Residual Calculator

In statistics, a residual represents the difference between an observed value (y) and the predicted value (ŷ) generated by a model. The fundamental formula for an individual residual is expressed as e = y - ŷ. When evaluating an entire dataset, analysts often look at the Sum of Squared Residuals (SSR), calculated as ∑(y - ŷ)². Squaring these values prevents positive and negative discrepancies from canceling each other out, giving analysts a reliable penalty metric for prediction errors.

Imagine you are analyzing the relationship between study hours and exam scores. You collect three data points (X, Y): Point 1 (1, 55), Point 2 (2, 70), and Point 3 (3, 85). Suppose your linear regression model generates the prediction equation ŷ = 15x + 40. For Point 1, the prediction is ŷ = 15(1) + 40 = 55, yielding a residual of 55 - 55 = 0. For Point 2, ŷ = 15(2) + 40 = 70, resulting in a residual of 70 - 70 = 0. For Point 3, ŷ = 15(3) + 40 = 85, giving a residual of 85 - 85 = 0. In this textbook scenario, the model fits the data perfectly, and the sum of squared residuals equals zero.

Always inspect your residual values for patterns rather than just looking at the final sum. A healthy regression model will show residuals randomly scattered around zero, indicating that a linear approach is appropriate. If your residuals display a clear curved or funnel shape, your underlying relationship may be non-linear or suffering from heteroscedasticity.