Math101 · Independent Ontario learning library
Math101Linear Regression
Linear regression fits a least-squares line to paired quantitative data and uses residuals to assess predictions.
A regression line summarizes an average linear pattern; residuals show what the line misses.
Model form
A fitted linear model is written
where $\hat y$ is the predicted response, $b$ is slope, and $a$ is intercept. The hat distinguishes a prediction from an observed value $y$.
The explanatory variable is $x$ and the response variable is $y$.
Least-squares criterion
For each observation, the residual is
Least-squares regression chooses $a$ and $b$ to minimize
Squaring prevents positive and negative residuals from cancelling and penalizes large errors.
Interpreting slope
The slope describes predicted response change for one additional unit of the explanatory variable.
Always include both variable units.
Interpreting the intercept
The intercept predicts $y$ when $x=0$. It is meaningful only if zero is realistic and reasonably near the observed range.
An intercept can be mathematically necessary for the line while lacking a sensible contextual interpretation.
Common mistakes
Interpreting slope without units or context. Name both variables.
Forcing meaning onto an unrealistic intercept. Check whether $x=0$ makes sense.
Calculating residual as predicted minus observed. Use observed minus predicted.
Calling $R^2$ prediction accuracy or causal strength. It summarizes variation explained in the sample.
Extrapolating far beyond the data without warning. Model form may not persist.
Quick self-check
- Are explanatory and response variables identified correctly?
- Does the scatter plot support a linear model?
- Is slope interpreted as predicted change with units?
- Are residuals patternless around zero?
- Is $R^2$ described accurately?
- Are outliers, extrapolation, uncertainty, bias, and causal limits addressed?
