Linear Regression OLS Calculator

OLS coefficients: estimating slope and intercept

Ordinary Least Squares (OLS) finds the slope and intercept of y = a·x + b that drive the sum of squared residuals to its minimum. Because there's a closed-form answer, you never have to iterate: a = Σ((xᵢ − x̄)(yᵢ − ȳ)) / Σ(xᵢ − x̄)² and b = ȳ − a·x̄. As long as the four classical assumptions hold — linearity, independent errors, constant variance (homoscedasticity), normal residuals — the Gauss-Markov theorem guarantees that OLS is the Best Linear Unbiased Estimator, or BLUE. Run it on the pairs (1,2), (2,4), (3,5), (4,4), (5,7) and you land on x̄ = 3 and ȳ = 4.4; the numerator comes to 11, the denominator to 10, so a = 1.1 and b = 4.4 − 1.1·3 = 1.1.

Applications

You'll find it as the baseline model in machine learning (sklearn.linear_model.LinearRegression, statsmodels OLS). Experimental physics leans on it to estimate Hooke's law F = k·x from spring measurements. Econometrics uses it for demand-supply elasticity or the Phillips curve, lab work uses it to calibrate analytical instruments, and it's the obvious choice whenever you forecast sales or costs from a single predictor.

FAQ

Why squared residuals instead of absolute values? Squaring gives you a smooth, convex loss that has a closed-form solution, and it punishes outliers harder, which is what you want when big errors are the ones that hurt.

What if Gauss-Markov assumptions fail? If the variance isn't constant, reach for weighted least squares or robust standard errors. If the residuals are correlated, you'll want GLS or a time-series model such as ARIMA.

How do I interpret the coefficients? a tells you how much y is expected to change when x goes up by one unit. b is the predicted y at x = 0, which often sits outside the range of your data, so treat any extrapolation there with caution.

Does OLS handle multiple predictors? Yes. The matrix form β̂ = (XᵀX)⁻¹Xᵀy extends the simple case straight into multiple linear regression.