Live model
Generate a reproducible normal sample and measure finite-run error
Use seeded Box-Muller draws to test sample mean, sample spread, one-standard-deviation coverage, and a decision-threshold CDF. The theoretical CDF remains visible so simulation variability is not mistaken for a change in the model.
| Standardized band | Count | Relative frequency | Record type |
|---|
Current calculation process
Formula, substitution, intermediate quantities, and check
Z1=sqrt(-2lnU1)cos(2piU2); Z2=sqrt(-2lnU1)sin(2piU2); X=mu+sigmaZ; SE(xbar)=sigma/sqrtn
Uniform pairs become two independent standard-normal values through Box-Muller. Welford's online update computes stable sample moments. The sample CDF and one-SD coverage are empirical; their theoretical targets are not estimated from the sample.
Use the simulator
Five steps for a reproducible normal experiment
- Set the theoretical location and scale. mu and sigma define the population being sampled.
- Choose trial count for precision. Mean Monte Carlo SE decreases as 1/sqrtn, not as 1/n.
- Record the integer seed. Reproducibility requires the seed and algorithm together.
- Choose a decision threshold. The empirical fraction below it is compared with Phi((x-mu)/sigma).
- Judge error relative to SE. Small decimal disagreement is expected; repeated multi-SE disagreement deserves investigation.
Five foundations
What a seeded normal sample demonstrates
1. Uniforms are transformed
Box-Muller maps two independent uniform values into two independent standard-normal values using radius and angle.
2. Scaling restores units
Multiplying Z by sigma and adding mu gives draws in the selected measurement unit.
3. Finite moments fluctuate
The sample mean and SD are estimators, not forced matches to the theoretical inputs.
4. Coverage is empirical
About 68.27% within +/-1sigma is a theoretical benchmark; one run will land nearby rather than exactly there.
5. Reproducibility is not validation
A repeatable sample verifies a computational path. It cannot show that real operational data follow a normal distribution.
Calculation anatomy
Symbols and default Monte Carlo path
| Symbol | Meaning | Rule |
|---|---|---|
| U1,U2 | Seeded uniform draws | 0<U1<=1, 0<=U2<1 |
| Z | Box-Muller standard-normal draw | mean 0, SD 1 |
| X | Scaled normal draw | mu+sigmaZ |
| n | Number of trials | 100-200,000 |
| xbar, s | Sample mean and sample SD | Welford update, n-1 variance |
| SE(xbar) | Mean simulation error scale | sigma/sqrtn |
Defaults generate 10,000 values as 50+10Z from seed 83,021. Each value updates mean and squared deviation once. The threshold comparison uses count(X<=62)/10,000 against Phi(1.2).
Deep analysis
Three checks beyond "it looks bell-shaped"
Mean error scale
Divide simulated minus theoretical mean by sigma/sqrtn. This standardized discrepancy is more informative than raw error.
Sample spread
The n-1 sample SD tests whether transformed draws have the requested scale. A generator can match the mean yet miss variance.
Local decision probability
Threshold CDF comparison checks the part of the distribution used by a decision, not only global moments.
Decision cases
Validation and small-sample examples
Forecast engine smoke test
A team expects N(50,10^2) inputs and uses 62 as a service threshold. It stores the seed and compares both moments and empirical CDF after a code change.
Minimum 100-trial run
At n=100, mean SE is sigma/10 and threshold frequencies can move several percentage points. The page allows this educational boundary but makes the error scale visible.
Terms
Normal-simulation vocabulary
- Box-Muller transform
- Mapping from uniform pairs to standard-normal pairs.
- Pseudo-random generator
- Deterministic algorithm producing a sequence that behaves like random uniforms.
- Welford update
- Stable online algorithm for sample mean and variance.
- Monte Carlo SE
- Expected sampling-error scale from a finite simulation.
- Empirical CDF
- Fraction of simulated values at or below a threshold.
- Coverage frequency
- Sample fraction inside a defined interval such as mu+/-sigma.
FAQ
Questions about normal Monte Carlo work
What does Box-Muller do?
It converts two independent uniforms into two standard-normal values.
Why use Welford's algorithm?
It computes moments online with good numerical behavior.
Should +/-1sigma coverage equal 68.27%?
Only in theory; finite samples fluctuate around it.
Why compare an empirical CDF?
It validates the specific decision threshold as well as moments.
Does changing the seed change theory?
No, only the finite sample.
Can simulation prove real data are normal?
No. It samples from an assumed normal model.
Limits and evidence
Simulation boundaries
- The generator is analytical and reproducible, not cryptographically secure.
- Normality is assumed by construction and not tested against observed data.
- The trial cap protects browser performance and does not define sufficient precision.
- Band rows are descriptive bins and do not replace formal goodness-of-fit checks.
- The same seed is reproducible only with the same generator and implementation version.
Sources and related tools