Find R2 Calculator

R-squared (R²) Calculator – Coefficient of Determination

R-squared (R²) Calculator

Calculate Coefficient of Determination (R²)

SST = Σ(yᵢ – ȳ)². Must be greater than 0.
SSE = Σ(yᵢ – ŷᵢ)². Must be 0 or greater, and not greater than SST.

What is R-squared (R²)?

R-squared (R²), also known as the coefficient of determination, 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 indicates the "goodness of fit" of the model – how well the regression predictions approximate the real data points. An R² of 1 indicates that the regression predictions perfectly fit the data, while an R² of 0 indicates that the model does not explain any of the variability of the response data around its mean. Our R-squared Calculator helps you easily compute this value.

Statisticians, data scientists, economists, and researchers in various fields use R-squared to evaluate the strength of the linear relationship between variables and the predictive power of their models. For example, if a model predicting house prices based on size has an R² of 0.75, it means that 75% of the variability in house prices can be explained by the size of the houses according to that model.

A common misconception is that a high R-squared value always means a good model. While a higher R² generally indicates a better fit, it doesn't tell you if the model is biased, if the data meets the assumptions of the regression, or if there are other variables that should be included. Always consider R-squared in conjunction with other model diagnostics.

R-squared Formula and Mathematical Explanation

The R-squared value is calculated using the Total Sum of Squares (SST) and the Sum of Squared Errors (SSE, also known as the Residual Sum of Squares). SST measures the total variability in the dependent variable, while SSE measures the variability that is *not* explained by the regression model.

The formula for R-squared is:

R² = 1 – (SSE / SST)

Alternatively, since SST = SSR + SSE (where SSR is the Sum of Squares due to Regression, or Explained Sum of Squares), R-squared can also be expressed as:

R² = SSR / SST

Where:

  • SST (Total Sum of Squares) = Σ(yᵢ – ȳ)²: The sum of the squares of the differences between each observed value (yᵢ) and the mean of the observed values (ȳ). It represents the total variance in the dependent variable.
  • SSE (Sum of Squared Errors/Residuals) = Σ(yᵢ – ŷᵢ)²: The sum of the squares of the differences between each observed value (yᵢ) and the corresponding predicted value (ŷᵢ) from the model. It represents the variance not explained by the model.
  • SSR (Sum of Squares due to Regression) = Σ(ŷᵢ – ȳ)²: The sum of the squares of the differences between each predicted value (ŷᵢ) and the mean of the observed values (ȳ). It represents the variance explained by the model. SST = SSR + SSE.

The R-squared Calculator above uses these formulas to find the R² value based on the SST and SSE you provide.

Variables Table

Variable Meaning Unit Typical Range
SST Total Sum of Squares Squared units of Y > 0
SSE Sum of Squared Errors (Residuals) Squared units of Y ≥ 0, ≤ SST
SSR Sum of Squares due to Regression Squared units of Y ≥ 0, ≤ SST
R² Coefficient of Determination Dimensionless 0 to 1 (can be negative rarely)

Practical Examples (Real-World Use Cases)

Example 1: Predicting Sales Based on Advertising Spend

A company runs a regression model to predict weekly sales (Y) based on advertising spend (X). After collecting data and fitting the model, they find:

  • Total Sum of Squares (SST) = 500 (squared units of sales)
  • Sum of Squared Errors (SSE) = 100 (squared units of sales)

Using the R-squared Calculator or the formula R² = 1 – (SSE / SST):

R² = 1 – (100 / 500) = 1 – 0.2 = 0.8

This R² of 0.8 means that 80% of the variation in weekly sales can be explained by the advertising spend, according to their model.

Example 2: House Price Model

A real estate analyst develops a model to predict house prices based on square footage. The analysis yields:

  • SST = 1,200,000 (squared units of price)
  • SSE = 480,000 (squared units of price)

R² = 1 – (480,000 / 1,200,000) = 1 – 0.4 = 0.6

An R² of 0.6 indicates that 60% of the variability in house prices is explained by square footage in this model. While useful, it suggests other factors also significantly influence house prices.

