Displaying Distributions – Qualitative Variables – Part 2

1. Displaying Distributions – Qualitative Variables – Part 2 Lecture 16 Sec. 4.3.3 Wed, Feb 11, 2004

2. Studies with Two Qualitative Variables • Typically, the purpose of studying two variables is to see whether there is a relationship between them. • Also, when working with qualitative data, percentages are the numerical measure of choice. • The next-most-common measure is frequency (or count).

3. Relationships between Two Qualitative Variables • Frequency table – A table where • The rows represent values of one variable, • The columns represent values of the other variable, • And the cells show the frequency of the row-column combinations of values. • A frequency table is also called a contingency table.

4. Example • Let the row variable be the student’s year in college. • Let the column variable be whether the student is from Virginia or is from out of state. • This will be a 4 x 2 frequency table.

5. Example

6. Frequency Tables • If there is a relationship between the variables, then perhaps it will be apparent from the table. • Perhaps not. • Do we see any relationship between year in college and state of residence?

7. Example

8. Example • See example on page 199.

9. Example • Is there any apparent relationship between academic performance and nutritional status? • It is hard to say (in my opinion). • A possible relationship is that students with better nutrition perform better academically.

10. Excel Bar Graph

11. Excel Bar Graph

12. The Marginal Distribution • Each variable has a marginal distribution. • To find the marginal distribution of a variable, find the total frequency of the cells for each value of that variable. • Then express each total frequency as a percentage of the grand total for all cells.

13. The Marginal Distribution of Nutritional Status

14. Example • The grand total of frequencies is 1000. • The marginal distribution for nutritional status is

15. The Marginal Distribution of Academic Performance

16. Example • The marginal distribution for academic performance is

17. The Marginal Distribution • The marginal distribution shows us the distribution of one variable independently of the other variable.

18. Conditional Distributions • In the example, • What percentage of all students are below average academically and have poor nutrition? • What percentage of students who are below average academically have poor nutrition? • What percentage of students who have poor nutrition are below average academically?

19. Conditional Distributions • The answers are • 70/1000 = 7% • 70/200 = 35% • 70/290 = 24%

20. Conditional Distributions • To get the conditional distribution of academic performance given nutritional status, • For each category of nutritional status (i.e., for each column), divide the various frequencies in that category by the total for that category.

21. Conditional Distributions • The conditional distribution of academic performance given nutritional status is

22. Conditional Distributions • The conditional distribution of nutritional status given academic performance is

23. Let's Do It! • Let's do it! 4.8, p. 203 – Beer Tastes. • Let’s do it! 4.9, p. 205 – About Your Class. • Use the data concerning year in college vs. whether in or out of state.

24. Assignment • Page 206: Exercises 12 – 17. • Page 249: Exercises 62 – 66.