Coefficient of Determination (R²) Calculator
Calculate R-squared
What is the Coefficient of Determination (R-squared)?
The Coefficient of Determination, often denoted as R² or R-squared, is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. It provides an indication of the "goodness of fit" of the model – how well the regression predictions approximate the real data points.
An R² value ranges from 0 to 1 (or 0% to 100%).
- R² = 0: The model explains none of the variability of the response data around its mean.
- R² = 1: The model explains all the variability of the response data around its mean.
In simpler terms, if a model has an R² of 0.70, it means that 70% of the variation in the dependent variable can be explained by the independent variable(s) included in the model, while the remaining 30% is due to other factors or random variability.
The Coefficient of Determination Calculator is used by statisticians, data scientists, economists, researchers, and anyone working with regression models to assess their explanatory power. While a higher R² is generally better, it doesn't necessarily mean the model is good or that the relationship is causal. It's important to consider other factors like the context of the data, the simplicity of the model, and residual plots.
Common Misconceptions
- High R² means a good model: Not always. A high R² can occur with a biased model or due to overfitting, especially with many predictors.
- R² implies causality: R² only measures the strength of association, not causation.
- R² is always between 0 and 1: While typically true for standard linear regression, R² can be negative for models that fit the data worse than a horizontal line (i.e., just using the mean of y). Our Coefficient of Determination Calculator can show this.
Coefficient of Determination (R²) Formula and Mathematical Explanation
The Coefficient of Determination (R²) is calculated using the sums of squares:
- SST (Total Sum of Squares): Measures the total variability in the dependent variable (y) around its mean (ȳ). It's the sum of the squared differences between each observed y value and the mean of y.
SST = Σ(yᵢ – ȳ)² - SSE (Sum of Squared Errors / Residual Sum of Squares): Measures the variability that is NOT explained by the regression model. It's the sum of the squared differences between the observed y values (yᵢ) and the predicted y values (ŷᵢ) from the model.
SSE = Σ(yᵢ – ŷᵢ)² - SSR (Sum of Squares Regression / Explained Sum of Squares): Measures the variability that IS explained by the regression model. It's the sum of the squared differences between the predicted y values (ŷᵢ) and the mean of y (ȳ).
SSR = Σ(ŷᵢ – ȳ)²
The fundamental relationship is: SST = SSR + SSE
The R-squared value is then calculated as the ratio of the explained variance (SSR) to the total variance (SST), or one minus the ratio of unexplained variance (SSE) to total variance:
R² = SSR / SST = 1 – (SSE / SST)
A Coefficient of Determination Calculator uses these formulas based on the provided inputs.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| R² | Coefficient of Determination | Dimensionless | Typically 0 to 1, but can be < 0 |
| yᵢ | Observed value of the dependent variable for the i-th observation | Depends on data | Varies |
| ŷᵢ | Predicted value of the dependent variable for the i-th observation | Depends on data | Varies |
| ȳ | Mean of the observed values of the dependent variable | Depends on data | Varies |
| SSE | Sum of Squared Errors (Residuals) | (Unit of y)² | ≥ 0 |
| SST | Total Sum of Squares | (Unit of y)² | ≥ 0 (0 if all y are equal) |
| SSR | Sum of Squares due to Regression | (Unit of y)² | ≥ 0 |
Practical Examples (Real-World Use Cases)
Example 1: House Price Prediction
Suppose you build a simple linear regression model to predict house prices based on square footage. You have the following data:
- Observed Prices (y): $300k, $400k, $350k, $500k, $450k
- Predicted Prices (ŷ) from your model: $320k, $380k, $360k, $480k, $460k
Using a Coefficient of Determination Calculator or by hand:
- Calculate ȳ (mean of observed prices) = ($300+$400+$350+$500+$450)/5 = $400k
- Calculate SSE = (300-320)² + (400-380)² + (350-360)² + (500-480)² + (450-460)² = 400 + 400 + 100 + 400 + 100 = 1400
- Calculate SST = (300-400)² + (400-400)² + (350-400)² + (500-400)² + (450-400)² = 10000 + 0 + 2500 + 10000 + 2500 = 25000
- Calculate R² = 1 – (1400 / 25000) = 1 – 0.056 = 0.944
Interpretation: An R² of 0.944 means that 94.4% of the variation in house prices is explained by the square footage in your model.
Example 2: Ad Spend vs. Sales
A company models its monthly sales based on advertising spend. After running the regression, they find:
- SSE = 500 (squared units of sales)
- SST = 2000 (squared units of sales)
Using the Coefficient of Determination Calculator with SSE and SST:
R² = 1 – (500 / 2000) = 1 – 0.25 = 0.75
Interpretation: An R² of 0.75 indicates that 75% of the variation in monthly sales can be explained by the advertising spend, according to their model.
