Coefficient of Determination Calculator
Calculate R² to measure regression model fit and variance explained
How to Use
- Enter your X values (independent variable) separated by spaces, commas, or semicolons
- Enter your Y values (dependent variable) separated by spaces, commas, or semicolons
- Ensure X and Y values have the same number of data points
- Click calculate to see the R² value and interpretation
- Review the correlation coefficient and model fit assessment
What is R² (Coefficient of Determination)?
The coefficient of determination, denoted as R², is a statistical measure that represents the proportion of variance in the dependent variable that can be predicted from the independent variable(s). It ranges from 0 to 1, where 1 indicates perfect prediction and 0 indicates no predictive value.
R² is commonly used in regression analysis to assess how well a model fits the observed data. A higher R² value indicates that a greater proportion of the variance in the dependent variable is explained by the independent variable(s).
Interpreting R² Values
- 0.90 - 1.00: Excellent fit - The model explains most of the variance
- 0.70 - 0.89: Good fit - The model explains a large portion of variance
- 0.50 - 0.69: Moderate fit - The model explains about half the variance
- 0.30 - 0.49: Weak fit - The model explains limited variance
- 0.00 - 0.29: Very weak fit - The model has little explanatory power
Note that interpretation of R² depends on the context and field of study. In some fields like social sciences, lower R² values are common and still considered meaningful.
R² vs. Correlation Coefficient
While both R² and correlation coefficient (r) measure the strength of relationships, they have key differences:
- Correlation coefficient (r) ranges from -1 to +1, indicating direction and strength
- R² ranges from 0 to 1, representing proportion of variance explained
- R² is always non-negative, while correlation can be positive or negative
- R² = r² for simple linear regression (one independent variable)
Limitations of R²
R² has several limitations to consider:
- Adding more variables always increases R², even if they're not meaningful
- R² doesn't indicate if the regression coefficients are biased
- High R² doesn't prove causation between variables
- R² doesn't indicate whether the independent variables are the correct ones
- Non-linear relationships may have low R² despite strong associations
Adjusted R²
Adjusted R² is a modified version that accounts for the number of predictors in the model. It penalizes the addition of unnecessary variables and can decrease when predictors don't improve the model fit.
For multiple regression models, adjusted R² is often preferred over regular R² as it provides a more accurate assessment of model fit when comparing models with different numbers of predictors.
Frequently Asked Questions
- What does an R² of 0.75 mean?
- An R² of 0.75 means that 75% of the variance in the dependent variable can be explained by the independent variable(s) in your model. This is generally considered a good fit, indicating that the model explains a large portion of the variability in the data.
- Can R² be negative?
- In simple linear regression, R² cannot be negative. However, in some cases of multiple regression or when using certain estimation methods, R² can be negative, indicating that the model fits worse than a horizontal line (the mean of the dependent variable).
- What's the difference between R² and adjusted R²?
- Adjusted R² accounts for the number of predictors in the model and penalizes the addition of unnecessary variables. While R² always increases (or stays the same) when you add predictors, adjusted R² can decrease if the new predictors don't improve the model sufficiently.
- How many data points do I need for R²?
- Technically, you need at least 2 data points to calculate R², but for meaningful results, you should have many more. The minimum number depends on your field and the complexity of your model, but generally 10-20+ observations per predictor is recommended.
- Is a higher R² always better?
- Not necessarily. While a higher R² indicates more variance explained, you should consider the context, field of study, and whether the model is overfitted. Sometimes a simpler model with slightly lower R² is more useful and generalizable than a complex model with higher R².