3 Displaying and Describing Categorical Data

Slide 3- 2 The Three Rules of Data Analysis The three rules of data analysis won’t be difficult to remember: Make a picture—things may be revealed that are not obvious in the raw data. These will be things to think about. Make a picture—important features of and patterns in the data will show up. You may also see things that you did not expect. Make a picture—the best way to tell others about your data is with a well-chosen picture.

Slide 3- 3 Case Study - Titanic At 11:40 on the night of April 14, 1912, Frederick Fleet’s cry of “Iceberg, right ahead” signal the beginning of a nightmare that has become legend. By 2:15 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.

Slide 3- 4 Data – passengers and crew aboard the Titanic SurvivalAgeSexClass DeadAdultMaleThird DeadAdultMaleCrew DeadAdultMaleSecond DeadAdultMaleCrew DeadAdultMaleCrew DeadAdultFemaleSecond AliveAdultFemaleFirst DeadChildMaleFirst

Slide 3- 5 Titanic (cont.) Each case (row) of the data table represents a person on board the ship. The variables are Person’s Survival Person’s Age Sex Ticket Class.

Slide 3- 6 Titanic – Variables (cont.) Survival Age Sex Ticket Class Dead or Alive Categorical Adult or Child Categorical Male or Female Categorical First, second, third, crew Categorical

Slide 3- 7 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?

Slide 3- 8 What are the Ws? Who What When Where hoW Why People on the Titanic Survival status, age, sex, ticket class April 14, 1912 North Atlantic A variety of sources and internet sites Historical Interest – see questions above

Slide 3- 9 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

Slide Frequency Tables: Making Piles (cont.) A relative frequency table is similar, but gives the percentages (instead of counts) for each category.

Slide Answer to first question on slide 3-7 What percent of people were in first class? What in second class? There were 325 people in first class. There were 285 people in second class. There were 2201 people on board. So, there were 325 / 2201 ≈ 14.77% of the people in first class and 285 / 2201 ≈ 12.95% in second class.

Slide What’s 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.

Slide The Area Principle The ship display violates the area principle: The area occupied by the graph should correspond to the magnitude of the value it represents.

Slide 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:

Slide 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:

Slide 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.

Slide 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:

Slide 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:

Slide Contingency Tables (cont.) Each cell of the table gives the count for a combination of values of the two values.

Slide Contingency Table (cont.) How many crew members died when the Titanic sunk? 673 What percentage of passengers were crew members and died when the Titanic sunk? 673 / 2201 ≈ 30.58% How many first class passengers survived? 203

Slide 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:

Slide Conditional Distributions (cont.) The following is the conditional distribution of ticket Class, conditional on having perished:

Slide 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:

Slide Conditional Distributions (cont.) Better yet – use side bar charts

Slide 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.

Slide 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:

Slide Recall – Conditional distributions

Slide 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 to survivors! There were 203 survivors in first class. There were 118 survivors in second class. There were a total of 711 survivors in the ship. So, the percent of survivors in first class is 203 / 711 ≈ 28.6% and 118 / 711 ≈ 16.6% in second class. Thus, the percent of survivors in first class were higher than in second class.

Slide Recall – Contingency table

Slide Change question: What percent of passengers in first class survived? What percent of passengers second class survived? There were 203 survivors in first class. There were 325 first class passengers. So, 203 / 325 ≈ 62.5% of passengers in first class survived. There were 118 survivors in second class. There were 285 second class passengers. So, 118 / 285 ≈ 41.4% of passengers in second class survived.

Slide What Can Go Wrong? Don’t 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.

Slide 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? —make sure your display shows what it says it shows.

Slide What Can Go Wrong? (cont.) Don’t confuse similar sounding percentages. Pay particular attention to the wording of the context. Don’t 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.”

Slide What Can Go Wrong? (cont.) Don’t overstate your case—don’t claim something you can’t.

Slide 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.

Slide Exercise 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)

Slide 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?

Slide Answers Exercise 3.25 First, get a table with marginal totals:

Slide Answer Exercise 3.25 (cont.) What percent of the graduates are white? There are white graduates and total graduates. ≈ 82.5% of the graduates are white.

Slide Answer Exercise 3.25 (cont.) What percent of the graduates are planning to attend a 2-year college? There are 42 graduates 42 / 325 ≈ of the graduates are planning to attend a 2-year college.

Slide Answer Exercise 3.25 (cont.) What percent of the graduates are white and planning to attend a 2-year college? white graduates are planning to attend 2-year colleges. 36 / 325 ≈ of graduates are white and planning to attend a 2-year college.

Slide Answer Exercise 3.25 (cont.) What percent of the white graduates are planning to attend a 2-year college? white graduates are planning to attend 2-year colleges. There are white graduates. ≈ 13.4% of white graduates are planning to attend a 2-year college.

Slide Answer Exercise 3.25 (cont.) What percent of the graduates planning to attend a 2-year college are white? graduates are planning to attend 2-year colleges. white graduates are planning to attend 2-year colleges. ≈ 85.7% of the graduates planning to attend a 2-year college are white.

Slide Exercise 3.27 – Seniors 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

Slide Answer Exercise 3.27 – Seniors Find the conditional distributions (percentages) of plans for the white students.

Slide Answer Exercise 3.27 – Seniors (cont.) Find the conditional distributions (percentages) of plans for the minority students.

Slide Answer Exercise 3.27 – Seniors (cont.) Create a graph comparing the plans of white and minority students.

Slide Answer Exercise 3.27 – Seniors (cont.) Segmented bar chart:

Slide Alternatively – side bar charts:

Slide 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% -These graphs show that race and plans for graduation are. There is little evidence of an association between race and plans for graduation.