Immaculata Week 2014 July 28—August 1, 2014

1. Immaculata Week 2014July 28—August 1, 2014 Statistics: Analyzing 2 Categorical Variables ELEMENTARY LEVEL • Session #1 • Presented by: Dr. Del Ferster

2. Some questions to get us started • Why are statistics significant? Why should we have young students be aware of statistics? • What kind of statistics can we consider with elementary students? • Why do many people who have studied statistics have “bad memories” of the subject?

3. What’s in store for today? • We’re going to spend time today on QUALITATIVE STATISTICS. • We’ll consider effective ways to summarize qualitative statistics. • We’ll build TWO WAY TABLES. • We’ll do an activity involving qualitative statistics that you might be able to adapt for use with your students.

4. Some Basic Definitions

5. Qualitative Variables/Categorical Variables • Qualitative variables classify the data into categories. • The categories may or may not have a natural ordering to them. • Qualitative variables are also called categorical variables. • EXAMPLES • Eye color • Political party • Gender • Do you smoke?

6. Quantitative Variables • Quantitative variableshave numerical values that are measurements (length, weight, and so on) or counts (of how many). • Examples: • How many are in your family? • How many cars do you own?

7. More on Quantitative Variables • We further distinguish quantitative variables based on whether or not the values fall on a continuum. • A discrete variable is one for which you can count the number of possible values. • How many siblings a person has • A continuous variable can take on any value within a given interval. • A person’s weight

8. Quantitative Variables • We’ll take a closer look at quantitative variables during our next meeting.

9. 1 Categorical Variable A look at ways to represent our data

10. Distribution of a categorical variable • The distributionof a categorical variable provides the possible values that a variable can take on and how often these possible values occur. • The distribution of a categorical variable shows the pattern of variation of the variable.

11. Example #1 • According to the Bureau of Justice, the following data represent the number of inmates by ethnicity in 2007.

12. Graphing Qualitative Data • Often, rather than simply presenting numerical values, we choose to graph our data. • When generating a graph of 1 categorical variable, we might consider the following types of graph. • Pie Chart • Bar Graph

13. Pie Chart • A pie chart displays the distribution of the qualitative variable by dividing the circle into wedges corresponding to the categories of the variable such that the angle of each wedge is proportional to the percentage of items in that category. • Pie Charts are easy to do in EXCEL. 

14. A Pie Chart for the Prison Data

15. A Pie Chart for the Prison Data (Using Percents)

16. Bar Graph • A bar graph displays the distribution of a qualitative variable by listing the categories of the variable along one axis and drawing a bar over each category with a height equal to the percentage of items in that category. • The bars should all be of equal width. • We could also do one using percents. 

17. Bar Graph for the Prison Data

18. Comparing 2 Categorical Variable How does one variable compare to another? 2 Way Tables

19. Comparing 2Categorical Variables • 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.

20. Comparing 2Categorical Variables • When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables. Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable.

21. Two-Way Tables • Two-way tables come about when we are interested in the relationship between two categorical variables. • One of the variables is the row variable. • The other is the column variable. • The combination of a row variable and a column variable is a cell.

22. Example #2 • Dr. F is hosting 38 of his friends to a cookout. Now, Dr. F. has limited cooking skills, so everyone is having a burger. However, he has bought sufficient tomatoes so that anyone who wants tomato on his or her burger will be happy. • The following slide details the results of his burger and tomato survey. • For the record….a good burger needs only 2 things…CHEESE….and KETCHUP! 

23. Column variable Cells Row Totals Column Totals Row variable Overall Total Burger/Tomato Two-Way Table • Let’s look at the components of a 2 way table

24. Example #3 • Dr. F. decided to survey a group of young adults, to determine whether they expected to be rich by the age of 30. • He decided to consider gender as one variable • The other variable indicates each participant’s expected likelihood of being rich (using the following options) • Almost no chance • Some chance, but probably not • A 50-50 chance • A good chance • Almost certain

25. The summary 2 way table

26. Marginal Distribution TheMarginal 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.

27. More on Marginal Distribution • 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

28. Marginal Distribution Examine the marginal distribution of chance of getting rich.

29. The need for more?? • Marginal distributions tell us nothing about the relationship between two variables. A Conditional Distribution of a variable describes the values of that variable among individuals who have a specific value of another variable.

30. Conditional Distribution • 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.

31. Conditional Distribution • Calculate the conditional distribution of opinion among males. • Examine the relationship between gender and opinion.

32. A chance to work on one together An example concerning marginal and conditional distributions

33. Setting up our problem • Enrollment of recent high school graduates. The table below gives some census data concerning the enrollment status of recent high school graduates aged 16 to 24 years.

34. Continuing our problem • How many male recent high school graduates aged 16 to 24 years were enrolled full-time in two-year colleges? • How many female recent high school graduates aged 16 to 24 years were enrolled in graduate schools? 890 366

35. Now the big challenge • See if you can use the skills developed in this presentation to complete the handout that Dr. F. will distribute. • Feel free to consult your notes, work together, or ask me if you get really stuck or frustrated. • RELAX…it’s just for FUN! 

36. Solution #1 • The marginal distribution of gender

37. Solution #1 • Graph of The marginal distribution of gender

38. Solution #2 • The marginal distribution of status

39. Solution #2 • Graph of The marginal distribution of status

40. Solution #3 • Conditional Distribution of Gender for each status

41. Solution #3 • Graph of Conditional Distribution of Gender for each status

42. Solution #4 • Conditional Distribution of Status for Each Gender

43. Solution #4 • Graph of Conditional Distribution of Status for Each Gender

44. Solution #4 • A Different Graph of Conditional Distribution of Status for Each Gender

45. Questions or Concerns?

46. Looking Ahead • Next time we’ll be looking at: • 1. Analysis of quantitative statistics • 2. We’ll consider linear regression (without having to actually calculate the equation of the regression line. • 3. We’ll also look at correlation