# 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.

> Canonical page: https://elysiatools.com/en/tools/central-limit-theorem-demonstrator

- **Category:** Data Visualization

- **Keywords:** central limit theorem, clt, sampling distribution, normal distribution, gaussian, statistics, probability, sample mean, standard error, monte carlo, histogram, data science, teaching, ap statistics

## Overview

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

1. **Source distribution** — pick a clearly non-normal population: Uniform(0,1), Exponential(λ=1), Poisson(λ=4), or Bernoulli(p=0.3).
2. **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.
3. **Number of samples** — how many sample means to generate (the histogram resolution).
4. **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.

## Inputs

- **Source distribution** (select)
- **Sample size (n)** (number): Draws per sample mean. Larger n → narrower, more normal sampling distribution.
- **Number of samples** (number): How many sample means to generate (histogram resolution).
- **Number of bins** (number): 40
- **Random seed** (number): Same seed → identical results (reproducible demos).
- **Decimal places** (number): 3

## When to use

- When teaching or studying introductory statistics, probability theory, or data science concepts.
- When demonstrating how sample size (n) affects the shape and spread of a sampling distribution.
- When verifying empirical sample means and standard errors against theoretical calculations for non-normal populations.

## How it works

- Select a non-normal source distribution, such as Uniform, Exponential, Poisson, or Bernoulli, to act as the population.
- Set the sample size (n) per mean, the total number of sample repetitions, and the number of histogram bins.
- Enter an optional random seed to ensure the pseudo-random number generator produces reproducible results.
- Generate the interactive HTML output to view the histogram of sample means, the overlaid theoretical Normal curve, and the comparison of empirical versus theoretical metrics.

## Use cases

- Creating reproducible visual aids for AP Statistics or college-level data science lectures.
- Simulating Monte Carlo sampling to understand the behavior of Bernoulli trials in A/B testing.
- Comparing empirical standard error formulas against simulated outcomes to reinforce theoretical proofs.

## Frequently asked questions

### 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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