Residual Calculator
Calculate residuals for linear regression analysis and assess model accuracy
Data Input & Regression
Data Points
Residual Formula
Individual Residual
e: Residual
y: Observed value
ŷ: Predicted value
Sum of Squared Residuals
Quick Example
Given Data
Points: (1, 4), (2, 7), (3, 5)
Model: y = 2x + 2
Calculations
Point 1: e₁ = 4 - 4 = 0
Point 2: e₂ = 7 - 6 = 1
Point 3: e₃ = 5 - 8 = -3
Σ(e²) = 0² + 1² + (-3)² = 10
Interpretation Guide
Positive residuals: observed value > predicted
Negative residuals: observed value < predicted
Smaller residuals indicate better model fit
Random residual patterns suggest good linear fit
Understanding Residuals in Linear Regression
What are Residuals?
Residuals are the differences between observed values and predicted values in a regression model. They represent the "error" or unexplained variation in each data point and are crucial for assessing model accuracy and validity.
Key Applications
- •Model validation and accuracy assessment
- •Identifying outliers and unusual observations
- •Checking linear regression assumptions
- •Comparing different regression models
Statistical Measures
Sum of Squared Residuals (SSR)
Total squared deviation from the regression line. Lower values indicate better model fit.
Mean Squared Error (MSE)
Average squared residual. Used to compare models and calculate RMSE.
R-squared (R²)
Proportion of variance explained by the model. Values closer to 1 indicate better fit.
Residual Analysis Best Practices
Pattern Recognition
Random residual scatter suggests appropriate linear model. Patterns may indicate non-linear relationships.
Outlier Detection
Large residuals (typically |e| > 2-3 standard deviations) may indicate outliers or model inadequacy.
Model Comparison
Compare SSR, MSE, and R² values across different models to select the best performing one.