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IBMathsDPApplications & Interpretation (AI)HLRevision Notes4. Statistics & ProbabilityCombinations of Normal Distributions & Sample Mean DistributionsCentral Limit Theorem

Central Limit Theorem (DP IB Applications & Interpretation (AI)): Revision Note

Dan Finlay

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DP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): HL

Central Limit Theorem

What is the Central Limit Theorem?

  • The Central Limit Theorem says that if a sufficiently large random sample is taken from any distribution X then the sample mean distribution X with bar on top can be approximated by a normal distribution

    • In your exam n > 30 will be considered sufficiently large for the sample size

  • Therefore X with bar on top can be modelled by straight N invisible function application open parentheses mu comma sigma squared over n close parentheses

    • μ is the mean of X

    • σ² is the variance of X

    • n is the size of the sample

Do I always need to use the Central Limit Theorem when working with the sample mean distribution?

  • No – the Central Limit Theorem is not needed when the population is normally distributed

  • If X is normally distributed then X with bar on top is normally distributed

    • This is true regardless of the size of the sample

    • The Central Limit Theorem is not needed

  • If X is not normally distributed then X with bar on top is approximately normally distributed

    • Provided the sample size is large enough

    • The Central Limit Theorem is needed

Worked Example

The integers 1 to 29 are written on counters and placed in a bag. The expected value when one is picked at random is 15 and the variance is 70. Susie randomly picks 40 integers, returning the counter after each selection.

a) Find the probability that the mean of Susie's 40 numbers is less than 13.

4-9-1-ib-ai-hl-central-limit-theorem-a-we-solution

b) Explain whether it was necessary to use the Central Limit Theorem in your calculation.

4-9-1-ib-ai-hl-central-limit-theorem-b-we-solution

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    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.