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Sampling & Bootstrap — CLT Explorer (with steps)

Recreate uniform, normal, exponential, and Bernoulli samples with a deterministic LCG, inspect point estimates, percentile/t/normal confidence intervals, and explore CLT histograms with fully documented steps.

Designed for classrooms: shareable URLs, CSV export, ad slot reservation, delayed Giscus comments, and related tools for probability and inference.

Other languages: ja | en | es | zh-CN

Controls

Distribution & statistic
Exponential parameter
Seeds & CLT
95% confidence intervals

Results summary

Point estimate
—
Sample mean
—
Sample median
—
Sample standard deviation
—
n / B
— / —
Theoretical mean / σ
— / —

  • Bootstrap percentile

    —

  • t approximation (mean)

    —

  • Normal approximation (proportion)

    —

How it's calculated

  1. Initialise the LCG (a=1664525, c=1013904223, m=232) with seed so the sampling stream is reproducible.
  2. Draw n= observations from and evaluate the selected statistic .
  3. Resample B= times, take Type-1 quantiles, and report the chosen intervals ().
  4. For the CLT explorer, use seed with K= standardised sample means, summarised as mean/variance ≈ .

Both the shareable URL and CSV encode the seeds and configuration for reproducible demonstrations.

Histograms

Bootstrap distribution

Shows the sampling distribution of the statistic across bootstrap replicates and highlights the CI span.

CLT standardised means

Overlaying N(0,1) reveals how quickly the empirical mean and variance approach 0 and 1 respectively.

FAQ

Why use the Type-1 percentile?

It uses floor((B−1)p) order statistics so the interval endpoints are transparent to students, mirroring textbook bootstrap explanations.

What benefits does the fixed LCG bring?

The same parameters as our probability simulator (a=1664525, c=1013904223, m=232) guarantee identical samples for a given seed, which is ideal for lesson plans, handouts, and remote verification.