Find Line of Best Fit with Step-by-Step Derivation
Enter X values (comma-separated)
Enter Y values (comma-separated)
Results
Regression Equation
Slope (b₁)
Intercept (b₀)
R-squared
Correlation (r)
Step-by-Step Derivation
Linear Regression Formula
ŷ = b₀ + b₁x
b₁ = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / Σ(xᵢ - x̄)²
b₀ = ȳ - b₁x̄
R² = [Σ((xᵢ-x̄)(yᵢ-ȳ))²] / [Σ(xᵢ-x̄)² × Σ(yᵢ-ȳ)²]
Linear regression models the relationship between independent variable X and dependent variable Y.
⚠Both datasets must have the same number of values. Outliers can significantly affect regression results.
What is Linear Regression?
Linear regression is a statistical method to model the relationship between a dependent variable (Y) and one or more independent variables (X). The goal is to find the line that minimizes the sum of squared differences between observed and predicted values.
Slope (b₁)
Measures the change in Y for a one-unit change in X. Positive slope means Y increases as X increases.
Intercept (b₀)
The predicted value of Y when X equals 0. May not have practical meaning if X=0 is outside the data range.
R-squared
Proportion of variance in Y explained by X. Ranges from 0 to 1.
Predicted Value
ŷ = b₀ + b₁x: estimated Y value for a given X using the regression line.
Linear regression finds the best fitting straight line through data points using least squares method. Equation: ŷ = b₀ + b₁x, where b₀ is intercept and b₁ is slope.
What is least squares method?▼
Least squares minimizes the sum of squared differences between observed y values and predicted ŷ values. This gives the "best fit" line statistically.
How to interpret regression coefficients?▼
Slope (b₁): change in y per unit change in x. Intercept (b₀): predicted y when x=0. R²: proportion of variance explained by the model.
What is R-squared?▼
R² (coefficient of determination) ranges 0-1. 1 means perfect prediction, 0 means no predictive power. R² = r² where r is Pearson correlation.
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