R-Squared Calculator

Calculate the Coefficient of Determination (R²) to assess how well your regression model fits the observed data.

Calculation Results

R-Squared (R²) is calculated as: 1 – (RSS / TSS). It represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

Actual vs. Predicted Values

Plot showing observed values against values predicted by the regression model.

Data Summary

Point	Actual (y)	Predicted (ŷ)	Residual (y – ŷ)

Summary of observed data points, model predictions, and residuals. Values are unitless/relative.

What is R-Squared (Coefficient of Determination)?

R-squared, often denoted as R² and 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. In simpler terms, it tells you how well your regression model fits the actual data points. An R² value of 1 (or 100%) indicates that the regression predictions perfectly fit the data, while an R² of 0 (or 0%) means the model explains none of the variability of the response data around its mean.

R-squared is particularly crucial for anyone performing regression analysis, whether in academic research, data science, finance, engineering, or social sciences. It helps in evaluating the goodness-of-fit of a statistical model. Users need to understand that R² is a unitless measure and its interpretation depends heavily on the context of the data and the model being used. A common misunderstanding is expecting R² to indicate whether the predictor variables are actually good predictors or if the model has been misspecified; R² only measures the proportion of variance explained, not the validity of the model itself.

R-Squared Formula and Explanation

The R-squared value is derived from the comparison of the total variation in the dependent variable to the variation that is not explained by the regression model. The core components are the Total Sum of Squares (TSS) and the Residual Sum of Squares (RSS).

R² = 1 – (RSS / TSS)

Let’s break down the components:

• TSS (Total Sum of Squares): This measures the total variability of the actual (observed) dependent variable values (y) around their mean ($\bar{y}$). It represents the total variance that the model is trying to explain.
  TSS = Σ(yᵢ – $\bar{y}$)²
• RSS (Residual Sum of Squares): This measures the variability that is *not* explained by the regression model. It’s the sum of the squared differences between the actual (observed) values (yᵢ) and the predicted values (ŷᵢ) from the regression model. These differences are called residuals.
  RSS = Σ(yᵢ – ŷᵢ)²
• ESS (Explained Sum of Squares): Also known as the Regression Sum of Squares, this measures the variability explained by the regression model. It’s the sum of the squared differences between the predicted values (ŷᵢ) and the mean of the actual values ($\bar{y}$).
  ESS = Σ(ŷᵢ – $\bar{y}$)²

It’s important to note that TSS = ESS + RSS. Therefore, R² can also be expressed as ESS / TSS.

Variables Table

Explanations for variables used in R-Squared calculation. Units depend on the context of the data.

Variable	Meaning	Unit	Typical Range
yᵢ	Actual observed value of the dependent variable for observation i	Dependent on data	N/A
ŷᵢ	Predicted value of the dependent variable for observation i by the model	Dependent on data	N/A
$\bar{y}$	Mean of all actual observed values of the dependent variable	Dependent on data	N/A
TSS	Total Sum of Squares	Squared units of dependent variable	≥ 0
RSS	Residual Sum of Squares	Squared units of dependent variable	≥ 0
ESS	Explained Sum of Squares	Squared units of dependent variable	≥ 0
R²	Coefficient of Determination	Unitless	Typically 0 to 1 (can be negative for poor models)

Practical Examples of R-Squared Calculation

Let’s illustrate with two examples.

Example 1: Simple Linear Regression (e.g., predicting sales)

Suppose we are trying to predict monthly sales (in thousands of dollars) based on advertising spend (in thousands of dollars). Our regression model gives us predicted sales values.

Inputs:

• Actual Sales (y): 10, 12, 15, 18, 22 (thousands of dollars)
• Predicted Sales (ŷ): 11, 13, 14, 19, 21 (thousands of dollars)

Calculation Steps:

1. Calculate the mean of Actual Sales: $\bar{y}$ = (10 + 12 + 15 + 18 + 22) / 5 = 15.4
2. Calculate TSS: (10-15.4)² + (12-15.4)² + (15-15.4)² + (18-15.4)² + (22-15.4)² = 29.16 + 11.56 + 0.16 + 6.76 + 43.56 = 91.2
3. Calculate RSS: (10-11)² + (12-13)² + (15-14)² + (18-19)² + (22-21)² = 1 + 1 + 1 + 1 + 1 = 5
4. Calculate R²: 1 – (RSS / TSS) = 1 – (5 / 91.2) = 1 – 0.0548 = 0.9452

Result: R² = 0.9452. This indicates that approximately 94.52% of the variance in sales is explained by the advertising spend in this model. This is a very strong fit.

Example 2: A Model with Poorer Fit (e.g., predicting student scores)

Consider predicting student test scores (out of 100) based on hours studied. The model doesn’t capture the relationship well.

Inputs:

• Actual Scores (y): 70, 75, 80, 85, 90
• Predicted Scores (ŷ): 75, 72, 85, 88, 95

Calculation Steps:

1. Mean of Actual Scores: $\bar{y}$ = (70 + 75 + 80 + 85 + 90) / 5 = 80
2. TSS: (70-80)² + (75-80)² + (80-80)² + (85-80)² + (90-80)² = 100 + 25 + 0 + 25 + 100 = 250
3. RSS: (70-75)² + (75-72)² + (80-85)² + (85-88)² + (90-95)² = 25 + 9 + 25 + 9 + 25 = 93
4. R²: 1 – (RSS / TSS) = 1 – (93 / 250) = 1 – 0.372 = 0.628

Result: R² = 0.628. This suggests that about 62.8% of the variance in student scores is explained by the hours studied in this model. While not terrible, it indicates that other factors significantly influence the scores, and the model could be improved.

