Topic 2.1: Introducing Statistics: Are Variables Related?

Learning Objectives:

Identify questions to be answered about possible relationships in data.
Distinguish between categorical variables and quantitative variables in two-variable settings.
Choose appropriate graphical displays for two categorical variables and for two quantitative variables.
Recognize the numerical summaries introduced for comparing two categorical variables and two quantitative variables.
Describe how apparent patterns and associations may or may not indicate a real relationship.
Use context to explain what kind of relationship a graph or table might suggest.

Essential Knowledge:

Apparent patterns and associations in data may be random or not.
A categorical variable takes values that are category names or labels.
A quantitative variable takes numerical values for a measured or counted quantity.
To study two categorical variables, begin with graphs such as segmented bar graphs or mosaic plots and then compare conditional relative frequencies.
To study two quantitative variables, begin with a scatter plot and then consider correlation, linear regression, and the coefficient of determination.
Visual displays and numerical summaries help decide whether variables appear to be related.
Different variable types call for different displays and summaries.

Key Vocabulary

Categorical Variable	A variable whose values are category names or labels.
Quantitative Variable	A variable whose values are numerical measurements or counts.
Relationship	An association or pattern between two variables.
Association	A pattern showing that one variable may be connected to another.
Segmented Bar Graph	A bar graph that shows the distribution of one categorical variable within the categories of another.
Mosaic Plot	A display for two categorical variables that shows category proportions with bar widths and segment heights.
Two-Way Table	A table that summarizes counts for combinations of two categorical variables.
Conditional Relative Frequency	A percent found within one row or column of a two-way table to compare groups.
Scatter Plot	A graph of paired quantitative data points.
Correlation (r)	A numerical summary that describes the direction and strength of a linear relationship.
Line of Best Fit	A regression line used to model the pattern in quantitative data.
Coefficient of Determination	A numerical summary that describes how much variation in y is explained by the regression model.

Video 1: Introducing Statistics: Are Variables Related?

[0:00 – 6:55]

1. [0:12] This video asks how we decide whether there is a relationship between two variables and between two variables.

2. [0:40] A categorical variable takes values that are category names or .

3. [0:48] A quantitative variable takes numerical values for a measured or quantity.

Age Group and Educational Attainment [1:02 – 1:27] The first example asks whether age group and educational attainment are related. In the video, age is grouped and educational attainment is listed by categories, so both variables are treated as categorical.

Variable	Categories Shown in the Video
Age Group	25 to 34, 35 to 54, 55 and older
Educational Attainment	No high school diploma, high school graduate, some college/associate's degree, master's degree or higher

4. [1:27] In this example, educational attainment is a variable.

5. [1:44] Age group is also treated as because the ages have been grouped into .

6. [2:06] To study two categorical variables, we start with representations and then use representations.

Two Categorical Variables [2:12 – 3:09]

Use graphs such as segmented bar graphs and mosaic plots to compare distributions visually.
Then use a two-way table to calculate conditional relative frequencies and compare percents.

7. [2:27] One graph used for two categorical variables is a segmented .

8. [2:54] Another display for two categorical variables is a .

9. [3:08] The numerical summaries shown in parentheses were calculated as .

10. [3:31] The video says people ages 25 to 34 are to drop out of high school than the other age groups.

11. [3:52] The 35 to 54 group appeared to have earned a master's degree or higher.

School Attendance and Math Scores [4:08 – 4:56] The second example uses a random sample of 11 students. For each student, the video records percent of school days attended and the number of Algebra 1 questions answered correctly.

Variable	Recorded for Each Student
Attendance	Percent of school days attended
Math Score	Number of Algebra 1 questions answered correctly

Sample size: 11 students

12. [4:41] In the attendance example, both variables are .

13. [4:56] For two quantitative variables, we again begin with representations and then use representations.

14. [5:13] The standard graph for two quantitative variables is a .

15. [5:35] One numerical summary mentioned for two quantitative variables is the correlation value .

16. [5:35] Another numerical summary is the equation of a line of .

17. [5:35] The video also names the coefficient of .

18. [5:56] At first glance, as attendance increases, exam performance also tends to .

Choosing Tools by Variable Type [6:04 – 6:36]

Two categorical variables: bar-graph-based displays plus conditional relative frequencies from a two-way table.
Two quantitative variables: scatter plot plus correlation, linear regression, and coefficient of determination.

19. [6:18] For two categorical variables, the two-way table is used to calculate conditional relative .

20. [6:36] For two quantitative variables, the video highlights correlation, linear , and the coefficient of determination.

21. Reflection: How does identifying whether variables are categorical or quantitative help you choose the correct graph and numerical summary?

22. Reflection: Why does this lesson warn that an apparent pattern or association may be random and not automatically meaningful?

Exit Ticket

A school counselor is investigating two possible relationships.

Study A: grade level (9th, 10th, 11th, 12th) and preferred after-school activity (sports, clubs, job, or home).

Study B: hours of sleep on a school night and quiz score for a random sample of students.

(a) For Study A, identify the type of each variable and name one appropriate graph and one numerical summary to explore whether the variables are related.

(b) For Study B, identify the type of each variable and name one appropriate graph and two numerical summaries to explore whether the variables are related.

(c) Describe one apparent pattern that would suggest a relationship in each study.

(d) Explain why an apparent pattern in either study might still be due to random variation.

Topic 2.1: Introducing Statistics: Are Variables Related?

Key Vocabulary

Video 1: Introducing Statistics: Are Variables Related?

Exit Ticket

Class Responses (Esc to close)