Statistics Foundation · Lesson 5.3
Least squares and residuals.
Least squares explains how a regression line is chosen. This lesson studies residuals, squared errors, the least-squares criterion, fitted-line properties, residual diagnostics, leverage and influential observations.
Lesson route
Move from fitting a line to judging the fit.
0–20 min
Why least squares is needed
Understand why many possible lines can pass through a scatterplot and why a fitting rule is needed.
20–45 min
Residuals as vertical errors
Define residuals as observed minus fitted values and interpret positive, negative and large residuals.
45–75 min
Sum of squared residuals
Learn why residuals are squared and how the least-squares criterion chooses a line.
75–105 min
Deriving the fitted line
Connect the least-squares criterion to the formulas for slope and intercept.
105–130 min
Diagnostics
Use residual patterns to detect nonlinearity, changing spread and unusual observations.
130–160 min
Limitations
Understand outliers, leverage, influence and why a low residual sum does not guarantee a good scientific model.
Mastery checklist
Students should understand what the fitted line minimises.
Define residuals correctly.
Calculate fitted values and residuals.
Explain the least-squares criterion.
Compute a simple SSE.
Understand why residuals are squared.
Connect slope formula to least squares.
Interpret residual plots.
Recognise leverage and influence.