Central Limit Theorem Simulator

Generate sample means and watch the sampling distribution form

Sample size n

Samples

Population

Central Limit Theorem Formula

Sample mean	xbar = sum(x) / n
Mean of sample means	mean(xbar) approx population mean
Standard error	SE = sigma / sqrt(n)

How to Read the Simulation

Each simulated sample produces one sample mean.
The histogram shows the distribution of those sample means.
Larger sample size usually makes the sample means cluster more tightly.
The shape often becomes more bell-like even when the original data is skewed.

Example: if n is increased from 5 to 50, the histogram of sample means usually becomes narrower because the standard error decreases.

Frequently Asked Questions

What does the central limit theorem say?▼

The central limit theorem says that the distribution of sample means tends to become approximately normal as sample size increases.

Does the original data need to be normal?▼

No. The central limit theorem can apply even when the original population is not normal, especially with larger sample sizes.

What is standard error?▼

Standard error is the standard deviation of the sampling distribution of the sample mean. It equals population standard deviation divided by the square root of sample size.

Why simulate the central limit theorem?▼

Simulation makes it easier to see how sample means become more stable and more bell-shaped as sample size increases.

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