Linear Regression III
Grade:
6th to 8th, High School
This e-example allows students to explore three methods for measuring how well a linear model fits a set of data points.
- Five points and a regression line are shown on the graph. Move the red points by clicking and dragging them to new locations. Change the y‑intercept of the regression line by clicking and dragging the blue dot; change the slope of the regression line by clicking and dragging any other point on the line.
- Pressing the Squares button shows the squared distance from each point to the line. This calculation can be used to estimate the "least‑squares regression."
- Pressing the Absolute Value button shows the vertical distance from each point to the line. This calculation can be used to estimate the "least‑squares regression."
- Pressing the Shortest Distance button shows the perpendicular distance from each point to the line. This calculation can be used to estimate the "least‑squares regression."
- In this task a linear equation is used to model a set of data. By
modifying the data points, explore how each of three methods—distance
squared, absolute value, and shortest distance—measures how well the
model approximates the data. How do individual data points contribute to
the error? How do these contributions differ among the three methods of
measuring the "goodness of fit"?
- How do the three methods compare when one of the points is far from the line and the rest of the points are quite close?
- For at least four different sets of data points, record the error
measured by the absolute-value and shortest-distance methods. Be sure to
use data sets that are quite different from one another in the number
of points that are close to and far from the line. What relationships do
you notice among the errors? (Hint: For each data set, try doing some
arithmetic with the errors measured by the two methods.
NCTM Standards and Expectations- Probability / Data Analysis and Statistics
- Data Analysis and Probability