Central limit theorem simulator
Paste population values and set sample size and repetitions.
Histogram of sample means
| From | To | Samples | Share |
|---|---|---|---|
| Fill in the fields above — the result is computed on this page. | |||
How to use this page
- Paste a finite population or empirical distribution.
- Set n, repetitions and seed.
- Compare the histogram with theoretical and simulated standard errors.
What this page does not do: Simulation does not prove normality for a particular small sample. Dependence, infinite-variance heavy tails and unrepresentative data violate the model.
Further reading
The central limit theorem says that, under its assumptions, the mean of large samples approaches a normal distribution even when the population is not normal. Its centre tends to μ and its standard deviation to σ/√n.
The simulator samples with replacement from the entered empirical population. This demonstrates convergence, but one simulation remains random and seed-dependent.
Frequently asked questions
Why use a random seed?
The same seed reproduces the experiment, making settings comparable.
Is n = 30 always enough?
No. Required size depends on skewness, tails, dependence and the analysis goal.
See also
A/B test calculator
Compare two proportions in an A/B test. Calculate conversion rates, difference, uplift, z score, p value and c
A/B test planning — sample size, MDE and duration
Plan a two-proportion experiment, estimate detectable effect and traffic duration.
ACF and PACF autocorrelation calculator
Calculate ACF and PACF across time-series lags. Review the chart, reference bounds and coefficient table.