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

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 $\overline{X}$ can be approximated by a normal distribution

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

• Therefore $\overline{X}$ can be modelled (approximately) by $\mathrm{N}\left(\mu ,\frac{{\sigma }^2}{n}\right)$

  • 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 $\overline{X}$ 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 $\overline{X}$ is approximately normally distributed

  • Provided the sample size is large enough

  • The Central Limit Theorem is needed