The Central Limit Theorem states that the the distribution of \bar{X} approaches normality as n increases, regardless of what the distribution of X is.

Stated more formally, for samples of size n drawn from a distribution with mean \mu and finite variance \sigma^2, the distribution of the sample mean is approximately N\left(\mu, \frac{\sigma^2}{n} \right) for sufficiently large n.

Stated more formally, for samples of size n drawn from a distribution with mean \mu and finite variance \sigma^2, the distribution of the sample mean is approximately N\left(\mu, \frac{\sigma^2}{n} \right) for sufficiently large n.

## Software/Applets used on this page

## Glossary

### central limit theorem

The mean of a random sample of size n from a distribution of mean m and variance v will be normally distributed with mean m and variance v/n, for large n.

### limit

the value that a function f(x) approaches as the variable x approaches a value such as 0 or infinity

### mean

the sum of all the members of the list divided by the number of items in the list.

### sample

A collection of sampling units drawn from a sampling frame.

### union

The union of two sets A and B is the set containing all the elements of A and B.

### variance

The square of the standard deviation; a measure of dispersion.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic |
---|---|---|---|

AQA A-Level (UK - Pre-2017) | S1 | Estimation | Central Limit Theorem |

AQA A2 Further Maths 2017 | Statistics | Central Limit Theorem - Extra | Central Limit Theorem |

AQA AS/A2 Further Maths 2017 | Statistics | Central Limit Theorem - Extra | Central Limit Theorem |

CCEA A-Level (NI) | S2 | Estimation | Central Limit Theorem |

CIE A-Level (UK) | S2 | Estimation | Central Limit Theorem |

Edexcel A-Level (UK - Pre-2017) | S3 | Estimation | Central Limit Theorem |

Edexcel A2 Further Maths 2017 | Further Statistics 1 | Central Limit Theorem | Central Limit Theorem |

Edexcel AS/A2 Further Maths 2017 | Further Statistics 1 | Central Limit Theorem | Central Limit Theorem |

I.B. Higher Level | 7 | Estimation | Central Limit Theorem |

Methods (UK) | M15 | Estimation | Central Limit Theorem |

OCR A-Level (UK - Pre-2017) | S2 | Estimation | Central Limit Theorem |

OCR A2 Further Maths 2017 | Statistics | Hypothesis Tests and Confidence Intervals | Central Limit Theorem |

OCR MEI A2 Further Maths 2017 | Statistics B | Sample Mean and Central Limit Theorem | Central Limit Theorem |

OCR-MEI A-Level (UK - Pre-2017) | S3 | Estimation | Central Limit Theorem |

Universal (all site questions) | E | Estimation | Central Limit Theorem |

WJEC A-Level (Wales) | S2 | Estimation | Central Limit Theorem |