Coefficient of Determination (R²) Calculator

R² is the proportion of variance in one variable explained by another.

How the Math Works

The Coefficient of Determination, or R-squared (R²), measures how well a regression model predicts an outcome by quantifying the proportion of variance explained by the independent variables. It is calculated by squaring the correlation coefficient (r), which ranges from -1 to 1, resulting in R² values between 0 and 1. A higher R² indicates that the model accounts for more of the data's variability, with R² = 1 meaning perfect prediction and R² = 0 indicating no explanatory power. This simple squaring relationship transforms the correlation's directionless strength into a clear percentage of explained variance.

Practical Applications

To apply this calculation, first determine the correlation coefficient (r) between your independent and dependent variables using statistical methods or software. Square this value to obtain R², then multiply by 100 to express it as a percentage. For example, if r = 0.8, then R² = 0.64, meaning 64% of the variation in the dependent variable can be explained by the independent variable. This helps assess model quality before making predictions or drawing conclusions about relationships in your data analysis.

Day-to-Day Use

In everyday life, R² helps you understand how much trust to place in predictions and trends. If a weather app claims temperature predictions with R² = 0.9, you know 90% of temperature variation is explained by its model, making forecasts reliable. Similarly, when evaluating investment performance, an R² near 1 between your portfolio and market index means your fund closely tracks the market. It also helps identify misleading correlations — like a high r between ice cream sales and drowning incidents (both driven by summer heat) with low R² when controlling for temperature reveals the true relationship.

Worked example

r 0.9 → R² 0.81 (81% explained).

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