Box-Muller Normal Generator
Transforms pairs of independent uniform variates into exact standard normal variates using polar Box-Muller trigonometric equations.
Statistics & Probability / Gaussian Distribution & erf
FREE TO EXPLORELABORATORY 15 · STATISTICS & PROBABILITY
Continuous Gaussian Distribution, Z-Scores, and Error Function Numerical Integration
Investigate the continuous normal distribution, standardize observations using z-scores, evaluate the Gauss error function erf(z) via numerical integration, and determine the Full Width at Half Maximum (FWHM) proportionality factor. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
FULL WIDTH AT HALF MAXIMUM (FWHM)—
SAMPLING TELEMETRY
Ready to sample with Std Dev σ = 1.0.
Adjust Std Dev σ, 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
Real physical measurements have bounded support, whereas the mathematical Gaussian distribution has infinite tails. The empirical regression of Full Width at Half Maximum against standard deviation σ yields a gradient of m = 2.3550, confirming the theoretical factor 2√(2 ln 2) with R² = 1.000000.
Statistical tools & sampling modules.
Standardized Z-Score Normalizer
Applies affine scaling z = (x - μ)/σ to map arbitrary normal observations onto the standard normal domain N(0, 1).
Gauss Error Function Integrator
Evaluates erf(z) = (2/√π) ∫_0^z e^(-t^2) dt using high-order adaptive Gauss-Legendre quadrature.
Gaussian Peak Profile Analyzer
Measures the peak amplitude f(μ) = 1/(σ√(2π)) and evaluates half-maximum inflection coordinates.
Empirical Rule Cumulative Auditor
Computes cumulative probability mass within 1σ (68.27%), 2σ (95.45%), and 3σ (99.73%) intervals.
FWHM Proportionality Regression Engine
Performs linear regression of peak width FWHM versus σ to verify the theoretical factor 2√(2 ln 2) ≈ 2.35482.
FOLLOW THE METHOD
A reason behind every step.
Investigate the continuous normal distribution, standardize observations using z-scores, evaluate the Gauss error function erf(z) via numerical integration, and determine the Full Width at Half Maximum (FWHM) proportionality factor. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 σ.
01
Initialize Normal Parameters and Density Grid
Set mean μ = 0.0 and initial standard deviation σ = 1.0. Generate a high-resolution grid of 2,000 points spanning x ∈ [-5σ, +5σ] to map the probability density function.
02
Execute Box-Muller Stochastic Sampling
Draw 50,000 independent normal variates via the Box-Muller algorithm: Z_0 = √(-2 ln U_1) cos(2π U_2). Verify sample mean and variance convergence.
03
Evaluate the Gauss Error Function and Normal CDF
Compute cumulative normal probabilities Φ(z) = (1/2)[1 + erf(z/√2)] across z ∈ {1.0, 2.0, 3.0}, verifying compliance with the 68-95-99.7% empirical rule.
04
Measure the Full Width at Half Maximum (FWHM)
Locate the peak density amplitude f(0) = 1/(σ√(2π)) and determine the two horizontal coordinates where f(x) = f(0)/2. Calculate FWHM = x_right - x_left.
05
Step Standard Deviation Across σ = 1.0 to 6.0
Increment the scale parameter across standardized settings σ ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the broadening of the density curve and corresponding FWHM values.
06
Conduct Linear Regression on Gaussian FWHM Scaling
Fit a linear regression model y = mx to the coordinate pairs (x = σ, y = FWHM). Confirm that empirical slope m = 2.3550 matches the theoretical value 2√(2 ln 2) = 2.35482.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Gaussian Distribution & erf mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 2.3550 · x
| # | Standard Deviation Scale σ (σ) | Full Width at Half Maximum (FWHM) (FWHM) | Standard Deviation σ | Full Width at Half Maximum (FWHM) | Remove |
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