# Understanding the Correlation Coefficient

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## Objective

SWBAT use technology to graph different data sets and use the correlation coefficient to assess the strength and direction of the relationship between two data sets.

#### Big Idea

Students "uncover" the meaning of the correlation coefficient (r) by graphing and examining a variety of data sets.

## Opening

5 minutes

In today's class, students will work to uncover the significance of the correlation coefficient. Rather than telling students that the correlation is a number between -1 and 1 and indicates the strength of a linear relationship, this lesson allows students to come to those conclusions on their own.

I begin class by talking to students about the previous lesson, How Much for a Used Car?, where students generated scatter plots by hand and worked on finding a line of best fit.  I let students know that today, they will use technology to graph some data and look at the regression line that technology can generate for a scatter plot.

## Investigation

30 minutes

Next, we take a look at Connect the Dots and plot the first data set together.  I have access to Chromebooks at my school and like to use an online graphing program like plot.ly to do this work. I find graphing programs like Desmos and plot.ly help create student engagement with math more than the graphing calculator. My students each have a gmail account through the school and I have them sign in to plot.ly with their gmail account.  When students are signed in, they are able to save the graphs they create.  From there we create the first scatter plot together and I show them how to fit a linear line to the data. Next, students can either save and print their graphs or find the graph that matches with the one they just created from this hard copy.  I find this is easier than having students sketch each graph. The main point is that students need to be able to see all the graphs with their corresponding correlation coefficients in front of them so later they can put them in order.

Next, I let students work on the remaining data sets on their own.  After they have plotted all 7 data sets and found the correlation coefficient for each one, I ask them to see if they can find any patterns. Some students will need more explicit instructions to put the graphs in order from the smallest to largest value of "r."

Students who work faster can go on to questions 4 through 6 where they are asked to alter the y values of the original data set to create different values for "r."

## Discussion + Closing

25 minutes

The main point of our discussion today is to get students to articulate that the correlation coefficient is a value between -1 and 1.  I want to elicit from students that a negative value of "r" means that the data is negatively correlated while a positive "r" indicates a positive association.  I also want students to articulate that the closer "r" is to positive or negative 1 the stronger the relationship, with a perfect linear relationship occurring at exactly 1 and -1.  Students should also be able to understand that the closer "r" is to 0, the weaker the association is between the variables.

We list all of these observations on the board and I have students write them in their notebooks so they can come back to the list.  I find later in the unit students often get confused between residuals and the correlation coefficient, so I want them to get something down in writing.

If we have time, I have students work on Questions 4 through 6.  Students seemed to struggle with this piece of the task the last time I taught it, and I think next time I would spend more time on these questions.  Students seemed unclear about what the task was asking them to do and uncomfortable with the idea of generating their own y-values to pair with the x-values.  Ultimately, I think this is a good task to get students to understand how changes in the data affect the correlation coefficient.

To close today's class, I want students to capture their learning in writing about the correlation coefficient. I ask them to complete an exit ticket to address the following prompt:

How can you use the correlation coefficient to understand the relationship between two variables?

## Citations

Connect the Dots is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.

Source: http://www.mathematicsvisionproject.org/secondary-1-mathematics.html