# Central Limit Theorem Calculator

URL: https://speedycalculator.com/calculators/statistics/central-limit-theorem/
Last updated: 2026-10-30

> The central limit theorem calculator describes the sampling distribution of the sample mean: from the population mean, standard deviation and sample size it gives the mean and standard error of x̄, the range holding 95% of sample means, and it can work backwards from a standard error to the population standard deviation.

## Inputs

- 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)

## Results

- Sampling distribution
- Mean of the sample means (μx̄)
- Standard error σ/√n
- Standard error with finite population correction
- Population standard deviation (σ)
- Variance of x̄
- About 68% of sample means fall between
- About 95% of sample means fall between
- Conditions check
- Population versus sampling distribution

## Formula

```
μx̄ = μ · σx̄ = σ ÷ √n · with finite population: σx̄ = σ/√n × √((N − n)/(N − 1)) · x̄ ≈ N(μ, σ²/n) for large n
```

## Explanation

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 nStandard error σ/√n95% of sample means within417.5± 34.30168.75± 17.15495± 9.801003.5± 6.864001.75± 3.43

## 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.

## FAQ

**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.


## Sources

- [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e)
- [Penn State STAT 500 – Applied Statistics](https://online.stat.psu.edu/stat500/)
- [Khan Academy – Statistics and probability](https://www.khanacademy.org/math/statistics-probability)