# R-squared

R-squared is the share of the variation in y explained by the model: 1 - (residual sum of squares / total sum of squares). For a straight line with one predictor it equals r squared.

Statistics (university) · Part 08: Correlation and regression · Card 5 of 9 · Last updated 2026-10-19

## Analogy

If R-squared is 0.6, the line accounts for 60 percent of why the y values differ.

## Example

**Share explained**

Total SS = 6
Residual SS = 0.64+0.36+1+0.36+0.04 = 2.4
R-squared = 1 - 2.4/6 = 0.60
Also r squared = 0.77^2 = 0.60

## In practice

It tells you how useful a model is for prediction, but not whether it is correct.

## Practice

### Flashcards

- **What is R-squared for one predictor?** r squared: the share of variation in y explained.

### Interview questions

**For the points (1,2), (2,4), (3,5), (4,4), (5,5), find the least squares line and R-squared.**

- Means 3 and 4
- Slope Sxy / Sxx = 6 / 10 = 0.6
- Intercept = 4 - 1.8 = 2.2
- R-squared = 0.60
- Residuals sum to zero

## Questions people ask

### Does a high R-squared mean the model is good?

Not necessarily. A model can have a high R-squared and still be wrong in form, as residual plots may show. Adding more predictors always raises R-squared, even useless ones, so adjusted R-squared, which penalises extra predictors, is used for comparing models.

### What is a good R-squared value?

It depends on the field. Physics experiments may reach 0.99, while predicting human behaviour with 0.2 or 0.3 can still be useful. Judge R-squared against what is realistic for the problem, and against how the model will be used, since a rough guide may be enough for one task and useless for another.

## Related concepts

- [Least squares regression line](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-8/statistics-university-part-8-concept-2)
- [Correlation coefficient](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-8/statistics-university-part-8-concept-1)
- [Slope and intercept](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-8/statistics-university-part-8-concept-3)
- [Residuals](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-8/statistics-university-part-8-concept-4)

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