Math 1 · Unit 10 · Lesson 3
Residuals, Checking the Fit
Every model is a little bit wrong. A residual is the actual value minus the predicted value for one point: the vertical distance from the point to the line. Positive means the point is above the line and the model guessed too low. A residual plot shows all of them at once: random scatter around 0 means a line is a good model, and a pattern means the data is curved. A big r does not mean a line is right, so use the scatter plot, r, and the residual plot together.
The big idea
Every model is a little bit wrong, and the residual measures how wrong for each point: the actual value minus the predicted value. On the graph it is the vertical distance from the point to the line. A positive residual means the point is above the line and the model guessed too low; a negative one means below, and too high. A good model has small residuals with no pattern. A residual plot shows them all: if they scatter randomly above and below 0, a line is a good model, and if they form a pattern, like a U or a hill, the data is curved and a line is the wrong family. A big r does not settle it. A line fit to the gnats has r of about 0.98, yet its residuals go plus, minus, minus, plus, plus, which is a curve. So decide with all three together: the scatter plot, r, and the residual plot.
The parts of this lesson
Each part is its own short page. Go in order, or pick the one you need.
Printables
One page with the residual, above and below, the residual plot rule, the tuition table, and the gnats surprise. Plus a worksheet with an answer key and a short lesson plan.
Where this goes next
NC Math 1 standards in this lesson: NC.M1.S-ID.6b (assess the fit of a linear function by analyzing residuals), NC.M1.S-ID.8 (use the scatter plot, r, and the residual plot together to decide whether a line is appropriate).