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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. You can get a better display of the maths by downloading special TeX fonts from jsMath. In the meantime, we will do the best we can with the fonts you have, but it may not be pretty and some equations may not be rendered correctly.

## 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.

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AQA A-Level (UK - Pre-2017)S1EstimationCentral Limit Theorem-
AQA A2 Further Maths 2017StatisticsCentral Limit Theorem - ExtraCentral Limit Theorem-
AQA AS/A2 Further Maths 2017StatisticsCentral Limit Theorem - ExtraCentral Limit Theorem-
CCEA A-Level (NI)S2EstimationCentral Limit Theorem-
CIE A-Level (UK)S2EstimationCentral Limit Theorem-
Edexcel A-Level (UK - Pre-2017)S3EstimationCentral Limit Theorem-
Edexcel A2 Further Maths 2017Further Statistics 1Central Limit TheoremCentral Limit Theorem-
Edexcel AS/A2 Further Maths 2017Further Statistics 1Central Limit TheoremCentral Limit Theorem-
I.B. Higher Level7EstimationCentral Limit Theorem-
Methods (UK)M15EstimationCentral Limit Theorem-
OCR A-Level (UK - Pre-2017)S2EstimationCentral Limit Theorem-
OCR A2 Further Maths 2017StatisticsHypothesis Tests and Confidence IntervalsCentral Limit Theorem-
OCR MEI A2 Further Maths 2017Statistics BSample Mean and Central Limit TheoremCentral Limit Theorem-
OCR-MEI A-Level (UK - Pre-2017)S3EstimationCentral Limit Theorem-
Universal (all site questions)EEstimationCentral Limit Theorem-
WJEC A-Level (Wales)S2EstimationCentral Limit Theorem-