How to Use This Coefficient of Determination Calculator
- Select Input Method: Choose whether you want to input "Observed and Predicted Values" or "Sum of Squares (SSE and SST)".
- Enter Data:
- If "Observed and Predicted Values": Enter your comma-separated observed y values and predicted ŷ values into the respective text areas. Ensure you have the same number of values in both lists.
- If "Sum of Squares": Enter the calculated SSE and SST values into their fields. SSE must be non-negative, and SST must be positive (or zero if all observed y are identical, though R² is less meaningful then).
- Calculate: Click the "Calculate R²" button.
- Read Results:
- Primary Result (R²): This is the Coefficient of Determination, shown prominently.
- Intermediate Values: SSE, SST, and SSR are displayed. If you entered observed/predicted values, the number of data points (n) and the mean of observed Y (ȳ) are also shown.
- Interpretation: A brief sentence explains what the R² value means in terms of explained variance.
- Chart: The bar chart visually represents how SST is divided into SSR (explained) and SSE (unexplained).
- Data Table (if applicable): If you entered observed and predicted values, a table shows intermediate calculations for each data point.
- Reset: Click "Reset" to clear inputs and results.
Use the R² value to understand the goodness of fit of your regression model. A value closer to 1 suggests a better fit, but always consider the context. Our Coefficient of Determination Calculator makes this easy.
Key Factors That Affect Coefficient of Determination (R²) Results
- Number of Predictor Variables: Adding more predictors to a model, even irrelevant ones, will generally increase R² or keep it the same, but never decrease it. This can be misleading, which is why Adjusted R² is often preferred when comparing models with different numbers of predictors.
- Model Specification: If the chosen model (e.g., linear) does not fit the underlying relationship well (e.g., if the true relationship is non-linear), R² will be lower. Using the correct functional form is crucial.
- Outliers: Extreme values (outliers) in the data can disproportionately influence the regression line and thus affect SSE, SST, and R². They can either inflate or deflate R².
- Range of Data: A wider range of values for the independent and dependent variables can sometimes lead to a higher R², as it might increase SST more than SSE.
- Sample Size: While not directly affecting the R² formula for a given dataset, with very small samples, a high R² might be obtained by chance. Larger samples give more reliable R² estimates.
- Underlying Relationship Strength: The most important factor is the actual strength of the relationship between the independent and dependent variables. If there's a strong, clear relationship, R² will likely be higher.
- Data Transformation: Transforming variables (e.g., using logarithms) can change the relationship and thus the R² value.
Frequently Asked Questions (FAQ)
There's no universal "good" R² value. It depends heavily on the field of study. In some social sciences or fields with high inherent variability, an R² of 0.30 might be considered useful, while in precise physical sciences, an R² below 0.90 might be poor. Context is key.
Yes. While R² is typically between 0 and 1 for standard linear regression fitted by ordinary least squares, it can be negative if the model fits the data worse than a horizontal line (i.e., just using the mean of y as the prediction for all x). This usually happens with poorly specified models or when predictions are made outside the range of data used to fit the model and the model is bad.
R-squared always increases or stays the same when you add more predictors to the model, even if they are not useful. Adjusted R-squared accounts for the number of predictors in the model and only increases if the added predictor improves the model more than would be expected by chance. It's generally better for comparing models with different numbers of predictors.
No. A high R² only indicates that a large proportion of the variance is explained by the model, but the model could still be biased, miss important variables, or be based on spurious correlation. Always check residual plots and the model's assumptions.
No. R-squared is based on the total sum of squares (SST) of the dependent variable. If the dependent variables are different (or transformed differently), their SSTs are not comparable, and thus their R² values are not directly comparable.
Multiply the R² value by 100. For example, an R² of 0.85 means that 85% of the variance in the dependent variable is explained by the independent variables in the model.
A low R² suggests that your model does not explain much of the variation in the dependent variable. This could mean the independent variables are weak predictors, the model is misspecified, or there's a lot of inherent randomness. Consider if other variables might be more relevant or if a different model type is needed.
Yes, for simple linear regression (one independent variable), the Coefficient of Determination (R²) is equal to the square of the Pearson correlation coefficient (r) between the observed and predicted values, or between the independent and dependent variables.
Related Tools and Internal Resources
- Simple Linear Regression Calculator: Use this tool to perform a simple linear regression and get R-squared along with other metrics.
- Correlation Coefficient Calculator: Calculate the Pearson correlation coefficient between two variables.
- Guide to Basic Statistics: Learn more about fundamental statistical concepts.
- Data Analysis Techniques: Explore various methods for analyzing data.
- Understanding Variance and Standard Deviation: Deep dive into measures of dispersion.
- Model Evaluation Metrics: Learn about different ways to evaluate statistical models beyond the Coefficient of Determination Calculator.