Central Limit Theorem Calculator
Use the central limit theorem to find the sampling distribution of the mean: standard error σ/√n, the z-score and the probability that a sample mean falls below, above or between values.
How it works
x̄ ~ N(μ, σ/√n) for large n · standard error σₓ̄ = σ/√n · z = (x̄ − μ)/(σ/√n) · P from the standard normal CDF Φ(z)
The central limit theorem states that the mean of a random sample of size n is approximately normally distributed with mean μ and standard deviation σ/√n (the standard error), whatever the shape of the population, once n is large enough — n ≥ 30 is the usual rule of thumb. Standardizing the sample mean with that standard error gives a z-score, and the normal CDF turns it into a probability.
Worked example
IQ scores have μ = 100 and σ = 15. For samples of n = 36 people, the standard error is 15/√36 = 2.5. The probability that a sample mean is below 105 is P(Z < (105 − 100)/2.5) = P(Z < 2) = 0.97725, about 97.7%.
Sources: Central Limit Theorem — OpenStax Introductory Statistics, chapter 7 · Central limit theorem — Wikipedia
Frequently asked questions
How large does n need to be?
About 30 for moderately skewed populations; more for very skewed or heavy-tailed data. If the population is already normal, the sample mean is exactly normal for any n.
What is the standard error of the mean?
The standard deviation of the sampling distribution of x̄: σ/√n. Quadrupling the sample size halves it.
Does the CLT apply to sums too?
Yes. The sum of n observations is approximately normal with mean nμ and standard deviation σ√n — equivalent to working with the mean.
What if I only know the sample standard deviation?
For large samples you can plug it in for σ. For small samples use the t distribution instead of the normal.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.