How to Use This R-squared Calculator

Our R-squared Calculator is straightforward to use:

  1. Enter Total Sum of Squares (SST): Input the calculated SST value into the first field. This represents the total variability in your dependent variable.
  2. Enter Sum of Squared Errors (SSE): Input the calculated SSE (Residual Sum of Squares) from your regression model into the second field. This is the variability your model couldn't explain.
  3. View Results: The calculator automatically computes and displays the R-squared (R²) value, along with the calculated Sum of Squares due to Regression (SSR). The results update as you type.
  4. Interpret R²: The R² value is shown as the primary result, indicating the proportion of variance explained by your model (e.g., 0.8000 means 80%).
  5. Examine Chart and Table: The chart visually represents how SST is divided into SSR and SSE, and the table summarizes the values.
  6. Reset: You can click "Reset" to clear the fields and start over with default values.
  7. Copy: Click "Copy Results" to copy the main results and intermediate values to your clipboard.

A higher R² value (closer to 1) generally suggests a better fit, but always consider the context of your study and other statistical measures. The coefficient of determination calculator gives you a quick measure of your model's explanatory power.

Key Factors That Affect R-squared Results

Several factors can influence the R-squared value obtained from a regression analysis:

  1. Number of Predictor Variables: Adding more independent variables to a model, even if they are not truly significant, will almost always increase R². This is why Adjusted R² is often preferred, as it penalizes the addition of non-significant variables.
  2. Model Specification: The form of the model (linear, polynomial, etc.) and the variables included directly impact R². A model that better captures the underlying relationship will have a higher R².
  3. Outliers: Extreme data points (outliers) can significantly distort the regression line and, consequently, the R² value.
  4. Sample Size: While R² itself isn't directly dependent on sample size in the same way p-values are, larger samples tend to give more stable estimates of the true population R².
  5. Variability of the Data: If there's very little variation in the dependent variable to begin with (low SST), it can be harder to achieve a high R², even with a good model.
  6. Context of the Study: In some fields (like social sciences), R² values of 0.3 or 0.4 might be considered reasonably good, whereas in others (like physics), R² values below 0.9 might be seen as poor.
  7. Non-linear Relationships: If the true relationship between variables is non-linear, a linear regression model might yield a low R², even if there's a strong relationship.

Frequently Asked Questions (FAQ)

What is a good R-squared value?
The definition of a "good" R-squared value depends heavily on the context. In precision-focused fields like engineering, values above 0.9 might be expected. In social sciences or fields with more inherent variability, values like 0.3-0.6 might be considered informative. There's no single threshold for a good R².
Can R-squared be negative?
Yes, R-squared can be negative, although it's uncommon. A negative R² means the chosen model fits the data worse than a simple horizontal line (the mean of the dependent variable). This often indicates a very poorly specified model.
Does a high R-squared mean the model is good?
Not necessarily. A high R² indicates that the model explains a large proportion of the variance, but it doesn't guarantee the model is unbiased, the coefficients are reliable, or that it will predict well out-of-sample. Always check residual plots and other diagnostics. Our linear regression calculator can help explore this.
What is the difference between R-squared and Adjusted R-squared?
R-squared increases or stays the same when you add more predictors to the model, regardless of their significance. Adjusted R-squared accounts for the number of predictors in the model and only increases if the added predictors improve the model more than would be expected by chance. It's often a more reliable measure when comparing models with different numbers of predictors.
How do I calculate SST and SSE?
SST is the sum of squared differences between each observed Y and the mean of Y. SSE is the sum of squared differences between each observed Y and the predicted Y from your model. You typically get these values from statistical software after running a regression analysis.
Can I use the R-squared calculator for non-linear regression?
The concept of R-squared is most clearly defined for linear regression. While pseudo-R-squared measures exist for non-linear models, the interpretation isn't always as straightforward as the R² from linear regression calculated here.
What does an R-squared of 0 mean?
An R-squared of 0 means that the independent variable(s) in your model explain none of the variability of the dependent variable around its mean. Essentially, the model is no better than simply using the average of the dependent variable as the prediction for all observations.
Is R-squared the same as the correlation coefficient squared?
In simple linear regression (one independent variable), the square of the Pearson correlation coefficient (r) between the independent and dependent variables is equal to R-squared (R²). However, in multiple linear regression (more than one independent variable), R-squared is not simply the square of a single correlation coefficient. You might find our correlation coefficient calculator useful.

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