Chapter 1: Exploring Data Section 1.1 Analyzing Categorical Data The Practice of Statistics, 4th edition - For AP* STARNES, YATES, MOORE

Chapter 1 Exploring Data Introduction: Data Analysis: Making Sense of Data 1.1 Analyzing Categorical Data 1.2 Displaying Quantitative Data with Graphs 1.3 Describing Quantitative Data with Numbers

Section 1.1 Analyzing Categorical Data Learning Objectives After this section, you should be able to… CONSTRUCT and INTERPRET bar graphs and pie charts RECOGNIZE “good” and “bad” graphs CONSTRUCT and INTERPRET two-way tables DESCRIBE relationships between two categorical variables ORGANIZE statistical problems

Individuals vs Variables Individuals: the objects described by a set of data Individuals can be people, animals or things Variables: any characteristic of an individual Variables can take different values for different individuals

Analyzing Categorical Data Categorical Variables place individuals into one of several groups or categories The values of a categorical variable are labels for the different categories The distribution of a categorical variable lists the count or percent of individuals who fall into each category. Analyzing Categorical Data Example, page 8 Frequency Table Format Count of Stations Adult Contemporary 1556 Adult Standards 1196 Contemporary Hit 569 Country 2066 News/Talk 2179 Oldies 1060 Religious 2014 Rock 869 Spanish Language 750 Other Formats 1579 Total 13838 Relative Frequency Table Format Percent of Stations Adult Contemporary 11.2 Adult Standards 8.6 Contemporary Hit 4.1 Country 14.9 News/Talk 15.7 Oldies 7.7 Religious 14.6 Rock 6.3 Spanish Language 5.4 Other Formats 11.4 Total 99.9 Variable Count Percent Values

Analyzing Quantitative Data Quantitative Variables: take on numerical values What’s the difference between categorical and quantitative variables? “who” is being measured vs. “what” is being measured Do we ever use numbers to describe the values of a categorical variable? Analyzing Quantitative Data

The distribution of a variable tells us what values the variable takes and how often it takes these values. Distribution

Here is information about 10 randomly selected US residents from the 2000 census. Who are the individuals in the data set? What variables are measured? Identify each as categorical or quantitative. In what units were the quantitative variables measured? Describe the individual in the first row. Example

Individuals: the 10 randomly selected U. S Individuals: the 10 randomly selected U.S. residents from the 2000 census. Categorical: state, gender, marital status Quantitative: number of family members, age in years, total income in dollars, travel time to work in mins. This person is a 61 year old married female from Kentucky who drives 20 minutes to work and has a total income of $21,000. She has 2 family members in her household. Answers

Analyzing Categorical Data Displaying categorical data Frequency tables: displays the counts of each category Relative Frequency table: shows the percents of each category Analyzing Categorical Data Frequency Table Format Count of Stations Adult Contemporary 1556 Adult Standards 1196 Contemporary Hit 569 Country 2066 News/Talk 2179 Oldies 1060 Religious 2014 Rock 869 Spanish Language 750 Other Formats 1579 Total 13838 Relative Frequency Table Format Percent of Stations Adult Contemporary 11.2 Adult Standards 8.6 Contemporary Hit 4.1 Country 14.9 News/Talk 15.7 Oldies 7.7 Religious 14.6 Rock 6.3 Spanish Language 5.4 Other Formats 11.4 Total 99.9

Analyzing Categorical Data What is the difference between a frequency table and a relative frequency table? When is it better to use relative frequency tables? Analyzing Categorical Data

Analyzing Categorical Data Displaying categorical data Frequency tables can be difficult to read. Sometimes is is easier to analyze a distribution by displaying it with a bar graph or pie chart. Analyzing Categorical Data

Analyzing Categorical Data What is the most important thing to remember when making pie charts and bar graphs? Why do statisticians prefer bar graphs? When is it inappropriate to use a pie chart? Hint: categorical vs. quantitative What are some common ways to make a misleading graph? Analyzing Categorical Data

Analyzing Categorical Data Graphs: Good and Bad Bar graphs compare several quantities by comparing the heights of bars that represent those quantities. Our eyes react to the area of the bars as well as height. Be sure to make your bars equally wide. Avoid the temptation to replace the bars with pictures for greater appeal…this can be misleading! Analyzing Categorical Data Alternate Example: The following ad for DIRECTV has multiple problems. See how many your students can point out. First, the heights of the bars are not accurate. According to the graph, the difference between 81 and 95 is much greater than the difference between 56 and 81. Also, the extra width for the DIRECTV bar is deceptive since our eyes respond to the area, not just the height. Alternate Example This ad for DIRECTV has multiple problems. How many can you point out?

