Multinomial Frequency Generator
Draws discrete observations across K categorical bins under specified null or alternative frequency distributions.
Statistics & Probability / Chi-Squared Goodness of Fit
FREE TO EXPLORELABORATORY 29 · STATISTICS & PROBABILITY
Pearson's Chi-Squared Goodness-of-Fit Test, Multinomial Cell Deviations, and Degrees of Freedom
Investigate Pearson's classical Chi-Squared goodness-of-fit test, formulate observed versus expected cell frequency deviations, evaluate multinomial asymptotic convergence, and verify the Chi-Squared scaling gradient. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EXPECTED PEARSON STATISTIC E[Χ²]—
SAMPLING TELEMETRY
Ready to sample with Factor k = 1.0.
Adjust Factor 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
Binning continuous data into discrete categories discards order information within bins and depends on arbitrary bin choices. The empirical regression of expected Pearson test statistics against discrepancy factor k confirms theoretical test scaling with slope m = 2.4000 and R² = 1.000000.
Statistical tools & sampling modules.
Observed-Expected Frequency Table
Logs observed cell frequencies O_i alongside theoretical expected frequencies E_i = n p_i.
Pearson Deviance Calculator
Computes standardized cell residuals (O_i - E_i) / √E_i and total Pearson statistic χ² = ∑ (O_i - E_i)^2 / E_i.
Cochran Minimum Count Auditor
Verifies Cochran's rule that all expected cell counts satisfy E_i ≥ 5 to ensure valid asymptotic convergence.
Chi-Squared p-Value Evaluator
Integrates the Chi-Squared probability density with K - 1 degrees of freedom to obtain exact tail p-values.
Goodness-of-Fit Regression Engine
Performs linear regression of expected test statistic versus discrepancy factor k to confirm the slope m = 2.400.
FOLLOW THE METHOD
A reason behind every step.
Investigate Pearson's classical Chi-Squared goodness-of-fit test, formulate observed versus expected cell frequency deviations, evaluate multinomial asymptotic convergence, and verify the Chi-Squared scaling gradient. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 factor.
01
Establish Null Distribution and Cell Bins
Configure a discrete uniform null hypothesis across K = 5 bins with equal probabilities p_i = 0.20 on the goodness-of-fit console.
02
Execute Primary Sampling under Null Model
Draw total sample size n = 500. Compute expected frequencies E_i = 500 · 0.20 = 100 for all i ∈ {1, ..., 5}. Record observed counts O_i.
03
Evaluate Pearson Test Statistic under H_0
Compute χ² = ∑_{i=1}^5 (O_i - E_i)^2 / E_i. Across 20,000 runs, confirm that the empirical mean test statistic equals degrees of freedom df = K - 1 = 4.000.
04
Verify Cochran's Minimum Expected Count Rule
Reduce sample size to test threshold regimes, verifying that when expected frequencies drop below 5 (E_i < 5), the Chi-Squared approximation degrades.
05
Step Discrepancy Parameter Across k = 1 to 6
Inject systematic probability deviations across standardized steps k ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the expected test statistic response.
06
Conduct Linear Regression on Goodness-of-Fit Scaling
Fit a linear regression model y = mx to coordinate pairs (x = k, y = E[χ²]). Confirm that empirical slope m = 2.4000 validates non-central Chi-Squared response.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Chi-Squared Goodness of Fit mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 2.4000 · x
| # | Discrepancy Scale Factor k (factor) | Expected Pearson Statistic E[χ²] (Pearson χ²) | Discrepancy Factor k | Expected Pearson Statistic E[χ²] | Remove |
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