Statistics Reference
How to Interpret R² (Coefficient of Determination)
R² answers one question — what share of the outcome's variation does the model account for? — and gets pressed into answering many it cannot: whether the model is right, whether the effect matters, whether predictions will hold. This guide works the number from raw data, shows what counts as “good,” and lists the four conclusions R² does not license.
The Variance Bookkeeping
R² = 1 − SSR/SST = explained variation / total variation
SST = Σ(y − ȳ)² (total, around the mean)
SSR = Σ(y − ŷ)² (residual, around the fitted line)
The benchmark is the laziest possible model: predict every y with the plain mean. SST is that model's total squared error. The regression's leftover error is SSR. R² is simply the fraction of the mean-model's error the regression eliminated — the same sum-of-squares partition that powers ANOVA, applied to a fitted line.
Worked Example: R² = 0.6 From Five Points
Fit x = 1…5 against y = 2, 4, 5, 4, 5 (the running example of the regression calculator):
- Fitted line: slope 0.6, intercept 2.2 → ŷ = 0.6x + 2.2.
- Total variation: ȳ = 4, so SST = 4 + 0 + 1 + 0 + 1 = 6.
- Residuals: predictions 2.8, 3.4, 4.0, 4.6, 5.2 leave residuals −0.8, 0.6, 1.0, −0.6, −0.2, so SSR = 0.64 + 0.36 + 1.00 + 0.36 + 0.04 = 2.4.
- R²: 1 − 2.4/6 = 0.6. The line explains 60% of the variation; 40% remains.
Cross-check via correlation: the same data give r = 0.7746 in the correlation calculator, and 0.7746² = 0.6 — the simple-regression identity R² = r² in action.
What Counts as “Good” Depends on the Field
| Domain | Typical working R² | Why |
|---|---|---|
| Instrument calibration, physics | > 0.99 | Deterministic mechanisms, tiny noise |
| Biology, process engineering | 0.6 – 0.9 | Real mechanisms plus biological variation |
| Human behavior, markets | 0.05 – 0.4 | Outcomes driven by many unmeasured factors; small edges still matter |
The right comparison is never against 1.0 but against the best achievable in the domain and the value of the decision the model informs. A stock-return model with R² = 0.05 is a fortune; a thermometer calibration with R² = 0.95 is broken.
Four Things R² Does Not Mean
- High R² ≠ correct model. A line fitted to a smooth curve can score 0.9+ while being wrong everywhere that matters; famous constructed datasets share identical R² with wildly different shapes. Plot residuals — patterns there reveal what the number hides.
- High R² ≠ causation. It quantifies fit, not mechanism; every caveat in the correlation-vs-causation guide applies unchanged.
- High R² ≠ good future predictions. In-sample fit flatters: with enough parameters R² can be driven to 1 on pure noise. Trust out-of-sample checks and adjusted R² when models grow.
- Low R² ≠ no effect. Effect existence is the slope's question (its confidence interval and p-value); R² only says how much outcome variance the effect moves. A reliable 2% improvement across millions of users has tiny R² and enormous value.
Try the Regression Calculator
Fit the least-squares line and read slope, intercept, and R² off your own data — then judge the fit against your field's benchmarks.
Frequently Asked Questions
What does R² actually measure?
The proportion of the outcome's variance that the model accounts for: R² = 1 - SSR/SST, where SST is the total variation of y around its mean and SSR is the variation left in the residuals. R² = 0.6 means the regression explains 60% of the variance in this sample and leaves 40% unexplained. In simple linear regression it also equals the squared correlation, r².
What is a good R² value?
It depends entirely on the field and purpose. Physics calibrations routinely demand R² above 0.99; models of human behavior often can't exceed 0.2-0.4 because individual outcomes are genuinely noisy, yet remain scientifically and commercially valuable. Judge R² against what is achievable in the domain and against competing models - never against a universal threshold.
Does a high R² mean my model is correct?
No. A high R² says the model tracks this sample's variation, not that the relationship has the right form or will generalize. A curved relationship fitted with a line can still show high R²; overfitted models show inflated R² that collapses on new data; and extrapolation beyond the observed x-range is unsafe at any R². Always inspect the residuals, not just the summary number.
Does a low R² mean the predictor doesn't matter?
No. A predictor can have a real, precisely estimated effect while individual outcomes remain mostly noise - a drug can reliably lower blood pressure by 5 mmHg (small R², clear effect) because people vary for a hundred other reasons. Check the slope's confidence interval and p-value for whether the effect exists; use R² for how much of the outcome it explains.
How are R² and the correlation coefficient related?
In simple (one-predictor) linear regression, R² is exactly the square of Pearson's r: a correlation of 0.7746 gives R² = 0.6. Squaring loses the sign - r = +0.77 and r = -0.77 both give R² = 0.6 - so report r (or the slope) when direction matters. With multiple predictors, R² generalizes but no single r corresponds to it.
Why does adding any variable increase R², and what is adjusted R²?
Ordinary R² can never decrease when a predictor is added - even a column of random noise soaks up a little sample variance. Adjusted R² subtracts a penalty for each parameter, so it rises only when a new predictor explains more than chance would. When comparing models with different numbers of predictors, adjusted R² (or out-of-sample validation) is the fair yardstick.