Topic 2.4: Representing the Relationship Between Two Quantitative Variables

Learning Objectives:

Identify explanatory and response variables in bivariate quantitative data.
Represent two quantitative variables with a scatterplot.
Decide when a scatterplot is the appropriate display for bivariate data.
Describe a scatterplot using direction, form, strength, and unusual features.
Write a contextual description of the relationship shown in a scatterplot.

Essential Knowledge:

A bivariate quantitative data set records two quantitative variables for each individual in a sample or population.
A scatterplot shows one numeric value on the x-axis and one numeric value on the y-axis for each observation.
An explanatory variable is used to explain or predict corresponding values of a response variable.
Direction can be described as positive or negative, depending on how y changes as x increases.
Form can be linear or nonlinear, strength describes how closely points follow a pattern, and unusual features can include clusters or apparent outliers.
A properly constructed scatterplot includes a clear title, labeled axes, and scales with tick marks.

Key Vocabulary

Bivariate Quantitative Data	Observations of two quantitative variables recorded for each individual.
Scatterplot	A graph that displays paired quantitative data as points on coordinate axes.
Explanatory Variable	The variable used to explain or predict the values of another variable.
Response Variable	The variable whose values respond to changes in the explanatory variable.
Direction	The overall trend of the points as x-values increase.
Positive Association	As one variable increases, the other variable tends to increase.
Negative Association	As one variable increases, the other variable tends to decrease.
Form	The shape of the relationship, such as linear or nonlinear.
Strength	How closely the points follow a clear pattern.
Unusual Features	Points or groupings that stand out, such as clusters or apparent outliers.
Linear	A pattern that follows a roughly straight-line trend.
Nonlinear	A pattern that bends or curves rather than following a straight line.

Video 1: Explanatory and Response Variables; Constructing Scatter Plots

[0:00 – 5:14]

1. [0:00] Topic 2.4 distinguishes between and variables and shows how to construct a .

Income, Attendance, and Achievement Context [0:31 – 2:49] The lesson begins with group-level trends in income, attendance, and math achievement. The speaker stresses that these averages describe patterns for groups, not individual performance or intelligence.

2. [1:51] School administrators also noticed an income gap, with higher-income areas tending to have fewer chronically absent students.

3. [2:20] Some systems hoped that improving might raise .

4. [3:06] The two quantitative variables in the sample are percent and .

5. [3:31] If attendance is used to explain performance, then attendance is the variable and questions correct is the variable.

6. [3:31] Memory trick: the variable is explanatory, and the variable is response.

Scatterplot Setup Rule [3:31 – 4:39]

Put the explanatory variable on the x-axis.
Put the response variable on the y-axis.
Use a scatterplot when each individual has two quantitative values.

7. [4:00] Because the data involve two variables, the correct display is a .

8. [4:39] In a scatterplot, each student is represented by one pair of .

9. [4:39] The x-coordinate uses the variable, and the y-coordinate uses the variable.

10. [4:39] A well-constructed scatterplot should include a , labeled , and scales with .

11. [5:14] Explanatory variables help trends in response variables.

12. Reflection: How do explanatory and response variables help you decide what goes on each axis of a scatterplot?

Video 2: Describing Scatter Plots

[0:05 – 6:38]

13. [0:19] A complete scatterplot description includes , , , and unusual .

14. [2:01] In a association, as x-values increase, y-values also tend to .

15. [2:20] In a association, as x-values increase, y-values tend to .

16. [2:35] For the attendance data, the direction of the association is .

17. [3:32] The overall form of the relationship appears approximately .

18. [3:44] One unusual feature is a student with especially low .

19. [4:05] When the points follow the pattern closely, the association is .

20. [4:35] A weak association has more around the model.

Scatterplot Description Checklist [1:49 – 5:47]

State the direction: positive or negative.
State the form: linear or nonlinear.
State the strength and note any unusual features.
Describe the relationship in context of the variables.

21. [5:27] A complete written description should also mention the of the variables.

22. [5:47] In sentence form, the relationship between attendance rate and exam performance appears , , and .

23. Reflection: Why is it not enough to describe a scatterplot only by saying it is positive? What else belongs in a complete description?

Exit Ticket

A teacher recorded the number of hours each student studied for a quiz and the score each student earned.

Student	Hours Studied	Quiz Score (%)
A	1	62
B	2	67
C	3	73
D	4	78
E	5	84
F	6	90

(a) Identify the explanatory variable and the response variable, and explain why.

(b) Explain why a scatterplot is the appropriate display for these data.

(c) If you construct the scatterplot, describe the likely direction, form, and strength in context.

(d) List two features that a correctly constructed scatterplot should include.

Topic 2.4: Representing the Relationship Between Two Quantitative Variables

Key Vocabulary

Video 1: Explanatory and Response Variables; Constructing Scatter Plots

Video 2: Describing Scatter Plots

Exit Ticket

Class Responses (Esc to close)