Central Limit Theorem: Sampling Distribution Simulator
Draw samples from any population and watch the sample means pile up into a bell curve.
Pick any population shape (even a lopsided one). Draw samples and record each sample's mean. The pile of means always forms a bell curve, and it gets narrower as the sample size grows. That is the Central Limit Theorem.
About the Central Limit Theorem: Sampling Distribution Simulator
Free central limit theorem: sampling distribution simulator. Draw samples from any population and watch the sample means pile up into a bell curve. Drag, change the sliders and see the result live. No sign-up, works on phone and computer. Built for math, the central limit theorem: sampling distribution simulator runs instantly in your browser: change a setting or drag an object and the result updates at once, so you learn by trying things out rather than only reading about them.
Draw samples from any population and watch the sample means pile up into a bell curve. Use it to explore math ideas at your own pace, then check what you found against the key ideas further down this page.
How to use the Central Limit Theorem: Sampling Distribution Simulator
- Use the controls to change Population, Sample size n, Samples per second. The simulation reacts instantly.
- Pick an option such as Uniform, Skewed, Two humps, Dice to switch modes or load an example.
- Press "Start drawing", "Draw 1", "Clear" to start, reset or change what is happening.
- Where you see a glowing handle, object, weight or atom, drag it with your mouse or finger. Everything responds in real time.
- Watch the readouts and graphs update as you experiment, and compare what you see with the key ideas below.
Things to try
- Pick Skewed and set n to 1, then 5, then 30.
- Draw 1000 samples and compare the measured spread with σ/√n.
- Try the Two humps population and see what happens to the means.
- Try dice with n = 2 and n = 10.
Key ideas you can learn
- The average of a large enough sample is roughly normal, even if the population is not.
- The mean of the sample means equals the population mean.
- The spread of the sample means is σ divided by the square root of n (the standard error).
- Bigger samples give a narrower bell curve.
Where this is used in the real world
Polls, quality control, clinical trials, A/B tests and machine learning all use sampling distributions to judge how sure we can be.
Who is this simulation for?
Algebra, geometry, trigonometry and calculus students, teachers preparing demonstrations, and self-learners who want to see the maths move.
For teachers: project it on the board, let students predict what will happen, then run it together. For students: change one thing at a time and write down what changes.
Frequently asked questions
Why does the bell curve appear?
Averaging many random values cancels out extremes, and the sum of many small random effects is close to normal.
What is the standard error?
The standard deviation of the sample means, equal to σ divided by the square root of the sample size.
Is the Central Limit Theorem: Sampling Distribution Simulator free to use?
Yes. It is completely free, with no signup, no download and no ads inside the simulation. It runs in your web browser.
Does the Central Limit Theorem: Sampling Distribution Simulator work on a phone or tablet?
Yes. It uses touch as well as the mouse, so you can drag objects with your finger. A larger screen makes the controls easier to see.