Classwork: Frequency Tables 1. Choose or generate a question that will result in a range of categorical data. Examples: What is your favorite ice cream flavor?, If you could be a superhero, what would your super power be?, etc. 2. Survey at least 25 people to gather data to answer your question. Record your responses. 3. Organize your data into a frequency table. 4. Create a relative frequency table of the results. 5. Create a bar graph of your data. (Don’t forget labels!) 6. Write a brief explanation as to why or why not a pie chart would be appropriate for your data. Classwork: Frequency Tables

Analyzing Categorical Data In the past, we have looked at data with one categorical variable. Now we will look at data with more than one categorical variable. Reminder: Explanatory Variable:Any variable that explains the response variable. Often called an independent variable or predictor variable. Response Variable: The outcome of a study. A variable you would be interested in predicting or forecasting. Often called a dependent variable or predicted variable. Analyzing Categorical Data

Analyzing Categorical Data Two-Way Tables and Marginal Distributions When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables. Analyzing Categorical Data Definition: Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable. Example, p. 12 What are the variables described by this two-way table? How many young adults were surveyed? How many females were surveyed? Young adults by gender and chance of getting rich Female Male Total Almost no chance 96 98 194 Some chance, but probably not 426 286 712 A 50-50 chance 696 720 1416 A good chance 663 758 1421 Almost certain 486 597 1083 2367 2459 4826 Alternate Example: Super Powers A sample of 200 children from the United Kingdom ages 9-17 was selected from the CensusAtSchool website (www.censusatschool.com). The gender of each student was recorded along with which super power they would most like to have: invisibility, super strength, telepathy (ability to read minds), ability to fly, or ability to freeze time. Here are the results:

Analyzing Categorical Data Two-Way Tables and Marginal Distributions Analyzing Categorical Data Definition: The Marginal Distribution of one of the categorical variables in a two-way table of counts is the distribution of values of that variable among all individuals described by the table. Why use the marginal distribution?: Percents are often more informative than counts, especially when comparing groups of different sizes. To examine a marginal distribution, Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. Make a graph to display the marginal distribution.

Analyzing Categorical Data Two-Way Tables and Marginal Distributions Analyzing Categorical Data Example, p. 13 Examine the marginal distribution of chance of getting rich. Hint: Marginal distributions are calculated in the margins! If there is no “total” row or column, make one! Young adults by gender and chance of getting rich Female Male Total Almost no chance 96 98 194 Some chance, but probably not 426 286 712 A 50-50 chance 696 720 1416 A good chance 663 758 1421 Almost certain 486 597 1083 2367 2459 4826 Response Percent Almost no chance 194/4826 = 4.0% Some chance 712/4826 = 14.8% A 50-50 chance 1416/4826 = 29.3% A good chance 1421/4826 = 29.4% Almost certain 1083/4826 = 22.4%

Analyzing Categorical Data Relationships Between Categorical Variables Problem: Marginal distributions tell us nothing about the relationship between two variables. Analyzing Categorical Data Definition: A Conditional Distribution of a variable describes the values of that variable among individuals who have a specific value of another variable. To examine or compare conditional distributions, Select the row(s) or column(s) of interest. Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). Make a graph to display the conditional distribution. Use a side-by-side bar graph or segmented bar graph to compare distributions.

Analyzing Categorical Data Two-Way Tables and Conditional Distributions Analyzing Categorical Data Example, p. 15 Young adults by gender and chance of getting rich Female Male Total Almost no chance 96 98 194 Some chance, but probably not 426 286 712 A 50-50 chance 696 720 1416 A good chance 663 758 1421 Almost certain 486 597 1083 2367 2459 4826 Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion. Response Male Almost no chance 98/2459 = 4.0% Some chance 286/2459 = 11.6% A 50-50 chance 720/2459 = 29.3% A good chance 758/2459 = 30.8% Almost certain 597/2459 = 24.3% Female 96/2367 = 4.1% 426/2367 = 18.0% 696/2367 = 29.4% 663/2367 = 28.0% 486/2367 = 20.5%

Segmented Bar Graph: For each category of one variable, there is a single bar divided into categories of the other variable. Why are they good to use? They are easy to compare! Forces you to use percents Segmented Bar Graph

The whole point of analyzing more than one categorical variable at the same time is to see if they are associated. What does it mean for two variables to have an association? Knowing the value of one variable helps you predict the value of the other variable. (Think about explanatory and response) Association

Example: A sample of 200 children from the United Kingdom ages 9–17 was selected from the CensusAtSchool website. The gender of each student was recorded along with which super power they would most like to have: invisibility, super strength, telepathy (ability to read minds), ability to fly, or ability to freeze time. Would you say there is association between the variables by looking at the two-way table? In other words, does gender explain super power preference? Female Male Total Invisibilty 17 13 30 Super Strength 3 20 Telepathy 39 5 44 Fly 36 18 54 Freeze Time 32 52 115 85 200

Let’s create the conditional distribution: a) Explain what it would mean if there was no association between gender and superpower preference? (b) Based on this data, can we conclude there is an association between gender and super power preference? Justify. Female Invisibility .15 Super Strength .03 Telepathy .34 Fly .31 Freeze Time .17 Male Invisibility .15 Super Strength .20 Telepathy .06 Fly .21 Freeze Time .38

Section 1.1 Analyzing Categorical Data Summary In this section, we learned that… The distribution of a categorical variable lists the categories and gives the count or percent of individuals that fall into each category. Pie charts and bar graphs display the distribution of a categorical variable. A two-way table of counts organizes data about two categorical variables. The row-totals and column-totals in a two-way table give the marginal distributions of the two individual variables. There are two sets of conditional distributions for a two-way table.

Section 1.1 Analyzing Categorical Data Summary, continued In this section, we learned that… We can use a side-by-side bar graph or a segmented bar graph to display conditional distributions. To describe the association between the row and column variables, compare an appropriate set of conditional distributions. Even a strong association between two categorical variables can be influenced by other variables lurking in the background. You can organize many problems using the four steps state, plan, do, and conclude.

Looking Ahead… In the next Section… We’ll learn how to display quantitative data. Dotplots Stemplots Histograms We’ll also learn how to describe and compare distributions of quantitative data. In the next Section…