The coefficient of determination, written R-squared or R², is the proportion of the variance in your outcome variable that your model explains. It runs from 0 to 1, so an R² of .70 means the predictor variables account for 70 percent of the variability in the outcome and the remaining 30 percent is unexplained. It is calculated as R² = 1 - (SS residual / SS total), and it measures how closely a model fits, not whether the model is correct.
What counts as a good R-squared depends entirely on your field
This is the question every supervisor gets asked and the one with no universal answer. An R² that would be dismissed in one discipline is a strong result in another, because the ceiling is set by how much random variation the outcome carries.
| R² | Explains | How to read it |
|---|---|---|
| .90 and above | 90%+ | Normal in physical sciences and engineering. In social science it usually signals an error: a predictor that contains the outcome, or a duplicated variable. |
| .50 to .70 | 50 to 70% | Strong for most applied and behavioural research. |
| .25 to .50 | 25 to 50% | Typical of a well-specified model in psychology, education, and management. |
| .05 to .25 | 5 to 25% | Common when predicting individual human behaviour. Perfectly publishable if the predictors are significant and theory-driven. |
So an R² of .20 is not automatically weak. Predicting whether one person will do something is inherently noisy, and a model explaining a fifth of that noise can still identify a real, useful relationship. Judge the value against comparable studies in your own literature, and say so in the text. A model with a modest R² and significant, theoretically coherent predictors is a better dissertation result than a high R² nobody can explain.
Where the number comes from
R² compares the errors your model makes against the errors you would make with no model at all. Predicting every case at the mean of the outcome gives the total sum of squares. Predicting from the regression line leaves the residual sum of squares. The ratio between them is the share of variation you failed to remove, so one minus that ratio is the share you explained.
Two consequences follow directly from that arithmetic. R² can never fall as you add predictors, because an extra variable can always fit a little more noise, which is why it cannot be used to compare models of different sizes. And in a simple two-variable relationship, R² is exactly the square of Pearson's r, which is where the name comes from.
Adjusted R-squared, and when to report it instead
Adjusted R-squared penalises the model for each predictor it contains, so it only rises when a new variable improves the fit by more than chance would predict. It can fall, and it can go negative. For a model carrying several predictors it is the value to report and the value to compare models on.
A large gap between the two is itself a finding: it means predictors are being carried that do not earn their place, and the model is fitting your particular sample rather than the population. With a small sample and many predictors, that gap is the first sign of overfitting.
R-squared against the correlation coefficient
They answer different questions. Pearson's r gives the direction and strength of a linear relationship on a scale from -1 to +1. R² drops the direction and reports the proportion of variance explained, from 0 to 1. An r of -.80 and an r of +.80 both produce an R² of .64.
The squaring matters more than it looks. A correlation of .50 sounds like half of something, but it corresponds to an R² of .25, so only a quarter of the variance is accounted for. Reporting both keeps that honest.
What a high R-squared does not prove
A high value tells you the model fits these data closely. It does not establish that the model is correctly specified, that the relationship is causal, or that the results generalise.
- It says nothing about bias. A model can fit well and still be systematically wrong, which is why residual plots and the assumptions behind the model have to be checked separately.
- It is not evidence of cause. Explaining variance and producing it are different claims, and the distinction is the one an examiner will press on. See why a strong fit is not evidence of cause.
- It is inflated by overfitting. Enough predictors will drive R² towards 1 on any dataset, including random noise.
- It is not comparable across outcomes. Two models predicting different dependent variables cannot be ranked by R².
Reporting it in a dissertation
Give the R², the adjusted R² for any multi-predictor model, and the F test of the overall model with its degrees of freedom, then state the proportion of variance explained in plain words. A standard sentence reads: the model significantly predicted job satisfaction, F(3, 146) = 18.42, p < .001, and explained 27 percent of the variance in scores (R² = .275, adjusted R² = .260).
Interpret the figure against your own field rather than a generic threshold, and never present R² as though it settled the question of cause. The wider conventions are in the APA reporting rules for a regression, and the choice of accompanying how effect sizes are chosen and reported matters just as much as the fit statistic itself.