This simulation illustrates least squares regression and how the least squares solution minimizes the sum of the squared residuals.
Type of Material:
This can be used either as a demonstration tool in class (lecture or lab) or as part of a homework assignment. It would be most effective to use it for both: to allow students the opportunity to explore the it themselves after the concepts have been introduced. There is a short audio file (plays on the web) that explains how to use the applet.
Browser required for the website. Works in Firefox, Safari and Google Crome. If using Internet Explorer, you must have Java installed.
Identify Major Learning Goals:
The applet demonstrates, in a visual manner, various concepts related to least squares regression. These include
• sum of squares
• the mean line
• how the line of best fit is determined
• how the line of least squares solution minimizes the sum of the squared residuals.
Target Student Population:
Introductory level statistics students (either college or high school) could use this to understand the basics of regression analysis. More advanced students studying regression would benefit from seeing the connection between the scatterplot of the data, the regression line, and the mathematical derivations of R^2
Prerequisite Knowledge or Skills:
Students should understand the slope-intercept form of the line, scatterplot, and residuals.
Evaluation and Observation
This applet is a great in-class or out-of-class tool for reviewing the concepts behind least squares regression. There is audio and written explanation for understanding the use of the applet.
For each use of the applet, the user can generate a unique set of points that are plotted on the scatterplot and a LS line is fit to those points. I ran through 20+ sets of these points and every set had a very distinct positive slope. It would be helpful to see examples with negative slopes, non-linear relationships, etc. It would also be helpful to incorporate the notation version of the formulas as well as the written-out versions.
Potential Effectiveness as a Teaching Tool
This allows students to discover why least squares regression is called least squares regression. Students can make and test their own conjectures about which line best fits the data, which makes it more likely they will remember the concept well past the assignment and/or the course. It also connects the idea of the sum of squares to the least squares regression, which often isn’t linked for students in a visual manner.
There could be more detailed steps for the students to follow to elicit “discovery” learning. The audio guides the student through the applet and definitions are written out on the website. However, there could be more detailed “questions” provided that the student needs to investigate.
Ease of Use for Both Students and Faculty
Easy to use and the concepts are clearly demonstrated by the applet. The design of the website is very clean and simple.
The engagement could be enhanced with an animation showing a good fit vs. a bad fit to the points. It is left up to the students to find the line, while an animation might give them a better idea about where to start.