Linear Regression and R² Explained

Linear regression fits a straight line through data to model the relationship between a predictor (x) and a response (y). R² tells you how much of the variation in y is explained by x.

Simple linear regression model

ŷ = b₀ + b₁x b₁ = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)² b₀ = ȳ − b₁x̄ R² = SSR / SST = 1 − SSE / SST SST = Σ(yᵢ − ȳ)² (total variation) SSR = Σ(ŷᵢ − ȳ)² (explained by regression) SSE = Σ(yᵢ − ŷᵢ)² (residual / unexplained)

Worked example

Temperature (°C) vs cure time (min): x: 150, 160, 170, 180 x̄ = 165 y: 12, 10, 8, 6 ȳ = 9 b₁ = [(150−165)(12−9)+(160−165)(10−9)+(170−165)(8−9)+(180−165)(6−9)] / [(−15)²+(−5)²+(5)²+(15)²] = [−45 + (−5) + (−5) + (−45)] / [225+25+25+225] = −100 / 500 = −0.20 b₀ = 9 − (−0.20)(165) = 9 + 33 = 42 ŷ = 42 − 0.20x R² = 1 − 0/SST = 1.00 (perfect fit in this example)

Regression assumptions

Linear: relationship is linear (check residual vs fitted plot)
Independent: residuals are independent
Normal: residuals are normally distributed (check normal probability plot)
Equal variance: residual spread is constant across x (homoscedasticity)

CQE exam tips

R² = 0.80 means 80% of y's variation is explained by x; the remaining 20% is unexplained.
A high R² does not guarantee the model is valid — always check residual plots.
Correlation (r) and regression slope (b₁) have the same sign but measure different things.

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