Cross-Tabulation Matrix Compiler
Sorts bivariate categorical observations into R × C contingency tables and computes row and column marginal sums.
Statistics & Probability / Chi-Squared Independence Test
FREE TO EXPLORELABORATORY 30 · STATISTICS & PROBABILITY
Contingency Table Analysis, Chi-Squared Test of Independence, and Cramér's V
Construct two-way contingency tables, compute expected marginal product frequencies, perform the Chi-Squared test of independence, quantify association strength via Cramér's V, and verify the association scaling law. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
SQUARED ASSOCIATION METRIC V²—
SAMPLING TELEMETRY
Ready to sample with Coupling k = 1.0.
Adjust Coupling k, 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
The Chi-Squared test of independence measures statistical association, but cannot establish directional causation. The empirical regression of squared Cramér's V against coupling factor k confirms theoretical association scaling with slope m = 0.0400 and R² = 1.000000.
Statistical tools & sampling modules.
Marginal Independence Evaluator
Computes expected independence frequencies E_{ij} = (R_i · C_j) / N under the null hypothesis of statistical independence.
Pearson Contingency Engine
Calculates the total independence statistic χ² = ∑∑ (O_{ij} - E_{ij})^2 / E_{ij} with (R - 1)(C - 1) degrees of freedom.
Yates Continuity Corrector
Applies Yates' continuity correction |O - E| - 0.5 to 2 × 2 tables to prevent Type I error inflation in moderate samples.
Cramér's V Effect Size Meter
Computes normalized association strength V = √[χ² / (N · min(R - 1, C - 1))] bounded strictly in [0, 1].
Contingency Association Regression Unit
Performs linear regression of squared Cramér's V versus coupling factor k to extract the slope m = 0.040.
FOLLOW THE METHOD
A reason behind every step.
Construct two-way contingency tables, compute expected marginal product frequencies, perform the Chi-Squared test of independence, quantify association strength via Cramér's V, and verify the association scaling law. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 factor.
01
Initialize Cross-Tabulation Dimensions
Set contingency table dimensions to R = 2 rows and C = 2 columns with total sample size N = 1,000 on the simulation dashboard.
02
Synthesize Bivariate Categorical Data under Independence
Generate 1,000 independent pairs with marginal probabilities p_row = [0.5, 0.5] and p_col = [0.5, 0.5]. Compute expected cell counts E_{ij} = 250.
03
Evaluate Pearson Test Statistic under H_0
Compute χ² across 20,000 runs under complete independence. Confirm that empirical mean test statistic equals degrees of freedom df = (2-1)(2-1) = 1.000.
04
Introduce Controlled Bivariate Association
Inject an association coupling bias into the diagonal cells. Compute Cramér's V = √[χ² / (1000 · 1)] to quantify effect magnitude.
05
Step Association Coupling Factor Across k = 1 to 6
Increment the coupling factor across standardized settings k ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the squared association metric V².
06
Conduct Linear Regression on Association Scaling
Fit a linear regression model y = mx to coordinate pairs (x = k, y = V²). Confirm that empirical slope m = 0.0400 matches theoretical quadratic coupling.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Chi-Squared Independence Test mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.0400 · x
| # | Association Coupling Factor k (factor) | Squared Association Metric V² (Cramér V²) | Coupling Factor k | Squared Association Metric V² | Remove |
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