The central limit theorem (CLT) is one of the most useful results in statistics: no matter how a population's values are distributed, the means of repeated random samples from it settle into a predictable, approximately normal pattern. This calculator turns that theorem into three practical numbers — the sampling distribution's mean and spread, and the probability that a sample mean falls in a range you care about.

How the central limit theorem calculator works

Given a population mean (μ), a population standard deviation (σ), and a sample size (n), the calculator computes the standard error of the mean, SE = σ/√n — the standard deviation of the sampling distribution of x̄. By the CLT, that sampling distribution is centered at μ and becomes approximately normal as n grows, which is what lets the calculator use the standard normal distribution to answer probability questions about x̄.

For the probability views, a sample mean threshold (or a lower/upper bound pair) is converted to a z-score using z = (x̄ − μ)/SE, and the corresponding area under the standard normal curve gives the probability.

Inputs and what they mean

Population mean (μ) is the true average of the entire population — often unknown in practice, but assumed known here so the theorem's behavior is clear. Population standard deviation (σ) measures how spread out individual values are in the population; a larger σ means individual measurements vary more, which widens the sampling distribution too. Sample size (n) is how many observations go into each sample mean — it has an outsized effect on precision because SE shrinks with the square root of n, so quadrupling the sample size only halves the standard error.

Limits and edge cases

Independent observations from a normal population give exact normal sample means. For other finite-variance populations the normal calculation is an approximation whose adequacy depends on the population and sample size; n=30 is not a universal guarantee. Compare at most eight positive whole sample sizes through 1,000,000.

Sample-size comparison with a fixed interval

Compare sample-mean uncertainty across sample sizes while holding the population mean, population standard deviation and event interval fixed. The population distribution itself does not narrow.

Independent observations from a normal population give exact normal sample means. For other finite-variance populations the normal calculation is an approximation whose adequacy depends on the population and sample size; n=30 is not a universal guarantee. Compare at most eight positive whole sample sizes through 1,000,000.