What does the Central Limit Theorem say?

The CLT states that, for independent identically distributed random variables with finite mean μ and variance σ², the distribution of the sample mean X̅ converges to N(μ, σ²/n) as sample size n increases — regardless of the population's original shape.

Why does the histogram look normal even for skewed populations?

Averaging cancels out extreme values. Each sample mean is the sum of n random draws divided by n; by the CLT, those sums converge to normal once n is large enough relative to the population's skewness.

How large does n need to be?

A common rule of thumb is n ≥ 30 for roughly symmetric populations. Highly skewed distributions (like exponential) may need n ≥ 100 or more before the histogram looks convincingly bell-shaped. You can experiment with this tool directly.

What is the standard error shown on the chart?

The standard error (SE) is σ / √n — the predicted standard deviation of the sampling distribution of the mean. As n doubles, SE shrinks by a factor of √2, so the histogram gets narrower.

Does the CLT require the population to be normal?

No. The CLT applies to any distribution with finite mean and variance, including highly non-normal ones like the exponential or bimodal distributions available in this demo.

What is the red curve on the histogram?

The red overlay is the normal PDF with the theoretical CLT parameters: mean = population mean (μ), standard deviation = σ / √n. When the histogram closely follows this curve, the CLT is in full effect.

Central Limit Theorem Demo

Name: Central Limit Theorem Demo
Availability: InStock
Author: Nham Vu

Pick a population distribution, set sample size and number of draws, then watch the sampling distribution of the mean become bell-shaped as the CLT predicts.

Simulation Settings

Population Distribution

Sample Size (n) 30

1200

Number of Samples 1000

1005,000

Histogram Bins 30

1080

Population Shape

Sampling Distribution of the Mean

Observed CLT Normal

Click Run Simulation to generate the sampling distribution.

What you are seeing

Set the controls and run the simulation. Each bar in the histogram represents how often a sample mean fell in that range across all repetitions. The red curve is the normal distribution the CLT predicts for those means. As n grows, the histogram and the curve align more closely.

Summary

Pick a population distribution, set sample size and number of draws, then watch the sampling distribution of the mean become bell-shaped as the CLT predicts.

How it works

Select a population distribution from the dropdown (uniform, exponential, bimodal, or custom).
Adjust the sample size n — larger n produces faster convergence to normality.
Set the number of samples to draw (up to 5,000 repetitions).
Click "Run Simulation" — the tool draws that many samples of size n, computes each mean, and plots the histogram.
A red normal-curve overlay shows the CLT prediction: N(μ, σ²/n).
Compare the histogram shape to the population shape shown in the left panel.

Use cases

Statistics students visualizing why sample means are normally distributed even from skewed populations.
Teaching hypothesis testing foundations — the CLT justifies z-tests and t-tests.
Demonstrating how sample size n controls the spread of the sampling distribution.
Exploring convergence speed for different population shapes (symmetric vs. highly skewed).
Quality control: understanding why sample mean control charts assume normality.
Data science onboarding — intuitive proof that averages behave predictably at scale.
Checking how many repetitions are needed before the histogram stabilizes.
Research review: sanity-checking simulation designs for Monte Carlo studies.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu