1. Sample Means
The mean of repeated random samples taken from a population. The CLT describes how these sample means are distributed.
Central Limit Theorem (CLT) Simulation
Created:November 5, 2024
Last Updated:August 7, 2025
This interactive simulation demonstrates the Central Limit Theorem (CLT), one of the most important concepts in statistics. The CLT states that when independent random samples are taken from any population distribution, the distribution of sample means will approximate a normal distribution as the sample size increases. This simulation allows you to explore this phenomenon by selecting different underlying distributions, adjusting sample sizes, and observing how the sampling distribution of means evolves. Whether you're a student learning statistics or a practitioner wanting to visualize this fundamental theorem, this tool provides an intuitive understanding of how the CLT works in practice.
Simulation
Click the "Start Simulation" button to start the simulation. The simulation will run until you stop it. You can also click the "Quick Demo" to generate 500 samples quickly and view the results instantly.
5100
SlowFast
Samples Collected
0
Population μ
0.000
Sample Means μ
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Current Sample x̄
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⏸️ Paused
Sample Means μ: ---
Should equal population μ
Sample Means σ: ---
Observed standard error
Expected σ (σ/√n): 0.0000
Theoretical standard error
Learn More
Understanding the Central Limit Theorem
Key Concepts
2. Sample Size
The number of observations in each sample (). Generally, the CLT begins to apply when , though this may vary depending on the underlying distribution.
3. Normal Distribution
The limiting distribution of sample means, characterized by its bell shape and symmetric properties.
4. Standard Error
The standard deviation of the sampling distribution, calculated as where is the population standard deviation.
Mathematical Foundation
For a population with mean and standard deviation , if we take samples of size , then as increases:
Where:
- is the sampling distribution of means
- is the population mean
- is the population standard deviation
- is the sample size