How to Use This R-Squared Calculator

1. Gather Your Data: You need two sets of values for the same set of observations: the actual, observed values (y) of your dependent variable, and the predicted values (ŷ) generated by your regression model.
2. Input Actual Values: In the “Actual Values (y)” field, enter your observed data points, separated by commas. For example: 5, 8, 10, 12.
3. Input Predicted Values: In the “Predicted Values (ŷ)” field, enter the corresponding values that your regression model predicted for each of those observations, also separated by commas. Make sure the order matches the actual values. For example: 5.5, 7.8, 10.2, 11.5.
4. Calculate: Click the “Calculate R-Squared” button.
5. Interpret Results: The calculator will display the R-Squared value, along with intermediate calculations like TSS, RSS, and ESS.
   • An R² value close to 1 indicates a good fit.
   • An R² value close to 0 indicates a poor fit.
   • An R² value can be negative if the model performs worse than simply predicting the mean.
6. Visualize: The chart shows a scatter plot of actual versus predicted values. A perfect model would have all points falling on the diagonal line (y = ŷ). The table provides a point-by-point breakdown of residuals.
7. Copy/Reset: Use the “Copy Results” button to easily save the calculated values. Use the “Reset” button to clear the fields and start over.

Unit Considerations: R-squared is a unitless metric. The input values (actual and predicted) should share the same units, but these units can be anything (e.g., dollars, kilograms, scores, counts). The calculator treats them as relative values for the purpose of calculating the proportion of variance explained.

Key Factors That Affect R-Squared

1. Model Complexity: As you add more independent variables to a regression model, R-squared will tend to increase, even if the additional variables are not truly significant. This is why Adjusted R-squared is often preferred for models with multiple predictors.
2. Sample Size: While R-squared doesn’t directly depend on sample size in its basic calculation, a small sample size can lead to a less reliable R-squared value. A high R-squared in a small sample might be due to chance.
3. Data Variability: If the actual data points are tightly clustered around the regression line, R-squared will be high. If the data points are widely scattered, R-squared will be lower.
4. Outliers: Extreme values (outliers) can disproportionately influence R-squared. A single outlier can significantly inflate or deflate the R-squared value, depending on its position relative to the regression line.
5. Variable Selection: The choice of independent variables is paramount. If the chosen variables have little to no actual relationship with the dependent variable, R-squared will be low.
6. Model Specification: Using a linear model when the true relationship is non-linear, or vice-versa, will result in a poor fit and a low R-squared value.
7. Measurement Error: Inaccurate measurement of either the dependent or independent variables can introduce noise, leading to a lower R-squared.

Frequently Asked Questions (FAQ) about R-Squared

Q1: Can R-squared be negative?

Yes. If the regression model’s predictions are systematically worse than simply using the mean of the dependent variable as the prediction, the RSS can be larger than the TSS, leading to a negative R². This indicates a very poor model fit.

Q2: What is a “good” R-squared value?

There’s no universal threshold for a “good” R-squared. It depends heavily on the field of study and the specific problem. In some fields (like physics or econometrics), R² values above 0.8 or 0.9 might be considered excellent. In others (like social sciences or biology), an R² of 0.3 or 0.4 might be considered significant. Always compare against established benchmarks in your domain.

Q3: Does a high R-squared mean the independent variables *cause* the dependent variable?

No. R-squared only indicates the proportion of variance explained. It does not imply causation. Correlation does not equal causation, and a high R-squared could be due to a confounding variable or coincidence.

Q4: How does R-squared differ from Adjusted R-squared?

Adjusted R-squared modifies the R-squared value to account for the number of independent variables in the model. It penalizes the addition of unnecessary predictors, making it a better measure for comparing models with different numbers of variables. R-squared will always increase or stay the same when a new predictor is added, while Adjusted R-squared may decrease.

Q5: Can I use R-squared for non-linear regression?

The basic R-squared formula (1 – RSS/TSS) can be used for non-linear regression as well, as long as RSS and TSS are calculated correctly based on the model’s predictions. However, the interpretation might need more nuance, and Adjusted R-squared is often more useful.

Q6: What are the limitations of R-squared?

Limitations include its insensitivity to model misspecification (e.g., assuming linearity when it’s non-linear), its tendency to increase with more variables (favoring Adjusted R-squared), and its inability to indicate whether the model is appropriate or if the predictor variables are truly significant or causal. A high R-squared doesn’t automatically mean a model is good or reliable.

Q7: How should I handle units when inputting data?

The R-squared calculation is unitless. Ensure that your actual and predicted values are in the *same* units. The calculator treats them as relative measurements. For example, if you’re measuring temperature in Celsius, both your actual and predicted values should be in Celsius.

Q8: My R-squared is 0.99. Does this mean my model is perfect?

A very high R-squared like 0.99 suggests an excellent fit, meaning your model explains a large proportion of the variance in the data. However, it doesn’t guarantee perfection. There might still be systematic errors not captured by TSS/RSS, or the model might not generalize well to new, unseen data (overfitting). Always consider other diagnostic tools alongside R-squared.