Central Limit Theorem Simulator

Draw thousands of samples from a non-normal population and watch the distribution of their means turn into a bell curve.

Set up the simulation

Population shape

Sample size per draw (n)

Population mean μ = 3, standard deviation σ = 3.

Predict the spread, score points

Before drawing, guess the standard deviation the sample means will settle at.

The population (not what CLT changes)

No draws yet

0 sample means (n = 5)

No draws yet

How to use this tool

Pick a population shape: uniform, skewed, or bimodal. None of them look like a normal distribution.
Choose a sample size n, the number of values averaged into each sample mean.
Press a Draw button to take that many samples and plot their means on the right.
Compare small n (1 or 2) with larger n (10 or 30) and watch the right-hand histogram change shape.

Formula Reference

Standard error of the mean

SE = sigma / sqrt(n)

What stays fixed

the mean of the sample means equals the population mean

What changes with n

the spread shrinks and the shape approaches normal

ArticleThe Central Limit Theorem, Explained With Worked Examples ArticleStandard Error ArticleThe Law of Large Numbers ToolConfidence Interval Simulator CalcStandard Error Calculator GlossaryGlossary: Central Limit Theorem

Frequently Asked Questions

Does the population itself become normal?+

No. The left-hand histogram, the population, never changes shape no matter how many samples you draw. Only the distribution of sample means, on the right, becomes bell-shaped.

How large does n need to be?+

There is no fixed cutoff. Very skewed populations need a larger n before the sample means look normal; for the exponential-like preset, n around 30 already looks close to bell-shaped, while n = 1 still looks skewed because a single draw is just the population itself.

Why does the spread of the sample means shrink as n grows?+

Averaging cancels out some of the randomness in each individual value. The formula sigma / sqrt(n) shows the spread falling in proportion to the square root of the sample size, not the sample size itself.

Why does this matter in practice?+

It is why methods built on the normal distribution, such as t-tests and confidence intervals for a mean, work reasonably well even when the underlying data are not normal, provided the sample is not too small.

How to use this tool

Pick a population shape: uniform, skewed, or bimodal. None of them look like a normal distribution.

Choose a sample size n, the number of values averaged into each sample mean.

Press a Draw button to take that many samples and plot their means on the right.

Compare small n (1 or 2) with larger n (10 or 30) and watch the right-hand histogram change shape.