The central limit theorem (CLT) is the reason the normal distribution appears everywhere in statistics. It says that if you take many random samples of size n from any population with a finite mean μ and standard deviation σ, the averages of those samples are approximately normally distributed – even when the population itself is skewed or lumpy – and the approximation improves as n grows.
The distribution of sample means is centred on the population mean, μx̄ = μ, but it is much narrower than the population: its standard deviation, called the standard error, is σ/√n. Quadrupling the sample size halves the standard error. This is what lets polls, quality checks and experiments estimate a population mean from a modest sample.
This calculator gives the mean and standard error of x̄, the ranges that hold about 68% and 95% of sample means, and a chart comparing a normal population with the much tighter sampling distribution. It checks the usual conditions (n ≥ 30, sample at most 10% of the population) and applies the finite population correction when you enter a population size.
How to use the Central Limit Theorem Calculator
- What do you know?
- Population mean (μ)
- Population standard deviation (σ)
- Standard error of the sample mean (σx̄)
- Sample size (n)
- Population size N (0 = very large / unknown)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Enter the population mean, standard deviation and sample size; the calculator divides σ by √n for the standard error and multiplies by √((N − n)/(N − 1)) when the population size N is given. In reverse mode it multiplies a known standard error by √n to recover σ.
Worked example
A population has mean 60 and standard deviation 35 and we draw samples of n = 49. The sample means are centred on 60 with standard error 35/√49 = 5. About 95% of the sample means lie within 1.96 × 5 of 60, between 50.20 and 69.80, and a sample mean above 70 (two standard errors) has probability 1 − Φ(2) = 2.28%.
How the standard error shrinks with n (σ = 35)
| Sample size n | Standard error σ/√n | 95% of sample means within |
|---|---|---|
| 4 | 17.5 | ± 34.30 |
| 16 | 8.75 | ± 17.15 |
| 49 | 5 | ± 9.80 |
| 100 | 3.5 | ± 6.86 |
| 400 | 1.75 | ± 3.43 |
Formula
μx̄ = μ · σx̄ = σ ÷ √n · with finite population: σx̄ = σ/√n × √((N − n)/(N − 1)) · x̄ ≈ N(μ, σ²/n) for large n
Worked example
Inputs — What do you know?: Population σ – find the sampling distribution; Population mean (μ): 60; Population standard deviation (σ): 35; Sample size (n): 49; Population size N (0 = very large / unknown): 0.
Results — Sampling distribution = x̄ ~ N(60, 5²); Mean of the sample means (μx̄) = 60; Standard error σ/√n = 5; Variance of x̄ = 25; About 68% of sample means fall between = 55 to 65; About 95% of sample means fall between = 50.2002 to 69.7998.
Key terms
- Sampling distribution
- The distribution of a statistic, such as x̄, over all possible random samples of the same size.
- Standard error
- The standard deviation of a sampling distribution; σ/√n for the mean.
- Finite population correction
- The factor √((N − n)/(N − 1)) that reduces the standard error when sampling without replacement from a small population.
- Independence
- Each observation must not influence the others – usually ensured by random sampling.
Frequently asked questions
What does the central limit theorem state?
Averages of random samples of size n are approximately normal with mean μ and standard deviation σ/√n, whatever the shape of the population, once n is large enough.
How large must the sample be?
n ≥ 30 is the usual rule of thumb. Symmetric populations need less; very skewed ones (incomes, waiting times) may need more.
What is the standard error for σ = 35 and n = 49?
35/√49 = 5. The sample means are centred on μ with a spread of 5 instead of 35.
Does the CLT apply to proportions?
Yes – a proportion is the mean of 0/1 values, so p̂ is approximately normal with standard error √(p(1 − p)/n) when np and n(1 − p) are large enough.
What is the finite population correction?
When the sample is more than about 10% of the population, multiply the standard error by √((N − n)/(N − 1)); it shrinks to 0 when you sample everyone.
Does the CLT make my data normal?
No. It describes the distribution of the sample mean, not of individual observations.
How do I get the population σ from a standard error?
Multiply by √n: a standard error of 5 with n = 49 means σ = 35. Choose the second option in the first menu.