Investigating Linear Relationships: The Regression Line and Correlation
Part Four - The Centroid and the Regression Line

The Regression Line

The Effects of Outliers

Correlation and the Regression Line

The Centroid and the Regression Line

In a scatterplot you have two variables, that is, two sets of data. These are represented on the graph below by x-values and y-values, on the horizontal and vertical axes. You can compute the mean of each set of data, that is, the mean of the x-values (x-bar) and the mean of the y-values (y-bar). The point (x-bar, y-bar) is called the centroid. Go to Questions.

Finding the Relationship Between the Regression Line and the Centroid 1. CLEAR the graph and plot two points.   • (a) On your own paper, determine the centroid of these two points.   • (b) On your own paper, compute the midpoint of the two points. Compare this midpoint to the centroid you computed. Explain the connection   .• (c) Click on SHOW CENTROID. Compare the coordinates of the centroid shown to the coordinates you computed in parts (a) and (b). Explain and resolve any differences.     2. CLEAR the graph and plot three points.   • (a) On your own paper, determine the centroid of these three points.   • (b) Click on SHOW CENTROID. Confirm that you get the same point as in (a)   • (c) Click on SHOW LINE. What happens?   3. Experiment with several scatterplots to find the relationship between the centroid and the least-squares regression line. Describe this relationship. Go to the Regression Line Applet

Go To Reflection Question

Reflection Questions For further extension and reflection, consider the following question: When modeling real data using the regression line, what is a meaning for the centroid? What mathematical ideas can be developed through these activities? What is the value of using computer-based tools in activities like this? What should be the role of the teacher during activities like this? What assessment tasks can be used with these activities?

References and Credits