Investigating Linear Relationships: The Regression Line and Correlation
Part One - The Regression Line

The Regression Line

The Effects of Outliers

Correlation and the Regression Line

The Centroid and the Regression Line

Interactive computer-based tools provide students with the opportunity to manipulate numbers easily and investigate the relationships between data points and sets. As students work with bivariate data in grades 9-12, they will be able to investigate relationships using linear, exponential, power, logarithmic, and other functions for curve fitting (See Related 9-12 Data Analysis & Probability Standard). Using interactive tools like the one below, students can investigate the properties of regression lines and correlation.

The interactive tool below lets you plot points and find a line that "fits" the points. The line is called the "least-squares regression line." In addition, the equation of the line, the number of points, and the Pearson's correlation coefficient are displayed at the bottom of the figure. Go to Questions

Getting to Know the Regression Line 1. Plot one point and then click SHOW LINE. Why do you think a line is not graphed? 2. CLEAR the graph and plot two points that have whole number coordinates. • On your own paper, find an equation for the line through these two points.   • Click SHOW LINE. Compare the equation for the line drawn to the equation you calcluated. Explain and resolve any differences. 3. CLEAR the graph and plot 3 points. Think about a line that "fits" these three points as closely as possible. • Is it possible for a single straight line to contain all three of the points you plotted? • On your own paper, sketch a line that you think best fits the three points. • Click SHOW LINE. Do you think that the line graphed fits the points well? How does it compare to the line you drew? 4. CLEAR the graph and plot several points. Think about a line that best fits these points. • Click SHOW LINE to see the "least-squares regression line" that fits these points. • What do you think will happen to the regression line if you plot a new point? Try it and find out. (NOTE: When you plot a new point without clearing the graph, then the new regression line is drawn automatically.) • Plot some more points and see what happens. Describe any patterns or trends that you see. 5. The line that the computer draws is called the least-squares regression line. It "fits" the data points according to criteria that you will learn about later. Roughly, the least-squares regression line is the line that minimizes the squared "errors" between the actual points and points on the line. This makes the line fit the points. Just to get a better feel for the regression line, try the following tasks. • Plot 4 points so that the regression line is horizontal. Do this in several different ways. • Plot 3 points (not all on a line) so that the regression line is horizontal. Go to the Regression Line Applet

Go To Reflection Question

Reflection Questions Can you think of an example of a real situation where finding a line that "fits" the data will be a useful thing to do? How do you think that the computer calculates the least-squares regression line? Go to Part Two

References and Credits