When you build a statistical model or plot a trendline through a scatterplot, you need a mathematical way to know if your line actually represents the data well. The Coefficient of Determination, almost universally known as R-squared (R²), is the metric that answers this question.
Our free online Coefficient of Determination Calculator instantly evaluates your dataset to find the R-squared value. In simple terms, R-squared measures the percentage of the variance in the dependent variable (Y) that can be explained by the independent variable (X). It is the ultimate “goodness-of-fit” measurement for any linear regression model.
How to Read and Interpret R-Squared Values
In standard linear regression, R-squared always yields a score between 0.0 and 1.0 (or 0% to 100%). The closer the score is to 1.0, the tighter your data points cling to the regression line, and the better your model is at predicting future outcomes.
| R-Squared (R²) Score | Percentage Equivalent | Real-World Interpretation of Model Accuracy |
|---|---|---|
| 0.00 | 0% Explained | Zero correlation. Your independent variable explains none of the variation. The model fails to predict the data. |
| 0.30 – 0.50 | 30% – 50% Explained | Weak to Moderate. Often acceptable in psychology or human behavioral sciences where data is highly unpredictable. |
| 0.70 – 0.90 | 70% – 90% Explained | Strong Fit. The industry standard for financial forecasting, engineering, and hard sciences. |
| 1.00 | 100% Explained | Perfect Fit. Every single data point falls exactly on the trendline without a single error. |
The Math Behind the Coefficient of Determination
To calculate R-squared, the calculator compares the accuracy of your regression line against a simple horizontal baseline (which is just the average of all your Y values). The formula is: R² = 1 - (SSres / SStot)
| Statistical Component | Formula Logic | What it Actually Measures |
|---|---|---|
| Sum of Squared Residuals (SSres) | Σ (Actual Y - Predicted Y)² |
The “unexplained” variance. It measures how far your actual data points are from your regression line. A smaller number means less error. |
| Total Sum of Squares (SStot) | Σ (Actual Y - Mean Y)² |
The “total” variance. It measures how far your actual data points are from a flat, average horizontal line. |
By dividing the unexplained variance by the total variance, and subtracting from 1, you extract the exact percentage of the variation that your model successfully explains.
For more advanced data analysis and hypothesis testing, utilize our Bonferroni Correction Calculator to prevent false positives, or our Variance Calculator to measure the raw spread of your datasets.
Frequently Asked Questions (FAQ)
Can an R-squared value ever be negative?
Yes. While R-squared is strictly between 0 and 1 in standard linear regression with an intercept, it can be negative if you use a non-linear model or force the regression line through zero. A negative R-squared means that your predictive model is actually performing worse than a simple horizontal line drawn through the average of the data.
What is the difference between R-squared and Adjusted R-squared?
Standard R-squared has a major flaw: it always increases (or stays the same) whenever you add a new variable to your model, even if that variable is total nonsense. Adjusted R-squared penalizes you for adding useless variables, providing a much more accurate score when dealing with multiple regression models.
Does a high Coefficient of Determination prove causation?
Absolutely not. The golden rule of statistics applies here: correlation does not equal causation. An R-squared of 0.99 only proves that the two variables move together in a highly predictable mathematical pattern; it does not prove that one variable is causing the change in the other.
Is R-squared the same as Pearson’s correlation coefficient (r)?
They are closely related in simple linear regression. The Coefficient of Determination (R²) is literally the square of Pearson’s correlation coefficient (r). If your Pearson’s r is 0.8, your R-squared is 0.64 (or 64%).