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

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

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

Example Let the row variable be the student’s year in college. 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. 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. This will be a 4 x 2 frequency table.

Example (frequency)Virginia Out of State Freshman00 Sophomore00 Junior00 Senior00

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

Example (percentage)Virginia Out of State Freshman00 Sophomore00 Junior00 Senior00

Example See example on page 199. See example on page 199. Nutritional Status PoorAdeq.Exc. AcademicPerformance Below Average Above903070

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

Excel Bar Graph

The Marginal Distribution Each variable has a 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. 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. Then express each total frequency as a percentage of the grand total for all cells.

The Marginal Distribution of Nutritional Status Nutritional Status PoorAdeq.Exc. AcademicPerformance Below Average Above Total

Example The grand total of frequencies is The grand total of frequencies is The marginal distribution for nutritional status is The marginal distribution for nutritional status is PoorAdequateExcellent 29%57.5%13.5%

The Marginal Distribution of Academic Performance Nutritional Status Total PoorAdeq.Exc. AcademicPerformance Below Average Above

Example The marginal distribution for academic performance is The marginal distribution for academic performance is BelowAverageAbove ?????????

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

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

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

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

Conditional Distributions The conditional distribution of academic performance given nutritional status is The conditional distribution of academic performance given nutritional status is Nutritional Status PoorOkayGood AcademicPerformance Below24%17%26% Average45%78%22% Above31%5%52%

Conditional Distributions The conditional distribution of nutritional status given academic performance is The conditional distribution of nutritional status given academic performance is Nutritional Status PoorOkayGood AcademicPerformance Below????????? Average????????? Above?????????

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

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