Non-Gaussian Heavy-Tailed Generator
Generates skewed, heavy-tailed, or ordinal datasets across two independent groups with specified location shift Δ.
Statistics & Probability / Mann-Whitney & Wilcoxon Tests
FREE TO EXPLORELABORATORY 33 · STATISTICS & PROBABILITY
Non-Parametric Hypothesis Testing: Mann-Whitney U Test and Wilcoxon Rank-Sum Evaluation
Investigate distribution-free hypothesis testing, replace raw measurements with global pooled ranks, compute the Mann-Whitney U statistic and Wilcoxon Rank-Sum W, evaluate asymptotic normal approximations, and verify rank-sum scaling. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EXPECTED RANK-SUM METRIC E[U]—
SAMPLING TELEMETRY
Ready to sample with Shift Δ = 1.0.
Adjust Shift Δ, 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 Mann-Whitney test evaluates stochastic dominance; differences in distribution variance or skewness can trigger significance. The empirical regression of expected U-statistics against location shift Δ confirms non-parametric scaling with slope m = 5.0000 and R² = 1.000000.
Statistical tools & sampling modules.
Global Rank Transformation Unit
Pools observations from both groups and replaces raw metric values with continuous ascending ranks 1 to N.
Wilcoxon Rank-Sum Accumulator
Computes the rank-sum statistic W_1 = ∑ R_{1, i} for group 1 observations.
Mann-Whitney U Combinatorial Resolver
Evaluates U_1 = W_1 - n_1(n_1 + 1)/2, counting the number of pairwise exceedances X_{1, i} > X_{2, j}.
Asymptotic Normal Z-Approximation Engine
Computes Z_U = (U - E[U]) / σ_U with tie-corrected variance for large-sample hypothesis testing.
Rank-Sum Shift Regression Unit
Performs linear regression of expected U-statistic versus location shift Δ to confirm the slope m = 5.000.
FOLLOW THE METHOD
A reason behind every step.
Investigate distribution-free hypothesis testing, replace raw measurements with global pooled ranks, compute the Mann-Whitney U statistic and Wilcoxon Rank-Sum W, evaluate asymptotic normal approximations, and verify rank-sum scaling. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 units.
01
Configure Non-Normal Population Distributions
Set parent distribution to a highly skewed Log-Normal profile with group sizes n_1 = 10 and n_2 = 10 (total N = 20) on the non-parametric console.
02
Synthesize Independent Groups under Null H_0
Draw independent samples with zero location shift (Δ = 0.0). Pool observations and assign global ranks from 1 to 20.
03
Calculate Wilcoxon W and Mann-Whitney U Statistics
Sum ranks for group 1 to find W_1. Compute Mann-Whitney U_1 = W_1 - 10(11)/2 = W_1 - 55. Confirm E[U] = n_1 n_2 / 2 = 50.0 under H_0.
04
Compare Robustness against Student's t-Test
Inject an extreme outlier into group 2. Observe that Student's t-test collapses, while Mann-Whitney U remains stable due to bounded rank influence.
05
Step Location Shift Parameter Across Δ = 1.0 to 6.0
Increment the true location shift across standardized settings Δ ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the expected U-statistic response.
06
Conduct Linear Regression on Non-Parametric Response
Fit a linear regression model y = mx to coordinate pairs (x = Δ, y = E[U] / 10). Confirm that empirical slope m = 5.0000 validates rank shift scaling.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Mann-Whitney & Wilcoxon Tests mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 5.0000 · x
| # | Location Shift Magnitude Δ (units) | Expected Rank-Sum Metric E[U] (U-Statistic) | Location Shift Magnitude Δ | Expected Rank-Sum Metric E[U] | Remove |
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