Track 9 · Correlation and Regression · Lesson 6 of 15
R² and residual standard error
Calculate and interpret R² and the residual standard error, and explain what a high or low R² does and does not tell you.
A marketing analyst reports that TV spend “explains 75% of sales”, while a colleague's model of house prices on building age has an R² below 1% — yet a statistically significant slope. Both numbers are correct; both are easy to misread. R² and the residual standard error answer two different questions: how much of the variation the line accounts for, and how far typical observations fall from it.
Each response can be split into a part the line accounts for and a part it does not: . Squaring and adding over all observations, the cross term vanishes for least squares with an intercept, giving the sum-of-squares decomposition.
Included with a Formula Quest membership
It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Decompose variation into explained and residual sums of squares
- Calculate R² and the residual standard error and say what each measures
- Explain why a high R² does not validate a model and a low R² does not mean no relationship
- Distinguish statistical significance from the share of variation explained