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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.

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## 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:

SyllabusModuleSectionTopic
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