Calculate the coefficient of determination (R²) and correlation coefficient (r) for your data sets
The Coefficient of Determination Calculator is a powerful tool that helps you understand how well your independent variable (X) explains the variance in your dependent variable (Y). It computes both the correlation coefficient (r) and the coefficient of determination (R²) with detailed steps and a calculation table.
This calculator is useful in regression analysis, data modeling, and statistical research, helping you make sense of the strength and direction of relationships between variables.
Instant R² and r Calculation: Get the coefficient of determination and correlation coefficient in seconds.
Step-by-Step Breakdown: Includes calculations for SSxx, SSyy, SSxy, r, and R².
Calculations Table: View data summary including sums of squares, cross-products, and individual terms.
Interpretation Ready: Provides exact values to help assess fit of regression models.
Reset Option: Start over with one click.
Decimal Precision Control: Choose rounding from 2 to 6 decimal places.
Copy & Save: Copy results or download as a formatted PDF for reports or presentations.
Accurate & Reliable: Eliminates manual errors in regression diagnostics.
Educational Value: Step-by-step computation is ideal for teaching and learning statistics.
Saves Time: Quickly analyze data relationships without spreadsheets or complex software.
Mobile Friendly: Works seamlessly on desktop and mobile browsers.
Free & Accessible: No installation, sign-up, or subscription required.
Academic Research: Determine how well variables are related in psychology, sociology, and education studies.
Business Intelligence: Measure how much sales performance is influenced by marketing spend.
Finance & Economics: Analyze historical trends between variables like GDP and inflation.
Scientific Experiments: Validate or reject hypotheses using linear regression metrics.
Data Science & Analytics: Validate model performance in machine learning pipelines.
Understanding R² and r is foundational to regression analysis:
Correlation Coefficient (r) indicates the strength and direction of a linear relationship. Ranges from -1 to 1.
r = -1 → Perfect negative correlation
r = 0 → No correlation
r = 1 → Perfect positive correlation
Coefficient of Determination (R²) explains how much of the variation in Y is explained by X.
R² = 0.80 means 80% of the variance in Y is explained by X.
This calculator helps evaluate whether your linear regression model is a good fit for your data and how confidently you can use it for prediction.
| Metric | Measures | Value Range |
|---|---|---|
| R² | Goodness of fit (linear models) | 0 to 1 |
| r | Strength & direction of relation | -1 to 1 |
| RMSE | Error magnitude | 0 to ∞ |
| MAE | Average absolute error | 0 to ∞ |
| Adjusted R² | R² adjusted for multiple variables | 0 to 1 |
R² is a statistical measure that indicates how much of the variance in the dependent variable (Y) is explained by the independent variable (X).
An R² of 1 means a perfect linear relationship; 100% of the variation in Y is explained by X.
R² is the square of the correlation coefficient (r²), so R² = r × r.
In theory, no. But a negative R² may occur if a model is forced through the origin or if there's computational error.
The calculator will show an error or return invalid results. Ensure both sets have the same number of values.
It means that 80% of the variation in Y can be explained by changes in X. The rest is due to unexplained factors or error.
To measure how well a stock’s return tracks a market index or benchmark.
Yes, to analyze how machine settings influence output characteristics.
You may get an R² value near 0 and an r value close to 0, indicating little to no linear correlation.
Yes, each step—from SSxx and SSyy to SSxy and r—is shown in detail.
An R² of 0.95 in a study of temperature vs. ice cream sales shows temperature strongly influences sales.
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