Free Math Simulation

Explore the Central Limit Theorem: Sampling Distribution

Draw samples from any population and watch the sample means pile up into a bell curve. Free, no signup, runs in your browser.

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Free to use
No signup
Nothing to install
Runs in your browser

Free to use

No signup

Central Limit Theorem: Sampling Distribution

Setting

Open the Central Limit Theorem: Sampling Distribution

Illustration. Open the tool for the real thing.

See it in action

The Central Limit Theorem: Sampling Distribution with sample data entered and the result showing.

Screenshot of the Central Limit Theorem: Sampling Distribution with sample data entered: The Central Limit Theorem: Sampling Distribution with sample data entered and the result showing. — Sample data entered into the Central Limit Theorem: Sampling Distribution. Open the tool to try your own.

Explained

What is the Central Limit Theorem: Sampling Distribution?

Draw samples from any population and watch the sample means pile up into a bell curve. It runs in your browser, so a change to a setting or a drag on an object shows up at once.

Polls, quality control, clinical trials, A/B tests and machine learning all use sampling distributions to judge how sure we can be.

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Key idea 1Key idea 2Key idea 3

3 steps to your result

Open the tool and follow along. Nothing to install.

In the Central Limit Theorem: Sampling Distribution Simulator, use the controls to change Population, Sample size n, Samples per second. The simulation reacts instantly.

In the Central Limit Theorem: Sampling Distribution Simulator, press "Start drawing", "Draw 1", "Draw 100", "Clear" to start, reset or change what is happening.

Change one setting at a time and watch which readout moves, then compare it with this idea: The average of a large enough sample is roughly normal, even if the population is not.

What you get

Key idea 1

The average of a large enough sample is roughly normal, even if the population is not.

Key idea 2

The mean of the sample means equals the population mean.

Key idea 3

The spread of the sample means is σ divided by the square root of n (the standard error).

Experiment 1

Pick Skewed and set n to 1, then 5, then 30.

Experiment 2

Draw 1000 samples and compare the measured spread with σ/√n.

Experiment 3

Try the Two humps population and see what happens to the means.

Used for

Pick one to see how it fits.

Use case

Math learners

Algebra, geometry, trigonometry and calculus students, teachers preparing demonstrations, and self-learners who want to see the maths move.

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How it compares

A plain look at this tool next to the usual alternative.

	This simulation	Textbook diagram
Interactive	Drag, tap and change settings	Fixed picture
Install	Nothing to install	Not needed
Cost	Free to use	Depends on the book

Questions people ask

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. The Central Limit Theorem: Sampling Distribution Simulator is free, needs no signup or download and runs in your web browser. It is part of the Math group of AZRS simulations.

Does the Central Limit Theorem: Sampling Distribution Simulator work on a phone or tablet?

Yes. The Central Limit Theorem: Sampling Distribution Simulator takes touch input as well as mouse input, so you can use its 3 adjustable settings with a finger. A larger screen makes the controls and graphs easier to read.

How do I use the Central Limit Theorem: Sampling Distribution?

1. In the Central Limit Theorem: Sampling Distribution Simulator, use the controls to change Population, Sample size n, Samples per second. The simulation reacts instantly. 2. In the Central Limit Theorem: Sampling Distribution Simulator, pick an option such as Uniform, Skewed, Two humps, Dice to switch modes or load an example. 3. In the Central Limit Theorem: Sampling Distribution Simulator, press "Start drawing", "Draw 1", "Draw 100", "Clear" to start, reset or change what is happening. 4. Start with this experiment: Pick Skewed and set n to 1, then 5, then 30. Drag any glowing handle, object or slider with your mouse or finger and the Central Limit Theorem: Sampling Distribution Simulator responds in real time. 5. Change one setting at a time and watch which readout moves, then compare it with this idea: The average of a large enough sample is roughly normal, even if the population is not.