What source distributions can I choose from?

You can select from Uniform (0–1), Exponential (λ=1), Poisson (λ=4), or Bernoulli (p=0.3) distributions.

How does increasing the sample size (n) affect the output?

A larger sample size narrows the sampling distribution, reducing the standard error and making the histogram shape more closely match the theoretical normal curve.

Why is there a random seed option?

The random seed ensures the pseudo-random number generator produces identical, reproducible samples, which is ideal for classroom demonstrations.

What metrics are compared in the results?

The tool compares the empirical mean and standard error of your generated samples against the theoretical mean and standard error.

Can I adjust the resolution of the histogram?

Yes, you can customize both the number of sample repetitions (up to 50,000) and the number of histogram bins (up to 120) to refine the visualization.

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Data Visualization

Central Limit Theorem Demonstrator

See the Central Limit Theorem in action: draw samples from a non-normal source distribution (uniform, exponential, Poisson, or Bernoulli), plot the histogram of sample means, and overlay the predicted Normal curve N(μ, σ²/n). Compare empirical vs theoretical mean and standard error, and watch convergence tighten as the sample size grows.

Details

What this tool helps you do

An interactive statistics teaching tool for intro stats / data-science students, A/B-test instructors and AP Statistics teachers:

Source distribution — pick a clearly non-normal population: Uniform(0,1), Exponential(λ=1), Poisson(λ=4), or Bernoulli(p=0.3).
Sample size n — how many draws go into each sample mean. Raise it to watch the sampling distribution narrow (spread ∝ σ/√n) and bell out.
Number of samples — how many sample means to generate (the histogram resolution).
Output — a histogram of the sample means with the theoretical Normal curve overlaid, plus empirical vs theoretical mean and standard error, and a rug plot of the raw source data.

Uses a seeded PRNG (mulberry32) so every run with the same inputs is reproducible — great for classroom demos where everyone should see the same picture.

Execution

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Fill in the form, run the tool, and review the result in one place.

Prepared example runs

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2 examples

Exponential source, n=30 → near-normal means

Draw from Exponential(λ=1) with n=30 per mean, 5000 samples. The skewed source produces a bell-shaped sampling distribution matching N(1, 1/30).

<div class="clt-wrap">…histogram of means with red normal curve overlay, empirical mean ≈ 1.000…</div>

Bernoulli source, small n vs large n

Bernoulli(p=0.3) with n=5 shows a lumpy sampling distribution; raising n to 50 smooths it toward normal — illustrating convergence.

<div class="clt-wrap">…n=50 bernoulli means histogram close to the normal curve, SE ≈ √(0.21/50)…</div>

Inputs

Set the required fields, then run the tool.

6 options

SettingsAdjust formats, ranges, numbers, and modes.6

Source distributionselectOptionalSample size (n)numberRequiredDraws per sample mean. Larger n → narrower, more normal sampling distribution.Number of samplesnumberRequiredHow many sample means to generate (histogram resolution).Number of binsnumberRequiredRandom seednumberOptionalSame seed → identical results (reproducible demos).Decimal placesnumberRequired

Result

Ready for a run

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Samples

Central Limit Theorem Demonstrator

What this tool helps you do

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Prepared example runs

Inputs

Result

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Key facts

Overview

When to use

How it works

Use cases

Examples

1. Visualizing Exponential Convergence

2. Demonstrating Bernoulli Convergence with Small vs Large Samples

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