Linear Regression Calculator

Choose a and b to minimise Σ(yᵢ − a − bxᵢ)². Closed-form solutions: b = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)², a = ȳ − b·x̄.

Interpretation: a 1-unit increase in x is associated with a b-unit change in the average of y. The intercept a is the predicted y when x = 0 — meaningful only if x = 0 is inside or near your data.

R² = 1 − SS_res / SS_tot is the proportion of y's variance explained by x. Equals r² for simple regression.

Residual standard error s = √(SS_res / (n − 2)). The typical vertical distance from the line — y is predicted with roughly ±2s accuracy for new observations.

SE of the slope: s / √Σ(xᵢ − x̄)². Determines how precisely we know b.

t = b / SE(b), df = n − 2. Tests H₀: true slope = 0 (i.e. x has no linear effect on y).

95% CI on slope: b ± t* · SE(b). If this interval excludes 0, the slope is significant at α = 0.05.

Linearity — the true relationship between mean y and x is a straight line.

Independence — residuals don't form patterns over time or grouping.

Normality — residuals are approximately normal (matters for small-sample p-values; the CLT helps for large n).

Equal variance (homoscedasticity) — residual spread is roughly constant across x.

Diagnostic: residual-vs-fitted and QQ plots reveal violations far better than R² alone.

Same data as the correlation calculator: x = 1..10, y = (2,4,5,4,5,7,8,8,10,12).

b = 73.5 / 82.5 ≈ 0.891. a = 6.5 − 0.891·5.5 ≈ 1.6.

Residuals: yᵢ − (1.6 + 0.891·xᵢ). SS_res ≈ 9.45. s ≈ √(9.45/8) ≈ 1.087.

SE(slope) ≈ 1.087/√82.5 ≈ 0.120. t = 0.891/0.120 ≈ 7.44, df = 8, p < 0.001. Slope is highly significant.

Prediction at x = 11: ŷ = 1.6 + 0.891·11 ≈ 11.4 (with a wider prediction interval than confidence interval because individual variability adds on top).

Extrapolation: predicting at x far outside the observed range, where the assumed linear shape may break.

Influential points: a single high-leverage outlier can pull the slope dramatically. Inspect leverage and Cook's distance, not just R².

Omitted variable bias: a strong relationship between x and y might really be driven by a third variable z. Simple regression can't detect this.

Reporting R² without checking residual patterns. A curved relationship can give R² = 0.95 and yet the line is the wrong model.

Confusing prediction interval (for one new y) with confidence interval (for the mean y at that x). Prediction intervals are always wider.