1. + Warm Up Which of these variables are categorical? Which are quantitative?

2. + Chapter 1: Exploring Data 8/27 Section 1.1 Analyzing Categorical Data The Practice of Statistics, 4 th edition - For AP* STARNES, YATES, MOORE

3. + Data Analysis RECALL FROM LAST CLASS…. Statistics* is the science of data. Data Analysis* is the process of organizing, displaying, summarizing, and asking questions about data. Definitions: Individuals* – objects (people, animals, things) described by a set of data Variable* - any characteristic of an individual Categorical Variable* – places an individual into one of several groups or categories. Quantitative Variable * – takes numerical values for which it makes sense to find an average.

4. + Data Analysis A variable generally takes on many different values.In data analysis, we are interested in how often avariable takes on each value. Distribution* – tells us what values a variable takes and how often it takes those values Variable of Interest: MPG Dotplot of MPG Distribution Example

5. + Introduction Data Analysis: Making Sense of Data In this section, we learned that… A dataset contains information on individuals*. For each individual, data give values for one or more variables*. Variables can be categorical* or quantitative*. The distribution* of a variable describes what values it takes and how often it takes them. Inference* is the process of making a conclusion about a population based on a sample set of data. Summary

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

7. + Section 1.1 Analyzing Categorical Data 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 Learning Objectives

8. + Analyzing Categorical Data Categorical Variables* place individuals into one of several groups or categories The values of a categorical variable are labels for thedifferent categories The distribution of a categorical variable lists the count orpercent of individuals who fall into each category. Frequency Table FormatCount of Stations Adult Contemporary1556 Adult Standards1196 Contemporary Hit569 Country2066 News/Talk2179 Oldies1060 Religious2014 Rock869 Spanish Language750 Other Formats1579 Total13838 Relative Frequency Table FormatPercent of Stations Adult Contemporary11.2 Adult Standards8.6 Contemporary Hit4.1 Country14.9 News/Talk15.7 Oldies7.7 Religious14.6 Rock6.3 Spanish Language5.4 Other Formats11.4 Total99.9 Example, page 8 Count Percent Variable Values

9. + Frequency Table*: displays the counts in each category Relative Frequency Table*: Shows the percents in each category How do we calculate the relative frequency? Frequency Table FormatCount of Stations Adult Contemporary1556 Adult Standards1196 Contemporary Hit569 Country2066 News/Talk2179 Oldies1060 Religious2014 Rock869 Spanish Language750 Other Formats1579 Total13838 Relative Frequency Table FormatPercent of Stations Adult Contemporary11.2 Adult Standards8.6 Contemporary Hit4.1 Country14.9 News/Talk15.7 Oldies7.7 Religious14.6 Rock6.3 Spanish Language5.4 Other Formats11.4 Total99.9

10. + Note: Roundoff Errors It’s a good idea to check data for consistency. The counts should add up to 13,838 ---the total number of stations. (They do) The percents should add up to be 100%....in this case, they add up to be 99.9%....why? Each percent is rounded to the nearest tenth. The EXACT percents would add to 100% but the rounded percents will only come close. ROUNDOFF ERROR*: The effect of rounding off results Roundoff errors don’t point to mistakes in our work, just the effects of rounding

11. + Analyzing Categorical Data Displaying categorical data Frequency tables can be difficult to read. Sometimes is is easier to analyze a distribution by displaying itwith a bar graph (bar chart) or pie chart. Frequency Table FormatCount of Stations Adult Contemporary1556 Adult Standards1196 Contemporary Hit569 Country2066 News/Talk2179 Oldies1060 Religious2014 Rock869 Spanish Language750 Other Formats1579 Total13838 Relative Frequency Table FormatPercent of Stations Adult Contemporary11.2 Adult Standards8.6 Contemporary Hit4.1 Country14.9 News/Talk15.7 Oldies7.7 Religious14.6 Rock6.3 Spanish Language5.4 Other Formats11.4 Total99.9

12. + Analyzing Categorical Data 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! Graphs: Good and Bad Alternate Example This ad for DIRECTV has multiple problems. How many can you point out?

13. + 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. Definition: Two-way Table* – describes two categorical variables, organizing counts according to a row variable and a column variable. Young adults by gender and chance of getting rich FemaleMaleTotal Almost no chance9698194 Some chance, but probably not426286712 A 50-50 chance6967201416 A good chance6637581421 Almost certain4865971083 Total236724594826 Example, p. 12 What are the variables described by this two- way table? How many young adults were surveyed?

14. + Analyzing Categorical Data Two-Way Tables and Marginal Distributions 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. Note: Percents are often more informative than counts, especially when comparing groups of different sizes. To examine a marginal distribution, 1) Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. 2) Make a graph to display the marginal distribution.

15. + Example) Find the percent of young adults who think they are almost certain to be rich by age 30? Young adults by gender and chance of getting rich FemaleMaleTotal Almost no chance9698194 Some chance, but probably not426286712 A 50-50 chance6967201416 A good chance6637581421 Almost certain4865971083 Total236724594826

16. + Young adults by gender and chance of getting rich FemaleMaleTotal Almost no chance9698194 Some chance, but probably not426286712 A 50-50 chance6967201416 A good chance6637581421 Almost certain4865971083 Total236724594826 Analyzing Categorical Data Two-Way Tables and Marginal Distributions ResponsePercent 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% Example, p. 13 Examine the marginal distribution of chance of getting rich.

17. + Analyzing Categorical Data Relationships Between Categorical Variables Marginal distributions tell us nothing about the relationshipbetween two variables. 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, 1) Select the row(s) or column(s) of interest. 2) Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). 3) Make a graph to display the conditional distribution. Use a side-by-side bar graph or segmented bar graph to compare distributions.

18. + Example) Calculate the conditional distribution of the opinion among the men… Find the percent of MEN who think they are almost certain to be rich by age 30? Young adults by gender and chance of getting rich FemaleMaleTotal Almost no chance9698194 Some chance, but probably not426286712 A 50-50 chance6967201416 A good chance6637581421 Almost certain4865971083 Total236724594826

19. + Young adults by gender and chance of getting rich FemaleMaleTotal Almost no chance9698194 Some chance, but probably not426286712 A 50-50 chance6967201416 A good chance6637581421 Almost certain4865971083 Total236724594826 Analyzing Categorical Data Two-Way Tables and Conditional Distributions ResponseMale 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% Example, p. 15 Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion. Female 96/2367 = 4.1% 426/2367 = 18.0% 696/2367 = 29.4% 663/2367 = 28.0% 486/2367 = 20.5%

20. + Analyzing Categorical Data Organizing a Statistical Problem As you learn more about statistics, you will be asked to solvemore complex problems. Here is a four-step process you can follow. State: What’s the question that you’re trying to answer? Plan: How will you go about answering the question? What statistical techniques does this problem call for? Do: Make graphs and carry out needed calculations. Conclude: Give your practical conclusion in the setting of the real-world problem. How to Organize a Statistical Problem: A Four-Step Process WRITE THIS DOWN IN THE FRONT PAGE OF YOUR NOTEBOOK!

21. + Section 1.1 Analyzing Categorical Data 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. Summary

22. + Section 1.1 Analyzing Categorical Data 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. Summary, continued

23. + Looking Ahead… 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…

24. + HOMEWORK: Pg. 7 #1-6 Section 1.1 #10-34 even