Students interpret the value of the correlation coefficient as a measure of strength and direction of a linear relationship.
Students explain why correlation does not imply causation.
New York State Common Core Math Module 2, Algebra I, Lesson 19
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These lessons introduces students to the correlation coefficient, a measure of the strength of a linear relationship between two numerical values. The focus of this lesson is on what the correlation coefficient (generally identified as r) tells us about the relationship between two numerical variables. Students use technology to determine the value of the correlation coefficient or r.
Lesson 19 Summary
Linear relationships are often described in terms of strength and direction.
The correlation coefficient is a measure of the strength and direction of a linear relationship.
The closer the value of the correlation coefficient is to +1 or −1, the stronger the linear relationship.
Just because there is a strong correlation between the two variables does not mean there is a cause-and-effect relationship.
Example 1: Positive and Negative Linear Relationships
Linear relationships can be described as either positive or negative. Below are two scatter plots that display a linear relationship between two numerical variables x and y.
Example 2: Some Linear Relationships are Stronger than Others
Below are two scatter plots that show a linear relationship between two numerical variables x and y.
Example 3: The Correlation Coefficient
The correlation coefficient is a number between −1 and +1 (including −1 and +1) that measures the strength and direction of a linear relationship. The correlation coefficient is denoted by the letter r
Properties of the Correlation Coefficient
Property 1: The sign of (positive or negative) corresponds to the direction of the linear relationship
Property 2: A value of r = +1 indicates a perfect positive linear relationship, with all points in the scatter plot falling exactly on a straight line.
Property 3: A value of r = -1 indicates a perfect negative linear relationship, with all points in the scatter plot
Property 4: The closer the value of r is to +1 or -1, the stronger the linear relationship.
1. The scatter plot below displays data on the number of defects per 100 cars and a measure of customer satisfaction (on a scale from 1 to 1000, with higher scores indicating greater satisfaction) for the 33 brands of cars sold in the United States in 2009.
a. Which of the following is the value of the correlation coefficient for this data set: r = -0.95, r = -0.24, r = 0.83, or r = 1.00?b. Explain why you selected this value.
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