Central Limit Theorem Calculator
See how the central limit theorem predicts the mean and spread of sample means drawn from a population.
What the central limit theorem says
The central limit theorem (CLT) states that if you repeatedly take random samples of size n from any population and calculate the mean of each sample, the distribution of those sample means will approach a normal (bell-shaped) distribution as n grows, regardless of the shape of the original population.
How this calculator works
Enter the population’s mean and standard deviation along with your sample size. The calculator reports the mean of the sampling distribution, which always equals the population mean, and the standard error, calculated as the population standard deviation divided by the square root of the sample size (SE = σ / √n). A larger sample size produces a smaller standard error, meaning sample means cluster more tightly around the true population mean.
As a general guideline, a sample size of at least 30 is often used as a rule of thumb for the sampling distribution to be reasonably approximated as normal, though this can vary with how skewed the original population is. This tool is for general statistics education.