The Correlation Coefficient
The mathematical purpose of this lesson is for students to find and interpret the correlation coefficient, and to use it to understand the strength of a linear relationship. The term correlation coefficient is introduced and is defined as a number that can be used to determine how well a line models the data. The work of this lesson connects to previous work because students plotted and analyzed residuals to informally assess the fit of linear models. The work of this lesson connects to upcoming work because students will use technology to compute the correlation coefficient and use it to describe the relationship between two variables.
When students sorted scatter plots, they were given the opportunity to analyze representations, statements, and structures closely and make connections (MP2, MP7). When students take turns with a partner, matching graphs of residuals to scatter plots that display linear models with the correlation coefficient, students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).
- Match the correlation coefficient to its appropriate scatter plot and linear model.
- Use the correlation coefficient to determine the goodness of fit for a linear model.
- Let’s see how good a linear model is for some data.
Print and cut up slips from the blackline master for the card sort activity. One copy is needed for every group of 2 students.
- I can describe the goodness of fit of a linear model using the correlation coefficient.
- I can match the correlation coefficient with a scatter plot and linear model.
A number between -1 and 1 that describes the strength and direction of a linear association between two numerical variables. The sign of the correlation coefficient is the same as the sign of the slope of the best fit line. The closer the correlation coefficient is to 0, the weaker the linear relationship. When the correlation coefficient is closer to 1 or -1, the linear model fits the data better.
The first figure shows a correlation coefficient which is close to 1, the second a correlation coefficient which is positive but closer to 0, and the third a correlation coefficient which is close to -1.
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