Coefficient of Determination Calculator (R-squared)

The coefficient of determination, commonly denoted as R², is calculated using the total sum of squares (SST) and the sum of squared regression (SSR), or alternatively through the residual sum of squares (SSE). The fundamental formula is expressed as R² = 1 - (SSE / SST), where SSE represents the sum of squared residuals (the differences between actual and predicted values) and SST represents the total sum of squares (the differences between actual values and the mean of the dependent variable). The resulting value typically ranges from 0 to 1, where 0 indicates that the model explains none of the variability, and 1 indicates perfect prediction.

Imagine you are analyzing the relationship between study hours (X) and exam scores (Y) for five students. Your data pairs (X, Y) are: (1, 55), (2, 60), (3, 65), (4, 70), and (5, 85). First, the calculator determines the mean of Y, which is 67. Next, it computes the total sum of squares (SST) by summing the squared differences between each Y value and the mean. Then, it fits a linear regression line to find the predicted Y values, calculates the residual sum of squares (SSE), and applies the formula R² = 1 - (SSE / SST). For this dataset, the resulting R² value is approximately 0.91, meaning that 91% of the variance in exam scores can be explained by the number of study hours.

A high R-squared does not automatically imply causation; always evaluate your regression model alongside domain expertise and scatter plots. Be cautious of overfitting when adding too many independent variables, as unadjusted R-squared will naturally increase even if the added variables lack true predictive power. When comparing models with differing numbers of predictors, consider reviewing the adjusted R-squared metric for a more reliable evaluation.