Definition: A residual is the difference between an observed y-value and the y-value predicted by a model, calculated as actual minus predicted.
What does a residual show?
It measures how far a data point lies from the fitted line or curve, vertically. A positive residual means the point is above the line, so the model predicted too low. A negative residual means the point is below.
How do you compute one?
Predict with the model, then subtract: residual = y(actual) – y(predicted).
Example. A line of best fit is y = 4.2x + 18, where x is hours of study and y is a score. A student studied 6 hours and scored 49. The prediction is 4.2(6) + 18 = 43.2. The residual is 49 – 43.2 = 5.8, so this student did better than predicted.
How are residual plots used?
Plot the residuals against x. If they scatter randomly around zero, a line is a good model. If they form a curve, a line is the wrong choice. Typical mistakes are subtracting in the wrong order, which flips the sign, and reading the residual horizontally instead of vertically.