Coefficient of Determination (R²) Calculator

Hypothesis Testing and Statistical Inference

Enter your data points below to calculate the Coefficient of Determination (R²), Adjusted R², Correlation Coefficient (r), and the linear regression equation. This tool helps you assess the 'goodness of fit' for your linear model.

Practical Examples

Explore these examples to understand how the Coefficient of Determination is used in different scenarios.

Strong Positive Correlation

Positive Correlation

A scenario where Y consistently increases as X increases, showing a strong linear relationship.

X: 1, 2, 3, 4, 5

Y: 2.1, 3.9, 6.2, 8.1, 10.3

Strong Negative Correlation

Negative Correlation

A scenario where Y consistently decreases as X increases.

X: 10, 20, 30, 40, 50

Y: 95, 82, 73, 66, 51

Weak or No Correlation

No Correlation

Data points are scattered randomly, indicating no clear linear relationship between X and Y.

X: 1, 2, 3, 4, 5, 6

Y: 3, 8, 5, 1, 9, 4

Real-World: Study Hours vs. Score

Real World

Analyzing the relationship between hours spent studying and the final exam score.

X: 1, 2, 3, 4, 5, 6, 7, 8

Y: 65, 68, 72, 75, 82, 88, 90, 95

Other Titles
Understanding the Coefficient of Determination (R²): A Comprehensive Guide
Dive deep into the concepts, application, and mathematics behind R-squared to master your statistical analyses.

What is the Coefficient of Determination (R²)?

  • Defining Goodness of Fit
  • R-Squared vs. Adjusted R-Squared
  • The Range and Meaning of R² Values
The Coefficient of Determination, commonly known as R-squared or R², is a key metric in statistical analysis, particularly in the context of regression models. It represents the proportion of the variance in the dependent variable that can be predicted from the independent variable(s). In simpler terms, R² measures how well the regression line (the model) fits the observed data. An R² of 100% means the model perfectly explains the variability of the response data around its mean, while an R² of 0% indicates that the model explains none of the variability.
Distinguishing R² from Adjusted R²
While R² is a useful measure, it has a limitation: it never decreases, even if you add irrelevant independent variables to a model. This can be misleading. The Adjusted R², on the other hand, accounts for the number of predictors in the model. It increases only if the new variable improves the model more than would be expected by chance. This makes Adjusted R² a more reliable measure when comparing models with different numbers of independent variables.

Step-by-Step Guide to Using the R-Squared Calculator

  • Inputting Your Data
  • Interpreting the Results Section
  • Using the Examples
Our calculator simplifies the process of finding R². Here's how to use it effectively:
1. Data Entry
You need two sets of data: the Independent Variable (X-Values) and the Dependent Variable (Y-Values). Enter these as comma-separated lists into their respective text fields. Ensure that both lists have the same number of data points.
2. Calculation
Click the 'Calculate' button. The tool will instantly perform the linear regression analysis.
3. Understanding the Output
The results card will display not only the R² and Adjusted R² values but also the correlation coefficient (r), the number of data points (n), and the equation of the best-fit line. This gives a comprehensive view of the relationship between your variables.

Real-World Applications of R-Squared

  • Finance and Economics
  • Medical and Social Sciences
  • Engineering and Quality Control
R-squared is not just an academic concept; it has powerful applications across various fields.
In Finance
Analysts use R² to measure how much of a mutual fund's or stock's movement can be explained by movements in a benchmark index, like the S&P 500. A high R² suggests the fund's performance largely mirrors the index.
In Social Sciences
Researchers might use R² to determine how well factors like education level and years of experience predict income levels. This helps in understanding the drivers of economic outcomes.

Interpreting R-Squared: Common Pitfalls and Best Practices

  • High R-Squared Doesn't Imply Causation
  • The Problem of Overfitting
  • Context is King
A high R² value is often seen as a good sign, but it's crucial to interpret it correctly to avoid common statistical fallacies.
Correlation vs. Causation
Remember that R² only indicates the strength of the association between variables; it does not prove that changes in the independent variable cause changes in the dependent variable. There could be lurking variables or a coincidental relationship.
What is a 'Good' R² Value?
There is no universal standard for a 'good' R² value. In fields like physics or chemistry, where systems are precise, you might expect R² values above 95%. In social sciences, where human behavior introduces more variability, an R² of 50% might be considered strong. Always evaluate R² within the context of your specific field and research question.

The Math Behind R-Squared: Formulas and Derivations

  • Total Sum of Squares (SStot)
  • Sum of Squared Residuals (SSres)
  • The R-Squared Formula
For those interested in the underlying mathematics, R² is derived from two key components:
Formulas
1. Total Sum of Squares (SStot): This measures the total variance in the Y-values. Formula: SStot = Σ(yᵢ - ȳ)², where yᵢ is each observed value and ȳ is the mean of Y.
2. Sum of Squared Residuals (SSres): This measures the variance that is not explained by the model. It's the sum of the squared differences between the observed values (yᵢ) and the predicted values (ŷᵢ). Formula: SSres = Σ(yᵢ - ŷᵢ)².
3. R-Squared (R²): The formula itself is R² = 1 - (SSres / SStot). This shows that R² is simply 1 minus the proportion of unexplained variance.