Yes, the coefficient of determination and R² are the same thing. The two terms are used interchangeably across statistics, machine learning, and scientific research. R² (also written R-squared or R2) is simply the shorthand notation for the coefficient of determination, representing the proportion of variance in a dependent variable that a model explains.
What R² Actually Measures
The coefficient of determination tells you how well your model’s predictions match the actual data. More precisely, it measures the proportion of the total variability in your outcome variable that your model accounts for. An R² of 0.75 means your model explains 75% of the variation in the data. The remaining 25% is unexplained, whether due to missing variables, randomness, or other factors.
The formula breaks down into a comparison of two things: how far off your model’s predictions are versus how far off you’d be if you just guessed the average every time. In formal terms:
R² = 1 − (sum of squared errors / total sum of squares)
The “total sum of squares” (SST) measures how spread out your actual data points are from their average. The “sum of squared errors” (SSE) measures how far your model’s predictions land from the actual values. When your model fits perfectly, SSE is zero, and R² equals 1. When your model is no better than guessing the average, R² equals 0.
You can also think of it from the other direction. The “sum of squares due to regression” (SSR) captures the portion of variability your model does explain, and the relationship SST = SSR + SSE always holds. So R² is equivalently the ratio SSR/SST.
Lowercase r² vs. Uppercase R²
There is one notation subtlety worth knowing. In simple linear regression, where you have a single predictor variable, the coefficient of determination is often written as lowercase r². In that context, r is the Pearson correlation coefficient between your two variables, and squaring it gives you the proportion of variance explained. If the correlation between hours studied and test scores is 0.8, then r² = 0.64, meaning 64% of the variation in test scores is explained by hours studied.
When you move to multiple regression with more than one predictor, the notation shifts to uppercase R². This “multiple coefficient of determination” works the same way conceptually but accounts for several predictors at once. Penn State’s statistics program notes that most statistical software doesn’t bother distinguishing between the two, labeling both as R². In everyday conversation and most published research, people treat R², r², and “coefficient of determination” as identical.
The R² Scale
The best possible R² value is +1, meaning the model perfectly predicts every data point. The worst possible value isn’t zero. It’s negative infinity. A negative R² means your model performs worse than simply predicting the average for every observation. This typically signals a fundamentally flawed model rather than a merely weak one.
What counts as a “good” R² depends entirely on your field. In physics or engineering, where relationships between variables are tightly controlled, R² values above 0.95 are common. In social sciences or medicine, where human behavior introduces enormous variability, an R² of 0.30 or 0.40 can represent a meaningful finding.
Adjusted R² Solves a Key Problem
Standard R² has a well-known flaw: it always increases when you add more predictor variables to a model, even if those variables are irrelevant. Throw in a random column of data, and R² will tick upward slightly. This makes it unreliable for comparing models of different sizes.
Adjusted R² fixes this by penalizing the addition of predictors that don’t genuinely improve the model. It factors in the number of predictors relative to the number of data points. If a new variable improves predictions enough to justify its inclusion, adjusted R² goes up. If not, adjusted R² actually decreases. This makes it a better tool for deciding how many variables your model really needs, helping prevent overfitting where a model memorizes the training data rather than capturing real patterns.
Where R² Falls Short
R² works well for linear models, where you’re fitting a straight line (or a flat plane in higher dimensions) through your data. For nonlinear models, it becomes unreliable. A study in BMC Pharmacology called R² “an inadequate measure for the goodness of fit in nonlinear models,” noting that the mathematical literature has recognized this limitation for decades. The problem is that R² can return misleadingly high values for nonlinear fits, suggesting a model is excellent when it isn’t.
For nonlinear models, information-based metrics like AIC (Akaike Information Criterion) or BIC (Bayesian Information Criterion) provide a much clearer picture of model quality. These metrics don’t share R²’s bias and are better at identifying which model among several candidates best represents the true underlying relationship in your data.
Even in linear regression, a high R² doesn’t prove causation, and it doesn’t mean your model is correctly specified. You can get a strong R² from a model that violates key assumptions or that fits the current data well but generalizes poorly to new observations. R² is one piece of the puzzle, not the whole picture.