Sampling Distribution Simulator
The Central Limit Theorem is stated so often that its actual claim gets lost. It does not say your data becomes normal. It says the distribution of the *sample mean* becomes normal — even when the population it came from looks nothing like a bell curve. This lets you draw from a deliberately skewed population, take means, and watch them pile up symmetrically anyway. It is the clearest demonstration in statistics and it takes about two seconds.
What this generator does
Draws a sample from the population shape you choose, takes its mean, and repeats. The means are histogrammed and the observed spread compared against one over the square root of the sample size — which is the standard error, and the reason sample size has diminishing returns.
How to use this tool
- Choose a population shape — skewed is the interesting one.
- Set how many values go into each sample.
- Run it, then increase the sample size and watch the spread shrink.
- Compare the observed spread against one over the square root of n.
Understanding the controls
- Population shape
- Skewed is the one worth trying: it looks nothing like a bell curve, and the means still come out symmetric. That is the whole claim of the theorem.
- Sample size
- How many values go into each mean. Quadrupling this halves the spread, which the numbers make plain.
- Repetitions
- How many samples to take. More gives a smoother shape without changing its width.
- Seed
- Any word reproduces the identical set of samples, so the same picture comes back when you show it to someone else.
Common use cases
- Teaching the Central Limit Theorem from data rather than assertion
- Showing why a bigger sample helps, and by exactly how much
- Explaining standard error to someone who finds the formula opaque
- Checking intuition about how skew affects an average
- Demonstrating why quadrupling a sample only halves the error
How this generator works
Each repetition draws a fresh sample from a population standardised to mean zero and spread one, then records its mean. The theoretical spread of those means is one divided by the square root of the sample size, printed beside the observed value.
Randomness and fairness
Every sample is drawn independently, so the histogram differs between runs while its shape does not. Seeded runs are reproducible and therefore explicitly not cryptographically secure; unseeded runs use the browser's cryptographically secure generator.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- The theorem describes the distribution of the mean, not of the data. This tool cannot make skewed data normal and does not claim to.
- Very small samples from a heavily skewed population stay visibly skewed — the theorem is asymptotic, and seeing where it fails is part of the point.
- The three population shapes are illustrative rather than a library of distributions.
- Samples are not retained between visits; a seed is the only way to recover a particular picture.
Privacy and your data
Samples are generated and summarised entirely in your browser. No settings or results are uploaded or stored.
Related generators
- Binomial Experiment SimulatorRepeat a fixed run of trials thousands of times and watch the count of successes form its distribution, with theory shown beside it.
- Survey Sample Size CalculatorHow many people you need to ask for a given margin of error, including the finite population correction most calculators leave out.
- Confidence Interval CalculatorTurn a count out of a sample into an interval, with a warning when the sample is too small for the usual approximation to be trusted.