Coefficient of Determination (R²) | Calculation & Interpretation
The coefficient of determination is a number between 0 and 1 that measures how well a statistical model predicts an outcome.
|Coefficient of determination (R2)||Interpretation|
|0||The model does not predict the outcome.|
|Between 0 and 1||The model partially predicts the outcome.|
|1||The model perfectly predicts the outcome.|
The coefficient of determination is often written as R2, which is pronounced as “r squared.” For simple linear regressions, a lowercase r is usually used instead (r2).
What is the coefficient of determination?
The lowest possible value of R² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its R² will be to 1.
More technically, R2 is a measure of goodness of fit. It is the proportion of variance in the dependent variable that is explained by the model.
Graphing your linear regression data usually gives you a good clue as to whether its R2 is high or low. For example, the graphs below show two sets of simulated data:
- The observations are shown as dots.
- The model’s predictions (the line of best fit) are shown as a black line.
- The distance between the observations and their predicted values (the residuals) are shown as purple lines.
You can see in the first dataset that when the R2 is high, the observations are close to the model’s predictions. In other words, most points are close to the line of best fit:
In contrast, you can see in the second dataset that when the R2 is low, the observations are far from the model’s predictions. In other words, when the R2 is low, many points are far from the line of best fit:
Calculating the coefficient of determination
You can choose between two formulas to calculate the coefficient of determination (R²) of a simple linear regression. The first formula is specific to simple linear regressions, and the second formula can be used to calculate the R² of many types of statistical models.
Formula 1: Using the correlation coefficient
Formula 2: Using the regression outputs
Interpreting the coefficient of determination
Another way of thinking of it is that the R² is the proportion of variance that is shared between the independent and dependent variables.
You can also say that the R² is the proportion of variance “explained” or “accounted for” by the model. The proportion that remains (1 − R²) is the variance that is not predicted by the model.
If you prefer, you can write the R² as a percentage instead of a proportion. Simply multiply the proportion by 100.
R² as an effect size
Lastly, you can also interpret the R² as an effect size: a measure of the strength of the relationship between the dependent and independent variables. Psychologist and statistician Jacob Cohen (1988) suggested the following rules of thumb for simple linear regressions:
|Minimum coefficient of determination (R²) value||Effect size interpretation|
Be careful: the R² on its own can’t tell you anything about causation.
Reporting the coefficient of determination
If you decide to include a coefficient of determination (R²) in your research paper, dissertation or thesis, you should report it in your results section. You can follow these rules if you want to report statistics in APA Style:
- You should use “r²” for statistical models with one independent variable (such as simple linear regressions). Use “R²” for statistical models with multiple independent variables.
- You don’t need to provide a reference or formula since the coefficient of determination is a commonly used statistic.
- You should italicize r² and R² when reporting their values (but don’t italicize the ²).
- You shouldn’t include a leading zero (a zero before the decimal point) since the coefficient of determination can’t be greater than one.
- You should provide two significant digits after the decimal point.
- Very often, the coefficient of determination is provided alongside related statistical results, such as the F value, degrees of freedom, and p value.
Frequently asked questions about the coefficient of determination
- What is the definition of the coefficient of determination (R²)?
- What is the formula for the coefficient of determination (R²)?
- How do I calculate the coefficient of determination (R²) in R?
You can use the summary() function to view the R² of a linear model in R. You will see the “R-squared” near the bottom of the output.
- How do I calculate the coefficient of determination (R²) in Excel?