A residual is the difference between an observed y value and the predicted value ŷ from the regression line: eᵢ = yᵢ − ŷᵢ. Residual plots (residuals vs. fitted values or vs. predictor) reveal whether the linear model is appropriate — random scatter around zero indicates a good fit, while patterns suggest the model is misspecified. The coefficient of determination R² = r² gives the proportion of variability in y explained by the linear model, ranging from 0 (no explanatory power) to 1 (perfect linear fit).
Generate residual plots from regression output in software and practice recognizing patterns: funnel shapes indicate non-constant variance, curved patterns indicate nonlinearity. Connect R² to correlation: if r = 0.8, then R² = 0.64 — 64% of variation in y is explained by x.
Once you have fit a linear regression line to data, the natural question is: how well does it fit? The residual for each observation is the answer to that question at a single point — it is the gap between what the model predicted and what actually happened: eᵢ = yᵢ − ŷᵢ. A positive residual means the true value was above the line; negative means it was below. Crucially, residuals are signed — they don't cancel each other out by accident, but they do cancel on average: in ordinary least squares regression, the residuals always sum to zero. This is not a coincidence; the regression line was chosen precisely to minimize the sum of squared residuals, and that optimization forces the sum to be zero.
The most informative diagnostic tool is the residual plot — a scatterplot of residuals on the vertical axis against fitted values (or against the predictor x) on the horizontal axis. If the linear model is appropriate, this plot should look like random scatter around the horizontal line at zero. No trend, no fan shape, no curves. Any pattern in the residual plot is evidence of a model problem. A curved pattern (like a smile or frown) indicates the relationship is not linear — you fit a line to something curved. A funnel shape (residuals spread out more as fitted values increase) indicates heteroscedasticity — the variance of the errors is not constant. Both problems violate the assumptions that make regression inference valid. Reading a residual plot is more important than memorizing any formula.
The coefficient of determination, R², answers the question: what fraction of the total variation in y does the model account for? To build the intuition, think about two extremes. If you ignored x entirely and just predicted ȳ for every observation, your total prediction error would be the total variability in y — called the total sum of squares (SST). Now imagine the regression model reduces that error by explaining some of the variation. The residual sum of squares (SSR) is the variation the model could not explain. R² = 1 − SSR/SST: the proportion of variability the model did explain. An R² of 0.64 means 64% of the variation in y is accounted for by the linear relationship with x; the other 36% is noise the model cannot see.
For simple linear regression with one predictor, R² equals r² — the square of the correlation coefficient you already know. This means R² inherits a clean geometric meaning: if r = 0.8, there is a strong linear relationship, and R² = 0.64. If r = 0.5, the relationship is moderate, and only R² = 0.25 of the variation is explained. The squaring is important — it strips the sign from r (direction doesn't matter for explanatory power) and always gives a value between 0 and 1. However, a high R² is not sufficient evidence that a model is good. A curved relationship can still have high R² while being fundamentally misspecified. Always pair R² with a residual plot inspection: R² measures how much variation is explained; the residual plot tells you whether the explanation is valid.