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Central Limit Theorem Simulator

Pick any population distribution, set a sample size, and draw samples. Watch the distribution of sample means gradually become bell-shaped, no matter what the original population looks like. This is the Central Limit Theorem in action.

Distributions

Population Distribution

μ=3.501.02.33.54.76.0Countn=5000

Sampling Distribution of Means (n = 30)

0.00.30.50.81.0Count

Click the "Draw samples" buttons above to start building the sampling distribution.

Controls

150100

Statistics

Population mean (μ)
3.5000
Population std dev (σ)
1.4434
Sampling dist. mean (x̄)
---
Sampling dist. std dev
---
Theoretical σ/√n
0.2635
Samples drawn
0

Step-by-Step Explanation

1. Population parameters

Distribution: Uniform [1, 6]\text{Distribution: Uniform [1, 6]}
μ=3.5000,σ=1.4434\mu = 3.5000, \quad \sigma = 1.4434
These are the true population parameters.\text{These are the true population parameters.}

2. Standard error of the mean

σxˉ=σn\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
σxˉ=1.443430\sigma_{\bar{x}} = \frac{1.4434}{\sqrt{30}}
σxˉ=0.2635\sigma_{\bar{x}} = 0.2635

3. Central Limit Theorem

Xˉ→dN(μ,  σ2n) as n→∞\bar{X} \xrightarrow{d} N\left(\mu,\; \frac{\sigma^2}{n}\right) \text{ as } n \to \infty
Xˉ≈N(3.50,  0.26352)\bar{X} \approx N\left(3.50,\; 0.2635^2\right)
The sampling distribution of xˉ approaches normal.\text{The sampling distribution of } \bar{x} \text{ approaches normal.}

Reference Guide

The Central Limit Theorem

If you take many random samples of size nn from any population with mean μ\mu and finite standard deviation σ\sigma, the distribution of the sample means will be approximately normal for large enough nn.

Xˉ∼N(μ,  σ2n)\bar{X} \sim N\left(\mu,\; \frac{\sigma^2}{n}\right)

A common rule of thumb is that n≥30n \ge 30 is large enough for most population shapes.

Standard Error of the Mean

The standard deviation of the sampling distribution (called the standard error) shrinks as the sample size grows.

σxˉ=σn\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

Quadrupling the sample size cuts the standard error in half. This is why larger samples give more precise estimates.

Why the CLT Matters

The CLT is the foundation of inferential statistics. It lets us use normal distribution tools (z-scores, confidence intervals, hypothesis tests) even when the population itself is not normally distributed.

Try the exponential or bimodal population in this simulator. The population looks nothing like a bell curve, but the sampling distribution of means converges to one.

Effect of Sample Size

As you increase nn, two things happen. The sampling distribution becomes more normally shaped, and it becomes narrower (smaller standard error).

Try setting n = 5 and then n = 50 with the same population. With larger n, the histogram of sample means is tighter around the population mean and looks more symmetric, even for highly skewed populations.