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

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

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 | - |