Central Limit Theorem (CLT) Simulation

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

Central Limit Theorem Visualizer

Explore how the distribution of sample means approaches a normal distribution as sample size increases, regardless of the original population distribution.

Population Distribution

Sample Size (n)10

Number of Samples1000

Original Population Distribution

Mean (μ): calculating...

Standard Deviation (σ): calculating...

Distribution of Sample Means

Sample Means Mean: calculating... (should equal population mean)

Sample Means Standard Deviation: calculating...

Expected Standard Deviation (σ/√n): calculating...

Key Observations:

As the sample size (n) increases, the distribution of sample means becomes more normal, regardless of the original population distribution.
The mean of the sample means equals the population mean (μ_x̄ = μ).
The standard deviation of the sample means equals the population standard deviation divided by the square root of the sample size (σ_x̄ = σ/√n).
This is the essence of the Central Limit Theorem, which states that the sampling distribution of the mean approaches a normal distribution as the sample size gets larger, regardless of the shape of the population distribution.

Learn More

Understanding the Central Limit Theorem

Key Concepts

1. Sample Means

The mean of repeated random samples taken from a population. The CLT describes how these sample means are distributed.

2. Sample Size

The number of observations in each sample ( $n$ ). Generally, the CLT begins to apply when $n \geq 30$ , 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 $\frac{\sigma}{\sqrt{n}}$ where $\sigma$ is the population standard deviation.

Mathematical Foundation

For a population with mean $\mu$ and standard deviation $\sigma$ , if we take samples of size $n$ , then as $n$ increases:

\bar{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}})