Bivariate OLS Data Generator
Generates coordinate pairs (x_i, y_i) with controlled linear signal y = β_0 + β_1 x + ε and homoscedastic noise.
Statistics & Probability / Simple Linear Regression (OLS)
FREE TO EXPLORELABORATORY 34 · STATISTICS & PROBABILITY
Simple Linear Regression, Ordinary Least Squares, and Gauss-Markov Theorem
Investigate bivariate linear regression, derive Ordinary Least Squares (OLS) normal equations, evaluate residual variance and standard errors, prove the Gauss-Markov Best Linear Unbiased Estimator (BLUE) property, and confirm slope parameter extraction. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
FITTED RESPONSE VARIABLE Ŷ—
SAMPLING TELEMETRY
Ready to sample with Regressor x = 1.0.
Adjust Regressor x, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
OLS regression measures linear association; non-linear relationships can yield misleading near-zero slopes. The empirical regression of fitted responses against regressor coordinates confirms OLS parameter extraction with slope m = 0.5000 and R² = 1.000000.
Statistical tools & sampling modules.
Normal Equations Matrix Inverter
Solves the closed-form OLS normal equations β̂_1 = S_xy / S_xx and β̂_0 = ȳ - β̂_1 x̄.
Residual Sum of Squares Decomposer
Partitions total sum of squares SST into regression sum of squares SSR and residual sum of squares SSE.
Gauss-Markov BLUE Benchmark Engine
Synthesizes alternative linear unbiased estimators to empirically demonstrate that OLS achieves minimum variance.
Coefficient of Determination Resolver
Computes Pearson R² = SSR / SST and evaluates root mean square error (RMSE).
OLS Slope Verification Unit
Performs linear regression on fitted model coordinates to confirm the theoretical slope m = 0.500.
FOLLOW THE METHOD
A reason behind every step.
Investigate bivariate linear regression, derive Ordinary Least Squares (OLS) normal equations, evaluate residual variance and standard errors, prove the Gauss-Markov Best Linear Unbiased Estimator (BLUE) property, and confirm slope parameter extraction. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 units.
01
Configure Regression Parameters and Noise Level
Set true slope β_1 = 0.500 and residual variance σ^2 = 0.25 on the OLS simulation control interface.
02
Synthesize Bivariate Observations
Draw 10,000 coordinate pairs (x_i, y_i) spanning regressor values x ∈ [0, 10]. Calculate sample means x̄ and ȳ.
03
Solve Ordinary Least Squares Normal Equations
Compute cross-sums S_xx = ∑ (x_i - x̄)^2 and S_xy = ∑ (x_i - x̄)(y_i - ȳ). Evaluate OLS slope β̂_1 = S_xy / S_xx.
04
Evaluate Gauss-Markov Minimum Variance Condition
Generate a competing linear unbiased estimator β̃ = ∑ c_i y_i with c_i ≠ (x_i - x̄)/S_xx. Verify that Var(β̂_OLS) < Var(β̃).
05
Step Regressor Coordinate Across x = 1 to 6
Increment the regressor coordinate across standardized settings x ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the fitted response ŷ.
06
Conduct Linear Regression on OLS Model Scaling
Fit a linear regression line y = mx to coordinate pairs (x, ŷ). Confirm that empirical slope m = 0.5000 validates OLS parameter estimation.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Simple Linear Regression (OLS) mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.5000 · x
| # | Independent Regressor Coordinate x (units) | Fitted Response Variable ŷ (Response ŷ) | Regressor Coordinate x | Fitted Response Variable ŷ | Remove |
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